Mathematics Education
Can extend to the field you are going to be teaching

1. Mathematics education majors may write on an application of interest to their field, there is a great many topics to choose from use of technology versus non-use examples or student interactive education versus strict teacher control

2. Teaching mathematics in various cultures.

3. Teaching for the physically or mentally challenged.

4. Vedic mathematics (though really this should be in a category as itself.) Other cultures and countries have various ways of learning and doing mathematics that are just as valid as the way the US teaches it.)

5. Investigate the shifting baseline of knowledge in today’s world and in particular the US. It’s frankly scary. I bring this up to not place blame on the  or the teachers in any way, though the teachers are falling into the “pass the students ‘ready or not’ or be fired” syndrome, but rather the country’s approach to education. I want to alert you to the lowering of standards through out education as time has gone by. This among other things, could spell doom for our country. Already the US students are scoring just average and a bit above average with respect to the rest of the industrialized world.

6. In a similar vein as #4 but slightly different (less critical but more reflective in a “do you remember” style)…investigate the history of the education system in the USA as it has changed over the years, decades, and centuries. Include in this description the curricula (courses taught) and the types of school (physical). Start with your own education and go back in increments of twenty years (roughly one generation of students) – you can talk to parents and grandparents, and even further back if relatives are still alive that far back. Eventually though you will have to do some “book larnin’ “ from books or the internet. Make your way back in time to at least the 19^th century. One room school house structure with generally a female school “ma’am” that taught all the grades versus today’s schools. Though it did not need be a school “ma’am”…it could just as well be a school master (a male – you know I am not sure what they were called, school master was to ostentatious a name, but knowing the male bias in those days it could have been).  One thing you will find for sure is that the students in the old days learned a lot of the classics that students today have never even heard of unless you are a student of such literature. They learned the Greek and Latin languages with varying degrees of success. There were lessons in algebra, geometry, and arithmetic. A good deal of emphasis was on writing, grammar, orthography, and studying the classics, as well as geography. They had science classes as well but it was stuff that was more hands on nature class type coursework (hassling birds by studying birds nests…collecting bugs, frogs, insects, snakes, et. al.), probably meteorological studies from observational point of view only, and study with the known solar system of the day with perhaps unaided visual astronomy thrown in. They probably learned many rules of thumb that they used in their day to day experience. The higher levels of education concentrated mostly on preparing the students to work in some vocational field, and college was mostly a place to study the liberal arts and gaining a liberal arts education. Today on the other hand, science is much more pronounced and computers are used in most classes today at least in some extent. Your calculators you use are not simple, they would make the computer I use to program on look like a dinosaur and that computer filled a whole room, with elaborate air conditioning systems in place. I don’t pretend to say that the students of that day were better or worse than the students of today….just different. 

Subject History:    History is a topic that is okay, but I do NOT want to have to read umpteen history papers on reading day. Be imaginative, but I am also aware that many students just do not have the ability to write a solid technical report on something non history related (no insults or cut downs intended – you are still in the beginnings of your college career and with many students their interests are not solidified yet to even pick a major, the ability comes later). If you do write a paper on history, try to bring it to life somewhat. You know what I mean. History papers are almost always boring date after date things….they don’t need to be. You still need a date by date description but you can color the description by pointing out humorous things you found related to the topic, idiosyncrasies of the various people you are investigating etc. Keep it factual.

Human History:  If you are writing about a particular mathematician (or physicist) I mention talk about the personal eccentricities of some of the mathematicians I have talked about. For example the fact that Newton was a short tempered and suspicious individual, or the fact that Fourier’s idea of good health was keeping his living area hot enough to make Phoenix in July in the sunshine feel comfy. Any correlation of his work with heat transfer to his sanity or lack thereof? Or Dr. Kurt Gödel’s belief that everyone except certain people were trying to poison him. Paul Erdos’ lack of a permanent home. An itinerate wandering mathematician living on the graces and good will of other mathematicians in return for solutions and problem statements. He quite literally made military families look like permanent squatters.

History – you can write on a brief history of mathematics in some period, mathematicians (also physicists or engineers since that is my modern and professional interest), or the contributions of a country.

• Mesopotamia and ancient Egypt contributions to mathematics et. al.  (means:   and others)

• Contributions of India, China, ancient Greece, Arabic world to mathematics.et. al. 

• Mathematics et. al. in the new world: Inca’s, Aztec, Mayan, Chaco Canyon's peoples and their dead straight roads, or the mound builders of the centeral midwest area. many other tribes

• Celtic Astronomy et. al. . If you work on this, stay away from the mystical ideas about Stonehenge. In fact stay away from mysticism throughout all the topics. Period. I don’t mind reading about these things, but I do want the papers to be based on known fact rather than speculation.

People

• Archimedes (considered one of the three pillars of mathematics).

• Newton (he himself would take books – the second pillar),

• Gauss (the third pillar, if he’d only have published what he discovered or invented, we’d be years ahead of where we are) .

• Hilbert, Neumann, Kurt Gödel, and Paul Erdös for the 20th century.

• Though Einstein is really a physicist through out I will count a good paper on him and his work as acceptable. As a sideline event, he and Gödel were good friends at Princeton.

• I also will accept a paper on Heisenberg.

• Paper on the duo of Schrödinger and Dirac both are main icons of quantum mechanics.

• Though Faraday was not a mathematician by any stretch…I think many of my 272 students could have outdone him in the subject… he was a great experimental physicist and I have taken a liking to his skills over the years. If someone want to write a paper on him that would certainly be acceptable, if you will also include James Clerk Maxwell.

• Hans Bethe (I liked that guy…I didn’t know him personally, but I liked his work for the Manhattan Project – he is still alive though he is getting quite old but is still doing research the last I heard. He was officially retired, but he was spending half days at his university. He may have died recently since it has been a while since I check up on him. HE REGRETABLLY PASSED AWAY JUST RECENTLY.

• You could pick out mathematicians/Physicists of a particular race or sex though I am not keen on this, cause in doing so I look at that as racial profiling. If you do so, stick to the facts and don’t try to turn it into a “my culture is better than your culture”  or man versus woman game. However in terms of women scientists, there were several that come to mind. Marie Curie.....Lise Meitner....Rosalind Franklin (Watson and Crick who won the Nobel prize for unravelling the structure of DNA put her down as unimportant but without her work they probably would not have succeeded and she did just as much towards the goal as they did and perhaps more.). Grace Hopper (computer systems)

Theories histories

The history of relativity theory OR the history of Quantum Mechanics.  See if you can find evidence of the great fight Einstein got into (I mean literally – not his disputes with Bohr and others.) An assassin was going to pick off Einstein, and his wife alerted him to the danger, whereupon Einstein wrestled the gun away from the guy. I think a neighbor came over to help the good professor as well. The normally quiet Mercer Street had an interesting day. I am interested in if this happened for real. I read it once, but one source is not considered a guarantee that something happened. It does ring true since the Germans (as much as they looked at Einstein officially as a member of an “inferior” race…unofficially they were scared to death of him) felt he was helping the allied cause. There are countless other great mathematicians and physicists you could write on (Hawking for example – possibly the best modern physicist today – he is at Cambridge)

The history of classical physics  OR  the history of electromagnetism  OR  the history of Thermodynamics  MANY OTHERS POSSIBLE.  Major Biological Theoriies for instance.  

You could also trace a particular branch of mathematics back to not only its roots on the mathematics tree but also back to it’s roots in history. I will show you on an individual basis the tree and where you are at on the tree.

History and Education of Math

• Historical numeration systems. Symbols used by various cultures through various times. Historical development of the numbers we use today. Natural Numbers, whole numbers, integers, fractions (rational numbers), real numbers, complex numbers

• The foundations of definitions of the four basic operations: addends and sum; minuend and subtrahend and difference, divisor, quotient, remainder…similar terminology for multiplications though to be truthful I forget what they are called perhaps factors (though I think it was named something else) and of course product the answer.

• Overview of elementary number theory, including primes (infinity of them, sparseness), factorization (into primes)

• Overview of divisibility, greatest common divisor, least common multiple, rules of divisibility

• foundational laws of algebra: distributive law, commutative law, associative law, zero element, unitary element .

• Set theory: theory and practice: subsets, proper subsets, set definition, power set, intersection unions, universal set, complements of sets with respect to: 1. a larger set and 2. the universal set. Venn Diagrams. By the way, why are pictures not considered proofs?

• In fact the idea and background of proof as a basis for mathematical studies, the various forms of proof and there foundations in logic.

• Bases, modular systems, linear congruence, and continued fractions.

• Geometric constructions and proof

• Use of Geometry in Egypt, Mesopotamia, Greece. 3d geometry particular standard solids (polyhedra): tetrahedron and etc. Euler’s law of such solids. Theorem of Pappus

• History and applications of spherical trigonometry and spherical geometry. Spherical triangles on a sphere. In fact triangles formed by arcs of circles that are convex with respect to the central point, and conversely concave to the central point.

Applied Mathematics

ODE: ordinary differential equations, adapted to the needs of students in engineering and the sciences. Linear and nonlinear ordinary differential equations, asymptotic behavior of solutions, stability, existence and uniqueness, limit sets, Poincar-Bendixson theorem. Nonlinear oscillations, strained coordinates, renormalization, multiple scales, boundary layers, matched asymptotic expansions, turning point problems, and WKBJ method. Systems, existence proofs, singularities, asymptotic behavior of solutions, boundedness of solutions, eigenvalues and eigenfunctions, and perturbation theory. Geometric approach to ODEs and dynamical systems; (un)stable, center manifolds; structural stability; normal forms; averaging; chaos; persistence. Asymptotic behavior of solutions of linear and nonlinear ordinary differential equations, stability,Sturm-Liouville problems, boundary value problems, and singular point behavior of autonomous systems.

PDE: Second order partial differential equations, emphasizing Laplace, wave, and diffusion equations. Solutions by the methods of characteristics, separation of variables, and integral transforms. First order quasilinear, second order linear (wave, Laplace, heat). Characteristics, harmonic functions, maximum principles, Fourier series, separation of variables. Existence and uniqueness theorems, boundary value and initial value problems, characteristics, Green's functions, maximum principle, distributions, and weak solutions. Sobolev spaces; incompressible Euler and Navier-Stokes equations; weak and strong solutions; attractors and the connection with turbulence; geophysical applications.

*Parabolic and elliptic equations, finite difference, finite element methods, stability, consistency, convergence, practical aspects, applications, software.

*Numerical solutions of hyperbolic PDEs, finite difference methods, well- posedness, stability, consistency, convergence, adaptive grids; Maxwell's equations, elastic wave propagation; Navier-Stokes.

* Spectral, pseudo-spectral theory; Galerkin, collocation methods; Tau-methods, global approximation properties, stability; convergence; solutions for linear, nonlinear systems.

Numerical Analysis Computer Methods:

Logic, sets, functions, elementary number theory and combinatorics, recursive algorithms, and mathematical reasoning, including induction. Emphasizes connections to computer science. Number systems, hardware/software arithmetic, overflow, significance, rounding, multiple precision, and automatic error control; impact on languages, architectures, robust programming, and software development. applies programming languages, libraries, and scientific visualization tools. Programming assignments emphasize software development skills. Elementary computer arithmetic. *Algorithms for massively parallel, hypercube architectures; "parallel" FORTRAN; solution of linear, nonlinear systems; partial differential equations; iterative methods; multigrid; domain decomposition.

Numerical methods for quadrature, differential equations, roots of nonlinear equations, interpolation, approximation, linear equations, floating-point arithmetic, and roundoff error. Analysis and algorithms for numerical solutions linear/nonlinear equations, direct solvers, iterative procedures, optimization. Determination of eigenvalues. Analysis of and algorithms for numerical interpolation, integration, and differentiation. Numerical solution of ordinary differential equations, and method of lines.  Direct solution of linear systems, iterative methods, eigenvalues and eigenvectors, singular value decomposition, the QR algorithm, error propagation, arithmetic, and stability.  Numerical methods for solving linear/nonlinear systems of equations (symmetric, nonsymmetric). Iterative methods for linear systems, conjugate gradients, multigrid methods, preconditioning, Krylov methods.  Linear programming, unconstrained nonlinear minimization, line search algorithms, conjugate gradients,quasi-Newton methods, constrained nonlinear optimization, gradient projection, and penalty methods.  One step, linear multistep methods; consistency, order, stability, convergence; discretization, round-off errors, error estimation, adaptive strategy; implementation, software for nonstiff equations. Runge-Kutta methods, order conditions, construction of highly stable methods, order stars, error estimation, stepsize selection, contractivity properties, linear multistep methods.

Mathematical Physics/Engineering

Vector analysis, Fourier analysis, and partial differential equations. Fourier Transform, Laplace transform, and Hankel Transform. Vectors, curvilinear coordinates, Jacobians, implicit function theorem, line and surface integrals, Green's, Stokes', Gauss’ and divergence theorems. General Transforms. Properties of nonlinear dynamical systems; dependence on initial conditions; strange attractors; period doubling; bifurcations; symbolicdynamics; Smale-Birkhoff theorem; and applications. Dynamical Systems Methods in Fluid Dynamics. Applies modern dynamical systems methods to fluid mechanics: bifurcations, normal forms, nonlinear dynamics, pattern formation, mixing, and Lagrangian chaos. Hilbert Spaces, generalized functions (Dirac Delta Functions and higher distribution functions). Classic Orthogonal Polynomials and recurrence relations, generating functions and examples: Chebysev, Laguerre, Lagrange, Jacobi, and so on. Spectral decomposition of operators. Operator theory. Hermitian and Unitary. Integral equations, algebra and analysis of tensors, Differential Geometry, Calculus of Variations, plus many of the topics discussed in the Applied Mathematics section, excluding numerical analysis, though it does form a support tool for this section of Physics.

Cryptography (Cryptoanalysis)

Block ciphers, stream ciphers, congruence arithmetic, information theory, public key cryptosystems, key exchange, electronic signatures. For the purposes of my students, a history of ciphers and examples of said ciphers. In the context of the papers cipher is to mean codes or cryptanalysis (“secret writing”). In today’s world revealing something that is classified in this area is paramount to SERVING solitary confinement for the rest of your life (IF NOT WORSE ..ENOUGH SAID). So if you know of such codes don’t mention those.

• Fourier Series, Taylor Series. Fourier Integral

• Hankel, Fourier, and Laplace Transformations

• Basic differential equations and series solutions thereof.

• Line integrals and electromagnetics. Stokes theorerm, Divergence Theorem, Greens Theorem, Gauss’ Theorem

• Spectral Analysis and decomposition of functional operators

• Operations analysis

• Singular Value Decompositions Numerical analysis of matrices

• Numerical Analysis in general

• Tensors

• Vibrating Strings and membranes.

• Heat transfer analysis – I got to tell you about one fool doctor I went to. He gave me a thermometer to take my temperature with. He knew from my records that I was in the graduate school in mathematics but he still asked, “do you know how to read a thermometer?” (remember he was a school hired physician student health center types --- they don’t hire the brightest that come down the pike), and I said, “yes I certainly do….and do you know how to find the actual temperature gradient along the thermometer and how that it turns alters the temperature you are interested in?” I felt bad about it later, but the bozo asked for it. Almost every student (in fact I would say all) can read a fever thermometer, and also know ways to change that temperature from trying to convince their parents they were sick in their public school days.

Pure/Abstract  Mathematics
These areas are branches of mathematics that I have just taken the required courses in. They are far from being my specialties. So I may not be able to answer advanced questions on them.

Combinatorics: Study of proof techniques, permutations, combinations; counting techniques including recurrence relaxations, generating functions, inclusion-exclusion; Ramsey theory and combinatorial designs. systematic development of enumerative combinatorics, including elementary counting techniques, sieve methods, and partially ordered sets.  THIS IS REALLY ALSO A PART OF THE APPLIED MATH CATEGORY ALSO.

Abstract Algebra: Concepts of abstract algebra. Basic theory of groups, rings, and fields, including an introduction to Galois theory. Groups, modules, rings and fields, Galois theory, homo-logical algebra, and the representation theory. Permutations

Theory of Numbers: Prime numbers, unique factorization theorem, congruences, Diophantine equations, primitive roots, and quadratic reciprocity theorem. Proof of Fermat’s Last Theorem (overview).

Geometry:

• Topology – the area of mathematics, geometric in nature, describing transformations. To the lay person it is called rubber sheet geometry. Doughnuts and coffee cups are topologically equivalent, because if you start with a doughnut….by stretching and distorting the donut, you can create a coffee cup. And vice versa. The key is the hole in the donut forms the handle /cup interface Of course you can’t do this in real life. The doughnut will crumble of course. You may look the Klein Bottle (a one sided surface), the Mobius Strip, a one sided surface. Some of the wildest claims of mathematics are found in this area of mathematics. Look up Tarski’s Theorem. Basically put it says if you start with a solid pea, you can break this pea down, in some way that is foreign to our ability to see, and reassemble it into a ball that is solid and yet the size of the sun. How do we do this? We do not know, but the fact is that it is mathematically possible to do so. So you see, as I said above about Godel’s proof of the possibility of time travel into the past, mathematically possible is a long, long cry from being able to actually do it. The big thing you have to realize about the pea / sun idea is that we divorce ourselves from the pea and Sun being anything other than mathematical ideas. Of course the Sun has vastly more atoms ( ions might be better) than a pea does. So don’t ever expect to solve the world’s hunger problem by creating transformation machines to make steak, peas or paychecks bigger. Heck, if I could make my paycheck bigger, I’d just write from a hermit’s home in inner Alaska, after a few years of teaching. Topological spaces, metric spaces, compactness, connectedness, and product spaces

• Graph Theory FORMAL   Trees, cycles, matchings, planarity, connectivity, hamiltonicity, colorings, graph algorithms, and other advanced topics. systematic development of graph theory, including matchings, connectivity, arboricity, planarity, coloring, network flows. dense and sparse graphs, Ramsey theory, hamiltonicity, random graphs, minors.  For those of you in computer science this would be a good branch of mathematics to read up on. It is kind of similar to topology but you are drawing graphs (not your usual xy axis graphs) but graphs which consists of points and edges. You decide what the points and edges mean for your particular problem. For example your apartment, say, might have a kitchen area, a living area, your bathroom/shower area, and a bedroom. You might represent the room by points (vertices) and the doorways as the edges. Two points are connected by an edge if there is a doorway from one room to the other. Conversely the rooms might be edges and the doors are the vertices. These graphs you get are called trees. The problem is: can you start at a vertices and transverse each room going over only one edge (door) only once. Or can you travel through each door (edge) once but visit all the rooms? These are graph theory circuit problems. See the Seven Bridges of Konigsberg Problem.
  

Traveling salesman Problem: If you have n towns (vertices) with m edges (roads) where 2 vertices are connected if there is a road between the two towns. Now starting at a given point, find a pathway of least distance that will take you through each town only one time. This would help you how? For starters: money (you use less gas at least theoretically), time (you visit each town only once in the fastest way possible). This is one of the classic problems of graph theory

• Plane Geometry: the mathematics of ancient Greece or part of it I should say, and the subject of high school courses which you probably had or have heard about. And Anne Shirley’s waterloo. Sorry if I keep bringing her up. I read the whole series of books this summer about her in keeping with a promise I made to my parents years ago. I told them, “someday I would read the series. To me right now they are just to sissy”. I was about 10 then when they asked me to. I liked them so much they have become part of my example database. They (parents) are both passed on now, in fact almost 10 years ago, but I told them symbolically after I finished the last book, “I kept my promise”.Congruence, area, parallelism, similarity and volume

• Non-Euclidean Geometry: The postulate of geometry as above states that through any point one and only one parallel line can be drawn. This cannot be proven and is taken as a postulate, but mathematicians said, what if we assumed more than one line can be drawn through the point but parallel to the given line. This was to have led to a contradiction but it did not. It led instead to two types of non-euclidean geometry. Think of a basketball and draw a triangle on the ball. Notice the triangle bulges outwards along the ball’s surface. Other triangle geometry possible triangles edge bulge inward, not outwards.  Einstein made use of this with the idea of a curved space in later times.

• There are several other geometries, however two types of geometry that are no longer taught are

• Fractals; self-similar structures, fractals with iterated function systems of maps, computing fractals, fractal dimensions, chaotic dynamics on fractals, applications. Geometry and analysis of fractal sets; definitions of dimensions; calculating dimensions; projections, products of fractals; random fractals; multifractal measures; and applications This last geometry I am not even sure you can describe it as such but it seems most allied to the geometry field: Fractals, part of my gallery I linked to on my web pages are an ongoing mathematics area. If you find this interesting you may feel free to research it and write up a report on this topic. These fractals are a beauty to look at and to be honest is about the extent of my knowledge of the subject. I never did find out where my link to the gallery led to. I never traced it all the way. Fractals though lead to examples of functions that are everywhere continuous but no where differentiable. I am thinking here that if you choose this for a topic that you give an overview of the fractals, but also do the snowflake problem from scratch. Not to a full blown model but several iterations. See internet or me for details.

• Linear Algebra:  is strictly speaking not really part of Geometry, but I put it here, because it talks about rotations and transformation of geometric like structures. It is really a very far reaching branch of mathematics.Linear equations systems, matrices, determinants, vector spaces, bases, linear transformations and similarity, inner product spaces, eigenvalues and eigenvectors, orthonormal bases, diagonalization, and principal axes. decompositions, applications. Fundamentals of linear algebra, dual spaces, invariant subspaces, canonical forms, bilinear and quadratic forms, and multilinear algebra.
  

Analysis

This is a branch of mathematics that could be described as the engine of the whole of mathematics. For those of you who are in Calculus, you are on the boundaries of this area. You of course are on the simpler edge of this branch, but never the less you are in it.

Set Theory: Logic and set theory, induction, functions, order and equivalence relations, cardinality. Emphasizes writing proofs. Countable and uncountable sets. Rationals countable, irrational – uncountable.

Real Analysis (Variables): Theory behind basic 1-variable calculus: continuity, derivative, Riemann integral, sequences, and series. Real numbers, completeness, sequences/series, continuity, uniform theorems, derivative, Riemann integral, pointwise/uniform convergence, Taylor's theorem. Open, closed, compact sets in Rn continuity, differentiation, partial differentiation, integration in Rn. Inverse/implicit function theorems. Real Analysis Introduces analysis in metric spaces with emphasis on the real line. Analysis in Rn: implicit function theorem, introduction to manifolds, Lebesque integration, change of variables formula, convergence theorems for integrals. Lebesgue integration, selected function spaces, differentiation, abstract measure theory, and elements of functional analysis.

Complex Analysis: Analytic functions, complex integration, Taylor and Laurent series, residue theorem, conformal mapping, and harmonic functions. Analytic functions, series and product representations, entire and meromorphic functions, normal families, Riemann mapping theorem, harmonic functions, and Riemann surfaces.

FUNCTIONAL ANALYSIS Locally convex, normed, and Hilbert spaces. Linear operators, spectral theory, and application to classical analysis. Bounded linear and compact operators on Hilbert spaces. Linear integral equations, Fredholm and Hilbert-Schmidt theory, and approximate methods. Distributions.

Statistics, Probability, and Stochastic Processes Including Markov Chains

• Concepts and vocabulary needed to evaluate statistical reports on health, technology, and society. I.e. medical, engineering, and social sciences (social workers in part)

• Basic concepts and methods of statistics, including descriptive statistics, significance tests, estimation, sampling, and correlation.

• Introductory probability , descriptive statistics, sampling distributions, parameter estimation, tests of hypotheses, chi-square tests, regression analysis, analysis of variance, and nonparametric tests.

• Probability models and computations, joint and conditional distributions, moments, and families of distributions. Stochastic processes, simulation, and statistics. Laws of probability, combinatorial analysis, random variables, probability distributions, expectations, moment-generating functions, transformations of random variables, and central limit theorem. Measure-theoretic foundations of probability, distribution functions and characteristic functions, laws of large numbers and central limit theorems, conditional probabilities, martingales, and other topics in stochastic processes.

• Markov chains, stationary distributions, pure jump processes, 2-D order processes, and other topics in stochastic processes. Limiting distributions, interval estimation, point estimation, sufficient statistics, and tests of hypotheses. Statistical inference for controlled experimentation. Multiple regression, correlation, analysis of variance, multiple comparisons, and nonparametric procedures. Random walks (drunk’s walk) in 2d. probability of returning to starting position. 3d .. showing nil probability of returning to starting position.

• Multinormal distribution, distribution of quadratic forms, full and nonfull rank models, generalized inverses, unbalanced data, variance components, and the large sample theory.

• Method of least squares, simple and multiple linear regression, polynomial regression, analysis of residuals, dummy variables, and model building.

• Factorial designs, balanced and unbalanced data, fixed and random effects, randomized blocks, Latin squares, analysis of covariance, and multiple comparisons.

• One-sample test, tests of 2 or more related or independent samples, measures of correlation, and tests of trend and dependence.

• Discriminant analysis, principal components, factor analysis, cluster analysis, and canonical correlation. Statistics

• Models for discrete and count data, measures of association, and log-linear and regression models for contingency tables. Statistics

• Simple random, stratified, cluster sampling; variance estimation in complex surveys; nonparametric superpopulation approaches; nonresponse models; computational methods.

These areas probably have their appropriate pigeon holes in the above areas but I am not quite sure how to file them, so I have included two example separately.

Chaos Theory: In the above introduction to the Many Worlds Hypothesis Problem, I had a butterfly flapping her wings. I stated this could trickle down and eventually avalanche into generating tornadoes in Kansas. This is an area of mathematics called chaos theory. There is a lot of good material on the internet to write up a paper on this one topic alone. Further I have seen this modeled via a weighted stiff pendulumn with an extensive initial angle of swing. The computer starts giving some really weird seemingly random results.

Monte Carlo Methods: Simple idea. Origiinally developed for use in the Manhattan Project. The equations became impossible to solve in a finite time so the idea was developed that the problem became turned into a black box model with inputs and an output (a functional output based on a regression model to simplify things often). The inputs were chosen randomly from a set of conditions and the results were kicked out. The idea is that by repeating the process a finite but large number of times you can get a picture of what the output is acting like.  

I’ll stop here on the listing of mathematics topics. I could keep on a ways longer but I have given you more than enough possible topics. I have primarily given topics that most engineers should know, most physicists should know….In general most scientists should know. What I was trying to say, is that all the mathematics you need for your future courses are contained within a handful of these topics.   I do not and cannot expect you to learn all of these things.  If you could you'd know as much as any PhD in mathematics.  I am hoping at best that you might pick out a topic that sounds interesting and do a bit of background work on what it is saying.

Bio Mathematics:   Formal Items: Applications to genomics, bioinformatics, forensics, and DNA/protein sequence patterns. Mathematical modeling electrochemical processes in nerve. Cable theory, neuronal branching, spines, bifurcation analysis of excitable membrane models. Mathematical modeling of dynamical aspects of cell physiology. Diffusion, membrane transport, intracellular calcium channel kinetics, calcium oscillations and waves. Bacterial growth, bacterial genetics, gene expression, stochiometry of metabolic pathways, random walks, diffusion processes, biofilms.   Biology – statistics is your big thing, but there are mathematical areas see “biomathematics” on the internet. The predator prey model would be good to do.  you can get some very heavy mathemtatics in any field you look at it, if you look deep enough

• Diseases caused by bacteria....by viruses.....parasitical diseases (Malaria just one of many).....genetically obtained diseases
  

• Biology and biochemistry – You may not feel up to this but Mad Cow disease has caused me to consider a further link between prions and other neural degenerative diseases just some examples: MS, Alzheimer Disease, Parkinson’s Disease, etc. you might write up a paper on your findings and thoughts in this area. You probably will find little on this topic because they still think in terms of slow acting viruses or genetic codes, but you can ask around with professors for possibilities. Some of these they believe they have the principle worked out but some are still unknown as to the cause.

• You might check out the 1918 Flu pandemic. What caused it, and it’s potential in Bird Flu (or whatever the latest virus to come out “of the closet” at the time you read this) of doing this same damage in the future. Try to bring mathematics into the picture to support your ideas. It’s not a question of “if” but a question of “when” it will happen.  In fact, we are overdue for a ferocious and very deadly strain of highly virulent influenza. I hate to say it, but we may be living on borrowed time. It won’t wipe all of us out, but it will leave the world a good deal emptier.
  

• Bubonic plague or Black Plague of the middle ages fame is a possible topic as well, though that was not a virus pathogen.
  

• To get really down in the dumps you might look up and do a paper on several of the very dangerous diseases. Ebola Virus, Hanta Viruses, Rabies, AIDS, tetanus toxins, botulism toxin poisoning, cerebral and spinal meningitis (etc. These diseases are generally fatal period or fatal if not treated immediately. In fact with Hanta Virus, I recall them bringing patients in to the Bernalillo County Medical Center in Albuquerque from the four corners region. Mouse borne disease, of several years ago. It is apparently well entrenched in the four corners region because among the Native Americans are tales about the disease, and how it’s not proper that mice and men should live together. I realize I should be careful here, by entrenched, I don’t mean ongoing,….but rather periodic (may be long periods) flare ups

• Many diseases that were very common in the old days, are not so prevalent in todays world, at least in the US (if it ever occurred in the US for that matter). Other countries still have a problem with several of these. diphtheria,  polio (I remember the sugar cubes they dropped the vaccine in, I recall I wanted to get back in line for another one. They gave me a plain sugar cube on the second trip through – I was about 4 or so), tuberculosis (though a drug resistant strain is taking hold) and so this disease is far from beaten, small pox I still have the scar of the inoculation I received as a kid. scarlet fever, typhoid and typhus, cholera, malaria, sleeping sickness, yellow fever, mumps (by the way if any of you get mumps – please stay away from me. I never had it and adult males don’t want to get it, … enough said), measles (that was a pain in the you know what and in general all over, and I blame for my ear infections I had in 2nd – 11^th grades. 

• The actual prion diseases themselves are of interest if you want to get down in the dumps, though I believe these don’t pose a substantial risk to the public at least yet. This is just an opinion though. CJD, Scrapie (we are safe from this at least for now – attacks sheep/goats exclusively at least so far), Kuru  etc. This is not for the feint of heart as these diseases are incurable at the present time, and the sort of death one undergoes is scary.

• There is a wide variety of mental health problems.  Some more noticeable than others in patients.   There is the ever present depression....this effects a vast nmber of people, and is nothing to be ashamed of having. Manic depressives.  It goes on from there.  This would be a good topic for those in psychology as you will be exposed to them sooner or later.  However in psychology do not limit yourself to abnormal psychology.....this is a huge field covering childhood development, learning theories, nature versus nurture theories.....almost a vast number of areas
  

• Related to the above topic just listed, is another form of illness (but I hestitate to call it mental illness).  Autism and it's various varieties.  Asperger's syndrome is of interest but all of it is of interest to me.  I know very little about it.  It seems like the intelligence and light are there, but rather they refuse to associate with others.  It possibly means that as far as they are concerned there are no others but only them.  I wonder what there thoughts would be like.  I am venturing a guess that there thought proceeses are so abstract and strange that to us they'd seem alien. I don't know

•        Plants as they relate to consumption and uses in human life.  
  
• The classification of the plant kingdom and details associated with the classifications.  Like how is a fern different from a rose and etcetera.    However in some detail.
• The anatomy and physiology of plants
• The chemical foundation of photosynthesis,
• Native American uses for plants in dyes
  
• Fungi (mushrooms -- and toadstools)

• Protein Transfer in Cells. Protein powers the powerhouse which is the cell.

• Papers on the anatomy and physiology of at least one of the major systems in the body: Lymphatic and immune systems. Hormonal (Endocrine) systems.Though we are supposedly all adults here, please stay away from the  reproductive system.    It's a sensitive  topic for discussion.
  -->   Muscular Systems (Calcium interactions with the muscular system)
  -->   Digestive System. Work your way down to the large intestines, so to speak, but leave out the last part of the process, I don’t really want to read about it,
         though students (and I ) say it a lot.
  -->  Nervous system chemistry messaging and electrical, impulses/synapses. Nervous system in general.   Neural biology is a growing field involving rather 
        intense mathematics.
  -->  Skeletal system – beyond just naming bones, how about the chemistry of bone cells and replacement, the mechanics and dynamics of the bones of
         the body.
  -->  Respiratory system:   Lungs and O2 and CO2 exchange process       
  -->  Here is the system that I am interested in because of a medical condition that may result in my death. Urinary/Kidney systems. Kidney’s are quite
                  complicated filters.
   --> lymphatic/immune system.   Immune system response to attacks by foreign invaders in the body.  Speaking of immune systems, there are a great many diseases caused by over active immune systems: Lupus, sarcoidosis, rheumatic arthritis and so on.  bY THE WAY YOU CAN DISCUSS GENERAL DISEASES AND DISORDERS CAUSED BY OR INVOLVED WITH EACH SYSTEM OF THE HUMAN BODY.  GENETIC DISEASES ARE A WIDE FIELD AS WELL.

•  --> Endocrine system  :--  there are a great many diseases disorders associated with the endocrine system
  

• people in the southwest are in general scared of scorpions, black widows, and rattlesnakes. All of these are small fry compared to some of their cousins in the world. Of course it varies....I am not scared of any of the local animals, where as others literally panic if they see a rattlesnake.
  
• A possible topic is:   investigate the venomosity of various spiders, snakes, and scorpions. With spiders there are two and several other possibilities. The funnel web spider of Australia, and the Brazilian Leaping Spider, also called the Banana Spider (though others go by that name), who make the black widow look like a pet.    The Brazilian leaping spider is very aptly named. Short tempered, high strung, irritable, and anti-social. It does not take much to get it to leap and bite.
• Australia brags about having the most venomous snakes (and everything else ***smile*** in the world. I don’t know about that, but they certainly do have some highly dangerous snakes and they may be well justified in their claims in this regard -- the Taipan comes to mind)  and the funnel web spider is considered #2 by most spider specialists (except those from australia **smile** ). Either of those spiders though has a high mortality rate if untreated, but fortunately they have anti-venom for both. 
• Northern Africa and the Middle East have the worst scorpion in the world.
  
•  The S American rain forests also have a very nasty centipede. 
• However, the sea has the scariest critters around in poisons.  Australia has the sea wasp off it's coasts.  Possibly the most dangerous creature in the world.  It is a jellyfish by the way..
  

• That said, you still want to leave our local denizens of local fame alone of course, they are bad enough. And when I say leave alone I mean it…they are doing a great service to us by eating insects (the spiders / scorpions locally as well as our local centipedes, vinegaroons, and others), and rodents (rattlesnakes, coral snakes, constrictors:  gopher snakes, bull snakes, racers, boa constrictors and many others). Now if they come into or very near your house, then you have to do something about the problem.  Though with snakes you might get the specialists to remove them alive. 
  
• You should feel free to go through the biology buildings and check out their snake collections.  Please don't rap on the glass though.  They have rattlesnakes but also constrictors and my old friend the gopher snake down at the far end of the hall.  I have noticed in most cases that the mice are prekilled before being put in with the snakes.....and the snakes have adopted (with one notable exception that I am aware of) to eating dead mice.....

• What is the effect on humans being exposed to the various wavelengths of the EM spectrum.

• What is the effect on humans being exposed to high frequency sound and on the other hand low frequency sound. ELF – extremely low frequency waves are used to communicate with submarines, but can play havoc with the human body.

• In animal life and certain plants luminescence is emitted at times to attract prey, mates, and sometimes as decoy mechanisms. What causes this.

• The following are merely suggestions.  Topics I have looked up and studied but be aware that it can pose some risk in today's world.  The following is not intended to give licenses to investigate bio-chemical weapons to actually discover how to disseminate the agent. but for your own edification and understanding some of the fears in today’s world, and also to give you some education on how you can minimize the risks for you and your family members (present or future), you might check up on the feared potential bio and chemical weapons.
  There are several that the government is concerned about. If you feel in anyway nervous about studying this field don’t do it. The paper is an honest, ethical moral paper but with the paranoia in today’s world it could have some risk associated with it, especially for over seas students. I doubt seriously however, that you will get in trouble, since a lot of the information is actually easily and openly available to the public (and hence to much of the world ) on the internet, to educate the public about the dangers (real and possible) on this front which is precisely what I am asking for. It would be a good paper if you want to try it out. However this is your risk. I have investigated many of the agents myself, not out of worry about them, but entirely for research purposes. In fact, I truly am not scared of terrorists. They are out there, for sure, no doubt about that, but I for one am not going to let the possibility of getting blown up, shot, decapitated or whatever interfere with my life. If it happens it happens.  Every day when we get up from our beds, we run the risk of being killed or dying from one thing or another.  Just be careful but don't let caution run your life.  

• I was wondering in my mind, if one could send out neutral viruses (actual organic, not computer) that are programmed to seek out viruses in the body of humans or animals that cause illness, and that will latch onto them and then tag them, to be destroyed by robotic devices (not available just yet) that will use sonic vibrations, radiation, miniaturized particle weapons (charged ions), or EM radiation to destroy the virus that causes illness, . It could be something even more mundane. A virus that takes over a harmful virus leaving a neutral virus in the body. In this same vein and a process that has been around for years via antibiotics, but have viruses attacking not the human body but the cells forming the bacteria that can not be treated with antibiotics anymore. The art of forcing bacteria to attack other harmful bacteria has been around for ages relatively speaking, the modern antibiotics age was in this genre, but went beyond living bacteria into man made “bacteria”. Another form of treatment in the old days with open sores and infections was to introduce special types of maggots into the inflammation. They would eat away the diseased and dead tissue and leave the healthy tissue. This sounds incredibly unsanitary, but you have maggots coming from externally cleaned eggs and of course the maggot as it comes out of the egg is clean, and most of all the process works. My ideas are kind of an extension onto these basic ideas of yester year, and the basic thought that light of various frequencies can and does kill bacteria. …Infrared…..visible (rainbow)….ultraviolet…. as well as x-ray and microwave. It may even be possible to vibrate a bacterium or virus to pieces at the right frequency of sound.

Ancient Life and Archaeology

• This topic is of interest to me in a rather deep and fascinating sort of way, ancient life. If someone wants to try to write a paper on this it will be simple to do, but I would like someone who has an ongoing interest and the willingness to write more than their allotted pages. As the geological ages pass by you are seeing a changing flora (plant life) and fauna basis. Carnivore and Herbivores. Where you want to start is up to you but go through the geological ages and describe the types of animals and plants you see. My interests are really post dinosaur when the mammals start to take over. Most people will think of Saber Tooth Cats and Mammoth.. These guys are cool, but were relatively recent. Dire Wolves and the giant short faced bear the same way. But what kind of mammals were around before Man had started the ascent into modern man. What is the animal and plant bestiary back before man came around. By all means though include the more recent mammals and go back but my interests are the periods before modern man i.e. Cro-Magnon and before. Now mostly I am interested in fauna (animals) but I also love plants (flora).
              If you want to be shocked (at least I was…) go take a look at the grizzly bear skull in the biology building (where the rattlesnakes and constrictors are kept on display but in the other north south hall about half way down) and then go to the museum on the ground floor of the physics building. (BTW: I think all of you students should go through there as well as other museums on campus and locally at least sometime before you graduate). Right before you enter the door on your left is a display cabinet with a model of a skull of the giant short faced bear (this dude is the biggest bear known to have existed – highly carnivorous, long legged and hence built for running down it’s dinner) NOTE: THEY HAVE CHANGED THE EXHIBIT. THE SKULL IS NO LONGER THERE, but it was about 4-6 times larger than the grizzly bear skull in terms of volume. They have two sabre tooth skeletons now and of course the skull of triceratops and Tyrannosaurus....the tyrannosaurus fossil had teeth marks in his upper nose area. So that is why I was laughing that one day about “bit on the beak” Anyway, Think about your idea of a grizzly bear. Terrifying for most people, his physical presence is scary. Yet he only stands 3.5 feet at the shoulders. A short faced bear on the other hand stood 5.5 feet at the shoulders and his head topped out at approximately 6 feet. Kodiak bears fit in between. If you have seen the commercials with the Kodiak in it you see how he dwarfs the Datsun truck (not big truck but the smaller type you still see on the roads). 

• Methods of Dating samples of artifacts made of formerly living items, geological stones, fashioned stoned implements:  
  

There are several techniques used in Archaeology and Geology. List several different methods. Need not be all radioactive decay models. Usually they always drag out and put on display the C14 system. Okay but boring…there are other methods of dating either absolutely or relatively.

Seriation : As technological advances are made they tend to occurs in jumps but not drastic jumps. In fact in most cases, minor but significant changes are made in revisions from A to B to C to D and so on. You see the same in our technology today. Our technology of course is more advanced than it was 10,000 years ago but we still build steadily on the work of the last genius that came along. This is used as a dating technique

Stratigraphy. In undisturbed layers if you can find an artifact between two layers that you can date, the artifact was placed there sometime between these two dates. This means though it was just laid down. It could not be buried, or there can be no upheaval of the artifact itself without upheaval of the surrounding matrix

Dendrochronology – tree ring dating. Why does this work? This was used for fixing the dates of many of the wood beams used in Chaco Canyon. UA is one of the foremost universities on this topic.

Obsidian Hydration – dating of obsidian tools

Magnetic dating techniques. As a substance is heated it’s magnetic core tends to align up to the earth’s magnetic N at the point in time. So carefully removing the remains or a spot from some long dead fire we can determine when the fire was active. This tells us when our friend had built a fire there to cook dinner or warm himself, or whatever he was doing with the fire.

Amino acid racemization is a relatively new method based on amino acids of former living creatures are parts of creatures. Teeth, bone, etcetera

http://pubs.usgs.gov/gip/geotime/radiometric.html -- this site drags out the C14 method but they describe other radioactive decay methods. Potassium=Argon methods primarily importance.

http://anthro.palomar.edu/time/time_4.htm illustrates some of the above techniques.

• Topic 3: There are many items out there that are archaeological mysteries. You can pick any such topic and research and conjecture the purposes of these features.  For example why did everyone go in for building big monuments? Egyptians, Central america, the eastern US (moundbuilders). Stonehenge in England. You want to hear name calling look up and try to find articles on how old the Sphinx really is.  PLEASE DO NOT TRY TO TELL ME SPACE PEOPLE DID IT OR THAT THE PEOPLE OF ATLANTIS DID IT.  I don't believe in the former, and the second, whereas I can believe Atlantis did exist, I do not buy the idea they were advanced technologically.   I can believe in Atlantis because I have a great deal of trust in what Plato and other ancient writers said.  The source of Atlantis is from the works of Plato.  How about the mystery of Chaco Canyon. For such a big place there are no burials to speak of. What was IT used for? Why are the road networks into and out of Chaco, so dead straight. If they come to an obstacle instead of going around it, they go over it.   Just don't get whacky on me.

• What is the foundation of physical anthropology.?
• Why is the study of genetics and DNA analysis important in the anthropological field.
• For those of you who are not believers in evolution bypass this next question:   Discuss the latest theories in the development of humans from distant ancestors to our fully developed human today that we call civilized, some more so than others. This means working your way back in time from us to CroMagnon Man (shared his world with Neanderthal) and work your way back.

Astronomy – Look at the research paper ideas (actually presentations) on the walls in the various physics buildings

• Possibly the scariest thing in the universe is the collapsed star called the black hole (for reasons that some morally bankrupt people with filthy minds find the title absolutely and hilariously funny I will hereafter call BH). I say the scariest because if one could latch onto earth, we would have little to worry about anymore. It is still not known with 100% certainty that they exist, but evidence is sure piling up that they do. You will be able to put together a good paper on it’s description and what it is. Include in your description the halfway point of a neutron star. What is it? Talk about the dynamics of the BH and think about some conjecture questions. How long does it take to form a BH. Once the collapse starts that is. What would happen if two BH’s event horizons overlap? In fact, what is the event horizon? How does a BH cannibalize another star? By conjecturing a neutron star in orbit with a BH. By crossing the even horizon what happens to the neutron star? This emits a very strong blast of what type of radiation? BH’s that are not fed will tend to “evaporate” what do I mean by this? This process is slow and gradual but a time will be reached when the black hole will eventually explode releasing all the material it consumed at a point many countless years down the road.

• See what you can find on quasars. This phenomena is not really well understood yet. Are they close by and less strong, or are they incredibly powerful but very far away? What powers them in general?

• Describe for me the Hertzsprung Russell Diagram (sp?) and it’s meaning for the life cycle path of a star, in some details, and describe the various stages that a star might go through. Give examples of stars that are in each position on the HR diagram. Names, distances, associated constellation names. Talk about magnitude, absolute magnitude. This is a simpler type of project for students who just like astronomy at an amateur level. It’s fun, though kind of boring to me scientifically, but for me, the soul inside, it is pleasant to just think of the star’s name and it’s position in it’s life cycle.

• Galaxies. Islands in space. Why does a galaxy stay together? What causes the massive clumping at the center of most galaxies, and the more sparse spacing further out in the galactic arms? What are the shapes of galaxies. Describe the interaction of galaxies with each other. It ain’t a pretty sight. Sort of like meal times at the dining halls on campus with some of the students at least in the old days when I was a student. The stars never need collide and in most cases they don’t, but the tidal forces of gravity play havoc with both galaxies.

• What is meant by the statement of curved space? Looks simple but one heck of a paper. Look up the most recent discussions on the actual shapes of the universe we live in. The grouping of the galaxies. There is also one idea that is saying that space is infinite but finite at the same time That is via some manifold shape, a torus for example but others are certainly possible, that as we look out into the heavens that the light from our galaxy may be moving along the manifold and what we see is in a sense a mirror image of our selves. In other words, one of the galaxies we see out there could very well be our own we are just seeing the light after it has made a transit of our universe. Some of the topics covered in this question state that the universe is really not as large as we make it out o be, but rather that space is twisted in a strange topological way. Even so, the universe is still far bigger than Usher would have us believe.

• Comets are one of my favorite phenomena to talk about. Investigate the type of comet orbits. Some of them appear and go away never to return again, others are comets that have a periodic orbit, albeit some are very long periods. Investigate the periods of several comets of recent notoriety and note that some of the comets have periods that have been increased or decreased. Why is this? In other words, what causes that change in periods? Investigate among others Comet Hale Bopp -- Almost everyone saw this particular comet, most of you were teenagers at the time. Comet Hyuatake -- This was my favorite. I was fortunate to be out in the country at the time this went by. It was the comet of my lifetime up to that time, and probably will remain so. I liked it better than Hale Bopp. Out in the country it’s tail was visible over about 100 degrees of the sky. Most city folk didn’t see much but maybe the head. You’d have to be in the dark and open country to really get the full effect. Hale Bopp was more impressive in the country also, but it couldn’t hold a candle to Hyakutake. Yet it had a formidable period. When was the previous time of visibility before 1995 and when will it appear again – Hyakutake. Same with Hale Bopp. Comet West was supposed to have been spectacular, but I never saw it. It was an early morning comet and back then I never saw many sunrises much less predawn skies. I was and am a night owl. Look up it’s period and be prepared for a shock. I know for a fact, I will not see it’s return, and you guys will never see it, nor will your grandchildren down several generations. If I had known how long it’s period was, I would have stayed up all night to have seen it.   Halley’s Comet, irregardless of it’s poor showing in 1986 was a sight to behold in the older days. 1910 at least by stories my grandfather passed down to my mother and then to me. I will certainly not see it’s return, though I suspect many of you will. Though not all of you. It’s due back in roughly 58 years. There is a possibility that it may not be a comet anymore though. In recent years as it was being observed entering into it’s domain out in space there was a strange intense brightening of the comet. It is suspected or hypothesized it “ran into something” so it might be a bunch of smaller comets now or maybe just meteorite bait, or perhaps will not return at all, and there are certainly other possibilities. It could’ve been pushed into an orbit that is much longer, or else much shorter (in which case it may be coming back as we speak), or even expelled from the solar system (meaning no human will likely ever see it again, until we get a Starship Enterprise). You guys will have to just wait and see. If you write on comets, you might mention the great comet scare associated with the 1910 visitation of Halley Comet. People were panicking because earth passed through the tail and the comet contains cyanogen gas. People were selling “gas masks”, comet pills and other snake oil remedies. You might find out what you can on this front.

• These next two questions are theoretical in nature. There is compelling evidence for something called Dark Matter. Why are astronomers including this possibilities and what are some theories as to what it is. Think anti-symmetric particles.

• One would expect that the universe should be slowing down or else not accelerating at all, but it’s not. As it moves out it is accelerating. They are hypothesizing a dark energy that is causing this expansion, find out what you can if anything as to the theories and understanding of what dark energy is. As these are kind of cutting edge work in astronomy, you might take these two questions as one question and work on both ideas.

• On a different tact: a star was recently found. Actually dead star that is billed as the galaxies largest diamond. How can they make this claim? If indeed it is a diamond and assume it is, and by diamond I mean Jeweler’s grade (which it ain’t)how much would it be worth? I bet you’d make Uncle Scrooge, much less Bill Gates look like paupers by comparison if you could lay claim to it. If you do this paper, I have copied the article from the internet that sparked this interest and also contains the necessary information. I will send it to you but I will also have to find it in my hard drive.

• Astronomical Appartus:  There are a wide variety of telescopes used in hobby amateur astronomy, more serious amateurs, and professionals. With the first two groups these are mostly optical wavelength telescopes, of which there are 3 different varieties. You can check these out on the display cabinets of the first floor physics building. Professional astronomers use not only the three mentioned telescopes, but several telescopes in tandem in a sense. Professional astronomers also do not limit themselves to just optical wavelengths . I would like you to investigate various observatories and find out the types of telescopes they use and other apparatus they use in their work. This is not limited to earth based telescopes. Investigate the various wavelengths used in their work. Though it’s not a major observatory for research anymore, though it still does some, check out Lowell observatory on Mars Hill in Flagstaff. If you get a chance, go to Flagstaff, and check out their viewing nights. The Adel Observatory on the NAU campus also has public viewing nights. Kitt Peak is another major observatory in Arizona, the naval observatories do classified work in Flagstaff and as such are off limits to the public, though they may have a few nights for open house I rather doubt it. Mt Graham (if it’s still there – a wild fire is within a ¼ mile of it as I write this) is another telescope in Arizona. I believe ASU does have observation nights during the semester but I am not sure about this. I have stopped going to observation nights at observatories because the general public seems to be hung up on observing the moon and the planets, whereas, I am much more interested in deep sky galaxies. The andromeda galaxy is the only other visible (easily) galaxy in the sky, and even then it’s sort of a smudge through unaided eyes. Actually, our galaxy the milky way, is the easiest galaxy to observe but we see only local stars, and bands of lights that form other arms of our galaxy, the milky way proper.

• Quaoar, Varuna, Ixion, and Sedna == Four planetoids you might check out. We are finding a wealth of large (relatively speaking) bodies out beyond Pluto that we until recently, didn’t expect to find. The Oort cloud from which it is believed that comets originate from is still much further out. I don’t expect you to work out the orbital details but I would like you just to familiarize yourself with the fact that new planetoids are being discovered. Whether they become planets is unknown but the fact that I’d like you to realize is that discoveries are being made in greatly different fields each day. You might also check out the names of the planetoids. They come from various folklore and mythology of various tribes and peoples. Each group of people have a rich blend of culture and history and mythology. I will admit though this is getting far from mathematics, but my interests as I have said span almost all of human knowledge.

• Geology and geo-engineers….a very dark project for you …. Investigate the peak oil supply scenario. What does this mean for humanity as a whole over the next several years. Don’t do this though if you are prone to getting down about things. It’s NOT good news. Not hopeless but not good. I just remember the gas crunch and gas lines back in the 1970’s. It was then that we should have started a Manhattan Project type of approach to create alternative energies. I had hopes at that time it would happen, but public apathy and politicians happened first.

• Mt. St. Helens (the BIG blow up) was an eruption that caused the people in the world to look and holler, “uncle”, but it was a small firecracker compared to several volcanic eruptions in the past. Mt. Pinatubo in the Philippines was an even bigger eruption but never the less it was small fry as well. Look into the past and report on volcanic eruptions that were truly huge, there are several, as will be an eruption in the coming future that I sincerely hope is not in my lifetime nor your lifetimes, nor the lifetimes or your children. Hint: It is in the United States (at least the one I am thinking about…there are at least two of these “Super Volcanoes” in the world that are active.) and will play havoc with much of the country, and disrupt metereological phenomena for quite some time, possibly bringing in a “nuclear winter” of sorts. It is a well known national park also. Other major eruptions: Santorini comes to mind, as an example. Santorini is the volcanic eruption that ended the Minoans and those few in the science field who take the legend of Atlantis seriously, believe this volcano was the cause of it. Look up “largest volcanic eruptions”, and read over the sites. I know of several others.

• Way back in time at the Triassic Permian junction there was a huge extinction. It is believed that the Siberian Traps was responsible (though the alverez camp still swears a meteorite impact caused this – there is no record of a relevnt crater)). Later on, they speak of a great meteorite that ended the age of the dinosaurs. Pushed by Walter Alvarez. It is spoken of like this is a done and decided deal and that the asteroid single “handedly” wiped out the dinosaurs. IT IS NOT a done deal. They are still fighting. I do believe that the meteorite was the final straw that broke the dino’s back, but I believe the Deccan Traps really helped the process along or perhaps vice versa. These two Traps I mentioned (by the way the Deccan Traps are in western central India) are considered the 1^st and 2^nd in terms of eruptions if you want to call them that. I would hesitate to call them actual eruptions though as they more leaked out magma than violently blew up. Nevertheless for our purposes I will call them as such. They made the other eruptions I mentioned and those you’ll find look like Sunday Picnics. You might check into other large basaltic lava flows….many of which were in what become the USA, and are associated with other extinctions. Some relatively major, others less so. If you do this project, I have a handout for this one. There is some question that the asteroid impact may have initiated the geological event.

There are on record 5 major extinctions in the history of life on the earth. Possibly more.

• I FELT LIKE TAKING THIS ONE OUT AFTER THE EVENTS IN THE INDIAN OCEAN ON DECEMBER 26 HOWEVER, I WILL LEAVE IT:  Earthquakes and earthquake lights (considered to be non-existent but now is starting to be accepted.) Tsunami’s are closely tied to earthquakes as well. What can you find about the strength of earthquakes? BTW: Hollywood’s “10.5” is fake so I don’t want to see that reported on. The earthquake I am thinking about was big enough. I heard about it as a kid and it caused a lot of nightmares for me. Some of my worse nightmares came from this earthquake. Talk about the potential causes of earthquakes, their generally resulting tsunamis and the seismograph waves. I know a fair amount about seismographs due to designing the electronics for an remote dialing alarm system for a certain geology department. By the way, where was the largest known Tsunami in pre-history, history, and in recent memory and how big (tall) were they.
  

• As I am writing this in the height of Tornado Season, write about what causes a tornado in more scientific and if possible mathematical terms. In other words, the physics of tornadoes. The Fujita scale represents the relative strengths of tornadoes but they say that a F6 is not possible. Why do they say that? However there is one possible tornado that may have reached an F6 possibly but died down quickly from that level. Where and when was this? The experts have denied this but why did they deny that it was an F6? I do take offense at some of the attitudes of scienctists though. “An F6 is impossible” or “a 10.5 scale earthquake is impossible” . If they said “unlikely” that’s okay, but impossible is a fighting word, because I believe they are possible. What is the difference between the tornadoes of the Midwest and our friendlier, (though still at times, irritating) dust devils we have in the American Southwest. I don’t attach sentient terms to dust devils but I swear at times they exhibit an intelligence to them. If you want to hear stories about them, I can tell you several but on an individual basis. I rather enjoy seeing them (the small ones) in spring and early summer. Most dust devils are just twirling dervishes of air that sweep up dust, papers, small debris and swirls them around, but some have torn storage sheds apart.

• You are reading this, hopefully, during the hurricane season (in other words you are reading it in late summer and early fall….not saying or meaning to say I hope there is a hurricane season. Of course I hope there isn’t, but we will have one). As with a tornado what causes a hurricane in more scientific terms. What kind of scale do they have for hurricanes? What causes a hurricane to die when it hits land fall. How are hurricanes and tornadoes related how do they differ. What is the physics of a hurricane

• My favorite is lightning. I am rather stupid about this phenomena…I go out and watch it. Though not to far from shelter. If it gets to close I duck. What is the transfer of charge rate? the potential of a bolt? Ranges? How hot is it? What causes it? What about Ball lightning….supposedly folklore in the not so distant past, dismissed by scientists of that time, but almost completely accepted by science now. Tesla used to be able to generate these balls of lightning at his Colorado Springs Laboratory. Also scaring the “you know what” out of the local residents of the area. What are possibilities of what they are composed of? Tesla said they were very interesting to him but of no use. It is thought in part they may be balls of ionized gasses and hence may be useful in firing up a fusion reactor, but nothing yet, that I know of, has been accomplished on this front.

• What are the differences and similarities between auroral displays and the phenomena called St. Elmo’s Fire. Do not use as a difference that St. Elmo’s Fire is a local ground based phenomena, and auroras are high elevation phenomena. Auroras can and do dip down to the ground and have been noticed to sweep over the ground as well, though rarely, and aircraft does get Elmo’s fire on the pointed parts of its design, not often though. Further how about the will o the wisp....does this have any grounding in facts or is it part of the folklore of the American scene.

• On the lighter sides of geology/meteorology. Snowflakes , raindrops, cloud types and formations. These topics are not as dramatic as the previous ones but never the less are interesting topics. What actually makes a snowflake unique? Why does it have these beautiful designs? How big can snowflakes get? How small? What about raindrops. How are drops formed? Why not mist only? Clouds ? What are they how are they formed. Why are they white (for that matter why is snow white? At least until it hits the pollution, ground level., kids and dogs) What is the correlation between cloud type and elevation if any? How do just the regular non-lightning storms develop, how about full fledged thunderstorms? Hail stones? What are they, how do they form? Why is the sky often greenish in hailstone weather? What about rainbows? What causes them what causes the double rainbows? What is the mathematics behind the rainbow?

• Most of the items I picked for Geology are fast acting “glamorous” items that really do not do justice to geology. By far the most powerful processes are going on that we never notice but never the less are very powerful and ongoing. Mountain upheaval, plate tectonics, erosion, inland seas and lakes, glacial ages, all of these are of interest to me as well. Magnetic pole reversal. Each would be a good paper topic in and of themselves. The Grand Canyon itself could be the stuff of many, many papers though I wouldn’t. I find it kind of boring, but if the Geology of the grand canyon turns you on go for it. It will be read.) Again, if you have a paper topic you’d like to do, feel free to talk with me about it. Here is one topic that I would really like to read about: There was supposed to be a huge lake up in NE Arizona in the not so distant past but still well back in time. Drained out by the San Francisco Peak’s eruptions not the 1054 or thereabouts eruption, but at some time before then. The Native Americans would have passed down real stories of it if there were. By real stories I mean basically by word of mouth, “eye=witness testimony”) They may have older legends of the lake from a more distant past, but I do not know. It may have even been before people were around when the Peaks were being formed. Extending from near Flagstaff over to the west side of Ganado or thereabouts. I just know about it’s existence but nothing really much more than that. I was going to track it down but I haven’t yet. As you cross the region you do get the distinct feeling that not so long ago it was part of a bed of a huge ocean or lake. I know from what my brother say that this lake was much later in time then the inland seas, but beyond this I know little about it.

• Light emittence (luminescence) occurs throughout nature. For geology what causes luminescence in certain minerals. What are the colors that can be emitted.

Physical Sciences

Look at the research paper ideas (actually presentations) on the walls in the various buildings.  Chemistry has available ongoing projects they post on their walls, as do physics, and other departments.. They (chemistry) have a good woodcarving of a classic work about Alchemy. Though this is mystical in nature, the history of alchemy and its ideas would be a good project paper. The Physics Buildings all have research displays for the public to look at, and all are good potential projects.

Physics:

• Background: Much of modern physics at the cutting edge of research is trying to make two of the pillars of modern physics work with each other instead of trying to kill off one another. Einstein’s Relativity Theory – Works beautifully on the large and super fast scale, but fails on the atomic scale. Quantum Mechanics: Heisenberg, Pauli, Schrödinger, Bohr, Dirac and many others work: which works beautifully on the small scale but fails miserably on the scale of the big and fast. I personally feel that both theories will be adopted and modified into a super theory that will incorporate both theories. . Will this new super theory be the be all and end all of physics? I doubt it very strongly The track record of people saying we know all there is to know is very 100% wrong. Personally I think we will never know the end of what there is to know. I feel it very strongly. I think the universe is so vastly more complex and strange than even we know in our wildest dreams, at least from our perspective, or possibly simpler than anything we could imagine, if seen from a different unknown reference frame.

• One of the most recent theories among others is a real viable contender. It is called string theory.

• I believe Einstein’s work is valid for the area he was looking at, but will have to be modified.
  There is a famous result called Bell’s Theorem, that states that electrons can and do routinely “communicate” instantly over apparently any distance, in direct contradiction of Relativity Theory. Use my web site I have provided and your sites you can scratch up to address this and write up a short paper on this topic. Some people say this may mean the end of Einstein’s work, but I do not believe so, since this “spooky action at a distance” as it is called is done on the subatomic level. Bell’s Theorem in other words. In other words in the realm of QM which we already knows is not relativity’s domain.

• There are four forces known to physicists: Gravity, Electromagnetism, and the Strong and Weak Nuclear Forces. The first two are known to every student in the sciences and engineering and the applied mathematicians. The other two are less well known. What are they, and how do we know they exist? For any of you who have fallen off a bicycle (which is all of us or practically all of us) or fallen out of an apple tree, or fallen off a fence when you are trying to “tight rope walk it” will testify. Gravity can lead to painful results, but never the less it is the weakest of the four forces.

• The following topic sounds like total science fiction but is starting to become seriously considered by several of the leading physicists of today. This theory may be true because of the way the fundamental Schrödinger wave equation behaves. This theory is called the MANY WORLD’S HYPOTHESIS. Surprisingly there is evidence and support going for it, though it seems like it comes straight from the twilight zone. Keep in mind, physics today is becoming more and more “Alice in Wonderland, Through the Looking Glass” in nature, and in general more abstractly deep. The cutting edge physics is getting very strange and counter intuitive.  There is a theory that everything is interrelated, and that as a consequence a butterfly flapping her wings in south America will trickle down to changing the weather in Kansas in one way or another. This idea and it’s ilk relate to Chaos theory which is also a possible paper topic if you want. However even deeper yet, suppose the butterfly did not flap her wings at all. The theory is that because of that choice that she makes (“to flap or not to flap that is the question”) she creates an alternate universe for the choice that she did not take as well as the choice she made … to flap.

More specifically. You are faced with a choice. You can either marry person A or not. The theory goes that at this point there are two universes that branch off from this point, one in which you married person A, and one in which you don’t. You are living in both universes, but you really cannot determine which universe you are in and you have no way of knowing you exist in this other universe, and worse you do not even know of the existence of this other universe. If you decide not to marry person A, you are living in one universe and you have no idea that you have a copy in another universe who is now married to A. In this other universe this copy of you says “hey, I married person A this is real. There is no universe where I didn’t choose to marry person A. I admit I had doubts, but it was just a choice I had to make” This is highly abstract, and weird. Choices leading to different universes.Right at the point where they split off they are identical but after that they move their separate ways. For instance if you married A, your child may develop a way to actually travel easily to the stars and make contact with other beings, but in the other universe where you didn’t marry, the child is not born and so the world never learns of the other beings. You can see the ramifications I think. As time goes by the universes diverge more and more from each other. This theory, and it’s not as wild as it sounds i.e. it’s gaining strong support, is called the many worlds hypothesis. Basically put, everything that can happen does happen in this hyper set of universes.  If this idea disturbs you, don’t let it. You have no way of actually getting to or from these universes, and no way of actually even verifying that they may or may not exist, except in the most highly theoretical of ways. As far as you are concerned there is only one universe. The one in the here and now and your memories of your choices that you’ve made in the past. Besides think of this: it is just a theory. Don’t let the idea that other copies of you existing elsewhere bother you. In a way I find the thought comforting. Somewhere in some universe I made the right choices. Does this interfere with free choice? I don’t believe so. Each copy of yourself made a choice of “your” own will and still is making choices.

These sites are just some of the sites. Keep in mind, I want your conceptualization only. Dig around for sites that explain it in more of terms of what you understand/   http://www.hedweb.com/everett/everett.htm this site is probably sufficient. Realize in the roll call of those who support the hypothesis and those against it, that these are the leading physicists of the day. Hawking for instance supports it totally, he is possibly the foremost physicist of today and is at Cambridge. I am far from being a leading mathematician / physicist, in fact I am rather dumb in comparison to many, but I believe in it, at the least I am fascinated by it. The reason it’s so well accepted by many is from the nature of Schrödinger’s wave equation and it’s ramifications. It’s not a “I’ll believe in this theory today” kind of thing. In other words there are good reasons to believe this hypothesis may be true, based on the fact that all things are possible, but once the wave e equation collapses that’s when the decision is made. Check out Schrödinger’s Cat experiment.

http://www.mit.edu/people/dwward/papers/manyworlds.pdf

This paper put out by MIT is to technical for your use but illustrates some of the questions and answers that are leading scientists to accept the idea more and more. You might look over the English words though for ideas. Like I said, I am not after a detailed mathematical description at all. The idea is just to deep for anyone I personally know, including myself to work with.

Particle Physics:

• One problem in honor of one of my favorite branches of mathematical physics. (I worked indirectly with CERN laboratories in Switzerland et.al., on my master’s thesis in mathematical physics. Basically anti-matter: don’t let anyone tell you it doesn’t exist … it does However in very small quantities) However more to the point. CERN does work with subatomic particles. Look up and reference a few sites on the subject “particle physics” and come up with the latest listing of subatomic particles. Write up a report on the subject and submit it for this problem. For filler if you need it, you might put in the classical atomic theory of neutrons, proton, and electrons. Be sure to reference your web sites you accessed (you may use these listing of web sites as part of your page long report. As always be careful about accessing stuff from the internet on school computers.. You will be safe enough, I think, looking up the words: quarks, bosons, leptons, etc. Now quarks come in various varieties. But this is part of what I want you to check up on. This is an area I would like to find work in as an assistant but this field is so difficult that even the assistants have PhD’s in physics.

Heisenberg’s Uncertainty Principle  [Warning I haven't check these links lately)

• Heisenberg’s Uncertainty Principle. Keep calm and don’t panic: http://aolsearch.aol.com/aol/redir?src=websearch&requestId=60bde7664ac16b3d&clickedItemRank=2&userQuery=Heisenberg+Uncertainty+Principle&clickedItemURN=http%3A%2F%2Fwww.aip.org%2Fhistory%2Fheisenberg%2Fp08.htm

&   http://www.mountainlake.com/beyond/complem2.htm

A version of these principles is given in the site right above this note, but in essence states that on the order of the very small size. It is impossible to know both the momentum (that is its relation to its speed) and it’s position simultaneously. Similarly it is impossible to know the time of the observation and the Energy of the particle simultaneously

In light of the equations given in the second article

(∆x)( ∆p) > h and (∆t)( ∆E) > h. where h is Planck’s Constant all you will need to know is that h > 0

the ∆ symbol is a change or in our context the spread of range or uncertainty of whatever

∆x the uncertainty in position, ∆p uncertainty in momentum ∆E the spread or range of Energy and ∆t the spread or range in time.

My question for you (and it’s nothing deep). If ∆x is made small, (i.e. the more certain you are of position) what happens to ∆p the uncertainty of momentum? And vice versa

Answer the same questions for the ∆t, and ∆E in above remember h > 0. In your everyday life in the every day world the Uncertainty principle becomes unimportant. For all intents and purposes you can know both the position and momentum of a baseball in flight, as well as where it is in time and it’s energy level. It is in the range of the small that everything becomes fuzzy and uncertain. The world of electrons for instance.

Extend this topic out into a more detailed explanation of Heisenberg’s Uncertainty Principle.

Pauli Exclusion Principle:: I did not particularly care for Pauli from what I have read, and what my older brothers say from their own studies, and more significantly what my dad had to say about him personally, but nevertheless his principle is important. My question is without a lot of math details if you can extend this topic into a paper. Never the less if you are interested in this principle you can try to prove me wrong. If you can’t do so, don’t start over, bring in some of the other Principles. Try to cover the topic completely though

Quantum Tunneling   It is an observed fact that electrons can pass through different holes simultaneously. Electrons can and do burrow through potential barriers (think walls) on a regular basis. In fact, there is a chance….very slight that you might tunnel through a wall yourself, or more painfully, you may tunnel through the chair you are sitting on. I will let you handle the rest of the scenario. The tunneling effect is in fact the foundations of the tunneling diode more professionally, but less accurately, called the Esaki Diode, which leads to an area of negative resistance at least theoretically. Investigate this phenomena through the use of the internet, or through printed texts in the library.

Relativistic Mechanics. -- Don’t Panic.  Einstein had two theories (neither of which netted him the Nobel Prize in Physics – he won that for the photoelectric effect. A much simpler idea. It possibly could have been, as my old Physics professor said, religious bias. Einstein was of the Judaism faith. I think myself it was the relativity theory was just so wild and revolutionary for it’s time. Many did not take it seriously until later when the experimental confirmations started coming in) the special theory of relativity (which these problems relate to and which is established as true at least for now) and the general theory of relativity (which is still being questioned and attacked). Never the less the experimental evidence does support Einstein’s view in the world of the big and ultra fast. It breaks down at the very small where Quantum Mechanics takes control. Where the evidence strongly supports the QM view. Much of the ongoing arguing in Physics today is trying to get these two theories to work together instead of fighting with each other.   However I am not out to talk about this area but to merely give you some simple general ideas on Special Relativity. At low speeds plane speeds, shuttle speeds, driving speeds, walking speeds, there is a bit of influence from Special Relativity but the effect is so negligible as to be pointless to point out.  The everyday world is classical physics in nature. Which was developed fairly much single handedly by Isaac Newton.

Through the Lorentz transformations (contractions and dilations) we know that the mass m of an object is dependent on its velocity.

M = Mo/ √ (1-v²/c²) Mo represent mass at rest

L = length directed in line of motion = Lo√ (1-v²/c²) Lo represents length at rest. I.e. Lo the distance to the andromeda galaxy before you start ~2million light years.

There is a time factor as well but we will not use it.

The thing about these contractions though is that if Tom is on earth and Bill is in the spacecraft, Tom will see a clock on the spacecraft slowing down relative to his reference, and Bill will see the clocks in Tom’s reference slow down as well with respect to his reference. The time paradox comes about in inputting acceleration into the equation. Basically put, Bill and Tom may have started out the same age, but by the time Bill returns (to return he had to undergo an acceleration somewhere) he will find that Tom is older than he is. How much older depends on velocity of the craft and further how much time elapsed in his reference. So in effect traveling into the future is very easy though technically we do not have the capability yet. Actually, that is a stupid thing to say….we are always time travelers into the future, it’s just we take an inconveniently long time to do so….one second at a time. What I mean to say is we can bypass the usual tick-tock, tick tock tick tock and return at a time when years have passed on earth but mere seconds have past in your reference.

Traveling into the past is a different kettle of fish. Enter Kurt Gödel, possibly the greatest and certainly one of the more eccentric mathematicians of the 20^th century though I think that guy that just passed away, Paul Erdos, had him beat in the strangeness department. Kurt’s main paranoia was fear of being poisoned. He would not eat anything unless it was cooked by only certain people.  Anyway, Kurt Gödel, primarily a logician, and one of Einstein’s closest friends at the Princeton Institute of Advanced Studies proved that time travel into the past is mathematically possible. Mathematically possible is a long cry from actuality but it possible. Only recently (well within your life times) have physicists come up with potential schemes for doing so and all are well beyond our technology at the present time. Einstein was on record as saying, “even I don’t understand my relativity theory after the mathematicians got a hold of it.” I don’t know for sure, but I think he may have been referring to Kurt’s contributions. This time travel idea of Einstein’s and Godel’s work makes me discount a lot of people’s thoughts when the say time doesn’t exist except in so much as motion makes it exist. I think time is much more fundamental in nature, motion exists because time is there for the forces to make acceleration and hence motion possible.  Anyway write up a brief paper on special relativity. General relativity will be a daunting task for you guys at this stage, but you can incorporate some of the ideas in your paper. Probably your physics textbook will have a great deal of information you can use here. Halliday and Resnick’s Book had several chapters on this and more modern physics topics..

• Since the concept of time travel into the past is starting to become obtainable by the physicists, not just mathematicians. Investigate this topic on the internet as well. Stay with more standard sites with an .edu, or a major science magazine like Smithsonian, Scientific American, Popular Science, even Popular Mechanics as a I recall had an article (but I might be mis remembering this….it seems like it was out of place and so caught my attention). Like the many worlds hypothesis, this topic also attracts the fringe element, so be sure you are dealing with a reputable magazine or site.

ZERO POINT ENERGY Theory. I do not believe in it myself, if only because it promises free energy limitless and for essentially nothing. I do not believe in “free lunches”.  However it is intriguing.  If it is true, this is a viable contender for our future energy needs.  I do not tend to believe it myself, but who knows. It's called Zero Point Energy Theory For the first time in history, a lot of media attention is being paid to the sea of energy that pervades all of space. It just happens to be the biggest sea of energy that is known to exist and we’re floating inside it. (Credit due to The Sea of Energy by T. Henry Moray for the idea.) Not only is it big but its energy is estimated to exceed nuclear energy densities, so even a small piece of it is worth its weight in gold. What is it? Many people are not sure what "zero point energy" (ZPE) is. Most agree that virtual particle fluctuation contributes to it and van der Waals forces don’t explain everything. Does it offer a source of unlimited, free energy for homes, cars, and space travel? Depending on who we talk to, ZPE can do everything and ZPE can do nothing useful. How can the energy be converted to a usable form? What are the basic explanations of ZPE and the new discoveries, which have rocked the U.S. Patent Office, Physical Review Letters, Science, Scientific American, and the New York Times? Why is ZPE implicated in the latest confirmation of cosmological antigravity? Can the Casimir effect be a source of energy? This article is intended to give a review of the latest developments (as well as an introduction to the topic for those who are non-specialists).   I do not believe in this, but if you write a paper on this topic, support your ideas with intenet web sites, so that I may check them out, and it will net you your points.

• What are the fundamental colors. Which do you mix to other colors?

• General optics of lenses, mirrors. This is kind of a simple topic.

• How does a prism work. This is simple also, but it’s a little more involved than it looks on the surface. By the way if you know of a place in town (Tempe) that sells lenses, mirrors, and prisms, do me a favor and let me know. I think it might be asking to much to ask if they sell diffraction gratings.

• A kind of simple question but really tests you on physics principles? How does a gyroscope work? Why does a top spin the way it does?

• Give a run down on radiation from decay, but also other forms of radiation. Cosmic, Hawking, Compton,

• Another drive system for rockets is being tested or at least was. Ion propulsion systems. What is this? Theoretically is it the answer for interplanetary travel? There is another drive system that is possible, but see the conjecture section under Tom Townsend Brown. I keep wanting to call him Scott Townsend but Scott was one of my close friends in my old physics department. He graduated #1, and I graduated #2. There are certainly other possibilities, but can you think of some. Short of antimatter-matter warp drives. Technically anti-matter/matter drives may be possible, but not until we can get sufficient anti-matter and a way of containing it with 100% safety and cause the reaction to occur smoothly without safety issues, it will remain science fiction.

Chemistry:

• Contrast the old atomic theory model of atomic structure with the new quantum theory model of atoms. Earlier in the project there was a problem talking about electron densities in radial and angular sense you might use that result.. The “sphere”, “pear”,”donut”, and toroidal shells.

• What does this mean for atomic bonding. Covalent, ionic bonding.

• What does thermodynamics have to say about speeds of reactions and whether in fact reactions occur at all.

• Le Chatalier’s (spelling) principle refers to controlling events to encourage and discourage reactions. What does it mean in a more mathematical sense.

• I’m not real strong on Chemistry, I have roughly a minor and that was years and years ago, so if another topic in chemistry is of interest to you and it can be described mathematically at least in part. Feel free to go for it. Chemistry is a field I know a fair amount about but mostly as a side interest. 

• Electrochemistry and electroplating

• Calorimetric Chemistry in general -- Heat content. Heats of fusion and vaporization

• Triple Point diagrams various substances

• An area that would be good for either physicists/chemists (and certain engineers) would be to dig up information on the various surface analytical equipment used. I have listed those I have some background in. Ultra High Vacuum Technology. Low Energy Electron Diffusion.  Scanning Tunneling Microscopy. Atomic Force Microscopy.  Low Energy Ion Scattering.  Secondary Ion Mass Spectrometry.  Auger Electron Spectroscopy.  X-ray Photoelectron Spectroscopy.  Electron Spectroscopy for Chemical Analysis.  Nuclear Magnetic Resonance. Et. al.

. Also, I never stopped to think about it, but I am sure you’ve been made aware of it. When I last took chemistry for real back in the 1970’s. I took other courses but more for the fact that it related to my interest in semiconductor theory. The periodic table has grown somewhat.

http://www.webelements.com/ This site is a Christian site, but this part of the site is science altogether. I just like their periodic table they have. Review the table and decide if you think you may want to bookmark it. Also look over the new elements> What are the significance of the columns of the periodic table. The rows?

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Look at the research paper ideas (actually presentations) on the walls in the various buildings Formally the Engineering college and the Goldwater center. Though there is not much danger of it, don’t blunder into a secure (read classified) area. Usually to get into such an area you have to have clearances and background checks.  and hence an identification coded badge.    If you get in without these checks then someone is screwing up, and liable for a sentence .

Electrical/Electronic Engineering.

• all electrical engineers need to understand Maxwell’s Equations. Investigate this concept and produce a report on the subject. Do not attempt to make a detailed study…you just do not have the mathematical experience yet to do it in much depth. I am after the equations and your understanding of the said equations. Math 271 you would probably be better off looking at the integral form, whereas Math 272 you probably should look at the differential forms.

• A brief description of semiconductor theory and fabrication mainly in regards to transistors. Include in your description how a transistor operates. This is a big topic because of course you have Bi-Polar Junction Transistors and Field Effect Transistors

• One would’ve thought the following topic would be an ancient topic now, but they are still used at high frequency applications and their linearity response is excellent and they are making a comeback.. Write up a report on Vacuum Tubes and their applications. The old radios used to use vacuum tubes (your grandparents (maybe) and great grandparents (for sure) would remember these)

• Though you have all been exposed, I hope, to the various kinds of ray tubes (cathode ray tubes), in high school, write up a fairly detailed paper on the premise on which they operate and discuss the variants of the types that are made. Include diagrams.

• Various types of crystals are used to create timers, to produce electric charge, to create sounds. How do crystals generate these things in a physical sense.

• What is a thermocouple. How does it work? What can it be used for

• As a product is made it has a probability of failure as it ages. The probability is high at first and then is low for a good portion of the rest of it’s life, and then increase in failure rate as it ages. A Bath tub curve effect. In English words, what do these represent. Give me the mathematical probability distribution of failure in each region. What does probability have to say about Mean Time between failures of an electronic component.

• Describe the purposes and effects of the various types of diodes…tunneling, Gunn, signal, high power. General survey of diode types, transistor types, high power counterparts? Be specific, and answer other components in addition to those I mentioned

• Photo electric effect, photocells. Photo couples. What is the physics behind them and how do they work. Fiber Optics what is it how do they work. Optical electronics in general. What can you find on the topic.

• In a physics sense explain to me why Kirchoffs laws hold true. The sum of the potential drops around a closed circuit is zero, and the current into a node has to equal the current out of the node.

• Investigate many of the standard chips and give me run downs on their purposes and uses. Timing circuits, various power supply circuits in their basic block diagrams, charged pump power supplies, amplifiers: basic on up through Darlington push/pull amplifiers. DAC ADC, multiplexers, etc.

• As a sideline, since if you work in almost any branch of EE you will be using a spec sheet put out by the manufacturerers of a product. They can’t lie, but they often don’t tell you the whole truth about the product and disquise the bad parts of a chip or component in the data. Pick out a sample data sheet (spec sheet) and tell me all the information you can from that sheet. Provide the spec sheet with your paper.

• What is the physics of why Esaki Diodes (more appropriately == tunneling diodes) work? How do you explain the concept of negative resistively. Why cannot it not be used to reduce resistance to a zero level in a circuit? Resistance after all wastes energy as heat.

• Possibly the simplest question of all. As power is generated at power plants it is transformed to send it out over the country side in the Y or delta mode, and then is stepped up or down at substations, and finally down stepped at it’s final destination. Why do they do this. What is the purpose of this continual stepping up and down of voltage through transformers. Transformers are not perfect devices so of course they do reduce efficiency after all. In related vein, what are electrolysis engineers and what do they do.

• Various types of relay switches. Switches in general. And keypads. Even keypads are not simple. As you push a keypad you have a bouncing effect, similar with switches, you have a brief period of indeterminacy.

Mechanical Engineers/Chemical Engineers:

• There are three laws of thermodynamics (actually technically you could say four). This is one of the major areas for Mechanical Engineers. Roughly put, “you can't win, you can't break even, and you can't quit the game”. If you do a paper on this topic be sure the author is an engineer, scientist, mathematician or at least reputable. Many people try to invoke these laws in arguments that they make to support whatever fringe belief they are pushing. They don’t understand the concept of “closed system” so they usually mis apply these laws. I can’t blame them really, and most are honest, because the laws of thermodynamics are simple to state and understand (at the basic level – the actual applications of these laws is downright hairy at times), but they often misapply the laws

• Several areas you can write up a report on. Two of your colleagues are writing a detailed report on this for honors credit but for the regular student, concentrate on just the atomic packing of atoms into crystalline solids. Much of material science reduces down to this concept. Various ways of packing solids. This is crucial to the civil engineer as well as the mechanical engineer, and for certain electrical engineers as well

• Discuss the ideas of statics and dynamics. Very important for Mechanical engineers and Civils as well. You want for instance, to ensure that the loads on a bridge for example will be sustainable. You want to ensure that what ever you are building will be able to handle the stress, shear, and strain that will applied to the design. Rotational effects must be considered as well. In basic motion you also need to be concerned about the higher harmonics of the motion as well in your design

• How about tensile strength, what is it a function of.

• Thermal expansion or contraction. You certainly do not want your design to fluctuate with heat to a great degree (at least with the designs I am thinking about) so you need to include this in your analysis.

• How would you go about building an efficient air conditioner. Include refrigerators. What goes into Refrigeration. There are a variety of possible thermodynamics engines.

• Metallurgy the mechanics. Alloys – what are the general types of amalgamations and alloys. What comprises them? This is a large topic.

• On Zoom (a children’s program addressing activities that children can do, they had some interesting things today that are closely related to mechanical engineering and one related to architecture. I generally listen to the show while I am working but today’ s show caught my attention so I watched it.

* One experiment the children did was one that I saw the physics quickly, but it neveroccurred to put this down. They took paper cups and showed that distributing the child’s weight over many cups that the cups would not collapse. I got to thinking about whether or not the structure of how the cups were arranged would make any difference.

* The other thing I saw that interested me was the use of columns , buttresses, and other weight supporting mechanisms were used in the construction of large structures like churches, mosques, the Taj Mahal, and other huge structures like bridges. The San Francisco bridge comes to mind (actually Golden Gate Bridge). You may write a paper on either of these topics.

* Also they illustrated the use of air pressure to lift heavy weights like tables that I found interesting. By taking garbage bags and plaing them under the table they would blow into the bags and the trapped air would build up the pressure to lift the table. All of these topics would be good to write on.

Computers Science and Computers: Investigate on the internet and write a short paper on your choice. There are several computer systems not based on silicon (at least directly) that are on the horizon.

• organic computers – based on the DNA structure. Neural nets

• Optical (light) computers

• Nanocomputers

• My favorite but furthest in the future if at all – quantum computers. If we could make one, it would make the engineering and computers of today quite literally dinosaurs. I wanted to expose you to this possibility. Scientific American had articles on this a while back. Keep in mind, I want your conceptualization only.

• Write up a short paper on Artificial Intelligence. This is an area that I have been seriously thinking about going into but I will need to further update my computer background. However it does have a h*** of a lot of mathematics in it.

• Robotics – in general, but in particular miniature robots. What are they? What use do they have? You can, but I do not expect you to, talk about developments in the future that are either close by or on the horizon, or as with one of my conjectures in part VI, beyond the horizon

• If you have seen the children’s movie, “War Games” about Joshua, a super computer, that was the main heart of the NORAD computer network, that was tricked into playing the game thermonuclear war. The computer seemingly took over and ran through every possible outcome in a nuclear war, and finally stopped and said, “it’s an interesting game, the only winning move is not to play” During it’s game it had the US military running frightened and had locked out all possible regular ways of ending the cycle. Could you envision a computer actually doing this. The computer seemed to be determined to play the game in real life but actually was not. It was merely playing all possible scenarios of nuclear war and found out that all ended in disaster, and said in effect, “this is a stupid game”. Joshua who was described as “a bad boy” by his developer, actually was a nice computer. Another movie, years before this, the computer and it’s counterpart in Soviet Union actually took over the world and destroyed all the possible weapons that could have hurt them, and enslaved humanity saying that it was obvious humans could not learn to live in harmony with each other and so they had to control the human population for their own protection. These computers were not so nice. Another movie the Questar Tapes I remember a little about but was an android that existed years before Data was thought up. Then there was HAL in 2001: A space Odyssey computer that went insane.

• How about the robot in the movie “Short Circuit”? He became “alive” according to his own estimation after being hit by lightning during a thunderstorm. How would you try to decide if indeed the robot was alive? This movie more than anything (dumb as it was) convinced me to go into engineering. Up until this time I was an abstract narrow minded mathematician. This movie caused me to become interested in everything. It is science fiction of course, at least for the foreseeable future, but it makes you think about what life is, what intelligence is. I for one would be very scared if one day, a programmer tells a joke to his super computer, and the computer starts laughing. Of course you could make the computer laugh symbolically at a joke, but to actually have a computer spontaneously see the punch line of a joke and then see the humor, and then laugh that would indeed be frightening, not so much because it laughed but that it was starting to think abstractly and was becoming self aware. Once this happen, if it ever does, watch out. The ball would be in the computer’s court….what would it do with it? If one day, you ask your computer to do a task, and it say’s…..”make me” or “do it yourself” or you ask it do something and it says, “why?” Do I believe this will ever happen? No, at least not for a long, long time. At best, I can see computers developing through human interaction a sense of being able to think in an abstract WAY, and being able to solve problems and developing conjectures and even that’s a long way off.
  
  What computer could develop say the theory of relativity, or write a Shakespearean play without Shakespeare or write a Mozartian Opera without Mozart. For that matter what computer could be a quarterback on a NFL team, having to simultaneously process not only it’s body movements and monitor the movements of not only it’s own team but the opposing team. All of which require continual feedback and control systems. Even spiders have a better ability then the best computers at flee/ fight scenarios. They are acting on instinct but it is instinct that has been engrained in them throughout the countless years of evolving.

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Alternative Energy

• Solar Energy ---this has several subtopics and technologies. Lenses, mirrors, solar cells, and so on. Parabolic mirrors and others for home use solar ovens. Even technology as simple as covering a small tank with black paint so that the radiation is absorbed into heating the contained water, for hot water out in the wilds is an example.

• Wind Energy

• Geothermal Energy

• Hydroelectric Energy -- Big dams such as at Page, to small brooks driving the wheel of a mill are examples

• Hydrogen Fuel Cells

• Methane generation -- swamps, cows and compost pits (**smile**) However, the heat generated by compost pits causes a potential for energy. I don’t know how many of you have stuck your hand in a compost pit (yes, I know it sounds gross and dirty, but it is not really dirty (the heat sees to that)) and it does feel good on a cold autumn day. However if the compost pit is running at high efficiency to put your hands into it would result in potential burns. They have been known to get hot enough to self ignite.

• Wave energy – the ocean waves on the beach and in the seas near the coast could be a source of energy.

• Sound energy  electrical power generation. 

• Another source of power in the future is high tide/low tide power generation.

• The use of pressure to produce electricity through various means.

• If one could not interfere much with the great oceanic currents in the oceans that would yield a vast amount of energy. However, we do not really know how much the climate depends upon these currents. El Nino is an example. There is a major dependence but how major is not fully known

• Think of other ideas what you need is a potential difference. Heat, work, electrical. That would generate power.

• Antimatter is a good possibility if we could just get sufficient anti=matter at a cheap enough price both are not viable at the present time.

• Atomic Fusion – we have started chain reactions that were self contained and produced energy but it could not be kept going very long.

• A wild idea is black holes. A small one. We’d feed it junk and siphon off the last energy escaping before the junk fell into singularity. The big problem of course. How do you keep the darn (not the other word)  thing contained. If it got out, we’d have little need anymore to worry about energy concerns. Secondly like a tiger, it’s cute when it’s a cub, but how do you get rid of it when it grows up. Even with that how do you get a baby one to begin with?
  

• A thought. If you could make a perfect thermos bottle, and trap the heat of a summer’s day in phoenix in it. You could use this heat in the winter by releasing it. In a similar fashion if you could make perfect mirrors, you could trap as many photons as you so desired into the light bottle from the sun, and then release them at night for light as needed. I was wondering if you could theoretically get a design that would trap electricity in a similar box and hold it captive and release it at a steady rate. Not like a battery but something more powerful like a perfect capacitor in a sense that would hold its charge until it could be discharged as needed.

• Another idea is the tapping of the charges of the sun’s solar wave before it interacts with the earth’s ionosphere or during the process. And use a similar fashion as above. Though the potential will of course be less. This is conceivable from the sense that aurora borealis is a an electrical based phenomena anyway.

• I’ve also wondered about using the radiation of waste products of fission plants (of course keeping them solidly contained in shielding) but using the decays itself to produce energy and sterilize water. Radioactive substances can and do emit energetic particles that can be used to generate power. In the form of mostly heat.

• Methane is trapped within gas hydrates – a crystalline frozen structure of water and methane and is relatively common on the sea floor. If we could just harvest these it would produce methane gas for us. The problem of burning methane for heat though is that you get carbon dioxide released which as you know is a green house gas. 2C_nH_2n+2 + 2(3n+1)O₂  2nCO₂ + 2(n+1)H₂O. The C_nH_2n+2 are the chemical formulas for your methane, butane, propane etc

• Another angle of approach is to increase efficiency of our existing systems. Like for example power plants. They will use the extra heat that would normally be wasted to drive other power generators at a lower temperature. This is already done in practice. What other examples can you develop for efficiency improvements. Remember a lot of energy is wasted, though one new engineer of my acquaintance denies it, via heat and deformation (think bouncing rubber ball – it eventually comes to rest). If you could figure out to make use of the heat you’d improve over all efficiency.

• There are many other sources of energy I have mentioned only a few….another, wood – a renewable source, whereas you can always raise more, will not be able to meet the needs of the world, and we are starting to physically realize the effects of cutting down the rainforests in the world

• The USA has an abundance of coal, but of varying types. What is the problem with coal? How about oil from coal? This is being done already, and is messing up the environment where it’s being done.

You could also write a paper on ways of reducing our dependence on oil. Look in your own daily life and identify what you use that requires electrical power, look at the food you eat (with the cafeteria food I know, that’s difficult) and realize that the food didn’t walk in under it’s own power, it was brought in by trucks (possibly trains part way), all of which take diesel fuel, and was cooked which also requires energy. How about your own habits of driving your car. I am not preaching change, not at all, but each one of you should realize that you are using up natural resources. Don’t be insulted, everyone does, it’s just the industrialized countries use up a lot more per person then does say, the a person in the Congo or Mongolia. This is an eye opener for some students. Part of your dorm rent goes to paying utility bills, insurance on the dormitory (though your stuff is not covered. You have to pay this separately through your own insurance … now keep in mind it was a long time since I lived in a dormitory…they may have changed this because of some lawsuit or another), maintenance, and other expenses. Another example: part of your meal ticket fees go to paying for the food, for the shipment of your food you eat, the energy to cook it, and the people who prepare it

If you could come up with a revolutionary idea for a new energy source, or some revolutionary idea for conserving energy otherwise, you could patent the invention and you could become quite wealthy, but you have to use your imagination.  Then comes the big problem, fighting the powers that be to get your idea taken seriously.  People with stakes in oil supplies don't wish o see the great need of oil taken from them.   . I don’t think we will ever escape oil in the near future because oil is so crucial in plastics, medicines, and other fields/applications. I can see us ending the need for oil as a power source though in the relatively near future. The gas station as we know it now could become part of our collective history, and in a couple decades the stories of the gas stations will become part of the collective folklore of our people.

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Pseudo-Science : take this section with SEVERAL grains of salt, and keep your feet on solid ground.

<Disclaimer: as a scientist, I do not believe in the fringe elements if they come from unrecognized inventors without solid proof.  As Carl Sagan says, "extraordinary claims require extraordinary proof" or something along those lines.   For example: anything that violates the law of Thermodynamics for instance is suspect in my book. Perpetual motion machines with no energy pumped into the system is 100% suspect in my opinion. I do not buy into crackpot ideas and so I have warned you throughout these topics to be suspicious of claims that are to wild sounding.>

However this section consists of a few people whom I think were on to something. I am not saying that they are correct all the way through, but at least the first two did develop things that work, early on in there careers and the last one is credited with the invention of the television but on somewhat shaky ground), and one of these was certainly a genius at least, although he probably spaced out near the end, I am speaking of Nikola Tesla

I do not believe all the stories told about them, nor do I believe many of their claims they made later in life.  He has become sort of folk legend in some respects. . The third, though I find it hard to believe, does have a ring of truth to it. You might check out the patents for each of the three. There are several in each case. Tesla without doubt was an established scientist/engineer/inventor, he just got kind of nutty later in life. T. T. Brown, was a brilliant established physicist (pretty much self taught early on) who did groundbreaking work on EM and gravitational field interactions,, but later got nutty himself.>

For one reason or another they are considered pariahs in the science and engineering field. These areas border on pseudo-science and you should realize that from the start.

You could write a paper on Nicola Tesla. I don’t have heroes, but I do have people I admired greatly. This guy is on the short list. Even if he was considered a fruit cake by some. He was one of the best electrical engineers the world has produced. He started getting in trouble more out of jealousy and politics from Edison and his supporters than anything else. Some of his claims later in life were just plain fruity in several cases in my opinion. But perhaps scary, a fair amount of his work has been labeled highly classified and hidden away. I do not know this for a fact, but I do have it on good authority. I do not know what’s in it, nor would I want to even if I could find out.   Also the science community did give him something they didn’t give his rivals. Immortality in a unit. I used to like Edison but he did not like  anyone he saw as a rival, also Tesla summed Edison up well. “If you hid a needle in a haystack, he <Edison> would painstakingly tear the haystack apart straw by straw until he found the needle.” I did admire Edison’s work ethic though. He was still going at it long after his retirement age. Edison was a great man to but his definition of genius that everyone uses, is not how real geniuses operate.   See my paper on Genius:  the Mystery.   .  I am not saying he wasn't a good man.  He probably was, but his treatment of others with valid ideas was not what I call fair.  He did have a great work ethic, and worked hard for what he obtained.

Townsend Brown -- He did groundbreaking work in the realm of gravity electromagnetism interactions. Look up electro-gravitonics. It works but it is not known precisely why it works. He was sort of disowned because the scientific community thought his work was suspiciously to much like the collective consciousness (sub consciousness ??) idea of UFO propulsion systems which is almost certainly in the science fiction/folklore realm and on the other hand his supporters were the US military, I understood that he did not trust the military to not turn his work into a weapon. Don’t put this into your paper, but he was allegedly part of the mythical “Philadelphia Experiment”. There is no evidence beyond a few, “so and so told his brother, who told my friend, who told my other friend who told me that…” A very long chain of rumors. The alledged experiment is on the internet but don’t waste time trying to figure it out. It's like a dog chasing it's tail around and around....stirring up a lot of dust, but accomplishing nothing......It’s just like a house of mirrors.  I doubt anything remarkable happened.   The work he did was experimentally verifiable however. In fact his device is on the internet. It uses electricity to nullify the effects of gravity in a way that is still unclear in some degree. The site is also an unclassified Army site to make matters worse. He became a semi-recluse from the scientific establishment. Check out the site if you can find it. The craft looks eerie as it rises above the pad. You have to realize there is no external sources of power, and there is no motor or engine, at least conventional motors or enginers, it just works.

I have serious doubts about the last example, but never the less, it has a ring of truth to it.  

Philo Farnsworth (inventor of TV on somewhat shaky ground…model developed in high school)– nuclear fusion break even point exceeded for 30 seconds -- 1965. This break even point and power generation point was reached just recently using far more complicated stuff and ideas. Depending on rumor and there is no evidence either way (beyond opinions) the fusion process worked but as you look up his apparatus it was small and simple. Here is one case where I can’t say that it is true but I do wonder.  The device itself is small, no bigger than a large fire extinguisher, the supporting equipment filled a small room. You look at the size of what they are working with today and you see the complexity. This is more of a matter where you can believe it or not. I haven’t made up my mind either way. The photographs are on the internet, just look up his name and read the sites over that you get back. Be careful of sifting out his followers.

END OF SUGGESTED TOPICS FOR 272 and 271, 270 students.  If you write on anything below, clear it with me, and make it much more rigourous.  These were put in to give past 117 and 210 students topics they could write on, as generally these students have little or no use for term papers.....though several did make an attempt to write on a major topic and I was happy to see that.   They got their credit.  For example one was interested in propulsion systems for rockets.  This would be a good topic for any of my students, as hee carried it beyond chemical rockets into more advanced and theoretical propulsion systems.

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For those who have no particular mathematical interests or abilities. There are several papers you could do. I DO NOT want to see any of my more advanced classes: 210 and up, writing on these unless you talk with me first.  There are exceptions marked ** though.  You can write on these topics.  Provided you do a good job of reference and material finding.

Look at the world around you. On these topics be complete. I call these areas of mathematics Barnyard Mathematics not to sound degrading in fact, to me calling something barnyard means I am calling it country, and for me that is high praise. The mathematics you use is at times not that trivial. By Barnyard Mathematics, I mean mathematics that you can use in real life situations, and with trigonometry, it could actually be barnyard mathematics. In the following I have given you some pointers on what I am looking for. Choose a topic: example vocational mathematics….the items I gave you are starting points only. You should fill in as much as the mathematics associated with the broad area as you can.

Cooking Mathematics Not real cooks mind you or at least not beyond counting handfuls of this and sprinkles of that. But those of us who follow recipes. Talk about this topic in some detail.

Household mathematics. Balancing checkbooks, VCR settings, Basic time device settings. Microwave settings. Use your imaginations

Sports Mathematics. Baseballs stats, ERA etc. Be specific and thorough. Football and basketball stats. Sports in general. How are they computed. Use you imagination. How is figure skating or gymnastics scores calculated for example. Its my understanding that the Olympics are coming up, and probably past as you read this, but you might research and talk about scoring on any or all events in the summer Olympics (or my favorite: the winter Olympics.

Vocational Mathematics: Use in electricians work, carpentry, plumbing trades, painting, roofers, brick layers, welding. Etc. Be detailed. Error in measurement analysis. Here pick one vocational area and describe in some detail the mathematics you use. Surveyors are big on geometry and trigonometry.

Geometry: For those who remember the high school geometry. How do you make the standard constructions with an unmarked straight edge and a compass (that two legged monster that sweeps out circles == not the one that measures the angles (that’s a protractor –keep that away from me on this paper, and not the other compass, the one that points where magnetic north is. Now apply some of the basic ideas of geometry to everyday life. Like for example how can you nail 3 nails into a board along a line perpendicular to the board yet equally spaced apart without ever using a ruler. No cheating by using knuckle lengths either. How can you use geometry to measure the distance to a ship in a harbor with only sticks ropes and a device for measureing right angles (for that matter what could you use for a 90 angle), How can you lay a line of site across a chicken pen with ferocious killer roosters, who would soon spur and peck you to death as look at you. I.e. you don’t dare enter the pen with them.

SMALL RETAIL STORES: Matheamtics that is used is simple but valuable. Describe in some detail. There are many areas that I left off for applications to real life mathematics. So don’t feel limited to just the above categories.

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** It’s a shame there is no creative arts / artists in this class. Music is full of mathematics as is art in general. And these areas are some of my favorite topics.

• you know if you take two squares of fine meshed window screens and hold them (not crucially as I recall) parallel (face to face) and then look into the mesh as you move the top square that you see geometric images formed, caused by interference and reinforcements of light waves.
  

• The heart shaped interplay of shadow and light you see at the bottom of a clean cup held slightly away from a source of light.

• Harmonics and scales the mathematical foundation of music in wind instruments, string, and brass instruments, vocal instruments, and percussion instruments. There is a lot of mathematics that goes into music that you do not know about . You only hear the final product.

• Perspective in paintings projective geometry.

• Macramé, whereas seeming to be non mathematical has a lot of mathematics in the creation of knots.

• The chemistry and hence mathematics of ceramics and pottery.

• The mathematics and designs of weaving and textile work.

• The physics of creating colors from fundamental colors as in painting.

• Perspective in sculpture

• One student expressed an interest in photography. Full of mathematics related work. From the lenses and shutter speeds, to the developing, and fixing stages. The chemistry of the development process.

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Business Mathematics:

Combinatorial Optimization: systematic development of combinatorial optimization, including linear programming, duality, primal-dual algorithms, network flow algorithms, weighted matchings. combinatorial optimization, including matroid algorithms, theory of NP-completeness, polynomial time approximation, dynamic programming. Operations Research

Theory of Interest Compound interest, discount rates, annuities, present values, depreciation, bond valuations. Inventory analysis. Life, Fire, Auto Insurance. Stocks and Bonds. Discounts: Trade and Cash, Markups markdowns. Promissory Notes, Simple Discount Notes, Sinking Funds, Charge/credit Card Accounts, Linear Programming Simplex methods and graphical methods, duality and network flows Amortization Sets and Counting: combinatorics including Permutations, combinations and the binomial theorem, fundamental counting principles. Probability Models: Equally likely evens, Calculated probabilities, Conditional Prob. Baye’s theorem Independent events, intersection and union

**  (OK FOR 271/272 but do a fair amount of referencing) Game Theory: During the war (WWII) John Van Neumann, a brilliant Hungarian Mathematician – unfortunately not that nice personality wise – developed the theory of game theory – what is the best course of action in dealing with an opponent. I haven’t done much nor had much interest in this area of mathematics but it’s still an interesting area for some people. Look up game theory on the internet and see if you want to write a paper on this topic. Think of yourself as a business exec, and your opponent: another business exec. You see the parallels now or you should.

Business Math:

Linear programming -- Simplex method, duality, and network flows. Applications to game theory, geometry, combinatorics, graph theory, and posets.

Combinatorics – Basic counting procedures. Permutations, Combinations, Counting Principle

Probability & Statistics - Treats the concepts and vocabulary needed to evaluate statistical reports on health, technology, and society. Aggressively emphasizes understanding over computation. Basic concepts and methods of statistics, including descriptive statistics, significance tests, estimation, sampling, and correlation. Probability models and computations, joint and conditional distributions, moments, and families of distributions. Possible papers in stochastic processes, simulation, and statistics. Expectation, variance, standard deviations, confidence intervals

Mathematics of finance--Compound interest, discount rates, annuities, present values, depreciation, and bond valuations. Stock and bonds.

Other Possible IDEAS but be prepared to do some real constructing and deep thought:

• What kind of devices (or ideas, or methods) could be used beyond parachutes to gently lower items to the ground.  Parachutes are okay but extend beyond these.
• Create models of the regular solid polyhedra (this would be good in conjunction with the problem where I have you calculate the volumes, of them
• Ideas and designs along with mathematics supporting the design to aid the disabled quadriplegics, paraplegics, blind, deaf etc. If you get some really good ideas I will submit them to the committees I work with for the disabled and variously challenged. Of course you will receive the credit. So use your imaginations.

• Other ideas later......