Research could include some interesting computer work if desired. Recent results indicate that any "reasonable" voting procedure must either be dictatorial or subject to strategic manipulation.
Many "possibility" theorems have been proved for voting mechanisms which satisfy relaxed versions of Arrow's axioms. How does one fit this model to real data?
How are the Lotka-Volterra models of competition and predation affected by the assumption that one species grows logistically in the absence of the other? A typical problem in this field would ask how to maximize the present value of discounted net economic revenue associated with the hunting and capture of whales. How does an optimal strategy vary with the number of competing whaling fleets? In , a Dutch mathematician L E. Brouwer proved that every continuous function from a n-cell to itself has at least one fixed point; that is, if f: There has also been much progress on the problem of computationally determining fixed points.
Joel Franklin, Methods of Mathematical Economics. Here N is the number of tumor cells at time t, K is the largest tumor size and b is a positive constant.
A thesis in this area would begin with an investigation of the mathematical properties of this model and the statistical tests for deciding when it is a good one. The thesis would then move to a consideration of stochastic models of the tumor growth process. Most defense spending and planning is determined by assessments of the conventional ie.
The dynamic nature of warfare has historically been modelled by a particular simple linked system of differential equations first studied by F. Lanchester Models of Warfare. The Fundamental Theorem of Algebra states that every non-constant polynomial with complex coefficients has a complex root.
Gauss was the first person to give a proof of this result; in fact, he discovered four different proofs. All known proofs require some complex analysis.
However, the theorem is one of algebra and a purely algebraic proof would be nice to find. Emil Artin has given one that's almost purely algebraic. Any text in complex analysis.
J Munkres, An Introduction to Topology. Serge Lang, Algebra for Artin's proof. A real number r is "algebraic" if r is the root of a polynomial with integer coefficients.
Thus every rational number is algebraic as are many of the more familiar irrational numbers such as the square root of 2 and the l7th root of 3. Liouville was the first to show explicitly that a certain number was not algebraic. Later in the l9th Century, proofs were discovered that e and pi are not algebraic. All these proofs are within the grasp of a senior mathematics major. Would you like to see epsilons and deltas returned to Greek , where they belong?
Your beginning calculus teachers only pay lip service to them anyway, fudging the definition of limit through phrases like "a tiny bit away" or "as close as you please. In some ways it leaps back in time past the 19th Century godfathers of modern analysis to the founders of calculus by introducing, but in a rigorous way, "infinitesimals" into the real number system.
Mathematics is not a static, immutable body of knowledge. New approaches to old problems are constantly being investigated and, if found promising, developed. Nonstandard analysis is a good and exciting example of this mathematical fact of life. Robinson, Non-standard Analysis H Jerome Keisler, Elementary Calculus The relation between fields, vector spaces, polynomials, and groups was exploited by Galois to give a beautiful characterization of the automorphisms of fields.
From this work came the proof that a general solution for fifth degree polynomial equations does not exist. Along the way it will be possible to touch on other topics such as the impossibility of trisecting an arbitrary angle with straight edge and compass or the proof that the number e is transcendental. Mathematicians since antiquity have tried to find order in the apparent irregular distribution of prime numbers.
Let PI x be the number of primes not exceeding x. Many of the greatest mathematicians of the 19th Century attempted to prove this result and in so doing developed the theory of functions of a complex variable to a very high degree. Partial results were obtained by Chebyshev in and Riemann in , but the Prime Number Theorem as it is now called remained a conjecture until Hadamard and de la Valle' Poussin independently and simultaneously proved it in However, Hilbert's proof did not determine the numerical value of g k for any k.
Peter Schumer, Introduction to Number Theory. Primes like 3 and 5 or and are called twin primes since their difference is only 2. It is unknown whether or not there are infinitely many twin primes. In , Leonard Euler showed that the series S extended over all primes diverges; this gives an analytic proof that there are infinitely many primes.
However, in Viggo Brun proved the following: Hence most primes are not twin primes. A computer search for large twin primes could be fun too. Landau, Elementary Number Theory, Chelsea, ; pp.
Do numbers like make any sense? The above are examples of infinite continued fractions in fact, x is the positive square root of 2. Moreover, their theory is intimately related to the solution of Diophantine equations, Farey fractions, and the approximation of irrationals by rational numbers.
Homrighausen, "Continued Fractions", Senior Thesis, One such area originated with the work of the Russian mathematician Schnirelmann. He proved that there is a finite number k so that all integers are the sum of at most k primes. Subsequent work has centered upon results with bases other than primes, determining effective values for k, and studying how sparse a set can be and still generate the integers -- the theory of essential components. This topic involves simply determining whether a given integer n is prime or composite, and if composite, determining its prime factorization.
Checking all trial divisors less than the square root of n suffices but it is clearly totally impractical for large n. Why did Euler initially think that 1,, was prime before rectifying his mistake? Analytic number theory involves applying calculus and complex analysis to the study of the integers. Its origins date back to Euler's proof of the infinitude of primes , Dirichlet's proof of infinitely many primes in an arithmetic progression , and Vinogradov's theorem that all sufficiently large odd integers are the sum of three primes Did you spot the arithmetic progression in the sentence above?
A finite field is, naturally, a field with finitely many elements. Are there other types of finite fields? Are there different ways of representing their elements and operations?
In what sense can one say that a product of infinitely many factors converges to a number? To what does it converge?
Can one generalize the idea of n! This topic is closely related to a beautiful and powerful instrument called the Gamma Function. Infinite products have recently been used to investigate the probability of eventual nuclear war. We're also interested in investigating whether prose styles of different authors can be distinguished by the computer.
Representation theory is one of the most fruitful and useful areas of mathematics. The development of the theory was carried on at the turn of the century by Frobenius as well as Shur and Burnside. In fact there are some theorems for which only representation theoretic proofs are known.
Representation theory also has wide and profound applications outside mathematics. Most notable of these are in chemistry and physics. A thesis in this area might restrict itself to linear representation of finite groups. Here one only needs background in linear and abstract algebra. Lie groups are all around us. In fact unless you had a very unusual abstract algebra course the ONLY groups you know are Lie groups. Don't worry there are very important non-Lie groups out there. Lie group theory has had an enormous influence in all areas of mathematics and has proved to be an indispensable tool in physics and chemistry as well.
A thesis in this area would study manifold theory and the theory of matrix groups. The only prerequisites for this topic are calculus, linear and abstract algebra. Sometimes, coming up with an effective topic for a college …. In the course of Nursing Training, students must complete several …. Writing an argumentative essay is quite challenging, especially if you …. How to get an A without even trying.
How did technology effect our learning. Digital is anew print. Paper books have recently become horse …. College students are those who attain their higher education as …. Middle School Essay Topics. Discursive Essay The main idea of writing discursive essays is ….
Of course, you want your topic to be impressive, but make sure you choose a subject area in which you feel comfortable working. If you attempt to write a dissertation based on a topic you are unsure of, it will show.
One of the most important concerns in choosing a thesis topic is that the topic speaks to an area of current or future demand. A good thesis topic is a general idea that is in need of development, verification or refutation. Your thesis topic should be of interest to you, your advisor, and the research community.
Admittedly, some thesis topics are more current and important than the others. It is a good idea to select something which will arouse interest of your committee and be a significant contribution to the corresponding subject field.
As you begin researching your topic, you may want to revise your thesis statement based on new information you have learned. This is perfectly fine, just have fun and pursue the truth, wherever it leads. Look through the suggested research paper topics and find one in a category that you can relate to easily. The Influence of Varying Cost Formats on Decision. Likelihood of Achieving WHO Leprosy Goals: An Expert Survey; Exploring the Link between .
Sep 01, · One of the most common questions I get asked is how to choose a thesis topic or research project. Unfortunately it’s not as simple as just “finding a gap in the literature”, and there are many complicating factors to consider. Topic Selection Guide: A List of Top Education Thesis Topics. Your education thesis topic may not be original, but it should be manageable and rich in available literature. Below you’ll find a list of educational topics broken up by major knowledge sections; such as education administration, classroom management, curriculum .