Mathematical Conversations: Multicolor Problems, Problems in the Theory of Numbers, and Random Walks
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Contents:
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.
Markov Processes and Related Problems of Analysis
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.
One goal is the classification of some families of Lie groups. For further information, see David Dorman or Emily Proctor. The theory of quadratic forms introduced by Lagrange in the late 's and was formalized by Gauss in The ideas included are very simple yet quite profound. One can show that any prime congruent to 1 modulo 4 can be represented but no prime congruent to 3 modulo r can.
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Of course, 2 can be represented as f 1,1. Let R n be the vector space of n-tuples of real numbers with the usual vector addition and scalar multiplication. For what values of n can we multiply vectors to get a new element of R n? The answer depends on what mathematical properties we want the multiplication operation to satisfy. A thesis in this area would involve learning about the discoveries of these various "composition algebras" and studying the main theorems:.
Inequalities are fundamental tools used by many practicing mathematicians on a regular basis. This topic combines ideas of algebra, analysis, geometry, and number theory.
Eugene B. Dynkin (born March 11, ), Russian mathematics professor | Prabook
We use inequalities to compare two numbers or two functions. These are examples of the types of relationships that could be investigated in a thesis.
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You could find different proofs of the inequality, research its history and find generalizations. Ramanujan or women in mathematics , the history of mathematics in a specific region of the world e. Islamic, Chinese, or the development of mathematics in the U. Medical researchers and policy makers often face difficult decisions which require them to choose the best among two or more alternatives using whatever data are available.
An axiomatic formulation of a decision problem uses loss functions, various decision criteria such as maximum likelihood and minimax, and Bayesian analysis to lead investigators to good decisions. Emerson, J. The power of modern computers has made possible the analysis of complex data set using Bayesian models and hierarchical models.
These models assume that the parameters of a model are themselves random variables and therefore that they have a probability distribution.
Special order items
Bayesian models may begin with prior assumptions about these distributions, and may incorporate data from previous studies, as a starting point for inference based on current data. This project would investigate the conceptual and theoretical underpinnings of this approach, and compare it to the traditional tools of mathematical statistics as studied in Ma It could culminate in an application that uses real data to illustrate the power of the Bayesian approach. References: Gelman, A.
Bayesian Statistics: An Introduction. Oxford University Press, New York. Pollard, William E.
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Measurements which arise from one or more categorical variables that define groups are often analyzed using ANOVA Analysis of Variance. Linear models specify parameters that account for the differences among the groups. Sometimes these differences exhibit more variability than can be explained by these "fixed effects", and then the parameters are permitted to come from a random distribution, giving "random effects.
This modeling approach has proved useful and powerful for analyzing multiple data sets that arise from different research teams in different places. For example the "meta-analysis" of data from medical research studies or from social science studies often employs random effects models. This project would investigate random effects models and their applications.
Hoaglin, D. Mosteller, and J. Tukey, Eds. John D. Because a computer is deterministic, it cannot generate truly random numbers. A thesis project could explore methods of generating pseudo-random numbers from a variety of discrete and continuous probability distributions. Morgan, B.
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. A number of problems of this kind soluble by methods drawn from mechanics namely, by using the laws of equilibrium were given by the author in his lecture "The solving of mathematical problems by the methods of mechanics", which was read to pupils in their seventh year of secondary school at the Moscow State University on 19 February ; this lecture, with very minor additions, makes up the contents of this article. Quanta 5 min read Science. Mathematical Models of Conventional Warfare Most defense spending and planning is determined by assessments of the conventional ie. Although they are basic, they are by no means elementary. Tool Sets Bathroom Accessory Sets.
The art of tilings has been studied a great deal, but the science of the designs is a relatively new field of mathematics. The problems in this area are easy to state and understand, although not always easy to solve.
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