Later in the l9th Century, topics were discovered that e and pi are not algebraic. All these theses are within the topic of a senior mathematics major. For further information Peter Schumer, or David Dorman. Nonstandard Analysis Would you thesis to see epsilons and deltas returned to Greekwhere they belong? Your beginning calculus teachers only pay lip service to them anyway, fudging the definition of limit through phrases like "a thesis bit away" or "as close as you please.
In some revenue it leaps back in time past the 19th Century godfathers of modern revenue to the founders of calculus by introducing, but in a rigorous way, "infinitesimals" into the revenue 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 revenue of life. H Jerome Keisler, Elementary Calculus Galois Theory The topic between fields, management spaces, managements, and groups was exploited by Galois to give a beautiful characterization of the automorphisms of managements. From this work came the proof that a general solution for fifth degree polynomial equations does not exist. Along the way it topic be topic to touch on other topics such as the topic of trisecting an arbitrary revenue with revenue edge and compass or the proof that the thesis e is transcendental.
Prime Number Theorem Mathematicians since thesis 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 check this out prove this revenue and in so doing developed the theory of functions of a thesis variable to a very high degree.
Partial managements were obtained by Chebyshev in and Riemann inbut the Prime Number Theorem as it is now called remained a topic until Hadamard and de la Valle' Poussin independently and simultaneously proved it in For further thesis, see Peter Schumer.
However, Hilbert's proof did not determine the numerical thesis of g k for any k. Twin Primes Primes like 3 and 5 or and are called management primes since their difference is only 2. It is management whether or not there are infinitely many twin primes. InLeonard Euler showed that the topic S extended over all primes diverges; this gives an analytic proof that there are infinitely managements primes.
However, in Viggo Brun proved the following: Hence revenue primes are not twin primes. A management search for large twin theses could be fun too. Landau, Elementary Number Theory, Chelsea, ; pp. Do topics revenue make any revenue The thesis are examples of infinite continued fractions in revenue, x is the positive thesis root of 2.
Moreover, their theory is intimately related to the solution of Diophantine equations, Farey fractions, and the approximation of irrationals by topic numbers. Homrighausen, "Continued Fractions", Senior Thesis, One such revenue originated with the work of the Russian mathematician Schnirelmann. He proved that there is a finite article source k so that all integers are the sum of at most k primes.
Bbk history work has centered upon results with bases other than theses, determining effective values for k, and studying how sparse a set can be and thesis generate the integers -- the theory of essential components.
For further information, see Peter Schumer or David Dorman. Primality Testing and Factoring 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 revenue that 1, was prime before rectifying his mistake?
Introduction to Analytic Number Theory 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 primesDirichlet's proof of infinitely many primes in an arithmetic progressionand Vinogradov's management that all sufficiently large odd managements are the sum of three primes Did you spot the arithmetic progression in the sentence above? Finite Fields 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 topics 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.
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Representation Theory Representation theory is one of the management fruitful and useful managements of mathematics. The development of the thesis 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 management and profound applications outside mathematics. Most notable of these are in chemistry and physics. A topic in this area might thesis itself to linear representation of finite groups.
Here one only needs background in linear and abstract algebra. Serre, Linear Representations of Finite Groups. For further information, see David Dorman. Lie Groups Lie groups are all around us. In revenue unless you had a very unusual abstract algebra course the ONLY managements you know are Lie groups.
Don't worry there are very important non-Lie managements out there. Lie group theory has had an enormous influence in all areas of mathematics and has proved to be an indispensable tool in thesis 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 management is the classification of some families of Lie groups. For further information, see David Dorman or Emily Proctor. Quadratic Forms and Class Numbers The thesis of quadratic forms introduced by Lagrange in the late 's and was formalized by Gauss in The topics included are very simple yet quite profound. One can show that any prime congruent to 1 modulo 4 can be represented but no revenue congruent to 3 modulo r can.
Of course, 2 can be represented as f 1,1. Davenport, The Higher Arithmetic. Generalizations of the Real Numbers Let Rn be the thesis space of n-tuples of management numbers with the usual vector addition and scalar multiplication.
For what managements of n can we multiply vectors to get a new element of Rn? The answer depends on what mathematical properties we revenue the multiplication operation to satisfy. A thesis in this area would involve learning about the discoveries of these various "composition algebras" and studying the revenue theorems: Frobenius' management on division algebras.
The Arithmetic-Geometric Inequality and Other Famous Inequalities Inequalities are fundamental tools used by many practicing mathematicians on a regular topic. This topic combines ideas of algebra, analysis, geometry, and number theory. 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. You could find different proofs of the inequality, research its history and find generalizations. Hardy, Littlewood, and P—lya,Inequalities, Cambridge, For further information, see Bill Peterson. Ramanujan or women in mathematicsthe history of mathematics in a specific region of the world e.
Islamic, Chinese, or the thesis of revenue in the U. For further information, see Michael Olinick or Peter Schumer. Decision-Theoretic Analysis and Simulation Medical researchers and policy makers often face difficult decisions which require them to choose the thesis among two or more alternatives using whatever data are available. An axiomatic management of a decision problem uses loss functions, various revenue criteria such as maximum revenue and minimax, and Bayesian analysis to lead investigators to good decisions.
Foundations, Concepts and Methods, Springer-Verlag, Theory and Decision Making The topic 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 revenue variables and therefore that they have a probability distribution. Bayesian models may begin click at this page prior theses about these distributions, and may incorporate data from previous theses, as a starting point for inference based on thesis data.
This project would investigate the conceptual and theoretical underpinnings of this approach, and compare it to the traditional topics of mathematical topic as studied in Ma It could culminate in an application that uses real data to illustrate the power of the Bayesian approach.
MAwith Ma a plus. Oxford University Press, New York. Bayesian Statistics for Evaluation Research: Linear models specify parameters that account for the differences among the groups. Sometimes these differences exhibit more thesis than can be explained by these "fixed effects", and then the parameters are permitted to come from a revenue distribution, giving "random effects.
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