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. 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. 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. 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, 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. Foundations, Concepts and Methods, Springer-Verlag, 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.
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.
Oxford University Press, New York. Bayesian Statistics for Evaluation Research: 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. MA , with a plus. 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. The art of tilings has been studied a great deal, but the science of the designs is a relatively new field of mathematics.
Some possible topics in this area are: The problems in this area are easy to state and understand, although not always easy to solve. The pictures are great and the history of tilings and patterns goes back to antiquity. An example of a specific problem that a thesis might investigate is: Devise a scheme for the description and classification of all tilings by angle-regular hexagons.
Roughly speaking, a contraction of the plane is a transformation f: With a little effort CF can even be made to look like a tree or a flower!! A thesis in this area would involve learning about these contraction mapping theorems in the plane and in other metric spaces, learning how the choice of contractions effects the shape of CF and possibly writing computer programs to generate CF from F.
Consider a population of individuals which produce offspring of the same kind. Associating a probability distribution with the number of offspring an individual will produce in each generation gives rise to a stochastic i. The earliest applications concerned the disappearance of "family names," as passed on from fathers to sons. Modern applications involve inheritance of genetic traits, propagation of jobs in a computer network, and particle decay in nuclear chain reactions.
A key tool in the study of branching processes is the theory of generating functions, which is an interesting area of study in its own right. Branching processes with biological applications. The Poisson Process is a fundamental building block for continuous time probability models. The process counts the number of "events" that occur during the time interval [0, T ], where the times between successive events are independent and have a common exponential distribution.
Incoming calls to a telephone switchboard, decays of radioactive particles, or student arrivals to the Proctor lunch line are all events that might be modeled in this way. Poisson processes in space rather than time have been used to model distributions of stars and galaxies, or positions of mutations along a chromosome. Starting with characterizations of the Poisson process, a thesis might develop some of its important properties and applications. Wiley, , Chapter 1. Two famous problems in elementary probability are the "Birthday Problem" and the "Coupon Collector's Problem.
For the second, imagine that each box of your favorite breakfast cereal contains a coupon bearing one of the letters "P", "R", "I", "Z" and "E". Now suppose that the "equally likely" assumptions are dropped. But how does one prove such claims? A thesis might investigate the theory of majorization, which provides important tools for establishing these and other inequalities. This is a modern topic combining ideas from probability and graph theory.
A "cover time" is the expected time to visit all vertices when a random walk is performed on a connected graph. Here is a simple example reported by Jay Emerson from his recent Ph.
Consider a rook moving on a 2x2 chessboard. From any square on the board, the rook has two available moves. If the successive choices are made by tossing a coin, what is the expected number of moves until the rook has visited each square on the board? Reliability theory is concerned with computing the probability that a system, typically consisting of multiple components, will function properly e.
The components are subject to deterioration and failure effects, which are modeled as random processes, and the status of the system is determined in some way by the status of the components. For example, a series system functions if and only if each component functions, whereas a parallel system functions if and only if at least one component functions. In more complicated systems, it is not easy to express system reliability exactly as a function of component reliabilities, and one seeks instead various bounds on performance.
Specifically, in order to be Riemann integrable, a function must be continuous almost everywhere. However, many interesting functions that show up as limits of integrable functions or even as derivatives do not enjoy this property. Certainly one would want at least every derivative to be integrable. The awkward case of 'his or her'.
Can you spell these 10 commonly misspelled words? Explore the year a word first appeared. See dissertation defined for English-language learners See dissertation defined for kids. Examples of dissertation in a Sentence He wrote his dissertation on an obscure 16th-century poet.
Recent Examples of dissertation from the Web The hackers used stolen account credentials to access university professors' accounts and allegedly stole journals, dissertations and electronic books in science and technology, engineering, medical and other fields.
His official biography lists a Ph. Ashley Volion, 34, works part time at the Advocacy Center of Louisiana and is currently working on her dissertation in disabilities studies. Her parents are models of achievement; her mother wrote her doctoral dissertation on infant malnutrition in rural Brazil, and her father is an economist who works on health policy in the developing world.
Francine Patterson was working on her doctoral dissertation on the linguistic capabilities of gorillas and in started to teach Koko sign language. Hindt most recently was accused of plagiarizing his doctoral dissertation at the University of Houston.
A thank you to fans who get the shaft," 10 June But their games last night devolved into dissertations on solo play while the other starters hung around the 3-point line like beggars hoping to cadge quarters from the stars.
First Known Use of dissertation Other Education Terms baccalaureate , colloquium , corequisite , monograph , pedant , practicum , survey course , thesis. Definition of dissertation for English Language Learners. Learn More about dissertation See words that rhyme with dissertation Spanish Central: Seen and Heard What made you want to look up dissertation? Need even more definitions? Get Word of the Day daily email!
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Example Dissertation Topics Choosing dissertation topics can be hard, so we\'ve provided a wide range of dissertation topic ideas for you and sorted them into subject areas; just browse the list of subjects below to find inspiration for your dissertation topic.
Thesis Topic Generation or How to come up with a Thesis Topic! It can be very helpful to choose a broad subject area at first. As you read and research that subject area, you will narrow down toward a thesis topic.
Dissertation definition is - an extended usually written treatment of a subject; specifically: one submitted for a doctorate. How to use dissertation in a sentence. Jun 25, · Subject – International Development Topic – Success and Failure of HIV/AIDS Micro-insurance in sub-Saharan Africa Abstract: The purpose of this dissertation is to outline the advantages and disadvantages of micro-insurance for HIV/AIDS afflicted societies in sub-Saharan and Southeast Africa.
Are you the one tossing a coin to know what the right dissertation subject for you is? Look through our review of dissertation subjects to find one that will suit you best! Find a Dissertation Your library may be able to provide access to the dissertation you're looking for. Whether you are looking for a specific graduate work, or want to find out what dissertations or theses have been written on a particular subject, ProQuest provides a variety of tools to help.