Pages: 79

No chapter abstract available.

Pages: 14

Teaching is an important part of being a professor. One has to gain the trust of the students by his/her knowledge and understanding in the subject and act as a role model to get the students through the challenges faced in the course. Two main values of good teaching are caring and accepting its responsibility. Teaching is an important part of being a professor. One has to gain the trust of the students by his/her knowledge and understanding in the subject and act as a role model to get the students through the challenges faced in the course. Two main values of good teaching are caring and accepting its responsibility. Hearing other students do problems, instead of just seeing the instructor do them perfectly in front of them, is very important for the students’ learning. In graduate level mathematics courses, which are theoretical in nature, it is useful to discuss the patterns and ideas behind the proofs.

Pages: 11

Writing a research paper is like writing a short story. In modern times, the value of the paper is determined by the level of the journal it is accepted to. Research requires patience and determination and to write well, one has to read well and observe. A good writer first sets the main plot and major events in the story, then writes the details. It is about knowing the beginning and the end and then filling the middle and sometimes the end changes based on the details in the middle. Separating ideas from the details is the key, especially in mathematics, in which details are very important. This chapter describes the structure of research papers in mathematics and the publication process. A paper begins with a title, which in a few words states the problem that has been solved and studied in the paper.

Pages: 11

The core mindset for all areas is the same and as a graduate student in mathematics you have the core mindset to be able to follow at least some pieces in every talk. It is good for the mind to listen to research talks and be able to pick up different ideas and ways of thinking. One can search the speaker’s name and find the paper among his/her latest downloads on the archive. Reading the introduction of the paper and finding and writing the ideas and main results, can greatly enhance your understanding of the talk. One can typically combine the two slots of time and aim to finish in 30 minutes. The talk should be in a way to invite others to work in the same area and maybe become your collaborators in a related problem and paper in the future.

Pages: 9

The process of writing a research paper in mathematics is like the process of building a house. One needs to build the foundation and sketch an overall structure, then place in it windows and doors and afterwards give it an interior design in order to make it suitable for the occupant. In contrast to taking a course and concentrating on one textbook, in research one finds many sources available for the same information. The key is to balance between pure and applied mathematics and work and be interested in both to ensure stronger results and more opportunities in the future. The chapter discusses the statement has to be perfect needs to be verified by the right reasoning or a reference to another paper or book needs to be made to justify its validity. In research one has to devote much time and effort, but also keep the cycle of writing active and have plans on what to work.

Pages: 29

An introduction is given here on how to use the computer program Latex to prepare research papers, dissertations and presentations in mathematics. Sometimes when there is an uneven balance between the text and equations on a page, Latex puts extra white space at the end of the page. One way to fix this is to type, raggedbottom after packages before begin document at the beginning of the file. As for preparing a dissertation, since dissertations often have multiple chapters with many pages in each, then one can use a separate document file for each chapter and then apply the “include” command. The best way is to work on each chapter separately as independent documents with the beginning and ending codes and when they are completed and have been compiled without errors, the “include” command in a new file. To compile the Beamer presentation, the same icon as in a regular document may be clicked on.

Pages: 125

No chapter abstract available.

Pages: 45

Laser technology has taken an important role in medicine, making many medical procedures possible and is widely used in various technological devices such as in telescopes and in regular devices for instance, shopping scanners to read bar codes or in CD and DVD players. The field of image processing employs lasers to produce images with high quality and in the field of remote sensing, lasers help to determine information about the atmosphere, such as the condensation of water in the clouds. Focusing the laser beam and intensifying its energy to a smaller point is essential in surgical procedures, for example, when some tumor cells need to be destroyed. Self-focusing of the laser beam has been studied in the literature and it describes the situation, in which the rays in the beam converge automatically when traveling in the medium because of the variations in temperature in the medium or the presence of obstacles in the atmosphere.

Pages: 32

The web is typically viewed as a directed graph with websites thought as vertices or nodes and links between them as edges. The power method and the idea of PageRank itself, were first introduced in 1998 by two researchers, S. Brin and L. Page, working at Google. In the context of web communities, if the number of links from an unrelated and lower ranked web community i going to a page in web community j, is less than the number of links connecting pages in community j, then link spam has likely occurred. One may follow the arrows on the graph and determine in how many steps the process can move out of state i and return to it. The greatest common divisor of the number of steps taken by different possible paths is referred to as the period of state i and when this number is 1, then state i is said to be aperiodic.

Pages: 45

The study of the heat and wave equations lie in the core of graduate level courses in partial differential equations. Originating from the wave equation, fluid models are extensively studied in the research community. Most of the work in the literature regarding the Navier-Stokes equations has been on the two-dimensional incompressible equations; however, the study on the three-dimensional Navier-Stokes equation and the equation in case of compressible flow are beginning to take shape. Brownian motion is continuous but nowhere differentiable and thus, the usual rules of integration can not be applied to stochastic integrals. The main tool in stochastic analysis is the Itô formula and is applied as follows. To introduce these techniques, we explain its first result, which is the Central limit theorem for stochastic Navier-Stokes equations. Contrary to large and moderate deviations, the Central limit theorem describes typical events.