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Mathematics Research Paper Topics

How Does One Do Mathematical Research?

❶Model theory has strong relations with algebra.




Homological characterizations of quasi-complete intersections , Jason M. Bridge spectra of cables of 2-bridge knots , Nicholas John Owad. Stable local cohomology and cosupport , Peder Thompson. Graph centers, hypergraph degree sequences, and induced-saturation , Sarah Lynne Behrens. Systems of parameters and the Cohen-Macaulay property , Katharine Shultis. Betti sequences over local rings and connected sums of Gorenstein rings , Zheng Yang.

Results on edge-colored graphs and pancyclicity , James Carraher. Well-posedness and stability of a semilinear Mindlin-Timoshenko plate model , Pei Pei. Embedding and Nonembedding Results for R. Periodic modules over Gorenstein local rings , Amanda Croll. Decompositions of Betti Diagrams , Courtney Gibbons. Closure and homological properties of auto stackable groups , Ashley Johnson. Random search models of foraging behavior: The Theory of Discrete Fractional Calculus: The background will serve to orient your reader, providing the first idea of where you will be leading him.

In the background, you will give the most explicit description of the history of your problem, although hints and references may occur elsewhere. The reader hopes to have certain questions answered in this section: Why should he read this paper? What is the point of this paper? Where did this problem come from? What was already known in this field? Why did this author think this question was interesting?

If he dislikes partial differential equations, for example, he should be warned early on that he will encounter them. If he isn't familiar with the first concepts of probability, then he should be warned in advance if your paper depends on that understanding. Remember at this point that although you may have spent hundreds of hours working on your problem, your reader wants to have all these questions answered clearly in a matter of minutes. In the second section of your paper, the introduction, you will begin to lead the reader into your work in particular, zooming in from the big picture towards your specific results.

This is the place to introduce the definitions and lemmas which are standard in the field, but which your readers may not know. The body, which will be made up of several sections, contains most of your work. By the time you reach the final section, implications, you may be tired of your problem, but this section is critical to your readers. You, as the world expert on the topic of your paper, are in a unique situation to direct future research in your field.

A reader who likes your paper may want to continue work in your field. If you were to continue working on this topic, what questions would you ask?

Also, for some papers, there may be important implications of your work. If you have worked on a mathematical model of a physical phenomenon, what are the consequences, in the physical world, of your mathematical work? These are the questions which your readers will hope to have answered in the final section of the paper. You should take care not to disappoint them! Formal and Informal Exposition. Once you have a basic outline for your paper, you should consider "the formal or logical structure consisting of definitions, theorems, and proofs, and the complementary informal or introductory material consisting of motivations, analogies, examples, and metamathematical explanations.

This division of the material should be conspicuously maintained in any mathematical presentation, because the nature of the subject requires above all else that the logical structure be clear. Thus, the next stage in the writing process may be to develop an outline of the logical structure of your paper. Several questions may help: To begin, what exactly have you proven?

What are the lemmas your own or others on which these theorems stand. Which are the corollaries of these theorems? In deciding which results to call lemmas, which theorems, and which corollaries, ask yourself which are the central ideas. Which ones follow naturally from others, and which ones are the real work horses of the paper?

The structure of writing requires that your hypotheses and deductions must conform to a linear order. However, few research papers actually have a linear structure, in which lemmas become more and more complicated, one on top of another, until one theorem is proven, followed by a sequence of increasingly complex corollaries.

On the contrary, most proofs could be modeled with very complicated graphs, in which several basic hypotheses combine with a few well known theorems in a complex way. There may be several seemingly independent lines of reasoning which converge at the final step.

It goes without saying that any assertion should follow the lemmas and theorems on which it depends. However, there may be many linear orders which satisfy this requirement. In view of this difficulty, it is your responsibility to, first, understand this structure, and, second, to arrange the necessarily linear structure of your writing to reflect the structure of the work as well as possible.

The exact way in which this will proceed depends, of course, on the specific situation. One technique to assist you in revealing the complex logical structure of your paper is a proper naming of results. By naming your results appropriately lemmas as underpinnings, theorems as the real substance, and corollaries as the finishing work , you will create a certain sense of parallelness among your lemmas, and help your reader to appreciate, without having struggled through the research with you, which are the really critical ideas, and which they can skim through more quickly.

Another technique for developing a concise logical outline stems from a warning by Paul Halmos, in HTWM, never to repeat a proof: If several steps in the proof of Theorem 2 bear a very close resemblance to parts of the proof of Theorem 1, that's a signal that something may be less than completely understood. Other symptoms of the same disease are: When that happens the chances are very good that there is a lemma that is worth finding, formulating, and proving, a lemma from which both Theorem 1 and Theorem 2 are more easily and more clearly deduced.

These issues of structure should be well thought through BEFORE you begin to write your paper, although the process of writing itself which surely help you better understand the structure. Now that we have discussed the formal structure, we turn to the informal structure. The formal structure contains the formal definitions, theorem-proof format, and rigorous logic which is the language of 'pure' mathematics.

The informal structure complements the formal and runs in parallel. It uses less rigorous, but no less accurate! For although mathematicians write in the language of logic, very few actually think in the language of logic although we do think logically , and so to understand your work, they will be immensely aided by subtle demonstration of why something is true, and how you came to prove such a theorem.

Outlining, before you write, what you hope to communicate in these informal sections will, most likely, lead to more effective communication. Before you begin to write, you must also consider notation. The selection of notation is a critical part of writing a research paper. In effect, you are inventing a language which your readers must learn in order to understand your paper. Good notation firstly allows the reader to forget that he is learning a new language, and secondly provides a framework in which the essentials of your proof are clearly understood.

My Symbolic Postmodern System Model. Every universal sentence implies each of its instances. But not every universal sentence is implied by the set of its instances—a point not found in Tarski A counterinstance of a universal sentence is a false instance.

Do Multiple Representations Need Explanations? Elementary school children, some of whom were nonnative speakers of English, learned to add and subtract integers in a discovery-based multimedia game either with or without verbal guidance in English or optionally in Spanish Groups Second Discourse on Perfect Ideas: In this discourse the Notion is considered as being related to the Groupoid.

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