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Boolean functions, theorems and technical application: Model theory is a branch of mathematical logics that deals with study of relations between the formal languages and their interpretations, or models. A model is a structure that gives meaning to the formal language sentences, and if it satisfies also a particular theory it is called a model of the theory.
Model theory has strong relations with algebra. Set theory is the branch of mathematics that deals with the general behavior of sets. The set theory lies in basis of the most of the mathematical disciplines, it has deeply influenced on the understanding of the subject of mathematics.
Recursion theory, also called computability theory, is a branch of mathematical logic that is in close relation with computer science and deals with the study of computable functions and Turing degrees, including also the study of generalized computability and definability. Proof theory is a branch of mathematical logics where the phenomenon of mathematical proof becomes an object of algebra or arithmetic.
The proof is usually presented as the structure of data, such as plain lists or trees, up to the hypothetic extremely complicated structures or machines which are constructed according to the axioms and rules of the logical systems. The proof theory can be viewed as a branch of philosophic logic where the main interest is in the proof-theoretic semantics.
Foundations of probability theory Probability theory is the branch of mathematics dealing with analysis of random variables, processes and events.
The sequence of repeated random events may keep within certain statistical patterns, in such way becoming predictable. Probability theory is applicable to many activities for which it is essential to consider the results of quantitative analysis of large sets of data. Ordinary differential equations Differential equation is an equation that connects the meaning of a certain unknown function in a certain points with the meaning of its various derivatives in the same point.
A differential equation contains in its form an unknown function, its derivatives and independent variables, but not any equation that contains the derivatives of an unknown function is a differential equation. It is also worth noting that a differential equation may not contain an unknown function, some of its derivatives and free variables at all, but it must contain at least one variable.
The topics that deal with the differential equations are the following:. Usage of Differential equations. Fundamental theorems of algebra, calculus, curves and projective geometry The topics that deal with the fundamental theorems are the following:. Fundamental Theorem of Poker.
Fundamental theorem of poker is a principle explaining the nature of poker and its main regularity, basing on that the right decision in this game is the decision that has the largest expected value, so a player should act as if they see all the cards of their opponents. However, to write a flawless research paper on any of the above mentioned topics a writer requires to:. Give a history and background regarding the development of the particular topic or theorem being discussed. Break the topics into sub topics to simplify the explanation of the topic and to help the readers understand it better.
Clearly and comprehensively elucidate the conclusion of the theorem or topic that is being discussed. Generally, the paper format for the mathematics research papers is more flexible than for other scientific fields, so you have a possibility to develop the outline of your work in a way you need for your topic in general.
However, there are several standard sections that must be included to ease the perception of your work: Background, Introduction, Body and Future Work or Conclusion, where you first give the description of the problem history, including its key notions, then present the specific results of your study and then providing the possible direction of the future research in your field. The methodology of mathematics in not a subject that is widely studied, but still there exist several issues that could help to develop the methodology of your own research.
The first issue is the necessity to express complicated relations symbolically, which could help to master the notions that could hardly be expressed in words.
The essential part of mathematics is abstraction that gives the possibility to codify out knowledge about several examples and thus to learn their common features. The same importance has the rigorous notion of proof which makes mathematics applicable and essential in physics, engineering, computer science etc. Considering these several key points of a research, each writer should himself define the own research strategy.
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We understand students have plenty on their plates, which is why we love to help them out. Let us do the work for you, so you have time to do what you want to do! Mathematics Research Paper Topics. The topics that deal with Homology Theories are the following: Unary and Binary Operations Research paper topics pertaining to Geometry: Euclidean Geometry Euclidean or elementary geometry is a geometric theory based on the system of axioms that was first stated by Euclid in the 3rd century BC.
Stereometry Stereometry is a branch of geometry that deals with the solid figures in space. 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. Bad notation, on the other hand, is disastrous and may deter the reader from even reading your paper.
In most cases, it is wise to follow convention. Using epsilon for a prime integer, or x f for a function, is certainly possible, but almost never a good idea. The first step in writing a good proof comes with the statement of the theorem.
A well-worded theorem will make writing the proof much easier. The statement of the theorem should, first of all, contain exactly the right hypotheses. Of course, all the necessary hypotheses must be included.
On the other hand, extraneous assumptions will simply distract from the point of the theorem, and should be eliminated when possible. When writing a proof, as when writing an entire paper, you must put down, in a linear order, a set of hypotheses and deductions which are probably not linear in form. I suggest that, before you write you map out the hypotheses and the deductions, and attempt to order the statements in a way which will cause the least confusion to the reader. A familiar trick of bad teaching is to begin a proof by saying: This is the traditional backward proof-writing of classical analysis.
It has the advantage of being easily verifiable by a machine as opposed to understandable by a human being , and it has the dubious advantage that something at the end comes out to be less than e. The way to make the human reader's task less demanding is obvious: Neither arrangement is elegant, but the forward one is graspable and rememberable.
Such a proof is easy to write. The author starts from the first equation, makes a natural substitution to get the second, collects terms, permutes, inserts and immediately cancels an inspired factor, and by steps such as these proceeds till he gets the last equation. This is, once again, coding, and the reader is forced not only to learn as he goes, but, at the same time, to decode as he goes.
The double effort is needless. By spending another ten minutes writing a carefully worded paragraph, the author can save each of his readers half an hour and a lot of confusion. The paragraph should be a recipe for action, to replace the unhelpful code that merely reports the results of the act and leaves the reader to guess how they were obtained.
The paragraph would say something like this: As in any form of communication, there are certain stylistic practice which will make your writing more or less understandable. These may be best checked and corrected after writing the first draft.
Many of these ideas are from HTWM, and are more fully justified there. Structuring the paper The purpose of nearly all writing is to communicate. Does your result strengthen a previous result by giving a more precise characterization of something? Have you proved a stronger result of an old theorem by weakening the hypotheses or by strengthening the conclusions? Have you proven the equivalence of two definitions? Is it a classification theorem of structures which were previously defined but not understood?
Does is connect two previously unrelated aspects of mathematics? Does it apply a new method to an old problem? Does it provide a new proof for an old theorem? Is it a special case of a larger question? Are your results concentrated in one dramatic theorem? Or do you have several theorems which are related, but equally significant?
Have you found important counterexamples? Is your research purely theoretical mathematics, in the theorem-proof sense, or does your research involve several different types of activity, for example, modeling a problem on the computer, proving a theorem, and then doing physical experiments related to your work?
Is your work a clear although small step toward the solution of a classic problem, or is it a new problem? 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.
Writing a Proof The first step in writing a good proof comes with the statement of the theorem. Write the proof forward A familiar trick of bad teaching is to begin a proof by saying: Specific Recommendations As in any form of communication, there are certain stylistic practice which will make your writing more or less understandable. Notation that hasn't been used in several pages or even paragraphs should carry a reference or a reminder of the meaning.
Writing a Research Paper in Mathematics Ashley Reiter September 12, Section 1: Introduction: Why bother? Good mathematical writing, like good mathematics thinking, is a skill which must be practiced and developed for optimal performance.
Some research papers by Charles Weibel. K-theory of line bundles and smooth varieties (C. Haesemayer and C. Weibel), Proc. AMS (to appear). 11pp. preprint,
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