# Writing mathematical papers

Replace it by "each" or "every", or recast the whole sentence" p. Once you have considered the structure and relevance of your research, you are ready to outline your paper. There may be several seemingly independent lines of reasoning which converge at the final step.

Avoid unnecessary notation. The overworked period is no worse than the overworked comma. What is the point of this paper?

Good mathematical writing, like good mathematics thinking, is a skill which must be practiced and developed for optimal performance. These may be best checked and corrected after writing the first draft.

## How to write a good math explanation

Have you proven the equivalence of two definitions? The double effort is needless. Neither arrangement is elegant, but the forward one is graspable and rememberable. Why did this author think this question was interesting? These may be best checked and corrected after writing the first draft. Which ones follow naturally from others, and which ones are the real work horses of the paper? You should take care not to disappoint them! The title is the first contact that readers will have with your paper. Which are the corollaries of these theorems? It would be truly surprising if yours were to come as a high school student! They are easy for the reader to overlook, and the oversight causes backtracking, confusion, delay. Mathematics has been accumulating for thousands of years, based on the work of thousands or millions of practitioners. The accepted format for research papers is much less rigidly defined for mathematics than for many other scientific fields. 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.

If he dislikes partial differential equations, for example, he should be warned early on that he will encounter them. Does is connect two previously unrelated aspects of mathematics?

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