The Difference Between "be" and "is" in math writing When I want to write a paper, two problems bother me in my math writing!
First: what is the difference between be and is or are in math writing? For example, when I want to begin a theorem, I do not know if I should start with "Let $X$ be $Y$" or "Assume $X$ is $Y$". In some situations, it is really difficult to decide whether be is better or  is.
Second: I do not know why but several times I used the expression as follows. For example, when I want to define Vandermonde matrix, I usually write as follows and I do not know which format is better or correct 


*

*The Vandermonde matrix is defined (as follows) or (as shown) or (in the following form) or (by) or (as below)


I would like to ask you to help me about these two difficult problems in my math writing. 
Thanks for any suggestions. 
 A: The use of "be" versus "is" is not a matter of mathematical style: it's basic English grammar. 
"Let" is often followed by a clause with a verb in the present subjunctive, which is identical to the infinitive form of the verb. A mnemonic might be the title of the Beatles song, Let It Be.
The clause that follows "assume," however, is either a statement that could be a sentence on its own or is a phrase without a main verb at all. For example, "Assume it is true," which contains the statement, "It is true." Another example is, "Assume the contrary," which you can interpret by inserting "is true" at the end. 
Regarding "as follows," this is very often perfectly OK to use. I would not use it to introduce something that was written in the form of a dictionary definition. "As shown below" works similarly. It's hard to give more specific advice without an example of the context. 
A: "Assume" depends on the context, while "let" does not depend on the context.
"let" describes a term or makes a statement that should be true.
Use of "assume" is often to divide a compound statement in case of deletion or to prove an assertion.
Ex.: Let $x$ be a real number. Assume $x$ is positive.
