http://www.billhowell.ca/Neural%20nets/Paper%20reviews/0_Nomenclature%20and%20symbols%20for%20mathematical%20verifications%20of%20peer%20reviews.txt


As a reviewer, I find that a step-by-step re-typing of a part of the paper as I have done below forces me to pay attention to details that I might otherwise skim over.  Even though this is perhaps too time intensive to apply to the full paper, by doing so over part of the authors' work, it gives me far grater confidence in the rest of the paper, which is read, but not analysed step-by-step. It also gives the authors a better idea of the weaknesses of the reviewer!

+----+
Nomenclature : 

//  denotes the vertical bar for "given" (vertical bar characters are not accepted by paper review system)
A    denotes [scalar, vector, matrix] 
A_T  denotes transpose of A, also transpose(A
//A//  denotes absolute value of matrix A (each element)
////A//// denotes spectral norm of A ////A////2
A_bar denotes an overscore on A
A_tilde  denotes the authors' use of tilde over a Matrix symbol 
*******  denotes start/end of topics & sub-topics
+-----+   denotes authors' steps in a [proof, development]
+----->   denotes start checks on specific steps by the reviewer (me) (additional ">" for sub-steps)
<-----+   denotes start checks on specific steps by the reviewer (me)
>> short reviewer comments
Greek and Latin symbols are written in short text form. 
These substitutions were necessary given the limited number of useful ASCII characters accepted by some paper review systems, the lack of superscript and subscript with simple text editors, and to make the text easier to use in software for symbolic expression processing (eventually - not ready yet at this time).  


Notation 
|x| is the absolute value vector of x, |x| = (|x1|, |x2|, . . . , |xn|)T 
||x||2  is the vector norm of x, ||x||2 = √Σni=1 |xi|2
I is the identity matrix
A > (≥)0 means A is a positive definite(semidefinite)matrix
A > (≥)B means A − B is a positive definite(semidefinite)matrix
A ≽ 0 means A is a positive(nonnegative) matrix, i.e, aij ≥ 0,
A ≽ B means the elements of matrices A,B satisfy the inequality aij ≥ bij
|A| is the absolute value matrix of A; |A| = (|aij |)n×n
(A) is spectral radius of A
λmax(A) means the maximum eigenvalue of matrix A
λmin(A) means the minimum eigenvalue of matrix A
ρ(A) is the spectral radium of matrix A
||A||2 is the spectral norm of matrix A. ||A||2 = √λmax(ATA)
Note : best viewed in text editor without automatic line wrap, constant font width

If A is an m × n matrix and B is a p × q matrix, then the Kronecker product A ⊗ B is the mp × nq block matrix:
                   a11*B   ...   a1n*B 
A ⊗ B   =    .           ...           .
                   a1n*B  ...    ann*B