hardmath

Knoxville, TN

uclue.com

Age: 61

Enjoys programming in Prolog.

Richard O'Keefe: "Prolog is an efficient programming language because it is a very stupid theorem prover."

6h
comment Why is Euler's Totient function always even?
Inversion map might be clarified to mean mapping to an additive inverse. This previous Math.SE Question would be worth adding as a link for the proposition about $Aut(Z_n)$.
6h
comment Metric on the Set of Binary rectangular matrices
Yes, and this depends only on the set being finite, not the particular set being binary matrices of a fixed size.
7h
reviewed Close If a and b are the two solutions to x^2 - x - 2=0, then a+b=?
7h
comment Metric on the Set of Binary rectangular matrices
Closely related, but arguably not quite a duplicate: Is it true that any metric on a finite set is the discrete metric?.
7h
revised Metric on the Set of Binary rectangular matrices
Added details for proof that on finite sets any two metric are strongly equivalent
7h
comment Metric on the Set of Binary rectangular matrices
By strong equivalence of metrics is meant that for two distance functions $d_1(x,y)$ and $d_2(x,y)$ there exist positive constants $c_1,c_2$ such that for all $x\neq y$: $$ d_1(x,y) \lt c_1 d_2(x,y)\; \text{ and } \; d_2(x,y) \lt c_2 d_1(x,y) $$
7h
comment Metric on the Set of Binary rectangular matrices
The reason I asked for a number of clarifications about your Question was to make sure the terms you were using ("metrics") were well-defined. You agree that by metric was meant a distance function. There is a widely adopted notion of equivalence of metrics which says two metrics are equivalent iff the topologies they give are the same (see link in Answer).
7h
comment What is the (currently) optimal root finding algorithm for multivariate functions?
Currently there is no "optimal" method for all equations. The practical choice of a Newton-Raphson root finder vs. derivative-free root finder will depend on the cost of function evaluations and partial derivative evaluations. Too little context is provided to nominate a preference among the widely studied alternatives.
8h
comment What is the (currently) optimal root finding algorithm for multivariate functions?
As written you are asking to solve one equation for more than one unknown. Is this really what you meant to ask?
8h
revised Help With Factoring Trinomials
Fixed/formatted per clarifying comment by OP
11h
answered Metric on the Set of Binary rectangular matrices
11h
comment maximum modulus principle question
For the last step, consider that $f+\overline{f}$ would be a real-valued holomorphic function on the interior of $\gamma$, so the real part of $f$ would be constant (and similarly for its imaginary part).
11h
revised maximum modulus principle question
Slight corrections
15h
comment Negative modulo operations
This is more of an issue with programming languages and implementations. There is no confusion when one is referring to equivalence classes of integers in elementary number theory. The operation you call mod can be defined for negative operands in various ways (and has been), but a survey of such approaches would be off-topic here.
15h
comment maximum modulus principle question
The case where $f$ has a zero inside $\gamma$ cannot be ignored, as for example $f(z)=z$ has constant modulus on the unit circle centered at the origin (of the complex plane). The trick is how to exploit the absence of an interior zero.
16h
comment How does $A_n$ look in Aut$(X)$?
Normally the "automorphism" terminology would be used only where $X$ has some algebraic structure and we restrict attention to permutations of $X$ preserving that structure. Here $S_n = Sym(X)$ or $Perm(X)$ would suffice.
16h
comment Metric on the Set of Binary rectangular matrices
@SkSarifHassan: Important clarification -- all binary matrices of all sizes, or just binary matrices of fixed size $m\times n$ ?
18h
comment Good Reference for Justifying (less well-known fields of) Math?
The title suggests an even broader scope than likely you intended, from reading the body of your Question. Perhaps "justifying the study of specialized fields of mathematics" would better frame your Question. To improve it you should give specific examples of "narrow" fields that you think need such justification, or examples of where justification has been satisfactorily provided for fields (in your opinion). Details will help your Question receive considered attention and not get closed as overly broad or primarily opinion based.
18h
comment Metric on the Set of Binary rectangular matrices
By metric do you mean a distance function that gives this (finite??) set a metric topology? Did you mean to create a topology on all sizes of binary matrices taken together?
18h
comment Eigenvalues and eigen vectors
To look at it another way, given a matrix $A$ with eigenvector $u\neq 0$, the corresponding eigenvalue is uniquely determined by the definition: $$ Au = \lambda u $$ Further, thus while two different eigenvectors $u,v$ might share a common eigenvalue, two different eigenvalues cannot share a common eigenvector.
1 2 3 4 5