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Mar
26
comment Has anybody ever considered “full derivative”?
@Anixx If you add an infinitesimal to rationals, you get formal Laurent series with rational coefficients. The Levi-Civita field is different in that it allows real coefficients and rational powers of $\epsilon$. But to define your full derivative in cases that aren't polynomials, you need a method for defining things like $sin(\epsilon)$. Hyperreal fields make this work perfectly, but physics.umanitoba.ca/~khodr/Publications/… suggests that the Levi-Civita field is probably good enough, at least for analytic functions (I haven't thought about it much).
Mar
25
comment Has anybody ever considered “full derivative”?
I think you misunderstand the notation in that PDF. $No(\omega)$ is actually the dyadic rationals thanks to the tree rank (see Theorem 15). $ No(\omega_1) $ is a hyperreal system assuming CH, but that has a lot of different flavors of infinitesimals, not just what you can get with reals and $\omega $.
Mar
25
comment Has anybody ever considered “full derivative”?
A side comment that has no bearing on this question: @Anixx, you can take $No(\omega)$, but as I said in answer to math.stackexchange.com/questions/1193422/… that won't give you the hyperreals.
Mar
25
comment Definition: Mathematical way to define the “left” and the “right”
IIRC if your alien is in a part of the universe made of antimatter, using the weak interactions would give them the opposite of the correct idea. I think Feynman made a joke about not shaking hands with an alien who presents the wrong one.
Mar
25
answered Definition: Mathematical way to define the “left” and the “right”
Mar
25
revised Why hyperreal numbers are built so complicatedly?
added order on Laurent series
Mar
25
comment Describe the diffrence between the following two problems and give an example of a physical situation which may be modeled by each equation
Have your lectures or textbook mentioned "forcing functions"? Also, do you know the graphs of the heaviside and dirac delta functions?
Mar
25
revised Describe the diffrence between the following two problems and give an example of a physical situation which may be modeled by each equation
latexified
Mar
25
answered Slice, projection, contour: A terminology question.
Mar
25
comment Why hyperreal numbers are built so complicatedly?
@RossMillikan, that field is the field of Formal Laurent series. It doesn't have the "elementary extension" property that we would like.
Mar
24
answered Extending the Ordinals analogously to the Integers
Mar
24
answered Why hyperreal numbers are built so complicatedly?
Mar
24
comment Why hyperreal numbers are built so complicatedly?
I think what you intended is a true statement about the hyperreals, so that's not really an avoided problem.
Mar
22
comment What function would do this
Every function on the integers of period 2 is a linear combination of the powers of the two second roots of unity. i.e. It's $a(-1)^n+b(1)^n$ for some $a$ and $b$. Matt Samuel's answer exhibits $a$ and $b$. Crash's answer elaborates on Henning Makholm's comment: saying what a function does on every input is defining that function. We consider $\cos(n\pi)$, $(-1)^n$, and "the function on the integers that outputs $-1$ when the input is odd and $1$ otherwise" to all refer to the same function.
Mar
22
comment Do any of these sequences have infinitely-many distinct iterates under run-length substitution?
@user48481 A run is a sequence of adjacent repeated values. I wouldn't have thought of this as a technical term until now, when I couldn't find this definition in a dictionary.
Mar
21
answered Dimension Theorem Corollary
Mar
19
answered Cross products and determinants in $\mathbb{R}^3$
Mar
15
answered countable subset of surreal games
Mar
15
comment Why can a Nim sum be written as powers of 2?
@BrianM.Scott Your first and third links are identical. www1.icsi.berkeley.edu/~ejf/pfiles/nimgroups.pdf is another PDF about this question (or at least a very related one).
Mar
15
answered Optimal strategy in a match picking game