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Mar
28
answered Is $540^\circ$ a straight angle?
Mar
27
comment Types of definitions
Compare "A field is a field of reals if it's dedekind complete." to "A field is a field of reals if it's isomorphic as an ordered field to the following: [dedekind cuts construction]". I think this is close to what Bitwise is getting at, but the distinction doesn't hold mathematical significance, to my knowledge.
Mar
27
comment How should one depict a sign graph/chart with only 1 critical point and one interval?
You could also show the whole real line and write "not defined". As long as it's clear what you're trying to represent, it should be fine.
Mar
26
comment Why hyperreal numbers are built so complicatedly?
@Anixx, Those expressions needn't even make sense for arbitrary real functions f. I can't explain this issue in a comment, but maybe I or someone else will write a new long answer to this question that clears this stuff up.
Mar
26
comment Why hyperreal numbers are built so complicatedly?
@Anixx For one thing, you can't define differentiability properly for all real functions without the transfer principle of the hyperreals. I suppose "is this function differentiable at x=17?" Would be an unanswered question.
Mar
26
comment Why hyperreal numbers are built so complicatedly?
@Anixx, I think you misunderstood my meaning. The purpose of the hyperreals is to have a framework for derivatives and integrals that let's you answer all the same questions as regular calculus, but with infinitesimal numbers, rather than epsilon delta definitions.
Mar
26
comment Negative infinity to square equals positive infinity?
I think you want parentheses around that $-\infty$.
Mar
26
answered Negative infinity to square equals positive infinity?
Mar
26
comment Describe the diffrence between the following two problems and give an example of a physical situation which may be modeled by each equation
The Heaviside distribution is discontinuous, and is a forcing function in this equation.
Mar
26
comment All Eigenvalues of the operator $L(v)= L^2(v).$
@TedMosby, Since the answer posted shows that it must be 0 or 1, your equation, which is also true, doesn't contradict that. -1 doesn't satisfy the quadratic so it's actually impossible, even though the cubic makes it a candidate.
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.