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Mar
2
revised Game-winning strategy
clarified some things about the Sprague-Grundy theorem
Mar
1
comment Game-winning strategy
You made a good observation about reinterpreting the game. +1 It turns out that after that observation, much is already known about this game/relevant algorithms. See my answer for more details.
Mar
1
answered Game-winning strategy
Feb
29
answered Why does an inconsistent linear system have no “real” solutions?
Feb
27
comment Seemingly contradictory results
@BradGraham, most often, "algebra" is related to finitary operations (look up "universal algebra", for instance). There are lots of infinitary operations like those described on the Wikipedia page for "divergent series" or "linear functionals on $\ell_2$" or even just the supremum of a bounded sequence of real numbers.
Feb
27
comment Seemingly contradictory results
"are there any other examples of seemingly contradictory statements?" You may as well be asking for a list of abuses of notation or confluences of terminology, which I think you can probably find online.
Feb
27
answered Seemingly contradictory results
Feb
26
comment A set of vectors in $R^2$ for which $x + y$ stays in the set but $\frac12x$ may be outside.
@improve since the book said "but leave the origin" and the question was about closure under addition, I didn't think that was worth mentioning here. But you're certainly correct.
Feb
26
answered A set of vectors in $R^2$ for which $x + y$ stays in the set but $\frac12x$ may be outside.
Feb
26
comment A continuous function with positive and negative values but never zero?
The key point is that the complex plane with 0 deleted is connected but the real line with 0 deleted is not.
Feb
26
comment A continuous function with positive and negative values but never zero?
@Joanpemo the set of $x+0i$ for real $x$ is commonly referred to as the real axis in the complex plane. When $x$ is positive, you have the positive real (semi)axis. I don't think this is controversial, and was the convention in my courses from precalculus through graduate complex analysis.
Feb
26
comment A continuous function with positive and negative values but never zero?
This can't happen for real functions since you can't go from positive to negative on the number line without going through zero. This can happen with complex functions because you can go around zero in the complex plane to go from positive real axis to the negative real axis.
Feb
26
comment How can I begin reading journals and papers?
Papers in most subsubfields of math require some relevant background both in general experience with formal mathematics and content knowledge. Sometimes an undergrad intro to combinatorics is enough, but often much more specialized knowledge is required. Without a particular paper you'd like to read, I'd say just start with undergrad/graduate textbooks first and then branch out into real papers.
Feb
25
comment Is there a systematic way to find bijective function between non-empty set $\mathbb{X}$ and Natural numbers?
I think you should clarify the context a bit more. Are you asking "for most things I encounter, how would I know if it's countable?" or are you asking "does every countable set have a computable bijection to the naturals?" (No), or something else? Are you following a book or a course or a YouTube video?, etc.
Feb
24
comment Why is $\frac{1}{\lambda}$ the inverse of $\lambda$?
The following is a pedantic observation: only nonzero elements of a field have a multiplicative inverse.
Feb
22
comment Theorems Implying their Own Generalization
Most of the time when a theorem implies it's own generalization in a straightforward way, it's called a lemma and the generalization is the theorem. One source would be even lemma that proves something for two (like the intersection of two open sets is open) which leads immediately to a theorem about finite many.
Feb
21
comment How to calculate the area of the visible parts of a 3D PieChart?
Are you doing this for math fun, or for a demonstration of your point? If the latter, I would recommend writing some code to count pixels of the image based on color. If the former, do you have exact specifications of the height of the cylinder and is the viewing angle always fixed?
Feb
20
comment Differentiation: Vectors or scalars
@James, if you're saying don't bring this stuff up to first year calculus students (and perhaps students learning multivariable calculus for the first time) who aren't already thinking about it, then I would agree. I was merely responding to "Why do you think addition of points makes no sense?" with an explanation about why it can be considered in some contexts to make no sense.
Feb
20
comment Differentiation: Vectors or scalars
@James, the distinction is actually pretty useful in clearing things up. In the 1-dimensional space of time, specific times (like dates) are like points and signed time intervals (e.g. +3 days, -4 hours) are like vectors. You can take the difference of dates to get a time interval, and you can add a time interval to a date, but you can't add dates together. See the answers to math.stackexchange.com/questions/645672/… for more info.
Feb
20
comment What does “Mathematics of Computation” mean?
Unfortunately, the journal Mathematics of Computation has articles that are more about "mathematics, using (computer) computation" than the usual English meaning of the phrase "mathematics of computation" you outline here.