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Sep
19
comment minimum distance between graphs of functions
by moving the point in the direction with the smaller angle to the connection line. Two tangents which are both orthogonal to the same line are of course parallel to each other. Of course this again only applies if the curves don't intersect.
Sep
19
comment minimum distance between graphs of functions
@Geokal: Maybe you should add a more complete explanation what distance you mean. Since the first (and — at least to me, and obviously also to Siminore — most obvious) interpretation doesn't apply, here's my next-best guess: Do you mean the minimal distance of the curves in the plane, that is $\min_{x_1,x_2}\sqrt{(x_1-x_2)^2+(f(x_1)-f(x_2))^2}$? In that case, as long as the curves have well defined tangents in the points realizing the minimum, the answer is even simpler: The connecting line between the points must be orthogonal on both tangents, or else you could make the distance smaller …
Sep
19
comment minimum distance between graphs of functions
Assuming the two curves don't intersect, here's a hint: If the lines are parallel, then what does it tell you about their slopes?
Sep
18
answered Is Fractal perimeter always infinite?
Sep
18
comment Common inadequate definitions
Actually the most common inadequate definition of primes I've heard (which also would include the $1$ as prime) is: "A prime number is a number which is divisible only by $1$ and by itself."
Sep
18
comment Where is the error on this approximation to $\pi$
@pion: For example the proof that $\pi<22/7$. Since it is easily verified that $22/7<3.15$, this also proves that $\pi<3.15$.
Sep
18
comment Where is the error on this approximation to $\pi$
The author of the web page does not proof his claim that the areas are the same. The disproof is simple: If the author of the web page were right, we would have $\pi>3.15$. However we know from established proofs that $\pi<3.15$. Therefore the author's unproven claim cannot be true.
Sep
18
comment Common inadequate definitions
I think that's partially because of bad naming. I have no idea who came up with the term "even" and "odd" function (I guess the intuition is that even powers of $x$ are even functions, and odd powers of $x$ are odd functions). I personally prefer the names "symmetric" and "antisymmetric" function, because that better (although still not precisely) describes what it means (invariant resp. changing sign under the operation $x\mapsto-x$). Those names also directly imply that they are both special, so that the majority of functions is neither.
Sep
18
revised A formula that defines constructible universe
fixed typo in title
Sep
18
answered Are “sum” and “product” defined when there is only one number?
Sep
17
comment Is $y(t) = t^2 + i\cdot t^2$ a periodic function?
I have no idea at all what you want to say with that latest comment, sorry.
Sep
17
comment Is $y(t) = t^2 + i\cdot t^2$ a periodic function?
Because for a given $a$, there's exactly one $t$ which fulfils the equation $t=-a/2$, namely the one which you get by dividing $a$ by $2$ and changing the sign. For example, for $a=1$, the only $t$ which fulfils $t=-a/2$ is $t=-1/2$, but e.g. $t=1$ doesn't fulfill that equation. Thus $t^2$ is not $1$-periodic. The same is true for any $a$, therefore $t^2$ is not periodic at all. The critical point is that for fixed $a$, the equation would have to be fulfilled for all $t$. There's no $a$ such that $t=-a/2$ is fulfilled for all $t$.
Sep
17
comment Is $y(t) = t^2 + i\cdot t^2$ a periodic function?
@mvr950: I'm not sure what you mean. The equation which would have to be true for arbitrary $t$ is $t=-a/2$. This equation is clearly not fulfilled for $t\ne-a/2$ because that's exactly what $t\ne-a/2$ means. But it would have to be fulfilled for all values of $t$ if $t^2$ were periodic.
Sep
17
comment Recapturing + on Natural numbers.
Note that one such partial order is known under the name "divides", and the corresponding min and max are known under the names gcd and lcm.
Sep
17
answered Some infinite dimensional spaces, their elements and completion
Sep
17
answered Is $y(t) = t^2 + i\cdot t^2$ a periodic function?
Sep
17
comment How do I simplify this limit with function equations?
Hint: l'Hospital
Sep
17
comment Derivative of $u(x) = x^T x$
@RodCarvalho: Your edit changed the lower-case into an uppercase $T$. Now it reads as a transpose. Which may or may not be what user24205 meant. The fact that this way $u(x)$ doesn't map to $\mathbb R^n$ speaks against this interpretation.
Sep
17
comment Derivative of $u(x) = x^T x$
What exactly do you mean with $x^t x$? Multiplying the vector with its $t$-th component? (BTW, you should use LaTeX formatting; in this case, it would just mean to surround your formulas by dollar signs.)
Sep
17
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