The only examples of this that I know of are the constant curvature surfaces, which can be locally modeled either by the Euclidean plane ${\bf R}^2$, the sphere ${\bf S}^2$, or the hyperbolic plane ${\bf H}^2$, for which we have classical formulae for the distance function.

But I don't know of any other examples. For instance, the distance function on the surface of the solid ellipsoid or solid torus in ${\bf R}^3$ looks quite unpleasant already to write down explicitly. Presumably Zoll surfaces would be the next thing to try, but I don't know of any tractable explicit examples of Zoll surfaces that are not already constant curvature.

As is well know, the homotopy groups of $\mathrm{colim} S_X = SP^\infty X$ give the homology of $X$ (this is the Dold-Thom theorem), and the homotopy groups of $\mathrm{hocolim} S_X = SP^\infty_h X$ given the stable homotopy of $X$.

Is there a model for $SP^\infty X$, the ordinary infinite symmetric product, as a homotopy colimit as opposed to a categorical colimit?

The motivation for this question comes from thinking about $\infty$-categories. In an $\infty$-category, one does not really have a good notion (at least not one that I am aware of) of strict categorical colimits. So I'm wondering if there is, nonetheless, some easily defined functor on the $\infty$-category of spaces which will let us calculate ordinary homology. In short, is there any $\infty$-categorical analog of the Dold-Thom theorem?

Life insurance proceeds paid to you because of the death of the insured person are not taxable unless the policy was turned over to you for a price. This is true even if the proceeds were paid under an accident or health insurance policy or an endowment contract. However, interest income received as a result of life insurance proceeds may be taxable.

However, when I Google "life insurance proceeds estate tax", lots of scary-looking legal advice comes up, suggesting complex estate tax planning. It's not clear to me whether this is a valid concern.

I'm not looking for a treatise on the estate tax; I just want to know: if my wife and I were to pass away, would our children have to pay taxes on our life insurance proceeds, or not?

$(x^2 + y^2/9)^ 2 ≤ z ≤ x^2y , $

$0 ≤ x $.

I know that I should use this formula,

$ \int\int ( z2(x,y) - z1(x,y) ) dx dy$

(where $z2 = x^2y$ and $z1=(x^2 + y^2/9)^ 2 $ .

Is there any substitution which I could use? Or anything else...because when I try to do it..it seems an endless battle.

:)

Commander's Strike
Melee weapon
Target: One creature
Attack: An ally of your choice makes a melee basic attack against the target
Hit: Ally's basic attack damage + your Intelligence modifier

And let my ally Rufus make a melee basic attack against the Zombie, does Rufus need to be within normal melee range of the Zombie? Or can Rufus make the attack from anywhere, since the power does not specify?

One of the player's in our group thinks that Rufus should get the attack regardless of positioning. The way I read this power is that I need to be within melee striking distance of the Zombie and Rufus needs to be within melee striking distance of the Zombie.

A followup question is on powers that redirect or force attacks. Can a power that forces a monster to switch targets choose a new target that is outside of the original range? For example, if the Zombie attacks me with a melee attack and I redirect it to a Skeleton behind me, the Skeleton is beyond melee striking distance of the Zombie. What happens? Does the attack fizzle and nothing happens? Was it an illegal redirect? Does the attack still go through?

For example, consider the following line from this table:

Sep 2, 2010 10,270.08 10,350.98 10,211.80 10,320.10 3,704,210,000

The last number 3,704,210,000 is the volume information. What does it exactly represent?

Many thanks

For example, consider the following line from this table:

Date        | Open      | High      | Low       | Close     | Volume
Sep 2, 2010 | 10,270.08 | 10,350.98 | 10,211.80 | 10,320.10 | 3,704,210,00

The last number 3,704,210,000 is the volume information. What does it exactly represent?

So let $f : (A,R) \to (A',R')$ a homomorphism, does this induce a homomorphism $(M,\in) \to (M',\in)$, such that the obvious diagram commutes? For this we have to check $G(x)=G(y) \Rightarrow G'(f(x))=G'(f(y))$ for all $x,y \in A$. Is this true?

If this works out fine and $R'$ is extensional, then $f$ extends to a homomorphism $(M,\in) \to (A',R')$. Is this extension unique? If yes, we have found a functor which is left-adjoint to the forgetful functor from relational classes to extensional classes.

If this does not work out, what about adding the assumption of transitivity? And if this also does not work, is there nevertheless some left-adjoint, which is then different from the Mostowski collapse?

$^1$ In $ZF$ we can't define pairs of classes, and perhaps the categories above are not well-defined. This does not affect the content of my question, which can be translated to a well-formed statement in $ZF$.

Can $G$ and $H$ be conjugate in $Sym(\mathbb{N})$?

It would seem strange if this happened, but I can't think of an invariant that would definitively distinguish $G$ from $H$.

What if $G$ only has finitely many orbits of size $n$ for each $n \in \mathbb{N}$? This would at least ensure that $G$ cannot be conjugate to one of its own subgroups.