I know that Mathematica allows for a lot of nice visualisations, so here I am. I was thinking if one could visualise (using Mathematica) the fact that a 3d (or simpler a 2d one) Torus doesn't have intrinsic curvature in contrast to a 3d sphere that does. In the 2d case, one idea to showcase this difference, would be by showing that a 2d torus can be mapped to a flat 2d surface, not possible for a sphere. The context is related to topology of phase space (of classical Hamiltonian systems). And this visualization idea originates from a recent post on physics SE, that can be found here, from which I quote:

If one takes the 2-dim torus and cuts it and unfold it, one gets the 2-dimensional cylinder . If further one cuts the 2-dim cylinder and unfold it, one gets a 2-dimensional flat surface. This means the (intrinsic) curvature of the 2-dim torus is zero and can be mapped into a flat space of the same dimension.

For the 2-dim sphere, this is not possible. There is no way it can be cut and mapped into a flat surface of the same dimension. It has (intrinsic) curvature non-zero and this is also a measure away from flatness (and also a measure of dimension correlation). One example is maps of earth (2-dimensional spherical surface) on a flat paper, one can see that the map contains distortions, since there is no mapping of a sphere into a flat surface.

3D torus:

enter image description here

3D sphere:

enter image description here

Does anyone have an idea as to how one could visualize the process of cutting e.g. the 2-dim torus (as elaborated in the quote), step by step, and showing that it reaches a 2d flat surface? Point being to show visually that a sphere has intrinsic curvature and a torus doesn't.

My wife and I keep a stick of butter in the cupboard rather than the refrigerator. Even with just the two of us, it rarely lasts very long, but I'd say it may go as long as a week without refrigeration before it's all gone. There have been times when I have told friends about this and they express concern due to the fact that butter is a dairy product. They cannot comprehend not refrigerating it at all times.

What allows butter to stay perfectly tasty and edible, even after a week outside of the refrigerator, where milk, sour cream, or cheese would be foul in a day or less?

I normally use SQL Server but for this project I'm having to learn DB2. If I can get the below code to work I'm set for the whole project. Below is a elementary example of what I cannot get to work. All I want to do is make a Created Global Temp Table (CGTT) and append data to it. The same code below will work if I use a Declared Global Temp Table, but for my purposes I must have a CGTT.

CREATE GLOBAL TEMPORARY TABLE TEST_TBL(
    KEY_ID BIGINT,
    SOMETEXT VARCHAR(10)
);

INSERT INTO USERID.TEST_TBL
VALUES(123456,'TEST TEST');

SELECT * FROM USERID.TEST_TBL;
SELECT COUNT(*) FROM USERID.TEST_TBL;

The above executes fine with no errors. When the insert is executed it even tells me "1 row updated". However, the select * and count(*) both give me 0 results; I'm wondering if this is a permissions issue, but can I have permissions to create a CGTT but not insert to it??

Any help provided would be most appreciated. Thanks in advance.

This has come up several times and thus far our DM has ruled against it. The question is: can a player take an immediate action against another player's or creature's immediate action, so long as conditions are met, like not being on their own turn?

For example, if one of the other players moved in to flank a red dragon and the dragon decided to tail whip [Tail Strike (immediate reaction, when an enemy moves to a position where it flanks the red dragon, at-will)], could I then, as a Bard, use Arrow of Warning?

Immediate Interrupt

Trigger: An enemy within range makes an attack roll against an ally

Target: The triggering enemy

Hit: 3[W] + Charisma modifier damage. The ally who was attacked by the triggering enemy can make an at-will attack against that enemy as a free action. The ally gains a bonus to this attack roll equal to your Wisdom modifier (minimum 1).

Our DM has said many times that you cannot take an immediate action against another immediate action. In the end, as a DM, it's his call anyway, but straight from the rules none of us can find a specific answer against it, so we would assume it is allowed.

Is someone who is wheelchair-bound obligated to say matir asurim or zokef kefufim if they cannot get up?

In the procedure prescribed for the first stage of a metzora's purification (Lev. 14:1-7), the Torah lists four items to be dipped into a mixture of blood and water: a live bird, a piece of cedar wood, a strip of crimson wool and a sprig of hyssop.

Rashi (to 14:6, citing Sifra) points out that while all four items are dipped, only three of them are attached together: the crimson wool is used to tie the wood and the hyssop, while the bird is just held along with them. And Negaim 14:1 adds that the tips of the bird's wings and tail are to be touching the bundle.

This sounds reminiscent of what we do when taking the Four Species on Sukkos; there too, three of them are bound together, with the esrog remaining apart but touching them. Does anyone suggest any reason for this commonality?

I'm trying to export a table of graphics into a gif animated file with just one repetition.

Export["C:\Race.gif", t, "AnimationRepetitions" -> 1] 

It doesn't work. The exported gif repeats and repeats over.

What should I do?

Here's something from Slonneger's "Syntax and Semantics of Programming Languages":

A variable may occur both bound and free in the same lambda expression: for example, in λx.yλy.yx the first occurrence of y is free and the other two are bound.

I assume the free variable is the y right after the λx. and the bound y's are the λy.y which I can sort of intuitively grasp. So ((λx.yλy.yx)a)b) would reduce to (yλy.ya)b) then to bba ? Can someone explain how this came to be? In the end it's the expression b twice. Can someone perhaps provide more examples of bound and free variables?

Is it normal for Mathematica to take over 30mins to try to compute the first eigenvector of a 100x100 matrix? The matrix is reasonably sparse - about 90% of the cells are 0s and each column sums to either 1 or 0.

The command I'm using is

m = ToExpression@Import@"http://pastebin.com/raw.php?i=yM4pUYQV";
N[Eigenvectors[m, Quartics -> True][[1]]]

My machine is an i5 3ghz and the process still isn't terminated yet. It's a quad core machine, but only one of the cores is being maxed out for over 30mins.

I have to generate current values between 0mA to 110mA. Can I generate current automatically using a micro-controller? (For example: 20mA , 30mA...) I need to keep the current stable for ms in frequency between 1Hz-100Hz.

If so, how?