rschwieb
Reputation
63,328
97/100 score
 22h comment Synthetic geometry , angles.I need some ideas Maybe others will disagree, but once you start talking about lengths and angles as real numbers, you're in metric geometry not synthetic geometry. Synthetic geometry only has the tools provided by incidence, congruence, and a few other things like that. No numbers assigned to anything. 1d answered nilpotent endomorphism on finitely generated modules over a domain 1d revised Equivalent properties of Von Neumann regular rings added 178 characters in body; deleted 8 characters in body; added 77 characters in body 1d answered Equivalent properties of Von Neumann regular rings 2d comment Definition of fixed point free relation Also please use a better title @SergFillipenko 2d comment Definition of fixed point free relation @SergFillipenko the graph of the line $y=x+1$. Is a straight line, but also has no fixed points. 2d comment How to rotate a 3D object, using only local x-, y-, and z-rotations, so that it always faces a camera at the origin And I think what you are describing is that you want the camera's line of sight to be normal to the plane of the sheet. Is this correct? If so, why don't you just consider the camera as pointing at the center of the default sheet, and then using a rotation around the origin to move the line of sight of the camera to point through your special point $(x,y,z)$ with the proper roll? The default sheet would be carried along by the transformation to remain oriented to the camera. 2d comment How to rotate a 3D object, using only local x-, y-, and z-rotations, so that it always faces a camera at the origin I have a hard time understanding your description. (Don't worry, this happens in a majority of people describing similar problems.) Your default sheet has its center at and is tangent to the point $(0,0,1)$ on the unit sphere? The only rotations about the center of this sheet that will leave the sheet tangent are ones that rotate around the $z$ axis and/or flip the $x,y$ plane. This seems relatively boring and unlikely to be what you meant. Can you help clarify? 2d revised Extracting the Axis a Quaternion is rotating around from the Quaternion itself Directly added 177 characters in body 2d answered Extracting the Axis a Quaternion is rotating around from the Quaternion itself Directly 2d comment $\Bbb R,\Bbb C,\Bbb H$ are the only complete normed division rings @MarioCarneiro The type of norm involved here is this flavor, not norms like those for Banach algebras. 2d comment $\Bbb R,\Bbb C,\Bbb H$ are the only complete normed division rings The Frobenius Theorem doesn't make any mention of norms. I thought that "normed" and "finite dimensional" were simply two different conditions that were sufficient to make the theorem, but perhaps every proof of Frobenius Theorem depends on manufacturing a norm out of a finite dimensional algebra, somehow. Certainly we all know the norms that exist on those three algebras. 2d revised When is an ideal also a ring, and could then be anything said about its relation to the original ring added 6 characters in body 2d comment When is an ideal also a ring, and could then be anything said about its relation to the original ring @Stefan I went ahead and incorporated some of our discussion into the answer since I am at a real computer now (and not on a mobile device.) 2d revised When is an ideal also a ring, and could then be anything said about its relation to the original ring added 1021 characters in body Feb 8 comment When is an ideal also a ring, and could then be anything said about its relation to the original ring @ZelosMalum You don't believe in rings without identity? or just in the name "rng"? Feb 8 comment When is an ideal also a ring, and could then be anything said about its relation to the original ring @ZelosMalum I think most people say "rng" as "rung," but my teacher joked it was pronounced "wrong." You might also find it amusing to know that semirings (rings where the underlying abelian group is relaxed to a commutative monoid, so there are 'no negatives') are sometimes called "rigs" (no n). Much easier to pronounce... Feb 8 comment When is an ideal also a ring, and could then be anything said about its relation to the original ring @Stefan : Yes, that is correct, $eR$ is not necessarily an ideal. Here, we have it as an assumption. If $I$ is a two-sided ideal with an identity $e$, then $I=eI\subseteq eR\subseteq I$, so $I=eR$. Similarly $I=Re$. Both $Re$ and $eR$ are the ideal $I$ under these assumptions. Feb 8 comment When is an ideal also a ring, and could then be anything said about its relation to the original ring @Stefan Oops, didn't realize I borked a dollar sign somewhere in my last comment. Fixing it here: "if $I$ is an ideal, and $x\in I$ and $S$ is any nonempty subset of and $R$ , then $Sx\subseteq I$ . Apply this to $I=eR$ and $S=(1-e)R$" Feb 8 comment When is an ideal also a ring, and could then be anything said about its relation to the original ring @Stefan if $I$ is an ideal, and x\in I$and S is any nonempty subset of$R$, then$Sx\subseteq I$. Apply this to$I=eR$and$S=(1-e)R\$