Steven-Owen
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 May 17 comment Calculating Topological Genus If you want to make your comment and answer, I can choose it as correct. Apr 24 comment Finding a solution to the recurrence relation $T(n) = T(n-1) + T(n/2) + n$ It's a fantastic answer. Apr 24 comment Finding a solution to the recurrence relation $T(n) = T(n-1) + T(n/2) + n$ "Getting the particular solution part is very easy." "But getting the particular solution part is very difficult." Apr 13 comment Do Groebner bases give the smallest generating set for Ideals? Where can I learn more about finding minimal generating set? Is there some notion of dimension analogous to linear algebra? Apr 6 comment Harris' AG ex 2.24: projective variety under regular map. First of all, thank you so much for your help. Second, Which book do you recommend for an absolute beginner? Lots of books seem to offer pretty convoluted definitions for concepts that I imagine have rich intuitive meaning and a lot of the beauty seems to be getting lost in commutative algebra (which I also have never studied). Apr 5 comment Harris' AG ex 2.24: projective variety under regular map. And defining it as $x_{n+i}^d-\phi_i$ would clearly not work, since our points on the graph don't satisfy this polynomial. I just cannot think of a way to get homogenous polynomials that would just carve out the image of the regular map. That is, is the bigger graph where we are just looking at the regular map between the two projective spaces? Apr 5 comment Harris' AG ex 2.24: projective variety under regular map. Sure sure. Restricting our attention to just the $\phi_i$ being homogenous and of the same degree, say $d$, how should I define the polynomials, since I cannot just use $x_{n+i}-\phi_i$ (as I would in the affine case) since this is not homogenous. Apr 5 comment Harris' AG ex 2.24: projective variety under regular map. I am not saying $\phi_i$ is a regular map, but $\phi=[\phi_0:\ldots:\phi_i:\ldots:\phi_k]$ is a regular map, right? I completely believe you that the machinery of schemes is useful here, but I don't think it's necessary given that it's given in chapter 2 of Harris book. Apr 5 comment Harris' AG ex 2.24: projective variety under regular map. How is that identity not of that form? Isn't the identity just that with $\phi_i([X_0:\ldots:X_k])=X_i$? If I were to assume all regular maps were of this form where each $\phi_i$ is homogenous, do you have any advice for the question? I agree with you that this form of a regular map does not cover all of them, as I've read in Harris's book. Apr 5 comment Given $\sum |a_n|^2$ converges and $a_n \neq -1$, show that $\prod (1+a_n)$ converges to a non-zero limit implies $\sum a_n$ converges. It may very well be, my teacher just put it on the problem set. He took problems from lots of different books without citing them, sometimes because he made minor changes. Mar 31 comment Problems with votes please complete the question Mar 29 comment A linear subspace of projective space Is it easy to show that the "biggest" linear proper linear subspace is a hyperplane? And thus, by what you've shown since it cannot be contained in that, it must be the whole thing? Mar 29 comment A linear subspace of projective space how do we know there cannot be more than one hyperplane? is the span of two distinct hyperplanes the whole space necessarily? Mar 25 comment Do complex eigenvalues of a real matrix imply a rotation-dilation? I see what you mean. My above method is correct too, right? Mar 25 comment Do complex eigenvalues of a real matrix imply a rotation-dilation? Are the two $\lambda$ different? Just say $v=cw$ where $c$ is real nonzero. Then we have that $av-bw=L(v)=L(cw)=c(bv+aw)$. Then what do I take the difference of? I am just saying $0=L(v-cw)=(av-bw)-c(bv+aw)=(a-bc)v-(b+ca)w=(a-bc)cw-(b+ca)w=(ac-bc^2)w-(b+ca)w=‌​(-b-bc^2 )w=0$. So $b=-bc^2$. If $b\neq0$ then $c^2=-1$, a contradiction to $c$ is real. Mar 25 comment Do complex eigenvalues of a real matrix imply a rotation-dilation? I don't know if what I said made sense, sorry. Could you just elaborate briefly on the implication from $b\neq0$ to $v,w$ being linearly independent over the reals. Mar 25 comment Do complex eigenvalues of a real matrix imply a rotation-dilation? Sorry, why must they be linearly independent over the real numbers? Is it because if they weren't then $cv=w$ for real $c$ and thus multiplying them by a complex number $\lambda$ just had the effect of a real scalar since it would $(av-bcv,bv+acv)=((a-bc)v,(b+cv)v)$ where $a,b,c$ are real numbers? Mar 23 comment Do complex eigenvalues of a real matrix imply a rotation-dilation? Thank you for the help. A couple clarification questions: 1) What do you mean in 5. by real and imaginary parts of $v$? Do you mean writing $v$ as $x+iy$ where $x,y$ are real vectors and looking at the span of ${x,y}$ with the scalar field being $C$? 2) In 6. when you say likewise for $-i(v-\bar v)$ are you just saying $A^n(-i(v-\bar v))= -iA^n(v-\bar v)=-i(\lambda^nv-\bar\lambda^n\bar v)$. So what? 3) In 6. again, what is this point in $K$? It seems like we are only talking about complex vectors. Mar 23 comment Do complex eigenvalues of a real matrix imply a rotation-dilation? I don't quite follow. We want to assume that $0\in int(K)$ so having this fixed point doesn't really seem to help. Mar 22 comment Is primitivity invariant under matrix conjugation. O ok. I see, sorry. Let me rephrase my question though.