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comment Algebraic multiplicity of an eigen value
Algebraic multiplicity is just the degree of the linear term corresponding to $\lambda$ in the characteristic polynomial. Every matrix is similar to an upper triangular matrix and the determinant is invariant under similarity. Is this not enough to prove the claim?
Jan
12
comment Homomorphism between homotopy groups of spheres induced by the fibration and the multiplication map of $SO(n)$
@archipelago: Did you read my answer? I gave a large class of examples where $\lambda=0$.
Jan
12
answered Homomorphism between homotopy groups of spheres induced by the fibration and the multiplication map of $SO(n)$
Jan
11
comment Conditions for the integral to equal zero
The integral is nonzero even when $f(x)=1$. If your asking if $f(x)$ has to be bounded when the integral is finite, this is also not true. Consider, $f(x)=x$.
Jan
11
comment Nonlinear eigenvalue problem $Ax = f(c) x$
I'm a bit confused about the set up. If the $v_{i}'s$ are eigenectors with eigenvalues $b_{i}$, are they not precisely the eigenvectors for $f(c_{i})=b_{i}$. Is the base field real or complex? Is $A$ nice in any way? (e.g. diagonalizable, hermetian, unitary, orthogonal, real self-adjoint)
Jan
11
comment $k$-jets , submanifolds
I think your observation is the key step. $J^{1}(n,1)$ should just be the dual of $\mathbb{R}^{n}$, $\Sigma^{n-1}$ is all nonzero functionals and $\Sigma^{n}$ contains just the $0$ functional. Are these submanifolds? For the second part, perhaps you could use this characterization of $\Sigma^{n}$ along with the definition of a Morse function.
Jan
11
comment What does $d\zeta_1\wedge\cdots\wedge d\zeta_n$ mean in the context of Cauchy formula (on polydiscs)?
@TedShifrin: You are absolutely right. I, admittedly, did not read the problem carefully and assumed (a dangerous practice for mathematicians) that the resulting manifold would be complex.
Jan
10
answered Examples of a Subbasis and Basis of a topology.
Jan
10
comment What does $d\zeta_1\wedge\cdots\wedge d\zeta_n$ mean in the context of Cauchy formula (on polydiscs)?
I'm not a great person to ask for a reference, as complex geometry is a bit outside my field of expertise. However, a brief account can be found in a book by M. Nakahara called "Geometry, Topology and physics. I think it's in Chapter 7, or somewhere around there.
Jan
10
comment What does $d\zeta_1\wedge\cdots\wedge d\zeta_n$ mean in the context of Cauchy formula (on polydiscs)?
Also, I don't think you want the wedge in the denominator.
Jan
10
comment What does $d\zeta_1\wedge\cdots\wedge d\zeta_n$ mean in the context of Cauchy formula (on polydiscs)?
It is the wedge product of forms. You can view $\partial_{0}P$ as a complex manifold. The $n$ fold wedge will give a measure a measure on this manifold which you can integrate against.
Dec
5
awarded  Yearling
Sep
30
awarded  Explainer
May
17
answered Why is abstract algebra so important?
May
17
comment Series of functions as Lebesgue integral with counting measure
you are integrating with respect to $n$, so you want to view $f_{n}(x)=: f(x,n)$ as a function of $n$.
May
16
comment Can one prove the existence of tensor product without explicitly constructing it?
I also said, I suspect. I am not certain it will work.
May
16
comment Can one prove the existence of tensor product without explicitly constructing it?
@Qiaochu Yuan Not true, there is a more general version which is applicable in any model category
May
16
answered Can one prove the existence of tensor product without explicitly constructing it?
May
16
answered Intuition and Motivation - Linear Operator $T - \lambda_k I$ ? [Lay P270 Thm 5.1.2]
May
16
comment Convergent subsequence proof
looks good to me