All Questions
12,778 questions
7
votes
4
answers
973
views
I was wondering if the set of singular loops is a (somewhere) submanifold of loop space?
The set of all smooth maps $S^1\to M^n$ ($M$ is a smooth manifold) is a generalized manifold(see http://ncatlab.org/nlab/show/smooth+loop+space).
I was wondering if the set of singular loops (maps ...
- 5,063
0
votes
1
answer
498
views
Quotient of \ell_1 by space of finite sequences
The following question came up during a reading of Rudin's functional analysis. I have not been able to find any information through searching online, but I apologise if the answer is obvious, or the ...
- 11
1
vote
2
answers
252
views
On bounded homogeneous connected domains of C^n
So let $D\subseteq \mathbb{C}^n$ be a bounded connected open set with a transitive action of its group of biholomorphisms (which we denote by $Hol(D)$). Note that I'm not assuming that $D$ is ...
- 7,609
0
votes
0
answers
819
views
Possible application of Rouche's theorem to aproblem of complex roots of polynomials
The following holds:
Let $P(x)$ be a polynomial in one variable $x$ of degree $3$ with complex coefficients
such that
a)
$$
P(-1)=P(1)=0
$$
Then
b)
the formal derivative $P^{'}(x)$ has a root in ...
- 1,641
2
votes
1
answer
208
views
Is there an elementary proof for preserving inequalities under the change of l_p metrics?
Here is what I mean exactly:
Let $A=(a_1,a_2)$ and $B=(b_1,b_2)$ be two points in the real plane (for simplicity, but general finite dimensions would also be nice), and define the $\ell_p$-metric as ...
8
votes
2
answers
865
views
frechet manifolds book
hi, does anyone know a good book or some lecture notes on the theory of frechet manifolds ?
1
vote
1
answer
299
views
Compact complex surfaces having infinitely many negative curves?
I am trying to find a (smooth) compact complex surface $X$ so that the set of irreducible curves $C$ on $X$ for which $C.C<0$ is infinite. Do any of you know of an example. Thanks.
- 117
4
votes
0
answers
238
views
dimension of induced comodule
Let $\pi : G \to H$ be epimorphism of Hopf superalgebras, where $G$ be an quantum super group of function on $GL(m|n)$, $H$ be an quantum group of function on $GL(m) \otimes GL(n)$; $W$ an finite ...
- 41
18
votes
1
answer
1k
views
Who introduced the notion of "stability" in numerical analysis?
I am preparing a lecture course on the applications of operator theory where I intended to make some numerical analysis application. I was wondering about this question while browsing the literature I ...
- 4,729
1
vote
0
answers
404
views
weakly conformal map
Maybe an easy topology excercise. Say u is a weakly conformal map from a region of complex plane C to C. Then $u_z*{\bar u}_z=0$. How to derive that u is holomorphic or antiholomorphic, i.e. $u_z=0$ ...
- 65
2
votes
1
answer
2k
views
Degree of holomorphic maps between compact Riemann surfaces
Can all nonzero degree map between compact Riemann surfaces (both genus >1 ) be deformed to holomorphic maps, if we can change the conformal structures on them? The simplest case: does there exist ...
- 65
7
votes
1
answer
1k
views
Helmholtz-Decomposition on compact Riemannian manifolds
For smooth domains $\Omega$ in $\mathbb{R}^n$ it is known that one can decompose vector fields in $L^p(\Omega)^n$, $1 < p <\infty $ into a "gradient"- and a "divergence-free"-part such that
$L^...
- 73
2
votes
1
answer
608
views
Stein's extension operator and wave front sets
Let $K\subset\mathbb{R}^d$ be a compact set with non-empty interior and Lipschitz boundary. In Section VI.3 of his book "Singular Integrals and Differentiability Properties of Functions", E. M. Stein ...
- 7,817
7
votes
2
answers
1k
views
A book on Banach Manifold for a Dynamicist
Hi all,
Could you give me a suggestion of suitable book about Banach Manifolds for someone that have background in functional analysis at the level of Conway's book and Do Carmo's book on Riemannian ...
user11178
3
votes
1
answer
429
views
de Rham cohomology class of diagonal
I post again a question I asked in the post by Descartes:
Since this is the topic on diagonal, I like to ask a question: Let $X$ be a compact Kahler manifold of complex dimension $n$, and let $\Delta ...
- 117
2
votes
1
answer
672
views
How to calculate Dr. Curt McMullen's expanding eigenvalues for totally degenerate groups?
What is required in order to derive the expanding eigenvalues of Dr. Curt McMullen's torus orbifold bundles over the circle and the corresponding totally degenerate groups, as presented in Section 3.7 ...
- 23
10
votes
2
answers
811
views
Classification of holomorphic disc bundles
I've had difficulty finding sources which treat the classification of holomorphic disc bundles over (compact and noncompact) Riemann surfaces. Note that by "bundle", I mean a holomorphic fiber bundle,...
- 1,221
2
votes
1
answer
949
views
Hereditarily indecomposable Banach spaces and Separable Quotient problem
A Banach space $X$ is called indecomposable if there exists no infinite-dimensional subspaces $M$ and $N$ such that $X = M \bigoplus N$. If every infinite-dimensional closed subspace
of $X$ is ...
- 515
5
votes
2
answers
3k
views
Diagonalization of a matrix of differential operators
Dear community,
i have a question regarding differential operators acting on vector valued functions and how to "diagonalize" them.
To explain my question i will use an example:
Let $V^k$ be the ...
7
votes
1
answer
1k
views
How to construct a scalar differential operator having the same spectrum as a non-scalar differential operator exploiting symmetries?
I am interested in eigenvalue problems for differential operators acting on one forms on closed two-dimensional manifolds and how they relate to eigenvalue problems of associated operators acting on ...
1
vote
1
answer
3k
views
Is point to set distance continuous?
Assume $\mathbf{d}:\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}_0^+$ is a metric such that the function $\psi(x)=\mathbf{d}(x,y)$ for any $y\in\mathbb{R}^n$ is continuous in the Euclidean ...
- 27
6
votes
0
answers
3k
views
Projective and injective tensor product
It is well known that for arbitrary Banach spaces $X$ and $Y$ we have that the dual space
$(X \hat{\otimes}_{\pi} Y)^* = \mathcal{L}(X, Y^*)$.
If we take $\ell^p$ and $\ell^q$ such that $p < q^{\...
4
votes
1
answer
1k
views
Hausdorff dimension of graphs .
Is there an easy way to calculate the Hausdorff dimension of the graph of a real "elementary" function, like $f(x)=\sin(1/x)$ ?
- 3,630
5
votes
2
answers
4k
views
finite codimension implies closed?
Let $E$ be a (complete) topological vector space, and $u:E\to E$ be continuous. Is it always true that if ${\rm Im}(u)$ is of finite codimension in $E$, then it is closed in $E$ or do we have to ...
- 53
2
votes
3
answers
489
views
harmonic 1-form with bounded energy on a strip in $\mathbb{R}^2$
Let $S=[-a,a]\times[b,+\infty] \mod \{(-a,t) =(a,t) \mid t \in [b,+\infty]\}$ be a strip in $\mathbb{R}^2$ with identified sides. Let $w$ be a real harmonic 1-form on $S$, which has a primitive $f$ on ...
6
votes
0
answers
733
views
$f(x) \ne g(x)$ but $f(f(x))=g(g(x))$ - is there a name/some discussion of this property?
In the context of iteration of functions I look at the eigenvalues of the associated matrixoperator/Carleman-matrix .
If a function $\small f(x)$ has a negative eigenvalue in its associated ...
- 5,342
3
votes
4
answers
514
views
Better terminology than "equivalence class of functions"
Let $X = C(\mathbb R)$ be the Fréchet space of real-valued continuous functions. For each $f \in X$ and each compact set $D \subseteq \mathbb R$, let $$[f]_D = \{ g \in X : \mbox{$g(t) = f(t)$ for ...
- 8,512
3
votes
0
answers
361
views
Is this an injective function ?
Hi all,
I got stuck with a problem that pop up in a paper about location of zeros for some analytic functions that I am working on.
The problem is the following:
Fix two arbitrary positive ...
- 2,044
3
votes
1
answer
624
views
How to calculate a Fredholm index numerically
How can one calculate the index of a Fredholm operator numerically ?
In numerically calculations one uses always finte dimensional spaces.
But linear operators on finite dimensional spaces have ...
- 2,753
6
votes
2
answers
979
views
Literature on behaviour of eigenfunctions under multiplication?
Dear community,
I would be happy about any literature or comments on the behaviour of the pointwise product of eigenfunctions of a self-adjoint operator with discrete spectrum, acting on a separable ...
- 199
6
votes
1
answer
482
views
Analytic functions with algebraic Taylor coefficients at some point.
This question just came to my mind when reading the question
When may Function (meromorphic) be expanded as power series with coefficients of integers
Suppose $f$ is an analytic function on some ...
- 2,810
1
vote
1
answer
247
views
Distance between lattices of invariant subspaces of matrices
For a linear transformation $A: C^n \to C^n$ let $Inv(A)$ be the lattice of all $A$-invariant subspaces. In work I.~Gohberg, L.~Rodman "On the Distance between Lattices of Invariant Subspaces of ...
- 95
2
votes
1
answer
1k
views
When may function (meromorphic) be expanded as power series with coefficients of integers?
Let $F$ be meromorphic function. With what properties may it be expanded as power series with coefficients of integers in such a form
$$
F=\sum_0^{\infty}a_i x^i,a_i\in \mathbb{N} \cup \{0\},\exists M ...
- 3,094
2
votes
1
answer
323
views
Recovering Schauder decompositions
The problem of Schauder decomposition of a given Banach space seems to play an important role in the geometry of Banach spaces, especially when one is interested in finite dimensional Schauder ...
- 23
6
votes
3
answers
4k
views
Universal property of blowups
Can anyone help me with a proof of the following claim (see for example the book Higher algebraic geometry of Olivier Debarre, proof of Proposition 1.43, page 31):
Let X be a complex manifold, and ...
- 117
0
votes
1
answer
454
views
Is this set of functions compact?
Let $\mathcal{F}$ be the set of continuous functions $\varphi$ from $\mathbb{C}$ to $[0,1]$ that satisfy $\begin{align}\varphi(z)=\frac{1}{2\pi}\int_{0}^{2\pi}\varphi(z+e^{i\theta})d\theta\end{align}$ ...
- 1
4
votes
2
answers
484
views
When is a metric space isometrically embeddable into some Banach space?
EDIT
Oops---I found the answer to the first question of mine here on Wikipedia---this is really classic material. I'll leave the question open for a bit, in case someone tells me something ...
- 28.6k
6
votes
4
answers
8k
views
Characterization of the non-negative definite functions $f(x,y)$
The common definition of the non-negative definite functions is as follows:
Definition 1: A continuous complex-valued function $f(x)$ is called non-negative definite, if for any real numbers $x_1,\...
- 1,649
9
votes
1
answer
456
views
Embeddings of Sobolev-Orlicz spaces
The Birnbaum--Orlicz spaces generalize the Lebesgue spaces (see http://en.wikipedia.org/wiki/Birnbaum-Orlicz_space for a precise definition). The space $L_\Phi(\Omega)$ is defined for convex functions ...
- 52.3k
17
votes
4
answers
10k
views
Analytic implicit function theorem
I'm looking for a proof of the analytic implicit function theorem (IFT). The only related proof I could find was the holomorphic inverse function theorem (by Henri Cartan). On Wikipedia, the analytic ...
- 183
4
votes
2
answers
917
views
Self-similarity of a dendrite fractal
The Julia set of the map $z \mapsto z^2+i$ is a dendrite fractal. I would like to know which affine maps (other than identity) map this region to a subset of itself. I imagine there are two three ...
- 22.8k
4
votes
1
answer
568
views
Crossed product of a non unital C*-algebra
Let $X$ be a locally compact space, and let $T:X\rightarrow X$ be a homeomorphism. Then \begin{align*}
&\alpha:C_0(X)\rightarrow C_0(X)\\\
&\alpha(f)=f\circ T
\end{align*}
is an automorphism. ...
- 43
4
votes
1
answer
525
views
An analytic subset as a singular homology class of a compact manifold
We know every differential manifold can be triangulable. Let $M$ be a compact complex manifold of dimension $m$ and V be an analytic subset of dimension $s$ of $M.$ If $V$ has no singularity then $V$ ...
- 750
1
vote
1
answer
124
views
Lifting infinitesimal deformations for coverings
Let $f:X \rightarrow Y$ be an (unramified) holomorpic covering map between two (maybe non compact) complex manifolds.
Q: Does every infinitesimal deformation of Y lift faithfully to an infinitesimal ...
- 11
2
votes
1
answer
535
views
about decomposition of a non-negative definite operators
Hello,
Many years before, I had the following problem.
We first give a definition. Given a non-negative definite real-valued definite matrix $n^2\times n^2$ matrix $M$, it is called separable if it ...
- 1,649
1
vote
1
answer
233
views
How to go from a potential resolvent to the associated operator
I am reading Link. The author appears to use the following fact:
Let $H$ be a Hilbert space. For every $\zeta \in \mathbb{C}\setminus\mathbb{R}$ we have a bounded operator $R(\zeta): H \to H$. We also ...
1
vote
0
answers
192
views
Holomorphic vector fields with growth conditions on $X_\mathrm{reg}$
Let $M$ be a complex manifold with a hermitian metric (volumes and distances will be wrt this metric). Let $X\subset M$ be a complex analytic subspace of $M$ and $Y\subset X$ an analytic set ...
- 1,205
5
votes
1
answer
510
views
The space $H(D)$ of holomorphic functions.
A very natural example of a nuclear Montel space is the space $H(D)$ of all holomorphic functions on the open disc topologized by the family of seminorms
$$p_n(f)=\sup\{|f(z)|\colon |z|\leq 1-\tfrac{...
- 51
1
vote
0
answers
235
views
glue together a sequence of holomorphic forms
hallo,
my problem is the following: i have a finite sequence of holomorphic $k-$forms $\alpha_{k}$, each defined on open subsets $U_{k} \subset M$, where $M$ is a complex $n$-dimensional manifold, ...
- 19
0
votes
2
answers
415
views
Commutative *-subrings of the noncommutative C*-algebra $B(l^2)$
A $\star$-ring is a ring with an involutive anti-automorphism. The simplest example of a noncommutative $\star$-ring is perhaps $B(l^2)$, the ring of bounded linear functions on the sequence space $l^...
user2529