All Questions
12,777 questions
15
votes
2
answers
2k
views
Range of completely positive projection
Let $A$ be a C*-algebra. Suppose that $P:A \rightarrow A$ is a contractive completely positive projection. Does the range $P(A)$ is completely order isomorphic to a $C^*$-algebra?
In the case where ...
- 1,222
1
vote
2
answers
733
views
Quantum Error Correction
One can correct the errors in a quantum channel iff the coherent information of the input state is not reduced by the channel. This is analogous to sending quantum entanglement through a channel. If ...
- 29
11
votes
2
answers
2k
views
How "generalized eigenvalues" combine into producing the spectral measure?
Hi... I am wondering how 'eigenvalues' that don't lie in my Hilbert space combine into producing the spectral measure. I study probability and I am quite ignorant in the field of spectral analysis of ...
- 333
0
votes
1
answer
330
views
Convex sets and projections
Hello!
I recently started (it's purely self-education) reading a "Mathematical programming and optimizations" book, did a vast part of the exercises related to the theoretical part and at one moment ...
10
votes
3
answers
3k
views
Complex manifolds where bounded holomorphic functions are constant
Liouville's theorem states that all bounded holomorphic functions on $\mathbb{C}^n$ are constant.
I'm wondering which connected complex manifolds have this property ?
Connected compact complex ...
- 205
1
vote
2
answers
682
views
structure of singular matrices whose entries have modulus one
Let $A$ be a $n \times n$ matrix all of whose entries has modulus 1.
Suppose the matrix $A$ is singular.
We will assume without loss of generality that all the entries in the first row and the ...
- 1,795
0
votes
2
answers
2k
views
fundamental solution of radial wave equation
i am trying to find resources on the derivation of the fundamental solution to the radial wave equation. any suggestions of or links to books, papers, and/or notes would be much appreciated. i have ...
- 1
4
votes
2
answers
1k
views
Trace space and Neumann boundary condition
In which sense is it possible to solve $\Delta u=0$, $\partial_\nu u=\phi$, for $\int\phi=0$ on a closed domain, say a ball $B^3\subset\mathbb R^3$?
For example would a $\phi\in L^p(\partial B^3)$, $...
- 2,041
4
votes
1
answer
645
views
Factorization in the Wiener algebra on the unit disc.
Consider the Banach algebra $W^+=\ell^1(\mathbb{Z}^+)$, viewed upon as the analytic functions $f$ on the unit disc $\mathbb{D}$ such that $$\|f\|=\sum_{k\ge0}|a_k|<\infty$$ where
$$f(z)=\sum a_kz^k$...
7
votes
3
answers
1k
views
Non-Borel subspace of Banach space
Let $X$ be a separable Banach space, $M \subset X$ a linear subspace. Must $M$ be a Borel set in $X$?
I believe the answer is "no," since I have seen authors who are careful to talk about "Borel ...
- 29.8k
2
votes
1
answer
5k
views
Pade approximant to exponential function
Suppose:
a) $p(z)$ is an even degree polynomial (of degree $k = 2j$) with real coefficients;
b) $p(0) = 1$;
c) $p(z)$ and $p(-z)$ have no roots in common anywhere in the complex plane;
d) $f(z) = ...
- 21
16
votes
2
answers
2k
views
The Cauchy–Riemann equations and analyticity
I would be interested to learn if the following generalization of the classical Looman-Menchoff theorem is true.
Assume that the function $f=u+iv$, defined on a domain $D\subset\mathbb{C}$, is such ...
- 22.3k
-1
votes
1
answer
2k
views
Absolute values and Frobenius norm [closed]
The Frobenius, or Hilbert-Schmidt, norm of an $n$ by $n$ matrix $A$ is defined as $\|A\|_2 = \sqrt{\sum_{i,j=1}^n |A_{ij}|^2}$. The absolute value of $A$ is the unique positive matrix $|A|$ satisfying ...
- 1
4
votes
1
answer
221
views
existence of charaterization of amenable groups by complementation?
Recall that we say that a closed space $F$ of a Banach space $E$ is complemented if there exists a contractive projection $P$ from $E$ onto $F$.
Do you know a charaterization of discrete amenable ...
- 1,222
3
votes
2
answers
1k
views
Do the Euler method's approximations always approach the true solution?
Let $B$ be a Banach space and $f : [0,+\infty)\times B \to B$ be a continuous function which is Lipschitz continuous in the second argument with Lipschitz constant $L$ (which does not depend on the ...
user5810
3
votes
5
answers
11k
views
What is the angle between two complex vectors?
Let $x, y\in R^n$ and $x, y$ are nonzero, it is well known
$\frac{x^Ty}{\parallel x\parallel_2\parallel y\parallel_2}(\parallel x\parallel_2+\parallel y\parallel_2)\le \parallel x+y\parallel_2$. How ...
- 1,858
10
votes
4
answers
783
views
Does a quantitative version of Fredholm theory exist?
Let $X$ be a Banach space and $K:X\to X$ be a compact operator. If $I+K$ is injective then it is onto and hence the inverse $(I+K)^{-1}$ is bounded. What kind of qualitative or quantitative ...
- 9,007
2
votes
4
answers
358
views
When do functions near F have zeros near a zero of F?
Consider a sequence of functions $F_n : \mathbb{R}^d \to \mathbb{R}^d$, a function $F: \mathbb{R}^d \to \mathbb{R}^d$, and an $\mathbf{x} \in \mathbb{R}^d$ so that $F(\mathbf{x}) = \mathbf{0}$. In ...
- 1,068
4
votes
3
answers
1k
views
Set of invertible operators in B(H) is connected. Is it true? Is there a reference?
Suppose $H$ is a Hilbert space, $B(H)$ is the algebra of bounded linear operators on it, $K(H)$ is ideal of compact operators in $B(H)$, $Inv(B(H)/K(H))$ is the topological group of invertible ...
- 1,284
19
votes
2
answers
2k
views
Complete metric on the space of Jordan curves?
I was interested in putting a complete metric on the space of Jordan curves. Say, just planar Jordan curves contained in $B(\bar{0}, 2) \backslash B(\bar{0}, 1)$ which separates $\bar{0}$ and infinity....
- 375
2
votes
0
answers
267
views
Finer properties of a sequence of harmonic functions
This was a question that arose for me when I was thinking about how one proves strong unique continuation for elliptic equations. I never could come up with a satisfactory answer.
Background:
When ...
- 2,299
41
votes
4
answers
16k
views
Product of Borel sigma algebras
If $X$ and $Y$ are separable metric spaces, then the Borel $\sigma$-algebra $B(X \times Y)$ of the product is the $\sigma$-algebra generated by $B(X)\times B(Y)$. I am embarrassed to admit that I ...
- 31.5k
7
votes
1
answer
577
views
Are the compact and Haagerup approximation properties equivalent?
The following essentially implies the equivalence of Anantharaman-Delaroche's compact approximation property (page 337 of Link) and the Haagerup approximation property.
Let $M$ be a type ${II}_{1}$ ...
- 7,067
6
votes
1
answer
453
views
The typical size of a random element in a Banach space
Let $X$ be a separable Banach space, and let $\mathbb P$ be a Radon probability measure on $X$ with zero mean and covariance operator $K : X^* \to X$. Let $x$ be an $X$-valued random variable with ...
- 8,512
4
votes
2
answers
2k
views
Are there any uses for complex sine? [sin z]
The sine function can take a complex argument. e.g. sin(x + iy)
But does it get used that way in any field? Either practical (e.g. electrical engineering) or in other fields of math? Naturally, I am ...
- 143
27
votes
5
answers
4k
views
What is the naming reason of poles in complex analysis?
A function $f: \textbf{C} \to \textbf{C}$ has a pole of order $k$ if $f(z) = \frac{g(z)}{(z-z_0)^{k}}$ where $g(z)$ is a nonzero analytic function. Why do we call it poles?
- 279
1
vote
1
answer
210
views
Is the metric obtained by altering the metric of a Hilbert space on a finite-dimensional subspace equivalent to the original one? [closed]
Suppose a Hilbert space W can be written as the direct sum (not necessarily orthogonal) of the closed subspaces H and V, where H is assumed to be of finite dimension. Define a new inner product via
||...
- 2,935
8
votes
2
answers
8k
views
Version of the Poincaré Inequality
Let $\Omega\subset \mathbb{R}^n$ open and bounded. The Poincaré inequality
$$\|u\|_p \le C \|\nabla u\|_p$$
($\|\cdot\|_p$ denotes the usual $L^p(\Omega)$-norm; the Lebesgue measure shall be used here)...
- 2,270
2
votes
2
answers
710
views
Are there good inequalities on the norm?
It's well known that in a Hilbert space, good inequalities exist concerning the norm due to the existence of inner product.Now let X be a general Banach algebra, are there good inequalities concerning ...
- 1,528
0
votes
2
answers
377
views
"Frobenius-finite" linear operators on a Hilbert Space
Let $H = L_2(S)$ be the complex Hilbert space over $S$ with the counting measure. (There might be another term for this concept, but) I define a continuous linear operator $L$ on $H$ with matrix ...
user5810
3
votes
2
answers
3k
views
Eigenvalues convolution-type operator
Let $J_1$ be the Bessel function of the first kind and let $H_1(x) = \frac{J_1(|x|)}{|x|}$ for $n = 1$. Define the operator $Tf(x) = (f * H_1)(x)$ from $L^2$ to $L^2$.
Since the $H_1$-function is the ...
- 455
0
votes
1
answer
4k
views
1
vote
1
answer
506
views
Bessel sequence, uniformly minimal, separated
Is every unit norm Bessel sequence in a Hilbert space a finite union of separated ones? Is every unit norm separated sequence a finite union of uniformly minimal (minimal with uniformly bounded ...
- 11
2
votes
1
answer
1k
views
Green's function for wave equations in R² or R³
Hello,
For almost one year, I am searching for the Green's function for wave equation in R² or R³ with some boundary conditions. As far as I know, when the boundaries permit the method of images, we ...
- 1,649
22
votes
5
answers
3k
views
Is $L^p(\mathbb{R})$ minus the zero function contractible?
Is $L^p(\mathbb{R}) \setminus 0$ contractible? My intuition says that the answer is yes, but I'm afraid that this is based on thinking of this as somehow similar to a limit of $\mathbb{R}^n \setminus ...
- 433
0
votes
3
answers
1k
views
Sobolev norm and Beppo-Levi norm
I've asked this question on math.stackexchange.com but I'm not satisfied by the answers I got, so I've decided to ask here instead. As always I apologize if my notation is not precise enough. I am a ...
- 661
0
votes
3
answers
2k
views
Higher direct image of coherent sheaf
Hi.
Can any one me say if there is a simple proof of this claim which i can prove it by localization and no easy technique of nuclear spaces...
Let $f:X\rightarrow S$ be an open, surjective ...
- 435
3
votes
1
answer
2k
views
Queries about the Skolem-Mahler-Lech theorem (integer zeros of exponential polynomials)
The Skolem-Mahler-Lech Theorem says that the integer zeros of an exponential polynomial are the union of complete arithmetic progressions and a finite number of exceptional zeros. http://terrytao....
- 1,795
3
votes
1
answer
675
views
Relation between entire function of exponential type and exponential polynomials
Is it true in general that the theory of entire function of exponential type and and that of exponential polynomials (with purely imaginary exponents) are analogous ?
Can one derive results about ...
- 1,795
2
votes
1
answer
286
views
Linear independence in the algebraic closure of $\mathbb{C}(z)$
Fix $N>0$. Let $b_i=(b_{i,1}, b_{i,2}, b_{i,3}, b_{i,4})$, $i=1,\ldots, m$, be distinct 4-tuples of integers with with all $0\leq b_{i,j}< N$. (The zero tuple is disallowed.)
Define $w_i=(\...
- 454
2
votes
1
answer
303
views
Proper sobolev spaces invariant under no-linearities
Let $f:H^s\to H^s$ at least continuous and not necesarily linear. Is there some kind of criterion or condition over $f$ that lets to ensure that $f({H^{s+k}})\subseteq H^{s+k}$?
- 151
1
vote
1
answer
630
views
Stuck on a convergence argument in $H_0^1(\Omega)$.
I'm trying to verify that a functional I have satisfies the Palais Smale condition for appliction of the Mountain Pass lemma.
However I've encountered this step along the way which seems clear to me ...
- 2,641
3
votes
1
answer
2k
views
fourier transform of radon measure
hi,
assume that I have a function $q$ which is a Fourier Multiplier of order zero, i.e.
$$
\left|\left( \frac{d}{dx}\right)^nq(x)\right|\lesssim \left(\frac{1}{1+|x|}\right)^n\quad \mbox{for all ...
- 979
2
votes
3
answers
632
views
How to find the almost period of an exponential polynomial
Let $u(t) = \Sigma_{k=1}^n c_k e^{i \lambda_k t} (c_k \in \mathbb C, \lambda_k \in \mathbb R) $ be an exponential polynomial of order $n$ with purely imaginary exponents. We can assume that the ...
- 1,795
129
votes
2
answers
16k
views
What are the shapes of rational functions?
I would like to understand and compute the shapes of rational functions, that is, holomorphic maps of the Riemann sphere to itself, or equivalently, ratios of two polynomials, up to Moebius ...
- 25.1k
38
votes
2
answers
13k
views
What, exactly, has Louis de Branges proved about the Riemann Hypothesis?
I know this is a dangerous topic which could attract many cranks and nutters, but:
According to Wikipedia [and probably his own website, but I have a hard time seeing exactly what he's claiming] Louis ...
- 1,990
3
votes
0
answers
649
views
Stability by flat base change of certain properties
Hi.
Let $f:X\rightarrow S$ be a surjective proper, open morphism of reduced or without embedded component complex spaces (or, in alg.geom, surjective proper, universally open morphism of excellent, ...
- 435
7
votes
3
answers
1k
views
Index of an Operator
I've been working through the heat-equation proof of the Atiyah-Singer index theorem. My question is what is the motivation for the definition of the index of an operator? I know there is the ...
- 205
0
votes
0
answers
362
views
Gradient of the energy functional in $H^{1,2}$-norm
I have to use estimates for the gradient of the energy functional on the free loop space of a fixed compact manifold $Q$. As such, one considers $H^{1,2}$-maps of the circle into $Q$. The energy ...
- 2,935
4
votes
1
answer
6k
views
Inverse of a function defined by an integral
Hi, I have a function defined by an integral as follows.
$$
z=f(w) = \int_0^w \frac{(\zeta-a_1)^{\alpha_1}(\zeta-a_2)^{\alpha_2}...}{(\zeta-b_1)^{\beta_1}(\zeta-b_2)^{\beta_2}...}\ d\zeta
$$
where $w$ ...
- 167