All Questions
12,777 questions
21
votes
0
answers
732
views
Closed connected additive subgroups of the Hilbert space
It is a classical result that a closed and connected additive subgroup of $\mathbb{R}^n$ is necessarily a linear subspace. However, this is no longer true in infinite dimension: a very easy example is ...
- 60.5k
1
vote
1
answer
1k
views
Laplace equation over concentric spheres
Is there a closed formula for the solution of Dirichlet problem ($\Delta u=0$) for annulus $r <|x| < R$, $x \in R^n$ (n>2), with two given boundary value functions, $f$ over $|x|=r$ and $g$ over ...
- 39
0
votes
1
answer
1k
views
Precompact set in L2 space?
Let A be a bounded interval in R. Suppose we have a collection of functions, such that
Each function is $\in$ $L^r(A)$, where r is any number $\in$ $[1, \infty]$,
The fractional derivative of order ...
- 11
2
votes
1
answer
466
views
What is the regularity of the argument of a complex function?
Let $\psi=f+ig=\rho e^{i\theta}$ be a complex function on some open subset of $\mathbb{R}^n$, where $f,g,\rho$ and $\theta$ are real-valued. I happened to find that the identity of differentiation for ...
- 305
1
vote
0
answers
195
views
Lower semicontinuity of Bregman distances/divergences
For a Banach space $X$ and a convex functional $J:X \to [0,\infty]$ (i.e. with values in the extended reals), consider the associated Bregman distance: For $x,y\in X$ and $\xi\in\partial J(y)$:
\begin{...
- 12.7k
4
votes
4
answers
1k
views
An example of a non-paracompact tvs (over the reals, say)
What is an example of a non-paracompact topological vector space?
I'm aware of this question, but I don't care if my tvs is locally convex. In fact the wilder the better. The only criterion is that ...
- 35.5k
5
votes
1
answer
1k
views
Lipschitz properties of minima/minimizers of convex functions of two variables
Suppose I have a function $f(x,y)$ from $\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ that is convex in both $x$ and $y$. Set
$g(y) = \min_{x} f(x,y)$
What I would like is for $g(y)$ to be ...
- 165
3
votes
1
answer
436
views
When does a mother wavelet generate a frame?
This question is about conditions on a mother wavelet that generates a countable familily of child wavelets via scaling and translation, that are both necessary and sufficient for the child wavelets ...
- 1,544
33
votes
4
answers
11k
views
Counterexample for the Open Mapping Theorem
I would like to ask a counterexample for the classical theorem in functional analysis: the open mapping theorem in the case that $Y$ is Banach, but $X$ is not Banach to show that the completeness of X ...
- 331
25
votes
5
answers
6k
views
When I can safely assume that a function is a Laplace transform of other function?
If I have a function and I want to represent it as being the Laplace transform of another, that is, I want to be sure that there is $\hat{f}(s)$ such that my function $f(x)$ can be written as:
$$f(x) =...
- 923
12
votes
0
answers
2k
views
Dolbeault cohomology of complex tori.
Let $T=\mathbb C^n/\Lambda$ a complex torus. It is completely elementary to prove that the de Rham cohomology of $T$ in degree $q$ is isomorphic to the $q$-th exterior power of the dual of $\mathbb C^...
- 7,902
-2
votes
1
answer
665
views
weak convergence
I know the following result is true in the case of strong convergence. But I don't know whether it is true in the case of weak convergence also.
Let $p>1$. Suppose that each $x_n$ is a non negative ...
- 779
2
votes
1
answer
1k
views
Range of the Radon Transform
Let us consider the Radon transform in two dimensions:
$$\tag{1}Rf(r,\theta):=\int\limits_{-\infty}^{\infty} f(r\cos\theta-t\sin\theta,r\sin\theta+t\cos\theta) dt,$$
where $r\in\mathbb{R}$ and $0\...
- 931
6
votes
2
answers
4k
views
Question about Schauder bases in C([0,1]).
I would like to check a statement about Schauder bases in $C([0,1])$ to be sure that I don't lie to my students on Monday. The statement(s) that I would like to check are:
The family of monomials $\{...
- 26.8k
29
votes
1
answer
4k
views
Furstenberg's Conjecture on 2-3-invariant continuous probability measures on the circle
Hillel Furstenberg conjectured that the only $2$-$3$-invariant probability measure on the circle without atoms is the Lebesgue measure. More precisely:
Question: (Furstenberg) Let $\mu$ be a ...
- 25.5k
12
votes
2
answers
754
views
Can the unit complex 1-dimensional disc be embedded isometrically into complex euclidean space?
Let $D$ denote the unit complex 1-dimensional disc, together with the hyperbolic metric $h_D=\frac{4dz\wedge d\bar{z}}{(1-|z|^2)^2}$of curvature $-1$. By Nash's embedding theorem, we can always embed ...
user2529
5
votes
3
answers
1k
views
Product of sine
For which $n\in \mathbb{N}$, can we find (reps. find explicitly) $n+1$ integers $0 < k_1 < k_2 <\cdots < k_n < q<2^{2n}$
such that
$$\prod_{i=1}^{n} \sin\left(\frac{k_i \pi}{q} \...
- 2,829
8
votes
3
answers
1k
views
Fourier dimension of the sum of sets
This question came up when my supervisors, my colleague, and I were considering arithmetic progressions in sets of fractional dimension. In particular, we were interested in "extracting" Salem sets ...
- 505
26
votes
3
answers
7k
views
Dual of bounded uniformly continuous functions
Let $(X,d)$ be a metric space, and let $C_u(X)$ be the Banach space of bounded uniformly continuous functions on $X$ (with the uniform norm). How can I characterize its dual space $C_u(X)^*$?
I ...
- 29.8k
3
votes
3
answers
2k
views
Analytic continuation via square of absolute value
Is the following fact true (I think that I can prove it but I don't trust
myself on these matters): let $f(z)$ be an analytic function defined on
some open subset $U$ of ${\mathbb C}$. Assume that the ...
- 7,037
32
votes
2
answers
4k
views
Are there non-reflexive vector spaces isomorphic to their bi-dual?
Let $V$ be an infinite dimensional topological vector space and consider the natural application $\iota\colon V\to V^{**}$. The space $V$ is said to be reflexive if $\iota$ is an isomorphism.
Are ...
- 7,902
0
votes
1
answer
263
views
Separability of inner product to a product of Minkowski function and norm
I’ve encountered the following assumption:
Let D be a set such that there exists a Minkowski function $f(u)$ on $\mathbb{R}^l$ and norm $g(v)$ on $\mathbb{R}^m$ such that
$\forall u\in \mathbb{R}^l, \...
- 1
19
votes
2
answers
11k
views
Meaning of $\Subset$ notation
The symbol $\Subset$ (occurring in places where $\subseteq$ could occur syntactically) comes up frequently in a paper I'm reading. The paper lives at the intersection of a few areas of math, and I ...
- 1,601
1
vote
0
answers
113
views
Unbounded Convex domain
Take an unbounded convex domain in C^n, with n>1. Suppose that it is Kobayashi hyperbolic. Is it true that it is biholomorphic to a BOUNDED convex domain? For n=1 it is true due to the Riemann mapping ...
- 11
0
votes
1
answer
296
views
Continuity of cylindrical functions.
Let $C_c^\infty(\mathbb R^n)$ be the functions from $\mathbb R^n$ to $\mathbb R$ with compact support, further let $X$ be a separable Hilbert space with a fixed orthonormal basis $(e_n)_n$. Define the ...
- 455
7
votes
1
answer
537
views
Algebraic topology for nonlinear compact operators
There are analogues of certain basic notions in algebraic topology in the theory of Banach spaces. For example, the Brouwer fixed point theorem generalizes to the Schauder fixed point theorem, and ...
- 6,870
1
vote
4
answers
614
views
Variants of point fixed theorem
Let $E$ be a dual Banach space and $C$ a nonempty convex weak* compact subset of $E$. Let $G$ be a group of weak* continuous linear isometries on $E$. Suppose that $g(C)\subset C$ for all $g\in G$.
...
- 1,222
9
votes
3
answers
1k
views
An analytic proof of the De Franchis theorem
The De Franchis theorem in its simplest form states that given two compact Riemann surfaces $\Sigma_{g_1},\Sigma_{g_2}$ where $g_1,g_2 > 1$, there are only finitely many non-constant holomorphic ...
- 1,159
16
votes
5
answers
6k
views
Complex Analysis applications toward Number Theory
I'm an undergrad who is taking a Complex Analysis Course mainly for its applications in number theory.
So I would like to ask some guidelines about which theorems/concepts should I focus on in order ...
- 321
6
votes
2
answers
1k
views
Embedding Theorem for big line bundles
Kodaira embedding theorem says that a positive line bundle is ample, i.e. high tensor powers are holomorphically embeddable into complex projective space of high dimension.
However, ampleness is not ...
user2529
4
votes
2
answers
2k
views
Inclusions of $C^{k,\alpha}$ spaces
When is $C^{k,\alpha}(\bar{\Omega})$ a
subset of
$C^{k',\alpha'}(\bar{\Omega})$?
Gilbarg and Trudinger says that "for the domains of interest in this work the inclusion will hold whenever $k + \...
- 1,771
6
votes
2
answers
909
views
Do maps have flows?
In A New Kind of Science: Open Problems and Projects(pg. 36).
How can one extend recursive function definitions to continuous numbers? What is the continuous analog of the Ackermann function? The ...
user37691
0
votes
1
answer
319
views
Hilbert space automorphisms realized as induced by transformations of some base-spaces
Following question may be soft. Fix abstract hilbert space H and consider any automorphism A in banach-spaces sence (i.e. no conditions on metric). Call A is realizable if exist measure space $(X,\mu)$...
- 789
8
votes
1
answer
431
views
Injectivity for bimodules and Hochschild cohomology
Let $A$ be a Banach algebra and let $X$ be an $A$-bimodule. Is there a notion of (relative) injectivity for $X$ which would imply that $\mathcal{H}^n(A,X)$ vanishes for all $n\ge 1$? Here $\mathcal{H}^...
user2412
-1
votes
1
answer
1k
views
relation between inclusion and embedding [closed]
Assume that $X$ and $Y$ are two Banach spaces, now we have that $X$ is included in $Y$, in the sense that $\forall a\in X$, we have $a\in Y$. Then can we get that $X$ is embedded in $Y$, namely, $\...
- 331
6
votes
0
answers
299
views
Spectrum of an operator arising in a dynamical problem
(Question edited according to Denis Serre comment).
While studying the action of dilating map of the circle on probability measures, I ran across the following operator:
$$\mathcal{K}^* : L^2_0(\mu)\...
- 14.4k
9
votes
2
answers
1k
views
Generalization of the positive semidefinite Grothendieck inequality
In a recent paper, S. Khot and A. Naor show a natural generalization of the positive semidefinite Grothendieck's inequality. Grothendieck showed that there exists a constant $K > 0$ such that for ...
- 28.6k
2
votes
2
answers
1k
views
Are coordinate functions on topological vector spaces always continuous?
Let $V$ be a Hausdorff locally convex topological vector space over the field $\mathbb{K}$.
Let $B$ be a subset of $V$ such that
$\;$ for all functions $c : B\to \mathbb{K}$, if $\displaystyle\sum_{...
user5810
5
votes
2
answers
3k
views
Algorithm for Weierstrass Preparation Theorem for Formal Power Series
The Weierstrass preparation theorem for formal power series rings guarantees that if a given formal series $f(z) = \sum a_k z^k \in R[[z]]$ where $R$ is a complete local ring with maximal ideal $M$ ...
- 51
13
votes
1
answer
682
views
How can one "see" the Hopf fibration in the space of lattices in the plane?
This question is inspired from Etienne Ghys's talk on Knots and Dynamics from ICM 2006.
The map $L \mapsto (G_4(L), G_6(L))$ gives a bijection between all lattices $L\subset \mathbb{C}$ (including ...
- 1,821
7
votes
3
answers
2k
views
How can one express the Dedekind eta function as a sum over the lattice?
The Dedekind eta function $\eta(\tau)$ can be regarded as a formula which assigns a number to a lattice $L \subset \mathbb{C}$. The algorithm is: rotate the lattice so that one of its basis vectors ...
- 1,821
6
votes
3
answers
3k
views
Can the adjoint of the exterior derivative in semi-Riemannian geometry be defined without the Hodge * operator?
The adjoint of the exterior derivarive is defined by
$\delta:=(-1)^k\ast^{-1}d\ast$,
but I need a way which avoids the Hodge $\ast$ operator.
Is there another definition?
For example, for ...
- 4,312
0
votes
1
answer
1k
views
Linear Mapping and integration
I have been reading the paper - "Introduction to Quantum Fisher Information".
In section 1.2 the author talks about the linear map $\mathbb{J}_D$, which he defines as follows:
Let $D \in M_n$ be a ...
- 145
1
vote
2
answers
331
views
factorisation of analytic functions
If I have an analytic function in plane $F(x,y)$ that is zero on a curve $y=f(x)$, is it true that
$F=(y-f(x))^n h$, where $h$ is nonzero on the curve? More general, can be somethink said about ...
- 11
3
votes
1
answer
499
views
methods for interpolating a function, holomorphic in the upper halfplane
Let $n,k\colon\mathbb{R}\to\mathbb{R}$ be real functions such that function $N$ given by $N(x)=n(x)-ik(x)$ is a holomorphic function in the upper half-plane. Also I know some additional properties of ...
- 1,284
13
votes
6
answers
3k
views
Sets with equal positive measure in every interval
Hi,
I want to write a proof that relies on the fact that:
There are Borel Sets $A$ and $B$ contained in $\mathbb{R}$ such that
$A \cap B = \emptyset$ and $\lambda(A \cap (x,y)) = \lambda(B \cap (x,...
- 133
7
votes
1
answer
2k
views
Solving the Beltrami Equation for a very simple Beltrami Coefficient
Let $\mu$ be a function on the complex plane with the property $\mu(z) = \overline{\mu(\bar{z})}$, such that $\mu(z) = \epsilon e^{-2\pi i \bar{z}}$ on the upper-half plane, where $\epsilon$ is a ...
- 245
7
votes
2
answers
2k
views
Brownian Motion Winding Number
Take a simple random walk $\gamma$ in the complex plane conditioned to start at point $a$ and end at point $b$. For this random walk, we can define the winding number $W_\gamma(a,b)$ around $b$ in the ...
- 4,952
6
votes
4
answers
7k
views
Why do we want to have orthogonal bases in decompositions?
In the decompositions I encountered so far, we all had orthogonal set of bases. For example in Singular Value Decomposition, we had orthogonal singular right and left vectors, in [discrete] cosine ...
- 497
0
votes
1
answer
758
views
Invariance of the cylindrical Laplace equation under conformal transform
hello,
it is often said that a conformal mapping preserves the Laplace equetion in 2D.
However, if this is true for the cartesian coordinates (x,y), where the laplacian is:
$$
\frac{\partial^2 \phi}{\...
- 167