All Questions
12,780 questions
0
votes
1
answer
234
views
A property of "Schwartz" quadratic forms
Consider $K(x, y)$, $f(x)$ Schwartz functions and $g(y)$ a tempered distribution. Suppose $$K(x, y) = K(y, x)$$ Define
$$h(t) = \int f(x - t) K(x, y) g(y - t) dx dy$$
It appears to me $h(t)$ is a ...
- 1,368
2
votes
1
answer
96
views
Are the real components of s-roots subharmonic?
Suppose $f(z)$ is an analytic function on a domain $D$ which maps negative axis to negative axis. For $s>1$ consider the function $$u(z)=\Re \sqrt[s]{f(z)}$$ with the branch cut along the negative ...
- 1,306
2
votes
0
answers
176
views
A limit involving a regularizing kernel
I am studying the following article by Benoit Perthame: http://www.mendeley.com/research/uniqueness-error-estimates-first-order-quasilinear-conservation-laws-via-kinetic-entropy-defect-measure/#
...
- 2,222
0
votes
0
answers
68
views
can we say fixed point existance of a set valued map over a compact set is homotopy invariant?
Consider two set valued maps over different compact sets as $F(\mathbf{x}):D\rightarrow\rightarrow D$, $G(\mathbf{x}):E\rightarrow\rightarrow E$ where $D,R\subset Y$. Assume there is a homotopy pair $(...
- 383
1
vote
1
answer
254
views
Extending linear operators to multi-linear ones
Suppose we are given a linear operator $L$ on a Banach space $X$. Is there any way to extend $L$ to a multi-linear operator $\mathcal{L}$ in such a way that
$$\mathcal{L}(x_1, x_2^*, \ldots, x_n^*) = ...
user7807
0
votes
1
answer
255
views
Average compared with discrete average for some $\lbrace -1,1 \rbrace$ polynomials
Let $k>0$ be a positive integer. Set $n=4k.$ Let $R(t)$ be a polynomial of degree $n-1$
with coefficients in $\lbrace -1,1 \rbrace$.
Consider the discrete average
$$
D(n,R) = \frac{\sum_{j=0}^{n-...
- 1,641
0
votes
1
answer
221
views
Sort-of extension of Young inequality to arbitrary measures
Hello folks,
Suppose we're working in ${\mathbb T}^q$, the $q$-times Cartesian product of $[-\pi,\pi]$ with the endpoints identified. So, $ L^p := L^p({\mathbb T}^q) $.
The Young inequality may be ...
- 1
1
vote
0
answers
215
views
Classification of Self similar sets
I am looking at self similar sets in $\mathbb{C}$ defined as the fixed set or a sequence of contractions or an iterated function system. I am currently trying to classify these sets by how they are ...
- 11
0
votes
0
answers
184
views
Extension of closed linear functionals...
If f is a closed linear functional defined on a dense subspace of a Banach space X, and, consider f1 which is an extension of f to X, is there a way to show that f1 is also closed without invoking the ...
- 55
1
vote
0
answers
153
views
The existence of the solution of the perturbed KdV Equation(semi-group operator)
Consider the perturbed KdV Equation$$u_t-6uu_x+u_{xxx}=\epsilon u,u(x,0)=f(x)$$where $f(x)=v(x,0)$，$v(x,t)$ is a soliton solution.$u$ satisfies the condition$u\to 0 $when $|x|\to\infty$
I want to use ...
- 61
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
2
votes
0
answers
238
views
Non-realizable CR structures?
Hill, Penrose, and Sparling have an example of a non-realizable CR structure, a 5-manifold $M^5$ that comes equipped with a "twisted version" of the Lewy operator for the quadric $Q^2$,
$v = \frac{1}{...
- 21
1
vote
0
answers
29
views
Counting variables to look for invariances/range conditions
A while back, I asked this question on m.se. I wasn't terribly happy with the answer, and when someone asked a very similar question which isn't getting any action, it got me thinking again. Let me ...
- 203
10
votes
0
answers
609
views
Asymptotic non-distortion of the separable Hilbert space
By the work of E. Odell and Th. Schlumprecht, we know that the
separable Hilbert space $\ell_2$ is arbitrarily distortable. But
I don't know if an "asymptotic" version of their result is true.
To ...
- 576
8
votes
0
answers
196
views
Parametrizing derivations from the algebra of smooth functions on a manifold to its dual
$\newcommand{\Der}{\operatorname{Der}}$
$\newcommand{\Real}{{\mathbb R}}$
(Disclaimer: I fear this question may be a bit too basic for MO, but in my defence I have essentially zero differential ...
- 25.8k
0
votes
1
answer
261
views
Flat locus of $S_{1}$-morphism
Hi, everybody.
Consider an ${\rm S}_{1}$- morphism $f:X\rightarrow S$ of reduced complex spaces. Assume that $f$ is open (universally open in Alg.geom), equidimensional with $n$-pure dimensional ...
- 435
0
votes
0
answers
143
views
description of a convex set of functions
Hi everyone,
I have a question about the characterization of a set of functions.
Let $\Phi$ a set containing all the functions $\phi(x): \mathbb{R}_+\rightarrow \mathbb{R}_{+}$ that satisfy the ...
- 69
0
votes
0
answers
173
views
Implications of complex solutions of Matiyasevich / Chaitin diophantine polynomials.
This is a shot in the dark: In twf:202, an isomorphism $T\cong T^{7}$ between binary trees $T$ and seven tuples of binary trees T^{7} is mentioned. The argument for this isomorphism starts with the ...
- 975
7
votes
1
answer
286
views
a.e. convergence of the powers of an operator built from rotations
Consider two numbers $a,b\in R/Z$ and some integer $p\geq 1$. Let $T:L^p(R/Z)\rightarrow L^p(R/Z)$ be the operator given by
$$T(f)(x)=1/2(f(x+a)+f(x+b))$$
For which values of $a,b$ do we have almost ...
- 18.7k
3
votes
0
answers
237
views
Monotonicity of a certain parametric integral
I would like to ask for some help (hints, ideas) in solving the following problem:
Given integer $n>0$ and real $\alpha>0,\beta>1$ we want to show, that
if we define for any $x\in\mathbb{R}...
- 696
4
votes
0
answers
487
views
Convolutions and Toeplitz Operators
Let be $d>0$ an integer number and consider the Cartesian product $\mathbb Z^d$ as metric space, with the distance between $x,y\in\mathbb Z^d$ given by $\|x-y\|_1=\sum_{j=0}^d|x_j-y_j|$.
Let be $...
- 2,044
8
votes
0
answers
605
views
convergence rate in Wiener's approximation theorem
Wiener has the following fantastic results about approximations using translation families:
Given a function $h: \mathbb{R} \to \mathbb{R}$, the set $\{\sum a_i h(\cdot - x_i): a_i, x_i \in \mathbb{...
- 1,839
2
votes
0
answers
296
views
Is the complex harmonic extension of a $\mathcal{C^r}$ map from $S^1$ to $\mathbb{C}$ is smooth upto the boundary ?
Suppose we have a map $ h : S^1\to \mathbb{C} $ that we know is a $\mathcal{C^r} $ map ( in the sense of a map between 1-manifold ( or in the sense of a $2\pi$ periodic map from $\mathbb{R}\to \mathbb{...
- 1,471
0
votes
0
answers
301
views
Lifting of product of a Banach algebra
Let $A$ be a non unital Banach algebra. The product induces a bounded linear map $T:A \otimes_{\gamma} A\to A$ where $\otimes_\gamma$ denotes the Banach projective tensor product.
A lifting of $T$ is ...
- 1,222
3
votes
0
answers
84
views
Application and relevance of Sobolev gradients
The Sobolev gradient concept has been developed in the 1970s, with a first publication in 1985, and an introduction can be found at: Ranka
I would like to learn how strong the impact of Sobolev ...
- 5,327
4
votes
0
answers
166
views
Relationship between sequential compactness of a convex set and its extremal points
Suppose that $X$ is a compact convex subset of a topological vector space. Suppose also that the extremal points of $X$ have the additional property that any sequence $x_n$ of extremal points has a ...
- 93
3
votes
0
answers
479
views
torsion freeness of tensor product
Hi.
Let $f:A\rightarrow B$ be a morphism of local noetherian rings, $M$ (resp. $N$) a $B$ (resp. $A$-)-module of finite type. We assume that $prof_{A}(M)\geq 2$ and $N$ is torsion free.
Then it is ...
- 435
0
votes
1
answer
297
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
2
votes
0
answers
304
views
Question in complex analysis arising from large $N$ gauge theory
This is a question in complex analysis that comes up in the treatment of large
$N$ gauge theory with gauge group $SU(N)$ in the 't Hooft limit where $N$ is taken to infinity with $\lambda=g^2 N$ fixed ...
- 5,546
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
11
votes
0
answers
310
views
Combinatorial Hilbert spaces
Any closed subspace $V\subset {\ell}^2(\omega)$ has associated to it a subset ${\cal S}_V$ of ${\cal P}(\omega)$, call it a combinatorial Hilbert space, namely the set of all supports of all vectors ...
- 17.6k
1
vote
0
answers
76
views
h-oscillating function
I need help understanding the following condition:
$u_h\in L^2(\mathbb{T}^d)$, $\|u_h\|_{L^2(\mathbb{T}^d)}=1$, where $h$ is the semiclassical parameter and $\mathbb{T}^d$ is the flat torus, is ...
- 11
0
votes
1
answer
412
views
An integral arising in statistics
The integral I need:
$$t(x)=\int_{-K}^{K}\frac{\exp(ixy)}{1+y^{2q}}dy$$
$K<\infty$, q natural number
For q=1 this integral is
$$\pi/2-\int_{Arc}\frac{\exp(ixy)}{1+y^{2}}dy $$
Where Arc ...
- 267
2
votes
1
answer
265
views
Multiple ergodic averages with varying number of terms
Hi. I've been stuck on the following question for some time.
Consider a sequence of functions $\left( f_n \right)$ from an ergodic space $\left( \mathsf{X}, \mathsf{S}, \mu \right)$ to $\left[ 0,1 \...
0
votes
0
answers
169
views
Is degree a "strict -transform" birational invariant for surfaces in the complex projective 3-space?
(Edit) My question is as follows: My previous question was about a kind of "strict-transform birational" invariant not birational invariants as usual. So I just delete the question since it ...
4
votes
1
answer
246
views
Are all continuous linear operators on the space of entire functions "simple"?
Let $\langle \operatorname{Ent},+,\cdot \rangle$ be the (complex) vector space of entire functions.
For all members $n$ of $\{1,2,3,...\}$, define $||\cdot ||_n : \operatorname{Ent} \to \mathbb{R}$ ...
user5810
1
vote
0
answers
335
views
Universally open morphism with reduced fibers.
Hi.
I asked in the last post if, for a flat morphism $f:X\rightarrow S$ of complex spaces with reduced fibers and $S$ reduced, $X$ is reduced or not. In the algebraic setting, Liu said that the ...
- 435
1
vote
0
answers
125
views
base change for distributions
For distributions on smooth manifolds one can consider the push-forward which is defined for proper maps, and the pull-back which is defined under certain condition on the wave front set see ...
- 2,649
1
vote
0
answers
247
views
Entire functions bounded on large wedges of $\mathbb{C}$ [closed]
Define $W : [0,\infty) \to 2^{\mathbb{C}}$ by $W(\theta) = \{r\cdot \operatorname{exp}(i\cdot t) : \langle r,t \rangle \in (0,\infty) \times (-\theta,\theta)\}$.
For what $\theta$ does there exist an ...
user5810
1
vote
0
answers
130
views
Sum of univalent functions
Let $f,g$ be a pair of univalent functions on a proper subdomain $\Omega$ of $\mathbb C$. Does their sum $f+g$ necessarily omit any complex value?
Similarly, can all holomorphic function be written as ...
- 329
6
votes
0
answers
354
views
Ordering of completely bounded maps
Let A be a C*-algebra, let H be a Hilbert space, and let $T:A\rightarrow B(H)$ be a completely bounded (cb) map (that is, the dilations to maps $M_n(A)\rightarrow M_n(B(H))$ are uniformly bounded). ...
- 18.7k
1
vote
1
answer
201
views
real-valued functions on the modular surface
How does one write down $\mathbb{R}$-valued functions on the modular surface? I am considering taking an arbitrary function on the upper half plane $f:\mathbb{H} \to \mathbb{R}$ and averaging over ...
- 22.8k
2
votes
0
answers
146
views
Subspace where an operator is positive
Given a self-adjoint operator $\hat{T}$ on a Hilbert space $\mathcal{H}$, and assuming it has a basis of eigenvectors $\{\phi_n\}$ such that $\hat{T}\phi_n=\lambda_n\phi_n$, one can consider the ...
- 2,358
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
1
vote
0
answers
87
views
Are morphisms of intersection graphs of circle packings harmonic?
Let $P$ and $Q$ be circle packings on compact Riemann surfaces (along with some Riemannian metrics) $X$ and $Y$. Let $f\colon X\to Y$ be a conformal map taking each circle in $P$ to a circle in $Q$. ...
- 3,079
3
votes
0
answers
251
views
What is the origin of the metrization problem for compact convex sets?
The following is an ``old question in analysis:''
Is it true that every perfectly normal compact convex subset of a locally convex topological vector space is metrizable?
Here perfectly normal means ...
- 3,547
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
1
vote
0
answers
125
views
Isomorphisms of group extensions arising from antisymmetric forms
Let $V,W$ be topological vector spaces and fix continuous antisymmetric bilinear forms $\omega_1:V\times V\to \mathbb{R}$, $\omega_2:W\times W\to\mathbb{R}$. Since $\omega_1$ is a 2-cocycle (in fact ...
- 1,411
2
votes
0
answers
114
views
non-closed weak graph limit of symmetric operators
Hi Everyone,
I was recently reading Reed & Simon's functional analysis textbook (the first volume), and it mentions casually on page 294 that weak graph limits of a sequence of symmetric ...
- 21
1
vote
0
answers
149
views
Banach spaces with simple best approximate solutions
Let $\langle V,||.||\rangle$ be a Banach space such that:
$\;\;$ for all continuous linear maps $\: L : V\to V \:$ and members $v$ of $V$, there exists a unqiue member $u$ of $V$
$\;\;$ that ...
user5810