All Questions
12,823 questions
6
votes
0
answers
639
views
Hilbert subspaces of indefinite inner product spaces
Let $E$ be a real linear space, endowed with a non-degenerate symmetric
bilinear form $(.,.)$.
Suppose that the [indefinite] inner product space $(E,(.,.))$
satisfies the following [sequential] ...
- 4,060
8
votes
4
answers
2k
views
An integral that somehow equals pi^2/6 and involves dilogarithms?
I am attempting to show that
$$ \sum_{k \ge 1}^\infty {k^2 x^k \over (1+x^k)^2} \sim (1-x)^{-3} {\pi^2 \over 6} $$
as $x$ approaches 1 from below. The sum can be approximated by the integral
$$ \...
- 14k
16
votes
0
answers
1k
views
Finite Rank Commutators
My former student Detelin Dosev and I are interested in classifying the commutators in $L(X)$, the bounded linear operators on the Banach space $X$ (see our joint paper on my home page or the ArXiv ...
- 31.5k
17
votes
5
answers
3k
views
Conditional probabilities are measurable functions - when are they continuous?
Let $\Omega$ be a Banach space; for the sake of this post, we will take $\Omega = {\mathbb R}^2$, but I am more interested in the infinite dimensional setting. Take $\mathcal F$ to be the Borel $\...
- 8,512
2
votes
3
answers
593
views
l^p space inequality related to compressed sensing
I'm trying to read Donoho's 2004 paper Compressed Sensing and am having trouble with a supposedly trivial statement (equation 1.2 on page 3).
He makes the sparsity assumption on $\theta \in \mathbb{R}...
- 198
8
votes
2
answers
1k
views
Example for an integral, rectifiable varifold with unbounded first variation
I'm just looking for an example of an integral, rectifiable varifold, which has no locally bounded first variation.
Recapitulation
for every $m$-rectifiable varifold $\mu$ exists a $m$-rectifiable ...
- 143
1
vote
4
answers
2k
views
Space of rapidly decreasing functions
Let H_n(x) be the Hermite Polynomials defined as in
http://en.wikipedia.org/wiki/Hermite_polynomials
The Hermite Polynomials form an orthonormal basis of the space of the rapidly decreasing
functions.
...
- 21
60
votes
8
answers
36k
views
Inverse gamma function?
This is an analysis question I remember thinking about in high school. Reading some of the other topics here reminded me of this, and I'd like to hear other people's solutions to this.
We have the ...
- 2,179
3
votes
6
answers
3k
views
Cone in a metric space
We know the definition of a cone in a Real Banach Space.
I want to know if there is any definition for a cone in an abstract metric space.
Have you ever seen such definition anywhere?
- 520
1
vote
1
answer
188
views
Integral determines function behaviour
Let us define:
$f(t) = t^{-1} \int_{\mathbf{R}^{3}} Exp[-\frac{x^2}{2t}] h(x) dx,$
for a real function h. What can I say about this function if I know that
$f(t) \rightarrow 1$.
I think that the ...
- 1,491
9
votes
4
answers
10k
views
Applications of Euler-Cauchy ODEs
The Euler-Cauchy ODE (2nd order, homogeneous version) is:
$$
x^2 y'' + a x y' + b y = 0
$$
Looking in various books on ODEs and a random walk on a web search (i.e. I didn't click on every link, but ...
- 26.8k
1
vote
1
answer
2k
views
Closed form of a nonlinear recurrence sequence.
The master theorem seems to fail on nonlinear recursive functions. Is there a standard tool for finding the closed forms of recursive functions of this form?
The question comes from trying to find ...
- 19
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
2
votes
3
answers
1k
views
Baire category theorem
Let's call the following conditions (1): $X$ is a complete metric space with metric $d$, $X = \cup_{n=1}^\infty A_n$. Let $\bar{A}$ denote the closure of $A$.
Let's call the following statement (2): ...
- 498
5
votes
3
answers
1k
views
Functional calculus for direct integrals
Suppose I have a direct integral of Hilbert spaces $H = \int^\oplus H_x dx $, and suppose I have an operator $T: H \to H$ which is decomposable, and so it can be written as
$T = \int^\oplus T_x$ for ...
- 3,897
52
votes
11
answers
25k
views
Does the exponential function have a (compositional) square root?
(asked by Nathaniel Hellerstein on the Q&A board at JMM)
Is there a "half-exponential" function $h(x)$ such that $h(h(x))=e^x$? Is it unique? Is it analytic?
Related question: Is there an ...
- 1,119
5
votes
0
answers
417
views
Direct integrals and fields of operators
Suppose we have a measure space $(X,\mu)$ and a measurable field of Hilbert spaces $H_x$ on it. We can form the direct integral ${\cal{H}} = \int H_x \ d \mu$, which is a Hilbert space.
Suppose now ...
- 3,897
19
votes
5
answers
2k
views
Are there any nonlinear solutions to $f(x+1) - f(x) = f'(x)$?
Are there any nonlinear solutions to $f(x+1) - f(x) = f'(x)$?
(Asked by bcross at math.iuiui.edu on the Q&A board at JMM.)
- 1,119
5
votes
2
answers
482
views
Asymptotics of iterated polynomials
Let the sequence $u_1, u_2, \ldots$ satisfy $u_{n+1} = u_n - u_n^2 + O(u_n^3)$. Then it can be shown that if $u_n \to 0$ as $n \to \infty$, then $u_n = n^{-1} + O(n^{-2} \log n)$. (See N. G. de ...
- 14k
9
votes
2
answers
1k
views
Borsuk pairs of Banach spaces
Given $X$, $Y$ two real Banach spaces, let's say that $(X,\ Y)$
is a Borsuk pair if for any continuous mapping $T$ : {$x$ $\in$
$X$ ; $||x||\leq1$} $\rightarrow$ $Y$ s.t. $T$ is odd on {$x$
$\in$ $X$ ;...
- 4,060
10
votes
1
answer
776
views
Saito-Wright definition of Rickart C*-algebras
A C*-algebra is Rickart if for each $x\in A$ there is a projection $p\in A$ so that
$R(x)=pA$.
Here the right-annihilator $R(S)$ of $S\subset A$ is defined
as $$R(S)=\{a\in A\mid xa=0\, \forall x\...
- 810
2
votes
3
answers
6k
views
Lipschitz functions in $\mathbb{R}^n$
Hello,
If $f:\mathbb{R} \to \mathbb{R}$ a differentiable function, it is very easy to find its Lipschitz constant. Is there any way to extend this to functions $f: \mathbb{R} \to \mathbb{R}^n$ (or ...
- 143
34
votes
8
answers
9k
views
When is a Banach space a Hilbert space?
Let $\mathcal{X}$ be a real or complex Banach space.
It is a well known fact that $\mathcal{X}$ is a Hilbert space (i.e. the norm comes from an inner product) if the parallelogram identity holds.
...
- 343
6
votes
3
answers
1k
views
How can I embed an N-points metric space to a hypercube with low distortion?
I have a N-point metric space defined by the pairwise distance matrix. I want to encode these N points with binary strings, i.e. each point will be mapped to a vertex in a hypercube. The lengths of ...
- 171
9
votes
4
answers
1k
views
Notation for eventually less than
Is there some existing notation for
\[f(n)\leq g(n)\] for sufficiently large n
Apart from just writing that itself?
I'm thinking of something compact like the ...
- 7,013
0
votes
1
answer
831
views
The eliminant of a system of differential equations
I am reading an old paper dealing with linear differential operators. At one point it refers to something it calls the "eliminant" of a set of linear differential operators. It seems that this was a ...
- 757
2
votes
1
answer
168
views
Local supporting points of Lipschitz functions
Let X be a separable reflexive Banach space and f:X\to\mathbb{R} be a
Lipschitz function. Say that a point x in X is a local supporting point
of f if there exist x^* in X^* and an open neighborhood U ...
- 31
3
votes
2
answers
416
views
Which Banach spaces have categorical duals?
I was looking carefully at all the definitions, trying to understand exactly what was going on in this question on categorical duals in Banach spaces. It seems that in the category of Banach spaces ...
- 26.8k
4
votes
2
answers
4k
views
Compact Convex sets and Extreme Points
There are examples that show the set of extreme points of a compact convex subset of a locally convex topological vector space need not be closed when the real dimension of the space is at least 3. ...
- 629
7
votes
1
answer
570
views
Categorical duals in Banach spaces
Near the bottom of the nlab page for Banach space I see "To be described: duals (p+q=pq)".
Are $(\mathbb{R}^n)_p$ and $(\mathbb{R}^n)_q$ dual objects in the closed symmetric monoidal category of ...
- 25.2k
2
votes
1
answer
1k
views
Convergence of a general Bertrand series
Is the sum $$ S= \sum_{n=2}^\infty \frac{1}{ \log^1n \log^2n \log^3n \cdots\log^{TL(n)}n} $$
convergent?
Here
$\log^i n$ denotes the $i$'th iterate of $\log$ (in base 2) of $n$, so $\log^2n$ ...
- 1,306
185
votes
19
answers
36k
views
How do I make the conceptual transition from multivariable calculus to differential forms?
One way to define the algebra of differential forms $\Omega(M)$ on a smooth manifold $M$ (as explained by John Baez's week287) is as the exterior algebra of the dual of the module of derivations on ...
26
votes
3
answers
2k
views
Universality of zeta- and L-functions
Voronin´s Universality Theorem (for the Riemann zeta-Function) according to Wikipedia: Let $U$ be a compact subset of the "critical half-strip" $\{s\in\mathbb{C}:\frac{1}{2}<Re(s)<1\}$ with ...
- 7,127
28
votes
7
answers
5k
views
Rolle's theorem in n dimensions
This looks like a statement from a calculus textbook, which perhaps it should be.
"Rolle's theorem". Let $F\colon [a,b]\to\mathbb R^n$ be a continuous function such that $F(a)=F(b)$ and $F'(t)$ ...
2
votes
1
answer
1k
views
How can we use the bounded convergence theorem in this proof of the Riesz Representation Theorem?
I'm studying the proof of the Riesz Representation Theorem as it appears in Ch. 6 of Royden's Real Analysis. When I looked on the web I noted there are a few different theorems that go by the name "...
- 466
9
votes
1
answer
996
views
Topological "Interpolation" ?
Let E be a normed space, and let $T$:E * $\rightarrow$ E * be
a nonlinear operator.
Suppose that :
1) $T$ is continuous from (E *, ||.||) to itself (i.e., it is norm-continuous).
and
2) $T$ is ...
- 4,060
4
votes
1
answer
479
views
Distributions as presheaves?
The yoneda lemma gives us a characterization of $Psh(\mathcal{C})$ that seems very similar to the theory of distributions. That is, we have a notion of representable presheaves, similar to ...
27
votes
29
answers
30k
views
Alternative undergraduate analysis texts
Other than the standard baby Rudin, Spivak, and Stein-Shakarchi, are there other alternative and comprehensive analysis texts at the undergraduate level? For example something that has general results ...
29
votes
15
answers
6k
views
Important results that use infinite-dimensional manifolds?
Are Banach manifolds (or other types of infinite-dimensional manifolds) just curiosities, or have they been utilized to prove some interesting/important results? Where do they turn up? Important ...
7
votes
3
answers
2k
views
What are some interesting sequences of functions for thinking about types of convergence?
I'm thinking about the basic types of convergence for sequences of functions: convergence in measure, almost uniform convergence, convergence in Lp and point wise almost everywhere convergence. I'm ...
13
votes
6
answers
3k
views
When does local invertibility imply invertibility?
Generally, local invertibility does not imply invertibility. However, for differentiable functions from $\mathbb{R}$ to $\mathbb{R}$ then surjectivity and local invertibility do imply invertibility.
...
- 26.8k
6
votes
1
answer
989
views
What is the "continuity" in "absolute continuity", in general?
The wikipedia article on absolute continuity gives a delta-epsilon definition for a measure $\mu$ defined on the Borel $\sigma$-algebra on the real line, with respect to the Lebesgue measure $\lambda$:...
- 208
4
votes
3
answers
6k
views
Advantages of a back-propagation neural network over other function approximation methods
Hello.
Let's say I have a set of input vectors $I = \{\mathbf{x_1}, \dots, \mathbf{x_k}\} \subset \mathcal{R}^m$ and a set of output vectors $O = \{\mathbf{y_1}, \dots, \mathbf{y_k}\} \subset \...
- 143
9
votes
4
answers
1k
views
Boundedness of nonlinear continuous functionals
Let $K$ be the closed unit ball of $C[0,1]$, and let $f$ in $C(K,\mathbb{\, R})$.
Is it true that there exists an infinite dimensional reflexive subspace
$E$ of $C[0,1]$ s.t. $f(K\cap E)$ is bounded ?
...
- 4,060
24
votes
5
answers
3k
views
Sheaves and Differential Equations
How do sheaves arise in studying solutions to ordinary differential equations?
EDIT: Is it possible to construct non-isomorphic sheaves on a domain $D \subset \mathbb{R}^n$ using solution sets to ...
- 22.8k
2
votes
2
answers
341
views
Closed forms for Monotonic polynomial recurrences?
I have a monotonic polynomial recurrence of the following form:
c_n = 1-p + p*(c_n-1)^2
This comes from the probability that a specific branching process (Galton-Watson) will be extinct before the ...
- 2,383
6
votes
1
answer
427
views
Subspaces of $L^{2}$
[In what follows $0^{0}$= 1 by convention.]
Is there some closed infinite dimensional linear subspace $F$ of $L^{2}(0,1)$
such that $\left\lvert f\right\rvert^{\left\lvert f\right\rvert}$ belongs to $...
- 4,060
19
votes
7
answers
2k
views
Generalizations of "standard" calculus
We have the usual analogy between infinitesimal calculus (integrals and derivatives) and finite calculus (sums and forward differences), and also the generalization of infinitesimal calculus to ...
- 6,792
2
votes
2
answers
317
views
Bibliography for topologies defined by a family of seminorms
Hello
I am trying to learn more about Fréchet spaces (in order to study the theory of distributions) and was wondering what people thought was the best resource.
Thank you very much.
- 143
11
votes
3
answers
4k
views
When does a real polynomial have a pair of complex conjugate roots?
Suppose we have a polynomial function $f(z)=a_0+a_1z+a_2z^2+...+z^n$ with each $a_i$ between 0 and 1. Is there a method to determine if $f$ has a pair of complex conjugate roots?
There are many ...
- 2,709