All Questions
12,823 questions
9
votes
0
answers
262
views
Semi-norms for Schwartz-Bruhat space over Q_p
I'm an analyst beginning to do some work over the p-adics, in particular work with spaces of functions from $\mathbb{Q}_p$ to $\mathbb{C}$. The Schwartz-Bruhat space in this case is given by the space ...
- 91
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
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
14
votes
6
answers
1k
views
Consequence of equidistribution or not?
Let $\theta \not\in \mathbb{Q}$. We know that $(n\theta)_{n \geq 1}$ is equidistributed modulo 1.
Let $\epsilon_n = \mathrm{sign}\bigl(\sin(n\pi \theta)\bigr)$ and $S_N= \sum_{n=1}^N \epsilon_n$.
I'...
- 2,829
10
votes
6
answers
6k
views
Fourier transform of (real) exponential
Is it possible to make sense, in distributional sense, of the Fourier transform of the exponential function (defined over the whole real line)?
- 101
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
3
votes
0
answers
2k
views
Multi-variate secant method for solving $F(x)=0$
The secant method for solving an equation $F(x)=0$ in one variable is much older than Newton's one. Recall that given two iterates $x_{k-1}$ and $x_k$, it provides an update $x_k$ by taking the ...
- 52.3k
6
votes
3
answers
1k
views
Pinching and positive definite matrices
A pinching over $M_n({\mathbb C})$ is an endomorphism $T$ where the $(i,j)$-entry of $T(M)$ is given either by $0$ or by $m_{ij}$, depending on the pair $(i,j)$. Let us say that a pinching is ...
- 52.3k
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 ...
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
$$f(z)=\sum a_kz^k$...
7
votes
3
answers
1k
views
Non-Borel subspace of Banach space
Let $X$ be a separable Banach space, $M \subset X$ a linear subspace. Must $M$ be a Borel set in $X$?
I believe the answer is "no," since I have seen authors who are careful to talk about "Borel ...
- 29.8k
16
votes
2
answers
2k
views
The Cauchy–Riemann equations and analyticity
I would be interested to learn if the following generalization of the classical Looman-Menchoff theorem is true.
Assume that the function $f=u+iv$, defined on a domain $D\subset\mathbb{C}$, is such ...
- 22.3k
21
votes
1
answer
17k
views
A nonlinear first order ordinary differential equation: $y(t)^n+a(t)\frac{dy(t)}{dt}=ba(t)$
I am stuck on solving an apparently simple ODE. I have checked numerous texts, references, software packages and colleagues before posting this...
$$y(t)^n+a(t)\frac{dy(t)}{dt}=ba(t)$$
If the RHS ...
- 313
-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
5
votes
2
answers
1k
views
Formula for n-th iteration of dx/dt=B(x)
Let $B(x)$ be infinitely differentiable with respect to $x$.
Drop the use of parentheses on $B$ to delimit the argument $x$
and use them instead to hold the order of the derivative with respect to $x$....
- 133
1
vote
1
answer
255
views
In 1D, is a $W^{1,p}$ function always Lipschitz, for $p\ge1$?
We know that Morrey's inequality says $W^{1,p} \subset C^{0,\gamma}$ for $\gamma = 1 - n/p$ where $n$ is the dimension. However, in 1D, following the proof of Evans "Partial Differential Equations" (...
- 45
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
2
votes
0
answers
560
views
Generalization of repeated error function integral
Is there a name for the following integral?
$f(x, y, n) = \int_y^\infty (t - x)^n \exp(-t^2) dt$
The parameter $n$ is positive. The first priority is integer $n$, but more generally the case of real-...
- 5,227
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
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
-1
votes
2
answers
4k
views
product of two riemann integrable is riemann integrable [closed]
first show you only need to consider squares of functions as
f.g = 1/4 [(f+g)sqr - (f-g)sqr].
show then that you only need to consider only positive valued functions becuase f(x).g(x)=|f(x)|sqr.
then ...
- 1
2
votes
3
answers
3k
views
Power series solutions for nonlinear ordinary differential equations - references
I'm having a hard time finding some references on series solutions for "nonlinear" ODE's, the most I could find was a small excerpt on Wikipedia.
https://en.wikipedia.org/wiki/...
- 579
12
votes
2
answers
1k
views
Logarithm of AM/GM ratio: $\sqrt{\log((x+y)/(2\sqrt{xy}))}$
Recently, while playing around with infinite-divisibility, i arrived at the following metric:
$$d(x,y) := \sqrt{\log\left(\frac{x+y}{2\sqrt{xy}}\right)},$$
defined for positive reals $x$ and $y$. ...
- 28.6k
1
vote
2
answers
301
views
finding solution to function$f^{n}(x)=f(x+k)
according to question
Finding solutions to $f'(x) = f(x + k)$
i ask generalization of this question
i am trying to find non-trivial functions $f \colon \mathbb R \to \mathbb R$ that $f^{n}(x) = ...
- 479
0
votes
2
answers
528
views
condition number
Hi
I have the following matrix
A=[a_11 a_12 a_13 1;
a_21 a_22 a_23 1;
.
.
.
a_n1 a_n2 a_n3 1]
I have seen that when some of a_ij are big for instance in the ...
- 15
15
votes
7
answers
6k
views
Freshman's definition of sin(x)?
I would like to know how you would rigorously introduce the trigonometric functions ($\sin(x)$ and relatives) to first year calculus students. Suppose they have a reasonable definition of $\mathbb{R}$ ...
- 23.3k
49
votes
2
answers
19k
views
Irrationality of $ \pi e, \pi^{\pi}$ and $e^{\pi^2}$
What is known about irrationality of $\pi e$, $\pi^\pi$ and $e^{\pi^2}$?
- 6,819
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
14
votes
2
answers
2k
views
The vector field of a given flow
Let $f:(0,1)\rightarrow(0,1)$ be a map with some regularity (${\mathcal C}^1$, ${\mathcal C}^2$, ${\mathcal C}^\infty$, analytic ?). We assume that $f(t)> t$ for every $t$, and that $f'> 0$.
...
- 52.3k
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
5
answers
1k
views
Finding solutions to $f'(x) = f(x + k)$
I'm trying to find non-trivial functions $f \colon \mathbb R \to \mathbb R $ that $f'(x) = f(x+k)$ with $k \in \mathbb R$.
For $k \le 0$, I've found functions based on $f(x)= e^x$, such as $f(x) = e^{...
- 71
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
4
votes
5
answers
891
views
Analytic hypoellipticity of linear ordinary differential operators
Let $P = a_n(x) D_x^n + a_{n-1}(x) D_x^{n-1} + \ldots + a_0(x)$ be a linear ordinary differential operator with polynomial (or real analytic) coefficients $a_j(x)$. Suppose that $a_n(x)$ doesn't ...
- 1,412
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
0
votes
1
answer
2k
views
Existence of solutions for differential equations
While learning differential equations, I was reading some notes, and it was mentioned that for Dirichlet BVP
$$x'' = f (t, x), \quad x(0) = 0 = x(1).$$ Suppose $f : [0, 1] \times \mathbb{R}\to \mathbb{...
- 169
10
votes
3
answers
2k
views
Non-vanishing of zeta(s), Re(s)=1, without complex analysis?
Say you are allowed to use Fourier analysis, complex variables, Euler-Maclaurin, etc., but no complex analysis - no holomorphic continuations, no definition of analytic function, and, in particular, ...
- 20.2k
24
votes
6
answers
3k
views
A gamma function identity
In some of my previous work on mean values of Dirichlet L-functions, I came upon the following identity for the Gamma function:
\begin{equation}
\frac{\Gamma(a) \Gamma(1-a-b)}{\Gamma(1-b)}
+ \frac{\...
- 4,671
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
21
votes
1
answer
3k
views
Intuitive Proof of Cramer's Decomposition Theorem
Cramer's decomposition theorem states that if $X$ and $Y$ are independent real random variables and $X+Y$ has normal distribution, then both $X$ and $Y$ are normally distributed. I've seen a few ...
- 4,952
16
votes
6
answers
4k
views
Polynomial positive on an interval
If $p$ is a polynomial with real coefficients and p(x)>0 on [0,1], then $p(x)=\sum c_{i,j} x^i(1-x)^j$ with $c_{i,j}$ positive. I know this is true but but I need a proof/reference. Thanks!
5
votes
1
answer
723
views
What is the advantage of inverting elliptic integrals?
In the case of the circle I can hardly make any conclusions from the integral $(1)$, most of the theorems come from geometrical considerations. It's not clear how to prove periodicity from this ...
- 1,412
1
vote
1
answer
285
views
Sobolev imbedding failure due to a kink in the domain
I'm looking for a simple example where an inequality of the form $||u||_{L^q} \leq C||u||_{W^{1,p}}$ fails for some $1 \leq q \leq p^*$ (ie. within the acceptable range for which the bound should ...
- 2,641
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
0
votes
2
answers
197
views
Correctness of equation for $\sum_{n} n!^s$
It's possible, that equation $\sum_{n} n!^s=1+2\sum_n (2n+1)!^s$ is correct for all $s \in \mathbb{R}$ with which sum $\sum_{n} n!^s$ is convergent?
I'm looking for closed formula of that sum and ...
- 125