# Tagged Questions

**4**

votes

**3**answers

1k views

### Does the derivative of log have a Dirac delta term?

Dirac writes down the following formula on page 61 of his "Principles of quantum mechanics":
$\frac{d}{dx}\log x = \frac{1}{x} -i\pi\delta(x)$, see http://adsabs.harvard.edu/abs/1947pqm..book.....D ...

**1**

vote

**2**answers

295 views

### Limit with theorem of dominated convergence

Let $f\in L^{2,s}(\mathbb{R}^3)=\bigg\lbrace u\bigg|\int_{\mathbb{R}^3}dx\,|u(x)|^2(1+|x|^2)^s<\infty\bigg\rbrace$ ($s>\frac{1}{2}$)
I have to calculate this limit
$$\lim_{|x-y|\to ...

**0**

votes

**1**answer

126 views

### Limit of a function in a weighted Sobolev space

I have a function $f(x)$ in the space $H^{2,-s}(\mathbb{R}^3)$; have this limit sense
$$\lim_{|x-y|\to 0} f(x)$$
? ($y$ is a fixed point)
If i have $f$ in $H^2$ I can say that
$$\lim_{|x-y|\to 0} ...

**2**

votes

**0**answers

124 views

### Extension divergence-free, curl-converging vector field

Hi.
Consider a smooth open Set $\Omega\subset\mathbb{R}^3$ and a bounded sequence of vector fields $(u_n)_n \in L^2(\Omega)$ having $0$ divergence. I know how to extend this sequence to the whole ...

**2**

votes

**2**answers

379 views

### Relationship between the derivative of a matrix and its eigenvalues

Is there any relationship between the derivative of a matrix and its eigenvalues? If, for example, the derivative is strictly positive definite, can I say that the eigenvalues are strictly increasing?
...

**6**

votes

**1**answer

223 views

### Poincaré lemma in infinite dimensions

Hi everyone,
Is the PoincarĂ© lemma true in infinite dimensions?
Here's a precise statement:
Let $X$ be a Banach (or maybe Hilbert) vector space, $U$ a simply connected open set in $X$. Is it true ...

**0**

votes

**1**answer

291 views

### Integral inequality

Let $X$ be the d-dimensional hypercube $X=[0,1]^d$ and let $f$ and $g$ be such that $f(x) = 1$ if $x \in A$ and $0$ otherwise, $g(x)=1$ if $x \in B$ and $0$ otherwise, where $A$ and $B$ are generic ...

**0**

votes

**0**answers

239 views

### Checking whether this would be bounded

It may be better to post this question here. Assume that $M$ is an $m$ by $m$ ($m$ is an even number) symmetric
positive-semi-definite matrix with exactly $m/2$ positive eigenvalues
and every entry of ...

**4**

votes

**2**answers

371 views

### Can we calculate the inner product of a semicontinous function with the Dirac delta function?

Dear all,
It is clear that if $f:R\mapsto R$ is a continuous function, than $< f, \delta_x >=f(x)$. Now, if $f$ is only semicontinous, can we say that $< f, \delta_x >=f(x)$? I think this ...

**0**

votes

**1**answer

380 views

### Calculating a distributional derivative

Suppose that we have a sequence of functions $u_j$ that are in $L^{\infty}(0,1)$. Then the sequence of maps $N_j(s) := \|u_j(s)\|^2$ are also in $L^{\infty}(0,1)$. Hence they give rise to ...

**0**

votes

**2**answers

115 views

### representation of compact supported distribution

Is this true?
Any compact supported distribution can be represented as finite sum of partial derivatives of functions.

**0**

votes

**1**answer

384 views

### Sequence of smooth functions converging to sgn(x)

I'm looking for a sequence of smooth functions $f_i(x)$ converging to Sign$(x)$, each of which additionally have the following property:
\begin{equation}
f_i(x_1+x_2) = g_i(x_1, f_i(x_2))
...

**22**

votes

**16**answers

2k views

### functions satisfying “one-one iff onto”

Hello Everybody.
I need some more examples for the following really interesting phenomenon:
A function from the class ... is one-one iff it is onto.
Some ...

**10**

votes

**1**answer

573 views

### A problem concerning $L^2([0,1]\times[0,1])$

Trying to solve a conjecture in differential geometry, I am leaded to the following problem (which may seem weird to a analyst). I wonder if anyone know some techniques that happen to solve it.
Let ...

**0**

votes

**3**answers

703 views

### How to isolate $f(x)$ in $f(x+a)=f(x)+a\times g(x)$?

$a \in \mathbb{R}$
$f:\mathbb{R} \rightarrow \mathbb{R}$
$g:\mathbb{R} \rightarrow \mathbb{R}$
For generic functions $f$ and $g$, how isolate $f(x)$ in the equation below?
$f(x+a)=f(x)+a\times ...

**2**

votes

**1**answer

482 views

### A singular value inequality

Let $s_1,s_2: \mathbb{R}^{2\times 2} \mapsto \mathbb{R}_+$,
$s_{1}\left(\cdot\right)\ge s_{2}\left(\cdot\right)\ge 0$, be the
singular values of a $2\times2$ matrix. Is it true that
...

**4**

votes

**1**answer

525 views

### Harmonic functions on the plane

I have a question regarding harmonic maps from all of ${\Bbb R}^2$ into a domain in ${\Bbb R}^2$. Before stating my question in full generality, let me ask a special case of the question first. Is it ...

**4**

votes

**1**answer

2k 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 $w$ ...

**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 ...