All Questions
25 questions
5
votes
1
answer
375
views
Looking for a counterexample: Conditioning increases regularity?
Let $p(x,y,z)$ be a joint density (over $\mathbb{R}^3$) under no smoothness or regularity assumptions, besides its existence. I am looking for a (counter)example where $p(y|x)$ is less regular than $p(...
- 51
7
votes
2
answers
2k
views
Method of characteristics for higher order PDEs in more than two variables
I am trying to understand the mathod of characteristics for solving partial differential equations. However, all the examples I found over the internet are for first order PDEs or for second order ...
- 8,998
23
votes
5
answers
2k
views
PDEs and algebraic varieties
Let $P$ be an order $d$ differential operator with constant coefficients and consider a PDE of the form $Pf = \delta$. Taking the Fourier transform of $P$ we get a degree $d$ polynomial whose zero ...
- 8,998
1
vote
1
answer
2k
views
Product of Dirac delta function
The following equation may be meaningful, but how can we make it well-defined
$$\delta(x-a)\cdot\delta(x-b)=0$$
Question: How do we defined this equation? Or more broadly define product between ...
- 13
1
vote
1
answer
197
views
Does convolution with heat kernel converge to pointwise evaluation?
Let $G(t, x) := \frac{1}{\sqrt{4 \pi t}} \exp\left( -\frac{x^2}{4 t }\right)$ for all $(t, x) \in (0, T) \times \mathbb{R}$ be the fundamental solution to the heat equation $\partial_tu = \partial_{...
1
vote
0
answers
119
views
On some integral equation
Let $M$ be the set of continuous and increasing functions $h: [0,1)\to\mathbb R_+$ s.t. $h(0)=0$ and $h(1-)=+\infty$. Consider the equation as follows:
$$
1-t= \int\limits_0^\infty p(x)\Phi\left(\frac{...
- 1,399
0
votes
1
answer
73
views
A solution satisfying an integral inequality is bounded [closed]
Let $y$ be a positive function and $c>0$. If $y$ satisfies the following integral inequality
\begin{equation}
y(t)+\int_{0}^{t}y(s)ds\leq c_{1}\int_{0}^{t} y^{\frac{3}{2}}(s)ds+c_{2}
\end{equation}
...
- 11
3
votes
0
answers
99
views
Definition clarification: "regular directed distributions"
(I asked this question on math.stackexchange (see here) but didn't receive any reaction, hence I try it here. If it does not fit within here, just let me know in the comments.)
In the definition of ...
- 1,171
0
votes
0
answers
76
views
Linear dependence of the derivatives of a vector valued function
Let $f:\mathbb{R}\rightarrow\mathbb{R}^5$ be an injective smooth function, and consider the function
$$
g:\mathbb{R}^5\rightarrow\mathbb{R}^5
$$
given by
$$
g(t_1,t_2,t_3,a,b) = f(t_1)+a(f(t_2)-f(t_1))...
- 8,998
0
votes
0
answers
71
views
Sufficient condition to be increasing, following a vector field
Let $f\in\mathcal{C}^1(\mathbb{R}^n,\mathbb{R})$ $(n\geq 1)$ be an observable, and let $v^1,v^2\in\mathcal{C}^1(\mathbb{R}^n,\mathbb{R}^n)$ be two vector fields such that for any $(x^1_t)_{t\geq 0}$ ...
- 449
6
votes
2
answers
326
views
Looking for references to study $U^p$ and $V^p$ spaces
I am studying some papers in the analysis of nonlinear PDEs and I am encountering the $U^p$ and $V^p$ spaces for the first time. Where can I find references more detailed than papers?
Edited
The ...
- 159
9
votes
1
answer
621
views
Uniqueness of solutions of Young differential equations
Consider the following one dimensional Young differential equation:
\begin{align*}
&Y_t=\int_0^t Y_s dX_s,\quad t\in[0,1];\\
&Y_0=0.
\end{align*}
Here the driving process $X$ is a bounded ...
- 931
2
votes
0
answers
77
views
How we can do the derivative for this equation w.r.t.to time t>0
Let $x\in[0,L]$ and consider the following equation,
$$\varepsilon \left( t \right)=\frac{1}{2}\int_{0}^{L}{({{\rho }_{1}}{{\left| {{\varphi }_{t}} \right|}^{2}}+{{\rho }_{2}}{{\left| {{\varphi }_{t}} ...
- 199
5
votes
0
answers
262
views
Weighted reverse Poincare inequality over a function class of neural networks
We consider a probability measure supported on the whole space $\mathbb{R}^n$, whose density is $p(x)$. We also consider a (one-layer) neural network function class $\mathcal{C}$, whose elements have ...
- 325
2
votes
1
answer
234
views
Counter example about blow-up solution of DEs
Let $f(\cdot)$ be a continuously differentiable function over $\mathbb{R}$, and $u\in L^2_{loc}(0,\infty)$, $a\in \mathbb{R}$, and $x(t)$ solves the integral of $$\dot{x}(t)=ax(t)+f(x(t))+u(t), \quad ...
- 395
10
votes
0
answers
845
views
Witt's proof of Gelfand-Mazur / Ostrowski's Theorem
Previously asked on Math Stackexchange without answers.
Background: As sort of a hobby, Ernst Witt gave extremely short proofs for famous theorems. This question is about his six-line proof of the ...
- 2,454
-1
votes
1
answer
136
views
An elementary question about integration by parts! [closed]
Let $f,g: R \rightarrow R$ be two positive increasing functions. Under what (non-trivial) conditions one can guarantee that $\int_{0}^{\infty}f'g dx\geq \int_{0}^{\infty}g'fdx$.
5
votes
2
answers
978
views
Symbol of the Laplace-Beltrami on $\mathbb{S}^2$
This question is about how the principal part (or symbol) is defined on a manifold?-I assume that the answer is: As in $\mathbb{R}^n$ using local coordinates, i.e.
A differential operator $P=\sum_{|\...
- 329
6
votes
1
answer
314
views
Generators of a convex cone defined by a differential inequality
Consider the cone of continuously twice differentiable functions mapping positive reals to itself (i.e., $f\in C^2(\mathbb R_{++})$ and $f\colon \mathbb R_{++}\to\mathbb R_{++}$) that satisfy
\begin{...
- 61
5
votes
1
answer
187
views
Getting out a system of linear ODEs by knowing the Magnus expansion
Assume we are given for a transition between two time points $t_0 = 0$ and $t_1$ a matrix relationship, eventually describing the solution of a system of linear with non-constant coefficients,
$$Y(t_1)...
- 749
0
votes
1
answer
195
views
Existence of bounded $n-$th derivative of the solution of differential equation
This question is the copy from mat.stackexchange.com here. I requestioned here due to the very limited responses there.
Let $\phi:\mathbb{R}\mapsto\mathbb{R}$ be the standard normal density, $$\phi(x)...
- 133
8
votes
3
answers
637
views
Method to compute fundamental solutions which are distributions
The Malgrange-Ehrenpreis theorem tells us that there is a fundamental solution for any linear differential operator of constants coefficients. The original proof was not constructive (it was based on ...
1
vote
1
answer
383
views
Solution of a PDE and its uniqueness
Hallo,
consider $f: U \times I \rightarrow \mathbb{R}$, where $U \subset \mathbb{R}^{n}$ and $0 \in I \subset \mathbb{R}$ be two open sets. I am looking for the solution $f$ of the following PDE
$\...
- 339
44
votes
10
answers
47k
views
Is square of Delta function defined somewhere?
I am wondering whether anyone knows if the square of Dirac Delta function is defined somewhere.
In the beginning, this question might look strange. But by restricting the space of the test functions, ...
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