# Tagged Questions

Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.

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### Existence and regularity for fractional Poisson-type equation

According to Theorem 2.7 in the paper https://arxiv.org/pdf/1704.07560.pdf, we have the following classical results.
Let $s \in (0,1)$ and $1<p<\infty$. Then for any $F \in L^p(\mathbb{R}^n)$ ...

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### Harmonic oscillator in spherical coordinates

It is probably the most well-known result in quantum mechanics that the harmonic oscillator can be solved by supersymmetry.
More precisely, the operator
$$-\frac{d^2}{dx^2}+x^2$$
can be ...

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68 views

### The continuity points of the derivative of the everywhere differentiable functions on a closed interval is dense [duplicate]

Let $f:[a,b]\to\mathbb{R}$ be everywhere differentiable. Prove that the set of continuity points of $f'$ is dense.

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### On the growth of functions and their integrals [on hold]

Given that $f(x)=o(g(x))$, does it necessarily follow that
$$\int f(x) \mathrm{d}x = o\Bigg(\int g(x) \mathrm{d}x\Bigg)$$ ?
If no, what conditions should $f$ and $g$ satisfy for this to be true ?

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129 views

### Is it a weaker condition for the symmetry of mixed derivatives?

Asked in MathStackExchange and I think it may be harder than usual problems.
Conditions:
$f:\mathbb{R^2}\mapsto\mathbb{R}$ has second-order derivatives on some neighborhood of the zero point.
$f_{...

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**1**answer

36 views

### Monotonicity given an implicit function containing a Measure integral

The following question seems simple but I am not sure how to handle it correctly because of the integral with respect to a measure. I would be very thankful for any reply.Cheers!
Knowing that $$f(\...

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305 views

### Number defined by a recursive binary sequence

In a math column in Scientific American many years ago, I encountered a peculiar binary sequence I describe below. Unfortunately I can't find a reference on this, so I would be grateful for any ...

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51 views

### Riesz measure of a smooth subharmonic function on a ball

Let $B(x_{0},r)$ be the ball of center $ x_{0} $ and radius $r>0$ in $ \mathbb{R}^{k} $ ($ k\geq2 $), and $u$ a subharmonic function on an open neighborhood of the closure of $B(x_{0},r)$. Let $\mu$...

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47 views

### Alternative characterization of epi-convergence

I am struggling with the proof of a property of epi-convergence.
We need the following definitions:
For a sequence of sets $(C^\nu)_\nu$ in $\mathbb R^n$, the outer limit is the set $\limsup_\nu C^\...

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111 views

### The perturbation of a convex function can also be convex?

$ W^{1,\infty}(D)\ni f:D\to\mathbb R, (x,y)\mapsto f(x,y)$, is a strictly increasing on both dimensions (i.e. if $x_1>x_2$ then $f(x_1,y)>f(x_2,y)$), lipschitz continuous function defined on a ...

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62 views

### Regularity in Orlicz spaces for the Poisson equation

I see the following Lemma in :Regularity in Orlicz spaces for the Poisson equation | SpringerLink:(2007)
$$\Delta u=f \quad \quad \quad \quad (1)$$
Lemma 2: There is a constant $N_1 >1$ so that ...

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105 views

### Resolving a linear recurrence inequality (updated with context)

Caveat: I have a research problem in the numerical analysis of fluid dynamics similar to this one. I'm following ideas in some research papers (e.g. this one that I asked about on MO), but I have an ...

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86 views

### Conserved Positive Charge for a PDE

Let $(x,t) \in \mathbb{R}^2$, $W(x)$ be a (smooth enough) real-valued function and consider the following partial differential equation for the real-valued function $U(x,t)$
\begin{equation}
\frac{\...

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72 views

### Example of a function satisfying certain conditions on its derivatives

I am searching for examples of a non-negative function $f \in C^1((0,1];C^\infty _b(\mathbb{R}^n))$ (where $C^\infty _b(\mathbb{R}^n)$ is set of all smooth functions with bounded derivatives, $f(t,x)$ ...

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63 views

### A community effort: equilibrium in quitting games

This thread is in the spirit of the polymath project:
a combined effort of the community to solve a difficult open problem.
It is an activity of the European Network for Game Theory
whose goal is to ...