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        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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        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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        1answer
        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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        1answer
        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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        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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        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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        1answer
        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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        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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        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 ...

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        山西福彩快乐十分钟
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