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        Questions tagged [real-analysis]

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

        0
        votes
        0answers
        41 views

        A hard to answer Digamma question [on hold]

        I just learning online obout Polygamma function, and I want to know what x equal to (and how to get it) when ψ(x)=1 and x>1.
        -2
        votes
        0answers
        40 views

        What is the relationship (mapping) from a reciprocal function 1/r to a exponential function exp(-r)? [on hold]

        The mathematical problem: Consider the mapping from $r$ to $u$. for large $r$ the theory suggests a formulation like $u=a_1 e^{-a_2 r}$, which means that the function decay exponentially. for ...
        1
        vote
        0answers
        53 views

        How to see the divergence of a series is not faster than some order? [on hold]

        $$ \sum_{m=1}^{n} m^{-1+1/p} \leq Cn^{1/p} $$ For $1<p<2$, I know the LHS is divergent but I can't see its speed of divergence is not faster than $n^{1/p}$.
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        votes
        0answers
        70 views

        Square of Jacobian

        Given a smooth vector-valued function $u:B_1(0)\subset \mathbb{R}^n\to \mathbb{R}^m$, one can look at the matrix \begin{equation} Q_{ij}= \sum_{k=1}^m \nabla_i u^k \nabla_j u^k \end{equation} This is ...
        -3
        votes
        0answers
        34 views

        How to find the volume using triple integral spherical coordinates? [on hold]

        A hemispherical bowl of radius $5 cm$ is filled with water to within $3cm$ of the top. Find the volume of water in the bowl? How can I find the volume using spherical coordinates? writing it like ...
        -1
        votes
        0answers
        70 views

        Counter-example of Subsequence Criterion? [migrated]

        The last argument shows that if $X_n\to X_\infty$ a.s. and $N(n)\to\infty$ a.s., then $X_{N(n)}\to X_\infty$. We have written this out with care because the analogous result for convergence in ...
        -2
        votes
        0answers
        40 views

        Closed union of all connected subsets that contain x [on hold]

        If Cx(S) is the union of all connected subsets of S which contain x, it is connected. I understand that, but what I don’t understand is that if S is closed, then Cx(S) is closed. Isn’t that like ...
        2
        votes
        2answers
        99 views

        Existence of solution to linear fractional equation

        We consider the equation $$?\sum_{j=1}^n \frac{\lambda_j}{x-x_j} =i$$ where $\lambda_j>0$ and $x_j$ are real distinct numbers. I want to show that if $\lambda_k$ is small compared to the ...
        4
        votes
        0answers
        93 views

        Approximation of a compactly supported function by Gaussians

        Let $f:\mathbb{R}\to\mathbb{R}$ be a smooth function whose support is a closed interval, e.g. $\text{supp}(f)=[a,b]$. Then $f$ can be approximated (e.g. in $L^2$) by a linear combination of Gaussian ...
        10
        votes
        1answer
        315 views
        +50

        A question concerning Lusin’s Theorem

        We consider only the set $M$ of a.e. essentially locally bounded measurable functions $[0, 1] \to \mathbb R$. Here $m(S)$ denotes the Lebesgue measure of $S$. Let $f$ be measurable. For every $e$ in $...
        4
        votes
        0answers
        166 views

        A simple proof of Jordan curve theorem [closed]

        I need a short proff of the Jordan curve theorem please. The one I have is 16 pages long and is for a little expo, so I need one a little shorter. Thanks
        3
        votes
        1answer
        77 views

        Equivalent notion of approximate differentiability

        Is it true that the definition of approximate differentiability presented here of a function $f: \mathbb{R}^N \to \mathbb{R}$ is equivalent to the following one? $$\lim_{r \to 0} \rlap{-}\!\!\int_{...
        -1
        votes
        0answers
        37 views

        Baseline measurements [closed]

        I need to compare 2 different results to determine best performance (best defined as lowest). Baseline measure is set for each level for expected performance. Baseline measure for each level ...
        1
        vote
        0answers
        33 views

        Formal justification of the Chaos game in the Sierpinsky triangle

        I want to justify why the Chaos game works to produce Sierpinsky triangle. I use a theorem taken from Massopoust Interpolation and Approximation with Splines and Fractals. Suppose that $(X,d)$ is a ...
        2
        votes
        0answers
        41 views

        Generalized definition of integrable condition on rough complex subbundle

        Assume object are smooth at first. If we consider real subbundle, we can define integrability in terms of parameterization or coordinate. A rank $r$ real subbundle $\mathcal V\le TM$ is called ...

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