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        Questions tagged [asymptotics]

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        Show that $\| v\|_{L^2(0,b)}=o(\| u\|_{L^2(0,b)})$ as $b \to \infty$

        Let $\alpha<0$ and let $u(x):= e^{\alpha x}$ for $x \geq 0$. I'm reading a paper which states that there are constants $d_{j}, \beta_{j} \in \mathbb{C}$ with $\beta_j>0$ such that if we define $...
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        0answers
        104 views
        +50

        Calculating $\int_1^{\infty}\frac{\operatorname{ali}(x)}{x^3}dx$, where $\operatorname{ali}(x)$ is the inverse function of the logarithmic integral

        It is well-known that we can compute the closed-form of the integrals $$\int_1^{\infty}\frac{\log x}{x^2}dx$$ and $$\int_1^{\infty}\frac{\operatorname{li} (x)}{x^3}dx,$$ where $\operatorname{li} (x)$ ...
        4
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        1answer
        165 views
        +50

        What work can be done to study the solutions of $\varphi\left(x^{\sigma(x)}\sigma(x)^x\right)=2^{x-1} x^{3x-1}\varphi(x)$?

        For integers $n\geq 1$ I denote the Euler's totient function as $\varphi(n)$ and the divisor function $\sum_{1\leq d\mid n}d$ as $\sigma(n)$, that are two well-known mulitplicative functions. We ...
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        75 views

        Coefficients of some infinite product power series

        Let $f(n)\colon \mathbb{P}\to\mathbb{R}_{>0}$, where $\mathbb{P}=\{1,2,\dots\}$, be some ''nice'' function such that $f(n) \to \infty$ as $n\to\infty$. For instance, $f(n)=1+\log(n)$ or $f(n)=n$. ...
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        2answers
        223 views

        Random Walk on Pentagonal Tiling

        I’ve recently been looking at closed walks on tilings of the plane in which the “player” can move from one tile to any of its edge-adjacent neighbors. In particular, I’m trying to find asymptotic ...
        3
        votes
        1answer
        70 views

        Asymptotic behaviour of function using Fox $H$-function representation

        In equation (9) of this paper, it is claimed that the limiting behaviour $$ \int_0^\infty \frac{1-\cos(kx_0)}{s+Dk^\alpha}dk \sim \frac{\Gamma(2-\alpha)\sin(\pi(2-\alpha)/2)x_0^{\alpha-1}}{(\alpha-1)D}...
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        82 views

        size of a set defined by divisor function

        After some computations, I guessed the following conjecture. How can I prove or disprove it? thanks! Let $$ A(k)=\#\left\{\left(t,\frac{k+t+a}{4t-1}\right):1\leq t\leq k,\ 1\leq a\leq k+t,\ a|(k+t)^...
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        0answers
        62 views

        Use of Asymptotics in Diffusion Maps

        Question for brevity: Suppose $\varepsilon >0$ is small and that $$ f(\varepsilon) = f_1(\varepsilon) + \mathcal{O}(\varepsilon^k) $$ where $f_1$ has order $\varepsilon^{-\delta}$ for small fixed ...
        3
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        1answer
        163 views

        Sum over reciprocal of primes times coefficient

        I would like to show that $$ \sum_{p\leq x} \frac{1}{p^{1+2/\log x}}\left(\frac{\log\left(x/p\right)}{\log(x)}\right)^2=\log\log x +\mathcal{O}(1) $$ What I have tried Since we know that $$ \sum_{p\...
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        0answers
        78 views

        $L^1$ norm of oscillatory integral operator

        My question is about the $L^1_x$ norm of an oscillatory integral like $$ \int_{\mathbb{R}^n} e^{i(y\cdot x+\lambda \phi(y))}f(y)dy,$$ where $\lambda \in \mathbb{R}$, $f\in C^{\infty}_c(\mathbb{R}^n)$ ...
        5
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        0answers
        80 views

        Given the Ricci decays rapidly to 0 at infinity, is the metric asymptotically flat?

        Consider the manifold $M=\mathbb{R}^3 \setminus B$ where B is the ball with radius 1. Let $f \in C^{ \infty}(M) $ satisfying: $$f = \frac{C(\theta, \phi)}{r} + O( r^{-2}) $$ Where $(r,\theta,\phi)$ ...
        4
        votes
        1answer
        155 views

        Count weighted integer compositions

        What is the asymptotic growth of the sequence $$a_n:=\sum_{k\geq 0} 3^k c_{n,k},$$ as $n\rightarrow\infty$, where $c_{n,k}$ denotes the number of integer compositions of $n$ with exactly $k$ many 2s? ...
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        2answers
        179 views

        Estimating the number of functions which are at most $c$-to-$1$ for some constant $c \ge 2$

        Notation: $[m] := \{1, 2, \dots, m \}$. How many functions are there $f: [a] \to [b]$? The answer is easily seen to be $b^a$. How many $1$-to-$1$ functions are there $f: [a] \to [b]$? Again the ...
        5
        votes
        1answer
        219 views

        Overconcentration of Poles on the Circle of Convergence of a Power Series with Bounded Coefficients

        Let $V$ be an arbitrary set of infinitely many positive integers, and let: $$\varsigma_{V}\left(z\right)\overset{\textrm{def}}{=}\sum_{v\in V}z^{v}$$ Let $T_{V}$ denote the set of all $t\in\left[0,1\...
        1
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        2answers
        57 views

        Controllability Gramian asymptotics for small times

        Set-up. Consider the following linear controlled system $$ \dot{y}(t) = A y(t) + B u (t), \ \ t \in [0,T], \ \ \ \ \ \ \ \ \ \ (1) $$ where $y$ is the state of the system, $y(t) \in R ^n$, $A \in R ...

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