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# Questions tagged [asymptotics]

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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 $... 0answers 104 views +50 ### Calculating$\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$\operatorname{li} (x)$... 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 ... 0answers 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$f(n)=n$. ... 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 ... 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 $$\int_0^\infty \frac{1-\cos(kx_0)}{s+Dk^\alpha}dk \sim \frac{\Gamma(2-\alpha)\sin(\pi(2-\alpha)/2)x_0^{\alpha-1}}{(\alpha-1)D}... 0answers 82 views ### size of a set defined by divisor function After some computations, I guessed the following conjecture. How can I prove or disprove it? thanks! Let$$ 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)^... 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 ... 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 $$\sum_{p\... 0answers 78 views ### L^1 norm of oscillatory integral operator My question is about the L^1_x norm of an oscillatory integral like$$ \int_{\mathbb{R}^n} e^{i(y\cdot x+\lambda \phi(y))}f(y)dy,$$where \lambda \in \mathbb{R}, f\in C^{\infty}_c(\mathbb{R}^n) ... 0answers 80 views ### Given the Ricci decays rapidly to 0 at infinity, is the metric asymptotically flat? Consider the manifold M=\mathbb{R}^3 \setminus B where B is the ball with radius 1. Let f \in C^{ \infty}(M) satisfying:$$f = \frac{C(\theta, \phi)}{r} + O( r^{-2}) $$Where (r,\theta,\phi) ... 1answer 155 views ### Count weighted integer compositions What is the asymptotic growth of the sequence$$a_n:=\sum_{k\geq 0} 3^k c_{n,k},$$as n\rightarrow\infty, where c_{n,k} denotes the number of integer compositions of n with exactly k many 2s? ... 2answers 179 views ### Estimating the number of functions which are at most c-to-1 for some constant c \ge 2 Notation: [m] := \{1, 2, \dots, m \}. How many functions are there f: [a] \to [b]? The answer is easily seen to be b^a. How many 1-to-1 functions are there f: [a] \to [b]? Again the ... 1answer 219 views ### Overconcentration of Poles on the Circle of Convergence of a Power Series with Bounded Coefficients Let V be an arbitrary set of infinitely many positive integers, and let:$$\varsigma_{V}\left(z\right)\overset{\textrm{def}}{=}\sum_{v\in V}z^{v}$$Let T_{V} denote the set of all t\in\left[0,1\... 2answers 57 views ### Controllability Gramian asymptotics for small times Set-up. Consider the following linear controlled system$$ \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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