Questions involving asymptotic analysis, including growth of functions, Big-O notation, Big-Omega and Big-Theta notations.

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2
votes
1answer
68 views
+50

Non linear second order ODE up to $O(\epsilon) $ for $v_{xx}-\left(v^{3}-v\right)-\varepsilon\frac{1}{2}\left(1-v^{2}\right)=0$

I really need help solving this : $$v_{xx}-\left(v^{3}-v\right)-\varepsilon\frac{1}{2}\left(1-v^{2}\right)=0 $$ With boundary conditions : $$ v(\pm \infty )=-1+\frac{1}{4}\epsilon $$ I need to ...
1
vote
0answers
28 views

Asymptotic Expansion Method for Pricing American Option

In this Article I faced with Asymptotic Expansion method for pricing American option. the price $P(S,t)$ of this option satisfies the partial differential equation (PDE): $${{P}_{t}}+(r-\delta ...
1
vote
1answer
39 views

Why is this true: $1- (1-1/n)^{\varepsilon n} \leq \varepsilon + \mathcal{O}(\varepsilon^2)$

In my lecture notes, the following is written: $$1- (1-1/n)^{\varepsilon n} \leq \varepsilon + \mathcal{O}(\varepsilon^2)$$ as $\varepsilon \rightarrow 0$ and $n$ some fixed constant (non-negative ...
0
votes
0answers
20 views

Continuous mapping theorem with density convergence

Let us consider a bivariate random variable $X\in \mathbb{R}^2$ with $pdf$ $f$. Also let, based on a sample of size $n$, let the the estimator of the density be $f_n(x)$ at $x\in \mathbb{R^2}$ and we ...
-1
votes
1answer
108 views

The asymptotic of the first Chebyshev function, using the Prime Number Theorem [on hold]

Using the prime number theorem, show that: $\vartheta (x) \sim x$ Where $\vartheta (x) := \sum_{p \le x} \log p$ Any help on this would be great, thanks in advance.
-2
votes
0answers
21 views

Big O solving problem [on hold]

show that $T(n) = O(f(n))$ by providing the appropriate constants $c$ and $n$ base $0$ such that $T(n) \leq c f(n),~ n\geq n$ (base $0$)? $T(n)=n^3 + 3n - 60$, $f(n)=n^3$ $T(n)=n log_2 (n) + 4 f(n) ...
2
votes
1answer
24 views

Is there a way to estimate the number of positive integers less than or equal to $n$ that have a given prime $p$ as a least prime factor

The probability that an integer $p$ divides an integer $x$ is $\dfrac{1}{p}$. From this article on almost prime numbers, the number $\pi_k(n)$ of positive integers less than or equal to $n$ with at ...
0
votes
1answer
70 views

How can I solve this recurrence problem?

Given a function $$ f(n) = f(5n/13) + f(12n/13) + n \;\;\;\;∀n \geq 0 $$ I would like to find a function $g(n)$ such that $f ∈ Ө(g(n))$.
2
votes
1answer
16 views

Show that $\log^{i} n \in O(n^{j})$ for $i,j > 0 $

I want to show that $$\log^{i} n \in O(n^{j})$$ I tried to apply L'Hospital and came up with the following: $$\lim\limits_{n \rightarrow \infty}{\frac{\log^{i} n}{n^{j}}} =$$ $$\lim\limits_{n ...
1
vote
1answer
32 views

What are conditions under which convergence in quadratic mean implies convergence in almost sure sense?

What are the conditions on the sequence on $\{X_n\}$ (apart from the degenerate random variable), under which it can be claim that $||X_n-X||_{L^2(\mathbb{R})}\rightarrow 0$ implies $X_n\rightarrow ...
1
vote
0answers
38 views

Asymptotic behavior of the minimum eigenvalue of a certain Gram matrix with linear independence

Consider the density matrices with the following spectral decompositions: $$\rho=\lambda_1|\nu_1\rangle+\lambda_{2}|\nu_2\rangle$$ and $$\sigma=\gamma_1|\omega_1\rangle+\gamma_2|\omega_2\rangle$$ such ...
0
votes
1answer
20 views

Asymptotic behaviour of Hilbert transform

Let $f$ be a bounded function on $\mathbb{R}$ with compact support include in $[-K,K]$. Show that $$ H(f)(x)=\frac{a}{\pi x}+O(\frac{1}{x^2})$$ where $a=\int f(t)dt$ and $H$ denote the Hilbert ...
4
votes
2answers
168 views

Show that if $\int_0^x f(y)dy \sim Ax^\alpha$ then $f(x)\sim \alpha Ax^{\alpha -1}$

Let $f$ be a real, continuous function defined on $[0,\infty)$ such that $xf(x)$ is increasing for all sufficiently large values of $x$. Show that if $$\int_0^x f(y)\,dy \sim Ax^\alpha \quad ...
0
votes
0answers
17 views

Estimating the derivative of the difference function from that of a function

Suppose that $f$ is a twice differentiable function in an interval $(N,2N)$. We write $f_1(n,h)=f(n+h)-f(n)$, i.e., $f_1$ is the difference function. Then, a proof I'm reading estimates that if ...
2
votes
0answers
27 views

An advection problem with weak diffusion in asymptotic analysis.

Consider the following advection problem with weak diffusion: $$ \varepsilon\partial_{x}^2 u=\partial_{t}u+\partial_{x}u, $$ for $−\infty < x < \infty$, and $t > 0$ where $u(x, 0) = ...
0
votes
2answers
19 views

exponential boundedness of components given exponential boundedness of the norm

Let $v:[0,\infty)\rightarrow \mathbb{R}^n$ be a function such that $\forall t\ge 0$, $v_i(t)\ge 0$ and $$ ||v(t)||\le \beta ||v(0)||e^{-at}, t\ge 0$$ with $\beta,a>0$ can I conclude that for all ...
1
vote
3answers
30 views

Is this proof for big-Oh of $(x+2)log(x^9 + 5)$ correct?

Is my proof that $(x+2)log_{2}(x^9+5)$ is $\mathcal{O}(xlog_{2}x)$ correct when x tends towards infinity? $\left | f(x) \right | = \left | (x+2)log_{2}(x^9 + 5) \right |$ $\leq \left ...
3
votes
1answer
64 views

Asymptotic behaviour of $\int_0^1 g(x)\exp(-nx)dx$ as $n\rightarrow\infty$

Let $g:(0,1]\rightarrow\mathbb{R}_+$ be an invertible monotonically non-increasing function that integrates to $1$ and has $g(1)=0$, $g(0)=\infty$; eg. $g(x)=x^{-1/2}-1$ or $g(x)=\ln(1/x)$. I believe ...
1
vote
0answers
18 views

does an exponential bound on a Lyapunov candidate implies asymptotic stability?

if I have a Lyapunov candidate $V:[0,\infty)\rightarrow \mathbb{R}$ and I'm able to show that $$ V(t)\le k e^{-\eta t} V(0),\qquad \forall t\in[0,\infty) $$ can I conclude something about ...
2
votes
1answer
46 views

Big-O vs. Best Big-O

Is there a difference between the method to find a big-O function and the method to find the best big-O function. Take for example the following function: $f(n) = 1 + 2 + 3 + ... + n$ It is easy to ...
3
votes
1answer
69 views

Steepest descent method with movable maximum

Suppose we want to find the asymptotic behavior as $n \rightarrow \infty$ of the integral $$\int_C \frac{dz}{z} \frac{e^z}{z^n}=\int_C \frac{dz}{z} \exp(z-n \ln z)$$ where $C$ is some contour in the ...
8
votes
5answers
174 views

The maximal size of between $\varphi(n)$ divided by $\lambda(n)$.

I want to find $$f(n) = \max\left\{\frac{\varphi(k)}{\lambda(k)} : 1 \leq k \leq n\right\}$$ In other words, I want to find the maximal value of $\frac{\varphi(k)}{\lambda(k)}$ when $k$ is ...
1
vote
1answer
43 views

How to prove that $\log(n)$ is $O(n^c)$? [closed]

What's a straightforward way to prove that? For any c>0.
0
votes
1answer
23 views

Why is this true - easy question concerning asymptotics of exponential

Suppose $\lambda > 0$ is constant as $t \searrow 0$. In my lecture notes it is written that $\left(1+\sum_{k=1}^{\infty} \frac{(-\lambda t)^k}{k!}\right) \lambda t = \lambda t + o(t)$ and ...
1
vote
2answers
60 views

Find whether $f(n) = o(g(n))$ or $g(n) = o(f(n))$

Find whether $f(x) = O(g(n))$ or $g(n) = O(f(x))$ where $$ f(n) = (\log n)^{\log n} \quad\quad\text{and}\quad\quad g(n) = 2^{(\log_2n)^2} $$ I found that $f(n) = n^{ \log {\log n}}$, ...
1
vote
0answers
21 views

Second order perturbed equation

I've been studying asymptotic behavior on Ordinary Differential Equations. While doing some excercises I found out one excercise which has had me thinking for a while, so I am asking humbly for your ...
3
votes
1answer
56 views

Reference for asymptotics on sum

Quite simply I'm looking for the large $m$ asymptotic behavior of \begin{equation} \sum_{k=1}^{m}{m\choose k}\frac{a^k}{k} \end{equation} where $a$ is a constant. This looks easy for someone who knows ...
0
votes
2answers
78 views

Summing divergent asymptotic series [closed]

I found the sine integral si to be $$Si (x)\sim \frac \pi 2+\sum _{n=1}^\infty (-1)^n \left(\frac{(2 n-1)! \sin (x)}{x^{2 n}}+\frac{(2 n-2)! \cos (x)}{x^{2 n-1}}\right)$$ Say I want to find ...
0
votes
1answer
20 views

Estimating size of partial euler product

What estimates are there for product over primes $p \leq x$ $\prod_{p \leq x}(1-\frac{1}{p^{r}})$ given $r$ is positive integer. Something better than $\prod_{p \leq x}(1-\frac{1}{p^{r}}) \leq ...
5
votes
1answer
640 views

Order of growth of the entire function $\sin(\sqrt{z})/\sqrt{z}$

Show that $$f(z)=\frac{\sin\sqrt z}{\sqrt z}$$ is an entire function of finite order $\rho$ and determine $\rho$. I observed that the two determinations of the square root differ only for the signum. ...
4
votes
1answer
454 views

Can a curve be an asymptote?

$f(x)=x^3+\frac{3}{x-1}$ This was the question given to me.I replied that $f(x)$ will have only a single vertical asymptote of $x=1$. My teacher told that there'll be be two asymptotes.One is the ...
0
votes
1answer
19 views

Calculate the asymptotic growth of a sum that contains log or binom

I'm looking for a basic explanation how to calculate the asymptotic growth of sums. Take for example this one: $\sum_{i=1}^{lg(n!)} 2^{n^2}$ or this one: $\sum_{i=0}^{n} {n\choose{i}}$ The ...
1
vote
1answer
48 views

Is $\sqrt{2/(27\pi n)}\sim n^{-1/2}$?

Is $\sqrt{2/(27\pi n)}\sim n^{-1/2}$? Since $$ \sqrt{\frac{2}{27\pi n}}=\sqrt{\frac{2}{27\pi}}\cdot\frac{1}{\sqrt{n}}\sim\frac{1}{\sqrt{n}}=n^{-1/2}, $$ I would say, yes, of course.
1
vote
1answer
31 views

Prove/disprove the following asymptotic bound

Indicating with $p$ and $q$ prime numbers, is it true that for $x\rightarrow\infty$ $$ \sum_{\substack{p\leq x \\ p\equiv 1 ...
2
votes
1answer
25 views

Convergence to a distribution implies convergence of a logarithm?

Let $X_n$ be a sequence of almost surely positive real-valued random variables s.t. $$\sqrt{n} \, \left( X_n -a \right) \to_D \mathcal{N} ( 0, 1)$$ where $\to_D$ denotes convergence in distribution ...
1
vote
0answers
10 views

Stationary Phase method with Singular test function

Consider the following integral $I(x,t) = \int_{-\infty}^{\infty}\{F(k)exp(it\psi(k)) \}dk$ with $\psi(k) = (k-k_0)(\frac{x}{t}) - (\beta(k)-\beta_0)$ where $\beta_0=\beta(k_0)$ and $F(k)= ...
7
votes
0answers
101 views

Heat equation asymptotic behaviour 2

Let $D$ be the domain defined as $D := \{ (x,t): t \in [0,1) , \; x < (1-t)^\alpha \}$. Let $u(x,t)$ satisfy the heat equation $u_t = \frac{1}{2}u_{xx}$ in $D$, with initial condition: ...
0
votes
1answer
37 views

Asymptotic value of a Cauchy Singular integral

Let, $\zeta(x,t) = A_0sin(k_0x)cos(\omega t) + \frac{2k_0A_0}{\pi} \{\int_{0}^{\infty}\frac{cos(kx)cos(\beta t)-cos(k_0x)cos(\omega t)}{k^2-k_0^2}dk\}$ Here $\beta ^2 = gktanh(kh)\ and\ \omega^2 = ...
1
vote
1answer
52 views

Estimation of a probability of marginal values of a random variable

My question is related with this question on combinatorics of 0-1-matrices from MO. Trying to obtain a (asymptotic) lower bound for $\alpha(n)$ by probabilistic approach (see, for instance, “The ...
1
vote
0answers
18 views

Stationary Phase method with Singular test function

I'm stuck at the following integral $I(x,t) = \int_{-\infty}^{\infty}\{F(k)exp(it\psi(k)) \}dk$ with $\psi(k) = (k-k_0)(\frac{x}{t}) - (\beta(k)-\beta_0)$ where $\beta_0=\beta(k_0)$ and $F(k)= ...
4
votes
0answers
33 views

Asymptotic behavior of many derivatives

To compute the residue of a pole of very high order $M$ at $z=0$, one needs to compute $\frac{d^M}{dz^M} g(z)$ Suppose that $g(z)$ is a reasonable but not trivial function, that itself may depend on ...
0
votes
0answers
6 views

Multiscale expansion: Higher harmonics for Higher order solution

The following is related to he topic of "Evolution Equations for Slowly Modulated Weakly Nonlinear Water Waves Over Horizontal Sea Bed" from Sec 13.2, Theory and Applications of Ocean Surface Waves, ...
2
votes
3answers
101 views

The asymptotic behavior of an integral

The integral in hand is $$ I(n) = \frac{1}{\pi}\int_{-1}^{1} \frac{(1+2x)^{2n}}{\sqrt{1-x^2}}\, dx $$ I dont know whether it has closed-form or not, but currently I only want to know its asymptotic ...
1
vote
1answer
37 views

How to solve this type of exercises $\sqrt{x^6+x^5-2x^3+O(x^2)}$

I have a simulation test with this type of exercise, asymptotic expansion: $$\sqrt{x^6+x^5-2x^3+O(x^2)}$$ with $$ x\rightarrow \infty$$ I have studied the theory of Landau's symbols but I have no ...
1
vote
2answers
78 views

When does $f\sim g$ implies $f'\sim g'$?

Given two $C^1$ functions $f,g:[0,+\infty)\to [0,+\infty)$ such that $f(x)\sim g(x)$ as $x\to\infty$, which good conditions guarantee that $f'(x)\sim g'(x)$? I thought that monotonicity of the ...
1
vote
0answers
37 views

Can we find the closed-form of the series?

I want to calculate the series $$ F(N,g)=\frac{1}{g^N}\sum_{m=0}^{N(g-1)}\Big(\sum_{i=0}^{[m/g]}(-1)^i\binom{N}{i}\binom{N-1+m-gi}{N-1}\Big)^2 $$ where $g=2,3,4,\cdots$, and $N$ is any positive ...
1
vote
1answer
265 views

Limit involving tetration

Let the notation be $a^{\wedge\wedge}b = \underbrace{a^{a^{\cdot^{\cdot^{a}}}}}_{b\,times}$ for tetration. My mentor conjectured the following: Let $n$ be a positive integer, then let $A(n)$ be ...
4
votes
0answers
125 views

Heat equation, boundary gradient singularity

Consider the Cauchy-Dirichlet problem for the heat equation in a non-cylindrical region $\Omega \subset \mathbf{R}^+ \times \mathbf{R}$: $\Omega = \{ (t,x): \; 0 \leq t \leq 1, \; x \leq ...
4
votes
3answers
131 views

What is $\lim_{x\to 0} \sum_{n=2}^\infty \frac{\sqrt{x}\ln n}{1+n^2 x}$?

What is $\displaystyle\lim_{x\to 0} \sum_{n=2}^\infty \frac{\sqrt{x}\ln n}{1+n^2 x}$ ? Find an asymptotic expansion of $\displaystyle \sum_{n=2}^\infty \frac{\sqrt{x}\ln n}{1+n^2 x}$ as $x\to ...
0
votes
3answers
55 views

Prove or disprove $f(n)$=$2^{n+1}$ is $O(2^n)$

I need to prove or disprove $f(n)$=$2^{n+1}$ is $O(2^n)$. I believe this statement is true, so I want to prove it. I know that $f(n)$ is $O(g(n))$ if there are positive constants $C$ and $k$ such ...