For questions that involve concrete approximations, such as finding an approximate value of a number with some precision. For questions that belong to the mathematical area of Approximation Theory, use (approximation-theory).

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2answers
17 views

Finding an approximate function using orthonormal basis

I'm trying to take a function in $C_0[0,1]$ space (let's call this $f(x)$) and trying to find the best approximate of $f(x)$ at $P_2[0,1]$ space (let's call this approximate $p(x)$). Note that $P_2[0,...
0
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1answer
49 views

Approximation of series using integral

In notes of statistical physics I found the following approximation $$\sum\limits_{n=0}^{\infty}F\left(n+\frac{1}{2}\right)\approx \int_{0}^{\infty}F(x)dx+\frac{1}{24}F'(0)$$ for $F$ such that the ...
3
votes
1answer
50 views

Proof of Stirling's Formula using Trapezoid rule and Wallis Product

I need a proof of stirling's formula which uses the riemann's sum and trapezoid approximation to come up with $ \frac {n!}{(n/e)^n \sqrt n}$ $ \rightarrow C$ where $C$ is derived from Wallis product. ...
0
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0answers
9 views

Approximation for $(p_1^2 + (1-p_1)^2)^n$ where $p_1 = 1-(1-(k/n))^N$

I am looking for an approximation for $(p_1^2 + (1-p_1)^2)^n$ where $p_1 = 1-(1-(k/n))^N$. We can assume that $n$ and $N$ are large and that $k < n < N$. Is there a simple approximation for ...
0
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0answers
27 views

How to remap continued fractions from $\mathbb{R}$ to a discrete set of integers

Assuming that I have a continuous fraction \begin{equation} x = a_0 + k_1 \cfrac{x_1}{a_1 + k_2 \cfrac{x_2}{a_2 + k_3 \cfrac{x_3}{a_3 + k_4 \cfrac{x_4}{a_4 \ddots } } } } \end{...
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0answers
16 views

Approximation of non-differentiable optimization problems with max function

The book by D. Bertsekas "Constrained optimization and Lagrange multiplier methods", Ch. 5.1.3 describes at p. 312 a method that is used to solve non-differentiable optimization problems by ...
0
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1answer
41 views

Comparing $f(x)$ and the $f\_approx(x)$ in Matlab, Octave or Mathematica/Wolfram

I have a function $f(x)$ and I wrote the approximation for $f(x)$ as $f\_approx(x)$ which is a simple algebraic formula with sums and products . I now want to study the 2 functions side by side and ...
0
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0answers
41 views

Getting the DFT of irregularly spaced points

I am trying to estimate the discrete Fourier transform of a discrete surface, $x:\{1,\dots,N\}\times \{1,\dots,N\} \to\mathbf{R}$, given a sparse set of samples on the grid. If we had all the ...
0
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1answer
18 views

Bin packing approximation algorithm

I know that bin packing cannot be solved in $\mathrm P$ unless $\mathrm P=\mathrm{NP}$, because we could solve partition problem. However, I do not see why this theorem is a collorary. There is ...
0
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1answer
47 views

Analysis of bisection search

http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-00sc-introduction-to-computer-science-and-programming-spring-2011/unit-1/lecture-3-problem-solving/ In the following video i'm ...
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9answers
265 views

How do I prove that $\sqrt{20+\sqrt{20+\sqrt{20}}}-\sqrt{20-\sqrt{20-\sqrt{20}}} \approx 1$

How do I prove that $$\sqrt{20+\sqrt{20+\sqrt{20}}}-\sqrt{20-\sqrt{20-\sqrt{20}}} \approx 1$$ without using the calculator?
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0answers
22 views

Aproximation for the variance (sum)

Given that we know The mean of a population $\tilde W(t) = \sum_{i=1}^{n}f_{i}(t)*W_{i}$ The variance of the population in the previous step $Var(0) = \sum_{i=1}^{n}f_{i}(0)*(W_{i}-\tilde W(0) )$ ...
1
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2answers
112 views

Compute $\lim_{n\to\infty} n \bigg( \sum_{k=0}^n f(\frac{k}{n}) - n\int_0^1 f(x) \, dx \bigg)$

The task is to show the following limit exists, and then compute it. Here, $\,\mathrm{f}:\left[0,1\right] \to \mathbb{R}$ is a continuously differentiable function. $$ \lim_{n \to \infty}\left\{n\...
1
vote
1answer
69 views

Sequence of polynomials converging to zero, but not uniformly on unit disc

I have been trying to solve the following without success so far: Show that there exists a sequence of polynomials satisfying $P_n(z)\rightarrow 0$ for every $z\in \mathbb{C}$, but the convergence is ...
0
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1answer
26 views

Calculating approximate growth from three numbers [closed]

I have a set of three numbers 3600, 5200,12000; how do I calculate an approximate 4th number ...
2
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1answer
37 views

A square-root approximation method that would halt on $\sqrt{378}$

Back in the early $'90$s, I used to program in a (now obsolete) scripting language called LOGO. Now, one peculiar glitch that I encountered at the time, was the interpreter halting on $\sqrt{378}$. ...
0
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1answer
20 views

Approximate perfect matching through MST

If I compute a minimum spanning tree T in a graph with an even number of vertices and T contains a perfect matching M (which is unique in this case), can I get some approximation guarantee on the ...
0
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0answers
29 views

On the inverse of the regularized upper incomplete gamma function

I'm interested to bound/approximate the the inverse of the regularized upper incomplete gamma function $Q^{-1}(a,z)$, where $Q(a,z) = \frac{\int_z^\infty t^{a-1} e^{-t} \mathrm{d} t}{\Gamma(a)} $. I ...
2
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0answers
16 views

Central Limit Theorem Heuristics

Surrounding the central limit theorem there exist several heuristics which say when a normal distribution is a reasonable approximation to the mean $\frac{X_1 + \cdots + X_N}{N}$ of $N$ independent (...
3
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1answer
56 views

Asymptotic vlaue of $ f(n)=\sum_{i=0}^n\lfloor \sqrt{i}\rfloor\binom{n}{i} $

Inspired by this question I tried to find an asymptotic formula for $$ f(n)=\sum_{i=0}^n\lfloor \sqrt{i}\rfloor\binom{n}{i} $$ With the observation: $$ f(n)=\sum_{i=0}^n\frac{\lfloor \sqrt{i}\rfloor+\...
1
vote
1answer
31 views

How to use Taylor's Theorem to obtain an upper bound for an error approximation

$e \approx 1 + 1 + \frac{1^2}{2!} + \frac{1^3}{3!} + \frac{1^4}{4!} + \frac{1^5}{5!}$ must find upper bound for this but I don't see what I should be doing. The remainder/error is given by $\frac{f^{n+...
2
votes
1answer
150 views

How does one show that $\cos {\left (\ln 2 \right )}\approx \frac{10}{13}$?

How does one approximate the value of something like this? Apparently Euler found the value of $\large \frac{2^i+2^{-i}}{2}\large $ [which equals $\cos {\left (\ln 2 \right )}$] to be close to $\...
2
votes
1answer
44 views

Maxima of $f(x)/e^x$ where $f(x)$ is an approximation of $e^x$ using Stirling's

Let $$f(x)=1+\sum_{n=1}^\infty\frac{x^n}{\sqrt{2\pi n}(n/e)^n}\tag1$$ and let $$g(x)=\frac{f(x)}{e^x}\tag2$$ If we plot $g(x)$ we get a graph that looks like this: Clearly there is a maximum at ...
2
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0answers
24 views

(Conceptual_Calculus) Differential Conditions v. Derivative Conditions

I have few questions regarding the reason we learn about the differential conditions in higher dimensions and dealing with multivariable calculus. In the context of optimization (e.g. finding ...
0
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0answers
21 views

How can we use the Lindley's method to approximate the following expression?

The Lindley's(1980) approximation is one of the most popular methods that is used to obtain Bayes estimates. In this method we need to maximum likelihood estimators(MLEs) of the unknown parameters. ...
-1
votes
1answer
26 views

Iteration: Approximation and Errors, finding all possible iterative arrangements

I am looking at a relatively simple problem to reiterate: $x^4=e^x$ I've found 5 different possible forms 1: $x_{r+1}=\frac{e^x}{x^3}$ 2: $x_{r+1}=(\frac{e^x}{x^2})^{0.5}$ 3: $x_{r+1}=(\frac{e^x}...
0
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0answers
15 views

Approximating $\chi_0$ pointwise with holomorphic functions

Define $\chi_0:\mathbb{C}\to \mathbb{C}$ as $$ \chi_0(z)=\left\{ \begin{gathered} 1 \quad z=0\hfill \\ 0 \quad z\ne 0 \hfill \\ \end{gathered} \right.$$ Does there exist a sequence of ...
0
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1answer
57 views

(Terminology_Taylor Series) “expand at $x_0$, evaluate at x, affine approximation”

I am reading one-variable calculus book where it explains Taylor series and little confused with the following terms: (1) Expand $f(x)$ at $x_0$ (2) Evaluate $f(x)$ at x (3) Best Affine, ...
21
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7answers
4k views

How to show this formula to get a square root of a number in “just few seconds” is true?

I don't remember in which topic I found it but I know it was there. And I still have not find a proof of this nice approximation. Let $x$ be a non perfect square number. If $y$ is the closer ...
2
votes
1answer
63 views

Taylor Series for a Function of $3$ Variables

The Taylor expansion of the function $f(x,y)$ is: \begin{equation} f(x+u,y+v) \approx f(x,y) + u \frac{\partial f (x,y)}{\partial x}+v \frac{\partial f (x,y)}{\partial y} + uv \frac{\partial^2 f (x,y)...
1
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2answers
42 views

Partial Derivatives Approximation

By definition we know the following: \begin{equation} \frac{\partial f(x,y)}{\partial x} \approx \frac {f(x+ \delta x,y)-f(x,y)}{\delta x} \end{equation} \begin{equation} \frac{\partial f(x,y)}{\...
2
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0answers
29 views

Chebyshev's inequality and quadratic function

I am trying to use Chebyshev's inequality in order to find sample sizes $n$ such that some condition is met with probability $p$ or larger. That is, find $n\in \mathbb{N}$ such that $\mathbb{P}(X_n &...
0
votes
1answer
23 views

finding a close enough point with implicit function theorem

Let $a_1, ...,a_n ,B\in \mathbb{R}^n$ , not all on the same plane. Prove that for a small enough neighborhood of zero $U$ and $\forall u_1,...,u_n \in U $ there is a point $C \in \mathbb{R}^n$ that $...
0
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0answers
25 views

Formal approximation for second-order ODE with varying coefficients

I have a differential equation of the form $$0=a+by(x)+cf(x)+z(x)f''(x)$$where the functions $y$ and $z$ are known and we want to find $f$. If $z$ is constant, i.e. $z(x)=Z$, it is straightforward to ...
2
votes
3answers
73 views

Finding an approximation of a function's root

I have the polynomial function $f (x) = x^5+2x^2+1$. I am trying to find an approximation to its root in $[-2,-1]$, with the precision of $0.1$, and with a minimal number of steps. The answer I was ...
0
votes
1answer
32 views

Behaviour of the Spectral Weight Function $\frac{\sin^2{(\pi f t)}}{(\pi f)^2}$

I'm looking into the properties of the so called spectral weight function $W_0 = \frac{\sin^2{(\pi f t)}}{(\pi f)^2}$. While not important for the question, this function is is encountered in the ...
0
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0answers
35 views

Maximum of Contour Line

Consider the potential $U(x,y)=ay^{2}+b(e^{x-y}-1)^{2}+c(e^{x+y}-1)^{2}$ where $a$, $b$ and $c$ are known constants. I want to move through a contour line of this potential $U(x,y)=k$, say $y=g(x)$. ...
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0answers
11 views

Estimate the drift and diffusion function numerically

I have a 1D problem as following $$\frac{\partial f}{\partial t} = \frac{\partial}{\partial x} \Big[ \frac{1}{2} \frac{\partial (g(x) f)}{\partial x} -\mu(x)f \Big]$$ I have a time-series of ...
0
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0answers
41 views

Fitting a continuous curve over a piecewise constant data

I have some measurements that are piecewise constant over a certain variable. For example, in the following image, the vertical axis represents the measurement data and the variable is on the ...
1
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1answer
39 views

Normal Approximation - how many bookings so probability for “overbooking” stays under certain value

I need some help with the following: A hotel has $r$ rooms. The probability that a guest who booked a room also appears (which means: no cancellation) is $p = 0.9$. I'd like to know how many rooms ...
0
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1answer
95 views

What is the value of $e^{-10000}$?

What is the value of $e^{-10000}$? We know that the function $e$ does not attain value $0$ anymore. But in R and Matlab the value of $e^{-10000}$ is given as $0$ which is not correct anymore. I ...
2
votes
3answers
108 views

Why does $(\sin x)^2=x^2$ and $\sin x=x$ in these contexts?

Contexts (it must also be noted that as $\delta t$ tends to zero, $\delta \theta$ also tends to zero): First context $$ \lim_{\delta t \to 0} \frac{-2v\sin(\delta\theta/2)^2}{\delta t} = \lim_{\delta ...
0
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0answers
75 views

What is the field of mathematics that describes the transition into statistical mechanics?

There are interesting changes that occur in a sample of interacting objects, such as gas particles, as you approach a statistically significant sample. The position or velocity of any given particle ...
0
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1answer
25 views

Central Limit Theorem and Normal Approximation

having started 'learning' all that is related to the Central Limit Theorem just one day ago, I am already a bit confused - maybe you can help me seeing through the cloud of misunderstanding. Let's ...
1
vote
1answer
52 views

When is $1-(1-p)^n \sim pn$

Let $0<p=p(n)<1$ with $p=o(1)$. For which $p$ is it true that $1-(1-p)^n \sim pn$? With $\sim$ I mean that they are asymptotically the same, so $\frac{1-(1-p)^n}{pn}\rightarrow 1$, or at least ...
0
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0answers
22 views

Spherical Wave Approximation

Lets suppose i have $ K $ data points $(r_i,\phi_i,\theta_i,p_i)$ and i want to approximate my data points with the following function: $$p(r,\theta,\phi) = \sum_{n=0}^{N} \sum_{m=-n}^{n} c_{n,m}h_n^{...
0
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0answers
33 views

The approximation formula $\left|\alpha -\frac{p}{q}\right| \le \frac{1}{\sqrt{5}q^2}$

I have seen a result about the approximation of irrational numbers and want to find its proof. Suppose $\alpha$ is an irrational number, then there are infinitely many integers $p,q$ with $(p,q)=...
1
vote
0answers
34 views

A geometric proof for the “small angle approximation” for the sine, cosine and tangent

How can be deduced the so called "small angle approximations" for sine, cosine and tangent, namely $\sin \theta \approx \theta$ $\tan\theta \approx \theta$ $\cos\theta \approx 1-\frac{\theta^2}{2}$ ...
2
votes
1answer
103 views

Calculating large exponential probabilities

Earlier today there was Youtube video attempting to solve a problem for a certain game. In it he tries to calculate the probability of certain events happening which narrows down to this equation: $P(...
0
votes
2answers
46 views

Perturbation: compute an approximation to the solution of the equation $y+\epsilon\sin y=x^2$

Compute approximation to the solution of the equation $y+\epsilon \sin y=x^2$ using perturbation method. Assume that terms involving powers of $\epsilon$ of order 3 or more can be ignored. So far I ...