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Questions tagged [approximation]

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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Low-dimension matrix approximation?

I have a matrix $\mathbf E_1 \in \mathbb R^{m\times n_1}$. I would like to approximate it with a matrix $\mathbf E_2 \in \mathbb R^{m\times n_2}$ with $n_2 < n_1$, where "approximate" ...
Mew's user avatar
  • 347
1 vote
1 answer
57 views

Is the reciprocal golden ratio well approximated by this exponentially sparse series of reciprocal Fibonacci numbers?

Let $1/\phi= \phi-1\approx0.618\,$ denote the reciprocal golden ratio and $\mathrm F(k)\;(k=0,1,...)$ the Fibonacci numbers, where $\mathrm F(0)=0,\mathrm F(1)=1,$ and $\mathrm F(k+1)=\mathrm F(k)+\...
John Bentin's user avatar
5 votes
1 answer
298 views

Is there a reason why the Maclaurin coefficients of the Riemann zeta function are asymptotically close to -1?

If I look at the numerical values of the Maclaurin series of the Riemann zeta function I see that they approach -1 extremely quickly. In fact, if I take $\zeta(x)=\sum_{n=0}^\infty a_n x^n$ then ...
Jean Du Plessis's user avatar
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0 answers
9 views

Question About Approximation in Formula For cGMP Synthesis

In this paper, it has a formula. $\alpha \propto \frac{1}{1 + (\frac{C}{K_{Ca}})^m}$, Where $\alpha$ is the rate of synthesis of the molecule cGMP, $C$ is the calcium concentration, and $K_{Ca}$ is ...
MeltedStatementRecognizing's user avatar
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0 answers
37 views
+50

Sobolev functions approximated by ridge functions

Let $f \in W^{k,2}(\mathbb{R}^d)$, a Sobolev space with smoothness $k$ and dimension $d$. We aim to approximate $f$ using ridge functions of the form $g(\mathbf{a}.\mathbf{x})$. Suppose the ...
Maths Freak's user avatar
0 votes
0 answers
23 views

Approximation of entropy of binomial distribution

The approximation of entropy of binomial distribution is: $$\frac1 2 \log_2 \big( 2\pi e\, np(1-p) \big) + O \left( \frac{1}{n} \right)$$ Based on my understanding, this approximation is for large n ...
MarcG's user avatar
  • 11
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0 answers
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Approximation or an analytical form of integrals

I'm working on a bayesian rule of wavelet shrinkage using the raised cosine distribution as a prior distribution for $\theta$. Its expression is given by: $$\delta(d) = \frac{(1-\alpha) \int_{\frac{-\...
juliana marchesi's user avatar
1 vote
1 answer
27 views

Local Linear Fit

What do you call it when you have a set of known $(x,y)$ data points and you estimate a $y$-value for a given $x$-value by performing a linear fit between its two known neighbor $(x,y)$ points? As an ...
user1228123's user avatar
3 votes
1 answer
93 views

Approximating the Prime Counting Function as $\pi(x) \approx \frac{x^2}{\ln\left(\Gamma(x+1)\right)}$

Approximating the Prime Counting Function as $\boxed{\pi(x) \approx \frac{x^2}{\ln\left(\Gamma(x+1)\right)}}$ Intro________________ In a unrelated topic I was viewing how the mechanical statistics ...
Joako's user avatar
  • 1,596
2 votes
1 answer
100 views

Adequate Root Finder To Compute The Quantile Function

The cumulative distribution function of the standard normal distribution $\Phi(z)=\displaystyle\frac{1}{\sqrt{2\pi}}\int_{-\infty}^z e^{-t^2/2}dt$ cannot be expressed in terms of elementary functions, ...
m-stgt's user avatar
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1 answer
52 views

Is every compact set equal to an intersection of nested bounded sets with smooth boundary?

Let $K$ be a compact set of (say) the plane $\mathbb{R}^{2}$. Do there exist bounded open sets $(U_{n})_{n\in\mathbb{N}}$ with $C^\infty$-boundary $\partial U_{n}$ such that $$\ldots\subseteq U_{n+1}\...
Calculix's user avatar
  • 3,386
0 votes
1 answer
39 views

Algorithm to compute a convolution recursively

Let $$ f(t) = \int_0^t k(t-s)g(s) \, ds. $$ Assume that $g$ is only given in a grid $t_j = j\delta_t$, and that we wish to compute similarly $f$ on the same grid. What's an efficient algorithm to ...
G. Gare's user avatar
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-4 votes
2 answers
91 views

Approximation for $n!$ using $\sqrt{2 \pi n}$

For Stirling's Approximation , I am using to seeing this written in the following notation: $$\ln(n!) = n(\ln(n)) - n$$ But recently, I saw this version (e.g. https://www.youtube.com/watch?v=...
wulasa's user avatar
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0 answers
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Can't confirm the descent direction at a point

Consider the problem \begin{equation} \underset{\mathbb{R}^2}{\text{min}} f (x) = \frac12x_1^4 + 2x_2^4 + x_1^2 - x_1 x_2 + x_2^2 . \end{equation} Suppose that the function $f$ is minimized starting ...
Superunknown's user avatar
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1 vote
0 answers
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Divergent Tail Sums of Approximations of Non-trace Class Compact Operators

I'm working on approximations of compact operators that are not trace class, and I'm looking for ways to provide meaningful approximation error estimates for truncated eigenfunction expansions. I ...
user avatar
3 votes
2 answers
91 views

$f=0$ on $\partial\Omega$ implies $f\in H_0^1(\Omega)$

Let $\Omega\subset\mathbb{R}^n$ be an open set. Let $f\in C(\bar{\Omega})\cap H^1(\Omega)$ with $f=0$ on $\partial\Omega$. Claim: Then $f\in H_0^1(\Omega)$ holds. Since $H_0^1(\Omega)$ is the closure ...
MaxwellDgt's user avatar
0 votes
0 answers
15 views

Differentiable of Moreau-Yosida

i read the next paper: "practical aspects of the Moreau-Yosida regulrization I: theoretical properties and the introduction" and in the first part it claims that for a given function f and ...
Chen's user avatar
  • 49
1 vote
0 answers
62 views

Estimation of a sum by an integral in Shor's algorithm

I am reading Peter W. Shor's original paper on his algorithm for integer factorisation. On p.17 of the paper he is trying estimate a sum by an integral. He claimed that the expression inside the ...
eurekamath's user avatar
1 vote
1 answer
177 views

Proving the high precision of the series correct to at least half a billion digits

I recently learned about this high-precision series. It is claimed that it is correct to at least half a billion digits. I am curious to know how it works. $$ \sum_{n=1}^\infty\frac{\left\lfloor n e^{\...
Pustam Raut's user avatar
  • 2,302
0 votes
0 answers
33 views

Approximating expression from Meiser, Vogts book on Stone-Weierstraß with Stirlings formula

I am trying to understand proof of Stone-Weierstraß in Meiser and Dietmar Vogts book "Funktionalanalysis". In one of his lemmas (page 22), he draws the conclusion, which I can't follow: $$\...
JohnBlue's user avatar
1 vote
0 answers
123 views

The connection between $\pi$, $e$ and $20$ [closed]

It's well documented that $e^{\pi} \approx 20+\pi$. This can be explained using the following series: $$\sum\limits_{k=1}^{\infty}\frac{8\pi k^{2}-2}{e^{\pi k^{2}}} = 1$$ The series is quickly ...
Darmani V's user avatar
  • 290
-1 votes
0 answers
40 views

Differential Approximation of $f(x,y)\:=\:x\cdot \:e^{x^2-y^2}$ - Multivariable Calculus Problem

I have a differential approximation of a multivariable calculus problem of my academy that I can't figure out. Here's the question: Given the result of the calculation of $3.02\cdot e^{3.02^2-2.9^2}$, ...
Yuval Yanay's user avatar
1 vote
1 answer
48 views

What are the Chebyshev sets for the taxicab metric?

A set of points $S \subseteq \mathbb{R}^n$ is called a Chebyshev set if the metric projection w.r.t $S$ is single-valued. That is, for every point $x\in \mathbb{R}^n$, there is a unique point $y\in S$ ...
Erel Segal-Halevi's user avatar
1 vote
2 answers
58 views

Solving $\frac{M e^{-M}}{1-e^{-1}} - \epsilon = 0$ for $M \in \mathbb{R}$

I am trying to solve the following nonlinear equation analytically: $$ \frac{M e^{-M}}{1-e^{-1}} - \epsilon = 0 \, , $$ where $ M \in \mathbb{R} $ and $ 0 < \epsilon \ll 1 $. A solution can be ...
Siegfriedenberghofen's user avatar
0 votes
1 answer
33 views

What is the definition of and quantifier order in consistency for one-step methods?

Consider the usual Cauchy IVP $y = f(x, y)$ with $y(x_0) = y_0$, satisfying assumptions of Picard's theorem in a rectangle $D$. A one-step method is essentially $$y_{n+1} = y_{n} + h \Phi(x_n, y_n; h),...
Linear Christmas's user avatar
0 votes
1 answer
59 views

Approximating asymptotically the Laplace inverse of $\frac{\exp\left(-\sqrt{s^2 + 1}\right)}{\sqrt{s^2 + 1}}$ for larger t

I was trying to find the behavior of the inverse laplace transform of the function $$F(s)=\frac{\exp\left(-\sqrt{s^2 + 1}\right)}{\sqrt{s^2 + 1}}$$ for larger $t$. So basically here is my approach: I ...
MB17's user avatar
  • 173
1 vote
0 answers
43 views

Sum of a Normal and a Chi2 distribution

I am doing some Monte Carlo exercises, and I have to derive the distribution of the following equation: $y= x + x \times u$ where $u\sim T_k$ is a Student $t$ with $k$ d.f. and $x\sim \sqrt{\frac{\chi^...
andrea's user avatar
  • 11
3 votes
1 answer
104 views

Question About Small Angle Approximation

I'm currently going through "An Introduction to Mechanics" by Kleppner and Kolenkow. On page 38 of the text, he glosses over the small-angle approximation of $sin(x)$ and $cos(x)$. ...
Ethan Chan's user avatar
  • 2,292
0 votes
0 answers
21 views

How to solve a paths optimization problem in node-labeled and weighted graphs

I am having trouble finding a way to optimize this problem in Python. Even if it is not exact, I am trying some heuristics. Let $G = (V,\mathbf{L}, E,\mathbf{W})$ be a graph where $V$ is the set of ...
The Bosco's user avatar
  • 1,965
1 vote
0 answers
28 views

Numerical Computation of the Gamma Function for large complex numbers

I'm looking for a method to numerically compute the Gamma function $Γ(z)$ for complex numbers of the form $$z= \frac{1}{2} + it,$$ particularly for large values of $t$. Does anyone know of any ...
Felipe Oliveira's user avatar
4 votes
2 answers
110 views

pow and its relative error

Investigating the floating-point implementation of the $\operatorname{pow}(x,b)=x^b$ with $x,b\in\Bbb R$ in some library implementations, I found that some pow ...
emacs drives me nuts's user avatar
0 votes
1 answer
33 views

Natural Log's Property Doesn't Transfer Over

I am trying to rewrite the summation of $\ln(x)$ equation into a continuous function using logarithmic properties. We already know that $\left(\sum_{n=1}^{x}\ln\left(n\right)\right)$ is just equal to $...
Monke's user avatar
  • 1
2 votes
0 answers
110 views

How to calculate an upper bound for $\sum_{k=0}^n \binom{n}{k} \frac{(-x)^k}{\sqrt{k+a}}$

As the title mentioned, I want to get a closed-form result of \begin{equation} \sum_{k=0}^n \binom{n}{k} \frac{(-x)^k}{\sqrt{k+a}}, \end{equation} where $x\in[0,1]$ is a real number, and $a$ is a ...
Jobs Adam's user avatar
  • 243
0 votes
2 answers
36 views

Logarithm approximation loses solution

I was doing an exercise in which I had to plot by hand a function. The function was $$f(x)=\ln\left(\frac{\sqrt[3]{x}}{3x - 1}\right)$$ which I rewrote into $$f(x)=\frac{\ln\left(x\right)}{3} - \ln\...
serax's user avatar
  • 135
0 votes
0 answers
46 views

How to approximate the integral $\int^{\infty}_0 x^{2a}\cos{(c_0 x^a)}\exp{(- c_1 x^{2a} - x)} dx$

For the integral $$ \int^{\infty}_{0}x^{2a}\cos\left(c_{0}x^{a}\right) \exp\left(-c_{1}x^{2a} - x\right)\,{\rm d}x $$ that is ...
LOREY CHU's user avatar
1 vote
1 answer
35 views

The operator norm regarding to the difference between a mollified function and the function itself

Let $\rho_\epsilon$ be a mollifier that has support in $B(0,\epsilon)$, define the operator $$T_\epsilon (f)=\rho_\epsilon*f-f,$$ for every $f\in L^2(\mathbb R^d)$, can we prove or disprove that the ...
Euler's little helper's user avatar
10 votes
4 answers
838 views

Approximating the length of a circular arc using geometrical construction. How does it work?

I was going through my Engineering Drawing textbook and came upon this topic. Using only a compass and a straightedge, one can supposedly approximate the length of a given circular arc by following ...
Aayush Dhungana's user avatar
8 votes
1 answer
276 views

Approximating $\log x$ by a sum of power functions $a x^b$

Let's approximate $\log x$ on the interval $(0,1)$ by a power function $a x^b$ to minimize the integral of the squared difference $$\delta_0(a,b)=\int_0^1\left(\log x-a x^b\right)^2dx.\tag1$$ It's ...
Vladimir Reshetnikov's user avatar
8 votes
6 answers
479 views

Approximate solution of $x^x=(x-n)^{(x+n)}$

Interested by this problem which ask for the solution of $$f_n(x)=x^x-(x-n)^{x+n}$$ that I rewrote as $$g(x)=x\log(x)-(x+n)\log(x-n)$$ After two series expansions, I obtained, for large $n$, as a very ...
Claude Leibovici's user avatar
3 votes
1 answer
95 views

rational complex minmax

As the title says, I wanted to understand the following part of a proof in Guttel (Corollary 2, page 6). $f$ is an analytic function and $r_m \in \mathcal{P}_{m-1}/q_{m-1}$ is a rational function with ...
jacopoburelli's user avatar
0 votes
0 answers
28 views

What is the maximum area that can be enclosed in a polygon formed by n wires? [duplicate]

The problem is that we have n wires of different lengths, i.e. $w_1,w_2,...,w_n$. The wires are aligned in a way such that they enclose the maximum area. What is that maximum area, or its best ...
Panda's user avatar
  • 101
1 vote
0 answers
27 views

Knapsack with fixed number of bins?

Constant: d, a fixed number of bins/sacks Input: $v_1,v_2,...,v_n$ item profits, $0<w_1,w_2,...,w_n\leq1$ item weights. Output: $B_1,B_2,...,B_d$ which are d subsets of $\{1,2,...,n\}$ s.t. they ...
alon's user avatar
  • 11
0 votes
0 answers
29 views

Asymptotic of fourier transform of the probability distribution of a stochastic process with finite jump

I'm trying to model an ensemble of particle that constantly feel a viscous drag $\gamma$ and for which, at a exponential rate $\mu$, two of them gain a velocity $\delta$ and $-\delta$. The evolution ...
Syrocco's user avatar
  • 243
0 votes
0 answers
27 views

Approximating binomial with normal distribution

You wanted to estimate how many packages were damaged in a shipment of $5000$ packages. You randomly selected $300$ packages and found that 16 of them were damaged. Determine the lower bound of the ...
Need_MathHelp's user avatar
0 votes
0 answers
84 views

Good approximation of $\sin(x)^5$ to use for ODE?

I need to find an approximate solution of $x'(t)= - \sin(x)^5$ with $x \in [0, \pi]$. I know there's no explicit solution, but I wonder if there are good approximations (whatever that means, let's say ...
tommy1996q's user avatar
  • 3,366
0 votes
1 answer
42 views

An explicit formula for orthogonal functions

I am interested in orthogonal functions for the inner product $$\int_a^b f(x)g(x) \alpha(x) dx$$ where $\alpha$ is a non-negative function. Given linearly independant functions $f_0, \ldots, f_\ell$, ...
Wirdspan's user avatar
  • 567
0 votes
0 answers
10 views

Approximation for the generalized First-Order Marcum Q-Function

I want to approximate the first-order Marcum Q-function $Q_{1}(a,b)$. However, the goal is to obtain an acceptable approximation for any values of $a$ and $b$. Is there perhaps a possible solution for ...
Henry's user avatar
  • 85
0 votes
1 answer
43 views

Correctly understand the implication of approximation ratio for the set cover problem?

I am currently reading this wikipedia article about the set cover problem and it said here that "it cannot be approximated to $\left[ {1 - o\left( 1 \right)} \right]\ln \left( n \right)$ unless $...
Tuong Nguyen Minh's user avatar
1 vote
1 answer
106 views

Assumptions made in Liouville-Green Approximation

Question: Consider $y'' + \frac{1-x}{x^3} y = 0$ Derive the first three terms in Liouville Green approximation. Hence obtain corresponding asymptotic approximations as x → ∞ of two linearly ...
vegetandy's user avatar
  • 305
1 vote
2 answers
63 views

Approximating $\#\{\pi \in S_n:\exists k,\pi(k)=k,~k\leq m\}$

Consider the permutations from $S_n$ and let $$ K(n,m)=\#\{\pi \in S_n:\exists k,\pi(k)=k,~k\leq m\} $$ be the number of such permutations which have a fixed point in the first $m$ positions. I am ...
kodlu's user avatar
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