For questions related to approximations

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Is $\frac{x^2(1-p)}{(n-x+1)p}\leq \frac{x^2}{(n-x)p}=O((np)^{-1})=o(1)$?

Is it true that for $n \rightarrow \infty$, $p \gg n^{-1}$, $0<p<1$ and $x=O(1)$, $$\frac{x^2(1-p)}{(n-x+1)p}\leq \frac{x^2}{(n-x)p}=O((np)^{-1})=o(1)?$$
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31 views

Function Approximation

I need to solve the following equation $$-\frac{\partial S(x,y,t)}{\partial t}=ax^2+bx\frac{\partial S(x,y,t)}{\partial x}+c\Big[\frac{\partial S(x,y,t)}{\partial ...
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19 views

Approximating real vectors in the unit hypercube by rational vectors in the unit hypercube

I am currently looking for an error bound for an problem where I have to approximate a real vector in the unit hypercube using a rational vector in the unit hypercube. Specifically: given a ...
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2answers
89 views

What is the sum that the square root button on calculator does so I can do it without the calculator button [duplicate]

I am not very good when it comes to Maths but the current work I am doing means I need to get better and quick. I have been teaching myself about areas, diagonals and square roots. However I am ...
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1answer
41 views

Understanding approximation $\sqrt{\frac{z}{z-2h}} \approx 1 + \frac{h}{z}$, $h>0, |z| \gg h.$

I am having troubles showing the above step regarding the approximation $$ \sqrt{\frac{z}{z-2h}} - \sqrt{\frac{z}{z+2h}} \approx \frac{2h}{z}, \quad h>0, |z| \gg h$$ given in an old exam. My ...
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2answers
36 views

Monotonically approximate $L^p$ function by step function

It is a classical fact that a $L^p(R^d)$ ($1\leq p<+\infty$) function can be approximated by step functions with compact support, but my question will be, can we require that the step function is ...
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58 views

Approximating a function $f(x)$ by an power function $x^n$ [on hold]

Let $x>0$ be a positive integer. Let $f(x) \leq x$ be any function such that $f(x)$ is an integer also. I am looking for a power function that approximates $f(x)$. That is, find a real $0 \leq n ...
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1answer
53 views

Meaning of $\sim$?

I often read $f(x) \sim g(x)$ and I wonder what the Standard Interpretation of this $\sim$ is. It seems to mean something like asymptotically equally distributed, something like $f(x)=g(x)(1+o(1))$. ...
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70 views

How to approximate large sum of exponential variables

Is there any way to approximate the following sum: $$ \sum_{i_1=1}^N \sum_{i_2=1}^N \cdots \sum_{i_k=1}^N \cdots \sum_{i_N=1}^N \exp(-r_{i_1} - r_{i_{k+1}} - r_{i_{2k+1}} - r_{i_{3k+1}} \cdots - ...
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2answers
52 views

Newton Raphson Method for double roots

I am currently working on Newton Raphson Method. I am kind of facing a problem how Newton Raphson Method work on more than second order quadratic functions with double roots. I have googled and ...
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1answer
31 views

A better approximation of $({n \over e})^n$ than by $ \Gamma(1+n-1/2)$ ? (Focus is “reverse” to the Stirling approximation)

In a formula in my self-study of a summation method based on the matrix of Eulerian numbers (which I thus call "Eulerian summation") I am considering terms like $$ \Big({n \over e}\Big)^n \cdot {1 ...
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1answer
24 views

Approximate a surface by set of points

Given a set of points $(X, Y, Z)$ obtained from the experimental data that can be considered as a 3D surface. What is the common approach to get an approximating function Z=f(x,y) that describes the ...
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1answer
32 views

Derivatives as Linear Approximations

I have always thought of the fact that a derivative is a linear approximation as being nothing more than that- an approximation. But is there an epsilon-delta meaning behind that? Is there a stronger ...
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1answer
21 views

How to approximate the bounding region of a 2d differentiable mapping locally?

I have got a differentiable mapping $f:\Bbb R^2 \to \Bbb R^2$, Is the image of $f$ of a very small convex subset (e.g., a unit square) around any point, a bounded region? If it is bounded, can I ...
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29 views

Computer Algebra Systems for Experimental Mathematics (especially Integer Relations with PSLQ)

I would like to use a computer algebra system to do some experimental mathematics, particularly Integer Relation problems using the PSLQ algorithm. I know that Maple has a PSLQ implementation, but ...
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11 views

spatial-partitioning based physical simulation

I've learnt that spatial-partitioning based physical simulation is kind of "approximate" computation. Is it because: since the whole space is partitioned into cells, and only the interactions of ...
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1answer
24 views

Approximating error using Taylors theorem

I have used a Maclaurin series for the function $f(x) = \cos(2x)$ and have successfully produced: $\dfrac{2^n cos(\frac{n\pi}2)x^n}{n!}$ Now I want to estimate the error in approximating $\cos(2x)$ ...
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2answers
106 views

Using series find $\int_0^1 \sqrt{1+x^4}\hspace{1mm} dx$ up to $2$ decimal places

I cannot figure out an aesthetic way to do this. Can someone give a beautiful solution to this ugly question? This is what I have tried yet. I used the fact that $$x = ...
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0answers
13 views

Is it possible to approximate or bound this nonlinear mapping L h(x)?

I wanted to make sure whether the following approximation is possible. I have a vector ${\bf{x}} \in \mathbb{R}^N$ where the elements of the vector $\bf{x}$ are random variables. There is a matrix ...
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14 views

Using LLL to get approximate rational representations of numbers

Does anyone understand how is the LLL algorithm implemented to obtained the values of $(x,y)$ for approximating $\pi$ in this portion of text?
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1answer
188 views

Approximate value of a slowly-converging sum of $\sum|\sin n|^n/n$

In this question on Math.SE there appears this sum: $$ S = \sum_{n\geq1}s_n, \qquad s_n = \frac{|\sin n|^n}{n}, $$ which converges very slowly. What methods would you suggest for evaluating it ...
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Nonnegative Matrix Factorization in Machine Learning

I have a matrix $X^{m\times n}$, and I need to factorize it into $W^{m \times p}H^{p \times n}$, $p$ is the number of factors s.t. $p << min(m,n)$, and $m$ is the number of variables while $n$ ...
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0answers
38 views

Application of Weierstrass approximation theorem

How to approximate a continuous function to a desired accuracy using a polynomial? Theorem: For any $\varepsilon > 0$ and $f \in C([a,b])$, there exists a polynomial $p$ such that $\sup_{x \in ...
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31 views

Help understanding this approximation

In a paper that I'm reading, the authors write:- $$N_e \approx \frac{3}{4} (e^{-y}+y)-1.04. \tag{4.31}$$ Now, an analytic approximation can be obtained by using the expansion with respect ...
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6answers
174 views

When the approximation $\pi\simeq 3.14$ is NOT sufficent

It's common at schools to use $3.14$ as an appropriate approximation of $\pi$. However, here it's stated that for some purposes, $\pi$ should be approximated to $32$ decimal places. I need an example ...
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2answers
32 views

Poisson approximation to binomial distribution: $f(x)\geq g(x)$ or $f(x) \leq g(x)$

We have a random variable $X$ which has a Binomial Distribution Bin(n,p) and a random variable $Y$ which has a Poisson Distribution Poisson(np). We are interested in $$f(x):=Pr[X \geq x].$$ For ...
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1answer
50 views

Double Integrals & Expected Value Monte Carlo Method

Tell me if I'm wrong Let $\Omega = [a,b]\times[c,d]\subseteq\mathbb{R}^2$, then $$ \iint_\Omega ...
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1answer
42 views

integral approximation (law of large numbers)

I am totally at a loss with this question and don't even know where to begin. Let $g:[0, 1]\rightarrow \mathbb{R}$ be a measurable and Lebesgue-integrable function. $U_1, U_2, \dots$ be a series of ...
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22 views

2-approximation for TSP-metric

I got stuck with the following question: Consider the following heuristic: Start with a tour containing only one vertex. At each step, find the vertex outside the tour with the lesser distance to ...
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1answer
61 views

A good function to fit this data

I'm computing the angle of intersection between to curves (the invariant manifolds of a dynamical system). I do this with a numerical algorithm, but I would like to fit a function with this data. ...
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229 views

$\pi^4 + \pi^5 \approx e^6$ is anything special going on here?

Saw it in the news: $$(\pi^4 + \pi^5)^{\Large\frac16} \approx 2.71828180861$$ Is this just pigeon-hole? DISCUSSION: counterfeit $e$ using $\pi$'s Given enough integers and $\pi$'s we can ...
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28 views

Is the following statement on the stability of the forward Euler method true or false?

My text asks whether the following statement is true or false: The forward Euler method for approximating the solution of $x'=\lambda x$ is stable for all $\lambda \in \mathbb R$ and all step ...
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1answer
32 views

Can we approximate $f(x) = \chi_{(0,\infty)}(x)$ by smooth monotone functions?

Given $f(x) = \text{sign}^+(x) = \chi_{(0,\infty)}(x)$, is it possible to a sequence $f_n$ such that $f_n$ is smooth, $f_n' \geq 0$ and $$f_n \to f?$$ In some sense of convergence? Preferably ...
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2answers
67 views

$(1-x)^y ≈ e^{-xy}$

Here is an approximation I often see in biology articles but don't really understand: $$(1-x)^y ≈ e^{-xy}$$ I think this $e^{-xy}$ closely approximates $(1-x)^y$ whenever $x$ is small. Can you help ...
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31 views

Approximation of measurable function by simple function

So a measurable function can be approximated by a series of simple function converging pointwise. The demonstration is easily understandable by taking a series of simple function where the value of ...
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27 views

Rational approximation or series expansion of $K_0$ and $K_1$ for small z

I'm looking for a series expansion of the modified Bessel functions of second kind $K_0(z)$ and $K_1(z)$ for $$|z|<5, ~~|phase(z)| < \pi$$ My $z$ can be described as $z = a\cdot \sqrt{ix}$, ...
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1answer
124 views

Lower bound for $(x^c-1)^{1/c}$

I have been trying to find a lower bound for $x>1$, $c>0$: $$ \Large(x^c-1)^{1/c} $$ My strategy is to find a lower bound for $(x^c-1)^{1/c}$ which can hopefully get rid of some of the $c$ ...
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1answer
114 views

Approximate Periodic Function by shifting Basis Functions

Given a periodic "Target Function" $F(t)$ a set of $N$ periodic "Basis Functions" $B_i(t)$ of arbitrary shape All functions are defined on the same interval $T$. I am allowed to shift ...
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3answers
48 views

Simplify function with polynomial via least-squares

I want to "adjust" (simplify) $f(x)$, a function, by $g(x)$, a polynomial, via least-squares. I want to write code for that. Apperently my code is issuing wrong results, so I was wondering if my ...
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1answer
22 views

Show that a subset $V \subseteq C[a,b]$ is a Haar subspace

Let $C[a,b]$ be the set of continuous functions on $[a,b]$, then a linear subspace $V \subseteq C[a,b]$ of finite dimension $n+1$ is called an Haar subspace iff one of the following equivalent ...
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127 views

Product of Gamma functions II

What is the value of the product of Gamma functions \begin{align} \prod_{k=1}^{10} \Gamma\left(\frac{k}{10}\right) \end{align} and can it be shown that \begin{align} \prod_{k=1}^{20} ...
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52 views

Cebysev (Tschebyscheff-) Approximation

I have to find the best Tschebyscheff-Approximation of $x(t) = \sin(t)$ on the Interval $[0,d]$ by a line where $0 < d < 2\pi$. But I have no idea how to perform such an approximation, do you ...
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1answer
58 views

Moving average as ODE

Is it possible to represent or approximate the moving average $m(t) = \frac{1}{w}\int_{t-w}^t x(\tau) d\tau$ of a function $x(t)$ as a set of ordinary differential equations $\dot{y} = \ldots$? I am ...
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1answer
44 views

How is this method for computing square roots a manifestation of Newton's method?

I discovered that we can use Newton's method to conveniently compute the square root of a positive integer. Newton's method stipulates that we keep iterating as such: $$ x_{n + 1} = x_n - ...
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22 views

Approximation of $C^\alpha(T^2)$ in $L^1$ by a $C^k$ function

Consider a Holder function $f \in C^\alpha(\mathbb{T}^2, \mathbb{R})$, $\alpha \in (0,1)$. I would like to approximate $f$ with $f_\epsilon \in C^k(\mathbb{T}^2, \mathbb{R})$, $k \in \mathbb{N}$, in ...
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1answer
22 views

Complete orthonormal system

Let $H$ a linear space with inner product. An orthonormal system $\{e_1, e_2, \dots \}$ is called complete in $H$ if $x=0$ is the only element that satisfies the relations: $$(x, e_n)=0, \ \ \ ...
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1answer
50 views

Explanation of a passage about a smooth approximation to $L^p$ function

I'm reading J.L Vazquez "Porous Medium Equation" book. In it, he says the following: We are given a function $a:\Omega \times (0,T) \to \mathbb{R}$ such that $a \geq 0$. We find a smooth ...
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1answer
47 views

Solve non-linear equations of 3 variables using Newton-Raphson Method iterms of c,s and q.

The three non-linear equations are given by \begin{equation} c[(6.7 * 10^8) + (1.2 * 10^8)s+(1-q)(2.6*10^8)]-0.00114532=0 \end{equation} \begin{equation} s[2.001 *c + 835(1-q)]-2.001*c =0 ...
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8 views

Confidence intervals for bernoulli trials over a cyclic time series

I have a time series with observations of 0 or 1 observed yearly for approx. 20 years. The time series is cyclic and I want to find a CI for the probability p over the cycle (mean). Unfortunately I ...
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25 views

Solve Van der Pol equation by Padé approximation

I want to solve the Van der Pol equation: $$f''+ \mu \, (f^2-1)f'+f=0, \quad f = f(t),$$ by Padé approximation. I know the solution should be the combination of $\sin{t}$ and multiplied by $\mu$, ...