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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41 views

Approximating the probability that a range bounds a given number, with very large numbers

Let $m$ numbers be chosen uniformly from $0,\dots,n-1$ without replacement and then sorted in ascending order as $\ell_0,\dots,\ell_{m-1}$. Let there be $b,e,x$ such that $0 \le b \le e \le m$ and $0 ...
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38 views

approximating Gegenbauer polynomials (or ultraspherical or Jacobi)

Looking for hardcore orthogonal polynomial people here... If we hold the degree $\ell$ constant and take the order $\alpha$ to infinity, the Gegenbauer polynomial $G_\ell^{(\alpha)}$ approaches the ...
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83 views

Calculate a 5x5 Vandermonde system for a 5 point mesh

This is problem 1.2 in Randall J Leveque's textbook, "Finite Di fference Methods for Ordinary and Partial Di fferential Equations". I'm struggling with how to actually do the computation, I'm not so ...
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32 views

any approximation for $\sum_{i_3=3}^{n-k+3}\sum_{i_4=i_3+1}^{n-k+4}\sum_{i_5=i_4+1}^{n-k+5}\cdots\sum_{i_k=i_{k-1}+1}^{n} 1, (n \gg k)$?

Is there any approximation for $$\sum\limits_{i_3=3}^{n-k+3}\sum\limits_{i_4=i_3+1}^{n-k+4}\sum\limits_{i_5=i_4+1}^{n-k+5}\cdots\sum\limits_{i_k=i_{k-1}+1}^{n} 1, \quad (n \gg k)$$ ? We know that ...
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78 views

Numerically integrating in to Chebyshev polynomial

I'm trying to find the Chebyshev interpolate for an ODE in a given interval. That is, given an ODE that looks something like: $$y'' = g(y) \ y'$$ I want to numerically integrate it inside the ...
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37 views

Zeta function universality: How to compute the shift parameter for simple functions?

I've come across Zeta function universality. For a nice function $f$ in a nice subset $U$ of the complex strip between real $0$ and $1$, one can find a real $t$, such the zeta function $\zeta$ shifted ...
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37 views

Approximating sums

I got a general question, that is motivated by a recent problem. So let me first describe the problem and then add the general part: I got a rather simple (using only basic elements) equation, which ...
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24 views

Iterative approximation of non-constant values in linear equation

The issue regards an algorithm for iterative approximation of unknown transaction values. For each iteration (each day), we are give the total revenue of all transactions for that day, and we have the ...
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83 views

Newton-Rhapson for reciprocal square root

I have a question about using Newton-Rhapson to refine a guess of the reciprocal square root function. The reciprocal square root of $a$ is the number $x$ which satisfies the following equation: ...
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143 views

Approximate convolution of independent Beta variates?

Is there a way to approximate the convolution of Beta variables? Specifically, I am trying to find an approximation to $g(x_0)$: $$g(x_0) = \int \delta(x_0-\sum_{i=1}^{n} a_i x_i) \prod_{i=1}^{n} ...
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128 views

Real approximation to the maximum using Laplace's method integral

The Laplace's Method states that under some conditions, it holds that: $ \sqrt{\frac{2\pi}{M(-g''(x_0))}} h(x_0) e^{M g(x_0)} \approx \int_a^b\! h(x) e^{M g(x)}\, dx \text { as } M\to\infty$ Where ...
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15 views

Can $u\in W^{1,2}_0$, $\Delta u\in L^2$, $u\geq 0$ be approximated by a sequence smooth function $u_k\geq 0$

Assume that $\Omega\in R^3$ is a bounded Lipschitz domain. $u\in W^{1,2}_{0}$, $\Delta u\in L^2$, $u\geq 0$. Is it possible to approximate u by a sequence of nonnegative smooth functions $u_k$, ...
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30 views

Convergence acceleration using rational approximation

How Pade Approximations accelerate the convergence of series??? Here I don't mean only the Pade Approximations, just in general how do the rational approximations contribute to series convergence ...
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20 views

How $(P_{r+1}-P_r)$ is maximized?

This is from DeGroot's "PROBABILITY and STATISTICS"(Second edition)(Cf. pages 87 to 92).I am rewriting the relevant stuff. Let $r$ be a positive integer and $r\leq n$. ...
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113 views

How many solutions to $x^3+y^3 = z^3\pm 1$ for $z$ less than a bound?

Assume $a,b,c, N$ as positive integers, let primitive be $\gcd(a,b,c) = 1$ and, $$a^2+b^2 = c^2\tag{1}$$ Supposing you want to know how many solutions there are with $c$ less than a bound $N$. ...
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29 views

Approximation of series

Can anyone provide an approximation for the series: $$\sum_{i=0}^{\infty}\frac{\left(1-\Gamma\left(i+1,\frac{b^{2}}{N_{01}}\right)\right)v^{-i-1}}{\mu_{1}N_{01}^{i}}\\\times ...
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69 views

Best simple exponential estimator

Given smooth enough positive $f(x)$. Find real $c$ such that $$\int_0^1 (e^{cx}-f(x))^2dx$$ is minimized. Note: it only looks innocent..
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44 views

Kalman filtering with angle measurements

I'm designing an Extended Kalman Filter which will take several types of measurements and try to estimate a location. One type of measurement is the direction to the location. I've thought about this ...
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173 views

Approximation of an exponential sum

Consider the flowing exponential sum with $x,\,y \ge 0$ and $c_i,\, d_i$ is a real number for $i=1,..,N$ $$ E=\sum\limits_{i = 1}^N \exp \left(- \left( x - c_i \right)^2 - \left( y - d_i \right)^2 ...
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91 views

Turn ugly series into a nice approximation

I am currently struggeling with a series(actually two). The problem is, that I can do nothing with them, since this expression is so ugly. I would love to hear about any kind of approximations that ...
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66 views

How do you interpolate the local maxima of a set of points in more than 3 dimensions?

I have a set of about 400 points each with 6 coordinates and one scalar value. How can I find out where the local maxima are?
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14 views

Fittig N-D periodic functions

Besides fourier series, are there any stable way to fit N-D nonlinear functions that demonstrate some degree of periodicity, sigmoid neural networks works poorly in this domain.
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164 views

Discrete approximation to a continuous probability density function

I want to approximate a continuous, finite probability density function, with a specified number $N$ of points, in the following way: If the pdf is 1-dimensional, defined over the section [0,1], then ...
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37 views

a question in projection

Let $V=L^2(\Omega)$, and $$k=\{v \in V ~s.t ~||v||_{L^2(\Omega)}\leq 1 \}$$ I need to find projection for any $u \in V$ on $k$. Please help me.I do not have any idea about this problem. I have many ...
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128 views

How to estimate linear operator?

Suppose the complex column vector $\mathbf{x}$ is linearly transformed by the complex matrix $\mathbf{T}$ into $\mathbf{y}$: \begin{align} \mathbf{y} = \mathbf{T}{x} \end{align} Assuming ...
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305 views

How to calculate probability with sigmoid output in feedforward neural network?

first of all I'm sorry for my not very skilled English, but I will do my best to explain my problem. I'm trying to create a feedforward neural network with one hidden layer (with probably arctan ...
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34 views

${{\left( 1-p \right)}^{n}}{{\text{ }}_{1}}{{F}_{1}}\left( n+1,n+a+1,\lambda T \right)\approx ?$

I am trying to approximate $$\frac{p\left( n+a,\bar\lambda T \right){{\left( 1-p \right)}^{n}}_{1}{{F}_{1}}\left( n+1,n+a+1,\lambda T \right)}{p\left( n+a,\bar\lambda T \right)}={{\left( 1-p ...
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314 views

Simpson's Rule derived from Trapezoidal Rule

I was just wondering if I could have some assistance in regards to the Trapezoidal Rule and Simpson's Rule. I have a question where it asks to generalize the Trapezoidal Rule to the case of ...
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43 views

Small Inhomogeneity of Differential Equation

Given a variable $x\in[-L,L]$ with $L\in \mathbb{R}$, first consider a generic homogeneous second order differential equation with potential $V(x)$: $$\left(\frac{d^2}{dx^2}+V(x)\right)f(x)=0$$ Let ...
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87 views

Smooth Approximation of $L^p$ function

Given a bounded domain $\Omega \subset \mathbb{R}^n$, is it possible to approximate every $L^p(\Omega)$ function (where $1\leq p < \infty$) by smooth functions $\mathit{C}^{\infty}$ ?
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22 views

Is there a way to estimate the range of fitting coefficients from only the data?

Considering an approximation $f$ for a set of $N$ data points $(x,y)$ using, for example, $M$ radial basis functions at arbitrary sites in the domain $f_i = \sum_{j=1} ^M c_j\phi(||x_i-x_j||)$ where ...
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80 views

Expansion in powers

Let $n=2k, k \in Z_+$. Let $$P_k\left(\frac{t}{\sqrt n}\right)=n!\sum_{\begin{smallmatrix} n_1+\ldots+n_k=n \\ ...
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266 views

Approximations of the incomplete elliptic integral of the second kind

For a calculation I am working on I need to determine the arc length $l$ of a part of an ellipse in terms of the major axis $2a$, the minor axis $2b$ and the angle $\phi$. I know that this is a ...
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121 views

Approximate CDF of the sum of a gaussian and a truncated gaussian

I am looking for a quick-to-compute approximation of the CDF of $X+Y$, where $X \sim N(0,\sigma_1^2)$ and $Y$ is a truncated gaussian, more specifically, a gaussian with mean $0$, standard deviation ...
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55 views

alternating series estimation with integral?

We know that there are some approximation like Abel's identity. If $\lambda_n$ is increasing and $$ C(x)=\sum_{\lambda_n\le x}c_n,\qquad(c_n\in\mathbb{C}) $$ Then if $X\ge\lambda_1$ and $\phi(x)$ has ...
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25 views

$i^{-k}\pi^{n/2}\Gamma(\frac{k}{2})/\Gamma(\frac{n+k}{2})\approx k^{-n/2}~~(k\rightarrow \infty)$

Here $i$ is complex number, $n$ is positive integer. Show that $$i^{-k}\pi^{n/2}\Gamma(\frac{k}{2})/\Gamma(\frac{n+k}{2})\approx k^{-n/2}~~(k\rightarrow \infty)$$ This question appears from Stein's ...
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100 views

Approximation of SDE

I have been struggling with the following problem: If you want to find a numerical result by simulating the paths of a stochastic differential equation, in particular a geometric brownian motion I ...
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75 views

Intuition for approximating Ei(x)

I'm working with a function that is defined to be the sum of a series, and I'd like to either find its values, or approximate them analytically: $F(x) = \frac{1}{w} ...
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26 views

clairification on standard deviation

I have a homework question that gives me a set of $x$ values and their respective $f(x)$ values and asks me to find the line which best fits the data. I have to do this by finding the estimated ...
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188 views

Chebyshev Rational Approximation

Find the Chebyshev rational approximation of degree 4 with m=n=2 for $f(x)=\sin(x)$ . Now I do have a program that can evaluate this But its asking me to find the coefficients of the Chebyshev ...
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63 views

How to approximate $\sqrt[3]{x}$ when $x$ is rational number

One wants to approximate the real value of $\sqrt[3]{x}$ when $x$ is rational number. One want to approximate to two decimal digits. Is there any way to do this quickly?
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135 views

Upper bound for L1-L2 optimization problem

I am interested in the following convex optimization problem: \begin{align*} \max & ||x||_1 \\ \text{s.t.} & ||x-a||_1 \le K \\ & ||b\circ x||_2 \le 1\\ & x \in R^n \end{align*} where ...
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193 views

Approximate function using non-orthogonal basis

I'm currently trying to wrap my head around somebody's (very concise) description of Finite Element Analysis (FEA): ...
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86 views

Rapidly convergent series for $\sum_{J=0}^{\infty} (2 J + 1) e^{-\beta J(J+1)}$ (rigid rotor)

I need to evaluate this series for arbitrary $\beta > 0$: $ Q = \sum_{J=0}^{\infty} (2 J + 1) e^{-\beta J(J+1)} $ Is it related to a known transcendental function? From the research I did, it ...
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184 views

Approximation algorithm for sine function

What is the most common approximation algorithm for sine function which can be implemented with self-organizing maps algorithm? I was thinking of least-squares approximation but I can't find a ...
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211 views

Bound on euclidean norm

Is it possible to find a suitable lower bound on $$\left|\left(tM+\sum_{k=2}^\infty\frac{(tM)^k}{k!}\right)\cdot b\right|$$ for $M$ as $n \times n$ matrix, $b$ as $1 \times n$ vector and for all $t$ ...
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223 views

Polynomial approx to the Normal density

I have found several polynomial some approximations to the Normal CDF$^{(1)}$, but my question is: are there good polynomial approximations to the Normal PDF? Thanks $^{(1)}$ For example, some are ...
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158 views

Solving or approximating an equation with radicals and arctan function

I have solved a differential equation recently, which left me with this whopper of inverse function to figure out. I know what $c$ is, I just haven't calculated its exact value based on the initial ...
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65 views

How can I compare two approximants to a bivariate function?

An application I'm working on has required me to find simple approximations to a rather complicated bivariate function $g(x,y)$ that also takes a long time to evaluate on the computer. Through sheer ...
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121 views

Worst-case error related to Cramer-Rao bound

I would like to understand the relation (if any) between the Cramer-Rao Lower Bound of estimation theory and the following simple definition of "reconstruction accuracy" which doesn't use any ...