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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310 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): ...
2
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540 views

Polynomial Interpolation and Error Bound

Problem: Use the Lagrange interpolating polynomial of degree three or less and four digit chopping arithmetic to approximate cos(.750) using the following values. Find an error bound for the ...
2
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145 views

Generating all positive integers from three operations

This question arose on sci.math, but since almost all the competent mathematicians there have migrated here, I thought I'd give the question a wider audience. Starting with an integer $t>2$, ...
2
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77 views

On bounding the average cost of top-down merge sort

Let $A_n$ be the average number of comparisons to sort $n$ keys by merging them in a top-down fashion (see any algorithm textbook). It can he shown that $$ A_0 = A_1 = 0;\quad A_n = ...
2
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81 views

Algorithm to predict next 3D points

For example, having this data: year x/y/z 2007 10/20/70 2008 20/10/70 2009 30/10/60 2010 40/10/50 2011 40/15/45 We want to predict what will be the x/y/z in ...
2
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149 views

approximation of the sum

I have difficulties to find an approximation formula (or bound from the below) for the following sum: $$ \sum_{k=1}^n\left( \frac{1}{35}\right)^{k-1}(n-k)!\left(k-\frac 32\right)!. $$
2
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142 views

Approximating sums like $\sum_{j=1}^n\sum_{k=1}^{\lfloor\frac{n}{j}\rfloor}\int_1^{\frac{n}{jk}}dx$

Can anyone tell me how to approximate the following functions? $f_3(n) = \displaystyle\sum_{j=1}^n\sum_{k=1}^{\lfloor\frac{n}{j}\rfloor}\int_1^{\frac{n}{jk}}dx$ $f_4(n) = ...
2
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185 views

calculate the rate of change

I am trying to calculate the change frequency for a set of data. Each bit of data has the date-time it was created. I would like to say for a specific set of data the change frequency is hourly, ...
2
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231 views

When is it valid to convert a function inside a probability integral to the indicator function?

I am faced with an approximation that replaces a probability density function with the indicator function and I am at a loss as to why this is valid. We want to model the lifetime $T$ of a website ...
2
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311 views

Multivariate B-Spline Derivatives

To construct a multivariate B-spline, we simply take the Kronecker tensor product between the univariate basis functions constructed for each individual dimension. What I'd like to know is how do you ...
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15 views

How can I express this in terms of Gauss-Hermite Quadrature?

I am having the following expression. This is the PDF of Nakagami-Lognormal Distribution. I want to express in terms of Gauss-Hermite abscissas and weights. How can I do it? ...
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26 views

Approximate $e^{0.01} \sin(0.02)$ by using its linearization

$f(x,y) = e^x \sin(y)$ $f_x = e^x \sin(y)$ $f_y = e^x \cos(y)$ $L(x,y) = f(0,0) + f_x(0,0)x +f_y(0,0)y$ Solving: $f(0,0) = e^0 \sin(0) = 0$ $f_x(0,0)x = e^0 \sin(0) \cdot x = 0$ $f_y(0,0)y = ...
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13 views

Alpha Max Plus Beta Min Calculation

I read about the Alpha Max Plus Beta Min algorithm described here. Here is a screenshot from the wikipedia page: I think understand what the algorithm is supposed to do. It makes an approximation ...
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19 views

Approximating number of partitions of $n$, denoted $p(n)$, by $p(n)\ge e^{c\sqrt n}$

I was to show that $p(n)$, the number of partition of a positive integer $n$, satisfy: $p(n)\ge \max_{1\le k\le n}{{n-1\choose k-1}\over k!}$, which was obvious because every unordered collection of ...
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33 views

$\int_{-\pi/2}^{\pi/2} |\vec{v}-\vec{V}|(\vec{v}-\vec{V}) d t$, with $\vec{v}=(\vec{i} \cos t +\vec{j} \sin t) (\sqrt{1-p^2 \cos^2 t}+p \sin t)$

I need to solve this integral: $$\int_{-\pi/2}^{\pi/2} |\vec{v}-\vec{V}|(\vec{v}-\vec{V}) d t$$ $$\vec{v}=(\vec{i} \cos t +\vec{j} \sin t) \sqrt{1-p^2 \cos 2 t+2 p \sin t \sqrt{1-p^2 \cos^2 t}}=$$ ...
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26 views

Quality of $E(f(X))\approx f(EX)+\frac 1 2 f''(EX)\sigma_X^2$ approximation

For convex $f$, we have Jensen's lower bound $Ef(X)\ge f(EX)$. What conditions do we need to put on $f,X$ so that the second order expansion in the title would be an upper/lower bound/good ...
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84 views

A double inequality for $\frac{\pi}{2}$

Approximating $\frac{\pi}{2}$ from above Since $$\left(\frac{\pi}{2}\right)^9\approx 58.220897$$ the root $$58^\frac{1}{9}\approx 1.5701$$ is not far from $$\frac{\pi}{2}\approx 1.570796$$ This ...
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20 views

Approximation of a hypergeometirc-like distribution

Fix $0<\varepsilon<1$. For $m\in\mathbb{N}$, let $$c_m=\max\left\{\frac{{m-1\choose s-1}{m\choose k-s}}{{2m\choose k}}:\;k=1,2,\dots,2m(1-\varepsilon)\;\mbox{and}\;s=1,2,\dots,k\right\}$$ Prove ...
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28 views

Intergral approximation?

For the integral: $$I=\int^{t/2}_{-t/2} e^{i(\omega_1-\omega_2)t'}dt'$$ What would be the lower limit on $t$ for the approximation: $$I\approx 2\pi \delta(\omega_1-\omega_2)$$ to hold? I would guess ...
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26 views

Unbounded approximation algorithm for minimum vertex cover

Suppose we find the minimum vertex cover of a graph by repeatedly choosing the vertex with the highest degree and delete all edges incident on that vertex, until there are no edges left. How can one ...
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20 views

Step in using Stirling's formula to get an upper bound

I'm having trouble seeing why the following holds. Given the conditions that $N=\binom{n}{2}$ and $m$ is a function of $n$ such that $N-m \to \infty$ as $n \to \infty$, why is it that $$(1+o(1)) ...
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42 views

How can I prove two empirically derived graphs are topologically equivalent?

I have two graphs that I've derived from an empirical data set and I suspect that they're topologically equivalent. It seems much easier to show that these graphs are not equivalent than to show that ...
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0answers
29 views

Approximating $\prod_{r=s}^t (1-b/r)$

I am currently trying to place an order of precision on the approximation $$\prod_{r=s}^t \left(1-\frac{b}{r}\right) \approx \left(\frac{s}{t}\right)^b$$ This follows because $$\prod_{r=s}^t ...
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38 views

Mixing Fuzzy Logic and Probabilistic interpretation of a dataset

A probabilistic data cloud is a set $M$ of data points $\{m_i\}_i$, where each data point $m_i$ is associated to an event $E_i$ expressing the set of the occurrences of $m_i$ in any possible ...
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0answers
56 views

Three almost-integers of the form $ce^{H_a+H_b}\approx 2^k\pm1$

The approximation $$H_n\approx log(2n+1)$$ http://math.stackexchange.com/a/1602945/134791 suggests that the harmonic number for composite odd numbers might be close to the sum of the harmonic ...
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25 views

Cubic curve as approximation of Euler spirals?

I was reading the wiki article about Euler spirals and I reached this passage: Unaware of the solution of the geometry by Leonhard Euler, Rankine cited the cubic curve (a polynomial curve of ...
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0answers
75 views

Series for Stieltjes constants from $\gamma= \sum_{n=1}^\infty \left(\frac{2}{n}-\sum_{j=n(n-1)+1}^{n(n+1)} \frac{1}{j}\right)$

Euler's constant has the following representations (Euler-Mascheroni constant expression, further simplification, http://math.stackexchange.com/a/129808/134791, Question on Macys formula for ...
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56 views

Why is $\sum{\frac{1}{(j/2)!}(\xi)^j}\approx\sum{\frac{1}{j!}(\xi)^{2j}}$?

Why is $\sum{\frac{1}{(j/2)!}(\xi)^j}\approx\sum{\frac{1}{j!}(\xi)^{2j}}$? This equation is used for solving the Schrödinger equation of the harmonic oscillator at one point in Griffiths, ...
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0answers
28 views

Softmax for Continuous Functions?

The softmax $\log \sum_{i=1}^n \exp(f_i)$ of vector $f$ is a smooth upper bound on $\max_i f_i$. However, the same cannot be said of $\log \int_{X} \exp(f(x))dx$ in relation to $\max_{x \in X} f(x)$ ...
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0answers
29 views

Asymptotic behaviour of generalized binomial coefficient $\frac{an(an-1)…(an-n+1)}{n!}$

Let $a\in(0,1)$. What is the asymptotic behaviour of $\frac{an(an-1)...(an-n+1)}{n!}$ as $n\rightarrow\infty$? It looks like it might be possible to express this in terms of gamma functions and use ...
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0answers
29 views

Normalization in least-p'th minimax algorithm

In the book "Practical Optimization: Algorithms and Engineering Applications", the least-$p$th minimax algorithm is presented, for approximation of the minimax optimizer (Alg. 8.1): $Loss_x(k)$ = ...
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0answers
36 views

Approximating $|1-e^{i\delta}|$

Actually this question is derived from the Rudin's Real and Complex Analysis's lemma 15.17. In the lemma, given condition is below; Let $h(z) \in H(\Omega)$ such that Re $h(z) = \log |1-z|$, |Im ...
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41 views

Uniform error of approximating the Heaviside function by a partial sum of its Fourier series

Suppose $f(x)=H(x-.5)$ where H = Heaviside function on $0<x<1$ is approximated by the first five nonzero terms of its Fourier sine series. Compute the uniform error (i.e maximum error, max p(x) ...
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36 views

Asymptotic binomial ratios

I am in need of asymptotic version of $$\frac{ \displaystyle \binom{n^{1-s}}{n^s}}{\displaystyle \binom{n}{n^{s}}}$$ where $n\in\Bbb N$ and $s\in\big(0,\frac12\big)$ and $$\displaystyle \frac{ ...
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0answers
35 views

Benefit of Padé approximation

I am currently starting to read papers about Padé approximation of the matrix exponential $\exp(A)$, namely $\exp(A) \approx P_{n,m}(A)Q^{-1}_{n,m}(A)$ I am now seeking for a good motivation behind ...
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49 views

If $0.2349$ is approximated to $0.2299$, what is number of significant figures in such approximation?

If $0.2349$ is approximated to $0.2299$, what is number of significant figures in such approximation? Let $x_T=0.2349$, $x_A=0.2299$, then absolute error = $|x_T-x_A|=0.0050=\frac{1}{2}\times ...
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0answers
52 views

Least-squares fit of a nonlinear (polar) system

I want to determine the six unknown coefficients (uppercase letters) of the model $$x=X_c+(Au+B)\cos(Cv+D),\\y=Y_c+(Au+B)\sin(Cv+D)$$ given a set of data $(x_k,y_k,u_k,v_k)$, by least-squares. As ...
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0answers
23 views

How is inequality/approximation obtained?

I am reading on combinatorics - probabilistic methods. In one particular problem I came across the inequality $$\binom{n}{k}(1-2^{-k})^{n-k} < n^k e^{-(n-k)2^{-k}}$$ I understand that ...
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0answers
28 views

Piecewise linear approximation of a (low order) trigonometric polynomial, with quantization

The Problem Given a trigonometric polynomial of order $K$: $$y(t)=\sum_{k=-K}^K c_k \ e ^ {j k \bar \omega t} \ , \qquad c_{-k}=\bar c_k$$ we want to find the best approximation to it using a ...
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0answers
35 views

Solving of numerical equation with integrals

Let's have equation $$ \cosh(2 \pi x) = \cos\left[\text{Re}\int \limits_{0}^{2 \pi}\sqrt{A - 2q\cos(2z)}dz \right]\times \cosh\left[ \text{Im}\int \limits_{0}^{2 \pi}\sqrt{A - 2q\cos(2z)}dz\right], $$ ...
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0answers
16 views

Least Square solution for approximating a sequence

Suppose I have a sequence of length N $a_1,...,a_N$ I want to approximate this sequence by $k^1,...,k^N$ where $k$ is my variable. What is the least square solution of this? is there a closed ...
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27 views

Trapezoidal Rule yielding the exact value of the integral

It is clear that if a function $f(x)$ is linear over the domain $a \leq x \leq b$, then one application of the trapezoidal rule, over the same domain, will yield the exact value of ...
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31 views

Please check this perturbation solution of polynomial root and truncation order.

I have a quintic polynomial where the coefficients depends on a parameter $c$, i.e. $$ a_0(c)+a_1(c)x+a_2(c)x^2+a_3(c)x^3+a_4(c)x^4+x^5 $$ I know that the roots of the polynomial are real and ...
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27 views

How to compare experimental data with teorethical prediction

I would like to know, what is the method to approximate experimental data to teorethical one. I have heard about polynomial regression. After calculating particular matrices and solving set of ...
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60 views

Is there an analytic approximation to integral of this form?

Started working on trying to find an analytical approximation to this integral and not getting very far. Any assistance or direction is greatly appreciated! Thanks Vince $$\int_{0}^{t} ...
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0answers
103 views

Derivation of approximation of Error function

In Abramowitz and Stegun there are some formulas for approximation of Error funtion. I am intrested in the formulas $7.1.25$ to $7.1.28$, here is one of them ($7.1.26$). ...
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0answers
74 views

Iterative algorithms to approximate unknown multivariate convex real-valued functions by a set of linear upper/lower bounds

I am looking for iterative algorithms that can approximate a multivariate convex real-valued function $f(\vec{x})=y,\, \Bbb{R}^n\rightarrow \Bbb{R}$. The function is not known beforehand, but it is ...
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0answers
91 views

Interesting approximation of distribution of numbers in a Farey sequence

I was investigating the distribution of the numbers in a Farey sequence and found some pattern. It is known that the number of elements in Farey sequence can be found using Euler totient function. So ...
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0answers
19 views

How do we derive efficiency from robustness in the virtual ant solution to the traveling salesman problem?

Using virtual ants/swarm intelligence to solve the Traveling Salesman Problem is an example of using a robust system to solve an efficiency problem. We normally think of robustness and efficiency as ...
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0answers
60 views

how to approximate compute p(a|b, c) , using p(b|c), p(a|*, c), p(a), p(b), p(c)

how to approximate compute p(a|b, c) , using p(b|c), p(a|*, c), p(a), p(b), p(c) p(a|*, c), * means anythins. p(a|*, c) = $\sum_i p(a|i, c) $ error = | p(a|b, c) - f(p(b|c), p(a|*, c), p(a), p(b), ...