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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1answer
40 views

Use stirlings approximation to prove inequality.

I have come across this statement in a text on finite elements. I can give you the reference if that will be useful. The text mentions that the inequality follows from Stirling's formula. I can't ...
9
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2answers
65 views

Existence of a function

I need some help: I am thinking about this problem. Any advice would be appreciated. Let's fix $\epsilon>0$. Does there exists some $f\in C^0([0,\pi])$ such that: ...
0
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0answers
33 views

What functions stem from Fourier Series with rational-only coefficients?

Given the Fourier series $$f(z) = \sum_{k=-\infty}^\infty c_k e^{ikz}$$ but with $c_k\in(\mathbb Q+ i\mathbb Q)$ instead of $\mathbb C$ (or even purely real), are the functions obtained this way in ...
4
votes
2answers
59 views

Iterative calculation of $\log x$

Suppose one is given an initial approximation of $\log x$, $y_0$, so that: $$y_0 = \log x + \epsilon \approx \log x$$ Here, all that is known about $x$ is that $x>1$. Is there a general method of ...
2
votes
2answers
34 views

How to choose the extent of approximation for this equation?

Given the curve $$y={1 \over{x^2-1}}-{1\over {x^2}}$$ Find reasonable approximations for the intersections of this curve with the straight line $y=Ax$ (a) when $A$ is a very small positive number ...
0
votes
1answer
31 views

functions that jump from zero to value k and then decay back to zero

I am looking for functions that are zero for all $x< a$ and then from a suddenly jump to a specified value $K$ and then drop back to zero either quickly or slowly and perhaps even not necessarily ...
2
votes
3answers
151 views

Is $e^{e^{2}}$ a relatively good approximation for $1000\phi$? [on hold]

Yesterday night, I found that $e^{e^{2}} \sim 1000\phi$, where $\phi$ is the golden ratio. I believe that it is correct to four decimal places. Would it be considered a relatively good ...
2
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0answers
52 views

Is there a closed-form solution (even approximated) to this inequality?

I have the following function: $f(x, \theta) = (1-\theta)(x+1)^{-\theta}\left[ \frac{2-2\theta}{1- 2\theta} (N^{1-2\theta} - (x+1)^{1-2\theta}) - (x+1)^{-\theta}(N^{1-\theta} - (x+1)^{1-\theta}) ...
1
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2answers
15 views

Approximation of derivative of discrete function

I have a function of which I only know the value of at some discrete points. Now I want to calculate the derivative of this function. The approximation of taking the difference of two consecutive ...
0
votes
2answers
34 views

Runge Phenomena and Taylor Expansion

From The Weierstrass Approximation Theorem Vs The Runge's Phenomenon: We contrast this to polynomial interpolation: this is a specific method for generating a sequence of polynomials that ...
0
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1answer
23 views

Fourier expansion of absolute value of a periodic function

For an arbitrary periodic function p(x), whose period and Fourier expansion might have been known in advance, how can we get the Fourier expansion/coefficients of |p(x)| from them? Or, if possible, ...
4
votes
2answers
44 views

Why can the elements of $L^\infty$ be approximated in $L^p$ by $C^1$-functions?

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain and $f\in L^\infty(\Omega)$. From which theorem does the existence of $(f_k)_{k\in\mathbb N}\subseteq C^1(\overline\Omega)$ with ...
3
votes
1answer
79 views

Can I find a good approximation of this function?

I am wondering, if I can find a good approximant for this function $$f(z)=\log \left[ \frac{1-z^2}{z \left(3-z^2\right)}\sinh \left\{\frac{z \left(3-z^2\right)}{1-z^2}\right\}\right]$$ assuming $z ...
4
votes
0answers
27 views

Shaking a box of rocks (Optimal Packing)

My coworker was telling me that when he plants seeds on his farm, he puts them all in a large container on the tractor and after a period of just driving, the seeds are more densely packed than when ...
1
vote
1answer
31 views

Imprecise logarithms that reference sets of numbers.

I apologize in advance if my question seems vague, I'm only in algebra II, so It may turn out that I lack the terminology to phrase my question correctly. Some background, we just finished our unit ...
0
votes
1answer
46 views

Approximation for a binomial coefficient sequence summation

What is a good approximation to $$\dfrac{{\binom{k}{i}}{\binom{k}{i}}(i-1)!}{\binom{k(k-1)/2}{i}}$$ $$\dfrac{{\binom{k}{i}}{\binom{k}{i}}(i-1)!}{(2^{(\log ...
0
votes
2answers
77 views

Is there a good approximation for this?

What is a good approximation for $\dfrac{k!}{\binom{k^2}{k}}$ as a function of $k$? Is there a $k_0\in\Bbb N$ such that for all $k\gt k_0$, ...
1
vote
1answer
31 views

Taylor Approximation of $\cos(0.02)$

Use a Maclaurin $(a=0)$ polynomial for $\cos{(x)}$ with $3$ nonzero terms to approximate $\cos{(0.02)}$. Also, use the Taylor Remainder Theorem to find a bound on the error $\left(\displaystyle ...
0
votes
1answer
23 views

How to calculate entropy from a set of samples?

entropy (information content) is defined as: $$ H(X) = \sum_{i} {\mathrm{P}(x_i)\,\mathrm{I}(x_i)} = -\sum_{i} {\mathrm{P}(x_i) \log_b \mathrm{P}(x_i)} $$ This allows to calculate the entropy of a ...
0
votes
0answers
23 views

Distribution of Difference of Ordered Values Drawn From A Normal Distribution

This question has come up at least twice now when I was trying to estimate something*. I could always write out the integral or find it computationally but I'm hoping someone will give me an exact ...
1
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1answer
47 views

Which “approximate” value of f(0.98) is this question looking for?

In a section of a calculus workbook dealing with local linearity and linear approximations of functions, the following question is posed: Consider the function f(x) = aln(x+2). Given that f'(1) = ...
0
votes
0answers
25 views

Approximate solutions to differential equations

If one has a differential equation for $y(x)$. If this differential equation has two solutions one for $x\ll a$ and the other for $a\ll x$, where $a$ is constant real value. My question is at what ...
1
vote
1answer
22 views

$T_2^4-T_1^4\approx4T_1^3(T_2-T_1)$ if $(T_2-T_1)/T_1$ small

I read, in a text of elementary physics, that $$T_2^4-T_1^4\approx4T_1^3(T_2-T_1)$$ if $\Delta T:=T_2-T_1$ "is small with respect to $T_1$". In a rigourous language, I suppose that it means that ...
1
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0answers
67 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 ($7.1.26$). ...
0
votes
0answers
19 views

Catenary with loads

I'm working on a program to do strength calculations on a 2D suspension cable, with equally distributed load (self weight) and vertical and horizontal forces at certain points A,B,C,...F along the ...
2
votes
6answers
1k views

How to prove this approximation for a logarithm? [closed]

I need to prove this approximation, but I am unable to conclude $$\log \left(1+\frac{1}{n}\right) \approx \frac{1}{n}$$
0
votes
0answers
68 views

Integration of a given Integral

Given the integral $$ \hat{\alpha}({r_{0}})=2\int^{\infty}_{{r_{0}}}\frac{dr}{r \sqrt{1-\frac{2M}{r}} \sqrt{\left(\frac{r}{{r_{0}}}\right)^{2}\left(1-\frac{2M}{{r_{0}}}\right) ...
5
votes
2answers
219 views

Numerical methods (for ODE/PDE) that could take approximate solutions/good initial guesses, and further refine it to an certain accuracy

I am currently playing with an old analog computer, which could solve time-dependent ODE/PDEs pretty fast, without time-stepping; thus there is no convergence issues caused by time-stepping because of ...
16
votes
0answers
157 views

Is there an integral for $\pi^4-\frac{2143}{22}$?

Ramanujan in his Lost Notebook (p.16, and a related one in Quarterly Journal of Mathematics, XLV, 1914) gave the very close approximation (by just $10^{-7}$), $$\pi^4 \approx ...
2
votes
1answer
35 views

Having trouble applying the formula for simpson's rule.

My professor provided following formula for simpson's rule: $$\int_a^b f(x)\;dx \approx \frac{h}{3} \sum_{j=0}^{n-1} ( f(x_{2j}) + 4f(x_{2j+1}) + f(x_{2j+2}))$$ Okay so my problem lies when solving ...
2
votes
0answers
74 views

Why the quadrature formula is exact one not an approximation?

I am reading this material on the algorithm of calculating the centroid of a polyhedron. I am confused by the last step of the deduction: The three coordinates of the centroid can be obtained: ...
0
votes
2answers
41 views

Proving Lower bounds on an Approximately Linear Function

We are looking for a lower bound on the function, $\frac{1.31}{e^{\frac{1.31}{x+1}} - 1}$ for $x \geq 2$. This function seems to behave linearly. We believe that the following statement holds: ...
0
votes
1answer
43 views

How to computer the proximal operator of a indicator function?

For $G_{1}(x) = I_{c}(x), c =\{x|Ax=b\}$, the Proximal operator is: $Prox_{\gamma G_{1}}(x) = Proj_{c}(x) = x + A^{T}(AA^{T})^{-1}(y-Ax)$ I hope to know, how to derivative this result. And, for ...
6
votes
5answers
1k views

Pythagorean theorem expressed without roots in an old Tamilian (Indian) statement

There's an old Tamil statement that predicts the hypotenuse of a right angle triangle to a reasonable level of accuracy considering it doesn't involve roots. This is how it goes: “Odum Neelam ...
0
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1answer
32 views

Approximate fraction of two integrals

could you propose a way to simplify or approximate (under some assumptions) $\bar{\eta}$ defined as below? $$ \bar{\eta} = \frac{\int f(t)dt}{\int\frac{f(t)}{\eta{(t)}}dt} $$ The $f(x)$ and ...
-1
votes
4answers
108 views

If $f(x)\ll1$ is it safe to assume that $f^{\prime}(x)\ll1$?

If $$f(x)\ll1$$ is it safe to assume that $$f^{\prime}(x)\ll1$$
1
vote
1answer
28 views

Integral approximation for alternating series

I can approximate the sum of $\frac 1 {n^2}$ using its integral. But what about $(-1)^n\frac 1 {n^2}$? Is it possible to approximate this using integrals? I want to know if there are other ways than ...
3
votes
1answer
48 views

Inverse of prime counting function

The prime counting function $ \pi (x) \approx \dfrac {x} {\ln(x-1)} $. This function returns the number of primes less than $x$. Note: $x-1$ gives a better estimate than $x$. How to find $x$ given $ ...
0
votes
1answer
27 views

Which polynomial has similar properties with Legendre?

I am looking for an kind of polynomial such as Legendre properties that polynomial sequence of orthogonal polynomials such as bellow image. Could you suggest to me one polynomial? Is B-spline correct? ...
3
votes
2answers
61 views

How to choose a left-add$(X)$-approximation with a certain property

Let $A$ be an artin algebra and $X,Y$ in mod-$A$. Suppose $0\rightarrow Y \stackrel{\alpha}{\rightarrow} X^n\stackrel{\beta}{\rightarrow} X^m$ is exact. Set $C:=Coker(\alpha)$ (as module) and ...
6
votes
1answer
84 views

For any smooth n-manifold $M$, construct a smooth map $f:M\to S^n$ which is not null-homotopic

PROBLEM: For any smooth n-manifold $M$, construct a smooth map $f:M\to S^n$ which is not null-homotopic ,or even of degree 1 The following is my idea: First, choosing an arbitrary open coordinate ...
0
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0answers
25 views

Ruling span derivation?

I have recently read a paper about the ruling span for electrical wires and they have an approximation that looks like it can be derived with mathematical intuition only. I'd like to find a derivation ...
3
votes
1answer
51 views

Minimizing Area by Approximation

Suppose I have an increasing step function $E_c$ given by $$E_c(\phi) = \sum_{i=1}^n E_i \theta(\phi - \phi_i),$$ where $\theta$ is the Heaviside step function and $E_i$, $\phi$, and $\phi_i$ are all ...
0
votes
1answer
24 views

Why is this true - easy question concerning asymptotics of exponential

Suppose $\lambda > 0$ is constant as $t \searrow 0$. In my lecture notes it is written that $\left(1+\sum_{k=1}^{\infty} \frac{(-\lambda t)^k}{k!}\right) \lambda t = \lambda t + o(t)$ and ...
1
vote
1answer
52 views

Approximating $\frac{(kn)!}{(n!)^k}$

Is there any approximations for the form $$\frac{(kn)!}{(n!)^k},$$ where $n$ and $k$ are positive integers? $n$ is not necessary much larger than $k$?
0
votes
1answer
12 views

Eliminating order notation in upper bound

I have that some value $E_i=\alpha^2\varepsilon_i^3+O(\varepsilon_i^4)$, where $\alpha>0$ is a fixed constant and for every $i$, $0<\varepsilon_i\ll1$. I would like to place an upper bound on ...
4
votes
0answers
35 views

Asymptotic behavior of many derivatives

To compute the residue of a pole of very high order $M$ at $z=0$, one needs to compute $\frac{d^M}{dz^M} g(z)$ Suppose that $g(z)$ is a reasonable but not trivial function, that itself may depend on ...
1
vote
0answers
23 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 ...
6
votes
0answers
82 views

Could this approximation be made simpler ? Solve $n!=a^n 10^k$

I need to find the smallest value of $n$ such that $$\frac{a^n}{n!}\leq 10^{-k}$$ in which $a$ and $k$ are given (these can be large numbers). I set the problem as : solve for $n$ the equation ...
0
votes
1answer
17 views

Ratio of step sizes in Richardson extrapolation for numerical integration

When using Richardson extrapolation for numerical integration, are there any criteria whether ratio between the steps should be or does it not matter what step size I use? For an integral I can write ...