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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0
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2answers
20 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
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1answer
18 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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0answers
26 views

approximate function with step functions [on hold]

I have a function $f:[a,b]\to \mathbb{R}$. I need to approximate it by a step function (as an histogram). I would like to minimize the $L^2$ distance between the two functions: $\int_a^b [f(x) - ...
0
votes
1answer
29 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
92 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
47 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
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1answer
26 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
52 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
59 views
+50

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
17 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
49 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
23 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
51 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
33 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
13 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
74 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
16 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 ...
0
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0answers
14 views

How should i apply Richardson Extrapolation?

I trying to understand how the Richardson Extrapolation works, and what it is good for. The internet has lots articles about the this, but they all seem to lack in what it is useful for. I wanted ...
0
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0answers
19 views

approximation of function by polynomials

Given a function $f \in L^2[a,b]$, it can be written as $f(x)=\sum_{n=0}^\infty c_nL_n(x)$. where $L_n(x)$ is shifted Legendre polynomial. I am taking the finite sum to approximate. If I take some ...
0
votes
2answers
19 views

Approximation Reasoning

I can't understand one step in the following problem. We start with a function $f(x)=x^\alpha$ on the interval $(0,1)$ where $\alpha>0$ is a constant. We pick two points $x_1<x_2$ from this ...
2
votes
0answers
15 views

Rate of convergence of a Weyl-Heisenberg (Gabor) frame expansion

If $\{g_{m,n}\}$ is a Gabor frame for $L^2(R)$, with window function $g$, and $f \in L^2(R)$, is there a bound on the approximation error of $f$ using a finite subset of the frame? That is, is ...
1
vote
0answers
13 views

Signs and value of higher order terms in the Taylor expansion of a strongly convex function

Say that I have a function $f: S \rightarrow \mathbb{R}^+$ such that: $S \subset \mathbb{R}^n$ is a closed convex set such as $S=[-10,10]^n$ $f$ is continuous and infinitely differentiable at all ...
1
vote
1answer
50 views

Prove that $y-x < \delta$

In Hardy's Pure Mathematics it says if $x^2<2, y^2>2, 2-x^2 < \delta,$ and $y^2 - 2 < \delta$, then $y-x<\delta$. I added the last two inequalities to get $(y+x)(y-x)<2\delta$. How ...
1
vote
1answer
33 views

error bound in function approximation algorithm

Suppose we have the set of floating point number with "m" bit mantissa and "e" bits for exponent. Suppose more over we want to approximate a function "f". From the theory we know that usually a ...
1
vote
1answer
34 views

approximation of $\pi$ by $\arctan$

Determinate the order n of the Maclaurin polynomial for $f(x)=4tan^{-1}x$ so that the remainader term $|R_{n}(1)|<0.000005$. Here $R_{n}(1)=\frac{f^{(n+1)}(c)}{(n+1)!}$ for some c between 0 and 1 ...
2
votes
1answer
42 views

Help me approximate this sum: $S = \sum_{j=2}^{N}{\frac{\ln \ln \ln \ j}{( \ln \ln \ j)^2}}$

I would like to figure out the asymptotic rate of growth for the sum $S = \sum_{j=2}^{N}{\frac{\ln \ln \ln j}{( \ln \ln j)^2}}$ in the limit of large $N$. Ultimately, I want to know if $S(N)$ is ...
22
votes
5answers
507 views

Why is it that this gives a good approximation of $\pi$?

At the end of a Physics examination, I decided to play around with my calculator, as I always do when I have time left over, and found that: $$\frac{1}{100} \cdot 11^{\ln(11)} \approx ...
1
vote
0answers
58 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 ...
2
votes
1answer
71 views

Approximation for probability of at least $t$ events

I'm reading through a paper, and they are discussing the approximate probability that $t+1$ out of $t^b$ events occur, where $b$ is a constant, and the probability of each event occurring is ...
0
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2answers
47 views

How can I approximate a decimal with two fractions where denominator is less or equal to $d$

I was looking for a way to approximate a decimal number with a fraction, whose denominator is less or equal to $d$. Basically, having a decimal $X$, I want to find two fractions such that ...
1
vote
1answer
21 views

Singular Value Decomposition for an image understanding

I'm trying to get an intuitive understanding of what an SVD decomposition does to an image. From my understanding, for an image $A \in \Bbb R^{m \times n}$, the singular values are the roots of the ...
0
votes
0answers
21 views

intersection of 2 paths

Could somebody please advise me on the best approach to take to find the intersection points of 2 paths. Each path is a data set of distance against time (I want to avoid curve fitting) and each path ...
0
votes
1answer
23 views

Approximating the circumference of given ellipse

Say we got the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{24}=1$, and the goal is to find the circumference using line integrals. So I parametrized the curve by $x=5\cos(t)$, $y=\sqrt{24}\sin(t)$. By ...
1
vote
1answer
19 views

How is optimal coordinates change chosen for Chebyshev expansion?

I'm looking into SLATEC implementation of Bessel function $J_0$ computation (readable in C in GSL), namely at its part for arguments in interval $(0,4)$. There a Chebyshev expansion is used, but the ...
1
vote
2answers
36 views

inverse complementary error function values near 0

$\newcommand{\erf}{\operatorname{erf}}\newcommand{\erfc}{\operatorname{erfc}}$Let's define for each $x>0$ $$\erf(x)=\frac {2}{\sqrt{\pi}}\int_0^xe^{-t^2} dt$$ and $$\erfc(x)=\frac ...
0
votes
0answers
29 views

Amenable groups are sofic

I am trying to understand proposition II.3.1 of http://arxiv.org/pdf/1309.2034v6.pdf, but I have some difficulties. I get the construction of the permutations $\sigma_{\gamma}$ for each $\gamma\in ...
3
votes
1answer
61 views

Higher Order PDE using Finite Difference

How to approximate higher-order partial differential equation using finite difference method? $$\frac{\partial^{2} y}{\partial t^{2}}+\frac{\partial^{4} y}{\partial x^{4}}=0$$
0
votes
0answers
19 views

System of Equations & Approximations

I am trying to derive Eq. (3.6) in the following thesis: http://drum.lib.umd.edu/bitstream/1903/14898/1/Khalil_umd_0117E_14726.pdf This is the equation I am trying to show: \begin{equation} ...
0
votes
1answer
21 views

Uniqueness of function approximation over three points?

Given a function $f(x)$, we want to approximate $f$ using $P(x)$, such that: $P(x_0) = f(x_0)$, $P(x_2) = f(x_2)$, $P'(x_1) = f'(x_1)$. Prove that such a $P$ is unique $\iff$ $x_1 \neq ...
0
votes
1answer
38 views

What can be said about $f''$ if the trapezoidal approximation is always an overestimate?

For any $a$ and $b$ the Trapezoidal approximation of the integral $\int_a^b f(x)\,dx$ is an overestimate. What can you conclude about the second derivative of $f$? I think it might mean that the ...
1
vote
1answer
38 views

Under which condition is $\hat\Sigma\approx\frac{1}{T-1}(X'X)$

Let $X$ be $T\times N$ random matrix. We are interested in the sample variance covariance matrix of $X$. It holds that \begin{align} ...
0
votes
2answers
34 views

Approximate summation of flooring function

I have this summation: $\sum_{k=0}^x \lfloor{\frac{k}{c}}\rfloor$ Do you have any ideas on any general expressions that can approximate this? P.S I know I can approximate it with a Fourier ...
0
votes
0answers
14 views

selecting points on a domain which represent the derivative of a function

I'm working on some algorithm part of which entails me to subdivide a domain based on the derivative of a function. Let's just consider the 1D case with a closed and bounded domain $[a,b]$ for a ...
1
vote
0answers
17 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 ...
4
votes
1answer
55 views

Is there an alternative to Taylor expansion of functions with more control over the error distribution?

As a physics student I see the Taylor series being (ab)used very often, mostly for the expansion of $\exp(x)$. The usual line of though is that the error (remainder) term of a high order is likely ...
3
votes
2answers
43 views

Asymmetric second difference quotient?

I need to find (approximate) the second derivative of a discrete function. Usually I would approximate the second derivative with $$f''(x)\approx\frac{f(x+h)-2f(x)+f(x-h)}{h^2}\tag{1}$$ In my case, ...
4
votes
1answer
38 views

Order of Growth of a Sum

Let $n>k$ and $r$ be arbitrary positive integers. Define $q=k/n$. I want to show that $$ \sum_{i=0}^{rk} \binom{rn}{i}q^i(1-q)^{rn-i}(rk-i)=\Theta(\sqrt{r}) $$ as $r\rightarrow \infty$. I've ...