## Square covered with circles November 30    2

I have a square 800x800 and i need to fully cover it with the least number of circles possible, each circle has a radius of 150. QUESTIONS: - What pattern would be the best to use? Clover, d ...

## Integral Apollonian circle packing with unique curvatures November 30    1

I was wondering if it is possible to construct an Apollonian gasket where every circle has a unique integer curvature.Take for instance the following gasket, defined by curvatures (−10, 18, ...

## What is the equation to evenly distribute circles in a spiral November 30    1

What is the equation to evenly distribute circles in a spiral? I have attached a picture to show what I am trying to achieve and need to know what the equation is for this.This packing is di ...

## Proof: At most 3 circles of radius 1/2 fit into the interior of a halfcircle of radius 1 November 30

It is a well known fact that at most 7 interior disjoint circles of radius 1/2 can be centered in a circle of radius 1; note that they don't need to be fully contained in the radius 1 circle.I am interested in a simple proof idea to show that at most

## Smallest-circle problem, but with circles instead of points November 30

I have a growing set of circles, each step I add one. I also need the smallest circle that contains all of the circles in my set. I found the wikipedia page about the smallest-circle problem and it states that there is a linear-time recursive algorit

## Showing a set has Jordan content November 30

The question is :Let $a<b$ and $f:[a,b]\rightarrow[0,\infty)$ be continuous. Let $D = \{(x,y)\in\mathbb{R}^2:\:x\in[a,b],\: y\in[0,f(x)]\}$. Show that D has content and $$\mu(D) = \int_{a}^bf(x)dx$$Here, $\mu(D)$ is the Jordan content of D.If I can

## Solve the integral equation with symmetric kernel November 30

I have the following integral equation with symmetric kernel $$g(x)=cos \pi x + \int_{0}^{1} k(x,t)g(t)dt$$ where $k(x,t)$ is a symmetric kernel given by $$k(x,t)= (x+1)t , 0 < x <t$$ and $$(t+1)x , t < x <1$$please ,help me I do not want

## Riemann Sum of $\int_{-1}^1x \ dx$ November 30

I am trying to find $\int_{-1}^1f(x) \ dx=\int_{-1}^1|x| \ dx$ using Riemann Sums. I have split $[-1,1]$ into two partitions, $P_n$ and $Q_n$, as the function is decreasing and increasing between $-1$ and $1$. So $P_n$ is a partition of $[-1,0]$ and

## Can we find a closed form for $\int\nolimits_{- \infty}^{\infty} \frac{\exp\left(-(a+bx)^2\right)}{1+\exp(x)}\mathrm dx$ November 30    4

Can we find a closed form for this definite integral: $$\int\nolimits_{- \infty}^{\infty} \frac{\exp\left(-(a+bx)^2\right)}{1+\exp(x)}\mathrm dx$$ The integral has a closed form when $a = 0$ and $b \neq 0$. (The integral diverges if $b=0$.) We have

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