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 ...
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? 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 ...
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
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
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
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
This is done in the solution of exercise in order to make it possible to do inverse Laplace transform. Though I am not sure how is that done, so here it is:$$\frac{s^2+3s+3}{2s^2+7s+7}=\frac{1}{2}-\frac{s+1}{2(2s^2+7s+7)}=\frac{1}{2}-\frac{0.5s+0.5}{
I missed a class this week in maths and been a bit lost since with Inverse Laplace, how do I go about finding the Inverse laplace of: $$\frac{(s+1)^3}{s^4}$$Do I simply expand the numerator? then inverse laplace? or is there a quicker way of doing it
Express your answer in terms of the error function: $$L^{-1}[\frac{1}{\sqrt{s^3+as^2}}$$Clue: $\qquad L[\frac{1}{\sqrt{t}}]=\sqrt\frac{π}{s} \qquad , \qquad s>0$Error function: $\frac{2}{\sqrt{π}}\int_0^te^{-w^2}dw$
Let $f_n$ be a sequence of nonnegative measurable functions which converge to $0$. If there exists an $M$ such that $$\int \sup f_1, ... , f_n \leq M$$ for all $n$, then $\lim \int f_n = 0$. Could someone give me a hint on this? I don't know how to s
I'm confused by some Category Theory notation but I give the whole question I'm interested in solving below for the sake of context. Also, I want to verify my understanding of the proper approach to a solution assuming I can figure out this notation.
Let $g$ be a non-negative integral function over $E$ and suppose $\{f_n\}$ is a sequence of measurable functions on $E$ such that for each $n$, $|f_n| \leq g$ a.e. on E. Show that $ \int$ lim inf $f_n \leq $ lim inf $\int f_n \leq$ lim sup $\int f_n
A complex Mixed Hodge Structure is given by a complex vector space $V$ together with a descending filtration $W$ and two ascending filtrations $F,\bar{F}$ that satisfy the condition\begin{equation} gr_F^pgr_{\bar{F}}^qgr_n^W(V)=0\qquad\text{if }n\neq
I want to deal with a convex constaint \begin{align} F(P)=P^{H}AP_{0}+P_{0}^{H}AP-P_{0}^{H}AP_{0}\succeq 0 \end{align} where $(\cdot)^{H}$ represents Hermitian transpose, $A$ is a positive definite matrix, $P_{0}$ is a given square matrix and $P$ is
Does any one have the paper?: the obstruction to splitting a mixed hodge structure over the integers I, Preprint, University of Utah, 1979, by James Carlson. I cannot find it on google.
I am trying to find $\int_{-1}^1f(x) \ dx=\int_{-1}^1|x| \ dx$ using Riemann Sums. I have split $[-1,1]$ into two partitions, $P_n$ and $Q_n$, as the function is decreasing and increasing between $-1$ and $1$. So $P_n$ is a partition of $[-1,0]$ and
Can we find a closed form for this definite integral: $$ \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