A team is to play $n$ games in a row.  In how many ways could they play this series of games without losing two games in a row.  There are $2^n$ total games to be played.  The number of games without consecutive losses would be $2^N$ - (Number of games with at least one consecutive loss).  I'm having a hard time counting the number of games with at least one consecutive loss.  In particular, I can't think of a way to count the number of games with $m$ consecutive losses.  Could someone provide some guidance?
I need to calculate no of possible substrings containing "00" as a substring. I know the length of the binary string.  Eg: for a string of length 4, possible substrings are: 0000 0001 0010 0011 0100 1000 1001 1100  I just need the number of possible combinations, not enumerate all of them.
When can I use a forward declaration? How many times has the matter in our solar system been recycled from previous stars? I've got a basic understanding of these facts:   The Universe is a little over 13 billion years old.  Our galaxy is almost that old. Our solar system is roughly 4.6 billion years old. The heavier elements (Carbon, oxygen, etc...) are only produced by stars.   When our solar system coalesced, the matter that makes up the planets came from a previous generation of stars.  (Meanwhile our Sun is busy making its own slightly-heavier elements, but those aren't ours.  Those belong to the next "generation" after our star dies.)  So we're all sitting on a rock whose matter was spewed out of a star (or stars) that preceded this solar system. But what I want to know is:  How many cycles of coalescing, fusing, nucleosynthesis, exploding -- and then repeating the cycle -- have this atoms in this ball of rock I'm standing on been through?    How do you know this?  Can you tell by the make-up of the matter in the solar system?  Looking around the Orion arm to see remnants of our former home? I am looking for a word that would describe someone who always goes out of their way to take their time and do a good job. Sort of the opposite of half-assing something.  I am looking for something that is more about a work ethic than a technically proper way of doing things. Something like canonical or protocol would not be a good fit. Why aren't my turtles breeding? No module named numpy I am trying to run Jupyter notebook from my Ubuntu subsystem in Windows 10. I've installed numpy, scipy, and pandas using the following commands:  pip install scipy numpy pip install pandas   When I load python in Ubuntu, there is no issue importing numpy in Ubuntu cmd.   But, when I run my Jupyter Notebook from Ubuntu terminal, and try to import numpy as np, or import pandas as pd, they report  ModuleNotFoundError                       Traceback (most recent call last) &lt;ipython-input-1-a9340201ed9f&gt; in &lt;module&gt;       5 import dash_html_components as html       6 import plotly.graph_objs as go ----&gt; 7 import numpy as np       8 from dash.dependencies import Input, Output       9   ModuleNotFoundError: No module named 'numpy'   Is there anyway to solve this issue? Thanks a lot. How do I enable or disable the global application menu? Usually I use beamer  with the Warsaw theme which puts theorems (and other things) in boxes. Now I switched my theme to Marburg but i still want to keep the boxes. How can one do that? Let $p&gt;3$ be a prime. Prove that $24 \mid p^2-1$.   I know this is very basic and old hat to many, but I love this question and I am interested in seeing whether there are any proofs beyond the two I already know. I'm a beginner with blender so this is probably a very basic question.  I have this basic model below. I've ended up with a lot of unwanted geometry that I want to simplify. I wanted to start by merging the two selected faces below but no matter what I try to delete or merge it just destroys the geometry.  How can I achieve what I want to do (which is combining all of the unwanted geometry, either starting with this merge or some other way?) Executing a certain action for all elements in an Enumerable How do I change the GRUB boot order? I have both Windows 7 and Ubuntu installed on a shared machine. Because a lot of the non-developers use Windows, I'd like to change the boot order to make it easier for them.  Currently the boot order looks like the following:   Ubuntu 11.10 kernelgeneric *86 Ubuntu 11.10 kernelgeneric *86 (safe boot)  Memory test Memory test Windows 7 on /dev/sda6   How do I change the default order so that Windows 7 is at the top of the list?   Windows 7 on /dev/sda6 Ubuntu 11.10 kernelgeneric *86 Ubuntu 11.10 kernelgeneric *86 (safe boot)  Memory test Memory test Reflection is quad dependent and looks non realistic Im rendering varous objects with a Sharp glossy effect but the reflections are quad dependent on the object (image below), meaning they behave separately within every quad making it look 8bit-like. Can anyone tell me why this happens?. (Im pretty sure subdividing would fix it but I want to know why I cant have a low poly object reflecting normaly.  The object has 2 modifiers: Subdiv Surface and Solidify. (Perhaps the Subdiv is causing it?). Help me you Blender savys I dont have a powerful CPU to test and test within every render would take me hours and hours.  Image: Taskbar > File Explorer > Context Menu failing me So I have the latest version of Windows 10, and this is a new PC.  File Explorer is in the taskbar.  When I open the context menu (right-click) on it, I get the list of folders as seen in the picture.  When I right click on any of them, I get the blank black box you see in the picture.  If I click on any of the folders, such as Downloads or Update.  Nothing happens.  They don't open.  The only way to use it is to click on the actual File Explorer icon and then navigate from there.  I've tried uninstalling things (not that I have much installed since it's a fresh install) and I've tried scans, etc.  This only happens with File Explorer.  Everything else works fine.  I can't figure it out. I am  with the Google login for Pokemon Go. Is there any way, be it contacting support or through Trainer Club, to either link my Google account to a new Trainer Club account or transfer my account to a Trainer Club account?  It seems like a better idea to have a Trainer Club account, so if there are Google login problems it won't matter.  Does anyone know if something like this is possible? After Hoth, why did Vader pursue the Millennium Falcon instead of Luke's X-Wing? In The Empire Strikes Back film, the opening crawl says:     The evil lord Darth Vader, obsessed with finding young Skywalker, has dispatched thousands of remote probes into the far reaches of space....   It has also been established in Return of the Jedi that Force users can sense each other:     Chewie whines nervously. Luke stares at the Huge Super Star Destroyer    that looms ever larger before them.      LUKE: Vader's on that ship.   If Darth Vader is obsessed with finding Luke and he can sense his presence, the obvious thing to do would be to follow Luke's X-Wing after the battle of Hoth rather than the Millennium Falcon. Why didn't he? Computing $\int_{0}^{\pi}\ln\left(1-2a\cos x+a^2\right) \, dx$ For $a\ge 0$ let's define $$I(a)=\int_{0}^{\pi}\ln\left(1-2a\cos x+a^2\right)dx.$$ Find explicit formula for $I(a)$.   My attempt: Let  $$\begin{align*} f_n(x) &amp;= \frac{\ln\left(1-2 \left(a+\frac{1}{n}\right)\cos x+\left(a+\frac{1}{n}\right)^2\right)-\ln\left(1-2a\cos x+a^2\right)}{\frac{1}{n}}\\ &amp;=\frac{\ln\left(\displaystyle\frac{1-2 \left(a+\frac{1}{n}\right)\cos x+\left(a+\frac{1}{n}\right)^2}{1-2a\cos x+a^2}\right)}{\frac{1}{n}}\\ &amp;=\frac{\ln\left(1+\dfrac{1}{n}\left(\displaystyle\frac{2a-2\cos x+\frac{1}{n}}{1-2a\cos x+a^2}\right)\right)}{\frac{1}{n}}. \end{align*}$$ Now it is easy to see that $f_n(x) \to \frac{2a-2\cos x}{1-2a\cos x+a^2}$ as $n \to \infty$. $|f_n(x)|\le \frac{2a+2}{(1-a)^2}$ RHS is integrable so $\lim_{n\to\infty}\int_0^\pi f_n(x)dx = \int_0^\pi \frac{2a-2\cos x}{1-2a\cos x+a^2} dx=I'(a)$. But  $$\int_0^\pi \frac{2a-2\cos x}{1-2a\cos x+a^2}=\int_0^\pi\left(1-\frac{(1-a)^2}{1-2a\cos x+a^2}\right)dx.$$ Consider $$\int_0^\pi\frac{dx}{1-2a\cos x+a^2}=\int_0^\infty\frac{\frac{dy}{1+t^2}}{1-2a\frac{1-t^2}{1+t^2}+a^2}=\int_0^\infty\frac{dt}{1+t^2-2a(1-t^2)+a^2(1+t^2)}=\int_0^\infty\frac{dt}{(1-a)^2+\left((1+a)t\right)^2}\stackrel{(*)}{=}\frac{1}{(1-a)^2}\int_0^\infty\frac{dt}{1+\left(\frac{1+a}{1-a}t\right)^2}=\frac{1}{(1-a)(1+a)}\int_0^\infty\frac{du}{1+u^2}=\frac{1}{(1-a)(1+a)}\frac{\pi}{2}.$$ So $$I'(a)=\frac{\pi}{2}\left(2-\frac{1-a}{1+a}\right)\Rightarrow I(a)=\frac{\pi}{2}\left(3a-2\ln\left(a+1\right)\right).$$  It looks too easy, is there any crucial lack?  $(*)$ — we have to check $a=1$ here by hand and actually consider $[0,1), (1,\infty)$ but result on these two intervals may differ only by constant - it may be important but in my opinion not crucial for this proof. How can we remove the launcher that come with Unity on this new Ubuntu version ? Deciding whether two metrics are topologically equivalent in the space $C^1([0,1])$ Consider the space $C^1([0,1])$ and the function $d:C([0,1])\times C([0,1]) \to \mathbb R$ defined as $d(f,g)=|f(0)-g(0)|+sup_{x \in [0,1]}|f'(x)-g'(x)|$. Decide whether the metrics $d$ and $d_{\infty}$ are topologically equivalent in $C^1([0,1])$ (where $d_{\infty}=sup_{x \in [0,1]}|f(x)-g(x)|)$  My attempt at a solution:  If two metrics are topologically equivalent, then they have the same convergent sequences. Honestly, I couldn't do anything. I am trying to define a sequence of functions $\{f_n\}_{n \in \mathbb N}$ such that $f_n \to f$ in, for instance, $(C^1([0,1]),d)$ but $f_n \not \to f$ in $(C^1([0,1]),d_{\infty})$. Could it be this two metrics are topologically equivalent? If this is the case, how could I prove it? If not, I would appreciate any hint to find an adequate sequence of functions that works for what I am trying to prove. I had Ubuntu 12.04 Beta installed on my computer. When 12.04 was finally released, I made a bootable USB Flash Drive using Startup Disk Creator.  Then I decided to check if this drive works properly and to reinstall the system on my desktop. I must add, my desktop behaves a bit strange when it comes to bootable USB's, it recognizes them as HDD.  In BIOS I changed priority of boot so USB Flash Drive (recognized as HDD) was first. Successfully booted, I installed Ubuntu. Everything worked fine, but...  Now I cannot boot from my real HDD. Every time I want to boot, I put the USB Flash Drive into my computer, boot, safely remove it and everything works.  What do I need to do to repair boot? I have attached an image as an example below. I want to twist a rectangle. Do I have to convert that rectangle into a curve or into mesh and then twist it with ctrl+T?   Please explain with an example.