There is a conclusion :Suppose $M$ is a connected topological manifold,then $M$ is path-connected.  We can define $S_p=\{q\in M:$there is a path from $p$ to $q\}$.Obviously,$S_p$ is nonempty since $p\in S_p$,If we prove that $S_p$ and $M\setminus S_p$ is open,then we can conclude that $M=S_p$ by connectness of $M$.  Suppose $q\in S_p$,then there is a coordinate chart $(U,\phi)$ around $q$ and $\phi(U)$ is open in $R^n$ for some $n$,how to prove $S_p$ is open by using the above fact?
prove every connected manifold is path connected manifold .       my thought:      connected space : Let $ X$  be a topological space. A separation of $ X $ is a pair $U, V$ of disjoint    nonempty open subsets of $ X $ whose union is $X$. The space $ X $ is said to be connected    if there does not exist a separation of  $X$ .      Components: Given  $X$ , define an equivalence relation on $X$ by setting x~y if there    is a connected subspace of $X$  containing both $ x$  and $ y$ . The equivalence classes are    called the components (or the "connected components") of $ X$.       Path Component: I define another equivalence relation on the space $ X$  by defining x ~ у    if there is a path in $ X$  from $ x$  to $ y$ . The equivalence classes are called the path components of $ X$ .       Theorem  : The path components of $ X$  are path connected disjoint subspaces    of $X$  whose union is $X$ , such that each nonempty path connected subspace of $X$  intersects only one of them.       thank you so much
Calling delete on variable allocated on the stack How to run applications as root? I am having some strange issue with Kate and Kwrite. When I click on Open File, it crashes with segmentation fault.   I am a complete newbie to Linux, and I think the issue is that I am not running the application as root.   How do I run applications as root in Ubuntu? Is it bad practice to do this? What is the purpose of the whole root thing, where even though we need to use root so frequently, it is not utilized as default? I'm using Ubuntu 13.04 on a Lenovo X220.  Why do I have so many .Xauthority.* files in my home directory? I.e  .Xauthority .Xauthority.0JW6UW .Xauthority.2HWRVW .Xauthority.2TA6VW .Xauthority.49F8VW .Xauthority.4E27UW .Xauthority.7CLTVW .Xauthority.8JGGXW .Xauthority.96SAVW .Xauthority.AOUFVW .Xauthority.ASDOVW .Xauthority.CGVJXW .Xauthority.E0ZSVW .Xauthority.E5VZWW .Xauthority.FE64UW .Xauthority.FIHDWW .Xauthority.HL45VW .Xauthority.HYCCVW .Xauthority.ILUEWW .Xauthority.JXJHVW .Xauthority.K1QJXW .Xauthority.KXSOVW .Xauthority.LEKEVW .Xauthority.M48WVW .Xauthority.M6QTVW .Xauthority.MW6NWW .Xauthority.N57TWW .Xauthority.O5HAWW .Xauthority.OIC5VW .Xauthority.P2RPVW .Xauthority.Q0FRVW .Xauthority.QW01WW .Xauthority.RDW5WW .Xauthority.T0TLWW .Xauthority.TSO4WW .Xauthority.U6R9VW .Xauthority.V3KAWW .Xauthority.W8Z1WW .Xauthority.XDLGVW .Xauthority.XOOHWW.Xauthority.XUICVW  They are all zero bytes except the first, with timestamps all over the place.  What are these, and are they safe to delete? How  can I delete a node after, for example, 2 weeks it has been created? A Putnam Integral $\int_2^4 \frac{\sqrt{\ln(9-x)}\,dx}{\sqrt{\ln(9-x)} + \sqrt{\ln(x+3)}}.$ This is a Putnam Problem that I have been trying to solve (on and off) for two years, but I have failed. I am in Calculus BC. This problem comes from the book "Calculus Eighth Edition by Larson, Hostetler, and Edwards". This problem is at the end of the first section of the chapter 8 exercises. Here's the problem:  Evaluate $$\int_2^4 \frac{\sqrt{\ln(9-x)}\,dx}{\sqrt{\ln(9-x)} + \sqrt{\ln(x+3)}}.$$  Please. Any help is very much appreciated. So are solutions. Thank you!  Edit: I like the solution given, but I was interested to see if there is any other way of doing the problem? I'm excited to see the results. I have an Android phone that connects with my computer via MTP. This works fine; I can see and transfer files with Nautilus. However, I often want to use the terminal to move large numbers of files, and I cannot seem to find the device anywhere in the filesystem tree. Nautilus reports the location as mtp://[usb:003,007]/, but it's not under /media or /mnt.  Does anyone know where it is? Whoa! What is this "Linked" thing? As per the title, what is this new feature? "Forgot" vs "Forget" Is the following correct, or is there more to it?  "I forgot his name" — I knew his name, but I forgot it.  "I forget his name" — I keep forgetting his name. Where using "forget" basically means that you tried to remember this information before as well, but you couldn't then either. What is the proof that the total number of subsets of a set is $2^n$? Solvability of $a \equiv x^2 \mod b$ How to achieve this smudge effect in photoshop? Normalizing point cloud data How I can get Digital Height Model (only object heights, DSM - DTM = DHM)?  Many applications can do this, but they convert it to GRID or TIN format. I was wondering if it was possible to keep the data as point clouds. Assume that given three predicates are presented below: $H(x)$: $x$ is a horse $A(x)$: $x$ is an animal $T(x,y)$: $x$ is a tail of $y$ Then, translate the following inference into an inference using predicate logic expressions and prove whether inference is valid or not (for instance, using natural deduction): Horses are animals.  Horses' tails are tails of animals. My thoughts: I am quite good at translating predicate logic expressions, but here I struggled to come up with formula for Horses' tails. My initial idea was to consider similar sentence such as &quot;w is a tail of a horse&quot; to form required inference, but it was not successful. Would be welcomed to hear your ideas about this task. How to find out the reason, why my PC can't run Windows 11? Windows 11 requires this features:  Processor 1 GHz or faster with 2 or more cores on a compatible 64-bit processor or System on a Chip (SoC) Memory 4 GB RAM Storage 64 GB or larger storage device System firmware UEFI, Secure Boot capable TPM Trusted Platform Module (TPM) version 2.0 Graphics card DirectX 12 compatible graphics / WDDM 2.x Display &gt;9” with HD Resolution (720p) Internet connection Microsoft account and internet connectivity required for setup for Windows 11 Home  I have a 15-ay108nh laptop. As far as I know, it supports all the required features. It has 8GB ram, 256GB storage, UEFI support, TPM 2.0 ( I enabled it in the BIOS ), DirectX 12 compatible graphics / WDDM 2.x, and 1080p screen. I installed the , but for some reason, I get this result:  How is it possible to find out exactly, why it fails? If nonnegative $f: [0,1] \rightarrow \mathbb{R}$ has a continuous $f''$, then $\int_0^1 \Big| \frac{f''(x)}{f(x)} \Big| \,dx >4$ If $f: D(0,1)\to \mathbb C$ is holomorphic, then there is a sequence $\{z_n\}$ in $D(0,1)$ such that $|z_n| \to 1$ and $\{f(z_n)\}$ is bounded Why are there two different hats for "Where in the World"? I am wondering why some got a moon and others got a sun as Where in the World hat for the same action:   vs.   And is it possible two collect both e.g. from different sites? I only get moons…  I know that there has already been some spoilering . L1 regression estimates median whereas L2 regression estimates mean? Need to get logfile smaller MySQL is giving weird result