## Splitting of cochains of loop space

Evaluation at base point induces a splitting of homology of free loop space $$LM$$ of a compact manifold $$M$$, i.e. $$H_*(LM)\cong H_*(M) \oplus H_*(LM, M)$$. Can such splitting be realised on cellular chain level, possibly given by some suitable Morse function on $$LM$$?

## How to move app/window to another space in mojave?

Sometimes row of spaces appears at top of the screen while dragging window to the top. It happened without activating mission control! Unfortunately I am not able to reproduce this behaviour…

## Start a window on top of a full screened space

How can I open a window on top of a full screened space?

I full screen a video in Safari.

I want to run some app that opens a little window on top of that full-screened video.

I don’t mind what technologies I use to do this (AppleScript, some bash commands, python, Swift, Xcode, whatever) just wondering how Apple makes it possible.

## Ubuntu freezes and notifies low disk space

I have Ubuntu installed via virtualbox on windows7. Ubuntu tends to frequently freeze. In addition in each boot there is a low disk space notification, which may be related. The storage is configured dynamically as far as i see. and i think i configured it to 10GB. Attached the setting image showing it has 10GB virtual size and 10GB actual size. Also attached disk analysis And attaching df-h (which i didn’t know how to paste here in a readable way)

In addition this is what lsblk outputs:

NAME   MAJ:MIN RM   SIZE RO TYPE MOUNTPOINT loop0    7:0    0 140.7M  1 loop /snap/gnome-3-26-1604/82 loop1    7:1    0 140.7M  1 loop /snap/gnome-3-26-1604/74 loop2    7:2    0   3.7M  1 loop /snap/gnome-system-monitor/70 loop3    7:3    0 294.2M  1 loop /snap/pycharm-community/121 loop4    7:4    0   2.3M  1 loop /snap/gnome-calculator/260 loop5    7:5    0  1008K  1 loop /snap/gnome-logs/61 loop6    7:6    0    13M  1 loop /snap/gnome-characters/139 loop7    7:7    0 295.9M  1 loop /snap/pycharm-community/123 loop8    7:8    0  53.7M  1 loop /snap/core18/941 loop9    7:9    0    91M  1 loop /snap/core/6405 loop10   7:10   0   3.7M  1 loop /snap/gnome-system-monitor/57 loop11   7:11   0     4M  1 loop /snap/gnome-calculator/406 loop12   7:12   0  91.1M  1 loop /snap/core/6531 loop13   7:13   0  34.8M  1 loop /snap/gtk-common-themes/1122 loop14   7:14   0   151M  1 loop /snap/gnome-3-28-1804/31 loop15   7:15   0 140.7M  1 loop /snap/gnome-3-26-1604/78 loop16   7:16   0  53.7M  1 loop /snap/core18/782 loop17   7:17   0  89.3M  1 loop /snap/core/6673 loop18   7:18   0   3.7M  1 loop /snap/gnome-system-monitor/77 loop19   7:19   0   151M  1 loop /snap/gnome-3-28-1804/36 loop20   7:20   0  14.5M  1 loop /snap/gnome-logs/45 loop21   7:21   0  35.3M  1 loop /snap/gtk-common-themes/1198 loop22   7:22   0  34.6M  1 loop /snap/gtk-common-themes/818 loop23   7:23   0  14.8M  1 loop /snap/gnome-characters/254 loop24   7:24   0  14.8M  1 loop /snap/gnome-characters/206 loop25   7:25   0  1008K  1 loop /snap/gnome-logs/57 loop26   7:26   0     4M  1 loop /snap/gnome-calculator/352 loop27   7:27   0 143.5M  1 loop /snap/gnome-3-28-1804/23 sda      8:0    0    10G  0 disk  └─sda1   8:1    0    10G  0 part / sr0     11:0    1    82M  0 rom  /media/yoni/VBox_GAs_6.0.4 

my questions:

1. Is the slowness and especially frequent freezing of the system relates to the low disk space?
2. If answer to 1 is yes, how do i increase disk space?
3. If the answer to 1 is no and i can ignore the disk space issue (or should i solve it anyway?) what can solve my Ubuntu from constant freezing?

## Shell script to calculate the total space consumed by a user by file-size

I’m working with a small bash code which is working fine but i’m just looking if there is better way to formulate this, In this code i’m looking for the Files between year 2002 and 2018 on the 7th column.

Below is the working code,

## Script:

#!/bin/bash # scriptName: Ftpcal.sh FILE="/home/pygo/Cyberark/ftplogs_3" AWK="/bin/awk" GREP="/bin/grep" USERS=""$AWK" '$  7 >= "2002" && $7 <= "2018"'$  FILE | "$AWK" '{print$  3}' | sort -u" for user in $USERS; do echo "User$  user " | tr -d "\n"; "$AWK" '$  7 >= "2002" && $7 <= "2018"' "$  FILE" | "$GREP" "$  user" | "$AWK" '{ total +=$  4}; END { print "Total Space consumed: "  total/1024/1024/1024 "GB"}'; done | column -t echo "" echo "==============================================================" "$AWK" '$  7 >= "2002" && $7 <= "2018"' "$  FILE" | "$AWK" '{ total +=$  4}; END { print "Total Space consumed by All Users: "  total/1024/1024/1024 "GB"}'; echo "" 

$sh Ftpcal.sh User 16871 Total Space consumed: 0.0905161GB User 253758 Total Space consumed: 0.0750855GB User 34130 Total Space consumed: 3.52537GB User 36640 Total Space consumed: 0.55393GB User 8490 Total Space consumed: 3.70858GB User tx-am Total Space consumed: 0.18992GB User tx-ffv Total Space consumed: 0.183137GB User tx-ttv Total Space consumed: 17.2371GB User tx-st Total Space consumed: 0.201205GB User tx-ti Total Space consumed: 58.9704GB User tx-tts Total Space consumed: 0.0762068GB ------------ snipped output -------------- ============================================================== Total Space consumed by All Users: 255.368GB  ## Sample data: -rw-r--r-- 1 34130 14063436 Aug 15 2002 /current/focus-del/files/from_fix.v.gz -rw-r--r-- 1 34130 14060876 Jul 12 2007 /current/focus-del/files/from1_fix.v.gz -rw-r--r-- 1 34130 58668461 Feb 23 2006 /current/focus-del/files/from_1.tar.gz -rw-r--r-- 1 34130 14069343 Aug 7 20017 /current/focus-del/files/from_tm_fix.v.gz -rw-r--r-- 1 34130 38179000 Dec 7 20016 /current/focus-del/files/from_tm.gds.gz -rw-r--r-- 1 34130 15157902 Nov 22 20015 /current/focus-del/files/from_for.tar.gz -rw-r--r-- 1 34130 97986560 Nov 4 20015 /current/focus-del/files/from_layout.tar  ## Sample Result: $   sh Ftp_cal.sh User  34130  Total  Space  consumed:  0.0808321GB  ============================================================== Total Space consumed by All Users: 0.0808321GB 

I’m okay with any better approach as a review process to make it more robust.

Thanks.

## Out of Space, Out of Space, Out of Space, Out of Space, . . . . .

Can someone tell me, succinctly, how to prevent this very irritating problem happening ever again? According to someone’s advice to another poster, I ran this command:

sudo find . -xdev -type f | cut -d “/” -f 2 | sort | uniq -c | sort -n

and got the following message (who could’ve predicted that?!):

sort: write failed: /tmp/sort7tteF8: No space left on device

I don’t even want to hear about inodes, or long lists of commands to put into the Terminal, etc. And I’m tired of scrolling down page after page, looking for a recent, relevant and, hopefully, useful answer in a forum such as this. I simply want my Ubuntu system NOT to run out of disk space when there is NO reason for it. Somebody help me, please! Thank you.

## Does non-zero winding number on a space curve imply a linked curve in the zero set?

Let $$f \colon \mathbb{R}^3 \to \mathbb{R}^2$$. I think I need $$f$$ to be real analytic, but maybe it’s not necessary. Let $$C$$ be a closed space curve in $$\mathbb{R}^3$$, which I might need to assume to be unknot. Is the following true?

If $$f(C) \subset \mathbb{R}^2$$ has a non-zero winding number around $$0$$, then $$f^{-1}(0)$$ contains a space curve in $$\mathbb{R}^3$$ that is linked to $$C$$.

I consider at as a generalization of Kronecker’s existence theorem.

If not true, what conditions am I missing to fix it? If true, where do I find a reference for this and its higher dimensional analogues?

## Spanning a finite vector space using different powers of a diagonalizable matrix.

D be an n x n real diagonal matrix. show that there exists a x $$\epsilon$$ $$\Bbb R ^n$$ such that

span{D$$^k$$x: 0$$\le$$k$$\le$$(n-1)} = $$\Bbb R ^n$$

if and only if the eigenvalues of D are distinct.

I can show that if D has distinct eigenvalues then $$\Bbb R ^n$$ isa direct sum of n eigenspaces and I can find the required x such that the condition holds, but i am unable to prove it the other way round.

## Does the orthogonal polar factor of the differential of a smooth map lie in a Sobolev space?

This question address a special case of this one.

Let $$\mathbb{D}^2$$ be the closed unit disk, and let $$f:\mathbb{D}^2 \to \mathbb{R}^2$$ be a real-analytic map, satisfying $$\det df>0$$ almost everywhere. Define $$U:= \{ p \in \mathbb{D}^2 \, | \, \det df>0\}$$

Let $$Q=Q(df)=df(\sqrt{df^Tdf})^{-1}$$ be the orthogonal polar factor of $$df$$; $$\sqrt{}$$ is the unique symmetric positive-definite matrix square root. $$Q$$ is well-defined uniquely on $$U$$-in particular it is defined a.e. on $$\mathbb{D}^2$$. The restriction $$Q|_U$$ is smooth.(even real-analytic).

Question: Does $$Q \in W^{1,p}(\mathbb{D}^2,\mathbb{R}^{4})$$ for some $$p \ge 1$$?

Discussion:

I think that the answer might be negative.

The only possible problem is when we approach critical points of $$f$$, where $$df=0$$. Indeed, the polar factor map $$\text{GL}_2^+ \to \text{SO}_2$$ can be extended to a smooth map from $$\text{GL}_2^+ \cup (\text{rank} = 1) \to \text{SO}_2$$. (In general dimension $$n$$, there is a smooth extension to $$\text{GL}_n^+ \cup (\text{rank} \ge n-1)$$).

However, the polar factor cannot be chosen continuously on all $$\text{GL}_2^+ \cup \det^{-1}(0)$$; there are singularities:

1. $$\lim_{t \to 0} Q(\left(\begin{matrix}t & 0 \ 0 & t\end{matrix}\right))=\text{sgn}(t) \left(\begin{matrix}1 & 0 \ 0 & 1\end{matrix}\right)$$, and

2. $$\lim_{t \to 0}Q(\left(\begin{matrix}0 & -t \ t & 0\end{matrix}\right))=\text{sgn}(t) \left(\begin{matrix}0 & -1 \ 1 & 0\end{matrix}\right)$$.

Since the sign function is not Sobolev, I tried lifting these singularities to create a singularity in $$Q=Q(df)$$ for some map $$f$$. My first attempt was to use only one of the two singularities above, but this failed- if assume $$df$$ is always of the form $$\left(\begin{matrix}t(x,y) & 0 \ 0 & t(x,y) \end{matrix}\right)$$ or always of the form $$\left(\begin{matrix}0 & -t(x,y) \ t(x,y) & 0\end{matrix}\right)$$, then it must be constant. Thus, it seems that in order to create a singularity, we somehow need to create a situation where $$df$$ interpolates between these two forms.

Comment: If we allow $$df$$ to switch orientations, then the orthogonal factor clearly does not need to be Sobolev. Even in dimension $$1$$, for $$f(x)=x^2$$, we have $$Q(x)=\text{sgn}(f'(x))=\text{sgn}(x)$$ which is not in $$W^{1,1}(\mathbb R,\mathbb R)$$. The purpose of this question is to determine if non-negligible singularities may occur when the map is a.e. orientation-preserving.

(Note that $$Q(\left(\begin{matrix}t & 0 \ 0 & s\end{matrix}\right))= \left(\begin{matrix}\text{sgn}(t) & 0 \ 0 & \text{sgn}(s)\end{matrix}\right)$$, but if we want $$\det \left(\begin{matrix}t & 0 \ 0 & s\end{matrix}\right)=ts \ge 0$$, we must force $$s,t$$ to switch signs “at the same times” (points), i.e. they should be “coupled” somehow, which explains why I chose to take $$s=t$$ in the examples above).

## “Insufficient Storage Space” problem with Titanium Root, preventing backup

I’m in a pickle I have no idea how to fix. I turned off the often suggested “Mount Namespace Separation” option in SU, and it doesn’t work with this issue. I uninstalled/ reinstalled Titanium Root, no luck. Restarted the device after toggling off the namespace option, no dice. I have no idea what to do. It’s not entirely a loss, as some games will backup just fine, just one brand is being stubborn. The Flutter series, where you collect butterflies and such. They’re huge files, and it backs them up for a long while before hitting the memory space issue. It is confirmed and set to the Titanium Backup file on my SD card, where I backed up other games. Any suggestions?