- www.youtube.com/watch?v=-_qNKbwM_eE Unsolved: Yang-Mills existence and mass gap by J Knudsen (2019). Gives 10 key points, but the truly hard ones are too quick. He knows the thing though.
Yang-Mills 1 by David Metzler (2011)
Source. A bit disappointing, too high level, with very few nuggests that are not Googleable withing 5 minutes.
Breakdown:
- 1 www.youtube.com/watch?v=j3fsPHnrgLg: too basic
- 2 www.youtube.com/watch?v=br6OxCLyqAI?t=569: mentions groups of Lie type in the context of classification of finite simple groups. Each group has a little diagram.
- 3 youtu.be/1baiIxKKQlQ?list=PL613A31A706529585&t=728 the original example of a local symmetry was general relativity, and that in that context it can be clearly seen that the local symmetry is what causes "forces" to appear
- youtu.be/1baiIxKKQlQ?list=PL613A31A706529585&t=933 local symmetry gives a conserved current. In the case of electromagnetism, this is electrical current. This was the only worthwhile thing he sad to 2021 Ciro. Summarized at: local symmetries of the Lagrangian imply conserved currents.
- 4 youtu.be/5ljKcWm7hoU?list=PL613A31A706529585&t=427 electromagnetism has both a global symmetry (special relativity) but also local symmetry, which leads to the conservation of charge current and forces.lecture 3 properly defines a local symmetry in terms of the context of the lagrangian density, and explains that the conservation of currents there is basically the statement of Noether's theorem in that context.
Millennium Prize Problem: Yang Mills Theory by David Gross (2018)
Source. 2 hour talk at the Kavli Institute for Theoretical Physics. Too mathematical, 2021 Ciro can't make much out of it.Lorenzo Sadun on the "Yang-Mills and Mass Gap" Millennium problem
. Source. Unknown year. He almost gets there, he's good. Just needed to be a little bit deeper. Play with physical addresses in Linux by 
Ciro Santilli 35 Updated 2025-01-10 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-01-10 +Created 1970-01-01Convert virtual addresses to physical from user space with
/proc/<pid>/pagemap and from kernel space with virt_to_phys:Dump all page tables from userspace with
/proc/<pid>/maps and /proc/<pid>/pagemap:Read and write physical addresses from userspace with
/dev/mem:In
v4.2, look under arch/x86/:include/asm/pgtable*include/asm/page*mm/pgtable*mm/page*
There seems to be no structs defined to represent the pages, only macros:
include/asm/page_types.h is specially interesting. Excerpt:#define _PAGE_BIT_PRESENT 0 /* is present */
#define _PAGE_BIT_RW 1 /* writeable */
#define _PAGE_BIT_USER 2 /* userspace addressable */
#define _PAGE_BIT_PWT 3 /* page write through */arch/x86/include/uapi/asm/processor-flags.h defines CR0, and in particular the PG bit position:#define X86_CR0_PG_BIT 31 /* Paging */What if Process 1 tries to access
0x00003000, which is not present?The hardware notifies the software via a Page Fault Exception.
When an exception happens, the CPU jumps to an address that the OS had previously registered as the fault handler. This is usually done at boot time by the OS.
This could happen for example due to a programming error:but there are cases where it is not a bug, for example in Linux when:
int *is = malloc(1);
is[2] = 1;- the program wants to increase its stack.It just tries to accesses a certain byte in a given possible range, and if the OS is happy it adds that page to the process address space, otherwise, it sends a signal to the process.
- the page was swapped to disk.The OS will need to do some work behind the processes back to get the page back into RAM.The OS can discover that this is the case based on the contents of the rest of the page table entry, since if the present flag is clear, the other entries of the page table entry are completely left for the OS to to what it wants.On Linux for example, when present = 0:
- if all the fields of the page table entry are 0, invalid address.
- else, the page has been swapped to disk, and the actual values of those fields encode the position of the page on the disk.
In any case, the OS needs to know which address generated the Page Fault to be able to deal with the problem. This is why the nice IA32 developers set the value of
cr2 to that address whenever a Page Fault occurs. The exception handler can then just look into cr2 to get the address.The algorithmically minded will have noticed that paging requires associative array (like Java
Map of Python dict()) abstract data structure where:- the keys are linear pages addresses, thus of integer type
- the values are physical page addresses, also of integer type
The single level paging scheme uses a simple array implementation of the associative array:and in C pseudo-code it looks like this:
- the keys are the array index
- this implementation is very fast in time
- but it is too inefficient in memory
linear_address[0] = physical_address_0
linear_address[1] = physical_address_1
linear_address[2] = physical_address_2
...
linear_address[2^20-1] = physical_address_NBut there another simple associative array implementation that overcomes the memory problem: an (unbalanced) k-ary tree.
A K-ary tree, is just like a binary tree, but with K children instead of 2.
Using a K-ary tree instead of an array implementation has the following trade-offs:
- it uses way less memory
- it is slower since we have to de-reference extra pointers
In C-pseudo code, a 2-level K-ary tree with and we have the following arrays:and it still contains
K = 2^10 looks like this:level0[0] = &level1_0[0]
level1_0[0] = physical_address_0_0
level1_0[1] = physical_address_0_1
...
level1_0[2^10-1] = physical_address_0_N
level0[1] = &level1_1[0]
level1_1[0] = physical_address_1_0
level1_1[1] = physical_address_1_1
...
level1_1[2^10-1] = physical_address_1_N
...
level0[N] = &level1_N[0]
level1_N[0] = physical_address_N_0
level1_N[1] = physical_address_N_1
...
level1_N[2^10-1] = physical_address_N_N- one
directory, which has2^10elements. Each element contains a pointer to a page table array. - up to 2^10
pagetablearrays. Each one has2^104 byte page entries.
2^10 * 2^10 = 2^20 possible keys.K-ary trees can save up a lot of space, because if we only have one key, then we only need the following arrays:
- one
directorywith 2^10 entries - one
pagetableatdirectory[0]with 2^10 entries - all other
directory[i]are marked as invalid, don't point to anything, and we don't allocatepagetablefor them at all
Their crash system does not have an amazing user interface.
Tested on Ubuntu 21.10.
After something crashes, look under
/var/crash for a crash file, which helps to determine which package to report under on Launchpad.E.g. a file
/var/crash/_usr_sbin_gdm3.0.crash makes you want to file the bug under gdm at: bugs.launchpad.net/ubuntu/+source/gdm/+filebugThen, while reporting the bug, you want to give the developpers access to that Ubuntu's crash report system has already uploaded the
.crash file. But you can't publicly upload it because it contains memory dumps and could contain secret information. The way to do it is to look at the ID under:sudo cat /var/crash/_usr_sbin_gdm3.0.uploaded.crash for you, so you just have to confirm it and give the ID on the ticket.You can view a list of all your uploaded errors at:and each of those contain a link to:which you yourself cannot see.
xdg-open https://errors.ubuntu.com/user/$(sudo cat /var/lib/whoopsie/whoopsie-id)https://errors.ubuntu.com/oops/<.uloaded error id>askubuntu.com/questions/434431/how-can-i-read-a-crash-file-from-var-crash asks how to read the
.crash files.Running:splits it up into a few files, but does not make any major improvements.
sudo apport-unpack /var/crash/_usr_sbin_gdm3.0.crash /tmp/appapport-retracesudo apt install apport-retrace
sudo chmod 666 /var/crash/_usr_sbin_gdm3.0.crash
apport-retrace -g /var/crash/_usr_sbin_gdm3.0.crashTried:but then
echo "deb http://ddebs.ubuntu.com $(lsb_release -cs) main restricted universe multiverse" | sudo tee -a /etc/apt/sources.list.d/ddebs.list
echo -e "deb http://ddebs.ubuntu.com $(lsb_release -cs)-updates main restricted universe multiverse\ndeb http://ddebs.ubuntu.com $(lsb_release -cs)-proposed main restricted universe multiverse" | sudo tee -a /etc/apt/sources.list.d/ddebs.list
sudo apt install ubuntu-dbgsym-keyringsudo apt update fails with:E: The repository 'http://ddebs.ubuntu.com impish-security Release' does not have a Release file.Bibliography:
- www.slipcue.com/music/brazil/brazillist.html cute site from the 90s
- learn-real.eu
- www.youtube.com/watch?v=6fo5NhnyR8I OpenAI realistic robot hand
One of the most beautiful things in mathematics are theorems of conjectures that are very simple to state and understand (e.g. for K-12, lower undergrad levels), but extremely hard to prove.
This is in contrast to conjectures in certain areas where you'd have to study for a few months just to precisely understand all the definitions and the interest of the problem statement.
Bibliography:
- mathoverflow.net/questions/75698/examples-of-seemingly-elementary-problems-that-are-hard-to-solve
- www.reddit.com/r/mathematics/comments/klev7b/whats_your_favorite_easy_to_state_and_understand/
- mathoverflow.net/questions/42512/awfully-sophisticated-proof-for-simple-facts this one is for proofs for which simpler proofs exist
- math.stackexchange.com/questions/415365/it-looks-straightforward-but-actually-it-isnt this one is for "there is some reason it looks easy", whatever that means
By Fred Sanger's group.
There are unlisted articles, also show them or only show them.