Ciro Santilli @cirosantilli 35

, which is quite little information! This idea is basically what is behind the importance of the ralationship between Lie group-Lie algebra correspondence via the exponential map. In the more general settings of groups and manifolds, restricting ourselves to be near the origin is a huge advantage.

Now suppose that we want to calculate

e^{1}

. The idea is to start from

e^{0}

and then then to use the first order of the Taylor series to extend the known value of

e^{0}

e^{1}

E.g., if we split into 2 parts, we know that:

e^{1} = e^{1 / 2} e^{1 / 2}

(3)

or in three parts:

e^{1} = e^{1 / 3} e^{1 / 3} e^{1 / 3}

(4)

so we can just use arbitrarily many parts

e^{1 / n}

that are arbitrarily close to

x = 0

e^{1} = (e^{1 / n})^{n}

(5)

and more generally for any

x

we have:

e^{x} = (e^{x / n})^{n}

(6)

Let's see what happens with the Taylor series. We have near

y = 0

in little-o notation:

e^{y} = 1 + y + o (y)

(7)

Therefore, for

y = x / n

, which is near

y = 0

for any fixed

x

e^{x / n} = 1 + x / n + o (1 / n)

(8)

and therefore:

e^{x} = (e^{x / n})^{n} = (1 + x / n + o (1 / n))^{n}

(9)

which is basically the formula tha we wanted. We just have to convince ourselves that at

lim_{n \to \infty}

, the

o (1 / n)

disappears, i.e.:

(1 + x / n + o (1 / n))^{n} = (1 + x / n)^{n}

(10)

To do that, let's multiply

e^{y}

by itself once:

e^{y} e^{y} = (1 + y + o (y)) (1 + y + o (y)) = 1 + 2 y + o (y)

(11)

and multiplying a third time:

e^{y} e^{y} e^{y} = (1 + 2 y + o (y)) (1 + y + o (y)) = 1 + 3 y + o (y)

(12)

TODO conclude.

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Cheap robot  Updated 2025-01-10  +Created 1970-01-01

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E. Coli genome starting point  Updated 2025-01-10  +Created 1970-01-01

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The conventional starting point is not at the E. Coli K-12 MG1655 origin of replication.

biocyc.org/ECOLI/NEW-IMAGE?type=EXTRAGENIC-SITE&object=G0-10506 explains:

This site is the origin of replication of the E. coli chromosome. It contains the binding sites for DnaA, which is critical for initiation of replication. Replication proceeds bidirectionally. For historical reasons, the numbering of E. coli's circular chromosome does not start at the origin of replication, but at the origin of transfer during conjugation.

If it is a bit hard to understand what they mean by "origin of transfer" though, as that term is usually associated with the origin of transfer of bacterial conjugation.

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Section header table  Updated 2025-01-10  +Created 1970-01-01

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Array of Elf64_Shdr structs.

Each entry contains metadata about a given section.

e_shoff of the ELF header gives the starting position, 0x40 here.

e_shentsize and e_shnum from the ELF header say that we have 7 entries, each 0x40 bytes long.

So the table takes bytes from 0x40 to 0x40 + 7 + 0x40 - 1 = 0x1FF.

Some section names are reserved for certain section types: www.sco.com/developers/gabi/2003-12-17/ch4.sheader.html#special_sections e.g. .text requires a SHT_PROGBITS type and SHF_ALLOC + SHF_EXECINSTR

Running:

readelf -S hello_world.o

outputs:

There are 7 section headers, starting at offset 0x40:

Section Headers:
  [Nr] Name              Type             Address           Offset
       Size              EntSize          Flags  Link  Info  Align
  [ 0]                   NULL             0000000000000000  00000000
       0000000000000000  0000000000000000           0     0     0
  [ 1] .data             PROGBITS         0000000000000000  00000200
       000000000000000d  0000000000000000  WA       0     0     4
  [ 2] .text             PROGBITS         0000000000000000  00000210
       0000000000000027  0000000000000000  AX       0     0     16
  [ 3] .shstrtab         STRTAB           0000000000000000  00000240
       0000000000000032  0000000000000000           0     0     1
  [ 4] .symtab           SYMTAB           0000000000000000  00000280
       00000000000000a8  0000000000000018           5     6     4
  [ 5] .strtab           STRTAB           0000000000000000  00000330
       0000000000000034  0000000000000000           0     0     1
  [ 6] .rela.text        RELA             0000000000000000  00000370
       0000000000000018  0000000000000018           4     2     4
Key to Flags:
  W (write), A (alloc), X (execute), M (merge), S (strings), l (large)
  I (info), L (link order), G (group), T (TLS), E (exclude), x (unknown)
  O (extra OS processing required) o (OS specific), p (processor specific)

The struct represented by each entry is:

typedef struct {
    Elf64_Word  sh_name;
    Elf64_Word  sh_type;
    Elf64_Xword sh_flags;
    Elf64_Addr  sh_addr;
    Elf64_Off   sh_offset;
    Elf64_Xword sh_size;
    Elf64_Word  sh_link;
    Elf64_Word  sh_info;
    Elf64_Xword sh_addralign;
    Elf64_Xword sh_entsize;
} Elf64_Shdr;

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Personal name  Updated 2025-01-10  +Created 1970-01-01

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The Godfather  Updated 2025-01-10  +Created 1970-01-01

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www.youtube.com/playlist?list=PLVV0r6CmEsFzCodipONmNiROhYCUWyz_U interview with Walter Murch by Web of Stories. He worked on sound design for all The Godfather films, and tells some interesting details about it. He's just super nice in general otherwise. E.g.:

www.youtube.com/watch?v=w4OK8__gfGo talks about the sound design for the restaurant scene in The Godfather Part II (1974)

Video 1.

Francis Coppola's Notebook on The Godfather

. Source.

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Symmetric and public-key cryptography  Updated 2025-01-10  +Created 1970-01-01

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Digital electronic circuit  Updated 2025-01-10  +Created 1970-01-01

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Gene  Updated 2025-01-10  +Created 1970-01-01

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Object hd  Updated 2025-01-10  +Created 1970-01-01

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Running:

hd hello_world.o

gives:

00000000  7f 45 4c 46 02 01 01 00  00 00 00 00 00 00 00 00  |.ELF............|
00000010  01 00 3e 00 01 00 00 00  00 00 00 00 00 00 00 00  |..>.............|
00000020  00 00 00 00 00 00 00 00  40 00 00 00 00 00 00 00  |........@.......|
00000030  00 00 00 00 40 00 00 00  00 00 40 00 07 00 03 00  |....@.....@.....|
00000040  00 00 00 00 00 00 00 00  00 00 00 00 00 00 00 00  |................|
*
00000080  01 00 00 00 01 00 00 00  03 00 00 00 00 00 00 00  |................|
00000090  00 00 00 00 00 00 00 00  00 02 00 00 00 00 00 00  |................|
000000a0  0d 00 00 00 00 00 00 00  00 00 00 00 00 00 00 00  |................|
000000b0  04 00 00 00 00 00 00 00  00 00 00 00 00 00 00 00  |................|
000000c0  07 00 00 00 01 00 00 00  06 00 00 00 00 00 00 00  |................|
000000d0  00 00 00 00 00 00 00 00  10 02 00 00 00 00 00 00  |................|
000000e0  27 00 00 00 00 00 00 00  00 00 00 00 00 00 00 00  |'...............|
000000f0  10 00 00 00 00 00 00 00  00 00 00 00 00 00 00 00  |................|
00000100  0d 00 00 00 03 00 00 00  00 00 00 00 00 00 00 00  |................|
00000110  00 00 00 00 00 00 00 00  40 02 00 00 00 00 00 00  |........@.......|
00000120  32 00 00 00 00 00 00 00  00 00 00 00 00 00 00 00  |2...............|
00000130  01 00 00 00 00 00 00 00  00 00 00 00 00 00 00 00  |................|
00000140  17 00 00 00 02 00 00 00  00 00 00 00 00 00 00 00  |................|
00000150  00 00 00 00 00 00 00 00  80 02 00 00 00 00 00 00  |................|
00000160  a8 00 00 00 00 00 00 00  05 00 00 00 06 00 00 00  |................|
00000170  04 00 00 00 00 00 00 00  18 00 00 00 00 00 00 00  |................|
00000180  1f 00 00 00 03 00 00 00  00 00 00 00 00 00 00 00  |................|
00000190  00 00 00 00 00 00 00 00  30 03 00 00 00 00 00 00  |........0.......|
000001a0  34 00 00 00 00 00 00 00  00 00 00 00 00 00 00 00  |4...............|
000001b0  01 00 00 00 00 00 00 00  00 00 00 00 00 00 00 00  |................|
000001c0  27 00 00 00 04 00 00 00  00 00 00 00 00 00 00 00  |'...............|
000001d0  00 00 00 00 00 00 00 00  70 03 00 00 00 00 00 00  |........p.......|
000001e0  18 00 00 00 00 00 00 00  04 00 00 00 02 00 00 00  |................|
000001f0  04 00 00 00 00 00 00 00  18 00 00 00 00 00 00 00  |................|
00000200  48 65 6c 6c 6f 20 77 6f  72 6c 64 21 0a 00 00 00  |Hello world!....|
00000210  b8 01 00 00 00 bf 01 00  00 00 48 be 00 00 00 00  |..........H.....|
00000220  00 00 00 00 ba 0d 00 00  00 0f 05 b8 3c 00 00 00  |............<...|
00000230  bf 00 00 00 00 0f 05 00  00 00 00 00 00 00 00 00  |................|
00000240  00 2e 64 61 74 61 00 2e  74 65 78 74 00 2e 73 68  |..data..text..sh|
00000250  73 74 72 74 61 62 00 2e  73 79 6d 74 61 62 00 2e  |strtab..symtab..|
00000260  73 74 72 74 61 62 00 2e  72 65 6c 61 2e 74 65 78  |strtab..rela.tex|
00000270  74 00 00 00 00 00 00 00  00 00 00 00 00 00 00 00  |t...............|
00000280  00 00 00 00 00 00 00 00  00 00 00 00 00 00 00 00  |................|
00000290  00 00 00 00 00 00 00 00  01 00 00 00 04 00 f1 ff  |................|
000002a0  00 00 00 00 00 00 00 00  00 00 00 00 00 00 00 00  |................|
000002b0  00 00 00 00 03 00 01 00  00 00 00 00 00 00 00 00  |................|
000002c0  00 00 00 00 00 00 00 00  00 00 00 00 03 00 02 00  |................|
000002d0  00 00 00 00 00 00 00 00  00 00 00 00 00 00 00 00  |................|
000002e0  11 00 00 00 00 00 01 00  00 00 00 00 00 00 00 00  |................|
000002f0  00 00 00 00 00 00 00 00  1d 00 00 00 00 00 f1 ff  |................|
00000300  0d 00 00 00 00 00 00 00  00 00 00 00 00 00 00 00  |................|
00000310  2d 00 00 00 10 00 02 00  00 00 00 00 00 00 00 00  |-...............|
00000320  00 00 00 00 00 00 00 00  00 00 00 00 00 00 00 00  |................|
00000330  00 68 65 6c 6c 6f 5f 77  6f 72 6c 64 2e 61 73 6d  |.hello_world.asm|
00000340  00 68 65 6c 6c 6f 5f 77  6f 72 6c 64 00 68 65 6c  |.hello_world.hel|
00000350  6c 6f 5f 77 6f 72 6c 64  5f 6c 65 6e 00 5f 73 74  |lo_world_len._st|
00000360  61 72 74 00 00 00 00 00  00 00 00 00 00 00 00 00  |art.............|
00000370  0c 00 00 00 00 00 00 00  01 00 00 00 02 00 00 00  |................|
00000380  00 00 00 00 00 00 00 00  00 00 00 00 00 00 00 00  |................|
00000390

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gvgai  Updated 2025-01-10  +Created 1970-01-01

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www.gvgai.net (dead as of 2023)

The project kind of died circa 2020 it seems, a shame. Likely they funding ran out. The domain is dead as of 2023, last archive from 2022: web.archive.org/web/20220331022932/http://gvgai.net/. Marks as funded by DeepMind. Researchers really should use university/GitHub domain names!

Similar goals to Ciro's 2D reinforcement learning games, but they were focusing mostly on discrete games.

They have some source at: github.com/GAIGResearch/GVGAI TODO review

A published book at: gaigresearch.github.io/gvgaibook/

From QMUL Game AI Research Group: