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How to compile a specific kernel module exactly same way as in running kernel

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I'm trying to compile kernel module "r8152.ko". My goal is that my compiled module is exactly the same as r8152.ko in /lib/module/\$(uname -r)

Here is what I tried:

Check kernel version

``````\$ uname -a
Linux LPT00102259F 5.15.0-56-generic #62~20.04.1-Ubuntu SMP Tue Nov 22 21:24:20 UTC 2022 x86_64 x86_64 x86_64 GNU/Linux
``````

`````` http://archive.ubuntu.com/ubuntu/pool/main/l/linux/linux-source-5.15.0_5.15.0-56.62_all.deb
``````

``````obj-\$(CONFIG_USB_RTL8152)    += r8152.o
``````

Copy .config, Module.symvers and run some build commands

``````p@LPT00102259F:~/Downloads/linux-source-5.15.0\$ make mrproper
CLEAN   arch/x86/tools
CLEAN   scripts/basic
CLEAN   scripts/genksyms
CLEAN   scripts/kconfig
CLEAN   scripts/mod
CLEAN   scripts/selinux/mdp
CLEAN   scripts
CLEAN   include/config include/generated arch/x86/include/generated .config .config.old Module.symvers
HOSTCC  scripts/basic/fixdep
HOSTCC  scripts/kconfig/conf.o
HOSTCC  scripts/kconfig/confdata.o
HOSTCC  scripts/kconfig/expr.o
LEX     scripts/kconfig/lexer.lex.c
YACC    scripts/kconfig/parser.tab.[ch]
HOSTCC  scripts/kconfig/lexer.lex.o
HOSTCC  scripts/kconfig/parser.tab.o
HOSTCC  scripts/kconfig/preprocess.o
HOSTCC  scripts/kconfig/symbol.o
HOSTCC  scripts/kconfig/util.o
HOSTLD  scripts/kconfig/conf
#
# configuration written to .config
#
SYSHDR  arch/x86/include/generated/uapi/asm/unistd_32.h
WRAP    arch/x86/include/generated/uapi/asm/bpf_perf_event.h
SYSHDR  arch/x86/include/generated/uapi/asm/unistd_64.h
WRAP    arch/x86/include/generated/uapi/asm/errno.h
SYSHDR  arch/x86/include/generated/uapi/asm/unistd_x32.h
WRAP    arch/x86/include/generated/uapi/asm/fcntl.h
SYSTBL  arch/x86/include/generated/asm/syscalls_32.h
SYSHDR  arch/x86/include/generated/asm/unistd_32_ia32.h
WRAP    arch/x86/include/generated/uapi/asm/ioctl.h
WRAP    arch/x86/include/generated/uapi/asm/ioctls.h
SYSHDR  arch/x86/include/generated/asm/unistd_64_x32.h
WRAP    arch/x86/include/generated/uapi/asm/ipcbuf.h
WRAP    arch/x86/include/generated/uapi/asm/param.h
WRAP    arch/x86/include/generated/uapi/asm/poll.h
WRAP    arch/x86/include/generated/uapi/asm/resource.h
WRAP    arch/x86/include/generated/uapi/asm/socket.h
WRAP    arch/x86/include/generated/uapi/asm/sockios.h
WRAP    arch/x86/include/generated/uapi/asm/termbits.h
SYSTBL  arch/x86/include/generated/asm/syscalls_64.h
WRAP    arch/x86/include/generated/uapi/asm/types.h
WRAP    arch/x86/include/generated/uapi/asm/termios.h
SYSTBL  arch/x86/include/generated/asm/syscalls_x32.h
HYPERCALLS arch/x86/include/generated/asm/xen-hypercalls.h
HOSTCC  arch/x86/tools/relocs_32.o
HOSTCC  arch/x86/tools/relocs_64.o
HOSTCC  arch/x86/tools/relocs_common.o
WRAP    arch/x86/include/generated/asm/early_ioremap.h
WRAP    arch/x86/include/generated/asm/export.h
WRAP    arch/x86/include/generated/asm/mcs_spinlock.h
WRAP    arch/x86/include/generated/asm/irq_regs.h
WRAP    arch/x86/include/generated/asm/kmap_size.h
WRAP    arch/x86/include/generated/asm/local64.h
WRAP    arch/x86/include/generated/asm/mmiowb.h
WRAP    arch/x86/include/generated/asm/module.lds.h
WRAP    arch/x86/include/generated/asm/rwonce.h
WRAP    arch/x86/include/generated/asm/unaligned.h
UPD     include/config/kernel.release
UPD     include/generated/uapi/linux/version.h
HOSTCC  scripts/bin2c
DESCEND objtool
HOSTCC  scripts/kallsyms
HOSTCC  scripts/genksyms/genksyms.o
DESCEND bpf/resolve_btfids
HOSTCC  scripts/sorttable
UPD     include/generated/utsrelease.h
HOSTLD  arch/x86/tools/relocs
YACC    scripts/genksyms/parse.tab.[ch]
HOSTCC  scripts/asn1_compiler
HOSTCC  scripts/selinux/mdp/mdp
LEX     scripts/genksyms/lex.lex.c
HOSTCC  scripts/genksyms/parse.tab.o
HOSTCC  scripts/genksyms/lex.lex.o
HOSTCC  scripts/sign-file
HOSTCC  scripts/extract-cert
HOSTCC  scripts/insert-sys-cert
HOSTLD  scripts/genksyms/genksyms
CC      scripts/mod/empty.o
HOSTCC  scripts/mod/mk_elfconfig
CC      scripts/mod/devicetable-offsets.s
MKELF   scripts/mod/elfconfig.h
HOSTCC  scripts/mod/modpost.o
UPD     scripts/mod/devicetable-offsets.h
HOSTCC  scripts/mod/sumversion.o
HOSTCC  scripts/mod/file2alias.o
HOSTLD  scripts/mod/modpost
CALL    scripts/atomic/check-atomics.sh
CC      kernel/bounds.s
UPD     include/generated/timeconst.h
UPD     include/generated/bounds.h
CC      arch/x86/kernel/asm-offsets.s
UPD     include/generated/asm-offsets.h
CALL    scripts/checksyscalls.sh
CALL    scripts/checksyscalls.sh
CALL    scripts/atomic/check-atomics.sh
DESCEND objtool
DESCEND bpf/resolve_btfids
LDS     scripts/module.lds
CC [M]  drivers/net/usb/r8152.o
MODPOST drivers/net/usb/Module.symvers
CC [M]  drivers/net/usb/r8152.mod.o
LD [M]  drivers/net/usb/r8152.ko
``````

Then I run modinfo on generated r8152.ko and check vermagic

``````vermagic:       5.15.64 SMP mod_unload modversions
``````

This is different from original file in /lib

``````vermagic:       5.15.0-56-generic SMP mod_unload modversions
``````

Why is that and what do I need to do so they are the same ?

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The first definition is given as a quotation in the book by Kuipers

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There is a similar, but more pithy definition, from Wolfram MathWorld

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This is related to the three-angle representation of an arbitrary rotation by Euler, roll-pitch-yaw, Tati-Bryan angles etc.

The second definition is from Wikipedia

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This related to the angle-axis representation of an arbitrary rotation. This version of the Rotation Theorem is found in many other places as well, including here on Math Overflow.

Both definitions are related to rotation, and the second at least has a reference to Euler 1776. The first is related to Euler axes and Euler angles, seems like the sort of thing Euler might have figured out, but Kuipers does not provide a reference.

Euler was notorious for not publishing much. Somebody, post Euler, has declared these theorems as Euler’s Rotation Theorem. Which one is it? Is there a more nuanced way to refer to them?

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There is a house with 100 rooms, and each room contains countably many boxes indexed with the natural numbers. Each box contains a random real number, which is the same over all the rooms (that is, box n contains the same real number in every room).

100 set theorists play a game. Each person will go into a unique room and open as many boxes as they like (perhaps countably many) as long as they leave at least one box in their room unopened. Then, each of them need to pick an unopened box in their room, and guess what real number is inside of it.

In order to win, 99 of them need to guess correctly.

The mathematicians can discuss a strategy beforehand, but after they go into their respective rooms, no more communication is allowed. What is a 100% winning strategy for this seemingly impossible task?

My question is why can the mathematicians agree on a specific representative for each equivalence class? I thought that the axiom of choice guaranteed a choice function existed, but didnt specify it. But for the solution to the riddle, they would need to know what the choice function is. Can someone explain?

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Exponentials of stochastic band matrices

This may be a duplicate, but I’ve done some searching and I can’t find exactly this problem setting, probably due to not knowing the right terminology for how to refer to the transition matrix.

I’m considering absorbing random walks $${X_t}$$ which have symmetric transition probabilities inside a window of size $$k$$. For example, on $${0, 1, 2, dots, 100}$$, with $$k=3$$ if $$X_t = 5$$
begin{align*} P[X_{t+1} = 4] &= P[X_{t+1} = 6]\ P[X_{t+1} = 3] &= P[X_{t+1} = 7]\ P[X_{t+1} = 2] &= P[X_{t+1} = 8]\ end{align*}

When the walk is within $$k$$ of the endpoint, the allowable transitions are truncated on both sides, e.g. if $$X_t = 1$$, $$P[X_t = 3] = 0$$.

I wanted to figure out the probability of the walk being absorbed at the upper endpoint, so I did some numerical experiments with walks on the integers from 0-100.
I calculated $$T^n$$ for the transition matrix $$T$$ and a large $$n$$. I then looked at $$(T^n)_{i, 100}$$ to see the probability of being absorbed at 100. It appears that the value is always equal to $$i/100$$, regardless of how I set the transition distribution or window-size.

The transition matrix for a walk on the integers in $$[0, N]$$ has the following properties:

1. $$T$$ is (row) stochastic
2. $$T$$ is a band matrix with upper and lower band size $$k$$
3. $$T_{i, j} = 0$$ if $$|i – j| > max (i, N – i)$$
4. $$T_{i,i + r} = T_{i, i – r}$$ (this is the main property I don’t know the name of)
5. $$T_{0, 0} = T_{1, 1} = 1$$

I would like a proof or counterexample of the following statement:
$$begin{equation*} limlimits_{n to infty} (T^n)_{i, N} = i/N end{equation*}$$

If the statement above is true, is it still true if the transition distributions are no longer left-right symmetric, but it’s still the case that $$E[X_{t + 1}] = X_t$$?

trouble with writing negative fractional exponential

Code:

``````begin{equation*}
v_o = (v_i - V_D) e^-frac{ t}{tau}
end{equation*}
``````

What I want:

What I am getting:

Erro in Ubuntu Server for service bind9

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If \$varphi\$ is a normal faithful semifinite weight, is \$eta_varphi(mathfrak{n}_varphicapmathfrak{n}_varphi^*)\$ dense in \$mathfrak{H}_varphi\$

Let $$M$$ be a von Neumann algebra and $$varphi: M_+to [0, infty]$$ be a normal, faithful semifinite weight. Consider its associated semi-cyclic representation
$$pi_varphi: Mto B(mathfrak{H}_varphi)$$
(see Takesaki’s second book, chapter VII for details).

Consider the associated map
$$eta_varphi: mathfrak{n}_varphito mathfrak{H}_varphi.$$

Is it true that $$eta_varphi(mathfrak{n}_varphicap mathfrak{n}_varphi^*)$$ is norm-dense in $$mathfrak{H}_varphi$$?

Attempt: If $$xi perp eta_varphi(mathfrak{n}_varphicap mathfrak{n}_varphi^*)$$, then for $$x,yin mathfrak{n}_varphi$$
$$0 = langle xi, eta_varphi(x^*y) rangle = langle pi_varphi(x)xi, eta_varphi(y)rangle$$
so that $$pi_varphi(mathfrak{n}_varphi)xi=0$$. Maybe this is sufficient to conclude that $$xi= 0$$? I feel like non-degeneracy of $$pi_varphi$$ is relevant.

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