• Log InLog In
  • Register
Liquid`
Team Liquid Liquipedia
EDT 09:52
CEST 15:52
KST 22:52
  • Home
  • Forum
  • Calendar
  • Streams
  • Liquipedia
  • Features
  • Store
  • EPT
  • TL+
  • StarCraft 2
  • Brood War
  • Smash
  • Heroes
  • Counter-Strike
  • Overwatch
  • Liquibet
  • Fantasy StarCraft
  • TLPD
  • StarCraft 2
  • Brood War
  • Blogs
Forum Sidebar
Events/Features
News
Featured News
Code S RO12 Preview: Maru, Trigger, Rogue, NightMare12Code S RO12 Preview: Cure, sOs, Reynor, Solar15[ASL19] Ro8 Preview: Unyielding3Official Ladder Map Pool Update (April 28, 2025)17[ASL19] Ro8 Preview: Rejuvenation8
Community News
Code S Season 1 - RO12 Group A Results (2025)3$1,250 WardiTV May [May 6th-May 18th]4Clem wins PiG Sty Festival #66Weekly Cups (April 28-May 4): ByuN & Astrea break through1Nexon wins bid to develop StarCraft IP content, distribute Overwatch mobile game29
StarCraft 2
General
QuickBooks Database Server Manager Not Running How does the number of casters affect your enjoyment of esports? Code S Season 1 - RO12 Group A Results (2025) quickbooks-database-server-manager-not-running quickbooks-has-stopped-working-^^[[855-749-2321]]
Tourneys
[GSL 2025] Code S:Season 1 - RO12 - Group A INu's Battles#12 < ByuN vs herO > [GSL 2025] Code S:Season 1 - RO12 - Group B GSL 2025 details announced - 2 seasons pre-EWC 2025 GSL Season 2 (Qualifiers)
Strategy
[G] PvT Cheese: 13 Gate Proxy Robo Simple Questions Simple Answers
Custom Maps
[UMS] Zillion Zerglings
External Content
Mutation # 472 Dead Heat Mutation # 471 Delivery Guaranteed Mutation # 470 Certain Demise Mutation # 469 Frostbite
Brood War
General
OGN to release AI-upscaled StarLeague from Feb 24 Battlenet Game Lobby Simulator [G] GenAI subtitles for Korean BW content BGH auto balance -> http://bghmmr.eu/ StarCraft & BroodWar Campaign Speedrun Quest
Tourneys
[ASL19] Ro8 Day 4 [CSLPRO] $1000 Spring is Here! Small VOD Thread 2.0 [Megathread] Daily Proleagues
Strategy
[G] How to get started on ladder as a new Z player Creating a full chart of Zerg builds [G] Mineral Boosting
Other Games
General Games
What do you want from future RTS games? Nintendo Switch Thread Stormgate/Frost Giant Megathread Grand Theft Auto VI Battle Aces/David Kim RTS Megathread
Dota 2
Official 'what is Dota anymore' discussion
League of Legends
LiquidLegends to reintegrate into TL.net
Heroes of the Storm
Simple Questions, Simple Answers
Hearthstone
Heroes of StarCraft mini-set
TL Mafia
Vanilla Mini Mafia TL Mafia Community Thread TL Mafia Plays: Diplomacy TL Mafia: Generative Agents Showdown Survivor II: The Amazon
Community
General
Elon Musk's lies, propaganda, etc. US Politics Mega-thread UK Politics Mega-thread Things Aren’t Peaceful in Palestine Russo-Ukrainian War Thread
Fan Clubs
Serral Fan Club
Media & Entertainment
Movie Discussion! Anime Discussion Thread [Books] Wool by Hugh Howey Surprisingly good films/Hidden Gems
Sports
2024 - 2025 Football Thread NHL Playoffs 2024 NBA General Discussion Formula 1 Discussion
World Cup 2022
Tech Support
Cleaning My Mechanical Keyboard How to clean a TTe Thermaltake keyboard? Logitech mx518 cleaning.
TL Community
BLinD-RawR 50K Post Watch Party The Automated Ban List TL.net Ten Commandments
Blogs
Info SLEgma_12
SLEgma_12
SECOND COMMING
XenOsky
What High-Performing Teams (…
TrAiDoS
WombaT’s Old BW Terran Theme …
WombaT
Heero Yuy & the Tax…
KrillinFromwales
BW PvZ Balance hypothetic…
Vasoline73
Test Entry for subject
xumakis
Customize Sidebar...

Website Feedback

Closed Threads



Active: 11312 users

Calculus 1 & Gandhi

Blogs > Laerties
Post a Reply
Laerties
Profile Blog Joined March 2010
United States361 Posts
Last Edited: 2011-06-24 06:48:18
June 24 2011 06:11 GMT
#1
This is my first time writing about myself + blogging so please don't hesitate to let me know what sucks.

So my primary reason for starting this blog is that I don't really have anyone who I can discuss my studies with. Now that its summer and I have hardly any obligations I've decided to dedicate most of my time towards studying things I'm interested in.


Anyway, I've decided to start off my studies by developing a strong understanding of Calc 1 and reading The Story of my Experiments With Truth by Gandhi, The Art of Happiness, by the Dali Llama, and Ethical Religion by Gandhi.

I took a calc class this year but it only got up to integration by u-sub, and it skipped over major things like the fundamental theorem…..Today I spent learning integration by parts & partial fractions. By parts is pretty easy, its just applying what I already know in a specific way but partial fractions is giving me a really hard time. I pretty much dicked around in math class up until the end of 11th grade, and my weak-ish understanding of basic concepts is making it more difficult than it should be. The textbook gives a nice table of how to set up the partial fractions depending on the denominator of the fractions but even with it, it takes me a minute to figure out what I'm looking at in the denominator. Also, why the hell is it that the coefficients on both sides of the equation need to be equal? When I look at examples it makes sense but I don't understand why its true…the whole thing just confuses me. I think I have most of the execution down though.

So i've read up to around page 160 in Gandhi's autobiography. Around January of this year I re-watched the movie Gandhi made in 1982 and I watched it like 10 times over the period of a month. Every time I watched it I began to realize more and more how fucking awesome all of Gandhi's principles are. Non-violence makes so much sense to me but I also recognize how hard it is to turn into action. So I've been reading his autobiography and I've begun to realize that his lifestyle was so much more radical than the one I'm used to. First of all he lived almost half his life celibate, made all his own clothes, fasted, took vows and constantly denied himself emotional pleasures. I really want to believe, and do to a certain extent, that his ideals are brilliant and genius but I sometimes think he might have just been crazy. He says that he receives freedom and pure happiness from restricting himself. I can see how disallowing your desires to influence you can be freeing, but it seems like he is just restricting himself in other ways. I just cant understand how denying pleasures can bring anyone happiness.

I've begun a few of my own 'experiments with truth' from reading his book. The first is, that I have stopped eating meat. Its been probably a bit over a month now since I've eaten any meat. It hasn't been that hard, and I do honestly feel better about what I'm doing. I claim to be stopping for moral reasons, but before I stopped I never felt guilty about eating meat. I think that from reading his book and considering the reasons for eating and not eating meat I have developed a stronger moral conduct in general. It's only been a few days since I started to try and actually develop a disliking for meat though. "Renunciation of objects without the renunciation of desires is short lived, however hard you may try". That quote really stuck with me, and I realized that to truly comply to my experiment I had to reject not just meat, but the desire to eat meat.

The other experiment I tried was to not lie, exaggerate or manipulate what I understand as truth in anyway. It has been realllllyy fucking hard. I seriously can't go a single day without letting one lie or exaggeration slip. I never thought that I was a liar before, but now I realize that I lie all the fucking time when communicating with other people. Gandhi saw truth as the end all be all, he once said "god is truth". He dedicated his entire life to following what he saw as truth. On a side note, thats also what's kind of awesome about math, its the only place where absolute truth exists, in every other field truth is relative. I've also been trying to talk less. Gandhi said in his book, and I agree, "Experience has taught me that silence is part of the spiritual discipline of a votary of truth". Also, "proneness to exaggerate to suppress or modify the truth, wittingly or unwittingly, is a natural weakness of man and silence is necessary to surmount it."

Gandhi spent most of his life reforming the bestial elements of his nature in order to develop moral and spiritual consistency. He said that this is what brings true happiness. I can't say I understand or relate to any aspect of that statement but I feel that it is worth the attempt, and my own endeavors have centered around understanding this statement.

Okay, so thats it. Sorry if this is terrible/boring/snobbish…

Happiness is when what you think, what you say, and what you do are in harmony.
Blisse
Profile Blog Joined July 2010
Canada3710 Posts
Last Edited: 2011-06-24 06:41:21
June 24 2011 06:40 GMT
#2
When you start actually learning about these people, like really studying, you sort of realize how amazing they are. Like Einstein or Newton. It still astounds me how much better they are than me, and how insignificant I will always be compared to the both of them.

Math is hard, but fun once you sort of understand it. And then you have to question, how the hell did these people figure out how to do integration by substition or integration by parts?

There are moments when you realize people are ridiculous and there are many many people that will stomp your ass.


Gandhi sounds like an amazing person, but don't press yourself to follow his words completely. Sure a lot of it is true and makes a lot of sense to you, but realize everyone is different, and if that's the way he wants to live his life, then so be it. Don't spend your existence dwelling on being inferior to them.
There is no one like you in the universe.
Laerties
Profile Blog Joined March 2010
United States361 Posts
June 24 2011 06:45 GMT
#3
On June 24 2011 15:40 Blisse wrote:
When you start actually learning about these people, like really studying, you sort of realize how amazing they are. Like Einstein or Newton. It still astounds me how much better they are than me, and how insignificant I will always be compared to the both of them.

Math is hard, but fun once you sort of understand it. And then you have to question, how the hell did these people figure out how to do integration by substition or integration by parts?

There are moments when you realize people are ridiculous and there are many many people that will stomp your ass.


Gandhi sounds like an amazing person, but don't press yourself to follow his words completely. Sure a lot of it is true and makes a lot of sense to you, but realize everyone is different, and if that's the way he wants to live his life, then so be it. Don't spend your existence dwelling on being inferior to them.


Thanks for the advice. I'm always thinking, how the hell did someone figure this out when learning a new concept. I guess thats what happens when you spend ALL YOUR TIME examining tangent lines and areas. Most of the time I feel like i'm just understanding how to execute the tasks of finding areas rates of change volumes etc rather than understanding what I am really doing and why it really works.
Happiness is when what you think, what you say, and what you do are in harmony.
GERMasta
Profile Joined October 2010
Germany212 Posts
June 24 2011 07:08 GMT
#4
Gandhi was an incredibly radical critic of modernity and most people don't really quite realize that, since he's mostly known for his advocacy of non-violence and his political successes against the British. But I would say that he was probably the most radical 20th century critic of modernity that I know of and I'm sure you will come to a similar understanding once you finish the books.

Having said that, many of his ideas I can certainly agree with (despite his criticism of liberalism), since he calls for being deeply self-reflective, being aware of our own values and motivations, leading a life of social and political activity (!), a life of relentless non-violence, consistency of values and austerity. He also emphasises our duties to others and to society, which I especially dig.

The other experiment I tried was to not lie, exaggerate or manipulate what I understand as truth in anyway. It has been realllllyy fucking hard.
You have to be careful with that once you hit a moral situation, since not telling lies could be overridden by higher order moral principles like, in this case, the principle of non-violence. You would certainly not want to tell the Nazis that you're hiding Jews in your cellar because that, even if that means you're telling a lie.
Abductedonut
Profile Blog Joined December 2010
United States324 Posts
Last Edited: 2011-06-24 08:08:28
June 24 2011 07:57 GMT
#5
On June 24 2011 15:40 Blisse wrote:
Math is hard, but fun once you sort of understand it. And then you have to question, how the hell did these people figure out how to do integration by substition or integration by parts?


Actually, things just seem out of reach when you have no idea how they came about. Discovering how to do integration by parts was trivial, because it's just the reverse of the chain rule. Check this out.

Let p(x) = f(x)*g(x)

Then p'(x) = f '(x)*g(x) + f(x)*g'(x) ( Simply by the chain rule, right?)

Solving for f(x)*g'(x)...

-f(x)*g'(x) + p'(x) = f '(x)*g(x)

-f(x)*g'(x) = - p'(x) + f '(x)*g(x)

f(x)*g'(x) = p'(x) - f '(x)*g(x)

Integrating both sides...

∫f(x)*g'(x) = ∫p'(x) - ∫f '(x)*g(x) ( But since ∫p'(x) = p(x) )

∫f(x)*g'(x) = p(x) - ∫f '(x)*g(x) ( But since p(x) = f(x)*g(x) )

∫f(x)*g'(x) = f(x)*g(x)- ∫f '(x)*g(x)

That's exactly the formula for integration by parts. This isn't a rigorous proof of course, but should give you an idea that discovering things sometimes only seems far if you don't have any clue as to how the people who discovered it might have discovered it. After you read that pseudo-proof it should be like, "wow that's easy!!"

On June 24 2011 15:11 Laerties wrote:
The textbook gives a nice table of how to set up the partial fractions depending on the denominator of the fractions but even with it, it takes me a minute to figure out what I'm looking at in the denominator. Also, why the hell is it that the coefficients on both sides of the equation need to be equal? When I look at examples it makes sense but I don't understand why its true…the whole thing just confuses me. I think I have most of the execution down though.


I'll teach you a secret about solving partial fractions that almost nobody knows. I'm not even sure if it's a legitimate method, but it works about 99% of the time. I have no idea where i learned this. Let's look at the partial fraction on this page:

http://www.purplemath.com/modules/partfrac.htm

So it says:

(3x + 2) / x*(x + 1) = A / x + B / ( x + 1)

(I'm not a teacher so I can't explain to you the ideas behind why this works.. but here we go...)

Look simply at...

(3x + 2) / x*(x + 1) = B / ( x + 1) ( so we're ignoring the A part. We will call this EQUATION1 )

Multiply both sides by (x+1) so it's...

(3x + 2) / x = B ( We will call this EQUATION2)

Now ask yourself, how could we have made the denomenator of B zero so that it was B / 0 in EQUATION1. By setting x = -1 so that it's B / ( -1 + 1 ). Now, let x = -1 in EQUATION2. If we do that then it's

( 3*(-1) + 2 ) / (-1) = B

So B = ( -3 + 2 ) / (-1) = 1 (You can check the website to make sure it's true.)

Now let's solve for A... Ignoring B, it becomes:

(3x + 2) / x*(x + 1) = A / x (EQUATION1)

Multiply both sides by x... now its:

(3x + 2) / (x + 1) = A (EQUATION2)

How could be have made the denominator of A zero in equation 1 zero? By letting x = 0. So, let x = 0 in equation2... we get...

(3*(0) + 2) / ( (0) + 1) = A

A = ( 0 + 2 ) / (1) = 2

So, A = 2 and B = 1... Finally we get...

(3x + 2) / x*(x + 1) = 2 / x + 1 / ( x + 1)

Partial Fractions will be VERY easy once you learn to use my method! And lastly, I shall part you with this amazing song!!

+ Show Spoiler +


*edit* Haha, I'm an idiot! "My" method is just the second method in that site I linked you..
Laerties
Profile Blog Joined March 2010
United States361 Posts
June 24 2011 08:06 GMT
#6
On June 24 2011 16:57 Abductedonut wrote:
Show nested quote +
On June 24 2011 15:40 Blisse wrote:
Math is hard, but fun once you sort of understand it. And then you have to question, how the hell did these people figure out how to do integration by substition or integration by parts?


Actually, things just seem out of reach when you have no idea how they came about. Discovering how to do integration by parts was trivial, because it's just the reverse of the chain rule. Check this out.

Let p(x) = f(x)*g(x)

Then p'(x) = f '(x)*g(x) + f(x)*g'(x) ( Simply by the chain rule, right?)

Solving for f(x)*g'(x)...

-f(x)*g'(x) + p'(x) = f '(x)*g(x)

-f(x)*g'(x) = - p'(x) + f '(x)*g(x)

f(x)*g'(x) = p'(x) - f '(x)*g(x)

Integrating both sides...

∫f(x)*g'(x) = ∫p'(x) - ∫f '(x)*g(x) ( But since ∫p'(x) = p(x) )

∫f(x)*g'(x) = p(x) - ∫f '(x)*g(x) ( But since p(x) = f(x)*g(x) )

∫f(x)*g'(x) = f(x)*g(x)- ∫f '(x)*g(x)

That's exactly the formula for integration by parts. This isn't a rigorous proof of course, but should give you an idea that discovering things sometimes only seems far if you don't have any clue as to how the people who discovered it might have discovered it. After you read that pseudo-proof it should be like, "wow that's easy!!"

Show nested quote +
On June 24 2011 15:11 Laerties wrote:
The textbook gives a nice table of how to set up the partial fractions depending on the denominator of the fractions but even with it, it takes me a minute to figure out what I'm looking at in the denominator. Also, why the hell is it that the coefficients on both sides of the equation need to be equal? When I look at examples it makes sense but I don't understand why its true…the whole thing just confuses me. I think I have most of the execution down though.


I'll teach you a secret about solving partial fractions that almost nobody knows. I'm not even sure if it's a legitimate method, but it works about 99% of the time. I have no idea where i learned this. Let's look at the partial fraction on this page:

http://www.purplemath.com/modules/partfrac.htm

So it says:

(3x + 2) / x*(x + 1) = A / x + B / ( x + 1)

(I'm not a teacher so I can't explain to you the ideas behind why this works.. but here we go...)

Look simply at...

(3x + 2) / x*(x + 1) = B / ( x + 1) ( so we're ignoring the A part. We will call this EQUATION1 )

Multiply both sides by (x+1) so it's...

(3x + 2) / x = B ( We will call this EQUATION2)

Now ask yourself, how could we have made the denomenator of B zero so that it was B / 0 in EQUATION1. By setting x = -1 so that it's B / ( -1 + 1 ). Now, let x = -1 in EQUATION2. If we do that then it's

( 3*(-1) + 2 ) / (-1) = B

So B = ( -3 + 2 ) / (-1) = 1 (You can check the website to make sure it's true.)

Now let's solve for A... Ignoring B, it becomes:

(3x + 2) / x*(x + 1) = A / x (EQUATION1)

Multiply both sides by x... now its:

(3x + 2) / (x + 1) = A (EQUATION2)

How could be have made the denominator of A zero in equation 1 zero? By letting x = 0. So, let x = 0 in equation2... we get...

(3*(0) + 2) / ( (0) + 1) = A

A = ( 0 + 2 ) / (1) = 2

So, A = 2 and B = 1... Finally we get...

(3x + 2) / x*(x + 1) = 2 / x + 1 / ( x + 1)

Partial Fractions will be VERY easy once you learn to use my method! And lastly, I shall part you with this amazing song!!

+ Show Spoiler +
http://www.youtube.com/watch?v=7eJoQWrhCtk


Very funny. Btw that method is only going to work completely for fractions that dont have a degree higher than 2 and have a lower degree numerator. So that is definitely not 99% of the time but w/e. It isn't the technical aspect or the use of proofs that goes over my head, its the principles that allow these things to work that I have a disconnect with.
Happiness is when what you think, what you say, and what you do are in harmony.
Abductedonut
Profile Blog Joined December 2010
United States324 Posts
Last Edited: 2011-06-24 08:14:38
June 24 2011 08:11 GMT
#7
On June 24 2011 17:06 Laerties wrote:
its the principles that allow these things to work that I have a disconnect with.


What does that even mean?

*edit* "have a lower degree numerator"

If the numerator is a higher degree than the denominator then you can just divide using long division and then only the remainder will be non-trivial to integrate.
Laerties
Profile Blog Joined March 2010
United States361 Posts
Last Edited: 2011-06-24 08:28:05
June 24 2011 08:26 GMT
#8
You can learn how to execute addition, but that doesn't necessarily mean you recognize or understand why its possible for you to add numbers in any order, or group them when adding more than 2 numbers *the commutative property*....same idea.

also ...your still going to run into fractions that have higher degree denominators, and your going to run into different forms of higher degree equations which require different set ups, and wont allow you to use the cover up method.
Happiness is when what you think, what you say, and what you do are in harmony.
Please log in or register to reply.
Live Events Refresh
Next event in 14h 8m
[ Submit Event ]
Live Streams
Refresh
StarCraft 2
Harstem 466
Lowko351
Hui .131
StarCraft: Brood War
Horang2 22861
Bisu 3071
Flash 1156
Shuttle 796
BeSt 635
Mini 411
Stork 410
Pusan 345
PianO 312
actioN 275
[ Show more ]
Soulkey 164
TY 107
Rush 96
Sea.KH 81
Mind 68
ToSsGirL 56
sSak 46
Aegong 35
Backho 31
Terrorterran 31
Shinee 26
Barracks 22
soO 15
sorry 15
yabsab 15
Shine 12
Sexy 8
Movie 8
Sacsri 8
IntoTheRainbow 3
Stormgate
NightEnD31
Dota 2
Gorgc2317
420jenkins1018
qojqva320
syndereN274
Counter-Strike
olofmeister2269
NBK_224
Super Smash Bros
Mew2King92
Heroes of the Storm
Khaldor237
Other Games
singsing3055
B2W.Neo1706
DeMusliM764
XcaliburYe649
hiko337
crisheroes272
ArmadaUGS107
SortOf104
djWHEAT90
Liquid`VortiX77
Organizations
StarCraft: Brood War
UltimateBattle 1236
StarCraft 2
WardiTV990
ESL.tv122
StarCraft: Brood War
lovetv 11
StarCraft 2
Blizzard YouTube
StarCraft: Brood War
BSLTrovo
sctven
[ Show 12 non-featured ]
StarCraft 2
• AfreecaTV YouTube
• intothetv
• Kozan
• IndyKCrew
• LaughNgamezSOOP
• Migwel
• sooper7s
StarCraft: Brood War
• BSLYoutube
• STPLYoutube
• ZZZeroYoutube
League of Legends
• Nemesis3512
• Jankos1366
Upcoming Events
Online Event
14h 8m
ShoWTimE vs MaxPax
SHIN vs herO
Clem vs Cure
SHIN vs Clem
ShoWTimE vs SHIN
SOOP
19h 8m
DongRaeGu vs sOs
CranKy Ducklings
20h 8m
WardiTV Invitational
21h 8m
SC Evo League
22h 8m
WardiTV Invitational
1d
Chat StarLeague
1d 2h
PassionCraft
1d 3h
Circuito Brasileiro de…
1d 4h
Online Event
1d 14h
MaxPax vs herO
SHIN vs Cure
Clem vs MaxPax
ShoWTimE vs herO
ShoWTimE vs Clem
[ Show More ]
Sparkling Tuna Cup
1d 20h
WardiTV Invitational
1d 21h
uThermal 2v2 Circuit
2 days
Chat StarLeague
2 days
Circuito Brasileiro de…
2 days
Afreeca Starleague
2 days
BeSt vs Light
Wardi Open
2 days
PiGosaur Monday
3 days
Afreeca Starleague
3 days
Snow vs Soulkey
WardiTV Invitational
3 days
Replay Cast
4 days
GSL Code S
4 days
ByuN vs Rogue
herO vs Cure
Replay Cast
5 days
GSL Code S
5 days
Classic vs Reynor
GuMiho vs Maru
The PondCast
5 days
RSL Revival
6 days
GSL Code S
6 days
Liquipedia Results

Completed

FGSL Season 1
PiG Sty Festival 6.0
Calamity Stars S2

Ongoing

BSL Nation Wars Season 2
StarCastTV Star League 4
JPL Season 2
ASL Season 19
YSL S1
BSL 2v2 Season 3
BSL Season 20
China & Korea Top Challenge
KCM Race Survival 2025 Season 2
2025 GSL S1
Heroes 10 EU
Asian Champions League '25
ECL Season 49: Europe
BLAST Rivals Spring 2025
MESA Nomadic Masters
CCT Season 2 Global Finals
IEM Melbourne 2025
YaLLa Compass Qatar 2025
PGL Bucharest 2025
BLAST Open Spring 2025
ESL Pro League S21

Upcoming

CSLPRO Spring 2025
NPSL Season 2
CSLPRO Last Chance 2025
CSLAN 2025
Esports World Cup 2025
HSC XXVII
Championship of Russia 2025
Bellum Gens Elite Stara Zagora 2025
2025 GSL S2
DreamHack Dallas 2025
FISSURE Playground #1
BLAST.tv Austin Major 2025
ESL Impact League Season 7
IEM Dallas 2025
PGL Astana 2025
TLPD

1. ByuN
2. TY
3. Dark
4. Solar
5. Stats
6. Nerchio
7. sOs
8. soO
9. INnoVation
10. Elazer
1. Rain
2. Flash
3. EffOrt
4. Last
5. Bisu
6. Soulkey
7. Mini
8. Sharp
Sidebar Settings...

Advertising | Privacy Policy | Terms Of Use | Contact Us

Original banner artwork: Jim Warren
The contents of this webpage are copyright © 2025 TLnet. All Rights Reserved.