Milnor's Lobachevsky Function

Definition

\[\text{Л}(x) := -\int_0^x \log \left| 2 \sin t\right|\;dt\]

Note that \[\begin{aligned} \text{Л}'(x) &= - \log |2 \sin x| \\ \text{Л}''(x) &= -\cot x \end{aligned}\]

The definition looks very strange. It's probably easiest to think of Л as being defined by the differential equation \[ \text{Л}''(x) = -\cot x \] subject to the initial conditions \[\begin{aligned} \text{Л}(0) &= 0\\ \text{Л}'\left(\frac \pi 2\right) &= 0\\ \end{aligned}\]

Useful Facts

A Circumradius Forumla

We begin with a seemingly-unrelated formula about the circumcircle of a triangle.

Let $t_{ijk}$ be a triangle with edge lengths $\ell_{ij}, \ell_{jk}, \ell_{ki}$ and angles $\alpha_{ij}^k, \alpha_{jk}^i, \alpha_{ki}^j$.
The radius of the circumcircle of $t_{ijk}$ is given by $R = \frac{\ell_{jk}}{2 \sin \alpha_{jk}^i}$.
Proof

We place our triangle inside of its circumcircle.

$c$ is the center of the circle. We draw a diagonal passing though $k$, and name the opposite point $r$. Note that angle $k-j-r$ must be a right angle, since the line $\overline{kr}$ is a diameter of the circle. We also place a point $s$ such that the angle $k-s-j$ is a right angle.

Note that angle $k-r-j$ must equal angle $\alpha_{jk}^i$ since both subtend the same arc between $k$ and $j$. Therefore, triangle $t_{isk}$ is similar to triangle $t_{rjk}$. In particular, this means that \[\frac {\ell_{kr}}{\ell_{kj}} = \frac {\ell_{ik}}{\ell_{sk}}\] Note that $\ell_{sk} = \ell_{ik} \sin \alpha_{jk}^i$. Furthermore, $\ell_{rk} = 2R$. Thus, we see that \[\frac {2R}{\ell_{kj}} = \frac {\ell_{ik}}{\ell_{ik} \sin \alpha_{jk}^i}\] Simplifying, we conclude that \[R = \frac{\ell_{jk}}{2\sin\alpha_{jk}^i}\]

Energy Derivative

Futhermore, let $\lambda_{mn} = 2 \log \ell_{mn}$. We define a function \[ f( \lambda_{ij}, \lambda_{jk}, \lambda_{ki}) = \frac 12 \big( \alpha_{jk}^i \lambda_{jk} + \alpha_{ki}^j \lambda_{ki} + \alpha_{ij}^k \lambda_{ij}\big) + \text{Л}( \alpha_{jk}^i) + \text{Л}( \alpha_{ki}^j) + \text{Л}( \alpha_{ij}^k) \]
\[\pd f {\lambda_{jk}} = \frac 12 \alpha_{jk}^i\]
Proof

Using the circumradius formula, this becomes a straightforward computation.

\[\begin{aligned} \pd f {\lambda_{jk}} &= \frac 12 \alpha_{jk}^i + \left(\frac 12\lambda_{jk} - \log |2 \sin \alpha_{jk}^i| \right) \pd {\alpha_{jk}^i}{\lambda_{jk}}\\ &\quad+ \left(\frac 12\lambda_{ki} - \log |2 \sin \alpha_{ki}^j| \right) \pd {\alpha_{ki}^j}{\lambda_{jk}} + \left(\frac 12 \lambda_{ij} - \log |2 \sin \alpha_{ij}^k| \right) \pd {\alpha_{ij}^k}{\lambda_{jk}}\\ &= \frac 12 \alpha_{jk}^i + \log\left(\frac {\ell_{jk}}{2 \sin \alpha_{jk}^i}\right) \pd {\alpha_{jk}^i}{\lambda_{jk}} + \log\left(\frac {\ell_{ki}}{2 \sin \alpha_{ki}^j}\right) \pd {\alpha_{ki}^j}{\lambda_{jk}} + \log\left(\frac {\ell_{ij}}{2 \sin \alpha_{ij}^k}\right) \pd {\alpha_{ij}^k}{\lambda_{jk}} \end{aligned}\] By the circumradius formula, each of those log terms is just $\log R$. (Equivalently, we could just use the law of sines here to observe that all of the terms are equal. It is not actually important what they are equal to). Thus, we have \[\begin{aligned} \pd f {\lambda_{jk}} &= \frac 12 \alpha_{jk}^i + \log R \pd{}{\lambda_{jk}} \left(\alpha_{jk}^i + \alpha_{ki}^j + \alpha_{ij}^k\right) \end{aligned}\] Since the sum of the angles in a triangle is always $\pi$, the derivative on the right vanishes. Thus, we obtain the desired equality \[\begin{aligned} \pd f {\lambda_{jk}} &= \frac 12 \alpha_{jk}^i \end{aligned}\]

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    ReplyDelete
  14. Hello everyone, i would like to share my experience with you all, I lost over 100 thousand dollars to all these fake so called BO merchants , after several attempts in trying to recover my money all efforts failed, i was looking through the page of the internet then i saw RECOVER COIN RESCUE TEAM they were recommended as a good and reputable company so i reach out to them, to my surprise i was able to to recover all my funds. contact: R E C O V E R C O I N @ R E S C U E T E A M . C O M or +1 775-372-9370.

    ReplyDelete
  15. This post is for persons seeking to recover all of their lost funds to online scams, for RECOVERY OF LOST BITCOIN, RECOVERY OF ANY LOST CRYPTOCURRENCY, if you are seeking--> RECOVERY OF LOST FUNDS TO RIPPERS IN FOREX TRADES AND BINARY OPTIONS CONTACT: B T C R E C O V E R Y 1 0 1 @ C O N S U L T A N T . C O M. I had my blockchain wallet spoofed by merciless rippers, they were able to get away with 4.0147BTC from my wallet , this made me very sad and depressed as i was desperately in need of help, i made my research online and came across this credible recovery agent.  The hack agency helped me recover all I lost and also reveal the identity of the perpetrators, that's why I'm most  appreciative and also sharing contact addr. For anyone in a similar situation, contact the recovery specialist with the address above or text via: +1 775-372-9370.   

    ReplyDelete
  16. You know how tricky and sophisticated cryptocurrency can be. I invested a lot in a project, mistakenly the devs were unknown. I planned to hold the coin long term but after 2 months the project was rug pulled, I lost about 1.3btc and 27 eth in total. After referrals and positive reviews I contacted NERDORACLE247@GMAIL.COM, to my greatest surprise he recovered it all from the block chain in less than 48 hours

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  17. I was lost with no hope for my wife was cheating and had always got away with it because I did not know how or always too scared to pin anything on her. With the help a friend who recommended me to who help hack her phone, email, chat, sms and expose her for a cheater she is. I just want to say a big thank you to AlbaniaspyhackerAtGmailDotCom. I am sure someone out there is looking for how to solve his relationship problems, you
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    ReplyDelete
  18. Do you need to hack into any, databaseserver spy on Facebook,Emails, Whatsapp, Viber, Snapchat, Instagram and many more.
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  19. The only way you can get your money is by hiring a professional recovery expert to help you break into the Firms database security system, using the information you provide, extract your file and get back your Bitcoins. You can achieve this by contacting this professional recovery agent: S W I F T B T C R E C O V E R Y 2 6 @ C A S H 4 U . C O M

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  21. It's a sad experience to lose your money to these wallets...I lost mine
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  25. It got to my notice that when ever my girlfriend is close to me she reduce the brightness of her cell phone while texting, its been 2 months since i noticed this but i never wanted to ask her cause she would end up deleting all, so i did it my way by hiring an expert whose email is spyexpert0@gmail.com this hacker helped in breaking into her phone and i was able to read all that she had been hiding, she really thinks she was smart after my tour on her phone i saw a lot of things that i could not believe my girlfriend does because i trusted her so much shout out to  spyexpert0 the great.

    ReplyDelete
  26. I googled a few recovery companies and after a few days of research I came across a crypto expert who introduced me to a programmer that helped me recover my lost coins. I was very skeptical as can be imagined but I’m glad to say my bitcoin was recovered and sent back to me after an hour. You should file a complaint directly to him at (M O N E Y R E C O V E R Y 0 0 1 @ S O L U T I O N 4 U . C O M) in case u need help recovering your lost transactions, I recently recovered mine within an hour.

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  29. Have you lost money to romance scams?  Have you experienced stolen funds from accounts and wire transfers? have you lost money while trading forex? Have you tried getting your money out from your binary account and was unable to place a withdrawal? Have you found your self wading through excessively complicated terms which might seem impossible to meet, all because you traded with an unregulated broker? Do not panic, I have good news for you. I lost over $200k to an unregulated broker who stocked my trading capital and denied me access to my account. But thank God I have recovered all my lost funds, through the help of a certified funds recovery expert. If my assistance is needed, kindly get in contact@ACTIVEBONORUM.COM  and I will guide you on effective steps to take and get back all your lost funds in just a few days.
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    ReplyDelete
  30. I made a deposit of $490,000 into this broker on 1st December, then I got a call from the account
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    ReplyDelete
  31. My experience with CryptoNx was completely unpleasant. They thought they would cease my hard earned profit after growing my account to over 120%. This started when I reached out to the account manager when I wasn't able to withdraw my money. He assured me a successful withdrawal within a week. 3 weeks went by and I was never able to reach him again. I knew I'd been scammed. So I started researching a credible wealth recovery agency that could get my funds as soon as possible without any stressful uncertain chargeback procedures. The best decision I ever made was to work with Sharon and his team from " activebonorum.com ". Unbelievable, They got penetration testers that recovered my funds including my profits in less than 10 days. Guys! I couldn't be happier. Please stay clear of these unregulated brokers.
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    ReplyDelete
  32. If you're going through issues with your credit report and upgrading your credit score, I will advise you to contact CLEVERHACKER.HACK@GMAIL.COM . He's capable of by passing into the 3 credit bureaus. He helped me delete an old eviction on my credit report and help me increase my credit score to 794. He has done a great job for me and i can share that he's a professional at what he does. You can also contact him directly on his number (803) 8145462. Thank me later

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  33. Please beware there are many fake cryptocurrrency investment platforms out there, I got scammed twice before I met a genius hacker who helped retrieve my bitcoin. The whole plan was so smooth I could not doubt it. Bitcoin is actually a great investment option but one thing I discovered over time is that it is not possible to mine bitcoin so don’t be deceived. I invested 15BTC on a particular website called FoxBtc, I monitored the profit yielding but got locked out of the account before I could withdraw. I went online and met this recovery expert at recoverywealthnow360 on google mail who helped me retrieve my bitcoins back in few hours. He saved my life , he can save yours.

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    ReplyDelete
  35. Please beware there are many fake bitcoin mining/clouding out there, I have been a victim once but now a certified blockchain consultant and I know better. The whole plan was so smooth I could not doubt it. Bitcoin is actually a great investment option but one thing I discovered over time is that it is not possible to mine bitcoin so don’t be deceived. I invested $25000 on a particular website called eurekaminingblock, I monitored the profit yielding but was told to open a new blockchain account to receive my payout. A public wallet was imported into the account and I was made to believe that was my profit. The bitcoin was labelled non spendable and it took me 2 years to be able to access it without the knowledge of the company. The non spendable bitcoin is the scam out there now and a lot of people are falling victim of it. I found Mr Dawson Wright on Quora who helped me access my investment together with the profit without the knowledge of the company. You can reach out to him also on fundsretrieval1 @ gmail . com if you need help, he is genius. Goodluck!
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    ReplyDelete
  36. WHY WOULD YOU NEED TO HIRE A HACKER??:
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    ReplyDelete
  37. FRESH AND UPDATED STUFFS WITH 98% VALIDITY RATE ( DUMPS/ DUMPS+ATM PINS /CVV & fULLZ /, CC TOP UPS, BANK LOGS, VERIZON, PAYPAL ACCOUNTS, EGIFT CARDS FOR WALMART AND AMAZON AND SPAMMING TOOLS FOR SALE) via Email: atmgeniuslinks@gmail.com / ICQ: https://icq.im/Anouymous_CC / Telegram: @Anonymous_Ccs

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    CCnum:: 4400663873545159
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    Expm: 06
    Expy: 19 Fname: Gary
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    Address: 5830 Elwynn Drive
    City: MILFORD
    State: OH
    Zip: 45150
    Country: USA
    Phone: 5138318649
    Email: gch619@fuse.net


    DEBIT : 432190 – 99, 445808, 450644, 450670, 451948, 454003, 458501, 471401, 472476, 486342, 503886, 522122, 528030, 529558, 531704, 531729, 539847, 545409, 551029

    FULLZ
    '''''''''''''
    First Name : Kathryn
    Middle Name : M
    Last Name : Ratledge
    Spouse Name : Ruth Ratledge
    Father Name : Thomas Ratledge
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    City : Greenville
    State : CA
    Zip Code : 29615
    Country : US
    Phone Number : 15629234492
    Credit Card Information :
    *********
    Card Type : Credit
    Credit Card Number : 4737033009002349
    Exp. Date : 02/23
    Name On Card : Kathryn Ratledge
    Cvv2 : 736
    ATM Pin: 5910
    Bank Name : Bank of America
    Routing Number : 122000661
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    Social Security Number : 340224188
    Birthday : 19
    Birth Month : 8/June
    Birth Year : 1980
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    ------

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    ReplyDelete

  38. Today Thursday 5th 2020 I found out about my cheating husband with the help of jeajamhacker@gmail.com

    ReplyDelete
  39. It can be stressful seeing as bitcoin is crypto currency but it is possible. In recent times, the tables have turned for internet scammers who think they can do what they like on the internet and get away with it. This is why I want to implore you to start a chargeback and reach out to R E C O V E R C O I N @ R E S C U E T E A M . C O M. they have an elite forensic team and agents on standby available to help you in your recovery process.

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  40. I paid russiancyberhackers@gmail.com upfront to get my wife's phone hacked and am happy russiancyberhackers@gmail.com is a man of his world. I didn't get disappointed but I got all I paid for within the space of 2 hours without trace.. people out there that has used russiancyberhackers@gmail.com to achieve one thing or the other  learn to come openly and appreciate good hackers like russiancyberhackers@gmail.com thank you

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  41. I noticed that my husband is having multiple affairs but I was never sure because I needed proof of what has been going on and am also happy I got the proof I have always wanted  with the help of jeajamhacker@gmail.com I got to access my husband phone remotely and that was how I knew that my thought has been right about my husband but didn't have any one to hack his phone. A very big thank you to the world for referring me to use jeajamhacker@gmail.com services and also with their good reviews about jeajamhacker@gmail.com I was moved to email jeajamhacker@gmail.con for help!

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  42. Have you ever gotten your bitcoins stolen from your wallet or invested in an ICO that turned out to be a scam, you are not alone because this happened to me too. I initially lost $90,000 in just 3 months from Cryptoallday, I contacted the authorities and they referred me to a bitcoin recovery company who helped me recover all my funds within less than 2 days... I'm speaking out to improve our awareness of these cryptocurrency thieves and help you if you have been a victim yourself. Simply send them an email bitcoinretrieval2018 gmail com

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  43. Be careful and stay away from binary trading if you do not know of a way to get your investments back after losing it, tried different recovery service providers and at the end I lost more than I invested, but at BITCOIN RECOVERY CONSULTANT, they can help you trade and still help you recover your investments from the broker you lost to, this is how I get my investments back so I have no worries because B T C R E C O V E R Y 1 0 1 @ C O N S U L T A N T . C O M, they have your back and make sure you never lose again.

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  44. russiancyberhackers@gmail.com always the best hacker to email when it comes to hacking a cheating spouse cell phone.

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  45. hello guys,i have just used hacking service of emrysghosthacker on gmal..i think this is the solution everyone out there facing any hacking related issue deserves, after been ripped of by many hackers out there, i now finally got a solution to my long time cheating husband’s phone..i can now view all his call logs, text, fb and all other social media conversations directly from my own phone..i recommend this guy as the best option now, his fast and reliable cuz he kept his promise with the information delivery and i promised him to share his reputation across online for getting this done for me atlast.Show less

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  50. I was scammed by a binary option website iq options to be exact. I lost about $153,000 to them. It was a really hard time for me because that was all i had and they tricked me into investing the money with a guarantee that i will make profit from the investment.. I was referred to Mr Gary, he’s a senior recovery specialist and he helped in recovering all my lost funds to me. I couldn’t believe at first that this was possible as i have lost all hope in contacting anyone on the internet, but i guess not everyone is bad after all. You can reach him on cybertechhub100 at gmail dot com

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  51. I was scammed by a binary option website iq options to be exact. I lost about $250,000 to them. It was a really hard time for me because that was all i had and they tricked me into investing the money with a guarantee that i will make profit from the investment.. I was referred to Mr Gary, he’s  a senior recovery specialist and he helped in recovering all my lost funds to me. I couldn’t believe at first that this was possible as i have lost all hope in contacting anyone on the internet, but i guess not everyone is bad after all. You can reach him on cyberhelp.me22at gmail dot com

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  52. Are you interested in the service of a hacker to get into a phone, facebook account, snapchat, Instagram, yahoo, Whatsapp, get verified on any social network account, increase your followers by any amount, bank wire and bank transfer.
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  53. Obrigado por compartilhar! Não conhecia todos, mas gostei de algumas mudanças, como opção por diferentes visualizações como espionar facebook

    ReplyDelete
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  55. Are you interested in the service of a hacker to get into a phone, facebook account, snapchat, Instagram, yahoo, Whatsapp, get verified on any social network account, increase your followers by any amount, bank wire and bank transfer.
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  56. HOW TO RECOVER LOST BTC I lost my bitcoin to fake blockchain.com impostors on Facebook. Hello I have to share my experience with a life saving group called alliedrecoverygroup from some blockchain scammers.
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  57. Do you need to spy on your spouse in any way and you do not know how? Do you need to track down you spouse phone,hack whatsapp, instagram, facebook and snapchat. chipcyberdog is here to help out, we do all works involving hacking tracking and spying under the supervision of the client ( we do not work outside the clients demand and supervision).
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  59. Are you in a financial crisis, looking for money to start your own business or to pay your bills?
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    ReplyDelete
  60. I am Joe Engressia, Certified Ethical Hacker (CEH) — Delivered By Leading Experts‎.
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    ReplyDelete
  61. Are you a victim of ;

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  62. The hack i purchased from jeajamhacker@gmail.com on April 10 2020 just expired today being December 1th 2020 it has really been an amazing journey so far i got to monitor everything my spouse has been doing on his phone for 9months and am really happy about the phone hack because with jeajamhacker@gmail.com the hack can not be traced. Thank you so much i never regretted my contact with you jeajamhacker@gmail.com.

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