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# Computability and Complexity 9-1 Computability and Complexity Andrei Bulatov Logic Reminder (Cnt’d)

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Computability and Complexity 9-1 Computability and Complexity Andrei Bulatov Logic Reminder (Cnt’d)

Computability and Complexity 9-2 Propositional logic Conjunctive normal forms Predicates, functions and quantifiers Terms Formulas

Computability and Complexity 9-3 Models (First Order Semantics) Let  be a vocabulary. A model appropriate to  is a pair M=(U,  ) consisting of the universe of M, a non-empty set U the interpretation, a function  that assigns - to each predicate symbol P a concrete predicate on U - to each function symbol f a concrete function on U the equality predicate symbol is always assigned the equality predicate on U

Computability and Complexity 9-4 Meaning of terms Let t be a term and let T be an assignment of variables in t with values from U if t is a variable, say, t = X, then is defined to be if then is defined to be

Computability and Complexity 9-5 A model M and a variable assignment T satisfy a formula  (written M, T |=  ) if if is an atomic formula, then M,T |=  if if  =  then M,T |=  if M,T |   if then M,T |=  if and if then M,T |=  if or if  =  X  then M,T |=  if, for every u  U, where is the assignment that is identical to T, except that if  =  X  then M,T |=  if, there exists u  U such that

Computability and Complexity 9-6 Examples A model of the vocabulary of graph theory is a pair G=(U,E), where U is a set and E is a binary relation, that is a graph A model of the vocabulary of number theory U = N, the predicate and function symbols have their usual sense U = Z, the predicate and function symbols have their usual sense U = Q, the predicate and function symbols have their usual sense U = R, the predicate and function symbols have their usual sense U = C, the predicate and function symbols have their usual sense U = {0,…,n-1}, the predicate symbols have their usual sense, the function symbols are interpreted by operations modulo n

Computability and Complexity 9-7 Types of First Order Formulas A formula  is said to be valid if M,T   for any model M and assignment T A formula  is said to be satisfiable if M,T   for some M and T A formula  is said to be unsatisfiable if M,T   for no M and T valid s a t i s f i a b l e unsatisfiable Formulas  and  are said to be equivalent,   , if they have the same satisfying models and assignments

Computability and Complexity 9-8 Valid Formulas Boolean tautologies Equality Quantifiers

Computability and Complexity 9-9 Models and Theories Let  be a vocabulary. For a sentence (set of sentences) , Mod (  ) denotes the class of all models satisfying  For a model (set of models) M, Th (M) denotes the set all sentences satisfied by M (each model from M ) It is called the elementary theory of M If then  is called a valid consequence of written

Computability and Complexity 9-10 Examples Let  be a first order description of a computer chip,  states that a deadlock never occurs  is the question “Can the chip be deadlocked?”

Computability and Complexity 9-11 Proof Systems A proof system consists of a set axioms a collection of proof rules |   A proof is a sequence of formulas, where every formula is Either an axiom or obtained from preceding formulas using a rule A theorem is any formula occurring in a proof

Computability and Complexity 9-12 Propositional Logic Axioms : the main tautologies Proof rules: substitution Let  and  be propositional formulas and let denote the formula obtained from  by replacing every occurrence of X with . Then  |  is a proof rule modus ponens ,  |   Theorems : the tautologies

Computability and Complexity 9-13 Predicate Calculus Let  be a vocabulary with plenty of predicate and function symbols Axioms: Proof rules : modus ponens Theorems : valid first order formulas AX0 Any Boolean tautology AX1 Any expression of the following forms: AX1a : t=t AX1b : AX1c : AX2 : Any expression of the form AX3 : Any expression of the form    X , with X not free in  AX4 : Any expression of the form (  X (    )  ((  X  )  (  X  ))

Computability and Complexity 9-14 Gödel’s Completeness Theorem Theorem Let  be a set of formulas and  a formula. Then    if and only if    Theorem Let  be a set of formulas and  a formula. Then    if and only if   

Computability and Complexity 9-15 Resolution Axioms : a set of disjunctions of atomic formulas (a set of clauses) Proof rules: resolution   P  ,    P   C  X  C , D   X  D  ,      C, D   C  D  (    is called the resolvent of  and  ) R R

Computability and Complexity 9-16 Satisfiability Instance: A conjunctive normal form . Question: Is  satisfiable? Satisfiability Instance: A conjunctive normal form  every clause of which contains exactly k literals. Question: Is  satisfiable? k- Satisfiability

Computability and Complexity 9-17 Theorem A Satisfiability instance, , is unsatisfiable if and only if    Theorem A Satisfiability instance, , is unsatisfiable if and only if    R Example : (1)(2)(3)(4)(5) Proof :

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# 750 - Q49 V42 - My GMAT experience

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Re: 750 - Q49 V42 - My GMAT experience [#permalink]

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03 Jan 2008, 11:38
Verbal

....ery good in verbal. Like all the international students, I struggled with SC. I practiced RC and CR with the 1000 tests (did 120 of each), and that was enough for me. I studied SC with the Manhattan book. I would like to stress here that this book was a BIBLE for me. SC was m...[/quote]

What do you mean saying 1000 test, is it book or software?
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06 Jan 2008, 02:30
kazakhb wrote:
Verbal

....ery good in verbal. Like all the international students, I struggled with SC. I practiced RC and CR with the 1000 tests (did 120 of each), and that was enough for me. I studied SC with the Manhattan book. I would like to stress here that this book was a BIBLE for me. SC was m...

What do you mean saying 1000 test, is it book or software?[/quote]

1000 tests are word documents with 1000 questions for SC, RC, CR. Do a search on the forum and you'll find the docs easily.
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20 Jan 2008, 11:58
Congrats!
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18 Feb 2008, 22:57
very impressive. good luck
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06 Mar 2008, 23:27
Nice job and good marks, thanks for providing us information.
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19 Mar 2008, 03:21
Hi Audio
No you are not sounding stupid thank you for your nice suggestion.
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21 Mar 2008, 01:37
Hi Audio CONGRATULATIONS from my side good job buddy.
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22 Mar 2008, 12:37
Thanks to everybody for the congratulations!

If there are any other questions I'll be happy to answer.
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25 Mar 2008, 01:37
Hi Audio,

Congratunation, you have a great score.
You said: "What I would also do is seriously browse this website for tips, there is some great advice out here. For example, recognizing the 3/2 structure is very helpful."

Do you mind providing it, or pasting a link here? Really appreciated!

Thank you,
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29 Mar 2008, 05:08
sondenso wrote:
Hi Audio,

Congratunation, you have a great score.
You said: "What I would also do is seriously browse this website for tips, there is some great advice out here. For example, recognizing the 3/2 structure is very helpful."

Do you mind providing it, or pasting a link here? Really appreciated!

Thank you,

If you do a search, I'm sure you'll find it. Plus if you browse in the "new to the verbal forum?" topic in the verbal forum, you should find it easily (the 3/2 structure is a structure you find in verbal questions). You'll also find other very precious tips.

Good luck!
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30 Mar 2008, 00:27
Thank you Audio,

Digging the forum sparked my thought after this post. I finished "New to Math forum". I am sure to finish Verbal forum aslo.

Regards,
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05 Apr 2008, 01:59
Hi Audio,
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30 Jul 2008, 07:00
Thanks very much for the tips.

As for a beginner, where would you recommend me to start?
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03 Aug 2008, 06:45
j8liu wrote:
Thanks very much for the tips.

As for a beginner, where would you recommend me to start?

It's been a while since I took that test, but I'll try to remember

First of all, my answer would be: it depends (great help isn't it??). The first questions you need to be asking yourself are: how good am I (honestly) in math? Am I a native speaker? For me the answers were "very good" and "no", so I knew I had to focus on verbal (scored 49 on math and 42 on verbal).

What I would do for starters is simply buy the Gmat book (the bible) and do a couple of exercises (I think there is some sort of review exam in there), just to get the feeling of the questions. You should do that because the data insufficiency questions can be a bit confusing in the beginning. Then after that I'd do one of the Gmat exams for the program you download on mba.com. That would be your first reality check. Depending on the kind of score you do, you will know where your strong and weak points are. Depending on that you should work on your weaknesses but not neglect your strengths. After that it's a question of having the right books and practicing.

In my opinion you don't need classes (unless you're either very rich or with a very inadequate background in math and verbal). Books are good enough.

Last point: I'm a pretty good test-taker, especially when it comes down to what I would call "speed" tests (a lot of short questions). I took a couple of tests like that in my life (including the CFA, which is an exam playing in a completely different league compared to Gmat), so I had experience with me. However, some people are either not good "speed" test takers, or simply have no experience. I would advise you then to practice in real conditions, with a timer. It does make a difference. I didn't do it because I felt quite comfortable about it, but if you're not, then you should do it.

Anyway, I hope this is clear enough for starters. Please ask if you have any other questions.
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03 Aug 2008, 10:52
Can you provide me with any suggestions for the last week preparation before the exam?? for both quant and verbal sections.

Thanks,
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08 Aug 2008, 16:20
bdumpala wrote:
Can you provide me with any suggestions for the last week preparation before the exam?? for both quant and verbal sections.

Thanks,

Well, I didn't really follow a particular scheme, I just practiced. That's the only secret I guess: you can plan as much as you like, but only practice will get you there IMO.
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04 Sep 2008, 06:48
Audio wrote:
Just wanted to share my GMAT experience. I just took the exam today, and I got 750.

My prep material was:

- Manhattan verbal books. I didn't open the RC and CR ones though. As a non-English native (although I have a very good grasp of English), SC book was essential. It really helped me improve my verbal skills.

- Math challenges. Great practice for maths. I recommend this to everybody, especially considering the price, best value for money I've seen. http://gmatclub.com/wiki/Tests

- OG ed.11. Good basic practice exercices.

- 1000 tests.

Background: I studied business in uni, with a strong emphasis on maths, chemistry and physics in the first years (commercial engineer, for those of you who understand what it means). So I've always been comfortable with figures and speed tests (did CFA 1st level too). I lived in several English-speaking countries, so I'm quite comfortable with the language, but I'm not perfectly bilingual. Some idioms were a bit tricky for example.

I'll split my explanations in maths and verbal.

Math

I did almost all the challenges, and did the difficult exercices in the OG. Maths was always going to be easier for me than verbal. I did have to polish my math theory, ecause I forgot a lot.

My recommendation would be the following for those of you who are not poor in maths (sorry if this sounds like I'm bragging): the OG math theory is sufficient to revise most of the principles you need to know. What really helped though were the "tricks", for example in order to add several consecutive integers, take the middle figure and multiply it by the number of figures, etc. This could sound like basic stuff for the math gurus, but I forgot most of this, and it's VERY VERY useful in a test where speed is essential. Anything that can make you save time is good, you'll defo need it. Surf on the math forum to check for this.

Also, knowing that you can use several methods to find answers was very useful - this might sound very stupid, but before going on the forum I never thought of replacing figures in the different results, etc. Try to identify the easiest method for each question. I still made that mistake in the exam, trying to solve a complicated problem without replacing, cost me 1.5 minutes, which was ridiculous.

Also, you HAVE to know the basic stuff inside out: 1 is not prime, 0 is even and isn't negative or positive, etc. The corrolary of this is: read the question CAREFULLY. If we speak about a POSITIVE INTEGER, you can forget about -0.5. Inversely, an integer can also be sq2. NEVER forget to test special figures such as 1 and 0 in the DS questions.

The best practice I found were the challenges, that are - I reckon - quite similar to what you are going to get on the GMAT. COngrats to the peopld who contributed to their creation btw, you guys are spot on. Well done. 79$for that kind of practice is ridiculously low (especially with the very high euro ), don't hesitate and get it! This is all I used for maths, and it was enough. I don't think I could increase my 49 without practicing like a nutter - not worth it. I think that was my limit. Verbal I'm not very good in verbal. Like all the international students, I struggled with SC. I practiced RC and CR with the 1000 tests (did 120 of each), and that was enough for me. I studied SC with the Manhattan book. I would like to stress here that this book was a BIBLE for me. SC was much clearer to me after that, as well as understanding the 3/2 concept. I would recommend all the international students to get this book. Last remark: I also don't believe in studying 6 months before the exam. Regular studying + 2 weeks intensive before the exam should do it (I took 2.5 weeks off, where I wrote a couple of essays and studied). Anyway, if you have questions, please don't hesitate! PS: sorry for the totally incoherent post structure, I can't be ars*d reading it again Hi Audio! Congrats on your score!!! I'm also international student. I'm strugling with GMAT for more than 2 years. I took my test and i got 480 and my goal is at least 700. I work full time and i think i don't dedicate enough time to prepare for this exam. After work i feel tired and spaced out. I'm thinking to quit my job so i can study efficiently 6-7hours a day. However, i wonder what did you mean by saying this "I also don't believe in studying 6 months before the exam"? What would be your advise to me? Thanks in advance. Intern Joined: 03 Sep 2008 Posts: 4 Followers: 0 Kudos [?]: 0 [0], given: 0 Re: 750 - Q49 V42 - My GMAT experience [#permalink] ### Show Tags 05 Sep 2008, 11:08 Hi, How many Perms & Combs questions did u find in GMAT?...There are roughly 2 months for me take the GMAT..Is OG 11 a good buy..Need your help..Thanks!! Senior Manager Joined: 13 Jun 2007 Posts: 410 Schools: Wharton, Booth, Stern Followers: 11 Kudos [?]: 81 [0], given: 0 Re: 750 - Q49 V42 - My GMAT experience [#permalink] ### Show Tags 20 Sep 2008, 13:43 sanb08 wrote: Hi, How many Perms & Combs questions did u find in GMAT?...There are roughly 2 months for me take the GMAT..Is OG 11 a good buy..Need your help..Thanks!! Ouf, I really can't remember, but my memory is very little (surprisingly enough). However maybe that's just pot luck. I would still recommend to study it thoroughly, just in case - you'll be kicking yourself if you get a lot of them. _________________ Wharton admits, join the rugby team!! It'll be by far the best experience of your MBA life Senior Manager Joined: 13 Jun 2007 Posts: 410 Schools: Wharton, Booth, Stern Followers: 11 Kudos [?]: 81 [0], given: 0 Re: 750 - Q49 V42 - My GMAT experience [#permalink] ### Show Tags 20 Sep 2008, 13:48 el1981 wrote: Audio wrote: Just wanted to share my GMAT experience. I just took the exam today, and I got 750. My prep material was: - Manhattan verbal books. I didn't open the RC and CR ones though. As a non-English native (although I have a very good grasp of English), SC book was essential. It really helped me improve my verbal skills. - Math challenges. Great practice for maths. I recommend this to everybody, especially considering the price, best value for money I've seen. http://gmatclub.com/wiki/Tests - OG ed.11. Good basic practice exercices. - 1000 tests. Background: I studied business in uni, with a strong emphasis on maths, chemistry and physics in the first years (commercial engineer, for those of you who understand what it means). So I've always been comfortable with figures and speed tests (did CFA 1st level too). I lived in several English-speaking countries, so I'm quite comfortable with the language, but I'm not perfectly bilingual. Some idioms were a bit tricky for example. I'll split my explanations in maths and verbal. Math I did almost all the challenges, and did the difficult exercices in the OG. Maths was always going to be easier for me than verbal. I did have to polish my math theory, ecause I forgot a lot. My recommendation would be the following for those of you who are not poor in maths (sorry if this sounds like I'm bragging): the OG math theory is sufficient to revise most of the principles you need to know. What really helped though were the "tricks", for example in order to add several consecutive integers, take the middle figure and multiply it by the number of figures, etc. This could sound like basic stuff for the math gurus, but I forgot most of this, and it's VERY VERY useful in a test where speed is essential. Anything that can make you save time is good, you'll defo need it. Surf on the math forum to check for this. Also, knowing that you can use several methods to find answers was very useful - this might sound very stupid, but before going on the forum I never thought of replacing figures in the different results, etc. Try to identify the easiest method for each question. I still made that mistake in the exam, trying to solve a complicated problem without replacing, cost me 1.5 minutes, which was ridiculous. Also, you HAVE to know the basic stuff inside out: 1 is not prime, 0 is even and isn't negative or positive, etc. The corrolary of this is: read the question CAREFULLY. If we speak about a POSITIVE INTEGER, you can forget about -0.5. Inversely, an integer can also be sq2. NEVER forget to test special figures such as 1 and 0 in the DS questions. The best practice I found were the challenges, that are - I reckon - quite similar to what you are going to get on the GMAT. COngrats to the peopld who contributed to their creation btw, you guys are spot on. Well done. 79$ for that kind of practice is ridiculously low (especially with the very high euro ), don't hesitate and get it!

This is all I used for maths, and it was enough. I don't think I could increase my 49 without practicing like a nutter - not worth it. I think that was my limit.

Verbal

I'm not very good in verbal. Like all the international students, I struggled with SC. I practiced RC and CR with the 1000 tests (did 120 of each), and that was enough for me. I studied SC with the Manhattan book. I would like to stress here that this book was a BIBLE for me. SC was much clearer to me after that, as well as understanding the 3/2 concept. I would recommend all the international students to get this book.

Last remark: I also don't believe in studying 6 months before the exam. Regular studying + 2 weeks intensive before the exam should do it (I took 2.5 weeks off, where I wrote a couple of essays and studied).

Anyway, if you have questions, please don't hesitate!

PS: sorry for the totally incoherent post structure, I can't be ars*d reading it again

Hi Audio! Congrats on your score!!!
I'm also international student. I'm strugling with GMAT for more than 2 years. I took my test and i got 480 and my goal is at least 700. I work full time and i think i don't dedicate enough time to prepare for this exam. After work i feel tired and spaced out. I'm thinking to quit my job so i can study efficiently 6-7hours a day. However, i wonder what did you mean by saying this "I also don't believe in studying 6 months before the exam"? What would be your advise to me? Thanks in advance.

Ok, I don't know how good or bad you are, but apparently you need to study quite a lot. If you can afford to quit your job, why not? However, I don't know your situation, but I think it's a very risky tactic. I mean, this is simply the first step of a VERY long process. Can you afford to quit your job for more than a year?

Otherwise, I'd just say: grab you balls in your hand and work it out. Nobody had it easy, studying for the GMAT while working. It all depends on how bad you want it.

Finally, what I meant by not believing in studying 6 months before the exam is that that's a very long period; there's a risk of getting bored / forgetting what you studied before.
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Re: 750 - Q49 V42 - My GMAT experience   [#permalink] 20 Sep 2008, 13:48

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# Limits for Excited ${{\boldsymbol q}}$ (${{\boldsymbol q}^{*}}$) from Single Production INSPIRE search

These limits are from ${{\mathit e}^{+}}$ ${{\mathit e}^{-}}$ $\rightarrow$ ${{\mathit q}^{*}}{{\overline{\mathit q}}}$ , ${{\mathit p}}$ ${{\overline{\mathit p}}}$ $\rightarrow$ ${{\mathit q}^{*}}$ X, or ${{\mathit p}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit q}^{*}}$ X and depend on transition magnetic couplings between ${{\mathit q}}$ and ${{\mathit q}^{*}}$. Assumptions about ${{\mathit q}^{*}}$ decay mode are given in the footnotes and comments.
VALUE (GeV) CL% DOCUMENT ID TECN  COMMENT
$\text{none 1500 - 2600}$ 95 1
 2018 AB
ATLS ${{\mathit p}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit b}^{*}}{{\mathit X}}$ , ${{\mathit b}^{*}}$ $\rightarrow$ ${{\mathit b}}{{\mathit g}}$
$\text{none 1500 - 5300}$ 95 2
 2018 BA
ATLS ${{\mathit p}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit q}^{*}}{{\mathit X}}$ , ${{\mathit q}^{*}}$ $\rightarrow$ ${{\mathit q}}{{\mathit \gamma}}$
$\text{none 1000 - 5500}$ 95 3
 2018 AG
CMS ${{\mathit p}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit q}^{*}}{{\mathit X}}$ , ${{\mathit q}^{*}}$ $\rightarrow$ ${{\mathit q}}{{\mathit \gamma}}$
$\text{none 1000 - 1800}$ 95 4
 2018 AG
CMS ${{\mathit p}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit b}^{*}}{{\mathit X}}$ , ${{\mathit b}^{*}}$ $\rightarrow$ ${{\mathit b}}{{\mathit \gamma}}$
$\text{none 600 - 6000}$ 95 5
 2018 BO
CMS ${{\mathit p}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit q}^{*}}$ ${{\mathit X}}$ , ${{\mathit q}^{*}}$ $\rightarrow$ ${{\mathit q}}{{\mathit g}}$
$\text{none 1200 - 5000}$ 95 6
 2018 P
CMS ${{\mathit p}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit q}^{*}}{{\mathit X}}$ , ${{\mathit q}^{*}}$ $\rightarrow$ ${{\mathit q}}{{\mathit W}}$
$\text{none 1200 - 4700}$ 95 6
 2018 P
CMS ${{\mathit p}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit q}^{*}}{{\mathit X}}$ , ${{\mathit q}^{*}}$ $\rightarrow$ ${{\mathit q}}{{\mathit Z}}$
$\bf{>6000}$ 95 7
 2017 AK
ATLS ${{\mathit p}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit q}^{*}}{{\mathit X}}$ , ${{\mathit q}^{*}}$ $\rightarrow$ ${{\mathit q}}{{\mathit g}}$
• • • We do not use the following data for averages, fits, limits, etc. • • •
$\text{none 600 - 5400}$ 95 8
 2017 W
CMS ${{\mathit p}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit q}^{*}}{{\mathit X}}$ , ${{\mathit q}^{*}}$ $\rightarrow$ ${{\mathit q}}{{\mathit g}}$
$\text{none 1100 - 2100}$ 95 9
 2016
ATLS ${{\mathit p}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit b}^{*}}{{\mathit X}}$ , ${{\mathit b}^{*}}$ $\rightarrow$ ${{\mathit b}}{{\mathit g}}$
$> 1500$ 95 10
 2016 AH
ATLS ${{\mathit p}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit b}^{*}}{{\mathit X}}$ , ${{\mathit b}^{*}}$ $\rightarrow$ ${{\mathit t}}{{\mathit W}}$
$> 4400$ 95 11
 2016 AI
ATLS ${{\mathit p}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit q}^{*}}{{\mathit X}}$ , ${{\mathit q}^{*}}$ $\rightarrow$ ${{\mathit q}}{{\mathit \gamma}}$
12
 2016 AV
ATLS ${{\mathit p}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit q}^{*}}{{\mathit X}}$ , ${{\mathit q}^{*}}$ $\rightarrow$ ${{\mathit W}}{{\mathit b}}$
$> 5200$ 95 13
 2016 S
ATLS ${{\mathit p}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit q}^{*}}{{\mathit X}}$ , ${{\mathit q}^{*}}$ $\rightarrow$ ${{\mathit q}}{{\mathit g}}$
$> 1390$ 95 14
 2016 I
CMS ${{\mathit p}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit b}^{*}}{{\mathit X}}$ , ${{\mathit b}^{*}}$ $\rightarrow$ ${{\mathit t}}{{\mathit W}}$
$> 5000$ 95 15
 2016 K
CMS ${{\mathit p}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit q}^{*}}{{\mathit X}}$ , ${{\mathit q}^{*}}$ $\rightarrow$ ${{\mathit q}}{{\mathit g}}$
$\text{none 500 - 1600}$ 95 16
 2016 L
CMS ${{\mathit p}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit q}^{*}}{{\mathit X}}$ , ${{\mathit q}^{*}}$ $\rightarrow$ ${{\mathit q}}{{\mathit g}}$
$> 4060$ 95 17
 2015 V
ATLS ${{\mathit p}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit q}^{*}}{{\mathit X}}$ , ${{\mathit q}^{*}}$ $\rightarrow$ ${{\mathit q}}{{\mathit g}}$
$> 3500$ 95 18
 2015 V
CMS ${{\mathit p}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit q}^{*}}{{\mathit X}}$ , ${{\mathit q}^{*}}$ $\rightarrow$ ${{\mathit q}}{{\mathit g}}$
$> 3500$ 95 19
 2014 A
ATLS ${{\mathit p}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit q}^{*}}{{\mathit X}}$ , ${{\mathit q}^{*}}$ $\rightarrow$ ${{\mathit q}}{{\mathit \gamma}}$
$> 3200$ 95 20
 2014
CMS ${{\mathit p}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit q}^{*}}{{\mathit X}}$ , ${{\mathit q}^{*}}$ $\rightarrow$ ${{\mathit q}}{{\mathit W}}$
$> 2900$ 95 21
 2014
CMS ${{\mathit p}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit q}^{*}}{{\mathit X}}$ , ${{\mathit q}^{*}}$ $\rightarrow$ ${{\mathit q}}{{\mathit Z}}$
$\text{none 700 - 3500}$ 95 22
 2014 J
CMS ${{\mathit p}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit q}^{*}}{{\mathit X}}$ , ${{\mathit q}^{*}}$ $\rightarrow$ ${{\mathit q}}{{\mathit \gamma}}$
$> 2380$ 95 23
 2013 AJ
CMS ${{\mathit p}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit q}^{*}}{{\mathit X}}$ , ${{\mathit q}^{*}}$ $\rightarrow$ ${{\mathit q}}{{\mathit W}}$
$> 2150$ 95 24
 2013 AJ
CMS ${{\mathit p}}$ ${{\mathit p}}$ $\rightarrow$ ${{\mathit q}^{*}}{{\mathit X}}$ , ${{\mathit q}^{*}}$ $\rightarrow$ ${{\mathit q}}{{\mathit Z}}$
1  AABOUD 2018AB assume ${{\mathit \Lambda}}$ = ${\mathit m}_{{{\mathit b}^{*}}}$, ${{\mathit f}_{{s}}}$ = ${{\mathit f}}$ = ${{\mathit f}^{\,'}}$ = 1. The contact interactions are not included in ${{\mathit b}^{*}}$ production and decay amplitudes.
2  AABOUD 2018BA search for first-generation excited quarks (${{\mathit u}^{*}}$ and ${{\mathit d}^{*}}$) with degenerate mass, assuming ${{\mathit \Lambda}}$ = ${\mathit m}_{{{\mathit q}^{*}}}$, ${{\mathit f}_{{s}}}$ = ${{\mathit f}}$ = ${{\mathit f}^{\,'}}$ = 1. The contact interactions are not included in ${{\mathit q}^{*}}$ production and decay amplitudes.
3  SIRUNYAN 2018AG search for first-generation excited quarks (${{\mathit u}^{*}}$ and ${{\mathit d}^{*}}$) with degenerate mass, assuming ${{\mathit \Lambda}}$ = ${\mathit m}_{{{\mathit q}^{*}}}$, ${{\mathit f}_{{s}}}$ = ${{\mathit f}}$ = ${{\mathit f}^{\,'}}$ = 1.
4  SIRUNYAN 2018AG search for excited ${{\mathit b}}$ quark assuming ${{\mathit \Lambda}}$ = ${\mathit m}_{{{\mathit q}^{*}}}$, ${{\mathit f}_{{s}}}$ = ${{\mathit f}}$ = ${{\mathit f}^{\,'}}$ = 1.
5  SIRUNYAN 2018BO assume ${{\mathit \Lambda}}$ = ${\mathit m}_{{{\mathit q}^{*}}}$, ${{\mathit f}_{{s}}}$ = ${{\mathit f}}$ = ${{\mathit f}^{\,'}}$ = 1. The contact interactions are not included in ${{\mathit q}^{*}}$ production and decay amplitudes.
6  SIRUNYAN 2018P use the hadronic decay of ${{\mathit W}}$ or ${{\mathit Z}}$, assuming ${{\mathit \Lambda}}$ = ${\mathit m}_{{{\mathit q}^{*}}}$, ${{\mathit f}_{{s}}}$= ${{\mathit f}}$ = ${{\mathit f}^{\,'}}$= 1.
7  AABOUD 2017AK assume $\Lambda$ = ${\mathit m}_{{{\mathit q}^{*}}}$, ${{\mathit f}_{{s}}}$ = ${{\mathit f}}$ = ${{\mathit f}^{\,'}}$ = 1. The contact interactions are not included in ${{\mathit q}^{*}}$ production and decay amplitudes. Only the decay of ${{\mathit q}^{*}}$ $\rightarrow$ ${{\mathit g}}{{\mathit u}}$ and ${{\mathit q}^{*}}$ $\rightarrow$ ${{\mathit g}}{{\mathit d}}$ is simulated as the benchmark signals in the analysis.
8  KHACHATRYAN 2017W assume $\Lambda$ = ${\mathit m}_{{{\mathit q}^{*}}}$, ${{\mathit f}_{{s}}}$ = ${{\mathit f}}$ = ${{\mathit f}^{\,'}}$ = 1. The contact interactions are not included in ${{\mathit q}^{*}}$ production and decay amplitudes.
9  AABOUD 2016 assume $\Lambda$ = ${\mathit m}_{{{\mathit b}^{*}}}$, ${{\mathit f}_{{s}}}$ = ${{\mathit f}}$ = ${{\mathit f}^{\,'}}$ = 1. The contact interactions are not included in the ${{\mathit b}^{*}}$ production and decay amplitudes.
10  AAD 2016AH search for ${{\mathit b}^{*}}$ decaying to ${{\mathit t}}{{\mathit W}}$ in ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 8 TeV. ${{\mathit f}_{{g}}}$ = ${{\mathit f}_{{L}}}$ = ${{\mathit f}_{{R}}}$ = 1 are assumed. See their Fig. 12b for limits on $\sigma \cdot{}\mathit B$.
11  AAD 2016AI assume $\Lambda$ = ${\mathit m}_{{{\mathit q}^{*}}}$, ${{\mathit f}_{{s}}}$ = ${{\mathit f}}$ = ${{\mathit f}^{\,'}}$ = 1.
12  AAD 2016AV search for single production of vector-like quarks decaying to ${{\mathit W}}{{\mathit b}}$ in ${{\mathit p}}{{\mathit p}}$ collisions. See their Fig. 8 for the limits on couplings and mixings.
13  AAD 2016S assume $\Lambda$ = ${\mathit m}_{{{\mathit q}^{*}}}$, ${{\mathit f}_{{s}}}$ = ${{\mathit f}}$ = ${{\mathit f}^{\,'}}$ = 1. The contact interactions are not included in ${{\mathit q}^{*}}$ production and decay amplitudes.
14  KHACHATRYAN 2016I search for ${{\mathit b}^{*}}$ decaying to ${{\mathit t}}{{\mathit W}}$ in ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 8 TeV. ${{\mathit \kappa}_{{L}}^{b}}$ = ${{\mathit g}_{{L}}}$ = 1, ${{\mathit \kappa}_{{R}}^{b}}$ = ${{\mathit g}_{{R}}}$ = 0 are assumed. See their Fig. 8 for limits on $\sigma \cdot{}\mathit B$.
15  KHACHATRYAN 2016K assume ${{\mathit \Lambda}}$ = ${\mathit m}_{{{\mathit q}^{*}}}$, ${{\mathit f}_{{s}}}$ = ${{\mathit f}}$ = ${{\mathit f}^{\,'}}$= 1. The contact interactions are not included in ${{\mathit q}^{*}}$ production and decay amplitudes.
16  KHACHATRYAN 2016L search for resonances decaying to dijets in ${{\mathit p}}{{\mathit p}}$ collisions at $\sqrt {s }$ = 8 TeV using the data scouting technique which increases the sensitivity to the low mass resonances.
17  AAD 2015V assume ${{\mathit \Lambda}}$ = ${\mathit m}_{{{\mathit q}^{*}}}$, ${{\mathit f}_{{s}}}$ = ${{\mathit f}}$ = ${{\mathit f}^{\,'}}$ = 1. The contact interactions are not included in ${{\mathit q}^{*}}$ production and decay amplitudes.
18  KHACHATRYAN 2015V assume ${{\mathit \Lambda}}$ = ${\mathit m}_{{{\mathit q}^{*}}}$, ${{\mathit f}_{{s}}}$ = ${{\mathit f}}$ = ${{\mathit f}^{\,'}}$ = 1. The contact interactions are not included in ${{\mathit q}^{*}}$ production and decay amplitudes.
19  AAD 2014A assume ${{\mathit \Lambda}}$ = ${\mathit m}_{{{\mathit q}^{*}}}$, ${{\mathit f}_{{s}}}$ = ${{\mathit f}}$ = ${{\mathit f}^{\,'}}$ = 1.
20  KHACHATRYAN 2014 use the hadronic decay of ${{\mathit W}}$, assuming ${{\mathit \Lambda}}$ = ${\mathit m}_{{{\mathit q}^{*}}}$, ${{\mathit f}_{{s}}}={{\mathit f}}={{\mathit f}^{\,'}}$ = 1.
21  KHACHATRYAN 2014 use the hadronic decay of ${{\mathit Z}}$, assuming ${{\mathit \Lambda}}$ = ${\mathit m}_{{{\mathit q}^{*}}}$, ${{\mathit f}_{{s}}}={{\mathit f}}={{\mathit f}^{\,'}}$ = 1.
22  KHACHATRYAN 2014J assume ${{\mathit f}_{{s}}}$ = ${{\mathit f}}$ = ${{\mathit f}^{\,'}}$ = ${{\mathit \Lambda}}$ $/$ ${\mathit m}_{{{\mathit q}^{*}}}$.
23  CHATRCHYAN 2013AJ use the hadronic decay of ${{\mathit W}}$.
24  CHATRCHYAN 2013AJ use the hadronic decay of ${{\mathit Z}}$.
References:
 AABOUD 2018BA
EPJ C78 102 Search for new phenomena in high-mass final states with a photon and a jet from $pp$ collisions at $\sqrt{s}$ = 13 TeV with the ATLAS detector
 AABOUD 2018AB
PR D98 032016 Search for resonances in the mass distribution of jet pairs with one or two jets identified as $b$-jets in proton-proton collisions at $\sqrt{s}=13$ TeV with the ATLAS detector
 SIRUNYAN 2018AG
PL B781 390 Search for excited quarks of light and heavy flavor in $\gamma +$?jet final states in proton?proton collisions at $\sqrt{s} =$ 13TeV
 SIRUNYAN 2018P
PR D97 072006 Search for massive resonances decaying into $WW$, $WZ$, $ZZ$, $qW$, and $qZ$ with dijet final states at $\sqrt{s}=13\text{ }\text{ }\mathrm{TeV}$
 SIRUNYAN 2018BO
JHEP 1808 130 Search for narrow and broad dijet resonances in proton-proton collisions at $\sqrt{s}=13$ TeV and constraints on dark matter mediators and other new particles
 AABOUD 2017AK
PR D96 052004 Search for New Phenomena in Dijet Events using 37 ${\mathrm {fb}}{}^{-1}$ of ${{\mathit p}}{{\mathit p}}$ Collision Data Collected at $\sqrt {s }$ = 13 TeV with the ATLAS Detector
 KHACHATRYAN 2017W
PL B769 520 Search for Dijet Resonances in Proton-Proton Collisions at $\sqrt {s }$ = 13 TeV and Constraints on Dark Matter and other Models
 AABOUD 2016
PL B759 229 Search for Resonances in the Mass Distribution of Jet Pairs with One or Two Jets Identified as ${\mathit {\mathit b}}$-Jets in Proton-proton Collisions at $\sqrt {s }$ =13 TeV with the ATLAS Detector
PL B754 302 Search for New Phenomena in Dijet Mass and Angular Distributions from ${{\mathit p}}{{\mathit p}}$ Collisions at $\sqrt {s }$ = 13 TeV with the ATLAS Detector
JHEP 1603 041 Search for New Phenomena with Photon $+$ Jet Events in Proton-Proton Collisions at $\sqrt {s }$ = 13 TeV with the ATLAS Detector
EPJ C76 442 Search for Single Production of Vector-like Quarks Decaying into ${{\mathit W}}{\mathit {\mathit b}}$ in ${{\mathit p}}{{\mathit p}}$ Collisions at $\sqrt {s }$ = 8 TeV with the ATLAS Detector
JHEP 1602 110 Search for the Production of Single Vector-Like and Excited Quarks in the ${{\mathit W}}{\mathit {\mathit t}}$ Final State in ${{\mathit p}}{{\mathit p}}$ Collisions at $\sqrt {s }$ = 8 TeV with the ATLAS Detector
 KHACHATRYAN 2016K
PRL 116 071801 Search for Narrow Resonances Decaying to Dijets in Proton-Proton Collisions at $\sqrt {s }$ = 13 TeV
 KHACHATRYAN 2016I
JHEP 1601 166 Search for the Production of an Excited Bottom Quark Decaying to ${{\mathit t}}{{\mathit W}}$ in Proton-Proton Collisions at $\sqrt {s }$ = 8 TeV
 KHACHATRYAN 2016L
PRL 117 031802 Search for Narrow Resonances in Dijet Final States at $\sqrt {s }$ = 8 TeV with the Novel CMS Technique of Data Scouting
PR D91 052007 Search for New Phenomena in the Dijet Mass Distribution using ${{\mathit p}}{{\mathit p}}$ Collision Data at $\sqrt {s }$ = 8 TeV with the ATLAS Detector
PR D91 052009 Search for Resonances and Quantum Black Holes using Dijet Mass Spectra in Proton-Proton Collisions at $\sqrt {s }$ = 8 TeV
PL B728 562 Search for New Phenomena in Photon + Jet Events Collected in Proton$−$Proton Collisions at $\sqrt {s }$ = 8 TeV with the ATLAS Detector
JHEP 1408 173 Search for Massive Resonances in Dijet Systems Containing Jets Tagged as ${{\mathit W}}$ or ${{\mathit Z}}$ Boson Decays in ${{\mathit p}}{{\mathit p}}$ Collisions at $\sqrt {s }$ = 8 TeV
PL B738 274 Search for Excited Quarks in the ${{\mathit \gamma}}$ + Jet Final State in Proton-Proton Collisions at $\sqrt {s }$ = 8 TeV
PL B723 280 Search for Heavy Resonances in the ${{\mathit W}}/{{\mathit Z}}$-Tagged Dijet Mass Spectrum in ${{\mathit p}}{{\mathit p}}$ Collisions at 7 TeV


text:
# Worksheet: Sum of a Finite Geometric Sequence

In this worksheet, we will practice calculating the sum of the terms in a geometric sequence with a definite number of terms.

Q1:

The sum of the terms of a sequence is called a series.

A geometric series is the sum of a geometric sequence; a geometric series with terms can be written as where is the first term and is the common ratio (the number you multiply one term by to get the next term in the sequence, ).

Find the sum of the first 6 terms of a geometric series with and .

• A
• B
• C
• D204
• E

Q2:

A geometric series has a first term of 3 and a common ratio of 5. Find the sum of the first 6 terms.

Q3:

Find, to two decimal places, the sum of the following geometric series:

Q4:

We can find a formula for the sum of a geometric series. Consider the series

Multiply the expression for by , the common ratio.

• A
• B
• C
• D
• E

So, we have and

The right-hand sides of the equations are very similar. Identify the terms that do \textbf{not} appear on the right-hand side of both equations.

• A
• B
• C
• D
• E

Now, consider the subtraction

Use the answer to the previous part to simplify the subtraction .

• A
• B
• C
• D
• E

Factor both sides of the equation.

• A
• B
• C
• D
• E

Rearrange the equation to make the subject of the formula.

• A
• B
• C
• D
• E

Q5:

There are two geometric series with a first term of 3 and whose first 3 terms have a sum of 21.

What are their common ratios?

• A3, 7
• B ,
• C , 3
• D2,
• E ,

Write an expression for the sum of the first terms of the sequence with first term 3 and common ratio 2.

• A
• B
• C
• D
• E

Q6:

In a geometric sequence, the first term is and the common ratio is .

Find the sum of the first 3 terms of a geometric sequence with and .

• A
• B410
• C
• D

Q7:

Find the geometric sequence given the first term is 324, the last term is 4, and the sum of all the terms is 484.

• A
• B
• C
• D

Q8:

Find the fourth term of the geometric sequence given by where is the sum of the first terms.

Q9:

Find the first three positive terms of a geometric sequence given their sum is 14 and the sum of their multiplicative inverses is .

• A , 6, and 24
• B , , and
• C , , and
• D , , and

Q10:

Find, in terms of and , the common ratio of the geometric sequence given the sum of the first ten terms is and the sum of the next ten terms is .

• A
• B
• C
• D
• E

Q11:

Find the geometric sequence and the sum of the first six terms given the second term is four times the fourth one, the sum of the fourth and seventh terms is 2 and all terms are positive.

• A ,
• B ,
• C ,
• D ,
• E ,

Q12:

Find the sum of the first 20 terms of the geometric sequence giving the answer to two decimal places.

Q13:

Find the sum of the geometric series .

• A
• B
• C
• D

Q14:

Find the sum of the geometric series .

Q15:

Find the geometric sequence and the sum of the first six terms given that the sum of the second and fourth terms is and the sum of the third and fifth terms is .

• A ,
• B ,
• C ,
• D ,
• E ,

Q16:

Find the sum of the first 7 terms of a geometric sequence given and .

• A
• B
• C
• D
• E

Q17:

Find the number of terms of the geometric sequence for which the sum is equal to 93.

Q18:

Find the least number of terms of the geometric sequence given the first term is 1 and the fourth term is 125, where the sum of the terms is 7,143.

Q19:

In a geometric sequence, the first term is , the common ratio is , and the last term is .

Find the sum of the geometric sequence with , , and .

Q20:

Find the geometric sequence and the sum of the first seven terms given , and all terms are positive.

• A ,
• B ,
• C ,
• D ,
• E ,

Q21:

Find the sum of the first 6 terms of the geometric series .

• A
• B
• C
• D

Q22:

Find the geometric sequence and the sum of the first five terms given the sum of the first three terms is 1 and the sum of the next three terms is 27.

• A ,
• B ,
• C ,
• D ,
• E ,

Q23:

Find the number of terms of the geometric sequence for which the sum is equal to 2,783.

Q24:

Find the sum of the first 6 terms of the geometric sequence .

• A
• B
• C
• D605

Q25:

Find the sum of the geometric sequence .


text:
# Clash between the “linktocpage” option of the hyperref package and the “\cftchapterpagefont” command of memoir class

I'm trying to produce a document using the memoir class and the hyperref package. Unfortunately, I get a lot of errors when I try to use the linktocpage option of the hyperref package with \renewcommand{\cftchapterpagefont}{\textrm}.

I guess the solution will be on the lines of this answer to a similar problem, but I haven't been able to figure it out. Here is a minimal working example:

\documentclass[openany,oneside,11pt]{memoir}
\NeedsTeXFormat{LaTeX2e}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Turkish character and language support
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage[turkish,shorthands=:!]{babel}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Required packages
\RequirePackage{amssymb,amsmath,hyperref,multicol,ifthen,graphicx,pifont,rotating,xcolor}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The  TOC
\cftsetindents{part}{0em}{3.0em}% Indendation and numwidth for part titles in the TOC
\cftsetindents{chapter}{1em}{2.0em}% Indendation and numwidth for chapter titles in the TOC
\setlength{\cftbeforechapterskip}{0pt}% No gap between chapter titles in the TOC
\renewcommand{\cftchapterfont}{\textrm}% Chapter title font in the TOC
\renewcommand{\cftchapterpagefont}{\textrm}% Clashes with the "linktocpage" option of the hyperref package
\renewcommand{\cftpartafterpnum}{\par % Thanks to https://tex.stackexchange.com/a/42500
\vspace{-2\baselineskip} \hskip -\memRTLleftskip\protect\mbox{}\protect\hrulefill\par%
\vspace{0.5\baselineskip} \hskip -\memRTLleftskip\protect\mbox{}\protect\hrulefill\par}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% hyperref setup
\hypersetup{%
unicode={true}, %
bookmarks={true}, %
allcolors={kmavi},%
pdfdisplaydoctitle={true}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The colors
\definecolor{kmavi}{RGB}{0,80,117}
\definecolor{mavi}{RGB}{0,115,168}

\begin{document}

\tableofcontents*

\part{My First Part}
\chapter{My First Chapter}
\chapter{My Second Chapter}

\end{document}

• I can't believe it, your suggestion is so simple and it works perfectly. Thank you very much! – Kutt Jun 9 '17 at 0:02
• Also load hyperref at the end of the preamble almost always, not somewhere in between. – user31729 Jun 9 '17 at 0:03

Since hyperref uses the page number as link anchor, \renewcommand{cftchapterpagefont}{\textrm} will fail due to that fact that is expecting an argument.

Use

\renewcommand{\cftchapterpagefont}{\rmfamily}

instead, which is a font switch not needing an argument.

\documentclass[openany,oneside,11pt]{memoir}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Turkish character and language support
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage[turkish,shorthands=:!]{babel}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Required packages
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{multicol}
\usepackage{ifthen}
\usepackage{graphicx}
\usepackage{pifont}
\usepackage{rotating}
\usepackage{xcolor}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The  TOC
\cftsetindents{part}{0em}{3.0em}% Indendation and numwidth for part titles in the TOC
\cftsetindents{chapter}{1em}{2.0em}% Indendation and numwidth for chapter titles in the TOC
\setlength{\cftbeforechapterskip}{0pt}% No gap between chapter titles in the TOC
\renewcommand{\cftchapterfont}{\rmfamily}% Chapter title font in the TOC
\renewcommand{\cftchapterpagefont}{\rmfamily}% Clashes with the "linktocpage" option of the hyperref package
\renewcommand{\cftpartafterpnum}{\par % Thanks to https://tex.stackexchange.com/a/42500
\vspace{-2\baselineskip} \hskip -\memRTLleftskip\protect\mbox{}\protect\hrulefill\par%
\vspace{0.5\baselineskip} \hskip -\memRTLleftskip\protect\mbox{}\protect\hrulefill\par}

\usepackage{hyperref}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% hyperref setup
\hypersetup{%
unicode={true}, %
bookmarks={true}, %
allcolors={kmavi},%
pdfdisplaydoctitle={true}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The colors
\definecolor{kmavi}{RGB}{0,80,117}
\definecolor{mavi}{RGB}{0,115,168}

\begin{document}


text:
# Problems with Black-Litterman: negative portfolio weights, and very poor returns

I am trying to implement the Black-Litterman model using own-defined views matrix (from consensus analysts). However, I have encountered the problems of negative portfolio weights in some periods, and some extreme weightings >1 (at most 200%). All of the results end up in underperformance relative to the benchmark (which is simply the market-weighted stocks without strategy, i.e. passive index). In one data set, the cumulative returns of the assets (before implemented in the model) generates superior returns relative to the benchmark, but when after implemented in the model, it underperforms.

Not sure what I am doing wrong, as I am following the instructions. At lest the BL main formula works as intended using examples from other data set or articles. Additionally, the model is also sensitive to changes in the inputs, such as the changes in the P-matrix and Q-matrix. Slightly changing the Q-matrix from negative to positive returns changes the results of the Black-Litterman model.

Could anyone guide me in the right direction?

• In BL not only do you specify your "views", you also get to specify the margin of error in your views (if I recall correctly it is the matrix $\Omega$). If these estimates are too small (i.e. if you attribute unwarranted precision to your views) you would get the effect that you are seeing. – noob2 Jul 2 '18 at 16:53
• @noob2 I have specified omega according to the classical formula that most uses., i.e. omega that is proportional to the views. – Mataunited17 Jul 4 '18 at 11:33


text:
# hboxplot

0th

Percentile

##### 2-D Generalization of Boxplot

If bin is an eroded hexbin object, i.e., an erodebin object, hboxplot() plots the high counts cells selected by erode(). By default, the high counts cells contain 50 percent of the counts so analagous to the interquartile “range”. The function distinguishes the last cells eroded using color. These cells correspond to one definition of the bivariate median.

Keywords
hplot
##### Usage
hboxplot(bin, xbnds = NULL, ybnds = NULL,
density, border = c(0, grey(0.7)), pen = c(2, 3),
unzoom = 1.1, clip ="off", reshape = FALSE,
xlab = NULL, ylab = NULL, main = "")
##### Arguments
bin

an object of class hexbin.

xbnds,ybnds

global x- and y-axis plotting limits for multiple plots.

density, border

arguments for polygon() each of length two, the first for the median, the second for the other cells.

pen

colors (“pen numbers”) for polygon().

unzoom

plot limit expansion factor when xbnds is missing.

clip

either 'on' or 'off' are the allowed arguments, when on everything is clipped to the plotting region.

reshape

logical value to reshape the plot although xbnds and ybnds are present.

xlab, ylab, main

x- and y- axis labels and main title

##### Details

The density, border, and pen arguments correspond to the polygon function calls for plotting two types of cells. The cell types, pen numbers and suggested colors are

 TYPE PEN COLOR cells of bin 2 light gray last eroded cells of bin (median cells) 1 black

The erode components of the hexbin objects must be present for the medians cells to plot.

When xbnds is missing or reshape is true, the plot changes graphics parameters and resets them. When xbnds is missing the function also zooms in based on the available data to provide increased resolution.

The zoom used the hexagon cell centers. The unzoom argument backs off a bit so the whole hexagon will fit in the plot.

Hboxplot() is used as a stand alone function, for producing separate legends .....

##### Value

invisibly, the hexViewport() used internally. Used to add to the plot afterwards.

##### References

see in grid.hexagons.

hexbin, erode, hcell2xy, gplot.hexbin, grid.hexagons, grid.hexlegend
library(hexbin) # NOT RUN { ## boxplot of smoothed counts x <- rnorm(10000) y <- rnorm(10000) bin <- hexbin(x,y) erodebin <- erode(smooth.hexbin(bin)) hboxplot(erodebin) hboxplot(erodebin, density = c(32,7), border = c(2,4)) hp <- hboxplot(erodebin, density = c(-1,17), main = "hboxplot(erode*(smooth*(.)))") pushHexport(hp) library("grid") grid.points(x[1:10], y[1:10])# just non-sense to show the principle popViewport() # }


text:
## Pappus’s theorem

Problem:

Given a point on the circumference of a cyclic quadrilateral, prove that the product of the distances from the point to any pair of opposite sides or to the diagonals are equal.

Proof:

Let a, b, c, d be the coordinates of the vertices A, B, C, D of the quadrilateral and consider the complex plane with origin at the circumcenter of ABCD. Without loss of generality, assume that the circumradius equals 1.

The equation of line AB is

$\left | \begin{array}{ccc} a & \overline{a} & 1 \\ b & \overline{b} & 1 \\ z & \overline{z} & 1 \end{array} \right | = 0$.

This is equivalent to $z(\overline{a}-\overline{b})-\overline{z}(a-b)=\overline{a}b-a\overline{b}$, that is,

$z+ab\overline{z}=a+b$

Let point $M_{1}$ be the foot of the perpendicular from a point M on the circumcircle to the line AB. If m is the coordinate of the point M, then

$z_{M_{1}}=\frac{m-ab\overline{m}+a+b}{2}$

and

$d(M, AB)=|m-m_{1}|=|m-\frac{m-ab\overline{m}+a+b}{2}|=|\frac{(m-a)(m-b)}{2m}|$ since $m \overline{m} = 1$.

Likewise,

$d(M, BC)=|\frac{(m-b)(m-c)}{2m}|$, $d(M, CD)=|\frac{(m-c)(m-d)}{2m}|$

$d(M, DA)=|\frac{(m-d)(m-a)}{2m}|$, $d(M, AC)=|\frac{(m-a)(m-c)}{2m}|$

and $d(M, BD)=|\frac{(m-b)(m-d)}{2m}|$

Thus,

$d(M, AB).d(M, CD)=d(M, BC).d(M, DA)=d(M, AC).d(M, BD)$ as claimed.

QED.

More later,

Nalin Pithwa

PS: The above example indicates how easy it is prove many fascinating theorems of pure plane geometry using the tools and techniques of complex numbers.

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text:
# History of Topics in Special Relativity/Lorentz transformation (squeeze)

## Lorentz transformation via squeeze mappings

A squeeze mapping relates blue and green parallelograms.

As already indicated in E:(3c) in exponential form or E:(6f) in terms of Cayley–Klein parameter, Lorentz boosts in terms of hyperbolic rotations can be expressed as w:squeeze mappings. Using w:asymptotic coordinates of a hyperbola (u,v), in relativity also known as w:light-cone coordinates, they have the general form (some authors alternatively add a factor of 2 or ${\displaystyle {\sqrt {2}}}$):[1]

${\displaystyle {\begin{matrix}&{\begin{array}{c}u=x_{0}-x_{1},\ v=x_{0}+x_{1}\\u'=x_{0}^{\prime }-x_{1}^{\prime },\ v'=x_{0}^{\prime }+x_{1}^{\prime }\end{array}}\\\hline (1)&(u',v')=\left(ku,\ {\frac {1}{k}}v\right)\\(2)&(u',v')=\left({\frac {1}{k}}v,\ ku\right)\\\hline &u'v'=uv\end{matrix}}}$

(9a)

with arbitrary k. This geometrically corresponds to the transformation of one parallelogram to other ones of same area, whose sides touch a hyperbola and both asymptotes. While equation system (1) corresponds to proper Lorentz boosts, equation system (2) produces improper ones. For instance, solving (1) for ${\displaystyle x'_{0},x'_{1}}$ gives:

{\displaystyle {\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime 2}+x_{1}^{\prime 2}\\\hline {\begin{aligned}x_{0}^{\prime }&={\frac {1}{2}}\left(k+{\frac {1}{k}}\right)x_{0}-{\frac {1}{2}}\left(k-{\frac {1}{k}}\right)x_{1}&&={\frac {x_{0}\left(k^{2}+1\right)-x_{1}\left(k^{2}-1\right)}{2k}}\\x_{1}^{\prime }&=-{\frac {1}{2}}\left(k-{\frac {1}{k}}\right)x_{0}+{\frac {1}{2}}\left(k+{\frac {1}{k}}\right)x_{1}&&={\frac {-x_{0}\left(k^{2}-1\right)+x_{1}\left(k^{2}+1\right)}{2k}}\\\\x_{0}&={\frac {1}{2}}\left(k+{\frac {1}{k}}\right)x_{0}^{\prime }+{\frac {1}{2}}\left(k-{\frac {1}{k}}\right)x_{1}^{\prime }&&={\frac {x_{0}^{\prime }\left(k^{2}+1\right)+x_{1}^{\prime }\left(k^{2}-1\right)}{2k}}\\x_{1}&={\frac {1}{2}}\left(k-{\frac {1}{k}}\right)x_{0}^{\prime }+{\frac {1}{2}}\left(k+{\frac {1}{k}}\right)x_{1}^{\prime }&&={\frac {x_{0}^{\prime }\left(k^{2}-1\right)+x_{1}^{\prime }\left(k^{2}+1\right)}{2k}}\end{aligned}}\end{matrix}}}

(9b)

The geometrical foundation of squeeze mapping (9a) was known for a long time since Apollonius (BC) and was used to generate hyperbolas by Speidell (1688) and Whiston (1710). Equation (9a-1) was implicitly used by Mercator (1668) and explicitly by Laisant (1874) and Günther (1880/81) in relation to elliptic trigonometry, or by Lie (1879-81), Bianchi (1886, 1894), Darboux (1891/94), Eisenhart (1905) as Lie transform[1] of w:pseudospherical surfaces in terms of the w:Sine-Gordon equation, or by Lipschitz (1885/86) in transformation theory. Equation (9a-2) was given by Reynaud (1819).

From that, different forms of Lorentz transformation were derived: (9b) by Lipschitz (1885/86), Bianchi (1886, 1894), Eisenhart (1905), trigonometric Lorentz boost E:(8a) by Bianchi (1886, 1894) and Darboux (1891/94), and trigonometric Lorentz boost E:(8b) by Eisenhart (1905). Lorentz boost (9b) was rediscovered in the framework of special relativity by w:Hermann Bondi (1964)[2] in terms of w:Bondi k-calculus, by which k can be physically interpreted as Doppler factor. Since (9b) is equivalent to E:(6f) in terms of Cayley–Klein parameter by setting ${\displaystyle k=\alpha ^{2}}$, it can be interpreted as the 1+1 dimensional special case of Lorentz Transformation E:(6e) stated by Gauss around 1800 (posthumously published 1863), E:Selling (1873), E:Bianchi (1888), E:Fricke (1891) and E:Woods (1895).

Rewriting (9a) in terms of w:homogeneous coordinates signifies squeeze mappings of the unit hyperbola in terms of a w:projective conic:

${\displaystyle {\begin{matrix}\left[u,v\right]=\left[{\frac {y_{1}}{y_{3}}},{\frac {y_{2}}{y_{3}}}\right]\quad \left(uv=1\quad \Rightarrow \quad y_{1}y_{2}-y_{3}^{2}=0\right)\\\left[k,{\frac {1}{k}}\right]=\left[{\frac {\alpha _{1}}{\alpha _{3}}},{\frac {\alpha _{2}}{\alpha _{3}}}\right]\quad \left(k{\frac {1}{k}}=1\quad \Rightarrow \quad \alpha _{1}\alpha _{2}-\alpha _{3}^{2}=0\right)\\\hline y'_{1}=\alpha _{1}y_{1}\\y'_{2}=\alpha _{2}y_{2}\\y'_{3}=\alpha _{3}y_{3}\\\hline y_{1}y_{2}-y_{3}^{2}=y'_{1}y'_{2}-y_{3}^{\prime 2}=0\\\hline uv={\frac {y_{1}y_{2}}{y_{3}^{2}}}=u'v'={\frac {y'_{1}y'_{2}}{y_{3}^{\prime 2}}}\end{matrix}}}$

(9c)

Such transformations were given by Klein (1871) to express motions in non-Euclidean space.

Furthermore, variables u, v in (9a) can be rearranged to produce another form of squeeze mapping, resulting in Lorentz transformation E:(5b) in terms of Cayley-Hermite parameter:

{\displaystyle {\begin{matrix}{\begin{matrix}u=x_{0}-x_{1}\\v=x_{0}+x_{1}\\u'=x_{0}^{\prime }-x_{1}^{\prime }\\v'=x_{0}^{\prime }+x_{1}^{\prime }\end{matrix}}\Rightarrow {\begin{matrix}u_{1}=x_{0}+x_{0}^{\prime }\\v_{1}=x_{0}-x_{0}^{\prime }\\u_{2}=x_{1}-x_{1}^{\prime }\\v_{2}=x_{1}+x_{1}^{\prime }\end{matrix}}\\\hline (u_{2},v_{2})=\left(au_{1},\ {\frac {1}{a}}v_{1}\right)\Rightarrow u_{2}v_{2}=u_{1}v_{1}\\(u',v')=\left({\frac {1+a}{1-a}}u,\ {\frac {1-a}{1+a}}v\right)\Rightarrow u'v'=uv\end{matrix}}\Rightarrow {\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime 2}+x_{1}^{\prime 2}\\\hline {\begin{aligned}x_{0}^{\prime }&=x_{0}{\frac {1+a^{2}}{1-a^{2}}}-x_{1}{\frac {2a}{1-a^{2}}}&&={\frac {x_{0}\left(1+a^{2}\right)-x_{1}2a}{1-a^{2}}}\\x_{1}^{\prime }&=-x_{0}{\frac {2a}{1-a^{2}}}+x_{1}{\frac {1+a^{2}}{1-a^{2}}}&&={\frac {-x_{0}2a+x_{1}\left(1+a^{2}\right)}{1-a^{2}}}\\\\x_{0}&=x_{0}^{\prime }{\frac {1+a^{2}}{1-a^{2}}}+x_{1}^{\prime }{\frac {2a}{1-a^{2}}}&&={\frac {x_{0}^{\prime }\left(1+a^{2}\right)+x_{1}^{\prime }2a}{1-a^{2}}}\\x_{1}&=x_{0}^{\prime }{\frac {2a}{1-a^{2}}}+x_{1}^{\prime }{\frac {1+a^{2}}{1-a^{2}}}&&={\frac {x_{0}^{\prime }2a+x_{1}^{\prime }\left(1+a^{2}\right)}{1-a^{2}}}\end{aligned}}\end{matrix}}}

(9d)

These Lorentz transformations were given (up to a sign change) by Laguerre (1882), Darboux (1887), Smith (1900) in relation to Laguerre geometry.

On the basis of factors k or a, all previous Lorentz boosts E:(3b), E:(4a), E:(8a), E:(8b), can be expressed as squeeze mappings as well:

${\displaystyle {\begin{array}{r|c|c|c|c|c|c}&(9a)&(9d)&(3b)&(4a)&(8a)&(8b)\\\hline {\frac {u'}{u}}={\frac {x_{0}^{\prime }-x_{1}^{\prime }}{x_{0}-x_{1}}}=&k&{\frac {1+a}{1-a}}&e^{\eta }&{\sqrt {\tfrac {1+\beta }{1-\beta }}}&{\frac {1+\sin \theta }{\cos \theta }}&{\frac {1+\cos \vartheta }{\sin \vartheta }}=\cot {\frac {\vartheta }{2}}\\\hline {\frac {u_{2}}{u_{1}}}={\frac {x_{1}-x_{1}^{\prime }}{x_{0}+x_{0}^{\prime }}}=&{\frac {k-1}{k+1}}&a&\tanh {\frac {\eta }{2}}&{\frac {\gamma -1}{\beta \gamma }}&{\frac {1-\cos \theta }{\sin \theta }}=\tan {\frac {\theta }{2}}&{\frac {1-\sin \vartheta }{\cos \vartheta }}\\\hline &{\frac {k^{2}-1}{k^{2}+1}}&{\frac {2a}{1+a^{2}}}&\tanh \eta &\beta &\sin \theta &\cos \vartheta \\\hline &{\frac {k^{2}+1}{2k}}&{\frac {1+a^{2}}{1-a^{2}}}&\cosh \eta &\gamma &\sec \theta &\csc \vartheta \\\hline &{\frac {k^{2}-1}{2k}}&{\frac {2a}{1-a^{2}}}&\sinh \eta &\beta \gamma &\tan \theta &\cot \vartheta \end{array}}}$

(9e)

Squeeze mappings in terms of ${\displaystyle \theta }$ were used by Darboux (1891/94) and Bianchi (1894), in terms of ${\displaystyle \eta }$ implicitly by Mercator (1668) and explicitly by Lindemann (1891), Elliott (1903), Herglotz (1909/10), in terms of ${\displaystyle \vartheta }$ by Eisenhart (1905), in terms of ${\displaystyle \beta }$ by Born (1921), w:Milne (1935) and Bondi (1964).

## Historical notation

### Apollonius (BC) – Hyperbola mapping

Halley's (1710) illustration of Apollonius prop. XII, showing ΔΕ·ΔΖ=HK·ΘΗ.
Halley's (1710) illustration of Apollonius prop. XIII, showing ΓH·HΘ=ΛK·KΔ (also equal to AΛ·ΛK and EM·MΞ), whereas AE·EZ is smaller.

w:Apollonius of Perga (c. 240–190 BC, and maybe other Greek geometers such as w:Menaechmus even earlier) defined proposition XII, 2. Book, which was translated into Latin several times by Giovanni Battista Memmo (1537), w:Federico Commandino (1566), w:Isaac Barrow (1675), w:Edmond Halley (1710). For example, the Halley translation reads:[M 1]

Let there be a hyperbola whose asymptotes are AB, BΓ, and let some point Δ be taken in that section, from which ΔΕ, ΔΓ are drawn to ΑΒ, ΒΓ; and let another point H be taken in that section, through which HΘ, HK are drawn parallel to ΔΕ, ΔΖ: I say that the rectangle EΔZ is equal to the rectangle ΘHK.
Let ΔH be joined, and A is connected to Γ. Therefore, since the rectangle AΔΓ is equal to the rectangle AHΓ, it follows that AH is to AΔ as ΔΓ is to ΓΗ. But AH is to ΑΔ as ΗΘ is to ΕΔ, and ΔΓ is to ΓΗ as ΔZ is to ΗΚ; wherefore as ΘΗ is to ΔΕ, so ΔZ is to ΗK: therefore the rectangle EΔZ is equal to the rectangle ΘHK.

A modernized translation was given by w:Thomas Heath as follows:[M 2]

If Q, q be any two points on a hyperbola, and parallel straight lines QH, qh be drawn to meet one asymptote at any angle, and QK, qk (also parallel to one another) meet the other asymptote at any angle, then HQ·QK = hq·qk. Let Qq meet the asymptotes in R,r. We have RQ.Qr=Rq.qr; therefore RQ:Rq=qr:Qr. But RQ:Rq=HQ:hq, and qr:Qr=qk:QK; therefore HQ:hq=qk:QK, or HQ.QK=hq.qk.

In the next proposition XIII, Apollonius showed that if a line is drawn parallel to the asymptotes, within the space between asymptotes and hyperbola, it must meet the hyperbola exactly once. In his demonstration, Apollonius used the previous proposition XII when comparing the area of several parallelograms whose sides are drawn parallel to the asymptotes.[M 3][M 4]

The ratios given by Apollonius:

${\displaystyle {\frac {\Delta \Gamma }{\Gamma H}}={\frac {\Delta Z}{HK}},\quad {\frac {\Theta H}{\Delta E}}={\frac {\Delta Z}{HK}}}$

represent an equation system that can be solved for ΔΕ, ΔΖ, resulting in the squeeze mapping:

${\displaystyle \Delta E={\frac {\Gamma H}{\Delta \Gamma }}\Theta H,\quad \Delta Z={\frac {\Delta \Gamma }{\Gamma H}}HK}$

producing ${\displaystyle \Delta E\cdot \Delta Z=HK\cdot \Theta H}$ in line with Apollonius result that rect. EΔZ is equal to rect. ΘHK. In case ΔΕ, ΔΖ, HK, ΘΗ are all drawn parallel to the respective asymptotes, it follows u'=ΔΕ, v'=ΔΖ, u=HK, v=ΘΗ, k=${\displaystyle {\tfrac {\Gamma H}{\Delta \Gamma }}}$ and therefore Apollonius result becomes equivalent to Lorentz boost (9a), signifying squeezed parallelograms located between the asymptotes and the hyperbola.

In general, the identity ${\displaystyle \Delta E\cdot \Delta Z=HK\cdot \Theta H}$ demonstrates the invariance of the area of all parallelograms that are constructed in line with the proposition XII, thereby representing all points of a hyperbola defined by HK·ΘΗ = const. That is, the invariant area HK·ΘΗ = const. together with const=1 gives HK=1/ΘΗ, which implies that ΘΗ is inverse proportional to HK. Thus when HK is increased into k·HK using some factor k, it follows that ΘΗ must be proportionally diminished into ΘΗ/k in order to preserve invariance of area.

### Mercator (1668) – hyperbolic relations

Mercator's (1668) illustration of AH·FH=AI·BI.

While deriving the w:Mercator series, w:Nicholas Mercator (1668) demonstrated Apollonius' proposition on a rectangular hyperbola algebraically as follows:[M 5]

${\displaystyle {\begin{matrix}AD=1+a,\ DF={\sqrt {2a+aa}}\\AH={\frac {1+a+{\sqrt {2a+aa}}}{\sqrt {2}}},\ FH={\frac {1+a-{\sqrt {2a+aa}}}{\sqrt {2}}}\\AI=BI={\frac {1}{\sqrt {2}}}\\1+a=c,\ {\sqrt {2a+aa}}=d,\ 1=cc-dd\\AH*FH={\frac {cc-dd}{{\sqrt {2}}*{\sqrt {2}}}}={\frac {1}{2}}\\AI*BI={\frac {1}{2}}\\\hline AH*FH=AI*BI\\AH.AI::BI.FH\end{matrix}}}$

It can be seen that Mercator's relations c and d implicitly correspond to hyperbolic functions cosh and sinh (which were explicitly introduced by E:Riccati (1757) much later). In particular, his result AH·FH=AI·BI and AH.AI::BI.FH, denoting that the ratio between AH and AI is equal to the ratio between BI and FH or ${\displaystyle {\tfrac {AH}{AI}}={\tfrac {BI}{FH}}}$ in modern notation, corresponds to squeeze mapping or Lorentz boost (9a) as well as (9e) in terms of η because:

${\displaystyle {\frac {AH}{AI}}={\frac {BI}{FH}}=1+a+{\sqrt {2a+a^{2}}}=c+d=\cosh \eta +\sinh \eta =e^{\eta }=k}$

or solved for AH and FH:

${\displaystyle AH=e^{\eta }AI}$ and ${\displaystyle FH=e^{-\eta }BI}$.

### Speidell (1688), Whiston (1710) – Hyperbola generation

Whiston's (1710) illustration of generating a hyperbola by parallelograms of equal area.

The case of squeezing a given square or parallelogram as a means to generate hyperbolas was discussed by w:Euclid Speidell (1688):[M 6]

[..] from a Square and an infinite company of Oblongs on a Superficies, each Equal to that Square, how a Curve is begotten which shall have the same properties and affections of an Hyperbola inscribed within a Right Angled Cone
[..] There is a Square ABCD, whose Side or Root is 10, let DB be prolonged in infinitum, and continually divided equally by the Root, or DB, and those Equal Divisions numbered by 10, 20, 30, 40, 50, 60, 70, &c. in infinitum: Upon these Numbers let Perpendiculars be erected, which call Ordinates, and each of those Perpendiculars of that length, that Perpendiculars let fall from the aforesaid Perpendiculars to the Side or Base CD (which call Complement Ordinates) the Oblongs made of the Ordinate Perpendiculars, and Complement Ordinate Perpendiculars may be ever Equal to the Square AD, which is easily done thus, for it is ${\displaystyle {\tfrac {100}{20}},{\tfrac {100}{30}},{\tfrac {100}{40}},{\tfrac {100}{50}}}$ &c. produces the Length of the Ordinate Perpendiculars
[..] all the Oblongs made of the Ordinates, and Complement Ordinates are each of them equal to the Square AD, which is here 100
[..] the like Demonstration serves for all the Oblongs or Parallelograms standing upon the Base CD, by the Tips or Angular Points of those Parallelograms, or from the Ends of all the Ordinates standing upon 20, 30, 40, 50, 60, 70, in infinitum, draw the Curve Line from A towards E, so shall you describe the Curve AEFGS [..].
This corresponds to squeeze mappings (9a) with u=v=10 and k=1,2,3,4,5,6,7,..., thus u'v'=uv=100.

In similar terms, w:William Whiston (1710/16) wrote:[M 7]

But it is to be acknowledg'd, that many Properties of an Hyperbola are better known from another manner of generating the Figure; which Way is this: Let LL and MM be infinite Right Lines intersecting each other in any Angle whatever in the Point C: From any Point whatever, as D or e, let Dc, Dd, be drawn parallel to the first Lines, or (ec, ed), which with the Lines first drawn make the Parallelograms as DcCd, or ecCd; Now conceive two sides of the Parallelogram as Dc, Dd, or ec, ed, to be so mov'd this way and that way, that they always keep the same Parallelism, and that at the same time the Area's always remain equal: That is to say, that Dc and ec remain always Parallel to MM, and Dd or ed always Parallel to LL; and that the Area of every Parallelogram be equal to every other, one Side being increas'd in the same Proportion wherein the other is diminish'd. By this means the Point D or e will describe a Curve-Line within the Angle comprehended by the first Lines;
This corresponds to squeeze mappings (9a).

### Reynaud (1819) – Hyperbola mapping

w:Antoine André Louis Reynaud algebraically expressed squeeze mappings by writing:[M 8]

"The system of equations ${\displaystyle (2)\ x={\frac {y'}{\alpha }},\ y=\alpha x'}$ determines all points of the curve ${\displaystyle S}$, because ${\displaystyle x'}$ and ${\displaystyle y'}$ being given numbers, each arbitrary value of ${\displaystyle \alpha }$ gives a point ${\displaystyle x,y}$ of this curve. The elimination of the indeterminate ${\displaystyle \alpha }$ between equations (2) will therefore lead to the equation ${\displaystyle xy=x'y'}$ of the curve in question. This curve is therefore a hyperbola related to its asymptotes ${\displaystyle xX,yY}$."
This is equivalent to (improper) Lorentz transformation (9a-2).

### Klein (1871) – Projective conic section

Elaborating on the w:Cayley–Klein metric, w:Felix Klein (1871) defined a w:projective conic in order to discuss motions such as rotation and translation in the non-Euclidean plane:[M 9]

{\displaystyle {\begin{matrix}x_{1}x_{2}-x_{3}^{2}=0\\\hline {\begin{aligned}x_{1}&=\alpha _{1}y_{1}\\x_{2}&=\alpha _{2}y_{2}\\x_{3}&=\alpha _{3}y_{3}\end{aligned}}\\\left(\alpha _{1}\alpha _{2}-\alpha _{3}^{2}=0\right)\\\hline {\frac {x_{1}x_{2}}{x_{3}^{2}}}={\text{invariant}}\end{matrix}}}
When the conic section is a hyperbola this is equivalent to squeeze mapping (9c). This becomes (9a) using ${\displaystyle \left[u,v\right]=\left[{\tfrac {x_{1}}{x_{3}}},{\tfrac {x_{2}}{x_{3}}}\right],\ \left[k,{\tfrac {1}{k}}\right]=\left[{\tfrac {\alpha _{1}}{\alpha _{3}}},{\tfrac {\alpha _{2}}{a_{3}}}\right]}$.

### Laisant (1874) – Elliptic polar coordinates

w:Charles-Ange Laisant extended circular trigonometry to elliptic trigonometry. In his model, polar coordinates x, y of circular trigonometry are related to polar coordinates x', y' of elliptic trigonometry by the relation[M 10]

${\displaystyle {\begin{matrix}x'=ax,\ y'={\frac {y}{a}}\\x'y'=xy\end{matrix}}}$

He noticed the geometrical implication that any elliptic polar system of coordinates obtained by this formula is located on the same equilateral hyperbola having its asymptotes as axes.

This is equivalent to Lorentz transformation (9a).

### Lie (1879-84) – Transforming pseudospherical surfaces

w:Sophus Lie (1879/80) derived an operation from w:Pierre Ossian Bonnet's (1867) investigations on surfaces of constant curvatures, by which pseudospherical surfaces can be transformed into each other.[M 11] Lie gave explicit formulas for this operation in two papers published in 1881: If ${\displaystyle (s,\sigma )}$ are asymptotic coordinates of two principal tangent curves and ${\displaystyle \Theta }$ their respective angle, and ${\displaystyle \Theta =f(s,\sigma )}$ is a solution of the Sine-Gordon equation ${\displaystyle {\tfrac {d^{2}\Theta }{ds\ d\sigma }}=K\sin \Theta }$, then the following operation (now called Lie transform) is also a solution from which infinitely many new surfaces of same curvature can be derived:[M 12]

${\displaystyle \Theta =f(s,\sigma )\Rightarrow \Theta =f\left(ms,\ {\frac {\sigma }{m}}\right)}$

In (1880/81) he wrote these relations as follows:[M 13]

${\displaystyle \vartheta =\Phi (s,S)\Rightarrow \vartheta =\Phi \left(ms,\ {\frac {S}{m}}\right)}$

In (1883/84) he showed that the combination of Lie transform O with Bianchi transform I produces w:Bäcklund transform B of pseudospherical surfaces:[M 14]

${\displaystyle B=OIO^{-1}}$
As shown by Bianchi (1886) and Darboux (1891/94), the Lie transform is equivalent to Lorentz transformations (9a) and (9b) in terms of light-cone coordinates 2s=u+v and 2σ=u-v. In general, it can be shown that the Sine-Gordon equation is Lorentz invariant.

### Günther (1880/81) – Elliptic polar coordinates

Following Laisant (1874), w:Siegmund Günther demonstrated the relation between circular polar coordinates and elliptic polar coordinates as[M 15]

${\displaystyle {\begin{matrix}x'=ax,\ y'={\frac {1}{a}}y\\x'y'=xy\end{matrix}}}$

showing that any elliptic polar system of coordinates obtained by this formula is located on the same equilateral hyperbola having its asymptotes as axes.

This is equivalent to Lorentz transformation (9a).

### Laguerre (1882) – Laguerre inversion

A transformation (later known as "Laguerre inversion") of E:oriented lines and spheres was given by w:Edmond Laguerre with R being the radius and D the distance of its center to the axis:[M 16]

{\displaystyle {\begin{matrix}D^{2}-D^{\prime 2}=R^{2}-R^{\prime 2}\\\hline \left.{\begin{aligned}D'&={\frac {D\left(1+\alpha ^{2}\right)-2\alpha R}{1-\alpha ^{2}}}\\R'&={\frac {2\alpha D-R\left(1+\alpha ^{2}\right)}{1-\alpha ^{2}}}\end{aligned}}\right|{\begin{aligned}D-D'&=\alpha (R-R')\\D+D'&={\frac {1}{\alpha }}(R+R')\end{aligned}}\end{matrix}}}
This is equivalent (up to a sign change for R) to a squeeze mapping in terms of Lorentz boost (9d).

### Darboux (1883–1891)

#### Transforming pseudospherical surfaces

w:Gaston Darboux (1883) followed Lie (1879/81) by transforming pseudospheres into each other as follows:[M 17]

${\displaystyle f(x,y)\Rightarrow f\left({\frac {x}{m}},\ ym\right)}$
This becomes Lorentz boost (9a) by interpreting x, y as light-cone coordinates.

Similar to Bianchi (1886), Darboux (1891/94) showed that the Lie transform gives rise to the following relations:[M 18]

{\displaystyle {\begin{aligned}(1)\quad &u+v=2\alpha ,\ u-v=2\beta ;\\(2)\quad &\omega =\varphi \left(\alpha ,\beta \right)\Rightarrow \omega =\varphi \left(\alpha m,\ {\frac {\beta }{m}}\right)\\(3)\quad &\omega =\psi (u,v)\Rightarrow \omega =\psi \left({\frac {u+v\sin h}{\cos h}},\ {\frac {v+u\sin h}{\cos h}}\right)\end{aligned}}}.
Equations (1) together with transformation (2) gives Lorentz boost (9a) in terms of light-cone coordinates.

#### Laguerre inversion

Following Laguerre (1882), Darboux (1887) formulated the Laguerre inversions in four dimensions using coordinates x,y,z,R:[M 19]

{\displaystyle {\begin{matrix}x^{\prime 2}+y^{\prime 2}+z^{\prime 2}-R^{\prime 2}=x^{2}+y^{2}+z^{2}-R^{2}\\\hline \left.{\begin{aligned}x'&=x,&z'&={\frac {1+k^{2}}{1-k^{2}}}z-{\frac {2kR}{1-k^{2}}},\\y'&=y,&R'&={\frac {2kz}{1-k^{2}}}-{\frac {1+k^{2}}{1-k^{2}}}R,\end{aligned}}\right|{\begin{aligned}z'+R'&={\frac {1+k}{1-k}}(z-R)\\z'-R'&={\frac {1-k}{1+k}}(z+R)\end{aligned}}\end{matrix}}}
This is equivalent (up to a sign change for R) to a squeeze mapping in terms of Lorentz boost (9d) where Darboux's k corresponds to a.

### Lipschitz (1885/86) - Quadratic forms

w:Rudolf Lipschitz (1885/86) discussed transformations leaving invariant the sum of squares

${\displaystyle x_{1}^{2}+x_{2}^{2}\dots +x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots +y_{n}^{2}}$

which he rewrote as

${\displaystyle x_{1}^{2}-y_{1}^{2}+x_{2}^{2}-y_{2}^{2}+\dots +x_{n}^{2}-y_{n}^{2}=0}$.

This led to the problem of finding transformations leaving invariant the pairs ${\displaystyle x_{a}^{2}-y_{a}^{2}}$ (where a=1...n) for which he gave the following solution:[M 20]

{\displaystyle {\begin{matrix}x_{a}^{2}-y_{a}^{2}={\mathfrak {x}}_{a}^{2}-{\mathfrak {y}}_{a}^{2}\\\hline {\begin{aligned}x_{a}-y_{a}&=\left({\mathfrak {x}}_{a}-{\mathfrak {y}}_{a}\right)r_{a}\\x_{a}+y_{a}&=\left({\mathfrak {x}}_{a}+{\mathfrak {y}}_{a}\right){\frac {1}{r_{a}}}\end{aligned}}\quad (1)\\\hline {\begin{aligned}2{\mathfrak {x}}_{a}&=\left(r_{a}+{\frac {1}{r_{a}}}\right)x_{a}+\left(r_{a}-{\frac {1}{r_{a}}}\right)y_{a}\\2{\mathfrak {y}}_{a}&=\left(r_{a}-{\frac {1}{r_{a}}}\right)x_{a}+\left(r_{a}+{\frac {1}{r_{a}}}\right)y_{a}\end{aligned}}\quad (2)\end{matrix}}}
Equation system (1) represents Lorentz boost or squeeze mapping (9a), and (2) represents Lorentz boost (9b).

### Bianchi (1886–1894) – Transforming pseudospherical surfaces

w:Luigi Bianchi (1886) followed Lie (1879/80) by writing the transformation of pseudospheres into each other, obtaining the result:[M 21]

{\displaystyle {\begin{aligned}(1)\quad &u+v=2\alpha ,\ u-v=2\beta ;\\(2)\quad &\Omega \left(\alpha ,\beta \right)\Rightarrow \Omega \left(k\alpha ,\ {\frac {\beta }{k}}\right);\\(3)\quad &\theta (u,v)\Rightarrow \theta \left({\frac {u+v\sin \sigma }{\cos \sigma }},\ {\frac {u\sin \sigma +v}{\cos \sigma }}\right)=\Theta _{\sigma }(u,v);\\&{\text{Inverse:}}\left({\frac {u-v\sin \sigma }{\cos \sigma }},\ {\frac {-u\sin \sigma +v}{\cos \sigma }}\right)\\(4)\quad &{\frac {1}{2}}\left(k+{\frac {1}{k}}\right)={\frac {1}{\cos \sigma }},\ {\frac {1}{2}}\left(k-{\frac {1}{k}}\right)={\frac {\sin \sigma }{\cos \sigma }}\end{aligned}}}.
Equations (1) together with transformation (2) gives Lorentz boost (9a) in terms of light-cone coordinates. Plugging equations (4) into (3) gives Lorentz boost (9b) in terms of Bondi's k factor.

In 1894, Bianchi redefined the variables u,v as asymptotic coordinates, by which the transformation obtains the form:[M 22]

${\displaystyle {\begin{matrix}\Omega \left(u,v\right)\Rightarrow \omega (u,v);\quad \Omega \left(u,v\right)=\omega \left(ku,\ {\frac {v}{k}}\right);\\k={\frac {1+\sin \sigma }{\cos \sigma }}\Rightarrow \Omega \left(u,v\right)=\omega \left({\frac {1+\sin \sigma }{\cos \sigma }}u,\ {\frac {1-\sin \sigma }{\cos \sigma }}v\right)\end{matrix}}}$.
This is consistent with one of the choices in (9e) where Bianchi's angle σ corresponds to θ.

### Lindemann (1890/91) – Weierstrass coordinates and Cayley absolute

w:Ferdinand von Lindemann employed the Cayley absolute related to surfaces of second degree and its transformation[M 23]

{\displaystyle {\begin{matrix}X_{1}X_{4}+X_{2}X_{3}=0\\X_{1}X_{4}+X_{2}X_{3}=\Xi _{1}\Xi _{4}+\Xi _{2}\Xi _{3}\\\hline {\begin{aligned}X_{1}&=\left(\lambda +\lambda _{1}\right)U_{4}&\Xi _{1}&=\left(\lambda -\lambda _{1}\right)U_{4}&X_{1}&={\frac {\lambda +\lambda _{1}}{\lambda -\lambda _{1}}}\Xi _{1}\\X_{2}&=\left(\lambda +\lambda _{3}\right)U_{4}&\Xi _{2}&=\left(\lambda -\lambda _{3}\right)U_{4}&X_{2}&={\frac {\lambda +\lambda _{3}}{\lambda -\lambda _{3}}}\Xi _{2}\\X_{3}&=\left(\lambda -\lambda _{3}\right)U_{2}&\Xi _{3}&=\left(\lambda +\lambda _{3}\right)U_{2}&X_{3}&={\frac {\lambda -\lambda _{3}}{\lambda +\lambda _{3}}}\Xi _{3}\\X_{4}&=\left(\lambda -\lambda _{1}\right)U_{1}&\Xi _{4}&=\left(\lambda +\lambda _{1}\right)U_{1}&X_{4}&={\frac {\lambda -\lambda _{1}}{\lambda +\lambda _{1}}}\Xi _{4}\end{aligned}}\end{matrix}}}

into which he put[M 24]

{\displaystyle {\begin{matrix}{\begin{aligned}X_{1}&=x_{1}+2kx_{4},&X_{2}&=x_{2}+ix_{3},&\lambda +\lambda _{1}&=\left(\lambda -\lambda _{1}\right)e^{a},\\X_{4}&=x_{1}-2kx_{4},&X_{3}&=x_{2}-ix_{3},&\lambda +\lambda _{3}&=\left(\lambda -\lambda _{3}\right)e^{\alpha i},\end{aligned}}\\\hline \Omega _{xx}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-4k^{2}x_{4}^{2}=-4k^{2}\\ds^{2}=dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}-4k^{2}dx_{4}^{2}\end{matrix}}}
This is equivalent to squeeze mapping (9a) as well as (9e) in terms of η with ${\displaystyle e^{\alpha i}=1}$ and 2k=1 .

### Haskell (1895) – Hyperbola mapping

w:Mellen W. Haskell applied the linear transformation

${\displaystyle \alpha '=k\alpha ,\ \beta '=k^{-1}\beta }$

in order to transform a hyperbola into itself.[M 25]

This is equivalent to Lorentz transformation (9a).

### Smith (1900) – Laguerre inversion

w:Percey F. Smith followed Laguerre (1882) and Darboux (1887) and defined the Laguerre inversion as follows:[M 26]

${\displaystyle {\begin{matrix}p^{\prime 2}-p^{2}=R^{\prime 2}-R^{2}\\\hline \kappa ={\frac {R'-R}{p'-p}}\\p'={\frac {\kappa ^{2}+1}{\kappa ^{2}-1}}p-{\frac {2\kappa }{\kappa ^{2}-1}}R,\quad R'={\frac {2\kappa }{\kappa ^{2}-1}}p-{\frac {\kappa ^{2}+1}{\kappa ^{2}-1}}R\end{matrix}}}$
This is equivalent (up to a sign change) to Lorentz transformation (9d).

### Elliott (1903) – Invariant theory

w:Edwin Bailey Elliott (1903) discussed a special cyclical subgroup of ternary linear transformations for which the (unit) determinant of transformation is resoluble into three ordinary algebraical factors, which he pointed out is in direct analogy to a subgroup formed by the following transformations:[M 27]

${\displaystyle {\begin{matrix}x=X\cosh \phi +Y\sinh \phi ,\quad y=X\sinh \phi +Y\cosh \phi \\\hline X+Y=e^{-\phi }(x+y),\quad X-Y=e^{\phi }(x-y)\end{matrix}}}$
The second line is equivalent to squeeze mapping or Lorentz boost (9a) as well as (9e) in terms of η.

### Eisenhart (1905) – Transforming pseudospherical surfaces

w:Luther Pfahler Eisenhart followed Lie (1879/81), Bianchi (1886, 1894) and Darboux (1891/94) in transforming pseudospherical surfaces:[M 28]

{\displaystyle {\begin{aligned}(1)\quad &\alpha ={\frac {u+v}{2}},\ \beta ={\frac {u-v}{2}}\\(2)\quad &\omega \left(\alpha ,\beta \right)\Rightarrow \omega \left(m\alpha ,\ {\frac {\beta }{m}}\right)\\(3)\quad &\omega (u,v)\Rightarrow \omega (\alpha +\beta ,\ \alpha -\beta )\Rightarrow \omega \left(\alpha m+{\frac {\beta }{m}},\ \alpha m-{\frac {\beta }{m}}\right)\\&\Rightarrow \omega \left[{\frac {\left(m^{2}+1\right)u+\left(m^{2}-1\right)v}{2m}},\ {\frac {\left(m^{2}-1\right)u+\left(m^{2}+1\right)v}{2m}}\right]\\(4)\quad &m={\frac {1-\cos \sigma }{\sin \sigma }}\Rightarrow \omega \left({\frac {u-v\cos \sigma }{\sin \sigma }},\ {\frac {v-u\cos \sigma }{\sin \sigma }}\right)\end{aligned}}}.
Equations (1) together with transformation (2) gives Lorentz boost (9a) in terms of light-cone coordinates. Transformation (3) is equivalent to Lorentz boost (9b) in terms of Bondi's k factor. Eisenhart's angle σ corresponds to ϑ in (9e).

### Herglotz (1909/10) – Special relativity

In relation to special relativity, w:Gustav Herglotz (1909/10) defined the Lorentz boost as follows:[M 29]

{\displaystyle {\begin{aligned}x&=x',&t-z&=(t'-z')e^{\vartheta }\\y&=y',&t+z&=(t'+z')e^{-\vartheta }\end{aligned}}}
This is equivalent to Lorentz transformation (9a) as well as (9e) in terms of η.

### Born (1921) – Special relativity

In the second edition of “Einstein's theory of relativity”, w:Max Born (1921) discussed the relation of the Lorentz transformation and the hyperbola:[M 30]

${\displaystyle {\begin{matrix}x'-ct'={\frac {1+\beta }{\alpha }}\left(x-ct\right)\\x'+ct'={\frac {1-\beta }{\alpha }}\left(x+ct\right)\\\left[\alpha ={\sqrt {1-\beta ^{2}}}\right]\\\hline \eta =x-ct,\ \xi =x+ct\\\xi \eta =(x-ct)(x+ct)=x^{2}-c^{2}t^{2}\\\eta ={\frac {1}{\xi }}\end{matrix}}}$
This is equivalent to Lorentz transformation (9a) as well as (9e) in terms of β.

## References

### Historical mathematical sources

1. Apollonius/Halley (1710), Prop. XII of book II on p. 114; Latin: "Si ab aliquo puncto eorum, qua sunt in sectione, ad asymptotos duæ rectæ lineæ in quibuslibet angulis ducantur, & ab alio quovis puncto in sectione sumpto ducantur aliæ rectæ his ipsis parallelæ : rectangulum sub parallelis contentum æquale erit contento sub rectis ipsis quibus ductæ fuerant parallelae.
Sit hyperbola, cujus asymptoti AB, BΓ sumatur in sectione aliquod punctum Δ, atque ab eo ad ΑΒ, ΒΓ, ducantur ΔΕ, ΔΓ; sumatur autem & alterum punctum H in sectione, per quod ducantur HΘ, HK ipsis ΔΕ, ΔΖ parallelæ: dico rectangulum EΔZ rectangulo ΘHK æquale esse.
Jungatur enim ΔH, & ad A, Γ producatur. itaque quoniam rectangulum AΔΓ aequatur rectangulo AHΓ; erit ut AH ad AΔ ita ΔΓ ad ΓΗ. sed ut AH ad ΑΔ ita ΗΘ ad ΕΔ, & ut ΔΓ ad ΓΗ ita ΔZ ad ΗΚ; quare ut ΘΗ ad ΔΕ ita ΔZ ad ΗK: rectangulum igitur ΕΔZ rectangulo ΘHK est æquale."
2. Apollonius/Heath (1896), Proposition 34 (Apollonius, Book II, Prop. 12).
3. Apollonius/Halley (1710), Prop. XIII of book II on p. 114-115
4. Apollonius/Heath (1896), Proposition 35 (Apollonius, Book II, Prop. 13).
5. Mercator (1667), prop. XIV, pp. 28-29. (He used this result to derive the Mercator series in prop. XV).
6. Speidell (1688), pp. 4-5
7. Whiston (1710/16). In the English version (1716) see pp. 16-17. In the original Latin version (1710) see pp. 16-18
8. Reynaud (1819), p. 247
9. Klein (1871), pp. 601–602
10. Laisant (1874a), pp. 73–76
11. Lie (1879/80), Collected papers, vol. 3, p. 389
12. Lie (1879/81), Collected papers, vol. 3, p. 393
13. Lie (1880/81), Collected papers, vol. 3, pp. 477–478
14. Lie (1883/84), Collected papers, vol. 3, p. 556
15. Günther (1880/81), pp. 383–385
16. Laguerre (1882), pp. 550–551.
17. Darboux (1883), p. 849
18. Darboux (1891/94), pp. 381–382
19. Darboux (1887)
20. Lipschitz (1886), pp. 90–92
21. Bianchi (1886), eq. 1 can be found on p. 226, eq. (2) on p. 240, eq. (3) on pp. 240–241, and for eq. (4) see the footnote on p. 240.
22. Bianchi (1894), pp. 433–434
23. Lindemann & Clebsch (1890/91), pp. 361–362
24. Lindemann & Clebsch (1890/91), p. 496
26. Smith (1900), p. 159
27. Elliott (1903), p. 109
28. Eisenhart (1905), p. 126
29. Herglotz (1909/10), pp. 404-408
30. Born (1921), pp. 179-180
• Apollonius of Perga (1896) [~200 BC], Thomas Little Heath (ed.), Treatise on Conic Sections, Cambridge University Press
• Apollonius of Perga (1710) [~200 BC], Edmond Halley (ed.), Apollonii Pergaei Conicorum libri octo, Oxoniæ
• Bianchi, L. (1886), Lezioni di geometria differenziale, Pisa: Nistri
• Bianchi, L. (1894), Lezioni di geometria differenziale, Pisa: Enrico Spoerri
• Born, M. (1921), Die Relativitätstheorie Einsteins und ihre physikalischen Grundlagen (2. edition), Berlin: Springer
• Darboux, G. (1883), "Sur les surfaces dont la courbure totale est constante", Compt. Rend., 12: 848–850
• Darboux, G. (1887), Leçons sur la théorie générale des surfaces. Première partie, Paris: Gauthier-Villars, pp. 254–256
• Darboux, G. (1894) [1891], Leçons sur la théorie générale des surfaces. Troisième partie, Paris: Gauthier-Villars This third part of his lectures was initially published in three steps: première fascicule (1890), deuxième fascicule (1891), and troisième fascicule (1895). The discussion of the Lie transform appears in the deuxième fascicule published in 1891.
• Elliott, E.B. (1903), "On ternariants for the special cyclical subgroup of linear transformations", Messenger of mathematics, 33 (1): 108–112
• Günther, S. (1881) [1880], Die Lehre von den gewöhnlichen und verallgemeinerten Hyperbelfunktionen, Halle: L. Nebert
• Haskell, M.W. (1895), "On the introduction of the notion of hyperbolic functions", Bull. Amer. Math. Soc., 1 (6): 155–159
• Herglotz, Gustav (1910) [1909], "Über den vom Standpunkt des Relativitätsprinzips aus als starr zu bezeichnenden Körper]", Annalen der Physik, 336 (2): 393–415, Bibcode:1910AnP...336..393H, doi:10.1002/andp.19103360208

### Secondary sources

1. Terng & Uhlenbeck (2000), p. 21
2. Bondi (1964), p. 118


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# Math Help - Continuous proof

1. ## Continuous proof

Prove: If f is a continuous function whose domain includes a closed interval [a,b], and L is a horizontal line, and (a,f(a)) is below L, and (b,f(b)) is above L, then there is a number x between a and b such that (x,f(x)) is on L.

2. Use the intermediate value theorem.

3. That theorem is not available to be used in the class I am in so is there some other way?

4. ## well see

in the given interval the function is continuous
now it means that there is no gap in the curve drawn(that is in your function)
now if one point is below the line and other is above the line it means that the function must cut the given horizontal line hence there is a point x,f(x) that must lie on line
physical example
(suppose you are suppose to draw a line using paint on floor while you were painting there was another line on the floor
1) one way is to keep on painting the floor and do not paint over the other line but doing this will make your line discontinuous hence to have a continuous line you will have to paint over the other line
same thing is happening in question that is to be continuous the function will have to intersect the line as one point it below and other is above so there will definetly be a point that lie on both of them i.e x,f(x) such that a<x<b. [here function can not take a jump as we are dealing with 2D graph]

5. The statement is intuitively true to me, but I need to come up with a mathematical proof, either using the epsilon-delta definition of continuity or something else. However, I can not simply cite the IVM.

6. Originally Posted by spikedpunch
The statement is intuitively true to me, but I need to come up with a mathematical proof, either using the epsilon-delta definition of continuity or something else. However, I can not simply cite the IVM.
Any solution of this will be equivalent to a proof of the intermediate value theorem.
Define $\mathcal{L}=\{x\in [a,b]:f(x)< L\}$.
That set is not empty and bounded above.
You can show that if $t=\sup(\mathcal{L})$ the $f(t)=L$.


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Poster Open Access

# How the stellar flicker noise affects the characterization of planetary transits ?

Sulis Sophia

Stellar activity is known to limit exoplanet detection and characterization. Among this activity, stellar convection (flicker'') evolves during the typical transit timescales (~ hours) and affects the inferred transit parameters.
We generated realistic simulations of transiting exoplanets based on solar HMI data. These simulations include planets from 1 to 10 Earth radii with different transit geometries. These simulations comprising hundreds of light curves are available to the community : https://doi.org/10.5281/zenodo.3686871
We analyzed the data using standard MCMC methods assuming the noise is white and Gaussian (WGN), or a Gaussian Process (GP). We show that, in both cases, the resulting planet parameters can be affected by biases,
which leads to biased planetary radius measurements. This demonstrates the need to develop robust stellar noise modeling to achieve PLATO's goal of characterizing exoplanets transiting solar-like stars.
Next steps of this study will be to investigate
i) how other noise sources (e.g., flares, spots and faculae) affect the inferred exoplanet parameters,
ii) which noise modeling allows to derive the most accurate transit parameters.

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# Chapter 12 - Rotation of a Rigid Body - Exercises and Problems - Page 348: 15

(a) The coordinates of the center of gravity are (6 cm, 4 cm) (b) $I = 0.0020~kg~m^2$ (c) $I = 0.0013~kg~m^2$

#### Work Step by Step

(a) We can find the x-coordinate of the center of gravity. $x_{cog} = \frac{(100~g)(0)+(200~g)(6~cm)+(100~g)(12~cm)}{100~g+200~g+100~g}$ $x_{cog} = 6~cm$ We can find the y-coordinate of the center of gravity. Note that the y-coordinate of the 200-gram mass is 8 cm. $y_{cog} = \frac{(100~g)(0)+(200~g)(8~cm)+(100~g)(0)}{100~g+200~g+100~g}$ $y_{cog} = 4~cm$ The coordinates of the center of gravity are (6 cm, 4 cm) (b) We can find the moment of inertia about the axis that passes through mass A and is perpendicular to the page. $I = \sum m_i~r_i^2$ $I = (0.10~kg)(0.10~m)^2+(0.20~kg)(0)^2+(0.10~kg)(0.10~m)^2$ $I = 0.0020~kg~m^2$ (c) We can find the moment of inertia about the axis that passes through masses B and C. $I = \sum m_i~r_i^2$ $I = (0.10~kg)(0)^2+(0.20~kg)(0.080~m)^2+(0.10~kg)(0)^2$ $I = 0.0013~kg~m^2$

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


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