tag:blogger.com,1999:blog-91538239467962707642024-02-19T03:27:13.654+00:00for the common peoplejoão pestanahttp://www.blogger.com/profile/03616394334438513692noreply@blogger.comBlogger19125tag:blogger.com,1999:blog-9153823946796270764.post-45548106778023984482016-06-05T18:48:00.003+01:002016-06-05T18:53:52.396+01:00Why you should not be surprised when you meet two people who share the same birthday<div class="separator" style="clear: both; text-align: center;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh0-TBUJl4CbCj1r6WBKq27n6nct-zET0OXKuR57IAtFFHkMF-X0jU3xcDVY3XVn43U_7h6gexkbGyoYRlqxZ43eivfUkMPpbW7eHL7j3gtms8EPGJh4vL_abW9T_fMsHiVqg5MKi1n1DU/s640/WHWZKYQSDR.jpg" width="640" /></div>
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This article is written with a coworker of mine in mind, given the disbelief shown when I presented what is know as <i>the birthday problem</i> or <i>the birthday paradox</i> — which is not a paradox, it just <i>feels</i> like it due to our weak probability intuition.<br />
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Let's start easy with the <i>pigeonhole principle</i> and a clear example of something everyone will surely agree with me — I hope. If we have 5 red balls and only 4 boxes to put them in (fig. 1), then surely one of the boxes <i>must</i> contain, at least, 2 balls. What does this have to do with people and their birthday? Well... Think of it this way. If we put 367 random people in the same room, then at least 2 of them must share their birthday. Like putting 367 red balls into 366 boxes, see?<br />
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<div class="separator" style="clear: both; text-align: center;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjzFsHlp2ZEfqx_uM0rUnvKuqBAxbiRjU7o-ZsNpg7LT5YMA_m8u0kNLmyVMAbyPbhINvMc6KkX6tv-IvdKZuChUxMR431VxjI4NOHhXEnuf_ISCVlapC0sk_6JsWm0uwYMf3CkuYV8KfQ/s1600/Untitled-2.png" /><br />
Figure 1 — Visual representation of 5 red balls and 4 boxes.</div>
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Before you start to curse my name, because surely all those people can have their birthday occur at any day — including February, 29 — so why should they necessarily overlap? Try thinking about the problem the other way around. Imagine how you can make sure no one shares a birthday! Easy, right? Just have all the people celebrate their birthday in sequence. Let's say, Mark's birthday is January, 1, Lisa's is January, 2, Brian's January, 3, ..., Sarah's December, 31, and so on until every person is assigned to a non-overlaping birthday. Then... What about the 367th person? What about John? As soon as John enters the room, he <i>must</i> share a birthday with someone, because he <i>must</i> have been born in one of the different 366 days that make up a full year.<br />
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It still doesn't feel right, does it? Of course that happens, if I assign a day per person, but surely we can reason that some of the days might not even be celebrated! And that's were your reason it's tricking you. You are not thinking about people having birthdays, but rather you're thinking about you and your own experience. In your daily life, birthday's are a special and not frequent event and double birthdays are even scarcer! In light of this, you think about all the empty days in which you don't know a single person who's celebrating their birthday on. When you meet someone new, most likely he or she will have his or her birthday in some of those days that are empty to you. You assume that, because birthdays are a rare event to you, then they surely must always be a rare event — and that's just not true.<br />
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Just to complete the above idea, as soon as you start to create empty days, you <i>must</i> create shared birthdays. If you move Lisa's birthday from January, 2 to January, 1, then no one celebrates their birthday on January, 2, but Lisa is automatically sharing January, 1 with Mark. Hopefully, by now, you agree with me that when the number of people is 367, there is a 100% guarantee that, at least, 2 among the 367 must share their birthday. I can tell you now that you only need 70 people to reach the 99.9% guarantee and just 23 people will suffice for those fifty-fifty odds.<br />
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Do you not believe me? Here comes the fun part! Let's prove it... mathematically. First, let's forget about February, 29 and assume each day is equally probable. The problem is as follows. We want to find the probability that among N people, at least, 2 share a birthday. Since there are only two possible situations: 2 different persons share a birthday (`A`) and 2 different persons <i>do not</i> share a birthday (`bar A`), among N people, we can analyse it either way. If `P(A)` is the probability of 2 people sharing a birthday, then `P(bar A)=1-P(A)` is the probability of 2 people <i>not</i> sharing a birthday, among N people. I shall use this form, as it makes the process easier.<br />
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We shall think in sequence and analyse each person individually, comparing him/her to all the others we've analysed before. Starting with person 1, the probability of his/her birthday not being the same as the people before is 1, because we haven't analysed anyone yet. Let's write it as `365/365`. Next, we analyse the second person and the probability that he/she doesn't share a birthday with person 1 is having been born in any possible day, except for person 1's birthday, so it becomes `364/365`. Again, using the same reasoning, the probability that person 3 doesn't share a birthday with person 1 and 2 is having been born in any possible day, except for person 1 and 2's birthdays, so it becomes `363/365`. Generally, we can say that `P(X)=1-(X-1)/365` is the probability of person X not sharing a birthday with all the X persons before him/her.<br />
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Now, we are in a position to (almost) obtain the result we want! The probability that among N people no ones shares a birthday is thus `P(bar A)=P(1)*P(2)*P(3)*...*P(N-1)*P(N)` and we can write it as a formula:<br />
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<div style="text-align:center;">
`P(bar A)=prod_(X=1)^N (1-(X-1)/365)=(365!)/(365^N(365-N)!)`
</div>
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You can test it yourself that for the values `N=1` and `N=366` we get the extreme results `P(bar A)=1` and `P(bar A)=0`, respectively, meaning that the probability no one shares a birthday among a group of 1 single person is 100% and the probability of no one sharing a birthday among 366 people is 0%, as explained in the beginning. If we calculate the complement of `P(bar A)`, then we get what we wanted all along: the probability of, among N people, at least 2 sharing a birthday `P(A)=1-P(bar A)`. Input the values `N=23` and `N=70` and we get, as promised, `P(A)=50.73%` and `P(A)=99.92%`, respectively.<br />
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Meeting someone who shares a birthday with you is one thing, but among a group of N random people, 2 people sharing the same birthday is another thing. That's why your reason tricks you. You fail to distinguish the two situations even though the first is somehow rare and the second is clearly not.<br />
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Just in case you're wondering how to calculate the likelihood of finding someone with the same birthday as you among N people, it's not difficult. We can do a similar reasoning as before, but instead of comparing will <i>all</i> previously analysed people, we compare only with the first — you. And thus, it becomes the probability of person 1 not sharing his/her birthday with you times the probability of person 2 not sharing his/her birthday with you, and so on. That is, `P(bar A)=364/365xx364/365xx364/365xx...=(364/365)^N` and again `P(A)=1-P(bar A)` is the probability of finding someone with the same birthday as you. So how many people do we need for the fifty-fifty chance? `N=253` will give you `P(A)=50.05%`.joão pestanahttp://www.blogger.com/profile/03616394334438513692noreply@blogger.comtag:blogger.com,1999:blog-9153823946796270764.post-58206666362346684472016-05-18T00:44:00.001+01:002016-05-19T02:27:11.389+01:00Sexuality explained through mathematics — a non-biased language approach<div class="separator" style="clear: both; text-align: center;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgDkUrpkQz8xn-vmTIHQS0OFYQv8oPQwACekgBqEnb_Qd27W4_8admkNrho4kOq7RhDK5P740RQuJ2k1qG5e-c59OFZBnoiNEHs2NNuwcQHUrse9ICBfiy9CfAHeS_TcV5PdSliR5IDlIU/s640/5104095BD6.jpg" width="640" /></div>
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So you think you've got sexuality pretty much figured out, right? If you're born a male, you're attracted to women and the other way around. If you've ever had sex, it's not a big mystery and the mechanics are quite simple. Right? Think again. Biologists and psychologist are still working their way into figuring out how it all works out. What do I think I do better? Nothing at all, to be honest. I'll just try and use the language of mathematics not to explain how and why things are the way they are, but rather to give you a glimpse of how complex the whole subject is.<br />
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Maybe, just maybe, you're a somewhat open-minded person and you allow yourself to accept the concept of homosexuality. Great! Does it help? Well... Yes and no. It complicates things on my end, but it does take you a step closer to a fantastic member of modern society. But now that we can work with both heterosexual and homosexual men and women, we have much more ways we can arrange people. Both men and women can either be attracted to their own gender or the opposite one. This give us 4 different sets in which we can arrange people into (table 1).<br />
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<table id="table_post">
<tbody>
<tr>
<td> </td>
<td>Man</td>
<td>Woman</td>
</tr>
<tr>
<td>Man</td>
<td>Homosexual</td>
<td>Heterosexual</td>
</tr>
<tr>
<td>Woman</td>
<td>Heterosexual</td>
<td>Homosexual</td>
</tr>
</tbody>
</table>
<div style="text-align: center;">
Table 1 — Assigning men and women to each other in couples and grouping them into two different sets.</div>
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Now for the fun part — the mathematics, of course! Let `m_i in M` be some random male-born person and `w_i in W` be some random female-born person each belonging to the set of all Men and Women, respectively. We can thus arrange each `m_i` and `w_i` into couples (a tuple of length 2) such as `(x_i,x_j)` where `x` can either be `m` or `w`. We can define as <i>homosexual</i> any couple that is fully contained within just one of the larger sets `M` or `W`, that is `(x_i,x_j) in W or M`. An <i>heterosexual</i> couple can then be defined as any couple that is not fully contained within just one of the larger sets, but can only be contained by the union of both, that is `(x_i,x_j) in M uu W`.<br />
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What if some person is attracted to both genders? Should we allow for bisexuality? Of course, definitely! Let's do it. Let's fix the first member of the couple and keep the second one as undefined. Assuming that the order of the pair does not matter, we can define a <i>bisexual</i> man or woman as the first member of either `(m_i,x_j)` or `(w_i,x_j) in M uu W`. But isn't the constrain to a tuple of length 2 too strong? I guess we are mostly a monogamous species, but nonetheless, some people do find themselves as an element of a tuple with length larger than 2. Under such conditions, it is certain that whenever `n > 2` two or more elements of the n-tuple has to be <i>bisexual</i>.<br />
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Let `Q = (x_1,x_2,x_3,...,x_(n-1),x_n)` be some n-tuple where `n` people are in a mutually consensual polygamous relationship such that `x_i in M uu W`. It should be obvious that at least two of the elements of the tuple must be <i>bisexual</i>. For example, we can have arrangements of 3 people in 2 different ways: `(m,m,w)` or `(m,w,w)`, because the order doesn't matter and we have only two different possible elements to fill three positions. In the first situation, the men must be bisexual and in the second situation, the women must be bisexual. I'm excluding the all men and all women tuples deliberately, because they add no complexity to the analysis.<br />
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Of course, the subject of bisexuality is more complex then either homosexuality or heterosexuality. While some really have no preference over gender, there are some who do have a stronger inclination towards on. While sexual arousal is mostly set specific in men, with both heterosexual and homosexual men responding mainly to respectively female and male stimuli; women show no such preference which suggests a natural bisexual tendency as described by Chivers ML, Rieger G, Latty E, Bailey JM. <i>A sex difference in the specificity of sexual arousal</i>. Psychol Sci. 2004 Nov;15(11):736-44.<br />
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Maybe we can define a coefficient of bisexuality, say `beta`, and allow it to run all the way from 0 to 1, where 0 indicates heterosexuality, 0.5 perfect bisexuality and 1 homosexuality. The range where `beta in (0,0.5)` shall be called imperfect bisexuality with heterosexual inclination and the range `beta in (0.5,1)` shall be the imperfect bisexuality with homosexuality inclination. But how to calculate? Well... I don't believe it to be something that is easily measurable, so we must use some proxy. What about the relationship history of an individual? It might just work! Let's define the homosexuality counting function as:<br />
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<div style="text-align:center;">
`pi_(x_i) (x_j) = {(1,if (x_i,x_j) in X),(0,text{otherwise}):}`
</div>
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Where `pi_(x_i)` counts in respect to person `x_i` and is 1 whenever `x_i` is paired with a `x_j` from the same larger set `X` which can either be `M` or `W` and 0 otherwise. Let's say we have `H={x_1,x_2,x_3,...,x_(t-1),x_t}` the ordered set of all people `x_i` has been in a relationship with (even if it was a one-time event) at time `t` and it can have any length `N >= 0`. Then, `x_i`'s bisexuality coefficient is simply:<br />
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<div style="text-align:center;">
`beta (x_i) = 1/N sum_(t=1)^N pi_(x_i) (x_t) and x_t in H`
</div>
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Or equivalently, it's simply the average of all `x_t in X` such that `x_i in X`. That is:<br />
<br />
<div style="text-align:center;">
`beta = text{all people of the same gender within the history} / text{all people within the history}`
</div>
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So what does all of this tell us? Absolutely nothing. Should we categorize people into distinct groups? Of course not! I just showed to you that bisexuality is a continuous function from 0 to 1 and thus cannot be split into finite groups unequivocally. Where along the line should you define the limits of each category? Why 0.32 and not 0.315?<br />
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Think we've got it all pretty much covered up? No. That's just half of it! What if you're a woman who is attracted to men? Are you heterosexual? Yes, you are. By our previously stated definition. But what if you were born with a male body? Does that make you homosexual? No, that's preposterous! In the same way that you as a person suffer zero change when you move from the inside of a Mercedes-Benz to a Tesla Motors vehicle, you suffer the same amount of change when you move from a male body into a female body and vice-versa: zero.<br />
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The body in which you are born, your sexual identity and your sexual orientation are completely different and independent things that have absolutely no influence over each other. Your body does not determine your sexual identity and even less influences your sexual preference. Your body is merely the vehicle that was assigned to you by default and sometimes mistakes do happen. Finally, I'd like to roughly quote John Oliver on this — <i>"sexual identity is who you are, sexual orientation is whom you love."</i><br />
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Let's do some more mathematics! If we define `theta = {XX,XY}` as the property of your body, then it can assume only one of two values — even if you decide it should change at some point — `XX` for female and `XY` for male. Your sexual orientation can be determined by the bisexual coefficient `beta`. What about your sexual identity? Should we fall victims of Occam's Razor and define it can also — like your body — assume only one of two possible values? I'll do that only for mathematical simplicity, but explicitly state that it is seldom as simple as that. Let's just say that your sexual identity is `omega={m,w}`.<br />
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Using our previously defined variables, we are now ready to poorly classify a person using a 3-dimensional function `P( theta , omega , beta )`. This is a very strange and curious function, because two variables are discrete and one is continuous within the range from 0 to 1. Again, and only for the sake of the argument, let's say `beta` can be mapped into a discrete function in such a way that the limits 0 and 1 correspond to <i>heterosexual preference</i> and <i>homosexual preference</i>, respectively, leaving every other possible value to assume the value of <i>no preference</i>. Now, our `P` function can assume any of only 12 possible values. Because I believe a graph would be too confusing, I'll display these in table 2.<br />
<br />
<table id="table_post">
<tbody>
<tr>
<td>Body (`theta`)</td>
<td>Identity (`omega`)</td>
<td>Preference (`beta`)</td>
<td>`P( theta , omega , beta )`</td>
</tr>
<tr>
<td>`XX`</td>
<td>`w`</td>
<td>`het`</td>
<td>heterosexual woman</td>
</tr>
<tr>
<td>`XX`</td>
<td>`w`</td>
<td>`hom`</td>
<td>homosexual woman</td>
</tr>
<tr>
<td>`XX`</td>
<td>`w`</td>
<td>`no`</td>
<td>bisexual woman</td>
</tr>
<tr>
<td>`XX`</td>
<td>`m`</td>
<td>`het`</td>
<td>heterosexual man</td>
</tr>
<tr>
<td>`XX`</td>
<td>`m`</td>
<td>`hom`</td>
<td>homosexual man</td>
</tr>
<tr>
<td>`XX`</td>
<td>`m`</td>
<td>`no`</td>
<td>bisexual man</td>
</tr>
<tr>
<td>`XY`</td>
<td>`w`</td>
<td>`het`</td>
<td>heterosexual woman</td>
</tr>
<tr>
<td>`XY`</td>
<td>`w`</td>
<td>`hom`</td>
<td>homosexual woman</td>
</tr>
<tr>
<td>`XY`</td>
<td>`w`</td>
<td>`no`</td>
<td>bisexual woman</td>
</tr>
<tr>
<td>`XY`</td>
<td>`m`</td>
<td>`het`</td>
<td>heterosexual man</td>
</tr>
<tr>
<td>`XY`</td>
<td>`m`</td>
<td>`hom`</td>
<td>homosexual man</td>
</tr>
<tr>
<td>`XY`</td>
<td>`m`</td>
<td>`no`</td>
<td>bisexual man</td>
</tr>
</tbody>
</table>
<div style="text-align: center;">
Table 2 — The possible values for `P` when mapped into a discrete domain.</div>
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The sole purpose of table 2 is to illustrate that no matter how you want to categorize people, in the end, the difference is negligible. I hope you are now capable of understand, by how ridiculous this article is, that you are only definable by your own identity and preferences. Your body has no influence whatsoever in the person that you are and is nothing but a mere shell for others to see and even that can now be changed. Sexuality is a complex issue and cannot and must not be expressed in a finite number of sets — that is, people are not categorizable. Every single person is unique and ever as interesting as the next one.<br />
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As a final note, for those of a more curious mind, you can search for terms like <i>pansexuality</i>, <i>androphilia</i>, <i>gynephilia</i>, <i>ambiphilia</i> and <i>polysexuality</i>. The way I see it is this: we'll create so many categories for people that, in the end, there is a category for every single one of us and then we abandon all categories, because we are already identifiable by other things — like our own name — that are singletons as well.joão pestanahttp://www.blogger.com/profile/03616394334438513692noreply@blogger.comtag:blogger.com,1999:blog-9153823946796270764.post-87631138148775654082016-01-11T03:33:00.001+00:002016-01-11T14:19:08.512+00:00The evil that bankers do — leveraging and the subprime mortgage crisis<div class="separator" style="clear: both; text-align: center;">
<img border="0" height="256" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh6ZkSUBMt5yyAKKqhNkJBxAYoQTXXT1SN4hSWqws7SOo6Du_ciMiBYO-dXnNutIwBcnHPz659c-OFqM-j9aWLEtyJG0YH3RHpQV2UqFKq-su4zA-7D8r7ftg0kZCS0wwHRLnstVtFJhnY/s640/RJQ5YA1291.jpg" width="640" /></div>
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I intend to continue on my <a href="/2016/01/everyone-wants-houses-until-they-dont.html">previous article</a>, but this time focus on the background gears that almost nobody noticed until it was too late. One might be tempted to ask such questions as why <i>did people applied for loans they knew they couldn't repay?</i> One might be equally tempted to blame the <i>evil</i> profit-seeking bankers for the whole crisis. Either way, only one thing is certain and that is the whole process involved millions of different individuals and possibly hundreds of different institutions.<br />
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We should start by understanding the concept of <i>leveraging</i>, which is a technique used to amplify gains — or losses, if things take a turn for the worst. Imagine that you have 100€ with you and while visiting your friend in another town you notice that the oranges there sell for just 0.5€ a piece while, back in your hometown, they sell for 1€. You seize the opportunity — a process called <i>arbitrage</i> — to buy all the oranges 100€ can get you — 200, if you're wondering. Back in your hometown, you go to the market and sell them for 200€ which, at the end of the day, represents a 100€ profit.<br />
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<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgILkjljj5cIpIype1B7j6Sv443S0W12dnca4M4gUGGALm8Di1BOibcAZOxpmD24EMXvojRPuHRcu_ymIP8LRNjeQKqujWgJJ-ntyrz3oboW4EUnoqsi7ULULcN4SEkDe5UfK5_iTKlhKY/s1600/Untitled-2.png" /></div>
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Having noticed what you have done, your friend also has 100€ with him and, since he's coming to visit you, he wants to make his trip profitable as well. Because he knows beforehand, he has time to go to the local bank and ask for a loan of 9 900€. Sum this up with the 100€ he already has and your friend has a total of 10 000€ to spend on oranges and he buys 20 000 units. He then arrives at your hometown and goes straight to the market where he sells the oranges for 20 000€. At the end of the day, he has paid back the 9 900€ to the bank plus interest — let's say 990€ at a 10% interest rate — and he's left with a total of 9 010€ profit — 100€ return from his own 100€ and 8 910€ return from the borrowed 9 900€.<br />
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<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiodwbDu_n_kB7jOVAzq12ixHpa4SDdHATOHc4RSBphRhHGw5b7J6d7oUdHAWigM5HFhlon9JymjTRvELCsorn-f8gi_qUiqSCxZxxgVbHNmIloPfH1jbM-haNUGXNLCWOra6eEnQKUMZo/s1600/1452498813_vector_65_06.png" /></div>
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As you can see, by using credit, your friend has amplified his gains compared to yours. This is the basic idea of leveraging and most institutions do this — including banks — in order to amplify their profits. Of course, this only works if what you get from the income exceeds the cost of borrowing the extra funds. In this example, you earn 2€ per every 1€ that you spend and, since each 1€ costs you 0.1€ to borrow, this translates into earnings of 0.9€ per 1€ borrowed.<br />
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Back to the main topic, the safest investment you can make is with the government. Believe it or not, it's even safer than banks' savings accounts due to the fact that even if the government lacks the funding needed to pay you back, it can just print more money and give it to you. Now, this doesn't hold true anymore for the individual governments of the euro-zone, but the European Central Bank will end up backing up those governments and it's true almost everywhere else in the world — including the U.S.A.<br />
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Because government bonds are a very low risk investment, it pays a very low rate of interest and it's not very attractive to serious investors that lust for riskier opportunities to earn more money in a shorter period of time. On the other hand, it's very attractive to banks and other credit institutions, because they can get money at a very low cost. That's precisely what banks did for two main reasons. The first reason is that it amplified their gains due to the leveraging effect. The second reason is that, with the extra funding, they could grant more loans. More loans translated into more cash flows from repayments and interest always amplified by the leverage.<br />
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<img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiXy8A9lb1rWZukkl692NQAaXaFY8yFna9q89UmE6ctc3IORxwP51FjBSlx99QpjPU8i1WVrbDEStk_PEdSqPiqmfLVe1BXS-Vw1Q19d3X4Q5e5PQcgL2tmRVzas87qJpgwOqztGu1ocpM/s200/Untitled-2.png" width="86" /></div>
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Since everything is going so well and banks are enjoying large profits, other investors want to enter the deal as well — it has a greater and faster return on investment that government bons. This gave banks an opportunity to further increase their gains by letting these investors into the game and they did it with the mortgage loans — the ones you, your friend and many other people signed with the banks. Because you signed a contract with the bank promising to pay, for example, 10 000€ plus interest, that contract is worth at least 10 000€ to the bank. You can read more about how and why this works in my previous article <a href="/2015/10/what-is-money.html">what is money?</a><br />
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The investment banks started to buy the mortgages from the banks that granted the loans. The lenders received a good price for the mortgages and now the risk of default belongs to the investors. These investment banks applied for credit in order to increase their gains through leveraging and bought a huge amount of mortgages from the lenders. Every month, the investment banks would receive very large amounts of money in the form of payments from the homeowners. The cash flow was good, but what was not good was the associated risk of default. In order to dilute the risk, the investment banks created bundles of mortgages which they called collateralised debt obligations (CDOs).<br />
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Each of these CDOs had to be filled with payments from the homeowners. The investment banks split each CDO into three parts that would be filled in sequencial order. When the payments from the homeowners finished filling up the first part, the second would begin to fill and whatever remained would go into the third part. You can picture this as putting coins into three piggy banks and you could only start the next one if the previous was full. Investing on the first piggy bank is safer than the second and the second safer than the third one — which is the riskier and thus provides a higher return on investment.<br />
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If some homeowners default on their mortgage payments, only the third piggy bank gets affected. The last one to suffer from defaults is the first piggy bank and is considered to be the safest investment while still providing a better return than government bonds. To make this piggy band even safer, investment banks could include an insurance on the investment for a fee — called the credit default swap (CDS).<br />
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You may have heard about those swaps before and not know what those were and it's basically this. If there isn't enough money from the homeowners in the first piggy bank and you bought the CDS, the investment bank will compensate you. Because of all this, the rating agencies gave the first piggy bank the highest rating they could and it attracted many investors.<br />
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The investment banks are making large amounts of money in profits as well as the investors that bought the CDOs. These investors are so pleased that they want to invest in more and the investment banks try to buy even more mortgages. The problem is that there are no more mortgages, because the all the homeowners that were considered safe already have a house. This is the turning point, as you can imagine. Because houses have been increasing in value, even if the homeowners default, the bank sells the house and gets the money back and so banks that grant loans start to relax their requirements and everyone is entitled to credit.<br />
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By now, banks started to grant loans to people who possibly couldn't afford to repay them and they didn't care too much, because they would sell the mortgage to the investment banks and then the risk of default would be theirs. These guys were making so much money that they became reckless and eventually people did start defaulting on their payments. In time, more and more households defaulted and the investment banks ended up with a portfolio of houses that decreased in value with each additional house they received.joão pestanahttp://www.blogger.com/profile/03616394334438513692noreply@blogger.comtag:blogger.com,1999:blog-9153823946796270764.post-80767959340516493052016-01-10T21:41:00.001+00:002016-01-11T14:17:09.796+00:00Everyone wants houses until they don't — the subprime mortgage crisis<div class="separator" style="clear: both; text-align: center;">
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I'll start by assuming that if you opened this article then you want to know how the Great Recession that began in 2007 came to be. It has been classified as the worst economic event to affect the entire world since the World War II. It's closely related to both the financial crisis and the subprime mortgage crisis that struck the U.S.A. at the same time. Due to the open markets that exist nowadays, there is worldwide freedom to trade, so any individual or institution can make investments in any other country — not just domestic investments. That is the very basic idea of how an event that should be contained within a single country can propagate worldwide.<br />
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Why is this such a big deal? Mainly, because it affected everyone worldwide from investors — individuals, governments and institutions — to households (or the <i>common folk</i>). You probably felt it too, even if you are not an investor and didn't contract any kind of loan — not even for a mortgage. Banks are often also investors and when an investment goes wrong, they lose money. Let them lose enough money and they stop having enough money to give back to their costumers — that is, you go to an ATM and are not allowed to make a withdrawal!<br />
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What is the solution for a bank that ran out of money? The government steps in and bails the bank out so you, as a costumer, can have your money back. And where does that money ultimately comes from? Well... from you, as a tax-payer. In order to be able to do this, the government has to increase its revenues and it may do so by raising taxes or lowering wages. That's why even if you had nothing to do with the problem, you still suffer its from consequences.<br />
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How did it all start and how did investors and individuals get involved with each other? It all comes down to mortgage loans. If you want to buy a house, but can't afford to pay it, you can rent a house — but it's not <i>your</i> house. You go to a bank and realize that you can get a loan to buy a house and then pay a monthly fee to the bank close to what you pay for renting a house. You figure that a couple years from then you actually own the house, so it's a pretty good deal. It's also an investment, because the house has value in itself while renting a house is basically <i>throwing money away</i>. That's the reason why the house effectively belongs to the bank until you finish repaying the loan — it's a collateral, should you default on your payments.<br />
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So you contracted a loan for your house and told your friends about it and they think it's a good idea for them as well. The housing market is increasing in value — so it's a very good investment. They also go to the bank, contract a loan and buy a house of their own. Usually, the banks make sure you are able to repay your debt — they require a proof of income and perform other risk measures on you. No matter how thorough their risk assessment is, every now and then someone will default on their payments. There is no problem there, because banks know this and have money set aside as a buffer for these events. If someone repeatedly defaults, the bank sells the house to get their money back.<br />
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If all works so well, then what went wrong? Imagine that many people start going to the bank for loans to buy their own house. The demand for houses is on the rise, their price is increasing and the economy is doing great — jobs in construction are plenty, for instance. Overall confidence is high and there is no reason to assume the worst. Banks keep on granting loans and people buy more and more houses.<br />
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The problem starts when there are no more people with a good credit rating — those whom the bank assess can pay back their loans, called <i>prime mortgages</i> —, but there is still a very high demand for credit and houses. Because the overall confidence is high, unemployment is low and the economy is doing great, banks start to reduce their requirements to get a loan — no proof of income, no downpayment and so on — called <i>subprime mortgages</i>.<br />
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Of course, eventually, these people will not be able to pay back their obligations and start defaulting. The bank keeps their house and sells them to get back the money owed, but there are too many people defaulting and the bank ends up with a lot of houses. Because the bank wants to sell them as quickly as possible, there is a great increase in the supply of houses that exceeds the demand and the prices start to drop drastically. The bank also loses money with every house that is sold below the value of the original loan.<br />
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Up to this point, people that defaulted on their payments lost their house and banks are desperately selling houses to get some of their money back. The value of houses drops drastically and those other individuals that are still able to repay their debt also start to default, because now their house is worth a lot less that the loan they have to pay. Because of all this factors, some banks start to have problems with having sufficient enough capital to give back to their costumers when they demand it.<br />
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Usually, when a bank is lacking in funds, it can also get a loan from other banks or investors, but the other banks are having a similar problem and investors are now afraid that the bank will default or go bankrupt, so they refuse to lend money to the bank. This is what happened to many banks worldwide that had to be bailed out by governments in order to guarantee that their costumers didn't lose all their money.<br />
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By now, you should already have a not so bad idea of how the <i>subprime mortgage crisis</i> came to happen in the U.S.A. and the only things I didn't mention was why the investors refused to lend money to the banks and how did the banks become so careless with their loans. These two factors are related to each other and also with the desire for banks to increase their profits while reducing their exposure to risk. I'll explain that in a <a href="/2016/01/the-evil-that-bankers-do-leveraging-and.html">follow-up dedicated article</a>.joão pestanahttp://www.blogger.com/profile/03616394334438513692noreply@blogger.comtag:blogger.com,1999:blog-9153823946796270764.post-34987734244244553582015-12-22T00:02:00.000+00:002015-12-22T00:18:19.624+00:00Carl Panzram's recipe for a sadistic serial killer<div class="separator" style="clear: both; text-align: center;">
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The reason I'm fascinated by deviant minds and individuals is because we learn very little — if anything at all — by observing what is the standard. Only by studying the unusual can one hope to understand what are the underlying mechanics. The story I bring can be true or a fiction of Panzram's constructed alter ego. Nonetheless, it offers an incredible insight into the construction of an individual.<br />
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In my life time I have murdered 21 human beings. I have committed thousands of burglaries, robberies, larcenies, arsons and last but not least I have committed sodomy on more than 1 000 male human beings. For all of these things I am not the least bit sorry. I have no conscience so that does not worry me. I don’t believe in Man, God nor devil. I hate the whole damed human race including myself.</blockquote>
Fear not, for those words were written by Carl Panzram, as in figure 1, while at the United States Penitentiary, Leavenworth in Kansas where he would end up killing Robert Warnke with an iron bar. For this crime, Panzram would finally receive the death penalty he wished — his own way of suicide. The sentence was carried in September, 1930.<br />
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Figure 1 — One of the papers where Carl Panzram wrote his autobiography and confession.</div><br />
The story begins in 1891 when Carl Panzram (figure 2) is born to a couple of german immigrants in the state of Minnesota. His father abandoned the family when he was 8 years old and by the time he was 11, he was sent to a reform school for stealing from his neighbour. He graduated after two years and about that moment he had the following to say.
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<blockquote>
After serving about 2 years there I was pronounced by the parole board to be a nice, clean boy of good morals, as pure as lily (...) I had been taught by christians how to be a hypocrite and I had learned more about stealing, lying, hating, burning and killing. I had learned that a boy's penus could be use for something besides to urinate with and that a rectum could be use for other purposes than crepitating. (...) I made up my mind that I would rob, burn, destroy and kill every where I went and everybody I could as long as I lived. That’s the way I was reformed in the Minnesota State Training School.</blockquote>
He was young and poor and so he learned how to ride and sleep on freight trains. On one of his travels, he recalls being gang raped by a group of transients he had invited to share the box he found. He was again raped when he tried to beg for food near a group of stray people in some mid-west town, but they got him intoxicated on whiskey and he only found what had happened when regained consciousness. I'll try and let him tell most of the story, because for a man without proper education I do believe he writes remarkably well.
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<blockquote>
I cried, I begged and pleaded for mercy, pity, and sympathy, but nothing I could say or do could sway them from their purpose. I left that box a sadder, sicker, but wiser boy. (...) These two experiences taught me several lessons. Lesson that I have never forgotten. I did not want to learn these lessons but I found out that it isn’t what one wants in this world that one gets. (...) Another lesson I learned at that time was that there were a lot of very nice things in this world. Among them were Whisky and Sodomy. But it depended on who and how they were used. I have used plenty of both since then but I have received more pleasure off of them since than I did those first times.</blockquote>
From then on he proceeds to live his life according to his motto <i>"rob 'em all, rape 'em all and kill 'em all."</i> By 1910, he had already tried the army and failed — he got imprisoned in military facilities for insubordination. This is when he claimed that no more goodness remained within him. He succeeded in robbing the house of the then secretary of war and future president William Howard Taft that had previously approved his army sentence.<br />
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Figure 2 — A portrait of Carl Panzram.</div><br />
From the spoils of the robbery, Panzram was able to purchase a yacht which marked the beginning of his killing spree. He started by hiring sailors as his crew which he intoxicated with alcohol in order to be able to sodomize, rob and kill them. He then proceeded to spread his legacy across different countries in Europe and Africa. While at the latter, he would target a very young boy around the age of 11.
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<blockquote>
He was looking for something. He found it too. I took him out to a gravel pit about one quarter miles away. I left him there, but first committed sodomy on him and then killed him. His brains were coming out of his ears when I left him, and he will never be any deader.</blockquote>
He finally returns to the United States where he spends most of his time incarcerated in different institutions. While at the Washington DC district jail, he befriended a young prison guard by the name of Henry Lesser — the only man in the world Panzram didn't want to kill — who insisted on him to write his life story and provided the writing materials.
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<blockquote>
You know that I spent several years in one of those places when I was a boy and the so called training that I received while there is mainly the cause of my being the degenerate beast that I am today. I have thought about that system of training young boys for all of my life and I know that the whole system is wrong. That system of beating goodness, religion and Jesus into boys in the 99 times out of 100 has the direct opposite effect of taking all of the goodness, kindness and love out of them and then replacing those with hate, envy, deceit, tyranny and every other kind of meanness there is.</blockquote>
Now that you know the story you may feel pity, anger, a mixture of both or even a whole panoply of feelings. You do have an opinion — at least, that's what I hope I induced upon you. We can analyse Panzram's life in different ways.<br />
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He may have been genetically predisposed to violent behaviour, had very high testosterone production or suffer from such a thing as the Klüver–Bucy syndrome. Some men are just pure evil psychopaths and nothing can be done to either alter or prevent their criminal tendencies. The question remains as <i>what if not?</i> What if we are really born blank canvas that can be painted anywhere from an ugly draft to a long admired masterpiece?
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Yes, hurry it up, you Hoosier bastard. I could hang a dozen men while you're fooling around.</blockquote>
Those were the last words that Carl Panzram offered humanity before he hanged. It's not possible to account for the veracity of all the crimes he claims to have committed. Maybe he was only a scared little boy who realised how fragile he was and no one ever vowed for him so he constructed his own strong and unforgiving alter ego to take vengeance upon the society he visioned as an entity who wronged him. Maybe he admitted and exaggerated his crimes to increase the inspired fear on others. Either way, it remains an almost unbelievable story of this personification of nihilism.joão pestanahttp://www.blogger.com/profile/03616394334438513692noreply@blogger.comtag:blogger.com,1999:blog-9153823946796270764.post-18178746605108259802015-12-02T23:55:00.003+00:002015-12-03T01:23:23.372+00:00Why women are diamonds and men chunks of coal<div class="separator" style="clear: both; text-align: center;">
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Before I begin, I feel I should disclosure that I am a man and a highly satisfied one. As a man, I adore women and that feeling extends beyond the sexual desire or need for female maternal attention. I believe that women are the true representative specimen of our species whilst men are nothing but a necessary satellite specimen that exists solemnly for one purpose — that of providing stronger disease resistance to future generations by means of genetic variability. By now, most men will have their ego boiling up in rage and stop reading. Nonetheless, if you do continue, I will provide the sufficient arguments and scientific evidence on which I'm supporting such claim.<br />
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Let us start with the basics. The human genome consists of 23 pairs of chromosomes. Your father contributes 23 chromosomes and your mother contributes the other 23, making a total of 46 chromosomes inside almost every cell in your body. This combination is crucial to provide genetic variability. Since they always come in pairs, there is room for some redundancy. Should one of the chromosomes be defective, the other one takes control and you have thus avoided disease. Even if you are not afflicted, thought, you carry the disease with you and might transmit it to your children unless you find a partner that can provide a non-defective chromosome. This happens for every chromosome except the X and Y ones.<br />
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<div style="text-align:center;">Figure 1 — Male chromosomes, The X (left) and Y (right) magnified about 10 000 times. © <a href="http://www.nature.com/nature/" target="_blank">Nature</a></div>
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Women are women because their 23th pair contains two X chromosomes whilst men are men because their 23th pair contains only one X chromosome and a tiny Y chromosome, as you can see in figure 1. This last one is responsible for the fetus differentiation into a male. Since X and Y are so different, even though they do share some genes, there is almost no room for redundancies. This means that men better receive the best X chromosome they can, because, if not, there is a high chance the Y chromosome doesn't have the information needed to cover for the malfunctioning one. Haemophilia is a very good example of such a situation. In sum, women have richer and more resilient genetics. The X chromosome represents about 5% of a human cell's genetic information in contrast with the less than 2% the Y chromosome represents.<br />
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Not only that, but Yamauchi, Y., Riel, J. M., Stoytcheva, Z. & Ward, M. A. presented their findings in a 2013 Science article with a self-explanatory title <i>Two Y Genes Can Replace the Entire Y Chromosome for Assisted Reproduction in the Mouse</i>. At least, for mice, the Y chromosome is rendered almost useless. The results cannot be directly extrapolated for other species such as us humans, but it is an amazing result nonetheless.<br />
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Another interesting finding is that it seems women have contributed more to the human DNA pool than men and this is very counterintuitive. One ought to believe that since each parent contributes half, then both male and female contribute on an equal ratio, but Lippold, S., Xu, H., Ko, A., Li, M., Renaud, G., Butthof, A., Schroeder, R. & Stoneking, M. seem to disagree. In a 2014 article published on Investigative Genetics entitled <i>Human paternal and maternal demographic histories: insights from high-resolution Y chromosome and mtDNA sequences</i> they present their results and provide some possible explanations for this.<br />
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Let us think for a moment on scarcity of resources. What's more valuable? Gold, diamonds, oil, sand or water? The first three are much rarer that the last two and thus are more valuable due to the greater amount of effort needed to get them. That effort includes the time spent just searching for gold, diamonds or oil and then the whole logistics of getting them from wherever they are.<br />
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Every organism that reproduces asexually is female. Now focus on us humans as a sexually reproductive species. There are two genders and each has its own features regarding sexuality and reproductive capability. Ignoring the fact that women are born with a huge, but mostly limited (between 1 and 2 million), number of eggs, these simply mature one — or two, rarely — at a time on a monthly basis. Men, on the other hand, can produce something like 1 500 sperm cells each second, according to Robert E. Braun in a paper entitled <i>Every sperm is sacred—or is it?</i> published in Nature Genetics 18, 202 - 204 (1998).<br />
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I believe it's time for some mathematics. Alice and Bob were born on the same day and reached sexual maturity at the same time — let's say on their 13th birthday. They will fall in love and marry and die on the same day — their 57th birthday. Assuming that neither menopause or andropause are reached, Alice and Bob were fertile for 44 years. I'm also assuming a constant gamete production rate for each of them — one egg per month for Alice and 1 500 sperm cells per second for Bob.<br />
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That means Alice produced one egg each 2 592 000 seconds. But by that time, Bob had already produced a total of 3 888 000 000 sperm cells. If Alice never got pregnant and produced exactly one egg every month, after 44 years she produced a total of 528 eggs. Bob, on the other hand, produced an astonishing 2 052 864 000 000 sperm cells. To put that in perspective and get you some feeling of how big a number that is, it's about 7 times the number of stars in the Milky Way.<br />
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My point is eggs are a scarce resource and, for that reason, women are very precious — as compared to men. Females are usually the limiting resource to reproduction in a species, because males often have a near to limitless supply of gametes. One man could fertilize all the women in the world and this fact gives women the ability to be picky when choosing a partner, because males will compete for their very rare eggs. Such competition will make some men more successful than the rest and thus create a greater reproductive variability. This idea is called Bateman's principle and is the reason why men are nothing but a satellite gender — they exist purely to provide genetic variability.<br />
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How successful can one man really be? Well... According to the article Chiara Batini et al. published on Nature Communications 6 entitled <i>Large-scale recent expansion of European patrilineages shown by population resequencing</i>, more than 60% of modern Europeans are the descents of just three men.<br />
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There is much more I would like to write about, but I feel this article has become long enough. If you're a man, make sure you treasure the women that are close to you — they truly are diamonds among us.joão pestanahttp://www.blogger.com/profile/03616394334438513692noreply@blogger.comtag:blogger.com,1999:blog-9153823946796270764.post-49468140735851883482015-11-21T16:08:00.000+00:002015-11-22T00:57:44.306+00:00A late night conversation about the nature of thought and sensory perception<div class="separator" style="clear: both; text-align: center;">
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This article is not something that I intended to write, but it was the fruit of a late dinner conversation among some friends. We started by discussing physics. As a physicist, I was trying to explain how the world fits together according to the most currently accepted theories. As non-physicists, my friends were always <i>"but why?"</i> and <i>"how does that happen?"</i> At a certain point, I had to realize and admit <i>"well... that's what every physicist wants to find out as well!"</i> I want to continue with this topic in a different article on when or if <a href="https://www.blogger.com/blogger.g?blogID=9153823946796270764#">we should ever stop</a> looking for answers. There will always be questions, but answers are harder to come by.<br />
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At a certain point, we started discussing vacuum chambers. Innocently enough, they asked me the question <i>"what does it remain when you remove all the air?"</i> For a vacuum chamber on earth, I argued, if it has a window and you can peek inside and see something, then you can be sure that radiation remains. The removal of air it's relatively easy and straightforward, just think vacuum cleaners. You might feel tempted to think that it's impossible, but imagine the following situation.<br />
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You're on a submarine going deep and deeper into the depths of the ocean. So long as the structure holds strong, it won't collapse over the immense pressure from the layers of water above. If there was water inside, it would be alright, but there isn't. The same things happens when you remove air from a chamber — all you need is a structure that can withstand the pressure from the layers of air above.<br />
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Now for the removal of radiation it's not as easy and that would require complete electromagnetic shielding in all possible wavelengths. If you know what a Faraday cage is, it would be something similar. And that's just to prevent the entry of new radiation, we still need to remove what remains already inside.<br />
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Driven by the arrow of conversation, another interesting question — so much I decided to write this article around it — came up. I will not disclosure what it is yet, but will create the context and let the question flow naturally as it did during our evening. As good inquisitors, my friends would not conform with the answers I gave them. I based them on the most accepted theories that are — despite the uncountable experimental evidence — and always will be theories.<br />
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Just as we've never seen an apple fall up, we take it for granted that if it falls it falls down towards the center of the Earth. Imagine thought that one day, one apple does fall up towards the sky accelerating away from the center of the Earth. Such a single event would suffice to invalidate many of our theories about how the gears of the universe roll. We can never say that an apple will always fall down. What we do say is that under certain circumstances, given all of our experimental data, we infer that it must fall down.<br />
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In the end, the universe is what each one of us perceives through our senses. They allow us to read the energy of electromagnetic radiation in which each different wavelength is associated with a different perceived color. Blue is more energetic than green and green is more energetic than red. Our brains interpret these differences in energy as what we call colors. You and I might perceive colors very differently, but so long as we agree on the names, there'll be no confusion.<br />
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Something similar happens for the air pressure differences we interpret as sounds or the specific molecules that trigger our noses to send a signal to the brain which interprets it as either a good or bad smell. Air itself has no proper smell — our brains ignore it — so we can know when particles that are not so common come up.<br />
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I see the brain as a processor of information that constantly arrives through our senses — and we do have more than the classic five. We don't have a temperature sensor. Instead, what we actually perceive as cold or hot is the rate at which our bodies lose or gain heat energy. The higher that rate is, the higher is our perceived sense of cold or hot. You can do a simple experiment to prove this at home as follows.
<br />
<blockquote>
Place a cotton napkin or towel and a spoon in the freezer for an hour. After the time has passed, pick them up. You can even use a thermometer, but you shouldn't need that to believe that they're at the same temperature. Now place each hand to the corresponding object and try to comprehend how you feel that one is colder than the other even though they're at the same temperature.</blockquote>
You've probably figured this out already. The metal in the spoon is a much better heat conductor than cotton which is actually a pretty good heat insulator — the reason we use it to make warm sweaters for the winter! That means that the rate at which your body loses heat to warm up the spoon is much higher that the rate for the cotton. Your body does not want to lose heat and so it gives you a warning that it feels really cold and you should take out your hand immediately.<br />
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What if somehow you lose all of the inputs? Imagine that all of your senses — not just the classic five — suddenly stop and you're left in the dark and cannot even feel pain. What happens to your brain? Does it keep working? Now this is the question that came up that night. I tried to argue that it's impossible for a brain to stop working, because there is always some input either as immediate or as a memory. Similarly, it's impossible to conceive a being with a brain — a cognitive processor — which never received a single input. The simple action of <i>waking</i> the brain is an input!<br />
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The problem here is that we've created an unnatural concept — that of absolute temperature. This is nothing more that the total energy a given group of particles has at a very specific moment in time. What would be more natural, as far as our body is concerned, would be the rate of change in temperature — how fast does the group of particles lose energy. They're moving fast, but slowing down.<br />
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I'll try to make my argument clearer by using an analogy with a <a href="https://www.blogger.com/blogger.g?blogID=9153823946796270764#">Turing machine</a> — that is, a computer. Imagine a processor without memory and accessories. That is much like a brain which never learned a single thing and has received no information from senses. Does it work? Unless you power it up, it doesn't, but powering it up is sending an electric signal (for a digital computer, that is). That constitutes information being entered and the processor might do something if it has any given set of instructions to perform when it powers up.<br />
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Our brains are hardwired to perform basic life supporting activities which need not to be learned. We have an innate ability to breathe and pump blood through our cardiovascular system. Everything else about the world around us must be learned through sensory information. We do need that first input — that kick off — nevertheless. If an input never happens, then nothing will work.<br />
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Even if we could build a loop that just goes on forever and somehow was built already performing that loop — that is, without an initial start input — there must be an equivalent information within the system that somehow says <i>"go on."</i> My point is this: without senses, nothing works and the coming alive is in itself an input. It may be an endless senseless loop, but it does need that startup instruction or something equivalent. If there exists a set of instructions that never gets that initial input, they're dead — an inanimate object no different from random clumps of matter. I think I finally understand what René Descartes really meant when he said <i>cogito ergo sum</i> — I think, therefore I am.<br />
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This was my conclusion from that very interesting late night conversation. These are my thoughts and the theory I've constructed in my head describing how the world I perceive through my senses works. What we all agree upon might be considered fact, but what we don't agree on might be called eccentric or mental illness. Who's to say what's correct? Our senses elude us all of the time and the concepts we create like absolute temperature add to that confusion.joão pestanahttp://www.blogger.com/profile/03616394334438513692noreply@blogger.comtag:blogger.com,1999:blog-9153823946796270764.post-30877672105708818492015-11-18T20:26:00.002+00:002015-11-18T20:34:39.814+00:00An obsessive-compulsive's view on the universe, people and the origin of life<div class="separator" style="clear: both; text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEil5Y1R8PpsPP45075xcBLqOYvAmBasmRXCNN_v62rtb3L3_GlfQZjRQPjJ8VICqsIS00P6YrflQG9msNXdvb9KPWiIiJWlhX1rPyEpcEDzL85AN1g3q_0ne6M7rMCDx8hyphenhyphenPU7xSwKhSTI/s640/4GVNJ3PV56.jpg" /></div>
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I must admit straightforwardly that I am, to a certain extent, an obsessive-compulsive. Though it's far from a serious case where I cannot leave the house without turning the knob precisely 13 times, it does causes me an itch or two from time to time. For instance, if I'm allowed to pick a number from a set of integers, I'll always choose multiples of 5 first, followed by prime numbers, odd numbers and only if no other choice remains, even numbers. This I do when I select the volume for the radio or the television and many other things. At first, it might seem that I like to go against the universe.<br />
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According to the <i>second law of thermodynamics</i>, the entropy of the universe as a whole must always increase. Entropy is the number of different ways you can arrange all the parts of a different system. Imagine that you own 2 pieces of LEGO, one red and one blue. Their entropy value is 1, because there is only one way you can arrange them. You can order the red one first or the blue one first, but that's the same order — just mirrored or rotated. Imagine that you break each LEGO in 2. You now own 4 pieces, two red and two blue and there are `C(4,2)=6` different ways you can arrange them. Split those again and you get `C(8,2)=28`. Notice how large the entropy of the system has increased by just splitting the original parts of the system in two two times! Another common definition of entropy is the measure of the disorder of a system and it must always increase.<br />
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I like to keep things in their place — properly organised — but the universe doesn't seem to like that. Am I trying to beat the universe at its own game? No, not at all or I'm failing miserably. The truth is that to create order, something else must become at least as disordered as the order I created. Imagine that you are trying to sort a 52 deck of playing cards in some order that you like. To do that, you are increasing the entropy of the universe by much, much more.<br />
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How? Every time you move, you waste energy which you have previously stored from food. That food — either animal or vegetarian — originated fundamentally from the Sun. Not all energy from the sun is converted into sugar by plants through photosynthesis, because some actually just warms up the plant. Your movements also push the air particles and increase their energy. Due to this, they collide even more with each other and break up and form new pairs. Remember how splitting 2 elements in two greatly increased the entropy of a system? Imagine how all the uncountable particles of air that split up increase the entropy of the universe!<br />
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We could say that everything in the universe acts as an entropy maximiser either willingly or unwillingly, like me. Take matter, for example, that wants to reach the lowest energy state it can. In order to do so, it must release all of its energy to elsewhere. If you drop a ball from yours hands while on the surface of the Earth, it will have its potential energy converted into kinetic energy during its travel towards the ground. While doing so, the ball loses some kinetic energy to the surrounding air and the remaining is lost on impact. Because the ball stood some height above the ground, it had a higher energy stored in itself — potential energy. At ground level, the potential energy of the ball is lower and is more stable — the ball will remain there. During its fall, the entropy of the universe increased.<br />
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Economists usually apply the same principle to people. They say that consumers are utility maximisers. There is some unknown function `u(x)` which is unique to every person and, given some information, it outputs the level of utility that person assigns to the information provided. We can say that if I prefer pears to apples, than my utility function given pears will have a higher value than if given apples.<br />
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<div style="text-align: center;">
`u( p e a r s ) > u( a p p l e s )`</div>
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Most of the time, we aren't aware of our own preferences. It's not hard to convince you that economists and economic models are at odds with the consumers' utility function. I believe we do maximise something whenever we perform a choice — even if it's not the best outcome possible. Given our knowledge at the moment and our expectations, we make an effort to select, from the set of all possible outcomes, the one that grants us our best chances possible. Imagine you can choose between two lottery tickets. The first grants you a prize of 100€ with 50% chance and the second the same prize, but with a 60% chance. Which one would you choose?<br />
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<div style="text-align: center;">
`u( l o t t e r y_2 ) > u( l o t t e r y_1 )`</div>
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If you're like me, you'll give a higher utility to the second lottery over the first one. Unfortunately, most of our decisions in life don't come out as easy as these two lottery tickets and so we are forced to extrapolate future results from current and past information. If we touched a poison ivy in the past, we're going to do avoid in the future. This happens, because we make use of information that it caused us harm in the past and we assume it will have the same result if we touch it again.<br />
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The whole point of this article is this argument. If the universe and all its contents are entropy maximisers, so are we — as part of the universe. The utility function is different and incoherent between people, but not the entropy maximisation principle. Willingly or unwillingly, we do increase the entropy of the universe more while being alive that if the whole set of particles that constitute us would on their own.<br />
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Although I would really like to call these ideas original, they are not. A final note on entropy and then I'll finish with some scientific literature for you to investigate on your own. It's possible that if you drop an entire bag of M&Ms to the floor, they will perfectly align in a circle. That is one possible final configuration of the act. The more likely scenario is one in which the M&Ms are all scramble in a seemingly random order.<br />
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Because the number of seemingly random configurations is much, much higher than the circle one, it is the one that is observed — because, you know, probabilities. The same principle applies to the energy configuration of the universe. It all scrambles because there are many more ways to randomly distribute it than to have it all organized somewhere as I would really like to.<br />
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Jeremy L. England published an article in <i>The Journal of Chemical Physics</i>, 139 entitled <i>Statistical physics of self-replication</i> in 2013. His idea is that life is a byproduct of maximizing the entropy or dissipation of energy. To the laws of physics, there is no difference between an inanimate object and a living organism — they're simply clumps of matter. The later one is much efficient at retrieving energy from its environment and dissipate it as heat. You should read <a href="https://www.quantamagazine.org/20140122-a-new-physics-theory-of-life/" target="_blank">this article</a> and in England's own words:
<br />
<blockquote>
You start with a random clump of atoms, and if you shine light on it for long enough, it should not be so surprising that you get a plant, (...)</blockquote>
It's both a beautiful and frightful thought to realize that perhaps when the universe finally reaches its end, that final state of maximum entropy, it may well be the state of ultimate order and everything is so well organized that it is indistinguishable from total caos.joão pestanahttp://www.blogger.com/profile/03616394334438513692noreply@blogger.comtag:blogger.com,1999:blog-9153823946796270764.post-21137020659436881162015-11-08T14:29:00.002+00:002015-11-08T14:32:18.650+00:00My time is different than yours<div class="separator" style="clear: both; text-align: center;">
<img border="0" height="256" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9Eb-USDf9ybPpoq7jgfXowy_M7mSVrnYjHQhSORZFKT3F6d_osGA2Pgf37Br6pUx2PfUpw201Ez0_CnBgpiPwgfsrCmHTT6tLJUhyQUKnOKDaRyhmns9RAnFaaGMFRZRwJkl50z3DHoY/s640/4EA8E81993.jpg" width="640" /></div>
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I will assume that you know the pythagorean theorem. You know that sing-along in which <i>the square of the hypotenuse is equal to the sum of the squares of the other two sides of a right triangle</i>. That literally means that if you draw 3 squares, each having a side equal to each of the 3 different sides of your triangle, and you sum the areas of the smaller 2 then you get the exact area of the bigger one!<br />
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<img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiDq0i4cSNVQGo7kukDLz6aGJdNo7kP3MLRTdakMUzEf_LiPpIo8TWtpliKL0qB1ryzEtEaRfqlKVwtE12PeH2CPoAAVORzfhsgxySe_WQQwwx9GzpkUsrz7cSvNlbuKoIITC69zhJHPI0/s200/Untitled-2.png" width="169" /></div>
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Figure 1 — A right triangle with each side labeled a, b and c.</div>
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If you want to be mathematical — and I know you do — we simply state the previous as the lovable equation `a^2=b^2+c^2` and you can check the figure 1 to identify each side (`a` is the hypotenuse). But we don't discuss mathematics today, even if it's geometry. I want to talk about the theory of relativity — yes, Albert Einstein's relativity!<br />
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Let's begin with a story and a challenge. There is a master clockmaker who is such an amazing craftsman that he has the godlike power to build identical clocks who always mark precisely the same time. His ability is such that he offers with each clock a lifetime warranty. You can get your money back if one of his clocks differs 1 second from all the rest. It's impossible to meddle with the mechanics. Can you think of any way of taking advantage of his generous lifetime warranty and get your money back?<br />
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It might surprise you, but the answer is yes. All you need to do is take the clock with you on a <i>quick</i> 31 thousand years ride at the speed of sound. Of course, if you can't wait that long, you can speed things up by speeding up! You only need 32 years orbiting the Earth at the International Space Station which has an average velocity 23 times higher than the speed of sound. I believe this last scenario is the most favorable one in order to deceive the master clockmaker.<br />
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For a long time, the Russian cosmonaut Sergei Avdeyev held the record for time dilation experienced by a human being. Due to his 747 cumulative days of space flight, he aged roughly 0.02 seconds less than a person who never left the Earth. According to Einstein's special relativity, the clock of a person with greater velocity ticks slower, but if you're thinking that's just a theory... Without taking time dilation effects into consideration, GPS and other systems alike would be useless. Satellites' clocks do run slower than earthbound ones and we need to account for the discrepancy in time measurement to avoid triangulation errors.<br />
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If I'm running and you're sitting or if I hike the mountains and you go swimming on the sea, our clocks run differently. The first effect is due to the different velocities and the second one is a consequence of gravity — or spacetime distortion. The bottom line is that my time is different than yours and I can show you why using only Pythagoras' theorem.<br />
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<img border="0" height="208" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgEWdQZDQxjGZHZIU4ipiML6BPgZkXRrnbJEQzSKCSVCtgj35yYOWQnroBu44L_gAXsudH-CV749-3LlseiLBRLNaqjhbKhB4ja_AwnuBoSSXUBrykwBqtcdVmovr6_zMMYJkf9b68zEm0/s640/Untitled-1.png" width="640" /></div>
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Figure 2 — Representation of a moving car at different positions and periods.</div>
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Suppose you're driving your car and you throw a ball into the air. If you can't imagine, just look at figure 2. As you move along, because your car is traveling, the ball goes up at different positions for each period. From your point of view, the ball only goes up, because both you and the ball are moving along with the car at the same velocity. For someone else who is just watching you pass by, the ball travels not a straight line up, but rather it describes a curve.<br />
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Long before Einstein, Galileo Galilei already described this type of event in his <i>Dialogo sopra i due massimi sistemi del mondo</i> (Dialogue Concerning the Two Chief World Systems) published in 1632 using ships as an example. The principle of relativity is far from a novelty. What Einstein added was the idea that the speed of light is always measured to be the same by everyone. It's as if light was the ball that you're throwing and it would always describe the same arch, whether you're standing still or inside a moving car.<br />
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<img border="0" height="272" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg2QSmklQEDL5lWcARRK8pQT7s3bWNL5X_MUfo3txgUmqFoVAE0Fe1OdOEqIaQMlrAltJqqTQE_7xVpXCxcI9Kr_aoQi2M97angN0BAy4D_CugCesYESFsVwxI6WMg1_cXjnEW3icn0r2w/s640/Untitled-3.png" width="640" /></div>
<div style="text-align: center;">
Figure 3 — Stationary train carriage with a light beam emitting at A and reflecting at B.</div>
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How can we then demonstrate the effect of time dilation using what we've discussed so far? It's quite simple and we need only one more example. Imagine that you're inside a train carriage as in figure 3 and that there is an emissor of light at the floor on point A such as a lightbulb and a reflector at point B such as a mirror. How long does the beam take to make the trip from A to B and back to A? It travels at the speed of light `c` and all we need to do is divide the total distance `bar(AB)+bar(BA)` by the velocity. Let's say that the height of the train carriage is `h` and thus `bar(AB)+bar(BA)=2h`.<br />
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<div style="text-align: center;">
`Delta t_(i n s i d e)=(2h)/c`</div>
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Now suppose that the carriage is moving at a constant velocity `v` and that a friend of yours is watching it pass by on the train platform. Because you're inside the train, nothing has changed. You still see the light beam travel from A to B and back to A in a vertical line and the time in which that happens is still `Delta t_(i n s i d e)`. Remember the example with the ball and the car? Well, it wasn't in vain. On the platform, as your friend watches the light beam go up and down, it does not describe a vertical line. It actually resembles a triangle as in figure 4.<br />
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<img border="0" height="272" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEicg-taG1bubMN_bLAY5F592zViCZuSkZYhrWv9t90DYLY23g57YPBMSTqrSWCxbiY_xIIzIe6MB902qqQCXozsfyozKqPCChZO-yfbPKwECHM8H9IVrLUuDEGmxqnnA69o-66UYG4VFhg/s640/Untitled-4.png" width="640" /></div>
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Figure 4 — Moving train carriage at a constant velocity.</div>
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Yes, this is the reason we can apply Pythagoras' theorem. So let's do it! The distance that the light beam describes from A to B is not just the height of the train carriage, but the hypotenuse of a right triangle in which the height is just one of its sides. The other side is the distance that the train traveled since the light was emitted at A up until the point it was received at B and that is simply `Delta x`. Let's call this distance `Z^2=(Delta x)^2+h^2` and it's larger than `h`.<br />
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<div style="text-align: center;">
`Delta x=v (Delta t_(o u t s i d e))/2`</div>
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If `Delta t_(o u t s i d e)` is the time measured by your friend on the platform that the light beam takes to complete the trip from A to B and back to A, then half of that is just the time to reach B starting at A. We want to relate your friend's time with your time `Delta t_(i n s i d e)`. Let's review what we know of the problem so far.<br />
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<div style="text-align: center;">
`{(Delta t_(i n s i d e)=(2h)/c), (Delta x=v (Delta t_(o u t s i d e))/2),(Z^2=(Delta x)^2+h^2):}`</div>
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If we write `h=(c Delta t_(i n s i d e))/2` and `Z=(c Delta t_(o u t s i d e))/2`, we can write the Pythagoras' expression as:<br />
<br />
<div style="text-align: center;">
`((c Delta t_(o u t s i d e))/2)^2=(v (Delta t_(o u t s i d e))/2)^2+((c Delta t_(i n s i d e))/2)^2`
</div>
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And solving that equation for `Delta t_(o u t s i d e)` yields the expression:<br />
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<div style="text-align: center;">
`Delta t_(o u t s i d e)=(Delta t_(i n s i d e))/sqrt(1-(v/c)^2)`
</div>
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Which I've used to determine the values I expressed in the fourth paragraph of this article. As you can see, you can relate the time experienced by a stationary observer — your friend on the platform — to that of a moving observer — you inside the train — by the relation between the velocity of the moving observer and the speed of light. You may notice that if the train stops `Delta t_(o u t s i d e)=Delta t_(i n s i d e)`, but as its velocity increases `Delta t_(o u t s i d e) > Delta t_(i n s i d e)`.<br />
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We can indeed get our money back from the master clockmaker by taking the clock on a fast trip, but we should check the warranty clause to see if it's not void for relativistic time dilations. I would guess that a master clockmaker would have this problem anticipated.joão pestanahttp://www.blogger.com/profile/03616394334438513692noreply@blogger.comtag:blogger.com,1999:blog-9153823946796270764.post-72242664301066926532015-11-03T00:03:00.001+00:002015-11-03T00:13:28.112+00:00How to get your mind blown with simple arithmetic<div class="separator" style="clear: both; text-align: center;">
<img border="0" height="256" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhK3jAin8XpcIsGmQL11RyMNSSWgBG8DJM36t_tTZYP7InIEl_LZGjtscPzEXRJqIAP2ba2P4mDO48Y1rVvSEPW3lu7OXalJ2svf0zTjNgTMYFj8Tk9J-OlhAqn5pR3FGU_znBZKW-15Hc/s640/75HN5HHXIE.jpg" width="640" /></div>
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Arithmetics is the oldest of all the branches of mathematics. It's the study of numbers and the basic properties among them such as addition, subtraction, multiplication and division — or simply addition if you read my <a href="/2015/10/if-you-know-how-to-add-you-know-almost.html">previous article</a>. Now I know that if you are literate enough to understand the article up to this point, you probably think you are good enough with these basic mathematics operations.<br />
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You know how to add, subtract, multiply and probably how to use a division algorithm. Even if you are somehow unable to perform these tasks, I know you can pick up a calculator and use it. The calculator does the math for you, but you still need to know what to input and have some sense of the expected result.<br />
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Deep inside of you, there's something working that tells you that you cannot have, as the result of a sum or multiplication, a smaller number than the ones used as inputs. This, I mean, for whole numbers — or, more precisely, the set of all natural numbers `NN`.<br />
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This happens because we use a numeral system that's mostly universal and feels natural. We use the decimal system with radix 10 which means we have 10 different symbols to represent 10 different quantities and then we repeat these symbols to represent even greater amounts. It feels natural because we are naturally born with ten fingers.<br />
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We think that mathematics is simple if it only consists of these symbols `n in {0,1,2,3,4,5,6,7,8,9}` and is complicated if it involves any other unusual symbol. Even letters from the alphabet, when used in mathematics, seem and feel weird mainly because we do not know what to do with them. Since they're not part of our base system, there's actually nothing we can do but to treat them as alien entities and keep them untouched to be replaced by some more meaningful symbol later — a number. We call these <i>variables</i>.<br />
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The decimal digits have very precise meanings as quantities in themselves. An empty park has `0` people in it until a lonely man comes for a walk and it becomes filled with `1` person. Some time later, a couple — that is, `2` different people — may also enter the park and the population is now of `3`. But we would know nothing if we were told that there were `A` people in the park.<br />
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There are infinitely many different numerical systems and some have been so influential in the past that they still persist nowadays. Haven't you ever wondered why we measure the degrees and time in multiples of 60? The Babylonians used a numerical system with radix 60. That is, they had 60 different unique symbols to represent quantities and only needed to repeat that for the quantity of 60 just like we do for the amount 10. We always start counting at `0` and then count the number of times we counted until `9` — that's what the `1` in `10` represents.<br />
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I could go on for a while about different numerical systems, but I'll move into the mind blowing part — as promised. We take the symbols `0...9` for granted and assume they're the only ones that are <i>mathematicable</i>. Whatever else is included in the mix must be a <i>variable</i> or have some very precise meaning like the symbol `in` which just means something <i>belongs to</i> or <i>is in</i>.<br />
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The beautiful thing — now I know I'm not on solid ground here — about mathematics is that it does not need and it does not depend on the numerical system that we choose to use. It always works and yields the same results. We do not have to conform to what everybody is using to and we are free to invent our own symbols to represent whatever quantities we desire. We can even have symbols for fractions, irrational or algebraic numbers — yes, bases like that are possible! Just think of the symbol `pi` that represents nothing more than a number just like `1` does. It sits somewhere close to `3` on the interval from there to `4` in our decimal representation.<br />
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For the prestige, I'm going to switch numbers and letters and perform mathematics. I'm going to keep the decimal system in order to keep the mind blowing contained. My ten symbols are `{O,A,B,C,D,E,F,G,H,I}` and everything else is nothing more than an entity which can be used as variables.<br />
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A classical example could be that Clara has the recipe for a lemon pie and it requires `A` lemon, `C` eggs, `BOO` grams of sugar and `BEO` grams of flour. Since she only has `BOO` grams of flour available, what are the remaining required quantities of the other ingredients to keep the proportions correct?<br />
<br />
First, we need to get the ratio between the available and the required flour `1=(BOO)/(BEO)=(BO)/(BE)=D/E`. Note that `1` is just my symbol to represent this ratio. I can use anything that's not part of my numerical system. Finally, we simply must multiply every quantity by this value. Clara will need `1xxA=D/E A=D/E` parts of a lemon, `1xxC=D/E C =(AB)/E` eggs and `1xxBOO=D/E BOO=(GOO)/E=AFO` grams of sugar.<br />
<br />
I used a fairly intuitive replacement and even `O` to represent the `n u l l` quantity. Everything had a straightforward relation, but imagine a system that's not base 10 and uses symbols that we seldom use or have never seen before! Mathematics will still work, but I'm not sure about our minds.joão pestanahttp://www.blogger.com/profile/03616394334438513692noreply@blogger.comtag:blogger.com,1999:blog-9153823946796270764.post-68267454355098753342015-11-01T22:22:00.000+00:002015-11-01T22:26:33.972+00:00Friends and Strangers<div class="separator" style="clear: both; text-align: center;">
<img border="0" height="256" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjO_1y5kwQlOHE8D0rtWwdvLBjdZ4WY48g3LXR-tD_qV1cjUojGmQ3NfhPuaVduOZFw_2OF0ZKlKsduSl-Ot0YHnqQ7kUcrdbjOL_l6iLsHheOVbkx7quoKAJTtYIc36BiYroCFLMv8qZg/s640/94356BMTJ5.jpg" width="640" /></div>
<br />
Many things can be mathematically proven even if, at first, it seems rather unlikely. How people connect with each other might fall into that category. It could be pretentious to suggest that something as stochastic — fancy word to say unpredictable — as human behaviour can be mathematically modelled. Though I'm a believer of such possibility, I'll try to remain neutral here and state only what is currently accepted.<br />
<br />
Think about how people connect. You have some easily countable number of friends and the remaining people are the non-friends — also known as strangers. It's easy to describe the relation between you and any given person at any given moment as either a friend or a stranger. That is, at any given moment, you have a fixed network that connects you to the rest of the people in the world. Of course, this may change over time.<br />
<br />
Now I know this might seem confusing and complex, but I assure you it's as simple as watching people pass by and simply categorize them as <i>"I know that person, it's a friend"</i> or "<i>I don't know that person, it's a stranger."</i> To illustrate my point, I present the image in Figure 1.<br />
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<img border="0" height="300" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhxnXqlMIzW7whCJ4eT75J4RXbhSmUvNLKWBfuQjE9G0D8khYeiiEoPQoGevDGKcNuElhsdqk235o3bbT-bLICyKPRj63Nmx6inElASf002zU8ymTo-CrImoi4_5I6H9BJA-esKgifNvLU/s400/Untitled-1.png" width="400" /></div>
<div class="separator" style="clear: both; text-align: center;">
Figure 1 — Diagram of someone's friendship network depicting 3 friends and 3 strangers.</div>
<br />
In the diagram, imagine you are the one portrayed in a black contour, those in blue are your friends and the ones in red are strangers. If the world had only this set of 7 people, it would perfectly describe your instant social network. Not portrayed are the networks of each of the remaining 6 people. You can imagine a similar picture for each individual. It may happen that someone is friends with everybody or friends with nobody.<br />
<br />
This is when mathematics enters the picture. The study of networks is a branch of graph theory, but it is also a pilar stone in sociology. Paul Erdős, Alfréd Rényi, and Vera T. Sós published what is called the <i>friendship theorem</i> in 1966. Though it relates directly with graph theory, it can be described in terms of friends and strangers — hence the name.<br />
<br />
It goes something like this. Imagine you're at a party and there is an unknown number of people attending. If you choose any pair of people at that party and it always happens that they share one and only one common friend, then there must be one person at the party which is friends with everybody. I would suppose this person to be the host of the party — also called the politician.<br />
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<img border="0" height="300" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEibnqpk-2fW6kIh9zeTPuqEol0EgXeOui-Vg1pzvy9Nt1ncY625yoR6fdem5WuWQW5m7X9piQgDJJuf6vvcHzlbg94vldZS2wPSFnK2R_YyGNB2visNzbN61lAgASafKKm7wt1aM-LOHQs/s400/Untitled-2.png" width="400" /></div>
<div class="separator" style="clear: both; text-align: center;">
Figure 2 — Representation of the friendship theorem with 9 people.</div>
<br />
Look at the Figure 2 and you can easily see that each individual shares exactly one common friend with a different individual. You, the host, lie at the center and are friends with everybody while still fulfilling the common friend property.<br />
<br />
There is another theorem in mathematics and it can be applied to the same line of thought we've been through. It's called the <i>theorem on friends and strangers </i>— a simple case of a more general theorem published by Frank P. Ramsey in 1930.<br />
<br />
Let's revisit the party scenario, but allow for only 6 people to attend. If we consider any 2 of them at random, they might be meeting for the first time — mutual strangers — or they might already have met before — mutual friends. What the theorem states is that there is always a group of at least 3 people at a party of 6 which are either all mutual strangers or all mutual friends among themselves.<br />
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<img border="0" height="300" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj41DW0d2vd7q9GFCIA_p6DiTYMPqvAVwKNvIRmScgdaSKvY2ZS7v3BQoMkcVvIGBlbvIPgDb45K1PVYlhamEcHMdoQkfMuHGSGMpxXcDmV79TYsjibKZG9HZNc4bPM9sdWdQJkDB5Mv8s/s400/Untitled-3.png" width="400" /></div>
<div class="separator" style="clear: both; text-align: center;">
Figure 3 — Possible connections between 6 people.</div>
<br />
If you see the Figure 3, we've a diagram that displays the total `C(6,2)=15` possible different connections among them. I didn't specify whether they're friends or strangers and that's why the lines are coloured black. If you try to create any friend or stranger connections among them, for example, by colouring them green and red respectively, you'll find that it's impossible not to create triangles. These triangles — either green or red — are the groups of 3 mutual strangers or friends the theorem mentions.<br />
<br />
We were able to calculate the total amount of connections among 6 people. Can we do it for the entire world population? Yes! We need just to use the `n` combinations of 2 where `n` is the total world population. Thus, `C(n,2)=1/2(n-1)n` is a positively growing second degree polynomial function of `n`.<br />
<br />
I should emphasize that this number means the total connections you have with all the people in the world either as friends or strangers. The number of possible ways you can <i>colour</i> each connection is much greater!<br />
<br />
Stanley Milgram conducted several experiments to determine the average path length for social networks. That is, the average number of people needed to go through to reach any given person within the social network. Imagine you want to meet someone who's friends with your best friend. The path length between you and that person is 2. You need to go to your best friend and only then can you reach the other person you want to meet. The result from what is known as the <i>small-world experiment</i> was that the average path length was about 6.<br />
<br />
This result would become associated with the six degrees of separation theory, a concept originated by Frigyes Karinthy in 1929. It states that you can reach any person within a network with 6 or fewer <i>friend of a friend</i> chains.<br />
<br />
Interestingly enough, a mathematitian called Manfred Kochen published a paper in 1979 entitled <i>contacts and influences</i> in which he stated the following:<br />
<blockquote class="tr_bq">
We noted above that in an unstructured population with `n = 1000` it is practically certain that any two individuals can contact one another by means of at least two intermediaries. In a structured population it is less likely, but still seems probable. And perhaps for the whole world’s population probably only one more bridging individual should be needed.</blockquote>
joão pestanahttp://www.blogger.com/profile/03616394334438513692noreply@blogger.comtag:blogger.com,1999:blog-9153823946796270764.post-77284078764395773212015-10-30T00:14:00.000+00:002015-11-13T01:17:50.581+00:00If you know how to add, you know (almost) everything about mathematics<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj-qeiT_PGquVPD4Papd83ai6fQdsseSUJvlX8KnWL8TccdkLSJzygQtWXf1iW3dYXusXiDelKA7T_g1huGvvjSU3CTZuudu4qbONHBEJ6H13GaZfpYFiPxKS2llsnpgHYOj6V8ybXrHiA/s1600/E13520154A.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="256" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj-qeiT_PGquVPD4Papd83ai6fQdsseSUJvlX8KnWL8TccdkLSJzygQtWXf1iW3dYXusXiDelKA7T_g1huGvvjSU3CTZuudu4qbONHBEJ6H13GaZfpYFiPxKS2llsnpgHYOj6V8ybXrHiA/s640/E13520154A.jpg" width="640" /></a></div>
<br />
Computers are essentially glorified calculators and all they know is how to add numbers in binary. From there, it's easy to do multiplication — or glorified adding —, subtraction and division — or glorified subtraction. To do a multiplication, you just add the same number over and over the times that you want. To do a division, you do the same procedure, but you remove a certain number over and over until you reach the point when, if you remove it one more time, your result becomes negative. You count the number of times that you did it and the remainder is... well, the remainder. But can we really subtract by adding? Yes! All the mechanical calculators did it — something called <i>the method of complements</i>.<br />
<br />
How does it work? It's very simple. Imagine you have two numbers `X` and `Y` and you want to subtract `Y` from `X`. Each number has a certain number of digits, lets say `N` and `M` respectively. So our numbers are `X = x_1 x_2 x_3 ... x_N` and `Y = y_1 y_2 y_3 ... y_M` where `x_i,y_i in {0,1,2,3,4,5,6,7,8,9}`.<br />
<br />
For example, if `X=1954` it means `x_1=1`, `x_2=9`, `x_3=5` and `x_4=4`. I want to subtract from that the number `Y=1756` which has the digits `y_1=1`, `y_2=7`, `y_3=5` and `y_4=6`. Usually, we'd do something like `X-Y=1954-1756=198`. We already know our final answer so now we just need to reach it using only the operation of addition.<br />
<br />
To do this we need to create `Z` as the complement of `Y`. What this means is that for every `y_i` ranging from `i=1` to `i=M-1`, we write down `z_i` as the distance from that digit to `9` and for the last `y_M` we do just the same thing for `z_M` but this time it's the distance to `10`. In our example, the complement of `y_1=1` is `z_1=8`, `y_2=7` is `z_2=2`, `y_3=5` is `z_3=4` and finally for `y_4=6` is `z_4=4`. Now we have our `Z=8244` as the complement of `Y`.<br />
<br />
I added this paragraph some time after writing the article to include a suggestion made by a friend of mine. He pointed out to me that this complement was a subtraction and I agree, but I refuse to call it as such. What I told him is that it's a fixed correspondence. A `1` will always be replaced by an `8` at any digit except the least significant one and will be replaced by a `9` at the later. We can work out the correspondence table 1 for future use.<br />
<br />
<table>
<tr>
<td>Original digit `y_i`</td>
<td>Most significant digits `z_1 ... z_(M-1)`</td>
<td>Least significant digit `z_M`</td>
</tr>
<tr>
<td>0</td>
<td>9</td>
<td>0</td>
</tr>
<tr>
<td>1</td>
<td>8</td>
<td>9</td>
</tr>
<tr>
<td>2</td>
<td>7</td>
<td>8</td>
</tr>
<tr>
<td>3</td>
<td>6</td>
<td>7</td>
</tr>
<tr>
<td>4</td>
<td>5</td>
<td>6</td>
</tr>
<tr>
<td>5</td>
<td>4</td>
<td>5</td>
</tr>
<tr>
<td>6</td>
<td>3</td>
<td>4</td>
</tr>
<tr>
<td>7</td>
<td>2</td>
<td>3</td>
</tr>
<tr>
<td>8</td>
<td>1</td>
<td>2</td>
</tr>
<tr>
<td>9</td>
<td>0</td>
<td>1</td>
</tr>
</table>
<div style="text-align:center;">Table 1 — Correspondence table for the most and least significant digits of Z.</div>
<br />
As a final step, to conclude our subtraction by addition, we need to add `Z` to `X` and ignore the first digit — which will be a `1`. That is, `X+Z=1954+8244=10198` and by ignoring the first `1` we get `198`. If you recall, it was our answer!<br />
<br />
Why does this work? This exploits a weakness in the machines that we build. In this sense, we can think like the machine and achieve the same result. Imagine that we have a calculator which can only display 4 digits. Our calculator can display exactly 10000 different numbers ranging from `0000` to `9999`. What would happen if we were to add `1` to `9999`? The answer is 10000, but the display can only show the last 4 digits so it discards the first and we'd see the result as `0000`.<br />
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<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgjhPXYDnc3wZrfZ7re4PuUT_xQKcyyHHfjClQxAOzV_t5L_zkCPQ_k39_mDtoAzlQfBpYoc42NCvvAifs05QebFxJAAcclv5MEYFExQVt7nC2rCphEtySHl4DBC6hXXq31XGOJ3k3D8A0/s1600/1446181559_calculator.png" /></div>
The calculator has a limit and we cannot use it to perfom calculations that will result in a number that's larger than 9999, because an <i>overflow</i> will occur and we'll only be able to see the last 4 digits of the result. If we were to perform `9999+9999=19998` in our special 4-digit calculator, we'd see `9998` as a result and be surprised at it — by adding two large numbers we got a smaller one!<br />
<br />
What we're actually doing is forcing the calculation to <i>overflow</i> exactly up to the point of the difference between `X` and `Y`. You can think of the complement `Z` as what it would take to make the calculator overflow — or reset — starting at `Y`. So whatever you add to it will yield the difference, because in our calculator `Y+Z=1756+8244=0000` (remember we disregard the first 1).<br />
<br />
I can give you another visual example of how this is supposed to work. Imagine that we have 3 identical glasses over some large container. The first is filled with some water and the second is also filled with water, but less than the first one.<br />
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<table>
<tbody>
<tr>
<td><div class="separator" style="clear: both; text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMYs3YYoZRRdt6G75lwxRL3y4DErcbANU5kr4ctmJHep-Nm5P-PNdaaIZDP0XraytGQcVDE1UrHapErfT7p8FxcBqJhypCq_Baqo4T2nKsaKXANFGDkjDBjZVFF0dLFvGVv7OewNtm_Hw/s1600/1.png" /></div>
</td>
<td><div class="separator" style="clear: both; text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiGIltEwOmIi0_FPqbTgVwDaYAgpz30usbRArONHReXYviOipVOfgBfeEW_IeLZppqXO6eK7yrC8fLvWGse81YHC_ayNYPFz5gYV4H-WSUiD48JnP8pB9YfFEuFq7Zg-VANuG7HHkBkpS0/s1600/2.png" /></div>
</td>
</tr>
<tr>
<td><div style="text-align: center;">
`X`</div>
</td>
<td><div style="text-align: center;">
`Y`</div>
</td>
</tr>
</tbody></table>
You want to find out the amount of water that the first glass has more than the second one. How would you do that? That's when the third glass enters the picture.<br />
<br />
<table>
<tbody>
<tr>
<td><div class="separator" style="clear: both; text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgQRm6zAcFVpuYNe6HtqhNs-rKQ7LNpfrUo-Lap1HGvRDTT9rrlprkNRRf9MfMsGFSTXaWuAxPbEhP6zQoXpjlhVKKXpboksp2OaRVZxGJqTIVgsN8drlmoN0nq2WeZDmq2hBFeo8WQIKQ/s1600/3.png" /></div>
</td>
</tr>
<tr>
<td><div style="text-align: center;">
`Z`</div>
</td>
</tr>
</tbody></table>
If it would so happen that it contained the exact amount of water that would fill up the second glass to the top and you were to pour ir over the first glass, you'd see the exact difference between the first and the second spill out into the larger container.<br />
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<table>
<tbody>
<tr>
<td><div class="separator" style="clear: both; text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjA-xOc-fKEd430sy80OvqPS1qANvWGap0BIp34fPoK26xuiAvPestUPTGUcJQrM33-ukepKdvHFyt2YJJdlBTBqxqBZEzwJBd7iVGTYDpxeZDnpEtzZ0UM8I91f3wKMU84QWzGC1b3Vls/s1600/5.png" /></div>
</td>
<td><div class="separator" style="clear: both; text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj93_6TV1r8XFfAWn5WsnMgG1aV3bnUlEn52NBO-5xrsOKM9i8qyPdrsZvl1aocLVMCdgHZaVRKzlAEGsVfdkbenrQsih7-cwwwgUbHYEAyayfwyRFuAtAGKhNXUA6WnwY37bwZlasBA_M/s1600/4.png" /></div>
</td>
</tr>
<tr>
<td><div style="text-align: center;">
Full glass</div>
</td>
<td><div style="text-align: center;">
`X-Y` (<i>spillage</i>)</div>
</td>
</tr>
</tbody></table>
From there, you'd just have to collect and measure the spilled water. It's simple! Now... Could you actually do it using only glasses of water? Here are some rules. You can somehow precisely collect the spillage. You can use how many glasses you need. You have no way of measuring water so you can only rely on overflows.joão pestanahttp://www.blogger.com/profile/03616394334438513692noreply@blogger.comtag:blogger.com,1999:blog-9153823946796270764.post-48127974659073174352015-10-24T21:46:00.004+01:002015-10-25T12:20:11.315+00:00On the probability of losing the EuroMillions transnational lottery<div class="separator" style="clear: both; text-align: center;">
<img border="0" height="256" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhhdgT9Udqq8s-SA9CYSfRj4pqQeRjQWRW7t_9hhO7jonL5ZR5IRfRmAnNTszcls4Uzad2t4VynyILUj1c6tlNO9h-hkGEP9t5EW6Pl-gPqdkdZC4hKj2600ZqVlwZGFANxwwG9a0uJEIU/s640/EFU9W9R2CN.jpg" width="640" /></div>
<br />
I want to talk about probability now. You cannot predict an outcome nor expect some sort of behaviour within some pattern based on past information. All you can do is estimate the likelihood of a certain event, but even if you can guarantee 99.9% of certainty, there is still the missing 0.1% playing against you. Probability is a way of organizing randomness.
<br />
<br />
The classical example is that of a perfectly balanced dice — a perfectly crafted cube marked with the number from 1 to 6 on each of its sides. If you roll and throw the dice, you do not know which number you will get faced up. You do know that you will get a number from 1 to 6 and that, because it's perfectly balanced, each number has an equal likelihood of coming up.
<br />
<br />
If we define 0 to mean that an event is impossible and 1 that some other event is certain, any number between 0 and 1 can only mean you are uncertain about the outcome of that event. This is called a probability and you can also imagine it as a percentage ranging from 0% to 100%. It means that if you repeat an event with a probability somewhere in between many, many times, you observe such event occurring with that frequency. The more times you repeat, the more this is true — something called the <i>law of large numbers</i>.
<br />
<br />
Think about the dice again and you can guarantee that some number will come up, you just don't know which one. We can add up all the probabilities to one, because one of them must occur for certain.
<br />
<br />
<div style="text-align: center;">
`P(1)+P(2)+P(3)+P(4)+P(5)+P(6)=1`</div>
<br />
As I said before, for a perfectly balanced dice, the probability of each number is the same. We can put it this way `P(1)=P(2)=P(3)=P(4)=P(5)=P(6)=P(n)` where `n` is an element of the set of possible results `N={1,2,3,4,5,6}` which means that `P(n)=1/6=0.1666...` or the probability of getting any particular number is about 16.7%.<br />
<br />
Let's go back to our subject. If the probability of rolling a dice and getting a 4 is 16.7%, what is the probability of not getting a 4? I believe you know the answer. Intuitively, you say that it's the probability of getting any number except 4, but instead of adding up all those probabilities we might do something smarter. Since we know that some number must come out, the probabilities all sum up to 1. If we remove from that the probability of getting a 4, we get the probability of getting any number except for 4. Put in another way, it's about 83.3%.<br />
<br />
<div style="text-align: center;">
`P("not "4)=1-P(4)=5/6=0.8333...`</div>
<br />
Can we do something more interesting that just playing with the odds of a regular dice? Of course! I want to know the probability of not winning the EuroMillions' first prize — a transnational lottery between several nations of Europe.<br />
<br />
Rather that add up all the immense possibilities of losing, I'm going to do the same thing I did before. If we find the probability of winning, all we have must do is subtract it from the certainty. The rules of the game are:<br />
<br />
<ol>
<li>The player must choose any 5 main numbers in the range of 1 to 50 and</li>
<li>Choose 2 more numbers, called stars, in the range of 1 to 11.</li>
</ol>
The important here is that the numbers do not repeat themselves and the order in which they come out doesn't matter, unlike in permutations. For small numbers, we may be able to count all the unique outcomes from a set of possible events regardless of the order in which they happen. Imagine that we draw simultaneously two balls at random from a jar of balls numbered 1 to 3 and add their values together. We can expect to draw the numbers `{1,2}`, `{1,3}` or `{2,3}`. Even if they come out as `{2,1}`, `{3,1}` or `{3,2}` it won't matter. It's the same result, because we get the values 3, 4 or 5, respectively in either case. That is, there are 3 unique possible results for the draw. How can we count this for larger combinations?<br />
<br />
If we count as 3 possibilities for the first ball and the remaining 2 for the second ball, we have a permutation with `3xx2=6` different outcomes — the order will matter. We must remove the repeating results. For this we have 2 different orders of each set of results so we divide the previous result by the number of repeating orderings and we get what we want. Luckily, there is a simple mathematical expression for this.<br />
<br />
<div style="text-align: center;">
`C(n,k)=(n!)/(k!(n-k)!)`</div>
<br />
The `C` stands for combinations and `!` means the factorial product `n! = n xx (n-1) xx (n-2) xx ... xx 2 xx 1`. From a set of different `n` elements we draw `k` elements at once and this expression gives us the total number of possible combinations regardless of the order in which they come out. For the previous example, we'd use it as `C(3,2)=(3!)/(2!*(3-2)!)=(3xx2xx1)/(2xx1xx1)=3`.<br />
<br />
For the lottery in question, we have a total of `C(50,5)=2" "118" "760` unique possibilities for the main numbers and `C(11,2)=55` different combinations of stars. To the find the total number of different draws we just multiply the two to obtain `2" "118" "760 xx 55 = 116" "531" "800`. This means that the probability of winning the jackpot is simply `1/(116" "531" "800)=0.0000000085813...` which means that the probability of not winning the jackpot is about `99.999999141865%`.<br />
<br />
Now I know you're thinking that there are more prizes besides the jackpot, but everyone wants the big one! Nevertheless, there is a simple formula that we can use to find the probability of getting any of the lesser prizes.<br />
<br />
<div style="text-align: center;">
`1/(P(m,s))=(C(n,k))/(C(k,m)C(n-k,k-m)) xx (C(z,t))/(C(t,s)C(z-t,t-s))`</div>
<br />
Where `n=50` is the total number of main numbers and `z=11` is the total number of star numbers, `k=5` is the number of choices for the main numbers and `t=2` is the number of choices for the star numbers. The only variables we are left with are `m` and `s` which are the number of main numbers and stars that we got right, respectively. So we get a nicer formula for `m in M={0,1,2,3,4,5}` and `s in S={0,1,2}`. I'm presenting all the information in Table 1.<br />
<br />
<div style="text-align: center;">
`1/(P(m,s))=(C(50,5))/(C(5,m)C(45,5-m)) xx (C(11,2))/(C(2,s)C(9,2-s))`</div>
<br />
<table id="table_post">
<tbody>
<tr>
<td>`m`</td>
<td>`s`</td>
<td>`1/(P(m,s))`</td>
<td>Prize distribution</td>
<td>Average Prize</td>
</tr>
<tr>
<td>0</td>
<td>0</td>
<td>2.6</td>
<td>0.000</td>
<td>0 €</td>
</tr>
<tr>
<td>0</td>
<td>1</td>
<td>5.3</td>
<td>0.000</td>
<td>0 €</td>
</tr>
<tr>
<td>0</td>
<td>2</td>
<td>95.4</td>
<td>0.000</td>
<td>0 €</td>
</tr>
<tr>
<td>1</td>
<td>0</td>
<td>4.3</td>
<td>0.000</td>
<td>0 €</td>
</tr>
<tr>
<td>1</td>
<td>1</td>
<td>8.7</td>
<td>0.000</td>
<td>0 €</td>
</tr>
<tr>
<td>1</td>
<td>2</td>
<td>156.4</td>
<td>0.065</td>
<td>10.82 €</td>
</tr>
<tr>
<td>2</td>
<td>0</td>
<td>22.8</td>
<td>0.180</td>
<td>4.07 €</td>
</tr>
<tr>
<td>2</td>
<td>1</td>
<td>45.6</td>
<td>0.176</td>
<td>8.08 €</td>
</tr>
<tr>
<td>2</td>
<td>2</td>
<td>821.2</td>
<td>0.023</td>
<td>20.26 €</td>
</tr>
<tr>
<td>3</td>
<td>0</td>
<td>327.0</td>
<td>0.037</td>
<td>12.26 €</td>
</tr>
<tr>
<td>3</td>
<td>1</td>
<td>653.9</td>
<td>0.022</td>
<td>14.73 €</td>
</tr>
<tr>
<td>3</td>
<td>2</td>
<td>11 770.9</td>
<td>0.005</td>
<td>64.19 €</td>
</tr>
<tr>
<td>4</td>
<td>0</td>
<td>14 386.6</td>
<td>0.007</td>
<td>105.57 €</td>
</tr>
<tr>
<td>4</td>
<td>1</td>
<td>28 773.3</td>
<td>0.007</td>
<td>213.39 €</td>
</tr>
<tr>
<td>4</td>
<td>2</td>
<td>517 919.1</td>
<td>0.008</td>
<td>4 813.08 €</td>
</tr>
<tr>
<td>5</td>
<td>0</td>
<td>3 236 994.4</td>
<td>0.016</td>
<td>80 099.60 €</td>
</tr>
<tr>
<td>5</td>
<td>1</td>
<td>6 473 988.9</td>
<td>0.048</td>
<td>468 741.33 €</td>
</tr>
<tr>
<td>5</td>
<td>2</td>
<td>116 531 800.0</td>
<td>0.320</td>
<td>46 466 582.27 €</td>
</tr>
</tbody></table>
<div style="text-align: center;">
Table 1 — Odds of winning, prize distribution and average prizes for the EuroMillions lottery.</div>
<br />
The value `1/(P(m,s))` should be interpreted as <i>approximately a 1 in</i> probability. The average prize displayed was taken from the <a href="https://www.euro-millions.com/odds-of-winning" target="_blank">euro-millions.com</a> website for the period between 10/05/2011 and 23/10/2015. If you add all the distributions, there is a 0.086 missing that is intended for the <i>booster fund</i>. As you can see, both `1/(P(m,s))` and the average prize grow extremely fast, so I took the <i>natural logarithm</i> — a very, very slow growing function — of each and plotted them against each other.<br />
<div class="separator" style="clear: both; text-align: center;">
<img border="0" height="300" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj1_FEHb77yCeBdYiLFDIgBh4kI51AFFJ36N8rDRjnFysytBf7yE2cQ_uF4AycURlylMfrbPRHRyn_68Xws5GzAc7-riHMnUOj6H2yRTeJEf38pHMrNXF66L0LPhyb_tGfe2mFIAtWWzU0/s400/figure_1.png" width="400" /></div>
<div style="text-align: center;">
Figure 1 — Prize and probability plot
</div>
<br />
As we expected, visible in Figure 1, the prize grows as the probability of losing increases. Even if we add up all the probabilities of hitting any of the prized combinations, that only amounts to about 7.81% leaving you with an astonishing 92.19% probability of remaining empty handed.joão pestanahttp://www.blogger.com/profile/03616394334438513692noreply@blogger.comtag:blogger.com,1999:blog-9153823946796270764.post-60065203018404431012015-10-22T00:12:00.000+01:002015-10-23T21:38:26.956+01:00How long would it take to reach the other side of the world?<div class="separator" style="clear: both; text-align: center;">
<img border="0" height="256" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEijchwDqPQPsUWIfJDbePnj-bpmqcBcv_dplhz6VyIvUOS3bCINu9H844x4GYeWiQkjWOIaKpZ0M033D7Gl44eJfPcbowqn_IIrlp8fc_8tNbw41WQD_EimXpA74rPoz04IBTSt-zEMKQQ/s640/K8W8LSVOVI.jpg" width="640" /></div>
<br />
Imagine that there was a straight tunnel passing right through the center of the Earth and it was large enough for you to fit in. For how long would you fall if you jumped right into it? The answer I'll provide might surprise you and I'll try to keep it as simpler as possible, but not simpler.<br />
<br />
What are the ingredients we need to cook up our answer? The most important one is imagination followed closely by intuition and, at a close third, some mathematics. Let us assume there is no air resistance, just to make things easier. The more real situation just implies a terminal velocity that would make the trip longer.<br />
<br />
It might seem plausible to expect that one accelerates until reaching the core of the planet and then decelerates for the rest of the journey, coming to a halt at the very edge of the other side of the planet. In this scenario, the highest value for the velocity would be achieved at the very center when the acceleration drops to zero. Doesn't this description resemble that of an oscillating spring?<br />
<br />
We're going to need a few tools from the trade such as Sir Isaac Newton's law of universal gravitation which states the following:<br />
<blockquote class="tr_bq">
Any two bodies in the universe attract each other with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.</blockquote>
What does this mean? It just says that if any object has a mass, than that object attracts other objects that have mass as well and that the closer they are to each other, the greater their attraction becomes. For example, when our moon was formed it used to be much, much closer to the Earth than it is now. Not only it appeared bigger and brighter in the sky, but the tides that it caused on our planet were enormous. Since then it has been slowly drifting away. The tides run shorter, the days grow larger and the skies become dimmer. This is all due to the fact that distance plays a very important role in the force of gravity — even bigger that the mass of the objects!<br />
<br />
If the mass of one object doubles then the gravitational force that that object exerts over others doubles as well — this means it's <i>directly proportional</i> —, but if the distance between both objects doubles then the force of attraction does not fall to a half... It falls to a quarter of what it was initially! This is what <i>inversely proportional to the square of the distance</i> means.<br />
<br />
So far, we've been describing a physics problem without recurring to mathematics. Unfortunately, we cannot go on like that <i>ad aeternum</i>. Let's us start by simply writing down Newton's law as many will easily recognize:<br />
<br />
<div style="text-align: center;">
`F_(gravity)=G(Mm)/r^2`</div>
<br />
The force `F` increases as any of the masses `M` (the Earth) or `m` (you) increases and decreases as the distance `r` increases. `G` is just some number that makes everything come out right in the units that we use daily to measure things such as distances and masses. Of course — and this is important — if we can write down the mass of an object, we can also define it's density `rho` as the ration between its mass and its volume `V` such that `M = rho V`.<br />
<br />
Why is this important? Because one of our biggest assumptions in order to solve this problem is that the density of Earth is constant — which is not. Can you remember how to calculate the Volume of a sphere? We're also assuming the Earth to be perfectly spherical — which again it's not.<br />
<br />
<div style="text-align: center;">
`V_(sphere) = 4/3 pi r^3`</div>
<br />
The next ingredient that we need also comes from Newton and is his second law of motion which was translated in 1729 by Andrew Motte as:<br />
<blockquote>
If a force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively. And this motion (being always directed the same way with the generating force), if the body moved before, is added to or subtracted from the former motion, according as they directly conspire with or are directly contrary to each other; or obliquely joined, when they are oblique, so as to produce a new motion compounded from the determination of both.</blockquote>
Imagine that you and your friends form a circle around a table and there is a blanket on top of it. It's very cold and everyone wants the blanket for oneself so every person starts to pull it their way. Nobody moves, because everyone is pulling in opposing directions and cancel each other out. Imagine now that your car won't start and you and your friends have to push it. You can only do this if everyone pushes in the same direction. Essentially, this is what Newton's second law of motion states — a change in the motion of an object is the result of all the combined `N` forces that act upon it within a given interval of time.<br />
<br />
<div style="text-align: center;">
`sum_(i=1)^N F_i Delta t = Delta p`</div>
<br />
A change in motion can also be stated as a change `Delta` in the momentum `p` of an object. That is, the <i>will</i> of an object to keep moving. Imagine a big truck that is at rest (not moving). Its <i>will</i> to move is none. Unless something acts upon it, it will remain at rest — Newton's first law of motion. If, on the other hand, it's going at a great velocity, then its <i>will</i> to keep moving is very large and it'll be very hard for you to force it to come to a halt. This is not only due to its velocity `v`, but also to its incredible mass `m`.<br />
<br />
The momentum of an object is proportional to both such that `p=mv`. Since the mass of an object can be considered constant, this implies that a change in momentum is due to a change in velocity or `Delta p = m Delta v`. The change in the velocity is what we usually call the acceleration `a`. Since there is only one force on our problem — that of Earth's gravity — we can put it simply as:<br />
<br />
<div style="text-align: center;">
`F_(gravity) = G(Mm)/r^2 = ma`</div>
<br />
Notice that the force of Earth's gravity is acting upon you and that force is the only source of change in your velocity — your acceleration. If you recall basic equation mathematics, you can cancel all the common terms that appear on both sides. In this case, we can cancel your mass — yes, the result is independent of it! This means that the travel time from one side of the planet to the other is the same for every single person. We're left with an expression for the acceleration that the Earth causes upon you:<br />
<br />
<div style="text-align: center;">
`a = G M_(Earth)/r^2`</div>
<br />
I'm now going to move on with the mathematics and feel free to skip this part if you're not inclined to follow through. It's actually not very important. We just have to replace Earth's mass with the statement that defines it as the product of its density and its volume.<br />
<br />
<div style="text-align: center;">
`a = G (rho V_(Earth))/r^2 = G 4/3 (rho pi r^3)/r^2 = 4/3 G rho pi r`</div>
<br />
I should mention in case you're wondering that `r` is the distance between you and the center of the planet and is our variable. It changes as you fall through the tunnel. We can do this, because it was also shown by Newton that any object with mass can be modelled as a single point with the same mass, placed at its center of gravity.<br />
<br />
We shall also consider that same center as our point of reference which means that when you fall towards it, you go from a higher point to a lower one — implying that the change in your position is negative. That is, your velocity is negative and thus your acceleration is negative as well. Don't worry if you don't understand and just ignore the minus sign I'm going to use from now on.<br />
<br />
The rest follows from basic calculus — we need to solve a differential equation. From basic physics, we know that acceleration is a change in velocity which is itself a change in motion. If we consider a straight line along the tunnel as the `x` axis, we can write the above expression in terms of the position along that line as:<br />
<br />
<div style="text-align: center;">
`a = (Delta v ) / (Delta t) = (Delta ( Delta x ) ) / ( Delta ( Delta t ) ) = - 4/3 G rho pi x`</div>
<br />
Let us just call `k^2` to the product of all that is constant and we make the expression much nicer. If `k^2=4/3 G rho pi` we can call upon our differential language and write everything simply as:<br />
<br />
<div style="text-align: center;">
`(d^2x)/dt^2 = - k^2 x`</div>
<br />
Which you might recognize as the equation that describes the harmonic oscillator. Your intuition that you would just bounce back and forth along the tunnel was right after all! The solution for that equation is an oscillating one — any sine, cosine or a combination of the two will do. Since we considered the center as our reference, we must use the cosine, because it starts at 1 and halfway through its period intercepts the zero point. Your position as you fall through the tunnel is given by:<br />
<br />
<div style="text-align: center;">
`x(t) = R cos(kt)`</div>
<br />
Where `R` is the radius of the Earth — your starting point relative to its center. How do we get to the travel time? We've all that we need right now. We just have to find the time `t` for which `x(t)` is zero and double that, because from `R` to `0` is just halfway through your epic journey!<br />
<br />
If you don't recall your trigonometry classes, a cosine function is null when its angle is a quarter of a circle — that is, `90` degrees or `pi/2` radians. We simply need to make sure that its argument is equal to that value:<br />
<br />
<div style="text-align: center;">
`kt = pi/2`</div>
<br />
After we solve for `t` and properly replace `k` we get what we wanted all along. And remember that `k^2` was the product of the constants, so `k` is the square root of that.<br />
<br />
<div style="text-align: center;">
`t = pi/(2k) = pi/2 sqrt(3/(4 pi rho G )) = sqrt((3pi)/(16 rho G))`</div>
<br />
Since you're asking for actual values, we must lookup the experimental vales for the Earth's density `rho_("average")=5514 kgm^(-3)` and the gravitational constant `G=6.67408(31)xx10^-11 Nm^2kg^(-2)`. The answer is `t=1265` seconds which translates to 21 minutes and some milliseconds. It would take you that long to reach the core of our planet. If we double that amount, we know that it would take you 42 minutes to reach the other side of the world.<br />
<br />
On the <i>American Journal of Physics</i> 83, 231 (2015), Alexander R. Klotz wrote an article entitled <i>The gravity tunnel in a non-uniform Earth</i>. He uses a more realistic model which takes into consideration the non-constant density of the Earth as well as the fact that it's not perfectly spherical. His result is an interval between 38 and 42 minutes for any given point on the surface of our planet.
<br />
<br />
Now I know that I quoted Douglas Adams in my previous post and that the final answer here is 42, but there is no correlation whatsoever. Or is it?joão pestanahttp://www.blogger.com/profile/03616394334438513692noreply@blogger.comtag:blogger.com,1999:blog-9153823946796270764.post-88638890939933342022015-10-21T00:05:00.001+01:002015-10-21T20:39:53.253+01:00What is money?<div class="separator" style="clear: both; text-align: center;">
<img border="0" height="256" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi9-jAqQCmfCnM9TLjIn5i3OEij7lb_BdXiJnLeL4GQxmTMhrhcmEEwXqO83YGn9694DFw_r7jsZL9TfvbFHcTMYZ83r628EhUynJ7QelvlIRC5hIq1hyphenhyphenLlUPrtcKDLbh3YgjapcPFBR4s/s640/numbers-money-calculating-calculation.jpg" width="640" /></div>
<br />
During my first year as a student of the Master in Economics, I had a course on the History of the Economic Thought and one of the assignments was to write about <i>what is money?</i> I had a very low grade. Nevertheless, I want to share my insights with you. I firmly believe it can be instructive to anyone. I started with an abstract — as any good essay must — and a quote. I'm skipping the introduction and minor sections.<br />
<br />
<div style="text-align: center;">
<b>Abstract</b></div>
<br />
Everyone knows what money is, even if they know nothing of financial, monetary or basic economics. In this essay, I explore an overview of money from its origins as an almost natural evolution from the barter system to the practically invisible medium of exchange that it is today. Then, I give a brief description of the modern creation of money. Finally, I try to get a sense of what money means besides its nature as an object of trade.<br />
<br />
<blockquote class="tr_bq">
This planet has — or rather had — a problem, which was this: most of the people living on it were unhappy for pretty much of the time. Many solutions were suggested for this problem, but most of these were largely concerned with the movement of small green pieces of paper, which was odd because on the whole it wasn’t the small green pieces of paper that were unhappy.<br />
<br />
<div style="text-align: right;">
— Douglas Adams, The Hitchhiker’s Guide to the Galaxy (1979), Introduction.</div>
</blockquote>
<br />
<div style="text-align: center;">
<b>The barter system</b></div>
<br />
In a subsistence economy, each individual consumes what he produces himself. But what if someone wants what another individual produces? Let us imagine we are born into a society which uses a barter system in which undesired goods or services are traded directly between agents for other desired goods or services.<br />
<br />
At first, it might seem simply enough. In fact, it’s the most basic system and — even in commercial economies — it happens all of the time. Individuals often trade commodities directly with each other, mostly between friends and family, but not restricted to those. Over the internet, an individual will easily find another which is interested in some direct trade.<br />
<br />
A barter system may be best indicated for closed communities that are self-sufficient, but it becomes rapidly cumbersome in a larger society. Let us suppose there is a village with five people and their possessions and desires are described in Table 1.<br />
<br />
<table id="table_post">
<tbody>
<tr>
<td><b>Name</b></td>
<td><b>Possession</b></td>
<td><b>Desire</b></td>
</tr>
<tr>
<td>John</td>
<td>Cows</td>
<td>Honey</td>
</tr>
<tr>
<td>Peter</td>
<td>Honey</td>
<td>Carrots</td>
</tr>
<tr>
<td>Sara</td>
<td>Chickens</td>
<td>Fish</td>
</tr>
<tr>
<td>Kate</td>
<td>Carrots</td>
<td>Milk</td>
</tr>
<tr>
<td>Adam</td>
<td>Fish</td>
<td>Pork</td>
</tr>
</tbody></table>
<div style="text-align: center;">
Table 1 — Description of the possessions and desires of some individuals.
</div>
<br />
Suppose now that Peter wants some carrots and Kate is the one who has some to trade, but she has no desire for the honey that Peter has. How can Peter acquire the carrots? He must trade his honey for milk with John and only then can he trade the milk for Kate’s carrots. But think about Sara who whats fish but Adam, who has fish, doesn’t want Sara’s chickens. Sara would need to search for some pork to trade with Adam and that could take a long time — long enough that Adam might have no fish left and must wait for the next fishing season.<br />
<br />
Sara’s solution would involve being in debt with Adam and promise to repay him at a later time — this is the basis of all modern currency, everyone is always in debt with everyone else.
The barter system is thus very restrict and does not allow for many things such as the measure of value and accumulation of wealth. In the first case, when Peter
acquired milk to trade with Kate, milk had become commodity money.<br />
<br />
<div style="text-align: center;">
<b>Commodity money</b></div>
<br />
In pre-revolutionary America, beaver fur, wampum, fish, rice and, in particular, tobacco have been used as commodity money, according to Rothbard (2002).
Another famous example is the economy of a prison, where there small society of inmates must abide to the laws of the prison and commodity money is a vital part of that. In Prisoner of War (POW) camps, Radford (1945) documented cases of cigarette currency that was subject to Gresham’s law, inflation, and especially deflation.<br />
<br />
Commodity money are goods that can be used as a medium of exchange, but also hold value in themselves and can be consumed. Some examples include gold, silver, copper, salt, shells, alcohol, cigarettes, cannabis, candy and cocoa beans. The individual recognises the utility in the commodity as constant and it gains several advantages over the barter system, because it creates a benchmark.
Take the economic principle of metallism, a term coined by Georg Friedrich Knapp to describe a monetary system in which the money derives its value from the type and quantity of metal of which it’s made.<br />
<br />
For example, a gold coin holds the same value either in coin form or if it’s melted. On the other hand, Friedrich Freiherr von Wieser says that even metallism isn’t as simple as it seems, because the actual value of the commodity on which the money is based is a duality between its industrial value and its exchange value, according to Von Mises (1954).
In January, 1999, Portugal was one of the first countries to adopt the new Euro currency. Even if coins and bills only began to be physically exchangeable in January, 2002, the Euro was already used in non-physical money exchange.<br />
<br />
Throughout the history of civilization, there have been several attempts to unify several countries into a single currency. Europe is no exception, with the 19th-century attempt called the Latin Monetary Union (LMU). According to Bae and Bailey (2011), the first treaty was signed between France, Belgium, Italy and Switzerland in 1865. Even if, at first, accomplished very little, it lasted for 60 years
and reached an end in 1925.<br />
<br />
<div style="text-align: center;">
<b>The gold standard</b></div>
<br />
The LMU was a gold standard monetary system in which the government sets a fixed conversion ratio for a direct conversion between currency and gold. The LMU began as both a gold and silver standard. Afterwards, the decreasing value in silver made the Union a <i>de facto</i> gold standard. This type of monetary system, even though it uses commodity money, has a fixed conversion ratio which makes it vulnerable to abuse. The actual value of the metal from which coins are made has fluctuations that depend on the overall available amount. As such, the value stamped may not be a true representation of the coin’s metal value.<br />
<br />
If, for example, there is a large supply of gold, the value stamped may be higher that the intrinsic metal value per its weight. According to McLeay et al. (2014b), 16th-century Spain experienced this problem of high inflation after the imports of large amounts of gold and silver from the newly discovered Americas.<br />
<br />
The LMU and other monetary systems alike are subject to attempts of abusing the system. Countries could, in theory, produce metal coins of a reduced purity and then trade them for standard coins, thus making a profit and, in fact, the LMU experienced problems like these, as described by Bae and Bailey (2011).
Despite the clear advantages of the gold standard, specially for unions like the
LMU that benefit from fixed international exchange ratios, nearly all of the modern
monetary systems are based on fiat money.<br />
<br />
<div style="text-align: center;">
<b>Fiat money</b></div>
<br />
Fiat money is money that is based on faith and is irredeemable. More precisely, its value is totally determined by some authority and has no use other than as a medium of exchange. Even metal coins can be fiat money, if its printed face value does not meet its real metal value. For that reason, it has a high risk of becoming worthless due to hyperinflation.<br />
<br />
According to Hanke and Kwok (2009), the first recorded hyperinflation occurred during the French Revolution — monthly inflation reached values as high as 143 percent — and the 20th century witnessed 28 hyperinflations — mostly due to the world wars and the communist era. One of the most recent and known cases is the Zimbabwe’s hyperinflation — it has become the first hyperinflation case of the 21st century — of 2007-2008, reaching peak monthly rates of 79.6 billion percent — despite this astonishing value, Hungary still holds the top position at the world’s hyperinflation
league table.<br />
<br />
The inflation came to an end when people stopped accepting the Zimbabwe dollar and it became worthless.
In general, if the issuing authority loses the ability to guarantee the value of its money, it loses all of its value. In reality, that may not always be the case, as Foote et al. (2004) noted. In northern Iraq, the Kurdish area used for its currency the so-called “Swiss dinars,” even though its value was not backed up by any authority.<br />
<br />
The regional government had no access to the Swiss dinars’ printing plates and it refused to adopt Saddam’s currency, thus fixing the supply of money. This gave the region some inflation stability, before the dinars began to fall apart due to overuse.
According to Schumpeter (1954), the idea of a monetary system based on paper
money and disconnected from gold can be traced to the ideas of David Ricardo (Proposals for an Economical and Secure Currency, 1816), but it has taken us quite some time to achieve that.<br />
<br />
<div style="text-align: center;">
<b>What is acceptable as money?</b></div>
<br />
We have been through an overview of what money is through a variety of situations, but money can be anything that someone accepts as payment for some good or service — it can even be just a debt, a promise of a future payment. McLeay et al. (2014b) suggest that money should meet three basic criteria, namely it should:<br />
<br />
<ol>
<li>Be a reliable <i>store of value</i>, some non-perishable good that retains its value over a reasonable amount of time;</li>
<li>Serve as a <i>unit of account</i>, that is, serve as an absolute quantizer for the prices of goods and services to be described in;</li>
<li>Be an accepted <i>medium of exchange</i> that people trust and hold on to in order to exchange it again later on.</li>
</ol>
As previously defined, this money can help resolve the problems of a barter system. Remember the village I described before. Sara wanted some fish from Adam, but she couldn’t afford it with the goods that she had nor by any kind of bartering with the rest of the people in the village. Her solution would be to enter in an agreement with Adam in which she would acquire a debt with him.<br />
<br />
Imagine now that, later on, Adam wanted some honey from Peter but Adam lacked the carrots that Peter wants. Adam could, for example, transfer Sara’s debt to Peter — because everyone trusts everyone in the village, Peter knows that Sara would always honor her debt, should he ever desire chickens. In this situation, Sara’s debt has become currency — an accepted medium of exchange that fulfills the three requirements mentioned above.<br />
<br />
So long as Sara lives and raises chickens, her debt will retain its value and it can even be used as a unit of account and a medium of exchange. Peter could say to Adam that 10 liters of his honey is worth 3 chickens from Sara. If Sara owed Adam for 10 chickens, he could split her debt with Peter and Sara would now owe Adam 7 chickens and Peter 3 chickens. This could carry on for a long time — between more people — and even though Sara backs up her debt, she hasn’t actually paid anything to anyone yet.<br />
<br />
This kind of debt exchange is behind the gears that drive the modern monetary system and what we are so used to call money.
Since modern currency is not tied to gold, the central bank won’t trade it for
anything other than more currency — should the banknotes suffer from wear and tear, for example. Money is just debt from the central bank to
the consumers.<br />
<br />
<div style="text-align: center;">
<b>The creation of money</b></div>
<br />
As McLeay et al. (2014a) put it, there is a misconception on how money is created. This misconception is as follows: every time someone makes a deposit in a bank, the bank uses it to make a loan for someone else.
In practice, thought, the loan is itself the creation of money and that power most often falls onto a commercial bank.<br />
<br />
This does not depend on the deposits that some particular commercial bank holds, but there are limits to the loans that it can make — or to the amount of money it can create. As I’ve explained before, modern money is debt, in this case to a particular bank. Money can thus be as easily destroyed as it can be created by using it to clear out the debt to the bank.
Bank deposits are debts from the bank to its costumers and bank loans are debts from the costumers to the banks.<br />
<br />
A bank cannot use its debt to someone to make a loan, it just creates a new deposit — with the same amount as the loan — and increases the overall available money — also known as broad money. The central bank then makes sure there is always enough money for everyone and supplies reserves to commercial banks as they provide loans.
If, in the village example, Sara were to pay her debt to everyone in the chickens that she owed, all money would be destroyed and the whole village would go back to the barter system.<br />
<br />
<div style="text-align: center;">
<b>The meaning of money</b></div>
<br />
Throughout this essay, I’ve been talking about what money is as a human creation in the economy. The criteria to define something as money is so flexible that most anything can be money. Money makes no sense without human society, be- cause it’s a consequence of the convergent ideas of different people. It may be hard for one person alone to fully understand the idea of what money is, because not
everyone sees it the same way. What is money? What does money mean to us?
<br />
<br />
To Mitchell and Mickel (1999), money is more than just an inert thing based on the idea of barter, but rather possesses subjective and affective meaning as well. This interpretation is not without foundation, because we are so used to hearing the stories of people who’ve done both amazing and horrible things in the name of money. Why is something not essential to our basic survival so important that it can even overcome our most primitive instincts?
<br />
<br />
Money was created, even if it seems to have spanned naturally, to ease trans- actions — both present and future. As a side effect of an accepted medium of exchange, that everyone takes for granted, people know that it is what everyone has come to value. It doesn’t matter where it came from: if you worked for it, inherited it or took it from someone else, because some merchant or bank will still accept it as credit — ignoring the fact that there are ways to discover if certain banknotes were stolen such as the serial number.<br />
<br />
Back to the Village example, imagine that Kate stole some of Sara’s debt from Adam. She would now have money to trade with someone else, ignoring that in such a small community she’d be easily caught.<br />
<br />
If we were to attribute an expected utility value to human life and money, there should be no argument against the fact that the first should be larger than the second or `E["human life"]>E["money"]`. The work of Fagley and Miller (1997) suggests that, when presented with a gamble, subjects were more risk adverse to- wards money and made riskier decisions when faced with human life. This is some- what counterintuitive, but it parallels with the findings14 that financial planners take more risks with their own money than with their clients’. We can put it as `E["own money"]>E["client money"]` and it seems that as the expected utility grows, so does the willingness to take risks.<br />
<br />
If we take a moment to think about it, one common idea is that wealthier people are less willing to part with their wealth than those who have substantially less. Whether this is true or a misconception is hard to tell. It remains to say that money means a lot to us. Even if we don’t understand it or its significance, we know its usefulness.<br />
<br />
<div style="text-align: center;">
<b>Final thoughts</b></div>
<br />
Money can have an overwhelming impact on our lives and we take great risks to have more of it. We have casinos and the lottery, because the idea of instantly becoming extremely wealthy is intoxicating to the point of making us forget our most basic survival instincts such as food and sex (Krueger, 1986 quoted by Mitchell and Mickel, 1999).<br />
<br />
Even if the concept of money emerged as a purely inert entity that gains its meaning from transactions — as it flows from one place to the other and back — it quickly became something that affects our own behaviour and emotions. The study of money should also include those factors, as it has become an essential part of any modern human society.<br />
<br />
We’ve seen the evolution of money from commodities to the hard to define thing
that it has become nowadays, mostly invisible and still everywhere around us —
we’re now allowed to carry huge amounts of currency summed up in a plastic card.<br />
<br />
<div style="text-align: center;">
<b>References</b></div>
<br />
Bae, K. H. and Bailey, W. (2011). <i>The latin monetary union: Some evidence on europe’s failed common currency</i>. Review of Development Finance, 1(2):131 – 149.<br />
Fagley, N. and Miller, P. M. (1997). <i>Framing effects and arenas of choice: Your money or your life? Organizational Behavior and Human Decision Processes</i>, 71(3):355 – 373.<br />
Foote, C., Block, W., Crane, K., and Gray, S. (2004). <i>Economic policy and prospects in iraq. Journal of Economic Perspectives</i>, 18(3):47–70.<br />
Hanke, S. H. and Kwok, A. K. F. (2009). <i>On the measurement of zimbabwe’s hyperinflation</i>. Cato Journal, 29(2).<br />
McLeay, M., Radia, A., and Thomas, R. (2014a). <i>Money creation in the modern economy</i>. Bank of England Quarterly Bulletin, 54(1):14–27.<br />
McLeay, M., Radia, A., and Thomas, R. (2014b). <i>Money in the modern economy: an introduction</i>. Bank of England Quarterly Bulletin, 54(1):4–13.<br />
Mitchell, T. R. and Mickel, A. E. (1999). <i>The meaning of money: An individual- difference perspective</i>. Academy of Management Review, 24(3):568 – 578.<br />
Radford, R. A. (1945). <i>The economic organisation of a p.o.w. camp</i>. Economica, New Series, 12(48):189–201.<br />
Rothbard, M. (2002). <i>A History of Money and Banking in the United States: The Colonial Era to World War II</i>. Ludwig von Mises Institute.<br />
Schumpeter, J. A. (1954). <i>History of economic analysis</i>. Allen and Unwin (Publish- ers) Ltd.<br />
Von Mises, L. (1954). <i>The Theory of Money and Credit</i>. Yale University Press.joão pestanahttp://www.blogger.com/profile/03616394334438513692noreply@blogger.comtag:blogger.com,1999:blog-9153823946796270764.post-51202473755875635982015-10-20T00:04:00.002+01:002015-10-20T00:08:12.745+01:00Rates of interest and Euler's number<div class="separator" style="clear: both; text-align: center;">
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<br />
In a <a href="/2015/10/how-to-use-credit-to-virtually-extend.html">previous post</a> I mentioned interest rates, but I didn't told you what they are. I simply assumed that you knew and, to a certain extent, I believe that's a reasonable assumption. This time I want to bring back the subject and focus just on what an interest rate is and not on what it's used for — which is probably what you know intrinsically.<br />
<br />
I'm a firm believer that if you can put a story behind your explanations you should do it. Single case examples go first and we end with a general statement that's universal. Imagine our dear friend Sarah has 1 € and she has the opportunity to double it after a year by investing it in a savings account. If that seems like it's not much, think again. It's a 100% rate of return on investment! Mathematically, we can write it as:<br />
<br />
<div style="text-align: center;">
`1 € + 1 € xx 100% = 2 €`</div>
<br />
This is what is expected to happen if the back credits the account once every end of year. It doesn't have to be like that and some banks do credit the accounts more often. Let's say that Sarah's bank credits her account every semester, everything else remaining the same. Will we get the same result? Let's see...<br />
<br />
<div style="text-align: center;">
`1 € + 1 € xx (100%)/2 = 1.50 €`</div>
<br />
<div style="text-align: center;">
`1.5 € + 1.5 € xx (100%)/2 = 2.25 €`</div>
<br />
Because we're being credited twice a year, the rate of return for each time is halved. On the first semester we get 0.50 € from our investment of 1 €, but on the second semester we receive 0.75€. This sums up to a total of 2.25 € and is bigger than the first example where the credit happens only once. What happens is that on the second semester, the amount already paid is considered reinvested and contributes positively to the final result. What if the bank paid every quarter year, would we get an even better deal?<br />
<br />
<div style="text-align: center;">
`1 € + 1 € xx (100%)/4 = 1.25 €`</div>
<br />
<div style="text-align: center;">
`1.25 € + 1.25 € xx (100%)/4 = 1.56 €`</div>
<br />
<div style="text-align: center;">
`1.56 € + 1.56 € xx (100%)/4 = 1.95 €`</div>
<br />
<div style="text-align: center;">
`1.95 € + 1.95 € xx (100%)/4 = 2.44 €`</div>
<br />
These numbers are rounded, but clearly the answer is yes! The more frequently the bank credits your account, the better is your return at the end of the year due to new credit being considered a reinvestment. Let's take a big leap here and ask ourselves what if the back paid us every second? Would we get an astronomically high rate of return? I'm going to give out the answer right way and prove it to you afterwards.<br />
<br />
The answer is a bit ambiguous, so let me clarify it. It always happens that, if you increase the frequency of credit, your final return is higher, but it has an upper limit. There is a number in which the return grows ever so closely with each increase in the credit frequency, but it cannot surpass it.<br />
<br />
We can see, from our previous results, that even if we always get a bigger value than before, the ratio between each successive increase diminishes. From 2 € to 2.25 €, we have a 12.5% increase, but from 2.25 € to 2.44 € that increase falls off to just about 8.51%. You can check for yourself that this pattern goes on to eventually reach a value very close to 0%. This is the scenario where the bank is crediting your account with micro-payments every second.<br />
<br />
It's time for the revelation of what that upper limit I told you about just before is. If you proceed endlessly the above calculation, each time increasing the frequency of credit from a quarter a year to monthly, weekly, daily, hourly ou per second, you'll reach an incredible number — Euler's number or `e`. The general form for the accumulation factor of a compound interest rate on an investment is well-known to many people as:<br />
<br />
<div style="text-align: center;">
`a(t)=(1+i/n)^(nt)`</div>
<br />
To find our answer — and the actual value of Euler's number — we just need to use it generally as we did before. We don't actually have to do it indefinitely, because in mathematics we can take the limit of a function as the variable tends towards infinity.<br />
<br />
<div style="text-align: center;">
`lim_(n->oo) (1+1/n)^n=e`</div>
<br />
Indeed, even if we do just some iterations of the previous method we do get a very good approximation. After 1000 steps — I used python, but it can easily be done in Excel or any other spreadsheet — you get the value `2.7169239322355936...` which is really close to the actual value of `e=2.718281828459045...` In fact, if you take the ratio of the absolute difference between the two over `e`, you'll find that the error is merely 0.04995% which makes it a pretty good approximation.<br />
<br />
This was just a complicated way to say that no matter how frequently your bank credits your account, you cannot surpass the value of Euler's number — a both irrational and transcendental number.joão pestanahttp://www.blogger.com/profile/03616394334438513692noreply@blogger.comtag:blogger.com,1999:blog-9153823946796270764.post-70392847457423141652015-10-18T00:58:00.000+01:002015-10-18T01:19:50.766+01:00The secrets of secrets<div class="separator" style="clear: both; text-align: center;">
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<br />
What is a secret? If you do a <a href="https://www.google.com/search?q=secret" target="_blank">Google</a> search, you'll probably encounter a reality show, a movie, a book or lingerie. That all seem good results — particularly the last one — but they ain't the definition of a secret, <i>per se</i>. It's a two syllable adjective and, according to <a href="http://www.merriam-webster.com/dictionary/secret" target="_blank">Merriam Webster</a>, it means "<i>kept hidden from others"</i> and <i>"known to only a few people."</i> Certainly, this feels much more correct.<br />
<br />
There are many times in which we wish to have information available to only a handful of people — or even just one more person — that we trust. The question is how to do that? There is a field of study that is dedicated to precisely that and it's called <a href="https://en.wikipedia.org/wiki/Cryptography" target="_blank">Cryptography</a>.<br />
<br />
Probably, you can recall a time in your life when you needed to pass a <i>secret</i> message to someone. For example, when you were at school. Imagine what would happen if the other kids — or even the teacher — caught your message instead of the intended recipient! Now... Do you remember the method that you used back then? I would bet my money that it was a simple letter swap method. You and your friend knew the right key to code and decode and an A could become a N and a N would become an A and so on to all the other letters in the alphabet.<br />
<br />
Just because I called it simple, it doesn't mean it's a weak type of encryption and someone could easily break your code. The real strength of any encryption depends on how long it takes to break. Yes, every encryption does break because, if it didn't, then decoding a message would be impossible. The single method that breaks all codes is called <i>brute force hacking</i> and it does just that. If you know how it works, you can program any <a href="http:/#">Turing machine</a> — computers, for example — to try every single possible combination of keys.<br />
<br />
I'll illustrate the whole method with an example. I have a message I want to pass to Sarah and we're the only two people that know the key. This key is presented in Table 1.<br />
<br />
<table id="table_post">
<tbody>
<tr>
<td>A</td>
<td>B</td>
<td>C</td>
<td>D</td>
<td>E</td>
<td>F</td>
<td>G</td>
<td>H</td>
<td>I</td>
<td>J</td>
<td>K</td>
<td>L</td>
<td>M</td>
</tr>
<tr>
<td>N</td>
<td>O</td>
<td>P</td>
<td>Q</td>
<td>R</td>
<td>S</td>
<td>T</td>
<td>U</td>
<td>V</td>
<td>W</td>
<td>X</td>
<td>Y</td>
<td>Z</td>
</tr>
</tbody></table>
<div style="text-align: center;">
Table 1 — Example of a letter swap encryption key.
</div>
<br />
To make it simple, all the letters are ordered but they didn't need to be. The ones in the top row switch for the ones in the bottom row and <i>vice-versa</i>. The process of encrypting and decrypting a message is pretty straightforward and follows these simple steps:<br />
<br />
<ol>
<li>Write down the message you intend to send;</li>
<li>Run your message against the key and switch all the letters accordingly;</li>
<li>Send your gibberish message to your intended recipient;</li>
<li>The recipient runs the message against the key you both share;</li>
<li>Switching back all the letters should reveal the original message.</li>
</ol>
<div>
The message I want to send to Sarah is the following.</div>
<blockquote class="tr_bq">
<span style="font-family: Courier New, Courier, monospace; font-size: large;">Let's meet tomorrow at 14 p.m. for coffee and smalltalk.</span></blockquote>
Notice that it contains several revealing elements that could be too revealing even when encrypted. I mean the numbers, the spaces, the apostrophe and the dots. I should either remove or switch these elements with words and also convert it all to uppercase, because my key has no lowercase correspondences. My message becomes now quite different.<br />
<blockquote class="tr_bq">
<span style="font-family: Courier New, Courier, monospace; font-size: large;">LETSMEETTOMORROWATFOURTEENPMFORCOFFEEANDSMALLTALK</span></blockquote>
It already resembles some sort of secret code, but it's still understandable. Let's run it against the key we have and see what comes out.<br />
<blockquote class="tr_bq">
<span style="font-family: Courier New, Courier, monospace; font-size: large;">YRGFZRRGGBZBEEBJNGSBHEGRRACZSBEPBSSRRNAQFZNYYGNYX</span></blockquote>
Success! I now have an unreadable message that I can send to Sarah and only her will be able to decode it. Or is it? What are the chances that, if fallen into the wrong hands, that person can decode the message?<br />
<br />
If the code breaker assumes that the code is based only on the 26 letters of the english alphabet and is a simple letter swap method, it means that it's simply a matter of finding out the right key for which the one in Table 1 is just one of many. Precisely, how many? Let's see...<br />
<br />
We have 26 available letters to fill the first cell on the table and we choose one. We are left with 25 letters to fill in the second available cell. We choose the second one and proceed until we have only one letter left for the last cell. It's a simple multiplication and we can write it as a factorial.<br />
<br />
<div style="text-align: center;">
`26xx25xx24xx23xx...xx4xx3xx2xx1=26!`</div>
<br />
Because we cannot repeat letters, the available choices get reduced along the line. This is known as permutations without repetition and calculated using the factorial function. This gives us the number of all possible keys which is just...<br />
<br />
<div style="text-align: center;">
`26! = 403 291 461 126 605 635 584 000 000`</div>
<br />
Now that's a really big number! Well... Yes and no. It's a big number of combinations for you to resolve on your own, but a computer can go through all of them in a relatively short time. Since we cannot make our messages indecipherable — it would render all this useless — by introducing a random element, we can only alter the complexity of the encryption process. In this sense, we can make the number of possible combinations so astronomically high that it would take the best computer a time larger that the age of the universe to go through all of them.<br />
<br />
You were also right to feel secure while sending your secret messages as a kid.joão pestanahttp://www.blogger.com/profile/03616394334438513692noreply@blogger.comtag:blogger.com,1999:blog-9153823946796270764.post-58365626191373156632015-10-14T20:13:00.000+01:002015-10-18T01:15:50.406+01:00How to use credit to virtually extend your total available budget vis-à-vis government bonds<div class="separator" style="clear: both; text-align: center;">
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<br />
Today, I bring the subject of economics to the table. As you may or may not know, a bond is like an IOU (I owe you). To put it in a very simple way, if my friend Peter is in need of some money, I can lend him some of mine and he will give me in return a promise of future payment — the IOU. This may or may not be subject to an interest rate. After some time has passed, the day comes when the bond Peter issued to me is maturing — reaching the end of its life. Peter will give me back the money he borrowed with the agreed interest, if any, and I will give him back his bond.<br />
<br />
If I have money stored which I don't use, it's better for me to invest it. Lets say that Peter and I agreed on a 1 000 € loan of 1 year at 2% interest rate. If I just had my money stored away, I'd still have the same 1 000 € after the year has passed. Since I invested it in Peter, I have now an extra 20€ and Peter was able to start his own business — or something like that.<br />
<br />
In a sense, that's how government bonds work. Because the expenditure of the government exceeds its revenue, it must issue bonds in order to fulfil the budget. When you hear the expression that the government debt is some percentage of the GDP (gross domestic product) what it means is that the government expects to have an expenditure that is larger that the revenue by that same amount. Lets say that the debt is 125% of the GDP and the GDP is 1 000 000 000 €, then the total expected expenditure by the government is 1.25 times larger. In this example, there would be a need to issue a total of 250 000 000 € in government bonds at some interest rate.<br />
<br />
But wait! Isn't a high public debt a bad thing? Well... Many economists disagree on that. The fact is that without issuing government bonds, there would be public sectors and services lacking funds to operate — like hospitals and schools — and government workers without wages. Anyone can buy government bonds and, if purchased in the country's own currency, it's free from credit risk. This happens because even if the government lacks the funds to redeem the bond at maturity, it can do so by raising taxes, issue new bonds to pay for the maturing ones or simply print more money (given that it has its own central bank). The return you get from government bonds may be lower that most investment opportunities, but it's essentially risk-free.<br />
<br />
I bet you're now wondering that what I mentioned above is very nice trick, but it probably won't hold up. Get a new loan to pay for the old one? Yes! That happens everyday and it's the basis of what I want to explain to you. I intend to show how someone can increase his or her total available budget by some amount almost permanently. The main idea is that, in a zero interest rate situation, you could ask for a new credit of 100 € maturing within a month and every next month you borrow another 100 € to pay for the previous one. It sounds like a lot of trouble, but if you think about it you just increased your total wealth by 100 €!<br />
<br />
Many credit companies that issue credit cards for consumption use have several ways of payment. The most typical is that you pay a yearly premium for the use of the credit card — something like 20 € — and than you can have fractioned payments or full repayment within 30 days. The fractioned payments have an interest rate associated with the ease of the financial burden, but the full repayment usually does not. How does this work? It's quite simple and I'll just illustrate with a numerical example.<br />
<br />
Lets imagine that Sarah has applied for a credit card with a 300 € line of credit and interest free if the she repays the full amount within 20 days of the invoice's issue. The invoice is issued on the 6th day of every month. If she is granted the application on the 1st and doesn't use the credit card in the first week, the invoice is issued with 0 € to pay. Then, on the 7th day, Sarah decides to use her credit card and buy a bicycle for 150€. After 30 days, the invoice is issued for the amount that Sarah used and she needs to pay it by the 26th. If she receives the invoice on the 6th, pays the 150€ back on the same day and on the next day she goes on a 2-day trip to the beach which also costs her 150€, by using the credit card to pay for it, Sarah has increased her total budget by 150€. So long as she keeps using the credit card for an expenditure of 150€ every month, her debt will continue to be pushed forward indefinitely in time.joão pestanahttp://www.blogger.com/profile/03616394334438513692noreply@blogger.comtag:blogger.com,1999:blog-9153823946796270764.post-79862682912791704152015-10-14T00:51:00.001+01:002015-10-18T01:10:55.491+01:00How to describe idiots in your facebook social network using mathematics<div class="separator" style="clear: both; text-align: center;">
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<br />
For my first publication, I thought of presenting some <a href="https://en.wikipedia.org/wiki/Recreational_mathematics" target="_blank">recreational mathematics</a>. This type of mathematics is not meant for research but instead used for entertainment, specially through the creation of puzzles and games.<br />
I won't spend much type describing what I'll do in this blog or try to justify it in any way. It may happen that this will be its only post or I may still be updating it years from now. I'll do it for as long as it pleases me.<br />
<br />
Now back to the problem at hand. The title is quite appealing! Wouldn't it be great if we could mathematically describe moronic people? This is one of the things that fascinates me the most these days — describing people and society objectively, using the concepts of the so called exact sciences.<br />
<br />
Let `I` represent the set of all the people that belong to your facebook friends (or your extended network — your facebook friends of friends). That is to say, people with whom it's likely you might exchange messages.<br />
Let `i` represent a single person within that set or, in other terms, `i in I`.<br />
If `T_0` represents the time that you last sent a message to the person `i` for which you did not receive either a reply or an acknowledgement (marked as <i>read</i>), and `t_i` represents the last time that same person `i` has been active, then we may simply describe the set of all idiots in your facebook social network as:<br />
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<div style="text-align: center;">
`X={i in I: t_i>T_0}`</div>
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Or put into words, `X` is the set of all people which are in your network such that the last time they were active largely exceeds the last time you sent a message to them.<br />
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The reason I consider those individuals to be idiots is that if someone intends to pretend he/she didn't notice the message, then he/she shouldn't also be active on facebook. Because the platform informs of such activity, people know that others will realize they've been there, conscientiously ignoring them.<br />
I suggest, at the very least, acknowledging the message by marking it as read. The difference is in admitting there is no intention of reply.joão pestanahttp://www.blogger.com/profile/03616394334438513692noreply@blogger.com