A T ,, It Q ' Co: Fill & Download for Free

Is using the name of the founder too limiting?Here are some companies whose experience suggests 'no':Arthur D. LittleMcKinsey and CoBain and CoAnderson ConsultingA T KearneyBooz Allen HamiltonPrice Waterhouse Consulting (PWC)McDonalds (Ray Kroc, normally credited with founding the chain, bought out the McDonald brothers)B&Q (If you're British, you'll recognise the name of the UK's equivalent to Home Depot)

Probably, Paul Erdos would have answered to your question, "Another roof, another proof".I'll attempt to present two ways how you can see it.First of all, there is a completely elementary proof which uses no more that the ring [math]\mathbb{C}[t][/math] is a Unique factorization domain (UFD).So assume that there are some solutions of the equation above in [math]\mathbb{C}[t].[/math] Choose a solution [math](a(t), b(t), c(t))[/math] such that the maximum of degrees of [math]a,b,c[/math] is positive and minimal possible among all solutions. Clearly, this choice ensures that [math]a(t) [/math][math][/math][math], b(t) [/math][math][/math][math], c(t) [/math][math][/math][math][/math] are co-prime.Then we have: [math]a(t)^3 =c(t)^3 -b(t)^3=\bigl(c(t) -b(t)\bigr) \bigl(c(t) -\omega [/math][math][/math][math] b(t)\bigr)\bigl(c(t) -\omega^2 b(t)\bigr),[/math] where [math]\omega [/math][math][/math][math][/math] is a third primitive root of unity.Now look at the factors [math]c(t) -b(t), c(t) -\omega b(t).[/math]Let's suppose that they have a common prime factor [math]q(t).[/math] Considering their sum and difference we conclude that [math]q[/math][math][/math] is a common factor of [math]c(t)[/math] and [math]b(t)[/math] as well. Moreover, [math]q(t)[/math] is a factor of [math]a(t).[/math] Thus [math]a[/math][math], b, c[/math] are not relatively prime. A contradiction.Repeating the same game with other pairs of factors we see that all three factors [math]c(t) -b(t), c(t) -\omega [/math][math][/math][math] b(t), [/math][math][/math][math] c(t) -\omega^2 b(t)[/math] are pairwise co-prime.Therefore [math]\begin{cases} c(t) -b(t)= d_1(t)^3 [/math][math][/math][math]\\ c(t) -\omega b(t) = d_2(t)^3 [/math][math][/math][math]\\ c(t) [/math][math][/math][math] -\omega^2 b(t)=d_3(t)^3 \end{cases},[/math] where [math]d_1, d_2, d_3 \in [/math][math][/math][math] \mathbb{C}[t][/math] are (pairwise) coprime.Note that [math]\omega^2 [/math][math][/math][math]+ \omega +1 =0.[/math]Multiplying the second equation by [math]\omega[/math] and the third equation by [math]\omega^2[/math] and adding all three of them we arrive at [math]d_1(t)^3 [/math][math][/math][math]+ \omega d_2(t)^3 [/math][math][/math][math]+ \omega^2 [/math][math][/math][math] d_3(t)^3 =0. [/math][math][/math][math][/math]Choosing [math]\eta_1 [/math][math][/math][math][/math] and [math]\eta_2 [/math][math][/math][math][/math] as any third roots of [math]- \omega[/math] and [math]- \omega^2[/math] and letting[math]a_1=d_1, [/math][math][/math][math]\; b_1 = \eta_1 d_2 [/math][math][/math][math][/math] and [math][/math][math] c_1 = \eta_2 d_2 [/math][math][/math][math] [/math][math][/math][math][/math] we get [math]a_1^3 = b_1^3 [/math][math][/math][math]+ c_1^3.[/math]By construction, at least one of [math]a_1, b_1, c_1[/math] is a nonconstant polynomial, and maximum of their degrees is smaller than that of [math]a[/math][math], b, c.[/math] This is a contradiction to the choice of [math]a[/math][math], b, c.[/math] [math]\blacksquare [/math][math][/math][math][/math]A more geometrical, but slightly less elementary solution for this problem will be:Let [math]a(t), b(t), c(t)[/math] as above of degree [math]n_1, n_2[/math] and [math]n_3,[/math] respectively, and [math]n = \operatorname{max}(n_1, n_2, n_3).[/math]Then we can define [math]A(u,v), B(u,v), C(u,v) [/math][math][/math][math][/math] as homogeneous polynomials of degree [math]n[/math] such that [math]A(u,1)=a(u), [/math][math][/math][math]\; B(u,1)=b(u), [/math][math][/math][math]\; C(u,1)=c(u).[/math]Then, by construction, [math]A(u,v)^3 +B(u,v)^3 =C(u,v)^3.[/math]Let [math]E=\{ (x:y:z) \in \mathbb{P}^2 [/math][math][/math][math]: x^3 +y^3 =z^3\}.[/math][math]E[/math] is a smooth curve of genus [math]1[/math], i.e. an elliptic curve.Now define a map [math]\varphi: \mathbb{P}^{1} \to E[/math] given by [math](u:v) \mapsto (A(u,v): B(u,v): C(u,v)).[/math]The map [math]\varphi[/math] is well-defined since [math]A(u,v), B(u,v), C(u,v)[/math] are homogeneous polynomials of the same degree which don't vanish simultaneously being relatively prime.Moreover, it is is non-constant and proper (since its domain is projective.) As the image of a proper map is closed and it's not a point, and [math]E[/math] is one-dimensional variety (irreducible) follows that [math]\varphi[/math] is a surjective morphism.As you probably know, [math]E[/math] is topologically a torus, i.e. it has genus 1 and there is one up to scaling differential form [math]\tau[/math] on it. This will imply that its pullback [math]\varphi^{*} \tau[/math] is a differential form on [math]\mathbb{P}^{1}[/math] which is impossible since it has genus [math]0.[/math] In more formal terms, a surjective map [math]\mathbb{P}^{1} \to E[/math] gives rise to an injection [math][/math][math] H^{0}(E, \Omega^{1}_{E}) \to H^{0}(\mathbb{P}^1, \Omega^{1}_{\mathbb{P}^1}).[/math] But this is absurd since the former is a vector space of dimension [math]1[/math] and the latter is a vector space of dimension [math]0.[/math] [math]\blacksquare[/math]

Q: For what reason(s) did Czechs and Slovaks go different ways and decide to separate?If it comes off as if I was too tired of explaining this, it is maybe because I am.Czechoslovakia formally split up on January 1. 1993The main reason for the break up were some disagreements between Slovaks and Czechs (not sure if foreingers are able to grasp that idea, but CzechoSlovakia was named CzechoSlovakia, because it was a land of Czechs and Slovaks. The idea of “Czechoslovaks” came from pro-czechoslovak, pro-union ideas and Czechoslovak nationalism)The main disagreements were of course mainly between the politicians, not very much between ordinary people. (No, we do not hate each other)Slovak & Czech politicians couldn´t really agree on anything at the begginning of 1990s.THEY COULDN´T EVEN AGREE ON THE OFFICIAL NAME of the country:Czech and Slovak Federal Republic ?Czech-Slovak Federal Republic ?Czech-Slovak Republic ?Czechoslovak Republic ?why can´t it be Slovak-Czech Republic? or Slovakoczech republic? or Slovak and Czech Federal Republic?So the politicians were like : “Nothing can be done about that. Let´s just break up.”So that happened in 1992, starting 1993 - the countries were already split.Now I can imagine you reading this like : “ But WHYYYYY WHYYYY Slovaks & Czechs couldn´t agree on anything whyyy whyyy whyyyyyyyyyyyyy?”… “But until then, they co-existed without any problems, whyyyyyyyy, EXPLAIN”Well, read carefully now :Originally , like, totally originally, Czechoslovakia was founded in 1918. It was founded as a common state of Slovaks AND Czechs.This, common state, only lasted for less than 75 years, yet it is so much imbeded in some foreigners minds that they still call us Czechoslovak or whatever.“But whyyyy whyyyyy you say less than 75 years? I can do math ! “Well, because during World War II, Czechoslovakia was divided for 6 years ! Okay? the 1993 split was not the first time.So, basically, the common state of Czechs and Slovaks was established and dissolved within just one century.You also need to understand, that Slovaks & Czechs, although they have some common roots, they had developed separately for about thousand years !So, logically, when they got together in 1918, it wasn´t just one “Czechoslovak nation” and cookie-pooping unicorns, but rather two nations.OFFICIALLY, Slovaks & Czechs were equal inside Czechoslovakia, but in REALITY, there were differences.There were twice as many Czechs as Slovaks.HISTORICALLY, Czechs fought against the influence of Germans & Austrians, while Slovaks fought against the influence of Hungarians (Magyars).Magyarization in Slovak lands was stronger than Germanization in Czech lands.… And, logically, Slovaks, in comparsion with Czechs, didn´t have as good chance to develop their language and culture.So this combined, meant that Czechs in common state were economically and culturally dominant.Now, Slovaks were unhappy, because they felt like all the decissions are made by Czechs, that Slovaks are always getting the short end of the stick.Czechs, were also unhappy, because they thought the poorer Slovakia was ruining their progress.If that makes you feel now like : “omg that sounds like they did hate each other! who you lying to ?” But actually no, they did not. (other countries: learn!)The reason why they still get along was the focus on the fact, that even though there are many differences, they still had many things in common!For example, the very similar language….. and similar Slavic rootsBut for now, we should focuse on the differences, that led to the split.During the communism (no, theres no communism today, seriously), the differences between opinions of Czechs and Slovaks did not even get a chance to come to the surface.November 17, 1989, the Velvet Revolution took place.It brought the collapse of the communist regime in Czechoslovakia.The first democratic elections were held and people chose their political representatives.The new politicians had to somehow settle on how this new democratic state would look like.And there, my friends, it became pretty apparent, that Slovaks and Czechs had quite different visions of the new country.The different views were for example regarding to the economic reforms or political arrangement of the country.Many Slovaks in this era wanted more autonomy for Slovakia. They called for decentralization of competences, so that Czechs would govern for Czech lands & Slovaks would govern Slovak lands.Czechs, of course, were okay with the centralization from Prague (today Czech capital), they were kinda willing to accept a bit of decentralization, but only to some specific extent.Annnnnnd there the politicians were negotiating for looooong months just about that, and of course, they reached NOTHING.In 1992, second elections were held.The winners of these elections - V. Mečiar in Slovakia & V. Klaus in Czechia - realized, (or became brutally honest about how it is) that they simply cannot find a compromise & settled on a break-up.!! BUT !! the politicians decided not to hold referendum & not to ask their people whether they actually want to split or nay.So until this day, we actually don´t know, how many Slovaks and how many Czechs would have voted for break-up and how many for common state. Everything you read “More people blah blah!!” “nooo but more people blah blah!!” are just some weird biased assumptions people make according to how they feel. Can you just accept we don´t know? Is - we don´t know an enough answer?In any case. The split was VERY PEACEFUL (other countries- learn!). and on January 1st, 1993, the people were celebrating New Year and New State.So yes, on new year´s eve, we are simultaneously celebrating our country´s birthday.Nowadays, Slovaks & Czechs have very good relations. Other countries with similar historic issues should really take an example here.