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HyperComplex Seminar
J. Czyż: Euler and Schrödinger equations: Turning points at entire complex mathematics and physics
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to invite the students to their mathematical seminars, and they both were deeply, profoundly interested in theology and philosophy, and a fascinated person for Professor Moreau. was Albert Schweitzer and Julian Wawrynowicz during his seminar in Bandlewo about 10 years ago led us to Gnieby where we could learn the spiritual and education heritage of Urszula, Sam Urszula Leduchowski. I would like to concentrate in my presentation on two equations when complex numbers play a visible role. First, this is Euler. equations, dated 100 to 1748, and second, the Venetian equation. divided by 126, a mass constant, Laplacian, minus nx to the same phase. coordinates, of course, and potential. . . And so I thought that complex imaginary identities are visible to all the viewers. and also participants online.
And I would like to say one thing before saying about complex numbers, a few words about numbers. I am, months ago it was. . . published in weekly Niedziela, my article Liczenie po polsku, Counting in Polish, where I explained the notion of numbers, numbers of a verb, and according to me, two numbers are meaning first namely 100 and 1000 and we know that 100 is so in Polish and in all Slavonic languages. And we write In Sanskrit and Hindu, we pronounce Shatam. In ancient Iranian, it was Satem. And some half of Indo-European language in a half, usually, East of the Indo-European languages. This new member. is pronounced similarly, and so that these languages are called a Saturn language. And this means that ordinary people in the European tribal groups count it up to.
and the as for the western in the Indo-European languages, they are called Cantum. Cantum means one hundred in every language. And as for thousand. . . It is quite different situation because we have in part, of course, stations in Russian and Ukrainian famous Tsyacha, but also English 100, German 100, Swedish 200. Why is that so? Because up to 1,000 people counting up to 1,000 belong to maybe set distinguished and small social groups. And let us come to complex numbers. And Almost all languages, and Polish is an exception, we say complex, in Russian complex. So, and we have complex, and Polish name, this Polone, is specific and accurate, because. . . If we literally translate it into French, it would be said that the number does not matter.
And as for Uloyone, the situation is most interesting, I think, because in all… Other languages, also called imaginary numbers, even in Czech, are imaginati. The difference is about Russian and Ukrainian, which are minima numbers, but the problem is in English language simulations. that the verb, the word and verb, uroionin, uroich, is specifically Polish and cosmic equivalence in other languages. Because, for instance, in Russian, as I know, mnimoje, dvanie, mnimoje, is close to, not to the imaginary level. And in Polish, the meaning is somewhat connected with the psychiatric context, yes? So, let us come to the question that mathematical approach about numbers. And I refer to the book of American authors, all the, Deubel and Oster, and Kolostek, Count of Examples in Analysis, Contra-Primerive Analysia in Russian edition, and what is the most, I think, natural and fundamental structure in mathematics.
Because referring to real numbers, the field of real numbers, we can introduce the notion of altered. . . of all the trees, native and older veneration was first defined in both. That older veneration demands a the leading to some of the strong sets until the axioms are that I think that it is understandable. Yes, that level of exploitation by man, this one, changed the vision that belongs to other parts. And then, the last condition. . . And to work with this problem, let us take the imaginary unit i. So suppose that it belongs to f-plug. So i, 3. belong to F as well, but what is I? To see it as minus I, so it must belong to F minus. That is a contradiction. So, second possibility is that I belong to F-.
So then, I3 and F-means that it is equal to minus I. And then I3. is equal to I, which belongs to F plus. In the traditional way. So, what I can express this fact that complex numbers are not self-sufficient in mathematics, on very elementary and basic level. Yes? So, and this is in contrast to real. . . to the field of the reals, yes? So the reals are, maybe say, much better than complex numbers. Yes, okay? Vocally? No? Ah, yes. Yes, okay. So, let us. . . oh but well with some players and maybe a little uh up is this committed by the camera so i must then let us say and can i just come to the history of mathematics in the last several centuries.
And what we can say about numbers, it is difficult to study all historical. . . cases and precedents, but I think that it may be said that before Euler and Euler equations, complex numbers in mathematics were episodic. It was, they were algebraic episodes when circumference, sound, force, all the algebraic situations were taken into account. And for instance, it is the It is almost sure that in the world of Geopantus, Cardano, Tartana, complex numbers more or less explicitly appear. But they were only isolated cases. And that's all. Leibniz, who lived and worked shortly before Euler, he wrote his general paper, short art book, Historia et omigo calculi differentialis, and he made a general sketch of about calculus, differentials, and about integration and differential calculus. But in his works, I found only cases of dealing with second-order equations and with specific coefficients.
And the great change was done when all the world and the world in the in the in Finitorum. And André Nail, while's editorial idea was to translate and edit this into English. And this is the content of the book, and as long as Schwingen volume, and that's it. Let me… the development in exact science, especially in mathematics, astronomy and mass physics, was initiated when the work of Copernicus, the revolutionibus, appeared. And then… The time period between the operation and the closure can be divided in two parts, about weeks 1960 and 1920. And they are quite different because it was a lazy development, mainly by, in Polish we could say, them ratnice. by numeric domain, and the heroes were Raticus, because Copernicus adjoined to the revolution for the Decimal Phase. tables of codes and codes of double sciences.
And RETIC made a jump forward because they jumped from four decimal places to 15th ones. And 20 years after the death of Reticus, it was so near the end of the 15th century, these tablets were edited. And then it was a problem how to multiply and divide, let's say, the two 15-cypher numbers. And then it was the work of Naepper to make a first approach to logarithm, and his work in that way. modifici logavit modum canonis descriptio. And his neighbor logavit were almost homomorphic, so that neighbor logavit of her. and more communication, and there was some correction. Corrected, yes. But literally, for the young Inca people, it was no problem about correction, because she used successively and effectively. in order to prove the Viscounts'explicit third law of movement of guns.
But Briggs observed that there is a following formula. Logarithms in decimal logarithms, Naples logarithm of 1, this is minus Naples logarithm of 1. And in this way, the number of tables of the decimal logons were recognized, but the problem was in the decimal. natural drug addicts, because it was recognized by some answers that Similar looks the graph of one by Earth and the field under. the derange of logarithms that look similar. But, and it was. . . And so, the period from this scale. . . Because, let me, here you have a period of some right mission analytics and we can say three years. Why? Because in this book of advanced, the historical development of calculus, the historical development of calculus, we can.
read at the beginning about John Naber and his political track contained proofs in Euclidean fashion that the poet was the Antichrist and that the world was due to end in the year 1700. And as for astronomer who developed Copernicus, Wajdu Tycho Brahe, he was in a duel, he lost a tip of nose and It was replaced by a piece of silver. And it was then maybe contained in a limited domain, tedious development, but after. . . of the BRICS, the Storm Development and Astronomy and Mathematical Analysis Institute. And so we can have the following stats. And if you need a time, those are open. and calculate power series of sine, cosine, but other mathematicians counted power series. And this domain was used by Bernoulli numbers and by some students of mathematics.
does not solve the solution of maculobin series for Tandem, for instance. So this developed was very strong and strong. Because on the other hand, this connection between fields after logarithms, natural logarithmic functions, and E2, it's diverged, it is integral, and this function has a. . . and the balance of rights, this relationship was not fully clarified until the time of the war. in the 18th century. And maybe this article. Maybe. Okay. So, we see that. . . So, it was a turning point, not a reflection, but what was the change after. . . after emerging Euler's introduction in Analysing Informatics. In the preceding state of mathematical analysis, this differential and and integral calculus, and all that was established, and all Euler's literature is not comprehensible to even qualified mathematics.
But Euler's book, Analysts in Influence, is clear and comprehensive. So it was, now it is a number of beginning of analysis as a mathematical as knowledge is as we learn and study it and to what our main point of this version then. Number E was established and as you see, the very brief tradition was adjoined to mathematics whose literature was built with very complicated formula. So And of course, then, the initial point of this development was astronomical, because it was well-known for its needs, trigonometric tables and logarithmic tables. discovered by Bessel and Jacobi in order to exclude a perturbation of Saturn or Hobbit by Jupiter. But let us pass to the desert preceding the 20th century and just to In quantum mechanics, they are similar.
If we look at the early 20th century and the first papers about quantum mechanics due to Bob's alphabet and sky zembel, they are is not really book for us, but if we take the book of Schrodinger in physical review, there is an undulatory text here. of the mechanics of atoms and molecules, it is a clear and we can point out analogy with the work of Euler. And for instance, here you see that this is time-dependent, and all spectra, yes, and this energy spectrum of hydrogen atoms, so we can. . . Note, it is clear to see introduction of complex numbers, the ordering load, and also the I-count. This is a spectral line of hydrogen and hydrogen atom, and see observed are. as I counted, 91, and implying from shredding equations, 57. And it looks very similar.
And what is Erkont Merhans I would like to. . . in the final part to say about quantum chemistry and Penrose ideas. And as for quantum chemistry, but I think that Professor Lucian Piera, from Faculty of Chemistry in Warsaw University, said. . . me what has run this development in quantum chemistry is that here we heard about 100 energy levels and values and eigenvalues and nothing. The time independent Schrodinger equations were in, it is out of theory. that one material, the other, that in the complex and imaginary numbers disappear, but when only ground energy level is taken into account by quantum chemistrers, and so, how long? About 3,000 electron systems can be recognized and studied and well-studied, well-approximated by quantum chemistry. If you would like to have more exact equations, then I'll put them.
at about 100, then we can use, we should apply relativistic improvement taken from, coming from Dirac equations, and then complex numbers reappear. And if we consider the case of When the systems are in magnetic fields, then also complex numbers reappear, and the number of electrons in systems in studies is about 100, but not more. But what is the present state of. . . development of quantum mechanics now, and about fantastic ideas of Roger Penrose. From a mathematical scientist's perspective, Robert Penrose believes that complex numbers will be crucial in a future development. plan of client of mathematical physics. And he believes that his twisters will play a crucial part in quantum chemistry, in quantum gravity, hidden dimensions, and Big Bang. And I doubt it. Why? Because it would be strange.
If complex numbers which turn out to be insufficient in mathematics, as I have thought, would be crucial for mathematical physics. So I think that other people are interested. in physics and mathematics may share such opinion. What do you think about the hyper-complex standards? Oh, because this is. . . domain which I am not familiar, but Pernos says about this phenols and other, and as for this, and of course, but let me say, quaternions are less sufficient than complex tumbler because they are green. non-commutative pieces. Octonions are non-associated. And because I don't know, I must admit, I don't know the hyper definitions of hyper complex numbers, but I think that this problem concerns such objects, which seems in some sense more artificial than complex numbers, than real numbers, of course.
And if we develop quantum mechanics, we can come back to real numbers in many times, as quantum chemistry and recent developments shows. .
S K R Y B O T
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