Mathematical Methods for Engineers and Scientists 2: Vector Analysis, Ordinary Differential Equations and Laplace Transforms
by K.T. Tang

Publisher: Springer
Number Of Pages: 339
Publication Date: 2006-12-19
ISBN-10 / ASIN: 3540302689
ISBN-13 / EAN: 9783540302681

Product Description:

Pedagogical insights gained through 30 years of teaching applied mathematics led the author to write this set of student-oriented books. Topics such as complex analysis, matrix theory, vector and tensor analysis, Fourier analysis, integral transforms, ordinary and partial differential equations are presented in a discursive style that is readable and easy to follow. Numerous clearly stated, completely worked out examples together with carefully selected problem sets with answers are used to enhance students' understanding and manipulative skill. The goal is to make students comfortable and confident in using advanced mathematical tools in junior, senior, and beginning graduate courses.

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Ali Tawafsheh

Es sei angenommen, es gäbe einen omnipotenten, allwissenden Gott. Dieser kreierte die Welt und greift von nun an vielleicht sogar noch ab und zu einmal in den Weltenlauf ein.

Doch wie unter „I. Schöpfungswahn“ bereits erläutert, besteht kein Grund für Gott dies zu tun. Weshalb sollte er schöpfen? Durch sein Allwissen bedingt wüsste er doch bereits, wie sich jedes einzelne Atom in diesem neuen Universum verhalten würde. Denn anders als der Laplace'sche Dämon befindet sich Gott außerhalb des Systems, das er betrachtet. Somit muss er sich nicht selbst mit einberechnen und kann deterministisch alles vorhersagen. Anzumerken sei, dass die Unschärferelation der Quantenmechanik dem Determinismus widerlegt, da ihr zufolge von keinem Teilchen der genaue Aufenthaltsort und die Bewegung exakt bestimmt werden können. Sie beruhen auf Wahrscheinlichkeiten und sind somit dem Zufall überlassen (Es handelt sich hierbei nicht um Messfehler). Doch da wir Gott Allwissen zuschreiben und er sich auch außerhalb der uns bekannten Physik bewegt, dürfte dies für ihn kein Hindernis darstellen.

Um zum eigentlichen Thema zurückzukehren: Gott weiß noch bevor er diese Welt geschaffen hat, wie sie zu jedem bestimmten Zeitpunkt aussehen würde. Somit gäbe es für ihn keinen Grund, überhaupt zu schöpfen.

Deshalb stellt sich mir eine Frage: Was wenn er die Welt gar nicht geschaffen hat?

Er hat sie nicht geschaffen, weil er immer noch daran denkt, was geschehen würde, wenn er sie auf diese Weise schaffen würde. Dies macht uns und die gesamte Welt zu einem Gedanken. Dies würde bedeuten, dass wir nichts als Gottes Pläne für eine Welt sind, die er vermutlich niemals schaffen wird. Man wird sich fragen, weshalb er überhaupt Pläne für eine Welt macht, wo er doch bereits weiß, dass er bereits weiß, wie es ausgehen wird.

Es ist wohl nicht so, dass Gott eine Wahl hätte. Allein seine Existenz und der Fakt seiner Allwissenheit lassen uns bereits auferstehen. Denn wenn die Bedingung seines Wissen erfüllt sein soll, so muss sich der Verlauf unserer Welt in den göttlichen Datenbanken wiederfinden. Gottes Existenz zwingt uns ins Leben.

Dies würde das Problem des fehlenden Grundes für eine Schöpfung, sowie die belegte Tatsache, dass Nichtsein dem Sein vorzuziehen ist, auflösen. Gott hatte keine Wahl.

Unglücklicherweise lässt sich dieses Theorem in keinster Weise verifizieren. Auch bleibt die Frage offen, woher ein solcher allwissender Gott kommen sollte. Der Grund unserer Existenz verschiebt sich nur zu der Frage nach dem Grund Gottes Existenz.

Deshalb wird diese Theorie im weiteren auch nicht weiter Erwähnung finden, aber verdiente es dennoch, angemerkt werden.

The Laplace Transform: Theory and Applications (Undergraduate Texts in Mathematics)
By Joel L. Schiff

Publisher: Springer
Number Of Pages: 233
Publication Date: 1999-10-14
ISBN-10 / ASIN: 0387986987
ISBN-13 / EAN: 9780387986982

The Laplace transform is an extremely versatile technique for solving differential equations, both ordinary and partial. It can also be used to solve difference equations. The present text, while mathematically rigorous, is readily accessible to students of either mathematics or engineering. Even the Dirac delta function, which is normally covered in a heuristic fashion, is given a completely justifiable treatment in the context of the Riemann-Stieltjes integral, yet at a level an undergraduate student can appreciate. When it comes to the deepest part of the theory, the Complex Inversion Formula, a knowledge of poles, residues, and contour integration of meromorphic functions is required. To this end, an entire chapter is devoted to the fundamentals of complex analysis. In addition to all the theoretical considerations, there are numerous worked examples drawn from engineering and physics.
When applying the Laplace transform, it is important to have a good understanding of the theory underlying it, rather than just a cursory knowledge of its application. This text provides that understanding.

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The concept of causality  is thrown away, this means that random happenings without any apparent causes are the norm. There need not be any cause for a particular effect. As maxwell put it -"the secret of the universe lies in the calculus of probablities" . Thus we cannot talk of a well defined postion of a partcle, only the probability with which it can be found there, any measurement of a particular variable (positon,velocity.etc) yields a result which is only one of the infinite possible states. Our perception of reality is only one of many possible versions of it.

 Surely this is counter intuitive. In the macro world, there does seem to be a strong cause-effect relationship as far as events are concerned. A ball moves-some one must have pushed it, the phone rings - some one must be calling me. Why this disconnect? The answer lies in scale. As we delve deeper and deeper into matter and observe smaller and more fundemental particles, the uncertainty induced by the 'wave' nature of matter becomes more pominent.So, while you can predict the orbit of the earth with a great degree of accuracy(orbits are chaotic so, complete accuracy of prediction is ruled out), it would be foolish to even try that for an electron. So all those pictures you see of atoms with well defined electron spheres and orbits are a bit of an exaggeration, reality is a lot fuzzier. It seems that nature is a lot subtler than it was thought to be.

However, in this case, I must say that I was a bit disappointed that they did not include the quote by Pierre-Simon Laplace (of Laplace Transformation fame) when speaking to Napoleon. Perhaps it was omitted because it is usually misquoted a tad and may not even have a perfectly accurate translation to English. However, since it is one of my personal favorites, I hope that you will read the details of the Laplace and Napoleon encounter on wikipedia.

I was also disappointed that they did not include any quotes from Robert Heinlein. So, I also hope you will click to read these quotes about religion from Robert A. Heinlein.

Thanks to rit for the link to the list that started this post!

Este é o terceiro ano que arrasto esta cadeira! No momento, é a unica cadeira que tenho para traz.

Ou me safo agora nos minitestes que valem metade da nota e depois o exame vale metade, ou depois terei de ir a exame de recurso. Tenho que fazer a cadeira este semestre! =/

Para este teste sai: integrais impróprios de 1ª, 2ª, e 3ª ordens, transformadas de Laplace (as invesas também) e as equações diferenciais de 1ª ordem. Tou lixado!

Cá vai disto!

Etrafımızdaki her türlü olayda kullanılan, 1763'de Thomas Bayes tarafından Royal Society dergisinde 'Essay towards solving a problem in the doctrine of chances' (rastlantısallık doktriniyle problem çözümü) makalesiyle ortaya konan ve sonrasında Pierre Simon Laplace tarafından geliştirilen ve çağdaş yaşamın içine dahil edilen olasılık kuramı...

Raslantısal bir olayın gözleminden önce, öne sürülen varsayımlara ilişkin olasılıkların değerlendirilmesine dayanan bu istatiksel çıkarım yöntemine örnekler verirsek;

Orhan pamuk'un, Nobel ödülü alma olasılığı %68'dir. piyasanın kar romanının talepleri karşılama olasılığı ise ;

1- nobel alırsa %55
2- nobeli alamazsa %90'dir.

buna göre 'kar' romanın zamanında talebi karşılama olasılığı nedir?

cevap bayes kuralıyla hesaplanır ve %66 olarak bulunur.

daha içimizden bir örnek verirsek;

herhangi bir blog'daki girdilerin %70'i ansiklopedik bilgilerden, %30'u ise geyik girdilerden oluşmaktadır. ansiklopedik bilgilerin %83'ü kurallara uygunken, geyik girdilerin %63'ü kurallara uygundur. 'rasgele' bir girdi seçilmesi halinde ise girdinin kurallara uygun olduğu gözüküyorsa, bu girdinin geyik girdi olma olasılığı nedir?

cevap: %25'tir.

bayes kuramının oldukça geniş bir alana hitap etmesine örnek olarak, pazarlamada, tüketici analizlerinde, siyasi ya da sektörel stratejik kararlar alınmasında, savaş stratejilerinde, yazılım teknolojisinde, hukukta, suçlu tespitinde, tıpta, fizikte, kimyada, biyolojide, şifreleme güvenliğinde veya bir online yemek sitesinden sipariş verdiğinizde ya da köşedeki marketten siparişlerinizde, kargo alımlarınızda hep şirketlerin sizin evde olup olmadığınıza, kredi kartı ile ödeyip ödemediğinize veyahut sizin gerçekten alıcı olup olmadığınıza dair maliyet analizi yapıp buna göre fiyatlandırma stratejisine giderek ulaşım giderlerini fiyata dahil etmesinde kullanılan yöntemlerden biri olarak örnek verebiliriz.

bayes kuramının önemine ilişkin iki tarihi örnek verirsek ise, ingiliz matematikçi alan turing'in nazilerin enigma şifresini çözmesinde kullanılması, bir diğer örnek ise, kalp krizlerini %50 oranında azalttığını öne sürerek bir ilaç ürettiğini iddia eden bir ilaç şirketinin aslında risk olasılığının yanlış olduğu yine bayes kuramıyla gösterilmiştir.

The point about zero is that we do not need to use it in the operations of daily life. No one goes out to buy zero fish. It is in a way the most civilized of all the cardinals, and its use is only forced on us by the needs of cultivated modes of thought.

Alfred North Whitehead

Leibniz called zero “a fine and wondrous refuge of the divine spirit.” But where does the idea come from? The history of the word may afford us a clue to this mystery. We receive the English word ‘zero’ from the French zéro which comes (along with ‘cipher’) from the Italian zefiro. The latter originates in turn from the Arabic sifr (from safira = “it was empty,” a translation of the Sankskrit sunya = “void” or “un-reality.”)

There are at least two distinct ideas of “zero,” each with more or less unique histories and origins. First, there is the modern, everyday notion of zero as itself a number (the additive identity, “nothing”) which is closest to the original meanings of the Arabic and Indian words. However, zero has a secondary meaning as well: there is a less “formal” idea of zero as simply a kind of positional place-holder (for example, to distinguish 4053 and 4503 from 453, where “0” stands for “none” in the tens’ or hundreds’ place.) Thus zero denotes both a digit and a value, but these two ideas are not completely separate. Neither zero has a simple, common or easily-described history.

What is certain is that, for over a thousand years, the Babylonians used a positional numbering system without zero. (Original texts from that era simply depend on context to resolve ambiguity.) There is certainly no evidence Babylonians felt any problems with the ambiguities which existed. Indeed, until around 400 BC, we do not find even spaces between numerals to denote positional difference (the earliest but debatable use of zeroes in Babylonia is in 700 BC, falling quite close to earliest use in Indian writing around 650 BC.)

When zeroes do arrive in the cuneiform script (in the form of double wedges, or triple hook symbols) they are almost only used between numerals -- that is, never at the end of a number (for example, 2015 but not 2150, which would often still be written 215.) Some researchers have suggested that these symbols were likely not intended by the Babylonian mathematicians to indicate zeroes as numbers, but were probably marks intending to clarify the interpretation of a number.

The length of time that mathematics was actively studied and investigated prior to the invention of zero suggests the discovery (or invention?) of “zero” is by no means obvious or natural. The late Hellene astronomer Ptolemy was one of the first Greeks to use “true” zeros (and these resulted from empirical measurements.) Even Fibonacci himself, who introduces Arabic and Indian numbering systems into Europe, still doesn’t treat zero as a “number” but as a “sign.”

Zero continued to be staunchly resisted by thinkers outside of the Muslim world until hundreds of years after Fibonacci. Still by the time of Cardan (who was solving cubic and quadratic equations without zeros) the value was not really a part of European mathematics. Only about 400 years ago does zero really come into widespread usage as a number in its own right, as a concept representing the amount “before” counting starts.

Does ‘zero’ somehow provide the basis for algebraic or “meta-structural” re-evaluation of the properties of numbers themselves? Laplace called zero “ ...a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity and the great ease which it lent to all computations put our arithmetic in the first rank of useful inventions.” The role of zero in the development of mathematics is incontestable. It is more than possible that Western society depends in many ways upon the uncanny and mysterious power of “nothing.”

While the origins of zero may remain clouded in mystery, a final question about the future of mathematics may perhaps be posed. What does this idea of zero still hide? What does this question of “nothing” evade?

What power could still lie stored within zero -- waiting within this mysterious void at the heart of number -- or rupture in the essence of value? What unpredictable and undreamt-of mysteries are waiting for mathematicians of the future?

Further Reading:

A. Aaboe, Episodes from the Early History of Mathematics (1964).
G. Ifrah, A universal history of numbers : From prehistory to the invention of the computer (London, 1998).
B. L. van der Waerden, Geometry and Algebra in Ancient Civilizations (New York, 1983).

Articles and other resources:

S. K. Adhikari, “Babylonian mathematics,” Indian J. Hist. Sci. 33 (1) (1998), 1-23.
S. Gandz, “A few notes on Egyptian and Babylonian mathematics,” in Studies and Essays in the History of Science and Learning Offered in Homage to George Sarton on the Occasion of his Sixtieth Birthday, 31 August 1944 (New York, 1947), 449-462.
K. Muroi, “Babylonian mathematics - ancient mathematics written in cuneiform writing” (trans. from Japanese), in Studies on the history of mathematics (Kyoto, 1998), 160-171.
G. Sarton, “Remarks on the study of Babylonian mathematics,” Isis 31 (1940), 398-404.

Understanding how randomly-moving objects arrive at a certain destination is no secret to scientists today. But no theory, until now, could predict the time it would take for an object to move between given addresses in a complex environment, like through the human body or the World Wide Web. Previous models only explained the passage of time when the event occurred in a homogenous environment, like in a vacuum or in a glass of water.

Could this mean we are going back to a Laplacian universe of sorts?

The following is Michel Serres's essay "Eternal Return" in Hermes IV: Distribution. Paris: Les Editions de Minuit, 1977. pp. 115-124. Original translation by Taylor Adkins on 10/10/07

Philosophers glorify Nietzsche for having suddenly rejoined the Greeks through their fulgurating intuition of the Eternal Return. Either from an ignorance of ethics or incomprehension of the general figure that this thesis takes in his philosophy, I reduce this to a vision of the world. Vision with the meaning of sight, and world with the sense of the world. All simply.

If time is considered in geometrical figures by optical interceptions and a mechanism of movements, the Eternal Return is cosmological. In that case, the solar system (and it only) has been calculated by Laplace. Celestial Mechanics and the Exposition of the System of the World established rigorously, for the first time, the mechanical invariability of the large axes for the planetary orbits. The stars turn forever. This eternal return reduces the world to the exclusion of the universe, and reduces mechanism to the exclusion of other sciences. Neither the Greeks nor the classical age ever obtained this demonstration. Conversely, the time that we consider is reversible.

If the time that one endorses is that of a formation, of bodies as spheres, and with which one tries to surpass mechanical reversibility, then, if there is return, it is cosmogonic. However, cosmogony enters science little by little around the middle of the 18^th century, with Thomas Wright and Buffon. If Laplace has erased the latter in the seventh note of the Exposition, the former has inspired Kant. Natural History and the Theory of the Sky marks the appearance of the Eternal Return in scientific cosmogony.

However, again, Laplace like Kant (at the origin of hypotheses that an ignorant history combines, but who have only this point in common) starts from a primary nebular state, the dissemination in any point of the space of a cloud of particles. Return to Epicurus, Leucippus and Democritus, reintroduction of atomism in the exact sciences again and again.

The Eternal Return as a vision of the world is a crushing intuition either from Laplace, if it is cosmological, or from Kant, if it is cosmogonic. And the return to the Ionian physicists is an accomplished gesture both by Laplace and Kant. Nietzsche got up at midday and his predecessors at dawn. Hence, it is shown that the world of the philosophers belongs to people who themselves rise late, in order to speak about the sun.

Kant takes the word cosmogony with the literal sense of production or genealogy of an order. Throughout the most general course of natural history, a distribution becomes systematic. The history of the world produces the order of the world. And this history is natural because the laws that work through it are interior to matter and space, and not exterior to them. In order for cosmogony to be the primary ground where one makes physical and mechanical laws function historically, without them exceeding the field of their jurisdiction. Leibniz produces the world through metaphysical mechanism. Kant produces it by a mechanical physics. He obtains a reciprocal application of exact knowledge over time. The law of the system, currently effective in the established order, is originally a law of distribution: displaced in time, it produces the contemporary system and the contemporary order and is not produced by them. It is not uninteresting to observe that before the birth of the century of its history the solution to some of its problems was indicated in the broadest order and the longest time. Better than the solution, the most general condition for a family of problems and solutions: the reciprocal application of the natural encyclopedic ensemble on a chronic line of formation. The application in one sense, and for a human time, names itself the history of the sciences; in the other, and generalized with the totality of possible time, it names itself a cosmogony. In the middle of the 18^th century, both disciplines appear at the same time. The first two examples of the operator (x-logy, x-gony) are the most general.

Time is no longer produced through a system, the system is through time. There was a time when the system had no place. This time one fears naming pre-systematic. It is cut out from its successor, this question is posed, already. It exists in parallel with the catastrophic hypotheses in order to explain the formation of the world (Buffon and the wrenching of the sun by the shock of a comet, of a torrent of matter, and of the discontinuist hypotheses to explain the appearance of the sciences). The idea of an epistemological cut is proper at these times, as in Kant, Comte and many others. The continuist hypothesis also exists. The quarrel over Greek science is only one example. Does the continuation of the Egyptians and the Babylonians, the harpedonaptes and the magi, intersect with these by a miracle? But, about that, we will speak elsewhere.

Cosmogony is the natural path which goes from the distribution to the system. The original distribution is a dissemination, a dispersion of atomic particles, in the manner of a Democritean chaos. A relatively homogeneous occupation of any space by a thin material of low density. The system, on the contrary, separates relatively dense clouds from the enormous and relatively vacuous lacunae. One already presents paradoxical definitions of order and disorder: chaos is relatively homogeneous, the system is the heterogeneous difference; where is the order? Geometrically, I want to say in position and site, the separated clusters are not disseminated more randomly—and, again, it doesn’t matter where, it is aleatory everywhere, it is the same homogeneity—, they are gathered and are bound in an exact area of space, according to the perspective of whether one draws a plane from our point of view by the crown of the Milky Way; prolonged ad infinitum, this plane appears to attract the greatest number of stars, all the more rare as some are very dispersed. Kant names this provision a systematic distribution. However, it is the same for the system of the world in the narrow sense: the spheres of planets are traced all the more closely to the vicinity of the celestial equatorial plane. They occupy, in the sphere, a very narrow crown. Hence a succession of analogies which reverse the opinion of those who believed in the words of those who announced the Copernican revolution as a decisive blow, a trauma driven to the heart of human narcissism. There are unnoticed analogies along the common plane of systematic distribution. The solar system finds itself everywhere. In the largest: the stars pile up around the common plane of our galaxy; the nonstellar spots are themselves milky ways, seen sometimes as circular and sometimes as elliptical, which proves that their components are ordered around a common plane. In the smallest: the matter of a star in the star is maximized in the vicinity of the equatorial plane of rotation, the satellites are distributed similarly around their planet, and the elementary particles around other atoms. Kant thus approximates the model of Böhr. What, then, is a system? Any provision of matter similar to the solar system, i.e. any provision similar to the system of the Earth and its satellite, the Moon. Narcissism is in excellent health, thank you. Consequently, and mechanically, a system is a system only because it is centered. Rotation is the systematic movement par excellence and the only one. Terrestrial movement has a geocentric center, the lunar movement has a center, the Earth, their common movement and that of the planets has a center, the Sun, its movement, real, the stars, seemingly fixed for the observer on a historical scale [1], revolve around a common center, each nebula has its own, as one can see it, without much else nearby. Consequently, any cluster is already proper in the mode of Böhr, and the total system of the universe refers to a single pole. There is a center of the universe. By this total system of recommenced analogies, one wonders if cosmogony is not in a way motionless in time, nearly stable, but only homothetic in space. Hypothesis, homothesis. The question here would be put: how the planetary diagram, that of the sun, but also mine, I am and have the moon, how this diagram, given at the level of the particles, ends up reproducing itself for the totality of the Universe. But it is already reproduced, from the large to small, from the system to the particle. One looks in the path of the plentiful multiplicity from the small systems to the largest unity. But this way, one has already traced it in the other direction. The former is none other than the small, hidden, the invisible lower part, bearing in itself, already, the centered system, structure, as one says, of the visible, in fact reproducing those. Homothesis, hypothesis. The disseminated atoms, the semina, are less corpuscles than munduscules. This immobility of reproduction is the theoretical inscription of the eternal return, for the first time. The origin of the macrocosm is the microcosm. Production and reproduction. That is false, of course, under the terms of the principle of Galileo: things do not reproduce in size in the same way. Few philosophical systems are informed by it. Space is heterogeneous to the laws. In short, what is a system, here? A planned unity, on a common plane, for space, and centered, with a common pole, for movement. All this without exterior. Isn't this definition, valid for a system of the Universe, valid universally, for any system? Universally, I understand by this, for our culture.

From the distribution to the system. The distribution is in a Democritean cloud. Chaos redistributes in a multiplicity of insulated clusters, making a quasi-vacuum around planets or future stars, by the double effect of Newtonian attraction and its opposite repulsion. Primary mechanical fault: if, indeed, the distribution, either in its atomic elements, or in the dispersed clusters, is devoid of initial speed, it can only gather, in the long term, into a single mass. Parti, as well, has a primary nebula. Laplace avoids this fault: his chaotic cluster is made of an elastic gas turning with the same angular velocity as its central condensation, under the terms of an original movement whose cause eludes mechanism or, at least cosmogony. His nebula is a nebula in the cosmological sense, not with the meaning of Democritus, but in the sense of Messier or Herschell. It is through this movement that the differenciation of planets is produced on the level of the solar equator. In other words, either the circular motion is primary, given, unengendered, perpetuating into a cosmological stability; or it is only secondary, generated by two forces starting from a stable distribution, when fatally, a second time, the cosmogonic eternal return enters into matter. If the nebula in Democritus is given by itself from a capacity to forming itself in a differenciated-centered [différencié-centré] system, the cosmogonic eternal return is already there. Laplace avoids this by reduction: it is given at the beginning with a nebula centered around the movement of rotation.

I said that, fatally, Kantian chaos would collect in a single mass. Theory of the Sky envisions this, but only at the end of the world. All of time from a differenciated, multiple and rotary system, cosmological time, which should be quasi-null from the perspective of the hypotheses, is widened by the enormous time of the world. The question is of evaluating the duration of the parentheses. A few seconds or a billion. Let us come to pulsation.

There exists, as we have seen, a geometrical center of the whole universe, a single pole for its movements. By the parallelism of space and time, this umbilical point is originary. Among primitive chaos, it is a matrix of the system. In order for it to be a geometrical and mechanical center, it is necessary that the greatest mass of the universe is amassed here. The differenciation and the relative density of the particles in a cloud are dispersed on a scale which presents a maximum. This heaviest atom attracts with it the strongest mass. Therefore, by sequence, the most powerful attraction, and so on. The pole is spatial, and the focus is mechanical, because the center is material. In a cosmogony of matter, movement, space, it is necessarily the maximum navel of formation, and any other local center relates and refers to it. The remarkable thing, for narcissism at least, is that we live within the nearest vicinity of all of this: since it is impossible for us to still see abstract chaos, far from its sphere of influence, we see only systems.

The formation of the world is thus not an isolated event, an instantaneous operation. It is ceaseless, it is a continuous formation. The center, constituted once and for all, diffuses its efficacy on the surrounding chaos. It forms a local, spherical system, occupied especially in its equatorial crown, where the local clusters turn around it with increasing eccentricities. At the limit, at the edge of the sphere, the planets become comets, and little by little, the comets escape from the path of system. Chaos still reigns at the exterior of the formed sphere. But, as attraction is unbounded, this edge is common to the perfect, cosmological system within the chaos, which is no longer primitive but indefinitely contemporary, this edge is itself effective, it extends, it gradually systematizes the Democritean distribution. "Creation is never terminated. It began one day, but it will never finish.” The innovation, the formation is instead the membrane between the form and the formless. Hence the expanding universe, the term is entirely Kantian. What is present and unceasingly effective: the center and the periphery, it is here and it is there. Hence the law of time: the closer the event takes place to the pole, the shorter it is, until it crushes the time of systematic formation; as it is narrow, more so is it long. We found a similar law in the hypothesis of the big-bang where the duration of production set out again in three states, a negligible fraction of a second, a few seconds, and a billion years. Consequently, the theory of the sky lengthens the duration with a deviation of the distance to the center—the formation increases its patience.

That is to say maintaining the state of the world at a time T, unspecified, in the continuation of the centuries. At this moment corresponds a distance D considered from the center towards the edge of the mobile crown. It is the moment of formation of local clusters on the periphery: there it is the youngest world. On the radial of a length D, by running from the edge towards the center, the worlds already born are increasingly old. And the oldest is in the vicinity of the pole. At a precise time, to which a precise length responds, this oldest world starts to die. The differenciated bodies pile up towards the center, fall, and are its ruin. But this aging and death which follow gradually reach extremely long distances. The pole is then, and at the same time, the navel of formation for the most remote periphery, and the vortex of death for a close sphere. The worlds formed for a time T are bordered by the ruins of the dead world and the chaos of formless nature. The system is a spherical, radially mobile crown, of which the interior has died and the exterior will be born. Expanding on one side, degradation on the other. Hence the model: the center, of life and death; the distribution, a solid crown, the system; a second distribution. Cosmogony, in fact, goes from one dissemination to another by way of a system. Cosmology is between two cosmogonies.

How does this occur in reality? Any body in space slows down in its rotational movement around the center, or by the residual particles of space, or by the actions of tides. Let us notice, with this passage, that with the Dialectic of Nature (p. 52) Engels glorifies Kant for “two brilliant assumptions:" primary chaotic distribution and the deceleration of rotation by the periodic pull of the seas. They are both principal requisites for the Return in Theory of the Sky. There is a cosmological eternal return, with the manner of Laplace, that in the condition of eliminating the physical constraints of the stars, considers them as points or mechanical solids. In the contrary case, Kant, I believe, had some hesitancy, a fear of perpetual movement of the first type. Consequently, in the long term, the fall of bodies on their central sun becomes inevitable. It accrues in a fulgurating way a cooled incandescence: extraordinary expansion, explosion, new dissemination in space of the material particles constituting the whole of these bodies. After a time when the cooling is achieved, we find the chaos of Epicurus. The first state, the last state. The process, then, starts again. Around the palingenetic center, the new distribution, resulting from the system, reforms a system, expands through its periphery, whereas the systems, arriving, time after time to their hour of death, turn over to the distribution. Second fault, to wait until the heat cools: it is the perpetual movement of the second type "the phoenix of nature burns only to revive from its ashes." Center, system, distribution. Center, distribution, system, distribution. Thus ad infinitum. A stone in the water, a maintained focus, circles, successive waves which are not arrested.

Both models are the projection of the sphere on its common level of systematic distribution. Each systematic crown is limited by two crowns of distribution, and vice versa. There are two fundamental states, according to whether in the vicinity of the center there is a system or a distribution. At the exterior, there is always distribution, stable, on standby for the action of the center, primary, eternal.

This universe of pulsations combines in a common fabric three exemplary concepts of the history of sciences. The fixed point, the fixed plane, and the cloud. They appear in this order in the course of the chronology, and they are even combined in this order in Kantian cosmogony. The cloud is quite primitive, but it breaks down instantaneously, as soon as the center, which is primitive, in its turn appears, including the remainder eternally. It is then ejected to the exterior, on standby as stock for all the new worlds to come. And reconstituted by the death of worlds while returning to the center. The Epicurean cloud is matrix and corpse, dustbin and rebirth. Absolutely primitive, absolutely last, and periodically interstitial. The center is generative, umbilical, productive, mortal. As for the crowns of systems, they are legalized by relative points, as dispersed centers referred primarily to the pole, the relatively fixed points and the plane respective to their distribution. It is on a common plane that one recognizes that a distribution is not a cloud, and it is at this common point that one recognizes that it is a system. The fixed plane is a transition, chronologically dated, between the fixed point and the cloud. That is true for Kantian cosmogony, and that remains true for the history of the sciences. This is what I wanted to demonstrate.

One then obtains the triangle according to:

D

C D

C S D

C D S D

C S D S D

C D S D S D

C S D S D S D

The distribution is the first state, it breaks down instantaneously, and the center is formed all at once. This center forms a system within its vicinity and the distribution remains outside. Then the crowns start to be produced by pulsations. The distribution cloud of the preceding system is the distribution stock of the following system. What remains stable: the center, the exterior distribution, the universal stock, ad infinitum. What changes is the triangle in the triangle. The universe is expanding but the model is quasi-stationary. Kant links, in one stroke, both fundamental cosmogonic models, by a curious pre-critical realization of the Transcendental Dialectic. Since each crown passes, by pulsation, from a phase of system to a phase of distribution and conversely, the Eternal Return is the operator and the engine of the expansion.

Poinsot, let us know it, established the statics of the solar world around the invariability of an equatorial plan. The common plane, in Kant, established as an intermediary between the dissemination and the center, marks the average concept of systematic distribution. The order is reversed, in celestial statics, and the plane and the couple relativizes the old royal function of the pole. But the end of the Memories of the Equator enlarges the question beyond the planetary system. Let us suppose for a moment that the fixed plane, eternal, invariant, is precisely subjected to variations. Like Bradley or everyone else, the swing of systematic constants leads Poinsot to find cosmogonic time. (I truly believe that it will be said one day of Bradley that, to have made the stars move, our final sitting, our reparable confidence, introduced us to the century of generalized suspicion, by a cosmogony at the time of formation, by this pre-systematic time of interpretation, and by the swing of the suns to the philosophy of universal attraction). Nevertheless, the fixed plane would move imperceptibly. If so, an action foreign to the system has efficacy over it. Our sun depicts itself as a sphere slowly revolving around some remote center. Then a very small couple alters the position of the general couple registered on the invariant table. The demonstration starts again, through repetition of the engine of the history of the sciences: a renewed swerve between invariance and variation. A new plane fixes by widening, a new center, a new major couple. And so on. Can one conceive an end of history? Yes, if there is an end of the space of the world, i.e. a total system, absolutely independent of an exterior action which could disturb its movements. Let us suppose that this exists. It joins a common center of gravity. It is the center of Kant. It is in rest. There is no reason why it goes here rather than there, no conceivable reason for Bradley that anything is there at all, since the system is without exterior. Then, through symmetry, all the forces are balanced: the great whole is perfectly motionless in absolute space. You deposit yourself now, at least in thought, at this point. You would not see any described surface that is not counterbalanced by another. You will thus be unable to determine any plane. If there is an absolute center, there is no longer a plane. At the limit, any cosmology disappears. The center can only be the origin of relative movement. Pass to the limit, to the end of history or to the end of space, and any science evaporates. It is thus only a science that is relative, from a determined cut in an exterior put outside of parentheses: only our system concerns us, it is relative, but infinitesimally altered, by our usage and from our science. Its quasi-invariance measures the solidity of knowledge, and its utility in relation to us. Positivism comes from being born. Poinsot teaches mechanism to a young polytechnician. August Comte, who, again, and like everyone else in the 19^th century, will announce an Eternal Return. Astronomy installs, for the century of history and thermodynamics, the most universal running-time meters: they are eternities.

[1] The idea of a common plane of distribution is borrowed by Kant from Thomas Wright: the idea of the movement of the stars is from Bradley. He never stopped celebrating Bradley.

One talks a great deal about the Copernican revolution. Has Copernicus established the heliocentric system? The response is evidently negative. Its reasoning could be only geometrical and kinetic. They were thus in dispute because of the equivalence of hypotheses, already stated by the Greek astronomers. Hence the hesitation of the classical age: Descartes, Pascal and Leibniz. These three, contrary to the legend, were not afraid of prison. Two among them, in effect, defended infinitism, just like Giordano Bruno who was burnt at the stake. If they feared the prison of Galileo, why weren’t they terrified by the butchery of Bruno? In fact, they applied the only scientific method, knowledge of the equivalence of hypotheses, geometrical and kinetic. The Copernican solution was elegant, but undemonstrated.

It was the same with Bradley. For those who inserted evidence of the phenomena of the aberration of the light, it was physically true that the sun was in the center. Not geometrically or mechanically, but as a physical phenomenon. However, two philosophers read Bradley: Kant and Comte. Even if one will forgive me, I will thus go back only to this time.

Géomètre de première catégorie, Laplace n’a pas tardé à se montrer un administrateur plus que médiocre ; de son premier travail nous avons immédiatement compris que nous nous étions trompés. Laplace ne traitait aucune question d’un bon point de vue : il cherchait des subtilités de partout, il avait seulement des idées problématiques et enfin il portait l’esprit de l’infiniment petit jusque dans l’administration.

(Napoléon Bonaparte)

Laplace (Pierre-Simon, marquis de), professeur, membre de l'Institut, législateur et ministre, né à Beaumont-en-Auge (Calvados), le 23 mars 1749, de "Pierre Laplace et de Marie-Anne Sochon", mort à Paris le 5 mars 1827 ; débuta comme professeur de mathématiques à l'Ecole Militaire ; fut admis à l'Académie des sciences en 1773, comme membre adjoint, et comme titualire en 1785 ; devint, sous la Révolution, en l'an II, professeur d'analyse aux écoles normales, puis membre et président du bureau des Longitudes ; était à cette époque très ardent républicain et adversaire de la dictature ; mais au 18 brumaire, il n'hésita cependant pas à se rallier à Bonaparte et fut nommé, le 3 nivôse an VIII, membre du Sénat conservateur, puis remplit pendant quelques jours, le poste de ministre de l'Intérieur, mais fut vite remplacé par Lucien Bonaparte ; devint vice-président du Sénat et chancelier en l'an XI, fut fait membre de la Légion d'honneur le 19 vendemiaire an XII, grand officier le 25 prairial suivant, créé comte de l'Empire le 24 avril 1808 et vota, en avril 1814, la déchéance de l'Empereur. Louis XVIII, reconnaissant, le nomma pair de France, le 4 juin 1814. Le Marquis de Laplace se tint à l'écart pendant les Cent-Jours ; reprit, sous la seconde Restauration, son siège à la Chambre haute, vota la mort du maréchal Ney, entra à l'Académie des sciences à sa réorganisation, en 1816, devint membre de l'Académie française. On a de lui : Exposition du système du monde ; Théorie analytique des probabilités, etc.

(Extrait du dictionnaire sur la Révolution et l'Empire du Dr Robinet)

Lien : Pierre-Simon Laplace sur Wikipedia

So, to whoever is wondering, here goes:

1. The Laplace transform is an easy way of solving differential equations. Complex engineering systems can have higher order differential terms in them. In the Laplace domain, this is reduced to a polynomial of that order, which, while difficult to factor (by hand) is much easier than the differential equation.
2. It turns convolution into multiplication. Electrical Engineering students everywhere cheer.
3. Stability is obvious. Once we find the roots of the denominator of the Laplace transform of a system, we know immediately if the system is stable or not.
4. Frequency response. The Fourier transform is a special case of the Laplace transform, so frequency response of a system or the frequency content of a signal can be determined. Systems can be designed by placing poles (roots of the denominator polynomial) and zeros (roots of the numerator polynomial) along the imaginary axis in the s-plane until the desired system response is obtained.
5. There are lots of books of tables allowing us to quickly perform the transform without doing too much math.

I'm sure there are other reasons, and engineers with different backgrounds will have more answers. Overall, the Laplace transform is a handy tool, but it is one of many. Controls engineers like it, but they use other tools as well. I'm a wireless and digital signal processing, so I tend to use the Fourier transform and the z-transform (the digital equivalent of the Laplace). If all you know is the Laplace, you really don't know much. But it is a very useful tool to have in the box.

If you do have a question such as this, feel free to ask. But no, I won't write the simulation for your Master's thesis, or Ph.D. dissertation, or do your homework for you. I will try to provide simple explanations of why things are done the way they are.

Pues bien, imaginemos que las piezas de ajedrez son las partículas elementales y recordemos al demonio de Laplace.

El mundo que nos ofrece el análisis retrospectivo es un universo laplaciano.

Es más aún que un universo laplaciano, porque es un universo estático y Laplace pedía saber la posición y la velocidad de los átomos.
Al demonio de Laplace, pues, le basta echar una ojeada al tablero del universo para conocer el pasado, porque sabe de qué modo se puede mover cada pieza.

También se podría decir que el universo que plantean los problemas tradicionales de ajedrez (los que se preguntan por el futuro) también presentan un universo determinista. Sin embargo, hay que hacer un matiz importante. Un problema de ajedrez tradicional plantea, por ejemplo: "Cómo dar mate en tres jugadas, pero no se pregunta: ¿Cuáles van a ser las próximas tres jugadas? Es decir, tiene en cuenta una intención: la de dar jaque mate. Porque es evidente que, si no se tiene esa intención, no se puede asegurar cuáles serán las próximas jugadas: alguien puede limitarse a mover una pieza adelante y atrás, por ejemplo.

El determinismo de los problemas de ajedrez tradicionales es, pues, un determinismo marcado por el teleologismo, por el fin que uno se propone (ganar la partida). Con ello se plantea la cuestión de las relaciones entre teleologismo y determinismo (algunos autores consideran incompatibles ambas posibilidades: un universo teleologista no es determinista y un universo determinista no es teleológico).
Volvemos al análisis retrospectivo. Nos hallamos ante un universo determinista que simboliza el planteado por Laplace. ahora bien, como ya se ha dicho antes, esa posibilidad determinista se da porque sólo hay una posible solución para cada problema.

Y detrás de este hecho, que sólo haya una solución, se halla una conciencia, en este caso la del señor Raymond Smullyan. Si no hubiese un ente consciente que se encargase de eliminar cualquier otra posible solución, sería altamente improbable que ante un tablero dea jedrez cualquiera, por ejemplo el de la partida número tres del match Karpov-Korchnoi en Baguio, se pudiese responder con una certeza del 100% a la pregunta: ¿Qué ha pasado en las tres últimas jugadas?
...
Ahora recordemos el principio de indeterminación de Heisenberg.
[Aquí acaba el texto abruptamente]