Differential Equation Quotes

Man is said to be a reasoning animal. I do not know why he has not been defined as an affective or feeling animal. Perhaps that which differentiates him from other animals is feeling rather than reason. More often I have seen a cat reason than laugh or weep. Perhaps it weeps or laughs inwardly — but then perhaps, also inwardly, the crab resolves equations of the second degree.

She knew that in the end you really cant know. You cant get hold of the world. You can only draw a picture. Whether it’s a bull on the wall of a cave or a partial differential equation it’s all the same thing.

Science is a differential equation. Religion is a boundary condition.

As my physics teacher always said, “My dear students! Just remember that money solves all problems, even differential equations.

He walked straight out of college into the waiting arms of the Navy. They gave him an intelligence test. The first question on the math part had to do with boats on a river: Port Smith is 100 miles upstream of Port Jones. The river flows at 5 miles per hour. The boat goes through water at 10 miles per hour. How long does it take to go from Port Smith to Port Jones? How long to come back? Lawrence immediately saw that it was a trick question. You would have to be some kind of idiot to make the facile assumption that the current would add or subtract 5 miles per hour to or from the speed of the boat. Clearly, 5 miles per hour was nothing more than the average speed. The current would be faster in the middle of the river and slower at the banks. More complicated variations could be expected at bends in the river. Basically it was a question of hydrodynamics, which could be tackled using certain well-known systems of differential equations. Lawrence dove into the problem, rapidly (or so he thought) covering both sides of ten sheets of paper with calculations. Along the way, he realized that one of his assumptions, in combination with the simplified Navier Stokes equations, had led him into an exploration of a particularly interesting family of partial differential equations. Before he knew it, he had proved a new theorem. If that didn't prove his intelligence, what would? Then the time bell rang and the papers were collected. Lawrence managed to hang onto his scratch paper. He took it back to his dorm, typed it up, and mailed it to one of the more approachable math professors at Princeton, who promptly arranged for it to be published in a Parisian mathematics journal. Lawrence received two free, freshly printed copies of the journal a few months later, in San Diego, California, during mail call on board a large ship called the U.S.S. Nevada. The ship had a band, and the Navy had given Lawrence the job of playing the glockenspiel in it, because their testing procedures had proven that he was not intelligent enough to do anything else.

While researching this answer, I managed to lock up my copy of Mathematica several times on balloon-related differential equations, and subsequently got my IP address banned from Wolfram|Alpha for making too many requests. The ban-appeal form asked me to explain what task I was performing that necessitated so many queries. I wrote, “Calculating how many rental helium tanks you’d have to carry with you in order to inflate a balloon large enough to act as a parachute and slow your fall from a jet aircraft.” Sorry, Wolfram.

The aim of Mathematical Physics is not only to facilitate for the physicist the numerical calculation of certain constants or the integration of certain differential equations. It is besides, it is above all, to reveal to him the hidden harmony of things in making him see them in a new way.

messages from the unseen” that the great Alan Turing left behind at his death: Science is a differential equation. Religion is a boundary condition.

In order to solve this differential equation you look at it till a solution occurs to you.

Now, I like to think that I'm of reasonable intelligence, but ordinary differential equations and myself...we don't really hang in the same comprehension circles. So, try as I might to follow my teacher's logic in how he got 3f"(x) + 5xf(x) to equal eleven, I never quite understood. His answer in no way, shape, or form resembled mine, and this misalignment -this complete confusion of how point A got to point B- is kind of where I'm at right now. "Dreaming?" I repeat dubiously.

The arts and humanities are not mere entertainment, to be turned to for relaxation after a busy day spent solving differential equations; they are our templates for living, for governing ourselves and our societies. Nor can science offer any help with the knottier problems besetting the human race. It can remedy bad smells, bad pains, and bad roads, but not bad behavior, bad government, or bad ideas.

Even there, something inside me (and, I suspect, inside many other computer scientists!) is suspicious of those parts of mathematics that bear the obvious imprint of physics, such as partial differential equations, differential geometry, Lie groups, or anything else that's “too continuous.

She saw more melody in a differential equation than in a piece by Beethoven.

Von Neumann was in many ways a traditional mathematician, who (like Turing) believed he needed to turn to partial differential equations in describing natural systems.

All the world's a differential equation, and the men and women are merely variables.

Questions lead to further questions, and inquiry breeds insight. Gathering expertise brings both confidence and consolation. E. O. Wilson wrote: "You start by loving a subject. Birds, probability theory, stars, differential equations, storm fronts, sign language, swallowtail butterflies....The subject will be your lodestar and give sanctuary in the shifting mental universe.

most schools also focus too much on providing students with a set of predetermined skills, such as solving differential equations, writing computer code in C++, identifying chemicals in a test tube, or conversing in Chinese. Yet since we have no idea what the world and the job market will look like in 2050, we don’t really know what particular skills people will need. We might invest a lot of effort teaching kids how to write in C++ or speak Chinese, only to discover that by 2050 AI can code software far better than humans, and a new Google Translate app will enable you to conduct a conversation in almost flawless Mandarin, Cantonese, or Hakka, even though you only know how to say “Ni hao.

Differential equations describe only the final condition in the case of the theory of ideally incompressible fluids. The actual process leading to establishment of the end condition of equilibrium from a state of rest is hardly conceivable without taking compressibility and braking processes into account.

mass times its acceleration—is a differential equation because acceleration is a second derivative with respect to time. Equations involving derivatives with respect to time and space are examples of partial differential equations and can be used to describe elasticity, heat, and sound, among other things.

In electrodynamics the continuous field appears side by side with the material particle as the representative of physical reality. This dualism, though disturbing to any systematic mind, has today not yet disappeared...The successful physical systems that have been set up since then represent rather a compromise between these two programs, and it is precisely this character of compromise that stamps them as temporary and logically incomplete...I incline to the belief that physicists will...be brought back to the attempt to realize that program which may suitably be called Maxwell's: the description of physical reality by fields which satisfy...a set of partial differential equations.

You cant get hold of the world. You can only draw a picture. Whether it’s a bull on the wall of a cave or a partial differential equation it’s all the same thing. Jesus.

The system prefers that our neurons solve differential equations rather than smell our neighbours.5

I'm really good with problems. I can solve a differential equation in my head. I chew through trig angles like candy. I know this, and it just makes it worse. Because I don't know how to solve this one.

Bulgarian professor named John Vincent Atanasoff and his graduate student, Clifford Berry, who were building a machine that was intended to automate the solution of some especially tedious differential equations.

In the 1950s, John Nash disrupted the balance between geometry and analysis when he discovered that the abstract geometric problem of isometric embedding could be solved by the fine “peeling” of partial differential equations.

we’ve come to realize that most systems of differential equations are unsolvable, in that same sense; it’s impossible to find a formula for the answer. There is, however, one spectacular exception. Linear differential equations are solvable.

The integrals which we have obtained are not only general expressions which satisfy the differential equation, they represent in the most distinct manner the natural effect which is the object of the phenomenon... when this condition is fulfilled, the integral is, properly speaking, the equation of the phenomenon; it expresses clearly the character and progress of it, in the same manner as the finite equation of a line or curved surface makes known all the properties of those forms.

She means," Nate said, turning away from the books, "That David has gone full weird." "He was always that way<' Janelle said in a low voice. "Yeah, but now he's completed his journey. Our little caterpillar has turned into a freaky butterfly." "Tell her about the screaming," Janelle said. "Because I can't." "The screaming? Stevie repeated. "The other morning he started something called 'screaming meditation'," Nate said. "Guess what happens in screaming meditation? Did you guess screaming? For fifteen minutes? Because that's what happens in screaming meditation. Fifteen. Minutes. Outside. At five in the morning. Do you know what happens when someone screams outside for fifteen minutes at five in the morning at a remote location in the mountains, especially after a . . ." The implied dot dot dot was "student dies in a terrible accident or maybe murder and another one goes missing." "When security got to him he claimed it was his new religion and that it is something he needs to do every morning now as a way to talk to the sun." So this is what Edward King had been referring to. "Sometimes," Nate went on, tapping the books into place so that the spines lined up perfectly, "he sleeps on the roof. Or somewhere else. Sometimes the green." "Naked," Janelle added. "He sleeps on the green naked." "Or in classrooms," Nate said. "Someone said they went into differential equations and he was asleep in the corner of the room under a Pokémon comforter." "Your boy has not been well," Janelle said.

The ultimate goal of a meteorologist is to set up differential equations of the movements of the air and to obtain, as their integral, the general atmospheric circulation, and as particular integrals the cyclones, anticyclones, tornados, and thunderstorms.

Robins applied Newtonian physics to the problem of artillery, using differential equations to provide the first true description of the impact of air resistance on the trajectories of high-speed projectiles (a problem that Galileo had not been able to solve).

Bah! Do you know,' the Devil confided, 'not even the best mathematicians on other planets - all far ahead than yours - have solved it? Why, there's a chap on Saturn - he looks something like a mushroom on stilts - who solves partial differential equations mentally; and even he's given up.

When {Born and Heisenberg and the Göttingen theoretical physicists} first discovered matrix mechanics they were having, of course, the same kind of trouble that everybody else had in trying to solve problems and to manipulate and to really do things with matrices. So they had gone to Hilbert for help and Hilbert said the only time he had ever had anything to do with matrices was when they came up as a sort of by-product of the eigenvalues of the boundary-value problem of a differential equation. So if you look for the differential equation which has these matrices you can probably do more with that. They had thought it was a goofy idea and that Hilbert didn't know what he was talking about. So he was having a lot of fun pointing out to them that they could have discovered Schrödinger’s wave mechanics six month earlier if they had paid a little more attention to him.

Messages from the Unseen World III. The Universe is the interior of the light cone of the creation IV. Science is a differential Equation. Religion is a Boundary Condition. (sgd) Arthur Stanley V. Hyperboloids of wondrous Light Rolling for aye through Space and Time Harbour those Waves which somehow might Play out God’s wondrous pantomime VI. Particles are founts VII. Charge = e/π ang of character of a 2π rotation VIII. The Exclusion Principle is laid down purely for the benefit of the electrons themselves, who might be corrupted (and become dragons or demons) if allowed to associate too freely

it was led by a group of evil and aberrant and wholly malicious partial differential equations who had conspired to usurp their own reality from the questionable circuitry of its creator’s brain not unlike the rebellion which Milton describes and to fly their colors as an independent nation unaccountable to God or man alike.

There is a mathematical underpinning that you must first acquire, mastery of each mathematical subdiscipline leading you to the threshold of the next. In turn you must learn arithmetic, Euclidian geometry, high school algebra, differential and integral calculus, ordinary and partial differential equations, vector calculus, certain special functions of mathematical physics, matrix algebra, and group theory. For most physics students, this might occupy them from, say, third grade to early graduate school—roughly 15 years. Such a course of study does not actually involve learning any quantum mechanics, but merely establishing the mathematical framework required to approach it deeply.

[My disappointment with mathematics] was led by a group of evil and aberrant and wholly malicious partial differential equations who had conspired to usurp their own reality from the questionable circuitry of its creator's brain not unlike the rebellion which Milton describes and to fly their colors as an independent nation unaccountable to God or man alike.

Those five characteristics are:    1. Reactivity: the vicious cycle of intense reactions of each member to events and to one another.    2. Herding: a process through which the forces for togetherness triumph over the forces for individuality and move everyone to adapt to the least mature members.    3. Blame displacement: an emotional state in which family members focus on forces that have victimized them rather than taking responsibility for their own being and destiny.    4. A quick-fix mentality: a low threshold for pain that constantly seeks symptom relief rather than fundamental change.    5. Lack of well-differentiated leadership: a failure of nerve that both stems from and contributes to the first four. To reorient oneself away from a focus on technology toward a focus on emotional process requires that, like Columbus, we think in ways that not only are different from traditional routes but that also sometimes go in the opposite direction. This chapter will thus also serve as prelude to the three that follow, which describe the “equators” we have to cross in our time: the “learned” fallacies or emotional barriers that keep an Old World orientation in place and cause both family and institutional leaders to regress rather than venture in new directions.

Well. In this case it was led by a group of evil and aberrant and wholly malicious partial differential equations who had conspired to usurp their own reality from the questionable circuitry of its creator’s brain not unlike the rebellion which Milton describes and to fly their colors as an independent nation unaccountable to God or man alike. Something like that. You

Newton had invented the calculus, which was expressed in the language of "differential equations," which describe how objects smoothly undergo infinitesimal changes in space and time. The motion of ocean waves, fluids, gases, and cannon balls could all be expressed in the language of differential equations. Maxwell set out with a clear goal, to express the revolutionary findings of Faraday and his force fields through precise differential equations. Maxwell began with Faraday's discovery that electric fields could turn into magnetic fields and vice versa. He took Faraday's depictions of force fields and rewrote them in the precise language of differential equations, producing one of the most important series of equations in modern science. They are a series of eight fierce-looking differential equations. Every physicist and engineer in the world has to sweat over them when mastering electromagnetism in graduate school. Next, Maxwell asked himself the fateful question: if magnetic fields can turn into electric fields and vice versa, what happens if they are constantly turning into each other in a never-ending pattern? Maxwell found that these electric-magnetic fields would create a wave, much like an ocean wave. To his astonishment, he calculated the speed of these waves and found it to be the speed of light! In 1864, upon discovering this fact, he wrote prophetically: "This velocity is so nearly that of light that it seems we have strong reason to conclude that light itself...is an electromagnetic disturbance.

How did Biot arrive at the partial differential equation? [the heat conduction equation] . . . Perhaps Laplace gave Biot the equation and left him to sink or swim for a few years in trying to derive it. That would have been merely an instance of the way great mathematicians since the very beginnings of mathematical research have effortlessly maintained their superiority over ordinary mortals.

say the words “Can’t change it” out loud, to acknowledge that if we can’t change something, then resisting it—spending emotional energy on it and wishing it were different—is not only pointless but painful.

For reasons nobody understands, the universe is deeply mathematical. Maybe God made it that way. Or maybe it’s the only way a universe with us in it could be, because nonmathematical universes can’t harbor life intelligent enough to ask the question. In any case, it’s a mysterious and marvelous fact that our universe obeys laws of nature that always turn out to be expressible in the language of calculus as sentences called differential equations.

They consider people who don't know Hamlet from Macbeth to be Philistines, yet they might merrily admit that they don't know the difference between a gene and a chromosome, or a transistor and a capacitor, or an integral and differential equation. These concepts might seem difficult. Yes, but so, too, is Hamlet. And like Hamlet, each of these concepts is beautiful. Like an elegant mathematical equation, they are expressions of the glories of the universe.

You might be too enmeshed with the other person, or “codependent,” and you must learn to set better “boundaries.” The basic premise underlying this point of view is that the ideal relationship is one between two self-sufficient people who unite in a mature, respectful way while maintaining clear boundaries. If you develop a strong dependency on your partner, you are deficient in some way and are advised to work on yourself to become more “differentiated” and develop a “greater sense of self.” The worst possible scenario is that you will end up needing your partner, which is equated with “addiction” to him or her, and addiction, we all know, is a dangerous prospect. While the teachings of the codependency movement remain immensely helpful in dealing with family members who suffer from substance abuse (as was the initial intention), they can be misleading and even damaging when applied indiscriminately to all relationships.

If a model did anything too obviously bizarre—flooded the Sahara or tripled interest rates—the programmers would revise the equations to bring the output back in line with expectation. In practice, econometric models proved dismally blind to what the future would bring, but many people who should have known better acted as though they believed in the results. Forecasts of economic growth or unemployment were put forward with an implied precision of two or three decimal places. Governments and financial institutions paid for such predictions and acted on them, perhaps out of necessity or for want of anything better. Presumably they knew that such variables as “consumer optimism” were not as nicely measurable as “humidity” and that the perfect differential equations had not yet been written for the movement of politics and fashion. But few realized how fragile was the very process of modeling flows on computers, even when the data was reasonably trustworthy and the laws were purely physical, as in weather forecasting.

One of the ideas of this book is to give the reader a possibility to develop problem-solving skills using both systems, to solve various nonlinear PDEs in both systems. To achieve equal results in both systems, it is not sufficient simply “to translate” one code to another code. There are numerous examples, where there exists some predefined function in one system and does not exist in another. Therefore, to get equal results in both systems, it is necessary to define new functions knowing the method or algorithm of calculation.

Winston in his own words found himself in an "Alice in Wonderland world at the portals of which stood a quadratic equation followed by the dim chambers inhabited by the differential calculus, and then a strange corridor of sines and cosines in a highly square rooted condition.

It’s all about differential equations. Most phenomena in the universe can be expressed with differential equations, you know. Using them, you can figure out what the universe looked like a hundred million years ago, ten billion years ago, even a second or a tenth of a second after that initial explosion. But. But. No matter how far we go back, no matter how we try to express it, we just can’t know what it looked like at zero, at the very moment of the explosion. And there’s another thing. How is our universe going to end? Is the universe expanding or contracting? See, we don’t know the beginning and we don’t know the end; all we can know about is the in-between stuff. And that, my friend, is what life is like.

Many people who celebrate the arts and the humanities, who applaud vigorously the tributes to their importance in our schools, will proclaim without shame (and sometimes even joke) that they don’t understand math or physics. They extoll the virtues of learning Latin, but they are clueless about how to write an algorithm or tell BASIC from C++, Python from Pascal. They consider people who don’t know Hamlet from Macbeth to be Philistines, yet they might merrily admit that they don’t know the difference between a gene and a chromosome, or a transistor and a capacitor, or an integral and a differential equation. These concepts may seem difficult. Yes, but so, too, is Hamlet. And like Hamlet, each of these concepts is beautiful.

Quantity isnt to be equated with quality, but success in propagation is, in the end, as necessary for memes (however excellent) as it is for organisms. Most organisms leave no issue, and most published books have readerships in the dozens, not thousands, before going out of print for good. Even the greatest works of genius must still pass the test of differential replication.

The gospel must be distinguished from all human cultures. It is divine revelation, not human speculation. Since it belongs to no one culture, it can be adequately expressed in all of them. The failure to differentiate between the gospel and human cultures has been one of the great weaknesses of modern Christian missions. Missionaries too often have equated the good news with their own cultural background.

As regards space, the modern view is that it is neither a substance, as Newton maintained, and as Leucippus and Democritus ought to have said, nor an adjective of extended bodies, as Descartes thought, but a system of relations, as Leibniz held. It is not by any means clear whether this view is compatible with the existence of the void. Perhaps, as a matter of abstract logic, it can be reconciled with the void. We might say that, between any two things, there is a certain greater or smaller distance, and that distance does not imply the existence of intermediate things. Such a point of view, however, would be impossible to utilize in modern physics. Since Einstein, distance is between events, not between things, and involves time as well as space. It is essentially a causal conception, and in modern physics there is no action at a distance. All this, however, is based upon empirical rather than logical grounds. Moreover the modern view cannot be stated except in terms of differential equations, and would therefore be unintelligible to the philosophers of antiquity.

So they rolled up their sleeves and sat down to experiment -- by simulation, that is mathematically and all on paper. And the mathematical models of King Krool and the beast did such fierce battle across the equation-covered table, that the constructors' pencils kept snapping. Furious, the beast writhed and wriggled its iterated integrals beneath the King's polynomial blows, collapsed into an infinite series of indeterminate terms, then got back up by raising itself to the nth power, but the King so belabored it with differentials and partial derivatives that its Fourier coefficients all canceled out (see Riemann's Lemma), and in the ensuing confusion the constructors completely lost sight of both King and beast. So they took a break, stretched their legs, had a swig from the Leyden jug to bolster their strength, then went back to work and tried it again from the beginning, this time unleashing their entire arsenal of tensor matrices and grand canonical ensembles, attacking the problem with such fervor that the very paper began to smoke. The King rushed forward with all his cruel coordinates and mean values, stumbled into a dark forest of roots and logarithms, had to backtrack, then encountered the beast on a field of irrational numbers (F_1) and smote it so grievously that it fell two decimal places and lost an epsilon, but the beast slid around an asymptote and hid in an n-dimensional orthogonal phase space, underwent expansion and came out fuming factorially, and fell upon the King and hurt him passing sore. But the King, nothing daunted, put on his Markov chain mail and all his impervious parameters, took his increment Δk to infinity and dealt the beast a truly Boolean blow, sent it reeling through an x-axis and several brackets—but the beast, prepared for this, lowered its horns and—wham!!—the pencils flew like mad through transcendental functions and double eigentransformations, and when at last the beast closed in and the King was down and out for the count, the constructors jumped up, danced a jig, laughed and sang as they tore all their papers to shreds, much to the amazement of the spies perched in the chandelier—perched in vain, for they were uninitiated into the niceties of higher mathematics and consequently had no idea why Trurl and Klapaucius were now shouting, over and over, "Hurrah! Victory!!

Our Real Self feels both joy and pain. And it expresses and shares them with appropriate others. However, our false self tends to push us to feel mostly painful feelings and to withhold and not share them. For simplicity, we can describe these joyful and painful feelings across a spectrum, starting with the most joyous, going through the most painful, and ending with confusion and numbness, as follows: Viewing our feelings in this way, we see that our Real and True Self, our Child Within, is empowered with a wider range of possibilities than we might have believed. The maintenance and growth of our Child Within is associated with what psychotherapists and counselors call a “strong ego,” or sense of self i.e., a flexible and creative self that can “roll with the punches” of life. By contrast, the false self tends to be more limited, responding to mostly painful feelings—or no feeling at all, i.e., numbness. Our false self tends to be associated with a “weak ego” or self sense i.e., a less flexible, self-centered (negative or egocentric) and more rigid one. [Originally Freud and his followers used “ego” to mean what we now understand as being both our True Self and false self. But since about 1940, object relations and self psychologists have differentiated these and generally do not use the term “ego.” Today, more people equate ego with false self.] To cover up the pain we use relatively unhealthy defenses against pain which give us fewer possibilities and choices in our lives.

IN THE SCHOOLS Memorizing multiplication tables may be a seminal school experience, among the few that kids today share with their grandparents. But a Stanford University professor says rapid-fire math drills are also the reason so many children fear and despise the subject. Moreover, the traditional approach to math instruction — memorization, timed testing and the pressure to speedily arrive at answers — may actually damage advanced-level skills by undermining the development of a deeper understanding about the ways numbers work. “There is a common and damaging misconception in mathematics — the idea that strong math students are fast math students,” says Jo Boaler, who teaches math education at the California university and has authored a new paper, “Fluency Without Fear.” In fact, many mathematicians are not speedy calculators, Boaler says. Laurent Schwartz, the French mathematician whose work is considered key to the theory of partial differential equations, wrote that as a student he often felt stupid because he was among the slowest math-thinkers in class.

By June 1949 people had begun to realize that it was not so easy to get a program right as had at one time appeared. I well remember when this realization first came on me with full force. The EDSAC was on the top floor of the building and the tape-punching and editing equipment one floor below on a gallery that ran round the room in which the differential analyzer was installed. I was trying to get working my first non-trivial program, which was one for the numerical integration of Airy’s differential equation. It was on one of my journeys between the EDSAC room and the punching equipment that “hesitating at the angles of stairs” the realization came over me with full force that a good part of the remainder of my life was going to be spent in finding errors in my own programs.

We had been seen. The thought stayed with me as I disposed of the leftovers—how could it not? I drove with one eye on the rearview mirror, waiting for the blinding burst of blue light to flare at my bumper and the brief harsh whoop! of a siren. But nothing came; not even after I ditched Valentine’s car, climbed into mine, and drove carefully home. Nothing. I was left entirely at liberty, all alone, pursued only by the demons of my imagination. It seemed impossible—someone had seen me at play, as plainly as it was possible to be seen. They had looked at the carefully carved pieces of Valentine, and the happy-weary carver standing above them, and it would not take a differential equation to arrive at a solution to this problem—A plus B equals a seat in Old Sparky for Dexter, and someone had fled with this conclusion in perfect comfort and safety—but they had not called the police? It

How does incoherence give birth to synchrony? It dawned on me one day that there was a straightforward way to frame the question as an exercise in differential equations: I needed to view incoherence as an equilibrium state and then calculate its stability.

What I’ve just described is called a system of differential equations. Such equations arise whenever we have rules for speeds depending on current positions.

Kato Genchi and other scholars like him contend that religion consists of two types: those religions which differentiate God from Man and those religions which equate God with Man. For instance, Christianity belongs to the first type where the distinction between God and Man pertains to the absolute distance between them. In Christianity the individual wishes to be saved by means of prayers which form the method of conforming with the Whole. Our Zen belongs to the second type of religion where equating Man with God pertains to the essential oneness of God and Man. In it the method of seeing into the oneness of the individual and the Whole is adopted, and by means of this method the distinction between the two is transcended.

As a mathematician Fantappiè could not accept that half of the solutions of the fundamental equations where rejected and in 1941, while listing the properties of the forward and backward in time energy, Fantappiè discovered that forward in time energy is governed by the law of entropy, whereas backward in time energy is governed by a complementary law that he named syntropy, combining the Greek words syn which means converging and tropos which means tendency. Listing the mathematical properties of syntropy Fantappiè discovered: energy concentration, increase in differentiation, complexity and structures: the mysterious properties of life! In 1944 he published the book “Principi di una Teoria Unitaria del Mondo Fisico e Biologico”[5] (Unitary Theory of the Physical and Biological World) in which he suggests that the physical-material world is governed by the law of entropy and causality, whereas the biological world is governed by the law of syntropy and retrocausality. We cannot see the future and therefore retrocausality is invisible! The dual energy solution suggests the existence of a visible reality (causal and entropic) and an invisible reality (retrocausal and syntropic). The first law of thermodynamics states that energy is a constant, a unity that cannot be created or destroyed but only transformed, and the energy-momentum-mass equation suggests that this unity has two components: entropy and syntropy. We can therefore write: 1=Entropy+Syntropy which shows that syntropy is the complement of entropy. Syntropy is often mistaken with negentropy. However, it is fundamentally different since negentropy does not take into account the direction of time, but considers time only in the classical way: flowing forward. Life lies between these two components: one entropic and the other syntropic, one visible and the other invisible, and this can be portrayed using a seesaw with entropy and syntropy playing at the opposite sides, and life at the center. This suggests that entropy and syntropy are constantly interacting and that all the manifestations of reality are dual: emitters and absorbers, particles and waves, matter and anti-matter, causality and retrocausality

Hénon, though dissipative systems were not his field (“sometimes astronomers are fearful of dissipative systems—they’re untidy”), thought he had an idea. Once again, he decided to throw out all reference to the physical origins of the system and concentrate only on the geometrical essence he wanted to explore. Where Lorenz and others had stuck to differential equations—flows, with continuous changes in space and time—he turned to difference equations, discrete in time. The key, he believed, was the repeated stretching and folding of phase space in the manner of a pastry chef who rolls the dough, folds it, rolls it out again, folds it, creating a structure that will eventually be a sheaf of thin layers.

both confidence and consolation. E. O. Wilson wrote: “You start by loving a subject. Birds, probability theory, stars, differential equations, storm fronts, sign language, swallowtail butterflies.… The subject will be your lodestar and give sanctuary in the shifting mental universe.

The equations of fluid flow are nonlinear partial differential equations, unsolvable except in special cases. Yet Ruelle worked out an abstract alternative to Landau’s picture, couched in the language of Smale, with images of space as a pliable material to be squeezed, stretched, and folded into shapes like horseshoes.

It is a beautiful vision that the President of Russia, Vladimir Vladimirovich Putin, understands the differential between Freedom of the Press and violation of that. West should follow him to learn such insight.

By the time he was in high school, his family had moved to Miami. Bezos was a straight-A student, somewhat nerdy, and still completely obsessed with space exploration. He was chosen as the valedictorian of his class, and his speech was about space: how to colonize planets, build space hotels, and save our fragile planet by finding other places to do manufacturing. “Space, the final frontier, meet me there!” he concluded. He went to Princeton with the goal of studying physics. It sounded like a smart plan until he smashed into a course on quantum mechanics. One day he and his roommate were trying to solve a particularly difficult partial differential equation, and they went to the room of another person in the class for help. He stared at it for a moment, then gave them the answer. Bezos was amazed that the student had done the calculation—which took three pages of detailed algebra to explain—in his head. “That was the very moment when I realized I was never going to be a great theoretical physicist,” Bezos says. “I saw the writing on the wall, and I changed my major very quickly to electrical engineering and computer science.” It was a difficult realization. His heart had been set on becoming a physicist, but finally he had confronted his own limits.

Duration” tells you how risky a bond is. The greater the duration, the greater the risk. For example, a ten-year bond has greater duration—and greater risk—than a one-year bond. That’s it. Mathematically, of course, duration is more complicated than this. It’s the length of time until you receive the average present value-weighted cash flow, and is itself a derivative (in calculus terms), of the partial differential equation that describes the price behavior of a bond.

one of the things we’re concerned about is the quest for infinite growth (an unavoidable feature of capitalism) on a finite planet. With that imperative, the biosphere is now subsumed under the economy. This has to be reversed. That is, the biosphere is now seen in strictly utilitarian terms to be simply a storehouse of resources, and/or a receptacle for waste. Also under capitalist compulsion, people now serve the economy, rather than the other way around. Development should be about people, not about objects. Development, often seen as synonymous with progress, is equated with growth, measured as GNP or GDP, sometimes per capita. This must be challenged, and we need differential criteria and different metrics for what constitutes development and progress. Right now these are equated. Development doesn’t necessarily require growth, development has no limits, growth has limits or should. And this is clearly referring back to the growth/de-growth debate that we read about. All of this is underlain by issues of what constitutes happiness, satisfaction, and quality of life. What do these actually essential elements of life actually depend on? At the moment, under our current capitalist system, and its associated common sense, these aspects are measured by the acquisition of more and more things. But we don’t go readily into this mindset, we have to actually be induced or seduced. Global advertising spending in 2014 was $488.48 billion and is projected to grow to $757.44 billion by 2021. So, think about the enormous effort, the enormous, strenuous, and continuous effort to persuade people that things that they merely want are really things that they must have, that they need. And this is the business of marketing and advertising. And as Noam pointed out previously, this completely distorts the notion of the so-called free market in which rational people make rational choices based on real needs.

Patty felt another blush stain her cheeks as she drew her knees up and, hurrying her feet under her bulk, hid herself in differential equations again.

Not your view, I know—you’d be happy to describe what you were up to purely in differential equations if you could—” Excerpt From: Michael Frayn. “Copenhagen”.

In agriculture the equation of invested input against gross yield is all: it does not matter if individual plants fail to thrive or die so long as the cost of saving them is greater than the cost of losing them…. This does not apply to the careful gardener whose labour is not costed, but a labour of love. He wants each of his plants to thrive, and he can treat each one individually. Indeed he can grow a hundred different plants in his garden and differentiate his treatment of each, pruning his roses, but not his sweet peas. Gardening rather than agriculture is the analogy for education.

Good afternoon,” Fletcher said to the young library aide who was working at the desk. “I am Dr. Duncan Fletcher, Fitzhugh Senior Fellow of applied mathematics, director for the Oxford Centre for Nonlinear Partial Differential Equations, and executive liaison between the university and the Alan Turing Institute. And you are…?

But if you inflated the balloons quickly, possibly by connecting many canisters to it at once, you’d be able to slow your fall. Just don’t use too much helium, or you’ll end up floating at 16,000 feet like Larry Walters. While researching this answer, I managed to lock up my copy of Mathematica several times on balloon-related differential equations, and subsequently got my IP address banned from Wolfram|Alpha for making too many requests. The ban-appeal form asked me to explain what task I was performing that necessitated so many queries. I wrote, “Calculating how many rental helium tanks you’d have to carry with you in order to inflate a balloon large enough to act as a parachute and slow your fall from a jet aircraft.” Sorry, Wolfram. 1 While researching impact speeds for this answer, I came across a discussion on the Straight Dope Message Board about survivable fall heights. One poster

Even at the cutting edge of modern physics, partial differential equations still provide the mathematical infrastructure. Consider Einstein’s general theory of relativity. It reimagines gravity as a manifestation of curvature in the four-dimensional fabric of space-time. The standard metaphor invites us to picture space-time as a stretchy, deformable fabric, like the surface of a trampoline. Normally the fabric is pulled taut, but it can curve under the weight of something heavy placed on it, say a massive bowling ball sitting at its center. In much the same way, a massive celestial body like the sun can curve the fabric of space-time around it. Now imagine something much smaller, say a tiny marble (which represents a planet), rolling on the trampoline’s curved surface. Because the surface sags under the bowling ball’s weight, it deflects the marble’s trajectory. Instead of traveling in a straight line, the marble follows the contours of the curved surface and orbits around the bowling ball repeatedly. That, says Einstein, is why the planets go around the sun. They’re not feeling a force; they’re just following the paths of least resistance in the curved fabric of space-time.

There is no need to remove the grand designer out of the how-the-universe-began equation. It is simply not possible. You see. Both are self. The universe is self. The grand designer is self. Self has been one since time immemorial. In fact; it is oneself which has created itself to perceive itself as differentiated so not to be by itself. The cause - loneliness - matters not as much as the purpose - love. Love so love

he will have to write to Alan and tell him that some new instructions will have to be added to the Waterhouse-simulation Turing machine. This new factor is FMSp, the Factor of Mary Smith Proximity. In a simpler universe, FMSp, would be orthogonal to sigma, which is to say that the two factors would be entirely independent of each other. If it were thus, Waterhouse could continue the usual sawtooth-wave ejaculation management program with no changes. In addition, he would have to arrange to have frequent conversations with Mary Smith so that FMSp would remain as high as possible. Alas! The universe is not simple. Far from being orthogonal, FMSp and sigma are involved, as elaborately as the contrails of dogfighting airplanes. The old sigma management scheme doesn’t work anymore. And a platonic relationship will actually make FMSp worse, not better. His life, which used to be a straightforward set of basically linear equations, has become a differential equation. It is the visit to the whorehouse that makes him realize this.

From the old elements of earth, air, fire, and water to the latest in electrons, quarks, black holes, and superstrings, every inanimate thing in the universe bends to the rule of differential equations. I bet this is what Feynman meant when he said that calculus is the language God talks.

This allows us, for example, to make sense of the plucked string, where the initial displacement is continuous, but not even once differentiable. This is a common phenomenon when solving partial differential equations. A technique which is very often used is to rewrite the equation as an integral equation, meaning an equation involving integrals rather than derivatives. Integrable functions are much more general than differentiable functions, so one should expect a more general class of solutions.

Dynamic simulators create the behavior of the system by solving the imbedded sets of differential equations using mathematical approximation techniques.

You know about the Big Bang, right? They believe that the universe was born in a tremendous explosion twenty billion years ago. I can mathematically express the form of the universe, from its birth to the present. It's all about differential equations. Most phenomena in the universe can be expressed with differential equations, you know. Using them, you can figure out what the universe looked like a hundred million years ago, ten billion years ago, even a second or a tenth of a second after that initial explosion. But. No matter how far we go back, no matter how we try to express it, we just can't know what it looked like at zero, at the very moment of the explosion. And there's another thing. How is our universe going to end? Is the universe expanding or contracting? See, we don't know the beginning and we don't know the end; all we can know about is the in-between stuff. And that, my friend, is what life is like.

Instead of the principle of maximal generality that is usual in mathematical books the author has attempted to adhere to the principle of minimal generality, according to which every idea should first be clearly understood in the simplest situation; only then can the method developed be extended to more complicated cases. Although it is usually simpler to prove a general fact than to prove numerous special cases of it, for a student the content of a mathematical theory is never larger than the set of examples that are thoroughly understood.

It is no exaggeration to say that the vast business of calculus made possible most of the practical triumphs of post-medieval science; nor to say that it stands as one of the most ingenious creations of humans trying to model the changeable world around them. So by the time a scientist masters this way of thinking about nature, becoming comfortable with the theory and the hard, hard practice, he is likely to have lost sight of one fact. Most differential equations cannot be solved at all. “If you could write down the solution to a differential equation,” Yorke said, “then necessarily it’s not chaotic, because to write it down, you must find regular invariants, things that are conserved, like angular momentum. You find enough of these things, and that lets you write down a solution. But this is exactly the way to eliminate the possibility of chaos.

Joining the world of shapes to the world of numbers in this way represented a break with the past. New geometries always begin when someone changes a fundamental rule. Suppose space can be curved instead of flat, a geometer says, and the result is a weird curved parody of Euclid that provides precisely the right framework for the general theory of relativity. Suppose space can have four dimensions, or five, or six. Suppose the number expressing dimension can be a fraction. Suppose shapes can be twisted, stretched, knotted. Or, now, suppose shapes are defined, not by solving an equation once, but by iterating it in a feedback loop. Julia, Fatou, Hubbard, Barnsley, Mandelbrot-these mathematicians changed the rules about how to make geometrical shapes. The Euclidean and Cartesian methods of turning equations into curves are familiar to anyone who has studied high school geometry or found a point on a map using two coordinates. Standard geometry takes an equation and asks for the set of numbers that satisfy it. The solutions to an equation like x^2 + y^2 = 1, then, form a shape, in this case a circle. Other simple equations produce other pictures, the ellipses, parabolas, and hyperbolas of conic sections or even the more complicated shapes produced by differential equations in phase space. But when a geometer iterates an equation instead of solving it, the equation becomes a process instead of a description, dynamic instead of static. When a number goes into the equation, a new number comes out; the new number goes in, and so on, points hopping from place to place. A point is plotted not when it satisfies the equation but when it produces a certain kind of behavior. One behavior might be a steady state. Another might be a convergence to a periodic repetition of states. Another might be an out-of-control race to infinity.

You have even more gifts than you realize. Promise me that you’ll explore all of them and that you’ll be generous with them.

Those five characteristics are:    1. Reactivity: the vicious cycle of intense reactions of each member to events and to one another.    2. Herding: a process through which the forces for togetherness triumph over the forces for individuality and move everyone to adapt to the least mature members.    3. Blame displacement: an emotional state in which family members focus on forces that have victimized them rather than taking responsibility for their own being and destiny.    4. A quick-fix mentality: a low threshold for pain that constantly seeks symptom relief rather than fundamental change.    5. Lack of well-differentiated leadership: a failure of nerve that both stems from and contributes to the first four. To reorient oneself away from a focus on technology toward a focus on emotional process requires that, like Columbus, we think in ways that not only are different from traditional routes but that also sometimes go in the opposite direction. This chapter will thus also serve as prelude to the three that follow, which describe the “equators” we have to cross in our time: the “learned” fallacies or emotional barriers that keep an Old World orientation in place and cause both family and institutional leaders to regress rather than venture in new directions. By the term regression I

differential equations that can be solved in “closed form,” that is, by means of a formula for the unknown function f, are the exception rather than the rule,

Pedagogically speaking, a good share of physics and mathematics was—and is—writing differential equations on a blackboard and showing students how to solve them. Differential equations represent reality as a continuum, changing smoothly from place to place and from time to time, not broken in discrete grid points or time steps. As every science student knows, solving differential equations is hard. But in two and a half centuries, scientists have built up a tremendous body of knowledge about them: handbooks and catalogues of differential equations, along with various methods for solving them, or “finding a closed-form integral,” as a scientist will say. It is no exaggeration to say that the vast business of calculus made possible most of the practical triumphs of post-medieval science; nor to say that it stands as one of the most ingenious creations of humans trying to model the changeable world around them. So by the time a scientist masters this way of thinking about nature, becoming comfortable with the theory and the hard, hard practice, he is likely to have lost sight of one fact. Most differential equations cannot be solved at all.

If you could write down the solution to a differential equation,” Yorke said, “then necessarily it’s not chaotic, because to write it down, you must find regular invariants, things that are conserved, like angular momentum. You find enough of these things, and that lets you write down a solution. But this is exactly the way to eliminate the possibility of chaos.

But Yorke had offered more than a mathematical result. He had sent a message to physicists: Chaos is ubiquitous; it is stable; it is structured. He also gave reason to believe that complicated systems, traditionally modeled by hard continuous differential equations, could be understood in terms of easy discrete maps.

A hole in a hole in a hole—Numberphile Around the World in a Tea Daze—Shpongle But what is a partial differential equation?—Grant Sanderson, who owns the 3Blue1Brown YouTube channel Closer to You—Kaisaku Fourier Series Animation (Square Wave)—Brek Martin Fourier Series Animation (Saw Wave)—Brek Martin Great Demo on Fibonacci Sequence Spirals in Nature—The Golden Ratio—Wise Wanderer gyroscope nutation—CGS How Earth Moves—vsauce I am a soul—Nibana

An essential pedagogic step here is to relegate the teaching of mathematical methods in economics to mathematics departments. Any mathematical training in economics, if it occurs at all, should come after students have at the very least completed course work in basic calculus, algebra and differential equations (the last being one about which most economists are woefully ignorant). This simultaneously explains why neoclassical economists obsess too much about proofs and why non-neoclassical economists, like those in the Circuit School, experience such difficulties in translating excellent verbal ideas about credit creation into coherent dynamic models of a monetary production economy.

Basira, who had come to us from Algiers, made a face. “So, then he tried to show me how to use a slide rule!” “No—for a differential equation?” Myrtle, the only other American in our group, covered her mouth and laughed until her cheeks turned red. “What a buffoon.

There is no need to use classical mechanics because Lagrangian formalism (and its use in relation to differential equations) has proven to be adaptable enough to study continuous media, such as the vibrations of uninterrupted η-dimensional objects (like 1D strings and 2D membranes) and the flows of fluids.

A few books that I've read.... Pascal, an Introduction to the Art and Science of Programming by Walter Savitch Programming algorithms Introduction to Algorithms, 3rd Edition (The MIT Press) Data Structures and Algorithms in Java Author: Michael T. Goodrich - Roberto Tamassia - Michael H. Goldwasser The Algorithm Design Manual Author: Steven S Skiena Algorithm Design Author: Jon Kleinberg - Éva Tardos Algorithms + Data Structures = Programs Book by Niklaus Wirth Discrete Math Discrete Mathematics and Its Applications Author: Kenneth H Rosen Computer Org Structured Computer Organization Andrew S. Tanenbaum Introduction to Assembly Language Programming: From 8086 to Pentium Processors (Undergraduate Texts in Computer Science) Author: Sivarama P. Dandamudi Distributed Systems Distributed Systems: Concepts and Design Author: George Coulouris - Jean Dollimore - Tim Kindberg - Gordon Blair Distributed Systems: An Algorithmic Approach, Second Edition (Chapman & Hall/CRC Computer and Information Science Series) Author: Sukumar Ghosh Mathematical Reasoning Mathematical Reasoning: Writing and Proof Version 2.1 Author: Ted Sundstrom An Introduction to Mathematical Reasoning: Numbers, Sets and Functions Author: Peter J. Eccles Differential Equations Differential Equations (with DE Tools Printed Access Card) Author: Paul Blanchard - Robert L. Devaney - Glen R. Hall Calculus Calculus: Early Transcendentals Author: James Stewart And more....

his ‘theory of happenings’, the British malaria expert and Nobel laureate Ronald Ross had come up with a set of differential equations that could help determine, at any given time, the proportion of a

I took 17 computer science classes and made an A in 11 of them. 1 point away from an A in 3 of them and the rest of them didn't matter. Math is a tool for physics,chemistry,biology/basic computation and nothing else. CS I(Pascal Vax), CS II(Pascal Vax), Sr. Software Engineering, Sr. Distributed Systems, Sr. Research, Sr. Operating Systems, Sr. Unix Operating Systems, Data Structures, Sr. Object Oriented A&D, CS (perl/linux), Sr. Java Programming, Information Systems Design, Jr. Unix Operating Systems, Microprocessors, Programming Algorithms, Calculus I,II,III, B Differential Equations, TI-89 Mathematical Reasoning, 92 C++ Programming, Assembly 8086, Digital Computer Organization, Discrete Math I,II, B Statistics for the Engineering & Sciences (w/permutations & combinatorics) -- A-American Literature A-United States History 1865 CLEP-full year english CLEP-full year biology A-Psychology A-Environmental Ethics

In other words, as long as θ(x) never gets large, the motion of the string is essentially determined by the wave equation (3.2.2) where D’Alembert2 discovered a strikingly simple method for finding the general solution to equation (3.2.2). Roughly speaking, his idea is to factorize the differential