Top 265 Differential Equations Quotes & Sayings - Page 4

It is clear that the building of models is not a purely mechanical process but requires skill of a high order - not merely mathematical skill but a sensitivity to the relative importance of different factors and a critical, almost an artistic, faculty in the selection of behaviour equations which are reasonable, tentative hypotheses in explaining the behaviour of actual economies.

A robber who justified his theft by saying that he really helped his victims, by his spending giving a boost to retail trade, would find few converts; but when this theory is clothed in Keynesian equations and impressive references to the 'multiplier effect,' it unfortunately carries more conviction.

Throughout the 1960s and 1970s devoted Beckett readers greeted each successively shorter volume from the master with a mixture of awe and apprehensiveness; it was like watching a great mathematician wielding an infinitesimal calculus, his equations approaching nearer and still nearer to the null point.

Any schemes - such as 'think of symmetry laws', or 'put the information in mathematical form', or 'guess equations'- are known to everybody now, and they are all tried all the time. When you are stuck, the answer cannot be one of these, because you will have tried these right away...The next scheme, the new discovery, is going to be made in a completely different way.

My brother is a genius. When we went to Italy, he was on the local television channel as a prodigy, who could solve very sophisticated mathematical equations. He was only seven or eight years old but he could solve mathematical problems for fourteen year olds.

People think six is a great many, when it's children. ...they don't mind six pairs of boots, or six pounds of apples, or six oranges, especially in equations, but they seem to think that you ought not to have five brothers and sisters.

Democracy turns upon and devours itself. Universal suffrage, in theory the palladium of our liberties, becomes the assurance of our slavery. And that slavery will grow more and more abject and ignoble as the differential birth rate, the deliberate encouragement of mendicancy and the failure of popular education produce a larger and larger mass of prehensile half-wits, and so make the demagogues more and more secure.

Standard mathematics has recently been rendered obsolete by the discovery that for years we have been writing the numeral five backward. This has led to reevaluation of counting as a method of getting from one to ten. Students are taught advanced concepts of Boolean algebra, and formerly unsolvable equations are dealt with by threats of reprisals.

The great paradox of the civil rights revolution is that instead of enforcing and expanding equality before the law, the revolution created differential rights based on race, gender and, any day now, sexual orientation. The great liberal revolution, centuries in the making, that brought forth equality in law has been overthrown. In its place we see rising a new feudal legal order of status-based rights.

It was basic research in the photoelectric field-in the photoelectric effect that would one day lead to solar panels. It was basic research in physics that would eventually produce the CAT scan. The calculations of today's GPS satellites are based on the equations that Einstein put to paper more than a century ago.

Arithmetic starts with the integers and proceeds by successively enlarging the number system by rational and negative numbers, irrational numbers, etc... But the next quite logical step after the reals, namely the introduction of infinitesimals, has simply been omitted. I think, in coming centuries it will be considered a great oddity in the history of mathematics that the first exact theory of infinitesimals was developed 300 years after the invention of the differential calculus.

The aim ... is to provide a clear and rigorous basis for determining when a causal ordering can be said to hold between two variables or groups of variables in a model . . . . The concepts refer to a model-a system of equations-and not to the 'real' world the model purports to describe.

The real truth that dare not speak itself, is that no one is in control, absolutely no one... Nobody is in control. This stuff is ruled by the equations of dynamics and chaos. Now, there may be entities seeking control - the World Bank, the Communist Party, the rich, the somebody-or-others - but to seek control is to take enormous aggravation upon yourself... It's like trying to control a dream.

I do not personally want to believe that we already know the equations that determine the evolution and fate of the universe; it would make life too dull for me as a scientist. ... I hope, and believe, that the Space Telescope might make the Big Bang cosmology appear incorrect to future generations, perhaps somewhat analogous to the way that Galileo's telescope showed that the earth-centered, Ptolemaic system was inadequate.

The truth shall set you free. When you hear the truth, it sets you free. So mathematics is truth. It adds up. There's no error. Only time there's an error is when man miscalculates his own problems or his own equations.

To most outsiders, modern mathematics is unknown territory. Its borders are protected by dense thickets of technical terms; its landscapes are a mass of indecipherable equations and incomprehensible concepts. Few realize that the world of modern mathematics is rich with vivid images and provocative ideas.

In our travels, we have come across many equations--math for understanding the universe, for making music, for mapping stars, and also for tipping, which is important. Here is our favorite equation: Us plus Them equals All of Us. It is very simple math. Try it sometime. You probably won’t even need a pencil.

The basic formulation, or bare-bones mechanics, of natural selection is a disarmingly simple argument, based on three undeniable facts (overproduction of offspring, variation, and heritability) and one syllogistic inference (natural selection, or the claim that organisms enjoying differential reproductive success will, on average, be those variants that are fortuitously better adapted to changing local environments, and that these variants will then pass their favored traits to offspring by inheritance).

What is it with science these days? Everyone is so quick to believe in it, in all these new scientific discoveries, new pills for this, new pills for that. Get thinner, grow hair, yada, yada, yada, but when it requires a little faith in something you all go crazy.' He shook his head, 'If miracles had chemical equations then everyone would believe.

India was China's teacher in religion and imaginative literature, and the world's teacher in trignometry, quandratic equations, grammar, phonetics, Arabian Nights, animal fables, chess, as well as in philosophy, and that she inspired Boccaccio, Goethe, Herder, Schopenhauer, Emerson, and probably also old Aesop.

Concern for man himself and his fate must always form the chief interest of all technical endeavors, concern for the great unsolved problems of the organization of labor and the distribution of goods-in order that the creations of our mind shall be a blessing and not a curse to mankind. Never forget this in the midst of your diagrams and equations.

I now want to tell three stories about advances in twentieth-century physics. A curious fact emerges in these tales: time and again physicists have been guided by their sense of beauty not only in developing new theories but even in judging the validity of physical theories once they are developed. Simplicity is part of what I mean by beauty, but it is a simplicity of ideas, not simplicity of a mechanical sort that can be measured by counting equations or symbols.

How many black holes have we been up close and personal with?" Kosta countered. "All sorts of odd things happen near the event horizon, from huge tidal forces to variations in time. Personally, I'm voting on it having to do with gravity, either a polarization of the fields themselves or else something related to the time differential." I didn't know physics had become a democracy," Hanan murmered.

In my own research when I'm working with equations, I never feel like I really understand what I'm doing if I'm solely relying on the mathematics for my understanding. I need to have a visual picture in my mind. I'm constantly translating from the math to some intuitive mind's-eye picture.

Millions of statements are made about the president every day on every subject and from every standpoint; threats of violence are not an integral feature of any one subject or viewpoint as distinct from others. Differential treatment of threats against the president, then, selects nothing but special risks, not special messages.

While physics and mathematics may tell us how the universe began, they are not much use in predicting human behavior because there are far too many equations to solve. I'm no better than anyone else at understanding what makes people tick, particularly women.

I understood Stephen's point of view because if you had been given a death sentence at the age of 21, would you find it easy to believe in a loving God? Also, Stephen's work was taking him into the depths of the universe, and it was, I thought, fairly understandable that there wasn't much room for God in his equations.

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.

As with Randall Terry and other anti-abortion leaders, women simply did not figure into [Roeder's] equations. If all the abortion providers were dead, the problem would be solved, and he'd never have to think about those who sought to end their pregnancies through illegal or dangerous means.

It required unusual inquisitiveness to pursue the development of scientific curiosities such as charged pith balls, the voltaic cell, and the electrostatic machine. Without such endeavors and the evolution of associated instrumentation, initially of purely scientific interest, most of the investigations that lead to the basic equations of electromagnetism would have been missed. ... We would have been deprived of electromagnetic machinery as well as knowledge of electromagnetic waves.

Extra dimensional theories are sometimes considered science fiction with equations. I think that's a wrong attitude. I think extra dimensions are with us, they are with us to stay, and they entered physics a long time ago. They are not going to go away.

I deliberately and consciously give preference to a dramatic, mythological way of thinking and speaking, because this is not only more expressive but also more exact than an abstract scientific terminology, which is wont to toy with the notion that its theoretic formulations may one fine day be resolved into algebraic equations.

The relativity principle in connection with the basic Maxwellian equations demands that the mass should be a direct measure of the energy contained in a body; light transfers mass. With radium there should be a noticeable diminution of mass. The idea is amusing and enticing; but whether the Almighty is laughing at it and is leading me up the garden path -- that I cannot know.

We can summarize electricity, magnetism and gravity into equations one inch long, and that's the power of field theory. And so I said to myself: I will create a field theory of strings. And when I did it one day, it was incredible, realizing that on a sheet of paper I can write down an equation which summarized almost all physical knowledge.

A common misconception about how things such as space shuttles come to be is that engineers simply apply the theories and equations of science. But this cannot be done until the new thing-to-be is conceived in the engineer's mind's eye. Rather than following from science, engineered things lead it.

This is often the way it is in physics - our mistake is not that we take our theories too seriously, but that we do not take them seriously enough. It is always hard to realize that these numbers and equations we play with at our desks have something to do with the real world. Even worse, there often seems to be a general agreement that certain phenomena are just not fit subjects for respectable theoretical and experimental effort.

[Mathematics] is security. Certainty. Truth. Beauty. Insight. Structure. Architecture. I see mathematics, the part of human knowledge that I call mathematics, as one thing - one great, glorious thing. Whether it is differential topology, or functional analysis, or homological algebra, it is all one thing. ... They are intimately interconnected, they are all facets of the same thing. That interconnection, that architecture, is secure truth and is beauty. That's what mathematics is to me.

Race is not the only differential in South Africa, in the new South Africa, where all schools are open, mother-tongue education is a very big issue. One of the main reasons why the dropout figure of black students, and the lower pass rate of black students in the present education system, and it was like that before, was that we didn't have mother-tongue education.

I have found that when calculating what no one has calculated before, like my observing sessions on the mountain, my mental acuity peaks. Ironically, these are the times that I would flunk the reality check normally reserved for mental patients and dazed boxers: What is your name? What day is it? Who is the president of the United States?... I do not know, and I do not care. I am at peace with my equations as I connect to the cosmic engines that drive our universe.

...the computer models are very good at solving equations of fluid dynamics but very bad at describing the real world. The real world is full of things like clouds and vegetation and soil and dust which the models describe very poorly.

Accordingly, we find Euler and D'Alembert devoting their talent and their patience to the establishment of the laws of rotation of the solid bodies. Lagrange has incorporated his own analysis of the problem with his general treatment of mechanics, and since his time M. Poinsôt has brought the subject under the power of a more searching analysis than that of the calculus, in which ideas take the place of symbols, and intelligent propositions supersede equations.

Evidence in support of general relativity came quickly. Astronomers had long known that Mercury’s orbital motion around the sun deviated slightly from what Newton’s mathematics predicted. In 1915, Einstein used his new equations to recalculate Mercury’s trajectory and was able to explain the discrepancy, a realization he later described to his colleague Adrian Fokker as so thrilling that for some hours it gave him heart palpitations.

Mathematicians are proud of the fact that, generally, they do their work with a piece of chalk and a blackboard. They value hand-done proofs above all else. A big question in mathematics today is whether or not computational proofs are legitimate. Some mathematicians won't accept computational proofs and insist that a real proof must be done by the human hand and mind, using equations.

Expected outcomes contribute to motivation independently of self-efficacy beliefs when outcomes are not completely controlled by quality of performance. This occurs when extraneous factors also affect outcomes, or outcomes are socially tied to a minimum level of performance so that some variations in quality of performance above and below the standard do not produce differential outcomes

The thing that got me started on the science that I've been building now for about 20 years or so was the question of okay, if mathematical equations can't make progress in understanding complex phenomena in the natural world, how might we make progress?

The illegitimate use of a state by economic interests for their own ends is based upon a preexisting illegitimate power of the state to enrich some persons at the expense of others. Eliminate that illegitimate power of giving differential economic benefits and you eliminate or drastically restrict the motive for wanting political influence.

My algebra was relatively poor. I found it very difficult to use equations that substituted numbers - to which I had a synesthetic and emotional response - for letters, to which I had none. It was because of this that I decided not to continue math at Advanced level, but chose to study history, French and German instead.

The complicated, ambiguous milieu of human contact is being replaced with simple, scalable equations. We maintain thousands more friends than any human being in history, but at the cost of complexity and depth. Every minute spent online is a minute of face-to-face time lost.

A study of interactions between women and obstetricians offers an explanation. It described three levels of increasing power imbalance: In the first, you fight and lose; in the second you don't fight because you know you can't win. However, in the highest level of power differential, your preferences are so manipulated that you act against your own interests, but you are content. Elective [meaning requested for no medical reason, not to be confused with needed surgery planned in advance] repeat cesarean exemplifies that highest level.

What counts, I found, is not what you cover, but what you uncover. Covering subjects in a class can be a boring exercise, and students feel it. Uncovering the laws of physics and making them see through the equations, on the other hand, demonstrates the process of discovery, with all its newness and excitement, and students love being part of it.

The policy that received more attention particularly in the past decade and a half or so has been the US cocaine policy, the differential treatment of crack versus powder cocaine and question is how my research impacted my view on policy. Clearly that policy is not based on the weight of the scientific evidence. That is when the policy was implemented, the concern about crack cocaine was so great that something had to be done and congress acted in the only way they knew how, they passed policy and that's what a responsible society should do.

Once you have an equalization instrument in place, as you have in Canada, there arise tremendous bureaucratic values - bureaucratic rent so to speak - in maintaining the system that you have. To shift to a system that paid the transfers directly to individuals, by having differential rates of federal income tax levied to adjust to provincial fiscal capacities, which would be my preference, you would have huge bureaucratic opposition. People would try to protect the rents they have in the current system of institutions.

My mother was the nicest person in the world. I still have people coming to me to say how she was so warm, generous, and kind-hearted. She never washed her dirty linen in public. She always maintained her equations with people.

Equations seem like treasures, spotted in the rough by some discerning individual, plucked and examined, placed in the grand storehouse of knowledge, passed on from generation to generation. This is so convenient a way to present scientific discovery, and so useful for textbooks, that it can be called the treasure-hunt picture of knowledge.

'Scientific' computer simulations predict global warming based on increased greenhouse gas emissions over time. However, without water's contribution taken into account they omit the largest greenhouse gas from their equations. How can such egregious calculation errors be so blatantly ignored? This is why man-made global warming is 'junk' science.

Because for me, equations and symbols aren't just a thing. They're a voice that speaks out about the incredible richness of nature and the startling simplicity in the patterns that twist and turn and warp and evolve all around us, from how the world works to how we behave.

It's a spectacular signal. It's a signal many of us have wanted to observe since the time LIGO was proposed. It shows the dynamics of objects in the strongest gravitational fields imaginable, a domain where Newton's gravity doesn't work at all, and one needs the fully non-linear Einstein field equations to explain the phenomena.

What is quite worrisome is the absence of analysis and reflection. Take the word "terrorism." It has become synonymous now with anti-Americanism, which, in turn, has become synonymous with being critical of the United States, which, in turn, has become synonymous with being unpatriotic. That's an unacceptable series of equations.

There is nothing that can be said by mathematical symbols and relations which cannot also be said by words. The converse, however, is false. Much that can be and is said by words cannot successfully be put into equations, because it is nonsense.

A mathematician either has a feeling for equations and an understanding and delight in it, not only in the purity of it, but in its beauty as well. I don't think that's something that you learn at school. I think you can get better in mathematics on a school level, but when you're talking about being a mathematician, I think that's definitely a gift of genes or whatever, you know? Whatever your pool is.