Discover the most popular and inspiring quotes and sayings on the topic of Nonlinearities. Share them with your friends on social media platforms like Facebook, Twitter, or your personal blogs, and let the world be inspired by their powerful messages. Here are the Top 100 Nonlinearities Quotes And Sayings by 92 Authors including Dean Koontz,Michael Nesmith,Rudolf Arnheim,Cynthia Weil,Heraclitus for you to enjoy and share.
Mr. Thomas, any scientist will tell you that in nature many systems appear to be chaotic, but when you study them long enough and closely enough, strange order always underlies the appearance of chaos.
Linear thinking typifies a highly developed industry. It starts to get these patterns built into it somehow. I'm not sure how that happens, but certainly you take a look at dinosaurs.
When a system is considered in two different states, the difference in volume or in any other property, between the two states, depends solely upon those states themselves and not upon the manner in which the system may pass from one state to the other.
My nature is to be linear, and when I'm not, I feel really proud of myself.
The only thing constant is change
The teacher manages to get along still with the cumbersome algebraic analysis, in spite of its difficulties and imperfections, and avoids the smooth infinitesimal calculus, although the eighteenth century shyness toward it had long lost all point.
This is a collaboration between a complex analyst, a dynamical system expert, and an arithmetical algebraic geometer. It sounds like a joke, a complex analyst, a dynamical system expert, and an arithmetical algebraic geometer walk into a bar ...
The principles governing the behavior of systems are not widely understood.
Their incredible determination creates the need for nonlinear thinking, combined with boundless energy. Champions also earn money in nonlinear ways. While the masses essentially trade their time for money, the great ones realize this is probably the worst way to acquire wealth. Using
Read Theodore Schwenk's marvelous book Sensitive Chaos (London, Rudolph Steiner Press, 1965),
Postulate 3. Assignable causes of variation may be found and eliminated.
Experience has shown repeatedly that a mathematical theory with a rich internal structure generally turns out to have significant implications for the understanding of the real world, often in ways no one could have envisioned before the theory was developed.
The more linear one tries to make the equation of planning , the more complex becomes it's algorithm
Analysis does not owe its really significant successes of the last century to any mysterious use of sqrt(-1), but to the quite natural circumstances that one has infinitely more freedom of mathematical movement if he lets quantities vary in a plane instead of only on a line.
We cannot forbear suggesting one practical result which it appears to us must be greatly facilitated by the independent manner in which the engine orders and combines its operations: we allude to the attainment of those combinations into which imaginary quantities enter.
The difficulty involved in the proper and adequate means of describing changes in continuous deformable bodies is the method of differential equations ... They express mathematically the physical concept of contiguous action.
Einstein's Theory of Relativity
People interpret things in a linear form but science proves otherwise.
All complex systems are hierarchical in nature, but also exhibit other patterns of regularity.
IN CHAOS THEORY, THE BUTTERFLY EFFECT IS THE SENSITIVE DEPENDENCY ON INITIAL CONDITIONS IN WHICH A SMALL CHANGE AT ONE PLACE IN A DETERMINISTIC NONLINEAR SYSTEM CAN RESULT IN LARGE DIFFERENCES IN A LATER STATE.
Physics is based on the assumption that certain fundamental features of nature are constant.
Digital mechanics predicts that for every continuous symmetry of physics there will be some microscopic process that violates that symmetry.
Given an approximate knowledge of a system's initial conditions and an understanding of natural law, one can calculate the approximate behavior of the system.
Considering the inconceivable complexity of processes even in a simple cell, it is little short of a miracle that the simplest possible model - namely, a linear equation between two variables - actually applies in quite a general number of cases.
Constants are widely known for the detestable practice of changing their values; we should prepare ourselves against the consequences of such fickleness
Instead of trying to specify a system in full detail, specify it only somewhat. You can then ride on the dynamics of the system in the direction you want to go.
Learning chiefly in mathematical sciences can so swallow up and fix one's thought, as to possess it entirely for some time; but when that amusement is over, nature will return, and be where it was, being rather diverted than overcome by such speculations.
Given certain known factors in an equation and the equation comprising a situation of absolute need - any form of need - you can predict the results. Leave a sick junkie in the back room of a drugstore and only one result is possible.
The algebraic sum of all the transformations occurring in a cyclical process can only be positive, or, as an extreme case, equal to nothing.
[Statement of the second law of thermodynamics, 1862]
Unpredictability is closely related to uncontrollability.
I regard it in fact as the great advantage of the mathematical technique that it allows us to describe, by means of algebraic equations, the general character of a pattern even where we are ignorant of the numerical values which will determine its particular manifestation.
How thoroughly it is ingrained in mathematical science that every real advance goes hand in hand with the invention of sharper tools and simpler methods which, at the same time, assist in understanding earlier theories and in casting aside some more complicated developments.
The Only Thing That Is Constant Is Change -
a constant cannot explain a variable.
The alternation of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.
We have usually no knowledge that any one factor will exert its effects independently of all others that can be varied, or that its effects are particularly simply related to variations in these other factors.
Explosive bifurcation is the sudden transition that wrenches the system out of one order, and into another.
Change leads to growth. Resistance leads to rigidity. Rigidity leads to ...
ahh, calculus. the mathematics of change
Some things that satisfy the rules of algebra can be interesting to mathematicians even though they don't always represent a real situation.
It follows that there are two different types of change: one that occurs within a given system which itself remains unchanged, and one whose occurrence changes the system itself.
Human history is highly nonlinear and unpredictable.
Higher degree of equations in maths and life have no real solutions.
Unlike us, machines do not have a 'nature' consistent across vast reaches of time. They are, at least to begin with, whatever we set in motion - with an inbuilt tendency towards the exponential.
The first idea is that human progress is exponential (that is, it expands by repeatedly multiplying by a constant) rather than linear (that is, expanding by repeatedly adding a constant). Linear versus exponential: Linear growth is steady; exponential growth becomes explosive.
If we pursue this matter further, we shall be told that the stable object is unchanging under the impact or stress of some particular external or internal variable or, perhaps, that it resists the passage of time.
It is more important to have beauty in one's equations than to have them fit experiment.
Very often in mathematics the crucial problem is to recognize and discover what are the relevant concepts; once this is accomplished the job may be more than half done.
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.
We humans think linearly but tech trends are exponential.
Inevitably, underlying instabilities begin to appear.
A Curve does not exist in its full power until contrasted with a straight line.
All models are approximations. Essentially, all models are wrong, but some are useful. However, the approximate nature of the model must always be borne in mind ...
Time is a series of fluctuating variables.
Nothing is constant except change
There is no doubt that we cannot do without variable quantities in the sense of the potential infinite. But from this very fact the necessity of the actual infinite can be demonstrated.
[All phenomena] are equally susceptible of being calculated, and all that is necessary, to reduce the whole of nature to laws similar to those which Newton discovered with the aid of the calculus, is to have a sufficient number of observations and a mathematics that is complex enough.
for any non-biological system, the distribution of output follows a power law curve.
If a self-organizing system becomes too static, it runs down; if it becomes too chaotic, it breaks apart.
I don't think that there is any hard and fast rule that says that documentary has to be linear at all.
Starting in the seventeenth century, the general theory of extreme values - maxima and minima - has become one of the systematic integrating principles of science.
The resulting, stable singularities now carry the name BKL in honor of Belinsky, Khalatnikov, and Lifshitz. A BKL singularity is chaotic. Highly chaotic. And lethal. Highly lethal.
You cannot run a linear system on a finite planet indefinitely.
I don't believe that life is linear. I think of it as circles - concentric circles that connect.
Algebra goes to the heart of the matter at it ignores the casual nature of particular cases.
Calm, focused, undistracted, the linear mind is being pushed aside by a new kind of mind that wants and needs to take in and dole out information in short, disjointed, often overlapping bursts - the faster, the better.
I was a linear thinker, and according to Zen linear thinking is nothing but a delusion, one of the many that keep us unhappy. Reality is nonlinear, Zen says. No future, no past. All is now.
We call it the zigzag theory. You want to find something that zigs and something that zags and blend them together to get a better combined performance.
In bodies moved, the motion is received, increased, diminished, or lost, according to the relations of the quantity of matter and velocity; each diversity is uniformity, each change is constancy.
Two possibilities present themselves for the analytical treatment of metrical geometry.
Models can easily become so complex that they are impenetrable, unexaminable, and virtually unalterable.
Baseball players or cricketers do not need to be able to solve explicitly the non-linear differential equations which govern the flight of the ball. They just catch it.
Mesarovic and Pestel are critical of the Forrester-Meadows world view, which is that of a homogeneous system with a fully predetermined evolution in time once the initial conditions are specified.
The orthogonal features, when combined, can explode into complexity.
If anything runs deeper than a mathematician's love of variables, it's a scientist's love of constants.
But the beauty of Einstein's equations, for example, is just as real to anyone who's experienced it as the beauty of music. We've learned in the 20th century that the equations that work have inner harmony.
Quantitative increase manifest itself outwardly
There are many levels of organization in nervous systems. Hence we aim to explain mechanisms at one level in terms of properties and dynamics at a lower level, and to fit that in with the properties at the higher levels.
Never, never, never think - that's one lesson you should have in life. Never think that lack of variability is stability.
Every existence above a certain rank has its singular points; the higher the rank the more of them. At these points, influences whose physical magnitude is too small to be taken account of by a finite being may produce results of the greatest importance.
It is by avoiding the rapid decay into the inert state of 'equilibrium' that an organism appears so enigmatic;
The fundamental laws necessary for the mathematical treatment of a large part of physics and the whole of chemistry are thus completely known, and the difficulty lies only in the fact that application of these laws leads to equations that are too complex to be solved.
Small islands of coherence in a sea of chaos can shift the whole system to a higher order.
A non-analogue image has an extremely compressed life. It starts as this and, in increasingly short time spans, becomes that.
One of the endlessly alluring aspects of mathematics is that its thorniest paradoxes have a way of blooming into beautiful theories.
The source of all great mathematics is the special case, the concrete example. It is frequent in mathematics that every instance of a concept of seemingly generality is, in essence, the same as a small and concrete special case.
In my paper the fact the XY was not equal to YX was very disagreeable to me. I felt this was the only point of difficulty in the whole scheme ... and I was not able to solve it.
I remember my friend Johnny von Neumann used to say, 'with four parameters I can fit an elephant and with five I can make him wiggle his trunk.'
The most complex object in mathematics, the Mandelbrot Set ... is so complex as to be uncontrollable by mankind and describable as 'chaos'.
Between the two. Harmony emerges from integration. Chaos and rigidity arise when integration is blocked.
All of audio as we know it is an attempt to be more and more perfectly linear. Linearity means higher quality sound. Hypersonic sound is exactly the opposite: it's 100 percent based on non-linearity.
The main concern of the study is with the outline of a theoretical system. Its minor variations from writer to writer are not a matter of concern to this analysis.
Not one is constant, but everything is changing.
There is also a rhythm and a pattern between the phenomena of nature which is not apparent to the eye, but only to the eye of analysis; and it is these rhythms and patterns which we call Physical Laws. What
One may characterize physics as the doctrine of the repeatable, be it a succession in time or the co-existence in space. The validity of physical theorems is founded on this repeatability.
True stability results when presumed order and presumed disorder are balanced.
The further a mathematical theory is developed, the more harmoniously and uniformly does its construction proceed, and unsuspected relations are disclosed between hitherto separated branches of the science.
Deep in the chaotic regime, slight changes in structure almost always cause vast changes in behavior. Complex controllable behavior seems precluded.
Vertical and horizontal lines are the expression of two opposing forces; they exist everywhere and dominate everything; their reciprocal action constitutes 'life'. I recognized that the equilibrium of any particular aspect of nature rests on the equivalence of its opposites.
It may happen that small differences in the initial conditions produce very great ones in the final phenomena.
For a finite-size system to persist in time (to live), it must evolve in such a way that it provides easier access to the imposed currents that flow through it.