Linear Equations Quotes

As Alaska zipped through something obvious about linear equations, stoner/baller Hank Walsten said, "Wait, wait. I don't get it." "That's because you have eight functioning brain cells." "Studies show that Marijuana is better for your health than those cigarettes," Hank said. Alaska swallowed a mouthful of fries, took a drag on her cigarette, and blew a smoke at Hank. "I may die young," she said. "But at least I'll die smart. Now, back to tangents.

You've never heard of Chaos theory? Non-linear equations? Strange attractors? Ms. Sattler, I refuse to believe you're not familiar with the concept of attraction.

Life is a linear equation in which you can't cross multiply! If you think you can do it, you can do it. If you think you can't do it, you can't do it. It's a simple formula!

And I think What does it matter that it is not a linear equation if any variable is raised to a power? We’re all just going to die anyway.

Like any good teacher, she tolerated little dissension. She smoked and talked and ate for an hour without stopping, and I scribbled in my notebook as the muddy waters of tangents and cosines began to clarify. But not everyone was so fortunate. As Alaska zipped through something obvious about linear equations, stoner/baller Hank Walsten said, "Wait, wait. I don't get it." "That's because you have eight functioning brain cells." "Studies show that marijuana is better for your health than those cigarettes," Hank said. Alaska swallowed a mouthful of french fries, took a drag on her cigarette, and blew smoke across the table at Hank. "I may die young," she said. "But at least I'll die smart. Now, back to tangents.

If he loved you guys, he wouldn’t hurt you.” He made it sound so simple, as if it were a mathematical equation. But the connection between pain and love wasn’t linear. It was a web.

Equations are classified by the highest power (value of the exponent) of its unknowns. If this is one, the equation is of first degree. If this is two, the equation is of second degree, and so on. Equations of higher degree than one yield multiple possible values for their unknown quantities. These values are known as roots. The first-degree equation (the linear equation): 3x – 9 = 0 (root: x = 3)

Smothered in numbers?” Sophie raised her eyebrows. “I didn’t know that could be fatal.” “It totally can,” Ava said. “Death by linear equations.

I had grown up thinking of life as a series of linear decisions that if made properly would land me on some distant safe shore where I would finally enjoy the fruits of my labor. Now that I was getting a glimpse of that shore I was struck by the inanity of such an equation. My mother was never going to get another chance to do anything else. She did not have the capacity for regrets, nor was she even able to enjoy the comfort of nostalgia or fond memories--her mind had leaked away too imperceptibly to allow for the clarity to look back on her life and wish she had done things differently. As I continued to worry over what sort of future I was setting myself up for, she seemed a painful cautionary tale that life was not a savings plan, accrued now for enjoyment later. I was alive now. My responsibility was to live now as fully as possible.

All right,” Malcolm said. “Let’s go back to the beginning.” He paused, staring at the ceiling. “Physics has had great success at describing certain kinds of behavior: planets in orbit, spacecraft going to the moon, pendulums and springs and rolling balls, that sort of thing. The regular movement of objects. These are described by what are called linear equations, and mathematicians can solve those equations easily. We’ve been doing it for hundreds of years.” “Okay,” Gennaro said. “But there is another kind of behavior, which physics handles badly. For example, anything to do with turbulence. Water coming out of a spout. Air moving over an airplane wing. Weather. Blood flowing through the heart. Turbulent events are described by nonlinear equations. They’re hard to solve—in fact, they’re usually impossible to solve. So physics has never understood this whole class of events. Until about ten years ago. The new theory that describes them is called chaos theory.

I've always been good at math. It's straightforward, black-and-white, right and wrong. Equations. Da thought of people as books to be read, but I've always thought of them more as formulas—full of variables, but always the sum of their parts. That's what their noise is, really: all of a person's components layered messily over one another. Thought and feeling and memory and all of it unorganized, until a person dies. Then it all gets compiled, straightened out into this linear thing, and you see exactly what the various parts add up to. What they equal.

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.

Another class of universal truths is the conservation laws, where the amount of some measured quantity remains unchanged no matter what. The three most important are the conservation of mass and energy, the conservation of linear and angular momentum, and the conservation of electric charge. These laws are in evidence on Earth, and everywhere we have thought to look—from the domain of particle physics to the large-scale structure of the universe. In spite of this boasting, all is not perfect in paradise. It happens that we cannot see, touch, or taste the source of eighty-five percent of the gravity we measure in the universe. This mysterious dark matter, which remains undetected except for its gravitational pull on matter we see, may be composed of exotic particles that we have yet to discover or identify. A small minority of astrophysicists, however, are unconvinced and have suggested that there is no dark matter—you just need to modify Newton’s law of gravity. Simply add a few components to the equations and all will be well. Perhaps

For S, the first piece of information in a list was always, and without fail, inextricably linked to the second piece of information, which could only be followed by the third. It didn’t matter whether he was memorizing Dante’s Divine Comedy or mathematical equations; his memories were always stored in linear chains. Which is why he could recite poems just as easily backward as forward.

Now, if on the one hand it is very satisfactory to be able to give a common ground in the theory of knowledge for the many varieties of statements concerning space, spatial configurations, and spatial relations which, taken together, constitute geometry, it must on the other hand be emphasised that this demonstrates very clearly with what little right mathematics may claim to expose the intuitional nature of space. Geometry contains no trace of that which makes the space of intuition what it is in virtue of its own entirely distinctive qualities which are not shared by “states of addition-machines” and “gas-mixtures” and “systems of solutions of linear equations”. It is left to metaphysics to make this “comprehensible” or indeed to show why and in what sense it is incomprehensible. We as mathematicians have reason to be proud of the wonderful insight into the knowledge of space which we gain, but, at the same time, we must recognise with humility that our conceptual theories enable us to grasp only one aspect of the nature of space, that which, moreover, is most formal and superficial.

The extraordinary value of the I Ching is that it reveals the secrets of dynamic natural law. Working with its changes opens up access to the middle level of the Positive Paradigm Wheel, the “e” energy layer of Einstein's Unified Theory. This middle level serves as mediating, two-directional gate-keeper between the ever-changing surface rim and the universal, timeless center. You can't get from here to there, except through the middle layer which, in Western thinking, is effectively taboo, buried in the inaccessible "unconscious." To the extent that natural law is a blind spot in the prevailing, linear and exclusively empirical paradigm, we are left powerless to move beyond the surface level of experience. The realm of light and conscience which rests beyond, on the far side of the dynamic energy level, remains functionally inaccessible. Moral codes promoted by religionists or politicians are sometimes equated with conscience. But they're no substitute for direct experience. Only by becoming intelligently competent in managing the subtle energies of the middle level is it possible to travel further inwards for the immediate, personal experience of inner light. When the middle level becomes clogged with painful memories, negative emotions and socially taboo urges, it becomes a barrier to deeper knowing. The Book of Change is indispensable as a tool for restoring the unnecessarily "unconscious" to conscious awareness, so that the levels of human potential can be linked and unified. In Positive Paradigm context, survivors who prevail in dangerous times aren't those with the most material wealth, possessions or political power. They're the ones who've successfully navigated the middle realm, reached the far shore of enlightenment and returned to the surface with their new information intact. Those who succeed in linking the levels of experience are genius-leaders in whatever fields they choose to engage. They're the fortunate ones who've acquired the inner wealth necessary to both hear the inner voice of conscience and act on the guidance they receive.

Each group was required to propose its own “fitness function”—a linear equation that it could use to measure its own impact without ambiguity. For example, a two-pizza team in charge of sending advertising e-mails to customers might choose for its fitness function the rate at which these messages were opened multiplied by the average order size those e-mails generated. A group writing software code for the fulfillment centers might home in on decreasing the cost of shipping each type of product and reducing the time that elapsed between a customer’s making a purchase and the item leaving the FC in a truck. Bezos wanted to personally approve each equation and track the results over time. It would be his way of guiding a team’s evolution. Bezos was applying

The entire company, he said, would restructure itself around what he called “two-pizza teams.” Employees would be organized into autonomous groups of fewer than ten people—small enough that, when working late, the team members could be fed with two pizza pies. These teams would be independently set loose on Amazon’s biggest problems. They would likely compete with one another for resources and sometimes duplicate their efforts, replicating the Darwinian realities of surviving in nature. Freed from the constraints of intracompany communication, Bezos hoped, these loosely coupled teams could move faster and get features to customers quicker. There were some head-scratching aspects to Bezos’s two-pizza-team concept. Each group was required to propose its own “fitness function”—a linear equation that it could use to measure its own impact without ambiguity. For example, a two-pizza team in charge of sending advertising e-mails to customers might choose for its fitness function the rate at which these messages were opened multiplied by the average order size those e-mails generated. A group writing software code for the fulfillment centers might home in on decreasing the cost of shipping each type of product and reducing the time that elapsed between a customer’s making a purchase and the item leaving the FC in a truck. Bezos wanted to personally approve each equation and track the results over time. It would be his way of guiding a team’s evolution.

liter. The correlation value, r, will be the same for either set of units.  Note however that the slope of the regression line won’t be the same since And the standard deviation parts of the equation still have units baked into them.     Correlation Takeaways We did a lot of looking at equations in this section.  What are the key takeaways? The

The key takeaway is that correlation is an understandable equation that relates the amount of change in x and y.  If the two variables have consistent change, there will be a high correlation; otherwise, there will have a lower correlation.    

Stephenson summarizes his communication policy as follows: Persons who wish to interfere with my concentration are politely requested not to do so, and warned that I don’t answer e-mail… lest [my communication policy’s] key message get lost in the verbiage, I will put it here succinctly: All of my time and attention are spoken for—several times over. Please do not ask for them. To further justify this policy, Stephenson wrote an essay titled “Why I Am a Bad Correspondent.” At the core of his explanation for his inaccessibility is the following decision: The productivity equation is a non-linear one, in other words. This accounts for why I am a bad correspondent and why I very rarely accept speaking engagements. If I organize my life in such a way that I get lots of long, consecutive, uninterrupted time-chunks, I can write novels. But as those chunks get separated and fragmented, my productivity as a novelist drops spectacularly.

Regression analysis mathematically describes the relationships between independent variables and a dependent variable. Use regression for two primary goals: To understand the relationships between these variables. How do changes in the independent variables relate to changes in the dependent variable? To predict the dependent variable by entering values for the independent variables into the regression equation.

Observed values of the dependent variable are the values of the dependent variable that you record during your study or experiment along with the values of the independent variables. These values are denoted using Y. Fitted values are the values that the model predicts for the dependent variable using the independent variables. If you input values for the independent variables into the regression equation, you obtain the fitted value. Predicted values and fitted values are synonyms.

This graph shows all the observations together with a line that represents the fitted relationship. As is traditional, the Y-axis displays the dependent variable, which is weight. The X-axis shows the independent variable, which is height. The line is the fitted line. If you enter the full range of height values that are on the X-axis into the regression equation that the chart displays, you will obtain the line shown on the graph. This line produces a smaller SSE than any other line you can draw through these observations. Visually, we see that that the fitted line has a positive slope that corresponds to the positive correlation we obtained earlier. The line follows the data points, which indicates that the model fits the data. The slope of the line equals the coefficient that I circled. This coefficient indicates how much mean weight tends to increase as we increase height. We can also enter a height value into the equation and obtain a prediction for the mean weight. Each point on the fitted line represents the mean weight for a given height. However, like any mean, there is variability around the mean. Notice how there is a spread of data points around the line. You can assess this variability by picking a spot on the line and observing the range of data points above and below that point. Finally, the vertical distance between each data point and the line is the residual for that observation.

Stephenson wrote an essay titled “Why I Am a Bad Correspondent.” At the core of his explanation for his inaccessibility is the following decision: The productivity equation is a non-linear one, in other words. This accounts for why I am a bad correspondent and why I very rarely accept speaking engagements. If I organize my life in such a way that I get lots of long, consecutive, uninterrupted time-chunks, I can write novels. But as those chunks get separated and fragmented, my productivity as a novelist drops spectacularly.

Any liquid or gas is a collection of individual bits, so many that they may as well be infinite. If each piece moved independently, then the fluid would have infinitely many possibilities, infinitely many “degrees of freedom” in the jargon, and the equations describing the motion would have to deal with infinitely many variables. But each particle does not move independently—its motion depends very much on the motion of its neighbors—and in a smooth flow, the degrees of freedom can be few. Potentially complex movements remain coupled together. Nearby bits remain nearby or drift apart in a smooth, linear way that produces neat lines in wind-tunnel pictures. The particles in a column of cigarette smoke rise as one, for a while.

The mathematician Stanislaw Ulam once said that calling a problem nonlinear was like going to the zoo and talking about all the interesting nonelephant animals you see there. His point was that most animals are not elephants, and most equations are not linear. Linear equations describe simple, idealized situations where causes are proportional to effects, and forces are proportional to responses. If you bend a steel girder by two millimeters instead of one, it will push back twice as hard. The word linear refers to this proportionality: If you graph the deflection of the girder versus the force applied, the relationship falls on a straight line.

But linearity is often an approximation to a more complicated reality. Most systems behave linearly only when they are close to equilibrium, and only when we don’t push them too hard. A civil engineer can predict how a skyscraper will sway in the wind, as long as the wind is not too strong. Electrical circuits are completely predictable—until they get fried by a power surge. When a system goes nonlinear, driven out of its normal operating range, all bets are off. The old equations no longer apply.

Each operation contributes to AES’s security in a specific way: * Without KeyExpansion, all rounds would use the same key, K, and AES would be vulnerable to slide attacks. * Without AddRoundKey, encryption wouldn’t depend on the key; hence, anyone could decrypt any ciphertext without the key. * SubBytes brings nonlinear operations, which add cryptographic strength. Without it, AES would just be a large system of linear equations that is solvable using high-school algebra. * Without ShiftRows, changes in a given column would never affect the other columns, meaning you could break AES by building four 232 element codebooks for each column. (Remember that in a secure block cipher, flipping a bit in the input should affect all the output bits.) * Without MixColumns, changes in a byte would not affect any other bytes of the state. A chosen-plaintext attacker could then decrypt any ciphertext after storing 16 lookup tables of 256 bytes each that hold the encrypted values of each possible value of a byte.

The more linear one tries to make the equation of planning , the more complex becomes it's algorithm

Each group was required to propose its own “fitness function”—a linear equation that it could use to measure its own impact without ambiguity. For example, a two-pizza team in charge of sending advertising e-mails to customers might choose for its fitness function the rate at which these messages were opened multiplied by the average order size those e-mails generated.

Without friction a simple linear equation expresses the amount of energy you need to accelerate a hockey puck. With friction the relationship gets complicated, because the amount of energy changes depending on how fast the puck is already moving. Nonlinearity means that the act of playing the game has a way of changing the rules. You cannot assign a constant importance to friction, because its importance depends on speed. Speed, in turn, depends on friction. That twisted changeability makes nonlinearity hard to calculate, but it also creates rich kinds of behavior that never occur in linear systems. In fluid dynamics, everything boils down to one canonical equation, the Navier-Stokes equation. It is a miracle of brevity, relating a fluid's velocity, pressure, density, and viscosity, but it happens to be nonlinear. So the nature of those relationships often becomes impossible to pin down. Analyzing the behavior of a nonlinear equation like the Navier-Stokes equation is like walking through a maze whose walls rearrange themselves with each step you take. As Von Neumann himself put it: "The character of the equation ... changes simultaneously in all relevant respects: Both order and degree change. Hence, bad mathematical difficulties must be expected." The world would be a different place—and science would not need chaos—if only the Navier-Stokes equation did not contain the demon of nonlinearity.

So which is more probable: That today's atheist apocalyptans are unique and right? Or that they are like their many predecessors—at the very least, in their motivations? If anything, the vehemence with which the believers in emergent complexity debunk all religion may betray their own creeping awareness of the religious underpinnings and precedents for their declarations. In fact, the concept of Armageddon first emerged in response to the invention of monotheism by the ancient Persian priest Zoroaster, around the tenth or eleventh century BCE. Until that time, the dominant religions maintained a pantheon of gods reigning in a cyclical precession along with the heavens, so there was little need for absolutes. As religions began focusing on a single god, things got a bit trickier. For if there is only one god, and that god has absolute power, then why do bad things happen? Why does evil still exist? If one's god is fighting for control of the universe against the gods of other people, then there's no problem. Just as in polytheism, the great achievements of one god can be undermined by the destructive acts of another. But what if a religion, such as Judaism of the First and Second Temple era, calls for one god and one god alone? How do its priests and followers explain the persistence of evil and suffering? They do it the same way Zoroaster did: by introducing time into the equation. The imperfection of the universe is a product of its incompleteness. There's only one true god, but he's not done yet. In the monotheist version, the precession of the gods was no longer a continuous cycle of seasonal deities or metaphors. It was nor a linear story with a clear endpoint in the victory of the one true and literal god. Once this happens, time can end. Creation is the Alpha, and the Return is the Omega. It's all good. This worked well enough to assuage the anxieties of both the civilization of the calendar and that of the clock. But what about us? Without time, without a future, how to we contend with the lingering imperfections in our reality? As members of a monotheist culture—however reluctant—we can't help but seek to apply its foundational framework to our current dilemma. The less aware we are of this process—or the more we refuse to admit its legacy in our construction of new models—the more vulnerable we become to its excesses. Repression and extremism are two sides of the same coin. In spite of their determination to avoid such constructs, even the most scientifically minded futurists apply the Alpha-Omega framework of messianic time to their upgraded apocalypse narratives. Emergence takes the place of the hand of God, mysteriously transforming a chaotic system into a self-organized one, with coherence and cooperation. Nobody seems able to explain how this actually happens.

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.