Exponential Function Quotes

The greatest shortcoming of the human race is our inability to understand the exponential function.

The greatest shortcoming of the human race is our inability to understand the exponential function.” —Albert A. Bartlett

Investing is a special thing. In terms of functionality, almost anyone can invest. But in terms of achieving the results of long-term profit and sustainable growth, only some people have the talent or skill sets for that. It’s like baseball for example… anyone can swing a bat at a ball. But only a few people make it to the big league, and even fewer become world champs. These days there are so many apps and platforms for individual investing, but that doesn’t mean everyone is achieving good results or ROI. There are great investors, good investors, and bad investors. A professional investor can achieve exponential growth and profit. A professional investor understands markets and industries and can account for both the traditional and the new.

Investing is a special thing. In terms of functionality, almost anyone invest. But in terms of achieving the results of long-term profit and sustainable growth, only some people have the talent or skill sets for that. It’s like baseball for example… anyone can swing a bat at a ball. But only a few guys make it to the big league, and even fewer become world champs. These days there are so many apps and platforms for individual investing, but that doesn’t mean everyone is achieving the same results. There are great investors, good investors, and bad investors. A professional investor can achieve exponential growth and profit. A professional investor understands markets and industries and can account for both the traditional and the new.

We have trouble estimating dramatic, exponential change. We cannot conceive that a piece of paper folded over 50 times could reach the sun. There are abrupt limits to the number of cognitive categories we can make and the number of people we can truly love and the number of acquaintances we can truly know. We throw up our hands at a problem phrased in an abstract way, but have no difficulty at all solving the same problem rephrased as a social dilemma. All of these things are expressions of the peculiarities of the human mind and heart, a refutation of the notion that the way we function and communicate and process information is straightforward and transparent. It is not. It is messy and opaque.

two of the most important function classes: exponential and trigonometric functions.

Based on the above analyses, it is reasonable to expect the hardware that can emulate human-brain functionality to be available for approximately one thousand dollars by around 2020. As we will discuss in chapter 4, the software that will replicate that functionality will take about a decade longer. However, the exponential growth of the price-performance, capacity, and speed of our hardware technology will continue during that period, so by 2030 it will take a village of human brains (around one thousand) to match a thousand dollars’ worth of computing. By 2050, one thousand dollars of computing will exceed the processing power of all human brains on Earth. Of course, this figure includes those brains still using only biological neurons.

Mosaic also marked a new stage in the evolution of the power law. Venture-capital returns are dominated by grand slams partly because of the dynamics of startups: most young businesses fail, but the ones that gain traction can grow exponentially. This is true of fashion brands or hotel chains as well as technology companies. But tech-focused venture portfolios are dominated by the power law for an additional reason: tech startups are founded upon technologies that may themselves progress exponentially. Because of his experience and temperament, Doerr was especially attuned to this phenomenon. As a young engineer at Intel, he had seen how Moore’s law transformed the value of companies that used semiconductors: the power of chips was doubling every two years, so startups that put them to good use could make better, cheaper products. For any given modem, digital watch, or personal computer, the cost of the semiconductors inside the engine would fall by 50 percent in two years, 75 percent in four years, and 87.5 percent in eight. With that sort of wind at a tech startup’s back, no wonder profits could grow exponentially. Mosaic, and the internet more generally, turbocharged this phenomenon. Again, Doerr grasped this better than most others. As well as working at Intel, he had known Bob Metcalfe, so he understood that Metcalfe’s law was even more explosive than Moore’s law. Rather than merely doubling in power every two years, as semiconductors did, the value of a network would rise as the square of the number of users.[70] Progress would thus be quadratic rather than merely exponential; something that keeps on squaring will soon grow a lot faster than something that keeps on doubling. Moreover, progress would not be tethered to the passage of time; it would be a function of the number of users. At the moment when Doerr met Clark, the number of internet users was about to triple over the next two years, meaning that the value of the network would jump ninefold, an effect massively more powerful than the mere doubling in the power of semiconductors over that same period. What’s more, Metcalfe’s law was not supplanting Moore’s law, which would have been dramatic enough. Rather, it was compounding it. The explosion of internet traffic would be fueled both by its rapid growth in usefulness (Metcalfe’s law) and by the falling cost of modems and computers (Moore’s law).[71] After listening to Clark’s pitch, Doerr was determined to invest. A magical browser that attracted millions to the internet had almost limitless potential. The price Doerr had to pay was secondary.

In the last centuries of the Exponential Age gender began to be liberated from biology, but that process wasn’t nearly finished when the Church War came. The worst cults in the war were also associated with gender oppression, so after the war the nascent Hives tried to purge all gender differences so abruptly that there was no time to come up with substitutes for all the other social functions gender used to have.

In the weeks and months after Immelt left GE in 2017, a parade of negative stories and embarrassing disclosures revealed major problems that sent the company’s stock into a long decline. Conversations about what happened inevitably shifted to blame, and Immelt was the obvious target. He had spent sixteen years at the top and, regardless of what Welch had left for him, he’d had plenty of time to fix it. But there was plenty of blame to go around. Perhaps most of it should be placed on the board of directors, the independent group that oversees the CEO. Board members claimed to have been unaware of problems and to have gotten bad guidance from external advisers, and they said they didn’t understand how the company went from good to bad seemingly overnight. Some directors had no experience in GE’s business lines, others had trouble staying awake during meetings, and many stumbled away from GE’s collapse wondering, How could we have known? It had been their job to know, however, and their job to ask the hard questions that weren’t fully answered, or were never asked at all. It was their job to oversee management, and it was their job to protect investors from fatal hubris. Still, the path ultimately leads back to Immelt. As chairman, he was also responsible for steering the board. There is no doubt that GE’s size and complexity, which grew exponentially under Immelt, made it difficult or even impossible to manage. The CEO of a company is responsible for its daily functions and for managing its operations, however vast. The chairman guides the board, which is responsible for overseeing management and the CEO. When the board chair and CEO are the same person, the top executive is essentially his own boss. It can only get worse with time if a chairman remakes the board to his own liking. Simply put, it is terrible governance to give so much power to a single person and so little voice to shareholders. That is one reason this governance structure has been slowly fading from corporate America since the Enron era.

am impressed with how well they do in a world where street crime seems to be increasing exponentially. There is an obvious economic relationship between the amount of “free crime” at the top of the economic food chain, and the increases in poverty, hopelessness addiction and street crime at the bottom. The police are doing a great job (on the streets) in my view. If memory serves me they responded to 35,000 calls in one year with their $40 million dollar budget. However, the public must know that they (police) rarely respond or get involved in high value economic crime, and often they tell victims of million dollar crimes, that their complaint is a “civil matter” and should be dealt with in the civil courts. They do a great job at the level of street safety and property protection, but at most crimes over a certain financial level, or complexity, they defer to others. Police do not appear to function in government buildings and office suites, like they do in the streets. My government has offices for high value economic crimes. These are commercial crime police units called the RCMP Integrated Markets Enforcement Teams. (RCMP IMET) By some coincidence they also operate on a budget in the neighborhood of $40 Million dollars…but $40 Million is what they have for the protection of the entire country of Canada. They handle perhaps a dozen cases a year, and we rarely hear of a successful prosecution. I believe it is intentional. No person power finds it wise…to investigate persons in power.

And while not owning assets has been standard practice for heavy machinery and non-mission-critical functions (e.g., copiers) for decades, recently there’s been an accelerating trend towards outsourcing even mission-critical assets.

Chapter 8: Exponential Functions Chapter 9:

The S curve is not just important as a model in its own right; it’s also the jack-of-all-trades of mathematics. If you zoom in on its midsection, it approximates a straight line. Many phenomena we think of as linear are in fact S curves, because nothing can grow without limit. Because of relativity, and contra Newton, acceleration does not increase linearly with force, but follows an S curve centered at zero. So does electric current as a function of voltage in the resistors found in electronic circuits, or in a light bulb (until the filament melts, which is itself another phase transition). If you zoom out from an S curve, it approximates a step function, with the output suddenly changing from zero to one at the threshold. So depending on the input voltages, the same curve represents the workings of a transistor in both digital computers and analog devices like amplifiers and radio tuners. The early part of an S curve is effectively an exponential, and near the saturation point it approximates exponential decay. When someone talks about exponential growth, ask yourself: How soon will it turn into an S curve? When will the population bomb peter out, Moore’s law lose steam, or the singularity fail to happen? Differentiate an S curve and you get a bell curve: slow, fast, slow becomes low, high, low. Add a succession of staggered upward and downward S curves, and you get something close to a sine wave. In fact, every function can be closely approximated by a sum of S curves: when the function goes up, you add an S curve; when it goes down, you subtract one. Children’s learning is not a steady improvement but an accumulation of S curves. So is technological change. Squint at the New York City skyline and you can see a sum of S curves unfolding across the horizon, each as sharp as a skyscraper’s corner. Most importantly for us, S curves lead to a new solution to the credit-assignment problem. If the universe is a symphony of phase transitions, let’s model it with one. That’s what the brain does: it tunes the system of phase transitions inside to the one outside. So let’s replace the perceptron’s step function with an S curve and see what happens.

There's something about Algebra, I just can't figure it out Polynomials, derivatives, quadratic equations, I see no absolute value in them A bunch of irrational numbers With square roots and exponential functions I'm still trying to see through the horizontal and vertical blurred lines This all reminds me Y I left my X-

In fact, as we looked at the history of larger cities, we found evidence of this in the way they were laid out and designed. Older towns once had a town square that operated as the functional center of the city. It was surrounded by the institutions of commerce and trade, law and government, the arts and places for dialogue. But alongside these key cultural institutions was always the church. The entire city would be situated around a center, and that’s what tied life in the city together. Again, the church was always an integral part of that connection. So we wondered: Was it possible to give new life to that long-lost vision of connection and unity, where multiple institutions came together for the common good of the city? Could it happen once again?

Exponential Backoff was a huge part of the successful functioning of the ALOHAnet beginning in 1971, and in the 1980s it was baked into TCP, becoming a critical part of the Internet. All these decades later, it still is. As one influential paper puts it, “For a transport endpoint embedded in a network of unknown topology and with an unknown, unknowable and constantly changing population of competing conversations, only one scheme has any hope of working—exponential backoff.

When assessing a startup for funding, investors typically categorize three major risk areas: Technology risk: Will it work? Market risk: Will people buy the product? Execution risk: Is the team able to function and pivot as needed?

what practical uses are there for functions? We can use them to establish a connection between the value of one physical quantity and another. For example, through experiments or theoretical considerations we can determine a function f(p) that links the air pressure p to the air density D. It would allow us to insert any value for the air pressure p and calculate the corresponding value for the air density D, which can be quite useful.

As a result I early asked the question, "Why should I do all the analysis in terms of Fourier integrals? Why are they the natural tools for the problem?" I soon found out, as many of you already know, that the eigenfunctions of translation are the complex exponentials. If you want time invariance, and certainly physicists and engineers do (so that an experiment done today or tomorrow will give the same results), then you are led to these functions. Similarly, if you believe in linearity then they are again the eigenfunctions. In quantum mechanics the quantum states are absolutely additive; they are not just a convenient linear approximation. Thus the trigonometric functions are the eigenfunctions one needs in both digital filter theory and quantum mechanics, to name but two places. Now when you use these eigenfunctions you are naturally led to representing various functions, first as a countable number and then as a non-countable number of them-namely, the Fourier series and the Fourier integral. Well, it is a theorem in the theory of Fourier integrals that the variability of the function multiplied by the variability of its transform exceeds a fixed constant, in one notation l/2pi. This says to me that in any linear, time invariant system you must find an uncertainty principle.

Altered Intercellular Communication: For the body to function properly, cells need to communicate. This happens constantly, with messages flowing through our bloodstream, immune system, and endocrine system. Over time, signals get crossed. Some cells become unresponsive, others become inflammation-producing zombie cells. This inflammation blocks further communication. Once this happens, messages can’t get through and the immune system can’t find pathogens.

as the nuclear physicist Al Bartlett warned, ‘The greatest shortcoming of the human race is our inability to understand the exponential function

the curve will never touch the x-axis (in other words: it has no roots),

water waves are actually trochoidal in shape,

You as the leader should have a Function Accountability Chart (FAC) to help CEOs and managers to see clearly the person who is accountable for each role and key positions. It should also show the metrics or key performance indicators assigned for each of the main functions of the business. Each item on the Profit & Loss, and Balance Sheet should be assigned to specific people who are accountable for these roles. Each team should know their responsibilities, accountabilities and who they are answerable

roots of functions

Exponential functions appear in many real-world situations: population growth, technological growth, product value over time, compounded interest, radioactive decay, statistical analysis, ...

Second only to guarding the freedom of exchange of information, acquire a firm basic knowledge in communications and in technology. Learn to use your native language well and precisely, because ours is and will become even more a communicating society. Equally important, learn the basics of the technological world. If you don't understand the laws of Newton, or don't know the difference between volts and amperes, you'll have little hope of understanding this next century. Trace the functions on a pocket calculator to make such concepts as exponential growth and sinewave oscillation less of a mystery.

By asking other people what their experiences were with a product and by asking them for their recommendations, we can access a vast amount of experience without having to spend time, money, or effort—or take the risks that we would by doing it on our own. So, we not only ask what product is best, but we ask, “What was your experience?” What should we watch out for and what tricks and tips should we follow? How can we best try it, fix it, and teach others to use it? Who has the best prices and delivery? Who is honest? All of this saves a tremendous amount of time. We could never function in the modern world without sharing experiences through word of mouth.

In this, the Age of Overload, the service that marketing provides is to make the decision process easier for the customer, every step of the way. SECRET Your new function is decision easification, which means easing the burden of the decision process in this overloaded world.

You are in the perfect position to exercise the new function of marketing: making decisions easier for your customers. You know the lay of the land better than anyone else. You also have a better perspective on what it takes to find, learn about, sort out, try, buy, and use your product than anyone else. You know best how your product fits into the marketplace. In fact, you could be the best guide through the burdensome decision-making process. In this Age of Overload, that’s exactly what the customer needs. The raw information she can get elsewhere. But what information does she need, when? What’s the sequence? What are the best sources of information from a customer’s point of view?

Hermann estimated that forgetting follows a roughly exponential decay function. We forget nearly half of the information we’ve learned within twenty minutes. After twenty-four hours, about 70 percent of it is gone, and a month later, we’re looking at losses of approximately 80 percent

In 2005, a computer expert named Ian Grigg, working at a company called Systemics, introduced a trial system he called “triple-entry bookkeeping.” Grigg worked in the field of cryptography, a science that dates way back to ancient times, when coded language to share “ciphers,” or secrets, first arose. Ever since Alan Turing’s calculating machine cracked the German military’s Enigma code, cryptography has underpinned much of what we’ve done in the computing age. Without it we wouldn’t be able to share private information across the Internet—such as our transactions within a bank’s Web site—without revealing it to unwanted prying eyes. As our computing capacity has exponentially grown, so too has the capacity of cryptography to impact our lives. For his part, Grigg believed it would lead to a programmable record-keeping system that would make fraud virtually impossible. In a nutshell, the concept took the existing, double-entry bookkeeping system and added a third book: the independent, open ledger that’s secured by cryptographic methods so that no one can change it. Grigg saw it as a way to combat fraud. The way Grigg described it, users would maintain their own, double-entry accounts, but added to these digitized books would be another function, essentially a time stamp, a cryptographically secured, signed receipt of every transaction. (The concept of a “signature” in cryptography means something far more scientific than a handwritten scrawl; it entails combining two associated numbers, or “keys”—one publicly known, the other private—to mathematically prove that the entity making the signature is uniquely authorized to do so.)