Daniel Fleisch - Collected Quotes

"Among the differences that will always be with you are the small overshoots and oscillations just before and after the vertical jumps in the square waves. This is called 'Gibbs ripple' and it will cause an overshoot of about 9% at the discontinuities of the square wave no matter how many terms of the series you add. But [...] adding more terms increases the frequency of the Gibbs ripple and reduces its horizontal extent in the vicinity of the jumps." (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)

"An understanding of complex numbers can make the study of waves consid erably less mysterious, and you probably already have an idea that complex numbers have real and imaginary parts. Unfortunately, the term 'imaginary' often leads to confusion about the nature and usefulness of complex numbers." (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)

"As the mechanical wave source moves through the medium, it pushes on a nearby segment of the material, and that segment moves away from the source and is compressed (that is, the same amount of mass is squeezed into a smaller volume, so the density of the segment increases). That segment of increased density exerts pressure on adjacent segments, and in this way a pulse (if the source gives a single push) or a harmonic wave (if the source oscillates back and forth) is generated by the source and propagates through the material." (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)

"Before considering the wave equation for mechanical waves, you should understand the difference between the motion of individual particles and the motion of the wave itself. Although the medium is disturbed as a wave goes by, which means that the particles of the medium are displaced from their equilibrium positions, those particles don’t travel very far from their undisturbed positions. The particles oscillate about their equilibrium positions, but the wave does not carry the particles along – a wave is not like a steady breeze or an ocean current which transports material in bulk from one location to another. For mechanical waves, the net displacement of material produced by the wave over one cycle, or over one million cycles, is zero. So, if the particles aren’t being carried along with the wave, what actually moves at the speed of the wave? […] the answer is energy." (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)

"But the presence of √−1 (the rotation operator between the two perpendicular numbe rlines in the complex plane) in the exponent causes the expression e^ix to move from the real to the imaginary number line. As it does so, its real and imaginary parts oscillate in a sinusoidal fashion […] So the real and imaginary parts of the expression e^ix oscillate in exactly the same way as the real and imaginary components of the rotating phasor […]" (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)

"So a very useful way to think about i (√−1) is as an operator that produces a 90◦ rotation of any vector to which it is applied. Thus the two perpendicular number lines form the basis of what we know today as the complex plane. Unfortunately, since multiplication by √−1 is needed to get from the horizontal to the vertical number line, the numbers along the vertical number line are called 'imaginary'. We say 'unfortunately' because these numbers are every bit as real as the numbers along the horizontal number line. But the terminology is pervasive, so when you first learned about complex numbers, you probably learned that they consist of a “real” and an 'imaginary' part." (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)

"That’s where boundary conditions come in. A boundary condition 'ties down' a function or its derivative to a specified value at a specified location in space or time. By constraining the solution of a differential equation top satisfy the boundary condition(s), you may be able to determine the value of the function or its derivatives at other locations. We say “may” because boundary conditions that are not well-posed may provide insufficient or contradictory information." (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)

"The 'disturbance' of such waves involves three things: the longitudinal displacement of material, changes in the density of the material, and variation of the pressure within the material. So pressure waves could also be called 'density waves' or even 'longitudinal displacement waves', and when you see graphs of the wave disturbance in physics and engineering textbooks, you should make sure you understand which of these quantities is being plotted as the 'displacement' of the wave." (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)

"The easiest way to think about the shape of a wave is to imagine taking a snapshot of the wave at some instant of time. To keep the notation simple, you can call the time at which the snapshot is taken t = 0; snapshots taken later will be timed relative to this first one. At the time of that first snapshot […] can be written as y = f(x, 0) […] Many waves maintain the same shape over time – the wave moves in the direction of propagation, but all peaks and troughs move in unison, so the shape does not change as the wave moves." (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)

"This equation is considered by some mathematicians and physicists to be the most important equation ever devised. In Euler’s relation, both sides of the equation are expressions for a complex number on the unit circle. The left side emphasizes the magnitude (the 1 multiplying e^iθ ) and direction in the complex plane (θ), while the right side emphasizes the real (cos θ) and imaginary (sin θ) components. Another approach to demonstrating the equivalence of the two sides of Euler’s relation is to write out the power-series representation of each side; [...]" (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)

"When you encounter the classical wave equation, it’s likely to be accompanied by some or all of the words 'linear, homogeneous, second-order partial differential equation'. You may also see the word 'hyperbolic' included in the list of adjectives. Each of these terms has a very specific mathematical meaning that’s an important property of the classical wave equation. But there are versions of the wave equation to which some of these words don’t apply, so it’s useful to spend some time understanding them." (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)

"Why are boundary conditions important in wave theory? One reason is this: Differential equations, by their very nature, tell you about the change in a function (or, if the equation involves second derivatives, about the change in the change of the function). Knowing how a function changes is very useful, and may be all you need in certain problems. But in many problems you wish to know not only how the function changes, but also what value the function takes on at certain locations or times." (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)

Frank Close - Collected Quotes

"Anyone who has played with Rubik’s cube knows that twisting the top clockwise and then rotating the right hand side to the back gives a different pattern than if you did the two operations in the reverse order. It is easier to see this with a die. If you rotate a die clockwise and then about the vertical, it will be oriented differently to the case where you had first rotated about the vertical and then clockwise. This is why matrices have proved so useful in keeping track of what happens when things rotate in three  dimensions, as the order matters." (Frank Close, "Antimatter", 2009)

"Dirac’s idea of the bottomless sea full of identical electrons also explains why all electrons and positrons created ‘out of the vacuum’ have identical properties rather than emerging with a random continuum of possibilities. Dirac also proposed that pro tons fill the sea, and today we recognize that their more basic seeds, the ‘quarks’ (which we shall meet in chapter 5), also satisfy the exclusion principle and fill an infinitely deep sea. It is the infinitely deep storehouse of the Dirac sea that provides us with the particles, and antiparticles, that we can materialize." (Frank Close, "Antimatter", 2009)

"In Dirac’s interpretation of the vacuum, if one electron in this sea were missing, it would leave a hole. The absence of a negatively charged electron with energy that is negative relative to sea-level, will appear as a positively charged particle with positive energy, namely with all the attributes of what was later called a positron. This was a strange idea, and quantum mechanics is still strange eighty years later; it was only in its infancy when Dirac made his proposal, which was a piece of radical genius." (Frank Close, "Antimatter", 2009)

"Mechanics is the science of motion. It describes how things move from om one point to another as time passes, the greater the distance moved each second so the greater is the speed. If something moving hits you, the impact will depend not just on how fast it’s travelling but also how massive it is. It is the momentum that matters: the product of mass and velocity. Mechanics also deals with energy, especially the energy due to motion, ‘kinetic energy’." (Frank Close, "Antimatter", 2009)

"Many phenomena require more than just real numbers to describe them mathematically. One such generalization of numbers is known as ‘matrices’. These involve numbers arranged in columns or rows with their own rules for addition and multiplication. Ordi nary numbers correspond to having the same number all down the top left to bottom right diagonal [...]." (Frank Close, "Antimatter", 2009)

"Schrodinger’s equation also explained why the orbital motion of electrons in atoms caused the spectral lines to multiply in magnetic fields. However, it gave no explanation for the electron’s own intrinsic ‘spin’. This known property of the electron had no place in Schrodinger’s theory. A more complete quantum mechanics, one that incorporated spin and applied at relativistic speeds, waited to be discovered." (Frank Close, "Antimatter", 2009)

"The first glimpse of the antiworld came not from experiment, a chance discovery, but from the beautiful patterns that Dirac had seen in his equations. As crotchets, minims, and semiquavers on a stave are mere symbols until interpreted by a maestro and transformed into sublime melody, so can arid equations miraculously reveal harmony in nature." (Frank Close, "Antimatter", 2009)

"The ‘Schrodinger Equation’ explained the behavior of electrons in atoms, and showed that in a hydrogen atom the electron is effectively moving with a speed of about two thousand kilometers a second. This is fast to our senses but is less than one percent of the speed of light. Schrodinger’s theory worked, and even today is widely applied to problems in atomic physics." (Frank Close, "Antimatter", 2009)

"There’s matter, like the electron; antimatter, like the positron; and then there are things that are neither matter nor antimatter. The most familiar example of something that is beyond substance is electromagnetic radiation. All electromagnetic radiation, from gamma rays through X-rays and ultra-violet to visible light, infra red, and radio waves, consists of photons of different energies. Matter and antimatter can cancel one another out, their annihilation leaving non-substance in the form of photons; if the conditions are right this sequence can happen in reverse where photons turn into pieces of matter and antimatter." (Frank Close, "Antimatter", 2009)

"Using matrices, Dirac was able to write an equation relating the total energy of a body to a sum of its energy at rest and its energy in motion, all consistent with Einstein’s theory of relativity. The fact that matrices keep account of what happens when things rotate was a bonus, as the maths was apparently saying that an electron can itself rotate: can spin! Furthermore, the fact that he had been able to solve the mathematics by using the simplest matrices, where a single number was replaced by two columns of pairs, implied a ‘two-ness’ to the spin, precisely what the Zeeman effect had implied. The missing ingredi ent in Schrodinger’s theory had miraculously emerged from the mathematics of matrices, which had been forced on Dirac by the requirements of Einstein’s theory of relativity." (Frank Close, "Antimatter", 2009)

George B Dyson - Collected Quotes

"An Internet search engine is a finite-state, deterministic machine, except at those junctures where people, individually and collectively, make a nondeterministic choice as to which results are selected as meaningful and given a click. These clicks are then immediately incorporated into the state of the deterministic machine, which grows ever so incrementally more knowledgeable with every click. This is what Turing defined as an oracle machine."  (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"If life, by some chance, happens to have originated, and survived, elsewhere in the universe, it will have had time to explore an unfathomable diversity of forms. Those best able to survive the passage of time, adapt to changing environments, and migrate across interstellar distances will become the most widespread. A life form that assumes digital representation, for all or part of its life cycle, will be able to travel at the speed of light." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"In our universe, we measure time with clocks, and computers have a 'clock speed', but the clocks that govern the digital universe are very different from the clocks that govern ours. In the digital universe, clocks exist to synchronize the translation between bits that are stored in memory (as structures in space) and bits that are communicated by code (as sequences in time). They are clocks more in the sense of regulating escapement than in the sense of measuring time."(George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"It is characteristic of objects of low complexity that it is easier to talk about the object than produce it and easier to predict its properties than to build it." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"Sixty-some years ago, biochemical organisms began to assemble digital computers. Now digital computers are beginning to assemble biochemical organisms. Viewed from a distance, this looks like part of a life cycle. But which part? Are biochemical organisms the larval phase of digital computers? Or are digital computers the larval phase of biochemical organisms?" (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"Life evolved, so far, by making use of the viral cloud as a source of backup copies and a way to rapidly exchange genetic code. Life may be better adapted to the digital universe than we think." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two faculties, which we may call intuition and ingenuity [...] (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"Monte Carlo is able to discover practical solutions to otherwise intractable problems because the most efficient search of an unmapped territory takes the form of a random walk. Today’s search engines, long descended from their ENIAC-era ancestors, still bear the imprint of their Monte Carlo origins: random search paths being accounted for, statistically, to accumulate increasingly accurate results. The genius of Monte Carlo - and its search-engine descendants - lies in the ability to extract meaningful solutions, in the face of overwhelming information, by recognizing that meaning resides less in the data at the end points and more in the intervening paths." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"Music allows us to assemble temporal sequences into mental scaffolding that transcends the thinness of time in which we live." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"One of the facets of extreme originality is not to regard as obvious the things that lesser minds call obvious," (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"Over long distances, it is expensive to transport structures, and inexpensive to transmit sequences. Turing machines, which by definition are structures that can be encoded as sequences, are already propagating themselves, locally, at the speed of light. The notion that one particular computer resides in one particular location at one time is obsolete. (George Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012) 

"Random search can be more efficient than nonrandom search - something that Good and Turing had discovered at Bletchley Park. A random network, whether of neurons, computers, words, or ideas, contains solutions, waiting to be discovered, to problems that need not be explicitly defined." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"The brain is a statistical, probabilistic system, with logic and mathematics running as higher-level processes. The computer is a logical, mathematical system, upon which higher-level statistical, probabilistic systems, such as human language and intelligence, could possibly be built." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"The good news is that, as Leibniz suggested, we appear to live in the best of all possible worlds, where the computable functions make life predictable enough to be survivable, while the noncomputable functions make life (and mathematical truth) unpredictable enough to remain interesting, no matter how far computers continue to advance."  (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"The fundamental, indivisible unit of information is the bit. The fundamental, indivisible unit of digital computation is the transformation of a bit between its two possible forms of existence: as structure (memory) or as sequence (code). This is what a Turing Machine does when reading a mark (or the absence of a mark) on a square of tape, changing its state of mind accordingly, and making (or erasing) a mark somewhere else." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"The genius of Monte Carlo - and its search-engine descendants - lies in the ability to extract meaningful solutions, in the face of overwhelming information, by recognizing that meaning resides less in the data at the end points and more in the intervening paths." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"The paradox of artificial intelligence is that any system simple enough to be understandable is not complicated enough to behave intelligently, and any system complicated enough to behave intelligently is not simple enough to understand. The path to artificial intelligence, suggested Turing, is to construct a machine with the curiosity of a child, and let intelligence evolve." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"The question of whether something is feasible in a type belongs to a higher logical type. It is characteristic of objects of low complexity that it is easier to talk about the object than produce it and easier to predict its properties than to build it. But in the complicated parts of formal logic it is always one order of magnitude harder to tell what an object can do than to produce the object." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"There are two kinds of creation myths: those where life arises out of the mud, and those where life falls from the sky. In this creation myth, computers arose from the mud, and code fell from the sky." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"Where does meaning come in? If everything is assigned a number, does this diminish the meaning in the world? What Gödel (and Turing) proved is that formal systems will, sooner or later, produce meaningful statements whose truth can be proved only outside the system itself. This limitation does not confine us to a world with any less meaning. It proves, on the contrary, that we live in a world where higher meaning exists." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"Nature uses digital computing for generation-to-generation information storage, combinatorics, and error correction but relies on analog computing for real-time intelligence and control." (George B Dyson, Analogia: The Emergence of Technology Beyond Programmable Control", 2020)

Clay Helberg - Collected Quotes

"Another key element in making informative graphs is to avoid confounding design variation with data variation. This means that changes in the scale of the graphic should always correspond to changes in the data being represented." (Clay Helberg, "Pitfalls of Data Analysis (or How to Avoid Lies and Damned Lies)", 1995) 

"Another trouble spot with graphs is multidimensional variation. This occurs where two-dimensional figures are used to represent one-dimensional values. What often happens is that the size of the graphic is scaled both horizontally and vertically according to the value being graphed. However, this results in the area of the graphic varying with the square of the underlying data, causing the eye to read an exaggerated effect in the graph." (Clay Helberg, "Pitfalls of Data Analysis (or How to Avoid Lies and Damned Lies)", 1995) 

"It may be helpful to consider some aspects of statistical thought which might lead many people to be distrustful of it. First of all, statistics requires the ability to consider things from a probabilistic perspective, employing quantitative technical concepts such as 'confidence', 'reliability', 'significance'. This is in contrast to the way non-mathematicians often cast problems: logical, concrete, often dichotomous conceptualizations are the norm: right or wrong, large or small, this or that." (Clay Helberg, "Pitfalls of Data Analysis (or How to Avoid Lies and Damned Lies)", 1995) 

"[...] many non-mathematicians hold quantitative data in a sort of awe. They have been lead to believe that numbers are, or at least should be, unquestionably correct." (Clay Helberg, "Pitfalls of Data Analysis (or How to Avoid Lies and Damned Lies)", 1995) 

"Most statistical models assume error free measurement, at least of independent (predictor) variables. However, as we all know, measurements are seldom if ever perfect. Particularly when dealing with noisy data such as questionnaire responses or processes which are difficult to measure precisely, we need to pay close attention to the effects of measurement errors. Two characteristics of measurement which are particularly important in psychological measurement are reliability and validity." (Clay Helberg, "Pitfalls of Data Analysis (or How to Avoid Lies and Damned Lies)", 1995) 

"Remember that a p-value merely indicates the probability of a particular set of data being generated by the null model - it has little to say about the size of a deviation from that model (especially in the tails of the distribution, where large changes in effect size cause only small changes in p-values)." (Clay Helberg, "Pitfalls of Data Analysis (or How to Avoid Lies and Damned Lies)", 1995)

"There are a number of ways that statistical techniques can be misapplied to problems in the real world. Three of the most common hazards are designing experiments with insufficient power, ignoring measurement error, and performing multiple comparisons." (Clay Helberg, "Pitfalls of Data Analysis (or How to Avoid Lies and Damned Lies)", 1995)

"We can consider three broad classes of statistical pitfalls. The first involves sources of bias. These are conditions or circumstances which affect the external validity of statistical results. The second category is errors in methodology, which can lead to inaccurate or invalid results. The third class of problems concerns interpretation of results, or how statistical results are applied (or misapplied) to real world issues." (Clay Helberg, "Pitfalls of Data Analysis (or How to Avoid Lies and Damned Lies)", 1995) 

References:
[1] Clay Helberg, "Pitfalls of Data Analysis (or How to Avoid Lies and Damned Lies)", 1995 [link]