Stanford iPhone Orchestra

I just want a cello.

The original article, with not enough pictures, is at:
Alice’s adventures in algebra: Wonderland solved – physics-math –
16 December 2009 – New Scientist by Melanie Bayley

The 19th century was a turbulent time for mathematics, with many new and controversial concepts, like imaginary numbers, becoming widely accepted in the mathematical community. Putting Alice’s Adventures in Wonderland in this context, it becomes clear that Dodgson, a stubbornly conservative mathematician, used some of the missing scenes to satirise these radical new ideas.

Even Dodgson’s keenest admirers would admit he was a cautious mathematician who produced little original work. He was, however, a conscientious tutor, and, above everything, he valued the ancient Greek textbook Euclid’s Elements as the epitome of mathematical thinking. Broadly speaking, it covered the geometry of circles, quadrilaterals, parallel lines and some basic trigonometry. But what’s really striking about Elements is its rigorous reasoning: it starts with a few incontrovertible truths, or axioms, and builds up complex arguments through simple, logical steps. Each proposition is stated, proved and finally signed off with QED.

For centuries, this approach had been seen as the pinnacle of mathematical and logical reasoning. Yet to Dodgson’s dismay, contemporary mathematicians weren’t always as rigorous as Euclid. He dismissed their writing as “semi-colloquial” and even “semi-logical”. Worse still for Dodgson, this new mathematics departed from the physical reality that had grounded Euclid’s works.

By now, scholars had started routinely using seemingly nonsensical concepts such as imaginary numbers – the square root of a negative number – which don’t represent physical quantities in the same way that whole numbers or fractions do. No Victorian embraced these new concepts wholeheartedly, and all struggled to find a philosophical framework that would accommodate them. But they gave mathematicians a freedom to explore new ideas, and some were prepared to go along with these strange concepts as long as they were manipulated using a consistent framework of operations. To Dodgson, though, the new mathematics was absurd, and while he accepted it might be interesting to an advanced mathematician, he believed it would be impossible to teach to an undergraduate.

Outgunned in the specialist press, Dodgson took his mathematics to his fiction. Using a technique familiar from Euclid’s proofs, reductio ad absurdum, he picked apart the “semi-logic” of the new abstract mathematics, mocking its weakness by taking these premises to their logical conclusions, with mad results. The outcome is Alice’s Adventures in Wonderland. Algebra and hookahs

Take the chapter “Advice from a caterpillar”, for example. By this point, Alice has fallen down a rabbit hole and eaten a cake that has shrunk her to a height of just 3 inches. Enter the Caterpillar, smoking a hookah pipe, who shows Alice a mushroom that can restore her to her proper size. The snag, of course, is that one side of the mushroom stretches her neck, while another shrinks her torso. She must eat exactly the right balance to regain her proper size and proportions.

While some have argued that this scene, with its hookah and “magic mushroom”, is about drugs, I believe it’s actually about what Dodgson saw as the absurdity of symbolic algebra, which severed the link between algebra, arithmetic and his beloved geometry. Whereas the book’s later chapters contain more specific mathematical analogies, this scene is subtle and playful, setting the tone for the madness that will follow.

The first clue may be in the pipe itself: the word “hookah” is, after all, of Arabic origin, like “algebra”, and it is perhaps striking that Augustus De Morgan, the first British mathematician to lay out a consistent set of rules for symbolic algebra, uses the original Arabic translation in Trigonometry and Double Algebra, which was published in 1849. He calls it “al jebr e al mokabala” or “restoration and reduction” – which almost exactly describes Alice’s experience. Restoration was what brought Alice to the mushroom: she was looking for something to eat or drink to “grow to my right size again”, and reduction was what actually happened when she ate some: she shrank so rapidly that her chin hit her foot.

De Morgan’s work explained the departure from universal arithmetic – where algebraic symbols stand for specific numbers rooted in a physical quantity – to that of symbolic algebra, where any “absurd” operations involving negative and impossible solutions are allowed, provided they follow an internal logic. Symbolic algebra is essentially what we use today as a finely honed language for communicating the relations between mathematical objects, but Victorians viewed algebra very differently. Even the early attempts at symbolic algebra retained an indirect relation to physical quantities.

De Morgan wanted to lose even this loose association with measurement, and proposed instead that symbolic algebra should be considered as a system of grammar. “Reduce” algebra from a universal arithmetic to a series of logical but purely symbolic operations, he said, and you will eventually be able to “restore” a more profound meaning to the system – though at this point he was unable to say exactly how. When Alice loses her temper

The madness of Wonderland, I believe, reflects Dodgson’s views on the dangers of this new symbolic algebra. Alice has moved from a rational world to a land where even numbers behave erratically. In the hallway, she tried to remember her multiplication tables, but they had slipped out of the base-10 number system we are used to. In the caterpillar scene, Dodgson’s qualms are reflected in the way Alice’s height fluctuates between 9 feet and 3 inches. Alice, bound by conventional arithmetic where a quantity such as size should be constant, finds this troubling: “Being so many different sizes in a day is very confusing,” she complains. “It isn’t,” replies the Caterpillar, who lives in this absurd world. Wonderland’s madness reflects Carroll’s views on the dangers of the new symbolic algebra.

The Caterpillar’s warning, at the end of this scene, is perhaps one of the most telling clues to Dodgson’s conservative mathematics. “Keep your temper,” he announces. Alice presumes he’s telling her not to get angry, but although he has been abrupt he has not been particularly irritable at this point, so it’s a somewhat puzzling thing to announce. To intellectuals at the time, though, the word “temper” also retained its original sense of “the proportion in which qualities are mingled”, a meaning that lives on today in phrases such as “justice tempered with mercy”. So the Caterpillar could well be telling Alice to keep her body in proportion – no matter what her size.

This may again reflect Dodgson’s love of Euclidean geometry, where absolute magnitude doesn’t matter: what’s important is the ratio of one length to another when considering the properties of a triangle, for example. To survive in Wonderland, Alice must act like a Euclidean geometer, keeping her ratios constant, even if her size changes.

Of course, she doesn’t. She swallows a piece of mushroom and her neck grows like a serpent with predictably chaotic results – until she balances her shape with a piece from the other side of the mushroom. It’s an important precursor to the next chapter, “Pig and pepper”, where Dodgson parodies another type of geometry.

By this point, Alice has returned to her proper size and shape, but she shrinks herself down to enter a small house. There she finds the Duchess in her kitchen nursing her baby, while her Cook adds too much pepper to the soup, making everyone sneeze except the Cheshire Cat. But when the Duchess gives the baby to Alice, it somehow turns into a pig.

The target of this scene is projective geometry, which examines the properties of figures that stay the same even when the figure is projected onto another surface – imagine shining an image onto a moving screen and then tilting the screen through different angles to give a family of shapes. The field involved various notions that Dodgson would have found ridiculous, not least of which is the “principle of continuity”.

Jean-Victor Poncelet, the French mathematician who set out the principle, describes it as follows: “Let a figure be conceived to undergo a certain continuous variation, and let some general property concerning it be granted as true, so long as the variation is confined within certain limits; then the same property will belong to all the successive states of the figure.”

The case of two intersecting circles is perhaps the simplest example to consider. Solve their equations, and you will find that they intersect at two distinct points. According to the principle of continuity, any continuous transformation to these circles – moving their centres away from one another, for example – will preserve the basic property that they intersect at two points. It’s just that when their centres are far enough apart the solution will involve an imaginary number that can’t be understood physically (see diagram).

Of course, when Poncelet talks of “figures”, he means geometric figures, but Dodgson playfully subjects Poncelet’s “semi-colloquial” argument to strict logical analysis and takes it to its most extreme conclusion. What works for a triangle should also work for a baby; if not, something is wrong with the principle, QED. So Dodgson turns a baby into a pig through the principle of continuity. Importantly, the baby retains most of its original features, as any object going through a continuous transformation must. His limbs are still held out like a starfish, and he has a queer shape, turned-up nose and small eyes. Alice only realises he has changed when his sneezes turn to grunts.

The baby’s discomfort with the whole process, and the Duchess’s unconcealed violence, signpost Dodgson’s virulent mistrust of “modern” projective geometry. Everyone in the pig and pepper scene is bad at doing their job. The Duchess is a bad aristocrat and an appallingly bad mother; the Cook is a bad cook who lets the kitchen fill with smoke, over-seasons the soup and eventually throws out her fire irons, pots and plates.

Alice, angry now at the strange turn of events, leaves the Duchess’s house and wanders into the Mad Hatter’s tea party, which explores the work of the Irish mathematician William Rowan Hamilton. Hamilton died in 1865, just after Alice was published, but by this time his discovery of quaternions in 1843 was being hailed as an important milestone in abstract algebra, since they allowed rotations to be calculated algebraically.

Just as complex numbers work with two terms, quaternions belong to a number system based on four terms (see “Imaginary mathematics”). Hamilton spent years working with three terms – one for each dimension of space – but could only make them rotate in a plane. When he added the fourth, he got the three-dimensional rotation he was looking for, but he had trouble conceptualising what this extra term meant. Like most Victorians, he assumed this term had to mean something, so in the preface to his Lectures on Quaternions of 1853 he added a footnote: “It seemed (and still seems) to me natural to connect this extra-spatial unit with the conception of time.”

Where geometry allowed the exploration of space, Hamilton believed, algebra allowed the investigation of “pure time”, a rather esoteric concept he had derived from Immanuel Kant that was meant to be a kind of Platonic ideal of time, distinct from the real time we humans experience. Other mathematicians were polite but cautious about this notion, believing pure time was a step too far.

The parallels between Hamilton’s maths and the Hatter’s tea party – or perhaps it should read “t-party” – are uncanny. Alice is now at a table with three strange characters: the Hatter, the March Hare and the Dormouse. The character Time, who has fallen out with the Hatter, is absent, and out of pique he won’t let the Hatter move the clocks past six.

Reading this scene with Hamilton’s maths in mind, the members of the Hatter’s tea party represent three terms of a quaternion, in which the all-important fourth term, time, is missing. Without Time, we are told, the characters are stuck at the tea table, constantly moving round to find clean cups and saucers.

Their movement around the table is reminiscent of Hamilton’s early attempts to calculate motion, which was limited to rotatations in a plane before he added time to the mix. Even when Alice joins the party, she can’t stop the Hatter, the Hare and the Dormouse shuffling round the table, because she’s not an extra-spatial unit like Time.

The Hatter’s nonsensical riddle in this scene – “Why is a raven like a writing desk?” – may more specifically target the theory of pure time. In the realm of pure time, Hamilton claimed, cause and effect are no longer linked, and the madness of the Hatter’s unanswerable question may reflect this.

Alice’s ensuing attempt to solve the riddle pokes fun at another aspect of quaternions: their multiplication is non-commutative, meaning that x × y is not the same as y × x. Alice’s answers are equally non-commutative. When the Hare tells her to “say what she means”, she replies that she does, “at least I mean what I say – that’s the same thing”. “Not the same thing a bit!” says the Hatter. “Why, you might just as well say that ‘I see what I eat’ is the same thing as ‘I eat what I see’!”

It’s an idea that must have grated on a conservative mathematician like Dodgson, since non-commutative algebras contradicted the basic laws of arithmetic and opened up a strange new world of mathematics, even more abstract than that of the symbolic algebraists.

When the scene ends, the Hatter and the Hare are trying to put the Dormouse into the teapot. This could be their route to freedom. If they could only lose him, they could exist independently, as a complex number with two terms. Still mad, according to Dodgson, but free from an endless rotation around the table.

And there Dodgson’s satire of his contemporary mathematicians seems to end. What, then, would remain of Alice’s Adventures in Wonderland without these analogies? Nothing but Dodgson’s original nursery tale, Alice’s Adventures Under Ground, charming but short on characteristic nonsense. Dodgson was most witty when he was poking fun at something, and only then when the subject matter got him truly riled. He wrote two uproariously funny pamphlets, fashioned in the style of mathematical proofs, which ridiculed changes at the University of Oxford. In comparison, other stories he wrote besides the Alice books were dull and moralistic.

I would venture that without Dodgson’s fierce satire aimed at his colleagues, Alice’s Adventures in Wonderland would never have become famous, and Lewis Carroll would not be remembered as the unrivalled master of nonsense fiction. Imaginary mathematics

The real numbers, which include fractions and irrational numbers like π that can nevertheless be represented as a point on a number line, are only one of many number systems.

Complex numbers, for example, consist of two terms – a real component and an “imaginary” component formed of some multiple of the square root of -1, now represented by the symbol i. They are written in the form a + bi.

The Victorian mathematician William Rowan Hamilton took this one step further, adding two more terms to make quaternions, which take the form a + bi + cj + dk and have their own strange rules of arithmetic.

Melanie Bayley is a DPhil candidate at the University of Oxford. Her work was supported by the UK’s Arts and Humanities Research Council

Fascinating story of karma-kickback – Read the entire piece at the site of a lawyer involved with the process:
Michael Geist – Canadian Recording Industry Faces $6 Billion Copyright Infringement Lawsuit
Monday December 07, 2009

The CRIA members were hit with the lawsuit [PDF] in October 2008, after artists decided to turn to the courts following decades of frustration with the rampant infringement (I am adviser to the Canadian Internet Policy and Public Interest Clinic, which is co-counsel, but have had no involvement in the case). The claims arise from a longstanding practice of the recording industry in Canada, described in the lawsuit as “exploit now, pay later if at all.”  It involves the use of works that are often included in compilation CDs (ie. the top dance tracks of 2009) or live recordings. The record labels create, press, distribute, and sell the CDs, but do not obtain the necessary copyright licences.

Instead, the names of the songs on the CDs are placed on a “pending list”, …

Over the years, the size of the pending list has grown dramatically, now containing over 300,000 songs. From Beyonce to Bruce Springsteen…

It is difficult to understand why the industry has been so reluctant to pay its bills.  …

The more likely reason is that the record labels have had little motivation to pay up.  As the balance has grown to over $50 million…

Having engaged in widespread copyright infringement for over 20 years, the CRIA members now face the prospect of far greater liability…

After years of claiming Canadian consumers disrespect copyright, the irony of having the recording industry face a massive lawsuit will not be lost on anyone…

[Editor’s Note: There really isn’t much more to this article of interest, unless you enjoy reading a pantheon to HP in disguise of a news story. I just thought that these first percentages were thought provoking. The article goes on for pages, and you can read it at:
IT PRO | HP: cleaning up IT’s dirty little secret By Matt Chapman, 13 Oct 2009 at 18:30  [End Editor’s note.]

Such an alarming figure isn’t going unchecked by an industry that now finds itself fighting to conserve materials, improve efficiency and recycle more of its products.

“People totally underestimate the amount this industry invests in research and development pushing the boundaries forward. You’ve only really got to look back a few years and see just how fast we’re printing now and how much the quality has ramped up in such a short period of time. The amount of investment that’s gone on there is staggering,” says Peter Mayhew, director at Lyra Research.

“It’s inevitable you now see that coming out through environmental initiatives.”

…

[Editors note: I can’t think of any reason that this is connected to Digital Cinema…it is just interesting. For months, the previous billionaires who had sold Skype to eBay were saying that they didn’t sell the code that makes Skype run. There was some consternation that this would cause the rediculously priced Skype to be worth even less, as eBay had been unable to create a software version in its stead.

So, the previous owners admit that the code is eBay’s, allowing eBay to sell a good portion of the company, and the original owers get part of the company back, but they had to pay cash for it. Sounds as convoluted as the p2p software that Skype is based upon.

The whole article is at: eBay and Skype founders settle lawsuit | IT PRO
By Nicole Kobie, 6 Nov 2009 at 17:26

End Editor’s note]

Silver Lake and Joltid have apparently agreed that Skype owns all the software which it previously licensed, giving it control over the software. On the other hand, Zennström and Friis will join Silver Lake, bringing a “significant capital investment” and being handed a 14 per cent stake in Skype.

That will leave Silver Lake and other investors with 56 per cent of the company, while eBay will keep 30 per cent.

It also means the previously agreed deal between Silver Lake and eBay will close at the end of this year, with the investors buying their stake from the online auction firm for $1.9 million, suggesting Skype is worth $2.75 billion.

“Skype will be well positioned to move forward under new owners with ownership and control over its core technology,” said eBay’s president and chief executive John Donahoe.

“At the same time, eBay continues to retain a significant stake in Skype and will benefit from its continued growth,” he added in a statement. “We look forward to closing the deal and focusing on growing our core ecommerce and payments businesses,”

Silver Lake managing director Egon Durban said his group was “very pleased” the legal battle was over. “We remain confident in a great future for Skype, and we look forward to working with Niklas, Janus and the other investors as partners to help the company achieve its full potential.”

“Hi; Yes, I’m a Consultant. Great, sure; I would like to work with you on this project. How much do I charge? It all depends. I use a Shrodinger Cat modification on Sartre’s Principle of No Regret.”

And so it goes. Except, now there is a tool. Science to the rescue. And just in time, it appears. More and more people are being given their Permission to be a Consultant papers.

Go to this site: FreelanceSwitch Hourly Rate Calculator

Please don’t waste your time reading any more of this article. All the intelligent stuff is at the above link.

It was 7:51 a.m. on Friday, January 12, the middle of the morning rush hour. In the next 43 minutes, as the violinist performed six classical pieces, 1,097 people passed by. Almost all of them were on the way to work, which meant, for almost all of them, a government job. L’Enfant Plaza is at the nucleus of federal Washington, and these were mostly mid-level bureaucrats with those indeterminate, oddly fungible titles: policy analyst, project manager, budget officer, specialist, facilitator, consultant.

Read the entire piece at:

Pearls Before Breakfast – washingtonpost.com

By Gene WeingartenWashington Post Staff Writer
Sunday, April 8, 2007; Page W10

Each passerby had a quick choice to make, one familiar to commuters …

On that Friday in January, those private questions would be answered in an unusually public way. …

The musician did not play popular tunes whose familiarity alone might have drawn interest. …

The acoustics proved surprisingly kind. Though the arcade is of utilitarian design, a buffer between the Metro escalator …

So, what do you think happened?

HANG ON, WE’LL GET YOU SOME EXPERT HELP.

Leonard Slatkin, music director of the National Symphony Orchestra, was asked the same question. What did he think would occur, hypothetically, if one of the world’s great violinists had performed incognito before a traveling rush-hour audience of 1,000-odd people?

“Let’s assume,” Slatkin said, “that he is not recognized and just taken for granted as a street musician . . . Still, I don’t think that if he’s really good, he’s going to go unnoticed. He’d get a larger audience in Europe . . . but, okay, out of 1,000 people, my guess is there might be 35 or 40 who will recognize the quality for what it is. Maybe 75 to 100 will stop and spend some time listening.”

So, a crowd would gather?

“Oh, yes.”

And how much will he make?

“About $150.”

Thanks, Maestro. As it happens, this is not hypothetical. It really happened.

“How’d I do?”

We’ll tell you in a minute.

“Well, who was the musician?”

Joshua Bell.

“NO!!!”

A onetime child prodigy, at 39 Joshua Bell has arrived as an internationally acclaimed virtuoso. Three days before he appeared at the Metro station, …

Bell was first pitched this idea shortly before Christmas, over coffee at a sandwich shop on Capitol Hill. A New Yorker, he was in town to perform at the Library of Congress…

“Here’s what I’m thinking,” Bell confided, as he sipped his coffee. “I’m thinking that I could do a tour where I’d play Kreisler’s music . . .”

He smiled.

“. . . on Kreisler’s violin.”

It was a snazzy, sequined idea — part inspiration and part gimmick — and it was typical of Bell, who has unapologetically embraced showmanship …

When Bell was asked if he’d be willing to don street clothes and perform at rush hour, he said:

“Uh, a stunt?”

Well, yes. A stunt. Would he think it . . . unseemly?

Bell drained his cup.

“Sounds like fun,” he said.

The article is long, but worth every minute…and be sure to block some time to read the accompanying piece referenced above.

Read the entire story at:

Tokyo University Researchers Using Ultrasound Technology to Enable Touchable Holograms | InteractiveTV Today

According to an article in MIT’s Technology Review publication, the touchable hologram’s visual component is generated by projecting an image from an LCD projector onto a concave mirror. A white paper abstract from the Tokyo University team behind the project explains its tactile (“haptic”) component as follows: “The Airborne Ultrasound Tactile Display is designed to provide tactile feedback in 3D free space. The display radiates airborne ultrasound, and produces high-fidelity pressure fields onto the user’s hands, without the use of gloves or mechanical attachments. The method is based on a nonlinear phenomenon of ultrasound: acoustic radiation pressure. When an object interrupts the propagation of ultrasound, a pressure field is exerted on the surface of the object. This pressure is called acoustic radiation pressure…The acoustic radiation pressure is proportional to the energy density of the ultrasound.

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The spatial distribution of the energy density of the ultrasound can be controlled by using the wave field synthesis techniques. With an ultrasound transducer array, various patterns of pressure field are produced in 3D free space. Unlike air-jets, the spatial and temporal resolutions are quite fine. The spatial resolution is comparable to the wavelength of the ultrasound. The frequency characteristics are sufficiently fine up to 1 kHz. The airborne ultrasound can be applied directly onto the skin without the risk of penetration. When the airborne ultrasound is applied on the surface of the skin, due to the large difference between the characteristic acoustic impedance of the air and that of the skin, about 99.9% of the incident acoustic energy is reflected on the surface of the skin. Hence, this tactile feedback system does not require the users to wear any clumsy gloves or mechanical attachments.” The Airborne Ultrasound Tactile Display is guided by a “vision-based hand tracking system,” the team explains, adding that “the tactile display exerts the radiation pressure on the user’s hands when they ‘touch’ 3D virtual objects.” A demo video of the new technology and the touchable holograms it enables is embedded above. More information on the project is available at: http://www.alab.t.u-tokyo.ac.jp/~siggraph/08/Tactile/SIGGRAPH08_abst.pdf

US court tells Microsoft to stop selling Word The patent infringement suit involves Microsoft’s use of XML.

By Stuart Turton, 12 Aug 2009 at 12:22

Microsoft has been ordered to stop selling Microsoft Word in the US, after finding itself on the wrong end of patent infringement suit.

The US District Court for the Eastern District of Texas sided with technology company i4i, which alleged that Microsoft had willfully infringed a patent relating to the creation of custom XML documents.

The software giant has been ordered to stop selling Microsoft Word – the cornerstone of its Office suite – in its current form within 60 days.

For a list of the fines that MS is ordered to pay, see the entire article at: US court tells Microsoft to stop selling Word | IT PRO

Fructose, unlike another sugar, glucose, is processed almost solely by the liver, and produces an excessive amount of triglycerides — fat which get into the bloodstream. Triglycerides can interfere with insulin signaling in the brain, which plays a major role in brain cell survival and plasticity, or the ability for the brain to change based on new experiences.

Results were similar in adolescent rats, …

Parent’s lab works … to examine how diet influences brain function.

…has been increasing steadily. High intake of fructose is associated with numerous health problems, including insulin insensitivity, type II diabetes, obesity and cardiovascular disease.

“The bottom line is that we were meant to have an apple a day as our source of fructose,” …

Exercise is a next step in ongoing research, …

Read the entire story at: Eating High Levels Of Fructose Impairs Memory In Rats

The physical properties of the matter we are creating are relevant to the conditions inside large planets, and we also hope that by studying it we can gain a greater understanding of what is going on during the creation of ‘miniature stars’ created by high-power laser implosions, which may one day allow the power of nuclear fusion to be harnessed here on Earth.’ [Other than that, it is a great day in toyland.]

Taken from an article in ScienceDaily: Transparent Aluminum Is ‘New State Of Matter’

The discovery was made possible with the development of a new source of radiation that is ten billion times brighter than any synchrotron in the world … [One wonders how bright that is in the SI standard of football fields.]

The Oxford team, along with their international colleagues, focused all this power down into a spot with a diameter less than a twentieth of the width of a human hair. [Ah~! Asked and answered – a football field to a human hair is recognizable at 10^-18]

Whilst the invisible effect lasted for only an extremely brief period – an estimated 40 femtoseconds – …

Professor Wark added: ‘What is particularly remarkable about our experiment is that we have turned ordinary aluminium into this exotic new material…

The researchers believe that the new approach is an ideal way to create and study such exotic states of matter…fusion. [Gotta include fusion.]

A report of the research, ‘Turning solid aluminium transparent by intense soft X-ray photoionization’, is published in Nature Physics. The research was carried out by an international team led by Oxford University scientists Professor Justin Wark, Dr Bob Nagler, Dr Gianluca Gregori, William Murphy, Sam Vinko and Thomas Whitcher. Adapted from materials provided by University of Oxford.

Read the entire article at: Transparent Aluminum Is ‘New State Of Matter’

See also: Matter & Energy

• Quantum Physics
• Physics
• Quantum Computing

Computers & Math

• Quantum Computers
• Encryption
• Hacking

Reference

• Mechanics
• Quantum number
• Nanowire
• Subatomic particle