Wednesday, February 4, 2009

A new historic physics record has been set by scientists for exceedingly small writing, opening a new door to computing‘s future. Stanford University physicists have claimed to have written the letters “SU” at sub-atomic size.

Graduate students Christopher Moon, Laila Mattos, Brian Foster and Gabriel Zeltzer, under the direction of assistant professor of physics Hari Manoharan, have produced the world’s smallest lettering, which is approximately 1.5 nanometres tall, using a molecular projector, called Scanning Tunneling Microscope (STM) to push individual carbon monoxide molecules on a copper or silver sheet surface, based on interference of electron energy states.

A nanometre (Greek: ?????, nanos, dwarf; ?????, metr?, count) is a unit of length in the metric system, equal to one billionth of a metre (i.e., 10^-9 m or one millionth of a millimetre), and also equals ten Ångström, an internationally recognized non-SI unit of length. It is often associated with the field of nanotechnology.

“We miniaturised their size so drastically that we ended up with the smallest writing in history,” said Manoharan. “S” and “U,” the two letters in honor of their employer have been reduced so tiny in nanoimprint that if used to print out 32 volumes of an Encyclopedia, 2,000 times, the contents would easily fit on a pinhead.

In the world of downsizing, nanoscribes Manoharan and Moon have proven that information, if reduced in size smaller than an atom, can be stored in more compact form than previously thought. In computing jargon, small sizing results to greater speed and better computer data storage.

“Writing really small has a long history. We wondered: What are the limits? How far can you go? Because materials are made of atoms, it was always believed that if you continue scaling down, you’d end up at that fundamental limit. You’d hit a wall,” said Manoharan.

In writing the letters, the Stanford team utilized an electron‘s unique feature of “pinball table for electrons” — its ability to bounce between different quantum states. In the vibration-proof basement lab of Stanford’s Varian Physics Building, the physicists used a Scanning tunneling microscope in encoding the “S” and “U” within the patterns formed by the electron’s activity, called wave function, arranging carbon monoxide molecules in a very specific pattern on a copper or silver sheet surface.

“Imagine [the copper as] a very shallow pool of water into which we put some rocks [the carbon monoxide molecules]. The water waves scatter and interfere off the rocks, making well defined standing wave patterns,” Manoharan noted. If the “rocks” are placed just right, then the shapes of the waves will form any letters in the alphabet, the researchers said. They used the quantum properties of electrons, rather than photons, as their source of illumination.

According to the study, the atoms were ordered in a circular fashion, with a hole in the middle. A flow of electrons was thereafter fired at the copper support, which resulted into a ripple effect in between the existing atoms. These were pushed aside, and a holographic projection of the letters “SU” became visible in the space between them. “What we did is show that the atom is not the limit — that you can go below that,” Manoharan said.

“It’s difficult to properly express the size of their stacked S and U, but the equivalent would be 0.3 nanometres. This is sufficiently small that you could copy out the Encyclopaedia Britannica on the head of a pin not just once, but thousands of times over,” Manoharan and his nanohologram collaborator Christopher Moon explained.

The team has also shown the salient features of the holographic principle, a property of quantum gravity theories which resolves the black hole information paradox within string theory. They stacked “S” and the “U” – two layers, or pages, of information — within the hologram.

The team stressed their discovery was concentrating electrons in space, in essence, a wire, hoping such a structure could be used to wire together a super-fast quantum computer in the future. In essence, “these electron patterns can act as holograms, that pack information into subatomic spaces, which could one day lead to unlimited information storage,” the study states.

The “Conclusion” of the Stanford article goes as follows:

The team is not the first to design or print small letters, as attempts have been made since as early as 1960. In December 1959, Nobel Prize-winning physicist Richard Feynman, who delivered his now-legendary lecture entitled “There’s Plenty of Room at the Bottom,” promised new opportunities for those who “thought small.”

Feynman was an American physicist known for the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the superfluidity of supercooled liquid helium, as well as work in particle physics (he proposed the parton model).

Feynman offered two challenges at the annual meeting of the American Physical Society, held that year in Caltech, offering a $1000 prize to the first person to solve each of them. Both challenges involved nanotechnology, and the first prize was won by William McLellan, who solved the first. The first problem required someone to build a working electric motor that would fit inside a cube ¹/₆₄ inches on each side. McLellan achieved this feat by November 1960 with his 250-microgram 2000-rpm motor consisting of 13 separate parts.

In 1985, the prize for the second challenge was claimed by Stanford Tom Newman, who, working with electrical engineering professor Fabian Pease, used electron lithography. He wrote or engraved the first page of Charles Dickens’ A Tale of Two Cities, at the required scale, on the head of a pin, with a beam of electrons. The main problem he had before he could claim the prize was finding the text after he had written it; the head of the pin was a huge empty space compared with the text inscribed on it. Such small print could only be read with an electron microscope.

In 1989, however, Stanford lost its record, when Donald Eigler and Erhard Schweizer, scientists at IBM’s Almaden Research Center in San Jose were the first to position or manipulate 35 individual atoms of xenon one at a time to form the letters I, B and M using a STM. The atoms were pushed on the surface of the nickel to create letters 5nm tall.

In 1991, Japanese researchers managed to chisel 1.5 nm-tall characters onto a molybdenum disulphide crystal, using the same STM method. Hitachi, at that time, set the record for the smallest microscopic calligraphy ever designed. The Stanford effort failed to surpass the feat, but it, however, introduced a novel technique. Having equaled Hitachi’s record, the Stanford team went a step further. They used a holographic variation on the IBM technique, for instead of fixing the letters onto a support, the new method created them holographically.

In the scientific breakthrough, the Stanford team has now claimed they have written the smallest letters ever – assembled from subatomic-sized bits as small as 0.3 nanometers, or roughly one third of a billionth of a meter. The new super-mini letters created are 40 times smaller than the original effort and more than four times smaller than the IBM initials, states the paper Quantum holographic encoding in a two-dimensional electron gas, published online in the journal Nature Nanotechnology. The new sub-atomic size letters are around a third of the size of the atomic ones created by Eigler and Schweizer at IBM.

A subatomic particle is an elementary or composite particle smaller than an atom. Particle physics and nuclear physics are concerned with the study of these particles, their interactions, and non-atomic matter. Subatomic particles include the atomic constituents electrons, protons, and neutrons. Protons and neutrons are composite particles, consisting of quarks.

“Everyone can look around and see the growing amount of information we deal with on a daily basis. All that knowledge is out there. For society to move forward, we need a better way to process it, and store it more densely,” Manoharan said. “Although these projections are stable — they’ll last as long as none of the carbon dioxide molecules move — this technique is unlikely to revolutionize storage, as it’s currently a bit too challenging to determine and create the appropriate pattern of molecules to create a desired hologram,” the authors cautioned. Nevertheless, they suggest that “the practical limits of both the technique and the data density it enables merit further research.”

In 2000, it was Hari Manoharan, Christopher Lutz and Donald Eigler who first experimentally observed quantum mirage at the IBM Almaden Research Center in San Jose, California. In physics, a quantum mirage is a peculiar result in quantum chaos. Their study in a paper published in Nature, states they demonstrated that the Kondo resonance signature of a magnetic adatom located at one focus of an elliptically shaped quantum corral could be projected to, and made large at the other focus of the corral.

Tuesday, February 1, 2011

The president of Egypt has suffered a “devastating blow” after the country’s army announced they would not use force against their own people, who continue to protest against the government tonight. The news came hours after six journalists who reported on the protests were released from custody.

Hosni Mubarak yesterday announced a new cabinet, which does not include several figures who protesters largely do not approve of. Analysts have, however, suggested little had changed within the government; many positions, they say, are filled with military figures.

In a statement broadcast on state media in Egypt, the army said: “To the great people of Egypt, your armed forces, acknowledging the legitimate rights of the people … have not and will not use force against the Egyptian people.” A BBC correspondent in Cairo said the announcement meant it “now seems increasingly likely that the 30-year rule of Mr Mubarak is drawing to a close.”

“The presence of the army in the streets is for your sake and to ensure your safety and wellbeing. The armed forces will not resort to use of force against our great people,” the statement added. “Your armed forces, who are aware of the legitimacy of your demands and are keen to assume their responsibility in protecting the nation and the citizens, affirms that freedom of expression through peaceful means is guaranteed to everybody.”

Earlier today, six journalists from the independent news network Al-Jazeera were released from custody after being detained by police. The U.S. State Department criticized the arrests; equipment was reportedly confiscated from the journalists.

Egyptian officials yesterday ordered the satellite channel to stop broadcasting in the country. Al-Jazeera said they were “appalled” by the government’s decision to close its Egyptian offices, which they described as the “latest attack by the Egyptian regime to strike at its freedom to report independently on the unprecedented events in Egypt.”

In a statement, the news agency added: “Al-Jazeera sees this as an act designed to stifle and repress the freedom of reporting by the network and its journalists. In this time of deep turmoil and unrest in Egyptian society it is imperative that voices from all sides be heard; the closing of our bureau by the Egyptian government is aimed at censoring and silencing the voices of the Egyptian people.”

On Friday, Wikinews reported the government had shut off practically all Internet traffic both out of and into the nation, as well as disrupting cellphone usage. A spokesperson for the social networking website Facebook said “limiting Internet access for millions of people is a matter of concern for the global community.”

A reported 50,000 campaigners, who are demanding the long-time leader step down and complaining of poverty, corruption, and oppression, filled Tahrir Square in Cairo today, chanting “We will stay until the coward leaves.” It is thought 100 people have so far died in the demonstrations. Today there have been protests in Suez, Mansoura, Damanhour, and Alexandria.

Speaking to news media in the area, many protesters said the new cabinet did little to quell their anger. “We want a complete change of government, with a civilian authority,” one said. Another added: “This is not a new government. This is the same regime—this is the same bluff. [Mubarak] has been bluffing us for 30 years.”

In Tahrir Square today, protesters played music as strings of barbed wire and army tanks stood nearby. Demonstrators scaled light poles, hanging Egyptian flags and calling for an end to Mubarak’s rule. “One poster featured Mubarak’s face plastered with a Hitler mustache, a sign of the deep resentment toward the 82-year-old leader they blame for widespread poverty, inflation and official indifference and brutality during his 30 years in power,” one journalist in the square reported this evening.

Wednesday, April 5, 2006

Australian Prime Minister John Howard has announced a five year plan costing AU$1.8 billion to address issues with the country’s mental health system. The plan follows a commitment made at the Council of Australian Governments (COAG) meeting in February.

The federal government will improve access to clinical and health services, increase the number of mental health professionals in Australia, create mental health work teams consisting of GPs, psychiatrists, psychologists and mental health nurses, provide respite services for people suffering mental illness and their carers, and introduce new programs for community awareness.

Mr Howard said the plan addresses issues which fall into its area of responsibility. He hopes that the states and territories will complement the federal government’s package by investing in supported accommodation, hospital and emergency services, crisis care services and the provision of mental health care in gaols.

Under the federal government’s plan, psychologists will play a greater role in the mental health system. From November, Patients will be able to claim a rebate from Medicare (Australia’s universal healthcare scheme) for the services of psychologists if they have been referred by a GP or psychiatrist. At present patients pay around $100 for a standard 30 minute consultation.

Mr Howard claims that there is an issue for mental health professionals in treating patients with a substance abuse problem and mental illness. To address this the government will provide extra funding for drug and alcohol rehabilitation services.

The government will increase the mental health workforce by funding an additional 400 mental health nursing and 200 clinical psychology places. 900 personal helpers and mentors will also be employed.

The government has promised increase funding for telephone counselling and suicide prevention services, living skills programs and additional support for those with a mental illness who are having difficulty finding or keeping work.

The government will also provide 900 personal helpers and mentors, increase funding for living skills programs and provide additional employment assistance to those who have difficulty finding or retaining employment due to their illness.

Friday, April 9, 2021

On Tuesday, North Korean government announced it would not participate in the delayed Tokyo 2020 Summer Olympics “to protect players from the world public health crisis caused by COVID-19”, the Associated Press reported. It is North Korea’s first official withdrawal from Olympics since their boycott of the 1988 Summer Olympics held in Seoul, South Korea, CBS News reported.

According to South Korea’s Ministry of Unification, this is the first time North Korea withdrew from a major international sports event due to concerns of an infectious disease. Several athletes from multiple disciplines did not participate in the 2016 Summer Olympics over concerns of the Zika virus, but North Korea did.

South Korean President Moon Jae-in had hoped the two nations could field a joint team similar to its unified Korean Peninsula team at the 2018 Winter Olympics, Al Jazeera reported.

While North Korea claims to have no cases of the coronavirus, professor of North Korean studies at Ewha Womans University Park Won Gon says it has shown “a coronavirus-related neurosis since it declared an emergency anti-virus system in January last year”, adding it was very unlikely the country would be able to procure enough vaccines for its population by July, when the 2020 Olympics are scheduled to take place.

In November, Daily NK reported North Korea had placed 81 thousand people, in quarantine excluding the soldiers in quarantine; and, by October 22, 32,011 people were placed in isolation. A source told Daily NK over 300 people had died in an isolation centre in Anju, South Pyongan.

Director of Asian Studies at Temple University, Japan Campus Jeff Kingston told CBS this decision was a “setback for diplomacy”. CBS also reported Seoul’s disappointment at the nation’s non-attendance, depriving South Korea a chance to discuss relations with North Korean dictator Kim Jong-un. Recently, North Korea’s vice department director of the Central Committee of the Workers’ Party of Korea and sister of Kim Jong-un Kim Yo Jong warned March 16 “[t]he peaceful spring days of three years ago are unlikely to return”, calling South Korean leaders as “all born with stupidity” and they “have become the dumb and deaf bereft of judgment”. The warning came after the United States and South Korea held joint military drills in March.

This also comes after North Korea tested two ballistic missile on March 24, described by former admiral of the Japanese fleet Yoji Koda as “just the beginning of North Korean fervour.”

North Korea participated in the boycotts of the 1964 Summer Olympics in Tokyo and 1988 Summer Olympics in Los Angeles.

Tuesday, April 5, 2005Terry Vo, the 10-year old Australian boy who had two hands and a foot reattached by surgeons after losing them in an accident, has had to have the foot re-amputated. He will be given a prosthetic foot in its place.

The operation to re-attach three limbs was thought to have been a first – but was ultimately unsuccessful, with the foot having died inside, and receiving insufficient blood supply following the surgery to reattach it.

“That would lead to the small muscles in the foot actually constricting, the toes bending over and a deformed …. foot that is sort of clawed over and doesn’t have good sensation,” said plastic surgeon, Mr Robert Love today, on Australia’s ABC Radio.

“Even if you can get all of that to survive, he [would be] worse off than having had an amputation.”

“What is very disappointing is that for the first two days after [the operation] the foot looked absolutely magnificent,” he said.

Terry’s hands were healing well, said the surgeon. The prosthetic foot would allow him to walk normally, since his knee was intact.

Monday, December 11, 2006

On December 7, BBC News reported a story about Dr James Anderson, a teacher in the Computer Science department at the University of Reading in the United Kingdom. In the report it was stated that Anderson had “solved a very important problem” that was 1200 years old, the problem of division by zero. According to the BBC, Anderson had created a new number, that he had named “nullity”, that lay outside of the real number line. Anderson terms this number a “transreal number”, and denotes it with the Greek letter ? {\displaystyle \Phi } . He had taught this number to pupils at Highdown School, in Emmer Green, Reading.

The BBC report provoked many reactions from mathematicians and others.

In reaction to the story, Mark C. Chu-Carroll, a computer scientist and researcher, posted a web log entry describing Anderson as an “idiot math teacher”, and describing the BBC’s story as “absolutely infuriating” and a story that “does an excellent job of demonstrating what total innumerate idiots reporters are”. Chu-Carroll stated that there was, in fact, no actual problem to be solved in the first place. “There is no number that meaningfully expresses the concept of what it means to divide by zero.”, he wrote, stating that all that Anderson had done was “assign a name to the concept of ‘not a number'”, something which was “not new” in that the IEEE floating-point standard, which describes how computers represent floating-point numbers, had included a concept of “not a number”, termed “NaN“, since 1985. Chu-Carroll further continued:

“Basically, he’s defined a non-solution to a non-problem. And by teaching it to his students, he’s doing them a great disservice. They’re going to leave his class believing that he’s a great genius who’s solved a supposed fundamental problem of math, and believing in this silly nullity thing as a valid mathematical concept.

“It’s not like there isn’t already enough stuff in basic math for kids to learn; there’s no excuse for taking advantage of a passive audience to shove this nonsense down their throats as an exercise in self-aggrandizement.

“To make matters worse, this idiot is a computer science professor! No one who’s studied CS should be able to get away with believing that re-inventing the concept of NaN is something noteworthy or profound; and no one who’s studied CS should think that defining meaningless values can somehow magically make invalid computations produce meaningful results. I’m ashamed for my field.”

There have been a wide range of other reactions from other people to the BBC news story. Comments range from the humorous and the ironic, such as the B1FF-style observation that “DIVIDION^[sic] BY ZERO IS IMPOSSIBLE BECAUSE MY CALCULATOR SAYS SO AND IT IS THE TRUTH” and the Chuck Norris Fact that “Only Chuck Norris can divide by zero.” (to which another reader replied “Chuck Norris just looks at zero, and it divides itself.”); through vigourous defences of Dr Anderson, with several people quoting the lyrics to Ira Gershwin‘s song “They All Laughed (At Christopher Columbus)”; to detailed mathematical discussions of Anderson’s proposed axioms of transfinite numbers.

Several readers have commented that they consider this to have damaged the reputation of the Computer Science department, and even the reputation of the University of Reading as a whole. “By publishing his childish nonsense the BBC actively harms the reputation of Reading University.” wrote one reader. “Looking forward to seeing Reading University maths application plummit.” wrote another. “Ignore all research papers from the University of Reading.” wrote a third. “I’m not sure why you refer to Reading as a ‘university’. This is a place the BBC reports as closing down its physics department because it’s too hard. Lecturers at Reading should stick to folk dancing and knitting, leaving academic subjects to grown ups.” wrote a fourth. Steve Kramarsky lamented that Dr Anderson is not from the “University of ‘Rithmetic“.

Several readers criticised the journalists at the BBC who ran the story for not apparently contacting any mathematicians about Dr Anderson’s idea. “Journalists are meant to check facts, not just accept whatever they are told by a self-interested third party and publish it without question.” wrote one reader on the BBC’s web site. However, on Slashdot another reader countered “The report is from Berkshire local news. Berkshire! Do you really expect a local news team to have a maths specialist? Finding a newsworthy story in Berkshire probably isn’t that easy, so local journalists have to cover any piece of fluff that comes up. Your attitude to the journalist should be sympathy, not scorn.”

Ben Goldacre, author of the Bad Science column in The Guardian, wrote on his web log that “what is odd is a reporter, editor, producer, newsroom, team, cameraman, soundman, TV channel, web editor, web copy writer, and so on, all thinking it’s a good idea to cover a brilliant new scientific breakthrough whilst clearly knowing nothing about the context. Maths isn’t that hard, you could even make a call to a mathematician about it.”, continuing that “it’s all very well for the BBC to think they’re being balanced and clever getting Dr Anderson back in to answer queries about his theory on Tuesday, but that rather skips the issue, and shines the spotlight quite unfairly on him (he looks like a very alright bloke to me).”.

From reading comments on his own web log as well as elsewhere, Goldacre concluded that he thought that “a lot of people might feel it’s reporter Ben Moore, and the rest of his doubtless extensive team, the people who drove the story, who we’d want to see answering the questions from the mathematicians.”.

Andrej Bauer, a professional mathematician from Slovenia writing on the Bad Science web log, stated that “whoever reported on this failed to call a university professor to check whether it was really new. Any university professor would have told this reporter that there are many ways of dealing with division by zero, and that Mr. Anderson’s was just one of known ones.”

Ollie Williams, one of the BBC Radio Berkshire reporters who wrote the BBC story, initially stated that “It seems odd to me that his theory would get as far as television if it’s so easily blown out of the water by visitors to our site, so there must be something more to it.” and directly responded to criticisms of BBC journalism on several points on his web log.

He pointed out that people should remember that his target audience was local people in Berkshire with no mathematical knowledge, and that he was “not writing for a global audience of mathematicians”. “Some people have had a go at Dr Anderson for using simplified terminology too,” he continued, “but he knows we’re playing to a mainstream audience, and at the time we filmed him, he was showing his theory to a class of schoolchildren. Those circumstances were never going to breed an in-depth half-hour scientific discussion, and none of our regular readers would want that.”.

On the matter of fact checking, he replied that “if you only want us to report scientific news once it’s appeared, peer-reviewed, in a recognised journal, it’s going to be very dry, and it probably won’t be news.”, adding that “It’s not for the BBC to become a journal of mathematics — that’s the job of journals of mathematics. It’s for the BBC to provide lively science reporting that engages and involves people. And if you look at the original page, you’ll find a list as long as your arm of engaged and involved people.”.

Williams pointed out that “We did not present Dr Anderson’s theory as gospel, although with hindsight it could have been made clearer that this is very much a theory and by no means universally accepted. But we certainly weren’t shouting a mathematical revolution from the rooftops. Dr Anderson has, in one or two places, been chastised for coming to the media with his theory instead of his peers — a sure sign of a quack, boffin and/or crank according to one blogger. Actually, one of our reporters happened to meet him during a demonstration against the closure of the university’s physics department a couple of weeks ago, got chatting, and discovered Dr Anderson reckoned he was onto something. He certainly didn’t break the door down looking for media coverage.”.

Some commentators, at the BBC web page and at Slashdot, have attempted serious mathematical descriptions of what Anderson has done, and subjected it to analysis. One description was that Anderson has taken the field of real numbers and given it complete closure so that all six of the common arithmetic operators were surjective functions, resulting in “an object which is barely a commutative ring (with operators with tons of funky corner cases)” and no actual gain “in terms of new theorems or strong relation statements from the extra axioms he has to tack on”.

Jamie Sawyer, a mathematics undergraduate at the University of Warwick writing in the Warwick Maths Society discussion forum, describes what Anderson has done as deciding that R ? { ? ? , + ? } {\displaystyle \mathbb {R} \cup \lbrace -\infty ,+\infty \rbrace } , the so-called extended real number line, is “not good enough […] because of the wonderful issue of what 0 0 {\displaystyle {\frac {0}{0}}} is equal to” and therefore creating a number system R ? { ? ? , ? , + ? } {\displaystyle \mathbb {R} \cup \lbrace -\infty ,\Phi ,+\infty \rbrace } .

Andrej Bauer stated that Anderson’s axioms of transreal arithmetic “are far from being original. First, you can adjoin + ? {\displaystyle +\infty } and ? ? {\displaystyle -\infty } to obtain something called the extended real line. Then you can adjoin a bottom element to represent an undefined value. This is all standard and quite old. In fact, it is well known in domain theory, which deals with how to represent things we compute with, that adjoining just bottom to the reals is not a good idea. It is better to adjoin many so-called partial elements, which denote approximations to reals. Bottom is then just the trivial approximation which means something like ‘any real’ or ‘undefined real’.”

Commentators have pointed out that in the field of mathematical analysis, 0 0 {\displaystyle {\frac {0}{0}}} (which Anderson has defined axiomatically to be ? {\displaystyle \Phi } ) is the limit of several functions, each of which tends to a different value at its limit:

• lim x ? 0 x 0 {\displaystyle \lim _{x\to 0}{\frac {x}{0}}} has two different limits, depending from whether x {\displaystyle x} approaches zero from a positive or from a negative direction.
• lim x ? 0 0 x {\displaystyle \lim _{x\to 0}{\frac {0}{x}}} also has two different limits. (This is the argument that commentators gave. In fact, 0 x {\displaystyle {\frac {0}{x}}} has the value 0 {\displaystyle 0} for all x ? 0 {\displaystyle x\neq 0} , and thus only one limit. It is simply discontinuous for x = 0 {\displaystyle x=0} . However, that limit is different to the two limits for lim x ? 0 x 0 {\displaystyle \lim _{x\to 0}{\frac {x}{0}}} , supporting the commentators’ main point that the values of the various limits are all different.)
• Whilst sin ? 0 = 0 {\displaystyle \sin 0=0} , the limit lim x ? 0 sin ? x x {\displaystyle \lim _{x\to 0}{\frac {\sin x}{x}}} can be shown to be 1, by expanding the sine function as an infinite Taylor series, dividing the series by x {\displaystyle x} , and then taking the limit of the result, which is 1.
• Whilst 1 ? cos ? 0 = 0 {\displaystyle 1-\cos 0=0} , the limit lim x ? 0 1 ? cos ? x x {\displaystyle \lim _{x\to 0}{\frac {1-\cos x}{x}}} can be shown to be 0, by expanding the cosine function as an infinite Taylor series, dividing the series subtracted from 1 by x {\displaystyle x} , and then taking the limit of the result, which is 0.

Commentators have also noted l’Hôpital’s rule.

It has been pointed out that Anderson’s set of transreal numbers is not, unlike the set of real numbers, a mathematical field. Simon Tatham, author of PuTTY, stated that Anderson’s system “doesn’t even think about the field axioms: addition is no longer invertible, multiplication isn’t invertible on nullity or infinity (or zero, but that’s expected!). So if you’re working in the transreals or transrationals, you can’t do simple algebraic transformations such as cancelling x {\displaystyle x} and ? x {\displaystyle -x} when both occur in the same expression, because that transformation becomes invalid if x {\displaystyle x} is nullity or infinity. So even the simplest exercises of ordinary algebra spew off a constant stream of ‘unless x is nullity’ special cases which you have to deal with separately — in much the same way that the occasional division spews off an ‘unless x is zero’ special case, only much more often.”

Tatham stated that “It’s telling that this monstrosity has been dreamed up by a computer scientist: persistent error indicators and universal absorbing states can often be good computer science, but he’s stepped way outside his field of competence if he thinks that that also makes them good maths.”, continuing that Anderson has “also totally missed the point when he tries to compute things like 0 0 {\displaystyle 0^{0}} using his arithmetic. The reason why things like that are generally considered to be ill-defined is not because of a lack of facile ‘proofs’ showing them to have one value or another; it’s because of a surfeit of such ‘proofs’ all of which disagree! Adding another one does not (as he appears to believe) solve any problem at all.” (In other words: 0 0 {\displaystyle 0^{0}} is what is known in mathematical analysis as an indeterminate form.)

To many observers, it appears that Anderson has done nothing more than re-invent the idea of “NaN“, a special value that computers have been using in floating-point calculations to represent undefined results for over two decades. In the various international standards for computing, including the IEEE floating-point standard and IBM’s standard for decimal arithmetic, a division of any non-zero number by zero results in one of two special infinity values, “+Inf” or “-Inf”, the sign of the infinity determined by the signs of the two operands (Negative zero exists in floating-point representations.); and a division of zero by zero results in NaN.

Anderson himself denies that he has re-invented NaN, and in fact claims that there are problems with NaN that are not shared by nullity. According to Anderson, “mathematical arithmetic is sociologically invalid” and IEEE floating-point arithmetic, with NaN, is also faulty. In one of his papers on a “perspex machine” dealing with “The Axioms of Transreal Arithmetic” (Jamie Sawyer writes that he has “worries about something which appears to be named after a plastic” — “Perspex” being a trade name for polymethyl methacrylate in the U.K..) Anderson writes:

We cannot accept an arithmetic in which a number is not equal to itself (NaN != NaN), or in which there are three kinds of numbers: plain numbers, silent numbers, and signalling numbers; because, on writing such a number down, in daily discourse, we can not always distinguish which kind of number it is and, even if we adopt some notational convention to make the distinction clear, we cannot know how the signalling numbers are to be used in the absence of having the whole program and computer that computed them available. So whilst IEEE floating-point arithmetic is an improvement on real arithmetic, in so far as it is total, not partial, both arithmetics are invalid models of arithmetic.

In fact, the standard convention for distinguishing the two types of NaNs when writing them down can be seen in ISO/IEC 10967, another international standard for how computers deal with numbers, which uses “qNaN” for non-signalling (“quiet”) NaNs and “sNaN” for signalling NaNs. Anderson continues:

[NaN’s] semantics are not defined, except by a long list of special cases in the IEEE standard.

“In other words,” writes Scott Lamb, a BSc. in Computer Science from the University of Idaho, “they are defined, but he doesn’t like the definition.”.

The main difference between nullity and NaN, according to both Anderson and commentators, is that nullity compares equal to nullity, whereas NaN does not compare equal to NaN. Commentators have pointed out that in very short order this difference leads to contradictory results. They stated that it requires only a few lines of proof, for example, to demonstrate that in Anderson’s system of “transreal arithmetic” both 1 = 2 {\displaystyle 1=2} and 1 ? 2 {\displaystyle 1\neq 2} , after which, in one commentator’s words, one can “prove anything that you like”. In aiming to provide a complete system of arithmetic, by adding extra axioms defining the results of the division of zero by zero and of the consequent operations on that result, half as many again as the number of axioms of real-number arithmetic, Anderson has produced a self-contradictory system of arithmetic, in accordance with Gödel’s incompleteness theorems.

One reader-submitted comment appended to the BBC news article read “Step 1. Create solution 2. Create problem 3. PROFIT!”, an allusion to the business plan employed by the underpants gnomes of the comedy television series South Park. In fact, Anderson does plan to profit from nullity, having registered on the 27th of July, 2006 a private limited company named Transreal Computing Ltd, whose mission statement is “to develop hardware and software to bring you fast and safe computation that does not fail on division by zero” and to “promote education and training in transreal computing”. The company is currently “in the research and development phase prior to trading in hardware and software”.

In a presentation given to potential investors in his company at the ANGLE plc showcase on the 28th of November, 2006, held at the University of Reading, Anderson stated his aims for the company as being:

• To build a transreal computer.
• To derive Maxwell’s equations and classical gravitation in transreal numbers so that they have no naked singularities (infinities affecting a neighbourhood of space).
• To collaborate with anyone who wants to unify Quantum Electro-Dynamics with gravitation.

To investors, Anderson makes the following promises:

• “I will help you develop a curriculum for transreal arithmetic if you want me to.”
• “I will help you unify QED and gravitation if you want me to.”
• “I will build a transreal supercomputer.”

He asks potential investors:

• “How much would you pay to know that the engine in your ship, car, aeroplane, or heart pacemaker won’t just stop dead?”
• “How much would you pay to know that your Government’s computer controlled military hardware won’t just stop or misfire?”

The current models of computer arithmetic are, in fact, already designed to allow programmers to write programs that will continue in the event of a division by zero. The IEEE’s Frequently Asked Questions document for the floating-point standard gives this reply to the question “Why doesn’t division by zero (or overflow, or underflow) stop the program or trigger an error?”:

“The [IEEE] 754 model encourages robust programs. It is intended not only for numerical analysts but also for spreadsheet users, database systems, or even coffee pots. The propagation rules for NaNs and infinities allow inconsequential exceptions to vanish. Similarly, gradual underflow maintains error properties over a precision’s range.

“When exceptional situations need attention, they can be examined immediately via traps or at a convenient time via status flags. Traps can be used to stop a program, but unrecoverable situations are extremely rare. Simply stopping a program is not an option for embedded systems or network agents. More often, traps log diagnostic information or substitute valid results.”

Simon Tatham stated that there is a basic problem with Anderson’s ideas, and thus with the idea of building a transreal supercomputer: “It’s a category error. The Anderson transrationals and transreals are theoretical algebraic structures, capable of representing arbitrarily big and arbitrarily precise numbers. So the question of their error-propagation semantics is totally meaningless: you don’t use them for down-and-dirty error-prone real computation, you use them for proving theorems. If you want to use this sort of thing in a computer, you have to think up some concrete representation of Anderson transfoos in bits and bytes, which will (if only by the limits of available memory) be unable to encompass the entire range of the structure. And the point at which you make this transition from theoretical abstract algebra to concrete bits and bytes is precisely where you should also be putting in error handling, because it’s where errors start to become possible. We define our theoretical algebraic structures to obey lots of axioms (like the field axioms, and total ordering) which make it possible to reason about them efficiently in the proving of theorems. We define our practical number representations in a computer to make it easy to detect errors. The Anderson transfoos are a consequence of fundamentally confusing the one with the other, and that by itself ought to be sufficient reason to hurl them aside with great force.”

Geomerics, a start-up company specializing in simulation software for physics and lighting and funded by ANGLE plc, had been asked to look into Anderson’s work by an unnamed client. Rich Wareham, a Senior Research and Development Engineer at Geomerics and a MEng. from the University of Cambridge, stated that Anderson’s system “might be a more interesting set of axioms for dealing with arithmetic exceptions but it isn’t the first attempt at just defining away the problem. Indeed it doesn’t fundamentally change anything. The reason computer programs crash when they divide by zero is not that the hardware can produce no result, merely that the programmer has not dealt with NaNs as they propagate through. Not dealing with nullities will similarly lead to program crashes.”

“Do the Anderson transrational semantics give any advantage over the IEEE ones?”, Wareham asked, answering “Well one assumes they have been thought out to be useful in themselves rather than to just propagate errors but I’m not sure that seeing a nullity pop out of your code would lead you to do anything other than what would happen if a NaN or Inf popped out, namely signal an error.”.

Wednesday, June 12, 2013

Two Australian Paralympic wheelchair basketball players, Shelley Chaplin and Leanne Del Toso, are cycling around Fiji to raise money for people in Fiji with disabilities. They hope to cover the route, which is roughly 500 kilometers (300 miles) long, in just ten days. They started on June 7, 2013 and plan to finish by June 16.

Along the way, they intend to do outreach, and mentor people with disabilities. They hope to raise A$13,000. So far, they have raised over A$12,400. They are using crowdfunding to finance their sporting event.

Del Toso suffered muscle deterioration in her legs and hands due to a degenerative neurological condition when she was 19, and rides her bike with the aid of orthotics. Chaplin was born a paraplegic, and is using a handcycle.

They won silver medals at the 2012 Summer Paralympics in London with the Australia women’s national wheelchair basketball team, commonly known as the Gliders.

Both also played for Victoria in Round One of the Australia Women’s Wheelchair Basketball League (WNWBL) competition last weekend. Victoria won all four of its games. They expect to be back in action again in Round Two in Perth on June 21–23.