Palo Alto mass killer of Ukulele Orchestra caught

I guess that many female readers would call this guy a heart-throb.



More importantly, however, Kevin Dahlgren (*1992) of Palo Alto is a mass killer who has murdered a Czech family of four in Brno, the second largest country in Czechia and the capital of Moravia (130 miles southeast of Prague).

A San Francisco Chronicle blog explains that he had some identity crisis and had to leave his family. The family hoped that his psychological state would improve overseas.

So he went to Czechia to teach English. Unfortunately, a family of four – the Harok family (my research using publicly available information only) – had to pay with their lives for the treatment of this guy. A mother (who is a teacher), a father, and two sons. The father (Martin Harok) and one of the two sons (Filip Harok – all the names are my research) were members of the "Ukulele Orchestra jako Brno" ("jako" means "as" or "as big as", in this case, and "jako Brno" is being used by all Czechs for something that is really big, almost like the City of Brno; it's a pun because these guys were from Brno; ukulele is a primitive musical instrument).




He has used several weapons (mainly knives) to kill the four people in their house (a photo gallery with the house; go to Google Maps: Brno-Ivanovice, Zatloukal's Street). Finally, he wanted to mask his acts – to make them look as an unlucky accident – so he put the house on fire, too. It's likely that police has a detailed documentation and witnesses for these events. After all, the four bodies were found on a pile in the garage by the family's neighbor.

It happened less than two days ago, on Wednesday night (Czech time). The guy managed to get to Vienna in some way and flew to Washington D.C. where he was caught an hour ago, with the help of the Austrian police and an FBI agent.

Unless I am confusing California with Chechnya – they sound similar, perhaps they are the same thing – I must ask: Why is California doing such things to us? [Just to be sure, this paragraph is a parody of the statements about Czechia after two Chechens terrorized the Boston Marathon. Nothing is meant seriously in this paragraph and I don't believe in the collective guilt linked to individual murders. It's annoying that a reader made it necessary for me to spoil this paragraph by this down-to-earth explanation.]

BTW California is going to ban fires. Maybe they should try to ban murders first.




The band "Ukulele Orchestra jako Brno" also has a YouTube channel with nine almost unknown videos. The first one, which is one year old, has collected about 5,000 visits and is the most visited one:



Fair music, somewhat amateurish, I would say. Some musicians from the video above are no longer alive, I think. Sad. The message is that you should better not attempt to learn English if you want to keep your chances to survive. [Spoilers: this is a joke, too.]

While the criminal only spoke English, his family actually had some Czech roots (links to Brno); Dahlgren is a name of Swedish origin, however. It's even claimed by ABC News etc. that he was a rather close relative of the murdered family. He was offering to teach English for $10 per hour on his Facebook profile (he joined Facebook less than 3 weeks ago!).

Most Czechs apparently want him to be tried in the U.S. because they believe that he could earn a harsher punishment in that way. For a murder of four, it's likely he would get a life in prison in the Czech Republic but speculations suggest that 15 years could be the final verdict, too.

It's being reported that Kevin Dahlgren despised the ugly and chaotic Western world. Not sure why he went to Czechia because we're a part of the ugly and chaotic Western world again and the houses (see the Google Maps links higher) are no worse than many houses in the U.S.

Read More

Does global warming cause tornadoes?

It was sort of inevitable that the deadly tornadoes in Oklahoma would ultimately be blamed on global warming and CO2 by someone. While most people – including those alarmed by "climate change" – reject this attribution, you can find pretty powerful people who promote this incredible link.



Senator Barbara Boxer was perhaps the most powerful person who enthusiastically supported the idea that the tornado outbreak was a message from Nature telling us to introduce new carbon taxes. She really sounds religious.

You may find lots of stories in the media that discuss a possible connection between tornadoes and the enhanced greenhouse effect. Thankfully, almost all of them (e.g. NY Daily News, Washington Examiner) say that there's no connection. But Barbara Boxer knows that such a connection would strengthen the case for the new taxes – so it must be a part of the consensus, right?

Without actually thinking about the science or asking researchers, leftwingers generally assume that whatever is convenient for their "cause" must be a part of the "scientific consensus".




But let us look at another magazine, National Geographic. It concludes by saying that a hypothetical influence of "global warming" on the frequency and strength of tornadoes could go in both ways and there is no evidence of a trend in either direction. They quote Roger Pielke Jr, among others. Still, the title asks whether there is a connection and it offers some ideas that could support such a link.

Which ideas?




Before they admit that it's always tricky to link a particular weather event to climate change, they offer these two paragraphs:

It sounds intuitive: Of course global warming should lead to more—and more powerful—tornadoes.

We're adding energy to the atmosphere by trapping heat with greenhouse gases, and tornadoes are the very picture of terrifying atmospheric energy.

What I find problematic is that it is not explained why these would-be arguments in favor of the connection are wrong. So many readers may just think that they're actual valid arguments in favor of the connection.

The quote above says that tornadoes are "the very picture of terrifying atmospheric energy" and we are adding it, so we are probably strengthening tornadoes. Is that a valid reasoning?

All forms of energy are convertible to each other – only the total energy is conserved. This is the statement known as the first law of thermodynamics. So adding energy in one form may increase the energy in other forms, too. However, there is also the second law of thermodynamics that says that you can't construct a device that does mechanical work by extracting the thermal energy (heat) from a colder object (of from an object indefinitely). In other words, the perpetual motion machine of the second kind is impossible.

(The perpetual motion of the first kind is a gadget that would create energy out of nothing and it is also impossible – because of the conservation of energy or, equivalently, the first law of thermodynamics.)

But the creation of a tornado out of the "global warming" would be exactly such a perpetual motion machine of the second kind! The reason is that whether we like it or not, a tornado is a place with concentrated "useful" mechanical energy – stored in the macroscopic ensembles of molecules – while a warmer atmosphere is the "useless" thermal energy i.e. energy stored in the chaotic motion of individual atoms. And one simply can't create the former (useful, concentrated energy) out of the latter (useless, chaotic energy).

Once the energy is converted to heat, i.e. universal chaos, it can't be converted to the "useful" mechanical forms anymore! Only if the temperature is non-uniform, the temperature differences may be exploited to do some mechanical work, like in "heat engines". But global warming creates no new temperature differences so it creates no new opportunity for such "natural heat engines" to produce new mechanical work!

Why is it so? Because the global warming is global, stupid. The widely discussed effect is caused by the increased CO2 concentration which adds something to the expected temperature at any place. But because the carbon dioxide molecules are almost instantly (within weeks at most) spread uniformly over the whole Earth's atmosphere, the greenhouse effect is equally strong everywhere. Well, there is some dependence on the latitude etc., and especially the feedbacks (e.g. the ice-albedo feedback) depend on the latitude, but this dependence is extremely slow. To extract a useful mechanical energy e.g. to create a tornado, the greenhouse effect would have to seriously change from one mile to another and it is clearly something that the uniform, "global" rise of the CO2 concentration isn't capable of doing.

In other words, the enhanced greenhouse effect doesn't change the magnitude of local non-uniformities, so it can't contribute to the tornadoes, either. The same comment is true not only for tornadoes but for any "special", localized weather event. These events don't really care about the overall temperature shifts and the overall temperature shifts (probably very modest ones) are the only thing that extra greenhouse gases may cause.

So at least in the leading approximation, CO2 isn't capable of changing the frequency of tornadoes – in either way. But you could hypothesize that there is some subtler, higher-order effect that causes such an influence, anyway. For example, the thickness of the troposphere (the circulating part of the atmosphere between the surface and the tropopause, about 10-20 km above the surface, where all the "complicated weather" takes place) depends on the overall strength of the greenhouse effect on the Earth.

It's plausible that as the troposphere gets thicker, there is more room for circulation and various manifestations of the circulation may strengthen. That's great but we should be interested not only about the Yes/No answer to the question "whether such an influence may exist" – in principle, almost everything influences everything else – but also about the estimated strength of such an influence. These estimates will be order-of-magnitude estimates of a sort; we won't discuss whether the effects boil down to wind shear, jet streams, funnel clouds, supercells, or other fancy concepts in atmospheric physics.

We should ask: by how many percent the frequency of tornadoes could rise or decrease (we really don't know the sign) if CO2 raised the global temperature by 1 °C?

That's a meaningful question, suggesting that we want to understand these things beyond demagogic slogans. So let's try to think a little bit. I proposed a mechanism suggesting that the strength or frequency of tornadoes could scale with (a power of) the thickness of the troposphere or, almost equivalently, with the total temperature increase caused by the greenhouse effect on Earth.

So how many percent may this unknown influence add? The key realization is that the greenhouse effect caused by the man-made CO2 is a small fraction of the overall greenhouse effect on Earth. The greenhouse effect on Earth is dominated by water vapor which adds over 30 °C to the temperature of pretty much every place on our blue, not green planet. Even if the whole 20th century warming was due to CO2 (which seems unlikely to me), we have only added about 2 percent to the overall warming caused by the greenhouse gases. We may add another percent, perhaps, but we're still talking about a few percent.

The thickness of the troposphere may have risen as a power law which also means by "several percent" and the same thing holds for all quantities describing the wind speeds, the total number of vortices, and so on, and so on. I actually think that this is an overestimate and because the troposphere is getting thicker, the atmospheric phenomena and energy are actually diluted into a larger volume so less is left for the near-surface weather. But let's be agnostic about the sign, exponents, and coefficients.

It's still true that these unknown influences only add or subtract several percent to or from the strength and frequency of the tornadoes we ultimately observe. And this is such a tiny change that it's ultimately undetectable. Why is it undetectable?

Because the number of tornadoes is notoriously variable. The interannual noise is so large that it totally prevents us from seeing a hypothetical signal that would only scale as a few percent. For example, look at this list of tornadoes in the U.S. spawn by tropical cyclones. The records (with lots of holes) begin around 1811. You see a few tornadoes a year, sometimes a dozen or two. There are exceptions like 115+ tornadoes in 1967 and 103+117 in 2004. It's pretty clear that in the past, the actual numbers were larger but people couldn't see everything due to the limitations of their observational technology.

At any rate, if the frequency of tornadoes grew by 2%, it would mean that there should have been just 113+ tornadoes and not 115+ tornadoes in 1967. Similarly for other numbers. The years for which the number of tornadoes are of order one are shifted by a tiny fraction of a single tornado. Try to statistically evaluate these chaotic tables in any way. It is absolutely clear that you couldn't possibly see a trend, whether it is a positive one or a negative one.

I believe it is a sufficient reason for us not to talk about such influences. Given the fact that we can't observe such an influence empirically, every proposed claim about such an influence has to suspected to be an artifact of errors, neglected terms and effects, and other things. We just don't know how large the influence is and what its sign is. Within the error margins, we observe the influence to be zero. So we should always act as if we were assuming that the influence is zero. Anything else amounts to bias – and violations of the presumption of innocence and other things.

Incidentally, I wrote about "noise" in the number of tornadoes on a given year. But the word "noise" is actually too disrespectful because what we really meant is "everything that has nothing to do with CO2". However, there could be – and there almost certainly are – many much stronger and signal-like influences on the frequency of tornadoes than the CO2 concentration. For this reason, it's already tendentious to pretend that the data are composed of a "signal" and of "noise", especially if we want to implicitly claim that CO2 is the only "signal" although it's almost certainly not the case.

Please, Ms Barbara Boxer and others, stop talking about these medieval hypothetical links that sound almost identical as the accusations against the witches in Salem, Massachusetts. Science supports none of your fantasies and every attempt by a person to rationalize such fantasies shows that the person lacks scientific integrity – and sometimes plain human honesty, too.

And that's the memo.

Read More

Augustin-Louis Cauchy: an anniversary

By the number of mathematical papers he wrote, Augustin-Louis Cauchy was second just to Leonhard Euler. As many college freshmen may testify, more theorems and concepts in mathematics were named after Cauchy than anyone else. And a conservative theoretical physicist shouldn't omit a CV of Cauchy because Cauchy was... well... very conservative!

He died on May 23rd, 1857, i.e. exactly 156 years ago. But before he managed to do that, he had to do many other things. For example, he had to be born – in August 1789, just a month after Bastille was stormed by a crowd on the street, a mess we often call the beginning of the French Revolution.




Louis François Cauchy, i.e. Cauchy's father, had really nothing to do with this mess. He was a high official in the Parisian police before the revolution took over (during the "New Regime"). During the Reign of Communist Terror in 1794, when Cauchy was five, the family had to move to Arcueil. Things got safer when Robespierre – a guy who worked to transform bourgeoisie to a gang of leftwingers – was finally executed in 1794 and the family could return. When Napoleon took over 5 years later, Cauchy's father returned to police.

He was working directly under another high-tier policeman called Pierre-Simon Laplace who is today known, ehm, as a top mathematician. Joseph Louis Lagrange was well-known to the Cauchy family, too.




In fact, Lagrange advised Cauchy's father to enroll his son into the Central School of Pantheon. And no, Lagrange didn't want Cauchy to learn some proper mathematics. Due to uncle Lagrange's advices, Cauchy was supposed to learn classical languages and humanities. And he has won many prices in Latin and humanities, indeed. But despite the attempts by Lagrange to direct young Cauchy to this sissies' stuff, he chose an engineering career. In 1802, he scored among top 1% of the applicants to École Polytechnique and was accepted. He had some problems with the military-style rules in the school but he finished the school when he was 18, with the highest honors, and continued with civil engineering at the School for Bridges and Roads.

When he was 21, he already started to work as an engineer as well as a manager at something that Napoleon intended to become a naval base, Cherbourg. He had enough time to work on mathematical papers, anyway. His first two manuscripts on polyhedra were accepted; the third one on conic sections was rejected.

When he was 23, he realized he was overworked and the engineering job sucked, so he returned to Paris. He formally remained an engineer but was working for the ministry of interior and was on an unpaid sick leave etc. More importantly, he would work on higher-order algebraic equations, symmetric groups, symmetric functions, and the other Galois-like stuff (Évariste Galois himself was just an infant at that moment!).

In 1815, Napoleon was defeated in Waterloo – just to be sure, the place is very far from the Perimeter Institute – and Bourbon king Louis XVIII led the restoration attempts. So at the Academy of Sciences, mathematician Gaspard Monge and thermodynamics pioneer Lazare Carnot had to be fired for political reasons while Cauchy could have been hired for the same reasons. ;-)

Cauchy accepted but because a big part of the academic establishment was already composed of politically correct, left-wing activists, the reaction of his peers to his acceptance was harsh. He earned many enemies. In 1815, Cauchy could finally quit the engineering job and take the professor chair at École Polytechnique after Louis Poinsot who left for health reasons. Before that, Cauchy had already proven Fermat's polygonal number theorems. Many liberal activists could have been fired from the Bonapartist school while conservative Cauchy – whom Wikipedia calls "reactionary" – could be promoted.

He was living with his parents when he was 28 but his dad found a wife for him – a babe from a family that published most of Cauchy's writings. They had two daughters. Incidentally, Cauchy had brothers who became lawyers and one of them partly a mathematician, too.

As a mathematical factory, Cauchy flourished in the mid-to-late 1820s because he was lucky to live in a conservative political atmosphere. In 1824, Louis XVIII died and was superseded by an even more right-wing king, Charles X, which made Cauchy even more happy and more productive. He was also teaching at several schools simultaneously.

Things changed discontinuously in 1830. Charles X had to flee and the leadership was hijacked by a non-Bourbon king Louis-Philippe. Riots involving ignorant students took place near Cauchy's home. It had to be really annoying. Cauchy's lust to publish papers went nearly to zero – it's similar to my year around 2005 except that I had to be satisfied with a "relative conservative" in the form of Larry Summers who was finally removed in 2006. ;-)

Cauchy went to exile. In Switzerland, they still wanted him to endorse the new regime. He refused so he lost all positions he had in France. He went to Turin, Italy in 1831 and became a foreign member of the Royal Swedish Academy of Sciences.



Prague in 1830

In 1833, Cauchy moved to Prague in my homeland, the Austrian Empire ;-), to become a tutor of Henri of Artois, a spoiled brat from an aristocratic family. Back in Paris, Cauchy was already a bad lecturer and his teaching style resembled that of Sheldon Cooper. These difficulties escalated with young Henri who had no respect for Cauchy, mathematics, or anything of the sort, so even though Cauchy took the job very seriously, it was a disaster. The two main results of this tutoring was that an irrelevant Henri became a life-long math hater; and Cauchy hadn't done any research for 5 years. In 1838, he returned to Paris with his family that had been accompanying him in Prague since 1834.

He couldn't return to teaching – formally because he refused to endorse the new regime – but he badly wanted to be formally recognized by the science establishment in Paris again. He decided to get "there" though the Bureau to Determine the Longitude. He didn't need the oath so he was elected but the king was refusing to approve him for 4 years in which Cauchy was getting no money and had no academic rights (e.g. submitting papers). After that, Cauchy was eliminated altogether and replaced by Poinsot. Note that the opposite replacement was discussed above.

Throughout the 19th century, France was converging towards the separation of state and church, entities that Cauchy always wanted to unify. He became an enthusiastic Jesuit and officer in various Catholic and Jesus Society institutes. His colleagues would have full mouths of non-discrimination and so on but when this top mathematician of the French history applied for an ordinary chair in mathematics, he got just 3 out of 45 votes. Leftwingers have been a biased scum for at least 150 years.

The pan-European revolution of 1848 was mostly good news for Cauchy. The oath was removed from the law and he could regain the professorship again. He died on this day in 1857, sort of respected again. His name is one of the 72 names inscribed into the Eiffel Tower.

Research

In his early career, he made lots of advances about the Appolonius problem of circles touching three other circles; Euler's formulae for polyhedra; he introduced the notion of convergence, and other things. He was quickly becoming a pioneer of mathematical analysis – that's the part of maths dealing with functions, sums, integrals, derivatives, and... (this is why it's more advanced than just calculus – which is otherwise almost the same thing) with all kinds of limits.

He also made contributions to physics – wave mechanics (Fresnel's wave theory) and elasticity (Cauchy stress tensor). Add various things about membranes, vibrations, and so on. I have already mentioned the Fermat polygonal number theorem.

When it comes to mathematical analysis, he really laid the foundations of the holomorphic functions of complex variables, e.g.\[

\oint_C f(z)\dd z = 0

\] if there are no singularities inside the closed contour \(C\). Seeds of this theorem already existed in 1814; the complete form was given in 1825. He figured out how to compute residues either from limits determining the Taylor expansion; or from the contour integrals. Pierre-Alphonse Laurent was the first man after Cauchy who contributed to this essential mathematical knowledge (Laurent series in 1843).

Cauchy tended to praise the mathematical rigor. Already in 1821, he was working with the infinitesimals – formally infinitely small numbers – but he was the first visible guy to have introduced the \(\varepsilon\)-\(\delta\) gymnastics (games with limits) as the rigorous incarnation of the infinitesimal numbers.

It's an excessive task to enumerate everything that Cauchy has done in mathematics. I find it sort of funny to copy-and-paste a list of insights named after Cauchy:

• Binet–Cauchy identity
• Cauchy's argument principle
• Cauchy–Binet formula
• Cauchy boundary condition
• Cauchy condensation test
• Cauchy's convergence test
• Cauchy (crater)
• Cauchy determinant
• Cauchy distribution
• Cauchy's equation
• Cauchy–Euler equation
• Cauchy functional equation
• Cauchy formula for repeated integration
• Cauchy–Frobenius lemma
• Cauchy–Hadamard theorem
• Cauchy horizon
• Cauchy's integral formula
• Cauchy's integral theorem
• Cauchy interlacing theorem
• Cauchy–Kovalevskaya theorem
• Cauchy matrix
• Cauchy momentum equation
• Cauchy–Peano theorem
• Cauchy principal value
• Cauchy problem
• Cauchy product
• Cauchy's radical test
• Cauchy–Riemann equations
• Cauchy–Schwarz inequality
• Cauchy sequence
• Cauchy surface
• Cauchy's mean value theorem
• Cauchy stress tensor
• Cauchy's theorem (geometry)
• Cauchy's theorem (group theory)
• Euler-Cauchy stress principle
• Maclaurin–Cauchy test

Let me remind you: this guy was voted as unworthy the chair of an ordinary scholar in mathematics by 42 out of 45 self-described "pro-Enlightment" researchers in mathematics. Certain left-wing portions of the Academia have been rotten for centuries.

And that's the memo.

Read More

Intriguing spectra of finite unified theories (FUT)

In November, I discussed FUTs (finite unified theories) which are \(\NNN=1\) supersymmetric grand-unification-inspired versions of MSSM with the additional constraint that the divergences already cancel at the level of the effective field theory. This finiteness boils down to the vanishing of the beta-functions, some anomalous dimensions, and some relationships between the gauge and Yukawa couplings.

This condition doesn't seem to be a "must" – the divergences may very well be taken care of by the high-energy phenomena (string theory ultimately takes care of all divergences so its approximations don't have to be finite by themselves) – but it is an aesthetically intriguing condition, anyway. Now, the same authors released a new paper

Finite Theories Before and After the Discovery of a Higgs Boson at the LHC (S. Heinemeyer, M. Mondragon, G. Zoupanos)

where they calculate some new predictions and intriguing details.




They focus on the third-generation fermions and their superpartners, the Higgs sector, and the gauginos. The nicest FUTs they consider boast names such as FUTA and FUTB – the latter seem particularly attractive. They also take some LHCb results into account. In these models, \(\tan\beta\) is typically rather large, \(\mu\) is almost necessarily negative.




The spectra seem very intriguing and consistent with everything we know. Unfortunately, they're inaccessible to the LHC – or marginally accessible – and perhaps even inaccessible to ILC/CLIC. I like the representative table of a FUTB model here:\[

\begin{array}{|l|l||l|l|}
\hline
m_b(M_Z) & 2.74 &&
m_t & 174.1 \\ \hline
m_h & 125.0 &&
m_A & 1517 \\ \hline
m_H & 1515&&
m_{H^\pm} & 1518 \\ \hline
m_{\tilde t_1} & 2483 &&
m_{\tilde t_2} & 2808 \\ \hline
m_{\tilde b_1} & 2403 &&
m_{\tilde b_2} & 2786 \\ \hline
m_{\tilde \tau_1} & 892 &&
m_{\tilde \tau_2} & 1089 \\ \hline
m_{\tilde\chi_1^\pm} & 1453 &&
m_{\tilde\chi_2^\pm} & 2127 \\ \hline
m_{\tilde\chi_1^0} & 790 &&
m_{\tilde\chi_2^0} & 1453 \\ \hline
m_{\tilde\chi_3^0} & 2123 &&
m_{\tilde\chi_4^0} & 2127 \\ \hline
m_{\tilde g} & 3632 && {\rm masses}& {\rm in}\,\GeV
\\ \hline
\end{array}

\] You see that the LSP is the lightest neutralino below \(800\GeV\). Staus are just somewhat heavier, \(900\GeV\) and \(1100\GeV\). Both sbottoms and stops fit the pattern that the lightest and heaviest one is at \(2500\GeV\) and \(2800\GeV\), respectively. The second lightest neutralino and the lightest chargino sit at \(1450\GeV\), the remaining four faces of the God particle find themselves above \(1500\GeV\) while the heavier chargino and the heaviest two neutralinos are above \(2100\GeV\). Finally, the gluino is above \(3600\GeV\).

Particularly the last figure is rather high (in a broader ensemble of models they analyze, the masses may go up to \(10\TeV\) or so). We would have trouble to see such a gluino for years. But this model or at least similar models may be right. From a theoretical viewpoint, I see absolutely no preference when I compare models with gluinos at \(1200\GeV\) and \(3600\GeV\). Some people become very emotional and start to say that one of them has to be right or wrong or its rightness or wrongness means something a priori. Well, it just doesn't. Nature doesn't give a damn whether it's easy or hard for us to observe the superpartners. Once we observe them, many new things start to be clear. If we don't observe them, we are still extremely far from ruling out supersymmetry – and nice special supersymmetric models such as FUTB in this paper.

Its not my – or other humans' – job to rate the beauty of the values of particle physics parameters that emerge from Nature's decisions. It's Her job. Nevertheless, I must say that I would find a spectrum like the table above – or many other tables – elegant. It would probably mean that all these obnoxious idiots who like to say bad things about SUSY could remain loud for many more years. That's an annoying vision from a personal viewpoint but it can't change anything about the reality and it is less important than the actual beauty and physical near-inevitability that is carried by supersymmetry at some scale. If the known – mostly theoretical – evidence makes two models equally plausible and elegant, then one is obliged to love both of them equally, regardless of the fact that one of them may be much more accessible to the experiments. I view this commandment as a part of the scientific integrity.

Read More

A proof of the Riemann Hypothesis using the convergence of an integral

Thursday morning update: After many hours, I decided that there is a critical error in the otherwise cleverly constructed proof. On page 138 (discussing Lemma 3), second part, he says "whence the function converges absolutely" essentially for any \(z\) with a real positive part. But it seems he hasn't really established that (except for circular reasoning) because if RH is false, and it may be false, the numerator \(|\psi(e^t)-e^t|\) goes like \(e^{at}\) for some positive \(a\) and the region of convergence is shifted by \(a\). So the "absolute" part of the convergence isn't correctly proven, it seems to me. Maybe it's enough to prove the "ordinary" convergence but I suspect that there could be a similar error in the \(g_1\) part of Lemma 3, too. Apologies if I am making a mistake.

Some people talk about the proof of "almost twin" prime integers separated by at most 70 million or something like that. I am not terribly excited by this result even if it is true. It's always more interesting to talk about somewhat promising proofs to the Riemann Hypothesis, not only because of the $1 million that will be given to the first person who solves the old puzzle.

Many people have thought that they had a proof but the candidate proofs have always failed so far. So you must understand it is extremely likely that we have another example of a failure here. But I am going to tell you, anyway. It would be great if some readers spend a sufficient time and energy by reading the paper. Please don't be repelled by the idiosyncratic Chinese English. Even I can recognize that it's not how a native speaker would formulate the ideas. ;-)

吴豪聪

That's his real name. Today, Hao-cong [first name] Wu [surname] of China sent me his new paper with a somewhat strange title (linguistically)

Showing How to Imply Proving The Riemann Hypothesis (PDF full)

published in the European Journal of Mathematical Sciences. How does the proof work?




It's likely that I won't quite reproduce everything that is needed for the proof in this blog entry even though I may try. Teaching things is the best way to learn them. ;-)

Wu elaborates upon some ideas initiated by Serge Lang, a famous mathematician. But that's the last comment about the sociological context. Now, let us look at the ideas which don't seem to require any esoteric new branches of mathematics.

The proof reduces the Riemann Hypothesis to a claim about the absolute convergence of an integral that is related to the Riemann \(\zeta\)-function in a simple way. Let's roll.




The function that Wu finds more convenient is called \(\psi(x)\), pronounce "psi of ex". It is related to the Riemann \(\zeta\)-function by the following identities\[

\eq{
\phi(s) &= -\frac{\zeta'(s)}{\zeta(s)} = \sum_{n=1}^\infty \frac{\Lambda(n)}{n^s} =\sum_p \frac{\log p}{p^s-1}=\\
&= s \int_1^\infty \frac{\psi(x)}{x^{s+1}}\dd x = \frac{s}{s-1}+s\int_1^\infty \frac{\psi(x)-x}{x^{s+1}}\dd x
}

\] where the sum over \(p\) goes over the primes \(2,3,5,\dots\). The first step you should be able to verify if you want to validate Wu's proof is that the identities above are satisfied if \(\psi(x)\) is defined as the manifestly convergent sum\[

\psi(x) = \sum_{p^m\leq x} \log p = \sum_{n\leq x} \Lambda(n)

\] where \(\Lambda(n)=\log p\) if \(\exists m\geq 1: \,n=p^m\) for a prime \(p\) and otherwise it is set to zero. Note that this \(\psi(x)\) is defined in such a way that for a large \(x\), it's expected to be very close to \(x\) because the "probability to be prime" \(1/\log x\) is cancelled by the factor \(\log p\) from the definition of \(\psi(x)\) – it's close enough already when we allow \(m=1\) only.

The second step is to realize that the presence of a zero or zeroes of \(\zeta(s)\) also implies (or would imply) a pole of \(\phi(s)\), the [minus] "logarithmic derivative of the \(\zeta\)-function", at the same location of the complex plane. To prove the Riemann hypothesis, it is sufficient to prove that \(\phi(s)\) has no poles for \[

\frac 12 \lt {\rm Re}(s) \lt 1

\] (in the "right half-strip", as I will call it) because the hypothetical "RH-violating" zeroes (and singularities) come in pairs symmetrically distributed relatively to the critical axis \(s=1/2+it\) for \(t\in\RR\). Note that \(\phi(s)\) has a pole (or would have a pole) even for a higher-order zero of \(\zeta(s)\).

The third step, and it's the only hard one, is to actually prove that one of the integrals involving \(\psi(x)\) used to calculate \(\psi(s)\) above\[

\int_1^\infty \frac{\psi(x)-x}{x^{s+1}}\dd x

\] is analytic in the right half-strip so it has no poles over there. Consequently, the \(\zeta\)-function has no zeroes in the right half-strip and, by the left-right symmetry, no zeroes in the left half-strip, either.

Wu reduces the claimed analyticity of the integral above to the absolute convergence (convergence even if the integrand is replaced by its absolute value) and uniform convergence (the speed of convergence may be taken to be \(\varepsilon\)-independent), \(\forall\varepsilon\gt 0\), of the integral\[

\int_1^\infty \frac{\psi(x)-x}{x^{3/2+\varepsilon}}\dd x.

\] It shouldn't be hard to see that the absolute and uniform convergence of the integral above (here) is enough for the analyticity of the previous integral, and therefore for the absence of the non-trivial zeroes. Note that the exponents \(s+1\) for \(s\) in the right half-strip and \(3/2+\varepsilon\) for a positive \(\varepsilon\) are the same objects.

So aside from the claims that should be straightforward, the beef of the proof should be the demonstration of the absolute and uniform convergence of the integral in the last displayed equation.

Note that Wu's approach is linked both to the "complex analytic" interpretation of the Riemann Hypothesis as well as the prime-integer-counting, "number-theoretical" interpretation. It's because sufficient experts know that the Riemann Hypothesis is equivalent to the statement\[

\forall \varepsilon\gt 0: \, \psi(x) = x+ O(x^{1/2+\varepsilon})

\] which says that if we accept that the probability for a "rough number \(x\)" to be a prime is \(1/\log(x)\), then the estimated number of primes up to \(n\) deviates from the actual one at most by a power law (that is producing the \(O(\dots)\) term above.

Proving the convergence

OK, so how does Wu want to prove the uniform and absolute convergence? He offers some introduction to the theory of functions of real and complex variables together with some lemmas that are not quite well-known and that may even be new. Finally, the proof boils down to the existence (for any \(s\) with a real positive part) of the Laplace transforms \(g_{1,2}(s)\) of a function called \(f_{1,2}(t)\) related to \(\psi(e^t)-e^t\) for the subscript \(1\) or its absolute value for the subscript \(2\).

If you quickly want to focus on claims related to the \(\zeta\)-function and ignore various theorems and lemmas about completely general functions and their convergence etc. (assuming that these things are harmless and perhaps known to you, explicitly or intuitively), you may find it helpful for me to say that only Theorem 5 (among 7 theorems) and Lemma 3 (among 3 lemmas) is what you want to read. If there is some circularity in Wu's argument (secretly assuming RH), it's probably somewhere in Theorem 5 or Lemma 3.

In particular, I believe that Theorem 5 contains the main trick that allows us to show the convergence in the right half-strip. This theorem claims the absence of poles (except for the \(s=1\) pole) of the function\[

\eq{
\Phi(s) &= \sum_p \frac{\log p}{p^s} = \phi(s)-\sum_p h_p(s),\\
|h_p(s)|&\leq B\frac{\log p}{|p^{2s}|}
}

\] On one hand, this capital \(\Phi(s)\) is shown to be rather close to the lowercase \(\phi(s)\), using an argument based on geometric series. On the other hand, the \(2s\)-th power of something appears in the difference between \(\Phi\) and \(\phi\) which makes \(\sum\log n/n^{2s}\) converge for \({\rm Re}(s)\geq 1/2+\delta\). So the coefficient \(2\) in \(2s\) here is the ultimate reason why the meromorphic character of \(\Phi(s)\) starts at \({\rm Re}(s)\gt 1/2\), how we get the one-half somewhere, and why the critical axis becomes a decisive boundary for the well-definedness of \(\phi(s)\), too.

I don't see any mistake so far but I haven't really devoured all the beef of the proof yet, either, so no complete confirmation from your humble correspondent yet. But it is apparently making more sense every minute!

See the previous TRF blog entries mentioning the Riemann Hypothesis.

Read More

Ask questions to James Hansen

Today, at 5 p.m. Boston Daylight Savings Time (11 p.m. Central European Time), James Hansen will give a talk over here.



Live Video streaming by Ustream

It's being claimed that you will be allowed to ask a question when he's finished.




First, people like James Hansen would make sure that the debate is over. And then they start the debate – so that no one inconvenient may really participate it.




I am not promising you anything, however. It's plausible that only convenient questions will be allowed. So Barbara Boxer will ask whether the Oklahoma tornadoes were caused by an SUV or by some beef steaks in McDonald's. Or some other sins against the glorious left-wing delusions that women and men of her caliber believe.

If you want to waste 73 minutes, feel more than free to watch a talk that Hansen gave yesterday in front of some people who think he is a "hero". Among other things, Hansen explains that he was skipping classes in the college because he didn't want to show how ignorant he was – which made him even more ignorant. But then he found the environmental movement and ignorance was transformed to a virtue.

Read More

Anthony Zee: Einstein Gravity in a Nutshell

Škoda is not just a carmaker; it is producing happy drivers. And you may see that even the engines in the factory are having a great time.

In the same way, Anthony Zee – as Zvi Bern noticed – decided to make many readers fall in love with the physics of general relativity by having written this wonderful tome, Einstein Gravity in a Nutshell. Bern said that the goal wasn't to create new experts but Zee corrected him that he wanted to make the readers fall in love so deeply that they may dream about becoming experts, too. And the clearly enthusiastic Anthony had to enjoy the writing of the book, too.

I received this large, almost 900-page scripture on Einstein's theory yesterday. Obviously, I haven't read the whole book yet but I may have spent more time with it than most readers (more than zero) so that I can tell you why you should buy it and what philosophy, style, and content you may expect.




It's a book addressed to a wide variety of readers, including very young ones (perhaps college freshmen and bright high school students) and amateur physicists. Experienced physicists and professionals may find some gems or at least entertainment in the book, too. Because of this goal, the book starts with elementary things such as the units including \(G,c,\hbar\) and Planck units, relativity even in classical physics, as well as basics of curved spaces, differential geometry, and so on.




The style is witty and somewhat dominated by words – and amusing titles. You may find lots of philosophical and historical remarks and stories from Anthony's professional life but the physics is always primary. And I mean physics, not rigorous mathematics. Tony is focusing on objects, phenomena, and their measurable and calculable quantities and the purpose of physics is to understand them and calculate them. So he spends almost no time with various picky issues – whether a function has to be smooth; whether one should use one fancy word from abstract mathematics or another. In fact, he considers the suppressed role of rigorous maths to be a part of the "shut up and calculate" paradigm that he subscribes to.

In some sense, you could say that the approach resembles the Feynman Lectures on Physics. It is very playful and the author is always careful to tell you think that are still fun and stop elaborating on details when he could start to bore you. So the book (probably) keeps its fun status at every place (it's true for the portions I have read). But Anthony Zee manages to penetrate much more deeply into general relativity with this strategy.

Once he goes through all the basics – which allow a beginner to start with the subject almost from scratch but which seem very entertaining for a reader who doesn't really need such introductions anymore – and he answers all the FAQs on tensors and lots of other things, he offers some of the simplest derivations of Einstein's equations and is ready to apply them.

It's useful to know what concepts are considered primary starting points by the author. I would say that Zee is elevating the concept of symmetries and the action – the latter allows us to formulate most dynamical laws in classical and quantum physics really concisely (although we know perfectly consistent quantum systems that don't seem to have any nice action; and the action always assumes that we prefer a particular classical limit of a quantum theory – and the classical limit isn't necessarily unique).

Concerning the applications, some of the historically important applications that were designed to verify the theory are suppressed. But you get very close to the cutting edge, including the general-relativistic aspects of topics that are hot in the contemporary high-energy theoretical physics and the cosmological/particle-physics interface. So you may actually learn advanced topics about black holes including some Hawking radiation (including the numerical prefactors of the temperature; but the author doesn't go extremely far here; note that amusingly enough, the Hawking radiation is even discussed in an introductory chapter); large and warped extra dimensions; de Sitter and anti de Sitter space including a discussion of conformal transformations (although it doesn't seem like a full-fledged textbook on AdS/CFT); topological field theories; Kaluza-Klein theory (with extra spatial dimensions) and braneworlds; Yang-Mills theory (there's lots of electromagnetism in the earlier chapters); even twistor theory; discussions on the cosmic inflation and the cosmological constant problem; and heuristic thoughts on quantum gravity (some of them are more heuristic than the state-of-the-art allows us; but Zee's philosophy is that textbook shouldn't be composed exclusively of the totally established stuff ready to be carved in stone).

Using lots of witticisms and clever analogies, Zee also proves some things you wouldn't expect – e.g. that Hades isn't inside the Earth. The equivalence principle is compared to the decision of all airlines, regardless of the size (and the size of their aircraft), to fly between two distant cities along the same path on the map. Witty and apt.

Anthony is convinced that most authors are explaining things in unnecessarily complicated ways – in some cases, perhaps, they want to look smart by looking incomprehensible. That's not Zee's cup of tea. He enjoys to simplify things as much as possible (but not more than that). And he loves to formulate things so that the reader is led to the conclusion that things are simple and make sense, after all. For example, there is a fun introduction to the least action principle (light isn't stupid enough not to know the best path) and we learn that "after Lagrange invented the Lagrangian, Hamilton invented the Hamiltonian". It makes sense, doesn't it?

There's a lot to find in the book. Some readers say that the book is less elementary than Hartle's book but more elementary than Carroll's. Maybe. Anthony is more playful and less formal but there are aspects in which he gets further than any other introductory textbook of GR.

The book is full of notes, a long index, and simply clever exercises. The illustrations are pretty and professional. If you are buying books to see photographs of attractive blonde women with toys, you won't be disappointed, either.

Because the book is really extensive and even the impressions it has made on your humble correspondent in the single day are numerous, I have to resist the temptation to offer you examples, excerpts etc. because that could make this blog entry really long by itself. Instead, I recommend you once again to try the book.

Read More