It’s 11/11/11 11:11:11.111111… UTC

November 11, 2011

In honour of this auspicious moment here is some fun binary maths.

What is this sequence 2,3,7,11,13,19,25,… ?

It is actually the sequence of binary dropped primes. If you want a longer list it is Sloane sequence A014580 These are the primes when you multiply in binary but always drop the carry bits. For example 3 x 3 = 112 x 112 = 1012 = 5. I dropped a carry that would normally add 4 to the answer so 5 is not a binary dropped prime. 9 is not prime (I will drop the “dropped” from now on) either, because it is 9 = 3 x 7.

A curious feature of these primes is that if you reverse the digits in binary you always get another prime so 11 = 10112 and 13 = 11012 are both primes. All primes except 3 must have an odd number of non-zero binary digits because otherwise they are a multiple of 3.

In honour of the date and time it seems appropriate to factorise some numbers that consist of all ones in binary

1111 = 11 x11 x 11

111111 = 11 x 111 x 111

1111111 = 1011 x 1101

11111111 = 11 x 11 x 11 x 11 x 11 x 11 x 11 x 11

Others that I missed are prime. You can probably see that a number in this form can only be a binary dropped prime it the number of digits is a normal prime, but the converse is not true because 127 = 11 x 13 in decimal terms

As well as multiplication you can also do binary dropped addition and this gives you a ring which is a unique factorization domain. Don’t worry about subtraction because it is the same as addition. You can also do modulo arithmetic with the rule that a = b (mod n) iff a+b is a multiple of n.

Are these useful or are they just a curiosity for setting math puzzles? In fact they can be used to generate interesting cyclic binary linear codes . Binary linear codes are sets of codewords consisting of sequences of bits that are closed under binary dropped addition. There are always 2k codewords for some k in such a binary linear code and if the block length is n it can be used to code k bits into n bits in a way that is easy to reverse and very efficient for error-correcting. Cyclic codes have the extra property that you can cycle the digits of the code by shifting them left and moving the last digit to the first. In a cyclic code, when you cycle a codeword in this way you always get another codeword in the set.

The simplest non-trivial cyclic codes are parity codes where the number of non-zero bits in a codeword is always even, In this case n = k+1 so encoding consists of adding a parity bit to a k-bit string. Now remember that a number with an even number of non-zero bits is always a multiple of 3 in binary dropped arithmetic, so can we generate other cyclic codes by using multiples of other numbers in binary dropped arithmetic?

(11 minutes and 11 .111 seconds to go, now observing a minutes silence…)

The answer is yes, but it only works when the numbers are factors of the numbers which are sequences of ones. This is because cycling the numbers is equivalent to working modulo such a number. So we can have codes with a block length that are a multiple of three where all codewords are multiples of 111. More interesting examples are codes with 7 bits using multiples of 1011 or 1101. This is the familiar Hamming code Ham(7,4) which is linked to the Fano plane and the octonions. Even better a string of 23 ones must factorize in binary dropped arithmetic into two number of 12 binary digits, because this would be needed for the Golay code, but it is 11.11.1111… so I will have to check the factorization another time.


HCP 2011: Will it Deliver?

November 10, 2011

The rumour mill is once again turning its rusty wheels, and there are suggestions that an interesting result will be revealed at Hadron Collider Physics conference in Paris next week. More on that in a minute.

You may think that things have been quietly lately but there have been a lot of workshops going. They have not been reported much but of course us bloggers have been trawling the slides for anything new and exciting. In case you want to search for anything we might have missed here is a convenient list of links:

One thing that turned up was an update to the Higgs -> WW analysis for ATLAS upgrading it from 1.7/fb to 2/fb, The effect is not terribly exciting, nothing has changed.
So now we are waiting for the HCP conference but not much is expected, or is it? The full schedule of talks can be found here. If this is to believed even the new update for H -> WW will not be shown. The only thing certainly new is the ATLAS+CMS combination of data shown at Lepton Photon nearly three months ago.
But then an organizer speaks of a last-minute talk being added and a comment over at NEW says “…or maybe something else violates CP at 3.5 sigma level.” So do we have a new rumour about – perhaps – a result from LHCb, or is someone just hyping the conference?
Apart from that the next big question is when will the next wave of Higgs results be revealed? They must have done more analysis at 2/fb, yet we have not had anything beyond 1/fb for the crucial diphoton search from ATLAS. I am sure they must have also looked at plots using 3/fb to 4/fb but nothing has been said, except a few vague rumours that I don’t find convincing.
Now they will be preparing the 5/fb plots that should be ready for approval in December. We may see them soon after but if the results are really so inconclusive we may have to wait for the 5/fb ATLAS+CMS combination. That means there may be nothing ready to show until Moriond in March, unless…
Rumour Update 24-Nov-2011: The rumour apparently concerns a measurement of ΔACP at 600/pb and will be shown in the last talk today at HCP11. This quantity is the difference between decays of a charmed D meson into Kaons or pions. It is not yet clear if the rumoured 3.5 sigma result is merely a signal of CP violation or a deviation from the standard model.

What is the Future for Particle Accelerators?

November 6, 2011

This year all physics eyes are on the Large Hadron Collider as it approaches its promised landmark discovery of the Higgs Boson (or maybe its undiscovery). At the same time some physicists are planning the future for the next generation of colliders. What will they be like?

The answer depends in part on what the LHC finds. Nothing is likely to be built if there is no sign that it will do anything useful, but decisions are overdue and they have to make some choices soon.

Hadron colliders

Accelerators like the LHC that collide protons are at the leading edge of the Energy and Luminosity frontiers because they work with the heaviest stable particles that are available. The downside of colliding protons is that they produce messy showers of hadrons making it difficult to separate the signal from the noise. With the Tevatron and now the LHC, hadron colliders have been transformed into precision experiments using advanced detectors.

One technique is to capture and track nearly all the particles from the collisions making it possible to reconstruct the jets corresponding to interesting high energy particles such as bottom quarks created in the heart of the collision. Missing energy and momentum can also be calculated by subtracting the observed energy of all the particles from the original energy of the protons. This may correspond to neutrinos that cannot be detected or even to new stable uncharged particles that could be candidates for dark matter.

High luminosities have been achieved making it possible to scour the data for rare events and build up a picture of the interactions with high statistics. As luminosity increases further there can be many collision events at once making it difficult to reconstruct everything that happens. The LHC is now moving towards a new method of operation where it looks for rare events producing high energy electrons, muons and photons that escape from the heart of the collision giving precise information about new particles that decayed without producing jets or missing energy. In this way hadron colliders are getting a new lease of life that turns them into precision tools very different from how they have been seen in the past.

So what is the future of hadron colliders? The LHC will go on to increase its energy to the design limit of 14 TeV while pushing its luminosity even higher over the coming years. Its luminosity is currently limited by the capabilities of the injection chain and the cryogenics. These could undergo an upgrade to push luminosities ten times higher so that each year they collect 50 times as much data as they have in 2011. Beyond that a higher energy upgrade is being planned that could push its energy up to 33 TeV. The magnets used in the LHC main ring today are based on superconducting  niobium-titanium coils to generate magnetic fields of 8.5 tesla. Newer magnets could be built using niobium-tin to push the field up to 20 Tesla to more than double the energy. If they could revive the tunnel of the abandoned SSC collider in Texas and use niobium-tin magnets it would be possible to build a 100 TeV collider, but the cost would be enormous. The high-energy upgrade for the LHC is not foreseen before 2030 and anything beyond that is very distant.  Realistically we must look to other methods for earlier advances.

Is the future linear?

The latest linear accelerator built to date is SLAC at Stanford with a centre of mass energy of 90 GeV. As hadron colliders reach their physical limits physicists are returning to the linear design for the next generation of colliders. When accelerating in a straight line there is no advantage in using heavy particles so linear colliders work equally well with electrons and positrons which give much cleaner collisions.

The most advanced proposal is the International Linear Collider which would provide centre of mass energies of at least 500 GeV with 1 TeV also possible. The aim of the ILC would be to study the Higgs boson and top quark with very high precision measurements of their mass, width and other parameters. This may seem like an unambitious goal but if the LHC finds nothing beyond the standard model in the data collected in 2011 this could be the best option. the standard model makes very precise predictions about the quantities that a linear collider could measure. If these can be checked, any deviations could give clues to the existence of new particles at higher energies. Such precision measurements have already been useful in predicting where the mass of the Higgs Boson lies, but once all the parameters of the standard model can be measured the technique will really come into its own. Finding solid evidence for deviations from the standard model would be the requirement to choose and justify the construction of the next collider at the energy frontier.

But there is an alternative. A new innovative design for a compact linear collider (CLIC)  is being studied at CERN and it could push the energy of linear colliders up to 3 TeV or even 5 TeV. The principle behind CLIC is to use a high intensity drive beam of electrons at lower energy to accelerate another lower intensity beam of electrons too much higher energy. Just think of how a simple transformer can be used to convert a high current low voltage source of electricity into a low current high voltage source. CLIC does a similar trick but the coils of wire in the transformer are replaced by resonant cavities. It is a beautiful idea, but is it worth doing?

The answer depends on whether there is anything to be found in the extended energy range. This is being explored by the LHC and so far nothing new has been seen with any level of certainty. There is still plenty of room for discovery but decisions must be made soon so the data collected in 2011 will be what any decision has to be based on.

It is going to be a hard choice. For me it would be swung towards CLIC if it could be the start of a design that could lead to even higher energies. Could the same trick be used a second time to provide even higher energies, or is it limited by the amount of power needed to run it? Do other designs have better prospects, such as a muon collider? Big money and decades of development are at stake so let’s hope that the right decision is made based on physics rather than politics.

Perhaps it is worth a poll. If it was a straight choice, which of these would you prefer to see international funds spent on?


10 reasons to buy into big science

November 5, 2011

When people hear the price tag for big science experiments like the Large Hadron Collider or the Hubble Space Telescope they wonder what the benefits are that justify the cost. I am not talking about projects with obvious potential benefits such as a fusion reactor. This is about pure science, why is it worth doing?  In fact there are lots of reasons so here is my list of the top 10, starting with the least important.

10 –  spin-off innovations: When scientists are asked to justify why the LHC was worth building they often roll out the list of technical innovations that have been invented at CERN; MRI scanners, touch sensitive displays and of course the world wide web. NASA has an even longer list from non-stick pans to velcro. This only makes number ten on my list because I think most of them would have been invented by industry anyway. The advantage of inventing them at CERN or NASA is that they are not owned by private companies. What would the web be like if it has been invented by a computer or telecoms company?

9 – National Prestige: New scientific discoveries can make big news stories and for many countries there is a lot of pride in being able to call in their own experts who have worked on the project to give their summary of what it means. Politicians love it.

8 – Entertainment: TV documentaries, science magazines, blogs etc, they are there because many people find big science entertaining.

7 – Employment: Big science projects employ lots of people, another secret favorite of politicians.

6 – International Cooperation: This is rarely brought up but it is very important. Science is a very international business that brings together people from different countries. They tend to put aside national differences because what matters to them is the science. The relationships last and carry over into industry and even politics.

5 – The development of hi-tech industries: Building an experiment like the Large Hadron Collider requires new technologies such as superconductors, cryogenics and large-scale computing facilities. These are subcontracted to private companies that develop new methods with applications elsewhere. It is very hard to quantify the benefit that this brings but in economic terms it could be worth a lot more than the money spent on experiments that push the limits of technology.

4- Education: Places like CERN are packed with young people and the directors like to brag about it to the point of being openly ageist as employers. This is good news (unless you are over 30 and interested in ajob at CERN) because it means that these people are learning new skills and going on to use them elsewhere in industry or other educational centres. Students and graduates at CERN or NASA have to learn how to do research  in physics, engineering and IT. In a world where science underpins the economies of developed countries it is an educational resource that no self-respecting nation can afford to miss out on if they want a prosperous future. Again this is rarely quantified but we hope that politicians who allocate the funds appreciate it.  In my opinion it is the top practical reason for funding big science.

3 – Inspiration: Big science inspires young minds

2 – Value for Money: When people quote the cost of something like the Large Hadron Collider in billions of dollars it certainly seems like a lot of money, but you have to remember that it is spread out over many years and many countries. The UK pays about £70 million per year for CERN. It is still a lot but it is a small part of the UK research budget and it brings all the above benefits. I have never seen a cost benefit analysis done on this basis but I bet it comes out as good value.

1 –  For the Knowledge: It has to be the number one reason for doing pure science, because we want to know and should know the answers to big questions. It is just part of what makes us human.