story - Twelve Labours - #08 Dark Horse Bookmakers

_{This puzzle is part of the ‘Twelve Labours’ series, but can be solved independently. Previous instalments can be found here: Prologue | 01 | 02 | 03 | 04 | 05 | 06 | 07}

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From the golf course Hercules sped as quickly as possible to Dark Horse Bookmakers, stepping inside as a clock chimed two o’clock. The whole place was buzzing with sound, as a group of ten or so punters watched the final moments of what appeared to be a thrilling race. As the leading horse crossed the line a big cheer went up – it appeared that all of them had won their bets...!

The owner, Diomedes, grinned and welcomed Hercules inside. “I’m afraid if you were hoping to bet on any of the races at Thessaloniki today, you’ve just missed out – that was the twelfth and final race. Windmill just pipped Chilli Sauce to the finish line – exactly as I knew she would...”

Diomedes gave a sneaky chuckle and proceeded round to the cashiers’ area. “So do you want to place a bet, young Hercules? The races at Miletus begin at two-thirty – there’s seven on the racecard there for today – or if you’re quick you could have a shot at the last race of the day at Athens, which starts in eight minutes.”

Hercules retrieved his mother’s betting slip from his pocket and looked down at it. “It’s the Athens race she wants – the two-ten – but...” – Hercules sighed – “...she’s forgotten to write down the name of the horse she wants to bet on.”

Diomedes’ grin grew wider and he beckoned Hercules closer, lowering his voice. “If I tell you the winning horses from the other races at Athens, perhaps you can work it out...” He passed Hercules a sheet of paper with the details of the six races for the day:

“Okay,” said Hercules, “Cheesecake has won two already, so she’s got form...” He stopped at the sight of Diomedes shaking his head and smiling.

“That’s not how it works,” Diomedes said, lowering his voice still further. “You see, here at Dark Horse we have a little... ‘inside information’ that we can use to our advantage. For any given race on any given racetrack across Greece, we can work out if it has a... er... foregone conclusion, shall we say? There’s a code, see, among the racing circles. If all except two horses in a race meet a certain criterion, that's a sign to us that the race is rigged, and the horse that meets that criterion in a very particular way is the one to put your money on. Now, it just so happens that all of the Athens races today are ‘predictable’ – and I hear you’re a whizz with puzzles, so if you want to get that betting slip of your mother’s submitted you’ve got five minutes to work out who to put your money on...”

TASK: Deduce the Greek betting syndicate’s method for rigging races. Identify the horse that will win the two-ten at Athens.

CSV version:

Race 1 (13:10),Race 2 (13:22),Race 3 (13:34),Race 4 (13:46),Race 5 (13:58),Race 6 (14:10)
CHEESECAKE,DISHCLOTH,CHEESECAKE,CHEESECAKE,DICTIONARY,CHEESECAKE
DUNGAREES,DICTIONARY,DISHCLOTH,DUNGAREES,DISHCLOTH,DICTIONARY
DICTIONARY,DUNGAREES,GOLDFISH,MAYONNAISE,ELEPHANT,DISHCLOTH

ELEPHANT,ELEPHANT,GOLDFINCH,MICROCHIPS,GOLDFINCH,DUNGAREES
MAYONNAISE,GOLDFISH,MAYONNAISE,MOBILE HOME,MAYONNAISE,GOLDFISH
MOBILE HOME,OLD HAT,MICROCHIPS,OLD HAT,PITCHFORK,MAYONNAISE
OLD HAT,PITCHFORK,MOBILE HOME,RICE PUDDING,RICE PUDDING,MICROCHIPS
RICE PUDDING,STRING BEANS,OLD HAT,STRING BEANS,STRING BEANS,STRING BEANS
STRING BEANS,TOMBSTONE,PITCHFORK,TOMBSTONE,TOMBSTONE,TOMBSTONE
VINTAGE CAR,VINTAGE CAR,TOMBSTONE,VINTAGE CAR,VINTAGE CAR,VINTAGE CAR

Answer

I observe that

in each race there are one or more letters occurring in exactly 8 of the 10 horses' names, and furthermore one way of picking them makes the letters spell out ATHENS. (Note that these numbers are consistent with the numbers of races specified in Thessaloniki and Miletus. I could quibble that "Αθήνα" is only five letters but I shan't.) It also happens that in the five races with known winners, the winner's name contains the letter in question. So far, so good. The first two races raised the hope that perhaps the winner was as simple as "the horse whose name has two of the relevant letter" but that isn't consistent with the other known results.

So we have to make sense of the following:

A MAYONNAISE  13 10  1/6
T TOMBSTONE   13 22  2/6
H CHEESECAKE  13 34  3/6
E CHEESECAKE  13 46  4/6
N VINTAGE CAR 13 58  5/6
S             14 10  6/6

and maybe also the following:

I WINDMILL     14 00  12/12
I CHILLI SAUCE 14 00  12/12 not this one

though

14:00 was the time of the end of that race. (So, metagaming this a bit, if we are supposed to be using this information then it must be the race number rather than its time that actually matters. The Athenian races run exactly every 12 minutes, so as far as those are concerned we get the same information from both.)

If

letter positions are relevant here, then WINDMILL has I in places 2 and 6, and CHILLI SAUCE has it in places 3 and 6, suggesting that it's the 2 rather than the 3 that's relevant. For that matter, all our Athenian winners seem to have the relevant letter rather early too. But the sequence 2,1,2,3,3 doesn't seem very promising.

Here is one possibility, which is at least consistent with everything we know:

the winner is the horse that has the specified letter earliest in its name.

In that case, the winner of the 14:10 will be

STRING BEANS.

quantum mechanics - Doubt in a certain equation of a research paper

In the given paper, I am stuck at equation (7). The equation that I am trying to solve for particle outside the well is : (1/g)(g'') + (1/(rg))*g' - (k_o)^2 = 0 where g = Radial wave function. r = Distance from origin.

The solution I arrived at is : AJ_0(ik_or) + BY_0(ik_or) where J ,Y are Bessel functions of first kind of order zero & second kind of order zero respectively.

My queries are : 1.) How can I reduce the solution in the form of equation (7) , i.e. , Psi = Ae^(-k_or) / (r^0.5) ?

2.) Am I solving the correct differential equation?

Link to research paper: abstract pdf

Relevant equations from this paper:

calculation puzzle - Labelling a graph with a partition of 100

Label the vertices of this graph with positive integers (repetitions allowed) whose sum is 100 in such a way that any pair of vertices are joined by an edge if (and only if) they have labels with a common divisor greater than 1 (i.e. they are not relatively prime).

Answer

My solution:

In text, that is represented as:

Starting from the lonely vertices and working clockwise: $1, 1, 42, 30, 7, 6, 5, 3, 3, 2$. For convenience, let us label these vertices A, B, C, D, E, F, G, H, I, J.

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Logic

• For every $K_n$ subgraph (set of n vertices in which all vertices are connected to each other), it's vertices must all share a unique prime factor that no other vertex outside this set has (Edit: OK this isn't always true, but it does help to use each prime to fill in each $K_n$).


• Notice the $K_5$ subgraph (CDFHI). To minimize the sum, assign a factor of $2$ to each.


• Then, each vertex of the nearby $K_4$ (CDFJ) can be assigned factor of $3$, and the two $K_2$s (DG, CE) factors of $5$, and $7$, in some order (It won't matter: C and D are both part of the $K_5$ and $K_4$).



• This gives a sum of $97$, but we cannot get to $100$ with the last two vertices.


• Trying another path, we give the $K_5$ vertices a factor of $3$ instead, and the $K_4$ a factor of $2$.


• This bumps the sum to $98$, where we can now assign $1$s to the lone remaining vertices.

integrable systems - R-matrix for spin chains

In algebraic Bethe ansatz procedure, one of the central objects is the R-matrix satisfying the Yang-Baxter equation, but all the papers/books give directly its expression without deriving it, so my question is How can i derive the R-matrix for XYZ/XXZ Heisenberg model?

Answer

This is essentially an answer to your questions R-matrix for spin chains, Elliptic R-matrix and Yang Baxter solution for XYZ model, $R$ matrix for XYZ spin chain, Algebraic Bethe Ansatz and $R$-matrices, which all basically ask the same question anyway.

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In short: to the best of my knowlegde, coming up with an R-matrix is an art, not a derivation. (Cf. the notion of a Lax pair in classical integrability.)

The quantum inverse-scattering method (QISM) was developed as a synthesis of the classical ISM, spin chains and lattice models. The best way to understand it is from this multi-topic point of view, rather than focussing just on spin chains. Faddeev's How Algebraic Bethe Ansatz works for integrable model [arXiv:hep-th/9605187] focusses on spin chains mostly, which makes several constructions --- such as the introduction of an auxiliary space --- appear somewhat ad hoc; at least it certainly felt so to me when I first read them. The vertex-model point of view makes these constructions much more natural; this is also why I organized my lecture notes A pedagogical introduction to quantum integrability, with a view towards theoretical high-energy physics, [arXiv:1501.06805] in the way I did.

Some more comments:

• 
  

  Once you know the Lax matrix (containing the vertex weights) of the six- or eight-vertex model you can solve for the R-matrix (solving the "RTT-relation" with $T=L$ for the case of one site), see e.g. Sections 9.6 and 10.4 in Baxter, Exactly solved models in statistical mechanics (or Appendix C in my lecture notes mentioned above).

  
  


• 
  

  Alternatively, you can look for solutions of the Yang--Baxter equation, and then interpret each R-matrix you get as a vertex model or see which spin chain it yields by computing the logarithmic derivative of the associated transfer matrix.

  
  


• 
  

  It might be instructive to read up on another example: Shastry's R-matrix for the Hubbard model. See e.g. Section 12.2 in Essler, Frahm, Göhmann, Klümper, Korepin, The one-dimensional Hubbard model [e-print].

  
  
  

What does negative energy imply?

I know energy comes in a number of forms and every form of energy is defined uniquely.

But if I had to give a broad definition of energy, I would define it as "ability of a system to perform work."

Using the above definition doesn't make any sense while trying to explain negative energy. So what does negative energy mean?

Answer

If positive work is "work done by the system" then negative work is just "work done on the system". The sign just tells if energy is added to the system or leaving the system.

photons - How does light travel?

How does light travel, does this not contradict the idea that going the speed of light stops time? Because if going the speed of light stops time and light goes the speed of light shouldn't it be dark?

electrical resistance - Does resistivity of material change under strain

As I am reading the document on physics of strain gauge, I encounter the following paragraph that I do not understand. How could resistivity change (delta rho) under strain/elongation? I always thought resistivity is a constant.