Poincaré Magazine - Issue 1

Feynman Problem

Feynman once posed this puzzle: the As all represent the same digit, and the dots each represent some digit, but not the same as the As.

What are the divisor and quotient?

The Answers at the Back of the Book more mathematical solution?

Ttion of the Poincaré Magazine.

–there Poincaré Magazine.content? ners, but there s lots of extensional content that s s mission, the aim of this

The Team

Joe Webb Ella Katz Editor-in-Chief Sej Patel Digital Media & Publicity Tom King Lead Writer Desync L Editing & Design

George Coote Milo Thomas

British Left Waffles on Falkland Islands

Disambiguating Language with Statistics

left was a past-

Fig. a The boring option

Fig. b Humanitarian aid

Ashumans, this ambiguity is rarely a humanChallenge (WSC) are an example of how it to refer to the trophy in the first sentence, but grammatical relations between the pronoun

brown suitcase because it was too big brown suitcase because it was too small

ple of part-ofin text-toresearch refuse fish is most commonly a noun, although it nn

probability of 3trigrams bump by on the the probability that it is able to estimate the joint probability of a list w1 wn tags t1 tn, written as P(w1 wn,t1 tn (Note that this mentions t0 t -1 in the i tn-1 represents the P(noun | noun, verb) is set to be A problem that arises is that the number of for calibrating the probabilities) using, in n

Complex Bashing

Eplex number o We will write XY to X Y

Proposition 2. Let A,B,C,D be distinct points. Then, AB CD if and only if the following holds:

Sometimes it is more convenient to write this as:

ing geometry problems with complex

Proposition 1. Let z and w be complex numbers. Then, the following are true:

(i) if and only if z is a real number.

(ii) if and only if z is purely imaginary.

(iii)

(iv) For any two complex numbers, conjugation is distributive over addition, subtraction, multiplication and division.

(v) We note that for | | =1, (iii) becomes ple for (iv) a,b,c,d

Proof. We want to show that AB_CD if a) as or

Proposition 3. The points A,B,C are collinear if and only if the following holds:

Proof.

Proposition 4. Let A,B,C,D be distinct points that lie on the unit circle. Then, ad+bc=0 if and only if

Proof. Assume AD lie

Combining these with Proposition 2, we

A,B,C are collinear, so we c

Proposition 5. Let A and B be distinct points on the unit circle. Let C be the foot of the perpendicular from an arbitrary point D to the line AB : Proof. Since lie on the unit circle,

Remark. As seen in Figure 1, point C

Foot of the Perpendicular

Problem. In ABC, the line passing through B is perpendicular to the line AC at E. This line meets the circumcircle of ABC at D. The foot of the perpendicular from D to the line BC is F. If O is the centre of the circumcircle of ABC, prove that OB is perpendicular to EF.

with elementary geometry to get a the line segment BC ABC. Proof.

We want to show E By sition 2, pressions for e f tuting for e f

Diagrams for the Problem

Noting that all lie on the unit circle, we can also compute Since ac + bd Remark.

tool in a geometer

Complex Numbers from A to ... Z –

All diagrams used in this article produced using GeoGebra

Pleiades

Jupiter & Moons

AstroSoc Photography

– Davina Nwachukwu & Baidandri Saha

Neutrinos

t be -antimatter CP (charge parity – for example, in the – but not to the So, where else can we search?

How Fluidity Can Save Physics

The Deep Underground Neutrino Experiment (DUNE) is one of the s most intense (anti)muon neutrino beam – each about

If neutrinos are seen to oscillate at a antineutrinos, this is a direct observation of CP violation!

is a bit of a misnomer –force isn a very neutrinos must be hitthe ultimate –s as if you re trying to hit a to see neutrino interactions, we mass changes as it propagates through space, causing the neutrino to change type

if neutrinos are seen to oscillate at then we

ploy -argon time-projection chambers technology which, as seen precisely capture neutrino interactions in –but we can Pandora

-algorithm approach where algorithms with wellthat we can obtain a measurement

Excitingly, my research is to recognition with respect to DUNE s

A charge-current electron neutrino event observed in the MicroBooNE far detector where the colour gradient corresponds to the energy deposition in the detector. Highly sophisticated pattern recognition software, such as Pandora, is required to extract the physics from these highly detailed and complex images.

into a recipe that a computer can –

Neutrino Cross Section Future in Prospects in Neutrino Physics,

The Next Generation of Large Neutrino Experiments

Understanding Our Place in the Universe

Twe exist? There are many ways

t see anti-planets orbiting anti-stars in antiout of about 1010–but this strange imbalance is one of the they apply to antimatter –t worry too much

imbalance we measure in experiments is way too small to account for the matter We perhaps there to measure a single process which can–say new – it has been twenty years since

We can t use the current experiments because we than a century to get to where we want – so -

DUNE is an experiment which points a high -power high-energy neutrino beam at a -ofthe- -argon time-That brings its own issues, not least, how to –pattern recognition is not easy, as we –– is going into the

The other experiment, Hyper-

or e certainly not as complex as the DUNE – Hyper-K can can –mature, well-operates in the same way as Super- -K has been running for twenty-exists (we use it for T2K) whereas DUNE t both experimental collaborations, but at s not bility of one of the experiments influencing

All right, so that s the sober, scientifically experiment is going to win? Well, the answer is Hyper-

Exploring Vertex Colourings by Sperner s Lemma

Definition 3

A simplicial triangulation is a type of gles in the triangulation cannot meet

Tgraphvertex, edge path -

Sperner s Lemma for Polygons

Sperner s Lemma for Polygons or the (mod 2) Sperner Lemma:

V = (v1,v2 vn) for in N , we represent the labels of the

Definition 1

}ee complete –xi xi+1 for some

Definition 2

A triangulation is a partition such tition of an original polygon is a set of polygons such that the union of the non-

Consider a graph with a boundary polygon P on n vertices. The graph is constructed by simplicially triangulating boundary polygons. Label the vertices either 0, 1, or 2 without restriction. Then, the number of complete adjacent pairs on the boundary subgraph has the same parity as the number of complete triangles. tition a polygon into triangles in a -green-blue triangles isgreen-

This graph includes six complete triangles, and six complete adjacent pairs on the boundary, which have the same parity. Here we associate the colours red, green and blue with the labels 0, 1 and 2 respectively.

Proof

are n triangles in the triangulation for some Ti for Let ei Ti

#complete trianglesTherefore, the parity of the total number of

#complete triangles Since the choice of (0,1) was arbitrary, we also

#complete triangles #complete triangles ing by three, since 3 Spernercutting

Su -winning paper, Rental armony: Sperner s Lemma in Fair Division. –

Knot or Not

1 will be through embeddings not

1. Knots apart, so what we knot theory, a branch of geometric 3 is a smoothly embedded2 through the subject, with a focus

3

is at least 4 crossings, then it must be the Let cate the change with blue next change with orange

Do you see how the orange arc is 1 isotopies

On the Total Curvature of Knots by John Milnor

blue orange the blue

Now we lift the orange arc up, twist,

2. Conway

We also re-position some of the arcs,

We are almost there, so the last few steps are left as an exercise for the

The simplest example of a nonone left--4 braids the action of permutations identity

K#L on a table, P

Q

see there is a path along the table from P to Q entirely within the

that knots do not cancel K L such that their sum K#L is picture5 of K#L follow-swallow surface, K L

K#L

K

Hence K symmetry of the argument, so is L K#L, where K is a nonguarantees that K#L is also noncomposite if it is the sum of two nonnot composite are the prime such as the trefoil, can only be

P Q P to Q which ignores L K

search for knot invariants to help crossing number is a simple example of an cal physics, particularly in quan-

Euler s Gem

Interactive Fourier Transform

FThe main purpose of this page is to present an form, so, here 3Blue1Brown, so please watch -time graphthe car in front of you the entire time, you can

re only thereof the general uncertainty principle 3Blue1Brown transform machine

diagrams used in this article produced using Desmos

Crossnumber

Across

Down

abc where a b, c, bc 10A (9A) A cube 7 a cube A multiple of 125 (9A)2

abc where c = b a 3 abcde, where a bc de is also a

(13A)2

A multiple of 13A but because it is so, because mathe-

Chess Puzzles

QED, as explained by Richard Feynman

century, three physicists ––with coherent theories to explain electron-

QED: The Strange Theory of Light and Matter. ––man tion my attempt at summarising parts of spoilers you concepts by Feynman himself, go man establishes is that, as far as the Q of QED –photomultiplier a machine which,

1) 2) 3) 4) 5) 7) e | 24

probabilities, let s see how we can We learn at school that the angle of Suppose we place a light any s -to-tail, for ll ror –– the -to-tail we get a approximate by saying that the angle the path of least time 1

This law it that the light is coming from the water that is famously in straight lines is similar to that of any

–familiar2

mirror where the arrows cancel out ly pointing in a similar

-light stopwatch will spin much more slowly than that of the bluefunctions the same as a sheet of inner outer the bubble s surface, each colour will

Focusing Lens

Let glass (your glasses) between the all the same – that is, the shortest So, there – so plenty of reason for you to go ing electronpart, so you can probably imagine what the rest of – Ella Katz

A Walk in the Quantum

Reinventing a World-Changing Algorithm

TM

now ocean of information that is the

The best analogy to this algorithm is At each step in the algorithm, each s time for some (We tween the websites the total amount of so

ll a has b c b c c to a d d to c in which case we utilise the adjacency matrix into the 4

Here, in A, rows represent to umns represent from 1s represent a example, since in our graph there is a b a (the tion A2,1

The Markov property: the future (Memorylessness) system is all that that our water-transferring system about anything else), allowing us to

degree matrix, which represent the we ll let deg(k deg(k

Again, formally, D where one can apply operations to it, such first, an initial state ll a is the first entry, b stochastic matrix you Elegantly, P AD –P x, it simulates the water-transfer P state, Px

c is twice as important as a d portant as b d to c patch c d, then d water d t be worthless

Google Matrix where N 1 is an N N E is a stochastic matrix for the What that amount of water from each remaining 1 –ll

what to set gorithm in The Anatomy of a Large-Scale Hypertextual Web Search Engine Brin tha they lly

Notice that we perform iterations until Gxn xn, the nth v we the meaning charac G – I

That to the branch that is Ergodic Theory We can guarantee that the stable chastic matrix is ergodic, meaning the

Irreducibility: is strongly connected ) Aperiodicity: The greatest common Ergodicity: irre ducib le aperiodic

re still using the Google probability 1– to teleport to any ular Python simulation of mine ga counts of a b c Which really isn -plus ics Drunkard s Walk, where the re so Transferring this to a more mathematicalD tion after the N

which physicists often treat the same as the root-meanDN

s Coin

gorithm for prime-factoring integers which you may or may not be familiar with rithms (not many though), with the next one on the popularity list being -

N ment for large N (in the case of one million boxes, it ics actually plays a role here, in is optimal ( bles picture output by a Python program s algorithm is possible, why information can ics, we can quantum coin s Coin is one that position of all of Poincaré let us not forget that Poincaré was a philosopher, mathematician, physicist, s a qubit

Coinmultiplication,

gate

Hadamard

why the wiggly -ness? Let single particle passing through both slitstern (explaining the wiggly-much larger than the classical whole experiment is a microcosm for communicate information much faster, s algorithm just

Quantum Simulation Emulation

guess we might as well try it, right? We

A directed graph representing a Web with eight websites

Exact (steady state) solution to the Web in Figure 1.

For reference, the green chart shows e g e is narrowly higher with g ephant in the room The uantum just one step s being s

Correspondence Principle, which says

Top: Classical walker Bottom: Quantum Walker

so, with the can s something the

– Joe Webb

Functional Analysis Previews

MA106 Linear Algebra1 means for a subset of Rn MA260 Norms, Metrics and Topologies or MA222 Metric Spaces

Functional analysis is the area of function spaces

topological vector spaces topological topology is yet, but really, all it means is that we open Most importantly, this means – it may be – some every point in the vector space at the same time practice, most function spaces (sets of normed

vector spaces notion of which are far closer Let V R (you V C C norm,

As a function satisfying the following

––a factor of a, we want its length to also increase by a factor of

v = 0 – we for each

w| as the v w n

tance in n semi-norm, which normed vector space open, continuous convergent ural ways open ball of centre vinV and radius r

We say that UsubV is open u U tor with |y

space x NinN such that space Cauchy such that |xnWe say that spaceCMA131 Analysis II2 f is C -

1

forms one of the archetypal spaces (it, many examples) you

The other type of space we will now inner product space Euclidean inner product from Aangle between two nonx, y n, by n V, inner product if on C again, in an analogous way to n

Conjugate symmetry –, this is simply symmetry, ––if

prescribe a lotHilbert spaces –s

2. 2.1

–completeness recurrent from MA260 Norms, Metrics and Topologies MA259 Multivariable Calculus , but here, you ll start to see the connection You ll also see some intersections with MA3H3 Set Theory

the only interesting application you ll see –Hilbert spaces ll

more generally to Banach spaces the usual concept of a basis they exist, but often it is extremely C continuous function , (most of which are continuous function can be written Fourier series

tool of Zorn s lemma3 lent to the Axiom of Choice, if you sence of the argument is to show that s lemma isn t

linear functionals These are continuous linear functions

scalars, usually or fmapsint is a linear functional on C

Riesz Repre-

sentation Theorem spectral theory –ory, with the particular example of Sturm–Liouville problems

theorem

Krein–Millman Theorem

The next part uses the Baire Category Theorem from MA260 Norms, Metrics and Topologies

tween Banach spaces that we weren t

MA3G7 Functional Analysis I, along with MA359 Measure Theory ally essential for pursuing any further

Among these are the Closed Graph Theorem, which states that the graph any continuous linear transformation between Banach

2.3 Fourth Year

MA433

Fourier Analysis MA4J0 Advanced Real Analysis

2.2

MA3G8 Functional Analysis II of the larger theorems in functional see is the Hahn–Banach Theorem, ar) function with a particular property by constructing this function only on a guaranteeing that this function must –showing part of the geometric Hahn–Banach

metric functions –as approximating a function as the sum Heisenberg s uncertainty principle

MA3G1 Theory of ), which a strictly formal meaning to objects such as the Dirac– function

Quantum Mechanics

MA4A7

Equations 3. 3.1 Hahn-Banach Theorem

Hahn–

– so the Banach limit of a nonis non-

with |y-

a there are two main it is not true that the Banach limit of a –limit –Banach is not we only generalisation of the limit in Functional weak convergence, where Banach–Alaoglu theorem ably be way too technical to un erstan –– George Coote

The Answers at the Back of the Book

"""Solves and prints the solution to Feynman's long division problem""" __author__ = "Joe Webb" __contact__ = "joe@wephy.com"

from itertools import product

def cond_test(num, A, state):

"""Tests whether the inputted number matches a particular state. A state is a string of 0s and 1s where 0 is a dot and 1 is an A""" return ''.join(str(int(i == A)) for i in str(num)) == state

def solve():

for divisor in range(100, 1_000):

# Test for a valid divisor, and simultaneously initialise A if not cond_test(divisor, A := str(divisor)[1], "010"): continue

# Let the quotient be xyAz, and then for each valid x for x in (digits := set(range(10)) - {int(A)}):

# Test quotient first digit * divisor = line 2 (and initialise) if not cond_test(l2 := x * divisor * 1_000, A, "0011000"): continue

# For remaining possibilities of quotient for y, z in product(digits, digits): # Cartesian product dividend = divisor * (quotient := int(f"{x}{y}{A}{z}"))

l4 = y * divisor * 100 # Initialise line 4 number l6 = int(A) * divisor * 10 # Initialise line 6 number

# Test all lines for validity if all(( # d represents the remainder throughout process cond_test(d := dividend, A, "0000100"), # Line 1 cond_test(d := d - l2, A, "000100"), # Line 3 cond_test(l4, A, "00100"), # Line 4 cond_test(d := d - l4, A, "00000"), # Line 5 cond_test(l6, A, "01000"), # Line 6 cond_test(d := d - l6, A, "0000") # Line 7 )): return (divisor, quotient) if

colours?

Primes are integers, so n Without loss of generality, let mial Q But |c – a a – b mn –obtain mn

G p – then (G, p p p k = |a H a,a 2 a k –p) forms a subgroup of G k s theorem, k G| = p –so p – nk n

Thus, ap –a nk (a k)n 1n p m m

But, by the well -

n + 1, we can n

But now, we n n Double again to reach 4n n 2n n n + 1, 4n + 2, 4n n + 2, 4n + 4, 4n n + 3, 2n n -

Particle Pals #1

– Milo Thomas

You – but there s – an article, some photos, –