Maths on Trial

Last week I had the pleasure to attend a talk by Leila Schneps on the mathematics of crime. A self-declared pure mathematician, Schneps recently became involved in studying the mathematics in criminal trials. Rather than focusing on the mathematics of forensic data, such as DNA, the talk was on the use of Bayesian networks in crime evidence analysis.

One of the most challenging tasks for juries in criminal trials is to weigh the impact of different pieces of evidence which may be dependent on each other. In fact, studies have shown that there are several fallacies that jury members fall into on a regular basis, such as overestimating the weight of a single piece of DNA evidence. This is because most people assume that the probability of an event A happening given that B happens [P(A|B)] is equal to an event B happening, given that A happens [P(B|A)]. However, this is NOT true: the two are connected by Bayes’ Rule.

For example, a forensic specialist may say that a piece of DNA is found in 1 in every 1000 people. The jury will take this to mean the the suspect must be guilty as he is that person out of 1000 (the chances of it being anyone else is so low). However, this is critically not true, as explained above.

Thus, Bayesian networks are a powerful tool to assess the weight of different kinds of evidence, taking into account their dependencies on one another, and what effect they have on the guilt of the suspect.

What are Bayesian Networks?

Bayesian Networks are a type of statistical model,  which organise a body of knowledge in any given area, in this case evidence of a criminal trial, by mapping out cause-and-effect relationships and encoding them with numbers that represent the extent to which one variable is likely to affect the other. These networks are named after British mathematician Reverend Thomas Bayes, due to their reliance on Bayes’ formula. This rule can be extended to multiple variables and multiple states, allowing complicated probabilities to be calculated.

Example

To illustrate an easy example, consider this Bayesian Network:

Let us say that the weather can only have three states: sunny, cloudy or rainy, that the grass can only be wet or dry, and that the sprinkler can be on or off. The arrows, which represent dependence, have been drawn in that way because if it rainy, then the lawn will be wet, but if it is sunny for a long time then this will cause us to turn on the sprinklers, and hence the lawn will also be wet.

By imputing probabilities into this network that reflect the reality of real weather, lawn and sprinkler-use behaviour, it can be made to answer questions like:

“If the lawn is wet, what are the chances it was caused by rain or by the sprinkler?”

“If the chance of rain increases, how does that affect my having to budget time for watering the lawn?”

I51G8WI2nsNL._SX334_BO1,204,203,200_.jpgn her presentation, Leila Schneps talked briefly about a book she had released entitled ‘Math on Trial’, which describes 10 trials spanning the 19th century to today, in which mathematical arguments were used (and greatly misused) as evidence. The cases discussed include “Sally Clark, who was accused of murdering her children by a doctor with a faulty sense of calculation; of nineteenth-century tycoon Hetty Green, whose dispute over her aunt’s will became a signal case in the forensic use of mathematics; and of the case of Amanda Knox, in which a judge’s misunderstanding of probability led him to discount critical evidence – which might have kept her in jail.”

After hearing Schneps transmit her passion and excitement, I am fascinated with this subject and can’t wait to get my hands on this book to learn more! M x

Benford’s Law

Benford’s Law is names after the American physicist Frank Benford who described it in 1938, although it had been previously mentioned by Simon Newcomb in 1881.

Benford’s Law states that in “naturally occurring collections of numbers” the leading significant digit is more likely to be small. For example, in sets of numbers which obey this law, the number 1 appears as the first significant digit about 30% of the time, which is much greater than if the digits were distributed uniformly: 11.1%.

In mathematics, a set of numbers satisfies this law if the leading digit, d, occurs with a probability:

P(d)=\log _{10}(d+1)-\log _{10}(d)=\log _{10}\left({\frac {d+1}{d}}\right)=\log _{10}\left(1+{\frac {1}{d}}\right).

Hence, if d = 1, then P(1) = log 2 = 30.1..%

The leading digits have the following distribution in Benford’s law:

Screen Shot 2016-09-27 at 2.57.40 PM.png
Source: Wikipedia
BenfordsLaw
Source: Wolfram MathWorld

As P(d) is proportional to the space between d and d+1 on a logarithmic scale, the mantissae of the logarithms of the numbers are uniformly and randomly distributed.

Applications

Benford’s law has found applications in a big variety of data sets, such as stock prices, house prices, death rates and mathematical constants.

File:Benford-physical.svg
Frequency of first significant digit of physical constants plotted against Benford’s Law | Source: Wikipedia

Due to this, fraud can be found applying Benford’s law to data sets. This is because if a person is trying to fabricate ‘random’ values to try to not appear suspicious, they will probably select numbers such that the initial digits are uniformly distributed. However, as explained above, this is completely wrong! In fact, this application of Benford’s law is so powerful that there is an “industry specialising in forensic accounting and auditing which uses these phenomena to look for inconsistencies in data.”

Datagenetics.com describes how:

“In 1993, in State of Arizona v. Wayne James Nelson (CV92-18841), the accused was found guilty of trying to defraud the state of nearly $2 million, by diverting funds to a bogus vendor.

The perpetrator selected payments with the intention of making them appear random: None of the check amounts was duplicated, there were no round-numbers, and all the values included dollars and cents amounts. However, as Benford’s Law is esoterically counterintuitive, he did not realize that his seemingly random looking selections were far from random.”

Sources: 1 | 2 | 3 | 4

 

After writing this I’ve realised that I touched on this law (in less detail) in a previous post during my Christmas series! M x

Choosing a Password

It has been argued that password systems are not a good way to authenticate. This is due to the fact that either they’re difficult to remember or they’re easy to remember, but therefore also easy to crack. So how do we choose a good password? XKCD posted this image suggesting a strategy for creating a password:

password_strength.png

This method is trying to eradicate to age old way of creating passwords that are, in fact, almost impossible for us to remember but relatively easy for a computer to crack.

The password suggested by XKCD (although now not a good password because everyone knows about it!) is practically resistant to the brute force approach, because, although it is composed of only lowercase letters, it is too long. Therefore, the method used to break this password would be the dictionary attack method. However, these words would probably not come up in a dictionary together as they aren’t usually associated to one another.

But what happens when this method (stringing together four words) becomes common practice? A method to combat this might be to look at the top 10,000 english words and try different combinations of these words until the password is found. Therefore, it is safest to always assume that the password cracker knows the method that you are using and so we must choose at least one uncommon word that is hard to guess, such as mirth, to include in the password. This will make it extremely difficult to crack.

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NEWS: New Twin Primes Found

PrimeGrid is a collaborative website with the aim to search for prime numbers. It is similar to GIMPS, which only searches for Mersenne Primes specifically. It works by allowing anyone to download their software and donate their “unused CPU time” to search for primes. PrimeGrid is responsible for many of the recent prime numbers that have been found, which includes “several in the last few months which rank in the top 160 largest known primes“.

On the 14th of September they announced their most recent discovery made by the user Tom Greer who discovered a new pair of twin primes. (Note that twin primes are prime numbers that differ by two.)

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The primes are “388,342 digits long, eclipsing the previous record of 200,700 digits”. These primes have been entered in the database for The Largest Known Primes, which is maintained by Chris Caldwell and are currently ranked 1st for twins and each are ranked 4180th overall.

Source

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MOVIE REVIEW: The Man Who Knew Infinity

MV5BMTU3Njg4MDM3OV5BMl5BanBnXkFtZTgwMjE5ODM3ODE@._V1_SY1000_CR0,0,674,1000_AL_.jpg‘The Man Who Knew Infinity’, based off of a book by Robert Kanigel with the same title, depicts the life of Srinivasa Ramanujan, a revolutionary mathematician who made extraordinary contributions to pure mathematics, specifically in mathematical analysis, number theory, infinite series, and continued fractions. The movie describes the story of how his work was recognised by G. H. Hardy, at the time fellow at Trinity College in Cambridge, resulting in Ramanujan moving to the UK to work and collaborate with Hardy and Littlewood, a colleague of Hardy. It then tells the story of his life and his difficulty to integrate in England due to the vast cultural divide between the British and the Indians. For example Ramanujan’s vegetarianism, due to religious reasons, was not understood resulting in him not being able to eat in the college cafeteria with all the other residents. His devout religiousness is displayed by the following statement made by Ramanujan, which is depicted in the movie:

“An equation for me has no meaning unless it represents a thought of God.”

The main mathematical discovery covered in the book was the partition formula. I found that the idea of partitions was explained simply, and therefore made the subject approachable to the general audience. However, the importance of this formula should have been highlighted more clearly; it was swallowed up by the dramatic ending of his illness resulting in his return to India after being appointed Fellow of Trinity College.

Overall, I really enjoyed the movie. Yes, it concentrated heavily on his personal life, rather than his mathematics (which as a mathematician isn’t exactly ideal) but we must remember that this is a movie, intended to entertain the general public, rather than a documentary. In terms of entertainment, I found it a highly enjoyable film to watch, and serves the purpose to popularise important figures in mathematics, which I think is really important in order to captivate younger minds and perhaps lead them to studying STEM subjects.

Have you watched the movie? If so, let me know what you thought! M x

 

Chladni Figures

Ernst Chladni was a German physicist who is sometimes labelled as the ‘father of acoustics‘. His work in this area includes research on vibrating plates and the calculation of the speed of sound for different gases.

One of Chladni’s greatest achievements was his invention of a technique to show the various methods of vibration of a rigid surface, such as a plate, which he detailed  in his book Entdeckungen über die Theorie des Klanges (“Discoveries in the Theory of Sound”) in 1787. His technique entailed:

drawing a bow over a piece of metal whose surface was lightly covered with sand. The plate was bowed until it reached resonance, when the vibration causes the sand to move and concentrate along the nodal lines where the surface is still, outlining the nodal lines.”

The patterns that emerge are beautiful and are now known as Chladni figures, although Chladni was building on experiments and observations by Robert Hooke in 1680 on vibrating glass plates.

Chladni also created a formula that successfully predicted the patterns found on vibrating circular plates.

Chladni’s discovery was extremely important as it inspired many of the acoustic researchers who later extended his work.

Once these patterns were well understood, they began to have many practical uses, for example violin makers use “Chladni figures to provide feedback as they shape the critical front and back plates of the instrument’s resonance box“.

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Isoperimetric inequality

The isoperimetric inequality dates back to the olden days, where there was a problem that asked: “Among all closed curves in the plane of fixed perimeter, which curve (if any) maximises the area of its enclosed region?” This is equivalent to asking: “Among all closed curves in the plane enclosing a fixed area, which curve (if any) minimises the perimeter?” The solution to this problem is expressed in the form of an inequality: the isoperimetric inequality.

The isoperimetric inequality is a geometric inequality that involves the square of the circumference of a closed curve in a plane and the area that it encloses in that plane. This inequality states that

4\pi A\leq L^{2},

where L is the length of a closed curve and A is the area of the planar region that it encloses.

Many proofs have been published for this inequality. For example, in 1938 E. Schmidt produced an elegant proof based on “the comparison of a smooth simple closed curve with an appropriate circle”.

An extension of this inequality is the isoperimetric quotient, Q, of a closed curve which is defined in the following way:

Q={\frac  {4\pi A}{L^{2}}}

Hence, it is the ratio of its area and that of a circle with the same perimeter.

The inequality highlights how  Q ≤ 1 and for a circle Q = 1.

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MATHS BITE: The Bridges of Königsberg

In 1735, the city of Königsberg in Prussia (which is now Kaliningrad, Russia) was divided into four sections by the Pregel River. These four sections where connected by seven bridges.

Source: Wikipedia

The Königsberg bridge problem asked whether there was a walk through the city that would cross each bridge once and only once.

In 1736 Euler proved that this walk was not possible, achieving this by inventing a kind of diagram called a network, that is made up of vertices (dots where lines meet) and arcs (lines). His solution was presented to the St. Petersburg Academy in August 1735, and was published in the journal Commentarii academiae scientiarum Petropolitanae in 1741.

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Source: Wolfram MathWorld

This was a novel idea; Euler discovered that the physical positions of the sections do not matter, but rather it’s the connections made by the bridges that matters. Hence, by having this intuition Euler opened up a new branch of mathematics: graph theory. Euler describes how:

“. . . this type of solution bears little relationship to mathematics, and I do not understand why you expect a mathematician to produce it, rather than anyone else, for the solution is based on reason alone, and its discovery does not depend on any mathematical principle.”

So why is such a walk not possible?

Euler showed that the possibility of this walk existing – called a Eulerian path – is dependant on the  degrees of the nodes. The degree of a node is how many edges touch it. He found that the necessary condition for this walk to exist is that should have exactly 0 or 2 nodes with an odd degree.

As all four of the nodes in the network of Königsberg have an odd degree, the walk does not exist.

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Spirals: 3D

As a continuation from my previous post on 2D spirals, I decided to discuss 3-dimensional spirals. With 2-dimensional spirals there are only 2 variables (r, for radius, and θ), whereas there is a third variable in the description of 3D spirals (h for height). Thus, all 2D spirals can be extended to the third dimension by adding this third variable in the z-axis. Below I will detail 3 different types of 3D spiral.

Helix

A helix can be seen as a type of spiral as it is a curve in 3 dimensional space. There are many different types of helices, for example a conic helix which can be described as a 3D spiral on a conic surface.

Helices can be either right-handed or left-handed, meaning that helices form ‘enantiomers’.

Two_Types_of_Helix.svg.png

Although there are many different formulae that produce a type of helix, the simplest parametric equations to produce one are:

x(t) = \cos(t),\,y(t) = \sin(t),\,z(t) = t.\,

Rising_circular.gif
Source: Wikipedia

Vortex

A vortex is a phenomenon in fluid dynamics where the flow of the fluid “is rotating around an axis line, which may be straight or curved“.

The shape produced by a vortex can be described as a spiral due to its curved shape.

vortex_schematic.gif

Rhumb Line

A Rhumb Line, or a loxodrome, is “is an arc crossing all meridians of longitude at the same angle, i.e. a path with constant bearing as measured relative to true or magnetic north.”

Source: Wikipedia

A rhumb line has an infinite number of revolutions as the separation between the lines decreases as it approaches the north or south pole (i.e. radius decreases); a rhumb line always spirals toward one of the pole. This is unique from the Archimedean spiral where the separation between the lines remains constant.

KUGSPI-9_Loxodrome.gif
Rhumb Line

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