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FTRP STEM

The implications of Prime numbers and the Riemann hypothesis on Asymmetric Cryptography


This essay was written by lower-sixth former Ishan Nathan, and shortlisted for the 2020 Fifth Form Transitional Research Project. The following provides a short abstract to the full essay, which can be found at the bottom.

Estimated read time of abstract: 2 minutes
Estimated read time of essay: 15 minutes

The academic study of prime numbers has been of mathematical interest for centuries and over time remarkable progress has been made in understanding the unique properties and patterns of these numbers. Over the last fifty years, the discovery of mathematical models has aided the progression of computer science. Whilst encryption, previously used for communication in the wars, has now been adopted into quotidian life. Mathematicians have discovered new methods for the secure transmission of information and have augmented them by introducing new messaging platforms using encryption algorithms based on prime numbers.  

It is widely accepted that Prime numbers are important in the field of number theory as they act as the “atoms of arithmetic”. They are defined as natural numbers greater than one, that are only divisible by exactly two numbers, one and itself. Mathematicians first studied primes explicitly in 300BC in Ancient Greek Mathematics, where Euclid proved that there was an infinitude of primes. Since then the understanding of primes has developed and the characteristics of primes enable it to have profound applications in security codes, blockchain analysis, cicada’s cycles, and Cryptography. Nevertheless, mathematicians do not understand the primes fully, due to their enigmatic behaviour whereby they appear to act randomly despite having some aspects of their behaviour which are predictable.  

Prime numbers and their application to modern-day life is not always apparent, as is their properties and patterns. Yet prime numbers play a fundamental part in our lives and act as a cornerstone for both: day to day messaging on encrypted platforms such as WhatsApp, and for consumers’ online e-commerce activities. The emergence of the internet has led to an increase in the number of online transactions taking place all over the internet on sites like eBay and Amazon, and modern-day cryptographic methods establish a mechanism for a secure form of communication. 

The RSA algorithm relies upon the quick speed for performing operations to determine large primes, and the computer-intensive reverse process in factorising large integers, in turn assuring the security of public-key cryptography. It is this high level of encryption that ensures the world of e-commerce to function protecting our sensitive information such as credit cards from the global market place. Yet what if there was a way to overcome this? 

The Riemann Hypothesis is widely accepted as one of the biggest mathematical unproven conjectures of our millennium. It is argued that the Riemann Hypothesis predicts the distribution of the primes and their unpredictable behaviour better than any other theorem. An abstract proof of the Riemann Hypothesis will undoubtedly enhance our understanding of primes and thus could lead to vulnerabilities within asymmetric cryptography. However, primes are special and they are like no other group of numbers. Despite mathematicians limited understanding of these numbers, a secure online communication network across the world has been created; just imagine the possibilities that could unravel when understanding the true enigmatic behaviour of prime numbers. 

To view Ishan’s full article, follow this link below.

“The implications of Prime numbers and the Riemann hypothesis on Asymmetric Cryptography” By Ishan Nathan
Categories
FTRP Law & Politics Social Sciences

Why did ‘Workington Man’ vote Conservative? An analysis of the factors contributing to the fall of the ‘Red Wall’


This essay was written by lower-sixth former Dominic Stagg, and shortlisted for the 2020 Fifth Form Transitional Research Project. The following provides a short abstract to the full essay, which can be found at the bottom.

Estimated read time of abstract: 1 minute
Estimated read time of essay: 16 minutes

Throughout modern British electoral history, the useof political stereotypes to identify potentially decisive voters has become increasingly common within the mainstream media, and influential in dictating the way in which political parties’campaign. In the 2019 UK General Election, this trend manifested itself as ‘Workington Man’-an older, white man who traditionally supported the Labour Party but voted ‘Leave’ in the EU Referendum in 2016. The Conservative Party’s substantial victory was characterised by the supposed fall of the Labour Party’s ‘Red Wall’, a term used to describe traditionallyLabour-supporting constituencies based in North Wales, Northern England,and the Midlands, regions in which ‘Workington Man’ is concentrated. This would therefore suggestthat‘Workington Man’was extremely significant in shaping the outcome of the election.

However, whilst the significance of ‘Workington Man’ was undoubtable, the reasons that caused this momentous shift from Labour to Conservative remain less obvious. It is true that the 2019 election was dominated and polarised by Brexit, yet the root causes of the breaking down of inherent social, political, and economic barriers between ‘Workington Man’ and the Conservatives would appear to be more complex and deep-rooted. Therefore, in this essay, I sought to gain a greater insight into the various short-term and long-term factors that contributed to ‘Workington Man’s’ disenfranchisement from the Labour Party, that ultimately caused the majority of such an electorate to vote Conservative in 2019. These are divided into three similarly important sub-sections: Brexit, in both its exaggeration of growing social polarisation and, more profoundly, in the way it offered political empowerment, as well as cultural issues and economic issues. I concluded that 2019 marked the culmination of the interaction between these separate but overlapping factors, a point in which ‘Workington Man’ and the Conservatives became politically aligned. Furthermore, I argued that ‘Workington Man’s’ reasons for voting Conservative were rooted not just in the Conservative’s appeal, amore recent phenomenon arising from both Brexit and the Conservative’s cultural shift, but also in a feeling of marginalisation from the Labour Party that can be traced much further back.

To view Dominic’s full article, follow this link below.

“Why did ‘Workington Man’ vote Conservative? An analysis of the factors which contributed to the fall of the ‘Red Wall’ in the 2019 UK General Election” By Dominic Stagg
Categories
Arts & Humanities FTRP Geography Law & Politics Social Sciences

Why did the UK withdraw from Singapore and Malaysia and what were the consequences for the region?

Gardens by the Bay is a nature park in Singapore


This essay was written by lower-sixth former Alexander Downey, and shortlisted for the 2020 Fifth Form Transitional Research Project. The following provides a short abstract to the full essay, which can be found at the bottom.

Estimated read time of abstract: 2 minutes
Estimated read time of essay: 12 minutes

When Harold Wilson took over as Prime Minister in 1964 from Alec Douglas-Home, he inherited a country riddled with financial difficulties. Macmillan’s supposed “Age of Affluence” left a remarkable balance of payments deficit of £400 million. The economic downturn was the trigger for calls for a review and a change in the way money was being spent and invested in foreign affairs and the military when the number of Brits at home who needed financial support grew.

Ever since the end of the Second World War Britain’s influence on the world stage had been in decline along with her empire. This led to Wilson taking the decision to continue with the post-war consensus idea of focusing on becoming a political power in Europe and adapting a role there rather than a worldwide role. Part of this meant reducing military commitments around the world, the term “East of Suez” was coined to refer to all British military bases and territories in the Eastern hemisphere, this included Malaysia and Singapore.

This region had a rather unique political situation due to the unique way in which Malaysia and Singapore were linked as well as Malaysia’s internal divisions. Following the decision to give independence to Malaya, Singapore, Sarawak, and Borneo forming the Federation of Malaysia, the internal politics of this new country were chaotic to say the least. The Malaysian-Chinese population were discriminated against by the Islamophilic regime leading to violent protests, Britain then feared they would be drawn into a Vietnam style conflict, especially when Singapore separated itself from the Federation forming its own sovereign state. The political tensions along with Britain’s changing international role were important factors in the decision to withdraw from Malaysia and Singapore. 

However, one can argue that the role of pressure groups in the UK were more important as they emphasised Britain’s changing role and the dangerous political atmosphere of the region at the time. Whilst the importance of the pressure groups is often overlooked, the main point they pushed was the economic situation and the cost of having military bases in the Eastern hemisphere, Wilson was aware of this, so the importance of the pressure groups was much less than the economic situation at home at the time. The consequences for the region have been, in the long run, intrinsically positive. Malaysia’s economy in particular initially suffered an economic downturn but both countries are now amongst the most powerful Southeast Asian economies and continue to grow. 

To view Alexander’s full article, follow this link below.

Why did the UK withdraw from Singapore and Malaysia in 1971 and what were the consequences for the region? By Alexander Downey
Categories
Arts & Humanities FTRP History Law & Politics

To what extent does Mao Zedong deserve his reputation as one of history’s most notorious dictators?


This essay was written by lower-sixth former Austin Humphrey, and shortlisted for the 2020 Fifth Form Transitional Research Project. The following provides a short abstract to the full essay, which can be found at the bottom.

Estimated read time of abstract: 2 minutes
Estimated read time of essay: 11 minutes

Mao Zedong was Chairman of the People’s Republic of China from 1949 to his death in 1976. Mao was a Communist revolutionary, described as having an ‘emphatic aura’ and ‘exuding overwhelming power’. He is known globally as an infamous killer, responsible for the deaths of millions, but can he be compared to the likes of Hitler or Stalin? Another question to consider is what makes a notorious dictator, and due to these factors we can determine whether or not Mao deserves his reputation. 

Firstly, we can examine the death toll of Mao. In 1958, Mao’s ‘Great leap forward’ killed approximately forty million people, by forcing peasants to stop work on farms and begin production of steel. Mao took over all agriculture in China, with no farming experience. He demanded that farmers kill sparrows, to stop them eating the crops. However, the sparrows were only eating pests, thus improving crop yields. Hence Mao’s arrogance and ignorance caused one of the most devastating famines in history.  

To compare Mao’s numbers here, we should look at Pol Pot, the former dictator of Cambodia. Pol Pot killed only two million people, which seems inconsequential compared to Mao. However with perspective, Pol Pot is responsible for the death of a quarter of his while country, while Mao only 6%. Therefore one reason for the extraordinarily high number of deaths is just because China’s population was so much greater than other nations’: 670million.  

Just examining the number of deaths may not be as important as analysing the intent behind them. Whilst the number of people Mao killed was almost double what Hitler and Stalin killed together, his primary intention was to increase China’s industry to make it a world superpower. This highlights Mao’s noble intentions whilst in power. 

On the other hand most would agree that Adolf Hitler’s intentions were horrific. When he murdered eleven million people in death camps, he singled out groups in society as second class humans, then purposely slaughtered them. Therefore, as death by ill-judgement is not the same as death by ill-intent, Mao doesn’t deserve to be compared with the likes of Hitler, who set out with the aim of genocide.  

Mao wasn’t completely innocent of malicious aspirations. In 1956 he launched ‘The Hundred Flowers Campaign’, which was an opportunity for everyone to present ideas on how to improve China. After a few months, the campaign stopped and anyone who criticised the government was persecuted. Many people, including former deputy chief of MI6 and British diplomat in China at that time: Sir Gerry Warner, believe that ‘The campaign was a deliberate attempt to flush out those who opposed Mao and Communism’. 

In conclusion, there are many ways to judge notoriety, the most important of which I believe is intent. Therefore due to Mao’s mass number of killings, but honourable intention on the whole, he deserves his reputation as one of history’s most notorious dictators, but falls short of the notoriety of those who intended to harm others. 

To view Austin’s full article, follow this link below.

“To what extent does Mao Zedong deserve his reputation as one of history’s most notorious dictators?” By Austin Humphrey
Categories
Arts & Humanities FTRP Geography

How can the Lego Group adapt to a world more focused on environmental sustainability?

This essay was written by lower-sixth former Sam Rooke, and shortlisted for the 2020 Fifth Form Transitional Research Project. The following provides a short abstract to the full essay, which can be found at the bottom.

Estimated read time of abstract: 1 minute
Estimated read time of essay: 11 minutes

The world is becoming increasingly aware of the impact that plastic has on the environment and its role in climate change. Therefore companies like Lego who’s brand relies on ABS Plastic could face major issues in the future if they do not find an environmentally sustainable method of operating. This research project investigates the impact of The Lego Group on the environment and its aims to become more economically sustainable. In order to do this it was important to understand the meaning and significance of environmental sustainability; why The Lego Group weren’t currently achieving this; how successful previous initiatives have been; and finally how achievable their future goals are.

To view Sam’s full article, follow this link below.

“How can the Lego Group adapt to a world more focused on
environmental sustainability?” By Sam Rooke
Categories
Arts & Humanities FTRP History

What was the most important initiative carried out in response to problems posed by the Crimean War in improving healthcare for infantry soldiers?

This essay was written by lower-sixth former Sebastian Evans, and shortlisted for the 2020 Fifth Form Transitional Research Project. The following provides a short abstract to the full essay, which can be found at the bottom.

Estimated read time of abstract: 1 minute
Estimated read time of essay: 9 minutes

This FTRP is about the main medical advances in healthcare for British infantry units during the Crimean war, which lasted from 1853 – 1856, and how these advances helped the British infantry in future wars fought by the British empire. The document talks about advances and improvements in hospitals, improvements in the soldiers’ diets, the introduction of ambulance trains and ambulance ships, and improving the medical staff working to save wounded soldiers. The document looks at not only the achievements of Florence Nightingale, but also other factors and initiatives that helped improve the chances of survival for sick or wounded infantry during the war. While all the initiatives mentioned were important in helping the infantry and improving healthcare, the single biggest one was undoubtedly the improvements to hospitals during the course of the war.

To view Sebastian’s full article, follow this link below.

“What was the most important initiative carried out in response to problems posed by the Crimean War in improving healthcare for infantry soldiers?” By Sebastian Evans
Categories
Economics FTRP History STEM

How Gambling in the 17th Century has shaped insurance markets in the 21st century

This essay was written by lower-sixth former Moog Clyde, and shortlisted for the 2020 Fifth Form Transitional Research Project. The following provides a short abstract to the full essay, which can be found at the bottom.

Estimated read time of abstract: 1 minute
Estimated read time of essay: 11 minutes

In 1654, the Chevalier de Mere, a French nobleman, posed the notorious ‘Problem of the Points’ to Blaise Pascal, an esteemed mathematician. The Problem of the Points concerned a game of chance containing two players with equal chances of winning any given round, and posed the question of how to split the stakes if one gambler has to leave the game prematurely. Despite several attempts, finding a definitive solution stumped even the greatest minds of the previous two hundred years, most notably Luca Pacioli (the ‘Father of Accounting’ ) in 1494 and Niccolò Tartaglia (solver of cubic equations and the first to apply maths to the paths of cannonballs, otherwise known as ballistics) in 1556. Even the great Galileo failed to discover a reasonable solution to the problem. Pascal was determined to find a logical and fair solution, and thus reached out to Pierre de Fermat, a brilliant mathematician himself. In their resulting correspondence, the pair developed the first explicit reasoning about what today is known as ‘expected value’ and laid the groundwork of probability, earning them both joint title of ‘the Fathers of Probability.’

Although it is easy to underplay the significance of this breakthrough as merely a clever, tidy solution, to appease opposing gamblers, in reality, it was truly revolutionary. It is difficult to understate how vast and significant the cognitive shift across Europe that occurred following this solution was. The notion that you can hang numbers into the future was alien to mathematicians merely years before this solution was proposed. Soon, others began to see the possibilities that this concept generated.

Within three years Christiaan Huygens adapted Fermat’s theory into a coherent pamphlet entitled ‘De Ratiociniis in ludo aleae,’ which was used as the standard text on probability for the next 50 years. Huygens attributed his developments to “some of the best mathematicians of France” (i.e. Pascal and Fermat). This text spread like wildfire among the academic community as it was evident that the new science of probability had the potential to transform the world. In the next few years, Huygens’ text was ripped out of the context of gambling and thrust into several aspects of life, including law and maths. In particular it was applied to a very different, brand new data set: mortality tables. Almost immediately, by using specific intricate data, insurance shifted from a form of blind gambling, based on hunches and guessing, to a remarkably accurate science.

It now is clear that this rapid chain reaction of discovery underpins all notions of mathematical ‘expected value’ and insurance came not from savvy merchants but from avid gamblers, eager to improve their craft.

To view Moog’s full article, follow this link below.

“How Gambling in the 17th Century has shaped insurance markets in the 21st century” By Moog Clyde
Categories
Arts & Humanities Classics English FTRP Poetry

To what extent can ‘The Tale of the Heike’ and ‘The Iliad’ be considered similar poems?

This essay was written by lower-sixth former Mattie Sutton, and shortlisted for the 2020 Fifth Form Transitional Research Project. The following provides a short abstract to the full essay, which can be found at the bottom.

Estimated read time of abstract: 1 minute
Estimated read time of essay: 13 minutes

‘The Tale of the Heike’, a Japanese tale of the fall of the Taira clan to the Minamoto, and ‘The Iliad’, the enchanting story of Achilles’ and the Greeks’ struggles against Troy, are two of the greatest epic poems to ever be written, yet from opposite sides of the globe. However, their geographical distance doesn’t necessarily mean they can’t share similarities and form a fascinating piece of literary discussion.

In this essay we look at both poems’ themes use of language and the tradition that surrounds them. We’ll delve to into the specifics of the Japanese and Greek, as well as taking a more holistic view of how the themes such as impermanence, glory, and the view of individuals work together to create the epics. Finally, sweeping from the Aeneid to Tolstoy, from the Hagakure to Bushido: The Soul of Japan we’ll consider the cultural impact of both epics and come to a conclusion over how similar these two pieces of awe-inspiring literature are.

To view Mattie’s full article, follow this link below.

“To what extent can ‘The Tale of the Heike’ and ‘The Iliad’ be
considered similar poems?” By Mattie Sutton
Categories
Independent Learning Assignment STEM

Can you Ever Truly Mix a Cuppa? – Spilling the Tea on Brouwer’s Fixed Point Theorem

This essay was written by upper-sixth former Ben Watkins, and a finalist for the 2020 Independent Learning Assignment. The following provides a short abstract to the full essay, which can be found at the bottom.

Estimated read time of abstract: 1 minute
Estimated read time of essay: 15 minutes

Is it possible that there are always two places on earth with the same temperature and pressure? How does the game show Blockbusters have any implications on algebraic topology? Can a general equilibrium ever be reached in an economy? Perhaps most crucially of all, can you ever truly mix a cup of tea? 

My ILA provides insight into Brouwer’s fixed point theorem, a theorem found in the field of algebraic topology. It uncovers how a remarkable and seemingly counterintuitive result in what is often considered to be an abstract field of mathematics can have such broad and pertinent results in the real world. However, this isn’t to say that this ILA doesn’t uncover the result of this theorem for the sake of the beauty of it as much as uncovering it for the sake of its applications. Indeed, Luitzen Egbertus Jan Brouwer himself (the discoverer of this theorem as well as often being called ‘the Father of Topology’) was very much an upholder of this mentality: that maths has great importance for the sake of maths itself. Philosophically, Brouwer was a neo-intuitionist, which means that he thought of mathematics as purely a mental phenomenon, the result of constructive mental activity rather than uncovering any principles of an objective reality. He is often quoted in saying that “The construction itself is an art, its application to the world an evil parasite.” 

To view Ben’s full article, follow this link below.

“Can you Ever Truly Mix a Cuppa? – Spilling the Tea on Brouwer’s Fixed Point Theorem” by Ben Watkins
Categories
Independent Learning Assignment Law & Politics Social Sciences

Pot-Luck Politicians, A Parliamentary Chamber from Sortition

This essay was written by upper-sixth former Tom Welsh, and a finalist for the 2020 Independent Learning Assignment. The following provides a short abstract to the full essay, which can be found at the bottom.

Estimated read time of abstract: 2 minutes
Estimated read time of essay: 1 hour

Sortition is the random selection of individuals, and in this paper I sought to understand its political application in legislatures before providing a potential application in the United Kingdom via a third House of Parliament. The motive being sortition’s inherent equality and true representativeness. 

In order to do so, an investigation of sortition’s use in history was made, before its role in political theory was considered. I then briefly looked at its current application in both juries and citizens’ assemblies, before considering sortition’s hypothetical applications and existing use outside of the United Kingdom. 

Once the theory was covered, I then took to explaining the functioning of the existing UK government before looking at current UK political participation in both formal methods (elections and referenda) and informal (social movements and pressure groups). 

Having discussing the underlying theory, and the use-case it was being applied to, a substantial portion of my paper attempted to outline a comprehensive description of why I believe sortition’s best application would be as an addition to the existing Parliament given the important role that both the existing Houses of Parliament play. That is not to say that I felt the chamber from sortition would have a small role to play – far from it. 

I then ended the paper with an overview of some of the potential issues that such an implementation might entail, with an attempt also being made to explain how they might be dealt with and why if they cannot, on balance, that is of little concern in any case given the many positives associated with sortition. Furthermore, if you have an interest in either Plato or Jean-Jacques Rousseau, the absolute ending of the paper comes in the form of a synthesis of both, achievable, in my opinion, through sortition. 

If I gone some way to perhaps intriguing you about my, perhaps controversial, proposition then do consider giving my ILA a read. In fact, even if I haven’t – undoubtedly it is not easy in a short abstract to fully convey the true nature of a piece of work – maybe consider giving it a read in any case. A word of warning though, perhaps read the paper one chapter at a time, as I apologise it is not exactly short. Nonetheless, if you do choose to read the full work, thank you. Yet more importantly, I hope I cause you, even if you disagree with my conclusions, to reconsider your own political views – indeed if you don’t already have any, that is fine too! 

To view Tom’s full article, follow this link below.

“Pot-Luck Politicians A Parliamentary Chamber from Sortition” by Tom Welsh