On Friday 30th September the School of Mathematics and Physics at the University of Lincoln hosted our first ever PGR symposium event. We invited alumni who are currently PGR students at other institutions to give a short (10-15 minute) presentation about their research projects alongside some of our own PhD students.

The first speaker was Chris Dickens who is currently in his final year studying for a PhD in the School of Mathematics and Physics. His research involves using density functional theory to investigate the formation of individual point defects and defect clusters in ternary semiconductor compounds.

In his talk, Chris spoke in particular about substituting iron for indium in CuInS_{2} and how this may introduce additional properties that are ideal for applications in photothermal energy conversion and photovoltaics. In the Q&A session he also explained how he chose the particular functional he used for his calculations.

On Friday 30th September the School of Mathematics and Physics at the University of Lincoln hosted our first ever PGR symposium event. We invited alumni who are currently PGR students at other institutions to give a short (10-15 minute) presentation about their research projects alongside some of our own PhD students. The speakers were:

Chris Dickens (University of Lincoln)

Harry Finch (University of Liverpool)

Sorcha Hulme (University of Liverpool)

Charlotte Vale (University of Surrey)

Nick von Jeinsen (University of Cambridge)

Reece Jones (NPL & University of Strathclyde)

George Bell (University of Lincoln)

Alek Radic (University of Cambridge)

The event was a great success, with some fantastic presentations and some exciting conversations about the future. Over the coming days we will be writing a short blog post about each of the speakers’ talks. Watch this space….!

On Friday 30th September we invited our alumni students to return to Lincoln for our first ever Maths and Physics Alumni Networking Event.

This was a fantastic opportunity for our graduates to show us what they’ve been up to since graduating, and for our current undergraduates to be inspired by what their predecessors have gone on to achieve.

We’d like to thank everyone who participated in the event! It was really great to see some familiar faces and to hear how well everyone is doing.

Some of the alumni even stayed to take part in our Welcome Week quiz!

The CCP9 (Collaborative Computational Project for the Study of the Electronic Structure of Condensed Matter) community conference took place between the 7th and 9th of September 2022 at the Crown Plaza Hotel in Manchester. The conference was attended by Professor Matt Watkins and Chris Dickens, a PhD student in the School of Mathematics and Physics at the University of Lincoln. The conference included a series of invited talks from more senior members of CCP9 as well as an array of 15 minute lightning talks from PhD and Postdoc researchers from across the country working in relevant areas. There was also a poster presentation where Chris presented work on copper-based ternary semiconductors.

Our 2022 UROS (undergraduate research opportunity scheme) projects are well underway. Our students are making good use of our drawing walls, and it looks like someone might be running an important simulation….

Last week we hosted Maddy, our final work experience student for 2022. Maddy had already told us that she was interested in quantum physics but when she arrived, we had a discussion and decided together that she would do some research into black holes.

Maddy spent the whole week reading not only about black holes, but about special relativity and general relativity as well. She even encountered some very hypothetical concepts such as ‘white holes’.

Maddy’s report that she wrote us is too long to post on our blog, but here’s what she had to say about her week of work experience with us:

“As well as learning about the nature and theories surrounding black holes, I have also gained an insight into the research process, where many sources can provide information that is unreliable. In addition, I have improved my understanding into some of the complex mathematics that goes into concepts such as general relativity, which contains complicated solutions which required the knowledge of high-level ideas in mathematics to grasp.

Throughout the course of this week, I was treated as a research student, which involved being given large amounts of independence surrounding what project I could choose to undertake, as well as how I would manage my time when carrying out this research. I greatly enjoyed this freedom as it allowed me to explore a rich topic that I had not yet seen in such a level of depth. I had not expected to have been able to carry out a project of my choosing, which came as a welcome surprise.

Before starting this work experience, I was interested in a career as a physics researcher, and this week has helped to reinforce this career path as highly interesting, engaging and something I would want to do in the future. This week has helped to further my curiosity surrounding the mathematics found in high level physics as it is intricate and something I had not yet fully encountered before this week.”

On the 12th July the School of Mathematics and Physics hosted more than 80 students from The Beauchamp College in Oadby near Leicester.

The schedule (see below) included lectures by Dr Simon Smith and Dr Paula Lins de Araujo, who talked about areas of mathematics related to their own research. The students were also given a campus tour, took part in an activity delivered by one of our undergraduate students Owen, and were given the opportunity to ask another of our undergraduate students, Leah, about her student experience.

Our visitors enjoyed their experience and we hope that some of them have been persuaded that a degree in mathematics is a great idea!

Many thanks to our students (Owen, Leah, Adam and Chris) for their help!

Recently two of our Physics with Philosophy BSc students Jodie and Olivia have completed a teacher trainings internship in a Lincolnshire school, where she delivered a lesson based around an experiment from her third year undergraduate dissertation project. Read more about their experience (written by Jodie) below.

—————————————————————————–

When the Lincolnshire SCITT (School centred initial teacher training) came to talk about their internship, I didn’t expect to be able to use my final year physics project during the internship to take a lesson in a school.

The internship allows undergraduate students to see what teaching would be like in their chosen subject, in my case Physics. Consisting of three weeks in a school, the internship provided opportunities to observe lessons, act as an informal teaching assistant and the chance to take a lesson. I was placed with fellow student Olivia at William Farr C of E school in Welton. The staff in the science department were eager to provide as many experiences and insights into the teaching life as possible, which without their open support and encouragement, we would never have had such an immersive and rewarding experience.

Returning to the subject of university however, my final year project was called “Visualising Acoustic resonance with lasers”. The brief was to design an outreach activity for A-Level students to teach them about the physics concepts involved in a particular experiment, linking elements of the A-Level curriculum to the activity in order for students to apply the knowledge attained in the classroom to a fun alternative experiment. The experiment in question consisted of a small mirror adhered to an elastic membrane to which reflected a laser onto a nearby wall. A speaker was placed under the elastic membrane so that when frequencies were played the membrane would vibrate and cause the laser to create visual patterns on the wall.

With the opportunity to take a lesson during the internship, it was the perfect moment to put the project to the test.

Ultimately the project wasn’t used with A-Level students and instead with a lively Year 7 class. They provided Olivia with the opportunity to introduce the basic concepts and terminology of waves before handing over to me to take them through the outreach activity I had amended in line with a KS3 Science textbook borrowed from one of the Physics teachers.

The students were very enthusiastic and had lots of questions to ask. At the end of the lesson, the students filled out feedback sheets which asked them if they enjoyed the activity, what they had learnt and what they would improve. I’m pleased to say that all students selected yes for the first question (thankfully) and they were all able to state something they had learnt, so we must have done something right! As for their suggestions for improvements they were all very kind and creative in their ideas and provided lots of food for thought going forward.

Last week we hosted a work experience student, Dexter Harland-Hackenschmidt, who is interested in mathematics and logic. We tasked him with a logic problem that we had seen on YouTube. The following is his report in the style of a blog post. Enjoy!

The video goes on to provide counter examples to options B, D and E, proving that these options cannot be concluded, as they are not true in all cases. After narrowing it down to options A and C, the truth table for implication (→) is invoked to argue that C would make (2) vacuously true (instead of false), and therefore that Pinocchio is not lying and thus the only viable answer, if any, is A. However, it is not made clear why the truth table for implication was used, and the video fails to explain other parts of the question. What does it even mean to conclude something in a logical sense?

An introduction to logic In order to fully understand the proof for this question we must first understand the basics of classical logic, which the video used in the proposed solution, and first order logic which allows the use of predicates which will be explained later. Classical Logic is built up of 3 core elements: propositions, logical connectives and ‘laws of inference’. Propositions are statements in classical logic which can only be true or false – not both.

Logical connectives connect propositions together and allow us to construct logical statements out of them. The first of the logical connectives used in this proof is conjunction, which is denoted by (ꓥ) and means “and”. For example, the two propositions “it is raining” (R) and “it is cloudy” (C) can be combined using the conjunction connective to form the statement “it is raining and it is cloudy” (R ꓥ C), and for this to be true both the individual propositions must be true as well. The second connective we shall introduce is disjunction, which is denoted by (V) and means “or”. For example, if the propositions “Steve has an apple” (A) and “Steve has an orange” (O) were connected by disjunction to form “Steve has an apple or Steve has an orange” (A V O), then if either of the individual propositions is true then their disjunction is true. The third connective we must define is negation, which is denoted by (¬) and means that if a proposition S is true then the negation of S (¬S) is false. The logical connective referred to as implication is denoted by (→) and means “if … then …”. For example “if Steve has oranges, then he has apples” (O→A). Finally, “if and only if” is denoted by (↔) and means that the propositions on either side have logical equivalence and can be used interchangeably. Basic sentences built out of atomic (i.e. irreducible) propositions A and B with the above connectives return different truth values depending on the truth values of the atomic propositions they act on. These can be summarised into a logical tool called a “truth table” (see Table 1).

Rules of inference allow for simple logical conclusions to be gathered from truth tables (shown below) for example Modus Tollens (if A→B is true and B is false then A is also false). In order to properly apply these connectives to the question being analysed in this blog post, first order logic must also be applied, since this helps in constructing more precise statements to rewrite the question in a logical form. This requires the use of two tools within first order logic, quantifiers such as ꓯ (for all) and ꓱ (there exists), and the use of predicates such as “green” to describe the properties of objects (i.e. hats). This can be written as g = green and h = hat such that g(h) means a hat that is green. These can be constructed into logical sentences such as ꓯh g(h), meaning all hats are green.

Applying first order logic to the problem In order to apply first order logic to the problem of Pinocchio, we must first rewrite the question using formal notation. In order to do this, we begin by defining the universe or domain of discourse which in this case would be all hats, whether they are owned by Pinocchio or not. Defined as such, let the set of all hats h_{i} be H= {h_{i}} (the subscript ‘i’ refers to the indices with which we label the hats; h_{i} could be any hat within the set, i.e. i = 1, 2, 3, …, N where N is the number of hats). We must also take care to define what the question means by “conclude”. We interpret this to mean that if an option can be concluded, then when Pinocchio makes the statement X the negation of that statement should imply that option for all possible cases. When this is the case, it is called a tautology. An example of a tautology within propositional logic is the logical inference Modus Ponens, which states that if Q → R is true and Q is true, then R is also true. The proof for this statement is demonstrated in truth table below (Table 2).

The final column in Table 2 is Modus Ponens and since it is true (blue) in all cases, then as described above it is a tautology. We will apply this to Pinocchio’s hats later on but first we must define some of the notation that we will use. Pinocchio will be referred to as ‘P’ and hats will be referred to as ‘h’. The predicate ‘g’ means “green” and the predicate p means “owned by P”.

We can then rewrite the question in formal logic…

Assume the following sentences are true:

If P says X then ¬X; ‘Pinocchio always lies‘

P says “ꓯh_{i} p(h_{i}) → g(h_{i})” ‘Pinocchio says, “All my hats are green” ‘

(Initially we attempted to use conjunction “p(h_{i}) ꓥ g(h_{i})” however writing the statement in that way meant that Pinocchio owned all hats in the universe!)

We can conclude from these two sentences that:

A) ꓱh_{i} p(h_{i}) ‘There exists a hat that is owned by Pinocchio‘

B) ꓱh_{i }ꓯh_{j }: g(h_{j}) ↔ i=j ‘There exists precisely one green hat that is owned by Pinocchio’

C) ¬ꓱh_{i} p(h_{i}) ‘There does not exist a hat that is owned by Pinocchio’

D) ꓱh_{i} p(h_{i}) ꓥ g(h_{i}) ‘There exists a hat that is both owned by Pinocchio and is green’

E) ¬ꓱh_{i} p(h_{i}) ꓥ g(h_{i}) ‘A hat that is both owned by Pinocchio and is green does not exist‘

(We use X to refer to “ꓯh_{i} p(h_{i})→g(h_{i})” for the sake of brevity)

The approach we took in our work was to note that if Pinocchio says X then X is a lie and thus ¬X is true, and if the negation of X implies any of the options is a tautology then the corresponding statement is a valid conclusion from (1) and (2) above. We created a test universe comprising of only two hats and examined the truth table (Table 3) to see if any of the statements were false in any of the models.

We therefore need to negate Pinocchio’s statement X (ꓯh_{i} p(h_{i})→g(h_{i})). The first thing to mention is that when negating a sentence that contains the quantifier ꓯ, the ꓯ is replaced with ꓱ. Then, the negation of an implication Q→R can be written as follows:

From Table 3 we can gather that (¬X→B), (¬X→ C), (¬X→ D), and (¬X→E) are not tautologies, since they contain some falsehoods (they are false in some rows or ‘models’). However, (¬X→A) is a tautology in this universe consisting of two hats as it contains only truths in every row (in every ‘model’).

Is this hold if we increase the number of hats arbitrarily? Yes! The way in which we can prove that ¬X→A is a tautology for any number of hats is with the use of the truth table for implication (see the first 3 columns of Table 2). Notice that whenever statement Q is false the statement Q→R is still true. This was mentioned in the video by Presh Talwalker but perhaps not explained fully. What this means is that when the ‘antecedent’ Q is false the ‘consequent’ R is vacuously true; because the statement holds no information it is impossible to prove false and is thus considered true within classical logic. We can apply this to the case of ¬X→A as it is only false in the case of ¬X being true and A being false. This allows us to prove A is a tautology in the following way. A is only false when P owns no hats, however, in that case ¬X is also false because P has to own a hat for ¬X to be true. Therefore whenever A is false ¬X is also false and the implication is true for all possible cases and A can be deemed true.

Conclusions

Although we came to the same logical conclusion as Presh did in his video, we have accomplished it in what we think is a more logical way. We have delved further into formal logic and have applied first order logic which was left unmentioned in the video. If, like us, you were initially confused by the proposed solution, hopefully this blog post has made both the question and the solution more clear.

Last week the School of Mathematics and Physics hosted 5 work experience students from Lincoln University Technical College, Branston Community Academy and William Farr.

The aim of the week was to give a taste of what it would be like to study Mathematics or Physics at the University of Lincoln (or elsewhere), and for the students to gain some experience related to working in a university.

On Monday the students were given a tour of the University of Lincoln campus and the Isaac Newton Building in which the School of Mathematics and Physics is based. We also gave the students a very useful talk about the university application process and finances.

“We received some insightful information about how to make your application stand out from other applications, how to know how much money you will receive from loans and how to use student loans when living independently.”

On Monday afternoon, the students met some of the academic staff who introduced themselves and spoke briefly about their careers and research interests. Finally, the students were shown the scanning electron microscope.

On Tuesday the students attend a workshop with Dr Claire McIlroy who spoke to them about types of flow, how fluid properties can lead to interesting flow phenomena, and how to describe this behaviour mathematically. In the afternoon the students performed experiments: by filming a droplet detaching from a faucet, the students were able to measure the viscosity of a range of fluids, including honey, ketchup and oil.

“Using a measuring tool in ImageJ we measured the diameter of the bridge of the fluid (the bridge is the part of the drop that gets smaller and smaller as it falls). We collected a series of pictures of a single drop just before it exits a pipette and at each point we measure the diameter of the fluid. Combining these results with the theory we saw in the morning, we calculated the viscosity of the fluid.”

On Wednesday morning the students joined Dr Matt Booth to discuss quantum physics and quantum mechanics, touching on wave-particle duality, Heisenberg’s uncertainty principle and even the Schrödinger equation! In the afternoon the students visited the undergraduate laboratory and performed some related experiments, including the diffraction of electrons by graphite.

” Overall on I thoroughly enjoyed Wednesday as I was able to see the theory and maths behind quantum mechanics but also experiments that can be used to demonstrate these effects.”

Thursday was astrophysics with Dr Phil Sutton, who introduced the idea of exotic planets orbiting multiple stars, which has previously been left purely to science fiction. However, scientists have now discovered such worlds. The students learned about recent discoveries and discussed them in the context of potential life. In the afternoon, the students were tasked with accessing the exoplanet archive data for over 5000 confirmed exoplanets and using Matlab to probe which exoplanets are most likely to host exomoons.

“Phil explained two of the main methods for examining exoplanets and the advantages and disadvantages of the methods – Radial velocity and Transit. Transits is the more efficient method for finding exoplanets; however a combination of both methods is needed to find the mass and length of the orbit.”

On Friday the students were given a tour of some student accommodation, which was very interesting. Then, to finish off the week, they were taken for lunch and then asked to write a blog style report about their week.

“This week was very fun and informative about what student life is like but also how to work up to a level that the professors and lecturers achieved. This week has definitely helped me make a decision on whether to go to university but has also helped me to expand my knowledge and has given me a wider view on my future and where university could take me.”