You might not have heard of Francesca Rust yet, because as she explains below she's only been cycling since last year – but VO2 Bikes, a new women's team, has already signed her up to race with them in 2015. That's a meteoric rise by anyone's standards, so expect to hear much more about her once the new season gets under way.
I asked Francesca if she would put together a guest blog recounting her first attempt at competitive cycling which, as is the case with so many new riders, was a time trial. With low entry fees and so many being organised by local clubs that have a real enthusiasm for bringing more women into the sport, time trials are one of the most accessible routes into racing, but the time trial world is a secretive one that can be daunting when a rider makes her first foray into it – Francesca is ideally placed to give other women who have been considering putting themselves to a test against the clock can expect to experience.
Francesca's own blog is here.
A bit of background about me, I only started cycling last year after being easily persuaded to do the London to Paris charity bike ride in aid of Ormiston. The 310 mile ride was quite possibly the best decision I have ever made as it gave me the platform to join my local cycling club, meet new people and join in the fun! I had one go at our 9.5mile timetrial last year of my make-shift TT bike. I just put clip-on bars onto my handlebars and suddenly I was "aero" like the rest. Result. I didn't do too badly (in my opinion – what really counts) but I was hooked. I was itching for more but had to wait for the spring to have a proper crack at it.
With winter over, I have spent most of this year having a go at my local club time trials, they vary between 5 miles and 21 miles, some flat and some "undulating"! I had set myself a goal of trying to win my club's TT league, but with few women turning up I was racing against myself most of the time but improving nonetheless.
Nearing the end of the year, I volunteered to help out at the E2/10 race with another lady from my club. This was the one down the A11 from Six Mile Bottom. Handing out rider numbers and chatting to a few, I was kind of gutted I couldn't do it. After a brief discussion with John from Chelmer, he told me that there would be another in a couple of weeks on the same course and I should enter it. Me being me, unsure of my own capabilities, I asked what the times were and if I even stood a chance! Thinking about it now though, who cares what my time is, I am racing against myself and it would be my first "proper" TT against cyclists outside of my club.
I literally had no idea how to do it – the application that is. The cycling, well it's a case of just keep pedaling! I was lucky that I had a number of guys in my club who were very supportive in helping me sign up – what sites I had to register on etc. Signed up via Cycling Time Trials, paid my sign on fee (much cheaper than sportives that's for sure), guessed my 10mile time – I had never done a 10mile, only 9.5 or 11. They use your best time on the distance to work out your handicap (I found out after).
20th September 2014 arrived. It was E2/10 day. I was incredibly nervous, but also very excited. I had two of my London to Paris cycling friends (Jo & Sue)coming to do it too which made the event even more exciting. They are both so very supportive and as we hadn't done this kind of TT before, I guess we were all in the same boat – SCARED. Arriving at the HQ, signed on, we decided we'll drive the course. It was a good decision as it made us realise just how long 5 miles out and back was, realise it's not a flat road – like it feels in the car doing 70mph and also terrify me with the amount of lorries and cars on the road. With anxiety already at peak, it started to rain. Oh joy! I'll be honest, between the three of us there were a couple of comments about still doing it – surely they'd cancel.
We got down to the start line to have a warm up. It was raining quite a lot now. Already soaked from standing in a puddle I pedalled off up the road, getting about 2 miles out then realising that I really need to raise my saddle. I was borrowing a bike from Aprire (London bike company) which I hadn't ridden before (error no. 1) getting back to my car and realising I didn't have any allen keys (error no. 2) I borrowed one off a nice couple and raised it slightly. I stopped to check it with one of the guys from my club also doing the TT and said I was starting in about 15 minutes so very little time to tweak or even warm up for that matter. Jo my L2P friend was also having technical difficulties – her gears weren't working. She was starting a minute after me so was having a bit of a flap too. Jon put me on his turbo and just said spin the legs. This is where I found out that you need different skewers to use on a turbo! I had about 9 minutes. Having a panic I thought ok I'll take a gel, this was probably more for the confidence boost than the actual energy as it wouldn't have fully kicked in by the time I was half way round.
Start line. Jo behind me. Jon supporting from the side. I pushed off up the A11. If I had a heart rate monitor on, I dread to think how high it peaked at that point. Surprisingly though, once I negotiated the transition onto the road, I secured my place in the worn line – on the road, not on the white line which I had originally thought was where you were supposed to ride. Overcoming that fear was a big change point. I was well and truly in the line of traffic. Most of the ride is a bit of a blur, all that was going through my head was my friend Bob saying I could do it in 23minutes. I joked at him and worked out what speed I would need to do to do it in that. Laughing I thought I would never be able to hold 26mph for 10 miles. Never. Coming off the A11 and onto the roundabout was a bit hairy to say the least. And inconsiderate driver undertook me and then stopped in front of me. I had to brake hard and then didn't have time to change my gears to pull away easily. I wasn't used to the gears being on the front – I normally ride a road bike with clip-ons remember! Pulling away again round the roundabout sitting upright was hard. Going back down the slip road to join the A11 again, 5 miles to go now. I reluctantly checked what average I was doing, 25mph ish. No way! This gave me a boost. I looked across the dual-carriage way and saw Jo in her bright green Chelmer kit. Another boost – my friendly fear was that she would overtake me. I was also closing on a guy in front. Just keep pedalling. I over took the chap. My average was going up. I could see the finish and just kept trying to push. With my legs not wanting to go any faster I just focussed on crossing the line. A lorry came by very close and blew me onto the white line. Panic and adrenaline kicked in and I dug in. legs burning and eyes (or visor) slightly blurry from the rain. Finish.
Thinking about it now, it was one of the best TT's I have done. It was out and back. Simple. It was scary, but there was so much support from everyone, the volunteers at sign-on, the guys at the push-off, and at the finish. They explained everything to me.
I came 4th with a time of 23.37. Sue came 1st and Jo came 8th (on a road bike). Unfortunately I didn't get a handicap win as I hadn't done a 10mile before so they use a rule figure which made me faster than others. Next time I might get it. But for me, finishing and overcoming the fear of a dual-carriage way TT was a win for me. I cannot wait until next year to do it again (when it's not raining)! I would highly recommend anyone to give it a go, you don't need a TT bike, you don't even need to be a member of a club, just register on Cycling Time Trials and find one near you, ask friends for advice and get pedaling!
Neutral Service will be featuring more guest blogs written by Francesca in 2015 as she begins her first full racing programme.


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Police Officer Who Starred In Crime Fighting Series Unmasked As Child Sex Offender
A 54-year-old police officer, who starred in the famed UK documentary, 'Bodycam Squads', is actually a pedophile.
Yes, the British cop admitted to a series of child sex offenses, sending shockwaves around the country.
Meanwhile, when authorities unmasked the 54-year-old co, they received a horrifying surprise. And that's because this man turned out to be a pedophile. According to reports, he tried to groom countless children online but was arrested on time.
Source: BPM Media
Lee Tatton starred on a famous UKTV's documentary, where he played the real-life role of a responsible officer. And that meant responding to crime in a timely fashion. But who knew what was really behind that mask?
Today, the Staffordshire officer's career is in the ruins, thanks to his gruesome ordeals. Furthermore, the police department describes the shocking moment when he confessed to the crime. And that includes a wide array of sex offenses, related to minors. The confessions arose at Birmingham's Crown Court, just this week.
The cop's own Operation Safenet squad carried out the arrests, leaving him shattered in regards to his heinous acts. On the other hand, this special task force is specifically designed for one reason. And that revolves around protecting vulnerable minors from all sorts of perverts as well as pedophiles.
Tatto received an immediate suspension from all of his duties and till today, continues his life on suspension, awaiting his final internal hearing.
During his court appearance, he pleaded guilty to several crimes. This includes engaging in sexual forms of communication with young children. The other offenses included taking part in indecent child photography. According to reports, Tatton will receive his sentence on November 2nd of this year.
Other reports indicated how the famous production company by the name of Raw Cut, kept shadowing his pedophile behaviors. These included numerous incidents. But in the end, they all went to vain. And that's when the 54-year-old former officer suffered from the consequences for his actions.

redpajama_set_name: RedPajamaCommonCrawl


A new, quick weight-loss & body-shaping plan for women.
Shape up, look younger, and live longer.
978-0-934232-48-7 • $19.95 • 242 pp.
A contributor to NBC's "Today" show, Dr. Altmann helps parents with friendly, direct guidance in an Q&A format. 978-1610021265 • $16.95 • 260 pp.
What if the air you breathe to live…becomes the air you breathe to die?
A Watcher agent, an Arkangel, and federal agent face off on the eve of Armageddon.
978-1640440074 • $17.99 • 300 pp.
A charming puppy's adventures take her across the country where she learns life lessons along the way.
978-1-943493-00-5 • $10.00 • 125 pp.
Sesame Street has aired on PBS for over 45 years! Let's discover the secrets to its success... 978-1-63491-438-3 • $30.95 • 133 pp.
"...a completely engrossing read: vivid and hard to put down."—Midwest Book Review.
The captivating story of a visionary inventor who helped spearhead today's digital display technology. 978-1-68409-510-0 $41.95 Hardcover • -508-7 $26.95 Paper • -509-4 $9.99 eBook • 298 pp.
In Antarctica, Periwinkle meets her cousins, learns to be brave, & realizes it's not what you look like that matters. 978-1-943198-03-0 • $16.99 • 48 pp.
Translating the Norse Myths, Kvilhaug reveals ritual and belief central to Northern and Western culture. 978-8792632746 • $79.99 • 712 pp.
Historical fiction adventure featuring an orphaned Chinese teen cub reporter who risks death to re-gain his honor. 978-0-9988034-1-8 • $12.95 • 292 pp.


Now comes Moderator Candy Crowley; the same Crowley that daily offers her opinions as news on Radio for ABC as each time a story about either Romney or Ryan must be "reported" one can hear the contemptuous and condescending sneer in her voice. However, this time it appears to be worse. After both the Romney and Obama camps agreed to abide by the debate format--and their lawyers had agreed--Candy Crowley decided to change it…seemingly on her own.
In order to appear not to be the obvious cause of this additional departure from sanity--if not the law--the Obama syndicate has yawningly announced in a rather feckless manner that it disagrees with Crowley on this…but 'what can one do'? ABC doesn't appear to care about any lawsuits that may result from this breach because if Crowley is successful in brow-beating and berating Governor Romney and if she is single-handedly able to resurrect the Obama campaign from its downward slide…they believe that Crowley's planned illegal and unethical behaviors--coupled with rampant voter fraud--just might win them and their fellow Marxists another 4 years! If not, the media will simply continue to threaten Republicans with the ACLU--who will do anything for Marxists and a few big $$bucks.


Large building fires are a relatively common occurrence in impoverished Bangladesh, due in part to lax regulations, and have killed hundreds in recent years.
"So far, 56 bodies have been recovered. The number could rise further as searching is still continuing," Julfikar Rahman, a director of the Fire Service and Civil Defence, told Reuters.
The fire broke out in a four-storey building on Wednesday night and quickly spread to nearby buildings in the Chawkbazar area of Old Dhaka, which dates back to the Mughal period more than 300 years ago.
Rahman said at least 50 people had been taken to hospital, some in critical condition.
Fire fighters fought for more than five hours to bring it under control, although it was still burning early on Thursday. It was not immediately clear what caused the blaze.
The Dhaka Tribune reported that the building housed a plastics warehouse and was full of flammable material.
"The cause of the fire is not yet clear," Rahman said.
He said firefighters struggled to find adequate water supplies and had to draw water from a nearby mosque to fight the blaze.
Building safety regulations are rarely followed in Bangladesh, where accidents kill hundreds every year.


Curator of Media Arts Richard Herskowitz, who also serves as Artistic Director of the Ashland Independent Film Festival, will present his selection of favorite short films from this year's edition of AIFF (April 12-16). Among the films shown will be Edge of Alchemy, the latest animated film by Stacey Steers, whose Night Hunter House was shown at the JSMA in 2012. Programmed by Richard Herskowitz, curator of media arts, Schnitzer Cinema brings the best of experimental, documentary, and arts-focused films and videos to the JSMA each year. Schnitzer Cinema is made possible in part with a grant from the UO Office of Academic Affairs.
Programs are free, with popcorn and refreshments provided!


Gibb Tools office in Aberdeen, responsible for selling the Teng range of tools and general consumables.
The role will involve driving the Teng Tools Sales Van around the exclusive area of Aberdeenshire, calling on garages, maintenance workshops and industrial customers and building up a regular round of clients with repeat business.
To join our friendly team the successful applicant must be self-motivated, fast thinking, with a good knowledge of garage hand tools and their operation and living in the Aberdeen area.


Eliza Douglas wears a Patrik Ervell shirt; Levi's Vintage Clothing 1947 jeans; Balenciaga sunglasses and shoes.
From left: Gosha Rubchinskiy x Adidas shorts; Prada socks; Balenciaga shoes; Douglas's own glasses (throughout). Calvin Klein 205W39NYC raincoat and pants.
From left: Burberry coat; Lanvin pants. AG T-shirt; Gosha Rubchinskiy x Adidas shorts; Prada socks; Balenciaga shoes.
Gosha Rubchinskiy vest and top; Gosha Rubchinskiy x Adidas hoodie and shorts.
Balenciaga shirt, jeans, and shoes; Douglas's own socks.
Prada sweater, shirt, and pants; Balenciaga T-shirt and shoes; Douglas's own socks.


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Mathematics MSci Hons (G101)
Mathematics MSci Hons - 2020 Entry
UCAS Code
Entry Year
A Level Requirements
AAA see all requirements
Full time 4 Year(s)
Mathematics forms the foundations of all science and technology and as such is an extensive and rigorous discipline. Our integrated Masters programme reflects this and offers a comprehensive, detailed study through excellent teaching, taking your academic work to an advanced level.
Maths is difficult to concisely define, but at its core it is the study of change, patterns, quantities, structures and space. This engaging programme, and our reputation for excellence in research, means that we are able to offer high-quality teaching delivered by academics who are leaders in their field. Our four year integrated Masters gives you the opportunity for deeper specialisation in maths and the chance to develop your research skills by undertaking a dissertation or industry research project.
During your first year, you will build on your previous knowledge and understanding of mathematical methods and concepts. Modules cover a wide range of topics from calculus, probability and statistics to logic, proofs and theorems. As well as developing your technical knowledge and statistical skills, you will also enhance your data analysis, problem-solving and quantitative reasoning skills.
In the second year, you will further develop your knowledge in analysis, algebra, probability and statistics. You will also be introduced to Computational Mathematics, exploring the theory and application of computation and numerical problem-solving methods. While studying these topics, you will complete our Project Skills module, which provides you with the chance to enhance your research and employment skills through an individual and group project. Additionally, you will gain experience of scientific writing, and you will practise using statistical software such as R and LaTeX.
Beyond second year, we offer a wide range of specialist optional modules, allowing you to develop and drive the programme to suit your interests and guide you to a specific career pathway. In third year, you will put the skills and knowledge you have gained so far into practice as your study begins to really focus in on advanced topics, such as Combinatorics, Bayesian Inference, Medical Statistics, and Number Theory.
In the fourth year, you will be able to widen your knowledge and skills by selecting from a pool of optional Masters-level modules. Some of these will be familiar from third year, while others will be new to this year. These advanced modules allow you to exercise what you have learnt during the programme, while expanding and evolving your skill set further.
You will also complete a major research project in either pure maths or statistics, guided by your interests and supervised by an active researcher, or undertaken as part of a collaborative industry project. This will cement your learning, provide you with valuable experience and position you for a career in maths or academia and research.
MSci Mathematics (Study Abroad)
In addition to our MSci degree, we also offer a MSci Mathematics (Study Abroad) programme. As well as studying at Lancaster, this programme allows you to spend a year gaining valuable knowledge and developing specialist skills at one of our partner institutions in North America. You get to experience an exciting new culture, explore another country and gain insight into academic life overseas, while expanding your professional network.
A Level AAA including A level Mathematics or Further Mathematics OR AAB including A level Mathematics and Further Mathematics
International Baccalaureate 36 points overall with 16 points from the best 3 Higher Level subjects including 6 in Mathematics HL
BTEC May be accepted alongside A level Mathematics and Further Mathematics
STEP Paper or the Test of Mathematics for University Admission Please note it is not a compulsory entry requirement to take these tests, but for applicants who are taking any of the papers alongside Mathematics and/or Further Mathematics we may be able to make a more favourable offer. Full details can be found on the Mathematics and Statistics webpage.
Students are provided with an understanding of functions, limits, and series, and knowledge of the basic techniques of differentiation and integration. Examples of functions and their graphs are presented, as are techniques for building new functions from old. Then the notion of a limit is considered along with the main tools of calculus and Taylor Series. Students will also learn how to add, multiply and divide polynomials, and will learn about rational functions and their partial fractions.
The exponential function is defined by means of a power series which is subsequently extended to the complex exponential function of an imaginary variable, so that students understand the connection between analysis, trigonometry and geometry. The trigonometric and hyperbolic functions are introduced in parallel with analogous power series so that students understand the role of functional identities. Such functional identities are later used to simplify integrals and to parametrise geometrical curves.
Convergence and Continuity
This module provides a rigorous overview of real numbers, sequences and continuity. Covering bounds, monotonicity, subsequences, invertibility, and the intermediate value theorem, among other topics, students will become familiar with definitions, theorems and proofs.
Examining a range of examples, students will become accustomed to mathematical writing and will develop an understanding of mathematical notation. Through this module, students will also gain an appreciation of the importance of proof, generalisation and abstraction in the logical development of formal theories, and develop an ability to imagine and 'see' complicated mathematical objects.
In addition to learning and developing subject specific knowledge, students will enhance their ability to assimilate information from different presentations of material; learn to apply previously acquired knowledge to new situations; and develop their communication skills.
An introduction to the basic ideas and notations involved in describing sets and their functions will be given. This module helps students to formalise the idea of the size of a set and what it means to be finite, countably infinite or uncountably finite. For finite sets, it is said that one is bigger than another if it contains more elements. What about infinite sets? Are some infinite sets bigger than others? Students will develop the tools to answer these questions and other counting problems, such as those involving recurrence relations, e.g. the Fibonacci numbers.
The module will also consider the connections between objects, leading to the study of graphs and networks – collections of nodes joined by edges. There are many applications of this theory in designing or understanding properties of systems, such as the infrastructure powering the internet, social networks, the London Underground and the global ecosystem.
Further Calculus
This course extends ideas of MATH101 from functions of a single real variable to functions of two real variables. The notions of differentiation and integration are extended from functions defined on a line to functions defined on the plane. Partial derivatives help us to understand surfaces, while repeated integrals enable us to calculate volumes.
In mathematical models, it is common to use functions of several variables. For example, the speed of an airliner can depend upon the air pressure and temperature, and the direction of the wind. To study functions of several variables, we introduce rates of change with respect to several quantities. We learn how to find maxima and minima. Applications include the method of least squares. Finally, we investigate various methods for solving differential equations of one variable.
Geometry and Calculus
The main focus of this module is vectors in two and three-dimensional space. Starting with the definition of vectors, students will discover some applications to finding equations of lines and planes, then they will consider some different ways of describing curves and surfaces via equations or parameters. Partial differentiation will be used to determine tangent lines and planes, and integration will be used to calculate the length of a curve.
In the second half of the course, the functions of several variables will be studied. When attempting to calculate an integral over one variable, one variable is often substituted for another more convenient one; here students will see the equivalent technique for a double integral, where they will have to substitute two variables simultaneously. They will also investigate some methods for finding maxima and minima of a function subject to certain conditions.
Finally, the module will explain how to calculate the areas of various surfaces and the volumes of various solids.
Integration and Differentiation
Building on the Convergence and Continuity module, students will explore the familiar topics of integration, and series and differentiation, and develop them further. Taking a different approach, students will learn about the concept of integrability of continuous functions; improper integrals of continuous functions; the definition of differentiability for functions; and the algebra of differentiation.
Applying the skills and knowledge gained from this module, students will tackle questions such as: can you sum up infinitely many numbers and get a finite number? They will also enhance their knowledge and understanding of the fundamental theorem of calculus.
Introducing the theory of matrices together with some basic applications, students will learn essential techniques such as arithmetic rules, row operations and computation of determinants by expansion about a row or a column.
The second part of the module covers a notable range of applications of matrices, such as solving systems of simultaneous linear equations, linear transformations, characteristic eigenvectors and eigenvalues.
The student will learn how to express a linear transformation of the real euclidean space using a matrix, from which they will be able to determine whether it is singular or not and obtain its characteristic equation and eigenspaces.
Numbers and Relations
The student is introduced to logic and mathematical proofs, with emphasis placed more on proving general theorems than on performing calculations. This is because a result which can be applied to many different cases is clearly more powerful than a calculation that deals only with a single specific case.
The language and structure of mathematical proofs will be explained, highlighting how logic can be used to express mathematical arguments in a concise and rigorous manner. These ideas will then be applied to the study of number theory, establishing several fundamental results such as Bezout's Theorem on highest common factors and the Fundamental Theorem of Arithmetic on prime factorisations.
The concept of congruence of integers is introduced to students and they study the idea that a highest common factor can be generalised from the integers to polynomials.
Probability theory is the study of chance phenomena, the concepts of which are fundamental to the study of statistics. This module will introduce students to some simple combinatorics, set theory and the axioms of probability.
Students will become aware of the different probability models used to characterise the outcomes of experiments that involve a chance or random component. The module covers ideas associated with the axioms of probability, conditional probability, independence, discrete random variables and their distributions, expectation and probability models.
To enable students to achieve a solid understanding of the broad role that statistical thinking plays in addressing scientific problems, the module begins with a brief overview of statistics in science and society. It then moves on to the selection of appropriate probability models to describe systematic and random variations of discrete and continuous real data sets. Students will learn to implement statistical techniques and to draw clear and informative conclusions.
The module will be supported by the statistical software package 'R', which forms the basis of weekly lab sessions. Students will develop a strategic understanding of statistics and the use of associated software, which will underpin the skills needed for all subsequent statistical modules of the degree.
This module builds on the binary operations studies in previous modules, such as addition or multiplication of numbers and composition of functions. Here,students will select a small number of properties which these and other examples have in common, and use them to define a group.
They will also consider the elementary properties of groups. By looking at maps between groups which 'preserve structure',a way of formalizing (and extending) the natural concept of what it means for two groups to be 'the same' will be discovered.
Ring theory provides a framework for studying sets with two binary operations: addition and multiplication. This gives students a wayto abstractly model various number systems, proving results that can be applied in many different situations, such as number theory and geometry. Familiar examples of rings include the integers, the integers modulation, the rational numbers, matrices and polynomials; several less familiar examples will also be explored.
Complex Analysis has its origins in differential calculus and the study of polynomial equations. In this module, students will consider the differential calculus of functions of a single complex variable and study power series and mappings by complex functions. They will use integral calculus of complex functions to find elegant and important results and will also use classical theorems to evaluate real integrals.
The first part of the module reviews complex numbers, and presents complex series and the complex derivative in a style similar to calculus. The module then introduces integrals along curves and develops complex function theory from Cauchy's Theorem for a triangle, which is proved by way of a bisection argument. These analytic ideas are used to prove the fundamental theorem of algebra, that every non-constant complex polynomial has a root. Finally, the theory is employed to evaluate some definite integrals.The module ends with basic discussion of harmonic functions, which play a significant role in physics.
Students will gain a solid understanding of computation and computer programming within the context of maths and statistics. This module expands on five key areas:
Programming and R
Numerical solutions of equations
Numerical differentiation and integration
Numerical solutions to ODEs
Under these headings, students will study a range of complex mathematical concepts, such as: data structures, fixed-point iteration, higher dimensions, first and second derivatives, non-parametric bootstraps, and modified Euler methods.
Throughout the module, students will gain an understanding of general programming and algorithms. They will develop a good level of IT skills and familiarity with computer tools that support mathematical computation.
Over the course of this module, students will have the opportunity to put their knowledge and skills into practice. Workshops, based in dedicated computing labs, allow them to gain relatable, practical experience of computational mathematics.
Linear Algebra II
Students will be provided with the foundational results and language of linear algebra, which they will be able to build upon in the second half of Year Two, and the more specialised Year Three modules. This module will give students the opportunity to study vector spaces, together with their structure-preserving maps and their relationship to matrices.
They will consider the effect of changing bases on the matrix representing one of these maps, and will examine how to choose bases so that this matrix is as simple as possible. Part of their study will also involve looking at the concepts of length and angle with regard to vector spaces.
Probability II
Probability provides the theoretical basis for statistics and is of interest in its own right.
Basic concepts from the first year probability module will be revisited and extended to these to encompass continuous random variables, with students investigating several important continuous probability distributions. Commonly used distributions are introduced and key properties proved, and examples from a variety of applications will be used to illustrate theoretical ideas.
Students will then focus on transformations of random variables and groups of two or more random variables, leading to two theoretical results about the behaviour of averages of large numbers of random variables which have important practical consequences in statistics.
A thorough look will be taken at the limits of sequences and convergence of series during this module. Students will learn to extend the notion of a limit to functions, leading to the analysis of differentiation, including proper proofs of techniques learned at A-level.
Time will be spent studying the Intermediate Value Theorem and the Mean Value Theorem, and their many applications of widely differing kinds will be explored. The next topic is new: sequences and series of functions (rather than just numbers), which again has many applications and is central to more advanced analysis.
Next, the notion of integration will be put under the microscope. Once it is properly defined (via limits) students will learn how to get from this definition to the familiar technique of evaluating integrals by reverse differentiation. They will also explore some applications of integration that are quite different from the ones in A-level, such as estimations of discrete sums of series.
Further possible topics include Stirling's Formula, infinite products and Fourier series.
Statistics II
Statistics is the science of understanding patterns of population behaviour from data. In the module, this topic will be approached by specifying a statistical model for the data. Statistical models usually include a number of unknown parameters, which need to be estimated.
The focus will be on likelihood-based parameter estimation to demonstrate how statistical models can be used to draw conclusions from observations and experimental data, and linear regression techniques within the statistical modelling framework will also be considered.
Students will come to recognise the role, and limitations, of the linear model for understanding, exploring and making inferences concerning the relationships between variables and making predictions.
Algebraic Curves
Students will be given a solid foundation in the basicsof algebraic geometry. They will explore how curves can be described by algebraic equations, and learn how to use abstract groups in dealing with geometrical objects (curves). The module will present applications and results of the theory of elliptic curves and provide a useful link between concepts from algebra and geometry.
Students will also gain an understanding of the notions and the main results pertaining to elliptic curves, and the way that algebra and geometry are linked via polynomial equations. Finally, they will learn to perform algebraic computations with elliptic curves.
Bayesian Inference
Bayesian statistics provides a mechanism for making decisions in the presence of uncertainty. Using Bayes' theorem, knowledge or rational beliefs are updated as fresh observations are collected. The purpose of the data collection exercise is expressed through a utility function, which is specific to the client or user. It defines what is to be gained or lost through taking particular actions in the current environment. Actions are continually made or not made depending on the expectation of this utility function at any point in time.
Bayesians admit probability as the sole measure of uncertainty. Thus Bayesian reasoning is based on a firm axiomatic system. In addition, since most people have an intuitive notion about probability, Bayesian analysis is readily communicated.
Combinatorics is the core subject of discrete mathematics which refers to the study of mathematical structures that are discrete in nature rather than continuous (for example graphs, lattices, designs and codes). While combinatorics is a huge subject - with many important connections to other areas of modern mathematics - it is a very accessible one.
In this module, students will be introduced to the fundamental topics of combinatorial enumeration (sophisticated counting methods), graph theory (graphs, networks and algorithms) and combinatorial design theory (Latin squares and block designs). They will also explore important practical applications of the results and methods.
Commutative Algebra
Students' knowledge of commutative rings as gained from their second year of study in Rings and Linear Algebra will be built upon, and an introduction to the fourth year Galois Theory module will be provided.
They will be introduced to two new classes of integral domains called Euclidean domains, where they have a counterpart of the division algorithm, and unique factorisation domains, in which an analogue of the Fundamental Theorem of Arithmetic holds.
How well-known concepts from the integers such as the highest common factor, the Euclidean algorithm, and factorisation of polynomials, carry over to this new setting, will also be explored.
Questions relating to linear ordinary differential equations will be considered during this module. Differential equations arise throughout the applications of mathematics, and consequently the study of them has always been recognised as a fundamental branch of the subject. The module aims to give a systematic introduction to the topic, striking a balance between methods for finding solutions of particular types of equations, and theoretical results about the nature of solutions.
While explicit solutions can only be found for special types of equations, some of the ideas of real analysis allow us to answer questions about the existence and uniqueness of solutions to more general equations, as well as allowing us to study certain properties of these solutions.
Generalised Linear Models
The concept of generalised linear models (GLMs), which have a range of applications in the biomedical, natural and social sciences, and can be used to relate a response variable to one or more explanatory variables, will be explored. The response variable may be classified as quantitative (continuous or discrete, i.e. countable) or categorical (two categories, i.e. binary, or more than categories, i.e. ordinal or nominal).
Students will come to understand the effect of censoring in the statistical analyses and will use appropriate statistical techniques for lifetime data. They will also become familiar with the programme R, which they will have the opportunity to use in weekly workshops.
Geometry of Curves and Surfaces
The topic of smooth curves and surfaces in three-dimensional space is introduced. The various geometrical properties of these objects, such as length, area, torsion and curvature, will be explored and students will have the opportunity to discover the meaning of these quantities. They will then use a variety of examples to calculate these values, and will use those values to apply techniques from calculus and linear algebra.
A number of well-known concepts will be encountered, such as length and area, and some new ideas will be introduced, including the curvature and torsion of a curve, and the first and second fundamental forms of a surface. Students will learn how to compute these quantities for a variety of examples, and in doing so will develop their geometric intuition and understanding.
The study of graphs - mathematical objects used to model pairwise relations between objects - is a cornerstone of discrete mathematics. As a result, students will develop an appreciation for a range of discrete mathematical techniques while undertaking this module.
Throughout the module, students will also learn about structural notions, such as connectivity, and will explore trees, minor closed families of graphs, matrices related to graphs, the Tutte polynomial of small graphs, and planar graphs and analogues.
While studying these areas, students will gain experience of following and constructing mathematical proofs, and correctly and coherently using mathematical notation.
Groups and Symmetry
Students will develop the knowledge of groups that they gained in second year during the Groups and Rings module. 'Direct products', which are used to construct new groups, will be studied, while any finite group will be shown to 'factor' into 'simple' pieces.
Situations will be considered in which a group 'acts' on a set by permuting its elements; this powerful idea is used to identify the symmetries of the Platonic solids, and to help study the structure of groups themselves.
Finally, students will prove some interesting and important results, known as 'Sylow's theorems', relating to subgroups of certain orders.
Students will examine the notion of a norm, which introduces a generalised notion of 'distance' to the purely algebraic setting of vector spaces. They will learn several quite natural ways to do this, both for vectors of any dimension and for functions. Focus will then be on the more special notion of an inner product which generalises angles at the same time as distances.
Once these concepts have been established, students will have the opportunity to study geometrical ideas like orthogonality, which can be seen to apply to much more general spaces than Euclidean spaces of three (or even n) dimensions, notably to infinite dimensional spaces of functions. For example, Hilbert space theory shows how Fourier series are really another case of expressing an element in terms of a basis, and how people can use orthogonality to find best approximations to a given function by functions of a prescribed type. Finally, students will look at some of the main results of linear algebra, which generalise very nicely to linear operators between Hilbert spaces.
Lebesgue Integration
Introducing the Lebesgue integral for functions on the real line, this module features a classical approach to the construction of Lebesgue measure on the line and to the definition of the integral. The bounded convergence theorem is used to prove the monotone and dominated convergence theorems, and the results are illustrated in classical convergence problems including Fourier integrals.
Among the range of topics addressed on this module, students will become familiar with Lebesgue's definition of the integral, and the integral of a step function. There will be an introduction to subsets of the real line, including open sets and countable sets. Students will measure of an open set, and will discover measurable sets and null sets. Additionally, the module will focus on integral functions, along with Lebesgue's integral of a bounded measurable function, his bounded convergence theorem and the integral of an unbounded function. Dominated convergence theorem; monotone convergence theorem.
Other topics on the module will include applications of the convergence theorems and Wallis's product for P. Gaussian integral, along with some classical limit inversion results and the Fourier cosine integral. Students will develop an understanding of Dirichlet's comb function, Archimedes' axiom and Cantor's uncountability theorem, and will learn to prove the structure theorem for open sets. In addition, students will be able to prove covering lemmas for open sets, as well as understanding the statement of Heine—Bore theorem, as well as understanding the concept and proving basic properties of outer measure. As well as understanding inner measure. Finally, students will be expected to prove Lebesgue's theorem on countable additivity of measure.
Likelihood Inference
Statistical inference is the theory of the extraction of information about the unknown parameters of an underlying probability distribution from observed data. Consequently, statistical inference underpins all practical statistical applications.
This module reinforces the likelihood approach taken in second year Statistics for single parameter statistical models, and extends this to problems where the probability for the data depends on more than one unknown parameter.
Students will also consider the issue of model choice: in situations where there are multiple models under consideration for the same data, how do we make a justified choice of which model is the 'best'?
The approach taken in this course is just one approach to statistical inference: a contrasting approach is covered in the Bayesian Inference module.
The aim of this module is to provide third year students with more options of applicable topics which draw upon second year pure mathematics modules and provide opportunities for further study. The theory of linear systems is engineering mathematics.
In the mid nineteenth century, the engineer Watt used a governor to control the amount of steam going into an engine, so that the input of steam reduced when the engine was going too quickly, and the input increased when the engine was going too slowly. Maxwell then developed a theory of controllers for various mechanical devices, and identified properties such as stability. The crucial idea of a controller is that the output can be fed back into the system to adjust the input.
Many devices can be described by linear systems of differential and integral equations which can be reduced to a standard (A,B,C,D) model. These include electrical appliances, heating systems and economic processes. The module shows how to reduce certain linear systems of differential equations to systems of matrix equations and thus solve them. Linear algebra enables students to classify (A,B,C,D) models and describe their properties in terms of quantities which are relatively easy to compute.
The module then describes feedback control for linear systems. The main result describes all the linear controllers that stabilise a (A,B,C,D) system.
Using the classical problem of data classification as a running example, this module covers mathematical representation and visualisation of multivariate data; dimensionality reduction; linear discriminant analysis; and Support Vector Machines. While studying these theoretical aspects, students will also gain experience of applying them using R.
An appreciation for multivariate statistical analysis will be developed during the module, as will an ability to represent and visualise high-dimensional data. Students will also gain the ability to evaluate larger statistical models, apply statistical computer packages to analyse large data sets, and extract and evaluate meaning from data.
Mathematical Education
Students will be given an opportunity to consider key issues in the teaching and learning of mathematics during this module. Whilst it is an academic study of mathematics education and not a training course for teachers, it does provide an excellent foundation for a PGCE especially in preparing students to write academically.
Having studied mathematics for many years, students are well-placed to reflect upon that experience and attempt to make sense of it in the light of theoretical frameworks developed by researchers in the field. This module will help them with this process so that as mathematics graduates they will be able to contribute knowledgeably to future debate about the ways in which this subject is treated within the education system.
Mathematical Education Placement
This module is run as a partnership between the Department of Mathematics and Statistics, the University of Cumbria and the Students' Union's volunteering unit.
Designed with employability in mind, this module is based on the Students' Union's Schools Partnership Scheme, which supports Lancaster students on 10-week placements in local primary and secondary schools.
Students will have the opportunity to take part in classroom observation and assistance, the development of classroom resources, the provision of one-on-one or small group support and possibly even teaching sections of lessons to the class as a whole.
Mathematics for Stochastic Finance
This module formally introduces students to the discipline of financial mathematics, providing them with an understanding of some of the maths that is used in the financial and business sectors.
Students will begin to encounter financial terminology and will study both European and American option pricing. The module will cover these in relation to discrete and continuous financial models, which include binomial, finite market and Black-Scholes models.
Students will also explore mathematical topics, some of which may be familiar, specifically in relation to finance. These include:
Conditional expectation
Stopping times
Black-Scholes formula
Throughout the module, students will learn key financial maths skills, such as constructing binomial tree models; determining associated risk-neutral probability; performing calculations with the Black-Scholes formula; and proving various steps in the derivation of the Black-Scholes formula. They will also be able to describe basic concepts of investment strategy analysis, and perform price calculations for stocks with and without dividend payments.
In addition, to these subject specific skills and knowledge, students will gain an appreciation for how mathematics can be used to model the real-world; improve their written and oral communication skills; and develop their critical thinking.
Medical Statistics: study design and data analysis
The aim is to introduce students to the study designs and statistical methods commonly used in health investigations, such as measuring disease, causality and confounding.
Students will develop a firm understanding of the key analytical methods and procedures used in studies of disease aetiology, appreciate the effect of censoring in the statistical analyses, and use appropriate statistical techniques for time to event data.
They will look at both observational and experimental designs and consider various health outcomes, studying a number of published articles to gain an understanding of the problems they are investigating as well as the mathematical and statistical concepts underpinning inference.
Metric Spaces
An introduction to the key concepts and methods of metric space theory, a core topic for pure mathematics and its applications, is given during this module. Studying this module will give students a deeper understanding of continuity as well as a basic grounding in abstract topology. With this grounding, they will be able to solve problems involving topological ideas, such as continuity and compactness.
They will also gain a firm foundation for further study of many topics including geometry, Lie groups and Hilbert space, and learn to apply their knowledge to areas including probability theory, differential equations, mathematical quantum theory and the theory of fractals.
Number theory is the study of the fascinating properties of the natural number system.
Many numbers are special in some sense, eg. primes or squares. Which numbers can be expressed as the sum of two squares? What is special about the number 561? Are there short cuts to factorizing large numbers or determining whether they are prime (this is important in cryptography)? The number of divisors of an integer fluctuates wildly, but is there a good estimation of the 'average' number of divisors in some sense?
Questions like these are easy to ask, and to describe to the non-specialist, but vary hugely in the amount of work needed to answer them. An extreme example is Fermat's last theorem, which is very simple to state, but was proved by Taylor and Wiles 300 years after it was first stated. To answer questions about the natural numbers, we sometimes use rational, real and complex numbers, as well as any ideas from algebra and analysis that help, including groups, integration, infinite series and even infinite products.
This module introduces some of the central ideas and problems of the subject, and some of the methods used to solve them, while constantly illustrating the results with exercises and examples involving actual numbers.
This module is ideal for students who want to develop an analytical and axiomatic approach to the theory of probabilities.
First the notion of a probability space will be examined through simple examples featuring both discrete and continuous sample spaces. Random variables and the expectation will be used to develop a probability calculus, which can be applied to achieve laws of large numbers for sums of independent random variables.
Students will also use the characteristic function to study the distributions of sums of independent variables, which have applications to random walks and to statistical physics.
This module gives students the opportunity to enhance their project skills, including both subject-related and transferable skills such as oral communication and scientific writing. They will work on mathematical document preparation and presentation, and with a statistical software package.
The module will revisit LaTeX and R, which will be used to prepare mathematical documents; text and mathematical symbols, displayed formulae, numbering, environments, lists, page and document layout, sections and table of contents, and tables and figures. In addition to this, students will undertake a short project, investigating a mathematical or statistical topic, and a group project, involving a group investigation of a mathematical or statistical topic under the direction of a supervisor.
Representation Theory of Finite Groups
Students will cover the basics of ordinary representation theory. Two approaches are presented: representations as group homomorphisms into matrix groups, and as modules over group algebras. The correspondence between the two will be discussed.
The second part is an introduction to the ordinary character theory of finite groups, intrinsic to representation theory. Students will learn the concepts of R-module and of group representations, the main results pertaining to group representations, and will handle basic applications in the study of finite groups.
They will also learn to perform computations with representations and morphisms in a selection of finite groups
Important examples of stochastic processes, and how these processes can be analysed, will be the focus of this module.
As an introduction to stochastic processes, students will look at the random walk process. Historically this is an important process, and was initially motivated as a model for how the wealth of a gambler varies over time (initial analyses focused on whether there are betting strategies for a gambler that would ensure they won).
The focus will then be on the most important class of stochastic processes, Markov processes (of which the random walk is a simple example). Students will discover how to analyse Markov processes, and how they are used to model queues and populations.
Modern statistics is characterised by computer-intensive methods for data analysis and development of new theory for their justification. In this module students will become familiar with topics from classical statistics as well as some from emerging areas.
Time series data will be explored through a wide variety of sequences of observations arising in environmental, economic, engineering and scientific contexts. Time series and volatility modelling will also be studied, and the techniques for the analysis of such data will be discussed, with emphasis on financial application.
Another area the module will focus on is some of the techniques developed for the analysis of multivariates, such as principal components analysis and cluster analysis.
Lastly,students will spend time looking at Change-Point Methods, which include traditional as well as some recently developed techniques for the detection of change in trend and variance.
This module is an introduction to elliptic curves, and hence students will have the opportunity to learn the basics of algebraic geometry. It also presents applications and results of the theory of elliptic curves and provides a useful link between concepts from algebra and geometry.
Students will look at how curves can be described by algebraic equations, and will develop an understanding of abstract groups, learning how to use them to deal with geometrical objects (curves). They will also investigate the way that algebra and geometry are linked via polynomial equations, performing algebraic computations with elliptic curves.
Assessing Financial Risk: Extreme Value Methods
Assessment of financial risk requires accurate estimates of the probability of rare events. Estimating the probability of such "extreme" events is challenging, as by nature they are sufficiently rare that there is little direct empirical evidence on which to base inference. Instead we have to extrapolate based on the past frequency of the occurrence of less extreme events. This module covers ideas from Extreme Value Theory which give a sound mathematical basis to such extrapolation, and shows practically how it can be used to give accurate assessments of financial risk in a wide range of scenarios.
In this module, students will study topics related to the understanding of special models to describe the extreme values of a financial times series, and they will learn to fit appropriate extreme value models to data which are maxima or threshold exceedance. Extreme value models will be used to evaluate Value at Risk and gain an understanding of the impact of heavy tailed data on standard statistical diagnostic tools.
Bayesian statistics is a framework for rational decision making using imperfect knowledge, expressed through probability distributions. Bayesian principles are applied in the fields of navigation, control, automation and artificial intelligence. The aim of decision makers is to make rational decisions that maximise some personal utility function which may represent quantities such as money which are related to the wealth of an individual.
Within the Bayesian framework, knowledge of the world, (the prior) is updated as fresh observations arrive to yield a posterior distribution which shows the revised knowledge. The evidence for the model is expressed by calculating a marginal likelihood. Future behaviour and the fit of the model are assessed using a predictive distribution. This includes sampling uncertainty and uncertainty of our knowledge.
In this module students will look at the posterior, the marginal and the predictive distributions for several one parameter conjugate models, and two families of multi-parameter fully conjugate models. The range of belief types that can be modelled by using mixtures of conjugate priors will be extended, and will also explore the use of non-conjugate formulations of models and use Monte-Carlo integration, importance sampling and rejection sampling for calculating and simulating from these distributions.
Clinical trials are planned experiments on human beings designed to assess the relative benefits of one or more forms of treatment. For instance, we might be interested in studying whether aspirin reduces the incidence of pregnancy-induced hypertension; or we may wish to assess whether a new immunosuppressive drug improves the survival rate of transplant recipients. Treatments may be procedural, for example, surgery or methods of care.
This module combines the study of technical methodology with discussion of more general research issues. First , the relative advantages and disadvantages of different types of medical studies will be discussed. Then students will explore the basic aspects of clinical trials as experimental designs, looking in particular at the definition and estimation of treatment effects. They will also cover cross-over trials, concepts of sample size determination, and equivalence trials. The module also includes a brief introduction to sequential trial designs and meta-analysis.
Computer Intensive Methods
Students will be introduced to Markov chain Monte Carlo methods and how to use them as a powerful technique for performing Bayesian inference on complex stochastic models.
The first part of the module looks in detail at the necessary concepts and theory for finite state-space Markov chains, before introducing analogous concepts and theory for continuous state-space Markov chains. In the second part of the course module the Metropolis-Hastings algorithm for sampling from a distribution known up to a constant of proportionality will be investigated.
In the third (and largest) part, students will take this knowledge and apply it to Bayesian inference as well as studying the Gibbs sampler. They will also examine the two most common Metropolis-Hastings algorithms (the random walk and the independence sampler). Examples will include hierarchical models, random effects models, and mixture models.
This module focuses on the kinds of statistical methods commonly used by statisticians to investigate the relationship between risk of disease and environmental factors.
You'll explore methods for the analysis of spatial data, including spatial point-process models, spatial case-control methods, spatially aggregated data, point source problems and geostatistics, and will spend time developing your skills performing similar analyses using the statistical package R.
Galois Theory is, in essence, the systematic study of properties of roots of polynomials. Starting with such a polynomial f over a field k (e.g. the rational numbers), one associates a 'smallest possible' field L containing k and the roots of f; and a finite group G which describes certain 'allowed' permutations of the roots of f. The Fundamental Theorem of Galois Theory says that under the right conditions, the fields which lie between k and L are in 1-to-1 correspondence with the subgroups of G.
In this module students will see two applications of the Fundamental Theorem. The first is the proof that in general a polynomial of degree 5 or higher cannot be solved via a formula in the way that quadratic polynomials can; the second is the fact that an angle cannot be trisected using only a ruler and compasses. These two applications are among the most celebrated results in the history of mathematics.
In this module students will learn techniques for formulating sensible models for data, enabling you to tackle problems such as the probability of success for a particular treatment, and how this depends on the patient's age, weight, blood pressure, and so on.
Students will be introduced to a large family of models, called the generalised linear models (GLMs), including the standard linear regression model as a special case, and will have the opportunity to discuss and investigate the theoretical properties of these models.
A common algorithm called iteratively reweighted least squares algorithm for the estimation of parameters will be studies. Using the statistical package 'R', you'll fit and check these models, and will produce confidence intervals and tests corresponding to questions of interest.
Students will have the opportunity to learn about Hilbert space, consolidating their understanding of linear algebra and enabling them to study applications of Hilbert space such as quantum mechanics and stochastic processes.
The module will teach students how to use inner products in analytical calculations, to use the concept of an operator on an infinite dimensional Hilbert space, to recognise situations in which Hilbert space methods are applicable and to understand concepts of linear algebra and analysis that apply in infinite dimensional vector spaces.
Industrial Dissertation
At the end of Year 3 students will complete a form stating their mathematical or statistical interests and based on that will be assigned a dissertation supervisor (a member of staff) and a topic. The dissertation may be in mathematics (MATH491), statistics (MATH492), or on an industrial project (MATH493), which is in cooperation with an external industrial partner. This depends on the degree scheme and choice.
In the first term there will be weekly supervisor meetings and students will be guided into their in-depth study of a specific topic. During the second term students will write a dissertation on what has been learnt and give an oral presentation. The dissertation will be submitted in the first week after the Easter recess. The grade is based 70% on the final written product, 10% on oral presentation, and 20% on the initiative and effort that is demonstrated during the entire two terms of the module.
Further information is available from the Year 4 Director of Studies and will be communicated to every Year 4 student at the beginning of Term 1.
Students will construct Lebesgue measure on the line, extending the idea of the length of an interval. They will use this to define an integral which is shown to have good properties under pointwise convergence. By looking at some basic results about the set of real numbers, properties of countable sets, open sets and algebraic numbers will be explored.
The opportunity will be given to illustrate the power of the convergence theorems in applications to some classical limit problems and analysis of Fourier integrals, which are fundamental to probability theory and differential equations.
Lie Groups and Lie Algebras
The theory of Lie groups and Lie algebras will be introduced during this module. The relationship between the two will be explored, and students will develop an understanding of the way that this forms an important and enduring part of modern mathematics and a great number of fields including theoretical physics. They will learn to appreciate the subtle and pervasive interplay between algebra and geometry, and to appreciate the unified nature of mathematics.
The abstract nature of the module will give them a taste of modern research in pure mathematics. At the end of the module, students will gain understanding of the structure theory of Lie algebras, manifolds and Lie groups. They will also gain basic knowledge of representations of Lie algebras.
In this module, students will learn how to use the likelihood function to obtain and summarise information about unknown parameters. They will calculate the likelihood function for statistical models which do not assume independent identically distributed data, and will learn to evaluate point estimates and make statements about the variability of these estimates.
Using the statistical package 'R',computational methods will be used to calculate maximum likelihood estimates. Time will be spent developing an understanding of the inter-relationships between parameters, and the concept of orthogonality. In addition to this, hypothesis tests will be performed using the generalised likelihood ratio statistic.
Longitudinal Data Analysis
Longitudinal data arise when a time-sequence of measurements is made on a response variable for each of a number of subjects in an experiment or observational study. For example, a patient's blood pressure may be measured daily following administration of one of several medical treatments for hypertension.
Typically, the practical objective of most longitudinal studies is to find out how the average value of the response varies over time, and how this average response profile is affected by different experimental treatments. This module presents an approach to the analysis of longitudinal data, based on statistical modelling and likelihood methods of parameter estimation and hypothesis testing.
Mathematics Dissertation
At the end of Year Three students will fill in a form stating their mathematical or statistical interests,and based on that they will be assigned a dissertation supervisor (a member of staff) and a topic. The dissertation may be in mathematics, statistics, or on an industrial project, which is in cooperation with an external industrial partner. This will depend on the student's degree scheme and personal choice.
In the first term they will meet their supervisor weekly and will be guided into their in-depth study of a specific topic. During the second term they will have to produce a written dissertation on what they have learned and give an oral presentation. The dissertation will be handed in on the first week after the Easter recess. The grade is based 70% on the final written product, 10% on the oral presentation, and 20% on the initiative and effort that the student has demonstrated during the entire two terms of the module.
Further information is available from the Year Four Director of Studies and will be communicated to every Year Four student at the beginning of term one.
In this module,students will construct Lebesgue measure on the line, extending the idea of the length of an interval. They will use this to define an integral which is shown to have good properties under pointwise convergence. Looking at some basic results about the set of real numbers, students will explore properties of countable sets, open sets and algebraic numbers.T
They will also have the opportunity to illustrate the power of the convergence theorems in applications to some classical limit problems and analysis of Fourier integrals, which are fundamental to probability theory and differential equations.
Operator Theory
Operator theory is a modern mathematical topic in analysis which provides powerful general methods for the analysis of linear problems, and possibly even infinite dimensional problems.
Early successes were in the solution of differential and integral equations. Now operator theory is also an extensive subject in its own right in the general area of functional analysis.
First students will review Hilbert spaces, before spending some time studying infinite-dimensional operators, notably the unilateral shift and multiplication operators, as well as basic concepts. They will then consider the criteria for invertibility of self adjoint operators, leading to the spectral theory of such operators.
Principles of Epidemiology
This module focuses on the basic principles of epidemiology, including its methodology and application to prevention and control of disease.The concepts and strategies used in epidemiologic studies will be examined, and students will come to appreciate a historical and general overview of epidemiology and related strategies for study design. They will also develop their knowledge of how to apply basic epidemiologic methods, and will gain confidence in assessing the validity of epidemiologic studies with respect to their design and inferences. Most inference will be likelihood based, although the emphasis is on conceptual considerations and interpretation.
The aim of this module is to develop an analytical and axiomatic approach to the theory of probabilities.
Students will consider the notion of a probability space, illustrated by simple examples featuring both discrete and continuous sample spaces. They will then use random variables and the expectation to develop a probability calculus, which is applied to achieve laws of large numbers for sums of independent random variables. Finally the characteristic function will be used to study the distributions of sums of independent variables, applying the results to random walks and to statistical physics.
In this module students will learn the basics of ordinary representation theory. Students will have the opportunity to explore the concepts of R-module and group representations, and the main results pertaining to group representations, as well as learning to handle basic applications in the study of finite groups. They will also develop their skills in performing computations with representations and morphisms in a selection of finite groups.
The first part of the module is an introduction to the ordinary representation theory of finite groups. Two approaches are presented: representations as group homomorphisms into matrix groups, and as modules over group algebras. The correspondence between both is discussed and special examples and constructions are studied.
The second part of the module concerns the ordinary character theory of finite groups, intrinsic to representation theory. The main objectives are to prove the orthogonality relations of characters and construct the character table of a finite group.
Stochastic Calculus for Finance
Stochastic calculus is a theory that enables the calculation of integrals with respect to stochastic processes. This module begins with the study of discrete-time stochastic processes, defining key concepts such as martingales and stopping times. This then leads on to the exploration of continuous-time processes, in particular, Brownian motion.
Students will learn to derive basic properties of Brownian motion and explore integration with respect to it. They will also examine the derivation of Ito's formula and how to apply this to Brownian motion.
Over the course of the module, students will also learn to justify and critique the use of stochastic models for real-life applications, and to use the stochastic calculus framework to formulate and solve problems involving uncertainty – a skill that underpins financial mathematics.
This module shows how the rules of probability can be used to formulate simple models describing processes, such as the length of a queue, which can change in a random manner, and how the properties of the processes, such as the mean queue size, can be deduced.
In Stochastic Processes students will learn how to use conditioning arguments and the reflection principle to calculate probabilities and expectations of random variables. They will also learn to calculate the distribution of a Markov Process at different time points and to calculate expected hitting times, as well as how to determine whether a Markov process has an asymptotic distribution and how to calculate it. Finally, they will develop an understanding of how stochastic processes are used as models.
Topology and Fractals
Fractals, roughly speaking, are strange and exotic sets in the plane (and in higher dimensions) which are often generated as limits of quite simple repeated procedures. The 'middle thirds Cantor set' in [0,1] is one such set. Another, the Sierpinski sieve, arises by repeated removal of diminishing internal triangles from a solid equilateral triangle.
This analysis module will explore a variety of fractals, partly for fun for their own sake but also to illustrate fundamental ideas of metric spaces, compactness, disconnectedness and fractal dimension. The discussion will be kept at a straightforward level and you'll consider topological ideas of open and closed sets in the setting of R^2.
Maths graduates are very versatile, having in-depth specialist knowledge and a wealth of skills. Through this degree, you will graduate with a comprehensive skill set, including data analysis and manipulation, logical thinking, problem-solving and quantitative reasoning, as well as adept knowledge of the discipline. As a result, mathematicians are sought after in a range of industries, such as business and finance, defence, education, infrastructure and power, and IT and communications.
The starting salary for many maths graduate roles is highly competitive, and career options include:
System Developer
A degree in this discipline can also be useful for roles such as Finance Manager, Insurance Underwriter, Meteorologist, Quantity Surveyor, Software Developer, and many more.
Alternatively, you may wish to undertake PhD level study at Lancaster and pursue a career in research and teaching.
Undergraduate tuition fees
For students starting at the University for the 2020 session, subsequent year's fees may be subject to increases. UK fees are set by the UK Government annually. For international applicants, any annual increase will be capped at 4% of the previous year's fee. For more information about tuition fees, including fees for Study Abroad and Work Placements, please visit our undergraduate tuition fees page.
Applicants from the Channel Islands and the Isle of Man
Some science and medicine courses have higher fees for students from the Channel Islands and the Isle of Man. You can find more information about this on our Island Fees page.
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Graduate Statistician status is awarded by the Royal Statistical Society to successful graduates who complete the required number of Part II statistics modules.
Further details about accreditations Learn More
Completion of this programme can lead to recommendation for exemption from some examinations with the Institute and Faculty of Actuaries.
The Institute of Mathematics and its Applications is a chartered professional body for mathematicians in the UK. Accreditation means that our degrees demonstrate both a high level of competency and professionalism in the area of mathematics.
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Which A Levels should I be studying?
Our minimum requirement for A Level applicants is that you should be studying at least three A Levels including A Level Mathematics. We encourage but do not require, you to study A Level Further Mathematics as well - our standard offer is slightly lower for those taking A Level Further Mathematics. If it is not possible for you to study A Level Further Mathematics, think about studying AS Level Further Mathematics - the style of mathematics in Further Mathematics, especially Further Pure Mathematics, is similar to university mathematics and will be excellent preparation for further study.
We do not usually accept vocational qualifications like BTEC or Cambridge Technicals unless you are also studying A Level Mathematics and A Level Further Mathematics. If this applies to you, then we would treat your vocational qualification as equivalent to a third A Level.
What about STEP?
The Sixth Term Examination Papers in Mathematics are a set of three papers, set by Cambridge Assessment and sat in June, which are designed to really test your problem-solving skills. Each paper lasts three hours and contains thirteen relatively long questions, all of which are optional - you answer as many of them as you wish, up to a maximum of six questions. You may enter any combination of the three papers - those applying to Cambridge will usually sit Paper 2 and Paper 3, while those applying to Warwick or Bath will usually sit Paper 1 plus, optionally, one or both of the other papers.
We value the way that STEP develops advanced problem-solving skills and all our standard offers include an alternative, slightly lower offer that includes a pass (Grade 3) in any STEP. Sitting STEP is optional, however, and if you choose not to sit STEP, this won't harm your chances of receiving an offer from us. If you sit STEP and do not pass, please do not worry - we will treat you exactly the same as if you hadn't sat STEP at all.
What about the Test of Maths for University Admission?
The Test of Maths for University Admission is a test, set by Cambridge Assessment and sat in November. Like STEP, the Test of Maths for University Admission is designed to test your problem-solving and readiness for university-level mathematics, but the format is very different - it is made up of two multiple-choice question papers, each lasting 75 minutes. Results are available at the end of November - you will receive a grade ranging from 1.0 to 9.0.
A strong performance in the Test of Maths for University Admission is very impressive. If you sit the Test and perform well, achieving a score above 4.5, you may well receive a lower offer from us. Sitting the Test is optional, however, and if you choose not to sit the Test, this won't harm your chances of receiving an offer from us. Indeed:
you do not need to tell us that you're sitting the Test;
if you sit the test, you do not need to tell us your result; and
a poor result will not damage your chances - we will treat you the same as if you hadn't sat the Test at all.
If you inform us of your Test result before receiving an offer from us and have achieved a score of 4.5 or higher, we will take this into account when we decide on your application, and we may well make you a lower offer than usual.
If you inform us of your Test result after receiving an offer from us and have achieved a score of 4.5 or higher, we will consider your result, and may well replace your existing offer with a lower one.
Our standard offer
The standard offers for courses administered by our Department are outlined in the table below.
STEP/TMUA
Including Maths at Grade A
Including Maths and Further Maths at least one at Grade A
Mathematics BSc
Mathematics BSc (Placement Year)
Mathematics MSci
Mathematics MSci (Study Abroad)
Statistics BSc
Statistics BSc (Placement Year)
Statistics MSci
Statistics MSci (Study Abroad)
Mathematics with Statistics BSc
Mathematics with Statistics BSc (Placement Year)
Mathematics with Statistics MSci
Mathematics with Statistics MSci (Study Abroad)
Financial Mathematics BSc
Financial Mathematics BSc (Industry)
Financial Mathematics MSci
Computer Science and Mathematics BSc
Computer Science and Mathematics MSci G100
G1G3
GCG3
G1GJ
G1GH
GN1J
GN1H
GG1K Not taken AAA AAB
GG1K STEP Level 3 +
or TMUA 4.5+ AAB ABB
Mathematics and Philosophy BA GV15 Not taken AAB ABB
Mathematics and Philosophy BA GV15 STEP Level 3 +
or TMUA 4.5+ ABB ABB
French Studies and Mathematics BA
German Studies and Mathematics BA
Spanish Studies and Mathematics BA GR11
GR14 N/A AAB
(Maths Grade A, Language Grade B) ABB
(Maths Grade A, Language Grade B)
Accounting, Finance and Mathematics BSc
Accounting, Finance and Mathematics BSc (Industry)
Economics and Mathematics BSc NG41
GL11 N/A AAB including Maths or Further Maths Grade A
BSc Theoretical Physics with Mathematics F3GC N/A AAB including Maths and Physics
MSci Theoretical Physics with Mathematics
MSci Theoretical Physics with Mathematics (Study Abroad) F3G1
F3G5 N/A AAA including Maths and Physics
A place for Laura
The diversity of opportunities at Lancaster has very much appealed to me during my time here. I have tried to make the most of these by taking part in everything from part-time work and volunteering to societies. As a Student Ambassador and Tutor, I've met new people and formed friendships that I otherwise may not have had. The part-time jobs have provided me with the extra income that I worried about having before I came to university, as well as many great opportunities to network and impress interviewers for roles.
Lancaster has helped me achieve all I have wanted and more. I have gained an insight experience at Deutsche Bank during the second year, allowing me to see how an investment bank operates - including experiencing the technology department of the trading floor of the company, which was such a surreal experience!
Last summer, I spent 10 weeks as an intern at the Royal Bank of Scotland in Edinburgh. This took place during the World Cup, which added an extra element of excitement and bonding between interns. I successfully obtained a graduate job offer. This meant that I had an offer for a job after university before even beginning my final year, which has been such a relief. This was all paid too, allowing me to save for my future.
My advice would be to grab every opportunity you can, as you never know where it may lead. This has led me to uncover key skills, such as work-life balance, time management and communication that I hope will serve me well in my future.
Laura Patterson
How you'll learn
Lectures will introduce you to course content and you will usually have four of these per week in each module. These are in large groups with fellow students from across the year group. Whilst the lecturer leads this form of teaching, we do encourage you to take part.
Our weekly workshops give you the opportunity to get enhanced support and guidance. You will work in small groups with specialist tutors. You will develop an understanding of module content and practise applying the skills you have gained.
In your first year, problem-solving classes will develop your skills to tackle university-style mathematics. Working in small groups, you will apply your mathematical knowledge to a set of problems.
Some practical work is undertaken in specialist computer labs. This involves working with statistical and mathematical software to develop skills and enhance your employability.
Completing assessments is a key part of your learning. Alongside main exams, you will also complete regular homework sheets for your modules. These allow you to track your progress and identify areas to work on with your workshop tutor.
Academic staff have dedicated office hours for student queries and for extra support. You will receive feedback on all work submitted and teaching staff are happy to discuss this with you if you have any concerns.
Academic tutor
When you start at Lancaster we will assign you an academic tutor, who will (where possible) remain your tutor for the duration of your studies at Lancaster. This is an academic member of staff who you will meet with once a term to check your progress and personal development.
Teaching office
This is your one-stop-shop for enquiries ranging from timetabling and exams to module choices and coursework. The office can also offer more general support, and if they don't have the solution then they will know who to contact.
Several of our degree programmes offer the opportunity to spend a year abroad studying at one of our partner institutions. Destinations vary each year, with past students staying in the USA, Canada, New Zealand and Australia.
We choose our partner universities to ensure that the course content of your third year complements the rest of your degree programme. You will study modules that link to those taught at Lancaster. You will have the opportunity to study specialist modules offered by our partner universities.
The modules that you study during your year abroad will count towards your final degree classification.
A year abroad will highlight to potential employers that you are flexible, adaptable and independent. You will also show cultural awareness and the ability to work with people from a variety of backgrounds.
Skills for your future
A degree in mathematics will provide you with both a specialist and transferable skill set sought after by employers across a wide range of sectors.
We are committed to developing your employability skills. Our dedicated Careers Officer works in partnership with the University's Careers Service to offer a range of workshops and talks. You can also access 1:1 appointments throughout the year through the University's Careers Service.
The education sector has an increasing demand for mathematics graduates to inspire the next generation of students. Our third year module in Mathematical Education provide an insight into what it would be like to complete a PGCE qualification after your degree.
Project Skills module
Our second year Project Skills module develops skills that will enhance your employability. This module includes coursework on scientific writing and using LaTeX software to prepare mathematical documents – complementing your pure mathematics and statistics knowledge.
Choosing a Placement or Industry pathway degree involves spending the third year of your four year degree working full-time in a business. Many students find that a placement year helps them to decide which career path they would like to take. The experience will give you a strong advantage when looking for employment after your degree.
Internship scheme
Undertaking relevant work experience while you are at university helps you to apply for graduate-level jobs. Through our Internship Scheme, you can apply for paid work placements. These give you the opportunity to practice the skills and knowledge learned during your degree. These opportunities can be both full and part-time, and range from 3 months to a year.
Accounting, Finance and Mathematics BSc Hons: NG41
Accounting, Finance and Mathematics (Industry) BSc Hons: NG42
Computer Science and Mathematics BSc Hons: GG14
Computer Science and Mathematics MSci Hons: GG1K
Computer Science and Mathematics (Placement Year) BSc Hons: GG1L
Economics and Mathematics BSc Hons: GL11
Economics and Mathematics (Industry) BSc Hons: GL12
Financial Mathematics BSc Hons: GN13
Financial Mathematics MSci Hons: GN1H
Financial Mathematics (Placement Year) BSc Hons: GN1J
French Studies and Mathematics BA Hons: GR11
German Studies and Mathematics BA Hons: GR12
Mathematics BSc Hons: G100
Mathematics (Placement Year) BSc Hons: G102
Mathematics (Study Abroad) MSci Hons: G103
Mathematics and Philosophy BA Hons: GV15
Mathematics with Statistics BSc Hons: G1G3
Mathematics with Statistics MSci Hons: G1GJ
Mathematics with Statistics (Placement Year) BSc Hons: GCG3
Mathematics with Statistics (Study Abroad) MSci Hons: G1GH
Mathematics, Operational Research, Statistics and Economics (MORSE) BSc Hons: GLN0
Mathematics, Operational Research, Statistics and Economics (MORSE) (Industry) BSc Hons: GLN1
Spanish Studies and Mathematics BA Hons: GR14
Theoretical Physics with Mathematics BSc Hons: F3GC
Theoretical Physics with Mathematics MSci Hons: F3G1
Theoretical Physics with Mathematics (Study Abroad) MSci Hons: F3G5
The information on this site relates primarily to 2020/21 entry to the University and every effort has been taken to ensure the information is correct at the time of publication. The University will use all reasonable effort to deliver the courses as described, but the University reserves the right to make changes to advertised courses.
In the event of a course being withdrawn or if there are any fundamental changes to your course, we will give you reasonable notice and you will be entitled to withdraw your application. You are advised to revisit our website for up-to-date information before you submit your application. Further legal information.
The amount of time you spend in lectures, seminars and similar will differ from year to year. Taken as an average over all years of the course, you will spend an average of 10.6 hours per week in lectures, seminars and similar during term time.
A broad range of assessments methods will be used throughout the degree. As a guide, 30% of assessment is by coursework over the duration of the course.
Our Students' Charter
We believe in the importance of a strong and productive partnership between our students and staff. In order to ensure your time at Lancaster is a positive experience we have worked with the Students' Union to articulate this relationship and the standards to which the University and its students aspire. View our Charter and other policies.
A top 10 university with a global reputation.
Our campus and the surrounding area is a great place to call home.
Your college will be your home away from home.
Career support for our students through university and beyond.
Build your global community on campus and around the world.


The following reflection on Genesis 15:18 comes from the handwritten notes in my ESV Journaling Bible, Interleaved Edition.
15:18: "I give this land": Or "I have given this land" (NASB; see footnote in ESV). In other words, there's a sense in which God has already accomplished what he promises to do in the present. God is already in the future, after all, so there's no doubt how his promise to Abram will turn out.
Do you see how encouraging this is?
When God says that something will happen, we can be confident that it will happen this way—indeed, from God's perspective it already has happened. When he makes this promise to Abram, he has already done everything necessary to ensure that Abram's descendants have received the land; because God is already in the future, seeing that his promise is fulfilled and waiting for Abram to arrive there himself.
By contrast, when I want to assure someone that I will fulfill a promise, I might say, "Don't give it a second thought. It's done!" But it's not done. In fact, I could drop dead before I fulfill the promise, no matter how sincerely I give my word.
This means that the blessing that awaits us, on the other side of whatever trial we happen to be going through at this moment, is as certain as the stars in the sky (v. 5).
This entry was posted on	April 10, 2019 at 1:44 pm	and is filed under Uncategorized.
Thanks, brother. I love you and appreciate your prayers.


text:
3.6.1.1 Bibliographical Information
Chapter Contents (Back)
Rosenfeld, A.[Azriel],
Picture Processing by Computer: Survey,
Surveys(1), No. 3, September 1969, pp. 147-176. BibRef 6909
Progress in Picture Processing: 1969-71,
Surveys(5), No. 2, June 1973, pp. 81-108. BibRef
Picture Processing: 1972,
CGIP(1), No. 4, December 1972, pp. 394-411.
Elsevier DOI BibRef
CGIP(3), No. 2, June 1974, pp. 178-192.
CGIP(6), No. 2, April 1977, pp. 157-179.
CGIP(13), No. 1, May 1980, pp. 46-79.
CVGIP(22), No. 3, June 1983, pp. 339-377.
CVGIP(30), No. 2, May 1985, pp. 189-231.
Image Analysis and Computer Vision: 1987,
CVGIP(58), No. 1, July 1993, pp. 85-135.
DOI Link BibRef
Image-Analysis and Computer Vision: 1993,
CVIU(62), No. 1, July 1995, pp. 90-131.
CVIU(63), No. 3, May 1996, pp. 568-602.
WWW Link.
UMDTR3733, January 1997.
WWW Link. BibRef
CVIU(70), No. 2, May 1998, pp. 239-284. BibRef
And: UMD--TR3861, January 1998.
CVIU(74), No. 1, April 1999, pp. 36-95. BibRef
CVIU(78), No. 2, May 2000, pp. 222-302. 0005
BibRef
WWW Link. Survey, Bibliography. Bibliography. The standard printed bibliography for computer vision. All of these also appeared as UMD reports. BibRef
Classifying the Literature Related to Computer Vision and Image Analysis,
CVIU(79), No. 2, August 2000, pp. 308-323. 0008
Earlier: UMD--TR4113, March 2000.
Price, K.E.[Keith E.],
Annotated Computer Vision Bibliography,
Online Book1997. BibRef 9700 USC Computer Vision Survey, Bibliography. Bibliography. The self reference to this work. Over 87,000 entries.
HTML Version. or
HTML Version. Also included in the Computer Science Bibliographies:
HTML Version. BibRef
Steiner, D.,
Annotated Bibliography of Bibliographies on Photo Interpretation and Remote Sensing,
Photogrammetria(26), October 1970, pp. 143-144. BibRef 7010
Foley, J.D.[James D.], Pizer, S.M.[Stephen M.],
Review of Graphic Languages (1972).,
CGIP(2), No. 2, October 1973, pp. 196-201.
Elsevier DOI 0501
Schrack, G.F.,
Computer graphics: A keyword-indexed bibliography for the years 1976, 1977, and 1978,
CGIP(14), No. 1, September 1980, pp. 24-79.
Computer Graphics: A Keyword-Indexed Bibliography for the Year 1979,
CGIP(15), No. 1, January 1981, pp. 45-78.
CGIP(18), No. 2, February 1982, pp. 145-187.
Hawkes, P.W.,
Electron Image Processing: 1978-1980,
Elsevier DOI BibRef 8201
Zhang, Y.J.,
Image Engineering and Related Publications,
IJIG(2), No. 3, July 2002, pp. 441-452. 0207
Zhao, X., Wang, T., Lu, H., Sun, X., Wang, X., Wang, F.,
A Bibliographic and Coauthorship Analysis of IEEE T-ITS Literature Between 2014 and 2016,
ITS(19), No. 9, September 2018, pp. 2751-2761.
IEEE DOI 1809
Collaboration, Productivity, Transportation, Social network services, Automation, Control systems, social network analysis BibRef
Chapter on Books, Collections, Overviews, General, and Surveys continues in
Dialogs, Discussions and Debates .
Last update:Jan 13, 2022 at 22:02:22


text:
Saint-Hilaire – miejscowość i gmina we Francji, w regionie Oksytania, w departamencie Aude. Przez miejscowość przepływa rzeka Lauquet.

Według danych na rok 1990 gminę zamieszkiwały 653 osoby, a gęstość zaludnienia wynosiła 28 osób/km² (wśród 1545 gmin Langwedocji-Roussillon Saint-Hilaire plasuje się na 444. miejscu pod względem liczby ludności, natomiast pod względem powierzchni na miejscu 308.).

Zabytki
Zabytki w miejscowości posiadające status Monument historique:
 opactwo św. Hilarego (Abbaye de Saint-Hilaire)

Przypisy

Bibliografia

Miejscowości w departamencie Aude

redpajama_set_name: RedPajamaWikipedia


After a short break we are back racing cyclocross in Canberra on 2nd July. During the break, it was great to see a good crew from Canberra making their way down to Victoria for #alpinecross. Lets just say Morto surprised more than a few riders with his 5th place in Elite Men at Mt Beauty. The following weekend Claire Aubrey and Aaron Thomson were again on the road, this time to Newcastle to successful defend the positions as leaders of the NSW State Series in extremely demanding conditions.
Round 5 of the CORC Cyclocross Series 2017, on Sunday 2nd July, will be held at Narrabundah Velodrome.
Please pre-register here to makes our event teams life easier on the day, it makes sure your name is spelt correctly and gives you priority seeding on the start line!
To continue to encourage women's participation in cyclocross, experienced local cyclocross rider Claire Aubrey will be hosting a pre-ride open to all registered women from 10:20 sharp. This will be a great opportunity to ride a lap of the course, practice the more challenging sections and get a few tips.


do you ever reprint shirts? i'd def buy this shirt too.
PLEASE reissue this shirt! Kids love it, adults love it, starts a conversation all the time. I've worn mine to threadbare and need a new one. I'll buy a dozen if necessary! Thanks.
I have worn out my copy of this shirt and need a new one. When will it come back around?????
Last week I visited a mate and he was wearing a shirt "Legolution".
I asked him where he got it and told me that he found out I should visit GLENNZ, 'cause his was a cheap knockoff.
Unfortunatly I can't find the shirt in your store (any longer). Will you be bringing it back?
Is there a link that I've missed?


Ring culture is a means of growing Tomatoes that allows fibrous feeding roots to develop within a bottomless 'ring' (where fertiliser is applied), whilst courser water-seeking roots develop into the ground, aggregate, 'Growbag' or Container.
Traditionally, the 'ring' or bottomless pot is about 8" (20cm) deep and 9-10" (22-25cm) in diameter, which is then placed on a shingle or gravel base. The bottomless ring is then filled with compost or John Innes Number 3, which is then moistened and planted up with the desired variety of Tomato.
The aggregate (shingle or gravel) base is then kept moist, with the rings being kept watered just sufficiently for the plants to grow.
Once water roots penetrated into the aggregate, no further watering of the rings is required, except when feeding the plants (liquid feed).
This method can easily be applied to Growbags and containers, with these acting as a replacement for the aggregate.
The main advantage of using Ring-culture with Tomatoes is that the Liquid feed is applied directly to the fibrous roots; aiding the development of the fruit; and is not diluted by daily watering, which can led to nutrient deficiencies within the plant.