Exploring sum-free sets

This post was contributed by Professor Sarah Hart of Birkbeck’s Department of Economics, Mathematics and Statistics and Head of the Pure Mathematics Research Group. Here, Professor Hart offers an insight into a major area of her research: sum-free sets

The set S = {1, 3, 5} is “sum-free”: if you add two numbers in S, the answer is outside S. So 1 + 3 = 4, and 4 is not in S.

There are many questions we could ask about sum-free sets. How big can they get? Are there infinite sum-free sets? The answer is yes – the set of all odd numbers is sum-free because the sum of any two odd numbers is even, and so lies outside the set. In fact this is also locally maximal (we cannot add any further elements to it while keeping it sum-free), because any even integer (see glossary below) is the sum of two odd integers so we cannot add any even integers to the set.

Having found this large sum-free set we might ask if we can divide up (partition) the set of integers into a small collection of sum-free sets. (Actually, the fact that 0 + 0 = 0 means we only look at the set of nonzero integers.) It turns out there is no way to partition the set of nonzero integers into a finite collection of sum-free sets, although there are partitions into an infinite number of sum-free sets.

Groups

Click the image to read Theorem of the Day's article on Professor Sarah Hart's work

Read Theorem of the Day’s article on Professor Sarah Hart’s work

The interesting thing about this set-up is that it can be generalised. The set of integers is just one example of a “group”, which is a set along with an operation which can combine two elements a, b of the set to produce a third element a*b.

For the integers, the operation is addition: two integers added together produce another integer, so a * b is defined as a + b. There are three rules that the operation must satisfy. One is “associativity”, which for the integers translates as the property that for any integers a, b and c, we have (a + b) + c = a + (b+c).

There are countless examples of groups, for example the set of symmetries of any shape. The operation is composition – so the combination of a rotation integer (see glossary below) and a reflection is the symmetry obtained by doing the rotation followed by the reflection. Unlike the group of integers, many groups are finite.

The notion of a sum-free set

The notion of a sum-free set can be generalised to any group, but we now talk about product-free sets because the notation is multiplicative. If G is a group, and the operation is *, then a subset S of G is product-free when a * b is not in S, for all a, b in S.

In the group of symmetries of the cube, for example, the set of reflections is product-free because the product of two reflections is a rotation. We can ask all the same questions. What is the biggest possible size of a product-free set? Or the smallest locally maximal product-free set? How can we partition the group into product-free sets?

My research – filled groups

I first started looking at these ideas ten years ago, in [1], but have returned to the subject recently with my PhD student Chimere Anabanti [2] – in particular we have been trying to find out more about small locally maximal product free sets. Along the way we have been able to answer a question [2] that has been open since the 1970s about so-called “filled groups” – ones whose locally maximal product free sets have a particular form.

Next steps – Solution-free sets

There is another generalisation of all this, and that is the direction I hope to move in next: we can think of product-free sets as sets having no solution to the equation a * b = c. So we can pick another equation and look for sets that don’t satisfy that equation. The general term for these types of sets is “solution-free sets”.

Professor Sarah Hart

Professor Sarah Hart

We can ask the same kinds of questions about them as for product-free sets. Examples include Sidon sets – the definition for the integers is a set where all differences are distinct; in other words there are no solutions to a – b = c – d when at least three of a, b, c and d are different.

It’s fun to ponder these ideas in their own right of course, but there are surprising and interesting links to other areas of mathematics. Product-free sets in certain groups give rise to a particularly nice type of error-correcting code. There are also applications to graph theory and to finite geometry. Sidon sets are used in research about the efficient design of sensor arrays.

Find out more

[1] Giudici and Hart “Small maximal sum-free sets”

[2] Anabanti and Hart “Locally maximal product-free sets of size 3” Preprint 10 on this page

[3] Anabanti and Hart “On a conjecture of Street and Whitehead on Locally maximal product-free sets”

Glossary:

  • Integer: A whole number (not a fractional number) that can be positive, negative, or zero e.g. -5, 1, 5, 8, 97, 3,043
  • Rotational symmetry: When an image is rotated (around a central point) so that it appears two or more times.
  • Maximal: A maximal element of a subset S of some partially ordered set (poset) is an element of S that is not smaller than any other element in S
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The proposed ‘right to disconnect’ after work hours is welcome, but not enough

This post was contributed by Professor Gillian Symon, member of the Digital Brain Switch project. Involving a multi-disciplinary team of UK researchers (including Birkbeck’s Dr Rebecca Whiting), the project explored the ways in which mobile communication technology  affects how we switch between different aspects of our lives. This article was originally posted on The Conversation on 23 March 2016.    

Changes proposed to France’s famously inflexible employment laws by French president François Hollande have prompted an outcry among students and unionists and even the barricading of schools by pupils. But among the raft of changes to working practices is the liberating notion that employees should have the right to disconnect: to ignore emails from employers during evenings and weekends so that time with friends and family is not affected by work distractions or feelings of guilt.

Limited interventions of this sort have been put forward in Germany and France before, but this is the first proposal that the right be enshrined in law.

There is much to like about it. First, it recognises the massive impact the widespread use of smartphones and tablets, Wi-Fi and high-speed mobile internet has had on our working lives. In as much as work emails, diaries and contacts are on a smartphone in our pocket, to some extent we are never truly “out of the office”. The proposal seeks to counter this in legislation, not to leave it to corporate custom and practice.

Second, the proposed legislation acknowledges the considerable research that suggests that we need to psychologically detach from work regularly, or risk becoming exhausted and losing our creativity.

Third and most importantly, it makes the employer at least partly responsible for managing this intrusive technology and its effects on employees. There is a recognised paradox, whereby technology allows flexibility over when and where we work, but at the same time acts as a leash that chains us to our (virtual) desks. For too long this has been seen as something employees themselves should manage.

The research into work-life balance my colleagues and I have conducted suggests that achieving the right balance has become another “life crisis”. It is one that is fed by endless media articles and self-help books, and one that is almost certainly unresolvable by the individual as so much of the pressure comes from bosses and colleagues at work. What we’ve found is that there needs to be respect for individuals’ chosen work-life boundaries at all levels within organisations.

So congratulations to the French for taking this particular taureau by the cornes. But is their proposed approach through new legislation the right answer?

It’s not easy, and often employers don’t make it any easier. wongstock/shutterstock.com

As far as it goes

There are three ways digital media and mobile technology have affected our lives that isn’t acknowledged by legislation, which is concerned only with time spent connected to work. In our research we’ve sought to highlight the creeping effects of “digi-housekeeping”: those endless technology maintenance tasks that we engage in – updating software, syncing devices, fighting technical problems – which often takes place outside of office hours and doesn’t appear on time sheets. None of this is accounted for by legislative approaches.

Nor does legislation address the way in which the use of social media for work may intrude into our privacy. When we blog and tweet for our employers, are we exploiting our personal identities for their ends? Are these additional tasks, and the need to maintain our digital presence online, causing us anxiety and increasing our workload without any formal recognition of the effort involved? These sorts of activities go beyond a concern with just maintaining a time boundary between work and life. They represent new tasks required to maintain our digital work lives.

What’s more, because the French legislation presumes an employee-employer relationship, it entirely ignores the anxieties of the self-employed, as those taking part in our research told us. While those working for themselves have always had to work hard, social media has put added pressure on them to be constantly online and accessible to maintain their business. We need more imaginative interventions that will address the needs of specific groups such as these.

What are 21st century working lives like?

The French legislation is important primarily because it makes clear the responsibilities of employers and organisations. However, it’s also rather a blunt-edged tool that doesn’t appreciate the intricacies of our online lives. Legislation like this enforces a strict work-life boundary that may be a thing of the past.

Read the original post on The Conversation

Read the original post on The Conversation

Our research collaborators kept video diaries that captured the complex circumstances of today’s workers in a more revealing way than traditional surveys can do. These video diaries suggest we might be making sense of our lives in radically different ways in the 21st century. We distinguish between online and offline lives rather than work and non-work hours, and we think more about how we prioritise time, rather than how we divide it.

To support flexible working, we may need flexible legislation that is based on other considerations than time alone, including where and how we work best. It’s very unlikely there will be a one-size-fits-all solution; researchers and policymakers are going to have to find more creative 21st century solutions for this very 21st century problem.

So the French government’s move to formally recognise the distraction caused by unfettered technology is welcome, but limited. To improve upon it, we need to understand much more fully the complexities of contemporary digital online lives, what boundaries people now find important, and how the law can best support them.

Find out more

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CEO rewards – More does not equal better

This post has been contributed by Dr Almuth McDowall, lecturer in Birkbeck’s Department of Organisational Psychology, with input from Paul Hajduk from PayData, Jonny Gifford from the Chartered Institute of Personnel and Development (CIPD), Dr Zara Whysall from Lane4 and Dr Duncan Jackson from Birkbeck. It builds on a recent practitioner report Almuth and colleagues produced for the Chartered Institute of Personnel and Development (The power and pitfalls of executive reward: a behavioural perspective)

CEO

The size of the packet

Chief executive officer (CEO) pay is a serious topic which requires serious and well-informed debate. In one way or another the size and makeup of CEO rewards affects everybody – either because it can be seen as part of the trend for wealth to be increasing concentrated in the hands of the few, or because the measures that drive the package do not necessarily serve the best interests of society as a whole.

The average UK CEO wage packet is now around the £5million per annum – imagine 100,000 £50 bank notes lined up neatly in a row to get an idea of what this means in reality. We witnessed ‘fat cat Tuesday’ in the first week of January; when CEOs had earned the average UK workers’ salary in just 22 hours.

Are CEOs worth the money?

The notion of the ‘fat cat’ and unfair pay gaps has been vehemently disputed as ‘pub economics’ by the Adam Smith Institute’s director Sam Bowman. The argument runs that organisations need to be profitable to survive so they can make a contribution to a nation’s economy; and that the value of a CEO is hard to quantify in an absolute sense.

Clearly, organisations need to be effective to survive. It is also arguable that the figurehead at the top contributes the most to the long-term success of the organisation. The symbolic value of the person at the top is great, and can make or break corporate success hence CEOs should be rewarded proportionally to their input. But does proportion equate to 180 times the average workers’ salary?

How can we explain the growth in CEO rewards?

In our report, Paul Hajduk of Paydata undertook a robust analysis of pay trends over the last few years. He set out to test whether the rise in CEO pay could be explained in the context of UK wealth creation. Yes, there is a widening gap between rich and poor, however the number of people in the 1 million plus earnings bracket has remained relatively stable. CEO rewards continue to grow, quite out of proportion to the rate of growth of high pay generally and also to the rather unsteady growth of the UK economy. So the growth in CEO rewards cannot be accounted for by wealth increase per se, can it be justified in terms of increased organisational performance? Apparently not. An interesting paper reveals that organisations with particularly highly paid CEOs are unlikely to be in the top 10 percent of high performing organisations [WSJ, 2015].

Paul says: “There would appear to be little to support the argument that high CEO pay growth is justified by how their role is often positioned, which is as wealth generating entrepreneurs. Yes, they may lose their job if things do not go well but they rarely lose much of their own money. We have yet so see if clawbacks built into reward arrangements will be truly effective in creating significant downside risk in CEO reward packages”.

This leads to the wider question of how organisational performance is benchmarked.

What is the link between CEO rewards and organisational performance?

Most organisations benchmark CEO success against hard measures such as profitability and productivity, the bulk of research in the field also concerns itself with financial indicators. Far fewer organisations use non-financial metrics such as staff health, safety or engagement measures in their annual reporting. However, it is important to consider the human aspects of performance and their link to organisational outcomes. One US study considered the characteristic of the CEOs of US basketball teams and the link to measures as wide ranging as external team reputation, winnings and fan attendance at matches (Resick, Whitman, Weingarden and Hiller, 2009). There is a shortage of parallel evidence in a business context which considers the relationship between differentiated measures of CEO performance, and the scope of their impact on organisations such as their members, including intangible assets such as motivation and engagement.

Are our reward structures creating the wrong kind of CEOs, or are our CEOs creating the wrong kinds of rewards?

Dr Zara Whysall from Lane4 says: “Our work with a range of organisations has shown that reward practice in organisations lacks an evidence-base, CEO reward practice appears no different. People tend to overestimate the motivating force of money in particular where rewards are delayed and not immediate, and we also do not pay enough attention to non-financial rewards and the impact they have an on organisation’s culture and ethos.”

There have been several high profile instances of rather dysfunctional examples of CEO stewardship particularly in the financial sector. It is therefore important to try and understand the influence CEOs have. Research shows us that powerful CEOs are good at negotiating rewards and clever at shining the spotlight on favourable indicators [Morse et al., 2014].

Time for a change?

There is a strong case for change in executive reward practice given that the justification for the maintaining the current status quo is at best dubious. However, the research detailed in our report shows that barriers are ingrained and institutionalised. Whilst there is body of people who support change and who state openly that the UK (and the world?) needs more considered, innovative and ethical CEOs in the future, there is less consensus on how such change can be brought about.

The vision for the future

London docklandsGiven the lack of evidence to support the ever escalating size of senior rewards, CEO salaries should be a smaller multiple of average earnings, with smaller bonus packages, and reduced long term incentives such as performance share schemes. But is it a more pressing question that in order for change to happen we need different people at the top?

There is the argument that any capping of rewards policy changes will negatively impact organisations’ ability to recruit and retain the quality of people needed to position themselves positively in the global market place. To counter this, it is informative to compare and contrast CEO reward in the most successful mission-led businesses. When profit distribution moves from being the primary motive the CEO reward package loses, in most cases, most of the upside variable pay elements and share-based pay disappears altogether. And yet these businesses attract and retain very able leaders who are maybe motivated by things other than the size of their reward package.

A fundamental shift in leadership practice might need to accompany these reward changes. There is ample evidence that shared leadership is better than top centric leadership [Wang et al., 2014]. It is also a fact that diversity at top levels does not mirror society at large. Only radical revision of selection, talent management and reward processes and structures will change the current status quo. Our report makes distinct recommendations for how to put this into practice.

But are our recommendations radical enough, or should we start again with a blank slate? Dr Almuth McDowall says: “This has been a fascinating and complex research project which we hope will offer a rich springboard for debate. There appears a cautious consensus that change is needed, yet a certain reluctance to challenge the current status quo. Do read our report, and let us know – are our recommendations radical enough?”

Find out more

References

  • Morse, A., Nanda, V., & Seru, A. (2014). Compensation Rigging by Powerful CEOs: A Reply and Cross-Sectional Evidence. Critical Finance Review, Vol. 3 No. 1, pp. 153-190.
  • Resick, C. J., Whitman, D. S., Weingarden, S. M., & Hiller, N. J. (2009). The bright-side and the dark-side of CEO personality: examining core self-evaluations, narcissism, transformational leadership, and strategic influence. Journal of Applied Psychology, 94(6), 1365.
  • Wall Street Journal (2015) How much the best-performance and worst-performance CEOs got paid. 25th June 2015
  • Wang, D, Waldman, D. A. and Zhang, Z. (2014) A meta-analysis of shared leadership and team effectiveness. Journal of Applied Psychology, Vol. 99 No. 2, pp.181 -199
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Should taxpayers fund university education at a time of crisis?

This post has been contributed by Dr Federica Rossi, lecturer in Birkbeck’s Department of Management, and Aldo Geuna, professor of Economics at the University of Torino, Italy. Their new book The University and the Economy: Pathways to Growth and Economic Development is published by Edward Elgar.

Higher-education-fundingWith the recent financial and debt crisis, the extent to which the public can afford to fund universities has become increasingly controversial: in the context of tight public budgets and widespread cuts to public spending, even in areas perceived as basic services to support the more vulnerable members of society, what reasons could there possibly be for continuing to fund a “luxury” like higher education?  Dr Federica Rossiand Professor Aldo Geuna consider the viability of publicly funded universities and their alternatives.

Some convincing reasons must exist if, as it emerges from data reported by the European University Association’s Public Funding Observatory a majority of 14 European countries, out of 24 for which data are available, have increased public funding for universities in the period 2008 -2012, right through the latest recession. Investment has increased in, among others, Austria, Germany, Poland, Belgium, the Netherlands and the Scandinavian countries. Ireland and the United Kingdom are the only countries in Northern Europe to have decreased investment, together with debt stricken countries in Southern Europe and some Eastern European ones.

The argument against: higher education generates mainly private benefits

The main argument against the public funding of higher education builds on the view that the benefits of higher education are enjoyed mainly by university graduates, in terms of potentially higher future earnings and lower probability of unemployment, rather than being shared across society. If the benefits of higher education are strictly private, then why should the general public pay for it? The cost of higher education instead should be met by the students who will directly benefit from it.

This argument is reinforced by the fact that information and communication technology makes it easy to disseminate and access knowledge: when students anywhere can access the knowledge produced by the best academics in the world at the click of a button, why should governments fund them to physically attend lectures at a local campus? Using openly disseminated materials provided by prestigious universities could provide a way for poorer students, who would not have the upfront resources to pay for university fees, to access the same or even better knowledge at a fraction of the price.

In reality, however, things are rarely this simple. And indeed several theoretical and practical arguments have been made to support the view that there are still good reasons for providing substantial public funding for higher education even (or maybe especially) at a time of crisis.

But: the private benefits of higher education are risky and with a delayed pay back

The opportunity cost of studying is high over the short term while its private benefits are risky and with a delayed pay-back. Given that young people often have no resources of their own, or are either unwilling to undertake the risk, or they have no access to private credit (as financial markets are imperfect), then without incentives, investment in education would be lower than optimal.

The only students who would invest in undertaking higher education would be those who are rich enough to pay upfront, ramping up social and economic inequality. This is even more undesirable when we consider that the private benefits of university education tend to be higher for those individuals who come from less well-off families who generally do not enrol at university.

So there is a general consensus that some form of government intervention is necessary in order to increase the number of students undertaking higher education, and to do so while preserving some form of equal opportunity of access.

One way to do this is to provide full public funding of higher education in order to make it completely free at the point of delivery. However, this is not the only possible approach, and indeed it is becoming less frequent: it has even been shown that blanket funding irrespective of an individual’s income could have regressive effects on the incomes distribution. In most countries, systems are in place according to which students contribute at least in part to the funding of their education, but grants and loans exist to support those students who lack resources of their own.

Recently, for example, the United Kingdom has introduced a loan-based system where students receive a loan from the government in order to fund their education, and pay the loan back once they benefit from higher salaries in the job market. Here, the government is prepared to sustain the risk of students not paying back their loans if their future income does not meet a minimum threshold or if students are impossible to track down – a risk which has an important systemic component (for example, both types of risk would be systematically increased by a recession, where graduates are less likely to reach the minimum income thresholds and also more likely to emigrate in search of better job opportunities).

If such systemic problems were to occur, the cost to the public purse could actually turn out to be quite high. Moreover, even if we accept that students should in part contribute to the cost of their university education, issues like how much they should contribute and whether this contribution should be linked to their ability to pay it, remain open to debate.

The social benefits of higher education are anything but negligible

It has been shown that both an increase in the share of graduate population and an increase in the growth rate of graduates generate a more than proportional increase in economic growth. This suggests that the broader public who have not attended university also benefit from having colleagues and fellow citizens with a higher level of education, thanks to the latter’s contribution to economic growth, which goes beyond their own individual productivity.

In particular, research has suggested that a more educated workforce:

  • has a positive effect on the productivity of colleagues with a lower level of education
  • facilitates and accelerates the adoption of existing technologies not yet implemented
  • is more likely to introduce product and process innovation and therefore to economically exploit radically new technologies – a particularly important process in economies which already operate on the technological frontier.

There are also important indirect effects. Education can improve citizens’ health, stimulate political participation and encourage a sense of civic duty and interpersonal trust, factors that are important for the competent functioning of economic institutions and their performance.

The presence of these external effects provides arguments in support of the general taxpaying public contributing to the funding of higher education, whose benefits are felt across society at all levels. This is one reason why public university fees are, and should be, quite low when compared to the average cost of a student’s university education.

What education, rather than just how much education, matters

If universities’ sources of funds are entirely private, this introduces incentives for universities to maximize enrolments in the short term by focusing on those disciplines for which demand is greater, in order to swell the number of enrolled students and thus increase revenues. Research has shown that privately funded institutions tend, overall, to focus on the most popular subjects. From the perspective of maximizing the contribution of higher education to economic growth, this is likely to lead to dynamic inefficiencies.

First, there is no linear correspondence between students’ demand for higher education and the actual labour market’s needs for specific competences, since student demand for courses is based upon subjective evaluations and incomplete information. This is the reason why, for example, the United Kingdom continues to pour public funds into the teaching of STEM disciplines which are perceived to have high economic importance even though student demand for STEM courses is low.

Second, there is a real difficulty of foreseeing what academic disciplines will turn out to be important in the future. How to anticipate, for example, which subjects will best support the educational needs of individuals and companies in 15 to 20 years’ time? The skills required by an economic system are the result of events that cannot easily be predicted, such as geopolitical changes or the emergence of new technologies. It is important to allow universities to continue to educate students in a broad variety of fields, keeping the system sufficiently flexible and open and allowing for possible adjustment in the event of unexpected changes – something that can be accomplished only if universities are at least in part free from the compelling need to passively respond to market demand.

9781782549482_4_1And finally…can technology be the answer?

While ICT is certainly helpful in broadening access to knowledge, there are numerous arguments that suggest that simply having access to knowledge does not equate gaining an education into a particular field or topic. Being able to access free university courses online does not substitute for the ability to attend a physical institution, for a number of reasons:

  • A certain level of previous education is often required in order to understand advanced knowledge. University institutions can provide the tailored training that weaker students (especially those from disadvantaged backgrounds, who may have the needed qualifications to enter university, but whose quality may not be as good) might need in order to bridge the knowledge gap that separates them from stronger students and allow them to complete their studies
  • The transmission of knowledge is enhanced by direct interactions with teachers
  • It has been shown that being part of a community of practice, though meeting and interacting with other students, enhances learning and motivation
  • Some of the benefits of higher education come from being part of a social community of students and from developing connections that continue after university

While some of these benefits could be re-created virtually (for example virtual communities of practice can be set up), overall it is unlikely that students can derive the same benefits from accessing materials that are openly available online as from attending university courses that are tailor-made for a specific and small group of students. The weaker students in particular are the least likely to have the ability to benefit from this, which invalidates to a large extent the argument that freely available online courses can provide an effective way for school leavers priced out of university to gain the same education as fee paying university students.

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