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In computer science the reachability is one of the fundamental problems taking its roots from the first undecidable decision problem in the computability theory - termination/halting problem in Turing Machine: "Given a description of an arbitrary computer program, decide whether the program finishes running or continues to run forever" or "Deciding, given a program and an input, whether the program will eventually halt when run with that input, or will run forever". In the modern world software is now everywhere (in almost all devices including phones, cars, planes, etc ). The solution of the reachability problem: "Deciding whether a particular piece of code will reach a bad state, can avoid some execution path or will eventually terminate" is the core component of the verification tools that can grantee the reliability of the code and correct functionality of complex technological devices. The proposed research of this project is mostly in the study of reachability problems for classical mathematical objects such as words, matrices, iterative maps and aims to get a progress with a solution of challenging and fundamental long standing open problems in mathematics and computer science, which also appear in the analysis of natural processes in physics, chemistry, biology, ecology, economics etc. The primary goal of this project is to demonstrate that it is possible to go significantly beyond known results related to reachability problems in matrix semigroups, iterative maps and related word problems by applying a combination of techniques from computational theory, number theory, algebra and combinatorics on words. Our principal objectives within this research programme are: identifying new classes with decidable reachability problems for words, matrices and maps, designing efficient algorithms for decidable cases and estimating their computational complexity. First, we propose to study generalized model that cover originally independent, but closely related open problems and investigate the reductions between them. Then we suggest following three approaches to get a better understanding of the core problems: investigation of topological properties of the reachability sets and their application for reachability analysis; translation of matrix reachability problems into combinatorial and computational problems on words; and the design of semi-algorithms for reachability problems in higher dimensions based on projection methods, where infinite reachability set can be mapped into various finite structures which preserve some of the reachability properties. The result of the project would be twofold. In relation to reachability problems for matrices and maps, we expect that new deep results related to open problems will be obtained by applying a combination of techniques from computational complexity theory, automata and formal langauges, algebra, number theory and combinatorics on words. At a more general level we expect to establish new direction of research connecting challenging problems in mathematics with theoretical computer science structures, methods and results. The list of indirect and long-term beneficiaries is not limited to developers of software verification techniques and algorithms, but also includes a variety of specialists in physics, chemistry, biology, environmental sciences and economics which require efficient tools for predicting the behaviour of the complex systems represented by matrices and matrix products.

Fullerenes are football-shaped cages of carbon atoms, for the discovery of which the British scientist Harry Kroto won the Nobel prize in 1996. Inside the cage is an empty space. Chemists and physicists have found many ingenious ways of trapping atoms or molecules inside the tiny fullerene cages. These encapsulated compounds are called endofullerenes. A remarkable method was pioneered by the Japanese scientists Komatsu and Murata, one of whom is a project partner on the current proposal. They performed "molecular surgery". First, a series of chemical reactions was used to open a hole in the fullerene cages. A small molecule such as water (H2O) was then inserted into each fullerene cage by using high temperature and pressure. Finally, a further series of chemical reactions was used to "sew" the holes back up again. The result was the remarkable chemical compound called water-endofullerene, denoted H2O@C60. Our team has succeeded in developing a new synthetic route which requires milder conditions and has improved yield for the production of H2O@C60. In addition we will encapsulate other small molecules in the fullerene cage, including ammonia (NH3) and methane (CH4). Molecules of ordinary water have two forms, which are called ortho and para-water, which are distinguished by the way the magnetic hydrogen nuclei point: in opposite sense for para-water, and in the same sense for ortho-water. In ordinary conditions, these two forms interconvert rapidly, and cannot be isolated. However, by trapping water molecules inside fullerene cages, the two forms are isolated and may be studied separately. We recently observed that these two forms of water have different electrical properties. At low temperatures, the two forms interconvert over a period of tens of hours. We will study the interconversion of the two forms of water, and develop a theory of why this conversion changes the electrical properties. In order to understand how these molecules behave, we will use several techniques. These methods include nuclear magnetic resonance (which involves a strong magnet and radiowaves), neutron scattering (in which the material is bombarded with neutrons from a nuclear reactor) and infrared spectroscopy (which involves the absorption of low-energy light waves). By combining the information from these different techniques, we will build up a complete picture of the quantum-mechanical behaviour of the trapped molecules. Since ortho and para-water have different electrical properties, we expect to distinguish between single H2O@C60 molecules in the ortho and para states, by measuring the electrical response of single molecules. This will be done scanning over a surface loaded with the fullerenes, using a very sharp tip. In this way, we hope to observe the ortho to para transition of single molecules - something that has never been done before. Although most of this project concerns basic science, this project could lead to technological and even medical advances in the future. For example, the ortho and para states of the individual H2O@C60 molecules could allow the storage of one bit of information inside a single molecule, without damaging it in any way. This might lead to a new form of very dense data storage. Since a single gram of H2O@C60 contains about 10^19 molecules, this single gram could in principle store 1 million terabytes of information, sufficient to store the DNA sequences of everyone on the planet (although it will be very difficult to store and retrieve this information). In addition, the quantum behaviour of the encapsulated molecules is expected to give rise to greatly enhanced magnetic resonance signals, leading to the possibility of greatly enhanced MRI images, with considerable medical benefits.

Doctoral Training Partnerships: a range of postgraduate training is funded by the Research Councils. For information on current funding routes, see the common terminology at www.rcuk.ac.uk/StudentshipTerminology. Training grants may be to one organisation or to a consortia of research organisations. This portal will show the lead organisation only.

Train speeds have steadily increased over time through advances in technology and the proposed second UK high speed railway line (HS2) will likely be designed with "passive provision" for future running at 400 km/hour. This is faster than on any ballasted track railway in the world. It is currently simply not known whether railway track for speeds of potentially 400 km/hour would be better constructed using a traditional ballast bed, a more highly engineered trackform such as a slabtrack or a hybrid between the two. Although slabtrack may have the advantage of greater permanence, ballasted track costs less to construct and if the need for ongoing maintenance can be overcome or reduced, may offer whole-life cost and carbon benefits. Certain knowledge gaps relating to ballasted track have become apparent from operational experience with HS1 and in the outline design of HS2. These concern 1. Track Geometry: experience on HS1 (London to the Channel Tunnel) is that certain sections of track, such as transition zones (between ballasted track and a more highly engineered trackform as used in tunnels and on bridges) and some curves require excessive tamping. This results in accelerated ballast degradation and increased ground vibration; both have an adverse effect on the environmental performance of the railway in terms of material use and impact on the surroundings. Thus the suitability of current design rules in terms of allowable combinations of speed, vertical and horizontal curve radius, and how these affect the need for ongoing maintenance to retain ride quality and passenger comfort is uncertain. 2. Critical velocity: on soft ground, train speeds can approach or exceed the speed of waves in the ground giving rise to resonance type effects and increased deformations. Instances of this phenomenon have been overcome using a number of mitigation measures such as the rebuilding of the embankment using compacted fill and geogrids, installation of a piled raft and ground treatment using either deep dry soil mixing or controlled modulus columns. The cost of such remedial measures can be very high, especially if they are taken primarily on a precautionary basis. However, many methods of analysis are unrefined (for example, linear elastic behaviour is often assumed or the heterogeneity of the ground, track support system and train dynamics are neglected), and conventional empirical methods may significantly overestimate dynamic amplification effects. Thus there is scope for achieving considerable economic benefits through the specification of more cost effective solutions, if the fundamental science can be better understood. 3. Ballast flight, ie the potential for ballast particles to become airborne during the passage of a very high speed train. This can cause extensive damage to the undersides of trains, and to the rails themselves if a small particle of ballast comes to rest on the rail and is then crushed. Investigations have shown that ballast flight depends on a combination of both mechanical and aerodynamic forces, and is therefore related to both train operating conditions and track layouts, but the exact conditions that give rise to it are not fully understood. The research idea is that, by understanding the underlying science associated with high speed railways and implementing it through appropriate, reasoned advances in engineering design, we can vastly improve on the effectiveness and reduce maintenance needs of ballasted railway track for line speeds up to at least 400 km/h.

Spintronics is like electronics except that it uses the spin of the electron (a quantum mechanical property that behaves like angular momentum and is closely linked to magnetism) as well as the electron's electric charge. Using spin and charge together could lead to computers that use much less energy, for example. Photo-spintronics adds light to the mix. This is very useful because light can easily carry information over long distances (think of optic fibres). Light and spin are also key to future quantum technologies such as quantum computing and quantum information. Our research is to find ways of using organic molecules, based on chains and rings of carbon atoms, in photo-spintronics. This is an exciting prospect because carbon has a low atomic number which reduces the chances of losing spin information, and because there are so many different organic molecules and ways of linking them that the opportunities to find new and useful phenomena are practically endless. Our plan is to study single molecules linking a semiconductor and a magnetic metal. Single molecule experiments are difficult but not impossible, and we have made them successfully in the past using a modified scanning tunnelling microscope. Single molecule studies have helped greatly in understanding molecular electronics because studying molecules individually reveals information that is lost when they are measured in a large group. Ours will be the first single molecule studies in photo-spintronics. We will create a population of excited electrons in the semiconductor by illuminating it and use the polarization of the light to control the spin of the electrons. We will then measure the current between the semiconductor and the ferromagnetic metal. If the current depends on the polarization of the light and the direction in which the metal is magnetized, that will be evidence that spin is being transported through the molecules. Once we show that we can make photo-spintronic measurements through a single molecule, we will investigate how the spin transport depends on the type of semiconductor, the metal, the voltage between the two (known as the bias), and the types of chemical bond between the molecule and the semiconductor and metal. This will show us how best to use organic molecules in future spintronic and photo-spintronic devices.

The societal needs such as helping elderly and rapid technological advances have transformed robotics in recent years. Making robots autonomous and at the same time able to interact safely with real world objects is desired in order to extend their range of applications to highly interactive tasks such as caring for the elderly. However, attaining robots capable of doing such tasks is challenging as the environmental model they often use is incomplete, which underlines the importance of sensors to obtain information at a sufficient rate to deal with external change. In robotics, the sensing modality par excellence so far has been vision in its multiple forms, for example lasers, or simply stereoscopic arrangements of conventional cameras. On other hand the animal world uses a wider variety of sensory modalities. The tactile/touch sensing is particularly important as many of the interactive tasks involve physical contact which carry precious information that is exploited by biological brains and ought to be exploited by robots to ensure adaptive behaviour. However, the absence of suitable tactile skin technology makes this task difficult. PRINTSKIN will develop a robust ultra-flexible tactile skin and endow state-of-the-art robotic hand with the tactile skin and validate the skin by using tactile information from large areas of robot hands to handle daily object with different curvatures. The tactile skin will be benchmarked against available semi-rigid skins such as iCub skin from EU project ROBOSKIN and Hex-O-Skin. The skin will be validated on at least two different industrial robotic hands (Shadow Hand and i-Limb) that are used in dexterous manipulation and prosthetics. The robust ultra-thin tactile skin will be developed using an innovative methodology involving printing of high-mobility materials such as silicon on ultra-flexible substrates such as polyimide. The tactile skin will have solid-state sensors (touch, temperature) and electronics printed on ultra-flexible substrates such as polyimide. The silicon-nanowires based ultra-thin active-matrix electronics in the backplane will be covered with a replaceable soft transducer layer. Integration of electronic and sensing modules on a foil or as stack of foils will be explored. 'Truly bottom-up approach' is the distinguishing feature of PRINTSKIN methodology as the development of tactile skin will begin with atom by atom synthesis of nanowires and finish with the development of tactile skin system - much like the way nature uses proteins and macromolecules to construct complex biological systems. This new technological platform to print tactile skin will enable an entirely new generation of high-performance and cost-effective systems on flexible substrates. Fabrication by printing will have important implications for cost-effective integration over large areas and on nonconventional substrates, such as plastic or paper. Printing of high-performance electronics is also appealing for mask-less approach, reduced material wastage, and scalability to large area. The proposed programme thus has the potential to emulate yet another revolution in the electronics industry and trigger transformation in various sectors including, robotics, healthcare, and wearable electronics.

This project is rooted in Ergodic Theory and Dynamical Systems, but also aims to have significant impact on Geometry. The main novelty will be to exploit techniques from Ergodic Optimization, a relatively new branch of Ergodic Theory and Dynamical Systems, to prove Optimal Hitting results in a symbolic dynamics context, and certain conjectures in Optimal Packing.

Energy Management of existing non-domestic buildings is wrought with many challenges, a number of which arguably exist due to the diversity found amongst individual buildings and amongst the humans who occupy them. Buildings are inherently unique systems making it difficult to generalize technology solutions for any individual property. Instead, to make robust investment decisions for the energy-efficient upkeep of a particular building requires some degree of tailored engineering and economic analysis. To understand why this is the case, one need only to consider the chain of questions one would likely need to address for decision-making in an arbitrary building. For instance, we might ask: what is the age of the building and the equipment currently installed in it? Does the heating system need to be replaced? If yes, is the current system a boiler, and if so, how efficiently does it perform? Would the building benefit from a new boiler or an electric heat pump? Would it benefit from replacing the heating distribution pipes? Do the cost / benefits of any of these technologies depend on government tariffs and subsidies? What is the risk faced if any available subsidies are cut in the future? How robust is either technology to the future price of natural gas and electricity? Would that risk be worth taking? Is it too expensive to even start thinking about the options and associated risks? How would a facility manager visualise the options available and possible spreads of benefits and risks for all these aspects? This project aims to respond to these challenges. Indeed, in order to make sound decisions on future building operation and technology investment, evidence shows that one needs adequate information on a number of engineering, economics, and social science matters pertaining to each individual project. To obtain this information has so-far been viewed as a costly exercise, and has contributed to the general perception that undertaking deep cuts to building energy consumption (achieving more than 15% in energy savings per investment) is an economically risky affair. This proposal is the first to develop and recommend an altogether new approach to performing building audits, energy simulation, uncertainty analysis, data visualization, and finally investment decision-making. It will lead to a marked reduction in the cost of acquiring information for robust retrofit and facility management decisions. The direct outputs of this project will be a series of software tools for three distinct but related purposes: (i) collecting building data on relevant uncertainty parameters (i.e., "what do we know now?"); (ii) propagating and quantifying uncertainty using building simulation models, measurements obtained from key monitored building sites, and cutting-edge statistical approaches (i.e., Bayesian analysis); and (iii) the display and interpretation of uncertainty. During the course of the project, workshops will be organised to lay out the current (uncertain) knowledge that has been, until now, largely undocumented in the buildings sector and inaccessible to the energy research community. This includes gaining understanding on the most common faults observed in managing conventional energy systems, and how spatial layouts in building evolve. The graphical presentation of risk information and understanding users' perception of uncertainty and risk will be key elements of these workshops and the research programme. Our software tools, user guidance, and numerical runs of test cases will be made available, as the web-based B-bem portal, via the University of Cambridge web site.

Doctoral Training Partnerships: a range of postgraduate training is funded by the Research Councils. For information on current funding routes, see the common terminology at www.rcuk.ac.uk/StudentshipTerminology. Training grants may be to one organisation or to a consortia of research organisations. This portal will show the lead organisation only.

Please refer to attached Royal Society application