rodym yn gwych.  Mae Sears yn canigam i ddiliadwch i ddiliadwch i ddiliadwch i ddiliadwch i annid y Sears yn unig mai ffordd ac yn ymwyfent yn y strategaeth Gwyl Lleid. Mae gennym mwren plastig yw prydau mewn ymniadau gŽserol mewn hyd yn 4 cwrdd. Rwyf yn y spentid ynghylch i gweithio yn cael weltydd a bhlygau gwneudol i gweithio'r ysgol, ac yn ystod i gynnigrwch mewn drwdyn nhw'r uchynedd i hyrwan o ddysbosiblwr, mae'n cyfwilwch gweithfeydd. Felly, mae'n gweithio £500,000 i gwyntio i gyd ddwyllwch gweithwyr around the coast and to develop policy to address marine plastics, which will involve working with community groups. We will host an international conference in 2019 to discuss improving our marine environment and protecting our wildlife, focusing on marine plastics. We are introducing a ban on the manufacture and sale of rinse-off personal care products containing plastic microbeads with the rest of the UK, which is expected to be in place 9 July 2018. We have also recently pledged our support of the Global Ghost Gear initiative to ensure that the issue of loss or abandoned fishing gear, often made of plastics, is addressed around the world. Maurice Corry I thank the cabinet secretary for her answer. It is clear that the UK Government is taking the lead in the fight against the scourge of plastic waste in our oceans. I am sure that we have all seen the scenes in Blue Planet recently in that documentary, which so vividly highlighted the damage plastic pollution causes to marine life. Will the cabinet secretary commit to working with the UK Government constructively in the light of the ban on microbeads and to consider how charges on single-use plastic items could help to reduce waste? Maurice Corry I thank Maurice Corry for that follow-up, although he will find that Michael Gove is on record as saying that there is a lot England could learn from the more ambitious approach that Wales and Scotland take on environmental issues. Of course, we support any initiative that the UK Government wants to take in respect of that. I hope that it will follow us in a number of the things that we are discussing. I am aware of the recent statement by Michael Gove on that. I am conscious that he is asking for a task force to look at environmental taxes. Maurice Corry will know that such taxes are not devolved to the Parliament, which is why we were more careful in our use of the word charges and levies rather than taxes. Of course, I welcome any moves. That is a global problem, and it will require a global effort. Kate Forbes As an MSP who represents east and west coastlines and who thinks that urgent action is required to reduce ocean plastics, does the Government agree with me that urgent action is required to crack down particularly on single-use disposable plastics such as straws, cotton buds and cutlery, which all have environmentally friendly alternatives, considering that the Government has taken the lead on changing behaviour through, for example, plastic bag charges? Kate Forbes Yes, I do. That is why we indicated in the programme for government that we were going to look very closely at single-use items. While coffee cups tend to be the one that most often refer to, there are a great many other items made of plastic, as we have all seen if we were watching Blue Planet, which can have an enormously damaging effect on our seas. I do think that all the options that are open to us should be looked at very carefully. That is why we flagged it up in the programme for government and we are considering a number of possible options that we can take forward in that context. Claudia Beamish I recognise the work that the Scottish Government, Zero Waste Scotland and others are doing on this very serious issue of marine litter and particularly plastics. Last week, along with members of all parties, I attended the Clean Beach Scotland reception, and it was truly inspiring what art can do to support communities and others in the work. Marine Conservation Society, Fidra, Scottish Fishermen's Federation and harbormasters from my region and elsewhere were involved, along with others. I wonder what work the cabinet secretary can highlight to ensure that all the partners are inclusively involved as we go forward. Claudia Beamish We are in an almost constant dialogue with just about all potential stakeholders. I too am struck by the extent to which this has become something that people understand at almost every level in society now. I would have to say that, even before Blue Planet, we were aware that, for example, many young children in schools have been coming to an understanding of the damage that this is causing. I know that schools such as Sunnyside primary school and ocean defenders have been set up, and I am aware that they have begun to work on a campaign of nace straw at all—plastic straws. That is one example of the many, many items that we could be looking at. We are working with absolutely everybody. The work that the various organisations that the member flags up is incredibly important. It needs to be backed up by both Government and global action, because, however strongly people feel about it and whatever the work that they do locally, without that global and Government back up, it will continue to not be sufficient. Kenneth Gibson Having hosted an event in Hullary last August on behalf of the X expedition, an all-female around Britain expedition examining plastic pollution in our seas, I am delighted that the Scottish Government is working hard to decrease such pollution. Will the Government fund fuller research into the impact of toxins found in plastic entering the food chain and how that can be diminished? The member will have heard my comments about single-use items and the work that is beginning to go into looking at them. Marine Scotland Science has already researched the uptake of toxic pollutants by microplastics and their bioavailability to species in the food chain. In a sense, that is also an answer to Claudia Beamish, because Marine Scotland is also working with MASTS, the Marine Alliance for Science and Technology for Scotland, co-chairing the microplastics group, which includes Herriot-Watt University, that has researched the impact of those toxins from microplastics. The member can be assured that we are aware of the concern, aware of the problem, and we are working to see what can be done to decrease that pollution. The constituency MSP for Sunnyside primary school, Eivor McKee. I am glad to hear that the cabinet secretary recognised the work of Sunnyside primary school ocean defenders in my Glasgow Provin constituency and the work that they are doing on their next draws a talk campaign. Does the cabinet secretary agree with me that this shows that everyone has a role in marine conservation, not just those constituencies that have beaches? I think that that is really important, because there will be a tendency on everybody's part to assume that it is those coastal communities that somehow have more responsibility than others, but the truth of the matter is that we know that it is not necessarily populations in coastal communities that can be creating the biggest problem for our seas. The health of our seas and oceans is absolutely essential for everybody's health and welfare, whether they live on the coast or inland. I want to reiterate how impressive Sunnyside primary is in a draw a talk campaign, because it does send out exactly the right message. It highlights something that is unnecessary and wasteful when it comes to single-use items, and it flags up that that is the kind of behaviour that must change. It is innovative and creative, and it is coming from young people who, of course, are our future. We are looking very hard where and in what way we will be able to legislate to reduce single-use items, such as the one that is being flagged up by Sunnyside primary school. I hope that the continued support from across the chamber is something that we can count on as we move forward. Question 2, Liz Smith. To ask the Scottish Government what action it is taking in light of concerns raised by COSLA regarding local authorities not being able to register all of their properties on the land register of Scotland by 2019. I should point out that this is perhaps a question that might have been more properly put to the Cabinet Secretary for Economy, Jobs and Fair Work, who is responsible for the land register. I can, however, advise the member that progress is being made by public sector bodies, including local authorities, to meet the 2019 target. A number of local authorities have voluntarily registered land assets, and Registers of Scotland have established a team of advisers dedicated specifically to supporting this task. We recognise that it is a challenging target, and we remain committed to working with local authorities on this matter. Liz Smith. I thank the cabinet secretary for her response, and I note to the comment about to whom it should be addressed. Nonetheless, COSLA has highlighted the issues with resources that can be leading to delay, meaning that that 2019 deadline will likely prove too much of a challenge given the number of titles that need to be added to the register. Could I ask the Scottish Government and her colleagues, if necessary, if it will provide assistance to help local authorities to progress with this matter? I am aware of the comments that are being made by COSLA in respect of that. The chamber should be aware that the Scottish ministers introduced a 25 per cent reduction in fees for voluntary registration in June 2015, which reduces the costs of registration for local authorities. Registers of Scotland have also extended their keeper-induced registration programme to include publicly owned land. It does not mean that the majority of not all local authority housing stock will be entered on to the land register by registers of Scotland with no cost or resource implications for local authorities. 2019 is a challenging target. We will hope to get as many areas of Scotland by that target date as possible. Question 3 has not been lodged. Question 4, Graham Dey. To ask the Scottish Government when it will bring forward legislation regarding the reintroduction of beavers. As the member will be aware, I set out the position on this on 11 December in answer to a written question. The date on which beavers will receive protection in line with the EU Habitats Directive and be placed on the list of protected species depends on the completion of the strategic environmental assessment process. The SEA was published for consultation on 12 December. It is expected that that will be completed and the Scottish statutory instrument laid in the first half of 2018. Graham Dey. I thank the cabinet secretary for the answer, but can I ask what progress is being made on a practical level developing an accompanying practical management regime? Will that be available for scrutiny alongside the secondary legislation and will there be a scheme in place to compensate anyone who is affected by serious and verifiable damage caused by beavers? There is currently good progress being made on the development of practical management arrangements, which were part of the agreement that was made at the start of the process. Our intention is that those arrangements should be sufficiently responsive and robust to prevent damage occurring in the first place. However, officials are indeed intending shortly to discuss with farmers and other land and fishery managers what sort of financial support may be appropriate where damage occurs and is attributable to beavers. Documents setting out the management regime and the associated financial arrangements will be made available for scrutiny alongside the secondary legislation when it is brought forward. David Stewart. Is the cabinet secretary aware of the trees-for-life assessment of beaver release in Strathglass in my region? Can I meet the cabinet secretary to discuss this issue further? I am always happy to meet David Stewart on whatever subject he wishes to raise with me. I am aware of the campaign that trees-for-life is running. I am conscious that there are issues in the members' part of the world in relation to beavers. I expect that that is exactly what he wishes to talk to me about. I will be happy to meet him. John Scott. The cabinet secretary is very aware of the illegal release of beavers in the take-catchment area, which in its own way is a wildlife crime. What steps is she taking to stop and discourage such illegal releases in the future of beavers and, indeed, other animals? John Scott may remember when I made the announcement in the first place that I was very clear that I would not tolerate continued illegal releases. The illegal release issue is—people need to remember this—criminal activity and needs to be treated extremely seriously. I am conscious that we have two major populations of beavers now in Scotland, one of which was there officially, the other that arose from an unauthorised release in the first place. Needless to say, it is the one that has arisen from the unauthorised release that is causing the greatest difficulty. That shows that, without proper planning and management in the first place, the release of any animal can create problems and the problems that are then created are more difficult to manage in the longer term. We will be taking decisive action when we understand that there are unauthorised or illegal releases. I hope that the member will take my absolute word that we mean what we say. Bill Kidd, to ask the Scottish Government what discussions it has had with the UK Government regarding the impact of Brexit on environmental legislation. We are in regular contact at both ministerial and official level with the UK Government on the potential impacts of the decision to exit the EU. I met with Mr Gove, Ms Griffiths, the Welsh Government's Cabinet Secretary for Energy, Planning and Rural Affairs and officials from the Northern Irish Executive in November, where I set out Scotland's commitment to the core EU environmental principles of precaution, prevention, rectifying pollution at source and the political pace principle. We met again last week, where I reaffirmed Scotland's ambition to not only carry over the status quo but to keep pace with Brussels to limit any potential divergence in standards. The Scottish Government is carefully considering whether any gaps could arise in existing domestic monitoring and enforcement powers that would need to be addressed to ensure that Scotland maintains high standards of environmental protection. I have asked the round table on environment and climate change for its views on where potential gaps may arise and asked them to provide a range of options and how best to fill them. Bill Kidd. I thank the cabinet secretary for her kind and comprehensive reply. Here is the political bit. Does the cabinet secretary have confidence that the UK Government has the ability or possibly the desire to deliver what has been dubbed a green Brexit and what is envisioned as the potential impact of Brexit on Scotland's environment? It is fair to say that the quality and depth of engagement by the UK Government since June 2016 makes it very hard to judge the degree of readiness or commitment, and therefore it is impossible to be absolutely confident about an answer to that. How far an ambition for a green Brexit is shared among UK ministers, I am afraid I cannot tell. However, the Scottish Government remains committed to engaging constructively and at meetings with UK ministers. I continue to press on matters of concern for Scotland's environment. Brexit must not provide an excuse to lower environmental standards. Current environmental standards should be maintained, where it is in our interest to set higher standards. We should be absolutely free to do that. Recent dialogue to be fair to the UK Government has been more constructive. However, there are still some unanswered questions, not least in relation to devolved powers, which are currently exercised in the framework provided by EU law. There is still no clarity on what type of future engagement in European agreements and institutions will be agreed, or indeed on future funding guarantees. Miles Briggs To ask the Scottish Government what action it is taking to reduce plastic waste in the marine environment. The member will have heard my initial response to his colleague at question 1. My answer to him would have been in the same terms. I say, Presiding Officer, that I am heartened by the significant interest in marine issues being shown by not just the Tory members but all members in this chamber. Miles Briggs The cabinet secretary for that answer. Can I ask this question specifically regarding any assessment that the Government has made with regard to sea bin technology, a floating debris interception device, powered by an underwater pump, which is now, I understand, being trialled in both England and 16 other countries. Does the cabinet secretary believe that the use of such technology, while not wishing to take away from any of the focus around prevention of plastic waste entering our seas in the first place, could actually play a significant role in reducing plastic waste from harbors, marinas and shallow coastal waters? Miles Briggs I thank the member for that question. It sounds like a promising piece of technology. We are embarking on a marine litter sink project. There is work being done with Arica having been identified as a case study area because there is very proactive community engagement there. I will ask a question as to whether this is a piece of technology that is being considered by that group. It may already be part of what they are looking at. Question 7, Colin Beattie. To ask the Scottish Government what its response is to a recent study that found that air pollution can increase the risk of birth defects. I am aware of the study that is referred to by the member. The Scottish Government recognises the impact that poor air quality can have on public health, especially for the young and old and those with pre-existing conditions. For that reason, we have in partnership with others developed the cleaner air for Scotland strategy. It sets out a series of actions for government, transport Scotland, local authorities and others to further reduce air pollution across all areas of Scotland. Colin Beattie. Would the cabinet secretary agree with me that Scotland has the opportunity to be a world leader in the reduction of air pollution, in particular through steps such as the introduction of low emission zones? I would certainly hope that Scotland, where possible, would become a world leader on this as on other things. We face air quality issues in parts of Scotland that are principally local in origin, but transboundary pollution is also an issue. We will continue to play our part to reduce the impact of Scottish emissions, recognising our international responsibilities and working in partnership with other countries to learn and share expertise, which is something that I am very keen to do whenever I can. That concludes Environment Questions. We move now to rural economy and connectivity questions. Question 1 has been withdrawn. Question 2, Tom Arthur. To ask the Scottish Government what action it is taking to improve rail services in the Renfrewshire South constituency. Minister Humza Yousaf. The Scottish Government is committed to improving rail services in Renfrewshire South. We have, through the ScotRail franchise, delivered enhanced passenger facilities such as the new waiting shelters, custom information screens, cycle parking and new ticket vending machines at stations in the constituency. In addition, the car park at Johnson station was expanded in 2015 to provide a further 80 spaces. In addition, more than £475 million is being invested in Ibello ScotRail's rolling stock, which will deliver major enhancements to train facilities and increase seating capacity by 23 per cent by 2019 to support growth in our railways. Tom Arthur. I thank the minister for that answer and I very much welcome the Government's commitment to invest in rail services in Renfrewshire South. However, my constituents are concerned that improved services could be threatened by the UK Government's proposal to cut funding for Scotland's railways by £600 million, which is why I raised the issue previously with the Cabinet Secretary for Finance, who informed the chamber that the UK Government had failed to give a satisfactory explanation. Can the minister update Parliament as to whether there has been any progress on the matter and what support, if any, he has had for other parties in this chamber? I thank the member for the question. There has been a further update from Her Majesty's Treasury, which still leaves a shortfall of more than £400 million. That is £400 million short of not what the Scottish Government asks for or demands, but it is coming directly from the industry. That is what the industry tells us. It needs maintenance, operations, renewals and enhancements on the network in the future. I am disappointed by the Tory response, which came back to me by defending its colleagues in Westminster, but I have to say from some of the other parties that there has been no response in almost three months when I wrote the letter to Labour's transport spokesman and not a single response. He never writes, never calls, but all members, regardless of party affiliation, should join with the Scottish Government. It would do well to stand up for Scotland's railways against those savage Tory cuts. To ask the Scottish Government how much Scotland's food and drinks exports were worth in the first nine months of 2017. In the first nine months of 2017, HMRC estimates show that Scotland's overseas food and drink exports were valued at approximately £4.3 billion, which is £4,300 million. Representing an increase of around £500 million, or 13 per cent, compared to the same period in 2016. 42 per cent of those exports were worth around £1.77 billion, were to the EU. For food exports alone, almost 70 per cent were to the EU. I apologise to the member. I think that I cut him off mid-flow. David Torrance, I thank the cabinet secretary for that answer. Can the cabinet secretary advise the Parliament as to the value of Scotland's farmed salmon exports, which countries are the best importers of Scottish farmed salmon? Does he share my concerns about such exports being held up at the borders due to a hard Brexit, a result that the Scottish salmon producers organisation says would be a disaster? Yes, the member is correct. Salmon exports are extremely valuable. They were worth £483 million in the first nine months of this year, an increase of 56 per cent in value from 2000, a really quite staggering increase, with France being the number one destination. The EU remains the biggest single regional market for our salmon, importing £215 million in the first nine months of this year. I share the concerns that Mr Torrance expresses. A hard Brexit risks access to Scotland's biggest overseas regional food and drink export market and could risk increasing the cost of exporting to the EU. The Scottish Government position remains clear that the whole of the UK should remain in the single market. If that is not possible, Scotland should, like Northern Ireland, be entitled to a special arrangement. That is essential in order to maintain a successful and sustainable aquaculture sector in Scotland. Indeed, for similar food sectors, Mr Torrance can be absolutely assured that I and my colleagues will continue to make these points to the UK Government at every available opportunity. Edward Mountain. I would like to refer members to my register of interests that I am a member of a farming partnership that produces food. The figures that the cabinet secretary replicated, and those of the latest figures in 2015, are that £1.8 billion of Scotland's export went to the EU, where £4.1 billion went to the UK—that is £2.3 billion more, cabinet secretary. Does the cabinet secretary believe that the UK single market in 2018 will remain more important to Scotland's food producers than the EU? Of course, all markets are important. That is why, in working with Scotland Food and Drink, we are going to be doing even more to build up further trade in the UK market. However, the threat to the existing market at the moment is to the EU. That threat is because of Brexit, and the SSPO, the salmon producers organisation, has put forward a very clear set of concerns that include an automatic inheritance, continuation of bilateral rights and obligations for the UK under existing international trade arrangements with third non-EU countries, clear and reliable legal redress and dispute resolution mechanisms, a definitive position on the jurisdiction of courts, a clear and definitive position on arbitration mechanisms, continued harmonisation of UK and EU laws and others that I cannot mention. None of those questions have been answered, and we are very close to Brexit, so the loss, the threat to the market that Mr Mountain seems to think is not of concern is a huge concern to the salmon producers, and the UK market will continue, and we will build on it further. To ask the Scottish Government when it last met the Secretary of State for Transport to discuss cross-border rail services. Keith Brown, the Cabinet Secretary for Economy, Jobs and Fair Work, met the Secretary of State for Transport on 3 July. I am currently in an exchange with Mr Grayling, the Secretary of State, regarding cross-border matters of mutual interest. In fact, today, later on, I will be signing off a letter in relation to the east coast partnership. I thank the minister for that answer when ministers next meet with the Secretary of State for Transport can ask him to raise the concerns of the hundreds of thousands of passengers who now use Lockerbie station in the south of Scotland. As a direct result of the UK Government's current franchise and delays in awarding new franchises, those passengers for Lockerbie see more trains pass through the station than stop there. They still do not have a direct service early morning to Edinburgh. There are no services to the capital from 10am to 2.30pm. They regularly suffer from overcrowding and they even find themselves banned from booking seats from Lockerbie to Edinburgh during busy periods. Will the minister raise the plight of those passengers and fight for extra services from Lockerbie station? I think that the member raises a good point. Just as a bit of context, it is probably worth noting that, as he alludes to the cross-border franchises, they are specified and awarded by the UK Government by the department. For Transport now, we have a generally good relationship with the DFT and we look to input that where we can. I will certainly take the points that he has raised. If he wants to formalise them and add to them in terms of future west coast partnership franchises, I will certainly do that. It is probably worth noting at this point that it is because of this Government's direct intervention that there are stops to Lockerbie and Motherwell on the west coast mainline that were not there before. Wherever we can input into strengthening those services, he has an absolute guarantee and an absolute assurance from me that we will make that case to the UK Government. Any cross-border services will require extensive community buy-in, therefore outreach will be needed to ensure transparency and accountability. One of the ministers could tell us how the Scottish Government is engaging with stakeholders and community groups to ensure that any cross-border real services meet their expectations. The member raises a good point. It is worth mentioning that those contracts are specified and awarded by the UK Government. We have some limited input in that. We have already started conversations in the west coast partnership with the three bidders that have been mentioned. We will continue to exchange with MSPs across the chamber. That is an open invitation to any MSP that wishes to write to me about what their expectations are of future cross-border franchises. I will certainly take that forward with the appropriate UK Government minister. To ask the Scottish Government what steps it will take regarding Northern Isles, Interisland ferry services, in light of the debate, on 6 December 2017. The 2016 S&P manifesto contained a commitment to, and I quote, take action to reduce ferry fares on ferry services to Orkney and Shetland. That Government is delivering on that commitment, and that is our priority. The Interisland ferry service, which the member talks about, of course, is the responsibility of the local authorities. There was never an agreement that the Scottish Government would automatically fund the council's financial ask. There is no provision within the draft budget for 2018-19. For that topic, we look to continue to work constructively on that issue. There is a window of opportunity for other political parties to engage constructively in that we would ask the question that if the funding is in the budget, will Rhoda Grant vote for it simply yes or no? Would, of course, suffice. Despite promises made to the Northern Isles council, the minister has confirmed yet again today for the second time that there is no additional money in this year's budget, not a penny for the Northern Isles interisland ferry services. Worse than that, there are huge cuts to council budgets making those services even more precarious. When will the Scottish Government implement the policy of fair funding for ferry services and stop playing politics with lifeline services? She is, of course, wrong on the premise of her question. There has been a cash increase in the draft budget to local governments, so they have been treated well. Now, there is a window of opportunity. Instead of reading from her pre-prepared script, she could have, of course, looked to answer my question, which was that if that funding is included in the finalised budget, will Rhoda Grant vote for the budget? She said she will not. Will she say from a sedentary position that she would not vote for additional funding for internal ferries? Well, there you have it. They are playing politics.  Liam McArthur Thank you, Presiding Officer. Three years ago, the transport minister's predecessor, Derek Mackay, informed me in this chamber that the provision of transport services should not place a disproportionate financial burden on any council, particularly with reference to revenue support for ferry services. Does Mr Yousaf believe that the lifeline internal ferry services in Orkney, which account for 14 per cent of OIC's total annual budget, represents a disproportionate financial burden? If not, why not? Of course, Orkney and Shetland Island Council receive additional funding through their special islands need allowance. The member is probably aware of that. Once again, I go back to the point that the promise made by the Scottish Government was to engage constructively in dialogue. We have done that, in fact, after the last meeting between myself, Derek Mackay and the leaders of both Shetland and Orkney Island Council, the leaders of those councils said that that meeting was constructive, that the engagement was positive. To the member, of course, there is a window of opportunity. On the fifth time of asking, would he vote for that budget if, of course, it includes funding for internal ferry services, he has refused to say that he would? Jamie Halcro Johnston Just last week, in response to the member for Orkney asking, I quote, when will the Scottish Government honour the commitment that it made in 2014 to provide fair funding for those lifeline services? The economy secretary responded that the commitment was made, and I quote again, as long ago as when I was transport minister and directly to the councils involved. Will the minister now accept that there is a clear commitment from the Scottish Government to deliver fair funding for Orkney and Shetland's internal ferries? Will he advise us on how the Scottish Government plans to deliver on that clear commitment? Jamie Halcro Johnston That was quite unbelievable, the hypocrisy of a Conservative member standing here, while they cut this Government's budget by £500 million over the next two years, demand that we reduce taxes and then demands that we put in funding for something that is not even this Government's responsibility. Does the member have no shame or no understanding, of course, of how budgets tend to work? Once again, I present to him with an opportunity, with an olive branch, if you will. If that money has been put in the budget by the time we get to the finalised budget scrutiny and process, will the member vote for that budget? Yes or no? So far, he has not said that he will. Jamie Halcro Johnston Minister, the questions are to you, not to the members in the audience. Jamie Halcro Johnston Question number six, Gordon Lindhurst. To ask the Scottish Government what action it is taking to reduce overcrowding on trains. Minister Humza Yousaf The Scottish Government is committed to significant investment in new and refurbished rolling stock in order to reduce overcrowding. Since 2007, 160 extra carriages have been introduced to the ScotRail fleet, with an extra 200 carriages to follow in the next 18 months. Over 475 million is being invested in ScotRail's rolling stock fleet during the franchise term. That includes new class 385s, which will provide almost 26 per cent more capacity. When they operate on seven cars and eight cars, that will be an increase of 44 per cent. When we introduce the 26 fully refurbished high-speed trains between Scotland's main cities from summer 2018, that will provide an additional 121 coaches, which is 40 per cent more seats. In the short term, the recent introduction of seven electric class 380 trains operating on the Edinburgh and Glasgow route will provide an increase of about 9,500 seats across the day. Gordon Lindhurst. I thank the minister for that answer, but what matters to commuters and those coming in to and from Edinburgh at the minute of this festive time of year is what is happening now. We have heard stories in recent weeks of customers fainting, bursting into tears and panicking as they are, and I quote, crammed in like sardines. The question is, what is the minister and the Government doing now to improve the rail passenger situation and make train services more efficient now? What I would say is that ScotRail is aware of the fact that during the festive period our trains are busy and they do what they can to strengthen services where possible. At the moment that they are using an automated passenger counting system, that gives them a more accurate idea of where some of the services have a heavier load. An example of that would be, for example, strengthening the 17 North Berwick to Haymarket weekday service from four carriages to six. Where they can strengthen services, they absolutely will. There is an understanding from passengers and commuters that, particularly during the festive periods, get towards Christmas in that last minute, Christmas shopping and people travelling rightly to winter markets and so on and so forth. There will be more passengers on our train service. I would say that, similar to my question, my answer to my colleague Jamie Halcro Johnston, I find it a tad bit hypocritical that the Conservatives stand here demanding more money for our railways while simultaneously cutting the budget by hundreds of millions of pounds for the railways. To ask the Scottish Government whether it will provide an update on the implementation of integrated smart ticketing on public transport. Smart tickets can now be used to make journeys on rail, on bus, subway, tram and air. Work is progressing with ferries. That means that Scotland has one of the most advanced smart integrated multi-modal public transport networks in the UK outside of London. Integrated ticketing between Glasgow Subway and ScotRail, as well as the introduction of multi-operator bus smart zones in Aberdeenshire, Dundee and more recently in Glasgow, with the hope that Edinburgh will soon follow in early 2018, have proven successful. Transport Scotland is now looking to expand the cross-mode interoperability for the full saltire smart card estate. In addition, we are working with the industry to support the contactless bank card payment system to bring more convenience to the travelling public. I thank the minister for that answer. A constituent of mine from Hamilton has raised concerns over the delays in implementing an integrated smart card system for our railways. His worry is that the ScotRail smart card will not be as functional as Oyster or perhaps the SPT Subway smart card and will have little advantage over paper tickets. I know that the Government has consulted recently and I wonder when the results will be published. While the aim of having one form of ticketing is a worthy one, is the minister satisfied that this proposal has been an effective use of money, especially when so many people are struggling to pay the increasing fares of transport operators such as ScotRail? Just on our final point, it is worth noting that this Government has capped any increase in fares, but the points raised by Monica Lennon on behalf of her constituent are a very fair one. The more integrated and seamless access to transport can be, the better for everybody. I should say that smart ticketing on ScotRail, the use of that has increased by 50 per cent in the past six months. 2.4 million journeys are now done on ScotRail using smart. More and more of ScotRail's ticketing options are coming on to smart. However, as I said to the member, I think that going by her supplementary, she probably agrees with this, that a lot more focus is now going on to EMV contactless solutions, which again will make that system even more accessible. I am more than happy to, of course, take further suggestions from Monica Lennon on her constituent and others. I would say that, as she has rightly alluded to, the consultation has just closed on this. Of course, I will keep it updated on the analysis. Thank you very much, minister and members. That concludes portfolio questions. We'll now move on to a statement on the Scottish Energy Strategy. We'll just take a few moments to change seats.


It's Wednesday afternoon, and we know what time that is. It's Hawaii, the state of clean energy time. My name is Mitch Ewan. I'm your host today. I'm the co-chair of our sponsor Hawaii Energy Policy Forum. We have financial support for this show from the Hawaii Natural Energy Institute. I'm very pleased to welcome my guest and good friend, Toby Kincaid. These are becoming a regular guest on my show, which is great because I love having him. He's always got something good to say. And Toby is an inventor, an author, and now a drum roll, please. He is now a publisher. And today on our show, this is a great honor for us on our show, he's gonna introduce his new industry publication, and it's called Green Hydrogen Today. So welcome, Toby, and congratulations on this latest episode in your very interesting life. So first of all, tell us something about Green Hydrogen Today. How did you come up with the idea? Well, Aloha, Commander. Good to be with you. Well, I've always loved magazines. I grew up with them. My father had Life Magazine and National Geographic. And I remember just hours and hours with magazines. And I was always struck by how the pictures would capture me. And for years now, I've been lamenting this green hydrogen industry is coming into focus. When is someone gonna publish a magazine? A Renewable Hydrogen Industry Magazine. So since I hadn't seen one, I thought, okay, that's it. Time for me to do this. So for the last six, seven, eight months especially, I've been going to conferences all over the world, well, virtually, and Mission Hydrogen at a Stuttgart as a great representation of what's going on with the European Union.  Well, now the European Union, why they don't think that the Green New Deal is a fantasy, they think it's a cornerstone. So they're whole, by 2050, they're gonna decarbonize their entire economy. So when I saw that, and then in the American space, the conferences put up in Berkeley by the Green Hydrogen Coalition, very well done, fantastic speakers. I was amazed at how comprehensive the industry has become. And then last month, just before the last time I saw you, I was at the Singapore Conference. This was the first annual Asia Pacific Summit for Hydrogen. And it was sponsored by the Sustainable Energy Council of the UK, and they did a marvelous job. I was floored, so I said, okay, that's it, that tears it. We've gotta do this magazine. So how did you come up with the name of Green Hydrogen today? Tell us. Well, that was kind of tough in the beginning because I've been contemplating this for a while and kind of building up steam in my boiler. And honestly, I didn't know what to call it.  Photovoltaic Hydrogen, I don't know, that's not very, that's not what we call it, clean hydrogen. Yeah, okay. See, some people refer to it that way. Do we call it renewable hydrogen? Well, yes, some of the Americans often do that. But in these conferences, the European Union, the Asia Pacific players, they're calling it green hydrogen. So I said, okay, that's it, green hydrogen. That's what we call the answer to what we've been looking for. How to solve global problems with a local solution.  If you were gonna build a pipeline, and you started at one end and then you're gonna go a thousand miles, but take a while. So maybe we put another crew at the other end and then we can work towards the center. The ancients did that a lot. But they also figured out, hey, if you put a crew every 20 miles, then you could do the whole thousand miles in the time it takes to do 20. So what I'm seeing is the whole world is kind of putting together their little part. And suddenly it's just like a crystal crystallized. It suddenly, it's manifest. And what astounded me is just the sophistication. In the Singapore conference, for example, I was struck by the Australians. And I have a little section later in hydrogen hubs. I'll tell you about that. But they are serious. They never once mentioned the United States or even Hawaii, they don't need us. What they're gonna do is they're gonna become the largest exporter of green energy as quickly as they can because the writing's on the wall and their customers, Singapore, Korea, Japan, Indonesia, India, all of their customers for liquid natural gas are beginning to ask them, hey, we wanna buy hydrogen now. Especially the Koreans, the Japanese and the Singaporeans. They are serious. So I thought, okay, that's it. We've got to put one place together where everyone can come and meet the industry and find out what's going on. Well, surely it was hard getting the name green hydrogen today. Surely that was taken, wasn't it? Well, you know, the interesting point. Well, at first I registered green hydrogen today and renewable hydrogen today because I wasn't sure. And then when I figured on green hydrogen, oh, gee, I sure hope it's there. And lo and behold, it was. So I thought, okay, that's mana from the above. So I'll do it. And I started then thinking about the magazine and the sections and I thought, well, there's so much information in this industry and it's so diverse. We're gonna have to create sections for each sector of the industry, green hydrogen law, green hydrogen finance, green hydrogen jobs. Then we go to cement and steel and heavy industries, aluminum, ammonia. All of these sectors whom you would never really have green hydrogen and making steel in the same sentence. But all of a sudden, now the Europeans and the Asian Pacific players are saying, yeah, that's exactly what we're gonna do. In fact, Korea, Ambassador Kim who did the opening ceremony, I was struck by what he was talking about. He was saying, look, we know our children are not gonna survive this world if we don't decarbonize. So we go back to your great phase, the decarbonizer. That's what we need, the decarbonator, you know, the carbonizer. So the world is doing it. And when I saw this response and how sophisticated everyone has become, I thought, okay, that's it. We've got to go forward. Well, how about walking us through your draft and it's actually your first edition? They're gonna be releasing it any day now. So how about giving everybody the preview and walking us through it? Oh, lovely, wonderful. Sure, first slide please. Okay, so here's our cover. And the theme is the world is organizing around green hydrogen. And to me, that's what I was struck by is just that we have coalitions, we have groups of industries now getting together. In the European Union, they're a big, on the big industries, the heavy industries, the ports, the airports.  And so that's what gave rise to that theme. So what's it gonna be a global publication that's not meant strictly for the US? That's right. And in the beginning, I originally thought this would just be a print magazine. But as I kind of went forward, I began to realize through some good counsel that actually the digital magazine has all, has so many tools and utilities that we can use to reach a global audience. So we need to do both, but first digital. What kind of tools are we talking about? Why is digital so much more attractive than the presentation? Well, I have a little bit of publishing in a former life in my background. So I had to kind of reacquaint myself with the terms of publishing, the slug, the bleed, the galley, the gutter, the creep, the orphan, the widow. So all of these are all things I had to remember and it could become a familiar with again. And the tools now, in the old days, we actually had to mechanically lay it out and you would take 10 people to have all of the functions, the writers, the layout artists and so forth. But now we have this wonderful company called Adobe and they make Photoshop, an illustrator and premium pro for video editing. And they also make InDesign, which is like a super word processor on steroids. And it's so professional. There's so many tools. It's like, I felt like, if you give me a strativarius, I'm not gonna make beautiful music just because it's strativarius. I actually have to have some skill in playing. So for the five weeks it took to put this magazine together, 80% of it was, ah, how do I get text to wrap around an object? Ah, so it was pulling teeth for sure. But those tools are there. And now looking at distribution, now there's some tools I never would have imagined. For example, I can acquire a license, have this digital magazine converted into an Apple app. And as such, I could sell it on the app store only for a few bucks or something or on the Google Play Store on the Android format. So if I can get this license and I will, then one option would be that we just posted up on the marketplaces that already exist where hundreds of thousands of people in 160 countries go to buy magazines or other media. So that's very exciting. Worldwide distribution almost of the click of a button. Yes, and for me it was like, well, if I build a website and then I send out a lot of emails, you know, no, no, no, no, no. The world is much more sophisticated now. So I'm catching up. Okay, well walk us through the content. Oh yeah, let's keep going. Next slide please. Oh, thank you. Okay, so this is kind of the first table of contents that I've tried to give a feel of all the different sectors. The fueling stations, aviation, the world is organizing and a brief history of hydrogen. I've tried to put a lot of different little things together. I've embedded now links. So when you do get it, you can click on anything and go directly to the right story you want. Or in the stories, I've also embedded as many links as I can. So for example, we have a page that I didn't show here, but towards the end, which is the stock page and I've listed the 15 prominent stocks in the hydrogen world. And so now you can click on that stock and it'll take you to the company. So if you want more information, very convenient, you can just say, oh, I wanna find out about plug power or ITM or sun hydrogen or any number of them, the gen cell, there's a bunch. So click on them and then you'll be able to see it. So for me, that's what the point of this is. I'm trying to create humbly a global app to kind of unite the entire green hydrogen industry. And what I saw, for example, in the Singapore event with the Australians that mentioned, they're very keen on this, is now we're going to see marketplaces emerge where they're talking about off-take contracts, off-take agreements where producers of green hydrogen now enter in a worldwide marketplace. And I believe it'll be actually the hard currency of the 21st century. Forget paper money, this is a commodity. This is something that has real value. It's energy and it's clean and it's potent. So I see the world really now beginning to gel in what in effect is the industrial revolution of the 21st century where we actually do exactly what you've been talking about, we decarbonize. And in that evolution, in that industrial revolution, we get everything we want. We get high power, we get safety, we get universal availability because it's based on water and renewables. And when you use it, it's 100% non-toxic. Wow, that's a message. All right, next though, let's keep going. Sure, what's the next one? Green hydrogen grid. Okay, this is the point. And what's to me amazing about this revolution is that we're standing on the shoulders of giants. I think Newton said that if I see further than other men, it's because I stand on the shoulders of giants. And that's how I feel. I'm standing on your shoulders. I'm standing on Stan Osserman's shoulders. I'm standing on Think Tech Hawaii's shoulders. And we're gonna come together with this amazing notion. You'll see in the lower right, you see Jules Verne, who was a prolific science fiction writer. He predicted almost every major kind of technological advance that we kind of deal with in our world. And he's saying back in 1874, hey, water is the new coal. It's the next coal. He's saying that in this wonderful mysterious island, he has this wonderful dialogue. And he's saying that he feels manufacturers will be powered by water in the future. And then another person is like, oh, come on, what are you talking about? Well, and then he goes on to explain it.  What do they call it, the smoker or smoky? Oh, they called it smoke. It's just smoke. That was a nickname for London at one point. We're going up to smoke. Oh my goodness. Well, that is true because back in the 1805 timeframe, that's when smog, you know, smoke and fog came out and acid rain because they were burning, you know, 14,000 little steam generators in the city limits, not to mention all the coal to heat the homes, the stately homes, you know, and all of that. So it's a smoker. But on this slide, what we're beginning to see is, you know, just a completely different attitude from anything else in how we can actually use renewables on site, create hydrogen fuel from water using an electrolyzer, store it safely. And then whenever you need it, you can burn it in turbines, you can use it in fuel cells. You can use it directly as a fuel for construction equipment and the farming equipment and of course trucks and semis and cars. So, but this gets to the whole theme of we're changing the world from CO2, from hydrogen stuck to a carbon, to hydrogen stuck to an oxygen. Door number two is better. Well, talk to us about aviation. Oh, let's go to that slide. That's amazing. This is totally- Now, Airbus has really done something spectacular. Because the European Union has said, we're gonna go for heavy industry, we're gonna re-look at all our ports and our airports, Airbus realized, well, hey, we could make green hydrogen on site, control our own fueling, control our own cost volatility because it's just a, there is no fuel cost, it's just the cost of hardware and that they can amortize. So what Airbus has decided is, and they announced to the world by 2035, they're gonna have these three aircraft, I'm only showing one, this is their blended body. This is for their long distance travel over 2,000 nautical miles. They've also designed a hydrogen powered turboprop for the small short hop carriers and then also a turbojet for kind of the mid distance. But this is their blended body concept and look at all the room inside, you can imagine it's, it's volumous. They have lots of room for fuel, cargo, passengers, all kinds of amenities. So this is what's coming and they've kind of thrown the gauntlet down to Boeing and everyone else who wants to talk about future aviation, it's definitely based on hydrogen because of the physics and now because of the economics, we can make the fuel. Green, clean and mean. Right on. The US Department of Energy held a workshop, gee, maybe just before Christmas on hydrogen at scale for aviation. It was, I think it went over three days and we had all the players there and man, they are working very hard behind the scenes now to evolve to hydrogen fueling for aircraft and for ground handling equipment. I mean, everything in the airport could be converted over to hydrogen and the thing that's driving it is climate change and decarbonizing, as we said, when we started out this interview decarbonizing the economy and they're really taking it seriously. So let's move on to the next slide which talks about hydrogen, green hydrogen hubs. Let's talk to us about that. Oh, wonderful. In the magazine, I did three hydrogen hubs, India which is a huge emerging market. Their goal is to convert 4% and that's ambitious within the next by 2030. And if they can achieve that, they will set the course because they're very savvy, they understand their own market and their own market demand is huge. India doesn't have a lot of petroleum so they import their fuel anyway. So they're beginning to realize like these other countries, hey, wait a second. We don't have to have a hole in the ground surrounded by men with guns. And if we don't have a hole in the ground that has any oil in it, what do we have to buy from somebody? Now the world economies are realizing that we can produce our own fuel, indigenously, right here with the renewable sources driving those electrolyzers making hydrogen fuel from water which you get mostly back when you use it.  So they're embracing that. The other hydrogen hub, Chile, I have an article about Chile, you would never, who would have ever thought, Chile, you know, South America, that little stripple land on the Pacific, you know, they're gonna be a major green energy exporter. Oh, you betcha. They have renewable resources better than anyone in the world just about. In the North, they have incredible solar, incredibly high, we call capacities. Meaning that they have many hours per day that they get direct beaming sun. Then in the mountains towards the center, they have lots of hydro, lots of micro hydro. And then in the South, you know, they have incredible winds. I mean, it just doesn't stop blowing. So they feel that they've tracked out their renewables. They have 70 times more renewable potential than their entire grid, just waiting to tap into. And they wanna make money. They've been, you know, victimized by the fossil fuel industry through the 18th, 19th, the 20th centuries. They've had a tough time. They've been abused. But they see their freedom. They see their self-determination. And that's what everyone should understand. That's what, that's the point. Everyone everywhere, relatively all at once, can have everything they would ever want if you have the right kit. Gotta have the right gear. But that's only it. If you have the right gear, you think hard about it. Sorry, didn't mean to interrupt you. No, no, no, no, please. You've been very long. So let's look at this. Oh yes, let's go to the next one. Sure, of course. Okay, this is just a quick emphasis on transition. Now you have capitalists who are worried about stranding assets. I get that. And you also have people who are worried about human assets being stranded. So we wanna create a job creation and education training program for green hydrogen. And so I registered those as well so that we could develop them and have people begin to list where their jobs are and so forth. So one point is each of these sections has its own website potential, right, URL? That's right. I've registered the site and I'll be building like a maniac for the next, well, year. But we'll just keep doing it. We'll just go one by, I'm hoping to do one a week. We'll see if I can. But the most important thing isn't how fast we do it, it's how well we do it. So I believe the more narrow the subject, the more accurate and interesting and comprehensive the information. So for aviation, we couldn't put that all in one magazine. There's just so much going on with drones. Now Plug Power has these great little stacks. They're looking at drones using fuel cells. Wonderful stuff. So I thought, okay, we have to put a .com website there that we can then flesh out all of these great details. So people could go to the websites and drill down if they really wanna get to it. I've just had a question come in from the audience. Oh, how wonderful. Yeah, the question is from one of our viewers, if hydrogen is being made with energy from a non-green fuel, how can you call it green hydrogen? Well, it's not. Yeah, let's talk about that. Most hydrogen in the world, like 98% of it, 75 million tons, is called gray hydrogen. Gray hydrogen means you're making it from methane. Now the big problem with that is you're gonna produce 10 times more CO2 than amount of hydrogen. So one ton of hydrogen, you're ending up with 10 tons of CO2. No good. That's gray hydrogen. So blue hydrogen is the idea of doing that, but they say, okay, we'll take the CO2 and we'll do CCS, carbon capture and storage. So we'll grab the CO2 and we'll stick it somewhere. So how about that guys? And they're like, well, okay, maybe. Well, that's blue hydrogen, but green hydrogen is very specific. It is using renewable energy and water. So there's no carbon atom involved. There's no pollution involved. There's no toxicity, no volatile organic compounds, no partially consumed hydrocarbons, no noxial stocks, no particulates, no mercury poisoning, all of that goes away. Exactly. You know, when you talk about climate change as a driver of this, you're right on. You know, what I heard in these seminars, what I got from these conferences is that people are worried about their children. They see the writing on the wall. If we don't evolve out of this fossil fuel, which fine, it worked for three centuries from coal, oil, gas. And some people say, well, it gave us all this wealth. Yes, but there was a big price. And there's a lot of inequity when some people have resources that everybody needs, which is energy. What's wonderful here is that with renewables and water and using green hydrogen technology, you could be in Timbuktu and be entirely energy rich. What brings me to our next slide, which is green hydrogen earth. Ah, yes. Oh, now here's a great quote again from Jules Verne. And what he's talking about 150 years ago is that the future of mankind cannot be based on fossil fuels. There's inequity. There's military stress. There's economic abuse. It just doesn't lend itself to a free and prosperous world. And we can't do it. There's no way fossil fuels could bring the entire population of the world above poverty. It cannot be done that way. You couldn't possibly distribute it and the cost would be more than all of the world's economies. But if we go with green hydrogen, we actually can coordinate everything. So for the grid, instead of a battery car, when you plug it in, it pulls on the grid instantly and the grid operators may not be ready for that huge load. But with green hydrogen, we use it on the off peak hours. That's where most of the renewable is. And that balances the grid. Now, all of these players, they get that, they know it, and they're acting on it. So I am astounded by why they're following your advice. Commander, they're listening to you. So listening to me, that's been a very good little walkthrough. It's not all the content. There's much more content in this magazine than I've seen. But let's go to the next slide because we're already thinking about the next issue, aren't we? Yes, absolutely. And the next issue is going to be green hydrogen in Hawaii. And the reason is you guys started it all in the modern world, in the modern times.  We are going to convert our economy to clean energy. That was 50 years ago. Everyone else kind of went, what? Are you kidding? Come on, yeah, yeah. Yeah, nice, I didn't know. And you guys have marched forward. You're the first in the world that I'm aware of that said, hey, by 2045, we're going to be clean. We're going to get this toxicity out. We're going to stop spending five billion a year on fossil fuels when we could spend four billion a year for the time that we're paying the leases. And much less, you'd have a thousand million, a billion dollars more money in your pocket if you go to a green hydrogen scenario. And we're making those cases. And in fact, the European Union and the Asia Pacific have found that low-hanging fruit and they're moving very quickly into market entry. So everything is moving. Here's our newest company in Hawaii. Next slide, please. Oh, yes, hydrogen Hawaii. I'm thrilled about this group. This is the future. And imagine the Pacific Rim, just your neighborhood. The Pacific Rim, how many islands do people live on in the Pacific Rim? Well, many. So what you see in this headquarters for hydrogen highway being built is that you're going to have all of the energy that you want and could use falling on you. Falls on your roof, it passes in the breeze, it's there. And what you're going to show the world is, hey, just simply reach out and grab some. And so I think it's beautiful. This is what their hydrogen station is going to look like. Pretty modern. Oh, it's going to be beautiful. Yeah. Well, let's go to the next slide. We only have about a couple of minutes left. So wind it up. This is nice. Stan Awesomeen wrote a beautiful piece here where you kind of, I asked him if he would kind of explain and open up a little bit about what we could do with green hydrogen in the next issue. Anywhere, a beautiful piece. And I paired it with this picture. And I wanted to put more writing around it and clutter it up. But I just saw the picture with his piece. And I went, oh, there it is. I don't need to fiddle with it. That's what I want everyone to see and to read and to know. That there is a future. It's a wonderful future. It's a future where all people can find dignity and self-determination. That's what it's all about. And that's what this matters. It's all about that last slide. It's all about green hydrogen today. So last slide, please. Well, thank you very much. Now, I would invite everyone listening and watching. Please send me an email, tobikincade at gmail.com. I'm still working on the website. I give you other emails, but just use that one for now. Send me just a ping or something. Yeah, I'd like to have a copy of the magazine and I'll send you a link and you can have a look at it. And I'd really appreciate any feedback, criticism, kudos, criticism being the best. That's how I learned. So that's how we get better. So I really would appreciate, and anyone who would like to submit an article, they can contact me as well and we'll get ready for the next issue. This is quarterly, so the next one comes out at the end of March. And I really hope everyone enjoys it. It's been an honor to work on it. Okay, so this is Hawaii, the state of clean energy. We've been talking to Tobikincade in his latest project, green hydrogen today. It's gonna be totally awesome. Moby, thank you so much for being on our show and keep in touch with us and give us updates as you go along. Maybe you can come in for little special additions. Why doesn't it fit you in? Be signing off until next Wednesday at Hawaii, the state of clean energy. And we're all in that canoe together. Don't forget that. Aloha.


Hi, in this lecture, let us learn an interesting and important theorem called Gashgorian theorem. Gashgorian theorem can be used to localize Eigen values of a given matrix. You may be asking why are we doing this theorem in between our discussion on power method? Well, Gashgorian theorem gives us how the Eigen values are located in the complex plane that can be used to verify some of the hypothesis of power method. Well, if you recall power method can be used to compute the dominant Eigen value and a corresponding Eigen vector of a given matrix. Well, this may be viewed as a disadvantage of power method because it cannot compute other Eigen values of a matrix when applied once. It only can capture dominant Eigen values. In fact, in the next lecture, we will see how to modify the power method to get other Eigen values also. But the fact is that when applied power method once, we get only one Eigen value and a corresponding Eigen vector. If we are directly applying the power method, then we will get the dominant Eigen value and a corresponding Eigen vector. However, it looks like a disadvantage. In many applications, we want only the dominant Eigen value of a given matrix. One good example is the well known Google search engine. Google search engine uses page rank algorithm to find the relevance of a site with the keyword given for searching. What Google search engine does is that for each site, it assigns a number called page rank score. Then it arranges all these numbers in the form of a matrix called Google matrix and Google will then compute a vector called page rank vector. The page rank vector is the Eigen vector with certain properties of the corresponding dominant Eigen value of the Google matrix. Therefore, Google does not need all the Eigen values of the Google matrix. It only wants the dominant Eigen value of the Google matrix. In that way, power method is highly preferred because the Google matrix in general can be with few billion dimensions. Therefore, if you go for computing all the Eigen values of the matrix, say for instance, if Google uses QR method which can give all the Eigen values of the matrix, then it will be very costly computationally. Whereas, power method will specifically capture only the dominant Eigen value. Therefore, power method is computationally very efficient in this case. Therefore, in fact, Google search engine uses power method. Now, we understood the importance of power method. The only disadvantage of the power method is when we want to understand the convergence of the power method, then we have to check certain hypothesis which are not practically possible to check. However, note that these hypothesis are not required if you want to just implement power method because to implement power method, we only need to give an initial guess vector which can be chosen arbitrarily. It is only that when we want to understand the convergence of the power method to the dominant Eigen value, then we need certain hypothesis. For instance, we need to know that the given matrix has a unique dominant Eigen value. And also, we need to know that the matrix has a complete set of Eigen vectors. That is, a set of Eigen vectors should form a basis for R n when the matrix is an n cross n matrix. Now, the question is can we somehow sense these hypothesis without explicitly knowing the Eigen values and the Eigen vectors? Well, the answer is may be possible. In certain cases, we can use the Gasgorian disc to conclude this. Now, I will introduce you to Gasgorian theorem through a couple of examples and then I will state the theorem. Let us first consider this 3 cross 3 matrix. First, we have to construct Gasgorian discs. Each Gasgorian disc is associated with a row of the matrix. Therefore, if we are working with a n cross n matrix, then we can construct n Gasgorian discs. How to construct that? Let us take the first row of this matrix. You have to take the diagonal element as the center of the disc and then take the modulus of all the non-diagonal elements of that row and then sum them up. That will be the radius which we will call as row 1 and the center is the diagonal element. In that way, the disc 1 which is coming from the first row is given by the set of all z in the complex plane such that mod z plus c is less than or equal to 1. Remember, these discs are constructed in the complex plane because we know that a matrix can have complex eigenvalues. In fact, Gasgorian disc theorem can localize the complex eigenvalues also. That is why we construct this disc in the complex plane. For the second disc, we have to take the second row of the matrix. Again take the diagonal element as the center of the disc and take the modulus of the non-diagonal elements of that row and sum them up. In this case, it sums to 0.35. So, you have to take these two numbers, take their modulus and then sum them. That gives you the radius as 0.35 and the center is 2. Similarly, the disc 3 is given by the set of all z in c such that mod z minus 5, 5 is the center is less than or equal to mod 0.2 plus mod 0.5. That sums to 0.7, which is the radius of the disc 3. So, this is the set of Gasgorian disc for the given matrix. Let us visualize this. The disc D1 is shown in the yellow color. The disc D2 is shown in blue color and the disc 3 is shown in green color. Now what the Gasgorian disc theorem says that all the eigenvalues of the matrix A will lie in this disc. In this case, in fact the Gasgorian disc theorem says that you will have exactly one eigenvalue. Let us call it as mu 1 that belongs to this set and another eigenvalue say lambda 2 lies in the disc D2 and the third eigenvalue say lambda 3 lies in the disc D3. They can be anywhere in the disc. Suppose if it is here, it means it is a complex eigenvalue. If it lies on the real line, it means it is a real eigenvalue. But what the Gasgorian disc theorem surely says that you cannot find a eigenvalue which is outside the disc. That is not possible. That is what the Gasgorian disc theorem says. This is what we mean by saying that the Gasgorian theorem localizes the eigenvalues. It means it just gives you a particular set in which all the eigenvalues lie. In this case, it is the union of this disc. In fact, let us see what are all the eigenvalues of the matrix A that we have taken. You can see that one eigenvalue is given by minus 2.9591 and so on. You can clearly see that it lies in this disc and the second eigenvalue that we have computed is 4.95 and so on and that you can clearly see that it lies in this disc. Whereas the third eigenvalue that we have computed is 2.0085 and so on and that lies in this disc. So, this particular matrix respects the Gasgorian disc theorem very well. Let us take another example. Here we have a 5 cross 5 matrix. Again you have to construct the Gasgorian disc. Since there are 5 rows, we will have 5 Gasgorian disc. This D1 is given like this where the center is minus 3 and the radius is obtained by summing the absolute values of these elements. Similarly, the second disc D2 has center 2 and the radius is obtained by summing 0.25 plus 0.1. The third disc is again the center is at 5 and the radius is 0.7 because you are summing the absolute values of these elements. The disc 4 has center at 3.5 and the radius as the sum of the modulus of these elements that adds to 1.35. Finally, the disc D5 has center at minus 2 and its radius is 0.5. Again let us visualize this disc. D1 is given like this, D2 is this and D3 is this disc, D4 is this and D5 is this disc. Now, Gasgorian disc theorem also says that you have precisely 2 Eigen values in this region. Why 2? Because this is the union of 2 discs. Therefore, 2 Eigen values lie in this set. Similarly, 3 Eigen values will lie in this set. That is what the Gasgorian disc theorem says. That is you can find 3 Eigen values lying in this set. Let us now precisely state the Gasgorian disc theorem. You are given a matrix A which is a n cross n matrix. First you have to find the Gasgorian disc. Each corresponding to a row of the matrix. Therefore, if it is a n cross n matrix, you have n Gasgorian disc. The radius is given like this and the discs are given like this. Now, the conclusion of the theorem is that each Eigen value of the matrix A lies in one of the discs Dk. It means what? You cannot find an Eigen value lying outside the union of this disc. That is what it means. The theorem also concludes that if you have some m number of discs whose union is disjoint from some n minus m number of discs, then precisely you have m Eigen values in the first set and the n minus m Eigen values in the second set. That is precisely what we have visualized here. You have m number of discs whose union is disjoint from n minus m number of discs. Here the m is equal to 2. So, you have 2 discs which are intersecting each other and their union is disjoint from another set of discs which consists of 3 discs here in this particular example. Therefore, you can precisely have m number of Eigen values in this set. Let us call this as R 1. So, R 1 will have 2 Eigen values in this example and this set can have 3 Eigen values. Let us call this as R 2. That is what the theorem says. Suppose among the discs, there is a collection of m discs whose union is disjoint from the union of the rest of all n minus m disc. We are calling the first set as R 1 and the second as R 2.  Let us prove only the first part of the theorem. Let lambda be an Eigen value of the matrix A and the and let V be the corresponding Eigen vector that we have chosen. Then you can write A V equal to lambda V. That is the definition of the Eigen vector. Also note that the Eigen vector should be non-zero. Now, let us take the maximum norm of the vector V and that is given by maximum of modulus of V 1, V 2 and so on up to V n. Let this maximum norm be achieved at the coordinate say R, where R is an integer between 1 to n. Then the Rth equation can be written like this. Why? Because I am just taking the Rth coordinate of this vector to the left hand side and writing this equation. Now, what we will do is we will take this term alone on the right hand side and keep all the other terms on the left hand side and then divide both sides by V R. That is we are taking this to other side that gives us A R R minus lambda into V R and then dividing both sides by V R.  Now, take the modulus on both sides and use the triangle inequality. We get modulus of lambda minus A R is less than or equal to this expression. Now, you see V R is the greater than or equal to all the other components V i i is equal to 1 to n. That implies mod V i divided by V R is less than or equal to 1. Therefore, each of this quantity is less than or equal to 1. So, you use this property and write this right hand side expression like this. You can see that mod lambda minus A R R is less than or equal to this, where each of this term is dominated by 1 here. Now, if you recall the disc D R is nothing but the set of all z in C such that mod z minus A R R is less than or equal to the radius of the earth disc is precisely given by this expression. So, that is what we want to prove for the first conclusion of the theorem. The theorem says that each eigen value of A lies in one of the discs. How we picked up that disc is by first picking up the maximum coordinate of the vector V and then we are trying to pack lambda into that disc D R, where the maximum norm is achieved. So, this completes the first part of the proof of Gashgorian theorem. We will not prove the second part. Now, let us see how to use Gashgorian theorem to check the hypothesis of power method. Let us again take our first example, where we have taken a 3 cross 3 matrix right. If you recall we have also constructed the Gashgorian discs for this matrix. Now, let us see whether we can get some idea about the hypothesis of the power method. Recall that the second hypothesis of the power method demands that A should have a complete set of eigenvectors. That is all the eigenvectors should be linearly independent right, but by seeing the Gashgorian disc you can see that one eigenvalue is sitting here, one eigenvalue is sitting here and one eigenvalue is sitting here. Therefore, this matrix has distinct eigenvalues. Therefore, all the eigenvectors are going to be linearly independent right. So, Gashgorian theorem gave us the information that all the eigenvalues of the matrix A are in fact, distinct. Now, let us check the first hypothesis, where we need to see whether A has a unique dominant eigenvalue. First observe that the dominant eigenvalue is coming from this disc right, because this point is 4.3 right and this is 5.7 right, the center is 5. Therefore, assuming that A has real eigenvalues, because we always work with real eigenvalues. In that way, the eigenvalue of A which is coming from this disc is going to be something between 4.3 to 5.7 and that is surely going to be the dominant eigenvalue. Why? Because we have to check this condition, lambda 1 should be strictly greater than lambda i for i equal to 1, 2 and so on. Here it is only 3 right. So, to check that what you have to do is, you see whether the disc which is lying on the negative side of the plane, when it shifted to the positive side, it should not intersect the circle which has the dominant eigenvalue. That is the idea. Let us shift this disc to the positive side and see, well the disc when you shift to the positive side, it is not intersecting the disc d 3 right. That shows that whatever may be the value that is coming from this part of the interval is surely going to be different from this part as well as this part. Therefore, from here you can in fact conclude that A has a unique dominant eigenvalue right. Let us take the next example where we had a 5 cross 5 matrix and here we have seen that the Gashgorian discs are given like this. You can clearly see that we cannot use Gashgorian disc theorem to check the hypothesis of the power method. Why? Because some of the discs are intersecting. For instance, the dominant eigenvalue is coming from this disc and it is intersecting this disc also. Therefore, it may happen that there is a eigenvalue with algebraic multiplicity as 2 or even 3 and also when you shift this disc to the other side that is when you take modulus of the eigenvalue which is sitting here, it may also coincide with the dominant eigenvalue right. Therefore, Gashgorian disc theorem cannot be used to verify the hypothesis of the power method. It does not mean that A is going to violate the hypothesis of the power method. It is only that the Gashgorian theorem cannot be used. Now, what is the idea should we give up or is there any other way to check the hypothesis of power method through Gashgorian theorem? Well, there is a nice idea that why not we apply the Gashgorian theorem to A transpose. Why? Because the eigenvalues of A and the eigenvalues of A transpose are one and the same. Therefore, we can also apply the Gashgorian theorem to A transpose and see if that gives us a good information. If so, then that can be borrowed to conclude how the eigenvalues are going to be located for the matrix A itself right. So, in this case what we have to do instead of taking the radius along the rows, you have to now take the radius along the columns that is take the absolute values of the non diagonal elements of the columns. Let us see in that way how the Gashgorian disc for A transpose are looking like. You can see that the disc D 1 is given like this. Again you can observe that the centers of these discs are not going to change, only the radii will change. For the first disc now you have to sum along the columns right. In the previous case we took the sum along the radius row and therefore, we had the sum as 1 for the radius of the first disc. Now the first disc will have radius 0.551 right. So, that is the advantage in this particular example it may be that A transpose may give a bad information then A sometime, but in this case A transpose is seems to be giving a better information than A. Similarly, for D 2 again you have to take the diagonal element 2 as the center of D 2 and then now the radius is computed along the column elements. In that way we got 1.3 for the disc 2 and similarly disc 3 has radius 0.55 again and the center is at 5 and similarly the other 2 discs. Let us see how they look like in the complex plane. You can see now that all the discs for A transpose are disjoint. Therefore, you can have the hypothesis of power method verified from the Gashgorian theorem applied to A transpose. You can in fact shift these 2 discs to the positive side and see whether they are intersecting the discs from where you get the dominant Eigen value. Note that the dominant Eigen value is coming from this disc right. Let us see how they look like when I shifted these 2 discs from the negative side to positive side. Why we are doing? Because the dominant Eigen value condition is checked with modulus of the Eigen values right. It goes like this. So, therefore, we have to check whether the absolute value of the Eigen values are going to coincide with the dominant Eigen value. That is why we are shifting this this to the positive side and now you can see that even when you shift them to positive side they do not intersect the discs from where you get the dominant Eigen value. Therefore, the Gashgorian theorem applied to A transpose gives us a good information and it tells us that the power method is going to converge. If at all you choose your x naught correctly again that is not a problem because what you need is that you have to choose your x naught such that c 1 should not be equal to 0. If you recall c 1 should not be equal to 0 what is c 1? c 1 is nothing but the coefficient of the first term in the representation c 1 v 1 plus up to c n v n right. This is not something very serious because we may choose c 1 v 1 plus 2 or 3 vectors and run the power method. If all these are going to converge to the same Eigen value then it is very likely that that is going to be dominant Eigen value because it is very unlikely that you will choose x naught 3 times for which all the 3 vectors will have 0 c 1 right. So, that may be very unlikely. In that way this is practically not very serious hypothesis whereas, the other two are really serious concern to check and Gashgorian theorem gives us a possibility that we may verify those hypothesis without having explicitly the idea about how the Eigen values are. Let us also state the Gashgorian theorem on A transpose. So, there is nothing new here you just have to apply the radius column wise now that is the only difference in this theorem when compared to the theorem that we have stated previously otherwise the conclusion remains the same. Well, before ending this lecture let us have another interesting application of the Gashgorian this theorem. If you recall in the last class we have studied the convergence theorem for iterative methods in particular for Jacobi and Gauss-Seidel method. We had given a necessary and sufficient condition for the convergence. What is that? That is using the spectral radius of the iterative matrix B that is the spectral radius of the iterative matrix should be less than 1. This is the necessary and the sufficient condition for the convergence of an iterative method right. What is spectral radius? Well if you recall the definition of spectral radius is nothing but the maximum over all the Eigen values lambda i right. In a way you can see that the spectral radius is nothing but the dominant Eigen value of the matrix B. Therefore, you can in fact compute the spectral radius using power method. If your purpose is to only see whether the Jacobian method or the Gauss-Seidel method is going to converge or any iterative method is going to converge or not then even you can go for checking the Gashgorian theorem. How you will do? For instance let us take the Jacobi method. The iterative matrix for the Jacobi method is given like this. You can see that all the Gashgorian disks of this matrix are having their centers as 0. In fact, Jacobi method for any system will have the diagonal elements as 0. You can verify that in this case what you have to do is you either go row wise and get the disks are also you go by column wise and see the disk whether all the disks are having their radii as something less than 1. If that is something less than 1 it means what all the Eigen values are going to lie in this disk right and if you know that the largest disk has its radii something less than 1 it means the spectral radius which is the dominant Eigen value will in the worst case be sitting somewhere here and that will be strictly less than 1 right. So that is the idea. Let us see how the Gashgorian disk is placed for the matrix BJ. You can see that the lost disk has radius more than 1. Therefore you cannot conclude the convergence of Jacobi method using the Gashgorian disk applied on BJ. Let us see whether we can conclude the convergence by applying Gashgorian disk on BJ transpose. Well you can see that when you take the disks with BJ transpose the radius of the first disk is the sum of these two numbers which amounts to be around 0.757 and the second disk as the radius something very near to 0.9 but still it is less than 1 and the third disk is pretty small. Its radius is only around 0.4. Therefore the Gashgorian theorem when applied to BJ transpose tells us that the Jacobi method is surely going to converge. In fact you can use the power method to compute the spectral radius also. In this case the spectral radius of BJ is 0.63825 and so on. Well the convergence may be little slow but still the Jacobi method converges. Without going to the explicit calculation of the Jacobi iteration we can understand either by using power method or more easily using the Gashgorian disk theorem we can conclude this. Let us go to see whether the same exercise can be done for Gauss-Seidel method. Well in the Gauss-Seidel method the Gashgorian disks for the iteration matrix BG is given like this and you can see whether you take row wise and apply the Gashgorian disk theorem or you take the disk column wise you can see that all the disks have radius something very less than 1. In fact you can see that the spectral radius of the iterative matrix of the Gauss-Seidel method is very very small that in fact shows that the Gauss-Seidel method in fact is going to converge very fast to the solution of this system. With this note let us end this lecture. Thank you for your attention.


Hey everybody, it's Greg here and at PhotonKina 2018 on September 25th, Panasonic has announced two full-frame mirrorless cameras. So let's go over some of the details. So you can see it's already posted on their website. It's going to be the Lumix S series and there'll be two full-frame cameras. So one will be a 47 megapixel S1R and the second one will be a 24 megapixel S1. Now they'll both have 4k 60 frames per second. They'll have a tilt-out screen. There'll be dual IS image stabilization in both and there'll be two card slots, one with an XQD card and one with an SD card, much like the Nikon D850 has. So those are the features. It'll have an L-mount compatibility with lenses from Leica and also there's going to be a new partnership with Leica, Sigma and Panasonic in all three of them creating lenses for this new L-mount mirrorless system. Now Panasonic will have three lenses on launch. They'll have the 24 to 105, the 70 to 200 and a prime 50 mil f1.4 and Leica right now has some lenses that will also fit as well as Sigma coming out with some and they say they'll have 10 lenses by 2020. Now these two cameras will not be available until 2019, somewhere around the March time frames. Alright so that's the official announcement. Two full frame mirrorless cameras from Panasonic, one 47 megapixels. The other is going to be 24. Let me know what you think about these upcoming cameras and lenses in this partnership. How do you think they're going to compete with Nikon or Canon or Sony? What do you think? Will the autofocus be as good? It's really hard to say at this point. Anyway that's the announcement. Thanks for watching this video and I'll see you in the next one.


I retraced my steps, hoping to find the path. But everything seemed unfamiliar. Even trying to go in the opposite direction did not feel right. I frantically searched for any recognizable landmark or signpost to guide me back to my group. With a mixture of anxiety and terror, my heart was pounding so hard in my chest that I could feel it in my ears. I frantically scanned the dense trees surrounding me, my eyes darting back and forth in a frenzy of fear. Every rustle of the leaves or snap of a twig sent shivers down my spine, as I desperately tried to catch a glimpse of the trail. Finally, my legs gave out from beneath me and I collapsed onto a fallen tree, gasping for air.