Good morning everybody. I think Abbey Williams, Director of the Institute for Global Leadership and Professor of the Practice of International Politics at the Fletcher School. It's very good to see all of you this morning and also welcome to all of those who are joining us online. We have some people who will be joining us later. I just want to say it is no mere formality when I say I'm delighted that my friend David Lambie, the UK Shadow Foreign Secretary, is here with us this morning to give a Dr. Jean, my Global Citizenship Award address. He's a dedicated and inspiring public servant with enormous integrity. David has agreed that he will take questions after his speech so we have two microphones in the eye and so you can just come up and ask questions. It also now of course gives me great pleasure to welcome the President of Tufts University, Tony Monaco. He is an accomplished leader, scientist and teacher with a strong commitment to academic excellence and a global perspective. And he has a really keen awareness of the transformative power of higher education in the lives of individuals and societies and during the past two years, these difficult years, he's led Tufts and the university community with competence and compassion. So it's a great pleasure to invite you President Monaco. Well thank you Abbey. Good morning everyone. It's a pleasure to welcome you to the Tufts Institute for Global Leadership, Dr. Jean Maier, Global Citizenship Award Lecture. I'm delighted that this lecture is being held in partnership with the Fletcher School of Law and Diplomacy.  The Institute for Global Leadership prepares new generations of critical thinkers who can provide the effective and ethical leadership needed to address the world's most difficult challenges. It also provides a forum for world leaders, renowned scholars, and seasoned practitioners to address topics of contemporary relevance. It's an honor to welcome the right Honorable David Lambie, Shadow Foreign Secretary of the United Kingdom. Both in and out of office, he has made distinguished contributions to social justice and human rights. It is most fitting that he is receiving a Dr. Jean Maier Global Citizenship Award today. This award was established in 1933 in honor of former Tufts President, Jean Maier, who believed that scholarship research and teaching should be dedicated to solving the world's most pressing problems. I can't believe that this award was established in 1933. This must be a typo. The Fletcher School was established in 1933. I think this award must have been established after Dr. Jean Maier was president. In this vein, the Shadow Foreign Secretary's topic, Ukraine and the Consequences for the West, is especially timely and of importance. We express our solidarity with the people of Ukraine and our deepest concern by those impacted by the tragedy of this war. Because of Tufts' deep and long history of global engagement, our university is playing an important role in helping to shape the world's understanding of the crisis, as well as supporting members of our community who have been affected by recent events. Before we begin the program, I would like to thank Dr. Abbey Williams and the team at the Institute for Global Leadership and the Fletcher School for organizing this lecture. It is now my pleasure to turn it over to Chichi Ikpiazu, who will formally introduce our speaker and present the Maier Award. Chichi is a senior majoring in international relations and Spanish from South Florida. She took the IGL flagship epic course on preventing genocide and mass atrocities in 2020. She is also a research assistant at the Fletcher School Center for Strategic Studies. Welcome Chichi and thank you everyone. The right honourable David Lambie is a member of parliament for Tottenham and the UK Shadow Foreign Secretary. He was elected as Labour MP for Tottenham at the age of 27 in June 2000. He is one of the leading agenda setters in the fields of social advocacy, diversity, and multiculturalism. He was Shadow Justice Secretary from April 2020 until November 2021. David Lambie served for nine years, 2001 to 2010, as a minister in the previous labour government, including as minister for constitutional affairs and minister for higher education. He continues to tackle the big issues facing society on a micro and macro scale and provide tangible solutions to bring about effective change. He led the campaign for the Windrush British citizens to be granted full British citizenship and has been at the forefront of the fight for justice for the families affected by the Grenfell fire. In 2017, he published the widely recognized review into the treatment of in outcomes for Black, Asian, and minority ethnic individuals in the criminal justice system. In 2020, he presented a TED talk as part of their latest countdown series calling for a global recognition that we cannot solve climate change without racial and social justice. He is the author of Tribes, A Search for Belonging in a Divided Society, published in 2020, which explores our innate need to belong and how globalization and digitization are deepening the divide in society. Today, I have the honour of presenting the right honourable David Lambie with the Dr. Jean Maier Global Citizenship Award. This award was not established in 1933, it was established in 1993, in honour of Jean Maier, 10th president of Tufts University, to challenge and inspire students and the community by bringing Tufts, distinguished scholars and practitioners whose moral courage, personal integrity and passion for scholarship exemplify his dictum that scholarship, research and teaching must be dedicated to solving the most pressing problems facing the world. David Lambie. Well look, thank you very much to the Institute for Global Leadership and the Fletcher School here at the Tufts University. I'm so great for you hosting me and hosting me to speak about foreign policy at this extraordinarily important time. And to be given the John Maier Global Citizenship Award is quite unexpected, but a wonderful wonderful things have done, so thank you very much indeed and my sincere thanks to Dr. Abby Williams for his continued support over so many years. I'm very very grateful indeed. In the late 1990s, I had my first taste of Massachusetts when I studied here just a short bus ride away at Harvard Law School. It was an inspiring time. I will never forget my first big exposure to the American Constitution. The first lesson that I learned was that democracy as proclaimed by America's founding fathers is always and always and always has been and always will be a work in progress. I also learned the great story of the 20th century and that was a story about how many different groups from working class people to people of color to women to LGBTQ plus found and fought hard to secure rights long denied their forebears. Back in the late 90s so much around us was changing. We were living in the wake of two liberal revolutions. The first was the social and cultural revolution of what's been called the Swinging 1960s. The second was the economic and free market revolution Satellite by Ronald Reagan and Margaret Thatcher in the 1980s. The Soviet Union had collapsed not long before. Communism and autocracy had capitulated to capitalism and democracy. Francis Fukuyama suggested this marked the endpoint of mankind's ideological evolution and the universalization of Western liberal democracy as the final form of human government. Progressives were winning the fight on both sides of the Atlantic. Bill Clinton was in the White House. Tony Blair had just won a landside victory for the Labour Party in the UK. The march towards a 21st century future was filled with hope. But as we reached the global financial crisis of 2008 that hope had started to evaporate. The twin liberal revolutions had come at a high price creating a hyper individualistic culture where rights overtook responsibilities where we could reach billions. People were constantly engaged on their smartphones but they had fewer meaningful connections. We got richer but inequality accelerated and the pursuit of profit was prioritized over democratic values. The age of individualism was defined by another paradox. The more atomized we became the more we sought belonging in tribal identities from the relatively benign to the outright destructive extremist Islamist behaviour. Far right terrorism organized crime gangs online our opinions did not gain the nuance that results from sophisticated debate and discussion. We gained access to infinite amounts of information but we lost the guard rails that sorted fact from fiction. Algorithms designed by tech companies to grip eyeballs pushed many of us to new extremes. The common ground upon which democracies depend began to crumble and malign actors including governments like Vladimir Putin's turned to ethno-nationalistic authoritarian politics and exploited our online spaces to interfere in our democracies with disinformation and lies. A broad Putin took advantage of unsuccessful Western interventions, the decline of American hedonary and a newly multipolar world. He invaded and still occupies part of Georgia. He annexed Crimea. He sought to carve out parts of eastern Ukraine. He used the strength of his armed forces to prop up the monstrous Bashar al-Assad who used chemical weapons against the Syrian people and he helped to drive a refugee crisis that reached Europe which was seized upon by hard-right populace to inflame new divisions between us and them. Meanwhile, authoritarians and their acolytes from Nigel Farage in the UK to Donald Trump in the US to Matteo Salvini and Marine Le Pen publicly expressed sympathies with Vladimir Putin as they rose to prominence in their own democracies parading their illiberalism as patriotism, pretending to be protectors of their nations while attacking the values of freedom, equality and democracy that they were founded upon and at the same time Putin saw we were in a cost of living crisis, a climate crisis, a global pandemic after years of sowing disunity in our democracies exploiting the vulnerabilities left by these two liberal revolutions. It's no coincidence that Putin saw this as our moment of maximum weakness and chose this as the moment to start his barbaric and illegal invasion of the sovereign country of Ukraine under the fog of disorder he thought he could act with impunity but the strength and the unity of the opposition Putin and faced shows in fact that he cannot remarkable and courageous the Ukrainian people in defence of their homeland has been quite extraordinary tougher global sanctions that many thought were possible unity within the European Union previously considered fractured a turning point in defence policy in Germany in Sweden in Finland and Poland NATO with more focus than we've ever seen since the Cold War and 141 countries in the United Nations General Assembly voting to condemn Putin's war of aggression despite the strong reaction that we have seen this is not the moment to be complacent it's the time for a radical rethink of foreign policy and to reboot our diplomacy Putin's invasion is shocking the images of tanks rolling across the borders of European nations reopens the deepest wounds of our continent's history many has said the world has changed changed forever on February the 24th and it did the horrific war in Ukraine is solely of Putin's making but it also highlighted contradictions in the West relationship with Russia as well as flaws in our broader foreign policy assumptions many in Europe believe the era of wars between states was over we reshaped our security our defence our intelligence and diplomacy to tackle very different threats allowing our core capabilities to dwindle just months before Russia's invasion Boris Johnson said that the era of tank battles on European soil was over now we see tanks rolling across the frontiers in Europe borders changed by force nuclear threats issued we must adjust our mindset and adapt our thinking for too long Western governments including Britain's conservatives believed they could ignore domestic policies which undermined our foreign policy we tolerated dependence on Russian oil and gas funding Putin's war chest regardless of his aggression and despite the urgent need to decarbonize dirty Russian money the loot of Putin's dictatorship was embraced from our football clubs to our politics oligarchs and kleptocrats use Britain's capital as both the hiding place and the service industry for their ill-gotten gains a spider's web of dirty money spread across London fueling crime on our streets making property unaffordable for most laundering reputations silencing critics and sustaining Putin's authoritarian regime this disregard for the contradictions in our policy has been exposed by this crisis we must now end the hypocrisy too often we saw the world as we wanted it to be not as it was some believe Putin could be moderated and influence by our engagement what the Germans called change through trade we have been repeatedly overly optimistic even naive particularly when we stood to profit even when Putin broke international law and invaded his neighbors our responses were weak the tame response to the seizure of Crimea in 2014 is one of the reasons we could not deter Putin this time around we must be finally realistic about the world view in the Kremlin we've long known that Putin saw the collapse of the Soviet Union not as a liberation but as a humiliation the catastrophe the consequences he told us time and again that needed to be reversed Putin seeks a sphere of influence a reconstituted Russian empire whether we like it or not Putin believes that domestic survival depends on total dominance of the political sphere the elimination of opponents and the fan of bigotry nationalism and nostalgia he will ruthlessly pursue Russia's interests as he sees them in zero-tongue get terms and he's taken lessons from the Arab spring seeing democratic revolutions as contagious when he saw the 2014 democratic revolution in Ukraine he feared the dangers of one in Russia as well it is time to understand Putin on his own terms but it's not only Britain's conservative government which made strategic mistakes on Russia Donald Trump's disastrous spell in the White House where he cozied up to dictators from Putin to Kim Jong-un while distancing the United States from its traditional allies in the European Union and institutions like NATO and the United Nations shows the danger of turning against the institutions that were created by the global community after the Second World War for too long also parts of the left even some members of my own party falsely divided the world into two camps America and the West on one side and victims on the other this has never been right but this view has now been exposed for all to see as a farce the rising aggression of countries including Russia China and Iran and in particular Putin's barbaric and illegal invasion of Ukraine are definitive proof of the world's wrongs that do not stem from Western actions we must confront our own historic mistakes but if we fail to see beyond them and falsely believe Western nations have nothing to contribute we miss the value of making common cause for people fighting for democracy around the world and we forget the value of the international institutions that arose to protect us all many people have drawn historical analogies with our current situation some have suggested we're entering a new cold war I think the Cold War analogy has its limitations the world today is far more interdependent and economically interconnected than it was in the days of the iron curtain unlike China Russia is not a serious economic competitor to the West it does not represent a coherent ideology like communism it is a nuclear superpower but it is a middling an unbalanced economy in free form with a leader clinging to a blood and soil nationalism of the past but there are some reflections that we can draw from the cold war that may be useful for the months and years ahead we need a patient long-term strategy and to equip ourselves for the task of a sustained confrontation not just with Putin but with Putinism and its imitators around the world dictatorships are no longer controlled by one bad actor in isolation but by interlink networks of illicit finance security services and peddlers of misinformation not only in one country or even in one region but across the world they aren't unified by one particular political ideology but the shared desire to hold on to power at whatever costs to enrich themselves and to counter this network of Putinism we must show that we can ditch short-termism on energy on economics on politics and on security that for far too long has dubbed our approach the first step is signaling this change and it should be a bang on all foreign campaign contributions from our politics saying very clearly that no to malign interference in our democracies and we must properly regulate big tech so that it's forced to quickly remove disinformation campaigns or face punishing fines we must also double down on unity our strength comes from our alliances rooted in common values not the transactional marriages of convenience or coercion which characterised Russia's alliances we must capitalise on the united economic front that has been forged against Putin in the Cold War there were mechanisms like COCON the coordinating committee for common export controls to sustain common approaches to export controls we should consider whether we need new structures to ensure that the UK the United States Europe Japan Australia Canada and other partners across the globe can maintain a common approach and we should seek to build the widest possible diplomatic coalition in opposition to this war this neo-imperialism is not just a challenge to the west if one sovereign UN member state can be carved up on a whim all states are threatened the Cold War also teaches us the imperative to manage the risks of escalation both lessons to learn and mistakes to avoid preventing a catastrophic conflict took strategy and resolve diplomacy and deterrence and even a little bit of luck even before this crisis we had already lost too much of the architecture of arms control built in the Cold War and post Cold War period such as the INF and open skies treaties we should maximise pressure on Putin and support the Ukrainians in their fight including with arms but also keep open channels of communication maintain military transparency and seek to avoid miscalculation NATO was right to rule out a no-fly zone which would bring Russia and NATO into direct conflict but Russia must know our absolute commitment to the principle that if a NATO ally is the victim of an armed attack each and every other member of the alliance will respond and we need to be ready for modern acts of aggression with accelerating and enhanced joint cyber defences among NATO member states and finally the Cold War teaches us that we must remain open to the Russian people ordinary Russians did not start this war many have courageously protested against it it takes real courage to challenge your government if you live in authoritarian state we must always distinguish between Putin and the Russian people and reach as many as we can with objective news allies should coordinate to get credible information to the Russian public through whatever means available with direct financial and diplomatic support to civil society and independent journalism we must think creatively about how to strengthen the voices of moderation and reform we must be a safe haven to Russians fleeing political persecution living in an age of authoritarians means reassessing our strategic priorities this must marks a turning point in Britain and for our allies after years of distraction and insularity Britain can carve a new leading position on the world stage first we strengthen our defences and lead the debate about the future of European security Britain has left the European Union the task now is to make brexit work on both sides of the channel it is time to leave behind the petty diplomatic spats with our neighbours pursued by Boris Johnson's UK government designed only to serve short-term domestic political instincts the British government must stop putting peace on the island of Ireland at risk with its reckless threats to the Good Friday agreement and we need a government that can rebuild relations of trust and mutual respect with our closest neighbours on the continent based on our shared values and common interests we need to end more than a decade of cuts to the army and rethink the assumptions of the integrated review the government has pursued an Indo-Pacific tilt but it must not do so at the cost of our commitments to European security a war ravages parts of our continent we need to put past brexit divisions behind us stop seeking rows with European partners and use this moment to explore new ways to rebuild relations with European allies through a new UK EU security pact and second we must sprint towards decarbonisation and end our dependency on dirty fossil fuels much of the funding for Putin's war machine has come from us and our partners running our industries heating our homes and filling our cars with oil and gas from Russia seven hundred million dollars per day from Europe we can revolutionise that if we have the will the UK government has said that the UK will end Russian oil imports to the UK by the end of 2022 and we support this but on its own this move will not shield us from rocketing energy prices our prime minister moves to fill the gap of Russian energy has so far been to look to new different authoritarians to go and buy our oil and whether it's Iran Saudi Arabia or elsewhere it's short termist it's ill-judged and it's not learning the lessons of Putin fossil fuels empower the worst sorts of dictators the only true form of energy independence is through clean energy and this is why a labour government in Britain would quadruple investment in a green recovery 225 billion pounds over the next eight years and third we must finally end our role as a facilitator of illicit finance and cleanse our society of dirty money not just from Russia but from corrupt corrupt elites across the world who have used Britain and our overseas territories to hide their ill-gotten wealth under our very noses and fourth we must restore our soft power because it's not only tyrants actions that we must change but the minds of their publics the united states and the uk together do so much good through the development that we lead across the world but Britain has stepped back from its former leadership cutting billions in aid and mismanaging the merger of our development and foreign ministries leaving them less than the sum of their parts facing the largest refugee crisis in Europe since the second world war the importance of humanitarian aid and long-term development could not be greater one of the uk's greatest exports is the bbc world service which plays a unique role both in delivering information to populations living in authoritarian regimes and embody the free speech and independent media that are the cornerstones of our democracies reaching nearly 400 million people per week in the first weeks of the Putin's invasion the bbc's russian language service audience tripled and has now been subject to new restrictions in russia but the fact that just 13 percent of russians see russia as the aggressor in putin's illegal war shows the scale of the task ahead a labor government would truly value the bbc world service along a refreshed british council and be a beacon for our values across the world i started this speech by saying my time in the united states taught me the great story of the 20th century is one of how minority groups gained rights through liberal democracy if this is true the story of the 21st century so far is a story of the reverse every year freedom house releases a report of the state of global democracy this year's report was titled the global expansion of authoritarian rule these times are dark but they're not without promise we should take encouragement from valium in putin's current failure to achieve his objectives in ukraine russia's huge poorly organized army being outlawed by ukraine's smaller but more skillful and determined troops and citizens because unlike the russians they actually know what they're fighting for it's the same thing that generations of british and american troops diplomats activists and ordinary people have struggled for the hope that our democracies are supposed to represent ukraine's formidable and courageous leader vladimir zelensky has called upon our collective conscience he has shown what it means to fight for a democratic nation state using ukraine's heroics as inspiration together britain the united states the european union and the rest of our allies and partners around the world have the chance to move past the age of authoritarians reaffirming our commitments to the values we share freedom democracy and the rule of law restoring the international institutions that spread them and giving hope to our nations once again thank you very much indeed thank you very much uh uh david for that erudite and thought for a broken speech so um the uh shadow foreign secretary has agreed to take questions so if you just go to the mics in the in the eyes introduce yourself so he knows who you are and pose your questions hi thank you so much for uh coming to speak with us today my name is in bolistan i'm an undergrad student here at tufts um it was disappointing to see the trump's administration move towards unilateralism and move away from the international community with pulling out of the paris climate agreement moving to withdraw from the world health organization especially in the face of the pandemic and the clear need for greater international cooperation but i think this moment with the international community's strong response to russia and condemning poons actions um and imposing economic sanctions on them has been encouraging and showing the power of multilateralism you spoke a bit about the uk's need for greater leadership uh in europe and for taking action on transnational issues but i was wondering if you think that this moment more in general for the international community has the potential to lead to increased effective widespread multilateralism um addressing transnational issues like climate change like uh invasion of sovereignty or do you think that it's more likely to just be more of a one-off moment thank you very much well that was a wonderful question um i'm very grateful um so here's the challenge um democracies this work in progress that i first fully came to understand and grip when i was at harvard law school can be a messy and difficult process here in the united states there are vexed different political traditions and ideas and ideals um all of us worry sometimes at the nature of that debate the tone of that debate the the partisan nature of what has gripped the united states in more recent times which is a departure from the ways that the political traditions in both the republicans and the democrats have been able to come together to find common cause and we too in the uk um have seen that as the traditional centre right uh traditions one nation traditions within the conservative party have been jettisoned and we have a more populism uh coming from borris johnson and indeed my own party um experiencing in its previous incarnation um an antisemitism that was lurking on the far left of the political spectrum and tradition um so democracies can be messy and they can be particularly messy when they come together nato uh an invention that was birthed in part by the british labour party in that at the period after the second world war and ernis bevin is this amazing alliance not a offensive alliance but a defensive alliance of countries coming together to say we will avoid war in europe by saying if one is attacked the other will come in support drawing and of course the important allies of the united states and canada and the european continent but look it's it's 30 countries there can be differences of opinion us from germany down to italy but it's what we have and it's the best that we have and as messy and difficult as that is this is a multilateral moment so you know what's interesting when you look at what's happened in ukraine and you reflect on cop 26 we could have gone further in glasgo and should have we should have met that multi multilateral moment that was presented in glasgo and if glasgo were tomorrow and we have the opportunity in egypt we must finally accelerate and address the huge issues that are presented by the climate emergency and i think the other challenge at the heart of your question is coming to terms now with this new multi polar world and if you like whilst there are some lessons that we can take from the cold war period i said that it had its limitations and the reason it has its limitations because really what we saw there was one great superpower the united states now we are living almost in a period similar to that period in the run-up to the first world war and just coming after it and in that period there were different powers and we're back there again china india the european union acting together russia the united states this is a multi polar world and we have to focus on our common interest and our values and maintain and renew those multi lateral forums that bring us together and that will need some new architecture certainly in europe the debate around european security the important intervention of germany particularly their change position on defense spending particularly will transform the european landscape but this is a pressured and challenging time because of course it's also right here in the united states and indeed perspectives in europe that this indopacific tilt the importance of the indopacific the importance of china requires us to operate along many many fronts but you know the challenges of dirty money just finally polluting our democracies interfering with our democracies requiring better regulation of the tech companies requiring much better regulation of things like tax evasion can only be done through multi lateral institutions and instruments so this is the moment and we must must meet the challenge i'm afraid the autocrats will win and create a much much more dangerous world hi good afternoon thank you so much for your talk i have two quick questions if i may one really follows up i think on what you just mentioned about the multi polar world that we currently live in and so while what's happening in ukraine i think has shown a remarkable we've seen a remarkable show of strength and unity of opposition in the eu and the west which i think was unthinkable even a few years ago and despite what you mentioned the vote in the general assembly i think we cannot talk about a united global community with regards to the position on on russia and if we look at the fact that both liz truss and sarge love off are in deli today speaking to prime minister moody i would like to hear your thoughts about the positions of states like india south africa and china who remain reluctant to condemn russia's actions and what that means for multi lateralism and what that means for a united position both within the west but as a global community my second question is in light of the limitations imposed on international action by the concern for escalating the conflict and you mentioned that the decision uh by nato uh not to consider a no fly zone was was the correct one i would like to hear what you think britain and other states can do to fulfill their responsibility to protect the people of ukraine and i apologize i didn't introduce myself my name's karen smith and i teach international relations at the university of leiden in the netherlands thank you very much well thank you very much um karen can i just deal with that last point you raised first we've got to do as much as we can do to protect ukraine i focus a lot obviously on politics and diplomacy but let's be absolutely clear that means with every military assistance and it has been very important to arm ukrainians to arm them with anti tank missiles to arm them with anti air missiles um to continue to do that in a coordinated way um both across nato allies and beyond um and to limit the ability for putin to wage war we've cut his reserves the sanctions has cut his reserves in half run on his currency um and hitting sectors that allow him to derisk his campaign which is why cutting him out of swift um putting sanctions on parts of the insurance industry which we don't talk enough about is so so important so we must continue to do that but it's my judgment that we mustn't walk blindly um into global conflict and clearly uh for nato allies to engage in direct confrontation uh with russia brings us into that global conflict so we must continue to arm ukrainians with the best equipment and equipment that they can use uh we must continue obviously to offer sanctuary to um those that are fleeing war and here i'm afraid very sadly the government of the united kingdom has not lived up to the expectations of the british people the country that offered sanctuary for ugandan asians the country that offered sanctuary for people fleeing the balkans the country that offered sanctuary for vietnamese boat people for sypriots and for others has not lived up to that promise and it harks back to that terrible period in the run-up to the second world war when jewish people were fleeing conflict and found the doors shut across so many countries in the world and i've been very critical uh on behalf of the british people of the government in relation to that india is a great great democracy it's an important democracy in the global community and i might say that i stand here with parents who um were born and raised in the country of gayana in south america and gayana is a country that was born out of colonialism uh both from former enslaved people from africa but also from indentured workers from india and my great-grandmother was one of those indentured workers from calcutta i've been to india many many times indeed india is a country obviously with an armed force much of its equipment has been supplied by russia i'm quite sure that in delhi they are looking at what is happening with those tanks that are running into the ground that are not properly equipped they're looking at the technology that's fluttering and not starting and they may well be reflecting on their proximity to russia over so many years um and whether really this is the part to be allied with um and i'm sure i know that intense conversations are going on with india as they reflect on this moment and for china for that country of so many people uh with a economy that's seen stuttering growth in this last period declining and falling a country that as we speak is struggling very badly with the pandemic it's a country that's going to need trade it's a country that takes very seriously its partnership with the european union and i hope that china makes the right call in the coming days and weeks as it reflects on the partnership that the global community has seen with russia and the immense concern that that is causing in countries like this and it also i think in china reflects on not just what governments do and their ability to sanction and work united but also what individual people do in free democracies who are now saying well actually look if nestle aren't going to pull out we're going to boycott nestle it reminds me of the boycotts in the period of the fight against apartheid in south africa that i was part of when i was younger that's the power of individuals seeking to exercise their ability to choose not to invest or not to buy that smartphone or whatever it is from china so this is a moment of course where i would like more people in that alliance against autocracy but i those conversations i suspect are going on in very very real time in those communities and let me just say one other thing it's important to recognize that there's a sort of outrage at what we're seeing this invasion of a sovereign democratic country but it is also important to say very very squarely that there are other conflicts in the world that we should be concerned with what's happening in afghanistan as people are starving to death we should be concerned what's happening in the yemen we should have raised much more noise about syria ethiopia is a real challenge and of course the plight of the palestinian people as well and the occupied territories and we have all got to reflect on the urgency of these other issues across the world that require our full attention at this time and also the campaign that people are waging in those countries for the ability to fight for their rights and their freedoms and their survivors and we should ally with them as well good morning uh mr lambi my name is jordan clubber i'm a second year graduate student at the flexor school um we've been discussing class about the unequal hierarchical status of black communities um in society um and we've looked at the in disappointment at people who are living and working and studying in the ukraine who are being denied access to the polish border or transport to the polish border um so i'm very interested to know if you've been able to comment or share thoughts on the intersection of racism and war um during this time you know i said that one of the things i learned when i was here in the united states is that great story of the 20th century where so many people at the beginning of the 20th century were denied rights certainly my ancestors it feels like your ancestors were denied rights we were colonized coming out of enslavement coming out of the scramble and the carve up of the continent of africa but we know too don't we that working people around much of europe and much of the settled economies of the united states and australia and canada and places like that were also denied rights and had to fight very hard to get those rights working class people and women because of the campaigning of suffragettes people like emeline pankhurst elinor roosevelt this country and others gaining rights but still so much to do never mind lgbtq communities deny that right to self actualize to be who you want to be and to be free of discrimination and that fight goes on but because of the work of martin luther king harvey milk emeline pankhurst and so many others those rights felt for the first time but my friend we know too that discrimination and racism is real and it's still with us you know i was looking at some open source intelligence that's freely available if you want to go and look at it that looked at some of the russian casualties during this crisis and this invasion of ukraine and it looked at the names of those russian soldiers who were lying in hospitals or who were ill or who had been killed i wonder if you know what the number one name is or that open source material because it might surprise some of you the number one name of those russian soldiers was muhammad it was muhammad because vladimir putin is using the young men from far flung parts of the remote as bekastan as it's done young Muslim boys often very poor fleeing poverty and he is sending them to their slaughter in ukraine and yes of course i have seen both in ukraine and in poland and of course i've had discussions with the ukrainian government i was glad that they responded issued a helpline to help particularly black and indian students african students out of ukraine by ringing this helpline to get out racism is real across the globe it's an issue it's a fight that continues of course it does not know it's not bound by countries and it's certainly not unique to the united states it's present all across the world and i think it's also important to remember that there are many communities and i you know one of the things that i've always tried to do is champion the rights of indigenous communities often get forgotten uh in their fight for freedom and their rights um in that discussion so yes it's real and yes we continue that battle thank you secretary lammy my name is joseph limb i'm a student of professor williams um part of the epic class of tufts i had two questions in particular and first congratulations on your government for securing the release of two prisoners in iran for living all that time in incarceration and i was wondering for the first question if you had played a personal role in the release and the creative diplomacy that the uk government took in releasing them and also in the spirit of creative diplomacy if it's the time now to express that spirit or the spirit of solidarity against russia in the context of the jcpoa the iran agreement um where the uk plays a role as the p5 plus one negotiator in negotiations um do you think that's further solidarity against russia would deter russia from making meaningful progress in denuclearizing iran and do you think there is a trade-off between the two thank you well thank you very much for that question and the situation in iran should concern us all i just a few weeks ago was with two young men um 123 124 i met these young men at a refugee camp in cali these young men have been caught up very sadly in a debate that's sometimes called fortress europe and a very vexed debate that we've had in the uk about refugees but i just want to talk about these two young men because who are these two young men that i met just three years ago well they're both ethnic jeez from iran who are terribly persecuted and treated in that country for being who they are and for wanting to follow the jewish faith and these young men had to flee they do not know what has happened to their parents and their loved ones they've made their way to the uk and i've continued to have contact with them because i met them in cali they're caught very sadly in a horrible web that's called the home office in the uk and it's failing to process people's right to claim refuge in our country properly and successfully but it's a reminder of the huge perils and problems in iran and what i was talking about when i was reflecting on authoritarian states and of course one of the central issues that's preoccupied the global community in dealing with vladimir putin is the fact that we are talking about a nuclear superpower and we've heard vladimir putin using nuclear threats in order to get his way in ukraine by the way when i was in kiv before the invasion in early january and indeed i've been to kiv before i was there just four years ago very good friends in kiv and what they say is we gave up our nuclear weapons 1995 we gave up our nuclear weapons they had a arsenal of 5000 warheads and they got a security guarantee from the us the uk and russia in order to go and some people say in ukraine if we hadn't given up our nuclear weapons we don't think we'd be in this position today and it's why we must stand with them so very firmly and it's really important because if we don't stand with ukraine then there will be other states that think well actually let's get some nuclear weapons that is the path that iran is on it's a dangerous path this enrichment of ukraine and it's it could escalate things terribly in the middle east because if there are we're to have nuclear weapons then there are other countries who would think well let's have some nuclear weapons as well it'd be a dangerous dangerous escalation we'd have nuclear weapons in the hands of an autograph by the way it's why it was such a horrendous mistake by donald trump to walk away from that agreement that had been rightly broken and it's my sincere hope that we can reach agreement once again in the coming weeks and now this is a matter of urgency you're absolutely right russia is part of that conversation and i said before you know we have to find ways despite war and conflict and we found them in the cold war to still be able to have the necessary conversations for the purposes of the global community we cannot fix climate change without china without russia without iran we have to put these issues above the day today and similarly we have to get agreement in the jocpa we've got to get agreement to limit and to move forward and to be able to ease the sanctions burden on the iran Iranian people in return for stepping away from that enrichment that has been happening because accelerated because donald trump took that disastrous decision thank you for the question thank you very much um dear mr lamby my name is high toe i'm currently a senior i'm a fourth year student at house university i study international relations and um thank you very much for your um answers regarding foreign policy issues i have one real quick regarding your uh record on education particularly educational policies particularly with your critique of oxford university for its anti-black racism which i tremendously admire you for at toughs the incoming class of 2026 11 percent of the admitted students from the united states identify as black or african-american albeit that is still lower than the percentage of 12.4 percent of black people overall represented in the united states but also at the same time although toughs and other institutions have been trying to make progress um waves of lawsuits have nominally come from asian-american communities who claim that their children are being discriminated against by affirmative action and in the uk of course home secretary priti patel um with paternal grandparents coming from india who had then lived in uganda has been one of the most vocal critics of um what would what she and what would many of her allies will consider as as as as the left that are that are discriminating against the nominally hardworking asian communities so to what extent do you think the labor party should be proactive in terms of responding to these potential critiques warranted or unwarranted thank you well look thank you very much um for the question let's just go back to the basics why is tough such a great university what makes it special what makes harvard law school down the way special what makes the school of orient and african studies where i went for my first degree special it's the fact that in these amphitheaters of brilliance and excellence great minds come together and those minds are a cross-section of ideas and thoughts there's something special about universities like this one because of that interdisciplinary coming together so you don't get a university until you have the social scientist sitting with the scientists um you know sitting with the philosophical theorist sitting with the design technician and applying that knowledge together that's what makes a great university the ability to bring these people from different disciplines together and what we're also learning is that it's even better when you bring people together from different social ethnic religious racial gender sexuality backgrounds it makes the conversation it makes the possibilities richer as a consequence so by the way my challenge which is now a few years old to give uh oxford and cambridge university's credit um was not just a challenge on the relation on behalf of black uk citizens at the time when i did the work when i looked at some of the colleges at oxford i found that there were working people there were more young people from two london boroughs the barn the boroughs of barnett and richmond these are pretty wealthy london boroughs there were more young people from these two london areas than the whole of leeds sheffield and manchester combined that wasn't about race that was about class never mind the fact that at the time there were more young people with the surname smith than there were african caribbeans and by the way let me be absolutely clear when i looked at the figures i was looking at the african caribbean figures and who were getting straight a's in their a-level exams which is the exam you take to get into and never there were hundreds of kids and they still were not getting in when i came to harvard what i found was a diverse place and it's that dive you know i was i was studying with veterans i was studying with asian americans uh more to do but that richness was very important and i'd come from so as before so if that's important and there is more to do that's why there of course there must be a role in different ways and different countries will apply this in different ways for affirmative actions some countries use it at the university stage other countries will use it in terms of access to housing or or earlier in the education process that is for different democracies to determine but i think that there is a role to level up the playground and the best way the playing field and the best way to see that is actually in the uk parliament the uk parliament i say this finally is making better decisions as a result of the many many more women that we have in the uk parliament and many of those women have benefited particularly my own party from um from the leadership of my party saying look we're not going to tolerate a situation where individual constituencies aren't selecting women we're going to have all women shortlist and i'm the david camera centre right him making decisions also effectively to put in affirmative action so that women and minorities could come through and you see that now on the conservative party benches so this ought not to be a partisan issue it really ought not to in the 21st century thank you very much for the question there are lots of questions but i think we we have to wrap up now thank you for being with us and i think on that optimistic note we can we can end the session and join me in thanking uh uh david lamby thank you very much


As the government of St. Lucia continues efforts to reduce impacts of climate change, the fishery sector has received much needed equipment. Over 200 VHF radios, valued at US$80,000, has been presented to fishers. More from Manasseh and Juan. The departments of fisheries and telecommunications in collaboration with the Global Environment Facility, Jeff and the Food and Agriculture Organization, FAO, through the Climate Change Adaptation in the Eastern Caribbean Project, CC for Fish Project, have procured a total of 200 very high-frequency radios for fisherfolk around the island. The VHF radios will enable fishermen to dialogue over substantial distances on marine radio channels, and is an invaluable fishing tool that alerts marine police during moments of distress. Two marine repeaters have also been purchased for the north and south of the island in an effort to re-initiate the use of VHF radios amongst fisherfolk. Acting Minister for Agriculture, Fisheries, Physical Planning, Natural Resources and Cooperatives, Honourable Harold Stanislaus, says the procurement of these equipment is in keeping with the objective of the CC for Fish Project to increase resilience and reduce vulnerability to climate change impacts in the Eastern Caribbean fisheries sector. The installation of the marine repeaters and distribution of the VHF radios are contributing to the second overracking goal of the National Adaptation Plan for St. Lucia, which is to accelerate the implementation of climate adaptation and risk reduction actions critical to safeguarding the country's socio-economic and environmental systems. Furthermore, the introduction of the VHF radios are consistent with the expected outcomes of the St. Lucia fisheries sectoral adaptation strategy action plan by strengthening preparedness for climate variability and extremes in the fisheries sector. Elementary Representative for Ansela Ray Canry's Honourable Dominic Fede expressed gratification on behalf of the fishermen of his constituency for the newly acquired equipment. For a long time now, you've heard stories about fishermen being lost at sea. We can't find them. There are a lot of ones. Their families go through the agony, the pain. And I think that this is going to help to address that problem. Really congratulations to the Ministry of Agriculture and Fisheries for a well thought out initiative. It is quite important what they're doing. And I supported wholeheartedly representing a constituency where many of the residents there do rely on fishing for their livelihoods. The license fee for the VHF radios have been waived by the Government of St. Lucia. From the Information Unit of the Ministry of Agriculture, I am Anisia Antoine reporting.


The development of catalytic Friedel-Crafts alkylations using Benzyl Dash, Propagyl, and Allyl alcohols or styrenes as substrates has gained attention over the last decade, with low catalyst loadings needed to produce a wide range of products. This approach is considered more environmentally friendly than classical Friedel-Crafts conditions that use toxic Benzyl halides. The review highlights different substrates and the first diastereoselective and inantio-selective Friedel-Crafts-type alkylations. This article was authored by Magnus Ruping and Boris J. Naksheim.


Welcome back math 241 lecture 46 out of Probably somewhere around 60 before we're all said and done. So we are in chapter 8 We will not go further than chapter 8. In fact chapter 9 starts math 242, which most of you will be Heading toward I would presume So We have quite a bit to do in chapter 8 still pretty loaded with things that we have not Encountered yet So let's go ahead today and also Friday in here. We'll do power series Still puts us just a little bit ahead. I guess according to the syllabus, but We're going to need it when we get to some of the later Parts of chapter 8 All right, so let's go ahead with power series Test grades by the way the those that I've graded I think I've get graded 10 Are pretty good couple of scores over a hundred Of course, you're only happy with that if it's on your paper Anybody That necessarily thrill you that you know person two seats down got a 102 Kind of thrill you I would think if you got one yourself All right power series So I intended to have all the tests graded, but I think I did a little more moving than grading yesterday. So I Got kind of closed out in available hours We will have a lot more series starting at zero now instead of starting at one one of the reasons is that we'll want a Some kind of a constant as the first term and instead of adjusting the exponent in here will Just start it at zero and we'll establish that first term as a constant. So notice the subscripts On the constants are different. So we're allowing for different coefficients on different terms as we progress and Then we're going to increase the power of x as we progress to the right. So it's kind of like an infinitely termed Polynomial is what we end up with So at n equals zero We get c sub zero x to the zero which is one So there's one of the reasons why we want to start at zero we get a constant n equals one c sub one x to the one n equals two c sub two x squared and so on so we get this polynomial looking thing We'll be able to deal with some some of the partial sums and we'll be able to get different Polynomials when we truncate, but if we let it run forever It is a power series, but it behaves in a lot of ways like a polynomial So we will determine When these things converge they'll converge for certain values of x I mean you can probably pick one off right away What is one value of x that would cause even this generic power series to converge? If x were what value? Zero if x were zero then all these terms disappear and it converges to c sub zero So you're going to get some values of x for which it converges and you'll see some Categories develop as we look at these problems some other values of x for which the series diverges Okay, there's another type let's get an example of this type in fact We've already seen this one can't get a whole lot simpler than this one So it's a good one to start with So we don't have any varying Coefficients as we go all we have is powers of x. We've already seen it So when x excuse me when n is zero x to the zero is one when n is one x to the first x squared x cubed Well, we say they converge for some values and diverge for other values What is this particular power series? What have we already classified this under that is if you were actually talking to me today Geometric and not all infinite geometric series converge when do they converge? Okay, so if the absolute value of the ratio is less than one We know that it converges What is the ratio on this particular infinite geometric series? X Because we're multiplying by x as we go So it will converge if the absolute value of x is less than one so we have a so-called interval of convergence from Negative one not including negative one All the way up to one but not including one So for any other value of x that we choose in this interval This is a convergent series for anything outside of this It diverges okay another type of power series and Really, this is a the first one that we looked at is a more Specialized version of the one we're about to look at so instead of just having Increasing powers of x as we progress to the right We have increasing powers of the binomial term x minus a So the first one ought to be c sub zero x minus a to the zero well, that's just c sub zero n is one When n is two n is three and we get a similar Type of expression except powers. It's a power series, but not a power series in x It's a power series in the binomial x minus a So why would this be a sub category of this? This is kind of a more generalized version this should work for any a value this only works for the a value of Zero right if a were zero in this one we'd really be back up here to this one So they're really all Power series of this type and if a happens to be zero it's a simpler looking power series now when we have varying Coefficients we're not going to have that nice luxury that we have here when the coefficients were all one where you can say okay What do I how do I progress from term one to term two term two to term three? We're multiplying by x we don't have that luxury here because of the differing Coefficients as we progress so this is not going to be unless it's a really specialized Problem it's not going to be an infinite geometric series. It is a power series. It's a power series in x minus a sometimes it is said to be centered x equals a and I think That will become more clear as we progress to this section why we would say that it's centered at x equals a All right, so there's the most generic version We'll see some like this, but it's the same thing except a is zebra. All right. We're going to look at Probably four examples maybe more. I don't know how many of those will get to today and then we'll kind of classify What happens in those? different problems into some Categories of answers and we'll see that all Types of answers will basically fall into one of three categories when we're done So the series n factorial x to the n now we think n factorial. That's getting larger as n gets larger x to the n May or may not be getting larger as x gets large as n gets larger because we really don't know what x is So the tendency is to see this and say well clearly that's divergent, but we don't know what x is So the question is When does this converge and we have no idea what after some examples? You'll see Solutions to problems as varied as they are Falling into categories, but we don't really know what's going to happen here So this is a test that has a lot of carry over throughout the rest of this chapter of the ratio test We're going to take the n plus first term divided by the predecessor the nth term Way out there to the right. We're going to see what we get We know that if L is less than one. What's our conclusion? Convergent L is greater than one Divergent and what happens if L is equal to one Okay, no conclusion. So the test fails All right for this problem the n plus first term would be what? okay x to the n plus one So everywhere. There's an end. We should now have an n plus one So there's the value of the n plus first term. I've got absolute value brackets started here. Do I really need those? Okay, well right unless x is negative so x is unknown so we better keep them around And you'll see in maybe not in this problem so much But is in in other problems the absolute value will certainly come into play So there's the n plus first term we want to divide it by its predecessor Which is the nth term which is normally the way it's Anded to us. What is in plus one factorial? How could we rewrite that? Good in plus one times in factorial I had a question after class yesterday About that if if that troubles you in any way just put in some numbers Eight factorial is eight times seven times six times five all the way back to one Which is really eight times what? seven factorial so Just enough to convince yourself that that is in fact true for n plus one factorial Just for the purposes of reduction x to the n plus one is x to the n times x to the one question Jacob Would in plus two factorial be in plus one times in plus two times in factorial There's a variety of things that could be So if you add in plus two Factorial if you wrote it out it would be that so you could say that is in plus two times what in plus one factorial or Depending on what you needed it for you could say it's in plus two times in plus one times In factorial that might be a little more advantageous Especially if you had one of these terms around numerator and denominator and you wanted to knock it out With a like term in the opposite position so it kind of just depends on what you need So back to the kind of planning ahead. We knew we had an in factorial here So it's advantageous to have one in the numerator as well We've got an x to the n down here So maybe it's advantageous not that we couldn't do the problem without it But it's nice to just kind of mark out like terms numerator and denominator And that's where we are in factorial over in factorial X to the n over x to the n in plus one is inside the absolute value brackets But it's unaffected by the absolute value so we can bring it out front So the x that remains Might be positive might be negative. So let's keep the absolute value notation around Does it really matter? What x is here? When n is allowed to increase without bound this term gets larger and larger and larger What is the only value for x? That would keep this thing from getting larger and larger and larger Zero so this only converges When x is zero for all other values of x this diverges So that's not actually a very interesting problem if it only converges for the single value of x equals zero But the reason for the example is to we're going to kind of see what Categories our answer falls into there's one of the categories It converges for a single value only in this case that single value is x equals zero Questions on that before it gets moved. So these kind of increase in difficulty as we go to I don't think any of them are Ridiculous there's one that's Kind of testy notice this starts at one so we're not going to have that Constant term we probably would want to start this one at one for what reason? Kind of problematic if we started it at zero because we'd have division by zero which would not be delightful So it n equals one we've got that to the first over one squared over two cubed over three and so on so it's kind of a Polynomial, but it's got an infinite amount of terms, so we want to do the ratio test to see when this converges If in fact it ever does I guess we could determine that now it would certainly converge at what x value? Three so if x were three this thing would clearly converge are there more values for which it converges? We don't know that and We'll see a problem that converges for all values That I think is our example that follows this so we do have some differing things that happen All right ratio test. What is the n plus first term? That work take the n plus first term divide it by the nth term you'll be experts at the ratio test Very soon if you're not already Should write this x minus three Okay, and then when we multiply by the reciprocal of the denominator And even if you did not do what Nicole said hopefully you would be able to say to yourself What's x minus three to the n plus one? Divided by x minus three to the end there's going to be one of them left in the numerator right because it's raised to one power Larger, but that's a cleaner reduction when you do separate it in the numerator In an n plus one we could in fact bring those out in front There's no question about them being positive when the end starts at one But the other which is x minus three is still variable We don't know if that's positive or negative or when it's positive and negative So let's leave it inside the absolute value notation what happens to n over n plus one as n gets infinitely large That's goes to one So one times the absolute value of x minus three is The absolute value of x minus three so there's our limit there's our L and We want to know where this converges So we would want L to be what? Less than one so there's our before this point. We've had in a lot of cases a numerical answer Yesterday we ended class. I think we ended class with an example that had e For our answer e being larger than one meant that the series Diverged here. We have absolute value of x minus three. This is only going to converge when that is less than one So that means that what is inside the absolute value notation? Is kind of trapped between negative one and one so that's the algebraic Solution to that absolute value inequality we can add three So in the middle we get x which is what we want Here we get four and here we get two So for all values of x between two and four This particular power series is a convergent power series So at this point in time we might make some adjustments to this but at this point in time the interval of convergence is That interval so for all values of x Anywhere from two to four but excluding two to four. We already knew one of those values, right? Didn't we already know three? From the original power series three was going to cause it to converge But apparently it's not just three. It's any value between two and four excluding two and four now If x were two What would this be? It'd be one and our answer To the ratio test the L value would be one now when the L value is one What does the ratio test tell us? It doesn't tell us anything. It doesn't give a conclusion So we probably ought to check out two Since it the ratio test fails to tell us what happens it to we ought to check that out ourselves Is the same thing true at four if x is four? The limit is one the ratio test fails to tell us what happens at x equals four We better check this out ourselves. So we're going to check out two and four separately So our end points x equals two and x equals four let's go all the way back to our original series So if x is two, I'll just go ahead and write it in two minus three to the end All over in well two minus three to the end is really what? negative one To the end Hey, that looks familiar Converge it because Okay, we've already done that right this is an alternating harmonic therefore it's convergent So even though the ratio test the limit was one fails to give us a conclusion We kind of do that by hand do that ourselves and We determined that it in fact converges at x equals two. Let's do the same thing at x equals four Go back to the original power series What is that one? Four minus three is one one to the end well one to the end is just one We know about that one, too. What do we know about one over n from n equals one to infinity? That diverges that is not alternating harmonic, but just the harmonic series so Initially we said our interval of convergence Based on the ratio test was from two to four excluding the end points Now we know we also include this In the area for which this particular power series converges. So our final interval of convergence is two To four we don't want to include four So there's the interval of Convergence I used the term centered at x equals a a while ago This should be centered at x equals three If that's the case right is our interval centered at x equals three That's right in the middle of two and four. So it is centered at x equals three and one more piece of terminology and that is the radius of convergence In this case is one so starting at that centering value which in this case is three We're going to go basically one unit to the left and one unit to the right So if we were swinging an arc of radius one, we would basically capture our interval of convergence So interval of convergence it is centered at the a value from the original series and the radius of Convergence how many units away from that centering value are we getting we're getting one unit away That okay All right. This will be our last example. I know there's quite a bit of time left But this one's a little bit of a battle This is called the Bessel function. There are some pictures of The Bessel function On page 595 kind of what the what this is being used to model the Bessel function Says that functions first arose when Bessel solved Kepler's equation for describing planetary motion Since that time these functions have been applied in many different physical situations including the temperature distribution in a circular plate and The shape of a vibrating drum head So I'm sure that piques your interest because that piqued mine But it's a neat function and we'll see something happen with this one that we haven't seen on another example So when does it converge? When does it diverge? So we're going to use the ratio test Can you see what a mess this might be? What is the n plus first term? negative one Which this really doesn't matter because we are taking absolute value So the alternating sine portion of the original and also this become irrelevant, but I'm going to go ahead and write it down X to the 2n should become x to the 2 in plus one Quantity n plus one that matters in this problem 2 to the 2n becomes 2 to the 2 times the quantity n plus 1 and This is kind of crazy looking in factorial squared becomes In plus 1 Factorial squared, so there's the n plus first term Everywhere there was an n we should have replaced it with an n plus 1 All right, let's go ahead in the denominator and write out the nth term the way it was given to us And let's see what happens Way out to the right as n gets infinitely large Before we start knocking out like terms Can I get rid of negative one to the n plus one and negative one to the end? I mean at best there they would be I mean it's going to be negative one right which is going to be done away with So there's no reason to carry those any further Let's leave things where they are right now x to the 2 times the quantity n plus 1 that would be x to the 2n plus 2 same thing down here 2 to the 2 times the quantity in plus 1 and let's write this as n plus 1 Factorial twice So that's in the numerator might be helpful I mean you might skip this step, but it might be helpful just because it'll get you thinking about Reductions or separations in the numerator that might be helpful to get rid of some things in the denominator Another absolute value bar over here All right, so in the numerator Let's go ahead and simplify some things as we go How might we want to rewrite this term? Okay x to the 2n times x squared and we want an x to the 2n to reduce with the other x to the 2n that's in the denominator similar fashion 2 to the 2n plus 2 ought to be 2 to the 2n Times 2 to the second while we're rewriting things. We've got an n plus 1 factorial and Another n plus 1 factorial. Let's rewrite both of those as Okay, so that's the numerator we want to multiply by the reciprocal of the denominator So here's our denominator. Let's just take things that are in the opposite position and Reverse those positions, so we're going to have an x to the 2n here 2 to the 2n In factorial in factorial good ol. Mr. Bessel now what Okay, now how many of those things can we bring out front that are not affected by absolute value We can bring out the one fourth and that n plus one is in the denominator, right? Two of them and we do have the square root of x squared. Well, we're really not worried about that either. Are we? I'll leave it, but it's it's really kind of does it's unaffected by the absolute value In is going to approach infinity any before we go to the next step any questions about the terms that Knocked each other out numerator and denominator or what remains Are you all right with that? What happens to this term right here as n gets infinitely large That goes to zero How about our old friend x here? We're really x squared if this term is going to zero For a single value of x that we choose this part out in front eventually overtakes this, right? So it'd be zero times whatever value we have here. It's not like this is Changing as this is changing and one could be getting smaller the other one getting larger. This is a fixed value for a Certain value of x so it looks like we're going to get zero That's a different kind of answer because we would expect this as the L value We would still have some work to do when is the limit the L value less than one When is this limit less than one? It's always less than one L which is zero Is always less than one so what does that say? About convergence for this particular power series Converge it for all values of x so the interval of convergence This one is still centered. Where is this centered and you could even go back to the original Version and see where it is centered. We didn't have an x minus a we just had an x, right? So it's centered at zero and what's the? radius of convergence Also infinite right is infinitely large so we have seen what types of solutions we have seen a single value of x as a solution it only converged at that one value and We have seen an interval two to four including two That was an interval of convergence and then we've seen one that can be centered for I mean convergent for all values of x that we choose So you'll see this Theorem and I think we've seen an example of each type in the book to this point For a given power series x minus a remember a could be zero and it'd be the simpler type There are only three possibilities for Convergence it converges at a single value namely the a value Converges for all values of x we just finished one of those and There is a positive number such that the series converges and this would be some interval Whether it includes or excludes the endpoints we have to kind of do those separately, but it ultimately ends up being some interval I think this is Probably let's not start another example, but let's deal a little bit further with this one I don't know how far back we need to go with that Let's go back all the way to the original just to make sure Let's see what and this is a good lead-in. I'm not trying to disuse up a few minutes of class That this has some nice carry over into what we do later in chapter 8 Let's look at a few of the partial Summs, so let's look at what the first term would look like Which in this case this would be in Equals zero right So when we enter in to this series try to generate the first term We'll enter in n equals zero and what do we get that to the zero? That x to the zero two to the zero zero factorial. I think we did we cover that did that come up yet? That zero factorial by definition is one Think when you put in n equals zero you're going to see that the first term is One so now we want to generate the first two terms So we're going to put in n equals zero Which we already have and also n equals one so it's going to be one What's the next term when n is one negative one to the first so it's going to be negative? x to the What power squared Right when n is one and when n is one, what's the denominator for? That's not a real complicated looking polynomial In fact this one isn't complicated either. I like this one the first one One if we were to graph that That'd be pretty simple. We could then graph this When the generating the n equals zero and the n equals one term What would that be if you graph that? Just the first two terms kind of a truncated vessel series Brabler opening down right shifted up one unit would be what that is What'd you say Jacob? Said it's fat kind of fat. Yeah, you weren't talking about me. Were you? Okay, I mean it's factual, but let's not put it out there on television Okay, so we went to n equals two Negative one to the two now. We're back to positive. So we all gonna are gonna alternate signs When in is to what do we get up here? X to the Four that's not a surprise is it because we have x to the two in so it's going to go up by multiples of two as we progress And when in is to what do we get in the bottom? That'd be four two to the fourth which is 16 We'd have two factorial two factorial is two squared, which is four sixteen times four sixty four that right Another polynomial If you would graph this one, I mean the original Bessel function looks kind of strange But I guess the reason for going through something like this. That's not really strange There's nothing strange about that one. There's really nothing strange about that one That's a fourth-degree polynomial Tell me something about fourth-degree polynomials kind of a any generic fourth-degree polynomial How many times might they hit the x-axis? Four right because they'd have four possible solutions How many changes in direction does a fourth-degree polynomial have Three right because you take the derivative which would be third degree and set the derivative equal to zero Which would have three possible solutions, so we all know what this is. That's we don't need to you know Debate this one, but this one although we don't graph a whole lot of fourth-degree polynomials a fourth-degree polynomial There's one two three four Probably is going to look something like that right and I started this kind of up one unit so there's the s sub three the seek the Sequence of partial sums going all the way up to n equals two Without generating this I think I have this one written down There's when n equals zero n equals one n equals two We would expect the signs to alternate. We would expect the power of x to be what x to the sixth You probably see why convergence happens because these values the denominator begins to overtake the numerator So that eventually the denominator gets so big that it really doesn't matter what the numerator is That's why this converges for all values of x But that's a sixth-degree polynomial Six possible roots five changes in direction Not anything we want to graph on a regular basis, but it's not ridiculous Some of these are pictured together on page 596 You'll see a lot of these sequence of partial sums graphed as separate truncating polynomials and you'll see some similarities you'll see that they overlap in a very significant way And as you go further out you'll see that they overlap Even more as you work your way out further because this term is really kind of in a sense almost insignificant But that's the interesting diagram We'll see the same phenomenon when we get to Maclaurin and Taylor polynomials that as you Get certain polynomials You'll see some overlap and then if you get a Taylor series and let it run You'll see some interesting phenomenon there. So we're truncating here But we are gathering an additional term as we go and we'll see that illustrated later in this chapter All right, we won't meet in this class Physically tomorrow, and I'll see you in here on Friday


Hello and welcome to the session. In this session we are going to discuss the following question and the question says that draw the vectors given by a is equal to the ordered pair 4 to b is equal to the ordered pair minus 3 5 on the grid. Let us start with the solution of the given question. We have to draw vectors a and b given in component form on the grid. Let us first draw vector a which is given by the ordered pair 4 to. We know that the components of any vector are given by the ordered pair xy where x is the horizontal component and y is vertical component. Now here in vector a horizontal component is 4 and vertical component is 2. Also both the components are positive so we take any point as initial point on the grid. So here we take this point as initial point. Since horizontal component is positive so we move 4 units to write horizontally. So here we move 4 units 1 2 3 4 and we reach this point. Again the vertical component is also positive. So from here we move 2 units up 1 2 and we reach this point and here this is the required terminal point. Also we have moved in upward direction so we mark this direction arrow on the line. It represents vector a which is given by the ordered pair 4 to. Now we have to draw vector b given by the ordered pair minus 3 5. Here horizontal component is minus 3 and vertical component is 5. Now again we take any point as initial point. Now horizontal component is minus 3 so from here we move 3 units to left horizontally as the horizontal component is negative and we reach this point and vertical component is 5. So from this point we move 5 units up because vertical component is positive 1 2 3 4 5 and we reach this point which is the terminal point. As we have moved in the upward direction here we mark the direction arrow. Now this is vector b which is given by the ordered pair minus 3 5. This way we have represented vector a and vector b on the grid. This completes our session. Hope you enjoyed this session.


What's going on? If you're new here, welcome back to another video. It's your girl Denise. If you guys are new to the channel, make sure you hit that subscribe button and join the family. Guys, today I have the most epic prank in mind. So guys, I don't know if you can hear, but Isaiah right now is rearranging his closet. He's in the bedroom while I do this intro, so I have to hurry up and do this new fast. So guys, today I am going to be manhandling my boyfriend to see his reaction. Now, I've been seeing this video trending on TikTok and stuff like that. I think the reactions are super hilarious, so I hope that Isaiah has an even crazier one. So guys, if you are ready for today's video, don't forget to smash that like button and comment team Denise in the comment section below because this one is going to be a banger. Let's get it guys. Are you in the family? Everything's all set up and ready to go.  So baby, if you're watching this, I'm sorry, but team Denise has to get you selected. Almost done.  Hi, Jason. What do you want?  You're being nice. What do you want? What do you want? What do you want? I just told you. No, you want something else. Are you hungry? You want food? Is that it? Well, yes, but that's not what I'm calling you over here for. Okay, I'll buy you food when I'm done. Give me like 10 minutes. No, come here. Can I have a kiss?   Fish oil?  We're not doing that. You're extra. We're not doing that. It's SPF gloss. Okay, well, I don't need SPF gloss. SPF? I don't need SPF. I don't need no gloss. I don't need none of that. Well, I want a kiss.  Why are you feeling a little sexy tonight? Sure, but you got to take that off because I'm not doing that. No. That's like, that's something else right there. But it's moisturizing. I promise you you're going to like it.  It's not sticky. It's sticky as hell because that's how sticky it was. Actually, it's sticky. No. Just try it. Babe, I'll give you a kiss. You just got to take that off unless you just want it on the cheek right now. That's all.  That's O.D. That's O.D. That's O.D. Cheap kisses are cheap. What does that even mean? That means it's not a full kiss. Okay. Oh, you said cheap kisses are cheap. Yeah. I thought you said cheap kisses are cheap. I'm like, what cheap is cheap? What cheap? All right, let's stop with your check. Okay, what's up? What are you doing? I'm wedgy at that, bro. Don't do that. Please don't ever do that again. I don't like that feeling. I don't like that. I don't like that. You know because you made me feel small. I don't like that. How? You scratched my leg, I think. I was trying to be sexy. No, bro. Don't ever do that again. I do that to you. You don't do that to me. No. Do what? I'm just trying to be sexy. No, I don't like that feeling. You made me feel small.  So it made me feel worse. I don't like when I can't feel the floor. I don't like that. Don't ever do that again. Okay. No.  Don't ever pick me up again. Babe, seriously? I don't care if we're in an emergency and you have to pick me up. Don't pick me up. Ever again. I'm sorry. I don't like that feeling. That made me feel small. I don't like that. I didn't mean to make you feel small. That made me feel weird, bro. Can I finish? No. See, I know. Let me go. Can you bring your leg up a little bit more? Let me go.  Let me go. No. Let me go.  I was just trying to be sexy in another way. What did I just tell you about? I don't like that feeling. I don't know what you're talking about. You're being really aggressive with me. No, I'm not.  I've never been picked up before in my life, let alone by my significant other. I don't like that feeling. You're making me feel very small right now. How am I making you feel small? I'm trying to process what just happened. Should I be more delicate? No. What do you mean no? Just don't do that. Why? Just don't do that. I don't like that feeling. You can have a hug. Okay. You can have a kiss if you take that off. I'm not kissing you with that. Okay. And on another note, again, don't ever do that again. Don't pick me up. Okay, I didn't pick you up just now. Don't slam me against the wall that I don't like that feeling. If anything, I'll do that to you and then that can escalate to something else. That's what I'm trying to do. No, bro. I'm uncomfortable with that. That's not going to work. That's not going to work. That just won't flow. Well, I'm sorry that made you feel uncomfortable. Okay. Thank you for your apology. Just don't do that again. That was weird. Nice hug. I'm processing that. Okay. A hug. Okay. I feel... All right, bro. Now you're going to squeeze my... I'm out, bro. Babe! Everything. 175 pure fat, baby. All right, Infinite Fam, so clearly that was a freaking success. Isaiah was super uncomfortable, which I am so sorry about that, babe. No, seriously. Don't ever pick me up again. Don't ever do that again. If I can't feel the floor, there's a problem. Oh, my gosh. I didn't think I would be able to execute it, guys, because honestly, Isaiah is pretty solid. Like, you have some nice muscle to you and I'm like... Nice fat. No. Isaiah is like 100% muscle. Like, I don't know why he acts like he's not. I'm not 100% muscle. Thank you for the compliment, but not at all. Okay, well, the point is you're really strong and you have a lot of muscles, especially in your legs.  Exactly. But that's what I'm saying. I'm going to get more water. I'm still processing. You guys did. Don't forget to smash that like button and comment team Janisse in the comment section below. I had to pull this W for you guys. But it's time for today's post notification shout out. Today's post notification shout out goes out to Selena Ramos. Thank you so much for your love and support. Isaiah and I love you. And if you want a post notification shout out, all you have to do is like, comment, share and subscribe and turn on those post notification balls so you notify whenever we post a new video. And with all that being said, we'll catch you in the next one.


Hello, contestants, and welcome to Scrum Mastery Challenge Lockdown! I will be your Games Master for these few weeks of fun and learning. I'm here to set your very first challenge. And to do that, I'm going to introduce you to our referee, Mr Paul Goddard, who should be just here. Thank you, Mr Games Master Serve. And a great pleasure to be your referee for this challenge. Now, contestants, you must listen carefully. I'm here to make sure fair game and the rules are understood. And I will be here to referee the game. But the winner of the game will be determined by the Games Master here on my right. I shall give you the first challenge, challenge number one. Listen carefully. You must introduce yourself as if you were meeting a new team for the very first time. And you only have 60 seconds to record this on video. The Games Master will judge the results and the most impactful video will win. The most impactful introduction will win, OK? Your video must be submitted to the Games Master by Sunday the 7th of June at 6pm UK time, which is GMT Plus 1. That's it. Thank you very much, referee. So, contestants, remember, email me a copy of your video by 1800 hours British summertime. Remember, you're representing your country. So, go bravely. Good luck.


Hello and welcome to the session. Today I will help you with the following question. The question says construct the following quadrilateral rectangle OKAY where OK is given to be 7 cm and KA is 5 cm. Now before constructing the actual figure, first let's have a look at the rough sketch of this rectangle OKAY where OK is 7 cm and KA is 5 cm. Now let's move on to the solution. We are given that OK is 7 cm and we know that OK is equal to AY since opposite sides of a rectangle are equal. So we have AY is equal to 7 cm. Now in the same way KA is equal to 5 cm and we know that KA is equal to OY since opposite sides of a rectangle are equal. So we have OY is equal to 5 cm. And we know that all the angles of a rectangle are right angles. And now we have got all the measurements of this rectangle OKAY where OK is 7 cm, KA 5 cm, AY 7 cm and OY 5 cm and all the angles are the right angles. So now we will construct the rectangle OKAY step by step. Now visualizing the rough sketch of this rectangle OKAY, we can very well say that a first step would be draw a line segment OK equal to 7 cm. So we have drawn line segment OK of 7 cm length. Now in the next step we will make angles of 90 degree measure O and K. So we have made angles of 90 degrees at O and at K. Now our next step is to locate the point A where KA is 5 cm. We do this by taking K as the centre and radius 5 cm and then we make an arc on the ray KZ. So we write with K as the centre and radius 5 cm we draw an arc on the ray KZ. So we have made an arc of radius 5 cm with K as the centre. Now let's mark this point as A where KA is 5 cm. Now likewise we locate the point Y where OY is 5 cm. So we write the fourth step as taking O as the centre and radius 5 cm we make an arc on the ray OX. So we have made an arc of radius 5 cm on the ray OX. Now let's mark this point as Y where OY is 5 cm. Now in the next step we will join the points A and Y. So as to obtain the rectangle OKAY. So our last step is join the points A and Y. Now on joining the points A and Y we get the required rectangle OKAY where OK is 7 cm, KA is 5 cm, Ay is 7 cm and OY is 5 cm. Hope you enjoyed the session. Have a good day.


Hello to Terrestrials and Extraterrestrials alike, I am Kim, this is Dustboats and Velikor, and today we're doing our February wrap-up video. So last month I read 14 books, and this month I read five, which is less. But to be fair, this month I made four videos, and last month I made one, which was really at the very end of January, and so technically kind of was in February. So in February I made five videos, and if we're being entirely serious, it takes me longer to write, film, and edit a video than it takes me to read a non-Melvelian book. Also my friend Allison and I did a rewatch of Veronica Mars, and that's three seasons in a movie, and so I blame her. Excuses aside, onto the books! First to the block was Good Omens by Terry Cratchit and Neil Gaiman. This is a delicious satire of the biblical build-up to the end of the world. The angels of heaven and the demons of hell are chomping at the bit for the Antichrist to set the end of days in motion. But it seems the demon in charge of keeping track of the Son of Satan, getting him properly hellishly educated, making sure he's adequately fed, and clothed, and tortured... Well, he might have possibly misplaced his charge. It is a ridiculous, convoluted, merry chase to the very end. Next up, The Daughter of the Pirate King by Tricia Levenseller. Levenseller follows the story of Alosa, the titular daughter of the pirate king, as she infiltrates a rival pirate ship by staging her own kidnapping. She is brazen and cocksure by day, silent and meticulous by night, as she searches the enemy's ship from stem to stern in search of a very specific treasure. Her only obstacle, the unfairly attractive and deceptively cunning first mate. If you like pirate superstition, convoluted ruses, and some badass broads, this might just be the book for you. The third in line was The Bone Witch by Rin Chepeko. I've posted my full review here. Short version. When a young girl brings her brother back from the dead, there are some unforeseen consequences. This one is for all the lovers of high fantasy, dark themes, nonlinear storytelling, and the lush tradition of Japanese geisha. Fourth is The Last Namsara by Kristin Sigarelli. It transports you to a world rich in mythology and follows the story of Asha, a young princess and dragon hunter who bears both the scars of dragon fire and the scars of a decade of her people's malice. Betrothed to a man who delights in cruelty, her father gives her a secret task and will reward her with her freedom if she brings him the head of the first dragon. But the tapestry of her world, the past, the present, and the truth is about to unravel, and Asha is the only one who doesn't see it coming. Asha's story is a tale of myths and traditions, of gods and heroes and monsters, and it's a story of love. And with a massive dose of psychology and politics in this book, how could I not love it? Last one. The Invisible Library by Genevieve Cogman barrels headfirst into an adventure more complex than I have ever seen before. Between punk zeppelins, a fey mafia, detectives, dragons, a glamorous cat burglar, all these things come together in an alternate London as our heroes work to solve a murder and to find a book. The Invisible Library exists between dimensions, per se, and seeks to acquire books that are unique to specific alternate worlds.  That's it, the five bucks I read this month. I am going on a mini-vacation in March, so maybe I can get my monthly book average up a little bit. What have you read this month? Did you read any of the books that I read this month because I want to talk about them? I don't know why that comes with this little dance. Let me know in comments because I love talking about books.