You have said from the beginning that that massacre had to be responded to, that Israel had to invade. But now it's been 100 days. It's 23,000 Palestinian civilian lives, according to reports. The hostages, 134 hostages, six Americans included, have been there for all this time. Things have changed. Doesn't Israel now have to wind this down? Isn't that what you want? Well, we all want to see this conflict end as quickly as possible. We all want to see the suffering of people who are caught in the middle and as quickly as possible. It's vital that Israel be able to do everything possible to ensure that October 7th never happens again. And it's made good progress toward that, toward that objective. That was an embarrassing and borderline incoherent attempt by Secretary Blinken to try to justify Israel's ongoing genocide in Gaza, which he supports by the way, hence why he's trying to justify it. And I mean, how exactly are they closer to ensuring another October 7th never happens again? What does that even mean? They have slaughtered 23,000 people, nearly 10,000 of which are children, by the way, and the remaining Gazans who survive their indiscriminate bombing campaign who have lost everything, including family members, are going to be more susceptible to radicalization than ever before. So it seems to me that they're no more capable of preventing this now than they were on October 6th, because so much more people hate them because of what they're doing in Gaza. And when he says that they've made good progress towards that goal, he's just lying. I mean, assuming that their goal was to eliminate Hamas and not the Palestinian people all together, there's a reason why they haven't done that yet, contrary to what he's saying And as Isshant the Roar of the Washington Post explains, Hamas is entrenched and difficult to defeat. Even after waging one of the most intense heavy military campaigns in recent history, Israel has only neutralized a fraction of the militant group's armed strength, and in the process it has ravaged the embattled territory where Hamas has held sway, displaced close to 90% of the population, flattened whole neighborhoods, triggered a sprawling humanitarian disaster and found itself flailing in a losing battle for global public opinion. So explain to me again how they're closer to that goal, Blinken. They're not.  But it gets worse because he tries to give us some optimism that things are getting better currently, if you could believe that. He actually says this, by the way. But as he says this, he's leaving out a crucial, crucial component to what's happening in Gaza. You say they've made significant progress but yet an important minister, Gallant, said they are about to attack Kanyunas in the south very strongly and there are people there who haven't even been moved back to the north. So we've seen now a phase down of the operations in the north. That's important. We've seen the withdrawal of a significant number of forces, of Israeli forces that were in Gaza, particularly again in the north, where they've achieved a lot. And as I said, we want to see this conflict come to an end as quickly as possible, consistent with Israel's objective of making sure that October 7th doesn't happen again. Spoken like a true NPC that's incapable of deviating away from his scripted dialogue tree, just insufferable and I cannot take the gaslighting. Now it'd be one thing if the fighting stopped altogether in the north and Gazans were able to return to their homes. But what is there to return to? He doesn't address this. They have nothing to go back to because what's left is uninhabitable as the UN's humanitarian chief put it. Quote, currently in Gaza, infectious diseases are spreading rapidly with little to no clean water in the region. While there are only a few doctors left at the few hospitals that are still open in medication and medical supplies are extremely scarce. Hospitals are only able to respond to the worst and most horrific emergencies with over 10 children having to have one or both legs amputated on average every day, often without anesthesia. Food is extremely scarce in Gaza due to Israel's blockade of food and destruction of agricultural land. Quote, famine is around the corner, Griffith said. A recent UN backed report by global food researchers found that the entire population is currently facing an acute hunger crisis while the entire population will be facing famine in the next six months if Israel's starvation campaign continues. But according to Blinken, you know everything is fine. It's going to be okay because the fighting has stopped in the north and they can go back home now to rubble and dirt and diseases. It's just it's so hard to hear these details and not feel furious when you hear the way that the Biden administration is trying to justify this. This is a genocide full stop funded by our tax dollars. And if Israel doesn't kill them directly with bombs, Gazans may die indirectly due to starvation and a lack of potable water also inflicted by Israel. Because remember, they're the ones in control of the water supply in Gaza. Minister Galant announced collective punishment at the start of this siege. And despite all of these facts, members of the Biden administration are pretending as if they're not privy to the same information that we're all privy to. They're pretending as if they don't see the same videos that we all see. And to make matters worse, members of Biden's administration is dismissing South Africa's genocide case against Israel at the ICJ case in point. As you know, you dismissed few days ago the case that brought by South Africa against Israel at the ICJ. Did you read the indictment? And if you did, do you believe that cutting off water, electricity and fuel on a civilian population does not constitute a war crime by itself? Nothing else? Yes, I read the indictment. And as I said, and we stand by what we said about this, we find it without merit. We find it counterproductive. And I'll leave it there. We believe the submission against Israel to the International Court of Justice distracts the world from all of these important efforts. And moreover, the charge of genocide is meritless. It's particularly gallant, given that those who are attacking Israel, Hamas, Hezbollah, the Houthis, as well as their support on Iran, continue to openly call for the annihilation of Israel and the mass murder of Jews. Brilliant big brand logic there. See, even though Israel is currently literally in the process of annihilating Palestinians and Gaza as we speak, well, Hezbollah and the Houthis say that they want to annihilate Israel. Therefore, they're the same thing. He acts like we're children, like we're not going to question what he says. And a lot of people will take what he says as if it's the gospel. But people with brains can see that he is lying. And furthermore, saying something is one thing, but doing something is an entirely different thing. Israel is doing the annihilation. Israel is doing the genocide right now. So regardless of what these other actors say they want to do to Israel, Israel is the one that is doing the thing that is bad right now that you are supporting and as infuriating as it is to see the U.S. just dismiss South Africa's legitimate claim of genocide against Israel. It's not surprising because, of course, they're going to say that because they're not merely defending Israel by dismissing this case, they're defending themselves as well because, of course, they are just as culpable as Israel here. Not only does the U.S. supply them with the bombs that they drop on the heads of children in Gaza, but they also provide them with cover on the U.N. Security Council and gaslight all of the public at Israel's behest, as you're seeing in this video. Former President Mike Pence even signed bombs Israel fired into South Lebanon. So, I mean, this is our doing. We're part of this now. So if Israel were to be held accountable by the international community, they won't be, the U.S. would also have to be held accountable as well as a co-conspirator, but that's not going to happen either. But the Biden administration is facing at least a minimal level of accountability in terms of him having to defend his administration's actions here or I should say inaction to a degree. But again, we're participating fully. But as Prem Thakr of the Intercept explains, in late December, 77 groups representing tens of thousands of lawyers, civil society leaders and activists from six continents filed an amicus brief in a lawsuit that Palestinian human rights organizations, residents of Gaza and U.S. citizens with family members impacted by Israel's ongoing assault brought against the Biden administration. In the amicus brief, which is an avenue for groups not directly involved in a lawsuit to give the court information to consider in its ruling, the organizations argue that the plaintiffs established that a genocide or serious risk of genocide of Palestinians in Gaza is occurring. They also argue that the U.S. is violating its duties under international law to prevent and not be complicit in genocide and that those U.S. failures contribute to the erosion of long and widely held norms of international law, including the Genocide Convention and Universal Declaration of Human Rights both established in 1948 in the wake of World War II. The lawsuit is headed for a hearing in the U.S. District Court for the Northern District of California later this month. Meanwhile, in an 84-page complaint brought by South Africa, Israel faces charges of genocide at the International Court of Justice at The Hague. Now, I wish I could tell you that I'm optimistic that these cases will bring justice, but I'm not because even if justice prevailed in the best-case scenario here, there still isn't going to be justice. Hell, even if Biden was literally found guilty of complicity with Israel's war crimes, nothing is going to come of that because in this country, we will go to the greatest lengths imaginable to protect our war criminals. And I say this because, as journalist Abby Martin points out, Bush passed a law known as the Hague Invasion Act, which says the U.S. can invade the Netherlands to prevent American personnel from being charged with war crimes. Yeah. And even in the absence of this law, I still don't expect any U.S. war criminals to ever be held accountable for anything ever because, I mean, if Henry Kissinger died without seeing a single day in prison, then U.S. officials can basically get away with anything, including literal mass murder. But that's not to say that Biden isn't being held accountable because he's already paid a large political price. Polls currently show that he is losing to Donald Trump in head-to-head matchups, which is terrifying.  Case in point. That's all right. That's all right. That's all right. Thank you. Look, folks, I understand their, I understand their passion and I've been quietly working, I've been quietly working with the Israeli government to get them to reduce and significantly get out of Gaza, using all that I can to do. He's trying so hard, yet he bypassed Congress to sell Israel more weapons that he knows they're going to use on innocent civilians, including journalists. And on that note, human rights groups like Freedom House, for example, don't think he's doing enough to address Israel's assassination and targeting of journalists because, predictably, his administration is pretending like there's no evidence that Israel is targeting journalists.   Now, Biden's liberal defenders defend him by pretending as if he's hopeless in this situation and there's nothing that he can do. But that is factually incorrect. Republican administrations, as immoral as they are, have literally done more to reign in Israel's brutality than Biden has done. As Shreeta Parsey explained in an op-ed for the nation, quote, in 1982, President Ronald Reagan was, quote, disgusted by Israeli bombardment of Lebanon. He stopped the transfer of cluster munitions to Israel and told Israeli Prime Minister Manasheem Begin in a phone call that this is a Holocaust. Reagan demanded that Israel withdraw its troops from Lebanon. Begin caved. 20 minutes after their phone call, Begin ordered a halt on attacks. So ask yourself, why won't Biden do the same thing that Reagan did? Why won't he just pick up the phone and tell them to stop? It's because he doesn't want to. That's why we call him genocide, Joe. He's not just getting strung along begrudgingly by Israel. He is a willing participant in their ethnic cleansing and genocide of the people of Gaza. He is culpable. He has blood on his hands. He is a war criminal. But even though he's still currently in the driver's seat, that could soon change if he doesn't meaningfully reign in Israel. Because as Akbar Shahid Ahmed argues in the Huffington Post, Biden's continued support of Israel risks the U.S. getting pulled into a broader conflict in the region if Israel's actions in Lebanon end up pulling in Iran or other countries. And I mean, if that happens, this will spiral out of control even more and all but guarantee his defeat in November because Americans don't want to see another war. And that means that if he loses, Trump is going to take over as commander in chief and pour gasoline on the fire lit by Israel and things will get exponentially worse fast. It's just such a hopeless situation here because the American people are saying one thing and politicians are all collectively choosing to just ignore what they want. And who cares if the American people were in support of this? Like it doesn't matter what public opinion says. Genocide is wrong period. Like we shouldn't have to gauge where the population is at when it comes to the mass slaughter of civilians.  I mean Jesus Christ, it's so frustrating. And it feels gross to even talk about genocide in the context of, oh, well, I wonder how this is going to affect Biden's presidential prospects. But I mean, you would think that Biden's ambivalence towards the suffering of innocent Palestinians would not extend towards his electoral prospects, but he genuinely seems uninterested and even trying to win back the people who he's lost who are outraged about his genocide in Gaza. But I mean, regardless of how this pans out, one thing is clear, the Biden administration's gas lighting of Americans is not working. And I expect him to continue to be ride or die with Israel and continue to gaslight us. But I mean, just stop. If you're going to keep doing this, stop lying to us. Because the people who support you, they're going to support you no matter what, even if you stop feigning concern for Gazans that you're supporting the slaughter of, just stop. Drop the pretense that you're concerned about civilian casualties because you're not. And I say that because that's what your actions dictate. You might say some really nice things and maybe caution Israel publicly that they should be a little bit more careful while not actually looking at what they're doing and seeing the evidence.


Welcome back to us here still on the search for human-humane architecture here in downtown Honolulu, which one could call a construct of Howley's Hawaii and that's the title of today's show and our expert is Dr. William Chapman who allows us to call him Bill from now on and Bill is highly educated at the most prestigious places in the world both nationally and internationally meaning at Columbia and at Oxford and he's currently a professor up at UH in Hawaiian Studies and at least as much also in the Department of Architecture where I'm teaching so thank you Bill for being here with us. Martin thanks for having me it's been really a while getting on the show and I'm glad you found they persevered and got me on so no thanks for pleasure and I just want to as a disclaimer right away I want to say that Howley Hawaii was definitely your title right? I'm very used to take the blame that's very innocent innocent yeah yeah no that's good but for let's say since this is an edu taining program and format let's inform the audience who might not be from here because this could be viewed and heard all over the world what in the world are Howley's? Well as you know and many people know Howley is a term that was used by the Hawaiian population to describe outsiders people who are coming from somewhere else and I understand it means without language really but also sort of without breath and soul to some degree as well so it's kind of remember this place for nobody I never met anyone who didn't speak Hawaiian language so that Howley stuck and that is still a term that's why they use today to describe people from the mainland of the US. I always use the even more catchy description which is that it means the one without breath and when Cook came he did not do what the Hawaiians did actually smell each other's breath when they welcome each other so they said he's the one without breath and how he ended is a is a story that most people know as well so but let's go the other way around what maybe Howley's did to Hawaii and but before we do that let's look at how people here were doing it before they got contacted which is actually another term right you get you get the pre-contact and the post-contact so this is an image here of because this is an architecture show here how people dwelled on the island pre-contact this is a probably early 20th century maybe late 19th century photograph of a of a Howley Peely there could be different kinds of vegetation used for the covering but Peely Grass which was a kind of grass that grew near the seashore kind of a stock thick stock grass not a grass that was used as we're more familiar with the grasses that are used for feeding animals which is all pretty much an import from other places when they brought it in for cattle grazing and things so this was a grass that Hawaiians would harvest and use for building houses particularly those near the shores if up in the mountains you might have other materials that were used in ferns and other things as well sometimes ferns were used to decorate them and they suited very much suited the needs of Hawaiians in a way buildings that could be replaced after storms or replaced periodically as they has maybe on the structure began to fail or the grass needed replacing these are all aspects to it and Hawaiians made permanent architecture they made mainly temples which were made out of fieldstone or rubble and but and those often included other components like wood components and things like that they had wind towers for telling the future and telling fortunes and speaking to the gods and they would usually have holly peely within them but basically most people lived in a form of holly peely so as most indigenous cultures they made it from scratch right here in particular because there was no other place to get it from right they have to basically do that and there was probably status represented in it as well the simplest ones were a little less than pup tents really just to two uprights forming a triangle more elaborate ones had a kind of bow shape to them and usually had a very low door I mean you could argue even that this door main already show the influence of Western civilization or Western influence just by the fact that's kind of a bigger door most of the doors and they really old representations of holly peely are quite close to the ground some people argue that them when you started getting vertical walls that that was a Western influence but some of the early images suggest that some of the buildings actually did have kind of vertical walls one of the interesting things is the whole way of men making them and tying them is very similar to the same craft involved in building canoes or adding outriggers to canoes or putting up mass and things like that kind of continuity lashing technique in the materials they're usually made out of Senate which is a kind of taken from coconut husk making a kind of cordage and then the cordage was used for tying and they didn't use really pins and they certainly didn't use metal because they didn't have any metal to work with little bits of iron in Hawaii at that time and then they got contacted right and everything changed and their entire life including architecture which is mostly an impression and embodiment of sociotel developments so how did that go well I think you know it varied like that picture shows there were Hawaiians living in grass houses well into the early part of the 20th century but I think by the 1920s they were really a rarity and remember coming across articles in the Hilo paper when they were building a rail link for the sugar plantation they'd come across an old holly peeling so really there weren't very many left there's still a kind of nostalgic element a little grass shack in Hawaii famous song as well but gradually Hawaiians had moved into buildings mainly made out of wood if you look at early images of Honolulu there were grass houses in Honolulu into a really late period into well into the 19th century quite typically they would take them with mud by that point because not so much to create a stronger roof but really to keep the animals they want a grazing moving animals at that point they wanted to keep them from chewing on the roof so they put mud on them protect them they also faced the importation of different kinds of insects and other pests they had rats already but they didn't have a lot of like that some of the termites and things like that we have now we're probably not here then and so once you started getting insect pests they became much less comfortable Hawaiians adapted them they put in windows some of the earliest representation show sort of vertical double-hung windows being put in at the ground level where they would open sideways like a slider provide ventilation and you could see that in response to European traditions they started big building really bigger hollies particularly for churches but eventually you started getting the use more of wood construction was really typical and stone construction stone construction they understood but they didn't understand it from a European point of view with a lot of set in mortar with walls with plates and then building roofs over these elements so some of the earliest Western settlers you know people like John Young over on the Big Island who had actually been more or less abducted by King Kamehameha to serve in his military and later serve as an advisor he built the kind of complicated compound over and on quite high on the Big Island and it was a kind of combination of a kind of a welch homestead and a and a Hawaiian house compound a little bit of both cultures it did get this kind of hybridity yeah yeah and maybe one keynote is as well plantation home and then maybe Dickey because you mentioned the roofs maybe you want to talk about these yeah well some of the earliest buildings somewhere actually there was actually a short period of building with adobe or mud brick construction there were an sizable number of people from Mexico and that area who brought this technology with them and one of the things that mud brick construction required was a kind of roof that would dramatically overhang it would protect it from the rain so it became kind of common and there was a examples you can still see at Waoli on the island of Kauai it's not mud brick it's actually stone construction with plaster over it but it has this very steep pitched roof with it's kind of a side that go like this kind of it's got a it's really a second double pitch really so architectural historians would call it a bell cast roof and kind of like a square bell and what this was was to provide a sort of space around the building to shed water Calma Capulia Church in Honolulu was very much the same kind of the construction and originally both of these buildings were probably covered with pili grass and then eventually they got covered with cedar shingle which was a were imported from the Pacific Northwest and they were used commonly but the pitch was necessary for the original covering with for the covering with the pili grass and the double pitch was because if you were to follow the angle of the steep roof down you wouldn't have room to walk underneath it you actually had the double pitch to give you a headspace so and then when they were covered with shingles they continued that tradition shingles don't require quite a steep a pitch but they also require a steep pitch and in fact at the time it's interesting that some of the famous architects you probably talked about other series CW Dickey and Hartwood some of these guys actually were involved in the restoration of these buildings in fact Hartwood worked on the violi mission in the 1920s and so he was familiar with this kind of construction and CW Dickey worked on the mission house in Honolulu in the 1920s when it was converted from having once been a kind of girl school for Hawaiian kids and then was turned into a museum so this is really part of this back to your term howly the kind of period in the 1920s when the kind of well-established Euro-American families were beginning to see their history in a kind of in terms we now think of as historic preservation kind of tell the story of pioneering life and they were really trying to tell that story not the story of Hawaiian life but the story of the little house in the prairie surrounded by the savages idea and so it's not surprising that CW Dickey and Hartwood and other proponents of this Ray Morris was another of this kind of Hawaii regionalist style that really gets started in the teens that really takes off in the 20s and 30s that that would have been a kind of signature element double-pitched roof became very important to them and so now everybody even my kids who grew up in Hawaii they all say hey dad there's a Dickey roof and so that's what you see yeah yeah before we go into our little mandatory break hardwoods by the way our friend Donna Hibbard has written books and write them is about hardwood and this mason did a really nice book on hardwood and there's one there's one project his latest work which is the board of water supply which basically breaks free from the traditional i love the border water supply building i think i don't know if you have a picture of it the show but it's probably my favorite building and i was a disconcerted the other day to hear that someone from the planning department at UH is being brought in to discuss how they might treat the area we call the capital district around the capital and they said it was out of alignment and so it needed to be taken out which is really disturbing oh my god but it may be queen's hospital well now we need a break to catch a breath and vent about that after the break okay back in a minute i've got the beagle sisters here with a healthy tip we encourage you to enjoy the food you eat this holiday season and keep it local and healthy yeah eat the rainbow in your rainbow and if you need any produce come to the red barn on the north shore aloha my name is richard emory host of condo insider more than a third of hawaii's population live in some form of association and our show is all about educating board members and owners about the responsibilities and obligations and providing solutions for a great association you can watch me live on thursdays 3 p.m to 4 p.m each week aloha aloha this is kaley akina with the weekly a hana kakou let's work together program on the think tech hawaii broadcast network monday's at 2 o'clock p.m movers and shakers and great ideas join us we'll see you then aloha hi i'm stacey hayashi and you can catch me on monday's at 11 on think tech hawai stacey to the rescue see you then okay we're back to bill with howley hawaii today and if we can get picture two and and three really quick after each other that's more the mainstream um one is picture two is is uh basically couple lay this is what we find today this is a very i like to call it allow myself to call it an invasive american this is not the best of the best of american because there was good stuff even suburban stuff the case study houses and things like that so this is the most watered down of the american dream and picture three which is also a permanent background is basically the high rise right so both these however we have to say in all these you know you find better buildings than than than worse buildings but still it's it's pretty much a an imported to say the least it allows us to say an invasive um fabric that's imported however if we get to the picture number four um uh this is we came from the border water supply and hardwood with your favorite building of his which you see in danger then we gotta really work on that and uh so that was segwaying into a modern era and that was basically sort of promoted by a rather tragic event in history right well my thought martin is really world war two i know you had to sort of brown on before but no this world war two is really the pivotal moment i mean i guess we could see heartwood's building as pivotal but it was already in the post war period and it was his effort to kind of come to grips with modernism where he came from really what you'd have to call a bozar architectural tradition which was based very much on a formal plan a sense of hierarchy you know it at the front door is all these things that architects were trained in whether they were in germany or whether they were in danmark or most famously in france the vehicle the bozar and so that building is neat because it has they sort of bristle a kind of a kibuzi influence modernism with a kind of chinese pavilion entrance area which is gives hierarchy to the building yeah he managed to kind of evoke something of asia as well and these gentlemen on some of that right in hawaiian architecture or architecture in hawai in the post war period and these two gentlemen and there are more but this is the three people persons man in the center of the picture are actually the next generation and they actually learned on the hardwood and they're learned on the diki right will you tell us who they are well we're going to talk about just three of them who i can identify um vladimir aspaf is grinning down below he's got the mustache peter wimbley's up behind him peter wimbley formed a firm that would become international and then to the right is um um um alfred pratt alfred pratt thank you and he was actually weirdly for a short period in prison during world war two just because he was german austrian to be to be sorry you're right but because they still were suspect right but by that point austria had become german and that's fair enough and my mother being austrian high mom that's i can totally relate to that i have posted german invasion right so anyway he um he would go on to design the arizona memorial in 1962 which is his most famous building but i think you had pictures of other buildings he did as well we can actually that's the second to last picture um is um i picked intentionally buildings because you were uh and this is good because it's actually even uh located very close to your favorite hardwood building and it and it applies a similar pattern which we can call bioclimatic because the fins of the of the board water supply are decorative but they're performatively decorative exactly because they actually cool the building and they block the sun during particular times of the day that's what the whole brisselet means you know it's unblind basically well i think you know both osipoff and osipoff had started doing fairly traditionalist buildings in the 30s he was doing kind of some of this regional stuff but kind of some of those houses would look like diki houses really some were kind of looked like suburban houses in california sort of um monterey style and things like that there's one in minowa that's very typical of that time period but i think in the post war period they really were trying to think of an architecture that would speak to hawaii so there you do have this group of people that were still interested in hawaiian regionalism and i think that's really important and all three of them addressed that in different ways um this picture by the way if we can go back for a second to the second to last picture is uh when we see our friend don hibbert here and uh one woman from the church and the guy to the right is my dear mentor robert mccarter who has written so many books about lucan franklark rides i think i met him when alba auto and many more and uh he was here so uh we gave him a tour and we heard the the woman from the church shared she said well it's recently getting getting hot in the church and we said has it always been like that she says no and we said what was different well there was some green that um the planters were rotting and um we took them down and this is almost similar to what you share with us the shocking news of some people thinking about doing something to the border water supply she shared with us well an architect is consulting in what they could do for a shading we just said we'll bring back what it was the original because there was some intent i mean vegetative facades for shading we wish would have that these days and there is actually you know tendencies in the in the world and contemporary modern architecture our students uh suggest that a lot so just just bring this back right i think probably the most famous i mean there's a kind of direct link between this maybe in the out rigger club by alba both of them are trying to come to grips with this new era of concrete construction kind of comes out of europe it comes out of the tradition of look who's the a and people like that but it definitely um was going to be the building material in the 1950s and certainly wood construction was other than for much smaller scale buildings like houses was not going to be popular and people were getting used i think during world war two one thing desoto may have mentioned was even that's when really some of the most voracious termites arrived in hawaii as a result of the shipping of materials during that time and um so that's that was a had an impact on it so they i think they thought like the the concrete would have a feeling of kind of permanence that was a kind of notion of modernism as well a kind of a time stopping quality there was an architecture that wasn't just in the sequence of new buildings it was a new architecture that was ending all the rules right and then this is a perfect or producer brings in the perfect picture of another project by aussie prophet embodies that perfect and again this commitment to trying to figure out how you make them kind of energy not energy efficient but comfortable basically taking advantage of the breezes taking advantage of vegetation trying to make them open kind of embracing the climate of hawaii which i know is a cause of yours right it is and these are the grandfathers of that movement but as you see even at the airport these days the the type if the late 20th century was the age of concrete i'm afraid maybe the early 21st century is the age of air conditioning and that there are some scary tendencies at the airport too to so to hawaii is that more and give it a give it some sheet rocky ornaments which by the way you tell me the hawaii you talked about symbolism but compared to other asian pacific cultures that you're an expert in the hawaii's were very relatively little symbolic right they were actually pretty rational in their in their gestures right well they certainly had representation and symbolism of stuff but i can see what you mean they were forced by necessity to some degree to be kind of practical with bitruvian and architectural right that's a great compliment and the last the last picture is now the third person who is pete wimily who is mostly and desoda and i got to talk about the hilton wine village and some of the next show so you know some of pete's work is which i feel is dismissed and maybe for some you know a good reason as being tiki architecture there's a lot of that i think he really was trying to go beyond tiki if you look at his earlier buildings i think he was trying to draw upon kind of the culture of the pacific he wasn't trying i mean the tiki movement is really something that comes out of california you know and it had to do with world war two as well it comes out of the broadway show south pacific it comes out of the bars that these guys had become familiar with when they were traveling around the beach culmer and all that and on the beach culmer it was in california who literally changed his name legally to beach culmer started this movement and it became very popular and then it kind of sort of was a sort of like a wave it's in some way starts in hawaii washes up in california with a serviceman after the war and then comes back to hawaii but i think wimbley was always trying to do something beyond that he was and uh locate something in place and that picture we saw just took this the other day at sunset and i was talking to rob here that um us having been in the desert and knowing the saguaro cactus if you cut a section through the horizontal section through the saguaro cactus you you see um or even when you look at it you see itself shading so through this undulation of shape there's always certain parts in the shade so it keeps it hydrated or prevents it from dehydrating and this building here and i have to give a compliment uh to kamehameha school in bob oda who is um uh you know anytime we are in touch i said i know bob you don't like the building because you constantly are going around and doing the spalling repair because these fins are so amazingly thin and they didn't have fiber reinforcement so it's always spalling off but they always go around and repair it and and i have to you know thank them for that because this would be a building that would be a pity to to see go because it's truly a marvel of of blend i think that's your point of of so these people came they're all let's say again asipov was russian but grew up in japan prize was austrian and wimbledon was american so they were from all over the place and they certainly brought the best from their culture right but different to these days where architects come also still richard meyer and other people come for the big developers and are less sensitive about uh the specificity of why and these guys were very very devoted and respectful and said we want to blend the best of what what we have what we bring with the best of what we find is that probably a couple reasons for that i mean one reason would be simply back in the 50s it was more of a commitment to come to y&b here you couldn't just kind of fax it off or send it as an attachment from your offices in new york right so you actually had to experience the place to some degree that probably had an impact too i'm looking at that picture in my mind now of the varsity building and i can't help but think that i'm also disappointed to have lost the cw diki design sort of deco sort of inspired varsity theater which was in the parking lot at that point that's it we run out of tragic loss it was and we take that as uh as motivation for doing another show bill yeah we do about something on preservation next time right thank you we do so much to talk about thank you very much for having been here much thanks


Hi, Sayada Hashimi here. I'm going to show you how you can build your first application with Visual Studio for Mac. In this video, we'll be developing an ASP.NET core razor pages web application. Let me quickly show you this website that we'll be creating here. So here I've got a website that displays the GitHub emojis that are available. You can see there's quite a few here. And this just calls into the GitHub API to get the information and image about the emoji. So this is what we're going to be developing today. All right, so let me go ahead and switch over to Visual Studio for Mac, and then we'll get started with this. The first thing we have to do is click on New Project. And this will bring up the New Project dialog. And the ASP.NET Core projects are under the .NET Core node. And we want to create a razor pages project here. So that's ASP.NET Core web app. I'm going to click Next. And then I'm going to give it a name. I'll say GitHub emojis. Then we'll click on Create here. All right, now our project has been successfully created. I'm going to go ahead and run this. And just to take a look at what we get by default and then go from there. All right, so here's what we get out of the razor pages template. We've got a kind of a basic website with a few different pages here. So let's go ahead and close out of this and go back to Visual Studio for Mac and start developing our project. Let me stop this. All right, the first thing we want to do is go and customize the layout page. The layout page contains a lot of shared components here. So you might have noticed we had a header at the very top of the application. There was also a footer and that sort of stuff. So we'll get rid of that. And the layout file also will bring in your CSS and JS files. This is a pretty basic application. So I don't need everything that's in here. I'm just going to keep the CSS file and get rid of everything else. And then for the content, I really just need render body. So I'm going to extract out render body, put it in here, and then just get rid of everything else that's here. Now that we're done customizing the layout file, what we need to do is create a class that will be responsible for calling the GitHub API for the emojis. Before we do that, let's take a look at the API that we're going to be calling here. Here I have the documentation for the GitHub API emojis endpoint. So here we can see that it returns a JSON dictionary of emojis that are available. So we've got the key. That's the text that you would use for the emoji. And then an image that represents that emoji. Here I've got the full result open to my browser. So this is what we're going to be doing. Let's go and create the GitHub emoji service class now. I'm going to switch back to Visual Studio for Mac. And the first thing that we want to do before we create our emoji service class, we want to create a class that will basically encapsulate the different properties that an emoji has. So I'll say new class, I'll name it emoji. And then we want to create two properties here inside this class. I'm going to use the prop snippet that's P-R-O-P. So when you invoke this, you'll type in P-R-O-P and then tab tab. Now I've created two different properties. One is called key. That's the actual emoji string. And then we've got the URL. That's the URL that the image is associated with that particular emoji. So now let me go ahead and, what I want to do now is I've already created a class that we can use for the GitHub API emoji service. I'm going to go into Finder and we're going to copy this class into the project. So let me copy this file and then I'll right click in Visual Studio for Mac and say reveal in Finder. And we just want to paste this here. When I paste this, keep an eye out to what happens inside the solution pad. All right, so I've pasted that file and we can see it's immediately available in the solution pad. So for .NET Core and ASP.NET Core projects, all files that are in the folder immediately appear by default. So let's take a quick look at this GitHub emoji service. It's pretty basic. We will call the GitHub API to retrieve the emojis. We're going to pass in a custom user agent that's required for the GitHub APIs. And then in Get Emojis From, we will convert the JSON dictionary into a list of emoji objects. All right, now what I want to do is I want to register this class as a service with ASP.NET Core. But before I do that, I want to create an interface out of this because I don't want to be passing around instances of this particular class, but instead of an interface. So to do that, I'm going to right click on the class and select Quick Fix, Extract Interface. Here we can see it's offered defaults for both the name as well as what items I want to include in that interface. I'm going to go with the defaults here. I can right click and say go to declaration if we're interested in taking a look at what was generated for us. I'm going to copy this name. And now we're going to go and register this class with ASP.NET Core. All right, so what I want to do is I want to register a Singleton class here for this GitHub Emoji service. Okay, so now we've registered our GitHub Emoji service with startup.cs and this will enable dependency injection. So now let's go and start customizing our index index page model. So I'm going to go into the page model and the first thing I'm going to do is I want to get a handle of this GitHub Emoji service. So I'm going to create a private field for this. Now I'm going to create a constructor for this index model class and then pass in the type that I would like to be injected. So here we've passed in iGitHubEmoji service into the constructor and then when this page model is created that will automatically be passed in through ASP.NET Core. All right, so now we've created the index model constructor here and we're accepting iGitHubEmoji service as a parameter. Since we've registered that in startup.cs and configure services that will automatically be injected through dependency injection. Now we want to go through and customize our onGit method here. All right, so now we want to move on to populate our onGit method here. When onGit is called we want to call the GitHub Emoji service that we just created to get the list of emojis and store that into a property. So first I'm going to go and create a property for this. Now let's go to customize the onGit method. Let's take another look at this GitHub Emoji service. So this is the function that we're going to be calling here. GetEmojis and it's async and it returns a task of iListEmoji. So going back to the page model I want to change the return type here and I want to replace this with async task. And now we want to call that method. All right, so now we've created our onGit method here. All this does is call the Emoji service and then assign the result to the emojis property here. So now we're going to go in and start editing the index.cshtml file. And we don't need any of this content here. So let's get rid of that. And we don't need this either. So let's go back. So here now what we want to do is we've made a property available on the index model class here. So emojis. So with razor pages, all the properties in those classes will be available in the razor page as well. So now I have access to the emojis property. So all we want to do is kind of loop through that. So to do that, so we're going to actually put that as a div with inside of another div. So let's start creating that. All right, so here I'm just going to use a razor for each statement to loop through the emojis. And then I'll create a div for each particular emoji here. The div for the individual emoji will have the image as well as the label underneath it. So now we've authored the razor and HTML portions of this application and this will render the emoji image as well as the key. So I'm going to go ahead and save this file with command S. I'm going to open the CSS file here. I actually, I have the CSS file already created ahead of time. I'm just going to go ahead and copy this to the right location. All right, so CSS files are and www.css by default. I'll replace that file. And one thing that I forgot about in the layout file was customizing the title. So I'm just going to have the title for this entire website to be GitHub emojis. So we'll save that. So let's go ahead and run our application and I'll do that with the play button. Okay, now that I'm looking at this, I realize I forgot to add the title here. So let's go back to the IDE. I'm going to add my H1, save that and then refresh the browser here. Okay, another thing that I should do here is I should also add a height and width. So yeah, I just added a height and width because we want this to be 128, 128 for each one of the images and then it will be laid out well when we load it up. So let me refresh this. Let's take a look. All right, so that's pretty much our first application. Let's scroll down and see if we can find any kind of interesting emojis here. You know, let me go and show you my, let me show you my daughter's favorite emoji here is the owl emoji. She really loves these owls. All right, so that's pretty much it for this video here. This is Sayed Hashimi with another video for Visual Studio for Mac tips and tricks. Please keep an eye out for more videos in this series.


No. No class on Tuesday. At least I think that's how it works here. I think finals week there's no class. I hope so. Okay. So yeah, there'll be no class.  And then if there's a real big problem with getting the solutions to the third test post, I'll just probably just put it in an email and just send it out to everybody as a PDF. Okay. But I'm hoping it won't have to go that far. All right. So why don't we continue talking about 5.2 here for a minute. I'm going to mention a few things about some of your problems. This I could just have you do yourself. And I'm going to talk about this a little bit. I did last time, but if you remember number 2B, and I only did part of this, but part of this problem, okay, this is just a part of the problem, not the entire problem, essentially amounts to you proving that 8 divides 8 to the 6th minus 1. We talked about this a little bit last time. So I guess the original problem was to show that 8 to the 6th is congruent to 1 module of some bigger number. But the whole point is that you break the modulus down into its pairwise relatively prime pieces, right, by factoring, and then you just have to do it, you have to prove this result for each of those pieces and then put everything together at the end. Okay. And I gave you a hint. I kind of just did this off the top of my head. It's a good hint because it works out pretty well. This, I'm going to tell you this just because I think this is actually a really pretty cool solution. Okay, so 8 to the 6th minus 1 is the same thing as, I may have written this for you yesterday, A cubed squared minus 1 squared, right, that's for sure, definitely true. This is a difference of squares, so you can factor this as A cubed plus 1 times A cubed minus 1. Yep, and I know I did this before. And I told you, some of you may have forgotten this, but I told you that both of these can be factored because both of these are, well, the first one is a sum of cubes and the second is a difference of cubes, right, because one is the same thing as one to the third. You can factor both of these and if you forgot how to do it, nice thing is we have the internet now. You can easily just go on Google and say factoring sum of cubes or difference of cubes. You can get all these formulas in, you know, three seconds, right. So this actually factors as, the first part factors as 8 plus 1 times A squared minus A plus 1. Let's see if I can squeeze this in here.  Okay. Okay, so what was the assumption in the problem, which I didn't write this down, but the assumption, somebody can tell me if I screw this up because my book is closed, is that in the beginning of the problem is that the GCD of A and 42 is equal to 1, right? I get that right. Now, and I think I mentioned this last time, but if A and 42 are relatively prime, that means that A can't be even because then 2 would divide A and 2, of course, divides 42. So the GCD couldn't be 1. It'd have to be at least 2. So A is odd, right? And again, I'm just jotting down notes here. So A is odd. Now, here's the idea. This is cool. All right. You want to prove that A divides A to the 6 minus 1. And we have A to the 6 minus 1 factored like this. See that? Well, if A is odd, then A plus 1 is certainly even, right? And A minus 1 is even. And now think about this. I want you to think about this for a second. What do we know about A plus 1 and A minus 1?  So these are consecutive even integers. Think about pairs of consecutive even integers. 0, 2, 6, 8, 14, 16, 22, 24. One of them is divisible by 4. You can prove that. And you should if you want to do it this way. If you just doodle some examples, you should convince yourself pretty quickly that if you have a pair of consecutive even integers, 1 is divisible by 4. And you guys actually should all be able to prove that. If you have basically what it boils down to is just if the integers even, start with the A minus 1, the smaller one. It's even. It's of the form 2k. Well, that k is either even or odd. If k is even, then it's of the form 4k. And then it's divisible by 4. If k is odd, then you add 2. And you get that it's divisible by 4. So one of them is divisible by 4. And again, this is something you guys should be able to prove. This is not that hard. So think about this. And I'm actually just, I'm not going to write anything else down. Think about this. They're both even. Both of these guys are even. And I'm telling you one of them, at least, well, it's not going to be both of course. Exactly one of them is divisible by 4. But the other one is still even. So you can extract an 8 from this expression. You see that? Slick. This is slick proof. So that's how you can do it. Oh, I don't know. This is probably, I mean, I don't know. I mean, I can't say that this is the best possible proof. But I can't imagine it gets much easier than this. This way you avoid using the division algorithm dividing by 8 and all that kind of nonsense.  Because you're trying to prove that 8 divides 8 of the 6 minus 1. Don't forget what we're trying to prove. One of these guys is divisible by 4. The other by 2. So we've got a 4k and say a 2l. So now we can get an 8 to be pulled out. So 8 divides the whole thing. Does that make sense? So fill in the details here. And yeah, the other part, I think, is just essentially just for Moz Theorem to get the other two factors. And then this is the one where you just kind of have to play with it a little bit. So you see kind of how to proceed. But I just wanted to show this because I thought this was kind of a neat way to do it. Okay. Any questions about this one? Okay. Now, I'm also going to talk about a couple more of these. And again, I apologize again for kind of screwing things up on Tuesday. But actually these problems all follow essentially the same format. And for the most part, there's not a whole lot going on with these. You just have to really notice one thing and then apply for Moz Theorem. And that'll solve almost every one of these. Almost every one of them. I mean, here's an example where you actually had to use some trick or you could get it to work out. But most of them just involve doing what I said. So let's see. Everybody have this down now? We good? Okay. So let me talk about, I think maybe I feel like I started talking about this on Tuesday. 4D, show that A to the ninth is congruent to A mod 30, right? For every integer A. Okay. Two parts to this. And what I'm going to do here really will apply to more than just this problem. We talked about this a little bit before. But first thing you want to do, and I know this is a little bit vague, but you want to take your congruence and you want to split it up into, okay, I'm just going to be informal here, into pieces by factoring the modulus. Okay. So what do I mean by that? I just mean, right. I think I said this to you guys on Tuesday. I'm not mistaken. This is an if and only if. 1, A to the ninth is congruent to A mod 2. 2, A to the ninth is congruent to A mod 3. And 3, A to the ninth is congruent to A mod 5. And I'm going to pause here because I want you guys to, I really encourage you to listen to what I'm going to say here in a second. This is almost as fundamental to doing these problems as the actual for mod's theorem is, really. Okay. And actually, I think it was you, Matt, that asked about this sort of idea on Tuesday. So think about this. Just go back to the definition. This is nothing deep, really. It isn't. Just go back to the definition of congruences. What does this mean? It means that 30 divides A to the ninth minus A. That's what that means. Well, 30 divides something. Sorry. Hang on. Okay. So if 30 divides A to the ninth minus nine, sorry, A to the ninth minus A, then all the factors are going to divide it as well. Two, three, and five, okay? And conversely, if two, three, and five divide A to the ninth minus A, then by a theorem that we have talked about before, their product, because they're pairwise relatively prime, we can squish them all together and the product's going to divide it, right? Okay. So all we have to do now is establish these three different conditions, okay? So what about number one? How do we do that? Well, these now you just apply for mod theorem two. That's really what it boils down to. Okay, so the first one or the corollary, right? A squared is congruent to A, right? Mod two. This is, actually it's a corollary, but this is for mod, for mod theorem. So when you have this, remember this, for any prime P in any integer A, A to the P's congruent to A mod P, right? So once you've got this, well, what can you say? I'm not going to go through all of this, but you can raise both sides to the fourth power, right? So what do we get? We get A to the eighth is congruent to A to the fourth mod two, right? Okay, but from here, right? This is where we started. We know that A to the fourth, if we score both sides, right? A to the fourth is congruent to A squared mod two. Does that make sense? So we know that A to the eighth is congruent to A to the fourth, which is congruent to A squared, which is congruent to A mod two. And so I'm going to stop with this one, but you can see how this is going to play out probably. Once you know that A to the eighth is congruent to A mod two, then A to the ninth is congruent to A squared, and A squared is congruent to A. Basically, the point is once you know that A squared is A, then A to everything is A. Basically, that's what it pulls down to. Yeah? It's easier to understand what was going on by instead doing if A squared is congruent to A, using the proper two E, we can multiply both of them by A, to get A to the third is congruent to A squared. I'm going to repeat that process, and the fourth is congruent to A to the fourth, which is A to the ninth. Yeah, I mean, it's, yeah, I mean, either one's about more or less equivalent. So, yeah, you could do that too, for sure. Yeah. Okay, so I'm going to sort of leave you to finish this yourself, but how do you do the mod three and the mod five? It's the same thing. You just use, for the three, just use the fact that A cubed is congruent to A by four months there. And for five A to the fifth is congruent to A. And then you just, you just, you know, kind of like this, you just kind of mess with it and it just falls out. And once you get all those three parts, then you get what you're looking for. Is this okay? Any questions here about this? About this particular problem? Yeah, okay. It's possible. Yeah, it's possible. Okay, are we okay then with this? Okay, so this is what you're doing for a lot of these problems is you, whatever your modulus is, you want to break it up, in fact, into powers of products of primes and then just attack each of those pieces individually using from us there. That applies to the vast majority of these. Let's see. So, for example, I'm going to do another one here. Six B is to show that A to the fifth and A have the same units digit. And this is going to follow the same, basically the same thing. I actually have to deal with something real quick. I'll be right back. Okay. So, and I think, yeah, there's no assumption on A here. This can be anything. Show that A to the fifth and A have the same units digit. So, this goes something like this. You have the same units digit, and we talked about this a little bit last time. If and only if. Okay, so remember what I said about the units digit? And it's just what this is congruent to mod 10, reduced between zero to nine, right? So, this just means if A to the fifth and A have the same units digit, that means A to the fifth is congruent to, say, C, right? Mod 10. And A is also congruent to C, mod 10. And here, C is just some integer between zero and nine. Okay, so remember, just think about this for a second. If we call C the units digit of A to the fifth, if we just call it C, right? It's some integer between zero and nine. That's all it is, right? Then A to the fifth is congruent to C, mod 10. And if A has the same units digit, then A is congruent to C, mod 10. So, what can we say about this? What can we conclude? A to the fifth, sorry, there's really, I really mean an if and only arrow here. If and only if. A to the fifth is congruent to A, mod 10. That's really what this problem boils down to now. And so, if A to the fifth and A are the same module of 10, that means they have the same units digit. So really, this is, now this is what you have to prove right here. A to the fifth is congruent to A, mod 10. Then you're done. And so, you go about this the same way we talked about with the other problem, right?  So, you just want to show that A to the fifth is congruent to A, mod two. And A to the fifth is congruent to A, mod five. Okay, and again, remember, and this is really easy. A to the fifth congruent to A, mod five is just for mass theorem, the correlator for mass theorem, right? So, we've grown to A, mod 10 eventually? Yes. So, we've broken into two congruences by factoring the modules. Yes. And then we have to show that that's been wrote to both of those. You have to show that A to the fifth is congruent to A, mod two and A to the fifth is congruent to A, mod five. Those are the two things you have to show. I apologize. No, it's okay. That's the idea, yeah. And mod two is easy because A squared is congruent to A, mod two by the correlator for mass theorem. And again, we remember when A squared is the same as A, then A to any power is the same as A. So, that's easy. The other part is just for mass theorem. That's easy. You're done. So, there's not a whole lot going on here.  Okay. Is this okay? Any questions? No? Okay. I'm going to talk about one more, and then I'm going to, I think I'm going to leave the rest to you to finish. But again, since this, you're not going to have solutions to this before the final, I thought it would be good to go through a few more than I normally would. So, 10A says that you're trying to prove something. So, it says to prove the following thing. Prove if A to the p is congruent to B to the p, mod p, then A is congruent to b, mod p. Okay. Sorry, this thing is, I've got to straighten this out. Okay. So, this is kind of interesting. It says that you can cancel the p out in some sense. P's a prime, of course, in this case. So, this is really all it boils down to. There's not a whole lot going on here. You are not, some of you might think that you want to just use the definition and go back to the definition of the congruence and say, okay, p divides A to the p minus B to the p, then p divides A minus B. Just do it directly, or try to do it just from the definition of congruence. You don't want to do that. You want to use the stuff that you have at your disposal from this section to do this. And so, what you want to note is that A to the p is congruent to A, right? Mod p. And B to the p is congruent to B, mod p. And this is just from Pramod's theorem, right? Prove that if A to the p is congruent to B to the p, mod p, then A is congruent to B mod p. So, and this is all there is to it. And you're invoking some theorems, an earlier theorem on congruences that basically say you can kind of change the order around certain ways. But A is congruent to A to the p, right? That's a property we know. We can flip the order. And by assumption, A to the p is congruent to B to the p, right? That's our assumption in the problem. We know that B to the p, over on the right-hand side, this is congruent to B mod p. And there you go. You're done. Use the transitive property of congruence and you get that A is congruent to B mod p. I have a question. Yes? What's wrong, but wasn't there also something in the theorem that you gave us that said A to the k is congruent to B to the k mod p? You could raise it to a power. You could raise both of them to a power. Yes. So couldn't you just use that? No. You're confusing with the converse. If A is congruent to B mod n, then A to the k is congruent to B to the k mod n. But it does not say that if A to the k is congruent to B to the k mod n, then A is congruent to B mod n. It's an if-then statement. It's not an if-and-only-if statement. Right? In fact, there was a question on the third exam that said true or false. If A squared is congruent to B squared mod n, then A is congruent to B mod n. False. Not true. Choose A to B4 and B to B2. Doesn't work. Yes? Right. No, no, I think that's just for the other part of the problem. Here you don't need it. You don't need it in this part. Because, yeah, what we used, A to the p can congruent to A and B to the p can congruent to B are true for any values of A and B. But the last part, I'm guessing, is where that comes into play. Probably. Okay. So really, you've got to, I think I've showed you all the tools, really, that you need to finish up the rest of the homework here. So I'm not going to do the rest of these for you. But it's really, these problems are not that bad once you just see what it is you're supposed to do. Okay. So, anyone want to ask any questions about some earlier stuff or would like to see a problem from, I don't know, maybe an induction problem or something like that. I mean, this is now, this is up to you guys now. Now, I've done my piece here. If you want to ask about stuff, feel free to. Yeah. Right. Yeah, I see what you're saying. I don't think that that is even going, I know what you're worried about here, but I don't think that's even going to come up on the exam. So the problems that you're going to have to use, I mean, really, these are just things that are going to follow from, you know, basic theorems in the section and such. And so, if there's any possible confusion, I'll make some note about it in the directions. But yeah, I know what you're asking, but it's not really going to come up. Yes. Do you just do, I don't know, like chapter one to number 10? It's an induction problem. Okay. Or less than what you're looking for. Okay. Let's see. What is the homework problem? Chapter one, number 10. Okay. Is it 1.1? Yeah, okay. Okay. How many pages is that? Yeah, I see it. Okay. So this is, yeah, page seven of your, your text. Okay. 10A. Yeah. So there are several different kinds of induction problems. You should be familiar with all of them, really. Okay. So it's less than or equal to 2 minus 1 over N. All right. Okay. Well, this is actually a good one, because this, what I'm going to do when I write the solution out here is, this is not going to be totally polished, but I'm going to do this in the way that I think your, the train of thought should sort of progress as you go through this problem. Okay. You're trying to prove this is true for every positive integer N. So remember what you want to do is, there are two steps that you need to follow here. So the first thing you need to, to establish is so-called base case, right? And that is you want to prove that, or verify that if you will. 1 over 1 squared is less than or equal to 2 minus 1 over 1. Right? That's the base case. N is 1, so there's only one term on the left-hand side, and so it just reduces to this. And what is this? This is just 1 is less than or equal to 1, which is, which is true. Okay. So this, this is really just, this first part really is obvious. And the second is the inductive step. Okay. The inductive step then is when you assume that whatever it is you're trying to prove is true for some positive integer N and you want to establish the same assertion for N plus 1. Right? Assume for some natural number N that 1 over 1 squared plus 1 over 2 squared plus on down to 1 over N squared is less than or equal to 2 minus 1 over N. Right? And you want to prove that this same inequality is true with N replaced with N plus 1, essentially. Right? So let's just, this is our inductive hypothesis. That's what this is called. Put a little asterisk next to this. And what we have to prove, this is what we're allowed to assume. What we have to prove then, yes. Well you never write assumed for every N and N because then you've assumed exactly what it is you're trying to prove. You never would write assumed for every N and N that this is true. Because it's like saying, prove for Maas last theorem. I'm assuming for Maas last theorem, therefore it's true. You don't do that. Okay. So what we have to prove is we just keep going. Right? Keep going. And instead of stopping at N, we stop at N plus 1 instead. Right? 1 over N plus 1 squared. And we have to prove that this is less than or equal to 2 minus 1 over N plus 1. Now the idea behind induction is that you want to take your inductive hypothesis and you want to mess with it until you get what it is you're trying to prove. Okay? So here's the idea really. What you're assuming for some N, this is your starting point and you want to get here. You don't want to start and some of you are doing this before. You're starting with what you want to prove and then doing some stuff to that. No, that's not right really. You start with what you assume and you get to where you need to go. Right? If you start with what you're trying to prove then you've tacitly assumed what it is you're trying to prove to be true. If you're working with it, maybe it's not true, maybe it doesn't even exist. That's what you have to establish. You can't get your hands on what you want to prove until you start with the inductive hypothesis and get there through logical steps. Okay? That's the thing I want to stress about induction. Do not mess with what you're trying to prove. Start with what you assume to get there. Okay? So that's the biggest slip-up I've seen when we were doing this before. So... One minus one. Yeah. Mm-hmm. You said that we were essentially trying to... Sort of. I mean, that's not exactly right, but... Okay, so... You're not... I mean, if you were just literally replacing in with n plus one, then you wouldn't have an n. Then you would have skipped from n minus one to n plus one. So I didn't really mean it that way. I just mean the statement, which is going up to n, instead go up to n plus one. So why are you adding an extra term on the left side if you don't have an extra term on the right side? Because... Okay. Look at the original statement here. Here's our original statement that we're trying to prove. So... If I were to call this inequality p of n. p of n. Well... Notice... Okay, this is probably a better way to say it. This guy right here and this guy right here are the same. So if I'm going to n plus one, it still has to have the same form. So this guy right here and this guy right here are going to have to be the same. I mean, imagine starting... Just imagine that n was four, right? Then the assertion would be one over one squared plus one over two squared plus one over three squared plus one over four squared is less than or equal to two minus one over four, right? So it's... Wherever you end up is less than or equal to two minus one over wherever you end it up. That's exactly what the assertion is saying. Okay? So what do we do? Well, we need to get to this point and this has a one over n plus one squared that this doesn't have. So why don't we just add one over n plus one squared to both sides of this, right? I'm not going to write that down, but that's what I'm doing right now. So I just took our inequality that we're starting with and I added one over n plus one squared to both sides of it. That's where I got... That's how I got from here down to here, okay? Well... Now this is why I said I'm going to do this a little bit more informally. What is it that we'd really like to be true in order to finish the proof here? Here, and this is where... I want you to listen to this for a second. Okay? This is important. This is what we have right now. So what we're trying to prove, we have the left-hand side of the inequality right here, but we do not have the right-hand side. This is different than this, okay? So what I would really like to show now, if possible, and think about this for a second, think about it. I really would like to show that whatever this is, is less than or equal to this. Because if I can do that, then think about it. Then we've got what we want on the left-hand side, less than or equal to this, and if I can show that this is less than or equal to this, then by transitivity, this is less than or equal to that. That's exactly what I want to show. That should be your thought process here at this point. So now your goal should be to prove that two minus one over n plus one over n plus one squared is less than or equal to two minus one over n plus one. Yes? Is it redundant or equal if you just say less than because there's no way those terms would be equals? Yeah, I mean, well, I mean, you're not really gaining anything by this, really. In fact, maybe they are the same because you can cancel out the twos, and it's not it. I mean, they're not, but let me put it this way. You're not gaining anything by doing that. You might just end up being wrong. So since you only have to prove less than or equal to anyways, why not just leave it there? You see what I'm saying? Okay. Okay, so this is now what you have to show is this. And now here's where this is what you would do now. This is what you would do on scratch paper, and then you would work your argument backwards in your solution. Okay? Well, what is this equivalent to? Well, the twos, we can certainly cancel the twos on both sides, right? So this is really equivalent to just minus one over n plus one over n plus one squared less than or equal to minus one over n plus one, right? And from here, basically what you're going to do is you're just going to keep sort of messing with this until it gets down to something that is obviously true, and then you work your argument backwards. Okay? Okay, well, maybe we don't want these negatives. So we can rewrite this as... Okay, sorry. I'm running out of room here. One over n plus one plus one over n plus one squared less than or equal to one over n. Okay, sorry, this is getting really... But you see what I did? I just brought the one over n plus one over to this side, the one over n I brought over to the other side. Well, that's not really going to get you anything here because the reciprocal of sum of two reciprocals, you can't just cancel the reciprocal out. One over one over a plus one over b is not a plus b. Okay, I think that's what you're saying. Just take the reciprocal and then I get n plus one plus n plus one squared less than or equal to n or something like that, but that's not the reciprocal when you have two fractions. No, no. So, I mean, for example, one is less than or equal to two. If you take the reciprocals, then you get... Then is one over one less than or equal to one half? No, it's not. Reciprocals in general are only going to flip things, but when you've got sums and things like this, yeah, you can't just take reciprocals and just hope things kind of work out. They generally don't. Okay, so what can we do with this? I don't know if you guys want me to finish this or not, I don't know. So, what you could do here is combine these into one fraction, right? Get a common denominator, just by multiplying the top and bottom of this guy by n plus one, by one over n plus one by n plus one, and then you can cross multiply from there. Okay, I'll go ahead and finish it since we're so close anyways. Okay, so what we had was, sorry, one over n plus one plus one over n plus one squared. What was it? Less than or equal to... What do we have? One over n, okay. Again, this is what we're ultimately going to try to prove this. Well, if we get a common denominator, this is the same thing as n plus one plus one over n plus one squared, less than or equal to one over n. Okay, see what I did here? I just multiplied the first fraction top and bottom by n plus one. Okay, and now let's just clear the fractions here. Now let's multiply everything through by n times n plus one squared, both sides. We can do that. It's positive. We don't have to worry about flipping inequality signs. Then what do you get? Well, the n plus one squared is going to cancel, and then you're still going to have an n here, and then it'll be n plus one plus one, right? Less than or equal to... On the other side, it'll be n plus one squared left over, because the n will cancel with the other n. Okay, so now you can see how this is going in your mind. Just keep going with this. n squared, okay, so this just becomes n plus two. So this is n squared plus two n, less than or equal to n squared plus two n plus one. See that? That is definitely true. How would a really good proof proceed then? Okay, here. Let me just... I want you guys to keep in mind this is important. I don't want to make sure you didn't forget this. I'm not saying that this is the right way to establish the proof. I'm saying this is what you do on scratch paper, and you say, I got to prove this. Well, this is equivalent. I mean, actually, in this case, you could do this, because these are all if and only ifs. In general, you may not be able to do that. But, well, what's this equivalent to? This really is equivalent to the following simple assertion, zero, less than or equal to one. And that is definitely true. So how would a nice proof proceed through this? We would start here, and we'd go backwards. Now, in this case, and I'm being a little more liberal now with the if and only if arrows, if you want to establish something, and everything you have is literally equivalent to what you're writing down after it, you can just do if and only, if and only, if and only, if and only, if and only. And get down to the end like this. That's fine. That's okay. So the proof really hinged on simply the fact that zero is less than or equal to one. That's it. Okay. Well, this one is a little trickier, maybe than some of the other ones were, but most, I would say, the majority of the induction problems are probably not quite this messy. Other questions?  I was just trying to see one that maybe wouldn't be as visible. Anything from the binomial? Okay. Binomial theorem section? I know. Okay. Okay. Sure. Let's see here. Okay. So I'll just remind you of this real quick. And this is the form the book gives it in. Okay. So yeah, just due to time here, I'm probably not going to do a long problem. I'm going to remind you of the binomial theorem and then kind of the thought process. I'm probably going to do a fairly short one this time. Okay. So binomial theorem says that a plus b to the n is equal to the sum from zero to n of n choose k a to the n minus k b to the k. Okay. So yes. Each term is just that n choose k. That's right. Yeah. I remember because I lost. Yeah. Yeah. That's right. Okay. 3D says to show that this might jog your memory. I may have even done this one for you before. I don't remember. But n choose zero plus... Oh, no. Let me see. Is that what I want to do? No. Sorry. Let's do 3B instead. Okay. Sorry. So minus n choose 1 plus n choose 2. This is really messy. Sorry. Minus. And then you just keep on going. And at the very end, it's plus minus 1 to the n times n choose n. And the point is that this is equal to zero. I guess. Right? Yes. And the point is to prove this by using the binomial theorem. Okay. Well, here's what you do. Here's the idea. Now, this one actually happens to be maybe a little easier than some of the other ones. But the idea here is look at your sum and all you're doing here is you're just trying to choose a and b appropriately so that when you expand out a plus b to the n using the binomial theorem, you get exactly that. Okay. And so, first of all, whatever your expansion is, this needs to be equal to zero. And remember, if you look at the binomial theorem, the coefficients of all these powers are of the form n choose k. Right? And all of these coefficients here are either 1 or minus 1. They're all either 1 or minus 1. Okay. So, yeah. So then, sorry. That's not what I meant to say. I mean, the coefficients actually go up there. I mean, we see these n choose k, these binomial coefficients coming up throughout the expression here. It's just that they also have either a 1 or a minus 1 on the outside. Okay. Any n features? No. I mean, if you look here, look at the n choose k and the binomial theorem, we have all of these things here except the signs are alternating. That's all I'm saying. The signs are alternating. Right? If we want this to be equal to zero, then we want, if we're going to use the binomial theorem and want everything to collapse to zero, then a plus b should be zero. Okay? And notice that we also don't have any a's or b's or anything. It's always just either 1 or minus 1. So the most natural thing to try first, then, is just to choose, for example, a to b1 and b to b minus 1. And then expand it out using the binomial theorem. And then just follow the reasoning here. Zero is certainly, whatever n is, zero is certainly 1 plus minus 1 to the n. You should all buy that. It doesn't matter what n is, right? 1 plus minus 1 is zero. Zero to any power of zero. And now this is where we use the binomial theorem. And now we're just, I'm just going to write all the terms out as we go up. Where do we start in the binomial theorem? Well, k equals zero, right? So it's n choose zero, 1 to the n minus zero times minus 1 to the zero. And what is, I'm not going to write that all down, but if you just follow along here, all you're doing is you're now using the right-hand side of the binomial theorem. The sum from zero to n, n choose k, a to the n minus k, b to the k. Well, we start with zero. a is 1 and b is minus 1. So this, since a and b are both 1 or minus 1, this product here has to be either 1 or minus 1. It has to be, always, okay? And so we're just going to be left with, in this case, because we're raising minus 1 to the zero, it's 1. So the coefficient is just 1 and choose zero. And that's exactly the first term that we want. What happens if we plug in 1 for k? Well, again, the a part, look at the binomial theorem. It's 1. It's always going to be 1. Don't think about that. If we plug in 1 for k for the b term, since b is minus 1, we're going to get minus 1 to the first, right? Which is minus 1. So then the second term will just be minus and choose 1. The minus comes from the fact that minus 1 is being raised to the first power. And then there's just, we just keep doing it, right? And this just continues all the way down to the end. And that's it. So all you're doing, once you identify a and b, you just plug them into the theorem and just write the expression out. Instead of the sigma notation, just write it out. And that's it. Okay, I think I probably have to stop because I have to hand your homework back as well before you go. Okay, so what I've decided to do is this, this homework. Man, I've been really nice to you guys lately because I have all this trauma occurred to me and I haven't had time to do my job as well as I should have. So I gave you a completion grade on this one as well. Okay? Now, before the final, you will have a salute. Some of these, I'm going to pick a few of these to write up solutions to. So you'll still have, so let me say it this way. There was another assignment I gave you a completion grade for, although online I wrote up solutions to three problems. Those are fair game in the graded homework portion of the exam. Okay, even though it wasn't graded, that's still fair game. All of these things that you see online that I have solutions for, those are fair game. Okay? All right. Thank you for joining us.