I'm Martian Joyner and we are navigating the journey. Today's journey is, it is a sad and it is a glorious journey. August 6, 8 and 9, 1945 are auspicious dates to include in your memory bank for you to hold on to, to remember. In the closing days of World War II, the world changed forever. August 6, the Americans dropped the bomb on Hiroshima. August 8, President Harry Truman signed the United Nations Charter, making America the first nation to complete the ratification process of the UN. And before the ink was dry, August 9, another exceptional day that we must remember was that the United States dropped a plutonium bomb devastating the city of Nagasaki. The Soviet Union declared war on Japan and thus the Cold War began. So, to talk about all of this and why we are commemorating August 9, is my dear, dear friend, Joanne Tachibana. Joanne is the President of the United Nations Association of Hawaii Chapter. Good morning, Joanne. Good morning, Martia. Thank you for the wonderful introduction and the overview of our commemorative celebrations. Together, more than a decade plus, it's been my privilege and pleasure to work with Martia to commemorate the often forgotten Nagasaki bombing on August 9. The community always celebrates August 6 with our sister city relationship and then they forget that the Americans bombed Nagasaki. And in between that bombing, August 8, which the day that the United Nations Association crossed the world celebrates, is when President Truman did sign that United Nations Charter. We say, wow, we've been World War I, World War II, and now we're going to have peace. We're going to gather all these wonderful nations and say, I want to work for peace, right? And this is the next stage, you know, with a bomb Nagasaki. So it is indeed a very troubling time and a significant moment. And so I have been a wonderful peace partner with Martia and we have constantly worked to bring attention to the second bombing of Nagasaki and this most significant time. And, you know, this year, we are so privileged because we have been, we are working with some wonderful peace partners and the Myokonomiya Hoji Temple on 2003, the One Avenue is where we're having our commemorative celebration on August 9 at 9 at 10 a.m. So we're hoping you can join us and we have some wonderful projects throughout the day. We have interstate members coming. And I guess Baker is going to be Dr. Maya Satorin and who is the consultant for the Obama Foundation. We'll be joined by the Council General of Japan and the Nagasaki Kenjin Kai, which is the group with Kibakusha, the people that survived the bombing of Nagasaki. And we have a wonderful program which Martia and I, our peace partner, Seeds for Peace, the Bahais, the Royal Wine Chapter and Nagasaki Kenjin Kai. And we have worked together to create this wonderful program for the public which everybody's invited to. And, you know, we are really pleased because the Royal Hawaiian Band has annually joined us to make this such a wonderful program. So, that's some highlights of that program on Friday, August 9. And, Joanne and I started this 1990, was it? I think that was 1990. It seems it's been a wonderful piece to you. When the Hibakusha of Nagasaki gave the bell to Honolulu, Stalingrad, which is now St. Petersburg, and Manchuria, the three cities that the Japanese bombed. And they gave us this lovely, lovely bell which is located in a beautiful, what was a little beautiful valley, reminiscent of Nagasaki. But now they built all kind of things around it so you lose the idea that it was a beautiful little valley. But they gave that bell to us and money to maintain it. So, from the time that bell was dedicated until now, that's been 1990, we might as well, if Frank Fosse was mayor and he did not like the idea of anything from Japan, he was still fighting the war. He did not want anything. So, he put that bell in storage, as well as the Hiroshima bell in storage. So, there they lie. It was a major fight to get them out. And then finally, City Council, don't ask me, oh, what was it? So, it was a long fight, a hard fight, but it was worth it. It's been worth every bit of it. Absolutely. And don't forget to mention that the city of Nagasaki was the first city in the world to have a bell that was dedicated to the city of Nagasaki. So, the city of Nagasaki was the first city in the world to have a bell that was dedicated to the city of Nagasaki. And don't forget to mention that the city of Nagasaki was very grateful to you and Val and how you got that award and that you have that symbol, right? Maybe talk a little bit about that. That's a really important part of their history. Yes. And I was given, like with Val and his wife, Francis, the Nagasaki Peace Prize, which was fabulous. And I have a replica of the bell that's absolutely darling. Yes. But the big prize was an all-expense trip to Nagasaki for both my husband and me. First-class air accommodations. The hotel was fabulous. It was, don't ask me the name of the hotel, but it was a French hotel. Okay. With the Japanese design, the Japanese interior, that culture where every little detail is taken care of, every day. Right. They paid for our food, our transportation, and to be at N Nagasaki on August 9th at that moment is unlike anything I have ever experienced. The bell ringing? When they, you know, on August 9th, to be there, to walk around the circles around the, what do you call it, round zero, and to hear the bells, Nagasaki bells ringing throughout the city.   There's nothing else you can say about that. Yeah. Yeah. And so that having that experience is what allows me to keep going, to honor the people of Nagasaki, to thank the people, the Hibak people, for what they've done for us. We can't imagine. Seven thousand degrees is what the temperature was when they dropped the bomb. Seven thousand degrees. And we're talking about 90 degrees of talk. You know, how dare we. Absolutely, absolutely. Yeah. Well, we really appreciate the fact that we, you know, that you and Val and Fran brought this back to Hawaii, and that we have been able to commemorate with our interfaith clergy and peace partners for this number of these past few decades. And I think as we work toward the 2020, which is the 75th anniversary of Nagasaki bombing, Hiroshima and the United Nations, you know, we were making some wonderful plans to celebrate. And the United Nations Association for Hawaii has initiated a project the last number of years to make people aware of the sunflower and its significance. And the sunflowers became a symbol of peace and nuclear disarmament when the Defense Ministers in Russia and the Ukraine met on a missile base in 1996. They were celebrated by scattering sunflower seeds and planting sunflowers everywhere, because under those to me before I knew this, that sunflowers remove toxins, including nuclear waste, soil and water. So this is, so we have brought sunflowers and awareness and seeds to, we have children youth day for the last 20 years. And on the first Sunday of the month, in Hawaii, the first to have commemorative days for children. And so we have to bring awareness. And now it's been for the Nagasaki ceremony. We are initiating awareness of the sunflowers and passing out seeds. And we worked with, we have a pleasure of Ceria Kumar, which it has his farm in Waianae. There's been a plot that there will be sunflowers looming for a commemorative celebration in 2020. And we're looking forward to actually having many plots of sunflowers all over the island and perhaps the state of Hawaii so that people will constantly remind them stop nuclear war and please work to make the world a better place and peacefulness. So for 2020. I have a suggestion. Okay. If we can be on good terms with the mayor. Mm-hmm. Mm-hmm. Yes. Let's do that. Let's do this. Okay. Now, what used to be Barber's Point was left in a mess, an absolute mess when the Navy moved out. Oh, the Kalalua area? So there is a plot of land that belongs to the city and it is earmarked for a part. The city has done nothing. Nothing. Okay. Okay. So let's see if we can get that plot of land to plant the sunflower seeds. Yeah. Yeah. Maybe because if they do a parking in a sunflower garden as part of that, that would be, you know, that would be symbolic. It would be wonderful. Let's work on that. Yes. Because if it, you know, it took them 20 years to get this little plot of land on Aloha's Drive in Waikiki. Mm-hmm. Mm-hmm. So who knows how, so if we start now, we might get this, you know. Yeah. I think that, you know, this is wonderful because new people are joining us and become aware. And then recently, the lemon tree project, which is the eco-op sustainability date, they plant lemon trees in schools and 75 schools of lemon trees get up carbon dioxide. And so we met and then so, we made a donation so that they will plant sunflowers around the lemon trees. So we're going to try to raise funds so that, you know, sustainable projects in schools, you know, the students, they have a curriculum and everything and sunflowers would be an added touch. So we're really looking forward to, you know, people would say, wow, that's so beautiful. You know, the sunflowers and Wailua recently write that. Oh, it's so gorgeous. Yes. Yeah. Yeah. So we're going to contact them to see if they would be a sustainable partner, you know. So they have the know-how. It's so exciting. Yeah, they have the know-how. Yeah. It's really exciting. Absolutely so exciting. So maybe with Ikeika being the chair of the council. Yes. And the mayor, both of whom term out next year. Maybe this could be their last, this, their going away present. Yeah, absolutely.  And then so I'm hoping, looking forward to the Council Chair Anderson will be able to even join us on this August 9th, you know. And I am so excited. I think we work so hard in this more awareness in the last year, this year, right? So I really, I'm so grateful that we have continued this work. Well, listen, Joanne, we need to take a break and we will be back in 60 seconds. Okay, sure. And let's talk about where we go from here. Okay? Okay. Thank you. Alrighty. Aloha. I'm Stan Osterman. Stan the energy man. Every Friday here on Think Tech Hawaii. If you're really interested in finding out what's going on in energy, especially here in Hawaii, but also all the way around the world and especially if it has to do with hydrogen. Look into Stan the energy man every Friday, 12 o'clock, Think Tech Hawaii. Be there. Aloha. Aloha. I'm Keisha King, host of At the Crossroads where we have conversations that are real and relevant. We have spoken with community leaders from right here locally in Hawaii and all around the world. Won't you join us on ThinkTechHawaii.com or on YouTube on the Think Tech Hawaii channel. Our conversations are real, relevant, and lots of fun. I'll see you at the Crossroads. Aloha. Aloha. I'm Marcia Joyner and we are back. We are navigating the journey. The journey through the bombing Nagasaki. And needless to say over the years I have learned so much, Joanne and I. Joanne, I guess. The one thing that I just discovered but you told me more about it was the Constitution that was written after the war and first by the Japanese and then of course the American woman that worked for MacArthur really wrote the final draft which I think was just a fabulous document. Because it replaced the Menji Constitution of 1889. Can you believe that? So what do you know about it? Right. Well, it was so significant and I know the Americans really encouraged the Japanese to make sure that there would be a normal war. They were so aggressive right there brutal to the countries they go to, you know, China and Burma and everything, right? So, I think they would never again raise arms against any other nation, right? So, I really felt that it was very significant that the Constitution that would renounce war and exercise take out the concept of war from all people. And so they started the reign of peace, right? And May 3, 1947 the Japanese post-war Constitution went into effect. Progressive Constitution granted universal suffrage, which is like amazing, right? Not doing it in places women voting in America was so hard. People don't realize how many women do not have the opportunity to vote. It took a long time for many, I apologize, third world countries to even accept that. Some women should be able to vote, right? And at the same time, the Constitution took away the powers that emperor Hirohito became a very symbolic rather than like the sun god but he became a symbolic power for the people to have an emblem, you know? And they had a bill of right. It's a abolished period. Outlawed Japanese right to make war, okay? Which is like really significant. Article 9, forbade the Japanese to wage war again, right? And there was even like political parties that base the premise on peace, right? So it was, you know, when you look back at the history, it was so significant, right? And the most important reform carried out with the American occupation was the establishment of the new Constitution. And a gentleman like Arthur, who really worked hard to rebuild Japan after the devastation of the bombings even in Tokyo Hiroshima and Nagasaki, right? He gave up much of his authority to the Japanese government, which indicates a lot of where, what role he played in making, imparting the Japanese back after being so, so devastated, you know? And then in September 1951, the United States and 48 other nations signed a formal peace treaty with Japan. April 28, 1952 the treaty went into effect in Japan so full sovereignty as the Allied occupation came to an end. And in general, my car there was a visionary and a peacemaker on the nose to so many. And I was very touched by reading the history of his contribution to not have the emperor was executed as a symbol of, you know, destroy Japan after him. Yes. And I have to tell you though that my husband was in the occupation of Japan. Oh, wow. He was quite young at the time. Early stages of the early stages of being in the Navy. Yeah. Yes. Very early. However, when we went back to as guests of the Japanese of Nagasaki he danced with he turned into that sailor again. He sang the Japanese songs. He danced with the all the ladies. It was unbelievable how he rolled back all those years and because he spoke so highly of being in Japan in 1952 you know and in the occupation and to see it then see the rubble and then to see it when we went back. It was unbelievable. Well, you know it. So watch this big guy turn into this young sailor. Yes. Peace is not so easy to accomplish but I think that Americans under MacArthur really has become a nation of friendship. Right. And then the United Nations of the UNESCO says that since war begins in the minds of men it is in the minds of men that the sense of peace must be constructed and that's a message that we really try to empower young people to be where you know who well when we think of this volunteer military that we have now and it's only one percent of the Americans that are that volunteer so you know while America loves war you watch all of the stuff that's on television and the war games they play and all that but when it comes to real fighting real war you see how they have pulled back from that I think. I hope. We hope. But listen Joanne tell us more about what we can expect on August 9th. First of all where is the temple it's such a beautiful absolutely gorgeous temple where is it? Okay they actually have a piece of temple they're located at the New Iwanu it's 2003 New Iwanu Avenue and Reverend Masayama Laura is a humble and Buddhist priest with a very peace making heart and you know I denounce the meaning of a proper singer and that's how we met him as he's signed up in Maria which he will be doing a Buddhist prayer for us and he will also be raising us up in Maria as part of our special program on August 9th this year this year and yes yes yes and so we hope you can join us as you mentioned earlier Dr. Marisa Taurang and we'll be welcoming a precious audience to be with a significant ceremony and we'll be joined by the Interstate clergy that we have the Hontakonganji and we have the Bahay and we have a Christian message from the St. Andrew Cathedral we'll also have we're looking we've invited from the Muslim community we and have peace makers from the United Nations Association we're still looking forward to having people be more aware of these commemorations because it's a part of history and how we can project for the future to normal wars and no more tunnel bombs that's kind of the message we want young people to be engaged in in the future right well we want also as we talk about the bells of Nagasaki and one of the things I'd like to leave with our audience and is that the bells were the Angelus which of course if you're Catholic you understand the bells are wrong three times a day and they were in the cathedral at the time of the bombing the bells were ringing and they were bombed and one bell remained and the story goes that the theory is that the bells should ring every day until the end of time so we tried very hard to ring bells until the end of time or at least our time so tell us tell us Joanne how how can we how can everybody participate it's open to the public and can they call you can they call you absolutely the number yes they can more information we'd be so delighted to hear from you yes my number I'm going to listen to the<PHONE_NUMBER> okay say it again<PHONE_NUMBER>87 7271 come on come on join a peace making celebration thank you I think we have less than a minute left but there's a gentleman that is filming because he's telling the story from Hiroshima and Nagasaki Nagasaki yes we're making a film called Sakura and Pearl and he will be interviewing Nagasaki Kinjin Kai members to and he will also be preparing that film documentary in December I'm guessing around December 7th around the commemoration of Pearl Harbor so thank you for the reminder because he will be there to introduce this part of the documentary film thank you so much Joanne and as always my love to you and thank you audience and we'll see you next time thank you


Iowa Governor Kim Reynolds endorsed Florida Governor Ron DeSantis for president. The endorsement could give DeSantis campaign a boost as he struggles to show progress against Donald Trump and the Republican primary.DeSantis has pinned his chances of emerging as an alternative to Trump alternative squarely on Iowa-E Reynolds is breaking with longstanding Iowa tradition to endorse him. The state's governor typically stays neutral before the caucuses that kick off the GOP nomination calendar, she announced. The endorsement at a Monday rally in Des Moines.


Hi everybody. This is Fabrizio Salzano, presenting for Product School. Let me start introducing myself quickly. I'm clearly an Italian professional from Accent, a computer scientist by background, and a passionate e-commerce in a bit testing. I also became recently proud that he too. Let me skip quickly to what is my experience. I'm a travel professional for more than 16 years, currently working at boogie.com. Prior to that, I've been part of the Expedia company. I run more than 500 bit tests, and that's why today we're going to talk about why you can always a bit test. Or as I say, in the line, a quick guide to a bit testing and more. Now let me start saying that this will be a very practical presentation for those looking for an intro to a bit testing, and it's possible alternatives if a bit testing is really not on the table. Now, everything starts with a definition or common sense based, as I say, definition of a bit testing. A bit testing is, to the core, a quick way to check in with your users to see what they really think about a feature or a change that you come up with. Why a bit testing? There are at least three reasons I can think of. The first one is that nobody can predict the future and not even your boss. The second is that is a good way to democratize learning and decisions. What I mean here is that giving exposure of how users react to your change enables everyone, regardless of their pay grade, of making a decision about the data. And the third is also enables you to have a progressive learning. Given the nature of a bit testing and the small steps approach. Let me tell you what are the building blocks of a bit testing. And so there are at least five main building blocks. There is an hypothesis, KPIs, type of test, the setup and the runtime or sample size. Let's go through them one by one. Starting from the hypothesis. The hypothesis is composed of four elements. What do you want to change? You can say those are the features that you are testing. Who is exposed to the change, your users, or user segment more specifically? What is the effect on the user that you expect or behavioral change? Last, not least, how do you measure the impact of the change? Typically KPIs. Let me give you an example. So let's say that you are adding a shopping cart on every page. Your user is all the returning customers. And the behavioral change will be for them to find with more ease the visitor than bookmarked properties. KPIs that probably you want to test is an increase in bookers, multi destination bookings, the cart interactions. This is what actually hypothesis look like one way. Second is KPIs. You can characterize KPIs in three buckets. Primary is actually the one that you will help you in making a decision. Second is supporting.  And last, not least, health metrics. Usually those are the metrics related to the system. Now here, the first step. It is important to define, in my opinion, for every experiment that you run, all of the above. So primary, secondary, and health metrics. And try to agree and keep that agreement and this consistency within your organization. So do not keep changing your KPIs every time unless it's strictly necessary. This will make your strategy much, much stronger. Let me give you an example. Again, the case of the cart. The primary, again, business metric will be net bookers. The secondary or supporting is the amount of users accessing the cart. So that's a behavioral metric. And the health metrics in this case, let's say those are system metrics, what is the uptime or the page load. Let's go on the third block, which is the topology of tests. This is a bit of a technical topic, but what is really important for you to know is there are three types of tests. A superiority test, which means that you expect your change to be better than what currently your customers are exposed to. A non-inferiority test, which means that your change will not be worse than what you're offering to the customers. And sometimes you also define by how much will not be worse. So what is the acceptable cost? And for your knowledge, it's also the third typology is equivalence test. An equivalence test is when it says it's not better, but it's not worse. It's really not as frequent to run this type of test. Now, the tip of this section is the following. Do choose a non-inferiority test for those changes need to happen. So think in terms of strategic value changes. In technology, there is a lot of re-platforming. So if the platform needs to happen, there is no point in hoping for a superiority test. Any non-inferiority test will do it. Otherwise, always go for a superiority test. The reason being that I don't imagine why you shouldn't really offer a better experience, a better product every time, over time to your users. Number four, test setup. Here's a bit of the bike of the test. Here you can define how many variations of your change you want to offer to the customers. Which court of users you're tracking, et cetera, et cetera. A couple of tips here. First of all, be aware of false positives. Now, a false positive means that you see an impact where actually there is none. So your metrics turn out to be conclusive in one direction. In the positive direction, I thought it's a false. It's not true. Now, this is much more common if you keep running consecutive tests, keeping base, not changing base and keeping testing small variation of whatever is your feature. So it's much better to pack all the variations in one experiment and run it if you can if you have the power in as a one single experiment. If you don't have the power to do this kind of experiments with many variations, then choose one variation stick to it. The second thing to design the test for decision making. And not only based on what you can do. So basically, what this means is that based on the data that your product test will produce, you should be able to really know what to do next. Now, last, not least is runtime and simple size. It's important to really calculate how many users will need to be exposed to your change in order to detect a significant impact on your main KPIs. The tip here is to always, always, always calculate the minimum sample size. It is not true that more data is always the better. And there is plenty of online calculators to do that. Now, there are some instance where actually you cannot be test. You can think of you are introducing too much friction to your users. Or let me put this differently. There are some instances where you should not be test. So first case is you are introducing too much friction to your users. Think of those B2B users. In our case, we, we know it works in the accommodation business. I season, our partners usually in summertime are very busy with the operational efforts, you could say, destructing them or disrupting them with a bit testing, usually not a good, a good idea. Another reason not to be test is when to test for the sake of testing. So you don't have a clear hypothesis or no idea on how to use the test results back to designing tests for decision making. And the last is for moral reasons, right? You should not be test testing a dark pattern. Think of the situation of hiding valuable information from your users. But I would say you should never apply this change in general, not only by a bit testing. Now, what if you actually you cannot be tested instead. And I can think of a few reasons. The most obvious. You lack the testing platform or the test itself is too expensive. Sometimes testing some features can be really expensive for business so they cannot afford it. The second most common is really when you don't have enough users as small sample size. The implication of having a small sample size means that your test might take too long months and maybe to reach conclusions maybe never. The third, you don't have the product yet. So you have nothing really to offer. And last, but it is very like uncommon is actually that the change you have in mind is really not good fit for the testing methodology. Now, what are you going to do if you fall in any of these buckets or other reasons that push you to not really use a bit testing. Let me introduce you to the pyramid of evidence. What the pyramid of evidence refers to is the fact that the base on the type of methodology that you choose to produce an evidence, you can actually stack them up. And each layer will be qualitatively better than the layer before. So, for instance, we know that randomized test or a bit testing is usually superior to an expert opinion, but this is kind of intuitive. Now, another takeaway from the pyramid of evidence is that not only we know that there is an increasing quality of each methodology. But there also there is usually in my experience any case in cost. Now, if you work in the tech industry, now, gathering an expert opinion, it costs usually nothing. Running a randomized test usually cost money, or at least the investment in the bit testing infrastructure, you know, the time of the developer. Let me then walk you through each level to give you some alternatives in case you don't have a bit testing. Let's not discuss meta reviews because meta reviews is actually reviewing a set of randomized tests. Very few companies can afford to do that. This means having a lot of a bit testing of the same typology around the same topic. So we will not I will not comment about that. So let's start from randomized tests. So I be testing what we talked about it. These were actually booking.com sits and is actually champion of a few things to keep in mind while I think about the bit testing. And also you would say best practices. It is good idea to always test a single change at the time. The reason for that is because if you test multiple things at the time you don't really know what is the contribution of each change to the KPS that you're measuring. The second is again, good practice to focus on a fast delivery means like roll out the test quickly and make the test again small so that actually you can learn fast and then reiterate on the learnings from the test. And this is actually also introduced the third tip iterate continuously keep iterating to keep improving your products, but beware of false positive. We mentioned what false positive is, and they're way more commonly you think. Now, slightly less qualitatively than the randomized test, there is a court research. Now, what is a court research. A court research means that we are investigating how different courts you can think of different groups of users multiple groups of users with similar attributes. behave with regard to a specific feature. You can think of those an example of two courts like maybe. Users that that like oranges and users that like so you isolate those two courts and usually you analyze and how they behave with regard to buying fruits as a feature. Now this is a the word research usually implies that is an observational study you don't really, you don't really experiment on the user just observing how they behave. And typically comes without randomization so you're not really randomizing how the court is composed, you're actually following a specific court of users that has a specific attributes. So it's actually quite biased. So, again, there is no intervention on experimentation just observation, no randomization. So you can identify what is called correlation between the user and the behavior, but not really causation. Now what this means in practice how you can call run a court research usually in an e-commerce business by running user search interviews. So you identify your bucket of users and then you ask them questions about specific feature or behavior that they are showing. Now one step below that there is the case research. Now what the case research is usually, again, word research so it's an observational study is more when you can think of a user individual or group individuals that submit a specific report. It's also called case reports. Now, again, a little bit less qualitatively less than court research. First of all, in volumes. It's not group users but single users. Same observation, no randomization attributes. An example of a case research again in e-commerce or many business today is comes from customer service tickets or guest reviews. So by analyzing them, you actually are researching through those cases to mine some learnings. One step below and this is I think is as obvious as it gets is the expert opinion. Now expert opinion on dismiss them too quickly. And of course you have to keep in mind that can be heavily biased because it's we came from randomized test to group of users. Court research to single reports which is the case research to a single user really asking an expert opinion. It is a fire starter not the end so it is good to trigger questions but won't say the end destination. An example again, spelling out the obvious. You could interview or engage with the business craft expert that is promoting an idea. I do think that they can they have their place and their time if you find the right individuals. Now, you could argue that what the pyramid of evidence is saying is that the higher you climb, the better is the view is that the case. So you everybody should do matter of use, if you want, or randomized test. Why would you ever go to a court case or expert level. Now, there are at least a couple of reasons right that we also mentioned them during this is this talk. The first one is that a bit as a bit testing can be quite expensive. Again, as we said, the platform itself or the test. The second reason is that a bit testing is not always advisable not technically possible. And this is back to where we're talking about one and not to a bit as a word where you are not able to be tested. Now, what is the pyramid of evidence really telling us is telling us that every approach has downsides and limitations. Now how you interact with this downsides and limitations up to you. What this means is that you can choose the right approach based on your context. And so what is the level of tech and quality level that you aim for you can afford. The second thing that is less obvious is that ideally you should walk the entirety of the pyramid of evidence bottom to top. Think of combining every layer again bottom up to build knowledge. So you can think of progressively invest in higher level of quality or you know technology investment, starting from an expert opinion. Then this could give you a lead into maybe analyzing some starting the case research. In this case research potentially could lead you to identifying cohorts of users that expose the behavior that you're looking for the insights from court research could lead you to design and a bit test that allow you to give the ultimate form of validation.  So the two options doesn't imply causation with correlation mean the two events happen at the same time. A causation means that one event is causing the other. And now I love always use this picture right we have this cat sitting on this roof bent exactly in the spot where the cat is clearly the cat didn't produce the band in the pillar. At the point there is correlation, the cat and the band are, you know, showing at the same time, but there is no causation so the cat didn't cause the the band. Why I'm saying that is because in the alternatives to be testing that I talk to you about. There's a correlation so there is an indication of two events happening at the same time, but there is no causation is not saying that one event is cause is causing the other. So be mindful in every non randomized methodology that you apply to not really make the mistake of confusing correlation with causation. And that's all folks. Thank you very much for your time.


Welcome back to my channel and welcome if you're new here.  I did my monthly Costco run, which was huge I also did my weekly shopping shopping for this weekend's camping trip Which you'll see that vlog coming out here shortly and kind of a stock up on some of the things that I just happened to run Out of so we have a massive grocery haul this week all the things I'm gonna share with you my meal plan what you can see or expect to see on a Monday for meal prep It's a busy day today. So let's jump right in to this week's grocery haul We are starting this week's Massive grocery haul off with a run to Costco what massive grocery haul doesn't start with a monthly trip to Costco So I have all the things you see here Plus over there is a case of water So that is one thing that is not shown but let's jump first into what we picked up from Costco The first thing I grabbed is this super cute denim jacket. I got this in an XL It was a bit big, but I wanted it oversized in case this winter I wanted to wear a cute hoodie under it, but you guys this cute jean jacket It's kind of a darker denim with the silver Embellishments it was ten dollars. So if you're making a Costco run highly recommend I mean ten dollars you can't beat that for a jacket and again, this is the Buffalo brand and again Costco $10. Also the coveted toilet paper and Paper towels all from Costco as well. So we went ahead and stocked up I think I have about four or five rolls of paper towels and literally one pack of paper toilet paper left So I was grateful that Costco was stocked up You guys know how my husband feels about these Marie calendars pot pies He's obsessed with these so I grabbed these for him. I do not eat these I don't even honestly really like them, but he likes to have them on those early early mornings If you guys didn't know my husband starts work at 3 30 in the morning a couple days of work a week So he just wants something quick and easy and this does the trick and he loves them So Costco's a great price eight of these for ten dollars Also, my husband as you know loves olives, so I went ahead and picked up this huge bag of olives It's less than five dollars I just throw it in a ziploc container or like a storage container and throw it in my fridge And it lasts forever I dump some of the liquid out and keep some of it in so that the olives stay nice and soft and moist and wet But yeah, you guys this is a killer deal at Costco. So I grabbed that Picked up some eggs. These are the Kirkland organic eggs $6 and 39 cents not bad for two dozen eggs This was the steel of the day So this is the simply fresh for fresh salads So these are those salads that are already in a plastic container all you do is throw your toppings on and your dressing 280 calories, of course the points were here on the screen for you, but this four pack of salads was $4 That's a dollar a salad and since we're heading out camping this week And I picked it up because we thought we could have it with our hot dog for lunch and kind of have a little bit more Balanced of a lunch while we're camping. So yeah, if you have a Costco $4 a dollar a salad restocks our heartburn medicine because killer price at Costco My husband and I both take one of these every single day So we go through this rather quickly, but it's only 1299 which is such a fantastic deal And then these are my very favorite makeup remover towelettes. I love these the Kirkland brand I just love these. They don't make my skin break out. They're fragrance free. They're awesome Actually get 180 of them for 1299 and there are a couple of packs of travel size, which I think has you know Yeah, right here 15 count travel packs I love those to throw in your suitcase or take with you on like a getaway weekend So love those and then of course I restocked my chosen foods mayonnaise You guys know this is our favorite mayonnaise and it's funny because my husband loves this mayo as well So best price at Costco lots of fresh fruit for camping and for the upcoming week I got this big pack of organic blueberries and I got the pack of organic strawberries. My gosh You guys don't these look delicious. I'm so excited that fruit is actually looking good again So excited pack of organic mushrooms We are gonna be making some campers packs for camping You guys will see all of this in my camping vlog But my husband wanted mushrooms in these little packs that we're making so I grabbed that and my husband lately has been Obsessed with bread and butter chips. I don't know what it is. He doesn't even like sweet pickles But he has been loving these and I love these on like my hamburger my sandwiches So four dollars at Costco for 64 ounces So we will forever have his bread and butter chips in our refrigerator Speaking of my husband, I grabbed this big pack of chips for him He takes one of these in his lunch every day these range from four to five smart points So great choice 15 dollars for this big pack of 54 bags So that will definitely last him quite a while and then I saw this and I had to get it So this is the prep works pro keeper for pieces. This is a produce keeper set. This is amazing you guys I always have my produce go bad, especially like watermelon that sits in its own juice or lettuce So I don't know if you can tell but there's a base down here and there is a grate So any moisture or liquid settles into this base Which is separate from the actual container and then there's an airflow vent as well to help bring air through and keep it dry I'm so excited and 25 dollars for four pieces with such a fantastic deal So when I unbox this to wash it, I'll show you guys what it looks like it may go on sale I wasn't sure but I had to pick it up because I've been wanting something for produce torch in my refrigerator And then lastly I picked up some lettuce a big bag of organic romaine hearts I thought I could chop this up for salads and mix that with baby spinach because I like to have baby spinach in my Smoothies as well. So I usually bag some of this up to throw in the freezer And then I thought I could mix it in for salads for the week So that is everything that I picked up from Costco So let's jump into my Walmart grocery pickup and what I grabbed from Trader Joe's. I told you this was a big haul All right, so here are those containers you guys. I'm obsessed and now I wish I would have bought two Because that would be even better. I do have some extra strawberries I'll just end up putting those in a different container, but here is the strawberry So what I like about it is I don't know if you guys can read this very well But it tells you whether or not there should be water because technically you throw your produce in here You wash it in this same container and if it says that there should be water here This one will show a little bit better. So this one is leafy greens So it says that there's water in event. So you actually wash your veggies in here The water will trickle down to the bottom if the vent is checked then you pop this air vent on the side So you can see how that's open and closed So it allows a little bit of air in there and then the water will settle to the bottom so cool So this is that entire pack of romaine Here are my strawberries my blueberries and that entire bag of spinach How fun so it looks like leafy greens You are okay with some water as well as an air vent, but berries you actually keep the little vent here So that's open and then that's closed leave it closed you guys These are awesome, and here's the rest of my massive grocery haul So I went to Walmart pickup and Trader Joe's so everything that doesn't say Trader Joe's on it is directly from Walmart Now this is for our camping trip this weekend as well as lunches and meals throughout the entire week Coupled with Costco we are set so let's jump in to this week's grocery haul first thing I picked up was a box of elbow macaroni My husband has requested his favorite pasta salad for camping Which is basically a macaroni salad with olives and green onions and all sorts of good stuff in their cheese cubes So I'm gonna be making that so I grabbed the elbows for that and as always I picked up my six pack of kombucha So there's three trilogy three ginger eight and three ginger berry You guys know I love kombucha whenever I do a Walmart pickup I always always try to grab this because it's a really good deal It's $15 for six bottles when normally they're about three to three fifty a bottle So killer deal at Walmart and then we needed some hot dog or I'm sorry hamburger buns for camping So I grabbed these great value hamburger buns. These are only a hundred calories That's actually the lowest calorie hamburger bun I've ever seen the ones at Fred Meyer where I normally shop for about 120 So excited to see that those were only a hundred calories and then I grabbed these big sub rolls We're making Philly cheesesteaks this next week And I thought that these would be a little more fresh and a little more fancy than a store brand store-bought roll This is directly from the bakery at Walmart For camping my husband requested French onion dip the deans in my opinion is the best So I grabbed that and then I also picked up some of the ballpark bun size hot dogs These are for lunches We're gonna do some hot dogs make it really easy have that with the pasta salad and some fruit So these are for him and then he also requested a little smoky So we like to throw these on a marshmallow skewer and throw these on the fire and they get nice and toasty They're really really good. So I picked those up and then for my husband's lunches next week He requested breakfast burritos.  They had no Already chunked sausage like sausage crumbles at Walmart at all. So I just grabbed this I'll just cook it down myself and make it into sausage crumbles also for the Philly cheesesteaks We needed some provolone cheese. So I grabbed that and then of course at Trader Joe's I grabbed the Trader Joe's light three cheese blend. I really really like this cheese It doesn't taste like a reduced fat cheese at all to me. So I always like to have that in my fridge I also always like to have the Trader Joe's organic shredded mozzarella cheese So I picked up another pack of that and then for my husband's breakfast burritos I grabbed the Fiesta blend cheese from Walmart For camping we decided we were going to make a spin on a s'more We're gonna do it in an ice cream cone with marshmallows and chocolate Well since he doesn't like chocolate I grabbed these caramel bits so that we can make his s'more cone with caramel So you guys will definitely be seeing this in my camping vlog that should be out not too long after we return from our trip So I thought these little caramel bits would be really easy just to throw in that cone Also at Trader Joe's restocked my apple gate smoked turkey bread I really really like this turkey breast and it's zero points on both blue and purple So I picked that up again just another thing I just like to have in my refrigerator for those times that I'm digging a sandwich or a wrap For the breakfast burritos some southern style hash browns just so I can fry those up and add those to those burritos Also some mild taco sauce I thought I was completely out of my thrive taco seasoning and I'm not so I'm so excited because it has much better Ingredients, but I'll probably just save this for something for my husband He likes to have burritos or tacos sometimes in his lunch So I'll save that for that and then I'm making these campfire packs Which basically is a foil pack that has potatoes and sausage and veggies So for those packs I picked up these sweet Italian chicken sausages from Trader Joe's I tried to get the Sam's Choice chicken apple because those are my favorite and they're only two points But they didn't have any so I opted for these. These are three points. They're really really delicious And then for hot dogs for me. I grabbed these organic grass-fed uncured beef hot dogs I'm weird about my hot dogs. So these are for me and then my husband will just have the regular ballpark For his macaroni salad We need cheese and I usually buy a big block But these were really really inexpensive at Walmart. Oh, and look they gave me reduced fat dang it I didn't mean to I didn't order reduced fat, but that's okay Nonetheless, we'll be using both of these blocks of cheese. There's another one here for that salad I just cut it up into cubes and mix it in with all the other goodness Also, I grabbed a parmesan cheese and I actually ordered the small container And they were out so they substituted with this big container Which is a great deal for two dollars and forty four cents for real parmesan. So that was a fantastic Substitution I also grabbed some sourdough English muffins. I'm going to be making breakfast sandwiches to take camping So all we have to do is warm them up. It makes it so easy So these are for my husband and I'll be using the Ezekiel English muffins for myself Almond milk the only almond milk that walmart had in stock was great value And this is the only one that I could find that did not have natural flavors and did not have carrageenan So great value unsweetened almond milk for the win And I also picked up just a small container of whole milk so that we have this on hand for recipes I have two or three recipes this next week that call for whole milk. So grabbed that and then Big surprise to no one. I picked up six more of the clio vanilla yogurt bars I think I still have five or six in my refrigerator, but Because i'm obsessed so I picked up six more of those this one was open Which isn't a big deal because the clio bar itself is wrapped on the inside. So grab those Also some dill relish for camping and also to make that salad I like to put dill relish in that macaroni salad just gives it that nice zing of pickles And then I grab four of the siggy's a four percent full fat yogurt I have two mixed berry and I have two vanilla there.  So I grabbed four of those because i'm out of yogurt for the week And then for my husband's lunch and other box of his little bites and party you guys know He loves these I don't really like these so it doesn't really tempt me having these in my pantry. So Otherwise, I don't know that I buy them, but he loves them and I throw one little package in his lunch each day And then for camping and also for a recipe. I grabbed the trader joe's organic corn chip dippers These are so addicting and delicious And they actually have pretty good ingredients the oil is not my favorite But overall for a chip they have good ingredients and it's organic So I know that the corn is organic non-gmo So grabbed that and we're going to have that with this french onion dip for camping And then again, I need it for an upcoming recipe and then to bring camping I grabbed these trader joe's pink and white. We both really like like the animal frosted cookies So I thought these would be a fun little thing to have on my splurge day while we're camping And then of course the restock on the yogurt pretzels for him. He loves these in his lunch I like these, but I don't like the point.  So at walmart I restock all my frozen veggies So I have two bags of the organic broccoli florets One bag of the mixed veggies and then one bag of the multicolor cauliflower blend I'm just about out of frozen veggies. So I'm going to throw these four bags in my freezer Grab some zucchini for those campers packs those foil packs and then just for snacking and salads Bell peppers for the packs as well as for salads and I have another recipe I think it's the philly cheese that calls for red bell peppers as well Some bananas for a recipe and then I like to really really ripen these up and throw these in the freezer for my smoothies Grab the pineapple. This is for a recipe meal prep I'm making like a pineapple chicken with the leftover delicious orange sauce from last week's meal prep If you haven't seen that video check it out because we saved some of the sauce and chicken for this week's meal prep So can't wait to share that with you guys on monday A watermelon because we thought what's better to take camping than watermelon. Am I right? So we grab that Snow peas for a recipe and again, no organic at Trader Joe's So I need to do some research to see if these should be eaten organic If you know, let me know down in the comments, but they didn't have it. So I just grabbed the regular Organic lemons for my water. I've been still on a lemon water kick I'm loving it a big red onion for those foil packs and a upcoming recipe Some green onions another thing that Trader Joe's does not have in organic So I don't know if that is something we should be purchasing organic or not But I grabbed them while I was there and last but not least some organic cilantro And this is for an upcoming recipe So that is it my friends Costco Trader Joe's walmart pickup Whoo, that was a grocery haul. So let's get into this week's meal plan and what you'll see on monday for meal prep So let's jump into this week's meal plan. As you guys know, this is my cariel meal planner I absolutely love it. They still do not have the spiral bound option on their website if you didn't know due to the coronavirus They actually had to eliminate their staff It is a very small business ran by a husband and wife and they just aren't able to make the Meal planners right now They do have a digital downloaded version if you want to take advantage of that My code will work for that as well and my code is down in the description box But without further ado, let's look at what we're doing for my meal plan So let's first go to the monthly plan. This is something I utilize every month I write down what I'm going to be making for the entire month. It's very easy for me to meal plan from there These recipes are from my meal plan service that I offer over on my facebook group So make sure you're part of that So I jot everything down there and then from there I utilize the weekly menu You can write down all of your meals I generally just track dinner because I do meal prep breakfast and lunch But I'll write down my dinners. There's an detachable grocery list I literally use that every single week. So I love my planner so much This is a total lifesaver for me. So this is this week's you'll see the grocery list is missing So this is the week of five six or I'm sorry six six through six twelve So today is saturday and we are currently camping. So I'm actually filming this video before we head out So it is basically my splurge day That's what we're doing on saturday sunday I'm going to be making a taco casserole. You will see that recipe in next week's what I eat today So stay tuned for that monday. I'm going to be making sesame chicken. I don't know. It just sounded really really delicious Probably pair this with some rice and some veggies Tuesday is sheet pan filly steak So I am making filly cheese steak sandwiches on a sheet pan crazy, right? Very very simple one pan cleanup will add it to a hoagie roll And that's what I bought those delicious bakery hoagie rolls for so I'm excited for that Wednesday, I'm going to be doing an instant pot chicken elfredo This is a very simple recipe because we actually are heading out camping again Next thursday, so I'm doing a simple recipe wednesday night to give me time to get ready to go camping So that is wednesdays thursday friday We're camping as well as into the following saturday can't wait stay tuned my friends for camping vlogs On the back of the weekly menu is where I plan out my meal prep It just makes it very easy to just do it right on the back. So for breakfast this week I'm going to be making a four flavor sheet pan pancake as you can tell I love sheet pans This is going to be a one big pancake four different ways So we'll have chocolate chip love strawberry banana all the things and you guys will be seeing this recipe along with my lunch and Snack in monday's video. So stay tuned Lunches is pineapple chicken stir fry So again, we're utilizing the leftover sauce and chicken from last week's meal prep And we're making a second meal with it So I love that we kind of got a two for one out of last week's lunch prep And by the way, that lunch was amazing. So super excited for that And then for a snack or sweet treat this week. I'm going to be making a blueberry muffin bread So we are going to combine blueberry muffins and bread and make a delicious sweet treat So stay tuned for all three of these recipes coming your way on monday for meal prep So again, that's my menu for the week. This is my cariel meal planner I will have the link down below to save you 10% off anything you pick up from their website And it's a great time to support such a small business Thank you for joining me on another weekly my w w grocery haul My friends did we have a haul? I'm so excited for camping I'm so excited for my meal plan and meal prep because that orange chicken bowl that I had this last week was Amazing and if that is in the indication of how good this week's lunch prep is going to be I cannot wait that meal plan and meal prep looks so incredibly delicious Down in the description box is the link for my website, which is where all of my recipes are The discounts and links to save you some money on my favorite things and the link to head over and join My facebook group. We'd love to have you be part of the community over there We're over 15 000 strong and it's just a wonderful supportive place to be So head on over and join us if you're new make sure you hit that little subscribe button and the bell located right next to it So you're notified when new videos are uploaded. You don't want to miss a single one Give this one a big thumbs up if you love grocery hauls leave your comments down below On what recipe you're most excited for have a fantastic weekend my friends and i'll see you all in my next video Bye guys


Hello and welcome to the session. This is Professor Farhad in which we would look at a compliance attestation report. Specifically, we're going to be looking at two reports. One is an examination and the other one is an agreed upon procedures. And what's neat about those compliance attestation reports is they summarize everything that we learn about the compliance attestation because that's the final product. Let's go ahead and get started. First thing I need to tell you just like all the other attestation reports, no standard report. The AI CPA provide examples and what I did, I took those examples and I showed you what would a report typically would look like. First, we have to have the title and the word independent has to be there. We are providing insurance. Then we have to have an appropriate address. See who are we addressing this report to nature of the engagement. Here we are examined. We are conducting an examination. We have examined Adam company's compliance with requirements listed in attachment one, whatever that attachment one during the period, January 1st, the December 31st, tax one identified the compliance matter. Again, it's an attachment one. Then we have to talk about the responsibility of the company management of Arab company is responsible for the compliance with the specified requirement. Our responsibilities to express an opinion. Remember, if we are conducting an examination, we have to issue an opinion on the compliance of Adam company. Then we list the procedures. What did we do? What is an examination? Well, it's conducted in accordance with the attestation standards established standards established by the AI CPA. Those standards require that we plan and perform the examination to obtain reasonable assurance whether Adam is in compliance and all material respect with the specified requirement reference above. Then we talked about what is an examination. It involves performing procedures to obtain evidence whether that did happen. The nature timing and extent selected depend on our judgment including an assessment of the risk of material non-compliance whether that's due to error or fraud. Again, this is an examination. Practically all examination sounds similar. We believe the evidence what that we obtain is sufficient and appropriate to provide to provide reasonable assurance for our opinion. Again, we need sufficient appropriate evidence because this is an examination and examination is basically an audit except we cannot call it an audit because we are not auditing. We are not examining historical financial statements. Then we'll have to let you know that this report does not provide legal determination. Our examination does not provide legal determination whether Adam company is in compliance or not. That's for the government or the appropriate legal authority to determine and our opinion based on our review based on our examination. Let's be specific. We cannot do any reviews for compliance. Adam company complied with all material respect. But again, this is not a legal determination. The next thing we're going to look at is an agreed upon procedures for a compliance attestation. Before we proceed any further, I have a public announcement about my company farhatlectures.com. Farhat accounting lectures is a supplemental educational tool. That's going to help you with your CPA exam preparation as well as your accounting courses. My CPA material is aligned with your CPA review course such as Becker, Roger, Wiley, Gleam, Miles. My accounting courses are aligned with your accounting courses broken down by chapter and topics. My resources consist of lectures, multiple choice questions, true false questions, as well as exercises. Go ahead. Start your free trial today. No obligation. No credit card required. So let's take a look at the agreed upon procedures. Again, we have to have a title and the word independent. We have to have an addressee, practically the same. And we have to look tell you what's the nature of the engagement. We have performed the procedures enumerated below. Now they will be listed here, which were agreed to by management and the board of directors of Adam company related to guess what we are reporting on Adam company's internal control over compliance. Remember in the examination, we did the examination for the compliance with the requirement. We could do a compliance on the assertion or we can do a compliance on the internal control over compliance, which requirement is listed in attachment one. So this is an agreed upon procedure in AUP and an AUP you can do an internal control over compliance and engagement of internal control over compliance. Adam's Adam company's management is responsible for its internal control over compliance. We have to say this, the sufficiency of these procedure is solely the responsibility of the party specified in the report. We are starting to say that this report is limited to specific parties and we're going to limit at the end the report, but this is we're starting to do so. Consequently, we make no representation regarding the sufficiency of the procedures enumerated below either for the purpose for which this report has been requested or for any other purpose. Just we're starting to limit this report. Now we would include all the procedures that we perform as well as the finding if there's any findings, especially findings of non-compliance and based on that we have to determine what should be, you know, what should we do? Then we have a little bit more about the report list of the procedures. It was conducted in accordance with AI CPA, AI CPA attestation standard. We were not engaged to and did not conduct an examination or a review. So we have to tell you this is not an examination. This is not a review and examination. We would have issued an opinion and a review. We would issued a conclusion. We're not doing so. Just just want to let you know. Therefore, we don't express an opinion or a conclusion. Also, we want to let you know had we performed additional procedures, other matter might have to come to our attention, but we did very specific thing. We followed the AI CPA and then agreed upon procedures with the client. Now, again, because this is an AUP agreed upon procedures, all agreed upon procedures, we limit the report because it's agreed between specified parties. This report is intended solely for the information and use of management of the management and the board of directors of Adam company is not intended to be and should not be used by any anyone other than the specified parties. And what do we do? We have a signature block. What should you do now? Go to far have lectures and work MCQs about attestation engagement, compliance, engagement, pro forma prospective and DNA socks. You have to be familiar with those concepts for the CPA exam. Don't shortchange yourself. This is important. Good luck. Study hard and of course stay safe.


We were looking at the proof of Flocke's theorem, recall the Flocke's theorem says that if you have a first order system dx by dt is equal to a into x, where a is a n by n periodic matrix with period t, then this system has at least one non-trivial solution chi of t, where chi of t satisfies this relation that chi of t plus the time period of the matrix capital T is equal to mu into chi of t. Note that mu in general is a complex constant. We had started by looking out at the proof of this theorem, so for that we introduce the notion of a fundamental matrix of a system and then we showed that if chi of t is a solution to this equation, then chi of t plus t is also a solution. So now recall that the definition of the fundamental matrix of our system is just that if we have n linearly independent solutions, if we place them column by side by side in a matrix, then that gives us the fundamental matrix of our system. So now let us look at these solutions chi of t and chi of t plus delta t. So suppose we have found n linearly independent solutions, n linearly independent solutions and I will call them chi 1 of t, chi 2 of t and so on up to chi n of t. We can form a fundamental matrix of our system and this would be phi of t which would basically just be all the chi's arranged side by side as a column. So chi 1, if I write it chi 1 written out as a matrix will have n rows and one column. So I will write it, write out the elements of chi 1. So chi, so this is all elements of chi 1, so this will be chi 1 1, so it is all the second element of chi 1 and then the nth element of chi 1. Similarly chi 2 the first element, chi 2 the second element and then the last one chi n the first element, the second element of chi n and the nth element of chi n. So we have our n by n system and this is the fundamental matrix of our system. We also know that if chi 1, chi 2, chi 3 up to chi n are solutions to our system we have just proved then that chi 1 of t plus t, chi 2 of t plus t up to chi n of t plus t each of these is also a solution to our system. We can form one more fundamental matrix and we will call this t plus t and this will be arranged by arranging each of those as columns just as we had done earlier. We took chi 1, chi 2, chi n up and arrange them as columns. Now I will take chi 1 at t plus t, chi 2 at t plus t each of these are also solutions. So I will arrange them as columns. So these are all at t and the corresponding ones here would be at t plus t. I am not writing the second one and so if this is a fundamental matrix then so is this because by definition the fundamental matrix consists of all the solutions to the system, all the linearly independent solutions to the system arranged as columns. Now you can see or I will assert that the relation between phi of t plus t is phi of t dotted with some n by n matrix c. This must be true. Why should it be so? Let us take one simple example to understand and you can immediately generalize that. So I will take a simple 2 by 2 example and then we can immediately generalize them to this n by n case. So let us take a 2 by 2 example. So my system is 2 by 2. So then we will have so dx by dt is equal to a of t dot x where a of t is a 2 by 2 matrix. So there will be 2 linearly independent solutions to this system in general. So I will call them chi 1 of t and chi 2 of t. The fundamental matrix phi of t for this system would be just a 2 by 2 matrix like before and we know that if chi 1, chi 2 are solutions then so are chi 1 at t plus t and chi 2 also at t plus t. So I can already get one more fundamental matrix at t plus t which is just the same things, this one. So you can see that these 2 are linearly independent. So I can write the column. The first column of chi phi t plus t, it is basically chi 1, 1 t plus t and chi 2, 1 t plus t. Now this is some vector. It is a solution vector to the 2 by 2 by 2 system and because this is a solution it can be expressed as a linear combination of the 2 linearly independent solutions that we have. So I expect this is a 2 by 2 system, there can be at most 2 linearly independent solutions. I have now 3 solutions. One solution is this and another 2 linearly independent solutions are this and that. So I can express this third one as a linear combination of the first and the second linearly independent solutions. That must be true. I will write it slightly down because there is no space. Basically I can take the fourth one and I should be in general again be able to express because this is also a solution, this fourth column of the equations. So in general I should be able to express it as a linear combination of 1 and 2. But the coefficients will in general be different. So this would be some alpha 2 into again chi 1, 1 of t into chi 2, 1 of t plus some beta 2 into chi 1, 2 of t into chi 2, 2 of t. What does this tell us? This tells us that I can place these 2 columns side by side and create the matrix phi t plus t and I can write the right hand side of these 2 equations as the product of 2 matrices. You can verify that this is true, you can verify that in this simple 2 by 2 example this is true. I am just placing the 2 column vectors side by side and creating a square matrix on the left and then on the right what I have is just this. And the matrix that you will have to multiply it with is alpha 1 beta 1 alpha 2 beta 2. You can check that this is indeed what you would get if you multiplied these 2 matrices and you would recover the equation that we had written in the last slide. So this you can immediately generalize this thing to the n by n case. And so I had asserted earlier that I will be in general able to write a relation like this which connects the phi the fundamental matrix at t plus t to the fundamental matrix at t and it multiplies it by a square matrix whose dimensions is n by n. So now you can immediately see that phi in the n by n case phi at t plus t we will in general be able to write it as phi of t dotted with a matrix c. You can see that because the determinant follows the rule the determinant of a dot b is determinant a into determinant b therefore you can see that determinant of phi of t plus t is equal to determinant of phi of t into determinant of c. You can see that because this is a fundamental matrix and this is also a fundamental matrix the determinants cannot be equal to 0 therefore determinant c in general is not equal to 0. Now what did we learn what do we gain by doing this? You will find that the solutions to this equation the solutions to our first order system are related to the eigenvalues of c let us see how. So we can see that from this matrix equation involving fundamental matrices we can obtain that c is equal to because phi is that has a non-zero determinant that means it is invertible. So I can in general write it as this is phi inverse t dotted with phi of t plus t and if I choose the time this is this is true expected to be true at all times. So if I choose small t is equal to 0 then this just becomes this and if you can adjust the initial conditions so that phi inverse of 0 becomes the unit matrix then the c matrix is just obtained by taking the fundamental matrix and evaluating it at the time period of the coefficient matrix. Now let us move onwards to completing the proof of the Floquet theorem we need this in order to prove the Floquet theorem let mu and s be eigenvalue and eigenvector of the matrix c this implies that c dotted with s is equal to mu dotted with s or mu s because mu is a scalar. Note that phi the fundamental matrix evaluated at time t dot s is a solution to our original system the system is dx by dt is equal to A into x how do we see that let us say let us take our simple 2 dimensional example. So the 2 dimensional example is just the 2 by 2 example is just this this is the fundamental matrix into let us say the component of the eigenvector in this 2 by 2 system is just s1 into s2 if you multiply this I will in general get a column matrix but note that I can write this multiplication as s1 times the first column of the square matrix plus s2 times the second column note that it is equivalent to writing like this what does this give us this tells us that this product of phi t dotted with the s vector is just a linear combination of those 2 vectors what are those 2 vectors each of them is a solution to our system by definition we had placed each solution as a column in the fundamental matrix. So each of them is a column in the fundamental matrix and what is this is showing us is that that this is just a linear combination of the columns if I have 2 solutions and if I take the linear combination then what I get is again a solution this is a property of the system. So, this is if I go back and substitute this this will be a solution to our system I can substitute this this will be a solution to my set of equations. So, therefore phi t the fundamental matrix dotted with s is a solution to our system. So, therefore we can set chi of t we had said that chi was a symbol for solutions to our system we have found one set of solutions. So, I can set chi of t to be equal to phi of t dotted with s. So, this implies chi of t chi of t plus t is equal to phi of I am just replacing with t with t plus t and this implies if I work on the right hand side. So, phi of t plus t is we have shown now into the matrix C this we have shown earlier here here we have shown. So, I am just using that. So, this phi of t plus t is being replaced by phi of t into C and this is chi of t plus t and now I can do this matrix multiplication later and put C inside the bracket and multiply it with s. C dot s we have said is equal to s is by definition an eigenvector of the matrix C with eigenvalue mu. So, this is equal to phi of t dotted with mu of s and this because s mu is a scalar. So, I can write this as phi of t dotted with s and from here we know that phi of t dotted with s is nothing but chi of t. So, this is mu times chi of t. So, we have proved the Floquet theorem. We have shown that if you have a solution to the linear homogeneous time periodic first order system dx by dt is equal to Ax then if chi of t is a solution then it satisfies chi of t plus t is equal to mu times chi of t. This was the assertion and we have just proved it. You can go over the various steps in case things are not clear you can go over it once more. So, this number mu which I had said could be possibly complex is called a characteristic number of the system. Now, what can we use the Floquet theorem for? We can use the Floquet theorem to come with a structure for what would be the solutions to a equation whose coefficients are time periodic. Let us see how. So, because mu is a characteristic number of the system, we will set mu is equal to e to the power rho i into t. In general, I would have n different solutions. So, there would be chi i, i going from 1 up to n. So, for each of such chi, there would be a corresponding mu. So, mu i, this is just a definition of rho i. We are doing this because we want to introduce an exponential in our final answer. So, I define this rho i by this equation that e to the power rho i into the time period of your coefficient matrix is equal to mu i or in other words rho i is equal to 1 by t log mu i. So, how what do we gain by doing this? So, let us say we have a solution to our first order periodic system and we know from Floquet theorem that it satisfies this. This implies I can write this as by the definition of rho i, I can write this as chi i into t. Now, if I take this equal to that, so I am basically using this equality and if I multiply both sides, so you can see what I am multiplying with. So, I am multiplying both sides by e to the power rho i t and then I am also multiplying minus rho i small t. So, I will get e to the power minus rho i t chi i r. Now, I can write this as we will show very easily that this is a function let us call it p i this is a vector. So, this is a function of time and we can show easily that p i of t is a periodic function of time. How do we see that? It is easy to see p i of t is just e to the power minus rho i of t chi i of. So, p i of t plus t is equal to e to the power minus rho i t plus t chi i of t plus t. I am going just going to work on the right hand side. So, this part. So, I can write this as e to the power minus rho i t into e to the power minus rho i capital t into chi i of t plus capital t and chi i of t plus capital t is just. So, chi i of t plus capital t is just e to the power plus rho i of capital t into chi i of t. If you substitute it here you will see that the plus rho i into capital t cancels out the minus rho i into capital t and we are left with e to the power minus rho i t into chi i of and this by definition is just p i of t. So, we have just managed to show that p i of t plus t is the same as p i of t. So, p i is a periodic function of. So, what do we gain by this? We gain that chi i of t plus t into e to the power minus rho i t plus t is equal to p i of t. Let us take this further. I can write this as chi i of t plus t into e to the power minus rho i t is equal to e to the power rho i t p i t. I have just multiplied both sides by e to the power plus rho i t. I have taken this equation. I have taken this equation and I have multiplied both sides by e to the power plus rho i into small t. This is what I have done. So, you will get this equation. But what is the left hand side? So, you can readily see that the left hand side is nothing but chi i of t and this is equal to the left hand side. So, we are basically using the fact that how do we know this? We know that chi i of t plus t is equal to mu times chi i of t and mu we had written it as e to the power rho i of t. So, this is how we are able to go from here to here. So, this is being used there. Now, what did we gain by doing all this? We gain something very interesting. We recall that chi i is a solution to our first order system dx by dt is equal to a into x. And so, chi i is one typical solution of this system. This analysis and this Floquet theorem is telling us at least one solution to this set of equation can be written in this form. Now, what is this form? This tells us that the solution to this set of equation can be written as e to the power rho i into t into pi into t, where pi is a periodic function with the same period as the matrix A, the same period t. Next, we will see how to apply this theorem to Matthew equation and what do we conclude about the solutions to the Matthew equation based on this theorem.


The study found that IT ambidexterity positively enhances the development of a digital dynamic capability, hospital department's knowledge processes, and ultimately leads to increased patient agility in clinical hospital departments in the Netherlands. This article was authored by Roger van de Wettering and Johan Versendel.