All right, very good morning. It is Friday 31st of May. Hope you're doing well. Quick overview then what we're going to cover. We're going to look at obviously the latest news about Trump and Mexico and that's created risk of market movement overnight and certainly it's going to be the focal point as we go into full UK and European market open. So we'll review that. We're also going to look at some Chinese data that came out overnight. Of course, it was the PMI data, Manufacturing and services and more broadly speaking, still remaining heavily in contraction for China in particular on their employment sub-constituent number. So we'll have a look at a couple of graphics to make sense of that. And then also a look at oil, some comments out of Clarida, who's the vice-chair at the Fed, which I think were quite important as well. So that's what's on the agenda. And then we'll hand over to Sam and he can look at the charts. So quick look at the general sentiment. As you can see here, we've got quite clear risk of tone to proceedings. The bund in fixed income markets has gapped up and is up about 33 ticks. You can see here, the US 10 year in the bottom right hand corner is up 13 and a half ticks with the pop, if you like, coming overnight session. If I bring up the SMP here and I start putting the SMP on a slightly shorter timeframe, you can see this is when the news came out last night and the SMP sold off immediately on the back of it. And again, we'll go into the headlines in more detail. In kind then, gold elevated and European stock futures, particularly the DAX will be an interesting one to watch. You can see here, the futures gap down, broke through yesterday's low and just coming up to the cash open in about 20 minutes. That'll definitely be one to keep an eye on. Just given the direct implications for the automakers, I believe I heard a snap this morning that there's eight EU manufacturers that operate with facilities within Mexico. And that does include VW being one. So definitely you'd expect that to weigh on the DAX, given the sector kind of profiling of that particular industry. Currency wise though, not too much reaction tends to be fairly isolated at the moment, at least, not really seeing much dollar reaction to the latest news. So the Dixie's flat and that is broadly reflected in both major pairs in euro dollar and cable. But otherwise, generally, risk of start to proceedings overnight reaction, if you like, to the headline of which came out, which was this. So let me read you through a couple of points here to surmise what's happened. President Trump vowed to impose a 5% tariff on Mexican goods until the country stops immigrants from entering the US illegally. The tariff would take effect on June 10th, quote, until such time illegal migrants coming through Mexico and into our country uppercase stop. Was what he tweeted yesterday. So, yeah, definitely something which reading between the lines here, of course, if you watched the rerun of his speech, I actually found it quite interesting. If you step away from markets for a second, you think about the political angle. You know, one of the things that's been and it's absolutely backfired for established political parties, in particular those of a more center ground trying to appease both sides of the argument. And taking Theresa May as an example, you know, sitting on the fence trying to compromise just is not cutting it at the moment with the just general electorate and listening to Trump. You know, he is a good salesman in that sense. The way he was talking it up, making America great again, obviously stopping full illegal migrants coming into Mexico by next week or so. That absolutely is never going to happen. But that doesn't matter at all. I mean, if you listen to him and the way he delivers it, I almost want to put on the manga hat. Because, you know, that's how well he spins it. But that's the point, you know, the point is all of this is a political ploy to try and garner, you know, kind of sentiment around the key issues. And remember, the two key issues for 2020 for Trump and a second term is trade and immigration. And it's those two things that are right at the epicenter of the main factors and forces moving markets at the moment. You've got trade with China. And don't forget we were looking at the stats yesterday in the briefing with Mexico, Mexico, an absolute key. That's the second biggest country of which the U.S. import goods from. And then immigration, of course, has kind of gone a bit quiet on that. But obviously it's one of the focal points of his policies. And as much as that went quiet after the government shut down on the border war funding, you can very much expect that issue to return to the table very quickly as he starts to kind of ramp things up. So, yeah, definitely this move, it dampens sentiment. The tariff man strikes again. And again, in a week where we've had concerns about the elevation and potential retaliation from China, who really have kind of changed their tone given what's been happening with the breakdown in the trade situation with the targeting of tech firms. This is just another kind of element on the global scale that's kind of dented sentiment. And so having a look here, the S&P 500, I know Sam will look at this in more detail, but there's obviously this is the longer daily continuation chart. This is the thing that we were looking at where you've got the 200 daily moving average on the red line. And then you've got that trend line as well. And we briefly, right at the beginning of the week, had a real test of that broke below it, but closed back above yesterday held above the level. But obviously there's latest development has helped bump things back down again. So technically speaking, does open up potentially a bit of clean airspace now for a pushback down. And technically 12, 37, 3 quarters. You've kind of got these previous levels here. Little choppy though, really that zone 37 to 22, I think looks like an area of stronger support. But don't forget that means we've got to move a decent 30, 40 points lower from where we are at the moment. Whether that happens today or not, I guess need yet to be seen. One thing I would say is that Americans would have been, you know, it was late in the day. The move didn't come until, you know, gone midnight, London time. So by that, what I mean is when Americans come in, they still probably need to price this in to some degree. So, you know, definitely would be mindful of that. So keeping on gold as well, of course, gold here. We're just, you know, we're coming back to the $1,300 handle again. I had a test at the R1, which is about one and a half bucks shy of the figure. And so definitely, you know, this kind of risk off trade on these latest fundamental developments, likely to fire things up a little bit in the gold market. And again, I'm sure Sam will be looking at this a bit more closely, just looking at that kind of trend line from the high of the year that was printed back in Feb, retested on the 14th of May. And that's coming in as kind of maybe a potential area to have a look at as we come up to the $1,300 psychological level. A few other things then, let's just have a look. Obviously, Mexican peso reacting immediately on the back of the latest news. The time stamp at the bottom is not London. Presumably this is, well, not sure what that actual time frame that is, but potentially local time. But moving 2.4%, I think the peso weakened on the back of this. I think this was after about a week ago that he had dropped the still tariffs, but certainly right back into the mix now. And so local assets are going to suffer. The other thing that's obviously not helping the situation for the open is the fact that this is a graphic of the Chinese manufacturing PMI data. So if you remember, one thing is here, what you have these various different coloured lines is the orange line here is the actual headline kind of composite PMI number. You can see we moved into quite deep contraction at 1.7 or 46 level before coming ramping back post the Lunar New Year holiday, but then we've fallen again to where we are kind of at the moment. So actually, this is this area here where I'm looking at the slightly lighter orange line. So again, dipped contraction, so excuse me, that is this line, contraction just below 50, we got back above it, flirted right on the level, but we've moved back into contraction is the point. The orange line at the bottom, this is new export orders. And one of the other ones that people are looking at quite closely is this is the factory employment gauge, which fell to its lowest since basically the financial crisis printing at 47. So definitely the Chinese economy still very much feeling the bite of this continuation from what really has been impacting their local economy through the last several months. So despite the recovery that we've had in Q1 in general global market sentiment and some stabilization in a few of the asset prices, one thing we have been seeing certainly over the last week or so is the yuan has continued is renewed its weakening kind of pattern, getting closer to that key threshold seven level and these kind of forward looking PMI indicators are tipping negative again and to points of which when looking at the sub components like employment as we've seen here, something that we haven't seen at levels in about 10 years. So what does that mean for general global growth concerns? Well, they have to be paired back again and this certainly has ramifications for the price of crude oil. Now it's been May, the month of May it being the 31st looks set to be the worst month for WTI crude since November. Having a look here, you can see really the market when it came raging back and the Fed came in to save the day, altered their communication to become dovish, the various different fiscal intervention on behalf of China, both government and central bank level created that really strong about turn and we had decent four months of persistent gains in crude but you can see here that the WTI monthly price change down heavily from what we've printed over the last couple of months and looking at the crude chart on a daily continuation, crude acts really nicely when it starts breaking some of these longer standing technical points of significance on the way up it was exactly the same playing round figures, the fibrid tracement of the sell-off in Q4 and also some of the previous price action and you can see here when the market really plummeted back on the 23rd of May consolidated exactly where we did before the breach of the technical levels I mean these levels here are not put on this morning, they've remained the case and you can see how the market kind of quick fire falls back down to the next point of significance and certainly we're in that test of that zone at the moment around that 55-75 that was the initial you can see here resistance the 38% fib of that Q4 sell-off acted as the support and platform then for the push that we saw in February-March and now consequently turns to a key area of support here where we're trading at the moment but as you can see going through May you know quite a sharp depreciation that we've had as the whole global growth concerns return and the trade war globally starts to sour as Trump starts to kind of up the up the ante as he goes into kind of full political kind of run into the election phase one thing that I think is important is this this was a comment out of this chap Richard Clarida who's the vice chair of the Fed and he said that the Federal Reserve could cut rates if the outlook takes an unexpected turn for the worse as he vowed to be nimble in ensuring the US expansion continues now if I was Donald Trump this is absolute music to my ears because it means then that basically I'm getting what I want which is I can play hardball and if the markets do respond negatively enough well the Fed are here quite clearly committing that they're going to step in and do what's necessary to continue the expansion of the economy IE not just hold interest rates as the current dot plot would suggest they're going to cut and that's definitely what the markets are pricing towards the end of the year and certainly that gives I think Trump wiggle room to be quite aggressive in his tone on this trade side because he's got the ultimate get out of jail free card which would be the Fed to help prop the market up and let's not forget Trump can then change the tone cut a deal not aggressive as complete complete stop on illegal immigration but he gets a little bit more border wall funding perhaps to appease then Congress to stop what he's doing and then net net he gains a concession and it's a political win and the markets then respond as he comes off the gas so to speak yeah I still you know I don't think more broadly speaking that although everything I've talked about in this briefing is quite negative even though I feel like the market does need to come back a little further particularly in the equity space I don't think this is the beginning of the end I actually think this is just part of the phase of a correction of which undoubtedly will turn obviously it's the timing which is key and I think Sam's going to look at some longer timeframes on say the equity markets of where could be quite interesting on those pullbacks but the point being is that from a fundamental narrative the powers that be let's say I Trump he knows what he's doing and he knows what he needs to achieve on a specific timeframe of which we as market participants are fully aware of is the point all right quickly the calendar what have we got now we've had the Chinese data of course come out so I wouldn't worry about German retail sales numbers that's really insignificant on a regular basis the German state CPI's will be coming out momentarily obviously as per usual these are littered throughout the morning could be quite interesting to keep an eye on I think remember last month I think they did jump up from their prior month and that did lead to some slight short-term appreciation in the euro at the time Italian GDP final readings so not to be I mean Italy in focus yes both politically yes has been in technical recessions so GDP quite interesting but these are final reading so shouldn't be anything too surprising going further forward mortgage approvals again not really a market move for sterling so I wouldn't worry about that as far as I'd say sterling risk around political developments on brexit I'd kind of classify those as very relatively limited at this point because where we're at at the moment is that really there's a bigger much more dominating macro theme which is the trade developments that we've discussed but also the fact that the timeline has kind of been drawn out now for this leadership process to take place going into the afternoon we get us personal income we've got core PCE data Canadian GDP Chicago PMI which is always quite interesting coming out to states University of Michigan but that's a final reading and then Baker Hughes recount for any of the oil traders speaker wise feds Bostick and feds Williams the latter of which is a voter and there will be prepared text it looks like and is speaking on monetary policy theories it could be quite an interesting one to look at typically Williams having a more hawkish stance on policy so yeah overall again I think it would be the European morning largely derived from the fallout of what's happened from overnight nothing too much here from a scheduled point of view on that basis it's definitely more of a probably US centric session once again with data but also how does the US stock market react to what we've had particularly with some of these levels that we looked at and the spools being taken out already how does the US react when they come in I think will be important for how the week finishes alright that is it for me I'll wish you a good day ahead and a good weekend thanks very much morning guys hope everyone is doing well and had a good week so far coming into last day of the month and week it will be interesting to see where some of these charts end up of course on the longer time frames I think worth talking starting I should say with the S&P let's put this on a longer time frame I mean you can see the move from last night keeping an eye on those lows just now especially with decks about to kick on in five minutes to keep an eye on exactly what happens around there on the longer time frame you can see here and it's got the 200 day moving average absolutely would be looking at that as well and seeing where we close the week and the month as mentioned you've also got the 27-22 area 37 as well that 15 point zone I think it's a pretty key one from a technical point of view I like the look of it as a place to get long I mean I do think Trump knows exactly what he's doing so the safest bet here may well be to just wait for a game-changing comment technical push higher and get in and hold your breath but looking here you've also got the fib from last year's low to this year's high comes in around there the 0.382 just below 27-22 as well you've got trend line from the previous all-time high that goes through that sort of area you can see here second third test breakthrough was in March and then that will come in in the mix there as well so 8 for March low 0.382 fib trend line from previous all-time high not a bad level technically to get in but of course if there's further developments like there were last night the technical levels aren't going to hold too firm but it's certainly something to keep an eye on maybe like Anthony saying not today but over the coming days or weeks as well but with the with the equities on their lows at the moment we're keeping an eye on the DAX of course to see how that does react looking at the Euro relatively quiet all week to be honest I mean the ranges of these moves have been very small especially from the London trading hours from a range perspective R1 and those highs looks okay I'll just be always slightly careful in these ones just to see how it gets to R1 so right now as it stands yes stays high and R1 really good resistance level risk reward looks good I'll just be keeping an eye how we react to this potential trend line if it finds resistance comes back down but then really pushes on very quickly well that would be the key that I don't necessarily want to get in this trade also above where we're trading around the R2 you've got some nice highs as well previously quite a lot of resistance nice resistance above where we're trading of course this market we just zoom to the left hand side and you can see this is the last Thursday's reaction on that multi-year low we're keeping an eye on that obviously into the close of the week if we were to have some strong dollar a strong dollar I should say today and break below that and close would be pretty key something to focus on again as we go to the back end of the session the pound having a look here it's all about for me anyway the longer term picture here if we just remove the pivots just to clean things up a bit you can see that 126 hand on the futures that we've been talking about the trend line from 2017 low that we had back in January and it's just hiding behind my camera at the moment so I just put that on you can see that's just been tested a little false break out of that also got the low that we had back on the 11th of December to key level key area where we close the week will be important but like with equities a fundamental development and that could be who's getting there in the hot seat at number 10 we could be absolutely key for deciding where this market goes rather than a technical level support but all things considered it's a good one right now that I would have marked up the yen yesterday well yesterday morning we had that break didn't we of the trend line which was a really good opportunity if you were trading at those hours of course with everything that happened yesterday equities while they recovered in the morning and into the back end of the session you can see the exact same well the opposite here move we then pushed back above and we're now trading on the highest we've been for quite some time but just be aware of these this area this is now 480 chart you can see not far from the high that we had back on the 13th and you also got potential trend line from starting on the 24th of March just above there so quite a key level to keep an eye on just above where we're trading here again on the yen against the dollar another save haven market to focus on obviously the yen being one of those but gold really did push push on as well quite again a bit of resistance above where we're trading you know noticeably just above where here you've got the 1300 handle for you know one there's some decent resistance we've just gone through what was the low of the 15th as well so along with those trend lines that from the previous highs do any of those come in around this area yes you can see you've got a potential one here that starts on the 25th to the 13th of this month we're keeping a cosine that definitely into the back end of the week I've said it all week as well with with gold it is a tricky market at the moment to really call the direction every time you think it's coming down you get that failed test and you know is it are we now trying to extend through I think maybe a break of this trend line could be that key to get us going in the right direction if you're a gold ball of course but as mentioned effect because he's do find support we could see a bit of a push iron of course with Clarida's comments overnight bit of dollar weakness may have well impacted that move is well and gold late yesterday having a look at at oil obviously a decent move yesterday to the downside and that's continued to an extent this morning already down 78 ticks from the open be keeping an eye on starting on the you have to wait for me to find the exact there we go low the fifth I've got written down a trend line from that not an ellipse just keeping an eye on this trend line that you can see we broke through if I get it right there you go we broke through yesterday evening you want a line in the sand for you know what can this market maybe turn into a day or you've got one there keeping an eye you can see that second test just above my camera now just above it there on the 23rd really good support yesterday before that overnight break through we came back to retest it really nice level you've got $56 on the futures just above there as well but it's keeping an eye that could be your your cue to maybe get out short if you're still really holding the position but really nice line in the sand for the travers to keep an eye on and of course those longer term levels that and talked about you know if we can you know keep pushing down you know where could be the the level where we start to find support in just purely from a support resistance point you have $55 12 which would I know be interesting to some people the $52 $59 you can see all these lows from here that certainly you know be a place where people might attempt to get in but you know things probably do have to change certainly with the equity markets for there to be a full on recovery of course these markets very correlated at the moment as well as usual any questions do do let us know have a quick look over at the DAX before just wrapping up just to see how it's opening with that increase in volume we can see just over the last few minutes is breaking that low there so keeping an eye then on US equities as well around those lows the NASDAQ down the S&P looking very identical if they were to break we see the idea of a big push lower to 27-22 40 point move you never know you never know but I hope you all have a good trading day and a great weekend from everyone at the amplifier team


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internal revenue service IRS tax news IRS provides revised answers that's a tongue twister right there people give that one a shot IRS provides revised answers move on over Peter Piper with your pickled pepper picking we have a new tongue tying Titan in any case IRS provides revised answers for 2020 recovery rebate credit on tracing payments IR 2022-24 February 2nd 2022 Washington the internal revenue service today updated it's frequently asked questions the FAQs on 2020 recovery rebate credit fs 2022-08 there's a link to that here this updated FAQ frequently asked question includes a revision to the information on tracing payments under topic F finding the first and second economic impact payments amounts to calculate the 2020 recovery rebate credit question 8 topic F update these FAQs are being issued to provide general information to taxpayers and tax professionals as expeditiously as possible more information about reliance is available there's a link to that here more information on the recovery rebate credit is available on irs.gov there's a link to that here as well irs FAQs so I'm just going to link on over here and see what they say with that FAQ this is f8 missing economic impact payment how do I request a payment trace to track my first and second economic impact payments updated February 2nd 2022 so remember with the with the economic impact payments you got basically three of them that we are looking into the first two were tied to the 2020 tax return and therefore if there were any problems with them you would need to remedy that typically with the recovery rebate credit at this point in time by filing the 2020 tax return and then the third one is going to be tied to the 2021 tax year if there were any problems with it then the remedy would generally be to file the 2021 tax return filing for the recovery rebate credit for the third payment of course in order to do that you need to verify what kind of stimulus payments were actually received so you might need to get that information if you don't know what that information is from the irs to make sure that you're reporting the items properly as you go forward so here we go a 8 you should only request a payment trace to track your payment if your online account shows a first or second payment was issued to you and you have not received it so in other words you can generally look on your account for example to see if the economic impact payments have going out you can basically log in to the irs portal and try to track the payments that went out but obviously the payments that went out on the irs's side are tracking the payments that they believe that they sent out now at some point in time if the payment didn't actually get to you if they sent like a paper check or something like that and the recipient didn't receive it then they should be able to figure that out at some point in time but it's possible that they sent the payment but it was never received or there could be fraud that took place and basically the payment went to somebody other than the participant through it should have gone to now in that event then of course there's kind of a problem because if you record that you didn't receive the payment on the tax return it's not going to it's not going to match what the irs is saying because they're going to say we sent the the payment in the tax return and so obviously there's going to be have to be some way to work out those kind of issues as they come up so it says do not request so once again it says here you should only request a payment to trace to track your payment if your online account shows a first and second payment was issued to you and you have not received it so if they say if you're looking and they say you shouldn't be requesting the track in other word generally just to see if the payment went out because you can basically look that up on your portal you're basically want to look at it if they say that they sent it out and you're saying you didn't receive it then you might the next step is to say well okay what happened to it where did it go what account did it go to how did you send it what address did it go to that's when it looks like to me you would be using this kind of trace thing do not request a payment trace to determine if you were eligible for a payment or to confirm the amount of payment you should have received how we process your claim will process your claim for a missing payment in one of two ways if a trace is initiated and the irs determines that the check wasn't cashed your irs will credit your account for that payment so obviously if the payment went out on the iris aside and you're saying yeah I didn't get it and they look at it they say oh the check is outstanding we sent a check maybe it went to the wrong place or got lost in the mail and they can tell that because it hasn't been cashed then that's one kind of scenario you would think would be remediable but the iris cannot reissue your payment instead you will need to claim the 2020 recovery rebate credit on your 2021 2020 tax return if eligible so in that event they're gonna sit they might say in there on their side they're gonna say okay we sent out the stimulus payment but it didn't get cash so apparently you didn't get it because it never actually left our account on the bank on the irs side of things we're not going to reissue it but we will recognize that it didn't go out so when you file the tax return for 2020 you want to you want to then calculate or claim the recovery rebate credit and that would be the way to remedy that that's my interpretation of their FAQ here note if you are filing a 2020 tax return before your trace is complete do not include the payment amount online 16 or 19 of the recovery rebate credit worksheet if you do you may receive a notice saying your 2020 recovery rebate credit was changed but an adjustment will be made after the trace is complete and it is determined that your payment has not been cashed you will not need to take any additional action to receive the credit if you do not request a trace or your on your payment you may receive an error when claiming the 2020 recovery rebate credit on your 2020 tax return so in that event if you if you're basically look up in your account and it's the irs says that they sent you the money and and you on your side are saying well no I didn't receive the money then you properly record the the recovery rebate credit on the on the tax return it's going to cause a problem on the irs side which could take some time to remedy especially in the current situation because there's going to be possibly a person that has to kind of get involved there because the machine the system is going to say yeah the thing that they claimed is not tying out to what we thought we paid them and therefore they're going to have to dig into it a little bit deeper so in that event if that was the case that's when you might want to initiate the trace which possibly could make things go you know a little bit faster as they as they got to determine the situation so since the payment was issued to you you may not be eligible for any credit if the check was cashed the Treasury Department's Bureau of the Fiscal Service there's a link to that here will send you a claim package that includes a copy of the cash check so obviously if you if you basically say hey look the thing I you didn't send me the check I never got the check and they're saying I know I sent the check and not only do I know I sent the check I see that it was cashed so now it looks like it was cashed and it might not have been cashed by you then the next step would be well was it you that cashed the check if it wasn't you that cashed the check then we still have a problem but it obviously if you did cash the check and you got the check and you know it ends at that point the next step would be well the iris looks on their side the check isn't outstanding the check was cashed the question now is is it a case of fraud someone else basically cashed the check or did you cash the check and get in the event in that event then you would have received the payment so follow the instructions the Treasury Department's Bureau of Fiscal Service will review your claim and the signature on the cancel check before determining whether the payment can be reversed and your recovery rebate credit adjustment start a payment trace so to start the payment trace you can call us at<PHONE_NUMBER> there'll be a link to this in the description so you can get to this FAQ if it's applicable to you mail or fax a completed form 3911 taxpayer statement regarding refund there's a link to that here to complete the form 3911 write quote II I P one in quote or II P two on the top of the form to identify the payment you want to trace complete the form answering all refund questions as they relate to your economic impact payment when completing line seven under section one check the box for quote individual in quote as the type of return entered quote 2020 in quote as the tax period do not write anything for the date filed signed the form if you file married filing joint both spouses must sign the form you will generally receive a response six weeks after we've received your request for a payment trace but there may be delays due to limited staffing I get up to date status on affected IRS operations and services there's a link to that here did not mail form 3911 if you have already requested a trace by phone you'll confuse the IRS if you if you ask them multiple times for the same thing generally if you mail or fax to form prior to the time frames above your request will not be processed until those time frames are met mail or fax to form to note do not send any anything other than a form 3911 to the fax numbers below so they got this information that's related directly to these forms and they'll be a link to this in the description so they've got the locations and then where you're going to mail it to so you got your table here and your contact information if this would be applicable to you under this FAQ


In this video, we're gonna introduce one of the most powerful proof techniques in all of mathematics. I mean, this technique is used all over the place in abstract algebra, number theory, topology, geometry. I mean, you can use this all the time. Whenever a set is indexed by natural numbers, you basically can use the principle of mathematical induction to help you prove things about the growth of that sequence of the index set with regard to that natural numbers. Basically, whenever the natural numbers come into play, it's very likely you might prove something using induction. Now, it does turn out there's commonly referred to as two different principles of mathematical induction. In this video, we're gonna refer to the first principle of mathematical induction. Sometimes it's referred to as the weak principle of induction, since the second principle is commonly referred to as strong induction. But honestly, this adjective first is generally omitted, and so we just call this the principle of mathematical induction for which people might just say the principle of induction or heck, people will just sometimes say by induction, right? One of my proudest moments as a graduate students when I was writing the proof of a theorem, right? So there was a theorem, and then my proof was the following. It follows by induction. Okay, that's like verbatim what I wrote down for this proof. I don't remember the exact theorem itself, but it didn't matter at the time. This was a homework exercise for a graduate class in abstract algebra. And my professor, when he wrote, when he read the proof, he gives me a nice looking red check mark to suggest that I got full marks on this exercise here. He accepted my proof. And the idea was that at this point, he trusted me as a mathematical student that I said it, the proof follows by induction. I didn't provide all of the details. I didn't provide any of the details. I just said that it could be done. And he trusted me as a student that yep, it does follow by induction. And I bet you could provide all the details that would be necessary, but I don't need to see these details. And since I believe that you can provide them, you don't need to see these details. It's a super, super proud moment. Now, for most of us watching this video, we're not at that moment in our mathematical careers. This might be our first exposure to mathematical induction. So let's actually talk about all the important steps that go into mathematical induction. I do want to mention before we start that though, in the previous video for lecture 19, we introduced the so-called well-ordering principle. And what we will actually see later on in this unit about integers is that the well-ordering principle actually provides to us a proof of mathematical induction. So because every set of every non-empty set of natural numbers has a minimal element, we actually get the principle of mathematical induction. So we can prove this as a theorem of that. We will postpone that proof until we're a little bit better at induction first, if that's okay. So what does the principle of induction tell us then? So let S of N be a statement about integers for some natural number N. So this is some statement, which we'll see some examples of this later on. Oftentimes these are equations, for example, about integers, but they're indexed by natural numbers, which remember, zero is our smallest natural number in consideration here. Now suppose that the statement S of N sub zero is true for some integer N sub zero. Now these statements do not necessarily have to be indexed for all natural numbers, but there's some first natural number that is considered, that the statement is true for that one, N sub zero. Now oftentimes, that'll be the number zero itself, because the statement will be true for all natural numbers. Sometimes N sub zero, so N sub zero could be zero when you're considering all natural numbers. But oftentimes N sub zero might be one because we're only considering the set of positive integers for which the statement is true. It might not be the case that the statement makes sense for zero, like if there's division involved, you can't divide by zero. So maybe zero is exceptional, but the statement's true for all others. But you are gonna see situations where this number N sub zero could be two, right? The statement could be true when N equals two, and we're just considering integers for which N is greater than equal to two. I mean, why not start at 17? There will be situations where a pattern might not begin until the 17th number, and then after that it follows, okay? So this first number is an integer, it doesn't matter which integer it is, but there's some initial value for which the statement is known to be true at the number N sub zero. It could be zero, it could be one, that's likely what it is, but it could be honestly anything, all right? And then, so we have that, so we have a statement that's indexed by natural numbers. We know the statement is true for a specific integer, which again, N sub zero is indexed, it's indexed by the natural number zero, it doesn't have to be zero itself. If for all integers K that are larger than or equal to N sub zero, if the statement S of K implies that S of K plus one is true, then, so if that's true, then S of N is true for all integers N greater than or equal to N sub zero. So let me try to unravel what's going on right here. So basically what we're saying here is we have these statements about integers, again they're indexed by natural numbers.  Then you have your next number which is N sub one, and which of course is just N zero plus one, just to be clear there. You have this next number N sub one, and this is gonna be our domino as well. And then you have another number N sub two, you have another number N sub three, and be well N sub three is just N sub zero plus three. That's all the number is here. And then this continues on for a while, okay? So this condition right here, if for all integers K, S of K implies S of K plus one is true, I want you to think about with our dominoes here, we have N sub K, and then the next domino is gonna be N K plus one. So what we're saying here is that, hey, if this domino falls over, that domino will fall over too. So when you push the domino over, it hits the next domino, which then hits the next domino, which then hits the next domino, which hits the next domino, and they're all knocking each other over. You know, you're watching this video, it's posted probably on YouTube. There's lots of other cool videos you can probably find on YouTube about domino sequences. Feel free to look up one, right? You know, these people who set up these domino arrangements, they push over the dominoes and this whole cascading domino sequence happens, they're cool to hold, to behold, right? There's two important things that you need when you have these domino sequences that make it cool. One, you have to have the case that when one domino falls, the next domino falls. If your dominoes are so far apart that when they fall over, kerplop, it doesn't hit the next one, that's a bummer when it comes to this domino sequence. So they have to be close together so that one will hit the next, okay? But then also what's important is that someone has to push over a domino. If you just see all of the dominoes completely standing up and nothing else is happening, you're like, well, that's kind of cool but it's really no different than watching a painting or a mosaic or something. It's when they fall over that you really see the art form happening. And so that's what an induction proof is. It's dominoes falling over. This first statement is that there is a domino falling over, the first domino. So one pushes the first domino, then they fall over. Then the second statement right here, which is commonly referred to as the inductive hypothesis, you have to prove that if S of K is true, then S of K plus one is also true. That is, if this domino falls, the next one will fall as well. They're close enough together to knock each other over. And as this is an arbitrary case right here, this will actually show all of them falling over at once. Induction is a very powerful proof technique because it actually proves an infinite number of statements about integer simultaneously. You prove that it's true at some point and then if it's true at some place in the middle and the next one falls over, then all of them have to be true. So an induction proof, again, we'll prove why induction is a valid proof technique later on in our lecture series, not in this video here. But what I can tell you is the strategy of induction proof. It always comes in three phases. There's the so-called base case. You have to prove that the statement is true for your initial value. Whether that's n equals zero and equals one and equals 17, it's true for some initial value because someone's got to start pushing over the dominoes. Second, you have this inductive hypothesis where you assume it's true for S of K because when you look at the statement right here, S of K implies S of K plus one. That is a conditional statement. S of K implies S of K plus one. How do you prove a conditional statement? Well, by the method of direct proof, you assume the hypothesis is true and then you conclude using laws of inference and things. You conclude that the conclusion is true, okay? So the inductive hypothesis is the assumption of S of K. So you assume S of K to be true and then you proceed to prove that S of K plus one is true. I call that the inductive step. So here you prove it for S of N zero. You assume S of K is true and then you prove S of K plus one is true. Those are the three steps to any good induction proof and I want to now show you a few examples in this video of how one can prove things by induction. Perhaps the classic example is the following. Prove the identity of one plus two plus three plus four plus five all the way up to plus N is equal to N times N plus one divided by two, okay? Now this is actually an identity we've already seen before in our lecture series, we've proven this true using combinatorial proof. So it might be interesting to compare this proof technique versus the combinatorial proof we'd seen previously. It's a very different technique and it still proves the statement true but in a very different way. And so we're gonna proceed to prove this by induction. When people write induction proofs, they often declare this. They don't always do it at the beginning but we're very much practicing. So I'm actually gonna tell you that we're gonna prove it by induction. But of course, as you get more experience with writing proofs and reading proofs, it turns out that these declarative statements about what proof technique you're using become happening less and less and less because it's understood that the audience that if I start describing a base case and I start describing an inductive hypothesis and inductive steps, then clearly I'm doing improved by induction. But for beginners who are not used to the template, it's sometimes useful to say it. But like I said with an example earlier, if I'm in a homework problem as a graduate student, there was one point I basically said, we prove the statement using induction and that was the whole proof. Sometimes that's all you say, it follows by induction. Again, we're not at that level yet but maybe one day we can be there. You can have that proud moment too. All right, so to prove this by induction, what we have to do is we have to provide a base case. So when you look at this thing, like what is the smallest integer for which this makes sense? You actually can get away with this identity being true for n equals zero but you have to make some sense out of the left-hand side. What does it mean from one plus two to plus three all the way up to zero? It feels a little exceptional. So if you wanna start with n equals one, I'm perfectly fine with that. A lot of people prove this identity with n equals one as the base case. But like I said, you could get away with n equals zero because when n equals zero, you can think of the left-hand side as this empty sum. There was nothing over there, empty sum. With the empty sum, it's understood to be zero. So the left-hand side is zero and then when you put zero in here, the right-hand side's equal to zero. Not a big deal but like I said, the most standard way people approach this is they prove this identity for positive integers and therefore our base case is gonna equal one. So we can declare that. We will show the identity holds for n equals one. Notice that the left-hand side of the equation will be one in that situation because if you take the numbers one plus all the way up to one, that's just the one itself. Which again, I would argue that that case is just as, when you write it like this, that's just the same thing as the empty sum. It's no more exceptional but like I said, we'll stick with n equals one here. So the left-hand side will just be the number one. On the right-hand side, you end up with one times one plus one over two, which gives you two over two, which equals one. And so this tells you that when n equals one, this identity holds. The left-hand side's one, the right-hand side's one, that then proves the base case. The base case might be trivial but it is necessary. If you don't have a base case, then the whole process might not get started. It's an easy step but it's a critical step for induction. Sometimes there's so easy people limit them that is bad practice. If you're gonna provide the details of an induction proof, make some statement about the base case, even if you're like, oh, the formula is trivially true for, is obviously true. It's clearly true when n equals one. You could say something quite pompous like that. It's like, duh, it's true for n equals one. I'm okay with that but you should say something, right? But of course, provide enough details for your audience so that it's appropriate. Then the second step is the inductive hypothesis. All this is, this is an assumption. We assume that the statement holds for some value k. Generally I will use a different symbol than n which would appear in the formula. So that's like, oh, this is some fixed arbitrary number. It's true at that step, okay? So then you just assume it. You assume the inductive hypothesis. Then in the inductive step, you're gonna prove that so because we've assumed, we assume the statement holds for S of k. We proved it holds for S of one. We're assuming it's true. Now we have to prove that the statement is true for S of k plus one. So how do you approach that? Well, I'm just gonna take the left-hand side of the equation for S of k plus one.  And this is then how you typically can then bring in your inductive hypothesis. When you look at this sum, look at the first k terms in the sum. By the inductive hypothesis, the first k terms is equal to k times k plus one over two. So I'm gonna substitute that in for the first k terms of my sum. And you should make mention that you're using your inductive hypothesis here. By the inductive hypothesis, you get that one plus two plus three plus k is equal to k times k plus one over two. But you also have this extra baggage of k plus one. But hey, let's just work with this for a moment. k times k plus one over two. I wanna add to it k plus one. I can find a common denominator times two over two here. We get the following statement. Now notice that both of the terms have a k plus one in it. If you factor it out, you'll be left with a k plus one times k plus two over two. And I'm gonna rewrite k plus two as k plus one plus one. Cause then this right here, this is the right hand side of the equation associated to S of k plus one. We started with the left hand side of the equation associated to S of k plus one. So those two things are equal to each other. So we've now proven the identity for n equals k plus one. That then means that the identity is true for all integers greater than or equal to one. And that's by induction. So we scream induction at the end of this proof. And that's then it follows by induction. This thing is in fact the case there. Now I do wanna make a comment about our commentatorial proof here that we had mentioned earlier. That when you look at this induction proof, this shows that the formula is true. But it doesn't at all tell you where the formula came from. That's the interesting thing about induction is that if you have a formula that you think it's true, you can prove it by induction and boom, there it is. As opposed to the commentatorial proofs we saw previously for which in the commentatorial proof it actually gives us not just that the identity is true. It gives us good reasons to why we think it's true because of its commentatorial relations. And so there's some benefits and weaknesses to induction because of that. It gives us no intuition to why it's true other than it's true. But the advantage is if you're like a researcher, a mathematical researcher and you're exploring some new formula, if you think the formula is true because you've collected data and it seems to be working for a while, if you can prove it by induction, you don't need a reason why it's true. It's like, oh, I now know it's true by induction. And then like all of the reasoning which you did secretly in your ivory tower doesn't have to be published because induction gives the truth of full of it, the truthfulness of it. And even though you sort of discovered it because of data collection or something, that actually is kind of an asset when you think of it that way. All right, so let's look at some two other induction proofs before we end this video. We're gonna go through these ones a little bit faster this time. Remember three phases, base case, inductive hypothesis and the inductive step. This time we're gonna prove that every integer of the form 10 to the n plus one power plus three times 10 to the nth power plus n is divisible to nine. And we're gonna do this for all natural numbers this time. So this will include zero as well. So our base case, we're gonna say that we prove this thing by induction. Okay, so we declare into the audience how we're gonna do it. Then for our base case, we take n equals zero and just look what happens when you plug in zero. You're gonna get 10 to the zero plus one power plus three times 10 to the zero plus five. Now, of course, zero plus one is one, 10 to the first power is 10. Here, 10 to the zero is one times three is just three. So now we have 10 plus three plus five, which is equal to 18, 18 is nine times two. So it is in fact the case that when n equals zero, that number is divisible to nine. So the base case is then satisfied. Base case is done. All right, now for our inductive hypothesis, assume that the number 10 to the k plus one plus three times 10 to the k plus five is equal to nine m. So we're saying that this number is divisible by nine and that's exactly this number when you plug in a k. So this gives us our inductive hypothesis. We're assuming it to be true. All right, so then of declaring to the audience right here, next we will show that the number 10 to the k plus two plus three times k to the 10 to the k plus one plus five is divisible nine. So I'm telling you what I'm about to do. And where do these numbers come from? Now the exponent of the 10 is k plus one and this exponent right here is k plus one plus one. I've already took the liberty of simplifying, no big deal. So let's start with the number. I wanna argue that divisible by nine. So this is where the magic thing comes into play. How do I show that this is equal to nine? Well, it takes a little bit of, I'm gonna show you what's done here, but you'll see that it's valid. Coming up with the exact calculations here can be a little bit more tricky. But the idea is like, what could you do with this? I have to somehow produce this number in my calculation cause I know that's divisible by nine. Well, you have these 10 to the k plus twos and 10 to the k plus ones, but here you have 10 to the k plus one and 10 to the k. If I could factor out 10, that might be helpful. Not sure what to do with the five yet, but these first two numbers are divisible by 10. You could factor out the 10 and therefore you're gonna get 10 to the k plus one plus three times 10 to the k. That's all times by 10 plus five. This is almost the number I'm looking for, but it's missing the five. So a nice little maneuver, what we can do here is we can add five, but we have to also subtract five so things are balanced. Distribute the 10 onto that negative five and then separate it. This is the same thing as taking 10 times 10 to the k plus one plus three times 10 to the k plus five, which notice that by the inductive hypothesis, I know that number in the parentheses is divisible by nine. You have a plus five, which is already there and then you have this minus 10 times five, okay? Now, like I said, by the inductive hypothesis, this number I know is divisible by nine. It's equal to nine M, which if you times it by 10, you're gonna get 90 M. But then what happens here? 10 times five is 50, five minus 50 is equal to 45, which hey, 45 factors is nine times five. You could factor out the nine so you get nine times 10 M plus five. So notice that this last equation, because this equation right here used, excuse me, this one right here used my inductive hypothesis. This last equation shows that the number was divisible by nine. So that then completes the inductive step. And so by induction, these numbers are always divisible by nine. I had all of the steps there. We had the base case, the inductive hypothesis, the inductive step. So we proved because we assumed S of k holds, we then proved that S of k plus one holds. And that's all that's necessary for the inductive proof. It then follows my induction that for every natural number, this bozo right here is gonna be divisible by nine. To complete this video, let's do one more example here. Again, we're gonna go through this one even faster. And also what's different about this one is let's do an inequality as opposed to an equation. The last one we did have to do with divisibility, but divisibility does turn into something about equations because the number equals nine times something. Let's try an inequality to show you how you can prove an inequality using induction. We're gonna prove that two to the n is greater than n for all natural numbers, which again, that includes zero here. So we'll prove the statement using induction. Let's consider the base case. Two to the zero is equal to one, which is greater than zero. So it's satisfied in that situation. Now some of you, when you think of the actual numbers, we erroneously think that doesn't include zero. So you might've started with one, which note here, two to the one equals two, which is greater than one. So it's satisfied there as well. So if you started at zero, started at one, you would both be true. I actually did two, because sometimes in induction proof, you actually have multiple base cases. I've now proven the base cases because what I wanna do going forward is I actually wanna assume that k is greater than one. I included zero because that is a natural number. I proved explicitly that it's also true for one. So then I can assume k equals one, that it's going forward that k is greater than one, okay? And that's what I'm gonna do with my induction hypothesis. I'm gonna assume that two k is greater than k for all k greater than zero. So I had to specifically prove it for zero and one if I wanted to be true for all natural numbers, okay? Because when you do your inductive hypothesis, the k is assumed to be larger than all of the base cases you've considered already. So I'm gonna be larger than one in this situation. All right, you could do larger than or equal to your base case of course, but I specifically do need that k is larger than one here. So that's at least two. And so how are you gonna prove this inequality? Let's play with around with this. Notice what we have to prove. We have to prove that two to the k plus one is greater than k plus one. That's our plan of attack. So what can we do with two k plus one? Well, I noticed that two k plus one factors as two times two to the k. I like that factorization because that's been the left-hand side of my inductive hypothesis. So then I can use my inductive hypothesis. Two times two to the k is larger than two to two times k, right? Since my inductive hypothesis two to the k is larger than k, that then gives you the inequality we're looking for, okay? But then also note here that k is greater than one, which means k is greater than or equal to two, like so. So what we get here is that two to the two times k, of course, this equals k plus k, but since k is larger than one, remember there, we can get another inequality here. It's greater than or equal to one in that situation. And then notice when we put this all together, we now have two to the k is greater than k plus one. All right? Now notice, yeah, you do have equality, but you also have it greater than or equal to. So that's more strict. The strictest symbol we used here was greater than, there was a strict greater than there. So that then when we put those together, we get that two to the k plus one is strictly greater than k plus one, finishing the inductive step. So we have all three cases, the base case, the inductive hypothesis, and then the inductive step. Since we finished all three phases, the result then follows by induction that this inequality is valid for all natural numbers. We'll do some more examples of inductive proofs inside of lecture 20. So take a look there if you wanna see some more examples, but we'll sign off for right now. See you next time.


It's the final of the Norbils Swedish FIM Speedway Grand Prix here in Holstervik, and what a line-up. Hansa Janowski off the inside in red. Freddie Lindgren gate two in blue. Gate three white, Ty Waffenthal in yet another final. And Bartosz Marslik going off the outside in yellow. We have the top two riders in the standings right now in Waffenthal and Lindgren next to each other on the grid. Freddie Lindgren of course would want to give the Swedish fans a little bit of consolation after Lindgren beat them earlier today in the football. They did funny enough, two nil it was. Freddie Lindgren if he could come up with a stunning performance now, I'm sure he would say, well there's no question he's going to send the Swedish crowd home very happy indeed. He wrote brilliantly in his last outing, here we go. And it's a great start for Janowski off the inside gate. Now Lindgren tries the inside run as Waffenthal goes around the outside. Janowski has closed the door, and now Lindgren is going to try the inside run. As Marslik comes through on Waffenthal as well, but watch out for Lindgren. Here comes Lindgren into second place, riding beautifully. Janowski has that lead now, looking super here. And then Marslik giving the better of Waffenthal. And now Waffenthal trying to put the squeeze on Lindgren. Oh my goodness me, Marslik's gone wide. And now Lindgren chasing Janowski down. Brilliant start from Janowski, all Waffenthal's out of shape. Very nearly ran in the side of Marslik there. Janowski the fast-starting Polish boy. Absolutely fabulous away from the inside. Lindgren on the inside, coming on strong. Waffenthal's out of the back and that is a shot. Marslik working overtime in third place. But Janowski with less than half a lap to go has written brilliantly. It's going to be a grub. Six straight wins on the night. That is a fantastic performance for Janowski. Lindgren holds second from Marslik, third. And that means that overall now, Ty Woefendern goes to Cardiff as leader of the world championship by an 11 point margin. Yep, fabulous night of speed. What a way to finish it here tonight. Janowski has been absolutely outstanding after a disappointing ride initially. He has come good 18 points in the next six races. You can't do better than that. You know, I'm amazing. That was a very hard beginning because I didn't score any points in my first run. But then we were still working and staying focused. And finally we scored very good points. So I'm very happy.


here we are at a vertical mine shaft it's a copper mine there is a piece of ladder here but i won't trust it it's just sitting there and the rungs are broken and doesn't look so good so we're going to just repel this first part and we'll see what shape the ladders are in after that if we need to we've got much more rope here we can drop down but right now we've just down to the first landing so we've put our anchor up on this one inch steel rod that's part of the um head frame here holding the timbers together and we backed it up to another one behind here so that should be good and this will give us enough length to come up and be able to get back up on stand on the platform here all right ready ready good luck be careful looks like everything's holding okay if you see a hawk coming after me holler at it because there's a big nest down here i know it's huge probably about a i don't know it's like three feet diameter of sticks okay i'm looking back up that's where that nest is and then look at all the branches that they've pulled in that have dropped off think of the work that that bird did on this nest i'll need to have all of those fall in so anyway i'm here at the top of a skip this looks like a dump pocket it's got a nice steel plate in the back there and the skip would come up and dump right down to that chute down there and then looking this way it goes down about another oh i don't know 70 60 70 feet and then it looks like it flattens out quite a bit because we're vertical right here and then it looks like it flattens out quite a bit maybe two or 45 or so so i'm one ladder section below that where the top of the skip rail curve like that and here's the bottom of that chute that was the dump pocket and here's where it comes out and now this must have been a haulage of some sort that bird has a lot of sticks doesn't he watch out behind you okay we're going to take it right off of the vertical shaft comes up to some big shoots there's a huge hanging wall on there right here it's like possibly an or pass and then a big chute behind here heavy duty stuff isn't it 250 s3 and we're going to walk gingerly because it's kind of dusty great powdery there's a ladder a well used ladder oh man that does go way up there i wonder what that is boy that's kind of a lot of use hasn't it some more chutes several chutes must be some big stoves up above us look out of the look out of the orchard we're whacking this post i'm in a big hole there all right i suppose that's so you can latch it and that's a heavy duty door that's full uh full dimension lumber look at the size of those two by fours big stuff i'm not sure what this thing is obviously got some purpose four chutes you can see part of the mineral vein up behind the the lagging there something tried to crash you there didn't it they must have taken this whole vein out yeah it could be looks brand new in there yep it's manly well the dust isn't kicking up too bad now look at this cribbing with the gobbling in it they stuck a ladder up there oh boy they used a ladder for part of it look at that pressure here oh yeah a lot of pressure oh yeah well look at that that's compressed look at this huge hanging wall it looks like it's caved in it you don't think so oh yeah maybe not it's like it ended there it's possible why the stuff's under some pressure isn't it that was a pretty wide vein that's gotta be 15 feet 12 15 feet from here oh yeah okay here we are back to the vertical shaft we came down those faces a couple of faces there's any dates anywhere all right let's go this way this looks like it splits already there's some kind of something in pencil there's more of the vein okay let's let's this probably goes around in the loop I'm guessing who knows a huge pillar in the middle here why these are all buckled too aren't they it's generally a pretty good shape though despite being under tremendous pressure buckled all these timbers so I wouldn't go doing pull-ups on one of those there's another chute so there are obviously some some stopes up above us that's got its gates in there I can't get it out yeah the amount of wood is crazy s50 250 s2 that's pretty neat it's like somebody broke through the wall here yeah I think this must look round I don't know we'll keep going here it definitely goes through look at the blue over there wow this is a pretty cool mine isn't it there was track throughout all this oh boy what do we have here the ladders and then we double compartment shaft look at this is natural timber oh yeah for a yep look at all the rounds up in there okay so this is the face of this I like the way it looks where they've fueled that bark almost looks like you would use that in a structure in a side of a nice house yep the log cabin let's go back and go through that hole I guess not an ore pass but a man pass look at the gob wall out of out of uh what you call it copper you don't see that too often do you so that ends there okay we've kind of looped back towards the main shaft now from over that way and that's straight in yeah big angle still this big pillar they left in here wow those are some hinges all right holy buck it's the size of those things go put your hand up there we it's gotta be they gotta be a foot on each side they gotta be a foot long at least huh wow open the door once in here nice box yeah there's a nice box you can live in here that was a just right box whatever that is I'll have to read it later I took a photo of it I'm packing I know it was like for a bearing you know for a seal and a shaft or something it's like a table and some uh yeah some pocket shells in the end of the table here this doesn't resemble the traditional fuse table but there is a spool here I mean a spindle that would have held a spool it looks like and then maybe they would have done their cutting on here for their fuses measured it out some more that's pretty nifty this is heading back towards the shaft now we looped around that center big um so here's the other way we looped around this big center part and this is heading back towards the the ladders in the shaft there's some load counts but I think first we have a ways to go down here don't we down here doesn't look too terrible doesn't this first ladder's pretty straight and about probably almost 20 feet I think we can do it ladder seems pretty good good looks like it goes straight down and flans all a little bit I'll hit the next ladder okay okay coming Julie watch that last step don't get your shoe on the nail though oh you did it see what I mean oh yeah I did the same thing one more step down looks like there was a another hoist down here or some sort to go from this this step down look at the big uh big hinges again but look at the big pin yeah that's cool oh my gosh want to open that up once see how it looks we're ladders oh boy oh and it gets deeper oh as far as I can see down there that's how long we've done so look what's right over here oh there's the skip car without any wheels but we can go look at it this is what we just came down don't ladders when they see that top ladder is vertical but from here you'd swear it was leaning back it's an optical illusion that happens like that when you're doing ladders and there's a dead bird over there oh yeah it's like a raven or a possibly a vulture no it looks like a raven than that so this is a skip car and the wheels are all gone all four of them but this would have run up the uh the track right behind us that we just came down it's nicely riveted it's still in beautiful shape except for the wheels are missing that's a pretty big one too actually okay let's go this way once yeah what does that say silver bar silver bar paint fruit it's showing fruit I know but it's a swab bucket I know well Bartlett pairs silver bar pairs that's a big bucket of pairs silver bar pairs there's a big spike next to it size that thing that's about uh I don't know how many penny that is 30 yeah well let's follow this looks like they had some ventilation down here you know there's some big mud in there I don't know what this was for maybe they just took that up here's a ventilation they got a well like a 10 or 12 inch round pipe yeah that's one of our cable guides yeah I don't know if it should be sitting there or not no probably not leaning up against it you fall on your toe that'll wreck your day another door big door they had this one sealed up pretty good with some rag stuffed in there another one with a big hole in it I think the hole is so you can reach through and you reach through here from the other side right oh here's a turning plate for the for this intersection instead of having a switch this is like a four-way intersection here that's just a short siding but it looks like it goes that way and to the left so so you can take your car and just spin it on this plate and you can redirect it in do any direction you want let's keep going straight this is a lot of mine isn't it yeah somebody's lunch yeah there's a piece of trackly in there is it standing on it oh on the top and on the side it's upside down that's the top that's the top some initials and then looting I've never seen anybody mark reel on the top yeah because it's gonna wear off maybe it's just a shipping deal it's like a little ground full there oh there's a nice uh carbide can still got the blue on it union carbide usually anyway well you wouldn't want this one to fall on you would you man that would hurt what does it say in there mine lamp oh yeah oh but the other side can you go spin it around there is the union carbide cool all right we're moving on past the union carbide can goes up into the stoves there's a big shoot we just passed a lot of track ties all the track has been taken out okay so far and some of these are uh natural timbers like this one and some are milled lumber there's another table up here or something it's the face over that way yeah another hole there's a little piece of ladder over there well I don't see any reason to go down this little hole we've got more ladders to go down possibly people have been digging their way in this more recently than when this mine was being officially worked doesn't it you see these little areas here yeah a lot of cool minerals in the world okay heading back towards the main shaft okay here's back to the turning plate we came from that way where the door is and let's take the right we didn't go this way yet oh look at the blue oops well that ladder doesn't go anywhere look at the minerals my goodness that's amazing in the colors it's almost black it's so blue down in there in that crevice all right let's catch up with you here's another turning plate huh what's this it's just kind of back the way we came from where this tangles all around doesn't it there's a small bucket with a tobacco can in it here's a ladder going back up well I think this is the uh that's the one we looked down earlier okay moving past that area we're the same shoots upon shoots upon shoots and then I see the I see the uh face over here now that's way up you can see up into the stove it's a clean mine isn't it yeah it doesn't really smell or anything no it's nice then here's the face of this all right let's go we gotta go uh we gotta go back to that plate we gotta go back to the main shaft and go the other way okay here we are back to this second turning plate that we saw we never went straight this way oh boy I feel air here cold air kind of wild mine isn't it here's another hoist uh support I don't know what happens after the end of this ladder can you get down there let's finish this out here there's more uh what does this say oh here really it's upside down pumpkin pumpkin huh oh look at that interesting gobbling well it's almost like a work of art all the different colors what a minerally area oh yeah a wood roller a wood cable guide that's cool yes it is is it still spin of course it does okay let's go back and go the other way from the main shaft and this time we're really going to do it there's another heavy-duty box here they must have kept something in there that they wanted secured like primers maybe something that they didn't want to blow okay here we are back to this first turning plate that we saw the door should be right here yep and then the uh the main shaft should be right up here somewhere yep there's the latch there's the skip car do we look this way okay there's the skip car there's the main shaft there's that door we opened now let's go explore this way I see some fuel cans one's got a pretty good label on it still aristo motor oil union oil of california it's pretty neat and it looks like a big carbide top right below you too to see union carbide on it they usually do that can is pretty cool you don't see a lot of cans like that with good labels on them i'm usually gonna rust it off or something there's another door it's almost identical to the one we came to does it open no but it's oh that looks like it's float in from somewhere something caved okay no more that way well we could go down if we wanted to okay want to can you open that again need help no i don't really got the door open okay here we go down we go that is pretty good so far okay looks like it gets deeper there she is that's a long ways on that's a lot of steps there it is and there's still a lot more down this way as far as you can see a lot of track in this one oh there's a pretty cool dynamite box here it's not complete but it's a Hercules it's an older one that has these steel straps on it and then look it has one it has a bar on it for a lifting bar yeah yeah that's pretty cool i don't know what that rack is for obviously for something is this a stemming rod yeah where what could be looks like it done it this is for packing their dynamite into the drill holes for stemming the hole so the charges doesn't come up back out the hole looks like it has a date on there April 15th of 1920 something i can't really read it we'll have to look at the picture that we just took kind of cool look left hand rules for Julie still this huge hanging wall there's a big chute gobbling wall yeah it does to me looks like it ends right there oh is that a crusty looking thing so i peeked around the corner there that's the end of that it doesn't do anything this is a big cave so i didn't want to go fooling around in there okay back to the other way this level isn't as big as the upper ones so far oh yeah look at that another door or a blocker down there here's back to the shaft oh yeah there's some more dynamite box parts that might be part of that one over that we were just looking at a mess of track here so i need a deeper you're going to mine's kind of the more messy they get with stuff laying around i suppose it's easier to pull things out of the upper levels track and stuff reuse it i had a chunk down here here's another big chute full of blue rocks another man way up to the next level or a sub level i don't know maybe just up into the stove because the ladder we came down was a lot longer than that one was and here's the end of this one so that was the third level that we were just down right right so i stopped a little ways down that last set of ladders tom's way down there and he's telling me to wait he's not sure it's there's a level plus the ladders are pretty beat up he's pretty far down there okay i'm all the way down into the sump which is the very bottom of this shaft and there's nothing down here you can see it has been flooded before look at the water line so right where i'm standing it's been about up to my head so that was a good 150 feet really you shine your light a little bit somewhere okay i don't know why they do that unless they follow the rain down here but it always seems like they make these shafts way deeper than what you know there's no level down here so okay back up we go looks like a little pulley there that the cable to ring the bell for the hoist operator would have run on that pulley all right here it comes from the bottom of the mine all the way down there and all I do was a sump now you know so that's going to do it for this one you guys thanks for watching we'll see you on our next adventure yeah


Hello, hi everyone. My name is Matt Lingard and I am chairing this session. Welcome to those of you I can see joining now and also a shout out to those watching this at some time in the future, who knows when. The title is on the screen now and hopefully you're in the right spot. The session is 25 minutes in total and our speaker, Marlise will be talking for about 20 and then there'll be an opportunity for you to ask questions. So please take advantage of the comments, post your comments, say hello and line up any questions you have and I won't waste any more of that 20 minutes. I'll hand over to Marlise now. Thank you. So welcome everybody to the session on from digital ethnography to exhibition exploring the entanglement of learning online work and life. My name is Marlise Gratian and this presentation is co created with Paula Shaw. She's unable to be with us today, but I've added her email into the PowerPoint as well and you will receive the PowerPoint after the session. So if there are any questions based on the presentation, feel free to reach out to us as well. So what did we do in terms of the research project that we undertook it's underpinned by the advanced he flexible framework. And we observed six higher education modules within the University of Derby over a 12 week period from February to May 2021. And these modules included different disciplines and levels of study, including to fully online modules to apprenticeships and to practical modules that transitioned to off campus blended learning due to the covert restrictions. The multidisciplinary research team of academics, students, media learning technologists and learning designers engaged with six module leaders and over 20 students in a series of in module vlog reflection reflections and captured module level learning analytics. Just a point to note and I think it's one of my the core messages of the presentation is around why would you use an ethnography as a method. I don't know if everybody's familiar with an ethnography so let me just go into that a little bit further. So ethnography literally means to write or represent a culture. This includes participant observation where the researcher participates in the society or culture being studied yet retains an analytical or observational position if you will. So that through reflection and analysis the ethnographer can describe and interpret the place, the people and its activities. An ethnographer looks for patterns describes relationships and picks meanings and ethnography is both spatial and temporal occurs in physical location or movement between physical locations and occurs over a period of time. The complexity of digital ethnography is so just to just step back a little bit so if you think about digital ethnography you're almost talking about being in a studying a place within a place because a student or a lecturer in a digital online learning community is within that digital space but they're also within their own space or within within the university space. And so digital ethnography is also spatial and temporal yet these spaces and time periods are complicated by the digital medium. Participants may access the virtual learning environment the VLE from a single location on a fixed computer or laptop via Wi-Fi or fixed ethernet connection. And participants may access the VLE from several locations on different fixed computers and or laptops. So either at work or via their mobile phone as well or at home and participants can access the VLE from their mobile phone moving between locations as well. Depending on the VLE participants can access the same space at the same time from different locations across the globe giving rise to unique contexts. Digital spaces at any one time participants will access a single VLE site plus they may have other digital spaces open to such as other browser tabs and apps. These digital spaces may access a combination of study work and personal spaces all at the same time. So when we're doing this ethnography and when we're trying to embed ourselves within that digital learning community there are spaces that will be open to a student or a lecturer that we as observers or researchers might not see or have access to as well. Time in digital space works differently as well in comparison to physical space. Once text images and videos etc are submitted to those spaces they become decentralized conversations, persistently retrievable and unless edited or deleted they remain in that same condition as they were first posted. This makes it easy to feel asynchronous conversations that happened in the past as events in the present time. So where in a physical space something would happen at one moment in time and that moment in time will not be repeated. Or you have to physically repeat it again within a digital space. These events have a longer temporal stamp if you will. So that gives a little bit of background on what ethnography is and specifically also how ethnography in that digital space can be quite complex. I'll go to the next slide. So the research focus for us was to when we observed these six modules to look at the entanglement of participants studies work and home life in relation to credit hours. And credit hours can then be divided up between on the one hand synchronous and asynchronous contact time and on the other hand independent private independent learning as well. In terms of the dissemination of the research. So there are we've the first phase of the research has now been disseminated and this presentation is part of that dissemination. But we've also launched a MOOC, a massive open online course, as well as a virtual exhibition. And I will go into the virtual exhibition in a moment. But just to mention that so Paula Shaw is the head researcher of this project and she's coordinated all the different teams with that have participated in the research. I was an observer in one of the modules and there and what the module that I observed was a dance module. And it was very interesting to on the one hand sit into a classroom space and observe the students and their tutor in how how they use these different spaces. So on the one hand, especially in dance, the physical space, you know, did they have enough room in their in their dorm or at home to perform the dance in some of the classes they went outside. But on the other hand, the digital spaces as well. So you could clearly see that some students were on their mobile phones, whereas other students used a laptop or desktop. You could see that in terms of interaction with certain media that made it easier or harder for them to to interact with the rest of the group. And so what we've done with the students is we've created a virtual exhibition and this is what I will be showing you today. And they've chosen from their video diaries, they've chosen elements that they want to highlight to you as a visitor to that virtual exhibition. So I'm going to talk through some of those but just to give you a sense of the space that that virtual exhibition is in. What you see now on screen is our atrium at the University of Derby. So this is a space where where basically is the the central hub of the university. All the classrooms and lecture halls are connected to the central space. And there are rooms for each of the modules. So we've got the operating department practice room. We have a room on senior leadership. We have a room on sports therapy, which was one of the other applied modules that we looked at. And there's the dance room. I'm just going to show one of the rooms just to give a little bit of a sense of what the exhibition then looks like. So in each of the rooms, there are artifacts and the artifacts are color coded. There is an amber color, which is what the researchers have observed. A green color is what the tutors of the module have observed. And then there's the red color, which are student observations. And if you go into those, there is either text or in some cases it's an image that you can then explore. And I welcome everybody to enter this virtual exhibition after the talk. Like I said, you will be able to find the link to the virtual exhibition in the PowerPoint. And I can at the end also copy the link over so that you have it in the chat as well. In the presentation, I will highlight a couple of artifacts for you. And I'll do that using the initial questions that we've asked in the research. And these initial questions come from the QA document that is referenced here and that you can access as well. The four questions are, is there a strong pedagogic rationale for whether sessions will be synchronous or asynchronous? How have learner analytics been used to enhance program session design? How learning analytics and are there opportunities for students to personalize their learning experience working at their own pace, accessing and revisiting bite-size asynchronous materials? So those are the four questions I will sort of illustrate with some of the artifacts from the exhibition. So the first question, was there a strong pedagogic rationale for whether sessions will be synchronous or asynchronous? What we noticed from a student point of view is that they had a physical or temporal reason why they would attend asynchronously or synchronously. One of the students highlighted that they were studying whilst at work as well. This was from one of the apprenticeships. And they were based in Egypt, so they also had a time zone difference to take into consideration. For them, it was really difficult to go to the tutorials and the synchronous sessions. And therefore for them, there was definitely a reason for the asynchronous materials. They were very useful. Other students really used it to manage their own time. So this student mentions, I found that waking up an hour earlier than I usually do, I'm really focusing on my studies before anything else has happened in the day is a great way for me to work. And this shows sort of how asynchronous can help as well with creating that flexibility. We'll go into flexibility a little bit more later as well. This was one of the tutor observations. So it is not necessarily a pedagogic reason for using asynchronous or synchronous, but it is a very practical one because they're saying that they, the students are telling me about their experiences within the clinical work placement. Lots of reflective pieces are being written. I'm going through and I've been commenting on upon some of them. Obviously this takes a huge amount of time, but I'm responding to at least one learner with one of their particular entries. It's really, really hard to run a module when there's so many other things to do within my work role. So what we see here is that the tutors as well have those two spaces, that work space that is digital and their physical temporal space. And they're struggling with that. Again, for us it was interesting with this question that to go back to the question, is there a pedagogic reason for using asynchronous or synchronous? Actually, from our observations and from the artifacts that we've collected, what we see is that it's more about time management and it's more about how do you manage those different spaces. The next question, how have learner analytics been used to enhance the program or session design? The same student that we looked at before when they were looking at studying whilst doing work. It's interesting how you plan, for example, your tutor sessions or your synchronous sessions. Because this is from our Blackboard system, we could see when students logged in and when most students were online. Using data like this to identify the best time to meet is definitely advisable. And it's something this lecture has done to later on accommodate the time zones. And one of the things we do do within the university as well as engagement checks, we do them at week three and week five. And so this lecture did do an engagement check at the beginning of the week. Three learners were picked up not having access to system for over seven days. And for one of them, it was actually 23 days where they've had to intervene. And so this is the type of data that has been used. We are actually quite surprised how little data is used. So that's definitely a focus for going forward. I will skip these slides on interaction and go to the third question. Are there opportunities for students to personalize their learning? There are two examples here. One is from the dance module and the other one is from the education module, if I'm not mistaken. In the dance module, there was one of the students who couldn't attend one of the dance classes. And normally in a dance class, you would have then missed quite a lot of your studying. But in this case, because all the materials were asynchronously available, she was able to pick that up really quickly. And she was quite pleased with that. And then this is someone who actually was working ahead because they also had their work and another study program that they wanted to get into. And so because of the asynchronous materials, they were able to work ahead. I am going to move at a quicker pace because I know that I'm running out of time. I just want to highlight one more time in terms of next steps what is available to you. So we've got the MOOC as well as the virtual exhibition. This is the beginning of the project. So as you saw on the beginning slide, this is a pilot project and we will continue this research. And one of the things, because you can see that from the examples from the artifacts I've shown, some of this might come across as very anecdotal. But if you do it at a larger scale, you do really start picking up on themes. And those themes are things that we can then use in our design processes as well. So I'm going to just show the next slide. Sorry, I'm struggling. I'm now struggling with the amount of screens I have on. So I want to thank our project sponsors. The project is funded by QAA and SingLink as you have seen has provided the virtual exhibition spaces for us. And that leaves me with a little bit of time for questions. If there are any, please let me know. And like I said, if there are any people who are watching this later on, please do feel free to raise any questions with myself and Paula Shull as well via email. Thank you. Thank you so much, Mali. That's fascinating. I must a personal thank you for the introduction to digital ethnography, which really helped this beginner over here. It's a really great presentation. And we've got a question from Emma, which will come up onto the screen and also request for the links to be shared, which I'm sure we can do as well. I'll let you read that question. So did I get any feedback on why it wasn't used? No, not really. We did have some structured interviews with the module tutors. I think we did notice a difference between the module tutors for the online modules who did use some of that data. But that's because they're more used to, I think, using the data to to see the students, if you will. We often in our learning design team talk about making the student visible through activities that then trigger data points as well. So I think they are a little bit more aware of that. I think in the COVID situation for some of the lecturers, we could really see them struggling with that online space. We could really see, for example, one of the lecturers commented on the use of cameras and microphones in synchronous sessions, for example. So they were trying to get students to get more engaged in that and they weren't really engaging. And I think what we've seen so far is we've observed what happened this iteration. And now, after this pilot study, what we'll be doing is looking at, OK, so do we want to put certain interventions into place? Maybe we want to train module tutors on the use of data and how to interpret that. Because I think, ultimately, that's my gut feeling is they don't necessarily know how that data translates into what you would normally do as a tutor in a physical space. So in a physical space, you don't have that type of data, but you have the faces in front of you and the question marks you can see by facial expressions, whether things landed well or didn't land well. You can, you feel it, you know, confusion in the room, if you will. And I think there's some training. There's a skills gap, I think, that certain module leaders have in that area. Yeah, I mean, I think you're absolutely right there, Mollies. And I think it's one that people would be interested in as well. Certainly talking to some of the staff we work with, they really are keen to get into this stage and understand it. But I think there's a lack of capability to be honest. Thank you so much. I don't think I've got any other questions in the chat for you and we are actually almost bang on time. So I think I will have to stop it there anyway. So thank you so much for your presentation, which was excellent. And, you know, everybody, thanks everybody for coming. But what I really want to say is how much we appreciate the time you'll have taken, I know, to submit this, to write it and to present it today. So it's much appreciated. And I'm sure the big people checking back online to watch it as well. So thank you very much. Yeah, thank you for having me. And I'll post the links. Shall I still post them in the chat? You'd better put them in Discord because as soon as we end here, the comments will go. So if everybody wants to look in Discord afterwards, then they'll be there. Great. Thank you very much everybody.