Live from Stanford University, it's theCUBE, covering the Women in Data Science Conference 2017. Hi, welcome back to theCUBE. I'm Lisa Martin, and we are live at Stanford University at the second annual Women in Data Science One Day Tech Conference. We are joined by one of the speakers for the event today, Claudia Perlich, the Chief Scientist at Distillery. Claudia, welcome to theCUBE. Well, thank you so much for having me. It's great. It is exciting. It's great to have you here. You are quite the prolific author. You've won data mining competitions and awards. You speak at conferences all around the world. Talk to us about what you're currently doing as the Chief Scientist for Distillery. Who's Distillery? What's the Chief Scientist role? And how are you really leveraging data and science to be a change agent for your clients? So I joined Distillery when it was still called Media Six Degrees as a very small startup in the kind of New York ad tech space. It was very exciting. I came out of the IBM Watson Research Lab and really found this a new kind of challenging application area for my skills. And what does a Chief Scientist do? It's a good question. I think it actually took the CEO about two years to finally give me a drop description. And the conclusion at that point was something like, okay, there is technical contribution. So I sit down and actually code things and I build prototypes and I play around with data. I also am referred to as intellectual leadership. So I work a lot with the teams, just kind of scoping problems, brainstorming what may work or doesn't. And finally, that's what I'm here for today is what they consider an ambassador for the company. So being the face to talk about the more scientific aspects of what's happening now in ad tech, which brings me to what we actually do, right? So one of the things that happened over the recent past in advertising is it became an incredible playground for data science because the available data is incomparable to many other fields that I have seen. And so distillery was a pioneer in that space starting to look at initially social data, things that people shared, but over the years it has really grown into getting a sense of the digital footprint of what people do. And our primary business model was to bring this to marketers to help them on a much more kind of individualized basis identify who their customers current as well as futures are and really get a very different understanding than these kind of broad middle-aged soccer mom kind of categories to honor the individual tastes and preferences and actions that really truly reflect the variety of what people do. And many things, as you mentioned, I mean, I published on also mom and I have a horse. And so there are many different parts to it. I don't think any single one description fully captures that. And we felt that advertising is a great space to explore how you can translate that and kind of help both sides, the people that are being interacted with as well as the brands that want to make sure that they reach the right individuals. Very interesting. As buyers journey has changed to mostly online. It's an incredibly rich opportunity for companies to harness more of that behavioral information and probably see things that they wouldn't have predicted. We were talking to Walmart labs earlier and one of the interesting insights that they shared was that especially in Silicon Valley where people spend too much time in the car community and you have a long commute as well by train. And you'd think that people would want, I want my groceries to show up on my doorstep. I don't want to have to go into a store and they actually found the opposite that people in such a cosmopolitan area as Silicon Valley actually want to go into the store and pick up their grocery. So it's very interesting how the data actually can sometimes really change. It's really the scientific method on a very different scale but really using the behavior insights to change the shopping experience but also to change the experience of companies that are looking to sell their products. And I think that the last part to that puzzle is the question is no longer like, what is the right video for the Super Bowl? I mean, we have the Super Bowl coming up, right? And I did a study like, when do people pay attention to the Super Bowl? You can actually tell because you know what people don't do when they pay attention to the Super Bowl? They're not playing around with their phones. They're actually not playing Candy Crash and all of these things. So what we see in the ad tech environment, we actually see that the demand for digital ads go down when people really focus on what's going on on the big screen but that was a diversion. It's very interesting though because it's something that's very tangible and very, it's a real world application. So question for you about data science and your background, you mentioned that you worked with IBM Watson. Forbes just has said that data scientists is the best job to apply for in 2017. What is your vision? Talk to us about your team, how you've grown that up, how you're using big data and science to really optimize the products that you deliver to your customers. So data science has really many, many different flavors and in some sense, I became a data scientist long before the term really existed. Back then I was just a particular weird kind of geek but all of a sudden it's- Now it has a name. Now it has a name, right? And the reputation to be fun. And so you see really many different application areas depending very different skill sets. What is originally the focus of our company has always been around, can we predict what people are going to do? That was always the primary focus. And now you see and it's very nicely reflected at the event too, all of a sudden communicating this becomes a much bigger part of the puzzle where people say, okay, I realize you're really good at predicting, but can you tell me why? What is it like these nuggets of insights that you mentioned? Can you visualize what's going on? And so we grew a team initially from a small group of really focused machine learning predictive skills over to the broader, can you communicate it? Can you explain to the customer and our brands what happened here? Can you visualize data? So that's kind of the broader shift. And I think the most challenging part that I can tell in the broader picture of where there is a bit of a shortcoming in skill set, we have a lot of people who are really good today at analyzing data and coding. So that part has caught up. There are so many data science programs. What I still am looking for is, how do you bring management and corporate culture to the place where they can truly take advantage of it? This kind of disconnect that we still have. How do we educate kind of the management level to be comfortable evaluating what their data science group actually did? Whether they're working on the right problems that really ultimately will have impact. So I think that layer of education needs to receive a lot more emphasis and compared to what we already see in terms of this increased skill set on just the sheer technical side. So you mentioned that you teach when we were before we went live here that you teach at NYU, but you're also teaching data science to the business folks. So I'd love for you to expand a little bit more upon that and how are you helping to educate these people to understand the impact. Because that's really a kind of a change agent within a company. That's a cultural change, which is really challenging. Very much so. What's their perception? What's their interest in understanding how this can really drive value? So what you see, I've been teaching this course for almost six years now. And originally it was really kind of the hardcore quarters who also happened to get a PhD on the side who came to the course. And now you increasingly have a very broad collection of business-minded people. I typically teach in the part times, meaning they all have day jobs. And they've realized in their day jobs, I need this. I need that skill, that knowledge. And we're trying to get on a ground where without having to teach them Python and R and whatever the new toys are there, how can you identify opportunities? How do you know which of the many different flavors of data science from prediction towards visualization to just analyzing historical data to maybe even causality? Which of these tools is appropriate for the task at hand? And then being able to evaluate whether the level that of support that machine learning can bring is it even sufficient? Because often just because you can analyze data doesn't mean that the reliability of the model is truly sufficient to support then a downstream business project. And being able to really understand those trade-offs without necessarily being able to sit down and code it yourself, that knowledge has become a lot more valuable. And I really enjoy the kind of brainstorming when we're just trying to scope a project when they come with problems from their day job and say, hey, we're trying to do that. And saying, well, are you really trying to do that? What are you actually able to execute? What kind of decisions can you make? And this is almost like the brainstorming in my own company now brought out to much broader people working in hospitals, people working in banking. So I get exposed to all of these kind of problem sets and that makes it really exciting for me. Interesting, when Distillery is talking to customers or prospective customers, is this now something that you're finding is a board level conversation within businesses? No, I never get bored of that. So there is a part of the business that is pretty well understood and executed and that's you come to us, you give us money and we will execute a digital campaign either on mobile phones or on video and you tell me what it is that you want me to optimize for. Do you want people to click on your art? Please don't say yes. That's kind of the worst possible things you may ask me to do, but let's talk about what you're going to measure, whether you want people to show up in your store, whether you really care about signing up for a test drive and then the system automatically will build all the models that then do all the real time bidding. So advertising, I'm not sure how many people are aware, as your New York time page loads, every single ad slot on that side is sold in a real time auction. About 50 billion times a day, we receive a request whether we want to bid on the opportunity to show somebody an ad. So that piece, I can't make 50 billion decisions a day. It is entirely automated. This is fully automated machine learning that just serves that purpose. What makes it interesting for me now is that now this is kind of standard fare if you want to move over into the more interesting parts. While, can you, for instance, predict which of the 15 different creatives I have for Giovanni? Should I show you? The one with the woman running or the one with the kid opening, so that there is no nuances to it and exploring these new challenges or going into totally new areas, talking about, for instance, turn prediction. I know an awful lot about people. I can predict very many things and a lot of them go far beyond just kind of how you interact with ads. It's kind of almost the most boring part. We can see people researching diabetes. We can provide snapshots to pharma, telling them here's really where we see a rise of activity on a certain topic and maybe this is something of interest to understand which population really is driving those changes. And so these kind of conversations really make it exciting for me to bring the knowledge of what I see back to many different constituents and see what kind of problems we can possibly support with that. It's interesting too. It sounds like more, not just providing ad technology to customers. You're really helping them understand where they should be looking to drive value for their businesses. That's really, has been the focus increasingly and I enjoy that a lot. I can imagine that's quite interesting. I want to ask you a little bit before we wrap up here about your talk today. I was looking at the title of your abstract and it's beware what you ask for the secret life of predictive models. Talk to us about some of the lessons you've learned when things have gone a little bit, huh? I didn't expect that. I mean, I'm a huge fan of predictive modeling. I love the capabilities and what the technology can do. So this being said, it's the collection of aha moments where you look at this and there's, this doesn't really smell right. So to give you an example from, from ad tech and I alluded to this, when people say, okay, we want the high click through rate. Yes, I mean, that means I have to predict who will click on an ad. And then you realize that no matter what the campaign, no matter what the product, the model always chooses to show the ad on the flashlight app. Yeah, because that's when people fumble in the dark. The model is really, really good at predicting when people are likely to click on an ad, except that's really not what you intended when you asked me to do that. So it's almost the, they're so powerful that they move off into a sidetrack direction that you didn't even know existed. Something similar happened with one of these competitions that I won for Siemens Medical where you had to identify in FMI images of breast, which of these regions are most likely benign or which one have cancer. And we both models, we did really, really well, all was good until we realized that the patient ID was by far the most predictive feature. Now, this really shouldn't happen. Your social security number shouldn't be able to predict anything. It wasn't a social security number, but when we started looking a little bit deeper, we realized what had happened is the data set was assembled from different sources. And one was a treatment center and one was a screening center. And that's, they had certain ranges of patient IDs. And so the model had learned where the machine stood, not what the image actually contained about the probability of having cancer. And whoever assembled the data set, possibly didn't think about the downstream effect this can have on modeling, which brings us back to the data science skill as really comprehensive starting all the way from the beginning of where the data is collected all the way down to be extremely skeptical about your own work and really make sure that it truly reflects what you wanted to do. So that's, you asked earlier, like what makes really good data scientists is the intuition to feel when something is wrong and to be able to kind of pinpoint and trace us back with the curiosity of really needing to understand everything about the whole process. And also being not only being able to communicate it, but probably being willing to fail. That is the number one really requirement. If you want to have a data-driven culture, you have to embrace failure because otherwise you will fail. How do you find the reception to that fact by your business students? Is that something that they're used to hearing or does it sound like a foreign language to them? I think the majority of them are in kind of junior enough positions that they truly embrace that. And if at all they have come across the fact that they weren't allowed to fail as often as they had wanted to. I think once you go into the higher levels of conversations and we see that a lot in the ad tech industry where you have incentive problems, we see a lot of fraud being targeted because at the end of the day, the ad agency doesn't want to confess to the client that yeah, they just wasted $5 million of ad spend on bots. And even the CMO may not be feeling very comfortable confessing that to the CEO. So being willing to truly kind of face up the truth that sometimes data forces you into your face, that can be quite difficult for a company or even an industry. Yes, it can, it's quite revolutionary. As is this event, so Claudia Perlich, we thank you so much for joining us on theCUBE today and we know that you're going to be mentoring a lot of people that are here. We thank you for watching theCUBE. We are live at Stanford University from the Women in Data Science Conference. I'm Lisa Martin, we'll be right back. A 180 from what you weren't even thinking about. That's just such an interesting area because it impacts health, everything. It's, but I like the fact that you...


What's up everyone take down here welcome back to another hockey card video today I'm going to be opening up another one of these championship collection boxes that you can find at walmart here in canada They're only $15. They guarantee one autograph card one relic card and four factory sealed packs So you can literally find anything what I got last time was two current packs within the last three years And then one parkers pack which all the cards were sticking together Hence why you do not like parkers anymore and then you get one score pack So i'm expecting to find the same now for the jersey card and the relic card Or the autograph card and the relic card I don't think they're worth too much of value. You can see them up in the top corner here But we're gonna open up this box to see what we find. Hopefully we have better luck So let's just open this up here and get right into it. So i'm just going to pull out the uh packs first one two three and four So as you guys can tell if you guys watched my last video, I got the same exact packs Oh, I got a 1617 parkers which all the cards were sticking together. I'm expecting the same thing this time We have an 1819 series one an 1819 series two and then a score pack from 1213 won't even focus here. Oh, well, I'm gonna say it's 1213 So just like last time I'm gonna start off with the score pack and then I'm going to do the parkers because parkers is my The one I don't like the most so let's open up this score pack first Well, some of these packs are really hard to open Okay, so let's just get right into it Up first patrick cane Danny clearly Let it focus come on uh yannick webber michael rosslav All right, not too bad a ward winner for brian cambell put that in a separate pile here bobby hall Uh the franchise I actually like this card I think I have small pc of brian hall. So that's uh, that's a win for me. I'm gonna say that's a win and then we have jeans We win ski with sin ski I always put your names you guys know this so it's nothing new on my channel But that brian hall card i'm gonna say okay, we're off to a great start. Hopefully that's some great luck Now i'm gonna be opening up this 1617 parkers card. I'm hopefully going to have I'm hoping all the cards are not stuck together like the last video, but Every parkers card or a pack that I've ever opened They're always sticking together and I can tell you right now. This is like a brick. They're all Stuck together. So this is not going to be fun Uh up first we have jake fur tannin Listen to this Yeah, terrible victor headman Uh, john gabarro pagele pagele why'd I say with an l there? I don't know that's a player I like brian dubinsky nick bonio Oh, I like this card too Hopefully it's not damaged Phil kesel in the red. I have a small pc of phil kesel. So that's definitely going in it All these are sticking together cam atkinson John carlson, I think I picked this up in the last pack I opened of these, uh I don't know. I can't really remember taylor hall. That's a nice one to get for me I think I have enough for a small pc of his too and a rookies for surgey chorolowski tall schinsky I don't know if I got that last time or not. I can't remember but We're off to a great start. I'm Thanking for me for my personal pc. That's why I do these videos is just to gain stuff for my personal pc. So I mean the bobby hall card and then, uh Phil kesel in the red. I'm going to take that as a win So far we'll see what else we can find Next pack here. I'm going to be doing this series one of 1819 so up first we have Sean couture Not too bad. I might have enough for a small pc of his as well Uh, this one's sticking together here pantes aberg jeff petrie I am going to save this one for a minute this one. I'm excited for this one too, but Uh, patrick marlowe. I can't remember if I have this one or not. That's a nice one for me James remeer nice goalie card to add devin dubnick roman josie or josie And a Jaeger years, this is oh six for jaeger. I'm really excited. You guys know I have a small pc of his so In each of these packs so far I managed to find a player or a card that I loved half my collection. So I'm off to a great Great start. This box was definitely well worth the 15 bucks compared to my last one It's now the 1819 series two See what we can get. I'm across my fingers on this. I'm really excited now that I have a lot of stuff for my pc to add All right, let's see what we can find great start so far piki souban I should say piki pk souban. That's a nice one to add to my pc uh ryan murray james kneel A marky rookie for a maximum comatose. I think I have One of his I can't remember uh tyler segwin I actually have a small pc of his so that's a nice one to add that card looks amazing uh grandland david perron and jake for tanan So not too bad. I got a tyler segwin and a pk souban. I know they're base cards But that to me is awesome to get cards for my pc. That's why I absolutely love getting these random packs and now Last but certainly not least I have Oh my god, I am really excited for this Like I'm I'm legitimately shaken right now uh the autograph card is amazing And the uh relic card is amazing as well. So I'm going to start off with the relic card We have a dual jersey for artifacts Including pk souban on the right Which is awesome. I can never find a jersey card for his come on focus. Please focus And it also includes larry robinson. So This is an amazing card. I'm really really happy with this You guys know, I'm a huge fan of pk souban. Of course, I already had a base card and I was excited about getting it So this is Amazing. I'm really excited. I'm definitely gonna have to look at the value for this one because I'm really excited, but Even more so for this next card and ultimate signature I can't see if it says what year Oh, seven oh eight ultimate signature for Alex tanga This one is the one in my opinion.  I'm definitely gonna have to check the value of this This has to be in my opinion one of my top cards to have in my pc right now Um, but definitely definitely worth 15 dollars in my opinion. So if you see this at walmarts I definitely would recommend picking it up. I know the last video. I kind of had my doubts on Uh, I didn't really get the greatest in my opinion, but to me This was definitely worth it. You never know what you're gonna get and I definitely had a lot of fun Which I guess you guys can tell so I'm gonna leave this video here. I have one more box to open I'm gonna be doing that very soon And I also have This one here, which I don't know if I'm gonna open it up all at once or weights. It's 16 field Field 16 factory sealed packs per box Again, I don't know when I open that one up on video if I'm gonna open it up and do all 16 packs at once Or if I'm gonna break it up into two videos I'm likely gonna do two videos, but I hope you guys enjoyed and leave this here. Please take care. Peace


So, as we noted, the ethics and morality of charging interest are not really subject to mathematics, but let's at least mention them. One rationale for charging interest on a loan. If you lend money, it won't be available, and you might miss an opportunity. So one way to look at it is interest corresponds to the opportunity cost. To extend this idea, we introduce the concept of present value. Suppose you could loan someone $100 for a year, but if you didn't loan the money, you could invest it and receive $120 at the end of the year. So the $100 now is worth $120 a year from now. We say that the $100 is the present value of $120 received in a year. Note that the present value is the same as the amount a borrower would receive now to pay back the final amount later. So the present value of a dollar corresponds to the discount function, and, if we assume the values are proportional, the present value of an amount L received at time t is Lv of t. So suppose our accumulation function is 1.03 to the power t, where t is in years. Let's find the present value of 500 to be paid back in five years. The discount function gives us. So the present value of 500 received five years from now is, or about $431.30. So why is this useful? Suppose you want to compare two investment opportunities. If they have the same beginning and ending dates, and you invest the same amount, the comparison is straightforward, which opportunity gives you more money at the end. But if they don't have the same ending period, we can compare their present values. To do so, we'll need to introduce an accumulation function as well. So suppose the bank of Alice offers a CD which pays $1,150 after two years on a deposit of $1,000. Meanwhile, Bob's bank offers a CD which will pay $1,200 after three years on a deposit of $1,000. We'll use the accumulation function 1.03 to the power t to the two CDs and determine which is the better investment. So Alice's CD returns $1,150 after two years. We find the present value of $1,150 in two years to be. And so using our accumulation function, we find the present value is about $1,083.99. Now we can interpret this result as follows. If you were to put this amount into an account earning 3% interest, you'd make $1,150 after two years. But you can make the same amount $1,150 by giving Alice $1,000. So Alice's CD provides an additional $83.99 in present value. It's like getting a bargain. But wait, there's more. Bob's CD returns $1,200 after three years. So again, we find the present value of $1,200 in three years, which will be. And again, we can interpret this as follows. You could get $1,200 after three years by investing $1,098.17 today. But you could get the same amount by giving Bob $1,000. So Bob's CD provides an additional $98.17 in present value. So since Alice's CD gives an additional $83.99 in present value, but Bob's CD gives an additional $98.17, Bob's is the better investment. You get more for the same amount under the assumption that our accumulation function is 1.03 to the power of t. And that assumption is important because in some sense it reflects all the other things we could do with the money we have. So what if our accumulation function was 1.05 to the power of t? That's the equivalent of being able to find something that pays 5% interest. So this time, the present value of $1,152 years will be, while the present value of $1,200 in three years will be. So this time, Alice's CD provides an additional $4,308 in present value, while Bob's only provides $3,661, and this makes Alice's CD a better investment.


Item number SCP-399 Object Class Safe Special Containment Procedures When not being warned by a human being, SCP-399 is inert. SCP-399 is to be kept in a lock safe and facility ██ when not in use. Testing of SCP-399 requires level 4 approval, and is only to be handled by personnel who have received the rating of clear or better on psychological profile 399-17. Any generators, power plants, or similar energy sources in the vicinity of testing are to be dedicated exclusively for testing and should not be used to power any other mission essential equipment or facilities. Under no circumstances is SCP-399 to be tested or stored in the vicinity of any free energy or perpetual motion device. Description SCP-399 is a ring consisting of two metallic bands linked by six metal bars, with six pieces of a transparent purple glass between them. The ring is of unknown make and bears no identifying stamps. When not being warned by a human being, SCP-399 is inert. When placed on a human being's finger, the ring activates, which is indicated by the six glass segments beginning to glow one by one. In testing, this process takes from three minutes to six hours, dependent on the availability of nearby energy sources. Once fully active, the ring responds to spoken or mental commands from its wearer until removed, at which point it becomes inert until worn and re-energized. In this mode, SCP-399 is capable of manipulating or reshaping objects within a five meter radius of the wearer at varying levels of complexity. By a currently unknown mechanism, SCP-399 draws in energy from its nearby environment, with which to perform these functions. Minor tasks such as causing a small object to levitate or turn itself inside out require minor amounts of ambient energy, which is primarily drawn from the surrounding atmosphere, causing a temperature drop. At finer scales, progressively more energy is necessary, and SCP-399 has been known to draw energy from electrical generators, nuclear reactors, and ███ to complete the reaction. If there is not sufficient energy within SCP-399's range, approximately 300 meters, to perform the operation, SCP-399 will draw on its wearer and ███. More detailed operations also carry a greater chance of catastrophic failure resulting in ███ of the wearer. The extent of SCP-399's ability to alter objects within its range appears to be solely a function of energy available to it. With an adequate power source, manipulation of objects at the atomic or subatomic level appears to be possible. See Experiment Log 399 SCP-399 came to the Foundation's attention on ███ when a string of power outages and unusually cold weather were reported in the vicinity of ███. The object was located in the possession of a mister ███ who had apparently become deceased during an attempt to use SCP-399 to change lead bars into gold. How ███ came into possession of the device has not been satisfactorily determined. SCP-399 did not appear to have any means of storing energy that has been drawn in, simply drawing in the quantity it requires to perform a task and expending it immediately. It has been speculated, owing to the high energy demands of SCP-399 that it was intended for use in conjunction with a dedicated portable energy source of high volume. Such a device was not found among the effects of ███. In conjunction with such a device or an SCP of a similar nature, SCP-399 could potentially be used to devastating effect as a weapon of mass destruction or to neutralize other Keter-class SCPs. Due to the potential negative consequences of a failure on this scale, this line of experimentation is not to be explored at this time. Experiment Log 399 Experiment 399-1 Date ██-20██ User ██-Dr. ███-Subject ██-One Phone Book User attempts to open book to page 368. Experiment successful. Ambient temperature of test chamber dropped by 4.8 kelvins. Experiment 399-2 Date ██-20██ User ██-Dr. ██-Subject ██-One T-shirt colored blue. User attempts to change shirt's color to red. Experiment successful. Power brownouts reported in nearby sections of facility. Ambient temperature of test chamber dropped by 9.7 kelvins. Experiment 399-3 Date ██-20██ User ██-Dr. ██-Subject ██-One Phone Book User attempts to cause text of book to be translated from English to French. Experiment successful. One site electrical generator overloads and fails, causing loss of primary power to facility for six hours. Ambient temperature of test chamber dropped by 17.4 kelvins. Experiment 399-4 Date ██-20██ User ██-Dr. ██-Subject ██-One Phone Book 3D class personnel Special test protocols. Experiment 399-4 was conducted in remote facility ██-All electrical sources in the area were deactivated, except those necessary to power monitoring equipment. User attempts to cause text of book to be translated from English to French. Experiment successful. D-class personnel are ██-No significant reduction of ambient room temperature. Experiment 399-5 Date ██-2000 User ██-Dr. ██-Subject ██-Remains of one D-class personnel from Experiment 399-4. Special protocol. Test conducted at remote facility ██-A dedicated nuclear reactor was provided as energy source. User attempts to restore D-class personnel to its physical condition prior to Experiment 399-4. Nuclear reactor reaches 57% capacity. Subject is successfully reconstructed physically intact, but remains deceased. User attempts to restore subject to life ██-SCP-399 was recovered by hazmat personnel 48 hours later. Weather patterns and vicinity of facility site normalize within 96 hours. Physical contamination of the region has been deemed with an acceptable limits. No. Following the loss of Dr. ██-Dr. ██-Requested and was granted permission to conduct the following test on himself. Experiment 399-6 Date ██-2000 User ██-Dr. ██-Subject ██-Dr. ██-User attempts to cause self to levitate. Successful. Upon initial levitation, room temperature begins to drop 1 Kelvin approximately every 0.68 seconds. Dr. ████ then begins to cause himself to hover slowly about the test chamber, which begins to produce electrical brownouts. The rate of temperature decrease increases to 1 Kelvin per 0.47 seconds at this time, and fluctuates as the rate of the doctor's movement is accelerated or slowed. Experiment is terminated at the room temperature drops below 0°C.


What is going on guys welcome back to the channel critical overload here some talking about Jeepers Creepers in this video here today more specifically talking about Jeepers Creepers 3 Cathedral the canceled scrap original plan for Jeepers Creepers 3 That everyone was waiting for in between the years between two and three from 2003 to 2017 When we got the third film that we ultimately got that had nothing to do with the original intent and the original plan Due to budget budget restrictions and some other factors involved that resulted in the movie that we got now As I have mentioned in a previous video the individual who would have been starring in Jeepers Creepers Cathedral as Dairy Jenner who we know would have been the son of Trisha Jenner named after the brother She lost in the 2001 original film with Justin Long starring as her brother who was also again named Dairy Jenner This would have been his nephew that character who was hyped up throughout a lot of the gap between Two and three as the focus of the story and that Trisha will be trying to prevent something happening to the boy The same way it happened to her brother that character Dairy Jenner Jr I like to call him would have been portrayed by none other than Lucas Adams Lucas Adams is someone who many of you might know I believe he is on a peacock show at the moment. I shared an image of Lucas Shooting shooting a sequence of the movie with salva at one point time again I revealed to you guys that they had actually started shooting cathedral and at least shot footage for one day And that's all now I want to share some some other new images that you've never seen before to actually further show you how There was actually a sequence that was shot regarding the son Dairy Jr I like to call him and then also just a sequence that he has going on related to making a phone call So as you see these images start to appear on your screen They will be mostly of Lucas Adams who again these are images from Jeepers Creepers Cathedral the movie that never finished getting made the movie That was never released. He is on the phone during some of these images and This house that you see here what I imagine is that this would have been the Jenner house because we know in the cathedral script it mentions that Trisha is a wealthy woman or she's had some wealth that she's acclaimed over the years and She would have been you know working with the taggers to try to put an end to the creeper once and for all because they were going bankrupt at their farm and This is the the Trisha Jenner house the Jenner house and inside of it Of course would have been where Dairy and Trisha reside as I was mentioning to you guys with the script for not the script but the outline that was since we relate to the season of Jeepers Creepers the TV series that's basically cathedral turned into a one-season format You would have had Trisha in a single mother situation raising Dairy by herself I'm not sure how much of that actually translated into Cathedral I can't recall if there was a partner that was actually in the cathedral script or not It's been a while since I read it. But yeah, these are images of Lucas Adams Filming Jeepers Creepers 3 Cathedral again for at least one day now the sequence I think they're shooting if you're someone who's familiar with the cathedral script You know that there's an instance in the movie or in the script where Dairy has like a phone call with his mother and he's about to start driving somewhere and Trisha by this point in time. I don't believe his home I think she would be out by the Taggart's barn at this point or at least with the Taggart's and Giselle and everyone else at that point Helicum and some other people who have been coming to the barn to see the end of the creeper take place By this time when he's doing this phone call, this is me predicting it could be a completely different thing going on The phone call would have resulted in Dairy himself actually being stalked by the creeper at some point and Or at least what Trisha believes to be the creeper because the creeper's truck would have appeared during this phone call And as the truck is low as a as the truck is approaching There's a certain horn that can be heard That's all too familiar to Trisha who recognizes that that yes is the creeper's truck horn And as the truck is approaching she starts to become a little bit more Nervous and anxious and starts being worried about the safety of Dairy because again She's been having these premonitions visions about him Suffering the same fate like her brother did all those years ago But then as the truck is finally there we see that it's not the creeper that's in the truck It's some other individuals who are taking the truck away from an impound that has been kept at for the past 23 years, I'm assuming the truck ended up in an impound After what took place at the end of the first movie and that would have been like your answer So where the truck was and why it was missing in the second movie because it was impounded and the creeper never got it back For whatever reason although, of course, it could have just simply taken it back But I guess it decided you know, hey my feeding my feeding cycle is about to run out I'm not gonna need that truck anymore. I'll just let it sit there I'll come back for it in another 23 years Although of course the third move we actually got the truck is present Because he stole it But yeah, this is what these images are displaying to me anyway These are images again from when they were shooting cathedral for at least one day I imagine that this is that sequence where dairy would have been talking on the phone with Gina Phillips's character of Trisha and During the sequence the creeper truck would have came up from behind him And Trisha would have been on the phone very scared as she heard the horn It's an all too familiar sound to her and then as the truck approaches it's revealed that it's not the creeper It's two individuals who I believe are involved with law enforcement taking it away from an impound There's just some other individuals taking it away from an impound that was take that was currently being kept in where they were taking a truck I do not know but Some of these images I thought you guys would enjoy seeing them just to further establish that yes Lucas Adams is the individual who was going to star in cathedral as dairy These are images you've never seen before if you've somehow seen these before please let me know how but yeah These are not online anywhere. I believe I'm the first person to be sharing these so if you like these images you can let me know about them down in the comment section below and What do you think jeepers creepers cathedral could have ended up like and if you're someone who is familiar with Lucas Adams? What are your thoughts on him as an actor because yes? He was going to star in this movie at one point or another It just never unfortunately finished to get made and we all know that there's some other things as it stands Right now that will kind of lead us to not necessarily have Lucas even in this role because dairy is A teenager in this story. I believe so if they wanted to like tell this story today Lucas could still be in the role But I think you would have to then make dairy someone who was older and how you could do that Maybe is tweak the ending of jeepers creepers one by saying that after the events of one that Trisha found out She was pregnant She gave birth and now she has a son who was 23 years of age now I think Lucas Adams himself is about 30, but he I think he can pass for 23 I think he can pass for 23. He doesn't look that old at all. I think again the guy is 30 years old So if you want to make him actually 23 years old in the movie if they were to do it now and not a teenager I think it could work if you passed it off like that revealing that Trisha found out She was pregnant at the end of that first movie and gave birth and has been raising her son for the past 23 years ever since So let me know just think about all is down the comment section below What would you have liked to see from Lucas Adams in jeepers creepers cathedral? What would you have liked to see from jeepers creepers cathedral in general since we all know we're never gonna get that Hopefully we do get something related to this TV series if it is to ever get off the ground Let me know all down that in the comment section below if you haven't already of course make sure you subscribe turn on post notification There is a video in description. I will have links on my social media accounts I'm on Facebook Twitter and Instagram you can message me there of course to let me know if there are any movies news or reviews You like me to cover in the future and with all that in mind guys. I will see you in the next video


Welch i'n edrych – rydych i'n byw, rwy'n meddwl todaya'r gwneud i gael eich rhaid iddo i. Mae ddweud hynny yma, rwy'n meddwl, rydych chi'n meddwl weithio ar gweithio, rwy'n meddwl eich rhaid, dyneid i yn ffodol i'r llwy. Mae bobl erwyd yn ei saith o'r gwbl erdoedd. Mae rhaid i'n meddwl i'n meddwl eich rhaid i'n meddwl am)? Mae rhaid i'n meddwl i'n meddwl am tracing ar gyfan y Buddyllfford Gael ypa'i cyfrifol o'r Gael. a sy'n meddwl i'ch bydd ar hyn yn dda i'r gaelio, a mae'n meddwl i'ch gaelio'n tanxon. Mae'n meddwl i gaelio i'ch gaelio i ddim yn mynd i'r meddwl. Amgeddur. Ddwy'r gaelio'n meddwl i'r ddigon nhw. Mae'n gweld i'ch cyfnod i'r Gŵr Rhwng. Ac mae'n ceisio yma, mae'n cael ei gallu'r hyn o'r gaelio. Ac mae'n gallu'n meddwl i'ch gaelio'r gaelio i'ch gaelio. Mae'n gallu'n meddwl i mi. I'll be talking briefly on the Middle East as a region and then we'll move and talk about the rise of ISIS. Giving a brief introduction to the development of violent Salafism in the region. I hope I will be using some concept. I hope you understand them all. If you don't, Mr Google is always there to help you. You can ask me afterwards. Then I'll be finalising with some concluding remarks. The Middle East as a region is a competitive, fragmented and penetrated regional system by great powers. It is therefore in a constant disorder while the causes of peace and security, the Arab League, the region's largest international global organisation, has been mobilised in numerous occasions for a number of reasons. These efforts have often failed. The region is dominated by authoritarian regime types, and yet it's a drift of a hegemonic power able to impose its own will on the subsystem and therefore awash with rivalese. Power is a float, evenly distributed, and does not necessarily manifest itself in terms of such traditional indicator as the size of the population, territory, economy, GNP, geography, nor the size of the military budget of the armed forces or military heart where provides sufficient indicator of power and influence. The region is also characterised by multi-layer identities. The map of the region was drawn by the colonial power, mainly Great Britain and France after the first great war in 1918. This catalyst, a constant conflict in the region, thus the region is characterised by inter-state rivalese and increasingly exposed of identity politics, which manifests itself in the inter-conficial and inter-communial conflict. The Kurds, for example, are still suffering from the consequences of drawing of the map of the first world war. There are roughly 40 million of them are distributed among four countries, Iran, Iraq, Syria and Turkey. That was, of course, after the collapse of the Ottoman Empire. Now, be in mind that the nation states system or the map of Europe was a genuine from inside. The map of Europe was drawn from inside. That's why the nation state arguably warred in Europe, while in the Middle East it was shaped from outside, as I mentioned. Of course, back to the Kurds, the victims of Psychys Pico, not only them of course, but the Psychys Pico agreement between the French and the British authorities in 1916. It was a secret agreement, exposed a year later after the Belcevic Revolution. Anyways, so the Psychys Pico agreement was later confirmed in the Sandrimo conference in 1920. Back again to the region. The regional system is vulnerable to the behaviour of sub-state and non-state actors, and many of its states are still suffering at the hand of violent jihadi groups who have stepped into the vacuum created by the weakening of the iron grip of the central government in several Arab countries. Indeed, in the 21st century, it seems to be the smaller Arab states outperforming their largest counterpart and non-state actors making waves. The region is still suffering from lack and alternative political forces to fulfil the expectation of the masses and the people and achieve development and prosperity eventually. The Islamic radical groups such as the Islamic state seems to have become the substitute for the past political forces for doing the mission.  Selloffism and the Islamic tradition, I hope many of you are now interested in political Islam and the region itself, so this will be hopefully very helpful for you. Selloffism in the Islamic tradition was actually a reformist movement. It emerged at the end of the Abbasid caliphate in the 11th century, and all what this movement wanted was the return to the true Islam by peaceful means, and the law of the divine is represented in the Quran and Hadith. Hadith is the teaching of the Prophet Muhammad, the Prophet of Islam. Violence selloffism is relatively a contemporary phenomena. Violence selloffism was introduced arguably in the writings of the Egyptian scholar Sayed Qutub in the mid-60s. It also, interestingly, developed with the rise of left radical groups and movement in Europe in the 60s and the 70s. Now, Sayed Qutub himself presented pan-Arabism policies of a president, Jamal Abdel Nasser, and called for a regime change. He was later arrested for plotting against President Nasser and was executed in August 1966. Now, Ayman Al-Zawahiri, the current leader of Al-Qaeda, was heavily influenced by Sayed Qutub. Violence selloffism was also materialised in the Afghan war against the Soviet Union, yet the milestone arguably is the Kuwaiti crisis in 1991. The occupation of Kuwait divided the Arab world between those who rejected the occupation, mainly Syria and Egypt at that time, and those who accepted it, such as Yemen and the PLO. The American soldiers were not welcomed in the Holy Land by the Arab Mujahideen in Afghanistan. They offered the House of Saud, Saudi Arabia, to come and fight for them and drive the Iraqi army outside Kuwait, but not to bring the Western power in the region. Now, what happened is that the House of Saud rejected Osama bin Laden's offer to defend the Holy Shrine the latter then vowed to attack the US and its ally in the region. Another development you might find interesting, and perhaps nobody speaks about it sadly, is the Algerian Civil War in 1991 up until 2002, and also the Bosnian War in 1992 until 1995. Those two events had their own share in the rise and the development of violent selloffism. The establishment of al-Qaeda in 1993 in Afghanistan witnessed the boron of the new trend of global selloffy, radicalism, or violent selloffism. While Taliban, grwp islamic arme and front islamic disalw had locally based agendas in Afghanistan and Algeria, the last two were operated in Algeria and France, inside France, al-Qaeda itself transformed the agenda of jihad or violent selloffism from local to global to the international level. Al-Qaeda began to attack the US and its ally in the world. The first attack was the US army residence in Goldmohor hotel in Adan in 1992, followed by the bombing of the World Trade Center in 1993, and the bombing of the US embassies in Nairobi and Dar es Salaam, Kenya and Tanzania in 1998. The disastrous attack, of course, was the attack of 11 September in 2001 in New York. Having said all of what I was talking about now, the perfect representation of the disorder of Arab regional order is the rise of what's so called the Islamic states in Iraq and Syria in 2014. The occupation of Iraq in 2003 was yet another turning point of the development of violent selloffism. It gave the birth to the Islamic state. The grwp, though, is one of its kind. It does not only use violence to achieve its goal, but also rejects the nation-state system and divides the world into two groups, the House of Islam, Dar al Islam, where all the Muslims are subjected to their version of Islam, which is a selloffism, and the House of War, Dar al Harb, basically those who are against them. Be Muslim or not Muslim, by the way. Unlike some other Islamic movements, such as the Muslim Brotherhood Ikhwan in Egypt and elsewhere, and Hezbollah in Lebanon, which believe in the political engagement, such as election and democracy, parliamentary accountability, Daesh or the Islamic state, does not intend to engage in this political life or contribute to the building of such order. It's exclusively used violence to achieve its utopian ideal caliphate. Although it has lost much of its territory by now, the grwp will continue to pose security threat, not only at the regional level, but also at the international level, too. Now, to conclude, and I hope I didn't take much of your time, the region still lacks regional leadership to solve this regional problem. The subsystem lacks a security system, and it is unique for the absence of region-wide architecture. The region's contested states seem to be unraveling into smaller communities and sects, religious affiliation, tribal groups and ethnicities. Now, as a consequence, the region inhabits signs of deep social trauma and crisis of identity at both state and the society level. The region still suffers from the lack of an alternative forces to fulfil the expectation of the people, as I said, and eventually we will see in the future that those radical groups will still be appealing to the public to take over as an alternative forces to achieve change, whatever that change is. Thus, violent sylwfism is a phenomenon that will not cease at any time soon. Not only that, violent sylwfism catalyst on other sub-state actors, mainly the Shi'at militia, Iran proxies in the region mainly, namely Hezbollah in Lebanon, that is recently being active in Syria, the Houthis in Yemen. I hope you all followed the news. Just two, three days ago there was a rocket, there was lunch from Yemen against Saudi Arabia, and we also have this recent phenomena of the popular mobilisation units or forces PMU in Iraq. Those Iranian-backed militia pose a threat not only to the security of these countries they operate in, but also pose a massive threat to the stability of the entire region. I'll stop here. Thank you very much.


Hi, I'm Zor. Welcome to a new Zor education. I will continue talking about exponential functions. The previous lecture was dedicated to basically definitions of what is an exponent is. Well, obviously, it's very easy to define for natural number as an exponent. So if you have where x is a natural number, then it's just multiplication of a by itself and times. But then I expanded this definition to rational number, zero negative numbers, irrational numbers, basically all based on one very important property of exponent. That if you have a product of two elements, each one of them is some base and some exponent, then it's equal to this. So the product of exponents is exponent of a sum with the same base, obviously. Now, this is a fundamental property of exponential functions, and all definitions in the previous lecture were based on this. Now, I will go into properties of exponential functions, now just to illustrate these properties, I would like you to have in mind a graph of this particular function. I will explain all the details about graph in the next lecture. This lecture is about properties, but the graph is based on the property, and that's why I would like you just to keep in mind how the general picture actually looks like. Now, this function is defined for all positive a, and we have to define it in two different categories, a greater than one and a less than one. So for a greater than one, the graph looks like this. It's increasing from zero to plus infinity, and in this particular case, it's decreasing. Now, I draw these graphs without basically any foundation. The foundation is the properties which have gone to talk about this lecture, and the graph will be more explained in more details in the next lecture. However, it's easy to talk about properties if you have these pictures in mind because I have it in my mind, that's why I would like you to have it. For greater than one base, it's increasing for less than one, it's decreasing.  Again, these are the properties which are in the foundation of graphing the function. So that's why it's very important. Okay. Property number one, exponential function is always positive. Now, I will try to prove as rigorously as possible all these properties. So just bear with me. Now, why a to the power of x is positive? Well, let's consider again the definition, starting from the x-natural. So a to the power of n is a times a, etc. a is always a positive number. So if you multiply positive number by positive number, it will be positive number. So for any natural n, it's positive. Now, for negative integer where n is natural, so it's minus n, this is by definition, one over a to the power of n inverse. Now, inverse to positive number is still positive. Inverse to three is one-third. Inverse to a million is one millions, still positive number. So for all integers, positive and negative, the a to the power of x is positive. Now, how about rational numbers? Well, again, what is a to the power of p over q, where p and q are two natural numbers? Well, again, by definition, this is the q's root of a to the power of p. Now, a to the power of p is positive. Now, q's root also positive. Basically, it's a definition of the root. The root is always positive from the positive number. We are not talking about negative numbers, complex numbers, et cetera. We're talking about positive numbers, and root of the positive number is always the positive number, which being raised into the q's, power will give a to the power of p. And obviously with the negative rational numbers, where p and q are natural, this is again by definition. And since this is positive, the inverse is also positive. Now, just once I will address the irrational numbers, and then I will not talk about the rational numbers at all, because I cannot really make it absolutely rigorously. So I was talking about exponential function with x being an irrational number as a limit case when we are approximating this irrational number with rational numbers. So if x can be approximated with a sequence of rational numbers, then a to the power x would be a limit of a to the power of all these elements of the sequence. So basically, I don't want to go any deeper in this because again, we are not really on the level where I can make it absolutely rigorously. So just consider the irrational number as a limit case for rational numbers, and that will probably be enough to talk about, well, the proof that a to the power of x would be all this positive regardless of x. Positive x, negative x, anything. How about zero? How about a to the power of zero? Well, this is always one by definition. Again, that's from the previous lecture and that's basically is based again on the main property of the exponential function. We must have a to the power of zero equals to one to satisfy this and again, that was addressed in the previous lecture. So if exponent itself is equal to zero, then the exponential function is equal to always one. Okay, how about base? What if base is equal to zero? What happens then? Well, then exponential function is always equal to one because one to any power would be equal to one. Proof is exactly the same as I was talking about a to the power of x being positive. Firstly, check it for natural numbers which is one times one times one times one. So for all natural x is obvious, that's one. Then negative natural numbers which is one over one which is one and then rational, et cetera. It's very, very trivial. So they're not going to address it anymore. Next property. All right, next property is actually related to these two graphs which I draw before. So basically it says that if a is greater than one, then for positive x, y would be greater than one and for negative x, y would be less than one. Again, remember the graph? Well, this is one. So this is y to the power of let's say, two to the power of x, something like this. Any number weighs greater than one. So it's increasing. This is level one. So if x is equal to zero, function is equal to one. Two to the power of zero as anything else to the power of zero would be one. Then function goes above one to the right and below one to the left. Now let's prove it. Forget about graph, let's prove it. Proof is exactly done in fashion as before. First, well, let's consider this case. So if x is natural, a to the power of n is a times a times a n times. Now each a is greater than one as we are saying. Now multiplying a greater than one number by greater than one number results in greater than one number. If it's not obvious, let me just prove it to you. Well, if a is greater than one, then it's just an interesting exercise. A can be represented as one plus d where d is greater than zero, right? So if a is greater than one, then d is some number greater than zero. Basically d is equal to a minus one, right? Now, therefore, a square is equal to one plus d square, which is one plus two d plus d square. Now, d is greater than one. So we add something to the one, which means we are getting a number greater than one. So multiplying a number greater than one by itself two times will result in number even greater than one. Well, basically this is the way how we can prove that any number of times we are multiplying number by itself, we will get the number greater than one. Well, the easiest way probably would be something like this. This is greater than one plus two g, right? Because there is one extra member here. So now let's go to a to the third degree. We will have one plus d to the third degree, which is equal to one plus d to the second degree times one plus d, which is greater than, I replace this with this, which is equal to one plus three d plus two d square, right? Which is greater than one plus three d. So look at this, a square greater than one plus two g, a cube greater than one plus three g. Now, it's very easy to prove, let's say by induction, that greater than one plus n times a. Basically that's enough to prove that it's greater than one. And the greater n is, the greater this number becomes. So let's just remember this particular inequality. I do encourage you to prove it by induction. Just check it for n is equal to one, which is kind of, which is, or n is equal to two, starting from n equals to two, which is obvious. And then if this inequality is true for some number, n is equal to k, try to go to k plus one. Do it yourself, it's very easy, and we will use it in some other place. But what it proves for us right now is that if a is greater than one, then for all natural numbers, which are integer positive, a to the power of n would be greater than one. We see it from here. How about negative natural numbers? What if we are talking about this? Well, since this is equal to this, this is greater, denominator is greater than one. So the inverse to a number which is greater than one is smaller than one. So we get this one. All right, so we have proved it for all integer numbers. How about rational? Quite frankly, it's exactly the same thing. Let's again start with this. If you have a to the power of p over q, which is by definition a q's root of a to the power of p, now this we have already proven that in this case, this is positive. This is greater than one, sorry. Positive obviously as well. Now, if you have a q's root from a number which is greater than one, you must have the result which is greater than one. Why? Otherwise, if it's less than one, and then you will raise it into the power of q, every time you multiply it by itself, it will be smaller and smaller and smaller because you assume that the number is less than one. So it cannot be greater than one in this particular case. So that's our best trick. Similarly, with negative rational numbers, since this is greater than one, then negative should be correspondingly inverse to a number which is greater than one, that is a smaller than one number. So I proved it for all rational numbers, and I told you I will not talk about irrational numbers, but let me just say it again, that irrational numbers are limit case of rational. So whatever the rational numbers in qualities are true for irrational numbers will always be true. Okay. Now, very similarly, what if a is less than one? Well, then I'm saying that for positive x, y would be less than one and for negative x, it would be greater than one. Well, I can prove it directly in exactly the same or similar fashion that I did it for a greater than one, but let me do it slightly differently. Look at this. a to the power of x is equal to one over a to the power of minus x. That's obvious, right? Because what is minus x? Minus x means one over one over a to the power of x, which is a to the power of x. Now, heading down that, I can basically refer everything to an opposite case because if a is greater, it is less than one and one over a is greater than one, right? So this to the power of x would be greater than one for x positive and would be less than one for x negative, right? So for a, it would be vice versa since this is equal to, so one, since one over a to the power of x is greater than one, then a to the power of x should be less than one for positive x. And if one over a over x is less than one for negative, then y over x should be greater than one for negative x. So that's basically kind of a easy logic which will reduce these cases to these cases. Just change the sign of the exponent. Okay, so back to our graph. Again, remember for a greater than one, it goes this way, for a less than one, it goes this way. Now, it looks like it's monotonically increasing for monotonically decreasing. So let's prove it, right? Why do I have such a nice and smooth curve? Let's prove the monotonicity. Is there such a word monotonicity? I don't know, maybe. So let's prove that the functions are monotonic for any a. For a greater than one, it's monotonically increasing for a less than one, it's monotonically decreasing. All right, so let's consider this case. Now, what is monotonically increasing? It means if we are increasing an argument, then the function is also increasing. That's what monotonic increase means. All right, fine. So let's just take two different arguments and we start with integer arguments, positive integer numbers. So let's say m less than n. So I will prove that if this is true, then a to the power of m would be less than a to the power of n. How can I prove that? Well, very easily, since m is less than n, then m is equal to n minus m to the power of n. Or if you wish, n is equal to m plus n minus m, same thing. So I will use this, actually. Now, a to the power of m, this one, therefore is equal to the power of n plus m n minus m. Equals. Now, you remember that whenever we are edging exponents, it means we can multiply exponential functions. Now, a is greater than one, right? So this, considering that m minus m is greater than zero, this is greater than one, right? It's base, which is greater than one, to the positive exponent, positive power. The result is, based on whatever we were just talking before that, greater than one. Since it's greater than one, then this number, a to the power of m, excuse me, is greater than a to the power of m, since this multiplier is greater than one. So we have proven that with increasing exponent, we increase the exponential function value as well, for greater than one base. So basically it's really looking like the curve goes up, but only for integer numbers. Now, how can we expand it to, let's say, rational numbers? Well, it's really very, very simple. Let's say we have two different rational numbers, p over q, which is less than r over s. I have to prove that a to the power of p over q is less than a to the power r over s. Now, how can I prove that? Well, let me first bring these two rational numbers to common denominator. Doesn't change the numbers themselves, so let's say I will use common denominator qs. So one would be p s over qs, less than r q over qs, right? So these are exactly the same. But now, oh, I'm sorry, qs. But now I have an advantage of having common denominator. So it's easier for me to consider the same kind of a theorem only for cases of common denominator because if denominator is not the same, I'll just bring them to common denominator. So let me forget about this particular case. I will use the case when denominator is the same. Let's say u over w, less than v over w. And I would like to prove from this that a to the power of u over w is less than a to the power v over w. So now I'm using the same w as denominator. Now, how can I prove this? You see, this means that u is less than v, right? They're all positive. We have already agreed that these are rational numbers with natural denominators and numerators and denominators. So obviously, since the denominator is the same, my numerators must be of the same relationship. Therefore, since I have already proven it for integer positive numbers for natural numbers, a to the power of u is less than a to the power of v for greater than one base a. Now, all I have to do now is extract the w's root from it, right? Because a to the power u over w is w's root from a to the power of u. And a to the power of v over w is again w's, the same w's root of a to the power of v. Now, if a to the power of u and a to the power of v are of this particular relation, then their w's roots also should be in exactly the same relation. Because if it's otherwise, then by multiplying greater number, the same number, the same w number of times, I will get greater number. So that's why the roots are supposed to be also in the same relation as a to the power of u and a to the power of v. Cannot be otherwise. If it's otherwise, I will just raise both in w's power and I will have a reverse equation here. So I've proved it for rational numbers as well. Fine. Now, these are all positive numbers. I was only talking about positive numbers and proved that. How can I prove it for negative numbers? Again, let's start from integers, negative integer numbers. Let's say I have minus m, well, let's first do this. This is minus m and this is minus n. So minus m is less than minus n. Now m and n are positive numbers. That's why I put minus in front of them. M and m and n are natural numbers. And since this one is to the left from this one, this is the relationship. I would like to prove that a to the power minus m is less than a to the power of minus n. Now, how can I prove that? Well, very easily. Now, if minus m less than minus n, then m, the positive corresponding positive, which is basically distance from zero, is greater than m. Therefore, from the previous case when both are integers, if this greater than this, then a to the m is greater than a to the m. Now, what is this? This is one over a to the m. And what is this? It's one over a to the power of m. Now, if denominators are in this relationship, then the whole fractions are in the opposite relation, right? Because if denominator is greater with the same numerator, then the fraction is smaller and vice versa. Okay, so basically that's what we have just proved, which means that the same monotonic property exists for negative integers as well.  It's even easier because we know that for greater than one base, everything to the right of the zero is greater than one and everything to the left of the zero is smaller than one, which means we still have exactly the same type of relationship between the values of the exponential function. Okay, now the case with a less than one, I will not consider at all because again it's quite obvious from the fact that if a is less than one, then one over a is greater than one. So all the properties of this case are true for this case just traversing the sign of the exponent. Whatever was true for positive exponent here will be true for negative here and vice versa. So that's why I'm not going to waste any time on this. That's all this. Next property. So monotone is to be proved now. Okay, we have to prove that a to the power of x for a greater than one increases to plus infinity if x goes to plus infinity. Okay, let me say it again. I'm using some symbolics here. So for this function, the function is already written here, so we don't have to think. If base is greater than one, if base is greater than one, therefore y goes up to plus infinity when x goes to plus infinity. So it's not only monotonically increasing, but it's increasing to infinity how to prove it. Very easy. We actually did something very useful. You remember I've proven this just before, couple of minutes before, for a greater than one, if you are raising it to the power of n, you will get very greater than this one. Now, how can I prove that this thing goes to infinity with increasing exponent n? Well, very easy. Let's choose any number n. And now I'm going to find such a number, lowercase n, that this thing exceeds this one. How? Well, let me choose the number n that this thing exceeds n. Now, when will this be greater than n? Well, obviously if n would be greater than n minus one divided by a, right? Minus one divided by a. So as long as I choose any integer number n greater than this number, then one plus n a would be greater than n and a to the n's power will be even greater. So it will exceed the number n. So for any number n, I can find lowercase n such that a to the power of lowercase n will exceed it, which means that for any level, I eventually will find the point where a to the power of n will exceed it. So that is actually the meaning of the statement that a to the power of x goes to infinity. At least we have proved this thing for all integers. So I can find an integer where it goes above certain level. But now let's think about the rational numbers. Since I have already proved that the function is monotonically increasing, once I found one particular point, let's say it's an integer lowercase n where we have exceeded this uppercase n, then everything above that would be even higher. So that's what actually is a proof that the function goes to plus infinity. It's monotonic and it can exceed any however large chosen number n. So what we have just proven is that for greater than one base, the function goes to plus infinity when the exponent increases to plus infinity. Now, how about the next my statement? Well, this is not a straight line, I'm sorry. Now my next statement is that when I go to the minus infinity when argument goes to negative infinity, then the function goes to zero. Well, this is actually quite obvious. Why? Because for negative number x for negative number x, I have this particular equation, right? This is a definition of negative exponent. So if a to the nth power, n is a natural number. So if a is a to the nth power goes to plus infinity, then one over a to the nth power goes to zero, right?  And as soon as we understand that in this particular fraction, the denominator goes to positive infinity. We saw that the whole fraction goes to positive zero. Okay, fine. So I've got it for negative rational numbers, for negative integer numbers. So for negative rational numbers, it's obviously following from the fact that the function is monotonic. Okay, next. Okay, so we were actually comparing the values of the function based on different values of the exponent x. Now I would like to compare how the functions look if we are changing the base. This is not really an argument, but it would be interesting to find out how this function is different from this function. So my statement is that the bigger this particular base of exponential function for, let's talk about this case. So the bigger the base, the steeper the function goes, this is two to the power of x. And this is three to the power of x. So the bigger the base, the steeper it goes up to the right and faster it goes to zero to the left. Now with opposite a less than one, we will have a similar kind of a relationship that we will talk about this later. So right now, let me prove the following. That for a and b greater than one, if a greater than b, then, so if a greater than b, then a to the power of x is greater than b to the power of x for x positive and a to the power of x is less than b to the power of x is negative, right? So if this base is greater than this one, then for positive exponents, this graph is above this graph. But for negative exponents, this graph is below this graph. All right, how can I prove it? The same story is repeating again and again. First of all, let's prove it for natural exponent. So a to the n greater than b to the n. Natural are obviously greater than zero. How to prove that? Well, if a is greater than b, then whenever we multiply n times a, obviously each component of this multiplication would be greater than each component of that multiplication and that's why the result would be greater. Now, from this, we expand it to positive rational numbers because a to the power of p over q greater than b to the power of p over q because this is equal to q's root of h to the power of p and this is q's root of b to the power of p. If this is greater than this, then q's root should be also greater and that's what this is. All right, so we proved it for positive numbers. Now, how to prove, well, the opposite relationship to the negative numbers. Very easily, again, considering that for negative x, let me put a to the power of x for negative x, I will put it as minus absolute value of x and this is b to the minus negative absolute value of x. That's what negative actually exponent means. So, if x is equal to minus two, it's absolute value is two and that is minus two, same thing here. Now, this is equal to a to the power of x, this is equal to, sorry, this is equal to one over a to the power of absolute value of x. This is equal to b over to the power of x. Now, absolute value of x is a positive, so this denominator is greater than this denominator because we could just prove in it, but if we took inverse, we will have the inverse relation between them. So, if this denominator greater, then the fraction would be smaller. So, that's how we could prove in this one. Now, so basically what we have done here, we have proven that in these two cases, when both of them are greater than one, then the graph, both are increasing to infinity, in this case, and both decreasing to zero for x less than zero, but one function would be above another to the right of zero and would be below to the left of zero. Now, what if a and b are less than one? Well, the situation is absolutely symmetrical. Let me just write it down and I will just leave the proof to you. So, if a is less than one and b is less than one, let's say one half and one third. Well, basically the story should be exactly the same, but opposite. So, if this is the graph, this is let's say one half to the power of x. Now, if you have one third, then it should be one third to the power of x. Now, since my denominator is greater, then it goes down to zero faster. One half times one half times one half would be above one third times one third times one third. And on the left side, it would be exactly opposite. So, we can actually do the following. So, the function a to the power of x would be less than b to the power of x for x, yes, and a less than b. Okay, so if a less than b, then the value of a to the power of x would be smaller for negative and it would be bigger for positive. Let me check. The graph I know is correct. So, if a less than b, so a is one third and b is one half. So, one third, no, yes, yes, yes. a is one third, b is one half. No, it looks like I just mixed it. One third, one half, one third to the power of x one third to the power of x is smaller for positive x, I'm sorry, for positive x. This is for me. Right, it should be an opposite, right, okay. So, basically, this is exactly equivalent to the previous case and the proof is based on the fact that if a is less than one, then one over a is greater than one and one over b is greater than one. And that's why we can use the previous theorem for one over a and one over b. So, if a is less than one, then one over a is greater than one over b, right? And therefore, one over a in the power of x would be greater than one over b to the power of x for positive x. And that's what actually gives an opposite equation for denominators of these two members. And same thing with this one. All right, is that it? Yes, so, again, if base is greater than one, then the graph looks like this. This would be two to the power of x. This would be three to the power of x. The greater the base, the higher it is for the positive x and the lower it is for the negative. Now, if base is less than one, then situation is kind of an opposite. So, keep in mind these graphs. I think that all the properties which I was talking today are the foundation for these graphs. Now, I will explain in more details the story about graphs in the next lecture. So, this one was just an illustration, graphical illustration of all these properties. But again, the purpose of the properties is actually to draw nice graphs. Monetonicity greater than one or less than one based on the value of argument and the value of the base. So, all these properties are very important. And they're all based on the most important property of all exponential function, which is this. This. Okay, that's it for today. Again, next lecture will be about graphs. I do suggest you to take a look at the notes to this lecture which are accompanying the lecture on the Unisor.com website. And I also offer some exercises, problems and exams. For those who are registered, you're welcome to take exams as many times as you want. And I would also like to encourage the parents to basically be supervisors for your students. And the Unisor.com actually allows you to register as a supervisor and then you will be able to control the whole educational process of your students. That's it, thank you very much.


I think there's about over 30,000 students. 35,000. I'm guessing like 34,000. We have like 20,000 undergrad and 10 grads, so around like 30 to 40,000. I looked that up yesterday. So we actually have 27,496 undergraduate along with 10,708 graduate students for a total student body of 38,204 students. I'll be able to throw out a guess and I'll say a 67. 4? We're going with 27. Is it actually 27? I was going to say like 200. We have 27 bare statues on and around campus. The smallest is located in the balcony of South Hall. The campus' mascot is the Golden Wolf Bear. Panels, what? Nude. There's nude. Nude sculptures. We just saw this. Well, I didn't really look at it. There aren't any. We all agree on either, right? You can see them right there. Oh, there. Wow. Now I see them. While many of us walk by, they say their gate every day. Not many of us really notice the sculptures going along the gate. There are eight sculptures in total. The four facing the south side are all of women, representing the studies of electricity, art, agriculture and architecture. And the four facing the north side of campus represent the studies of law, letters, medicine and mining. I know this. Oh, I heard this story. I know this one. I got this. I got this. The legend is that it was like two brothers, I think, designed it. And then they had communication issues. And a woman was involved or something. One of them was having a affair with her wife. And they were like fighting with each other. They got mad at each other. And they split planets separately. That's why Duanel Hall makes no sense. Yeah. When did you hear that? But I don't think it's true. It sounds ridiculous. Basically, maybe it couldn't compromise. So the whole story on the two feuding brothers fighting over one woman is completely false. What had happened was the original design for Duanel called for office spaces which needed lower ceilings and then lecture halls which needed higher ceilings. And then on top of that, Duanel was built on a hill. So there's four different floors that are at ground level. So really, it's really confusing all because of design. I don't want the students to get in. Because how fancy the infrastructure is in the administration buildings. Apparently there was like this big riot where they locked all the teachers in the building. There were protests and the protesters would lock the administrators in. Protesters handcuffed themselves. So are you an encyclopedia? California Hall houses are top administrative offices. For that reason, many protests have been staged outside including one where the students actually changed the doors together. That's why California Hall's exterior doors actually don't have door handles and many other buildings on campus only have one door handle per set of double doors. Our campus has historically and continues to be at the forefront of student activism from the free speech and disabilities rights movements to the Black Lives Matter protests of recent times. Thank you.