Hello everyone. Welcome to the NPTEL course on groundwater hydrology and management. This is week 12, lecture 5. We have come to the last lecture of this NPTEL course and I really hope that this journey that you took in the course has sensitized you on groundwater hydrology created an awareness of what are the parameters involved in groundwater management and the data that is needed. While speaking about data, that is what is the focus for our lectures in the current week and the last week. So we will focus again on this little bit on the data part. What do you do with all this data?  So what we have seen is there is a lot of data that comes into the database because we have started with a less understanding of groundwater hydrology. Then we establish the water balance from which you collect different data and put it into the water balance. But the water balance is a time-snap-stamped event. For example, if you say precipitation, it is an annual precipitation or monthly precipitation you use and then you use a monthly groundwater level, monthly ET, monthly soil moisture etc. What has happened is there is no connection between this timestamp and the next timestamp or the previous one, t equal to 0, t minus 1 or t plus 1. So this is where you are in need of a simulation model that can capture all this data and then evolve, automate from one timestamp to the other. But before that is done, there is something called a conceptual model. Conceptual model aggregates all the data that you collected from the field into one pictorial diagram and from the picture you do get a better understanding of what is happening in the real-life scenario because for example, you look at groundwater level and quality separately in a database. When you bring them in a conceptual model, then you understand the groundwater level is declining and the aquifer type is changing as the groundwater level declines and that is why the groundwater quality is bad. Unless you bring three data sets, already I said water level, the aquifer type and the water quality. So unless you bring all these three different data and marry them together as one conceptual model, it is difficult to understand what is happening and that is what the conceptual model helps in clarifying by bringing all the data together. So we have collected hydroclimate data ranging from rainfall and I have shown you about temperature, wind speed and other things, but rainfall, storage, river discharge, soil moisture, we have looked at data on geological conditions from the lithograph and more old data which include your stratification, layering of your aquifers and also we looked at high permeable, low permeable rocks, solids, etc. Porosity, all these terms are the geologic conditions because it is the base rock that actually disintegrates into soil. So when it disintegrates, you have the porosity and other things formed and sedimentation also occurs due to geological influences. Then we had collected groundwater which is level, the groundwater level before which we collected the climate and the geological setting, the layering of the aquifers, then groundwater data is collected. So we set up the water budget, didn't we? We just said, okay, this is the water level, we also use grace here for a small example and then we said for a rainfall, for the ET losses, the storage is declining or increasing, okay? So that is what we set up as a water budget. But the water budget was only for one time stamp and it is not giving us a full picture of a longer time series. More of the time series, how do these data react or interact between each other is not clear, okay? For example, how much of rainfall is actually getting into the groundwater and how much of that groundwater is going into base flow, all these need to be calculated and that is where a model actually helps to break these boundaries and then bring them as one entity. For example, water which comes as rainfall, is the same water which infiltrates, pushes the water into pore space and then goes into a groundwater comes back to base flow, etc. But we need to keep a check on that water always so that we know the volume and movement of water, right? So conception models are starting point for models. We use them as leverage point where you start using them for understanding or setting up a hydrological model. Any groundwater model would ask for a conception model starting. This is very similar to how you learn to solve physics problems and mathematics problems, right? When you are taught these, what do you say? Read the question, take the data from the question and draw it. For example, you have a train moving 100 kilometers per hour. How many stations would it pass before it stops in a zero speed? Those kind of questions or cannonball is fired and with a particular velocity it goes and then it makes a trajectory and falls down. And what is the energy differences at each stage? So you would draw it, you draw an angle and all these things. Similarly, when you collect all this data, you create a conceptual model which can be used to better understand the need of the model. Let's take an example here from the SWAT database. So this model is actually taking all the data from different sources. For example, VEM is your elevation data. How your elevation changes influences the rainfall conversion into runoff and the runoff going into infiltration and groundwater. So that changes based on the slope and angle of the slope. And then you have your land use map, especially if it influences your groundwater recharge, infiltration and ET. Because if there's more ET, there is less water going in and less water stored in the groundwater. We're just going to focus on the groundwater storage as is groundwater class. Then you have a soil map which can be of different soil types in a given location and every soil would behaves differently for a particular slope and rainfall. For example, there are some soils which are hydrophobic, which actually do not repel water. So if you have such a soil, your model will say no, water should infiltrate, but it doesn't infiltrate. So these kinds of things are built in the model when you have data for it. And last, of course, your weather models, etc. So all these can go into a database SWAT is a hydrological model for surface water, but I'm just using this diagram here because it gives a clear picture of you collect all this database together, pile them up, and then give it to a SWAT database. So for you, it will be a groundwater database. And then some parameterization happens, some categorization happens. You tweak the parameters, you tweak the hydrological conductivity, the permeability, the thickness of the aquifer, all these parameters you tweak. And then finally, calibrate the model and validate the model, you'll get outputs as a layer or output tables and charts. So where does the conceptual model come? Okay, so does it come here in a surface water model? As I said, it's not much necessary, but for groundwater models, it is necessary. Let's take an example from Shirasadmi and Pratapar 2016. What you see here is we collected different ground realities and aquifer types from bore logs. So we take, for example, distance and a depth. So for a one kilometer radius, for example, we have taken all the locations of the bore hole. And we know that when we drill the bore hole in this location in Mukteshwar, there was a lease and then some phyloid quarts and then bedrock. Okay, so there were different layers, but whereas in Maiorite was only one or two layers. So this was, for us, it was easy to differentiate because we had this location of the logs. And then we put it in a diagram where you have the axis as the depth. Okay, from zero level could be the bottom most zero. And from there, the bedrock, you can build up the elevation profile and how the aquifer material changes. And also with some rainfall and river network, we know that water flows down and forms into a stream. So water from here comes down and Maiorite also comes down to this drainage point. And how that drainage point looks down is also a question where there's a lot of interactions. This conceptual model example is purely given to show you only above the geological setting, not much, because this is what you would then think within yourself and say, should I have four layers or three layers in my model, etc. So for example, if I know that here there is only two layers, and then just 50 meters, 100 meters out, I have three to four layers, then what happens is there is a dilemma how much layers I want, or can I simplify all of this into one layer? And that is what happens in a conceptual model. Initially, you put everything in the paper and you draw the model. Then when you want to export this finding into the groundwater model, the next step. So the first step is data collection. Second step is creating the conceptual model. And third step is converting the conceptual model into a simulation model. So this third step is where your inferences from the conceptual model will help. The most important inference is how many layers. Looking at this geology type, we could clarify that. Oh, there is some differences in the layers. So let's club them because there's not much difference between a niche and a phylite, for example. For example, and I'm saying that if there is not much difference, then I club both of them as a niche. Okay, G and EIS is this niche. So what happens here is you are clubbing layers. Why would I club layers? Because the model runs faster, you have less anisotropy and heterogeneity to account for, because you assume that all are safe. So now the model would be more homogeneous and more isotropic. So your parameters also will be shared between these two equally. And that could be a good point for the modeling part because the models take a lot of band space, like a lot of computing power. And also it may crash if it is too much layers, because there's too much interactions you want to simplify. As I said, in the modeling part, you can make a model as complex as possible to mimic nature. Or you could make it less complex, simplified, but run well. So that is where you have to have a balance and give and take a policy so that it's a win-win for everyone. So think about it. But this conceptual model does help you in thinking that way of where should I put my recharge structures, where should I put my layers, how many layers, and even monitoring stations. Let us look at another groundwater model, conceptual model, where you could see that water is flowing from top to bottom. The stream is flowing. And while the stream is flowing, some water is being lost into the banks, river banks, and then it comes back. So this is kind of a losing stream and then gains downstream. So when the water comes here in the top, let me put the pointer. So when water comes down, you see that the part of the stream is losing water because the water goes into the ground part and into this ground part, both the sides of the water go. And that is being recorded in the wells that have been kept three meters away from the river bank, okay, or the river water perimeter. So three meters, two meters. So there are three pismeters and the depth of the pismeters through 3.58 meters. And we have a screen size of 0.76 meters to just think about one layer they want to monitor. So here's the point, water goes in, mixes with the groundwater aquifer and then comes back out here as a gaining stream. So this analogy and hydrology could be understood well when we draw it because when I draw it, I know there are trees. I can put the specific tree types and how much water they would take thereby I am planning well ahead on the water uptake of the plants and the water quality change because when water goes into the ground, it gets filtered out. And when it comes back out, there is some better quality of water. So this aspect can be learned from the groundwater conceptual model. And now when I put it into the modeling sphere where you have, for example, mod flow simulation model, you would notice that my understanding would help in putting some parameters in the model and also fine tuning the results when I know it's not happening. For example, in the gaining and losing phenomena is not caught by the model, the simulation model. And it's asking me to change the river discharge or the well height. Then I say, no, no, that is not possible because as per the conceptual model and as per my first field work, the field and conceptual model, this is not correct. At the end of the day, the model is still a model. It tries to mimic nature. It cannot mimic fully. So you are the best person to tell the model what is right and wrong because you will be having, for example, a field visit. And you would have seen the recharge and discharge zones in the field. So you will be in a better position to tell or find you the model. So this is the example of conceptual models. And these conceptual models are then fed into a modeling software. After the conceptual model setup of models are in a 3D environment. Some models are 1D, but nowadays because computing power is available, free open source models are available, 3D is kept. Because in a 3D, you also look at the spatial lateral movement and downward movement along with the 3 axis. I showed you the 3 axis. For example, you would have it like this. So 1 going down, 1 going x and y. But the x and y may be similar to each other. So you can make it 2D or 1D if only one vertical you want to look at. It's all simplifying your model depending on the data you have and the complexity you want to show in your modeling exercise. However, 3D is better and 3D is the best because at least it gives you the three dimensional availability and shows how water can move from x plane to y or y to z. And then from vertically to laterally and vice versa. So for all this, I would support to run a 3D model which is a higher advanced level of groundwater management and understanding. So let's see how a 3D model is set up. First, the conceptual model is done. As I said, your field work is taken, your data collection is put in and here you see in a mod flow, setting mod flow is what I'll be using to show you which is called modular flow and that is one of the leading groundwater models in the world because it is open source and a lot of people use it and a lot of forums are there to help you out with mod flow. It's a very, very good model that can actually simulate groundwater hydrology based on your inputs. So what is the first thing I'll say? It's a conceptual model where you have agreed to tell the model that it is only five layers, one to five and all of them are equally spaced, which never happens. Even a cake layer, you'll see it's not the same thickness everywhere but that is what this model is saying or at least visually that it is the same and there are different widths given here for the layers. Then once you set up the layers, then you put where the wells are, where the wells because the wells will give you the groundwater level in the layer. So then you divide your boundary into grills and each point in the grid either has data or doesn't have data. So you can choose the size of the grid based on your data availability and how fine you want to see the model. Again, understand that when you over parameterize it or over grid the model, then it takes long time to run and sometimes it is not needed to do that. So each grid I'm taking out. So this is the mod flow grid as columns and rows. I'm just taking one single cell out. So how does any model model the groundwater flows? For that one cell, it will look at how much is incoming, outgoing water and then based on the soil type and the demand and the corner depression, the water movement is given. Either it moves x direction, y direction or z direction, what is the velocity? Based on the velocity, the other neighboring cells get activated and then the water comes in from the central cube, central grid. So this is how one water moves from one grid to the next grid. We discussed about water coming from precipitation into the ground, into the aquifer. Something similar here is happening but there are other parameters also working at time t equal to 0. So at time t equal to 0, suppose my water is in the center block and at time t equal to 1, it moves forward to this block. And like this, it moves the movement is monitored by the modeling software and at the end it gives you the net moving velocity and net direction and the volume change because of this movement. So this is how a paid version of ModFlow looks like. It is called the GMS software provider and you can see on the left hand side, there's all these data that can come in and put on top of this model including groundwater levels, the thickness of the aquifer, rainfall, evapotranspiration rates from your copying calendar, etc. What you also see is they have reduced the number of wells when the family is around but the other regions is having one or two more layers. So based on the data, the number of layers is fixed and from the conceptual model also. But again, as I said in the upper elevation regions, you can see some other layers happening but it doesn't make much difference because there is not much water movement on top. It may be assumed to be of one particular type. Let's look at some of the results that comes out of the groundwater model. You can look it as a 3D mission to see how the water moves in and out or a planal or 2D. So this is a 2D plane top. So from the top, you're looking at the model and what it says is there is a hydraulic head high on the north and then there is a movement due towards the south 2010. This is kind of five years old but the science and information is the same. You set up the model, you give it rainfall and other data then you see how the water moves from one cube to the other and that aggregate movement will give you the hydraulic head distribution or groundwater level recorded in the WRI's website. So what do you see here is a hydraulic head is formed and then it slowly moves down from north to south, the river flows. I didn't put the river network in because that would kind of not show you the grids. So you can see the grids as boxes. So normally it is not a box, it is an uneven surface. So groundwater moves from top to bottom and then suddenly in this point within the medium range which is green, you see some red regions and those could be because of the how much you spend, how much you get from these countries and then water just comes down. You could see that there are some piezometers also monitored by different faculty. So all faculties can take part in a research and change it to based on their research question. And what we see here is a seasonal difference. So that is where the time stamp comes. You see water moving from north to south but then water also moves along the region, along the time scale. So from November to February, May and August you see a different change in the groundwater movement. You can see the hydraulic head, high head varies and there are some localized hydraulic heads created because of the ecosystem that is working in that region, which is trees and rocks which enable more infiltration and then losing and gaining streams. So basically this is just to show you how a groundwater model will give you output and what you could use the groundwater output for. So with this, I would like to conclude today's lecture and in fact it is the last lecture of the series where we looked at a very important part of groundwater hydrology management. Mostly the hydrology was understood using the scientific background and some management scenarios were discussed where both the government and the public can have a win-win situation not exploiting groundwater. Remember that for the agricultural productivity increase in India, groundwater created a very vital role and so we need to respect and conserve it so that our development is sustainable. We have analyzed key concepts in this 12-week extent and we also expressed or analyzed more focused groundwater relationships and analyzed different concepts on groundwater quality, groundwater availability and how they mix and match between each other. We have established groundwater budgets. As I said, the groundwater budgets could be very, very complex or simple based on the data that we have and since we have had good data in the recent years, there are good budgets that will be set up. Once you know the groundwater budget, it is as important to let the budget move for which you need groundwater parameters. So for example, if you have a water level in a well, unless you know the hydraulic conductivity around the well, the water cannot pass. It will just be stuck inside the well. You could see most probably in large dug wells because there will be cemented on the sides or there will be big rocks which prevent the water from moving. So once the water budget is set up, the key groundwater parameters are analyzed, hydraulic conductivity, thickness, specific yield, retention, etc. Then we looked at where can we get these data, which kind of platforms that we can use and WRIS website has been very helpful if giving the groundwater data, which is monitored by the CGWB. So now the CGWB is giving water level data and location. Using the location, you could create a conceptual model based on the data you are having and from the data and from your conceptual model, there could be multiple, multiple new revelations you can take or we refine your research question to answer a very focused discussion or focused groundwater problem. So I hope that this course has led for a better understanding of the conceptual part of groundwater and groundwater movement, which constitutes groundwater hydrology. Please remember India is the highest groundwater user in the world. So there is always a need for better groundwater managers and capacity built on groundwater conservation. Conservation happens with better understanding which can lead to better management and sustainable use of groundwater. Again, there are multiple more data and parameters that need to be discussed, but for a beginner's class, I think we have covered very lot and I hope you use it well for your exams and most importantly in your future situations where you have to address groundwater. With this, I would like to conclude the final lecture on groundwater hydrology and management in detail course. Thank you all.


Hi guys, my name is Shachi and welcome back to my channel. So I am a travel and a visa coach and on this channel you would have seen videos on the US visa process, how to prepare for the interview and also how to clear it. First of all, thank you so much for the overwhelming response on that video series. I am so happy and so grateful that it has managed to reach and help so many of you. If you still haven't checked it out, then click on the link above and make sure that you watch it after this video. Also if you want to book a one-to-one consultation session with me for a more personalized coaching, you can do that as well by clicking on the link which is mentioned right in the screen right now and also mentioned in the description box below. This video is going to be a year 2020 updated video about the US visa process, in particular for the B1, B2 visa category. Now there are a lot of questions which get asked commonly in the YouTube comments section and also in my Instagram DM and I feel that some of these questions are actually critical and they are actually sort of the deciding factor whether your visa gets cleared or not. So in this video I have picked not only the commonly asked ones but also the crucial ones and I am going to answer some of them. Let's get started. So the first question is, how soon can I apply again if my visa is rejected? So officially you can apply after three working days. That means that if your visa gets rejected today, you can apply for the next interview slot after three working days. Of course this is subject to availability of the interview slot in the center that you have chosen. Now there is a misconception that you need to wait for a significant period of time before you apply again like three months, like six months, nine months, sometimes even a year but this is not true at all. If you are confident about your application and you feel that you are well prepared for the interview, you can apply immediately, you do not have to wait for anything. On the other hand if you feel that you know you need a little bit more time to figure out your mistakes, to correct them and to prepare better for the interview, then feel free to take a gap of about 15 days or even a month and then apply again. Other way if you want to understand why your application was rejected and what could be the possible reasons then I have a separate video on that. I am going to leave the link for that about, make sure that you check it out. So the next question is, can I change information in my DS160 when I apply for the second time? There are some people whose visa has got rejected earlier and they are applying for their second or sometimes even for the third time. Now you need to understand that DS160 is an extremely important document because this is a document that the visa officer or the embassy officer is going to have in front of him or her when you appear for the interview. So the information that you put in your DS160 not only needs to be accurate but it also has to be consistent which means that you essentially cannot make any material changes or any significant changes in your DS160 unless those changes have truly happened. So the information you fill in your DS160 whether it is for the second time or for the third time should remain consistent from the beginning because of course they are going to have copies of all your previous DS160 as well. So one example for this and which I see very commonly is people change their salary, their monthly salary or their monthly income from their first DS160 to the other. Now if this change has truly happened and you have a supporting document for it, it's completely fine. If not you need to maintain that consistency. The second area where I see this mistake made very commonly is name of the contact person in the US. So first DS160 you have mentioned that you have a relative in the US and you have given their name and contact number then the same information needs to be maintained in your consequent DS160s as well unless there has been some change. So in short you cannot really make much changes in your DS160 unless the change has truly happened and you have the supporting document for that. If you want me to proof read and review and check your DS160 before you hit that submit button then feel free to book a consultation session with me. You know that the link is in the description box below. I also have a video which talks about the five count mistakes that people make in DS160 and how you can avoid it. Make sure that you check out that video I'm going to leave the link for that above. The next question is is there any particular number that I need to show in my bank account or is there any particular balance that I need to have in my bank account. Now this question is more relevant for people who are going for self funded visas or whose trips are going to be self sponsored and the answer is that no there is no particular number or there is no universal number which is applicable for everybody and you know which guarantee is a US visa. There is a very common misconception that you need 10 lakh rupees of balance in your bank account to ensure that you get a US visa. This is just you know misinformation it's not correct at all. The amount of bank balance required actually varies with each person. It depends upon your profile, it depends upon the purpose of the visit, on the duration of your trip, on the people who are travelling with you and so many other factors. So it obviously varies with each person and really important to understand what is the correct bank balance that is applicable for you. So I actually have a separate video which talks about this question in depth and which helps you calculate what is the funding that you should be showing in your bank account. So I am going to leave the link for that above, make sure that you check it out. The next question is how do I justify my purpose of visit. Nurse is a very common problem for people who apply for the B1 visa category. B1 visas are basically visas which are work visas so here you would normally go from your company for a short project, for a conference or for a business meeting. And here a lot of times even though the purpose is valid the visa gets rejected because the purpose was not communicated clearly. So if you fall under the B1 visa category then make sure that you prepare for this purpose of visit thoroughly. You need to ensure that the importance of your work is communicated. It is important that the embassy officer understands that you need to be physically present there to do the work and the work is not something which can be done or a video call or even over an email. It is also important to make sure that it doesn't sound too technical or too complicated which might result in the essence of the purpose itself getting lost. So I have worked with people from MNCs who travel to US in the B1 visa category and I help them define their purpose and also communicate it more clearly. So if you fall under this category and you are facing a similar issue then you can book a one-to-one consultation session with me. You know where to find the link for that. Moving ahead the next question is I am not comfortable in English what do I do? So a lot of you have messaged me saying that you are not comfortable giving the interview in English and what are the options? The US visa process actually gives you an option to hire a translator. So if you are not comfortable in English and you would like to communicate in Hindi you can hire a Hindi to English translator who will be present with you during the interview. This has to be booked well in advance so keep that in mind. Coming to the impact of this on the interview there is no direct relation whether hiring a translator will impact your interview in a positive or a negative manner. That is not true. It again depends on the rest of your profile, your purpose of the visit and everything else. But end of the day be prepared to answer this question that if you cannot communicate in English how will you manage in the USA and how do you plan to travel to the USA? Which means that if you are not able to speak English and you are not comfortable in English then how will you manage in the USA and how will you communicate there? So let me take one last question and this one is what do we do if we do not have any travel history? Now obviously travel history is something which is beneficial and it definitely helps your profile especially if you are applying for the tourist visa that is the B2 category. But it's not like people who do not have travel history do not get their visa approved. There are people who do not have any travel history and they still get their US visa. So what I would want to say is that if you do not have a travel history, if you haven't traveled out earlier there is no point worrying about it. Instead focus on the rest of your profile, in particular focus on your ties to the home country and what are your reasons to come back to India. This needs to be very strong, you need to have really strong reasons and really strong factors which show that you will come back to India and you have enough ties in your home country. Many times this becomes the deciding factor between whether the visa gets approved or rejected. So guys there are more questions but I don't want to include them in this video otherwise this is going to become way too long. So I will do a part 2 of this FAQ video and you know get the rest of the questions in there. In the meanwhile if there is something that you would like to ask something which is bothering you and troubling you feel free to leave them in the comment section below. You can also reach out to me via Instagram DM, my handle is right below. Don't forget to subscribe and hit the bell icon so that you don't miss out the part 2 of this video and all the other videos which I am going to be filming on the US visa process. In the meanwhile if you feel that you need a more personalized approach and a more one to one coaching to help you get through this US visa journey feel free to reach out to me, you can reach out to me via Instagram through my website or you can also directly book a session with me. The links for all of this is going to be in the description box below. So I guess that's all for this video wishing you all a very happy 2020 and all the very best for the US visa process. I'll see you in the next one signing off for now, bye.


Hello and welcome to the Grand Line Review, your source for everything One Piece. Today is a very exciting day for the Encyclopedia as we are going to be examining one of the undisputed most powerful devil fruits in the series, the Goro Goro Nomi. The Goro Goro Nomi is a logia type fruit that allows its user to conjure, manipulate and become electricity. It was consumed by the M&M of One Piece, called Inel, and first showcased in all of its terrifying glory during the Sky Pier arc. This fruit takes its name directly from any Japanese word, Goro Goro, which is an onomatopoeia used to represent the rumbling of thunder. And as a result, both Viz and Funimation, both throughout the reasonably, chose to dub the Goro Goro Nomi as the rumble rumble fruit in English. And all right, let's begin scaling the mountain of stupidity that this fruit represents in terms of sheer power and versatility. First and foremost, it should still probably be stated that this is indeed a logia fruit, and so the user essentially becomes electricity, which means that they are entirely intangible unless faced with C stone, Haki, or its natural counter, rubber. Logia fruits are pretty rare in the grand scheme of things, so I do feel like it's important to point that out every time because it's a ridiculous advantage in 99% of situations in the One Piece world, and probably closer to 100% in the real world. But in this case, the user is able to become and generate an incredible spectrum of electricity. As demonstrated in the series, the user can conjure anywhere between one to 200 million volts in any single attack. But it should also be stated that there's nothing really to say that the user can't just conjure less than one million volts as well. I mean, if the user just needed some sweet, sweet 240 volt power to run a kettle or something, then that would surely be achievable. Well, you know, whatever voltage you use in your country be it 110, 220, or any other crazy number, it doesn't matter all that much so long as the user of the fruit can achieve it. Although the maximum is definitely capped at 200 million volts. And look, it should be stated that when it comes to electricity, volts aren't exactly all that. And the real damage is going to be caused by the current entering and frying a particular thing or being. However, high voltage is needed to get around resistance and allow current to, you know, do its thing. So I guess maybe think of voltage as a gun and current as a bullet. Now, sadly, we have no real indication of the amps the user of the Gorogoronomy is capable of. But by using this handy guide, along with the visual impacts in the series, it could easily be up to 10 amps, which makes this fruit capable of causing mass amounts of death almost effortlessly, even if most of the victims struck in one piece do survive anyway. It goes well, manga. But essentially what we have is one of the most incredible offensive abilities in the entire world of devil fruits. You know, just throwing a lightning bolt at someone because why not? But of course, this is certainly not the only way to make use of the fruit. For example, it is also entirely possible for the user to heat up the air around them to such an extent that it creates a thunder clap, which can be used to neutralize enemy attacks and create a powerful shockwave. That's also not the end of temperature potential either, as the user can raise the temperature of any conductors such as gold to the point where they melt and in the specific case of gold, the user can then mold it as they please via the art of electromagnetism. And of course, the abilities of the Gorogoronomy can also be used to maximum effect in the extreme long range, as demonstrated by the iconic El Thor attack in which the user materializes electricity in the sky above their target and then directs it straight down at that poor, poor individual, causing the immediate destruction of anything in its path. But as awe-inspiring as that is, we don't quite need to be so boring about it either, as the user is entirely capable of sculpting their own electricity into highly complex shapes, which in the series are shown to be mainly animals, but really, if you can conjure this level of detail, then really, I don't think there's any limit to the electrical sculptures that you could create. I mean, if you wanted to strike down your enemy with a giant effigy of condoriano, then that's well within your power to make happen, as is the ability to completely transform yourself into the guise of a thunder guard, which is a super important point, really. Despite the amount of low gear users we've been shown in the series, they very rarely display their ability to physically morph themselves into an entirely new body. Although I guess that both times I can recall it happening, it resulted in a near instantaneous smack down from Luffy, so it's used to just really cool but debatably effective at higher levels of combat. But combat isn't everything, and it would be foolish to ignore the sheer speed of travel that the Gora Gora Namia allows the user to attain. Now, sadly, there's no simple answer to the question of how fast is lightning, because lightning itself is a sequence of events. But I guess what we're focused on here is what's called the return stroke, which is estimated to be able to travel at roughly 100,000 kilometers per second, which is roughly a third of the speed of light. But we do run into problems there because there is a maximum distance that lightning can travel before dissipating, which in the real world is between eight to 16 kilometers from a storm, which I imagine is why Enel needed the Arch Maxim to travel to the moon. With that said though, the user can essentially teleport themselves within an immediate radius of 16 kilometers at most. And once that's happened, then they can simply just do it again. And if that action were to take exactly one second each time, then you'd be able to get from one coast of the United States to the other in under three minutes. Furthermore, the fruit also has some nice surprise uses, such as very potentially saving you from death in the very odd circumstance that you succumb to it. Basically, the user of the Gora Gora Namia can use their electricity to jumpstart their heart upon entering cardiac arrest, making them even more difficult to deal with than they already are. And with that, let's examine the use of the Gora Gora Namia as catered to our Lord and Savior, God Enel. First of all, let's address it with all of the aforementioned abilities. Enel, quite understandably, began referring to himself as a God. And even believing it because that's just what this fruit does.  And they'll never express to any sort of strain when using the fruit, meaning that all of these devastating attacks can be summoned at no real cost to the user. But Enel wasn't happy at just that because he went to the trouble of constructing the Ark Maxim to produce apocalyptic scale power. And combining that Ark with the Gora Gora Namia was actually a really brilliant thing to do. And I don't think anybody ever gives Enel the credit he deserves for his ridiculous knowledge in the realm of engineering.  So not only that, but his Haki was enhanced heavily by the Gora Gora Namia, which allowed him to listen to electrical sound waves that traverse the air, essentially combining with observation Haki to give him an omniscient radar, further enforcing his belief of Godhood. However, Enel's issue is that he relied pretty much entirely on the instant benefits of the fruit, which, while great, led him to grow arrogant. And that's why he was defeated at the first sign of resistance to his abilities in the character of Monkey D. Luffy. I feel like if Enel had bothered to train seriously, then he would have been one of the most fearsome presences in the One Piece world. And before anyone tries to tell me that Enel wasn't weak, consider this. He was gifted the only devil fruit in existence so far that has been said to make the user practically invincible, and he had an incredible natural talent in observation Haki, and yet he's still lost to a pre-timeskip, pre-Haki, pre-gear Luffy. Yeah, the dude was pretty weak. Some other miscellaneous things to consider when becoming a lightning human. Whether it's particularly useful or not, there is the potential for the use of the Gora Gora Namia to create the phenomenon of ball lightning, which can be several meters in diameter. As for how one would go about creating it, I can't really tell you, and neither can anyone at this point in human history. But if you were so inclined for a fun arts and crafts object, then it's very important to know that it is entirely possible to petrify lightning by making it come into contact with sand, which creates a kind of glass known as fulgurite. Very delicate, but very, very cool. And I know that I'd probably spend far, far too much time trying to create these. To put it simply, what we have here is by far the most overpowered fruit examined in the Encyclopedia thus far, and quite possibly the most overpowered fruit that we will ever examine. I'm not gonna get into the debate about the Gora Gora Namia versus the Pika Pika Namia here. It's an interesting discussion, but one best for another video. But for now, this fruit for all intents and purposes does transform you into a god-like being. And the power granted to you seems to come at no cost whatsoever. You're just instantly one of the most powerful people on the planet, and with some serious training, then the potential to become a Yonko or even the pirate king is well within reach. And yes, it does have a weakness to rubber, sea stone, and armor, mentality, but the latter to apply to all devil fruits and encountering someone with a body made of rubber just isn't a common enough occurrence to detract from this fruit-selling features. And so it receives a perfect score, and there should not be a single person in this world who would hesitate to eat it. And with that, we are going to commit the Gora Gora Namia to the devil fruit Encyclopedia. Next week, we'll be changing up the pace and slowing things down a bit. The fair bit, actually, as we examine the ever-crazy Noronoro Namia. If you enjoyed this video and the content this channel produced in general, then please do consider donating to the Grand Line Review Patreon because the support of all of your amazing people is what continues to make this channel possible. Also, do check out my Teespring store if you're interested in shirts, hoodies, and other miscellaneous items with the proceeds going directly to support the channel as well. And if you'd like to join the fun at any time, then please do head over to my Discord server where a wide array of shenanigan retakes place on a daily basis. And finally, please do comment with your thoughts on the Gora Gora Namia. This has been the Grand Line Review, and I'll see you next time.


Hello, hello, writers! I'm Kristin Kiefer, author of fantasy fiction and creative writing resources, and you are listening to the Well-Storied Podcast, where I share insights, encouragement, and actionable advice designed to help you craft sensational novels and build your best writing life, always in 30 minutes or less, so you can get back to writing, of course! Ready for the show? Let's get talking! Hello again, writers, and welcome back to the Well-Storied Podcast. I'm your host, Kristin Kiefer, and today is Friday, July 31st, 2020. Today's episode of the podcast is titled 33 Ways to Write Stronger Characters, and it is a newly revised and updated version of an article that first appeared on the Well-Storied blog on June 11th, 2015. Today's article includes plenty of links that lead to articles containing additional information about what we're going to talk about today. So, if you would like to check out those links and read along as you listen in, please visit www.well-storied.com slash stronger characters. Without any further ado, let's dive in. No matter how much tension you pack into the pages of your book, readers won't care about your story if they don't first care about your characters. After all, there's nothing page turning about a predictable protagonist playing out a predictable plot. If you want readers to invest in your characters and their journeys, then you must craft characters that readers can believe in, characters so well-developed and realistic that they seem to spring to life on the page. But crafting a cast that moves beyond caricatures and cardboard cutouts can be difficult. What steps can you take to develop complex and compelling characters today? Here are 33 tips you won't want to miss. Tip number one, establish their role. Before you can effectively develop a character, you must determine the role they'll play in your story. Will they serve as a protagonist or antagonist? A sidekick, mentor, or love interest? Some other sort of secondary or tertiary character? Tip number two, define their narrative purpose. No matter the role they play, every character in your story must serve a purpose. If you can remove a character from your manuscript without impacting readers' understanding of your plot or story world, then that particular character doesn't add value to your story. Give them purpose or let them go. Tip number three, construct their identity. Defining the basic demographic markers that determine your character's identity can help you develop their worldview and experiences and help readers understand who your character is, of course. Consider your character's age, gender identity and sexuality, race, religion, ethnic background, level of education, and other common identifiers. Tip number four, consider their name. Though they may seem simple, names can hold great power. Consider using your character's name to showcase the era in which they live, to hint at their ancestry, establish a naming system for your fictional world, or otherwise lend additional depth to your story. Tip number five, establish their appearance. Helping readers visualize your character by establishing their height, build, and coloring is important. However, don't forget to consider other insightful elements of your character's appearance, such as their mannerisms, gait, physical tics, and body language. Tip number six, develop their personality. Your character's personality is more than just a list of positive and negative traits. By exploring how the characteristics that define your character impacts their voice and lived experiences, you allow your character's personality to breathe life onto the page. Tip number seven, determine their flaws. To be imperfect is to be human. By giving your character a moral shortcoming, negative character trait, quirk, fear, bias, and or limitation, you develop a realistic character with whom readers can relate. Never mind the conflict that a good flaw can create. Tip number eight, identify their false beliefs. The human experience is rife with internal conflict. Doubts, fears, flaws, and regrets can all lead characters to formulate false beliefs about themselves and the world that impact nearly every aspect of their characterization. Tip number nine, consider their worldview. A character's worldview can be defined as their breadth of knowledge and perspective concerning the world in which they live, which impacts how they think and interact. Elements like your character's upbringing, education, religious and political beliefs, relationships, and societal influences can all impact their unique worldview. Tip number 10, give them a voice. A character's voice is a unique way in which they engage with the world as determined by their worldview, personality, upbringing, and other key characterization elements. A character's voice commonly manifests in their mindset, speech patterns, vocabulary, opinions, internal narrative, body language, and mannerisms. Tip number 11, define their quirks. Everyone has strange qualities and habits. Defining the quirks that make your character unique can help further distinguish them in reader's minds as well as lend to their relatability and or serve as a key character flaw. Tip number 12, establish their relationships. Relationships are a defining aspect of our lives. Whether good or bad, past or present, determining your character's most important relationships can help readers better understand your character's world. Tip number 13, define their everyday reality. Consider what your character's life is like before their story begins. How do they spend their days? What are their daily joys and trials? Establishing this reality is key to laying the foundation for your character's first appearance on the page. Tip number 14, determine what they want. When your character first appears on the page, they'll likely be dissatisfied with their life in some way or become dissatisfied due to early events in their journey. Consider what your character believes will resolve this discontent. What do they want to have or achieve to lead a more satisfied life? Tip number 15, determine what they need. Often what a character wants isn't actually what they need to resolve the issues they're experiencing in their life. If this holds true for your character, then determine what they need to realize or achieve to lead a happier life. Tip number 16, give them a goal. A character's goal determines the arc of their story, prompting the actions they'll take and the conflicts they'll encounter. Defining your character's goal can help you address your story's plot with confidence and clarity. Tip number 17, define their motivation. Conflict means little without emotional context. To encourage readers to invest in your character's journey, determine why your character wants to achieve their goal. What will push them to act despite obstacles and hardships? Tip number 18, establish their history. Well-developed characters have lives that extend beyond the confines of the page. By giving readers a glimpse of your character's backstory, you can lend context to their characterization and the conflicts they experience throughout their journey. Tip number 19, define their ghost. A ghost is an aspect of a character's past, often a form of guilt, grief or grievance that haunts them throughout their journey. Determining the ghost that haunts your character can inform many aspects of their characterization, from their goals and motivations to their personality, worldview, flaws and false beliefs. Tip number 20, establish their desires. Hope and longing are powerful human emotions. Consider what your character desires for their life. What do they envision for themselves in an ideal world? What dreams do they wish could come true? And why do they feel their dream is unattainable, at least in the present moment? Tip number 21, give them interests. All work and no play makes Jack a dull boy. By giving your character interests that extend beyond the goal they're working to achieve, you lend additional depth and realism to readers' understanding of your character's life. Tip number 22, consider what they love. There's no motivator more powerful than love. By defining who or what your character loves most in the world, you reveal to readers what they're willing to fight for. Tip number 23, give them agency. Personal agency is the ability to exert power over one's life. Though your character may find themselves in less than ideal circumstances from time to time, allowing them to exert at least some small measure of control over their life is essential if they're to helm their journey and or experience growth. Tip number 24, make them relatable. Readers don't need to find characters likable to care about their stories, but they do need to relate to them in at least some small way. Consider what makes your character relatable no matter the role they play in your story. Tip number 25, make every character complex. It's easy to lean into well-worn tropes when developing secondary and tertiary characters who don't receive much time on the page. To ensure these characters feel just as realistic, consider one or two details you can slip into the text that hint at their richness and complexity. Tip number 26, showcase their emotional landscape. One note characters will always fail to come to life on the page. Allow readers to see your character as fully realized by showcasing what makes them laugh, cry, see the cower, and otherwise experience a wide array of emotions. Tip number 27, make them sweat. Handing your character's success on a silver platter is a sure-fire way to cheapen their development. By giving your character numerous obstacles and uncertainties to conquer throughout their journey, you create opportunities to introduce readers to new facets of their characterization and growth. Tip number 28, allow them to fail. Failure reveals who we are at heart. Allowing your character to stumble and fall throughout their journey gives them the opportunity to learn from their mistakes and develop as human beings. Tip number 29, allow them to suffer. When your character encounters conflicts and failures, resist the urge to immediately resolve these issues. By forcing your character to dwell in their negative circumstances and emotions for more than just a brief moment, you give your character the opportunity to prove their mettle while lending a sense of realism to their journey. Tip number 30, take them to their happy place. While allowing your character to sweat, fail, and suffer is important, don't pack your story full of so much conflict that readers never get to see what your character is like in happier times and moments of peace. Readers and characters alike need these small moments of joy and reprieve. Tip number 31, push them to their breaking point. Happier times aside, while developed stories require conflict. Ask yourself what line your character would refuse to cross to achieve their goal, then figure out what circumstance would force your character to cross it anyway. Bringing your character to this breaking point will be one of the most important and impactful moments in their journey. Tip number 32, give them a refuge. When all seems lost, your character needs a safe haven, a person, place, or even a memory from which they can draw much needed strength. Where will your character find this refuge? What comfort or words of wisdom will encourage them to rise from the ashes of their experiences? Tip number 33, define their story. If you've harnessed the tips in today's episode, then you have developed complex, compelling characters and explored key ways to bring those characters to life on the page. But at the end of the day, well-developed characters can't stand on their own two feet. Readers may not care about your story if they don't first care about your characters, but they also won't care about your characters if those characters fail to take them on a journey. You already have what you need to develop your character's stories. You've defined their wants and needs, their goals and motivations, their failures and breaking points. Now it's time to draw upon all of the elements you've developed to weave a story that readers won't soon forget. This is what makes for a strong character. Not a thousand tiny details, but one grand story of was, and is, and will become. So tell me, writer, what journey will your characters experience? Are you ready to map out your character's story? Get started today with the Pre-Write Project. By taking time to develop your story before putting pen to paper, you'll not only write your first draft with clarity and focus, you'll avoid countless unnecessary hours spent patching up pesky story issues. By completing the activities and guided questions in this 143-page digital workbook, you'll create a reference guide to use as you write and edit your story, helping you avoid the messiness of writing by the seat of your pants and instead embrace the power of drafting with a plan in place. If you would like to learn more about the Pre-Write Project and grab your copy today, simply visit www.well-storied.com-pre-write. Now here's to crafting incredible characters and their incredible stories. Thank you for listening to today's episode of The Podcast Writer. I hope you found it helpful to your writing journey. If so, make sure to subscribe to the podcast so you never miss a new episode and to give the podcast a quick rating or review. Doing so goes a long way toward helping the podcast reach new writers and lets me know that you're enjoying what I'm creating. You can also give me a shout out directly on Instagram at christen underscore keeper. For additional guidance as you work to craft sensational novels and build your best writing life, be sure to head on over to www.well-storied.com where I share blog posts, workbooks, e-courses and other helpful resources for writers. Again, that's w-e-l-l-s-t-o-r-i-e-d dot com. Thank you again for tuning into today's episode, my friend. Until next time, happy writing!


I mean, we really only know about the irregular migration situation in North America and in Western and Northern Europe, and nowhere else in the world. And we have irregular immigrants in any country in the world, whether it's Russia, whether it's India, whether it's South Africa, whether it's Jordan, whether it's Malaysia, any country you can think of, we have irregular migrants, but we haven't got good or data at all. There was very obviously irregular migrants evade registration, authorities, border controls. The reason for that is that they are not permitted to migrate from one country to the others. So they do this by circumventing the law and hide away from the authorities. So we do not have data, other than apprehension data, at points where they are apprehended on irregular entry or in the country for staying without permission, for working without permission. So we have apprehension data, but not registration data. And they are invisible to the authorities, basically. We have heard about methods, we have tried methods in different countries, but none of this is particularly convincing or plausible because research tells us that irregular migrants use such a variety of strategies, including disguising themselves as perfectly legitimate residents of a certain country, in particular by using passports of citizens of that country that we simply cannot identify them easily. We cannot estimate them easily and we don't yet have methods distracting these type of populations from the registered population in the country. It seems to me that there is a huge potential in using this data for migration in general. I can't quite see how we can identify from this data those who are regular and those who are irregular. If I look at a Facebook analysis of Facebook users where people declare where they were born and where they are based at the moment, then this allows us to immediately see the migration. But nowhere this data is telling us whether people have been migrating regularly or irregularly. So I can't quite see how this can enhance our understanding of irregular migration and data. I believe we already have the data that we need in order to measure regular and safe migration, which is internationally comparable data on regular migration. That means visa issued to third country nationals, residence permits issued to third country nationals, work permit issues, and if we have this data and can identify the trend, whether that's increasing or decreasing, then it tells us whether globally we are moving towards more facilitation of migration. If however, we would see this decreasing, certain countries or regions issuing fewer visa to workers, students, family members, businessmen, then we could conclude that the facilitation of migration is actually decreasing, which very often leads to more irregular migration. The demand for migration isn't decreasing. The fewer legal migration channels we offer, the more irregular migration we've got. So if we look at the development of issuing of visa, then we can see to what extent migration is facilitated or sort of made difficult. Governments need to agree on one common definition for all the phenomena and for all the types of migration that we want to measure. Who is what? What is migration? How many months in what country is considered transit migration, temporary migration, permanent migration, in order to globally mainstream data that then becomes comparable? And at the moment, despite UN, IOM, ILO, UNHCR efforts, there is still enormous discrepancy. There is even more discrepancy when it comes to the point of technically conducting data and turning administrative data into statistical data. So on that basis, there is still a lot to be done.


Being able to come and meet other people that are doing good work that is inspiring is really like replenishing to your soul. TicTac is for me really a place where I can meet a lot of really cool, interesting people who know about technology, who know about society, who know about public interests that are really useful in my work. I get to meet a lot of new people and see how they're innovating in their own spaces, their own countries. It's a great learning curve and I hope that this event keeps on happening every year.


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Let's look at Euler's totion function and some of the characteristics we saw. So the totion by definition is the count of numbers less than n which are relatively prime with n. So you need an algorithm that if you have some value n that you look at the values from 1 up until n minus 1, check which of those are relatively prime with n and then count those numbers and you get your answer of the totion function. But there are some shortcuts. So as we saw in the example we had the totion of for example 24, you need to manually go through and count the numbers and look at them. If n is 1, the totion of 1 we always get 1, if n is a prime number, some prime number p then using this approach because we have a prime number p, its divisors are 1 and p, so all the numbers less than p, 1 up to p minus 1, will be relatively prime with it. So therefore the totion of any prime number is simply p minus 1. It's easy to calculate. If it's not a prime number, unless it matters some of our other criteria, it's generally hard to calculate. That is not only hard for you to calculate manually, it's hard for a computer to calculate. That is if n is a very large number, again we talk about hundreds of digits long, then it's hard for a computer to calculate what is the totion of n in some reasonable time that it takes forever. There are no known algorithms for calculating it unless we have one of these shortcuts and we'll see that that's used as one of the security features. Another shortcut is if n, the totion of n where n is made from multiplying two prime numbers p and q, then the totion of n is the totion of p multiplied by the totion of q and the totion of p, if p is a prime number, is p minus 1, so that's this part here, and the totion of q, if q is a prime number, is q minus 1. So another shortcut we have is if we know n is made up by multiplying two primes together, then we can quickly calculate the totion of n. So an example, let's just scroll back and see some of our prime numbers. So some of the prime numbers here, let's say we have 191 and 79, let's choose two prime numbers, 191 and 79, I choose two prime numbers, choose n is the multiplication of those numbers and I need my calculator, I have a calculator, what is it, 191 times 79, 15168, thank you, 15089. So in this case the totion of 15089, I challenge you in the exam to calculate it manually, that is look at all the numbers up to 15088 and determine which ones are relatively prime with 15089, it will take you a long time on a piece of paper, a computer can do it because it's not so large, but the shortcut is since we know n is the multiplication of two primes, the multiplication of two primes, in fact the answer is the totion of P multiplied by the totion of Q in this case, which is the totion of 191 multiplied by the totion of 79 and because they have prime numbers, it's 190 times 78, whichever whatever that value is we'll calculate in a moment. So if we do know that n is made up by multiplying two primes, we can calculate easily using this formula. Similarly if we know that n is a prime number, easy to calculate, otherwise hard to calculate and especially for very large numbers, impossible to calculate in a reasonable time, there are no known algorithms that we'll calculate within a reasonable time. So 190 by 78, 190 by 78, 14820, so there are some properties of the totion function that are in fact used in the encryption algorithms. Any questions on how to determine the totion of some number, Euler's totion, we've just got two theorems to present and then we're done. So they are the shortcuts so I call them that we are useful. We're going to present two theorems and again they are used in different ciphers. I'm not going to attempt to prove them or explain how they're derived, we're just going to present them. You see in an exam they are given I think in most cases, so present them and give some examples of how they're used. When we look at our ciphers after the midterm, we'll see where they become important, these theorems. Let's just introduce them. First one, Fermat's theorem and in fact it can be viewed in two different forms, we'll go through both. If we have a prime number P and some integer, some positive integer A that is not divisible by P, then Fermat showed that if we take A and raise it to the power of P 1 and mod by P, the answer is 1. That is we have a prime number 11 for example, P is 11 and some integer A which is not divisible by P, so we have some A being in simple case 4, then 4 to the power of 11 minus 1, 4 to the power of 10, mod 11 will be 1. It's all that tells us. It's not hard to rearrange that or to consider a variation of that which is sometimes useful. If P is prime and A is any positive integer, then it also turns out that A to the power of P equals A in mod P, so that is if P is a prime number. All we'll do is give an example to show, well not to prove it's true, but show one or two cases just to see how it's used. So an example, let's say for example we have in mod 7, so if the question is 5 to the power of 7, mod 7, what is the answer? You can manually calculate this. There are small numbers, you can use a calculator to calculate it. If there were big numbers, your calculator would not handle them. So in the exam if I give you much larger numbers, even if you have a calculator, it would not handle them. In this case, easy. But if you see that it matches or fits the format of Fermat's theorem, then you take advantage of that. So let's look at the Fermat's theorem which says in the second case, if P is a prime number then some integer to the power of that prime number, mod that prime number is simply that integer. A to the power of P, mod P equals A. That's what that tells us. A to the power of P is equivalent to A when we're using mod P as the arithmetic. And that's what we have in our example. We have A equal to 7, A equal to 5, P equal to 7. A is 5 in this case, if we match it to that theorem. P is 7. So we could write it in the format of, what do we get? A to the power of P. And according to Fermat's theorem, if we're using mod P, that should be equivalent to A. And when I write 5 to the power of 7 mod 7, then another way to write it is A to the power of 7. And in here brackets mod 7. So 5 to the power of 7 mod 7 will be, what's the answer? 5. We have 5 to the power of 7 is equivalent to 5 when we're using modular arithmetic mod 7. So we can use Fermat's theorem here if we have some problem that fits the format. So you can use it as a shortcut to find the answer. Of course, in this case you could have used your calculator and found what is 5 to the power of 7. You find it is, I've done it before, 5 to the power of 7 is 78,125. Then you mod by 7 and you get 5. But if it's in the format's theorem, then you immediately know it's 5 is the answer. This becomes useful again when we deal with large numbers. A large number, hundreds of digits long to the power of another large number, very hard to calculate and then do the modulus. Unless it's matching this structure, in which case we can immediately get the answer. So that's just an example of how we can apply Fermat's theorem. It has two formats and it depends upon which one's more convenient to use. They both hold. The last one is Euler's theorem and it makes use of Euler's totient function. For every a and n that are relatively prime, a to the power of the totient of n is equivalent to 1 in mod n. And another way to express that, and I think this is missing, still for positive integers a and n, they still need to be relatively prime. I don't know why I haven't said that. So still relatively prime but it's just another way to write this. a to the power of the totient of n plus 1 equals a in mod n. Again, no attempt to explain or to derive this. We will see it later when we look at RSA and public key cryptography or asymmetric cryptography and how it's used to provide the features of security in ciphers. One example, a couple of examples on that one we got. Let's say 1024 mod 11, a simple case. And you could solve this manually quite easily but we could use Euler's theorem here and note what do we have. 1024 is in fact 2 to the power of 10. So it's also 2 to the power of 10 mod 11. And also note that the totient of 11 equals 10. 11 is a prime number. The totient of that will be 10. So what we have is 2 to the power of the totient of 11 mod 11. And now compare that to Euler's theorem. 2 to the power of the totient of 11 mod 11 should be what? Just a simple example showing that if we can express our question or our problem in the format of either Fermat's theorem or Euler's theorem then we may be able to solve it faster than having to manually calculate the exponential and calculating the modulus. 2 to the power of the totient of 11 mod 11. Fermat's, Euler's theorem says a to the power of the totient of mod n is 1 if a and n are relatively prime. Is 2 relatively prime with 11? Yes it is because their greatest common divisor is 1. Therefore this fits the format of Euler's theorem. Therefore the answer is 1 because we, just as a reminder, Euler's theorem says a to the the totient of n equals 1 when we mod by n. Assuming a and n are relatively prime. You can use this to solve, as we've said, to solve large problems, exponentials and modulus. When we have larger numbers. The other form, one more example, using the second form here, a if we have a to the totient of n plus 1 if we mod by n again assuming a and n are relatively prime we should get a as an answer. Let's try one. 5 to the power of 9 mod 24. Let's calculate it first in the full approach. I'll use my calculator to calculate that and then we'll check whether it fits the format of Euler's theorem. So 5 to the power of 9 mod 24 using my calculator after the scale. So I can do modulus 5 to the power of 9 mod 24 according my calculator if it calculates manually is 5. 5 to the power of 9 is 1953125 and then if you mod by 24 the answer is 5. And we see that because it fits Euler's theorem, the second form. Because we see, recall that the totient of 24 we calculated before the break was 8. So in fact what we get is 5 to the power of the totient of 24 plus 1 mod 24 which fits the structure of Euler's theorem, the second form. a to the power of the totient of n plus 1 equals a in mod n assuming a is relatively prime with n. In this case 5 is relatively prime with 24. That was using this equation. So the point is you can use these theorems to solve some problems as opposed to manually calculating or fully calculating first the exponential and then the mod. You'll see in exams it's faster to use the theorem than to try your calculator. Especially if you've got a large number, large numbers here. You find your calculator will not be accurate or it will start to round once you get to 12 digits. When you've got a large number with a large exponent the result will be very large and your calculator may not handle it. And hence using one of the Euler's theorem or even Fermat's theorem can be useful. I'll talk about the exam in a moment. So yes I'll answer that. Let's finish on this topic so we can talk about the exam. Any questions on what we've gone through today? Not about the exam but what we've gone through in modular arithmetic. We went through the first four operations. We haven't gone through exponentiation and logarithms and we will not go through them in detail. Exponentiation is simple. It's just repeated multiplications. So the same rules apply as our normal arithmetic. Logarithms which are the inverse operations of exponentiation Again we need to do some... We had the multiplicative inverse, the additive inverse. We need to do some operations to calculate a logarithm. It's much more complex and we'll go through them after the midterm. We'll not be covered in the exam. It needs a bit more time and we need to go through some other examples beforehand. But we will cover those last two operations at a later date. So we're going to finish here. So we've gone through the operations for modular arithmetic up the division. And importantly we've introduced things like relatively prime Euler's totient function and two theorems were presented or given to you that may be useful. Any questions on this topic before we stop?