to today's lecture. So just to share a quick recap, we are in module 5 and till now we have been learning about the fundamental principles of passive microwave remote sensing and we have been trying to understand what is a forward model and a few terminologies like emissivity and how they are related to forward model. So what we will do is this is the third lecture of the fifth module and in this lecture we will try to understand how passive microwave radiometer data is helpful in measuring precipitation. So before that I hope you remember the forward model that we discussed and we were also trying to understand the generic expression for top of atmosphere brightness using transmissivity of atmospheric path, surface brightness temperature, downwelling radiation that is scattered by the surface and upwelling radiation from the atmosphere. So you do not have to learn this expression by heart. So just take a look at the diagram towards the left side of the screen where ocean is shown, atmosphere is seen and if this is the radiometer which is looking down at the surface, there can be upwelling atmospheric TB, upwelling atmospheric TB. The atmosphere themselves they can contribute to downwelling atmospheric TB which again is getting scattered by the surface, the surface here being ocean and which is redirected to the radiometer which is on border platform which is a satellite and the surface themselves they can contribute to surface emission that is TB due to surface emission. So now we know these terms and of course transmissivity of the atmospheric path if given we can use this expression that is the generic expression for a forward model to estimate the top of atmosphere brightness that is what is the value that radiometer sees when it is looking down at the surface of the earth. Now I have taken ocean because as discussed in the previous lecture, oceans to a radiometer they provide a cold background creating contrast and making measurements about precipitation easier. So we will come to that in a short while but please make sure to remember this diagram so that further discussions are easy to understand. A quick question, assume what you see here is a radiometer, I am asking you what is r, what is the footprint size? Given the height, assume a wavelength of 1 centimeter is being used and assume the value of d is 1 meter. You can assume this value as theta the angle the angle with which the radiometer is looking down at the surface of the earth. So how do we solve the simple question? We can easily write sin theta is equal to wavelength by d, isn't it? Look at the diagram sin theta equal to lambda by d which means I can easily write r as nothing but 2 times h multiplied by lambda by d, isn't it? I am using the triangle for r by 2, so r is nothing but 2h multiplied by lambda by d. So when you plug in the values you will get 17 kilometer as the answer, simple question. For us to estimate what is the footprint size if the value of h is given, lambda is given and d is given, okay? See, when it comes to passive microwave remote sensing there are its own pros and cons that is advantages and disadvantages exist for passive microwave remote sensing and what you see in your screen is just an attempt to summarize the plus points as well as the disadvantages of passive microwave remote sensing. For example, just like the calculation you have done a passive radiometer will often have larger field of views which means the instantaneous field of view can be of the order of kilometers. It is not going to be crisp and clear like what you get in optical remote sensing, okay? And of course it does have a strong advantage for the study of weather for atmospheric sounding as we discussed in the last lecture because microwaves have the inherent ability to penetrate through clouds which means if there is a cyclone happening in the Bay of Bengal, microwave remote sensing can help us see through the cyclone core, okay? The eye of the cyclone, the vertical profile can be seen using microwaves, alright?  Radiance is linearly related to temperature and on the disadvantages of course when we look at polar orbiting satellites discontinuous temporal coverage can also be observed at low latitudes and we already know that when microwave brightness temperatures are being used for measuring precipitation over land because the emissivity of land is going to be highly variable in nature. It is going to vary due to the spatiotemporal variations in soil, vegetation, roughness because of which land offers a hot background for the radiometer which means due to the differing emissivity over land estimating precipitation using passive microwave remote sensing is challenging. I will not say it is difficult but it is challenging. We will see that shortly. So with these points in mind let us now learn about what are the applications of passive microwave remote sensing in hydrology and among the n number of applications we will try to focus on the ones that are relevant for hydrology and let us start with precipitation. When we talk about visible and infrared images precipitation is estimated by observing the cloud top temperature. You remember the diagram I showed you from INSAT 3D satellites which was very similar to the diagram you see when news is being read out especially in the weather section. So when we talk about visible and infrared images because they cannot penetrate through clouds what happens is that precipitation is estimated by observing the cloud top temperature and then they are correlated with ground-based rainfall measurements. But in the case of microwave measurements they are able to penetrate clouds and especially the frequencies below 100 gigahertz let me write that down below 100 gigahertz it is best suited for measuring precipitation because they coincide with the atmospheric windows. Remember atmospheric windows not a new term for you. So the frequencies which are less than 100 gigahertz they coincide with the atmospheric windows and hence they are suitable for the study of precipitation and at higher frequencies say greater than 50 gigahertz microwaves which propagate through the atmosphere they are affected by scattering, scattering due to ice particles and at lower frequencies there is thermal emission from liquid water droplets which tend to dominate the atmospheric effects. Now whether the microwaves that are propagating through the atmosphere whether it undergoes scattering or it does not. So all these are going to depend upon the properties of cloud you know their shape their size and so on. So in a nutshell over ocean the microwave emissivity of rain is nearly 0.9 the emissivity of ocean is much lower say nearly 0.5 and oceans they offer a cold background which enables any change in emissivity to be quickly measured through the brightness temperature and therefore it is easier to estimate precipitation over oceans by a radiometer. Let me reiterate oceans cold background and the emissivity of rain is higher so the cold background of oceans are offering some kind of contrast for a radiometer to easily estimate precipitation and in the case of land surface because there is you know because the land surface offers a hot background it is highly challenging to try and you know see for a contrast because it does not exist. So over land surface what happens is that scattering of microwaves by frozen hydrometeors is used to measure rainfall. So hydrometeors the term as such it means precipitation sized particles. So over land surface to estimate rainfall there are many physical models or empirical models to relate the scattering signature to the surface rainfall rates. Now please remember that the TB that is the brightness temperature that gets registered by a downward viewing space bond radiometer it is going to have an indirect relationship with the rainfall rate. The relationship is not going to be direct it is going to be indirect and therefore the underlying physics for rainfall retrieval they are completely different when the surface is land and when the surface is oceans. So there are you know land algorithms as well as ocean algorithms to quantitatively estimate precipitation by a radiometer using TB over land and over ocean. So two classes two broad classifications of algorithms exist and let me reiterate because you know the highly non-homogeneous land surface background it tends to have varying emissivity values. So what happens is that these varying emissivity values they add clutter to the emission from rainfall. thereby making it extremely difficult to detect the signals rainfall signals. And therefore the land rainfall algorithms using passive microwaves they have been traditionally dependent on empirical relationships utilizing the ice scattering property at 85 gigahertz channel let me write that down 85 gigahertz channel. Quantitative estimates of passive microwave rainfall retrieval indicate that although the retrieval over ocean surfaces are performed with acceptable accuracy over land retrieval based on ice scattering at 85 gigahertz continues to remain ambiguous. So this is an active area of research sharing just for information. And if we move forward we can see that there are emission based methods as well as scattering based methods. And assume a satellite bond radiometer is operating at frequency say below 50 gigahertz below 50 gigahertz. And it is capturing information while flying over an ocean say the Pacific Ocean. Remember ocean offers a cold background to a radiometer. So what happens is that the emission from the precipitation sized particles that is hydrometers can be easily detected because the cold background offered by oceans they create a contrast. Again over land it may not work because land does not offer a cold background. In some time we will discuss about the specific frequencies used by satellites to measure rainfall. But for now I want you to understand that there are two methods to estimate precipitation when it comes to passive microwave brightness temperatures. They are the emission based method and the scattering based method. So when I talk about scattering based method assume the same satellite bond radiometer which is operating at frequencies now above 50 gigahertz. So what happens is when the microwave frequencies are above 50 gigahertz now no longer is emission dominating. Now scattering tends to dominate. So I am trying to explain to you at what frequencies does emission dominate and at what frequencies does scattering dominate. So microwave frequencies above 50 gigahertz now no longer is emission dominating but now the scattering tends to dominate. So in the screen in front of you have tried to depict graphically what is meant by emission and scattering. Now the upwelling radiation that is emanating from the surface of the earth it will get scattered away from the field of view of the radiometer. And there are measures wherein we can use more than one channels more than one frequency to generate something known as a scattering index to detect the presence of precipitation. We shall discuss about these once we look at the case studies. But for now let me try to summarize the different satellite missions that contain or that operate using a radiometer. We can have the special sensor microwave imager which is on the defense meteorological satellite program. Then we have the advanced microwave sounding unit, the scanning multi-channel microwave radiometers, tropical rainfall measuring mission, advanced microwave scanning radiometer earth observing system, global precipitation measurement mission. So what are they? We can get microwave brightness temperature from the data products of SSMI, AMSU, SMMR, TRMM, GPM and AMSRE. And each of these use different frequencies for example the SMMR that is the scanning multi-channel microwave radiometer they use frequencies of 6.6, 10.7, 18, 21 and 37. Whereas the SSMI they use these frequencies that is 19.3, 22.3 and so on. Whereas AMSRE they use a different set of frequencies, different set of microwave frequencies and each of these can give us microwave brightness temperature using which we can estimate the rainfall rate. Now you may remember that in one of the earlier modules we discussed about TRMM sensors. The different sensors which are onboard TRMM satellite which is precipitation radar then there is also a TRMM microwave imager. So in the last module we spent a lot of time discussing about what is precipitation radar but now I want to highlight that TRMM also has a microwave imager wherein information is obtained in these frequency channels in horizontal and vertical polarization. H stands for horizontal and V stands for vertical polarization. You can see that passive microwave frequencies they often result in high instantaneous field of view. For example, 10.65 vertical channel it has IFOV of 59 kilometers you know 59 kilometers. So I am speaking about the major and minor axis of the ellipse which is the instantaneous field of view when seen through each of these channels. Now IFOV here stands for effective field of view. Similarly, you know as I mentioned earlier that the same satellite is carrying both a radar as well as a radiometer. Radar termed as precipitation radar abbreviated as PR and microwave imager abbreviated as TMI which stands for TRMM microwave imager. And both TMI and PR they are going to collect information in different channels.  Red shows the swath of TMI taken during one particular day and what you see black is the swath of PR orbit. So typically if you want to do a comparative assessment of the rainfall that is estimated from TMI that is microwave imager and the rainfall that is estimated using PR that is precipitation radar. The first step that is performed is something known as data colocation that is locating one within the other. Data colocation just to highlight that the swath of microwave imager and precipitation radar are different in TRMM data products. When we talk about GPM that is global precipitation measurement mission, GPM also has a microwave imager which is operating in different channels. Here also H stands for horizontal and V stands for vertical polarization. And as we discussed earlier the choice of which channel has to be selected for study of precipitation it is not random. You know if 23 vertical channel is chosen it is chosen for a particular reason. Remember the discussion we had on resonant frequency and the water vapor absorption lines. So the GPM microwave imager operates on several channels in vertical and horizontal polarization as you can see from this slide. Typically the GMI instrument has 13 channels wherein each channel is sensitive to a particular frequency of microwave. So the channels are so chosen that few channels are sensitive to heavy rainfall, few channels are sensitive to moderate rainfall. Again there are few channels which are sensitive to both snow and ice within clouds as well as there is sensitivity to water vapor and snowfall. So when multiple channels are involved in GMI it ensures that different ranges of possible precipitation types are being measured. So now I am curious to see how data from GMI would look like. So what you see in front of you is TB that is microwave brightness temperature in Kelvin from GPM GMI. It was taken when it was passing through the hood hood cyclone which occurred on 9th October 2014 at 06 UTC. You can see that the colors represent the temperature in Kelvin. So the range is from 200 to 300 Kelvin and towards the left side you see the figure captured in 37 V that is 37 Gigahertz in the vertical polarization and towards the right side you see the data that has been captured in 89 Gigahertz frequency for the vertical polarization. So this is how the data is going to look like and you can clearly see the parts of the cyclone, is it not. So as discussed earlier microwave frequencies they possess penetration capabilities which is why we use them to study more about cloud properties. And since the last two decades a great progress has been made in microwave based retrieval of precipitation. But remember that with the widespread acceptance of microwave based precipitation products it has also been known to contain uncertainties that is how do we trust the rainfall we get from microwave based sensors. And there are many studies which are conducted to estimate the uncertainty of microwave based precipitation products. For example, you may have the reference rainfall say they are from Indian meteorological department, rain gauges. You may have the rainfall from a microwave radiometer and we can easily compare to estimate the accuracy of microwave based precipitation products. So let us have a quick look into the same. I hope you remember the contingency matrix and measures like the probability of detection, the falls alarm ratio etc which can be derived from the contingency matrix. So shown here is the comparison of three satellite based precipitation products with respect to the reference precipitation data. So the 2A12, this is a name given to the TMI that is TRMM microwave imager 2A12 algorithm. It is also known as the Goddard profiling algorithm which estimates hydrometeor profiles by matching the observed TB with those from a pre-existing database of simulated hydrometeor profiles. So typically a Bayesian inversion scheme goes into creation of this product. I will not get into the details but just for you to understand this product comes from microwave imager. And when we talk about the 2A25, it is the name given to the algorithm 2A25 algorithm which is obtained from precipitation radar. And when we talk about 2B31, it is a combined algorithm which merges the information from both TMI as well as PR. So let me write combined here. Combined. So the guiding principle in the design of 2B31 combined algorithm has always been to merge the information content from both TMI as well as from PR. So what you see here is a comparative assessment of the rainfall over Mahanadi river basin of India. The colors note the values. Say you can see the values of probability of detection here from all the three sources, false alarm ratio from all the three sources, critical success index and miss. I hope you remember the derivations, the expressions to calculate POD, FAR, MIS etc which we have already covered. So just to show you that this is one type of analysis wherein we compare the precipitation that we get from passive microwave sensors. And we try to relate it to the reference precipitation just to do a comparative assessment of the accuracy of precipitation. Now again this is to share that there can be studies that try to use the orbital data for analysis and studies are also conducted to evaluate the gridded precipitation products. Just to let you know that a combination of TB values from different channels can also be used to check their relationship with near surface rainfall rate. So we have discussed about TRMM. Similarly, we can have comparative studies for GPM microwave imager with respect to the reference precipitation as well. In addition to the satellites that were mentioned, we also have the time resolved observation of precipitation structure and storm intensity with a constellation of small sacks abbreviated as tropics which is proposed to be launched in the year 2022. So for more details about tropics, I would urge you to visit this particular website as highlighted. So, this constellation of small satellites are aimed to provide measurements over the tropical region and each cubesat in tropics, it shall consist of a high performance radiometer which shall provide temperature profiles using seven frequencies near the oxygen absorption line, three channels near the water vapor absorption line, imaginary in a single channel near 90 GHz for precipitation measurements and a single channel at 206 GHz for cloud ice. And just to let you know that it is going to be launched in the year 2022. So what we will do is by now, I am going to assume that you as an audience for this course have been diligently following the tutorials, the exercises that were shared using Python. And so what we will do is, let me show you how to open and view a sample file for tropics which was downloaded. So as before, I have opened the Jupyter notebook and I am going to import certain libraries. The tropics data was made available as a netcdf file .nc as extension. So I am going to import the netcdf4 for plotting. I am going to import matplotlib.fiplot as plt and I am also going to import glob and numpy. Now I need to list the path. For that I am going to use glob.glob. This is for listing all the files which are having an extension of .nc netcdf. I need to view the latitude, longitude and the microwave brightness temperature pertaining to a particular file from tropics. So let me use nc.dataset command and remember each of these files are going to have a lengthy nomenclature. The file name is too long. These are the files which were downloaded as the sample. So what I will do is I do not want to make any mistake when I am typing the file name. So let me copy the lengthy file name, use it in Jupyter notebook so that there is no syntax error .nc. So now nc.dataset reads the file and then it is stored in DAT. So if you see the file name is present, the dimensions are present and most important the variables contained in the file are listed here. So now what I will do is I want to see how to extract the variable pertaining to the microwave brightness temperature. So let me follow a simple set of commands for the same and also I want to be aware of the latitude and longitude pertaining to each measurement of TB. So I am going to use for I in range 1, 329 using nc.dataset. I want to extract the values for brightness temperature as well as latitude and longitude. So let me extract the variables one by one. You may be wondering why I am using temp bright e underscore k because that is the name with which TB is saved in the file. So I am going to use the same name and I am going to extract the value corresponding to temp bright e underscore k which is going to be stored in TB. Similarly, there is a particular name with which the latitude and longitude are saved in the file. I am going to use those names to extract the value of latitude and the value of longitude. Please make sure that you do not make any syntax errors due to spelling mistakes long. Yes, let me also try to extract the brightness temperature values. So you may have you may be wondering why I am using 329 because 329 files were present in the folder which was downloaded as a sample data. I can get to display the value of latitude which is present as a mass variable. You can see the value of latitudes here. Similarly, I can display the value of longitudes. The value of longitudes and finally, I can also see the values for microwave brightness temperature in Kelvin. There you go. So what did we do? We have tried to take a sample file from tropics data and we have tried to extract the value of latitudes, longitudes and microwave brightness temperature. Remember there are n number of files present in this folder. I am showing you how to use simple set of commands in Python to extract the values of latitude, longitude and the variable of interest for me which is the microwave brightness temperature in Kelvin because this module is on passive microwave remote sensing. As we are talking about past existing as well as futuristic missions for study of precipitation using radiometers, I will also take the liberty to introduce you to Tempest which is a short form for temporal experiment for storms and tropical systems Cubesat constellation, Tempest. Now it is a Cubesat project of Colorado State University and the aim of Tempest is to demonstrate the ability to monitor the atmosphere with small satellites. Again to get more details about the upcoming Tempest mission, I would urge you to visit this website. So the radiometer onboard shall consist of a 6 U Cubesat and Tempest D shall provide passive millimeter wave observations using a compact radiometer. For more details about Tempest, I would urge you to visit this website where all the technical details about the mission are made available. So to summarize as part of this lecture, we were trying to have focused discussions on how passive microwave remote sensing can be used to estimate precipitation and what are the kind of studies that are being carried out to validate the precipitation measurements using reference data. And we also discussed about futuristic missions like Tempest and tropics and also we saw how to use a simple set of commands in Python to view the variables in sample data. So let me hope that you could find this lecture interesting and I will meet you in the next class. Thank you.


I think that's about it, well, that's exactly what I'm talking about. Okay, I understand. Okay, so welcome back everybody. So today is the lecture three, and this is a lecture about Judah's flow, even on manifold without conjugate points, particularly sources. So, lecture three. In this lecture, we mainly focus on measure of maximum entropy via tax and silver measures. I will construct a measure of maximum entropy. Let me just say what this is about. Basically, okay. So, we support that we have mg is our manifold with our conjugate points. There is the usual, I mean, we prove in the first lecture that the universal power X universal power is a copy of R and if any of them is a family. This is a copy of R. And, you know, we want to talk about maximum entropy possible there's a notion of entropy that I mean, I suppose I must people know the definition of the topological entropy for the dynamical system in general, but in this case of geodesic flow. I just want to use something specific like it's not the definition of entropy but it's a formula that is specifically do as it flow. So, I want to start with this. I mean, I mean, you didn't know what it was, but it was already done a negative curvature by money that you have this one that you call the topological entropy extra of the geodesic flow. So, you know, we change the number of the location of the flow is just the limit when are those infinity of one over our love of volume of the P. Okay, so this is the topological entropy for the geodesic flow respect the semantic where for any being in the universal color in it. Okay, this is the definition but it's a characterization and it's going to be a geodesic flow and this is what I want to be using.  Yeah, and you're already from this. You can see that in services. Corollary or a negation energy is a surface, the compact surface of genius at least two, at least two, and it's true that the topological entropy is positive. Why, because if I'm on the surface of games at least two, I know there is a magic of negative coverage inside and the magic of negative coverage and we know that the ball they grow exponentially. The things are comparable, like the second one of the problems are comparable. So the volumes of this ball is comparable to the volume of the ball in negative culture. So basically, the pool is that there exists G zero of negative coverage. And you know, by attributes of the matrix, the volume in G of the R is comparable to volume in G zero of B. Okay, and then you see that when you have a background of negative coverage, you always have questions to entropy. And now there is a usual, the very well known rational principle, rational principle. Because the topological entropy is a supremum of all metric entities. So it's top of the team is a supremum of each new P one. The meal is the one in the end. And probability measure. Okay. And this quantity here is a metric entropy. I mean, I'm not going to write the definition, but today we come to the metric entropy for a specific for a specific measure that I will construct and then you will see the definition if you haven't seen it before. So it's a measure. So today's lecture is construct a major meal. That we like to see people. And that man is called me as a man. Okay, so I will not. There's too much time and so construction of the major. So, yeah, so the rest of this talk will be just going to attempt to measure and then calculate the entropy of that measure. Exactly. So some notation. So I like this for this was in the city. I just write it by age.  And I mean, it is very standard that I'm not going to prove it here but there is a isomorphism between gamma. And in the first fundamental group of M. Like correspondence like elements of the fundamental group apps on the universal cover as, as a reference and you can write them. As a compact portion by gamma. So come as a group of isometrics and to start. So, see what we want to build is we want to build a person measure that is in pain by the field as. So remember, so I'm just going to tell you the. So, first of all, you want to remember this was last week we define our competition of the universal cover with a boundary. Okay, you take your design grade. Okay, the equivalence classes of your design grade, you want to construct a major that is invariant by the geodesic flow. And if you take a set, you let it by the geodesic flow. This is that they have the same measure. So in particular, this measure should project to the boundary that you could project this measure, you know, you could project this set to the boundary. Basically, what I'm saying is that you have the space of geodesics, which is, you know, one to one correspondence means. Okay. So what I want to say is that if I have an invariant measure here, he should define a measure on the space of geodesics, which is just the product of boundary minus the devil. So, invariant measure defines a measure on the product of boundary on the, you know, measure on x square minus the diagonal, because it just project to the boundary. So what we're going to do is that we will construct a measure on the boundary, and then we will now want to, we want to project it to the unit line and boundary of the universal curve. And how to do that is this part is absolutely one measure so first of all there is this point kind of series point kind of series. And it's defined by this quantity, so you pick it in X and S in R, you define this quantity, P, P, S is just the sum along the isometric group minus S distance between P and gamma. This is a point kind of series.  If you take any pair of points, any super points, P1, P2, PR, P3, if you take the geodesic joint in these guys, you can find a fix, you can find the existing R positive, so that if I take a R neighborhood of this geodesic, R neighborhood of this one, I covered the whole, the whole triumphant. And this is true in negative curvature, and it is also true here because of the, of the existence of background magic of negative curvature, because if it is true in negative curvature, you can take any geodesic, you can find the corresponding correspondence between this and the other geodesic in no point get points and you can easily see that this is true. So, Kornier, that Gromov hyperbolic, this is the kind of hyperbolicity that you have in negative curvature, and here you have it in the case of no point get points with a background magic of negative curve. And the Gromov hyperbolic implies that this guy, this point characteristics is finite when S is bigger than H, if S is bigger than the topological entropy is finite, and this guy diverges if S is less or equal to H. So, how do we use this? Will you just point characteristics now to define the family of measures? So, we define the given, this is the usual, you know, when you want to define invariant measures, you define dirac deltas along with orbits, and then the limit measure you get the limit will be an invariant measure. We're doing something similar here along the common orbit, an orbit of an element by the isomac group. So, given P, X, in X, and S in R, let's say S bigger than H, we define new P, X, S to be one over in X, X, S times the sum, gamma in gamma of exponential of minus X between P, gamma, X, and the dirac delta along the lines. So, you're just taking the dirac deltas along, along the orbit of X by gamma, and you normalize by this one. It's not difficult to see that this measure, any limit of this measure when S goes to H will be a measure that is supported in the boundary, because of this way, this way it's going to infinity. So, if we take a compact set on your universal cover, the measure of that set is going to zero as S is converting to H because of this. But, more precisely, we prove this, we prove this proposition, there exists, there exists a sequence S k converging to H, such that for all P in X, new P, X, S k is converging in the weak start to new P, and moreover, we have these three properties. These three properties are very important. I mean, each of them tells you something about the measure we are going to do that in. So, new P, P in X is gamma equilibrium, gamma equilibrium meaning that if I take a border set in the boundary, so as I said, this any limit will be a measure supported in the boundary because if you take a compact set as I said, this way it's going to zero. So, you can think of this measure being a measure defines only in the boundary. So, if you take any border set, you have new gamma P, gamma A equals new P A for all gamma, for all isometric. Okay. So, it's going to be very useful to the measure we construct, for the measure we construct to be to be invariant by the geodesic flow, because know that as a method, they, they commit with the geodesic flow. As a method means that it's a geodesic, so you commit with the geodesic flow. Having this property will help to define the geodesic that is invariant by the geodesic flow. The second property is also very important, which are that these measures I'm not saying a lot. They are absolutely consistent with respect to each other. And the density is given by a Boozman function is exponential of minus H, B P, U Xi. So this is for Xi in boundary events. This is also very useful. This is like, so remember, B P Q Xi, where the Boozman function attached to the geodesic going to be desired. So, so this is die. And this is B, there is this geodesic, it defines a vector here, and then you take the Boozman function, you know, the Boozman function was defined as B P Q Xi, because B P Q, where V is this vector, is the limit when G goes to infinity of distance between C, B, T, Q minus Q. And define this in this lecture. These functions are very useful here as they define the density of these back sensorial measures. And the last property is also very useful, it tells you that the measure that we will build is fully supported. The support of new P is about 100. So this will be used to prove invariance, this will be used to calculate entropy, and this is used to prove that the measure we will construct is fully supported. And this measure will be measured in the open set. Right, so let's prove this two properties here. We need to talk about limit, weak style limit for this measure. We need to see, first of all, this measure should be, you know, the client measure and then like it gives the client measure to the whole compactification of the universal cover. And then you can, if you fix now a compact set, you can find, you can find this subsequent. But now there is another trick, I mean, it's not really, we will do to say that this same subsequent war works for all elements in the universal cover. But let me just say that how, why these families of measures. These guys, new PXSK of X union the boundary is finite. Why, because if you look at the definition of this measure, if you look at the definition actually it is. If you look at the definition, you just use kind of triangle in this one here, exponential of minus S distance between X gamma P gamma X. This guy is that's an exponential of S distance between X times the distance exponential of minus X distance between X gamma X. And here also you have similar bounds to this triangle and equality exponential of minus S distance between PX exponential of minus S distance between X and gamma X. So if you sum over all, and you divide by PXX what you will get here will be one. What you would get here will be one and what you would get here would be, would be, yeah, what you would get here so we'll be one because by definition of the point that I say you divide by this quantity. What I'm basically saying is that new PXS of X union the boundary is less or equal to exponential of S distance between X and exponential of minus X distance between X. Okay. So, so, so this guy being compact, they said being compact so you have at least a subsequent that converge. So if you fix, fix K compact. So there exists, SK, converting H to that new PX, SK is converging in the week start to new P, so they are defined in P as a limit. So for all P in K, all P in K, this is just using the fact that the space of probability measures compact or finite measure in this case we just normalize, but it's a compact, and then you have a you have a compact in the week start topology so you have a you have a limit. If K is compact, you have a limit for HP, for if K is compact, you can have a limit for, for all points in that compact set. But now you want to replace this set K to X. Here you just use the gamma values of these measures. So basically what I'm saying is here by definition here. So by definition, new PXS at A is the same as no gamma PXS gamma A. Just by definition here. So, you see that these measures are gamma equivalent. So, new PXS is gamma equivalent. So it's implied that if a subsequence works for a compact set, it works for gamma of that compact set for gamma is in top. Same here is so same subsequence works for gamma K into K for every gamma. So you have a formula you have and you can write M as the union of K of gamma K gamma and gamma. So you have one subsequent that works for all. And also the limit measure will be gamma equivalent because the sequence would. So A is gamma equivalent because new PXS K is. So it's gamma equivalent. Just by construction. Okay, this is by construction. So this proves that this path. Now to prove this path, this uses something that is not very trivial in not conjugate points. And if you can see it very natural. So, let's write again the definition of the measure. So it would be, you write, you know, PXS is one over PXS. It's now gamma and gamma for exponential of minus S, P, gamma X, and then delta, gamma X. What you do is you add and subscribe to here. Distance between Q and comma X. So you can get a new PXS is one over PXS. And the sum gamma and gamma. So this guy exponential of minus S distance between Q and comma X, and not exponential of minus S. Sorry, I'm forgetting H story. There is an H. Oh, it's not H story. This is S story. So, this is S and S before which hours. So this is now distance between Q, gamma X, minus distance between P, gamma X, and delta, gamma X. So I just add and subscribe this term. What I'm telling that it's kind of easy in negative curvature is the fact that this quantity here is converging to the movement. Let me go for that. So think of this is what you have here. You have P here, and you have Q here. Okay, you have gamma. The orbit. I did not say it, but this guy, the accumulation set the accumulation point of the orbit of any points from the input to cover by his book is, is the whole boundary. This guy is going to be the boundary. So you can think of this gamma X as a point that is converging to something in the boundary. The thing of this is gamma X is a point that's going to the boundary. The thing is, you're measuring. You are taking the difference of distance when this point is going to the boundary. The boundary. Okay. And, and see that if you think of the definition of the, of the postman function, this is exactly going to converge to this guy, which is the movement function.  So you can see that this distance. So, in the, in the non-quadricate points, it requires a two-perch space, you can just see that the negative curvature of the whole sphere, there are limits of spheres. So you can just think of this point, you take a G circle or sphere around this point, and that guy is converting to the whole sphere when gamma is gone. Gamma X is converting to psi. So, this difference, which is this quantity is redefined to movement function. So the limit when Z tends to die of distance between Q, Z, Z minus distance between P, Z, this is B, P, Q, Z. This is what I'm saying basically. This equality is true. We proved it in, I should know how it's done. So, yeah, I'm going to go to the proof, but this is what we use to see that this is going to be the movement function. The picture is creating a curve so that it's a, it's a, it's a fact.  Or another way to see here also is that the movement function depends much more easily on B. If you have this function, it's a movement function, but it depends on B. If you prove that it's continuous, you can also have it. Yeah, so that's just, it's a proposition. So, yeah, so basically that will give you, that will give you part two, because what you have here is a movement function. You have to make this new P, new P, new P, okay. And the last part here, you see, you're supposed by contradiction, suppose that there is a, suppose that the support is not the whole thing. That means that there is new in X, open Z, new P of new zero. So in particular, in particular, new gamma P, gamma U equals zero for all gamma. Okay. Because we have gamma equivalent. Okay, this is going to be an issue this day. Okay. And if you use B now, you see that this implies that new P at gamma, because they are absolutely different. These measures. So in particular, new P. This is for all gamma. So this is something I didn't say, but it's true that this actually is transitive in the boundary, but this is something I, I didn't say it, but okay, I should write here the union, gamma U of gamma and gamma is zero. And this gives a contradiction that these measures are completely respect each other because this would give zero to the full boundary. I did not say it, but the action of gamma in the, in the, in the boundary is transitive. If you take an open Z, you're going to cover all the, you're going to cover all the, you're going to cover the whole boundary. So, this is new P of boundary of X is zero. So you have these three very important properties. Okay, so, yeah, maybe I can erase it now. So, yeah, so what we want to do now we want from this measure, we want to build from these families measures we want to build a measure on the unit in a bundle that is in vain. So one way away to do it is, you know, so now I have this family new people. So they satisfy these three properties. And so now I can define, define, define new bar on the product of the boundary minus a diagonal, which is like the space of your desert by the new bar. So you put the density here. So I will say what is that. Minus. H. I will say what it's going to be. So where the desire is going sometimes that some people call the promo product of the whole spherical distance, which is like this. If I have X union, the boundary of X, I have two points like eta. I have a point somewhere. I look at the whole sphere as a P. And then the whole sphere so that is a psi and eta going to be, and I take a geodesic connecting these two guys. And this distance is beta. So there are some technicalities here in non conjugate points because I might have multiple geodesics joining side to be tough. But you can prove that if you have also another geodesic joining side to be tough. Just do this inside the same. Okay, you can prove that it's not very difficult. You just use the ingredients of this. Yeah, so that's this quantity here. You can also define it by who's my function. This is another definition. You say plus. Yeah, this is the same definition geometrically this is the point. And Yeah, so you have this measure and this is very nice actually because it's a product measure and that's this key in what is going to come next over in the next lecture, but that this measure has a product stretch because it's given by I just an exercise. Take that this guy is going to buy it. Take the bar. Because remember, we want to build a measure that will be very, very good. Yeah, so. And this again another technicality that we are facing in this bar in this case of geodesic. So the problem with that point get points is what I just said here, if you define this map. So he defined from sx. This is the unique and your bundle to the boundary of x square minus the diagonal. So what does this do it just takes the needs be minus the class that's just defined by. So I have the, I just take the two end points. So this is the class and this is the minus. But I'm saying, even in no, no, no positive curvature, this map is not, is not bijected. It's not a bijection. But think of, think of just this simple case. You have this simple surface right here you have negative curvature here you have negative curvature, and you have here the curvature is zero. So all these geodesic here that have closed geodesic. If you look at them, they stay at a boundary distance. Even if you can think about it is there a boundary distance. So they define the same end points. So you might have this picture here. So he is not a bijection. So, and to project this measure to the unit engine bundle, we would like to use this monkey. I mean, if it was a bijection, or we would say your family is this term of expensive. Which means that you don't have this feature. You could just project it using the monkey. Using the inverse of the map. Okay. But here we can do that. But what we could do, which is, you know, very technical, is that this is what we do in our paper. Yeah, in our paper is that there is a measurable choice of an inverse. And that measurable choice that choice is going to buy it.  So this choice we have to do it globally in a measurable way. I mean, in a gamma where it was born to live in the measure that we're building. If I'm going to do it. There is a, there is a measurable choice of an inverse. So I mean, so this uses abstract type of unbalanced and stuff, but you can look at the papers. There's a measurable choice of an inverse. So that's just right. And that is gamma. You know, gamma in the sense that, you know, if you do a minus one. Oh, if you do this if you do gamma, gamma xi gamma gamma minus one gamma, but you get is minus one. So, once you have that, you cannot put it is measured to the unit and you might want to do this. We take for a measurable set in the unit and you might want to take me to love. Is, you take the, where minus the diagonal and the lines of a intercept. The length of a intercept means E minus one. So this is this choice. Okay. So this defines a measure on the unit tangent bundle. So maybe that's X. And maybe it's going to end by construction. So maybe by its flow invariant. It's been done by the geodesic flow. And, and moreover, you know, see that this measure is a finite means on compact sets. So if I take here a compact set. So, it's a compact and saying that we told us okay is fine. Why is that because look at the only way that this quantity. Blow up to infinity is that this guy blow up to infinity. The problem of product is going to infinity. And this problem of product is going to infinity only when Xi and data like Xi is approaching it. So if that is approaching data, you see that the problem of product is going to infinity, but that's not going to happen when you fix a compact set. When you fix a compact set these two. So the, the, the projection they are, you know, you have this one is uniform by something depending on the format. So once you have this, and me told up is that my variant imply that it goes through the portion. Me told up.  New one. Do you need to change a battle of your money. Okay, because remember the money for was just combat coaching by gamma. Something that's invaded by the group of eyes. Okay, 15 minutes. All right, so I have to prove now that this measure has. So if you don't know yet what the metric entropy is, don't worry too much, I will, you will see something that, that defines it. So, of me you is H. I'm sorry, I might be very. So there is this. Again, before we do this, there are some preliminaries. So, see that there is. But is that given. Yeah, so see that.  And to cut the measure of some of these sets, what we do usually is we project them to the boundary, and then we look at me bar on the boundary. So we're going to define it is this kind of projection. So, even given side, the boundary of x. We define this kind of projection. The boundary in PRP, which is from x minus p to the boundary. So this map is this mean this. This is a publication. This is I hear, and this is, if you take a point. So, you take it on inside x, and you send it to the other side of the boundary, and that gives you this. Similarly, here, this mark. This is a useful tool, you know, we do project the sets to the boundary to be able to calculate the measures. If I take P here, and X here, X, another point. I also draw this here does it. And then this is PR this point here is PRP x. Yeah, that isn't just defined. And this is back. This is why I mean, I don't want to write it. You will see that so that for all x in x bar, so I should have been using using. This guy is a publication. And P in x. There is an answer that this guy contains an open set. PR x of B contains an open set. You project a ball to the boundary. This is an open set. This is almost trivial by again, using most correspondence. So, yeah. This is just saying that remember how open says I define the boundary to find an open set, I fix a vector in an angle, I fix an angle epsilon. I send this, this is the boundary. So this defines an open set. So remember the open set, remember this map that P, which is S, P, X, the boundary. So the topology is given by the, you know, for this map is from a map of this. So this is how open sets are defined.  Because you can do it connected culture. And you do it. And after that, there is this position that measures now is that so tells you that for all row bigger than these are. You just say that if you project these guys because all these sets, they contain an open set, you just proving that this open set is not like getting smaller like there is a, you know, there is a lower bound in the measure of the open set that you get from projecting a board that is big enough. Is that one over me. I should just say this new P of the R side, the X row. This guy is comparable to comparable. I mean there's a constant. You know, from inside is comparable exponential of minus H distance between the X. And you have the same thing. So that is in the boundary. You have also the same thing when you take X under. So I should say, this is true for all. Okay, I should put here X. See, is that new P of the X row is also comparable to exponential of minus H distance between PX. The constant you have might be different from here and there. Maybe I want to do this. I have just done minutes to. So, let me just say in words why, by this is true. For instance, see that B and C, you just get them from, from a. To just a sketch because I'm running out of time and I just want to, I really want to prove that the measure has put entropy bed. So, a is B and C. Why you just use the property that I proved. So you use this property. This guy is exponential. So if you have, if you have. The opposite A will be just given by the interval of exponential of minus H. B, B, U, Z, B, U, Z, A. And you can see that. You can just see that this is the same as, you know, writing, you do this. I said I wasn't going to do the proof but now I'm doing B, B, U, Z. This is of the same order plus distance between. I add and subscribe. Oh, I should. I did something wrong here. So you add and subscribe and see that this is of the order of a constant times the measure you add and subscribe the distance or this is of the order of distance between. So you, what you have is a constant times exponential of minus a distance between. I don't know why I was here where I was using X here. So, I mean, I might have been doing this proper because you add and subscribe this, this quantity this distance and then you see that the other quantity you have will be uniform about it. So a implies B and C, but to see to get a. So I will not do the proof of a, but I really want to have. Or maybe I should just do it and then next lecture I will start by proving that then for being full. Yeah, that's, I think I saw five minutes into two small. Okay, I mean, I would just prove a, yeah. And then in the next lecture. We will just need 10 minutes to prove that. Let's try to prove a is that this says they are not. So first of all, there is this fact that we started with that. So fix the compact set K, compact set X for row bigger than R. R is a fixed radius, and this R will be given by depending on the most correspondence that the distance you have, you know, the area having the most level. There is, there for every accident, there is an excellent way to go. Sorry, there is absolute positive sense that if you pick, if you pick me, I will draw a picture for this. And X in the closure. So you have the C epsilon. I will say what I was going to be the subject. So for being for some being as X, X, and C epsilon is this guy, right? This is being this is being and C epsilon is this open set you get by taking an angle around me. See, absolutely for some being. So the picture for this, but is that. So, this is what I'm saying, you have your partification. And you have a second K. Okay. So this is X bar. And you have X point is in X. You want to find a vector at X. You want to go around that point. So for instance, if I take this year, there are the people that is back to that goal, right in the middle of K. If I project. So, so now, absolutely, absolutely be here. The accident will be here. So, obviously, if I project this set here, it's going to contain CV actually. And this big set here. So this big set here. Will be PX of B. Yeah, just take a back to the book. Yeah, and then, yeah, this is the idea to get this fight. And from this. We know that because the measure is not singular for every accident that I just have that might depend on that. This is how C of C epsilon B is bigger than L. Because these are open sets. This is how open sets are defined. This is how the topology of this guy defined.  It tells me that you'll find this, you'll find this. You find this lower bound and the main. The idea is to get to get to this part, what you use is just that this guy contains C epsilon B, and you use the, how to call it, the gamma. This guy, no P, no gamma P, gamma C epsilon B is equals no P of gamma C epsilon B by gamma covariance. So this is gamma covariance. Basically, you have this set here and you have another point. If you project it, that projection will contain one of these, one of these gamma C epsilon. And this guy, they have the same measure that is bounded by below by L. So in particular, you get new P of P R X B P rho is bigger than L. Because it contains, it always contains these sets. So yeah, I'm just out of time. I thought I would. We are just left about, you know, adding these pieces to, to calculate the measure of the entropy of this measure that you constructed. And we will see that it has full entropy like entropy of this measure. So I think I'm going to stop here. So next lecture, what I will do is, I will first use the first 10 minutes to do this estimate. And then calculate the entropy of the measure using what we have, just these three properties are enough. And, and then, yeah, and then I will prove some other properties like mixing and the city of this measure. And then, yeah, let me see what more we can do. So I was up here for today. If there are questions, so I'm going to ask to answer questions. I'm sorry. Can you explain more about the choice, measurable choice of the inverse of E. Yes. So, this is really using the, this is really using the hand banner. So, yeah. So I think. So maybe at the beginning of next lecture, I will start by explaining this, this choice, but it is based on these abstract results of hand banner. And, and I'm sure there's enough time to do it all, but I will, I will, I will talk about it and at the beginning of next lecture. Thank you very much. A question. Okay, thank you. So, see you. So not that I'm sure there is a kind of time for the lecture is at 2pm Italian time, 2pm 04. So the time is due and we will have two hours.


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Again, YouTube, welcome to Brain Power, brought to you by Lauren Askins and... Cody. Askins. He's my husband. Every Monday at two o'clock? I'm his wife.  I'm actually his wife. You would bring that back up again, live on camera to start today's show. Have to. Have to. What's up? You guys excited for Brain Power? Of course. You ready to jump in? If you have procrastination, please give this video a thumbs up if you have ever procrastinated. If you have procrastinated, share it on Facebook.  You said that. I'm going to thumb it up. I'm going to thumb it up. That's just human nature. Also, just to start out, make sure you subscribe to our channel so that way you know when we go live or when we post a new valuable video on Facebook or YouTube and all of that good stuff. So tomorrow may never come, bum bum bum bum. This is true. Tomorrow, there's always tomorrow. Wrong. There's not always tomorrow. There is today. There is the now. So today, stop making excuses for why you can't get it done and start focusing on the reasons why you must make it happen. So. I like that. Good little insert dagger kind of gut wrenching because I'm going to be honest, I used to be and still kind of am a really bad procrastinator and that's why when I was writing this topic today, I was like, I got to practice what I preach. So here we go. We are no longer procrastinators. We're going to be taking action, massive action on our massive goals and get into it. I like it. Thought so far. I like it. This is a good topic. I had no clue what we were talking about before right before we literally went live, which is the way I like it because it's off the cuff. But this topic is important because in the insurance business, this applies. Absolutely. In our relationship, this applies. Right. In everyone's world, social, personal, work-related, whatever, it 100% applies. Yeah. So stop procrastinating and here are the things that I have. I have three things that I want to talk about today real quick. Keep it short, brief on this Monday so you can get back to hustling. But I want you to take these three actions so that way you start procrastinating and you can reach whatever goals are important to you. Step number one, stop trying to be perfect. Nobody's perfect and you can learn from your mistakes. So. You know the old saying? What? Done is better than perfect. True. Because nothing you ever do will ever be perfect. Right. Nobody's perfect. Nothing's perfect. Completely. There's always room for improvement so stop trying to be perfect and telling yourself I can't mess up so I'm not going to do it. Right. Wrong. You're going to learn. If you do it, you're going to learn something if you do make a mistake. So that way you know for the future. Completely. So. That's good. Step number two, prioritize things that are truly, truly, truly important to you. And that goes to your goals. If your goals are really important to you, you're going to do whatever it takes to reach that goal so you're not going to want to procrastinate. You're not going to want to put it on the back burner. Each day, Cody's great about writing his goals down. Y'all know that. What's up Sterling? Hi Sterling. Thanks so much for joining. Cody's really great about writing his goals down every morning. So maybe the night before you plan out your day and you think about what do I have to do tomorrow to make sure I wake up and I'm not stressed about all the things I have to do. You know exactly the moment you wake up. I'm getting up. I'm going to the gym. I'm making breakfast. I'm headed to work. And these are the things that I'm going to do as soon as I get to work. So. I agree. I think by doing that every morning, what's up Dylan? What do you mean? By jumping in. That literally keeps my eye on the prize so to speak. It reminds me of, hey, here's what's important to get done today. Right now. Yes. I'm writing down sponsor goals, ticket goals, all these other things. And I think that's one of the more important than just writing goals down just to write them down. It reminds me of what's important. Every single day. Every single week. All week long. Every month. It keeps me on purpose.  Say you write down your goals every single day. Yeah. You know your goals. But do you know what you need to do on the next day to get you closer to reaching those goals? Completely. That's a good point. Yep. Seriously. Just having the baby steps to get to the big steps. So it's just a really big thing. Today's live video is going to be short and sweet. We're actually going to head to commercial. I know all of you hopefully know by now that we are putting on the 8% nation conference. That's right. It's for people, by the people, to help them grow and become better at whatever it is you're doing. By the people here. By the people. All right. So today just to recap a little bit. So far we've talked about procrastination. Stop trying to be perfect. People don't do stuff because they think it needs to be perfect. Right. Being perfect keeps you from taking action and doing anything. Nothing's going to be perfect. So put that out of your mind and just do it. Oh, we got it. And then step number two, prioritize things. If it's important to you, you'll do it. Remind yourself of what's important and then go do it. I think that's good so far. So far so good. All right. So far so good. DJ, are you ready for commercials? Should I keep going? What are these for? Oh, well, these are really good excuses that people tell themselves to procrastinate. Why don't we grab two before break? Okay. While we're drawing our own, while Cody is drawing the excuses here, y'all can leave you a comment about what your excuses are, your best procrastination excuse you've told yourself in the comments. Leave that in the comments for me. Cody, what's- I need a- Blank minute nap. I'm really guilty of telling myself, okay, I need a 22-minute nap and then I'm going to go do this. If you do say some weirdness, okay, I need 20 minutes. I'm like, 20 minutes ain't going to do it, right? It's really weird, Bob. If I take a nap and it's 90 minutes, it's two hours. That's why I don't take naps during the week. So, exactly. Because my day's gone. Don't take a nap. You don't need a nap. That's why you got the nighttime to nap. You've got eight hours in the nighttime to nap. Exactly right. During the day, you're grinding. Okay. Why is that- Dude, I guarantee you- Reince, did you take a nap this weekend, bro? One. I know I did, man. I did, too. We all did. It was a weekend. DJ probably did. Dylan did. Greg, do you nap? Nope. No. You didn't nap? Greg's just- Greg's a worker, man. Greg's a worker. Bethany did nap. Bailey? I don't take naps. No, come on, man. Bailey always makes his bed and he didn't take naps. You did, man. Yeah. Okay, see? It was about 80% nap. That's strong for naps. That's strong in favor of naps. Big no nap guy. Big nap guy. Derek, big no nap guy. I told the nap this weekend, Jeremy, we got more people in the office tuning in. That's funny. That's awesome. Okay, Cody, draw another one. What's another procrastination excuse? I'll just watch one more episode of- Leave your favorite TV show in the comments. Who doesn't that hit? Exactly. That's pretty good. Hi, Lisha. Does this happen to me? Yes. Really? Just one more episode of Impractical Jokers before we go to bed? Yes. Or when you were watching Breaking Bad- Oh, no, no, no, no. The most recent one was Suits. Suits. I'll just watch one more episode of Suits. Or it's almost- Oh, there you go. Joel, I can do it tomorrow on YouTube. I love it, brother dude. 100%. Some people will put off what they know they need to do today until tomorrow because it's easy. Hey, now I'll do it tomorrow. I promise. I guarantee you I'll do it tomorrow. Everybody's so bad about that. So bad. Myself included. I used to be way worse at it. I still have my moments, but I used to be really bad at it. Okay. So- It's good. John, we only have two more and then we'll go to commercial. So you all can learn about the 8% nation. Go ahead and do both? Yeah. Or we can save one until after we get back. We'll go out with friends and work tomorrow. So this kind of like- Okay, okay, okay. Your friends invite you to dinner, but you know that you don't have enough appointments set for tomorrow. So you're going to say, oh, I'll just make it up later on in the week. I'm going to go get a beer with my friends. No, don't go drink the beer. Do what you need to do to get your goals. That was one of the things last night. I knew I'd do a live Q&A on video every single night. You wanted to go out to my parents waiting to get home until almost 8.30.  I still did it. I still did it. Well, shoot. That's what you do. You tell people you're going to do something. And you do it. And then they're shocked if you don't. And that's the kind of people you want to surround yourself with. Like all these awesome people on Facebook and YouTube. Like Joe? Joe. Joe on Facebook. He's in our group. He's everywhere. He would have told me if I- he'd have said some if I didn't do it. So this is true. Okay. We're going to head to commercial break. We'll do our last procrastination excuse when we get back from break. Ray Lewis is in this commercial. He was just inducted into the sports hall of fame. Is that what it's called? NFL Hall of Fame? That's right. Something like that? He was. I just know he's a stud muffin. NFL Hall of Fame. And he is going to be at the Abrescent Nation Conference. And you talk about someone who didn't procrastinate. That is Mr. Lewis. That's his dude. We will be back from commercial break right after this. Whatever legacy you're going to leave, leave your legacy. Figuring he has nothing to lose. McAllister coming down the sideline. And McAllister is going to glue us through a block as well. 37 to go. He has that strength. He has that athleticism. He has that speed. And he was flat out the tackling machine. And here they come. We can get rid of them. Facebook and YouTube and Instagrammers. Oh. You know, you can't forget about the Instagrams. I forgot about them. Come back. We are talking about procrastination for those of you that have just joined in. Leave your best procrastination excuse in the comments.  And share it if you've ever procrastinated. So. Joel. Joel. You hit it on the head. I can do it tomorrow. So we have one last excuse that I came up. And then I'll give my final tidbit on how to not procrastinate any longer. Let me check Facebook 12 times first. Oh. We all are guilty of scrolling on either social media platform platforms. Plural. There's lots of them out there to just see what's going on in the world. So we don't have to keep doing what we're supposed to be doing. So. Yeah. That's a big one. That's good. Either you scroll in the morning rather than writing your goals down or getting your day in check or you scroll in the nighttime before you fall asleep or however many times during the day when you're supposed to be working. So. Let's get to the root of this though. Like. Let's dive a little deeper. Why do people. Why do we naturally. Why are we natural procrastinators. We naturally procrastinate because there's some lack of something whether that be a lack of self confidence. There's a fear. Like. Rance is leaving the room. He's like, Hey, I'll finish these lead orders later. Seriously though. Okay. Seriously though. There's a lack of self confidence in what they're doing because they're afraid to make a mistake like I talked about in the beginning or there is the fear of not doing it right. I just said that one. What's the other one I was just saying? I just lost my whole train of thought to take notes out of it. Like, well, I mean, I believe that we procrastinate because we don't because it's something that we don't want to do. We put it off because it's the root of it is it's something that we do not want to get handled right now. We don't want to accomplish it. We don't want to do it, but deep down we know we should do it. But are you really passionate about it? If you don't want to do it. I've got I've got I've got I've got stuff every day that I have to do that I may not be super passionate about. At the end of the day, if I want to be better, if I want the company to grow, if I want this conference to be amazing, if I want, you know, to be the best husband ever. I want it last year. Hopefully I'll win the husband of the year again this year. I've got to do things that I may not be always super passionate of myself always super passionate. I mean, am I right? We do things. All right. Facebook, YouTube, Instagram. Tell me why people procrastinate. Why? What's up, Kim? Why do they procrastinate? Sterling. You two are quiet. Facebook is destroying today.  Give me an example. What do you got? I'm a player. What's that mean? I play a player for real. I don't know. Probably not, but player Sterling. What's that mean? Okay. Anyway, so I don't know. You don't want to do it. If we wanted to do it, we wouldn't put it off. I also think that stress is a procrastinator. I think it's one of the biggest reasons people don't do what they're supposed to do because they're stressed. So say they're worried about their finances. Yeah. They're worried about, I don't know, something else. They have all these stresses on them and then they come into work and they see this 100 people that they're supposed to call today or what not, and then you get even more stressed. So then I find myself scrolling on Instagram and shopping.  So Sterling says we are looking for results at the end of the day. You have to produce. I say I don't do something because I already know the outcome. But do you really know the outcome? Joel says I already have enough in the pipeline for now. It's like, oh, I'm okay. It's settling. It's settling. It's settling because you don't want to do it. So you talk yourself into making yourself feel better about it. Hey, I'm fine. I've got enough for now. And for me, it wasn't always I didn't want to do it. It's that I thought everything was a light switch that I could turn on and off and that I could just do it whenever I felt like it. Why did you think that? I could achieve whatever I wanted. Did so many people tell you that you were a light switch? So you can just turn it off and on? And so you started to believe that? I think, deep down, it was an excuse. I'm so confident myself that I truly believe that it is the truth. But I use an excuse to just make myself feel better about myself than I should have, because I should have just done what I needed to do. Yeah. I think that's also a very good point. So are you tough enough on yourself to look in the mirror and tell yourself that you did everything that you could that day to get you where you really want to be?  Like this whole, the whole weight loss, the whole weight loss thing in industry is just, I can't help but laugh all the time, because everybody's like, hey, dude, I'll start eating good tomorrow and then the next day. I'll start eating really good on Monday. And then the whole week goes by, man, that was a horrible day, but I promise, I'm serious now. I will do really well on Monday. And then it's like, oh, no, man, today was busy. It's not my fault, but I promise I'll work out tomorrow. I'll really get started and I'll get serious on Monday. Well, this is a prime example of me. So my goal is to get abs, but yesterday I decided to take a two and a half hour nap instead of meal prep. So talk about procrastination. That's a good nap, dude. That's a long freaking nap. I nailed it on the head. I don't think we did it either, but I was exhausted. His mother's birthday party was this past weekend, so we were up a little later and dancing. And I go to bed at nine o'clock at night, so that was late. It was after midnight for me, which is rare. Okay, so going back to, we digress, going back to procrastination, you know what you need to do if you really just are organized and create a schedule for yourself throughout the day. It's going to help you reach those steps into not procrastinating. So I feel like if you find lack of motivation, if you're lacking the motivation to do whatever you're really passionate about, you have to set the small goals so that way you can clearly see the results. Like if you're not going back to Cody's weight loss thing, if you're not seeing results, people get frustrated and throw in the towel. So just like phone calling or setting appointments for insurance agents or whatnot, if you're not seeing the results of the appointments, you get frustrated and you don't want to do it. Take a deep breath. Don't get frustrated. Know how many people you're supposed to call in a day to get you to where you want to be. Know your numbers. Know whatever it is you need to know. That's why we break stuff down now. That's exactly why you break stuff down. 100%. So that was your third point. If you like motivation. Yep. Make sure you clearly see how to set some, I like that. Set some targets. Set some goals. Have some numbers. Have a schedule. Try to hold yourself accountable in some way. Try to reward yourself. Absolutely. People are less likely to keep going if they don't see results. So you have to see results to get to the big stuff. It's true. So don't procrastinate. Do your laundry every day. Make your bed every day. Don't procrastinate and just get it done. Per Bailey. Per Bailey. We have an employee, Bailey. He's our team member. A team member. And a beast. He is our team member. A dude. The man. He is a software genius. Now you're talking. From employee to software genius. He is. I like that. But he makes his bed every single day. He doesn't procrastinate about it. He starts his day off by being successful. It's on his daily to-do list. It's just what he does. And I think it's spreading in the office like wildfire. Because now every morning I make my bed. More consistently. Because I think of Bailey. You notice how you added that more consistently. As if it doesn't happen every day. More consistently. Than it used to be. Than it used to be. Yeah. I like it. It's true. Final thoughts for procrastination? Stop. Try to be perfect. Just stop freaking doing it. You're going to learn from your mistakes. Just summarize the whole show. Don't do it. Just do it. As Nike would say. Don't procrastinate. Just do whatever it is. And don't procrastinate. Don't do it. 8% commercial. If you haven't bought your tickets to 8% nation. I don't know what you're waiting for. 8% nation.com. If you have any questions, call into the office. Yeah. The office number is listed in the comments below. It's 83340 agent. 83340 agent. Right here. That's one of the things that people think. A lot of people say, hey, you know what? I'll buy my ticket later. Yeah. Dude, we may not have a ticket later. And I might not have an awesome hotel at the fair lane for you. We have a great deal. It's kind of amazing. We have a hotel, any room, $100 off. You can hang out with all of us in the same place. Every team member will be there. Mr. Fryer says, hey, Cody, not sure if you're a member, but I was the viewer that had an interview with a big insurance company. It went great. Thanks for your advice and training. Congratulations. Awesome, awesome, awesome job from Mr. Fryer on YouTube. That's awesome. Joel, thanks for all you do. You got it, brother. Thank you for all you do. Thank you for all the great things you do. So what we didn't do is we didn't give you an excuse to procrastinate. No. I think a lot of times what happens too is we know we need to do something, but we're scared of it.   If it's the right thing to do, it's the right thing to do. Because what happens is when you think it over, or you talk to your family, or you take weeks to make a decision, the fear sets in and grows, and then you end up making the wrong decision. If it's the right decision, then make it immediately. Same thing, because you'll procrastinate less if you act immediately rather than later. Act now, think less. Done. Boom. Boom. There's the nugget of the night. Take us out. Act now, think less. Here we go, 8% nation. Don't procrastinate on your tickets. Thank you all so much for watching. Can't wait to see you in Nashville in October. See you next Monday. Have a great week.