with chapter 11, right? So chapter 11, so we were looking at, let me just project the, yeah, so we were looking at chapter 11 versus seven onwards. So 7th, 11, he's saying, you know, he's not at all inferior to the most eminent or, you know, the ones, the apostles whom you think are very good of great caliber. We are, you know, we are not at all, I'm not at all inferior to any of them, okay? And he goes on to say about a speech. Yes, he was not like them. You could not speak in a polished way, sophisticated way. They were all trained orators, whereas he was not. So he says that. And then he also, truthfully, he says, you know, though I might be untrained in maybe the way to communicate through speech and all that, but I'm not so when it comes to knowledge, okay?  Did I sin in humbling myself and not really taking money from you for my rifle expenses when I came and shared the gospel? And he says, I robbed other churches, taking wages from them to minister to you. And he also mentions the Macedonian church.  I did not, I was a burden to no one. I did not want to be a burden to anyone. I did not take money from you. What I lacked financially, speaking, what I lacked, the brethren who came from Macedonia supplied. So Macedonia region, Philippi and, you know, the surrounding regions, they, he mentions about that church, you know, those churches inverse, sorry, in chapter eight, he talks about how these churches were actually poor churches, but they gave, right? So saying, you know, these churches actually gave, right? They gave. And in a sense, it seemed as if I was robbing other churches. You know, it's not saying, you know, it's not like he actually robbed, but he's saying it seemed like that. I was robbing other churches in order to minister to you. Like, you know, he's saying I was taking wages from the other churches in order to minister to you. So I was a burden to no one. I did not, you know, I did not take any expense from you. So we see the same thing, Paul writing to the efficient church and he says, you know, I coveted no one silver, gold, clothing, and you know that I worked with these hands in order to provide for the need and so on. So, and Paul also says, you know, I will continue to do this. Okay, verse 12, okay. I will continue to do this just a minute, sorry. So Paul says, you know, I will continue to do this. I will continue to live in this manner, right. And he says, you know, I'll make this my boast, okay.  So verse 12 onwards, he says, what I do, I will also continue to do that I may cut off the opportunity, okay. So we need to see that carefully. I may cut off the opportunity from those who desire an opportunity to be regarded just as we are in the things of which they boast.  They are boasting about, they are even saying that, you know, they're comparing themselves to Paul and the ministry team. So they are boasting about this.  So now he's saying, you know, I will not give them an opportunity to boast. I'll cut off the opportunity, right. I will give them, I will not give them an opportunity to boast like that because this is how I'm living, right. I'm living, I'm not being a burden to you. And this is how I'm ministering in my ministry. I'm not coming and demanding anything.  Okay. So he says, you know, I will give them, I will not give them an opportunity to boast because in this area they cannot boast because I know that they are not living like that. They are not living like that, right. And verse 13, for such are false apostles. The way they live, false apostles are like that. In other words, he's saying, you know, these are false apostles.  They're talking about bringing, sharing another spirit. They're bringing another gospel, right. So he's saying for such a false apostles, even their lifestyle and everything, such are false apostles. Deceitful workers, okay. So they are workers, but they are working with lies and deception, wanting to cheat people, one pretending to do something, but actually wanting to, you know, defraud or cheat people, deceitful workers, transforming themselves into apostles of Christ, okay. So he's saying such are false apostles. Now, these false apostles, their lifestyle is different, their message is different, such are they, right. And he's saying, no wonder, why?  Because Satan himself transforms into an angel of light. Okay, so these are false apostles.  Like they are motives are not pure. The way they live is not pure. And they like Satan, how he transforms himself into an angel of light. You know, outwardly, the appearance wise, it's like an angel of light. People are drawn, saying, wow, wonderful. But, you know, the apostles are like that. Transforming themselves into apostles of Christ. Verse 14, and no wonder for Satan himself transforms himself into an angel of light. So just like how Satan does transform himself, so also the apostles, false apostles, they transform. Verse 15, for it is no great thing if his ministers also transform themselves into ministers of righteousness, whose end will be according to their works, meaning they are actually serving Satan. Okay, he's saying his ministers, he's referring to these false apostles as Satan's ministers, right? And what are they doing? They are transforming themselves into ministers of righteousness. They are trying to portray themselves and make themselves look as if they are ministers of righteousness, but their end, the end result will be according to their works. Their works are denying it, right?  So, yeah. So in a sense, he's warning the people, warning the church against the false apostles, right? He's warning them against the message they bring, he's warning them against the manner of the spirit in which by which they minister, he's warning them against the gospel they bring, you know, what they are sharing, and the kind, the Lord, the Jesus himself, you know, what are they portraying, who are they portraying, right? So all that he wants them about, and so he exposes their ministry, right? Verse 16, I say again, let no one think me a fool, if otherwise at least receive me as a fool that I may, I also may boast a little. What I speak, I speak not according to the Lord, but as it were foolishly in this confidence of boasting. Seeing that many boast according to the flesh, I also will boast.  For you put up with it if one brings you into bondage, if one devours you, if one takes from you, if one exalts himself, if one strikes you on the face. To our shame, I say that we were too weak for that. But in whatever anyone is bold, I speak foolishly, I am bold also. So here again, he's talking about, you know, what has actually happened about these false apostles. He's saying, you know, I'm boasting.  So please receive me as a fool, okay? It seems foolish, but he's saying just listen, receive me as a fool, okay, no problem.  And what I'm speaking and saying, I'm not, you know, saying according to the Lord in the sense that it's not inspired, but what is happening, I'm just declaring to you, okay? Since that many boast according to the flesh, I also will boast for, and he's saying, you know, many people are boasting and it's all the boasters according to the flesh. He's referring to the false apostles. He's referring to those in the Corinthian church, some of them in the Corinthian church also, you know, who are boasting. So he's saying, so I will also boast, okay? I will also talk about this, okay? Now, he, earlier he talked about how they, how the false apostles, you know, he talks about certain things, right? He describes them. He says they are deceitful workers who three things he mentions earlier. Sorry, yeah. So they're deceitful workers who conceal the truth, misrepresent the truth. So he's saying they are deceitful, right? Secondly, he says they are unrighteous, right? They disguise in order to portray themselves as apostles of Christ, but actually they are not. Just like how Satan transforms himself into an angel of light, they do that also. And their reward will be according to their works. Okay, here he continues to describe the false apostles. And he, you know, this verse 20 talks about that. And so let's look at some of the things that he says, okay? Verse point number four here, he says they bring into bondage. Okay, that is one of their traits. They bring into bondage. He says, for you put up with it, saying, you know, you are tolerating that, but this is what is happening, right? If one brings you into bondage, bondage to sin, bondage to false doctrine, false teaching, they are doing that. And he also says they devour, if one devours you, now you're tolerating that. Also, they are devouring you. So devour means to literally, the word use there means to waste or forcibly take away or ruin you. Okay, so, you know, in normal English, it just means that, you know, one is completely eating you up, right? Devouring you. So they are wasting you. They are forcibly taking away certain things from you.  So we see that false apostles, they devour, they do this. And he says, again, if one takes from you, so they are literally, they are taking, they are taking what belongs rightfully to another person. So they take away that, they either manipulate, exploit through lies, they are taking away, you know, what belongs to them, material things and finances and all that, they are taking away from you. And then it says, if one exalts himself, right? So the false apostle exalts oneself and not the Lord Jesus, right? They exalt themselves in the name or in the, what is the reason? They are ministering the gospel and they say that they're ministering the gospel and in doing so, they exalt themselves. They talk highly about themselves. They talk highly about their accomplishments. They're exalting themselves, right?   So he's saying, you know, you put up with it.  We won't do it, right? We are too weak to live like that. Yes, we are weak people. In what way we can't, we can't do this. We don't want to do this. Yeah, so that, do you have a question? No, sir, you said it was an accident. Okay, okay, no problem. Sorry, that's fine. So we're saying, you know, we are too weak to do this. Yes, we are weak people, you know, because earlier he says, no, you are saying that this person is weak. You're saying that Paul is weak. Apostle Paul is weak. You're saying that, you know, these people are weak.   It's literally like hitting you on the face. Yeah, we are too weak to do this, right? In other words, he's saying, you know, we've never done that and we'll never do this and we don't do that.  See the difference, right? They are not actually apostles of Christ. See the difference, okay? And here, from verse 22 onwards, he talks about, he's again comparing him personally and he's comparing him with the other apostles and then he goes on to talk about some of the things that he personally has undergone in ministry, what he underwent in ministry, right? So here, he's sharing that. He's saying verse 22 onwards, are they Hebrews? So am I. Are they Israelites? So am I. Are they the seed of Abraham? So am I. Are they ministers of Christ?  I am more. And he goes on to talk from here till I think it's verse 33, right? Verse 33, he talks about his experience, okay?  He says, so am I. And so he, the question is, the answer to that is yes, you know, they are Hebrews and he's saying, so am I, I'm also a Hebrew. Are they Israelites? Well, I'm also an Israelite. Are they seed of Abraham or their descendants of Abraham? Like in naturally speaking, yes, so am I. I'm also a descendant of Abraham. Are they ministers of Christ? So then he goes on to say, you know, yes, I speak, I am more than, they are more of a minister of Christ than they are, okay? If you think that these are qualifications, right? They are physical birth and they are background and they are descended from the Jews. Jews from the Israelites and, you know, from Abraham, saying, yes, I qualify. I also qualify, okay? But he goes on to talk about some of the other qualifications, he say, are they ministers of Christ? And he says, I am more than that. I am more, you know, I'm a servant of Christ, I am more than that, okay? And he goes on to say what his qualifications are to say that he's more of a minister of Christ than them. What are his qualifications? Very interesting to see that, right? Verse 22 onwards, 233, okay? So he says, I am more in labors, more abundant in stripes above measure, in prisons more frequently in deaths often. From the Jews, five times I received 40 stripes minus one. Three times I was beaten with rods, once I was stoned. Three times I was shipwrecked, a night and a day, I have been in the deep. In journeys, often, in perils of waters, in perils of robbers, in perils of my own countrymen, in perils of the Gentiles, in perils, peril meaning danger, right? In perils in the city, in perils in the wilderness, in perils in the sea, in perils among false brethren. In weariness and toil, in sleeplessness often, in hunger and thirst, in fastings often, in cold and nakedness. Beside the other things, what comes upon me daily, my deep concern for all the churches. Who is weak and am I not weak? Who is made to stumble and I'm not, I do not burn with indignation. If I must boast, I must boast in the things which concern my infirmity.  In Damascus, the governor, under Eriters, the king was guarding the city of the Damascenes with a garrison, desiring to arrest me. But I was let down in a basket through a window in the wall and escaped from their hands. So he's describing all the dangers, all the difficulties and all the persecution and everything that he went through as a minister of Christ, as a servant of Christ. And the word he uses there, as a minister, is a humble servant. The Greek diaconas means a menial worker, someone who doesn't mind getting his hands dirty, a menial laborer. So he says, this is how I was, this is how I ministered, and therefore these are my qualifications or credentials to be the minister of God. So he's saying I will boast concerning my weakness, like feebleness, he uses the word infirmity and the Greek word means feebleness of body or feebleness, weakness of body and mind. He says, and then he says, I was let down in a basket in Damascus and people were guarding and I escaped in that manner. So this is how I lived, this is how I ministered. Okay, so he continues in the same manner in chapter 12 where he's talking about his ministry, but he's also, now he goes, first he starts with all the difficulties, all the challenges and all the discomfort that he has faced in ministry and has continued to minister, right? Not only facing that, he's continued to minister despite that and for him, for Paul, that's a qualification. He says, I minister as a servant and as a servant, I have undergone all these difficulties and for me, it's nothing, it's like, for me, this is just a qualification, which is just like my credentials in ministry. Okay, that's how he considers it, right? And then he, sorry, in chapter 12, he goes on to talk about some of his spiritual experiences, some of the encounters or one of them, he talks about that and also, he goes on to talk about the signs of the apostle. He talks about that. He talks about signs and wonders and so on, right? As much as the supernatural was part of his life, he says, these were also part of my life, these things that happened, these were also part of my life and these qualify me to be a servant of Christ, to be a minister of Christ, that is who I am. Okay, so he's comparing that with how the false apostles lived their lives. So they were living not as servants, but really to take from people. They were living to forcefully take, to cheat people, to feed on people, to take what belonged to people. Whereas Paul is saying, I did not even take any money from you, even for my rightful expenses, right? So which is fine, this is the way the Lord instituted it, that those who share the gospel must live by the gospel in the sense, in another place where Paul says, those who are taught the word, let them share in everything, in material things with the ones who teach them. So this is how God instituted it, but then he's saying, and he's going beyond it, and he's saying, I did not take anything, I decided that I will take care of my needs or others were giving, and I received that and I did not take from you. So probably Paul knew some of these problems and he didn't want anyone to blame his ministry. Even when it comes to money being taken from here to Jerusalem, to the poor in Jerusalem, he puts in some of those safety measures. He says, we will make sure that this goes and reaches safely and it's used in the right manner, but you choose, you will take that person, whom you choose and you appoint so that there's no pointing of fingers, there's no blame or anything, there's no blaming, right? So we will make sure that that is used in the right manner. So probably Paul felt that this kind of thing was there in the church, so he's taking all these precautions. So yeah, so chapter 11, we see some of these things. So we get an insight into the kind of difficulties it was there during, which was there during Paul's time. We know that travel was not safe, travel was not as comfortable as it is now, right? They had to travel on horses or donkeys and if it was land, they had to travel on ships and boats if it was sea and it was not sturdy, it was not the safest ones, safe modes of transport because well, they were not as sophisticated as they've lived as they are now. So if there was a storm, if there was something there, they would, they had every chance of losing their lives, but on top of that, he was also being hunted down, right? He was his own countrymen, we say, we mentioned that, right? In perils of my own countrymen, in perils of robbers, so there were other human beings, other people who were out to get him, maybe Robin and also, you know, all these things were happening. So he lists down all that. So we get an insight into the difficulties of doing ministry, right? All the difficulties that were there during this time and despite that, he went on these missionary journeys. So when we think of missionary journeys, it's, you know, today when we say, okay, we go from here to the other place and we think of taking a train or a plane or whatever, it's so comfortable, right? And it's safe and so on, but it wasn't so. And so we are able to, you know, appreciate the kind of zeal, the kind of ministry that Paul did and definitely by the grace of God, like he himself, you know, he himself accepts it because of the grace of God, nothing else. But the fact is that he was willing to labor even more, right? Okay. So any questions on further questions on chapter 10, chapter 11, before we move on to chapter 12 or anything that you noticed, that you saw, which was very, very, you know, something that spoke to you, something that, some of the things that you noticed you could share about that as well. Chapter 10 and chapter 11. So for me, you know, the takeaway is that, you know, he's describing the false apostles in chapter 11 and also to a certain extent in chapter 10, yeah, chapter 10 and 11, right? So those are some takeaways, you know, some lessons that we should not even entertain such thoughts to live like that. Those are some things to completely avoid. And chapter 10, we see a very good explanation about the kind of weapons that we have to live and to fight the spiritual warfare and the kind of weapons and the kind of effect or the kind of results it will actually bring, casting down arguments and strongholds and everything that exalts itself against the knowledge of Christ and bringing every thought captive to the obedience of Christ. So, you know, we have these spiritual weapons, the Word of God, the blood of the Lamb, the name of Jesus and a lot of business that he's covered us with and all that, right? So the thing is for us to live that kind of a life, to live that spiritual life, to live that overcoming life with the spiritual weapons, right? So that is something, yeah. Paul lived a life to be an example to the unbelievers, to the unbelievers and to the believers as well, right? Okay. Anything else, any other things that you noticed? What is your takeaway? What is your learning? Kannan said anything that you noticed, anything that you learned? Okay. So then let's move on to chapter 12. And chapter 12 also, he's talking about how he ministered as an apostle and some of the details he shares. So let's look at that. This is chapter 12. It's just coming up on the screen, I think. Yeah, yes, Aaron. So even though he was in the flesh, right? So that's also a very important point that we live in the flesh. We live as human beings, but we don't have to fight. So which means there is an option. See, you can choose to fight a carnal fight, okay? According to the flesh you can fight, right? Which means you, you know, you lash out in anger or you know, the thing is many of these things, let's say, a spiritual attack could come through people, right? Right through a false apostle, through, you know, Satan will use anyone, right, to come against us. So when we say, you know, I'm fighting a spiritual battle, even though I'm a human being, the options that are available for me are to fight a carnal battle, to fight a spiritual battle. So when I find a carnal battle, that will be against the person, against the person whom Satan is using as a tool, that fight will be against that person. But when I find this, fight a spiritual battle, it'll be against the spirit behind, which is causing that confusion or against the enemy, the actual enemy, the source, which is actually influencing or manipulating the person, right? The spirit behind the person. So, or the spirit that is causing this kind of confusion or whatever. So the spiritual battle is actually that, right? So yeah, when you say, he chose to war with the, you know, in the spirit. So that means that he chose. Yeah, there was an option, but he chose. So he chose to not, he chose not to fight the person or the people, but the spirits that were, you know, influencing, directing them, controlling them. Yeah. Yeah, yes, Kiran. So spiritual side, we can be strong. Yeah, when we fight that battle, we will be strong. These are very, nothing compared to carnal weapons, right? The spiritual weapons are of God. That's what we see. One of the description, the way in which these spiritual weapons are described, that is in chapter 10 and verse 4, the weapons of a warfare are not carnal, but mighty in God for pulling down strongholds, which means that, hey, these are godly weapons, right? So there's no way that these weapons cannot be effective in the battle. These are godly weapons. So the only way it can be ineffective in the sense, if I don't use it right, or if I don't continue using it, or if I out of fear or discouragement, I stop using these weapons. That's the only way these weapons will be ineffective or the spiritual battle will be ineffective because these weapons are mighty in God, and they do all this, they're mighty in God for pulling down, casting down, and bringing to captivity, all these things happen, right? Okay, so we have some more time. We'll start with chapter 12, and we might have to continue in the next class. So we'll start with chapter 12. Firstly, he talks about the encounter that he had. Let's look at that. It is doubtful, not profitable for me to boast, I will come to visions and revelations of the Lord. I know a man in Christ who 14 years ago, whether in the body I do not know, whether out of the body I do not know, God knows, such a one was caught up to the third heaven, and I know such a man, whether in the body or out of the body, I do not know, God knows, how he was caught up into paradise and heard inexpressible words, which it is not lawful for a man to utter. Of such a one I will boast, yet of myself I will not boast, except in my infirmities. Before I might desire to boast, I will not be a fool, for I will speak the truth, but I refrain, lest anyone should think of me above what he sees me to be or hears me to be. So Paul is saying, I will boast of this person who has had this kind of experience. He's talking about himself. It's just a way of explaining. I will come to visions and revelations, and then I know a man in Christ. He's referring to himself, but he's talking about himself in this manner. So he's saying, although boasting will not achieve anything good or useful, but I want to share this spiritual experience. So he says he was caught up to the third heavens. He's referring to himself as a man in Christ. He says he was caught up to the third heaven. And about this experience, he's not sure whether it was in his body or out of body. So whether he, like his body was here, and then he was lifted up in the spirit to the third heavens, he's not sure. Or whether he physically, he was lifted up, he was taken there. That also he's not sure. But he says whether in the body or out of the body, I do not know, but this is what happened. Had this experience, out of body experience, or the spiritual encounter. And he heard words which it says, it was not permitted to repeat these words, something that he heard there. And is not given permission to speak these words. It's not permitted for one to. It's not lawful for a man to utter here on earth. So heavenly words that he heard, now we don't know what is it that he saw and heard, but he was caught up in paradise and heard inexpressible words. That is what he says that he had this experience. And he says, you know, but of myself, I will not boast. And he kind of puts it in a very humble way so that people don't, you know, look beyond him. They don't consider him to be something larger than life, bigger than life. And so he talks about, you know, you don't see him talk about his spiritual experiences in any of the other apostles, I mean, sorry, epistles, right? See, in any of the other letters, he doesn't mention any such experiences. The first time he chooses to write to the church and share about his spiritual experience, right? And so he says that, yeah, I will not boast about it, but this is what happened. Okay. And he says, I'd rather choose to boast in my infirmities.  Okay. And later on, he, or in the following verses, he talks about why he will boast, the reason for his boasting and what it accomplishes, right? Okay. Let's look at verses seven onwards. And lest I should be exalted above measure by the abundance of the revelation. So which means that he received, this was one such thing. He received many such revelations, right? So this was one such experience. He heard words, not permitted, not lawful for a person to utter in the earthly realm. He saw this. So he's saying, you know, lest I should be boasted above measure by the abundance of revelations. You know, we know that God revealed to him about the cross, the depth of revelation, which was, which was not there in the previous times, in the previous dispensation. God chose him to reveal, right? Reveal to the, sorry, to the world the kind of revelation that he, the mysteries that were revealed to him, right? So we see all that happening. And Paul says that lest I should be exalted above measure by the abundance of revelation. A thorn in the flesh was given to me, a messenger of Christ to buffet me, lest I be exalted above measure. Concerning this thing, I pleaded with the Lord three times that it might depart from me. And he said to me, my grace is sufficient for you, for my strength is made perfect in weakness. Therefore, most gladly, I will rather boast in my infirmities that the power of Christ may rest upon me. Therefore, I take pleasure in infirmities and reproaches in needs, in persecutions, in distresses, for Christ's sake, for when I am weak, then I am strong. So this is the, so he gives the reason for boasting in his infirmities and because of the Lord's assurance that his grace is sufficient and his strength is made perfect or his strength is available and just suited and just right to take care of that weakness. So this, we will again revisit seven onwards. We will look at in our next class for there's some more that we need to consider because he talks about the thorn in the flesh. So what is the thorn in the flesh? Why does he say that he boasts in his infirmities? We will look at it in depth in the next class. Okay, so just a minute, sorry. Okay, so we'll stop here. You have a good weekend. God bless. We'll meet again.


Hello. So, in the last capsule we can completed the proof of the theorem of equidistribution of Hermann Weyl. As we saw it is a sharpening of the Kronecker's theorem. Now we will see certain ramifications of equidistribution modulo 1 of Hermann Weyl's theorem. So, we may have suspected that the result of Hermann Weyl has some connections with probability theory. In view of the fact, the Cesaro sums that you saw, the averages that you saw gives the expectation of the random variable f of x. And this together with the uniform distribution lurking in the background may have led you to believe that Hermann Weyl's theorem itself would be a special case of a more general phenomena. This is so and we shall demonstrate this through some examples. We cannot carry this to completion because that will be another course altogether. Here what you will give you is only a trailer of the full movie. So, first we define the notion of a normal number. So, let us look at numbers in the interval 0 to 1 and each real number in the interval 0 to 1 can be written in decimal form x equal to 0. a1, a2, a3, etc. where a1, a2, a3 are the digits of the number. Let us pick up some digit say 7 and let us ask how many times does a digit 7 appear in the string a1, a2, a3, etc. Let us look at the first n digits a1, a2, a3 up to an. In the first n digits, let kn be the number of times the digit 7 appears. And then let us take the ratio kn by n and let us ask for the limit kn by n as n tends to infinity. First of all will this limit exist and if so what is the value of the limit? This limit would be the asymptotic relative frequency with which the digit 7 appears. We say that the number is normal with respect to the decimal system if all digits not just 7, but all digits must appear with the same relative frequency same asymptotic relative frequency 1 by 10. So, this limit must exist and it must be equal to 1 by 10 and it must be 1 by 10 not only for 7, but for all the 10 digits. Then you say that the number x is normal. Now we state the theorem of Borel. The set of numbers in the interval 0, 1 that are not normal has Lebesgue measure 0. In other words, almost all numbers are normal. This is a celebrated theorem of Borel. Of course, there is nothing special about the decimal system. You could work with any radix. For example, you could work with binary systems and you can ask what is the asymptotic relative frequency with which the digits 0 and 1 appear and they are supposed to appear with asymptotic relative frequency half each. Then you will say that x is normal in the binary system. You can work with hexadecimal system, there is radix 16 if you like whatever. So, that is the first observation, but let us for simplicity work with a decimal system because that is the most familiar one and the arguments will go through and you will figure out how to modify the arguments appropriately for other radix. As an exercise, I want you to come up with a number x for which this limit does not exist. In other words, the limit itself does not exist. For example, suppose I take a rational number say 0.333333 infinitely many strings of threes is obviously not normal because the because the digit 3 appears with asymptotic relative frequency 1 and all of the digits appear with asymptotic relative frequency 0. So, rational numbers are unlikely to be normal except very special ones and I will leave it to you to figure out which rational numbers are going to be normal. So, please determine all rational numbers that are normal. So, let us look at the Borel's theorem in the light of what we are done. How is it very similar to Weyl's theorem on equidistribution model 1? So, to bring out the similarity, let us define a transformation from the closed interval 0 1 to the closed interval 0 1. What is this transformation I am talking about? Take the number x multiplied by 10 and take the fractional part. Multiplication by 10 simply shift the decimal point to the next and then I am knocking off the integer part and I am taking the fractional part. So, Tx of 0.a1a2a3 will be 0.a2a3a4 etc. Now, let us assume that this number x is irrational because I already told you that very few rational numbers are going to be normal and I am going to find out all of them. So, we may as well assume that x is irrational. Rational numbers anyway have Lebesgue measures 0 and so, the digit a2 will be 7 if and only if Tx lands up in the sub interval 7 by 10 8 by 10. Now, remember that if you take a number between 0 and 1, the first digit after the decimal point will be 7 if and only if that point lies in the interval 7 by 10 8 by 10. I just taken 7 for simplicity and for illustration purposes you can take any other digit if you like. So, a2 will be 7 if and only if Tx lies in this interval. Similarly, a3 will be 7 if and only if T squared x lies in the interval 7 by 10 8 by 10. So, we have a nice recipe for deciding whether the digit a4 will be 7 or not that will depend on whether t cube x lies in that interval. So, now it is very clear what we must do our kn by n the number of times the digit 7 appears in the first n strings is simply the arithmetic mean of the characteristic function evaluated at x with the characteristic function evaluated at tx plus zeta plus characteristic function evaluated t to the power n minus 1x where chi is the characteristic function of this interval 7 by 10 8 by 10. So, this open interval is the interval i and I have taken the characteristic function of that interval. We are asking the question whether this limit 1 upon n times chi of x plus zeta plus chi of t to the power n minus 1x whether that is going to be equal to 1 by 10, but 1 by 10 is a integral from 0 to 1 chi x dx. So, this is the equation that we are looking for and the question is whether this equation is valid for almost all x in the interval 0 to 1. Now, again the first thing to do would be to replace this characteristic function chi by integrable function f and ask the more general question whether limit as n tends to infinity 1 upon n f of x plus f of tx plus zeta plus f of t to the power n minus 1x whether that is equal to integral 0 to 1 f of x dx for almost all x. When you formulate this problem in this language, then the formulation shows that both Hermann Weyl's theorem and Borel's theorem are probably special cases of a more general phenomenon. This is indeed the case and the general phenomenon is encapsulated by what is called as a strong law of large numbers in probability theory and even beyond this we can go and there is a theorem called Burkov's individual ergodic theorem in dynamical systems. We are talking about measure theoretic dynamical system. The latter arose from the study of recurrence in differential equations quantifying a famous theorem of Poincare called the Poincare recurrence theorem. So, these are matters that are studied in measure theoretic dynamics and the Burkov's individual ergodic theorem arose from the study of large particle systems in statistical mechanics. What happens is that if you look at this particular equation, if you think of t as a transformation of space, the t was a transformation of some space and what we are done is that we have taken a characteristic function of a certain set A and the transformation takes a certain point and it looks at the orbit x t x t squared x t to the power n minus x dot dot dot that is the orbit of x. The successive hits if you like if you think of dart games. So, in the first n iterates in the first n trials, how many times do you hit the bullseye the set that you are chosen and you want to understand asymptotic relative frequency with which that happens and you have taken the characteristic function of the set and you computed this averages. So, in some sense this average that you see on the left hand side should be thought of as time averages and the average that you see on the right hand side should be thought of as space averages. So, this equality is an equality between time averages and space averages and for what transformations does this happen, those transformations are called ergodic transformations and that is exactly the way in which ergodicity was studied as phenomenon in statistical mechanics, whether time averages and space averages are equal or not. Now, you may want to consult the books of Patrick Billingsley. Patrick's Billingsley's book on measure and probability, it contains a proof of the Borel's theorem on normal numbers, a fairly elementary proof of Borel's theorem. Here elementary does not mean easy, it means he does not use a sophisticated machinery. On the other hand, Patrick Billingsley has written a beautiful book called ergodic theory in information. It is a very delightfully written book, there you will find these things discussed in the light of ergodic theory, time averages and space averages and ergodic transformations and so on. Of course, I must also mention the classic book of Hardy and Wright on number theory and where you will see a discussion of both Kronecker's theorem as well as Borel's theorem on normal numbers. With these general comments we must close this module on Fayer's theorem and we must move on to the next part of this course namely Fourier transforms. So, now the next few lectures will be on the second part of this course. So, we are leaving Fourier series for some time and we are passing on to Fourier transforms. Thank you very much.


In last few lectures, you have seen how a technology platform like NAPA, Nucleic Acid Progruble Protein Arrays could be so useful to perform high throughput assays for the proteins without having the purified proteins available to you. Just by taking simple cDNA, you can express the proteins on the chip and use them for different type of applications. Today's lecture, Dr. Josh Lebert is going to continue discussion about NAPA technology and mainly emphasis will be on one of the applications about how to use these arrays for doing research on tuberculosis. As you know, mycobacterium tuberculosis, it really affects large population in whole world and especially more relevant in Indian context when we have advent of several resistant strains of mycobacterium, especially MDR strains of TV. So how to use these array platforms to do some sort of novel biomarker based projects using NAPA technology. So let's welcome Dr. Josh Lebert to discuss about applications of NAPA for screening mycobacterium tuberculosis. Okay, so I showed you this. So this is how we analyzed the data. So let me remind you again that this little array up here is multiplexed and this array down here is all the same proteins but all as individual spots. And so we spent a lot of time trying to figure out how do you represent these data. It's always fun when you're doing an experiment the first time and you realize that no one has ever come up with a way to show what you're looking at because it's so new. So what we ended up deciding to do is this kind of ball and stick model where this ball is representative of the mixed spot and each of these individual circles here represents the five proteins that are expressed in that spot. And then the color here means that we detected signal at the mixed spot and color at any one of these spots means that when we tested the individual spot it also gave color. So if you look at something like this one right here that suggests that we detected M6, we detected this spot and probably it was the adenovirus E1A protein that was responsible for that signal. That would be our best guess, right? Here's another one and probably that protein is responsible for that. Here's one that was kind of weak and notice that we don't see any of these five spots lighting up. So that becomes a little bit of a question mark. So what would you call that? Thinking back to our statistics from yesterday, it's a possible false positive, right? The mixed spot says there's a spot there but when we went to confirm it we couldn't find it. So that might be a false positive. We don't know. It could be that we just didn't get good detection here, alright? Here's another one. Also just a little bit esoteric here but we actually look at two qualities of our spots on the arrays. We look at the spot on the array itself, the blue, and we also look at the signal intensity of this area around the spot which we call the ring. And a lot of our features have that sort of ring intensity which is another sign of a very strong response. So if we see a ring we usually think it's a very strong positive. So we went through and we did some protein interaction studies using this approach. We also, this array comprises a variety of viral proteins from common viruses that people are infected with and then we probed it with serum and that, for each person would tell us which viruses that person has had before, right? So that's shown here and yes, yes, oh how do we get the value for that? Yeah, that's a little complicated. What we do currently is that we have a software application that pulls up the images and the investigator doesn't know what the spots are but we have a five star scale for scoring a spot from one to five based on essentially the size and the look and we have images for each of the five and the investigator looks at the image and then he scores it or she scores it from a one to five. And you do it unbiased so you don't know which proteins are which. You just score the ring size. It's not as precise as an instrument doing it but it's not bad. And so a very narrow spot would be like a one, bigger would be two, three would be a pretty good size, four would be like spilling into the neighboring spots and five would be, you know, huge. So kind of the size and intensity of the spot of the ring. Yeah, all right, so I think here you can see that would be like a five, that ring right there whereas this one probably would be a one or a two. So there you can see that the spot there you can see a little bit of the ring right there.  So you get used to it after you do a lot of these. So here you can see some examples. So each column is a serum sample. Each row is a multiplex spot and then the little nodes around it indicate which proteins were individually detected down here, right? If you look at this guy here, it's sort of a standard what you'd expect. You've got a strong signal in the spot and you also have a spot signal for that for one of the proteins in the five, right? So presumably this is the one that gave that signal. Then here's another example where we had a strong spot signal and there were two positives in that in that midge. So that's something that you have to keep in mind is that up until now I've been acting like within every mixed spot there's only one target. It's certainly possible that there may be more than one. And obviously you hope that the mixed spot will give you that signal. Here's another one. So what's the concern on this one? It's a false positive, right? So this one, the major spot gave us, and it's interesting that the false positives often are weaker than these guys. It's a little bit weaker. And then what's that? False negative, right? So the individual spot gave us a signal, but the master spot, the mixed spot did not, right? And so this is the one that we worried about the most. And we looked for these and I think we found a few percent where that happened. But the vast majority of the time, the mixed spot was sufficient to find whatever was present in the individual spots. Because if the two methods agree well, then you feel a lot more comfortable using the mixed NAPPA, the multiplex NAPPA, over the standard NAPPA because it's so much less expensive. And so that's, this is kind of the strategy. You produce a mixed NAPPA array up here, you get good signal for all these proteins, you show that you have good correlation. You screen for antigens that are detected in the patient, let's say, and you see a number of spots that light up. You compare array to array to make sure that you're getting consistent results. This is all sort of quality control stuff. And then after you've got these hits, you take the individual spots that lit up and you break them down into individual proteins the next day to verify that those signals are real and to deconvolute which spots were positive, okay? And so this is kind of the summary of what the mathematics works out, right? So in the old way, to do 10,000 proteins, we would have to use five slides. And then after we did those five slides, we would have to come back the next day and use a sixth slide to confirm the hits and make sure that they were real, right? Using the mixed NAPPA, we can do all of the spots on one slide and then we just have to come back the next day and do a second slide to confirm that it's real. So here we have to do a total of six, here we have to do only two. So there's a lot more work on this side than this side. And of course, if you're trying to save money, this is definitely a cheaper way to go and yet you can get roughly the same numbers, right? Okay, so I don't have to tell you guys this, right? People, this is a big issue, it's a huge health issue all over the world, including in this country. One of the big challenges in some parts of the world is the co-infection of HIV with TB. And the diagnostic methods that are available for TB are already limited in standard TB, but it's particularly limited in the context of HIV where the common symptoms and the common molecular studies don't always apply. And so we were interested in studying a little bit whether or not we could identify good biomarkers for the detection of active TB, particularly in an HIV-positive population. Okay, so these are some of the things that I think you probably already know, those of you who are aware of a little bit of the TB issues. Antibiotrophils from patients can be very different from patient to patient. Not that many antigens have been reported. Sensitivity and specificity are not ideal. And even these that have been reported, many of them have not confirmed in other studies. We almost all the studies so far have been in HIV negative individuals.  So we had done a study together on a funded grant to look at the whole TB proteome, screened with serum and all that. And for a variety of complicated reasons, that particular study could not be used. There were some issues with mixed up spots on the array. It just that, you know, normally we don't do this, but this was one case where it was a problem. And so we reached the end of the study and we couldn't use the data. And yet we really wanted to do this study and we felt responsible to the agency for publishing it. And so we needed to repeat it, but we were out of money. And so we were really operating on a shoestring. We had like no funds at all. And we had to figure out a way to study the TB proteome with no money. And so that was where the idea for multiplex NAPA came up. I was racking my brain. How can I do this inexpensively? And so it was for that reason that we decided to try the multiplex NAPA. Because that way we could get the entire proteome of TB in one quarter of an array. In fact, one array would be able to do four proteomes at once. So that really lowered the cost. And of course, the other problem, besides the lack of money, was that we were almost out of serum. So we had very little serum to test. And of course, we needed that to do the study. All right, so this is how we outlined the study. For complicated reasons, we decided not to use five proteins per spot. We used three proteins per spot. Part of that was, if you remember when I started, I showed you that the number of spots that you can use depends a bit on the expected hit rate, right? And we thought that with TB, particularly in the population from South Africa, where we were getting the samples, the hit rate was likely to be higher than just 5%. In part because that population is also co-infected by other mycobacteria. And those would cross-react with the mycobacteria proteins from TB. So assuming that there was a higher hit rate, mathematically it made more sense to do three spots rather than five, okay? So we had three proteins per spot, right? And then we screened that with TB sera and positive disease and healthier individual disease compared that. When we got a hit, we took that hit and we divided it into three individual spots in that secondary verification and deconvolution array. We did this separate array and then from that, we got individual proteins that were positive. And then we did a third level of validation by testing different samples on a verification array. And then finally, we took those individual spots to ELISA. So once again, a multi-tiered set of experiments to make sure that whatever hits we observe really makes sense, okay? And so this is kind of a summary of that. In phase one, we screened 4,000 antigens, which is the entire proteome of TB. Phase two, we did a deconvolution on what turned out to be about 870 antigens. We did those 870 antigens on an independent set of samples. And then we did a ELISA verification on the best 40 of all those, all right? And then this kind of gives you a breakdown. Our samples, I think I have a better slide for that part. Yeah, this is a breakdown of the samples that we had. So we were looking at samples that came from the US and samples that came from South Africa. We had TB patients and controls. We also had, of these 66, a fraction of them had HIV. Of these 68, again, a fraction of HIV. So we had patients who had both who were either HIV positive or HIV negative, and did or did not have TB, if that all makes sense. So it's kind of a complicated clinical design because we were looking at two factors, TB and HIV, and two different countries. And this is how we broke the samples into all the different studies. This is published last year in MCP. So I won't go through all the numbers here, but you generally get the idea. So this is what the array mix looked like when we printed it, the total protein. So again, we're using the HD NAPPA and multiplexed. So we've combined the two technologies I told you about into one experiment today. And so these are all the proteins. We also had a subset down here of individual proteins and viral genes. That did two things for us. It gave us a reference spot for individual proteins. It also gave us some positive controls so that we could make sure that everything was working because we knew that most people would have a response to some of these common viruses. Okay, and then this is just to show you that we got good expression on the arrays. So the dotted line is the cutoff line. Everything below that line was considered to be absent on the array. There was a signal there, but it wasn't real. All of this up here shows that by and large, the vast majority of proteins were well expressed on the array and easily detectable. And of course that means that we have a good chance at detecting an immune response. Okay, and then this is just once again to show you that, okay, I should have mentioned this back here. This, what you're looking at right here is one quarter of an array. So every array had this set four times. And what that meant is that then we could screen four patients per array using special chambers that isolated each of the four chambers. So this, what we did here is we compared one of those subarrays to another subarray to make sure that from subarray to subarray, they were reproducible. And that's what this is. This is within a slide, and then this is looking at two different slides. Again, to make sure everything is aligned right. And then when we did our screening experiment, once again, we wanted to make sure that we were getting the same result day to day. And so here you see two different arrays, two different days. And again, you're getting very good alignment there. So this is kind of what you end up with. Here were three spots that were detected by this particular patient. So here's M1, here's M3, here's M2, right? Then we come over to M1, and then on the next day, we break it down into individual proteins. And what we could see is that it was protein number three here that was the protein that lit up for M1. For M2, it was protein number four, this protein here that lit up. And for M3, it was this protein here that lit up. So we were able to deconvolute the results in the next day. And then when all said and done, and we analyzed all the data for all the patients, there were some trends that were interesting. Here's a protein that's clearly showing a preponderance of signal in TB positive over TB negative in an HIV negative population. That's exactly the kind of marker we were looking for. Here's another marker, again, showing a strong signal in the, and even a difference in overall median signal in TB positive, in this case, for HIV positive. You can sort of see a difference even in the HIV negative population, which is what we were really trying to get to. And then by combining those markers together, we were able to get this AUC curve, which is a measure of biomarker quality. And you can see the curve is not at its strongest in the HIV, in this South African HIV positive population. But we were really aiming for this HIV negative population. And that looks pretty good. So I think that is what I've got here today. Maybe I can stop and take questions. So if I were to develop a clinical test right now, we would probably take the top antigens, the overall performing antigens, make more robust assays. Because these assays are academic lab grade assays. They're not commercial grade. Make them more robust. And then I would probably test that product in each population separately in order to derive what the cutoff value should be. That's the approach I would take if I were going to commercialize it. All 40 were tested with ELISA. Yeah, and these results are ELISA results. But they're pretty high confident with these. Yeah, we're pretty high confident with these. But they're not, I mean, if you look at the sensitivity, the sensitivity here is not outrageous. Because there are negatives, TB negatives that have signal. Now, as you know, this is challenging. Because what we call a TB negative patient might not be a TB negative patient. A lot of these patients have had TB before. And they don't have it right now. Or they have other mycobacteria or all kinds of other things could be going on. Not as true for the US population as it is for this population. But nonetheless, it can still be true. And so you do the best you can. But yeah, so I would probably include multiple antigens just because of this issue that there is a little bit of overlap. In today's lecture, you have seen that a new technology platform like NAPA can make research into new areas so easy and so useful. Imagine that expression and purification of all the MTB clones could have been so difficult. But now, because Dr. Laberit's group has access to all these clones for mycobacterium strains, it is very easy for them to prepare the chip which contains all the genes of mycobacterium strain. And therefore, now one could use this technology platform to screen the patient serum samples who are affected from mycobacterium. Of course, you have seen that there is a need to look into various type of controls, people who are never having infection, people who are having latent infection, active TB. And then also, if you could add the sample population affected from the various type of resistant strains, those could provide very novel information and probably new insights for the clinicians to treat these deadly diseases. We'll continue our discussion about use of NAPPA technology and other protein array platforms, especially for the drug screening in the next lecture. Thank you.


Alright, it's a busy week for the markets, but it was also a busy weekend for Jim's garden. Oh, okay, let's talk about that. Alright, so a lot of people I posted some pictures. You know, I was away for three weeks. And when you come back from a garden, a garden's a mess. And I had to restore order. Order was complete.   But my piece is from the garden. I go and I weed. I get up early and I weed. All I ask not to be disturbed. Wouldn't mind a corona midday. But what people who do gardening know, the secret to gardening is the joy of weed. The joy of making it neat. And for me, that's just pure pleasure. I obviously cut. What I do is I work on getting all the blow branches. They just sop up things. You don't need all the tomato plants. I have a potato plant that I moved over. I should never have touched it because I crushed the leaf. But I actually put some chicken manure on my peppers. I'm not happy with the way the peppers are growing. I have a whole box that I didn't even show of some strawberry blight that is just driving me nuts. You have to rip out my strawberries. I did because of the three-week lag, did not get the lettuce in time, which was a major problem because I love the lettuce. I'm going to have to pull the lettuce out this weekend. I invite people to tell me what I should plant at this late date because my temptation is to plant more beans. By the way, I've got a pea thing that people aren't talking about. I put pea plants all the way up in front of my fence and they're right against the lavender where all the bees are and the bees are doing extraordinarily well. I just hope the zucchinis and the kukes don't strangle them. The zucchinis and kukes can strangle any plant. We love the lavender picture on Twitter. Well, you know, the bees are important and to get bees, you need something that drives them. For new gardeners out there, lavender is a certain thing. I know people say I should get a hive. The kids won't let me get a hive. But lavender puts you on the flight path.  We got stock tips, gardening tips from Jim. I'm going to make pickles, Bruce Kamich, thank you. I'm going to make pickles this year. I'm still going to make my sauce. I made about 60 jars last year of leases, none better. It was Jim's none better, but I don't want any conflict in the kitchen. We're making the sauce again. I use everything from the garden, whether it be the onion, the garlic, I didn't mention I'm growing some garlic, whether it be the tomatoes, whether it be the peppers, everything from the garden, all organic and natural. I do vegetarian. I do some with cream. I do some with wine. And it's great joy when it happens. And I love the can. Yes, it's called tomato sauce in my house because I, you know, tomato gravy is when I make, when I sell in Philadelphia, I all proceeds go to various charts. That's wonderful. Jim, will we like... I might do a stand this year. I might do a stand.  Lisa did not want me to do a stand this year because what we do is we've been making the tomato sauce, tomato sauce and not doing the second boil, so you have to freeze it. And she said too many people would think that you can just keep it out. You have to freeze my sauce. I'd like this is a big disputed home, believe me. I want that second boil. I don't have it. Well, we look forward to more of your gardening pictures on Twitter. We fight about this stuff because it heats up the kitchen really badly. Thank you. Jim Cramer, thank you so much as always. For more of the stocks you mentioned, head back to TheStreet.com.


Obviously, that was a scary moment in pre-game and seeing him get on the board and cart it off. Certainly, the defensive staff gave Frank Bush a lot of credit. Those guys rallied. But thankfully, Dean's all right. He's back with us. Here, Coach, what was your vantage point on the fumble where you all were driving and getting up front down there? Yeah, so it was fourth and five. Like I said, I decided to go for it three timeouts, knowing that if we didn't get it, we could hold up to a field goal and still have a chance to go in with a touchdown and felt good about it. I thought the game went on. Dez settled down and delivered the ball. Drake, we got to do a better job with the ball security. They made a play and popped it out. And we got to do a better job there. Chance going into possibly win the game or at least tie it. Yeah, I thought he was amped up early. We were aggressive early. There was a couple other things I thought he was pressing. But he did a nice job settling down, D-led. There was going to be something he was going to have to fight through. He's not scared of the moment. He's poised. Like, again, it wasn't perfect. There are a lot of things that we got to look at the tape. I like what he's made up of. I think it says a lot about him as we finished the game. We were able to run the ball really well, which took some stress off the passing game. I don't know how the yards per clip, but you certainly feel it out there. And then, like I said, they made a play. We got to do a better job with the ball security at the end. But there's a lot to coach off of. Do you think the defense was OK? I don't know if it was unsettled, D-led. I mean, we were pretty sure he was OK, so it wasn't. But as you settle in, we were doing a decent job early to start up in the run. They made a few plays against some of the zones. And I thought we'd, Frank, those staff did a nice job adjusting. They hit the shot play. And they were driving. They could have had us in the defense bowed up, ripped the ball out, and we were able to stay in it. No. Human nature would be the most serious. Well, we called them up and we got in the warm-ups. We said our prayers for them and tried to update them and just be honest about the situation. You know, you were sitting there and everybody knew it. I mean, you were running the stadium solid. It was a scary moment for them. Thankfully, they're Saints doctors. Our doctors did a hell of a job and kind of evaluated and will continue to monitor them. But yeah, Michael, I don't think I've ever been a part of something like that. What kind of change could be what you were trying to call a bit early to settle down?  Got the screen to prove it, I believe was his first completion. Then he kind of got going and settled in there. And there's a lot of things we can correct over the tape. Yeah, like I said, I told Desmond yesterday, whether he threw for 400 yards or threw for 100, what I'm going to define his career is a lot to learn from. The things that I saw him operate procedure-wise, he had a lot of command. Now that we get to the next step is we got to continue to find solutions and make more plays. We got to get on the other side of this, Mike. You know, it was another close game. Had our chances, didn't get a ton, though. I thought he had plenty of instinct. I mean, a fourth and five, and it's just one play, but that's got to have it. I mean, you talk about got to have it plays. Critical third downs, fourth downs late.  I didn't see him flinch, you know, delivering the football in there. I got to watch the tape to see. There's a couple other things. You know, like I said, we were aggressive. He'll learn from certain things. He takes some risks, throw it away. He'll learn from some of that stuff. But I thought as the game went on, Michael, he settled down. It was obviously desperation time, though. He'd lose track of the time. No, no. So what happened was, you know, those are what you want to call them, hook and ladder plays, or we call them throwback plays. Yes, for him, whether it's in the return game or whatnot. I mean, you're kind of out of the Hail Mary range. Five seconds, no timeouts. So when they, and Davis was the one that stepped up. So he actually made a very instinctive play. You don't want to end the ball with it right there. But what he's trying to do is we're setting up a tunnel back. He's trying to get him to go back. So he was over there directing him. In hindsight, he was running out of bounds, but he kind of had an improv there as Davis kind of spooked him on something. So that's kind of what happened there. Yeah, I learned he's not scared of a moment. And we'll continue to improve. It was his first start against good secondary. You know, they were, Jeff, they were packing the paint. You know, we threw some things early. We got the double move. We got the penalty, but that's who that defense is. I mean, they're, they play that press quarters and they get in there and they're grabby and it's a veteran, they're savvy, you know, May and Matthew, it's not their first rodeo. But I thought as he progressed, I thought he settled down. And then there were some things that we were getting an advantage running it that it just made too much sense when you're, we were getting six yards of clip, it felt like. And so we tried to pound it in the red zone. And we were able to do that.  We get going with some momentum. I delivered a strike to Pruitt. Got it, got us the first down. I thought, again, we get into that fourth and five. I got to look at some of the tape. Certainly wasn't perfect, but I like this boys at least. At your point, you decided to go down the left. Sure. Yeah, just kind of open it up, see what it was about. We had CP out there and they did a nice job covering it. Tried to get them to show that we would be aggressive because no one, they're probably going to try to pack the paint. That was the thought there. Now, what about in the first two positions, that was 10 out of the five? Well, no, the other one, that's what I thought he was a little amped up. He took a shot down the field. I thought we had something underneath. Ironically, it was a play, we hit Pruitt later in the game where we thought Pruitt was open underneath. Again, he don't want to take their initiative away, but that's why I say he came out amped up. And then as he settled in, he was able to correct that, which plays that stuff in the right direction. What kind of learning curve or progress curve would you foresee over the last three games here from the other team? What do you expect from that? Yeah, you expect progress. We're going to have another big challenge on the road. Short week, Baltimore played yesterday, we got to turn around. We'll be out there Tuesday. It's one day shorter, an opponent that we're not really familiar with. Another conference, supposed to be cold and all the other things, it'll be good for him. Test, short week, could defense, go on the road and see if we can make the improvements. He did. What were you trying to do? Well, some of those was a personal package. Figured they'd come out and try to run the ball. And they did. And so just trying to get it just bigger in there and this different package that we ran. Mike had other packages. Whether we, as the game went on, they had the lead, we got to it a bunch, but that's it. Mike just trying to get the, play the strengths of all three of those guys. How did you see guys on my team? I thought it was good. Jeff, you know, as we get back in there, I didn't notice anything. Like I said, his demeanor, great, his interaction with the players. And it's a huge learning experience.  Certainly didn't go well early. And to his credit, he settled down, the team settled down. I seen the rally around him. He's an instinct to play smart, hard to tackle. He's the first guy to bring him down.  And I thought, you know, you feel him. He's another volume guy that, that as the game goes on, I mean, he, those guys don't, don't like tackling him. He made a good play on the two point, able to bounce it, show that burst to get to the front pylon. I really excited about him. He's been big for us. I'd have to check. Yeah, I don't have an update on that. Oh, I'm sorry. You said Huntley. Yeah. That's something lower body. Let's see the MRI didn't look good though. I'm sorry. I thought you were talking about somebody else. Sorry, but Caleb Huntley. No, no, Caleb Huntley is what you said. Yeah. Yeah, you know, going into this week, it was about, you know, communication and being able to get in and out of the huddle and working efficiently with that and handling the environment, which I thought we did a really good job of as an offense, getting on the ball, you know, not letting that playcock run down on us, getting the upfront communicated. So from that standpoint, I feel like we did a pretty good job of. And then obviously the execution part upfront, you know, they did a heck of a job blocking in both the run game and the pass game for me. And then, you know, you talk about the pass game from my standpoint. Obviously, you know, not where we want it to be.  In reality, all you got to do is just give them a chance. So that'd be something I'd look back on and be able to work on.  So obviously, there's a lot of emotions going on through that. But just being able to settle in and get going, obviously, that first drive coming out, you know, not being able to move the ball and then, you know, settling in, I think that really helped.  What blitzed a lot earlier? We're kind of like forcing you to make reads. What was your case then? Yeah, you know, they did a good job in the back end. I felt like, like I said, our O-line did a phenomenal job of the run block and the pass block. And then, you know, I think at all it comes down to is winning one-on-one matchups on the outside.  Like I said, just got to give the guys to make a chance. No, no, no. Other than the music loud right there under our hotel on that street, you know, I was able to sleep pretty well.   And obviously that's not how it went. When we came back off the sideline, it was like, okay, deep breath. Let's go back out there to work.  Obviously, the emotions are down because obviously we didn't get a win, didn't play as well as, you know, I wanted to from a self-scouting standpoint. But you know, I think what this team has shown over this year is that we're fighters. And that's something to look forward to for, you know, the next three games and then the next season is that this team, no matter, you know, whether we're up or down by a whole bunch, this team's gonna fight till the end, which is something that you can't take for granted. Yeah, I saw him laying on the ground for a split second too long than what I thought. So I sprinted over there to run and pick him up because, you know, at the end of the day, he's gonna look at that and, you know, think the game that if he would have caught that and we would have kept going, we would have scored or whatever it may have been, that, you know, he thinks that that one play right there is what made and broke the game, which is absolutely not true at all. You know, Drake made a lot of great plays in that game. Obviously, that one didn't go the way we wanted to.   And so, you know, I'm just excited to get back to work and come in tomorrow. Yeah, it's huge. And those runs are huge for us. And I know, I think there was a stout or something that I heard when we came in, a rookie minicamp that, you know, he was, you know, leading the NCAA in broken tackles and this and that. And, you know, he's not a speed guy.  You know, it's gonna take more than one guy to tackle him and bring him down.  Yeah, there wasn't much confusing out there. You know, what we saw is kind of what we got. I think we prepared pretty well as an offense. And, you know, I think at the end of the day, like I said, it's just about execution. You know, for the most part, we executed in the run game.   It comes down to, you know, just being able to move the ball and keep a flow to the game. And that flow is obviously put in the part of the pass game in there, which obviously I didn't do well today. Yeah, that's just my leadership. That's how I've been since, you know, I've been in high school. But, you know, I think that, you know, a lot of times, and obviously quarterback has a high role leadership.  And that kind of has a flow of how the offense goes or how the team goes. So, you know, no matter what the situation, whether three and out or their turnover, but a touchdown, just keep those guys up, keep them up in a good standpoint. So we know that next drive we go out there and go to work. You feel like the guys respond? Yeah, you know, I feel like they respond to me pretty well. Obviously, you know, they saw my preparation throughout the week and saw that, you know, we were being efficient in and out of the huddle, you know, not setting on calls, not doing any of, you know, what a typical rookie you would say would do.  How would you grade yourself? C minus the D. Like I said, I thought that, you know, it was really good in the execution of coming out the huddle, getting the play call, getting to the line of scrimmage with plenty of time on the playcock. And then, you know, post snap, felt like it could obviously been better. Yeah, just same thing every week. Just going out there warming up, getting my mind in the right place, knowing, you know, every time I walk out there for the first time, checking where the play clocks are at, no matter if I'm starting or not. So, you know, like I said earlier in the week, my routine hasn't changed that much, and I'm just going out here and doing the same thing. No, so I did that last week. That was the first time they were able to come down there last week, and then they were able to come down this week, obviously, for the first start. So, yeah.   So, when I walked back in the locker room, he said, hey, look down at me, he said, hey. You know what you did? He said, well, I said, what? He said, you're starting. You know, you went out there and played an NFL game, and that's not something to take for granted, that's not something to look back on.  What do you expect of yourself in the next three games? Obviously, to be better, to go out there and win, to go out there and lead the offense and execution, whether that's third down, whether that's red zone, you know, whether that's base down, just go out there, not be perfect, give guys and make a chance, like I said earlier, and just be able to keep the tempo, keep pressing the offense, to keep pushing them, to keep moving forward. Yeah, obviously, better in the passing game.  So, like I said, like I just said, keeping the ball moving forward, that's our ultimate goal, that's what Art talks about every single week, you know, taking no negative plays, and so that's something obviously we'll look into and keep trying to better. Yeah, no, you know, it wasn't necessarily to prove something. Like I said earlier, you know, when we looked in the week, it was about winning one-on-one matchups on the outside, we got those one-on-one matchups, but like I said, at the start of this, I didn't get the guys a chance. Try to make a perfect ball, try to do this, try to do that, win in reality, all I gotta do is give them a chance. Thank you. Thank you.


presentation is like the opposite of those where I try to put everything on the slide so I'm sorry. I'm just gonna jump straight in and I hope that I can summarize my ideas briefly enough yet still with enough substance to give us a constructive starting point for meaningful discussion and in distilling my argument down I've had to make a lot of generalizing statements the most blatant of which I've marked with asterisks I do not claim that there are no exceptions I'm certainly not the first to put forward aspects of this argument what I'm hoping to do however is to draw everything together into a workable model of practice that is not grounded in the discipline's normative crisis mode of operation so normative salved salved mode this is my first attempt then at articulating an enchantment model for archaeology I would like to put forward to you a series of cases that underlie my argument for the enchantment of the archaeological record case one archaeology has inherent in it sources of enchantment archaeologists but in fact everyone everywhere are literally atop untold histories all of the time histories that we've never seen before that we may know nothing of and that can thus surprise us and transform us in interminable ways the very nature of archaeology as a subject that's open to interpretation as new techniques and voices and intellectual frameworks etc. are introduced furthers this facility for surprise and transformation case two yet the typical methods we use as professionals tend to revolve around a crisis model driven by the sector's normative preservation paradigm and conservation ethos this model erroneously got guises archaeology as a non-renewable resource presuming there's some version of the past that can be saved in perpetuity case three such endangerment narratives are not only debilitating for archaeologists themselves but they have little to no appeal to wider audiences the crisis model blinds us and broader publics to new futures and different interpretations it fetishizes authenticity leading to cynicism false consciousness and nihilism in the face of its inevitable futility it betrays profound weaknesses and professional interpretative aptitude and begs for a new moral for the discipline my proposal is that archaeologists have the capacity to enchant the world at large and we can do this via the wondrous affordances of the heritage record itself teased out through number one fundamental changes to our primary recording systems and our higher order interpretations that are grounded in effective methodological practices number two facilitated agonistic dialogue building on the thinking of shantel move and Nicole Diffel and three cultivating both craft skills and creativity to make these previous two points realizable amongst both archaeologists and their audiences I'd like to suggest that in the context of archaeology and heritage enchantment is generated by what I will loosely call emotive engagement I defer to the definition of emotion and affect and feeling used by weatherl at al 2017 and I call attention to their assertion that quote emotion is action oriented it pushes people to do things as I interpret this emotion is enacted in the body it propels the body forward to act in some fashion enchantment then can spur change in the world what is crucial for my argument is that everyone has the aptitude to be inspired to feel to be emotively engaged moreover this it can be deliberately fostered even with people who might otherwise be resistant dozens of tested conceptual and methodological frameworks for this emotive fostering exist yes some of them are flawed many mistakenly assume emotion should only lead to positive affective outcomes and indeed most emotion assessment tools are actually embedded with this positivity bias nevertheless there is a huge body of evidence that indicates that emotive engagement leads to very powerful effects including remembrance attachment restoration learning care and resistance to hegemony leading to socio-economic political change further more the scholarship shows that people generally want to explore complex and controversial topics in cultural heritage context and major research endeavors across multiple continents demonstrate that people do not do not expect cultural institutions to be neutral but rather quote to have a social responsibility to take a leading role in inspiring people social and political activism in order to help bring about change and hope from lynch Bernadette Lynch however the evidence also suggests that professionals often wrongly judge their audiences underestimating their capacity to respond to debate to be challenged this is arguably the greatest problem we face as professionals because to fulfill our social responsibility for inspiring activism we need to invest in fundamental structural change yet only a minority of professional bodies are really prepared to accept the risks of such change and where participatory or inclusive models of practice have become standard the majority do actually follow a quote-unquote charity model which hasn't really changed institutional structures nor afforded true social justice or radical trust because these institutions don't actually have the means to deal with agonism or the frictions that come with true democratic debate so there are many different ways that I and others have been trying to tackle these issues with primary field work teams in commercial archaeology context and in research context and I wanted to just end by attending to one and the EU funded emotive project where we've been enrolling digital media in explorations of how affective engagement with heritage might produce care social conscience and civic welfare I'm not naive to the many problems of the digital and I really don't suggest that we must necessarily deploy digital technologies to achieve a motive impact but the digital does offer many means to subvert expectations and so can create the perfect condition to foster enchantment or to further enchantment and emotive we've been experimenting with this kind of subversion in various settings with different enchantment objectives in mind what our experiments have exposed is precisely those structural barriers that make pursuing activist goals in heritage almost impossible so in one case and you can just taking a tiny excerpt above where we've been developing a model for facilitated agonistic dialogue the seeming irreconcilability of professional views with those of wider publics is captured in the polar opposite responses that we received to a dialogue around mental health most of our users demonstrated the effect in a tech affective engagement that we had had hoped for so one says and you don't expect on a tour to get this level of debt with strangers or even with your family that's fantastic that experience would stick with you for a pretty long time whereas the institutional response portrayed the risk aversion that sabotages most activist participatory projects in this case this person said it could be quite damaging to people and then they go on to list all the many possible ways from mental health perspective also it'll destroy their political perspectives also it might get physical when they start getting passionate about it so there obviously there's so much to say here but I'm going to end by stating that I think enchantment needs to be fostered not just amongst why public audiences but most urgently amongst archaeologists and heritage professionals themselves I believe our professional practices regularly breed disempowerment stagnation and actually complete misconception of the heritage record and this doesn't need to happen flipping our model around through an enchantment let approach can I think expose archaeology for what it truly is an infinitely inspiring resource with a capacity to create a better world now and in the future thank you for listening


How's it going everyone Caleb here with the late and sports cards? We are gonna be working on the prism soccer hobby choice break This is number six random team style. Thanks again everyone for joining and good luck here on the 20 box case break Here are the results from that random one more time Appreciate you guys joining Can always feel free to drop an email and chat if you guys need to exchange contact information or something like that But also delete the message after a couple of minutes as well So you don't get any spam What's up Jeremiah? How's it going man? What's happening? You were going down All right guys here. They are good luck and thanks again. Let's see what we got here. All right All right, Silver's Wisa Pedro Tariq Lampti That's gonna be choice not numbered on the tri-color Jeffrey Schloep And autograph this time Man City is gonna be Paulo Zabelta Scopato not numbered flashback signature. Very cool. Here you go Man City And we have a no Mike. I want to hear about it though. Ashley Barnes. That's 12 of 29 This one is brutal for you. I'm sorry Five left on definitive and then after definitive. It's that we got top sterling after that guys some great great stuff coming up Just a couple of baseball spots left this evening a couple of them. I Want to hear about it Mike Hugo l'arise Marcus Rashford Jihazi and Saar tri-colors Autograph this time Aaron Cresswell signature scope not numbered And James Madison red to 29 Nice one there on the red Leichester City What the heck Mike? What I've heard about some wacky stuff happening on that auction house lately. I Don't know man Roberto Firmino choreo Joe Wilcock Yeah, we have a five years Corteur nice one people Corteur Chelsea good goalie there on the auto and it is a tiger stripe on the Aaron one Bussaka tiger stripe nice one for man you there in Juan Bussaka It is Matt's fault. It's certainly Matt's fault schlager It's certainly Matt's fault that it sold for too cheap It's certainly Sean's fault as well.  Congrats Very nice it for man or excuse me a totingham. There you go Knew the team just said the wrong thing. That's sick. We'll mag that Congrats totingham nice tiger Great player to get there Great player you go spurs Can't do a whole lot better than a son on your tiger stripe congrats Afinia That's Peru try color And they've been techie try color Carl Darlo signature scope auto Newcastle United and Antonio on the red to 29 You don't say Tottenham. I've heard Tottenham and Tottenham Yeah, I've heard I've kind of heard it all That's still sleep top. Oh, yeah Ben Chilwell, sorry Been techie for tons triory Cisoco and Harry Wings autograph now this one for Tottenham There you go It's like tater top. Yeah. Yeah. Okay, cuz Stani eight of nine nice one. I'm agreeing like Chester. There you go Yeah, that's true. It is more like a tater top 620 Mike it makes no sense. It makes no sense Mike Danny Welvick Yori Tealman's Autograph this time Emilio Martinez signatures auto for Ashton Dia and Then we have another tiger Trin cow this time nice one there congrats on that one Wolfsburg Trin cow very nice Couple of great tigers so far. We'll take those Madness Mike madness absolute madness empty It was graded. Yeah, I don't remember what the grade was virtual Van Dyke for Charleston Jadon Sancho Todd Cantwell Tri-color and Konate tri-color. We have autograph of John and McGinn Ashton Dia and Another tiger stripe Emiliano Martinez another one there. Jeez tiger stripes everywhere Congrats, holy cow. Those are flying out of this case These are nuts I'll tell you a funny story later tonight on this product reminder. Please do not let me forget Yeah, this is every single car Jordan Henderson who chitch Adam Webster and a Federico Fernandez Autograph this time is Deco on the flashback auto Chelsea Nice one there. This has been a firebreak guys No way another tiger stripe Mason whole gate this time Everton Tiger stripe after tiger stripe after tiger stripe. You think they weren't good We'll certainly take them though empty Guys, there will not be a recap on this break. This is a very quick break to go and wash through Help us make up a couple of minutes that way as well Sergio Conos tri-color Rodrigo Moreno not bad and autograph this time is Isaac Hayden Newcastle United on the auto And a red John McGinn 29 there Fair schlegger fair Tiger stripe you guys see it peeking out Ajeer Ben May we have a Chelsea autograph first on the flashback Gianfranco Zola Awesome card.  Nice one And tiger stripe once again from the case Kenny McLean tiger stripe very nice Norwich City No, it's they I mean it's exclusive to choice, but they are they are definitely they pop a more than a case that's correct Nice Steve. Yeah, this has been fireman They still sell well with that being said they definitely do Cool. Yeah, these boxes are pretty loaded up man choices good stuff. Let me go back here guys. Hold on Focus on it. There it goes Donker there cliche Sergio Regine Share and autograph this time is gonna be another Gianfranco Zola Feels like it's like we just got that one we did in a tyranny eight of nine nice one arsenal Quarantirany there you go eight of nine very solid coming Just like it feels like we just got that in fact we did just get that Happens man. It happens And I'm like Lovanna. There he is Thomas party arsenal Isaac Hayden on the autograph for Newcastle United Isaac Hayden And Thomas party that is gonna be red to 29 nice color match their arsenal We'll take that party party box Not too bad Exactly Mike exactly empty That's the thing about it It just doesn't make sense like the price of a sterling boxes Fairly similar and that's one of the best things you can get Kai hovers Pablo Mari Georgino. We got a points card here Points to 50 random to all the ends Of course you got points They had to do it to us man, and a new mofuepo. It's gonna be read to 29 Random that at the end guys. I think they can toss points in choice. That's brutal. You get one hit Nick Pope Lewis dunk The cell zone in a shell can say low man city on the tricolor Ali Watkins this time autograph for Aston Villa Ali Watkins and Tiger stripe of Jean-Jo Selvi Nice tiger another one on the Tigers tonight. Yeah, Mike. It's just you just have to not think about it I would say forget it and move on.   It's not bad You go and green Jordan Henderson nice one Liverpool six of nine There you go. Hey T hop. What's going on? What's happening man? Are you doing tonight? read of Mars Paul polka Hakeem Zaik Erin Cresswell signatures auto West Ham United and a Karmie Chuka Chuka Mouika 29 yeah, that's not even close It's a difficult Just keep looking at it, but it still isn't coming to me at all What's up Jonathan It's like Chuka Mouika or something It's got a lot of a lot of syllables Nathan Redmond a boomyang maybe some Sanchez tricolor Luke Thomas tricolor And autograph this time is gonna be in McHale Antonio West Ham United You go West Ham And we've got another Tiger this time Tiger is kensima Watford on the Tiger nice rookie Tiger There you go Watford awesome two boxes to go How you doing Jonathan? Holy cow Brad's a bit loud Ben Gibson John McGinn tricolor Edison Kovani tricolor redemption Wow No way wingman Sun redemption signature prism choice boom spurs big card Huge spurs break here congrats Very very nice once again spurs Denny M. There you go Denny That's nice Patrick Bamford That's to 29 Spurs break Doing good Jonathan doing good Thanks for asking man appreciate Spurs break Very nice guys You know James brilliance a Zeus Sebo Diallo Konga and we have five of five Lee Dixon club legends for arsenal nice gold scope auto There you go very very nice And a Wilfred Zaha red to 29 Nice ending man really good stuff at the end guys. Thank you again very much We'll have them all well protected and right out to you Let me do the random for the 250 points. I'm not gonna do a recap That's a really quick one to go on through Okay, just one sec Set up this random Remember, it's 250 points. I Have the 20 names back in here the name at the top will grab the points. Let's roll the dice 10 times name at top grabs the points One two three four five six seven eight nine and ten Suwan P. Congrats on the points.  Hope you did enjoy it