We're listening to the Naked Bible Podcast. To support this podcast, visit nakedbiblepodcast.com and click on the support link in the upper right-hand corner. If you're new to the podcast and Dr. Heizer's approach to the Bible, click on newstarthere at nakedbiblepodcast.com. Welcome to the Naked Bible Podcast, Episode 186, Farron and Audrey, Discovering Mercy. I'm the layman, Tray Strickland. He's a scholar, Dr. Michael Heizer. Hey, Mike, it's nice to be in the same room with you and Farron and Audrey in Rhode Island of all the places. Woohoo. Yeah, you were saying the other day that you never thought you'd see Rhode Island. So what do you think? Well, yeah, I never thought I would ever be in Rhode Island and here I am. And you get to see fall? Yes, I actually get to see color on the trees and it was pretty cold where we ate. We had a good time then. Can you tell everybody what we did last night for dinner? Oh, yeah. We took a little road trip to get here and the four of us had dinner at George's of Galilee. And for those of you who have read my fiction, The Poor Tent in particular, which is the sequel to The Facade, that is the restaurant where Brian and the Colonel sort of have their meeting of the minds that the second book pivots on. So we thought, hey, it's on the way. Why not go to George's for dinner? The Colonel was not there. No, fortunately. But the lobster and clam chowder was and it was pretty good. Yeah, it was good stuff. Although you made me crack some lobster claw for you because you didn't want to put it in the work. Right. That's right. Well, you know, what can I say? That's why I rarely order that kind of thing. I hear you.  You and me both. Well, we're glad that not only that we could make the trip, you know, and have some have some FaceTime and spend some time together with Fern and Audrey, but we wanted to take the opportunity to talk to them again and both give something of an update because they have lots of new things to say, things that are going on, things that have developed for them. And also to really sort of get into some specifics that part of the reason we're together at this point was again, I spend, I think, how long has it been? It was three days, but we've been doing that those three days for like, yeah, since 2010, seven years. And we want to talk about a few of the things, you know, just sort of extract a few things that were discussed in our group this time and share them with listeners. So we thought this would be a good time to do that. So let's just jump in. We guys have a new website and more importantly, a new nonprofit entity. So you've sort of gone to the next level, making a transition. So let's open with that. Tell us about that. Yes, we have Discovery Mercy is our nonprofit that we work out of. We currently are waiting for a tax exempt status that is six months that we're waiting for that. So it should be any day now. And we decided to through the encouragement of, you know, folks like you, Mike and others that said, you know, you really need to move into a nonprofit. And so we have and in that transition, we did do a website not to expose anything of what really we do. But when folks come to see us, they always ask us, well, you know, where are we coming to? What does it look like? And so we came up with a website that kind of shows where they'll be staying about the heart of the ministry, what we're doing with the ministry, what it will look like for them. So they're not totally caught off guard. So the website really is for that point. It does. I mean, we took a look at it and, you know, kudos to Joe and his wife for putting that together. Joe and Siobhan did the work that went into the website. And, you know, it's it's a beautiful site and it does give some information just a little bit. But yeah, you're right. It's it's sort of a basic, you know, visual orientation gives a little bit of the details about the accommodations and whatnot. Yeah, Audrey spent hours working on a format and a visual presentation so that the survivor, the person coming to see us would be very comfortable in what they saw. And she has spent a lot of time helping even with creating the atmosphere at the office and the home where they'll stay for safety and so they can begin their journey. Audrey, do you want to add to that? Any, any specific thinking that went into that? It was it was surrounded by the touching the heart of a survivor so that they knew what discovering Mercy's heart is. Because Fern and I really want them to know that they're coming to a place that they'll be loved and not judged and that what they've always wanted their heart would be known, seen and loved. One of the three reasons why we even chose the name Discovery Mercy was because we took the beginning letters in Mercy and we capitalized it, discovering me in Mercy. Because one of the things in trauma based mind control, the first thing that we helped them do is to normalize the dissociation so that they're not afraid of themselves. And so the discovering me in the midst of God's mercy was really important to us. And that's why we chose that name and the whole heart behind the nonprofit, the name, the place they stay, the website really was for the heart of the survivor. Right. Even in the logo, the logo has a heart that's half hidden and that's for survivors. They know their hearts there and they just can't find it all. And that's what we want them to do is find their heart. So the site is discoveringmercy.org. For those who might not have heard the previous episodes or maybe they did and without the necessity of going back through previous discussions we've had. Why don't we just, for those people and again for the sake of review, just sort of overview what it is that you do, you know, defining your trauma based mind control, dissociation, some of the terminology that gets tossed around. I don't know if we want to go into distinguishing any of that from sort of pop cultural portrayals of it because recently there have been a couple and people listening to this might hear the conversation today and think, oh, that's like the movie split or oh, that's like stranger things. You know, so if you want to make some distinctions there, we can.  And what do we mean by survivor? I guess to start with would be what is dissociation and that is a disconnection between a person's thoughts, memories, feelings, actions or sense of who he or she is. This is a normal process for everyone. Everyone experiences dissociation like mild common dissociation includes daydreaming or highway hypnosis or getting lost in a book or a movie. That's what everyone has and in a survivor's case it's much more organized and complex than just daydreaming on the highway. So trauma-based mind control is we are identifying a person or started as a child that they have lived through a lot of trauma and the trauma was purposefully perpetrated so that that child could be controlled through behavior. They can be conditioned to respond and obey to the perpetrator and that's what we're calling trauma-based mind control. I think that. So it's like an intentional wiring of thought or behavior, all those sorts of things. There is thought and intention behind that. And as we've discussed before, that intentional perpetration really came out of the 50s and we do have some clients that would probably have been earlier than that so I don't want to just stop at 50s. But it is the researchers of the educators, psychologists, psychiatrists. It's not just backyard Satanism. It's much more organized. It is for research purposes. Yeah, and again, for some listeners that will strike a chord with them. Research that was, again, maybe to say begun in the 50s might again be a bit anachronistic, but it usually gets associated with something like MKUltra, which wasn't just one program. It was dozens, maybe even hundreds of programs. And that was exposed in the 70s through the church hearings in Congress. And most of the records of that were destroyed, but we know about it because there were a number of boxes that didn't make it to destruction. And that research, though, was passed on. You had, again, academics, you had practitioners in the fields of psychiatrists and psychology, and they had graduate students. And the graduate students were interested in this area of research. And by this time, those graduate students have now had graduate students.  For instance, Annie Jacobson in her book on phenomena gets into it a little bit, you know, she mentions this, this part of the whole history of the US government's involvement in what, again, would broadly be termed psychic phenomenon. But since this is dealing with the mind, you know, she drifted over there a little bit. But, you know, there's some sort of, you know, official and covert, you know, use of this like there has been since the 50s, but you have a lot of people in these fields who are just interested in, you know, this area and everything associated with it. It never really went away. It just sort of actually multiplied and kind of spun out of control. I think you could even say, because it's not sort of governed by one agency or, you know, one funding source, you know, that sort of thing. So you guys are dealing with people that have been victimized and associated with that sort of stuff. And so I like your phrase, this isn't backyard Satanism. This is done by a lot of really smart people who have resources and a history of research, which means a history of techniques, you know, behind this. Now, we should say on the heels of that, one of the things I think that makes, at least in my exposure to this area, as far as a sort of Christian ministry that touches this, and there's not a lot of that to begin with. But one of the things that makes you different is that you actually spend a considerable amount of time in the academic research, you know, the journal literature. You're in contact with professors and experts that, again, for lack of a better way of saying it, this is their bread and butter academically. So you pay a lot of attention to real peer reviewed research and that and you use that you don't shun it.  So can you describe that a little bit you know your your relationship to the research. Yeah, I think it's really important.  While Audrey and I sport no mental health credentials at all. It was really important for me as I got into working with survivors that I noticed that the routine prayer ministry approach just didn't help. It wasn't you're working with someone who the conditioning to this child, the child first all did not have the opportunity to grow and develop in safety and connection and family and attunement in relationship at all. So there's this huge disconnect just from another human being. And so to bring them into a prayer ministry setting and and go into a relational thing. Let's get to the memories and let Jesus come in and speak. There was no foundation for relationship at all like that. However, a huge percentage 98% of the people that come to see us are born again. They love Jesus very much. And but all of the trauma the folks in the trauma inside of the person didn't know how to do relationship. So that sent us into Daniel Segal's work on interpersonal neurobiology. He's the father of that. That's saying basically every child in development synchronizes with the nervous system of the mom and the dad.  The brain is developing in a certain way. And so we noticed that that attachment, the attunement needed to develop a healthy child was purely absent. So Daniel Segal's work of interpersonal neurobiology. Bonnie Badanock takes a lot of the interpersonal neurobiology, some more attachment theory. And she writes a book to therapists and it's called the brain savvy therapist or the brain wise therapist. She's got a workbook with the two. And she is explaining to the therapists who have been trained typically to say, just integrate these, you know, just the child's regress, get them integrated. Bonnie's saying something totally different with the research of Dan Segal, Pat Ogden, Janina Fisher. And she's giving it to the therapist and saying, you know, these little kids that grew up in trauma, developed in trauma really think differently in their world. They can't just come alongside and do a worldview from a normal development. So we, we noticed that this was happening to our folks in front of us. And we went in and the attachment theory research and the body somatic theory research. And it was interesting how I would say that the Lord showed us that this is what we needed to do. And we just started doing this at about seven, eight years ago. And when we found these current research books, these books that came out of research, Bonnie Badanock's, Janina Fisher has a great book, Healing the Fragmented Cells of Trauma Survivors. It's not totally exactly what we do, but there is a lot of strong stepping stones that will help people have a paradigm shift to walk into helping a survivor rather than the old historic model of going through the memory, stirring up the darkness and having Jesus come in and speak truth. So we have seen huge results with the survivor who has been taught not to think when they understand what the neurobiological responses in their brain because these folks have been taught to understand what their brain's doing. They've been programmed into certain wave states, you know, like delta, theta, alpha, beta.  So they get, when we're talking about the brain development and that there's someone stuck in limbic, which is the back of their brain, they understand it. I know that prayer ministers or people hearing us might think, oh my gosh, I've got to be a PhD to help a survivor, but you really don't.  Yeah, and they may not have had words that I was in delta wave, brain wave, but they understand that what they're in right now presenting to us feels like this. And oh my word, that's the words for it. And so even like everyone knows that a soldier that goes to war has post-traumatic stress and when he comes back, sounds will, you know, set that off. And everyone knows that that's a psychological thing and an emotional thing that needs to be worked through. That really is how we normalize it for a severe trauma survivor because it's the same. While there are spiritual dynamics that go with it, it's really neurobiologically based because researchers were researching on the brain to create whatever they wanted. You know, in the wider field of let's just be real wide here like religious studies. I mean, you'll run into things like, okay, the Eastern Yogi, that again, they train themselves, you know, through meditation or whatever technique to, you know, again, using broad terminology to move in and out of altered states, you know, different states of consciousness. And this has been known for decades, you know, that this can be done.  And, you know, for whatever purpose, be in one or not the other. And that's really what, again, in very lay terms, and again, probably committing some imprecision here, but that's essentially what you're talking about. So you're talking about a sort of a phenomenon that has been recognized for a long time in other areas of research. It just so happens that there's an overlap with all this awful stuff. And again, the very manipulative kind of thing, you know, that people have been exposed to just broadly.  They're cases that aren't sort of trained, you know, intentionally, but because they've suffered repetitive trauma in some other way, that you get the same sort of thing going on. Yeah, I think that's key to say that dissociation is on a spectrum. So we have someone who has suffered trauma based mind control, that's going to be the most severe spectrum, but actually anyone that has gone through even an episode of sexual trauma or any kind of trauma, what we do even helps them. It really is across the board. I think overall human beings need to understand their own makeup and how they are human. And I think that's an important piece of this as well. Now, Audrey mentioned, you know, some of the supernatural aspects. So what is the relationship between trauma based association and again, what would, what many listeners I'm sure are thinking of in terms of sort of traditional, you know, demonization, deliverance ministry, because what we're describing to this point isn't that, but there is a relationship. So can you talk about that a little bit? Yeah. And a survivor, you know, if we give an example of a survivor that has been as a child brought up to be made to split her mind into into different neurobiological thoughts and moods, which was what we would call a part. So in a sort of an alter in another terminal, another part of the person. And so in, in a setting where there's a programming session, there's a bunch of kids and someone is going to get traumatized.  And so she takes that place and the whole way they're heckling, are you want this?  So this is your life. And so she goes through the trauma and it's horrendous.  So she'll reach for a cosmic entity to help her get through that moment. Right. So like instead of crying out to God, it can be a different entity or something that she's been taught to reach for. Something that has power when she's powerless. And then that and growing up as a child, God wasn't there for her. So what's more powerful? I'll take that. Yeah. See, and from the 17 year old there has already practiced this many, many, many times.  So Trey, if you need to edit it out, you just, so the child is on an altar, being raped, going to be raped. And she does not even have the words for this. There's no words for this. And they have her ask for, go ahead and ask for Jesus to come. So someone comes out dressed as Jesus and then he rapes her. Well, that didn't work so well for her. So the next time it's same ritual, same thing. They said, call for Lucifer, who is your father. Well, so she learns even though it's so abhorrent for her heart, there is so much sensory distortion. And she can't even have a reasoning mind. It is survival now. That's a different part of her brain. She's in survival now and she just wants to live. So she'll go reach through the defilement and then yuck of the darkness power. And she'll grasp on to Lucifer, her father. So I think it's important here because when, you know, when a lot of Christians sort of read about the occult or Satanism or something, they're reading about people who willfully, you know, want to tap into supernatural powers because, you know, this is going to give me power. It's going to give me XYZ. That same sort of wanting, for lack of a better term, wanting this attachment again for whatever purpose, you know, so you're describing a child is going to have that mechanism to reach for that thing, so that, you know, they can get like material wealth or prestige or what, you know, the typical occult Satanist kind of mentality, but they're doing it because they want whatever is being done to them to stop. And so you do that repeatedly and it sort of trains them to attach themselves or some part of themselves to that thing. You know, let's pause there. When there, a survivor is reaching for an entity, she or he is choosing life. They don't want to die. They're choosing to live and choosing life is Christ. They don't know it at that moment, but that's what their heart is. They want to live.  And so why not? But it really is out of a place of love for them, their own heart to choose to live. And now look what you just brought up, Mike. That's what we're talking about. See, while that ritual sounds horrible, and it is horrible, it's at most of us recoil inside. When we hear this level of trauma all the time, and this is what a survivor's lived through. But if the perpetrators were after the attachment tear to put in the belief that the child is not redeemable, because if you can get that set into a three year old by the time they're eight and you need to start building assassins, they already believe they're evil. So to start killing other little kids or whatever along the way, they're going to be taunted that they've already attached to Lucifer, to the dark side, to this. So it wasn't about that they wanted to, they brought a Luciferian Richelin or a Satanist Richelin to practice Satanism. It was for the goal to tear attachment and distort the identity of the child so they have no understanding of imaging God. None. Yeah. For our listeners, you've already heard some references again to supernatural beings, imaging God. Again, we're going to get at some point in our discussion today. Hey, how does my content, what does that do to help? How did you all sort of run into each other and connect and it actually amounted to something? You're getting a little bit of a glimpse of that at this point. It's really kind of interesting to me that, again, going back to pop culture, other than the two things that I mentioned, you know, the movie split and then Stranger Things. And Stranger Things especially, it's well done in a lot of aspects, but they actually took a lot of, you know, they don't have a lot of the one character, Eleven, she's a survivor and she has these powers and whatnot, but they actually took some of what was done to her when you're watching the show. You don't have to have her have flashbacks so that you know, like, what in the world is going on with this kid? And they took that, you know, from MK Ultra Research. I mean, they took it right out of that.  She's still conflicted, but, you know, she knows if she doesn't obey and she doesn't do this, what's going to happen to her and all this other stuff. So they took that right out of the research. It's the same thing like with Captain America Winter Soldier. There, it's a little more visceral because you're dealing with an adult probably, you know, Hollywood gets away with more there, but it's the same kind of thing. You know, what you actually see, maybe not the way it's portrayed in terms of the outcome, the powers and all that and sort of the more fantastic elements of science fiction. But there is some of that going on in the real world. And what's done to these people is sort of just like sucked right off the page of what you would read in a, in an MK ultra study. So that, yeah, that stuff actually happens to people and it's perpetrated, you know, against people, children for, you know, these quote unquote research purposes. And what you guys do is essentially deal with the aftermath, trying to help people process that and recover. Yeah. Well, I do appreciate the analogies to the the movies and the shows that are out there. I do want to say I've never saw those movies and shows. And I just want to say that for every one of the folks that come to see us, I would say that trauma survivors, those who have survived trauma based mind control, this purposeful perpetration, for me, they are the most loyal. They are the most gifted folks in the body of Christ. They have taught me more about how to love. And I've been in the church for almost 40 years. And for me, you know, when I think of David's Cave of Adolam, all the distressed and that, but they're the mighty men for me and those I minister to, I would have them stand with me anytime in that. So I don't, I know that they've come out of the trauma, but I always, I only see the heart and I think Audrey's right there with me. I only see the heart of Jesus and how much they've loved. And they've taught me a whole lot about that. So I just want to say, sometimes we say, well, what are the folks, you know, like really, because you only have a backdrop of a movie or something.  And when you always stay connected to the heart of them, even the places and the moments of messy is really just communication. It really isn't, it's not behavior as far as aberrant behavior, it's communication. And so when you can stay connected to the heart, you understand the communication of some of the behavior. Well, let's transition a little bit then. Again, referencing, you know, just broadly the people that you've come into contact with, that you've worked with, how do they find you? I mean, how do survivors sort of find Fern and Audrey and get started? How does that happen? Well, before the podcast, it was really word of mouth and who knew us and knew of other survivors and they just gave us, gave them our name. But we have had referrals from the podcast quite frequently from Trey and you, Mike. And so it's probably at least one or two a week sometimes. Yeah, I can, I mean, I don't know how many emails Trey gets to refer, but that's not very far off. I mean, one or two a week, I mean, it's a good number, you know, when you would put our people contacting each of us, you know, separately together. Yeah, we do offer a free 50 minute Skype time with folks that connect with us if they'd like that. And usually, Mick Lack has been really gracious in covering that for us and meeting that. And we, there was over nine hours in just last month that we spent time with folks that came through the Fern and Audrey podcast from you and Trey. So is that at least right now? Is that typically how the process begins for somebody? Typically, yes. We do have, it is all by word of mouth. The website is not even presented as what we do. It's really just showing you when you're going to come to see us. This is where you'll be. This is what you'll find. So we're doing no marketing at all. We have no strategy for marketing. When, like after the procedurally, okay, someone says, yeah, I'd like to have a Skype call and you have the Skype call. And I would imagine that's the point where you have to start talking to the person about making appointments and whatnot. And can you just describe that a little bit? Yeah, that's when it comes to the tough part because there is a fee involved with coming to see us. And let's just talk about that a little bit since that is one of the questions. Walking into a nonprofit, we are asking for folks to help us with that. But the fee is $120 for a 50 minute session. If someone's going to be traveling to come see us, 50 minutes is not a place to start. Typically, we have folks come for three to five days of ministry if they're coming from any significant, you know, place to fly into CS. So if they're doing a four day intensive, they're going to do, that's going to be six hours a day. So that's 24 hours of ministry. And the overnight stay is minimal. It's $25 a night. And we ask them if they would please just bring their own prayer or minister person, their support person. The ministry is, we didn't want them just there by themselves. We wanted someone else with them. They're looking at attunement, attachment, relationship, and for the journey for them. So you're looking at for four day intensive, just the ministry. That's not them getting there, running a car, flying in or whatever. It's a little over $3,000 for four days a week. Yeah, which is why, you know, we started the GoFundMe campaign, which in, you know, when you guys get your tax exam status or even now, you know, we can start phasing that out. So people who would would be donating to this. And there have been a number of people in our audience that have donated through McLot, you know, the, the GoFundMe campaign to Fern and Audrey. That's going to, you know, transition over to your site because you're able to take donations at your site now. Correct. And so it'll be a little, it'll be a little less clunky. Yeah. But, you know, that it's been helpful. We'd like to, of course, see, see more of that done. We'd like to max out your time, you know, and do as much of it as, as we can. But this is what both of you do full time. This is not a side hustle. This is a little part time evening job. This is, here's how we get our income or not kind of situation. Correct. Yeah. This, this is our, what we spend our life doing. Yeah. And for a survivor of four day intensive time, they typically will need to come back two or three more times in a year for, for them to get their mind back and know their own heart. It after decades of trauma, it's, it's a journey. It's not a sprint. Right. Right. It's not a quick fix by, by any stretch of the imagination. One of the things this might relate a little bit to some of what we, we've already talked about, but this past week during our discussions, I, I, one of the, one of the more helpful sort of phrases or characterizations of what you guys do versus again, what might be called a more traditional deliverance ministry or even exorcism that sort of thing is that I don't know which of you set it and it might have been one of the other people in the room.  And just frankly, scriptural truth, you know, speaking truth to them. So that that's a dramatic difference in how you are sort of sitting face to face with a person what what's actually happening between you. So, you know, the picking fights is, you know, kind of the Hollywood ish characterization of dealing with this demonized person. And that isn't it at all. I mean, that that's just not what's going on here. No, no, I think I'm just going to go into the example of a ministry session we had recently. And the the gal has come to see us for about four years now and she really can process well. So she sits down and she smiles at us and she says, I want to talk about my mom.  I think I want to kill her. And we smile back and we say, so you feel everyone who's involved in that emotion and that thought process.  That's the Christian.  And as giving communication, there's something here. It's time for you to look at this. So it's not about we're resolving an issue that she's got a conflict with her mom present day. It's a language that she feels it.  I don't like my mom. Where do you feel that we're just letting her experience that language, that emotion. So that then she starts feeling and seeing the separateness of each of those pieces. Okay. And while we're unpacking that she has a three hour session. So what I'm going to tell you in like eight minutes is going to three hours session. So we start identified and she's the one identifying. I can feel this. This is different. She may even have shifted into a little girl along the way because that's who held the trauma, the little girl. When the little girl came out and was talking with us, everyone else back here that had all the other emotions were there too. They were working together. It's relational. We're talking. It's relational. There's some laughter with it because it's just the little girl made a joke. It just was just relational. Meanwhile, everyone in the room is feeling an edge. Audrey and I and the gal we're working with. And the gal says, I do feel it. And Audrey said, the cosmic is there. Yes. She goes, yes, I feel it. Wasn't addressed. We evaluated the mindset of what she had about it. It's rather looking at her when she said that saying, you know, come out of her or whatever. We're staying because this is the point. Remember, imagine that when this connection was made, the overwhelm of that little girl, there was no reason. Prefrontal cortex wasn't online. The front of the brain wasn't online. She was back here. So the decision was made back here in limbic to get safe. It was a survival moment. Right now, she's feeling the option. She understands the mindset. That mindset was familiar to her. She thought it was hers. And the mindset is disdain for humanity. Just kill, steal, destroy. And so we just stayed there and we were tracking with this little girl. Meanwhile, the current age gal was there too. And they're all talking to us about, this is what this feels like. I didn't realize that wasn't me. So a new brain pathway is developing that has never developed because they always went into limbic and connected to be able to kill. And that's how we do it. We work with someone. We let them feel that and the strong sense of themselves connected in Christ. That's when it's just over. They connect there and they go, I don't want this anymore. I don't need this. Yeah, we stay in love and darkness doesn't want anything to do with love. So they back away with the choice she made to stay in love. And that's why with traditional prayer ministry, the picking of a fight like you said, Mike, is it's a power encounter. It's trying to match the power of an intelligent evil. And that's not who we are as humans. Now just imagine if when, because we all felt it, we all sensed the mindset, we all could recognize, we could all discern it. If I would have shifted and then kind of settled into the warfare stance. And you know, you the Lord Jesus revealed you to us now, we're in authority in the name of Jesus Christ. And I would have moved into that. You essentially move on to that turf. Yeah, it's a fear response of a prayer minister and they you that prayer minister just walked into darkness's realm. Yeah. Well, yeah, it's it's having the other it's having the the supernatural side pick the fight and you say, OK, let's do that.  Exactly. So you see in relationship and attuning with the fact that there was memory of why she hated her mom. There was real memory.  She's getting hurt right alongside of me and she's the one hurting me. So it's a right response. We helped the little girl grieve that she was powerless. She didn't she made a decision herself. She didn't want to connect to that anymore. And so there was an understanding of all of this convoluted emotional package that she sat with at the beginning of our session. She had understanding to the whole thing and she could settle in all aspects of her that got through that trauma could settle into the identity of I belong to God. Would it be fair to characterize it this way? So instead of let's just take this this other this part again, the one that wants to kill instead of viewing that as a demon to be cast out. Instead, what what you're doing is you're helping, you know, you're, I don't know how to your patient or the person that that's come to you. You're helping them instead develop a new pattern of recognizing that Jesus is superior to that. They are actually loved by him. He's not their enemy. In other words, all these these thought patterns and they want that they develop a pattern of choosing that instead of the other. Is that what's going on? I would say that that is what we're doing. But we demonstrate it. We don't use words. Okay. And we go back to the words they have given us about their words about their relationship with God. And we say, you see, the love was always there. So it's like a process of helping them understand and affirming the delegitimization of the thing that seeks to manipulate them or how they were taught and what they were grasping for kind of kind of a shift of allegiance or a shift of understanding. Yeah. And I want to bring back in the point of imaging and how in trauma-based mind control, they're dehumanized so that they don't recognize that edge is an intelligent evil. We're helping them understand what it is to be human where they can image out of who they were originally designed to be and not in this place of always in a defensive posture in darkness's realm. I think that's really key. This idea of being dehumanized and trauma-based mind control, you could hear just by the few stories that we've told and other things that you know that these little children were not allowed just to be human. They had to be superhuman. They had to be able to get the strength of a cosmic to kill. They had to be able to endure not being able to breathe or they had to know how to stop their body from having an arterial bleed or whatever. They had to know how to psychically manage everything. And even at the very basic place is to not have emotions like humans do. So this dehumanization is really one of the goals. And so I see where we connect strongly with divine counsel stuff is this idea of imaging and just what it is to be human, the position of being a human. While those words are just real small, their divine counsel brings a whole concept of just depth of community, family. This is why everything is that God did so much more. The divine counsel metaphor and theology that you guys have found the most useful that has the most impact is really the family stuff, the relational stuff, the imaging stuff. As opposed to this judicial kind of, okay, we're in the counsel courtroom, whatever, law court, and now we're going to sort of declare a verdict and tell what's his name to get lost or something like that. That setting and that metaphor is more confrontational as opposed to this other stuff because the other stuff really is attachment to the right things and the right person, building the right loyalties, thinking the right thoughts about Jesus, about God, and so on and so forth. And having emotions is okay, grieving loss is normal, wanting to be with another human being in relationship is exactly what God's design is for us. That is foreign to someone who has undergone this level of trauma. You know, if that's the case, you can see how someone who, again, in sort of a traditional prayer ministry approach, there is an assumption, at least a bit of an assumption, maybe a huge assumption on the part of, you know, maybe a pastor, prayer minister, whatever, that the person they're trying to help even is able to comprehend this, or even find it attractive at any level and they just sort of don't. But I think we have a series of things, you know, again, from this past week that I wrote down that I wanted to talk about. We might have touched on this, but what might surprise people about survivors, I think you talked a little bit about that already that, you know, who they are as, you know, as people despite again some of these other, right word manifestations that, you know, these these bad relationships that they might have built up in their lives. Anything else that you sort of took away, or heard this week may or maybe even going back to the North Port because that was your, in terms of sort of being out in the public. That was the first time that, you know, we tried that. Either that or again, sort of the meeting, you know, the last couple days, anything else you would want our audience to sort of know about either survivors or what you do, some of the obstacles maybe before we wrap up. I have two things going on in my mind. First of all, I just want to say that in typical prayer ministry coming out of the church for survivors in particular, which is very rare, because most modalities don't include the association of survivors. They look at this as it's a psychological and a spiritual component. And I think we have to look and be a thinking people to be able to see the research that has come out in regards to trauma, not psychology in regards to trauma and what trauma does to the brain. And we have to realize that trauma affects the organic brain. And we have to be aware of how that does that. And so it is going to be a paradigm shift to be able to minister to these people in that way. We're not abdicating that any kind of spiritual involvement.  Let me give you an example. When you have a child that's under duress, and they they're told to get away from that duress by going into a castle and getting safe. And now that child, anytime there's a hint of duress, physically, emotionally, mentally, relationship, rupture, whatever, they'll just go into the castle. Now they're sitting in front of a prayer minister, and now we're taking them into the dimensions to get their parts back. Do you see where we just took them? We didn't take them into a spiritual realm. We took them into their psychological phenomenology in their own head, their own picture of what's going on. And that's what's going to happen every time when we immediately jump right into a spiritual dynamic, because you see that they had rituals or something done. It's going to harm them. So I really want to encourage... There's been whoever's doing this to them and training their mind in the process has already anticipated that sort of thing. I'm wondering if that's the case, Mike. I'm wondering if that's the case. So I just want to... One of the things that you heard me this weekend was I was concerned that people would hear our approach and they would say, well, it's all psychology. It's not anything based in biblical stuff. It is very much based in biblical stuff and the foundation that Jesus Christ is the healer. But we're letting that person think and use their mind to get back to the place of who they are designed to be. And that's a thinking, reasoning, human being that has emotions and volition. Now, before we do wrap up, you've had a pretty considerable response of people from this audience who want to be trained. Or I want to learn how to help Fernanandri or I want to learn to do what they do. And you've sort of taken steps to go down that road to develop something. So let's talk about that. Yeah, right now Fernanandri are in the process of making a curriculum and the gal that helps transcribe the podcast is coming alongside of us. Kudos to Brenda. So we're the mouth and she's the pen. So that is coming along so that we can have times that people who want to know how to work with people and love them and not harm them can do so. And I do want to say we did keep a list. We have a list of 35 of you folks that say, hey, when you get this ready, if you're going to do anything, let us know. We do have that in our database. And if there's others, let us know because we are further in the process than last time we did the podcast. And it's first and foremost on our list to get this curriculum finished. Do you have any idea of a timetable? Did Brenda give you any idea of a timetable? There's no timetable. We are really working hard on that. Brenda, for those of you who know her, Brenda is very organized. And so I was just wondering if she has sort of laid that out for you already. She's probably not going to transcribe this part. We love you, Brenda. Thank you. Well, that's good. So yeah, thanks again for sharing some of your time with us and hopefully this is a both maybe in some respects, a good introduction to what you two do. We've had prior episodes, episodes, what 68 and 149 are sort of the key ones. We've had, you know, other ones with with Beth. But if you want to sort of zero in on Fern and Audrey, those are the two episodes. I think 149 Beth is in that one as well. Yeah, that was the follow up to the earlier one. But those are the two key episodes. If you want to learn more Naked Bible Podcast episodes 68 and episode 149. So again, for those who haven't listened to them, this is I think is a sufficient introduction. If you have listened to those, then you've got a good update. Things are progressing here. And one more time, the new website. DiscoveryMercy.org. Right. And if you land on that website, you will be able to contribute directly to Fern and Audrey's ministry. We will certainly, you know, give an announcement when they, you know, get the final approval of their tax exempt status. We will leave the GoFundMe campaign running because McClot does have tax exempt status currently. So just to heads up when they get their own tax exempt status, we'll phase one out and, you know, direct people to the other. Yeah, and absolutely be looking forward to Naked Bible Podcast with some more ideas on how to support the survivors. And also we would encourage churches. If you know somebody or have somebody in your community or congregation, please contact Fern and Audrey on their website, discoveringmercy.org. And yeah, I think you said in passing this week that there are several churches that are sort of waiting again to get your exempt status to jump in and help. So yeah, absolutely. Go ahead. They are. That's great. We'll keep a lookout for more. We always enjoy having Fern and Audrey on the show. We appreciate what y'all do. And we appreciate everybody out there supporting us and them. And with that, Mike, I just want to thank everybody for listening to the Naked Bible Podcast. God bless. Thanks for listening to the Naked Bible Podcast. To support this podcast, visit www.nakedbibleblog.com. To learn more about Dr. Heiser's other websites and blogs, go to www.brmsh.com.


In our quest to uncover the hidden truths about who we are as a people, we must consider that ancient knowledge that has us scratching our heads in bewilderment at times must have come from an advanced past. The Sumerians, for example, depict a planet that has long been lost to us. In fact, these people not only depict but also describe the planet referred to as Nibiru, home of the Anunnaki in fantastic detail. Though some early translations of the text proved inaccurate, the overall view remains and the inaccuracies are from the time and scale of the planet rather than it actually existing. For example, the early translations describe a 3,600 year old orbit when in fact that orbit is closer to 20,000 years as hypothesized by Caltech. The ancient Sumerians describe the home of the gods, a failing planet with a burning moon for a sun with a highly elliptical orbit. Some refer to it as a comet-like planet that enters the solar system at regular, yet massive gap intervals and reeks utter chaos to the solar system as it approaches the sun, causing cataclysmic occurrences everywhere. In the past it is known that where the asteroid belt is today was once a planet and that was smashed to bits by an ancient collision, probably from the approach of this planet, same with Uranus. Uranus is on its side as a planet. This suggests overwhelmingly that something is affecting its very being. The fact it's on its side may be the result of a dense planetary system that is perturbing the massive planet and its entire system of moons and rings. Wait to hear this. For years astronomers looked at Uranus with amazement as what they were observing were the unusual signs of an unsteady orbit. Uranus has a wobble and the cause must be something incredibly dense with enormous gravitational pull up to 20 times that of the Earth. In 1834 German astronomer Peter Andres Hansen allegedly suggested to a colleague that two planets were needed to explain oddities in the motion of what was then the farthest known planet Uranus, oddities that led to the discovery of Neptune in 1846. Two years later French astronomer Jaques Bonnet claimed that Neptune also stumbled along its orbit, hinting that a hidden planet must have been causing Neptune to speed up and slow down as it ran around the Sun. Another anomaly regarding Neptune is the fact that it is not frozen compared with Uranus. Yet, Neptune is farther from the Sun. This would suggest that at the other end of the spectrum there must be another heat source that is keeping Neptune from freezing. And indeed if Sumerian accounts are correct and this wandering planet has a sustainable atmosphere with its own heat source then maybe Nibiru is having these types of effects. In the early 1900s, Boston-born polymath Percival Lowell got into the planet hunting game. It's not clear if Lowell was the first person to use the phrase planet X but he certainly popularized it. Lowell calculated where planet X should be based on observations of Uranus and Neptune. Thirteen years after Lowell died in 1916, American astronomer Clyde Tombo picked up the torch. Using Lowell's calculations as a guide, Tombo's systematic observation led him to Pluto in 1930, close to where Lowell predicted planet X would be. Another anomaly regarding Uranus is the fact that it orbits the Sun in a separate direction from the other planets. It is thought this could be a direct result of the planet being on its side but no one has explained this phenomenon at all. The rings of the frozen world are also incredibly anomalous apart from it looking like a giant cutting tool as observed by Voyager 2. We can't see the rings from any of Earth's ground-based telescopes. We know however from Hubble observations that Uranus has outer and inner rings and get this. Scientists can't understand what materials they are made from. Ryan Cox states, the rings of Uranus are not like any material we have ever encountered before. The crazy thing is that two moons seem to be orbiting the rings. In fact, this is so anomalous that the shepherd moons seem to be somehow maintaining them, leading to alternative theories regarding what they actually are. Perhaps they have artificial origins. Now get this. The official explanation of what these are, are as follows. A shepherd moon, also known as a herd moon or a watcher moon, is a small natural satellite that clears a gap in planetary ring material or keeps particles within a ring contained. The name is a result of the fact that they limit the herd of the ring particles as a shepherd. Due to their gravitational effect, they pick up particles and deflect them from their original orbits through orbital resonance. Does that sound like a natural occurrence to you guys? The fact of the matter is that the more we discover, the more we are noticing that what we assume to be a true account as to how the solar system works, was in fact just more inaccurate guesswork based on the understanding we had at the time. Nowhere does it say this can't be challenged because nowhere is this presented as fact. The more we learn, the more we realize we really just don't know yet. The question of Nibiru existing is one of those questions that seems to have all the evidence necessary to prove it to be real. All we are lacking is the modern observation that confirms what the ancients seemingly already knew and the near 300 years search for the elusive planet X continues to fascinate us in modern times. Uranus is on its side, so is its entire system. Neptune has another heat source other than the sun and the ancient Sumerians tell us in no uncertain terms that this thing is real and it's even inhabited. The Anunnaki are actually said to be our creators. They came to the earth to mine gold deposits so they could release this gold into the planet's atmosphere to protect it from the sun. It's astonishing guys and it appears to be true. This answers can be found in Peru where there are ancient signs of activity on the mountain tops. Thousands of man size holes are carved into the barren rock. The strange holes stretching for a mile over uneven mountain terrain were here for so long that the local people have no idea who made them or why. Funny thing is no one really saw the big picture until the area was seen from the air. It does look like a search was here in remote times. Maybe they were prospect holes for gold. They are close to the Nazca Plateau where there appears to be an effort to signify the return of something. Maybe this is where the Anunnaki mined their gold. Maybe. Just maybe. That's all from us for the moment guys. You can let us know below what you are thinking about. Comments below and as always thank you for watching.


the first two lectures on more basic material, which I think will serve as a foundation for all the speakers of this conference. So I will go a little bit fast, because it should be stuff that you know you've heard of, and then you can dig deeper if there are things that I mentioned today that you don't already know. By the way, I've written a book with Jean-Philippe Bouchoque that came out about a year and a half ago. If you want to buy a nice hard copy, you can. It's quite expensive, as all academic books. I think it's available on the web illegally for free. I don't care. Go ahead and download it, if you can find it. I'm not collecting any royalties, anyway. OK, so but a lot of what I'm saying is in this book, and there's a lot more in the book than what I will be able to say in these six sessions. OK, so I want to talk about random matrices. And why are random matrices more interesting or at least different than random numbers? If I give you a random number, suppose I give you a random number 0.7, you can't do much with 0.7. It's just a number, you don't know the distribution, you don't know the mean, et cetera. Whereas if I give you a very large matrix, matrix has n square elements. And if n square goes to infinity or for physicists or applied people, very large say n is a 1,000 or a million, well, you easily get millions of entries or thousands of billions of entries. And so you can do statistics just directly on the entry of a matrix. And then you have phenomena that happen such as self-averaging that certain quantities that you can compute on one single matrix because a matrix is so large that these quantities actually tell you something about the ensemble of matrix that is withdrawn from. So this is why I find random matrix theory interesting. And this concept of self-averaging, the mathematician will talk about concentration of measure, which means that the probability of something is extremely concentrated on one point. And for all practical purpose, it becomes non-random. So the kind of matrix I'm going to talk about are going to be self-adjoint. So I have a matrix A that will be equal to its adjoint. I will talk mostly, I'll do everything with real matrices. But since I only talk about distribution of eigenvalues and outliers, all this is the same whether the matrix is real or complex. So actually, I will use a transpose because I will be thinking about real matrices. But everything I say will be true for emission matrices where you need to take the complex conjugate when you take the transpose. And so these matrices have real eigenvalues. And this is one reason we study them because non-self-adjoint matrices are not always diagonalizable, the eigenvalues can be complex. But with these matrices, we're safe. All the eigenvalues are real. And we're going to talk about, I'm going to define moments of matrices. So I'm going to have a moment operator or an expectation value operator, which I will call tau, which tau of a matrix will be simply 1 over n trace of the matrix. And since these matrices will be diagonalizable, I can always look at this in terms of the eigenvalues. And this will be 1 over n sum over k of lambda k. So this is kind of an expectation value. And the point is that another thing I want to say is that the matrices we're going to look at will have a spectrum of eigenvalues that is bounded. So there'll be a largest eigenvalue and a smallest eigenvalue. And even as n goes to infinity, the eigenvalues are bounded. And so all the moments exist. So I can define then the moment of a, so I can define tau of a to the k. And this I will call the kth moment of a matrix. So here, I'll be sort of going back and forth between sort of averages and specific matrices. So this is the moment k of this specific matrix. But all these things will converge in the large n limit. So this is why I'll be a bit sloppy sometimes. When I say mk, do I mean the kth moment of that specific matrix, which really means 1 over n trace of that matrix to the power k, which is really the same as 1 over n sum over l of lambda to l to the power k. So this is a specific moment of that specific matrix. But when the matrix is extremely large, this actually converge to the moment of the distribution. Yes. And this is one of these quantities that I call self-averaging. So one very important tool to study random matrices, especially bounded random matrices, it sounds very strict to say that I'm looking at bounded object. But it turns out that most random matrices that we want to encounter are actually bounded. So we'll talk about the Wigner ensemble, where the distributional eigenvalues is a semicircle, and it's strictly bounded. Unlike the Gaussian, for instance, if you look in classical property distribution, one of the most natural distribution you encounter is the Gaussian. And the Gaussian is not bounded. So the tools introduced like the Stilge's transform is not very good at dealing with unbounded objects. But for random matrices, the spectrum turns out to be very often bounded. And therefore, the Stilge's transform is a very strong tool to use with bounded objects. So what is the Stilge's transform? Well, before I define it, I will define the resolvent. So the resolvent is a matrix function. So we'll call g of z for a certain matrix A. And it's just the matrix z identity minus A inverse. So it's a function of a complex variable z. And it's a matrix function. So if A is a random matrix, this is a random function. So it's a random matrix that depends on a complex parameter. But this will be very useful because this object, if you look at it in terms of the eigenvalues of A, you can write it as sum over L of vLvL transpose z minus lambda L, where lambda L and vL are the eigenvalues and eigenvectors of the matrix A. So what it is, it's a sum of rank one object of projectors with a pole at the corresponding eigenvalue. So if you zoom in on this pole, you'll be able to find a projector on the eigenvalue. So this will be useful. I'll come back a lot to this resolvent. But for now, I just wanted to take the trace of this object. And again, my traits are always normalized. So define the Stilge's transform g of z as the normalized trace of this resolvent matrix, or which is the same as 1 over n trace of z identity minus A inverse, which in terms of the eigenvalues of A is the same as 1 over n sum over L of 1 over z minus lambda L. So it's a random function because A is a random matrix. It's eigenvalues are random. But the point is that almost everywhere this function will converge. When the matrix is large, this actually becomes a deterministic function. So we'll see that everywhere except where their eigenvalues is. So how do you see that this function g of z, this function of complex variable, converges to a deterministic function? Well, let's look at it. It's an erase. So let's look at this random function for z outside the real line. So I'll look at z equals x, a real number, minus i eta. So I put myself below the real axis. So I take some complex z. And I give it a negative imaginary part. And I looked at the value of this function. So I can just write g of z, which is the same as g of x plus minus i eta. And it's 1 over n sum over L of 1 over x minus lambda L minus i eta. And as usual, when you have a complex number denominator, it's better to use a conjugate and to put the complex number on the numerator. So I get 1 over n sum over L of x minus lambda L.  The eigenvalues are eigenvalues. Now, let me just focus on the imaginary part. So this is real. This is imaginary. And this is a real number at the bottom. So I can now try to understand. I could do the real part, but I'll just focus today on the imaginary part of this object. And so I have that the imaginary part of g of x minus i eta is just 1 over n sum over L of eta over x minus lambda L square plus eta square. Now, if you look at this object here, this ratio, that's what it's called the Cauchy kernel. So up to a factor of pi, which is normalization. So let me write this this way. So it's 1 over n sum over L of k eta of x minus lambda L. And then I need to put a pi. And I always forget where to put the pi. Just see my notes because I'll always get this wrong. So I need to put a pi here. And so I define this object k eta as 1 over pi eta over, as a function of, say, y, y square plus eta square. And this is a Cauchy kernel. What it looks like, it's a function that's centered at 0, has power dot tails, the maximum and the power dot tail, and has with eta. And it has integral 1. That's why I call it a kernel. So this is an object that this is just an object of with eta and integral 1. And so what I'm doing here, what I'm saying is that g evaluated at x and with imaginary part is essentially the convolution of the discrete distribution of the lambdas. So think of my matrix. So these are to look at the eigenvalues of my matrix. It has some eigenvalues lambda 1, lambda 2, lambda 3, t-t-t-t-t-t-t-t-t-t-t-t-t-t-t-t-t-t-t to lambda n. So it has n eigenvalues. This is kind of the empirical distribution I have for each eigenvalue. And these are random. And then I convolut this with a kernel of with eta. So basically over each eigenvalue I do a little kernel like this and then I sum this. So I'll get a smooth curve, which is basically just a convolution of this discrete set of eigenvalues and this kernel. Now the point is that if eta is large enough, so I must say as well that so lambda, so typically there's many smallest eigenvalue which I'll typically denote lambda minus and the largest eigenvalue lambda plus, which are really the not lambda minus and lambda plus are more like the edge of the spectrum, where lambda one is the smallest eigenvalue on that particular sample. So the notation is slightly different, but I'm abusing it a little bit by saying the lambda minus and lambda one are essentially the same as this problem. So what I'm saying is that since all my eigenvalues are bounded, I have typically a distance one over n between eigenvalues, okay? So that's depending on the density, but if the density, and again, there are also other cases where I could have a direct mass, but for now I'm not considering this. So I'm supposing I have all my eigenvalues are spread out, then because they're spread out over a finite interval, the typical distance between them is one over n. And so if I have eta, that's much greater than one over n, then actually the picture is more like this. So every kernel encompass quite a lot of eigenvalue, actually infinitely many as n goes to infinity. And so by sort of the law of large numbers, this convolution of this random distribution of eigenvalue with a broad kernel gives a deterministic answer, okay? So this is the reason why for eta much greater than one over n, I could claim that g of x minus i eta which I would say for a given matrix A converges to g mu if you want of x minus i eta where this is, or it's equal to its average expectation value. It doesn't depend on the sample, okay? So this is very important self averaging property of the shield just transform. And this is true, well, it's true only away from the real axis, it's very important that there's an image of the real axis. And it's important that there's an imaginary part. And it's important that the imaginary part is big enough but as n goes to infinity, the imaginary part could be as small as I want. So if I fix eta and I have a sequence of matrices that are bigger and bigger and bigger, at some point the matrix will be big enough that eta will be considered small enough and this thing will converge, okay? Now, the other thing that's nice about the fact that n goes to infinity, I can, if n goes to infinity, I can always find an eta that's much, much smaller than one but much, much bigger than one over n. So for instance, typically in application, you would say if you take eta to be of order one over square root of n, okay? So I'm gonna pick an eta that's, again, much, much bigger than one over n.  So when I'm, well, I'm just gonna write it. The argument is a bit weak here but row of lambda, but maybe I should have written this before. Okay, so, sorry, my argument is a bit clumsy. Go back a little bit. So what I'm saying is that when eta is large enough, when eta is large enough, I don't see the individual eigenvalues. I just see the distribution and the distribution I'm claiming, the distribution of eigenvalue of one sample is the same of the distribution of eigenvalues of the ensemble. And so when I compute this still just transform at a certain value of eta, I get the convolution of the density of eigenvalues with this kernel and now that I have this expression, I can let the kernel size to be as small as possible so it becomes a Dirac. And in that limit, I just, I get the limit where eta goes to zero but you really have to think that eta goes to zero in this way. So eta must stay much greater than one over N and but it may become extremely small, so for instance, and this is a good choice, eta goes, eta be eta is one over square root of N, then you can let eta goes to zero and then you get the density. So I have a factor of pi here and so you get pi times the density. Okay, so this is a very important formula that I've learned in physics but it took me a while to realize where it came from. It's called the, let me write it a little more. So I'm saying two things. Let me write gN for a fixed size N. I'm saying two things. I'm saying that this converges to a fixed function. Okay, for, so for, if I exclude the real axis, okay, I could actually compute it outside where the other eigenvalues but I've only shown you. So if I take a complex Z away from the real axis then these still just transform, compute on one sample converge to the still just transform of the ensemble. And the other thing I've shown here of eta goes to zero plus of this function, g of x minus i eta, I get pi rho of x. And this is called a, the plem, oh, let's forget how to write this word. This name is plemelge formula. So it says that the still just transform encodes the, encodes the density, okay? Imaginary parts, sorry, exactly. Okay, and the real part will converge to the Hilbert transform. Okay, but I don't have time to show. Okay, I'm all right. Okay, so those are basic facts that I want. But I really, because for practical application it's actually, it actually matters that essentially what I want to say is that adding an imaginary part to the still just transform sort of blurs. It's, so I think that the real important thing is that adding an imaginary part to the still just transform is like convoluting with a Cauchy kernel. And if the imaginary part is big, you can do with a broad Cauchy kernel.  It gives you back the density, okay? And so therefore you could also, so once you have this density, you can rewrite the still just transform then as a density. Okay, so this is the limiting still just transform. You can write as an integral, say if the spectrum goes from lambda plus to lambda minus. Okay, so you can write it in integral form. So the still just transform. So here maybe I should rewrite the definition. GN is for a given is one over N sum over L of one over Z one number L. So this is for one matrix and it converges to this, this thing that only depends on the density of eigenvalues. Okay, another few things that you could say. And again, on the real axis, this is not true. On the real axis, this is a random function. If you pick a Z on the real axis that's close to an eigenvalue of that particular matrix, you can have a huge number. So this is a real function. Actually on the real axis, this is a real function that has jumps. I mean, it's extremely wild. If I sort of wanted to plot it, it would behave like this. It would have poles everywhere. It sort of goes like this. So it sort of diverges to infinity up and down every time there's an eigenvalue, okay? Where typically something like this has a branch cut. So instead of having a large amount of poles, like it has n poles and n goes to infinity, this function with poles, again, on the real axis, it doesn't converge, but outside the real axis, actually all these poles as seen from above, seen from far away just look like a branch cut, okay? And this is quite important. So this kind of formula gives you a branch cut between lambda minus and lambda plus, but elsewhere on the complex plane, it's perfectly analytic. And in particular, it's analytic at z goes to infinity. So as z goes to infinity, this behaves as one over z, okay? So densities are normalized. So when z is extremely large, this becomes just one and this tends as one over z, okay? And actually you can do an expansion and you also, and again, you could also show this sort of convergence as a power series at infinity. It's another way to see this convergence of this complex function. It's very simple to see that if you take this formula and you expand it in powers of z at infinity, just write the result, you get the g of z equals sum L equals zero to infinity mL, mL over z to the L plus one. So you recover the moments. Remember, mL is just one over n sum of lambda, sort of mixing k's and L, okay? So these are the sample moments of that particular matrix. And this converges to, so this would be a gn, it converges to gz, which actually I'll write the same expression, well, except that here maybe I put a little n, this is for that particular matrix. And then here, these moments converge to the moment of the distribution of the eigenvalues. In particular, this would be one over z plus m1, one over z square plus m2, zq plus order one over z4. So g is analytic at infinity, and it's actually the moment generating function of the moments of this density row, okay? So I'm gonna look at one particular case. So a random matrix you should all know, be familiar with is the Wigner ensemble, which I will write as, I'll write the matrix x is a Wigner matrix, I'll write as h plus h transpose over square root of 2n. And h here is a iid matrix that's not symmetric.  So the Wigner ensemble is a symmetric matrix. So the way I build my symmetric matrix is by taking a non-symmetric matrix and adding its transpose, some normalization factor of square root of 2n. And in the matrix, I could take it normal zero one, but I'll take it normal zero sigma squared. So I have a parameter sigma squared. But a unit Wigner matrix would have elements here, normal zero one. And the moments, yes, the mean is zero and the variance is sigma squared. That's why I chose this normalization here. Okay. And now I would like to compute, if I can compute a still just transform of this ensemble, I will, I can get the density just at looking at the imaginary part, okay? And again, I won't do the computation, but what you can do is you can look at this matrix. So again, I'm all thinking about this in large n limit, but I could look at the one one element in the corner of that matrix. So again, what I want to do is compute the resolvent Z identity minus X, take the inverse. This is a resolvent, this is a resolvent matrix. And then take the normalized trace of this object. So I need to invert a large matrix. If I take the one one element and I use Schwarz complement, and then I do some averaging, I won't do the computation. I quickly get an equation that looks like this take the expectation value of one over G of Z equals Z minus sigma square expectation value of G, okay? G of Z. Now, and I'll also use one more trick here. I've assumed that if I look at the element one one and look at the rest of the matrix, the rest of matrix is also a bigger matrix of size N minus one and the Wigner matrix of size N minus one is really the same thing as a Wigner matrix of size and as N goes to infinity and N and N minus one are the same thing. So this is how I get such an equation, okay? There are many, many ways to get an equation you can but anyway, okay? And since I argue that if Z is not on the real access or it's at least not on where the eigenvalues are if Z is far from any eigenvalue on a real access or elsewhere in the complex plane then the function G of Z converges to the non-random and so I can get rid of the eigenvalues and I get the following expression that one over G equals Z minus sigma squared G, okay? So this is the equation that that's satisfied by the Stieljitz transform of a Wigner ensemble, okay? And this is a, it's a very simple quadratic equation in G so I can just solve it, okay? As two nice things, also later it will allow me to compute the inverse but let me first show you the solution which you should all know as well at least you should know the density. So the Stieljitz transform of the Wigner ensemble you have to be slightly careful because it's a quadratic equation, you'll get two roots and only one is the physical one and the physical root depend on Z so you have to be careful how you write it but if you do things carefully you get that for a Wigner you get this expression let me write it as Z times one minus square root of one minus four sigma squared over Z squared over two sigma squared, okay? So this is a Stieljitz transform of the Wigner ensemble with variance sigma of variance sigma squared and if you put sigma squared equals one you get the unit Wigner ensemble. I've written it this way because you see there's a square root and the square root is singular when the inside crosses zero and I want this to be analytical for Z big so when Z is big, this is small and this is an analytical function so this is why I put it this way so if you look at it in the complex plane you get two numbers, you get two sigma here minus two sigma and two sigma you'll get a branch cut here but outside the branch cut the function so for Z large, you're outside of the branch cut this is perfectly analytical function that you could expand at infinity so this is why I chose this if you write Z squared minus then you have to, it's never clear when you should take the plus sign or the minus sign but if you put one over Z squared in the square root then you get an expression that's correct everywhere and analytical everywhere except on the real axis between minus two sigma and plus two sigma, okay? And from this, this still just transformed you can recover the density so I said that I take the imaginary part of G of X minus I eta I will get pi times the density of eigenvalues at X, okay? And you see that this thing for this to have an imaginary part, here actually I need to be on the branch cut, okay? And you quickly convince yourself that row of lambda must be equal to it's four lambda, no so lambda squared minus four sigma squared square root over two pi sigma squared, okay? So this is the Wigner's then this if you draw this distribution it's a semicircle that goes from minus two sigma to two sigma and it's properly normalized and everything, okay? So this is the Wigner's semicircle law that you get for the Wigner ensemble, okay? And it has the properties that we want namely that it's analytic elsewhere everyone under complex plane except where they're eigenvalues and it's real and outside with another important thing I need is that if I call this lambda plus and it's called lambda minus beyond lambda plus it's actually real I can look at it on a real axis but this function doesn't make sense on a real axis here but outside for Z real and greater than lambda plus this function makes sense and it's monotonically decreasing as all still just transformed should be. Remember maybe it's easier to if I look at the integral form it's easier to see if I write G of Z as integral row of lambda Z minus lambda from lambda minus to lambda plus the lambda if I look for Z greater real and greater than lambda plus this is a monotonically decreasing function that tends to one over Z, okay? So here, so this is row of lambda and if I plot G of Z or G of X so G of Z on the real axis it starts with some value and then decrease as G of X goes to one over X, okay? Which means that it's invertible that's very important that this function if I go from X to infinity up to lambda plus this is a monotonic function I can invert this function and invert is very, very useful that's why I kept this formula here because I can read off the inverse just from this formula. So the inverse G of Z is Z of G. So Z of G is just one over G plus sigma squared G, okay? So the inverse of the still just transformed for real values beyond and I'll call this number this quantity G star, okay? So if you want G of Z is monotonic so G becomes smaller and smaller and smaller so Z of G increases as I increase G and so this is well-defined for G between zero and G star, okay? So there's a neighborhood of zero where this function is invertible, okay? Why is that important? So I wanna finish very, very quickly with something that should take many hours to explain the concept of free-ness which, okay? So one thing I forgot to mention but I'll be looking a lot at matrices that are rotation invariant, okay? What's a random matrix that's rotation invariant? It means that if I take say a matrix A and I consider a random rotation O A O transpose that these two are equal in law meaning that the probability of a matrix A is the same probability as any rotation of the matrix A. This would say that the ensemble A is rotation invariant, okay? And I will, so concept of free-ness is a, well, it's a concept that it's used if you deal with sort of algebras of non-commuting objects there's a notion of independence for non-commuting object that's very powerful, it's called free-ness and I'm not gonna go into the algebra of it but it turns out that large rotation invariant matrices are free. So I would say that A and B are free with respect to one another and I'll tell you why it's useful but I just wanna define A and B are free if they're large matrices. In practice, it's only true at infinity, okay? The animations of size n, if notion of free-ness doesn't exist is not true at size n but asymptotically as n goes to infinity is what I'm gonna say is true and at least one is rotation invariant. Okay, so for instance, if I take A plus a random matrix a random matrix OB or transpose by multiplying by random rotation matrix this I made B explicitly made B rotation invariant, okay? And so any matrix A could be a fixed matrix or it could be a Wigner matrix or it could be in the next hour we'll discuss about Wishart matrix it could be any matrix and then I take B again, any matrix it could be random, it could be fixed but I multiply by random rotation by here O is a rotation for another language taken from the hard measure on the orthogonal group and of course this is true as well for unitary matrices if I'm looking at complex admission matrices, okay? So if A and B are free or if they have these two property if they're large and at least one of them is rotation invariant that I can make explicitly then I can compute this object C the sum of the two matrices, okay? At least I can say many things about the matrix C which would be the sum, okay? So remember in classical probability distribution if I have a random number A and a random number B if I take this, if I say C equals A plus B so these are classical random numbers I can say something about C if A and B are independent, okay? And then C will be the convolution of the law of A the law of C will be the convolution of the law of A and the law of B, okay? For non-commuting object there's a similar notion, similar but different it gives different results and gives different law of large numbers many different things are different and this is called the notion of free probability but what I wanna say is that it applies to large matrices where one of them is rotation invariant and one thing I can say there's a function R that we'll define in a moment I can say that R of the matrix X will be equal to R of the matrix A plus R of the matrix B, okay? So there exists a function that characterizes the spectrum of the matrix C and this function is additive, okay? And this is extremely useful because if I can go from the spectrum to R and from R to the spectrum while knowing the spectrum of A and B if I take a rotation invariant combination of A and B I sum them I can compute the spectrum of the sum, okay? And the reason I insist on the fact that the still just transform is invertible is because the function R is directly related to the inverse of the still just transform, okay? So I wanna finish and then we can go for the coffee break with this fact. So this function, this wonderful function that's additive for free, for additional free objects is defined as R, let me write as R of G is the function Z of G. So it is Z of G is the inverse of G of Z. So I think the limiting still just transform on the real axis but away from the eigenvalues it's invertible and I call this function Z of G and it turns out this function always has a pole it always starts as one over G. So the R transform actually you remove this pole minus one over G and then it becomes an analytical function it's regular and zero, okay? So let's do this just for the Wigner ensemble. For the Wigner ensemble I have the Z of G is one over G plus sigma squared G and so the R transform I just get rid of the one over G. So I have that R for a Wigner matrix X of G is simply given by sigma squared G, okay? This is obviously the simplest R transform, okay? Every other random matrix ensemble will have a much more complicated and but this is natural because the Wigner ensemble is a bunch of Gaussian random numbers and the central limit theorem on Gaussian random numbers and it turns out that if you sum a lot of random matrices you end up with a Wigner matrix. So there's a central limit theorem that I won't go into the detail but if you add a bunch of randomly rotated matrices and you rescale them to keep the variance constant basically you'll get rid of, you'll start maybe with a very complicated R function but by adding and scaling and adding and scaling you get a central limit theorem and you converge to this, okay? And then from this, once you know the R transform you can just go back here, add back the pole at one over G, invert the function and you get the still just transform of the ensemble C and then you look at the imaginary part of that on the real axis and you get the density. So this is how you compute say sums of random variable. That's one more thing I wanna say. I won't use it for the addition but in the next hour I'll talk about multiplication. It's called a subordination relation. So this equation here, well, this combined with this I can rewrite it that if I look at the still just transforms I put G of C of some variable Z. This is just, it's really just some algebra. I take this formula and this formula, I recombine them. I get this nice formula that G A of Z minus R B of G C of Z. Trust me, you can do it as an exercise. Try to combine this and this, okay? And you have that the still just transform of the matrix C at some point is equal to the still just transform of the matrix A but at a different point. And of course this different point it's a nasty implicit equation because the point at which you need to evaluate depends on the R transformer B and but also on the still just transform that you're trying to get to, okay? So this by itself, well, it's useful but not super useful. Now what turns out if I go back to my big resolvent. So I had this matrix, big G of C which was defined as Z identity minus the matrix C inverse. This is a random object, okay? This is a n square dimensional object. So it's random in no way converges to anything in particular. So now I explicitly need to take the expectation value but if I look at this expression here of C which is given by A and B I can take the expectation value with respect to this object here. B could be itself random or just expectation value with respect to these random rotation and has this beautiful relation that's the same as this but now for a large matrix but in expectation value at the expectation value of this resolvent matrix G C of Z equals the resolvent matrix of A which I assume A is fixed. If it's not fixed then it's conditioned on A, okay? Evaluate it, it's the same expression, okay? So what does it mean? Well, there are two consequences. Well, GA is a matrix built from A. So it commutes with A. It's a multiple of identity. Well, this is for C but for A is the same thing, same expression. So it's a multiple of identity, the matrix A some inverse. So this is a matrix, it's a large matrix that commutes with A. C is a matrix that's built from A and some random stuff. So it's usually C doesn't commute with A but in expectation value it does. So one beautiful thing and it sort of makes sense just why the C in expectation value commutes with A because so again, the random matrix C doesn't commute with A. I add something random but think in the basis of A. So the basis where A is diagonal, C is something diagonal plus something random but something random that's rotation invariant cannot have off-diagonal elements on average. Of course on one particular realization it will have off-diagonal elements but on average there's kind of a symmetry. Anything that you put off the diagonal you could flip some axis and you would put the same thing with the minus sign. So the average of the off-diagonal elements are zero. So it's important the expectation value of C conditioned to A is in the same basis of A. And this is a set, well this state is more than that but what's implied here is that, and why is all this important because I'm gonna talk about eigenvector problems and if I'm gonna try to look at eigenvectors of matrices, remember that this big matrix G, it's a projector on each of the eigenvectors. And so I'll need to compute expectations of projectors and I'll need this formula. So definitely on Wednesday, there's a similar formula for matrix multiplication that I'll show later. And if I wanna look at eigenvectors of matrices that have been multiplied, this expression is extremely useful. And I think I should stop here and ask for questions or a coffee break. Thanks. Any questions or you can ask them here. Jean, any questions from the... There were no questions. I think you're very clear. Okay. Sorry, I was a bit fast but this is sort of material that people should know in random matrix theory. I think it was really a great introduction for everything we're going to do. Actually, we will know much more about free probability with Kami, right? So... Okay. All right, so we have a coffee break upstairs on the terrace, I think. And we start, let's be there around 1040 or, okay. See you.


Hello and welcome to the session. In this session we will discuss a question which says that the table below gives deterrence of consumption of coal in the generation of electricity in a thermal power station in each quarter for the years 1995 to 1997. To calculate four quarterly moving averages, show these moving averages on a graph. Now before starting the solution of this question we should know a result. And that is the procedure for calculating four yearly moving averages. For this we will draw a table and in this the values for the particular years will be given to us and we will find the four yearly moving total, the four yearly moving average and the four yearly moving average centre. Find four yearly moving total. For this we will add the values of these four years from in between the second and third years. So we will add these four and we will place the table in between second and third year plus B plus C plus D here in between second and third year and let it name as A1. Next four years, that is we will add these four values and we will place the total in between third and fourth years. So we will place the total of these values which will be B plus C plus D plus E here and let it name as A2. Next year is taken into account. Now we will calculate for yearly moving average. Now for calculating this we will divide four yearly moving totals that is A1, A2, A3 and A4 by 4 and we will get four yearly moving averages. So for this it will be A1 by 4 and let it be M1. Here it will be A2 by 4, let it be M2 and A3 by 4, let it be M3. Now we will calculate four yearly moving average centered. For this we will add against the third year. So we will write here M1 plus M2 whole upon 2 and let it be B1. It will be M2 plus M3 divided by 2 and place it against the fourth year. So it will be M2 plus M3 whole upon 2 and let it be B2. M4 whole upon 2 and let it be B3. Now for the graph B1 the fourth year with B2 with B3 bring to a line. We will work out as a key idea for solving now we will start with the solution. Now where this table is given to us and we have to calculate four quarterly moving averages. Now using this result which is given in the key idea we will make a table for the given data. We have made a table in which the first column shows the years, second, quarters, third, all years and turns and we have to find four quarterly moving total, four quarterly moving average and four quarterly moving average centered. In total we will add the values for the first in between plus 14 plus 12 equal to 47. Now next we will leave the first value and we will add the values for the next four quarters. That is we will add these four values in between third and fourth quarter. So here it will be 14 plus 12 plus 11 plus 13 which is equal to 50 in between fourth and 1995 and first quarter of the year 1996. So it will be 12 plus 11 plus 13 plus 19 which is 55. These four values and place the total in between second and first quarter will be equal to 61. Now next we will add and place the total in between second and third quarter. Adding one we will get 67 and on the next step we will add quarter. So it will get 70. Now in the next step of year 1996 and first quarter of year 1997 and the total of these four values will be 74. Then in between first and second quarter we will place the total in between second and third quarter. So it will be 80 in total by four. So in this case it will be 12 by 4 which is equal to 11.75 and for this it will be 50 by 4 which is equal to 12.5 and here it will be 55 by 4 which is equal to 13.75 and therefore it is 15.25 75 70 by 14.5. 74 by 4 is 18.5 and therefore it is 19.25. And now for the fourth part we will move in every center and we will divide them by two and place this value against the third quarter. So it will be 11.75. Calculating will give 0.125 which is equal to 12.1. Similarly for the next we will add and divide by two it will give 13.125 and we will place this value against the fourth quarter. So it will be 13.125. Now next we will add these two values and divide them by two and we will place this value against the first quarter of the year 1996. And calculating this will be 14.5. Now next divide them by two and place this value against the second quarter of the year 1996 and it will give 16. And next we will use and divide by two and place this value against the third quarter. So it will give 17.125. Now taking these two values we will get 18. Now next we will take these two values and we will get 18.875. And now next we will take these two values and we will get 19.625. Now we will round up all these values. So this will be 13.1. It will be 14.5. This is 16 and this will be 17.1. This is 18 and it will be 18.9 and this will be 19.6. The actual and the trend values on the graph. The graph that is the first part which is the coin used into the third quarter with 12 on the graph. That is the third quarter. We will plot all these points on the graph. We will get the graph that is the third quarter with so this is the required point on the graph. The quarter with 13.1. So this is the required point that is the fourth quarter of the year 1995 with 13.1. The quarter of the year 1996 with 14.5. The quarter of the year 1996 with 16. The third quarter with 17.1. The quarter of the year 1997 with second quarter of the year 1997 with 19.6. All these points on the graph. We will get the graph to show the given question and that's all for this session. Hope you all have enjoyed the session.