Chapter 27 of the Stillwater Tragedy. Mr. Taggett, in spite of the excellent subjection under which he held his nerves, caught his breath at these words, and a transient pallor overspread his face as he followed the pointing of Richard's finger. If William Durgin had testified falsely on that point, if he had swerved a hair's breath from the truth in that matter, then there was but one conclusion to be drawn from his perjury. A flash of lightning is not swifter, then was Mr. Taggett's thought in grasping the situation. In an instant he saw all his carefully articulated case fall to pieces in his hands. Richard crossed the narrow room and stood in front of him. Mr. Taggett, do you know why William Durgin lied? He lied because it was life or death with him. In a moment of confusion he had committed one of those simple fatal blunders, which men in his circumstances always commit. He had obliterated the spots on his clothes with red paint, when he ought to have used blue. That is a very grave supposition. It is not a supposition, cried Richard. The daylight is not a plainer fact. You are assuming too much, Mr. Shackford. I am assuming nothing. Durgin has convicted himself. He has fallen into a trap of his own devising. I charge him with the murder of the mule, Shackford. I charge him with taking the chisel and the matches from my workshop, to which he had free access, and I charge him with replacing these articles in order to divert suspicion upon me. My unfortunate relations with my cousin gave color to this suspicion. The plan was an adroit plan, and has succeeded, it seems. Mr. Taggett did not reply at once, and then very coldly. You will pardon me for suggesting it, but it will be necessary to ascertain if this is the cask which Durgin opened, and also if the head has not been repainted since. I understand what your doubt implies. It is your duty to assure yourself of these facts, and nothing can be easier. The person who packed the meat, it was probably a provision dealer named Stubbs, will of course be able to recognize his own work. The other question you can settle with a scratch of your penknife. You see, there has been only one thin coat of paint laid on. The grain of the wood is nearly distinguishable through it. The head is evidently new, but the cask itself is an old one. It has stood here these ten years. Mr. Taggett bent a penetrating look on Richard. Why did you refuse to answer the subpoena, Mr. Shackford? But I haven't refused. I was on my way to Justice Bemis's office when you knocked. Perhaps I am a trifle late, added Richard, catching Mr. Taggett's distrustful glance. The summons said to a clock, remarked Mr. Taggett, pressing the spring of his watch. It's now after three. After three? How could you neglect it with evidence of such presumable importance in your hands? It was only a moment ago that I discovered this. I had come here for Mr. Perkins's office. Mr. Perkins had informed me of the horrible charge which was to be laid at my door.  I think it unsettled my reason for a while. I was unable to put two ideas together. At first he didn't believe I had killed my cousin, and presently he seemed to believe it. When I got out on the street, the sidewalk lurched under my feet like the deck of a ship. Everything swam before me. I don't know how I managed to reach this house, and I don't know how long I have been sitting in a room upstairs, when the recollection of the subpoena occurred to me. I was standing here, dazed with despair. I saw that I was somehow caught in the toils, and that it was going to be impossible to prove my innocence. If another man had been in my position, I should have believed him guilty. I stood looking at the cask in the corner there, scarcely conscious of it. Then I noticed the blue paint on the head, and then William Durgen's testimony flashed across my mind. Where is he? cried Richard, turning swiftly. That man should be arrested. I am afraid he is gone, said Mr. Taggett, biting his lip. Do you mean he has fled? If you are correct, he has fled. He failed to answer the summons today, and the constable sent to look him up has been unable to find him. Durgen was in the bar room of the tavern at eight o'clock last night. He has not been seen since. He was not in the yard this morning. You have let him slip through your fingers. So it appears, for the moment. You still doubt me, Mr. Taggett? I don't let persons slip through my fingers. Richard curbed an impatient rejoinder, and said quietly, William Durgen had an accomplice. Mr. Taggett flushed, as if Richard had read his secret thought. Durgen's flight, if he really had fled, had suggested a fresh possibility to Mr. Taggett. What if Durgen were merely the pliant instrument of the cleverer man, who was now using him as a shield? This reflection was precisely in Mr. Taggett's line. In absconding, Durgen had not only secured his own personal safety, but had exonerated his accomplice. It was a desperate step to take, but it was a skillful one. He had an accomplice, repeated Mr. Taggett after a moment. Who was it? Terini. The man who was hurt the other day? Yes. You have grounds for your assertion? He and Durgen were intimate, and have been much together lately. I sat up with Terini the night before last. He acted and talked very strangely. The man was out of his head part of the time. But now as I think it over, I am convinced that he had this matter on his mind, and was hinting at it. I believe he would have made disclosures if I had urged him a little. He was evidently in great dread of a visit from some person, and that person was Durgen. Terini ought to be questioned without delay. He is very low and may die at any moment. He is lying in a house at the further end of the town. If it is not imperative that I should report myself to Justice Beemis, we had better go there at once. Mr. Taggett, who had been standing with his head half-bowed, lifted it quickly as he asked the question, Why did you withhold Lemuel Shackford's letter? It was never in my possession, Mr. Taggett, said Richard, starting. That paper is something I cannot explain at present. I can hardly believe in its existence, though Mr. Perkins declares he has had it in his hands, and it would be impossible for him to make a mistake in my cousin's writing. The letter was found in your lodgings. So I was told I don't understand it. That explanation will not satisfy the prosecuting attorney. I have only one theory about it, said Richard slowly. What is that? I prefer not to state it now. I wish to stop at my boarding-house on the way to Terini's. It will not be out of our course. Mr. Taggett gave silent acquiescence to this. Richard opened the scullery door, and the two passed into the court. Neither spoke until they reached Lime Street. Mrs. Spooner herself answered Richard's ring, for he had purposely dispensed with the use of his pass-key. I wanted to see you a moment, Mrs. Spooner, said Richard, making no motion to enter the hall. Thanks, we will not come in. I merely desire to ask you a question. Were you at home all day on that Monday immediately preceding my cousin's death? No," replied Mrs. Spooner, wonderingly, with her hand still resting on the knob. I wasn't at home at all. I spent the day and part of the night with my daughter Mary Anne at South Millville. It was a boy, added Mrs. Spooner quite irrelevantly, smoothing her ample apron with the disengaged hand. Then Janet was at home, said Richard, called Janet. A trim, intelligent-looking Nova Scotia girl was summoned from the basement kitchen. Janet, said Richard, do you remember the day about three weeks ago that Mrs. Spooner was absent at South Millville? Yes," replied the girl, without hesitation. It was the day before—and then she stopped. Exactly, it was the day before my cousin was killed. Now I want you to recollect whether any letter or note or written message of any description was left for me at this house on that day. Janet reflected, I think there was, Mr. Richard, a bit of paper-like. Mr. Taggart riveted his eyes on the girl, who brought the paper, demanded Richard. It was one of the Murphy boys, I think. Did you hand it to me? No, Mr. Richard, you had gone out. It was just after breakfast. You gave it to me when I came home to dinner then. No, returned Janet, becoming confused with a dim perception that something had gone wrong and she was committing herself. I remember I didn't come home. I dined at the Slocums. What did you do with that paper? I put it on the table in your room upstairs. Mr. Taggart's eyes gleamed a little at this. And that is all you can say about it, inquired Richard, with a fallen countenance. Janet reflected. She reflected a long while this time. No, Mr. Shackford, an hour or so afterwards. When I went up to do the chamber work, I saw that the wind had blown the paper off the table. I picked up the note and put it back, but the wind blew it off again. What then?  Oh, Mr. Richard, have I done something dreadful? Dreadful, cried Richard, Janet, I could hug you. Oh, Mr. Richard, said Janet, with a little coquettish movement natural to every feminine thing, bird, flower, or human being. You always have such a pleasant way with you. Then there was a moment of dead silence. Mrs. Spooner saw that the matter, whatever it was, was settled. You needn't wait, Janet, she said, with a severe mystified air. We are greatly obliged to you, Mrs. Spooner, not to mention Janet, said Richard, and if Mr. Tagget has no questions to ask, we will not detain you. Mrs. Spooner turned her small, amiable orbs on Richard's companion. That silent little man, Mr. Tagget, he doesn't look like much, was the landlady's unuttered reflection, and indeed he did not present a spirited appearance. Nevertheless, Mrs. Spooner followed him down the street with her curious gaze, until he and Richard passed out of sight. Neither Richard nor Mr. Tagget was disposed to converse, as they wended their way to Mitchell's alley. Richard's ire was slowly kindling at the shameful light in which he had been placed by Mr. Tagget, and Mr. Tagget was striving, was only partial success, to reconcile himself to the idea of young Shackford's innocence. Young Shackford's innocence was a very awkward thing for Mr. Tagget, for he had irretrievably committed himself at headquarters. As Richard's lay in ire was mingled a feeling of profound gratitude. The Lord was on my side, he said presently. He was on your side, as you remark, and when the Lord is on a man's side, a detective necessarily comes out second best. Really, Mr. Tagget, said Richard, smiling, that is a handsome admission on your part. I mean, sir, replied the latter, slightly nettle, that it sometimes seems as if the Lord himself took charge of a case. Certainly you are entitled to the credit of going to the bottom of this one. I have skilfully and laboriously damaged my reputation, Mr. Shackford. Mr. Tagget said this with so heavy an air that Richard felt a stir of sympathy in his bosom. I am very sorry, he said good-naturedly. No, I beg of you, exclaimed Mr. Tagget. Any expression of friendliness from you would finish me. For nearly ten days I have looked upon you as a most cruel and consummate villain. I know, said Richard, I must be quite a disappointment to you in a small way. Mr. Tagget laughed in spite of himself. I hope I don't take a morbid view of it, he said. A few steps further on he relaxed his gait. We have taken the Hennessy girl into custody. Do you imagine she was concerned? Have you questioned her? Yes.  I think Mary Hennessy an honest girl. She's little more than a child. I doubt if she knew anything whatever. Durgin was much too shrewd to trust her, I fancy. As the speakers struck into the principal street through the lower and busier end of which they were obliged to pass, Mr. Tagget caused a sensation. The drivers of carts and the pedestrians on both sidewalks stopped and looked at him. The part he had played in Slocum's yard was now an open secret and had produced an excitement that was not confined to the clientele of Snelling's bar-room. It was known that William Durgin had disappeared and that the constables were searching for him. The air was thick with flying projectures, but none of them precisely hit the mark. One rumor there was which seemed almost like a piece of poetical justice, a whisper to the effect that Roland Slocum was suspected of being in some way mixed up with the murder. The fact that Lawyer Perkins, with his green bag streaming in the wind, so to speak, had been seen darting into Mr. Slocum's private residence at two o'clock that afternoon was sufficient to give birth to the horrible legend. Mitchell's alley, said Mr. Tagget, thrusting his arm through Richards and hurrying on to escape the Stillwater Gaze. You went there directly from the station the night you got home. How did you know that? I was told by a fellow traveller of yours and a friend of mine. By Jove, did it ever strike you, Mr. Tagget, that there is such a thing as being too clever? It has occurred to me recently. Here is the house. Two sallow-skinned children with wide, wistful black eyes who were sitting on the stone step, shrily crowded themselves together against the door-jam to make passageway for Richard and Mr. Tagget. Then the two pairs of eyes veered round inquiringly and followed the strangers up the broken staircase and saw one of them knock at the door which faced the building. Richard's hasty tap, bringing no response, he lifted the latch without further ceremony and stepped into the chamber. Mr. Tagget, a pace or two behind him. The figure of Father O'Meara slowly rising from a kneeling posture at the bedside was the first object that met their eyes. The second was Terini's placid face, turned a little on the pillow. The third was Brigitte, sitting at the foot of the bed motionless, with her arms wrapped in her apron. He is dead, said the priest softly, advancing a step towards Richard.  He wanted to see you, Mr. Schachford, but you are not to be found. Richard sent a swift glance over the priest's shoulder. He wanted to tell me what part he played in my cousin's murder, said Richard. God forbid, the wretched man had many a sin on his soul, but not that. Not that. No, he had no hand in it, no more than you or I. His fault was that he concealed his knowledge of the deed after it was done. He did not even suspect who committed the crime, until two days afterwards, when William Derrigan, Richard's eyes lighted up as they encountered Mr. Tagget's. The priest mistook the significance of the glances. No, said Father O'Meara, indicating Brigitte with a quick motion of his head. The power soul does not understand a word, and even if she did, I should have to speak of these matters here and now, while they are fresh in my mind. I'm obeying the solemn injunctions of the dead. Two days after the murder, William Derrigan came to Tarini and confessed to the deed, offering to share with him a large sum in gold and notes, if he would hide the money temporarily. Tarini agreed to do so. Later, Derrigan invited to him his plan of turning suspicion upon you, Mr. Shackford. Indeed, I was directly charging you with the murder, if the worst came to the worst. Tarini agreed to that also, because of some real or fancy dingeery at your hands. It seems that the implement which Derrigan had employed in forcing the scullery door, the implement which he afterwards used so mercilessly, had been stolen from your workshop. The next morning, Derrigan put the tool back in its place, not to know on what other disposition to make of it, and it was then that the idea of shouldering the crime upon you entered his wicked heart. According to Tarini, Derrigan did not intend to harm the old gentleman, but simply to rob him. The unfortunate man was awakened by the noise Derrigan made in breaking open the safe, and rushed into his doom. Having then no fear of interruption, Derrigan leisurely ransacked the house. How he came across the will, and destroyed it with the idea that he was put in this state out of your possession. This and other details I shall give you, by and by. Father Amir paused a moment. After the accident at the mill, and the conviction that he was not to recover, Tarini's conscience began to prick him, when he reflected on Miss Slocum's kindness to his family during the strike, when he now saw her saving his wife and children from absolute starvation. He was nearly ready to break the oath with which he had bound himself to William Derrigan. Curiously enough this man, so reckless in many things, held his pledged word sacred. Meanwhile his wavering condition became apparent to Derrigan, who grew alarmed, and demanded the stolen property. Tarini refused to give it up, even his own bitter necessities had not tempted him to touch a penny of it. For the last three days he was in deadly terror, lest Derrigan should rest the money from him by farce. The poor woman here knows nothing of all this. It was her presence, however, which properly prevented Derrigan from proceeding to extremities with Tarini, who took care never to be left alone. I recollect, said Richard, the night I watched with him he was constantly expecting some one. I supposed him to be wandering in his mind. He was expect in Derrigan, though Tarini had every reason for belief in that he had fled. Mr. Taggett leaned forward and asked, when did he go, and where? He was too conned to confine his plans to Tarini. Three days ago Derrigan came here and begged for a portion of the banknote. Previously he had reclaimed the whole sum. He said the place was grown too warm for him, and that he had made up his mind to leave, but Tarini held on to the money, having resolved that he should be restored intact to you. He promised Derrigan, however, to keep his flight secret for three or four days, at the end of which time Tarini meant to reveal all to me at confession. The night he sat with him, Mr. Shackford, he was near a break in his promise.  His agony, lest he should die or lose his senses before he could make known, the full depth of Derrigan's villainy must have been swamped and terrible.  The money is hidden somewhere under the mattress, I believe, a better man than Tarini would have spent some of it, added Father O'Meara, waving a sort of benediction in the direction of the bed. Richard did not speak for a moment or two. The wretchedness and grimness of it all smote him to the heart. When he looked up, Mr. Taggett was gone, and the priest was gently drawing the coverlet over Tarini's face. Richard approached Father O'Meara and said, When the money is found, please take charge of it and see that every decent arrangement is made.  I am a Protestant, but I believe in any man's prayers when they are not addressed to a heathen image. I promise Tarini to send his wife and children to Italy. This pitiful, miserable gold, which costs so dear, and is worth so little, shall be made to do that much good, at least. As Richard was speaking, a light footful sounded on the staircase outside. Then the door which stood a jar was softly pushed open and Margaret paused on the threshold. At the rustle of her dress Richard turned and hastened towards her. It is all over, he said softly, laying his finger on his lip. Father O'Meara was again kneeling by the bedside. Let us go now, whispered Richard to Margaret. It seemed fit that they should leave the living and the dead to the murmured prayers and solemn ministration of the kindly priest. Such later services as Margaret could render to the bereaved woman were not to be wanting. At the foot of the stairs Richard Shackford halted abruptly and oblivious of the two children who were softly chattering together in the doorway caught Margaret's hand in his. Margaret Tarini has made a confession that sets at rest all question of my cousin's death. Do you mean he? Margaret faltered and left the sentence unfinished. No, it was William Durgen. God forgive him. William Durgen, the young girl's fingers, closed nervously on Richard's as she echoed the name and she began trembling. That, that is stranger yet. I will tell you everything when we get home. This is no time or place. But one thing I must ask you now and here.  Yes, said Margaret. That is all I care to know, cried Richard. That consoles me. And the two pairs of great inquisitive eyes, looking up from the stone step, saw the signorina standing quite mute and colorless with the strange gentleman's arms around her. And the signorina was smiling. End of Chapter 27 Chapter 28 of the Stillwater tragedy. This LibriVox recording is in the public domain. The Stillwater Tragedy by Thomas Bailey Aldrich Chapter 28 One June morning, precisely a year from that morning when the reader first saw the daylight breaking upon Stillwater, several workmen with ladders and hammers were putting up a freshly painted sign over the gate of the marble yard. Mr. Slocum and Richard stood on the opposite curb stone, to which they had retired in order to take in the general effect. The new sign read, Slocum and Shackford. Richard protested against the displacement of its weather-stained predecessor. It seemed to him an act little short of vandalism, but Mr. Slocum was obstinate and would have it done. He was secretly atoning for a deep injustice into which Richard had been at once too sensitive and too wise to inquire. If Mr. Slocum had harbored a temporary doubt of him, Richard did not care to know it. It was quite enough to suspect the fact. His sufficient recompense was that Margaret had not doubted. They had now been married six months. The shadow of the Tragedy in Welch's court had long ceased to oppress them. It had vanished with the hasty departure of Mr. Taggett. Neither he, nor William Durgen, was ever seen again in the flesh in Stillwater. But they both still led and will probably continue for years to lead a sort of phantasmal, legendary life in Snelling's bar room. Durgen in his flight had left no traces. From time to time, as the months rolled on, a misty rumor was blown to the town of his having been seen in some remote foreign city, now in one place and now in another, always on the point of departing, self-pursuant like the wandering Jew. But nothing authentic. His after fate was to be a sealed book in Stillwater. I really wish you had let the old sign stand, said Richard, as the carpenters removed the ladders. The yard can never be anything but Slokum's yard. It looks remarkably well up there, replied Mr. Slokum, shading his eyes critically with one hand. You object to the change, but for my part I don't object to changes. I trust I may live to see the day when even this sign will have to be altered to Slokum, Shephard, and Son. How would you like that? I can't say, returned Richard Laffane, as they passed into the yard together. I should first have to talk it over with the Son. End of Chapter 28. End of The Stillwater Tragedy by Thomas Bailey Aldridge


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I'm Dan Bredney, I'm from the Philosophy Department and McLean Center. One of my colleagues in the Philosophy Department once wrote that a good life is a life lived in multiple communities. By that measure, Bill Meadow led a very good life. Susan, children and grandchildren, friends, music, neonatology, the McLean Center, soccer, tennis, and other things. Many, many communities. I wanna talk today about the two communities in which I knew Bill, soccer, and the McLean Center. In each he was a teacher, teacher of the girls on the soccer team, of the fellows in case conference, and frankly, my own teacher. Now, when I say that Bill was a teacher, I mean something that might embarrass him and he'd surely deny it with one of his characteristically pungent phrases such as, bite me. I will get to my analysis of that phrase a little later. But in fact, Bill was a teacher in the old-fashioned German sense of bildung. He helped people to grow, to become grown-ups. Well, good things start with soccer and that's where I first came to know him. In a sense, I'm talking to you here today because my daughter, Allie, Bill's daughter, Beanie, and John Lantos' daughter, Emma, were childhood friends and eventually became teammates on the Red Dots, their traveling soccer team. Bill coached that team. He had earlier coached an AYSO soccer team until it became so successful, they were in effect the 1927 Yankees of Hyde Park AYSO, that the powers in charge urged Bill to do something else specifically to start a traveling team. Now, some of you might know this, a traveling, a middle school traveling soccer team is its own rather distinctive form of life. Sunday after Sunday, no matter what the weather and God's arranged it so that the weather is always terrible, all games are played on cold, rainy, windswept days. Sunday after Sunday in abysmal weather. You go to Jackson Park and you try to navigate around the bird droppings and you try to calm the other team's parents who have come from the suburbs and are frozen with terror of the south side of Chicago and run to their mini-vans the moment the final whistle blows. What else you get in your own car and you drive? And you drive and you drive. You drive until you reach some woe-begotten place like Tinley Park or St. Charles or Naperville or Libertyville or some other suburbanville and of course there, the weather will be worse. I learned things from being a soccer team dad that I might never otherwise have known. I learned what the off-sides rule is. I learned I could criticize the refs only so far when dad got ejected from the sidelines and sent to the parking lot for being excessively vociferous. I learned that Coach Bill was quite a good coach. So far as I could judge, he was a good tactical coach. He understood what our girls could and couldn't do. The team was weak offensively but strong defensively and so as soon as we got a lead, Bill would have everyone go into defensive mode, huddle on the back lines, control the ball, went in trouble, kicked the ball out and kick it out so as far as you can so the time is killed while the ball is being retrieved. But Bill also taught the kids to grow up. He had a few phrases that were legendary within the team and it's worth looking at them a moment. One was a phrase that all of you from Case Conference have heard, that's the I love you butt phrase. As you know, the butt is inevitably followed by some very mild criticisms such as in Case Conference, that's the stupidest thing anyone has ever said in the history of the world. In soccer, what followed the butt would be something such as don't you ever ever again make that kick with your left foot? In soccer, and I think most of the time in Case Conference, Bill was correct. That kick should not have been made with a left foot. But of course the main thing is that the sentence started with I love you. Even on those woe-begone suburban fields in that awful weather, I'm convinced that the girls heard that in his voice. Bill really did care for them. He showed it in trivial ways, such as buying them ice cream after the games, but mostly he showed it in how attentive he was. He was remarkably there and focused. And his focus was on teaching the girls not just this or that soccer skill, but on how to play together, how to watch out for one another on the field, that is how to become more mentally. Bill called various other things from the sideline, but the one that stuck with me was calmly, calmly. Calmly, calmly. He'd say it in the heat of the game as the girls were either back on their heels against a more powerful opponent or getting themselves together to pressure a weaker one. Either way, and the life lesson is obvious, calmly, calmly is the right way to proceed. It's a pretty good mantra. As should be clear, I'm invoking the traditional conceit that teams sports build character. Now, with some teams and some coaches, this is plainly false. With Bill and the Red Dogs, I think it was really quite true. I don't wanna overstate characters built by a thousand small steps. Still, they have to be the right steps, and some steps are more important than others. Excuse me, I'm very, very glad that Bill was my daughter's coach. Now, soccer Bill was coach Bill. The team families bought him a jacket with that name on the back, he often wore it to case conference. The traveling team ended more than a decade ago, but Bill and John and I and a few other dads, I don't know, is Joel Roth in the audience? Did Joel come? We've continued to meet for dinner every few months. It was a big part of our lives, that team. So, soccer is what led me to the McLean Center. A dozen years ago, our girls were playing Winter Soccer at an indoor rink down in Dalton, and in case you don't know about this sort of activity, indoor soccer is roller derby without skates, painted on green, painted concrete. It's a sport only an orthopedic surgeon could love. So, one day in Dalton between games, Bill and John asked me if I wanted to join the McLean Center. A couple of days later, I spoke with Mark, who agreed that it'd be worth a try. So, that's how I started attending Case Conference, and that's how I came to know Bill in a different community. At Case Conference, Bill, as always, was a teacher. He knew when to remind the fellows that a case was exemplary in the sense that it represented a type of case, a type of ethical dilemma. They were likely to come across often in their careers, so he would make the case a teaching moment. He also taught them that clinical ethics is about much more than applying the rules of some reigning ethical model. Slow code, so all of you remember this, was part of that kind of teaching. Somebody's grandfather is dying. The family is having a hard time accepting that fact. They want their grandfather to be a full code. A full code would be futile and cruel. What do you do? In Case Conference, someone would say, well, that we should let the family know that if grandpa arrests, a code will not be done. Bill would look at that person with a kind of, really? That's what you think, look? And he'd say, this family is having a very bad day. Why do you want to make it worse? He'd urge the family to be told something like, we'll do everything we can, everything that seems appropriate, and then if the patient arrests, you do a slow code. Initially, I was aghast at this. Eventually, I came around. I came to see that for Bill the issue was, what does the family need? That was the lesson he was trying to get across. So that brings me to the building in Case Conference. Teaching the fellows to think more broadly about the family is not just giving them another item on their clinical checklist. It's part of deepening them as human beings who happen to be medical clinicians. At Case Conference, Bill often described himself as a consequentialist. Now, I'm a philosophy professor. I regularly teach undergraduates about the philosophical doctrine of consequentialism. In fact, Bill was very much not a consequentialist. Consequentialism is the moral theory that says that the right thing to do is to produce the best consequences overall and generally. It's true that Bill cared about consequences. That part is accurate. But he couldn't have cared less about the overall and generally. What he cared about was the thoroughly local in particular. He cared about this family in front. The education he was trying to give the fellows, and since I was at the table, he was trying to teach me as well, was to become the kind of person who cared about this family here and now. When Bill got sick four years ago, he couldn't come to Case Conference. But then he started to get better. Gradually, he got stronger and started attending again. We would sit next to one another at the back end of the room. Like middle school students, we'd whisper to one another. Initially, his voice was weak and he didn't say much, but as he got stronger, Bill Speak came back.  I knew, however, that he was back when in response to what he took to be an especially unfamily-centered assertion, he blurted out sort of full voice, then ricocheting around the room, bite me. Bill said, bite me more than once in Case Conference. These two were teaching moments. Each fall, I watched as the new fellows' heads snapped to attention. There we'd be, 20 or so people, seated at a conference room, talking about someone else's tragedy. Bite me meant stop it. It meant grow up, become a grown-up, stops talking abstractly about medical ethics, focus on what this family here and now needs. Now, Bill would probably say something profane, you know, guess what he'd say. If he heard me put things in the following way, but he taught me that what's important in moral philosophy is what's important in just about every human interaction.  I mensch mitmenschen. Clinical ethics is applied moral philosophy.  That's what Bill was always trying to teach people, whether in soccer or in Case Conference, how to be a human being among human beings. Over the last few years after Case Conference, Bill and I would walk back across campus.  What's one of the high points of my week? Wednesday nights, my wife would ask how's Bill and I'd recount what we talked about. I miss him. Thank you.


A high-profile MP is being accused of bullying and destroying the reputation of a woman who's claimed political interference in the public service, Erin Lee. And I really wish I was stronger, but I can't. I'm sorry. Erin Lee took on the government. Now she claims she's paying the price. Erin Lee had repeated competence issues. Erin Lee is just a bully and to pick on somebody that's just an ordinary citizen I think is really below the belt. The Cabinet Minister attacked her after she went public with claims his colleague David Parker pushed officials into hiring Labour Party activist Claire Curran to look after his personal political agenda. I didn't enter that sort of boxing ring. I'm not a politician so it's a really, really hard blow. I've never tripped up. I've never stuffed up. I've done the job.


Hey there friends! How's it going? My name is Kevin and today we're playing My Sims, which looks like an abomination to be honest. I'm sure it'll be good. It's just it looks kind of scary. I'm telling you, they are way too happy. They're definitely not My Sims anyway, despite the title. Oh, I need a non-shock.  I'll be back when I get my non-shock. Uhhh... Fine. Let's use a non-shock. I've never played any of these My Sims games. I don't think I've ever played a Sims game on the Wii, so I'm very interested in how this is going to be. My town is simply going to be called Feck. They couldn't think of a name and they just got frustrated and were like, Feck. And then the guy was like, okay, Feck it is. It's a pretty unique town to tell you the truth. I'd expect it with a name like that. This is how Feck used to be. Full of life, personality and happy Sims. And then I came along. Oh yes, I had to make a sim. Fantastic. This is going to be clunky as a heck, isn't it? Oh, sorry for using the heck word. I don't normally pull that out, but I'm just, you know, really... Why did I not read how to do this? Why can I never just stop and read? Why are my eyes packed, man? I already looked deranged and I'm not even trying. Oh my God. Hey, I'm an escaped prison convict. Okay, he looks so feckin' scary, like some sort of predator for sure. Okay, that's somehow even better. I love this. He's up to something. I was about to say his eyes are too far apart, but I think that's the least of his worries, to be honest. Select a voice and then the picture. Okay, juxtap! That's fitting, I think. What's his name? Sim. I think that's actually very fitting. Just Sim. No one bothered to name him because nobody cared. Welcome to Feck. Let's...  Welcome to Feck. Let's go to Town Hall and let Mayor Rosalyn know that you're here. Oh my God, this is so scary. I'm not even walking. This is just him. Why is he walking like that? Don't waver me. You don't know the danger you're in. Oh, I can control him now. Fantastic. He's walking into furniture here while they're teaching me how to play.  That must be such a scary sight from her point of view. Well, if it isn't Sim. Oh yeah, that was me talking. I forgot he sounded like that. Oh my God, he's looking for his first victim. You can see it in his eyes. They're giving me a free house apparently, which I'm really grateful for. I hope it's nice.  You're going to be my first victim. I'm going to clean up this town one way or another.  What's wrong with you? Did you actually realize what I looked like? Don't act embarrassed. She's like, oh no, there's no house here. There was totally a house here earlier.  Well, you're a good builder, so let's try building it now. She's just making me build houses for them. This is your workspace for your house. You can stack blocks here up to two stories high. I'm good to go. I don't need to know more. What is the purpose of this? I don't need to know more.  And we're going to need lots of chimneys through the crematorium. Look at the roof.  Fantastic. I thought the house was going up in flames. I was like, probably for the best. Oh, it's not cute, geeky, studious, spooky, tasty, but it's a whole lot of fun. Well actually, no, it's one-fifth of a whole lot of fun. All right, I think we're done. Oh wait, no, no, no, we're not actually. All right, done. Let's see what the mayor has to say about that. She's going to love it. Wow, that's quite a house.  Like, it's just starting to drooling, just slumping down. Look at him. He does have a pecking clue. Like, even the way he runs, you can tell there's something off. Sure, like I'm 100% positive they've got the wrong guy here. They've already some epic builder, but I don't think he's even human. She told me to get some apples, so I'm just going to town, just knocking down the trees instead of shaking them.  Oh, this one's dying. We can chop that down. I'll just take a bit less pleasure in it. Wait, stomp. What does that do? Why can't I stomp on it? Wait, no, don't plant a tree. That's the opposite of what I want to do. What is he doing now? He's using a metal detector, even though you can clearly see that they're there. Alright, we're just going to move on from that and chop down some more trees. If I were a betting man, I'd say there was more chance of this guy chopping down people. Look at that house. Can I go inside? Yeah, enter. I want to see what it looks like on the inside. Oh, wow. I did a great job on the inside without even realizing. God, I'm talented. I'm walking into walls again, just another quirky part of his personality. He sounds like a child and everything with that voice.  Where do I find more apples? Maybe I shouldn't have knocked down all the apple trees. Why is there music coming from your house? Keep it down. This is the neighborhood watch. As in, I watch the neighborhood at night through your windows. Hi, maybe lock your door next time. Yo, Sim, I love apples. Okay, is this that all we're going to talk about? Could I have some of these apples? Did he just wash the sink after I used it? That's a bit rude. He doesn't have a bed. I wanted to go for a nap. Sleep until night. I don't even know this person. I just go to sleep in their living room. Your couch is horrible for sleeping on. Oh, wait, this is also getting me red essence. This is way easier than collecting apples. Give me red. Give me red. Get out of my way. I need red. Jesus Christ, you just know by the look on his face, he's about to murder him. Bob and Norpin. Yeah, Bob and Norpin. Hand over your reds. There we go. I got enough. All right, great. I'm out of here. See you later, Patrick. I'll be back when I need to sleep or bully someone. I can fish. Oh, fantastic. Do I have them? Oh, yes. I got, I got fish. I got, I got fish. Okay, I'm done fishing. I don't like it. Jesus, I'm not going in there. They might put up a fight for to pick on defenseless people, you know? I really think she gave me this task just to keep me out of the way because I can't find any apples.  Look at them all. There's sunglasses on and everything. Can I sleep until morning? I should have slept in that other guy's house. I know, but there's just a spooky vibe around town. I know I'm probably the one giving it off, but it's just too much to handle. Okay, now I'm all wet again. That's his version of fishing, I suppose. I slept until morning, but it didn't work. I think he might not be tired enough. Ah, a bit of a nightmare. Did you dream about yourself? There's a reason you have no mirrors in this house. All right, we have to find apples. It's really important for the mayor. Oh, no, and the menu, he looks even worse. I have five. How do I get more? I might try just planting some and see who let work. Okay, that's what he does with stomp. He just stomps it back into the ground. He has some serious power, even more reason to fear him. Ah, yes, I'm doing my part for Team Trees. I'm the one who chopped them all down in the first place. Some of them were old and dying. I was doing a good deed. I just disguised it to seem cool in front of you. Yeah, I bet that worked.  Okay, apple. Fantastic. I think the trees are nervous. They're shaking around me. I guess they know what I did to the previous trees. Oh, I'm fertilizing the tree. Apparently it gives a burst of essence, but it also damages the tree. But yet here we are fertilizing anyway. Okay, Mayor, I got ten apples. Your important task is complete. She's like, yeah, we need lemons. Go get some lemons. Just got to keep them busy, keep them busy. We'll use them later to build something for the town hall. What, a pyramid of apples? We built you a workshop. You wouldn't leave me build it myself, would you? After you saw what I'm capable of.  Why do you keep getting shocked? You clearly are filled in with the situation. Looks like we got to be in the workshop, too.  All right, this one I need privacy with, so just an old shack will do quite well, I think. No windows, please. We're done. How exciting. It's just a box. I built a chair. Is that what you wanted? Are you happy now?  Good thing he breeds down everyone's necks. I feel like I should keep an eye on my door when I sleep, so I'm just going to place that in front of the door. I want to make something myself. I want to just freedom to make something out of that anger essence that I gathered. What were you doing back there? Why are you in my house? All right, into the workshop. Let's see what I can make out of all this anger. I hope it's a boxing bag shaped like Patrick. I'm going to make it out of anger. I don't understand what. Are you just going to make it out of pure anger?   I love this. It's like some cursed totem pole.  Okay, yeah, just put some of that onto it for no reason whatsoever. It's just floating there. And one of these fish that I found earlier. God damn, I'm good at modern art. You like this, Patrick? It's made out of your anger. You must cover all of the stairs with blocks. Oh, just this one. Okay, that's fine. I can do that. There we go. I put a little fish on it. Okay, we're done now. Can I put this in Patrick's house? Just to taunt him? Or can I only go in my own house? Oh, he's my next door neighbor, the poor fecker. Loser. All right, can I put this out? No, don't throw it away. Oh, I can't place it. All right, Patrick, I'll see you around sooner rather than later. Sleep with one eye open, punk. You never should have crossed me. He's like, I didn't do anything. I love it when the bathroom doesn't even lead to my door. It's just completely off-center. But that is the least of my house's worries, to be fair. Wait, I can paint with my essences? How do I paint out of anger? My whole house is made out of anger. It just makes flames. Fantastic. God, that is the tackiest thing I've ever seen in my life, but we're done. It looks like a 12-year-old's house. Now the anger sculpture can go over here. Oh, my God, it's on fire and everything. He's definitely trying to summon something. He's probably trying to summon Jim. Which room is the bathroom? Gonna be probably there. I'll put a window into the living room. It's 2% tasty and 3% fun, apparently. I don't know what's tasty about this. I don't know, maybe the walls remind you of barbecue sauce. Give me apples, I need them. I've got a plan. See, I gave the village a blessing with all these apples. I'm so good to them. They're actually blessed to have me. Well, blessed and cursed. It's a blurst. My blurst sim.  I wanted to surround his house with apple trees. Gonna drown myself in the lake now that I can't plant apples around his house. Imagine he looked out and you could just see this head looking up at him. Oh, my God.  Look at him. He's just stalking the village. What did the mayor want me to do next? I actually cannot remember. No tasks. Well, guess I'll start trespassing. Let's find someone's house. Hello. Oh, the mayor wants to see me, apparently. I think that was just an excuse to get me out of your house. But still, I'll go see her. I like how it gives me the tip when I'm already in the town hall. Like, oh, don't forget to go see the mayor. I could use that tip earlier. I really need a podium so I can practice my important speeches. Okay, again, I have no idea why I'm doing this for you, but I'll do it. I don't understand. A podium? What do I have? I have wood to make a podium and I have apples. Will that help at all? Yes. Yes, it will for some reason. Some apples on the side to give you a bit of protection against would-be assassins. By adding apples to the podium, you've made it tastier. Sims that like tasty things will like the object more. I don't think anyone's gonna like this object more. Oh, I hope I can make stuff and just give them to people and then they'll put them in their houses. I'm gonna make so many Patrick angry sculptures. There's your apple podium. I hope you like it. Oh, Sim, it's perfect. She's just too afraid to say anything else. Look at that. That is just awful. I'm so happy. Why don't you check on Poppy? I tried to visit her and she just kicked me out.  She's too scared. She's probably filed her a straining order in the meantime. I'm gonna make Patrick a bed because he doesn't have one and I feel bad for my enemy. He must cover all the stairs, but I've covered them all with apples. I don't have anything else. Okay. He won't take it made out of wood either. He's too picky. I can't do this. Okay. There we go. I made him a table.  Come here, Patrick. I got a gift for you. Why you would answer the door? I have no idea. All right. Give stuff. Oh, I even get to place it. Oh my God. This game is amazing. Here's your new dining table, Patrick. I'm gonna put it here in the way of that sink. He seems to like using that sink a lot. He doesn't know how to make of it. I mean, I don't really know either. Oh God, you can still use the sink, but now I'm trapped inside it.  Oh wait. Hold on. I can get out of this site. All right. You can still make it into your sink, Patrick.  Make him angry. I need some anger points to stay perfect.  Thank you. All right. I think we've made him angry enough for one day. I think we're gonna end it there. But if you want to see more of this, please let me know because this game I imagine has some serious potential to feck around with. Like the village is literally called feck. Come on. That's gotta be a sign. That's such a coincidence. But I do hope you enjoyed. I appreciate you watching as all those folks. I do hope to see you next time. But bye for now.


What I wanted to do in this second part of these lectures is to add some more tricks and techniques to your bag of tricks. So we are going to do like a replica analysis of this of this problem. The goal is again to find the minimum, the minimum loss, but using a different, different set of techniques which eventually will lead to the same result but in a larger, you know, in a larger space on the average, a loss and I will also mention a couple of problems that we have with this with this derivation. So the idea to compute the minimum is to set up a stock Mac problem with a fictitious temperature. So we imagine that we have a parameter beta larger than zero which is the inverse temperature or problem. We define a function Z, let's say, Gibbs Boltzmann partition function where vector X live on on the sphere of radius and, and we integrate over the Boltzmann weight exponential of minus minus beta H of X, where H is our loss loss function. We know, and we are going to redefine mathematical mean as the minimal value of H of X subject to the constraint that X leaves on the sphere. And this is this minimum we assume that is achieved at some value X mean of the vector of solution. What is the reason why we introduce this, this object, well it is because in the limit for for large, large beta. We can apply the Laplace method Laplace approximation method to this to this integral. And we know that in the limit beta to infinity, this integral will be dominated by. And will behave as exponential of minus beta evaluated at E mean, exactly, which gives us a prescription to evaluate E mean. So the minimal loss is given by minus the limit beta to infinity of one over beta log of this partition function Z. So this is the recipe for computing the minimum of a function of several, several variables. And of course, in presence of randomness. This is this Z is a random is a random function so we need to compute a suitable statistics of this of this Z, or in particular, of log log Z. This is a random random object. Okay. So, this is random variable. So what we are going to study two objects. So the average of E mean, which will bring us with which will connect us to the previous result that we obtained using RMP techniques, but this this problem is also very very rich, because we are able to estimate also the full distribution of E mean, not just its average value. So the full distribution, at least in in a large variation setting so for large, for large and we are proceed by different, you know, gradual steps. So number one, this is what we're going to discuss now to compute the average value of E mean, of course, we need to compute the average value of log Z, where Z is this integral. Already the, the source of all our problems because we are unable to compute the average of log Z directly. And so, to compute the average of log Z, we use a heuristic method that was developed over the years, which is called the replica trick. This method is non rigorous. It is based on some on some heuristic. I mean it is based on an exact result, but there is a catch. So the idea is that the average of log Z can be written as limit and to zero of one over log of the average of Z to the end. Okay, the, this is an exact, an exact identity, how you prove it, well, you expand this for small n, this gives one plus n log Z. And then you take the average of this, which becomes one plus n average of log Z. And then you have a log of one plus something small, and to first order in little and this is just the coefficient of, of N. Okay. Well, most of you will not need this type of derivation, but it was just for completeness. Okay. So this is an exact identity, it is called the replica identity, because the way we are going to use it is by assuming first that little n is an integer. So when, when you assume that little n is an integer what you're computing essentially is the average of a product like a replicated product of your, of your integral here. And, and, you know, the power of an integral is just a larger integral. And that's, that's the reason why we can, we can compute this type of objects. Of course the mathematical subtleties arise because we will need to then compute the limit for little n going to zero so in the vicinity of zero whereas we will have a result for ends that are very far away from zero. And we will find only on the, on the integers. Okay. So our hope is that by closing the eyes, everything will will will work. And I'm sorry to report and this is a very difficult moment for me that this will probably not be the case in this, in this situation so we really need mathematicians, I said it. Okay. So there are a lot of mathematical subtleties concern concerning this, this trick, but, okay, let's let's proceed as if they did not exist. We have our object here. So our average of even becomes minus the limit beta to infinity of one over beta. We have the average of log Z, and we use the replica identity to write this as one over and log of the replicated partition function said to the end. That's, that's the formula that we are going to to use. Okay. So now we can proceed for a broader class of loss function, and then specialize it to our own loss function. So imagine that our broader class of loss function can be written in this in this form, vk of x squared, obviously restricted to x square equal to n. So the properties that we require of vk. So the vk is, we assume that our Gaussian distributed independent, and we have that they are centered. So the expectation of vk is zero, and then they have a peculiar covariance structure. So vk evaluated in x, a times be, sorry, the L evaluated in x, be this covariance is diagonal. The coefficient is given by the dot product of a xa and xb divided by n. Okay, so in, in, in our case the the function f. Now we're progress this problem, the function f is just linear. So it is Sigma square plus you, and the derivation is included in the in the handout it is in equation in equation 18. So if you, if you replace by vk of x, the, the, the linear, the linear system structure that we started from, you can compute the covariance and you will, you will observe you will prove that the covariance is in this form with a function f that is linear. Okay, but in general, we can, we can do this, this procedure for much larger much broader class of, of loss functions.  So what we have to compute now. So we have our object Z, which is the integral dx over x square equal to n exponential of minus beta over two. So minus beta, the loss function, but the loss function is written in this generalized form. Okay, so the first, the first thing to do to massage this object before we raise it to the power and is to use the so called Hubbard strata knowledge identity, which is a fancy name for a Gaussian integral. So you get du exponential minus one of you square minus I root be you why is equal to square root of two pi exponential of minus beta of y square. This, this identity is used every time you have a term raised to the power to in the exponent to lower the power to from two to one. So, you have this identity where you have something raised to the power to, and you, you rewrite this as something raised to the power one, but the price to pay is that you add one extra integration over the auxiliary variable you. Okay. So to do that, of course we need to use the Hubbard strata knowledge identity multiple times one for each terms for each component of of x here. So what we get here is that this will become an integral over a vector you this vector you will leave in on R to the M. E to the minus one of you transpose you. So this is the, the quadratic quadratic version of this object the multidimensional version of this, this object. And then we have the integral over DX exponential of minus I square root of beta. And then we have the summation k one to M of UK times vkx. So using this this trick we have linearized the dependence on vk in the in the exponent at the expenses of introducing a number of extra integrations over over you. Okay, so now we, we have to raise this Z to the power little and, and then we need to take the average over the disorder, which is in in the in the case. Okay. So, we need to take the average of Z to the end, where we assume that little and is an integer. Okay, so we need to replicate this integral little and times. So we get that this DU integral becomes replicated several times. So now we have the you with the next index a, where a goes from one to little and. And we have the constant to pie raised to the power and over to then we get exponential minus one of summation over a from one to little and.  So this the dot product. So it is essentially you a transpose if you want you a, and then we need to replicate also this V integral, right. So we are replicating this integral. And we get. Sorry, this one is not V is X. So we get product for a from one to little and of the X a. And then we have a product for K from one to M. So I'm bringing this down downstairs for for convenience. And this is half of what of the average of exponential minus I root beta summation a from one to end you a component K. vk of X a. The average over the distribution of of the V's in in doing this the step I of course exploited the fact that the V case are all independent. So I can move the average from outside to inside. Okay. Good. So now we have the replicated partition function. And we are the next thing to do is to perform the average over the V case. Which where of course we are going to use this, this condition, which I just about to erase where the V case are independent and Gaussian distributed with with this particular covariance structure. Okay. So, yeah, I have no choice. Sorry. Okay, so this by this notation, I mean the case, the case entry of the replicated vector UA. Okay, this is the notation that I'm using. So what we need to compute here is the average over the exponential of a Gaussian number, right, because the V case are Gaussian. And this is a sum of Gaussian Gaussian variables. Okay, so essentially what we what we need is to compute the average of exponential of Z, where Z is Gaussian with a particular covariance structure. So I'm going to use this result that the exponential of Z average over a normal distribution with me new and variance sigma square is exponential of new plus sigma square over two, which is a result that I put in a question 12 of the end out. So, so what is Z in our case, Z in our case is this exponent here. So it is minus I root beta summation over a of you a K. of X, a. So what is the, what is the average of Z. Well the average of said is minus I root beta summation over a UA component K average of the K of X, a by linearity. So this is of course zero, because we assume that our V case mean zero, and what is Z square.   So we get summation over a and be you a K, you be K. And then we have the average of V K X, a V K X, be, which is precisely the covariance structure that we assumed from the beginning, which is diagonal. And, and with with the pre factor that is this function of the, of the dots of the dot product. Okay. So, now we can continue from from from here, the result of this average is nothing but exponential. The value is zero exponential of sigma square over two. The, the variance of this. So we get the product K one to M of exponential minus beta over two summation. And then we have B, you a K, you be K. And then we have this function of X a dot X B over N. Okay, I'm using just the sigma square that I've computed there and there is this factor of two of one of that comes from from here. But now we can perform the integrals over you. Okay, so this term. So let me just, let me just erase here again. Good. So we have that. And then to the end average is the integral over the X a, and then we have an integral over you one. You, and each one with a factor of two pie to the M over two. Yes. So this is the X a. And then we have an integral over the U one over two pi to the M over two, the U and over two pi to the M over two.  Okay. How do we do that. I can just write this as minus one over two, the vector of the use. So the, the row vector of the of the use that multiplies a matrix. And all these is multiplied by a column vector of the use. So what is the, what is this matrix that connects the row vector and the column vector of the use. Well, in, in this term here, this is, this is just a quadratic, a pure quadratic term. So the matrix that is in between is just the identity. Right. We have just the identity matrix in little and dimension. And here we have a term that connects components of different type type A and type B of the vector of the vector you. Okay. So the object here is beta, because there is a minus beta over two of what of F x a dot x B over. So, do you agree with this with this rewriting. I'm just rewriting all the terms that depend on on you in this exponential form where you have a vector of you, vector of you, and in between, we have a matrix that must have this this form that connects the different entries of these two vectors it is just a just a rewriting of the integral that we that we had. So rewriting is very nice, because this integral here becomes a Gaussian multivariate Gaussian integral in little and dimensions. So the result of this multi dimensional integral which is also included in the, in the handout is the determinant of this matrix in this and raised to the, to the power of the size of the vectors. So the result of this integral is just the determinant of the identity matrix in n dimension, plus beta, our function of x a our covariance function of x a dot x B over and all raised to the power minus m over two. And of course, no in this case, the constants are fine because I included them in the, in the measures. Any questions about this. Yeah. What is the difference between the function and the loss function. And the loss function. That's as that's a great observation so it is a another quintessentially bad choice of notation because I called f of no in that case it's correct right so no no. So the loss function. So this this F, this F is the generic function that represents the covariance between the vk, vk values. Okay, in that in for our specific problem, this f of you is sigma square plus you, but we can we can do the derivation for a jet for a generic covariance, F. So we are going to specialize this F to this value only at the end of the calculation. Okay. Is this what what you were asking is. Okay, okay. Any other, any other question. This, this one. This, this one. Yeah, this one is, is, is a matrix of size little and by, by little and which is indexed by a and B. So this is a matrix and we sandwich it between the two vectors, the column vector and the row vector of size little and I don't, I don't know a way of writing it more, more clearly than that. Okay. Next, we do a change of variables, because you see everything, everything in here now depends on the specular combination the dot product of vectors of type a and vectors of type B so replicated different corresponding different replicas. Okay, X, X, A and X B. Okay, so it is natural to introduce a matrix Q of size little and by little and with entries Q AB, which are one of our n X a dot X B. Okay. So this is a matrix for which we can say something about the diagonal, easily. The diagonal is one over N X a modulus square, but we know that X module square must be equal to N, because this is a constraint problem on the sphere. So this matrix Q will have one one one one on the diagonal. Okay, that's that's the space over which we are. We are integrating. We use an identity that I reported on the, on the handout on equation 19. So that's an identity, and the identity is as follows if you have an integral over vectors X one X and of a function of the matrix X transpose times X. Where is the N times N matrix. So X is the matrix whose columns are the vectors X. Okay, that's, that's the left hand side of the identity. This is equal to a certain constant that is known explicitly depends on capital N and little and capital N is the size of these vectors little and is the number of the actors you are integrating over. This is equal to an integral over a certain domain of matrices Q. The number of matrices of Q times the determinant of Q raised to the power and minus and minus one over two in the key and the domain, the of Q is the matrix matrices that are non-negative, non-negative definite of the form such that, and of course, the integral runs over the X one square equal to N. Okay, so essentially this, this theorem establishes the Jacobian of the change of variables that we are that we are after and this Jacobian is just given by a determinant raised to some power that some exact power, and there is a pre factor that we can that is known exactly. Okay. This is an identity, it's a it's a theorem. So we can just apply it to our setting. So for getting constants, we have that Z to the end, the replicated partition functions, others is now equal. That's, that's precisely the setting that we have that we have here on on the left, where we have little and an integration in little of size capital N. So that's that's precisely the left hand side of the identity. So we can rewrite this object, modular constants that anyway we know as an integration over non-negative definite matrices with once on the diagonal of the determinant of Q to the power minus N plus one over two. Let me rewrite everything in this form, exponential minus N over two, phi N of Q. So all the terms that depends that have a capital N in front, I put, I put them here. So including the determinant of Q to the power N over two. So this function phi N of Q is given by what? Well, you will have a log that type of term. You will have minus log that of Q that comes from this term here. Plus alpha. Remember that alpha is capital M over N larger than one. And we assume, we assume that we are taking the we will take the limit and to infinity. In such a way that M over N is fixed plus alpha log that of identity plus beta. Let me call this f hat of Q. So the f hat of Q is essentially the function f evaluated on the matrix, on the matrix Q. Okay. Sorry. This is in the sense that the element Q, the element Q AB is the one that is in the position AB, and it is determined by this equality. Yeah, so so each element of f hat. Yeah. Okay, so with the choice, of course, if you sigma square plus you, or, you know, F, or F hat, equal to sigma square plus you, then we are back to our initial progress this problem but this, this expression is valid in general for for this type of functions that are expressed as sums of functions so you can pick a polynomial polynomial of higher order for example and the derivation would carry over in the same in the same way. The next step, of course, of this derivation will be to evaluate this integral, this matrix integral for large N using a subtle point or steepest descent type of argument. So we will be looking for an extremum of this function phi N over a specific set of a suitable set of matrices. Okay, so we will take the limit N to infinity of this capital N to infinity first.  And with all these operations in whatever order suits are suits as best for getting any type of mathematical rigor. In the end, we will, we will hopefully land to a nice result. Okay, so we have to take three limits capital N to infinity little n to zero beta to infinity. Okay. So well I think I can stop here. Yeah. Yeah, this result is not not exact in in any sense of the word exact. And under any choice of the word exact this result will not be exact. But, I mean, we can rename the word we can give a new meaning to the word exact and and that's, that's what we're going to explore. Yeah. Yeah, so.  But but then, you know, if you want to put it on a mathematically rigorous way that's that's a much harder, much harder problem you usually this gives a good heuristics on the on the solution that the then people who do like rigorous stuff would, you know, have as a starting because if you know what you need to prove at the beginning of, of, of your work is better than not knowing what you need to prove. So, Exactly. So after after you've done this, this calculation, then you need to start again with a completely different way and try to prove that this, this result is rigorous. So what, what is the source of non rigor, many things, we are interchanging limits freely. So we are taking limits and to infinity, you know, in in in different order than than was initially conceived, and also the fact that the replica is correct as as a calculus identity if you manage to extract the analytic continuation of your set to the end in the vicinity of little and close to zero. But what we are doing is to compute that to the end for little and integer and integer 123. So, unfortunately, the distance between one and zero is enormous. This type of problems. It's really enormous. I have a question about the very beginning of the computation. Yeah, I've seen that the partition function that you've written is basically a spherical integral. Yes. Okay, the integral is e to the minus beta, a function which is the most quadratic in X in the vector X. Is that right. At most loss function of the procrastos problem. It is quadratic in v of x. It is quadratic in vk of x. Yeah, if you if you take me to be a x plus in our case in our case. Yes. Then I guess that the sphere is such spherical integral with the of an exponential that has at most a quadratic stuff at the exponent can be computed, maybe not in a different way for sure. If you if you convert the spherical constraint into a Gaussian integral that you can do. It should give an analogous result to what you have without the need to use replicas. Does it work. So the I mean I'm not 100% convinced about this. This argument is because of the positivity constraint on the a transpose a matrix that appears in when you expand the quadratic form. So I think that that that will get in will get in the way of your of your argument, but we can we can discuss. Okay, I see, I see. So I'm not 100% convinced that you can avoid replicas and do an exact calculation on on this. Okay. Yeah, we, I guess, when you turn the spherical constraint into a Gaussian integral that you have minus square norm of X, and that should keep everything convergent, provided that the Lagrange parameter they say auxiliary Lagrange parameter that you introduced to enforce the constraint is bigger and smaller the second value depending on the smaller depending on the symmetry that you have. Okay, so you will have a minus. And that that might constitute like it might truncate the range of your integral. Yeah, yeah, for sure, which, which will create a sort of error function type of integral in the large dimension. All right, and that that I think is, is, is a source of the problem that you cannot. It will not lead to a convergent integral over over the full over the full. Yes. Okay, so you cannot exchange the integrals over the constraint with the with the integrals over K over X. That's that's my, that's my intuition. So relax now.


Solar Orbiter is a mission that we hope to launch in 2018 and it's a collaboration between NASA and ESA, which is the European Space Agency. In fact, Solar Orbiter will be going extremely close to the Sun and it will also be orbiting out of the ecliptic, which means that we'll be able to look at the polar regions of the Sun.


فشرف بل اشتغالي بالعلم ولا تبغي به ما عشت يا دا دا دا لا وياله من شراف عظيمي براثز و سيستز الله سبحانه وتعالى يبتلي عبادة الله تبارك وتعالى يتستي ساوانت بالسراء والضراء الله تستي with good الله تبارك وتعالى يتستي with hardship And Allah تبارك و تعالى ت تخقك بج проектك منها shop لخذ حياتك وراءها And Allah تبارك و تعالى يتRealize youه By reducing your wealth All of these are Tests Allah تعالى will also test you By giving you wealth Allah also tests you By giving you children Allah test you Even with a good Allah Test you Why does Allah do that الله تبارك وتعالى يتكلم في الكثير من القرآن why he does that to his servants قال تعالى الله says وبلوناهم بالحسنات والسيئات لعلهم يرجعون we have tested them with good we have tested them with hardship and suffering and pain لعلهم يرجعون so they can run back to us so they can turn back to us so they can repent and that's why Allah سبحانه وتعالى does that he said in another place in the Quran ولنوذيقنهم we will make them taste we will make them taste الله تبارك وتعالى he said من العذاب الأدنى the punishment the health fire the suffering the pain الله تبارك وتعالى he says we will make them suffer before the bigger punishment comes to them دون العذاب الأكبر before the great punishment comes to them why لعلهم يرجعون so they can turn back to Allah سبحانه وتعالى الله also says سبحانه وتعالى ولقد أهلكنا ما حولكم من القرى وصرفنا الآيات لعلهم يرجعون الله تبارك وتعالى he said we destroyed the villages that were around it الله تبارك وتعالى he said we diverted them from our vases لعلهم يرجعون so they can turn back to us so they can come to their senses again الله also says سبحانه وتعالى ظهر الفساد في البر والبحر بما كسبت أيدي الناس ليذيقهم بعض الذي عملوا لعلهم يرجعون the suffering, the pain the hardship has become apparent on the face of this earth it has become apparent on the earth and it also has become apparent on the ocean and the sea why بما كسبت أيدي الناس because of that which the people's hands have put forward الله says ليذيقهم so I can make them taste بعض الذي عملوا only some of that which they have done I am only doing this to them so they can taste some of the consequences of their actions not all of their actions but some of their actions لعلهم يرجعون so they can come back to their senses so they can come back to الله تبارك وتعالى oh brothers and sisters running back to Allah is of two types والفرار نوعان running back to Allah is of two types فرار السعادة فرار الأشقياء the running back of the righteous people the good doers and the next one is or the second type is the running of those who are the criminals and the ones who have put forward only evil actions الله تبارك وتعالى when it came to the commentary of the ayah and surah the fiftieth ayah ففروا إلى الله run back to Allah with repentance run back to Allah by asking him for forgiveness عبد الله بن عباس he said on that ayah فروا منه إليه run from him to him سبحانه وتعالى وعملوا بطاعته and do the righteous actions and that which he has commanded you سبحانه وتعالى the running back of the righteous people is that they run from Allah to Allah سبحانه وتعالى they also come with righteous actions in obedience to him سبحانه وتعالى it is the one that the messenger used to do in his application اللهم إني أعوذ برباك من سخطك وبمعافاتك من عقوبتك وأعوذ بكاملك لا أحصي ثاناء عليك أنت كما أثنيت على نفسك أولا I seek refuge in your anger from your pleasement أولا I seek refuge in your punishment from your forgiveness أولا I seek refuge in you لا أحصي ثاناء عليك I cannot give you praising enough أنت كما أثنيت على نفسك you are praised in that which you have mentioned of yourself that is enough for praise for you so this dua it teaches us that the messenger used to make which is what to run from Allah to Allah what does it mean to run from Allah to run from the things that Allah has prohibited you from سبحانه وتعالى brothers and sisters there is the other type of those who run who are they they are the criminals they run away from Allah but they don't run to Him سبحانه وتعالى they turn their back to Him سبحانه وتعالى they want to run away from death they want to run away from the punishment of the Hellfire Allah said about them قل إن الموت الذي تفرون منه فإنهم ملاقيكم ثم تردون إلى عالم الغيب والشهاد فينبئكم بما كنتم تعملون قلوا لهم محمد إن الموت الذي تفرون منه من which you guys are running away from it will come your way and you will taste it and you will go through it ثم بعد ذلك you will go back to the one who knows of everything he knows the unseen everything to him is knowledge and he will inform you of every single thing which you have done Allah also said about them the wrong doers and the criminals the ones who run away from Allah but they don't run to Him Allah said about them يقول الإنسان يومئذ أين المفر كلا لا وزر إلى ربك يومئذ للمستقر ينبأ الإنسان يومئذ بما قدم وآخر بل الإنسان على نفسه بصيرا ولو ألقى معاذيرا الله سرنا آية يقول الإنسان the person will say أين المفر where can I run to he wants to run away from Allah but he doesn't want to run to Him he wants to run away from Him that is the running away of the criminal the one who hasn't worked for meeting Allah all my brothers and sisters run back to Allah by coming with what بطوبة نصوحة by coming back with good repentance don't be from those who turn away from Allah الله تبارك وتعالى told us those who turn away from Him who do not repent who do not run back to Him Allah said to us ومن أعرض أين ذكرين those who turn away from my remembrance فإن له معيشة he is going to have a hard life he is not going to live a good life الله تبارك وتعالى he promised that and he mentions it in another آية الله سيث من عامل صالح من ذكر أو أنثا وهو مؤمن فلنحي أنه حياتا طيبة the male and the female believers الله تبارك وتعالى he said about them who come with the iman that was requested from them الله تبارك وتعالى he says I will give them a good life in this world which they will live in I ask Allah سبحانه وتعالى that He makes us from those who run from Him to Him سبحانه وتعالى and that Allah تبارك وتعالى also makes us those who do not turn away from Him and submit to Him as a slave should do so الله مغفلنا ذنوبنا وإسرافنا وثبت أقدامنا وانصرنا على القوم الكافرين سبحانك بحمدك أشهد أن لا إله إلا الله ستغفر كواتوب إلي


Another important thing is if the implant goes under the muscle or over the muscle and talking about determining your cleavage. So let's look at a couple of things. One is this is when you have your pectoralis muscle the implant goes underneath it. You can see the muscle the implant is right here underneath it and just always remember the implant is partially covered by the muscle and the lateral aspect is usually not it's called a dual plane implant. DUAL dual plane implant because part of it is under the muscle part of it is not. The other type is when it goes on top of the muscle. I would say 90% of the surgeons probably go dual plane under the muscle and there is a 10% where they like to go above the muscle or just under the fascia but most people do a dual plane. Super athletic people might want to have it on top of the muscle but there is a caveat to that if you're on top of the muscle there's a higher risk of capsule or contracture always remember that. The other thing is you can see this is the chest bone that's the sternal width so technically you can't get closer than this because you're not you don't want to divide these muscles too much or else the implants will converge together.


The pulling, sailing ceremony on of the immediate past Chief of the Navos staff at NNS Quora Parade Ground saw the former Chief of Navos staff describe the Niger Navy as professional and vibrant. He also described his experience as rewarding. He said the secret to his achievements during his regime was teamwork and professional conduct. As you have observed, for the past two and a half years that you have been monitoring, most importantly we are out of the parity prone countries in the world. A record which has not been achieved for the past 27 years. Illegal oil bunkering through the instrumentality of the question that could have been held have been brought to its lowest level. The Niger Navy has also liberated very highly on technology for its operations which has made it more result-oriented and cost-effective. In the sense that we don't need to deploy most of our capital shifts out there at sea in the world to keep pushing footprints for maritime offenders. From the comfort of all the operational commands, the Niger Navy can vector any maritime offending ship or platform or human beings, be them pirates, sea robbers and all that to be able to undertake arrests and prosecute them to the instrumentality of the former act that suppression of piracy of other maritime forces act of 2019 which was promulgated as expressed by the imprisonment of 12 pirates in 2021, July 2020 precisely.


What's going on guys, today I am back with another Destiny 2 video and today I have what in my opinion are the top 5 weapons for PvP, but before we get into the video guys stay tuned until the end if you want to win some silver. Also guys if you do enjoy the video leaving a like seriously does help me out and if you want to be a part of the most active Destiny channel on YouTube be sure to subscribe if you haven't already. Ok so in my opinion PvP isn't an easy thing to play, there are so many good players out there who run rampant in PvP and it makes it hard for noobs like us to enjoy it, it seriously does, but in today's video I bring you 5 weapons which in my opinion will definitely up your PvP game, even if you ain't a noob though some of these weapons surely you will enjoy using and cause havoc with. Now a couple on this list you will have heard of but some you probably haven't so let's go ok so in at number 5 we have the Meda Multi Tool which you've definitely heard of, this exotic scout which you turn from Destiny 1 has been sending a lot of PvP players crazy and actually how good this thing is, a lot of the maps suit scout rifles due to the long line of sight points to many other maps in the game, a weapon like the Meda Multi Tool is more or less unbeatable at range unless you come up against someone else using it. Although many people are complaining it's OP I actually don't think it is, there are many many characters for the Meda, it's just in my opinion being overused, when you get team shot with this thing it's over, it's as simple as that and team shots and Destiny 2's PvP go together like bread and butter. Now this weapon is easy for everyone to get, it's rewarded for completing the Devrim K weapon quest under EDZ so go do that if you haven't already. Ok so in at number 4 we have the Last Hope Sidearm, now this thing guys it is unstoppable at close range, nothing can outplay you even if you have the slightest bit of decent accuracy, this thing is truly incredible it's a 3 round burst sidearm and damn it does do that damage, range obviously as it should be ain't that great but if you're quick about the map and know all the short routes you can be seriously affected with this. I would suggest you using this though as intended as a secondary, this paired with the likes of Meda would be unstoppable. It's obtained from engrams I've had two I had one drop from failsafe and the other for leveling up my clan and getting an engram from Hawthorn but yeah guys this is a serious contender try it out if you have it. At number three we have the Cuboid, man oh man where to start this rare AR is easily one of the best ARs in the game for PvP there was no doubt about it the range this thing offers is on another level it's also crazy at close range it's actually people the first weapon I've ever gotten a run as a medal of medal with. I actually have that video on my channel I recorded live or playing PvP survival it's actually a great great watch that video will be found linked within a video description if you do want to check it out but trust me guys do not overlook this because it's a rare it's an amazing weapon in PvP if you get this to drop and it drops quite often please try it out in PvP I guarantee you won't be disappointed. In at number two we have the Uyule's gift this AR being an energy weapon it fires like a laser beam hardly any recoil on it at all and it hits like a bullet that great range to and can easily outplay most of the weapons in the game personally I didn't like it at first as a weapon in that number one spot took my heart but this is by far the second best AR in the game in my opinion one which drops quite often from legendary engrams too and to be honest you've probably already obtained this before but trust me guys this being an energy weapon it makes it a lot easier on you and your loadout setup as most ARs in the game are kinetic you don't get many options to pair amazing ARs with amazing kinetics with this you do the options of kinetics you can pair this with are crazy so get experimenting guys it would definitely up your game if you do this thing trust me people will not let you down in at that number one spot and in my opinion the best PvP weapon in the game is the origin story AR now I've already made a video on this stating how in my opinion it is the best AR in the game and that will be found also linked within the video description and I still think it is even though I've used many many other weapons since first playing with this this thing just feels right using something about the weight land sharks in PvP just feels great it's powerful he has great range more or less unstoppable at close to mid range and AR definitely worth you using in PvP now this is obtainable from Savala after you initially complete the story and then go on to complete two strikes after that he will have a choice of three weapons for you please pick this one if you've already passed that stage of the game and didn't choose this you can still get it from Savala's Vanguard engram so get grinding people it's definitely worth it and guys those are my best five PvP weapons in the game obviously though this is my opinion if you use something which you think would contend for a place on this list be sure to let me know within that comment section I absolutely love experimenting with new things within PvP and also guys if you want to win some silver simply drop a like in a video and leave a comment down below winners are picked from my discord server so go join it via the link within the video description if you haven't already but guys I hope you enjoyed the video thanks as always for stopping by and hopefully I will see you on that next one


No That'd be floaty juicy content Thank you, Rosos, and pay attention and follow us for your Twitter account If your nose is running and your feet are smelling, you're upside down And today, we are... I'm ready now Today, we're ready to do a trailer What's in that cup? Yerim It is Called Gulal Right? That's pretty easy, right? Gulal Gulal Sure And it's from the director of... Your mom? Yeah No, gang's waspher, it's the... Ah, yeah Kashiya He's part of Queen and Udan He's a very highly... Like, I feel like a lot of people that like love cinema He's like... When we say we love Vishal Bhardwaj They say he is even better And obviously we've seen very little of his work But, here we go The one who has the most power and power The one who has the most power and power Yes We're not just doing it I even make a sound And if someone is doing a show, I give him a tip Okay, stick To her What else does she have? I have a motorcycle Motorcycle? Yeah Car, bungalow, motor car Money What else does she have? Huh? To me To the elections You like that song? Yeah To the hospital party If you want to fight, you have to fight at a higher level What's the difference between college elections? Youth Youth changes the society But what do you want to change? What do you want to do? Ritvi, you're the one who has the most power You know everyone Yeah, we missed The red face because I don't understand What that is Me too But I'm sure it's something that India would know Or the line that it referenced and said this is a You know, referencing a famous line In an Indian film You know, it felt very Vishal Vardwaj It did, didn't it? I think they're very similar directors They're very gritty directors Exactly They're all about Also letting They're writing But they're acting as well They're acting Kind of lead the story Which, you know, we love That style of Cinema It looked really good because it looks old It looks older than it is Old trailer It looked really riveting It did And I'm betting it's only like early 2000s I'll check I think it said 2009 Which would make it 10 years Yeah And I also like that I'm not recognizing a lot of the faces that were in there Which was always exciting to watch People I haven't seen before Yeah In their roles Hold on, I want to Look it up real quick It's a law student witnesses Deception, homicide and crime After being elected as general secretary of his college Okay Weird It's a weird synopsis Right Got a two and a half hour run time Pretty standard Hey It's that guy K.K. Menon Wow Do you recognize I do It's the uncle It's the uncle From that dance that we did He's a freaking shape shifter That's what they say He is a freaking shape shifter When we saw that dance Which was fun And it was cut up Sorry Had to do it But You totally didn't recognize him It doesn't look And I didn't recognize him in here either Not one moment That guy is a shape shifter Yeah Wow Okay now My interest level in this has just gone up He's also in Black Friday I know we We have that We have that Yeah we do There it is Is he on the cover of it? If that's him I guarantee that's not him No but this is My interest level has just Shot through the roof I think that's him That's him That looks like him right Yeah but not really Yeah Wow He can make his face change A little bit Wow And I believe this is the exact same director too I think Yeah But yeah Holy cow Wow So please let me know Where this falls One in terms of Say the director's name for me Yeah as far as On your eye Cush up Obviously gangs We're two We know But Yeah In terms of his work Where does this lie Right And also in terms of Because apparently We need to We really need to get into it Severely overlooked to this gentleman Yeah I mean because we loved him in Header Yeah but he wasn't In my top five of that film No And we really Maybe it's one of those Ranbeer things We're like We didn't realize how good It was Very well be If you don't know what a Ranbeer thing is It's when Gully Boy We said Gully Boy He was fine If you were new That That young actor Pretty good But it was only a bot Who stole the show Well it was And it was But we had no clue about We had no idea How big of a transformation That was for him So And I'm wondering I'm betting he's like Similar to Can't believe I'm forgetting His name now The guy from Billy Billy the bad guy VJ Who was also in Gully Boy Yeah He might be That type of actor Character actor Who always comes in Yes But he might be bigger than VJ No I can't remember his name VJ What Yeah I'm losing my mind But I'm guessing He's similar to that But the trailer Looked really really good to me Really good It reminded me of a Vishal film And it got Because on Twitter If you don't follow us on Twitter Go follow us on Twitter Because I asked people Because I specifically asked the other day Okay Be closing Oh you're reacting to this And you know You're missing all these trailers So I'm like Okay what trailers are we missing Right this was one of the ones At the top That makes sense So thank you for the recommendation VJ Ross Thank you I was going to chew my own head off I'd like to see that But you know It looked really good Really interesting Yeah I'd like to get into this director That actor I really want to get into this actor This is enough This is the third time We're introduced to him in something And didn't know it was him I just And that is That says a lot about that guy When an actor is so good That you don't recognize them In trailers repeatedly I mean but come on We've done that with a lot of people Yeah but this in particular Especially a film like Header that we love so much And I can I didn't love Header I can see his face clearly In that film So can I And we just don't recognize him When he appears elsewhere It doesn't look like him I know And how big of a star is Yeah I know he's not like an S.R.K. or anything like that No But when he Like what is his Or is he just like a side character actor Like Paul Bay Right That's who I He's one of my favorite Character actors out there Most people don't know his name No but actors know him Here in LA Everyone's seen a film of his Correct He's one of those types of actors So is he that echelon? Same thing with John C. Riley Everybody knows that The caliber of that guy is extraordinary But he's not an A-list actor He's not a household name People would recognize him But they'd see him and say That's the guy from They wouldn't know his name He'll get a knife Exactly But they wouldn't know his name Not Gangster New York Oh man Or He's done so You know what else is Really good at his? But you need to know Their story some Is this the Laurel and Hardy Filming did Where he played Oliver Hardy Did you see it? I saw it when I Took my trip to Calcutta It was on-flight I watched it Was it good? I remember we saw it And we were like Oh you want to see it But then it didn't get Nominated for anything It was kind of lost It's slow It's great Yeah I remember Really good And John C. Riley And the actor whose name I'm forgetting who plays Stan Laurel Are Extremely, extremely good The makeup is great They transformed him Into Oliver Hardy So anyway It's that caliber of actor Like all of the actors In the industry Know that person by name And respect their work But a lot of other people don't Is that Who this man is Because we have no clue The synopsis they give On this little thing Because it's not the official Right Trailer The one with the Ingo Shubs Because the original didn't have Ingo Shubs Okay They said Gula A movie of sheer brilliance Brilliant direction Hard hitting script And fantastic cast Performance Awesome That's all we need If it has all of those things It should be a really good film