welcome back everyone thank you for joining in again we started to look at the missionary journey of Apostle Paul so I have a colorful picture and that's what I wanted to show you earlier so let me show that picture once and then we will get back into our study so here it is this is a little more colorful so hopefully that is it'll be interesting for you to see okay it's too tiny you know are you able to see or it's too small just to make a note of the same cities that I mentioned earlier you know the and the path to it because all that is happening is in the same region the Mediterranean region and a little bit of Europe so this is all like your Jerusalem Judea then your Syria then you go on to this portion here which is Asia Minor or what we are calling like the Mediterranean region and then you know you will see the journey move on to Macedonia okay so we will read about this place called Macedonia and then from further out will be Europe so I just want you to have a picture or an idea of where these things are taking place let me share one more map that brings more clarity to all the areas this one is a good one yeah okay so get like a mental photograph of this this is what you need for the Book of Acts all the areas that we will cover there'll be three missionary journeys that Paul will undertake you know moving around these places so that will carry on as we read through the Book of Acts but if we have a picture of this map then we will understand the Book of Acts so I'm repeating myself Jerusalem Judea remember Jerusalem Judea Samaria that region here going to Syria Syria a Tarsus Silesia is right here and I was mentioning about Asia Minor so this whole entire thing is Asia Minor where is Tarsus in present day times it's Turkey okay Turkey now this is this would be your Asia Asia Minor Asia region and then we will see slowly like after the first missionary journey Paul will then go to Macedonia for the second missionary journey so where is Macedonia from Asia he'll move on to the next region here this would be Macedonia and Macedonia there will be all these important places Philippi Thessalonica we will then talk about another region this is the Akea region where you have a very key city known as Corinth okay so Corinth we'll talk about that place so these regions will be covered in the different missionary journeys in Asia there is this city called Ephesus which is very very key in the region of Galatia this city called Colosse this is important so just have an idea of where these places are we'll talk about the individual journeys later so towards the end of Paul's life you know after he does his ministry here he will need to go to Italy to Rome so remember we said Europe parts of the Mediterranean and then Europe so he'll cover the Europe region and this map will give you a better understanding of how we will proceed later so if you have any questions you can actually ask I think maybe our brothers from Africa have a much better idea of where these regions are so let me stop sharing the map for now okay all right so let's get back into Acts 13 we now know about this island of Paphos and what exactly happens there and I told us about the spiritual opposition right so think about this like even Paul when he went to preach Satan was opposing him in various ways so right now it's a sorcerer who's engaged in evil practices who is the source behind the hindrance in the life of this man called Sergius Paulus he's a procuncel who's a procuncel procuncel was a dignitary or he was somebody of a high position or high rank in Paphos at that time so he's an intelligent man and he was open to the gospel but what was the problem spiritual hindrance remember in the book of Corinthians that's why Paul wrote because Paul understood all these things he wrote about in I think it's first Corinthians 4-4 where he says you know the Satan or the demonic powers they caused blindness spiritual blindness and that's exactly what they experienced in Paphos but once this procuncel and his work was dealt with it was easier to proclaim the gospel okay alright let me just correct myself I'm always mixing up first and second second Corinthians yeah correct second Corinthians 4-4 where it says the God of this world has blinded the minds of them which believe not okay so how does blindness come spiritual blindness through demonic forces and that's what we looked at so let me just pause I know I'm saying so many things but where are you at are there any thoughts so far or any comments please feel free to share and then I'll continue yeah so I just want to know about the we saw a brief introduction of Paul and his journey and all the things so is it available in that revival visitation like the complete thing is available over there or is there any other book we can really study so yeah so in the revival's visitations there's a brief introduction Jeffina there's a brief introduction to the life of Paul any other book I think we'll have to form a picture based on all our readings so you can refer to enduring word from David Kozik he shares some insights about the life of Paul and then of course you know revival's visitations and additional reading that will shed light for you or you may also study the book of Galatians the book of Corinthians to give that additional clarity about the life of Paul so we'll have to study that's when we'll gain a better picture is that okay okay great thanks Jeffina yeah anything else class alright so as Paul moves on his first missionary journey I think it's time for us also to move on so let's move forward we've seen now that in pathos the procuncel believe moving on to verse 13 let's let's start there and it's quite a long passage here hey maybe two of us can read so somebody can read from verse 13 to maybe 20 23 maybe and then from 24 till the end of the passage another person can read verse 41 another person so 13 to 23 somebody 24 to 41 another person chapter 13 was 13 to 23 now when Paul and his party said say it from pathos they came to Perga in Pamphylia and John departing from them written to Jerusalem but when they departed from Perga they came to Antioch in Pesadia and went into the synagogue on the Sabbath day and Saturday and after the reading of the law and the prophets the rulers of the synagogue sent to them saying man and brethren if you have any word of exhortation for the people say on and Paul stood up and motioning with his hands said men of Israel and you fear God listen the God of this people Israel chose our fathers and exalted the people when they dwelt as strangers in the land of Egypt and with an uplifted arm he brought them out of it now for a time of about 40 years he put up their ways in the wilderness and when he had destroyed seven nations in the land of Canaan he distributed their land to them by allotment after that he gave them judges for about 450 years until Samuel the prophet and after a word they asked for a king so God gave them Saul the son of Kish a man of the tribe of Benjamin for 40 years and when he had removed him he raised up from them for them David asking to whom also he gave testimony and said I found David the son of Jesse a man after my own heart who will do all my will and from this man said according to the promise God raised up for Israel a savior of Jesus thank you Jeffy now another person can continue from here verse 20 verse 23 onwards from this man said according to the promise God raised up for Israel a savior Jesus after John had first preached before his coming the baptism of repentance to all the people of Israel and as John was finishing his course he said who do you think I am I am not he but behold there comes one after me the sandals of whose feet I'm not worthy to lose men and brethren sons of family of Abraham and those among you who fear God to you the word of it this salvation has been sent for those who dwell in Jerusalem and their rulers because they did not know him nor even the voices of the prophets which are read every Sabbath have fulfilled them in condemning him and though they found no cause for death in him they asked Pilate that he should be put to death now when they had fulfilled all that was written concerning him they took him down from the tree and laid him in a tomb but God raised him from the dead he was seen for many days by those who came up with him from Galilee to Jerusalem who are his witnesses to his people to the people and we declare you we declare to you glad tidings that promise which was made to the fathers God has fulfilled this for us their children in that he raised up Jesus as it is also written in the second Psalm you are my son today I have gotten you and that he raised him from the dead no more to return to corruption he has spoken thus I will give you the sure mercies of David therefore he also says in another Psalm you will not allow your holy one to see corruption for David after he had served his own generation by the will of God fell asleep was buried with his fathers and so corruption but he whom God raised up so no corruption therefore it will be known to you brethren that through this man is preached to you the forgiveness of sins and by him everyone who believes is justified from all things from which you could not be justified by the law of Moses we were therefore less to what has been spoken in the prophets come upon you behold you despise us marble and perish for I work a work in your days a work which you will by no means believe though one where to declare it to you but should we continue till the end I think let's talk about this first and then we can continue yeah yeah thank you thanks okay okay so this is the sermon which is delivered by Paul and he is speaking to the people in Antioch of Sidia so we saw Paul and in verse 13 it says Paul and his party remember I kept saying that the limelight will move to Paul so till now who was the key person Barnabas Barnabas brought Paul and Barnabas was the one who introduced Paul to the church of Antioch now suddenly it is Paul and his party and Luke is writing also like that he's saying you know that Paul is the key person who's leading the team whereas Barnabas is in the team but Paul seems to have the importance they set sail from Paphos we saw about Paphos and then came to Perga in Pamphylia we saw that region and now something very key is happening John departing from them returned to Jerusalem now this incident though Luke doesn't talk much about it it's a it's an incident that Paul did not like okay remember earlier we saw John was with them when he started off when Paul and Barnabas started the missionary journey this John is now going back to Jerusalem now we may ask the question the journey has just started they only went to the island of Paphos and preached to the procuncel there how come this man John is returning so quickly hardly anything has been covered in the missionary journey so we can only speculate there are no concrete scriptures to tell us why John went back see he started in the missionary journey but some people say that he was probably homesick he was not very mature and then yeah right after one place he was homesick whereas others say that he was probably not very serious about these missionary journeys he had some unfinished tasks back home now we don't know the exact reason why this John went back but why I'm saying this is significant is because this creates a rift between Barnabas and Paul see the personality of Paul we will notice that he he is extremely hardworking extremely passionate and you know he is a very very focused person and so when he sees somebody like John a younger minister who he's not able to keep focus beyond one you know visit of one city or one region he's very upset with this John and he doesn't he takes a you know like he disliking towards this John but what about Barnabas you know there's a stark contrast between the personality of Paul and Barnabas Barnabas is more of the accommodating kind who probably thinks about why John has done this and what could have been the reason Paul is not like that he's a task master and he doesn't like any excuses so when John returned this later we'll see that it actually causes a rift between Paul and Barnabas and their personalities so that also awakens us to the fact that today when we work with one another you know God doesn't transform us and make us all of the same you know personality and style of ministering he allows us to be ourselves and when we are ourselves in group ministry sometimes it's not so easy because we find people making decisions which we would never make but the thing is we must understand how to work with people we'll see that there will be a rift between Barnabas Barnabas and Paul but later on you know at some point Paul will recognize Paul will sort of you know his personality will be a little more toned toned down and then he'll be willing to accommodate John Mark he'll say hey bring John because he'll be useful for the ministry so there are there is a journey of maturity journey of change as an individual that Paul will also go through Barnabas will also go through but this John going back home was not a good thing for Paul and Barnabas and their you know relationship in some ways okay fine now we are at verse 14 they departed to Perga they came to Antioch remember I told us don't get confused there are lots of Antiochs so this Antioch is the Antioch of it's as Pisidia but I saw some place that you know it's it's pronounced as Sidia so that's why I was saying Sidia so they come to Antioch of Sidia and they went there to the synagogue on the Sabbath day so think about this they knew where to find people people would be at the synagogue on the Sabbath day that's where they go and then they start to read and you know they start to speak so finally you know they are speaking Paul stood up and he is speaking to the men of Israel because obviously you will find Jews over there and so he's talking to the men of Israel and he says you who fear God listen they are all people in this in the synagogue to know about God that's why they are there now he starts narrating remember this this kind of a preaching we have seen this earlier it's nothing new Peter used to preach like this he used to go back to the fathers the patriarchs Abraham Isaac the journey the wilderness Moses David why because we are speaking to the Jews they need context and so it's really important for the minister to understand where the people are coming from even Paul recognizes here are Jews who are well versed in the scriptures they know about the history of Israel so he starts there and he says look you know how we dwelt as strangers in the land of Egypt and with an uplifted arm God brought us out and then he also talks about how you know for about a 40 year period of time we were in the wilderness because you know because of the various things that happened and then finally you know God brought us into the land of Canaan and you know the journey goes on he talks about the churches Samuel and then he talks about David who's chosen as king and then you know he continues why David because from David he's going to the seed of David who is that the Lord Jesus Christ so ultimately the sermon is about Jesus but all this building up context is to proclaim Christ so when we are saying that we are preaching we've got to preach the truth about the Lord Jesus it's it's always about the life of Jesus it's always about the deity of Jesus that we must come back to and that again is the motivation that Paul has so he's bringing the conversation or he's bringing the the focus to the Lord Jesus so from David he comes to David son of Jesse he talks a little bit and then he moves on he says the man's seed according to the promise God raised up for Israel a savior that's the center of this sermon a savior Jesus then he he talks about who this Jesus is he says how John the Baptist said that you know he is only there preaching baptism the baptism of repentance but even a man like John the Baptist who was honored among the people said that he was not worthy to lose the sandals of Jesus so again the focus is back to Jesus Jesus the seed of David Jesus greater than John the Baptist okay so then he is continuing you know he's going on talking and he says family of Abraham and those among you who fear God to you the word of this salvation has been sent so you see key themes in what he's talking he's saying Jesus is deity Jesus is honored among men Jesus is the one who brought salvation okay and he points out and he says that don't you know that like everyone was actually talking about Jesus and you know he talks about how the Lord Jesus he was raised by God from the dead and he finally you know he just moves on to the fact where he says that God did not allow him to see any form of corruption and so Jesus was resurrected from the dead so he mentioned salvation but again verse 38 he says forgiveness of sins okay so this is what Jesus brought for us salvation forgiveness of sins and he tells them now after preaching about Jesus he tells them what they need to do verse 39 he says and by him everyone who believes is justified from all things from which you could be justified by the law of Moses so Jesus is being proclaimed and then what Jesus did for us is being proclaimed what happened because of that salvation forgiveness is being proclaimed and then Paul is actually communicating to these Jews and saying that if you believe because everyone who believes is justified so we can now be justified okay and he's giving them the option of receiving salvation being forgiven being justified so this is a really a preaching session which is going on in Antioch of Sidia in one of the synagogues where the Jews are so now we can continue another person could kindly read the remaining portion here in Acts chapter 13 verse 42 verse 52 so roughly about 10 verses but we would require one more individual to read please starting from verse 42 yeah so when the Jews went out of the synagogue the Gentiles begged that these words might be preached to them the next Sabbath now when the congregation had broken up many of the Jews and the devoured proselytes followed Paul and Barnabas who speaking to them persuaded them to continue in the grace of God on the next Sabbath excuse me all almost the whole city came together to hear the word of God but when the Jews saw the magazine the multitude they feel they were filled with envy and contradicting and blessed and blessed farming they opposed the things spoken by Paul then Paul and Barnabas grew bold and said it was necessary that the word of God should be spoken to you first but since you reject it and judge yourself unworthy unworthy of everlasting life behold we turn to the Gentiles to the Gentiles for says the Lord has commanded us I have sent you as the light to the Gentiles they should be that you should be for salvation to the end of the earth now when the Gentiles had had this they were glad and glorified the word of God as they and as they had been appointed to internal life believed and the word of God was being spread throughout the region but the Jews stirred up the devout and prominent women and the chief men of the city raised up prostitution against Paul and Barnabas and expelled them from their region but these shook off but they shook off the dust from their feet against them and came to Econium and the disciples were filled with joy and with the Holy Spirit amen thank you Lubega thank you for reading through that passage okay yes jeffina you have a question okay maybe not so I'll go ahead and explain we saw that Paul preached to the Jews and now what's happening is there are Gentiles who are also remember we mentioned that proselytes so you have people from other communities who were devout followers of the God of Israel they are known as proselytes so there are Gentiles who begged for these words to be taught to them the next Sabbath and so the next Sabbath it happened we find that Paul goes up and there was a huge gathering so Luke describes it as the whole city came together to hear the word of God but you see this whole city coming together proved like a threat to the Jews because they were all about politics they were all about you know power games and power tactics so when they saw that there was such a huge following for Paul they felt threatened that what if these guys you know gain power in that place maybe they've come to elevate themselves and become the authorities of that place so you know these kind of thoughts cause them to be envious about Paul and Barnabas and so what did they do they started contradicting they started blaspheming Paul and they opposed the things that were spoken by Paul so just some time ago there was an openness right and they were willing to listen but now there is an opposition and because of that you know Paul speaks to them and he says that we are now at verse 46 then Paul and Barnabas grew bold okay so what is the result of experiencing persecution or opposition boldness that's what you see throughout the book of acts they were never scared or you know they they did not back down because of the rising opposition but with God's wisdom boldly they continued to proclaim so they were bold and they said it was necessary that the word of God should be spoken to you first who's you first Jews but since you reject it since the Jews are rejecting it and judge yourselves unworthy of everlasting life behold we turn to the Gentiles so they they are now in a way forced to take the gospel to the Gentiles because the Jews are not accepting of the gospel so based on the based on remember we had mentioned earlier the prophetic word which even Ananias had received so Paul states here in verse 47 I have set you as a light to the Gentiles that you should be for salvation to the ends of the earth so based on the word regarding the life of Paul he is now moving in that direction so acts 13 when we started according to the word of God for the things that I have called set aside Paul and Barnabas for the things that I have called them to do so in line with the purpose of God Paul is now journeying he went he tried preaching to the Jews they're not listening but according to the word on his life that he has to be a light to the Gentiles now what is he doing he's actually moving on to preaching to the Gentiles so it was 48 says now the Gentiles heard this they were glad and glorified the word of the Lord as many as had been appointed to eternal life believed so there was a move among the Gentiles and the Gentiles started to believe God so there was opposition but the word of God did not you know stop impacting lives we see a different community which is the Gentiles being receptive in Antioch of Sidia and then what happens over there there is obviously opposition was rising and opposition got pretty bad the Jews stirred up devout and prominent women so it's like an uprising that is starting against Paul and Barnabas intentional one so what do the Jews do they go to the who's who of the city and they instigate them against Paul and Barnabas and because of this it's so unfortunate that you know they Paul and Barnabas went there to do the ministry but you know they are now being expelled when they went to do a good thing they expelled so what what Paul and Barnabas doing remember when one was not welcome into the city earlier Jesus also said just shake off the dust from your feet what a shake of the dust from your feet it's a it's a it's an expression right it's it's a manner of speech so it doesn't literally mean that they shook the dust off their feet but in hermeneutics we understand that we have to interpret this basically they they said okay fine if you don't want us preaching here fine we'll just move on and the rejection which came to Paul and Barnabas as people of God they they just based on that let go of this community and said fine we won't minister to you anymore and so they shook the dust off their feet against them and they came to the next city which is Iconium and we will read more about what happened in Iconium later on but look at this persecution is always going on but what do we read in the midst of persecution generally we read the ministers were bold and then it also says the disciples were filled with joy and with the Holy Spirit so wow all this is existing together so it's not like a perfect world right like there's no opposition the surroundings are so conducive for Paul and Barnabas to preach the gospel and so you know things are moving forward smoothly no things are not moving smoothly and yet the gospel is spreading people are filled with joy and the Holy Spirit is doing his work so I'll stop here jeffina you have some questions please go ahead you can ask yes pastor I think I didn't turn on the mic last yeah so in verse 31 we see he was seen for many days by those who came up with him from Galilee to Jerusalem who are his witnesses to the people so I just want to know whether it's talking about the disciples or who it's actually talking about okay verse 31 of yeah correct he was seen for many days by those who came up uh with him from Galilee to Jerusalem who are his witnesses to the people so this one jeffina it's talking about the resurrected Christ yeah okay uh so what what is your question I mean I understand it's talking about the resurrected Christ yeah uh so who are these witnesses is it only his disciples or okay okay got it uh yeah I will tell you there is a verse actually yeah first Corinthians 15 6 where uh that verse it says uh I'll just read the NIV version which I readily got here it says after that he appeared to more than 500 of the brothers and sisters at the same time most of whom are still living though some have fallen asleep so 500 brothers and sisters it says which is likely see brothers and sisters is like a term used for believers uh which was used so those who believed in Jesus I think he just showed up to them it says 500 people in first Corinthians 15 6 those are the people okay is there any idea on how many days faster after resurrection he showed up to people how many days okay how see after many days uh okay right so just one second see uh the death of Jesus happened at the time of uh we are aware about the um the celebration or the observance of the Passover okay now Passover to Pentecost Passover to Pentecost is 50 days um so we know that Jesus was with them and then after he ascended it was roughly about 10 days before the baptism in the Holy Spirit so calculating like that Passover to Pentecost and then Jesus ascended there was 10 days duration before the uh uh Acts chapter 2 incident happened so 40 days is the calculation Jefina 40 days Jesus was uh walking around in his resurrected body thank you pastor uh I have two more questions yeah sure yeah so it says uh I don't know which verse it says uh you will not allow your holy one to seek corruption there is a verse in that so it talks about the physical body being corrupted yes you're right you're right okay so in verse 42 we see uh so Paul preached to the Jews and then the Gentiles are actually begging so I believe maybe they heard him speaking so why they asked him to preach one like words might be preached to them the next Sabbath like why the Sabbath is actually mentioned is it okay so yeah that question actually I don't have a definite answer to it but as you're stating maybe Sabbath to Sabbath is when they had large gatherings I feel so one Sabbath Paul minister to the Jews and maybe the obviously the Gentiles would have a different congregation they cannot be part of the Jewish group right so they are inviting Paul to a separate uh session and they may have thought that they need a large gathering uh and they need enough time so they could have picked the next Sabbath I I don't exactly know actually yeah sure thank you thank you any any other questions or thoughts all right so I think it's pretty good so far we've seen how the ministry has happened in Antioch of Sidia and Paul and Barnabas now move on to Iconium so we will look at Iconium remember the first missionary journey is still on it's not over yet okay till they make a full circle and then they come back to Antioch that's when we'll say that the journey is over journey has just started and unfortunately even in the very first place uh you know they're not not really the first place Paphos is where they ministered and then now in Iconium but there is opposition earlier through that sorcerer bar jesus and now through the Jewish Jewish notable people Jewish people of the city of Iconium they are having opposition but praise god you know they are bold they continue to minister they are thoroughly equipped right with the power of God with the word of God so nothing is really stopping them and you know they are carrying on as per the call of God on their lives so let's maybe stop here and we will pick up from Acts chapter 14 for the rest of Paul's missionary first missionary journey in our next class so at this point we can close with a word of prayer I would like to request someone to please go ahead and lead us in prayer let's pray heavenly father we come to you under the name of Jesus we thank you for this day thank you for the precious that we had today we thank you for your powerful truths the way you transform a person is beyond imagination Jesus but we thank you for the call that you have placed in our life as we are reading about acts and the life of Paul's Jesus but I just pray that it will stir up our desire it will stir up our passion to work more for your kingdom to move more super naturally and to not to just listen to the classes but to actually put it into practice in our life so that we can win souls for you Jesus filled us with the boldness and the confidence that the early church had so that we can move my team for your kingdom I give all the classmates and pastors and saints here have blessed them all in the name of Jesus for all the glory and honor in Jesus name I pray amen amen thank you jeffina thank you everyone god bless you have a good day and a very blessed weekend bye for now thank you thank you bye


What's up everybody? Jason here for Jaspys case breaks.com. This is Jaspys for the collectors repack This is a one-box break ran on teams number eight and as you can see in the item description guys This is a Repack product from Jaspys Made by Jaspys of course includes 25 or 30 cards per box a lot of graded cards rookies and Etc. Nothing too crazy, but a fun break for collectors who would like to just add to a PSA BGS or SGC card to their collection and again guys there is a hit list promo So at the end of the break we will take all hit list customer names and randomize them the top two names will win $50 each So example if you have three spots two of them go hit lists you get two spots into the entry and Then of course since there is only 12 boxes guys most spots only have a few pot hit possibilities Of course hip possibilities may run out before a team before we get to the end of the box run by a team on the card So just be aware of that and of course we go by what's on the card team-wise MLB for officially licensed anything Unlicensed slash collegiate cards we go by the label that was placed on the back of the card And again you get two random teams from the list below So what we're gonna do here guys is here all the customer names that are in this break Let's alphabetize. I'm sorry an alphabetize them, but let's double them up Let's roll it roll the dice and we got ourselves a five and a one six times one two three four five six Yes, sir. It's already on the schedule Sean O'Leary down to Barry Roberts That was a six six times here one two three four five Six down of the chargers. All right, so actually this is what I'll do is I'm gonna show you guys the way it came out Like this and then I'll alphabetize it by customer name right and then I'll show you guys quickly who you guys got just so it's a little bit easier that way I know some people got multiple spots and you do get two random teams each So Anthony snow. There's your teams are bugs Broncos India's raised Cowboys Raiders Ray Roberts, you have athletics and chargers Brandon with the Bangles Mavericks Eagles Mets Chad Los Balmojo Lakers and football team Chris Walker calves and Rockies Daniel Hayes You have Marlins Packers Twins Seahawks Reds D-backs Nationals from Pelicans Daniel Katz with the Yankees Brewers Ravens Phillies Edwin bucks Giants San Francisco Giants Eric got the Astros and Jets Greg with the 76ers Padres. However, the he braves John the McFlay with the nuggets in New York Giants Matt McLaughlin Magic and pistons Matthew Pacifico With the Knicks and Tigers Nick Stover. You have the Mariners Bulls Angels Thunder Wars Steelers PJ You have the Jags Red Sox others not listed Orioles Texans Bears Cubs are sorry Bears Kings and Cubs Rosario with the Suns and Rangers Shown all the way down here. I'm sorry not all the way down here up to here Yeah, the Spurs Dodgers Hawks Bills Colts Blue Jays Royals St. Louis Cardinals Saints T-Wills Patriots and Grizzlies and Steve Wright You have the Chicago White Sox Chiefs Hornets and Clippers So I don't I don't think anybody's really trading for teams and this right so what I'll do is I'll Probably I'll probably Highlight or something Whatever teams and whatever spots do get hits and that's how we'll separate it a little bit later So again, I don't think anybody's gonna trade. So this is just for Hey for purchase of purposes here for the sword doesn't ship or so Do that let's close the and I just grab the random box. I don't know Joe's been randomizing them or not, but Just grab random boxes There you go box nine. I Believe we're pretty much running down to the last couple cases. It looks like Think you only have what 12 cases. I believe yeah, so we're pretty much almost there That's kind of hard to see your teams there, but of course One through them Paper here and reuse that probably and it looks like it's just one All the empty boxes right there Perfect All right, guys So what I'll do is I'll have I'll just grab the cards with one hand And then I'll highlight the next one. Oh Little pony that is a rookie sizable signatures RPA to 99 of Christophe's for zingas Very nice one right there So that would be New York Knicks Yeah, this is box 8 J. Mac Why is that so Knicks? That's Michael Pacifico And we have a autograph patchbook collection definitive Yeah, I don't I don't think we've been going an older because I've seen box two over there I seen like a different number. I think they're all just random From what I know of no one told me that they were one through 12 You get me because I've seen box two back there. So we already did box two a long time ago They were just numbered by 12 Chris sale one out of five Very nice definitive booklet Yeah, I mean I just grabbed it. I didn't even notice the number Yeah, I don't think we were going in order This one actually is a little tight. I don't want to get stuck here. Give me one second guys There you go autograph book card Chris. Yeah, out of five So that was uh going to Boston, I believe Red Sox R PJ then we have a Andre Johnson 2003 tops pristine and that must be a rookie right out of 1499 Yes, sir nine five So that is Houston Texans PJ I'm speaking of PJ. What's going on, brother? Hell, yeah, man. I just pulled it for you right now Next one we got is a Mark Ingram 2011 score rookie that was obtained in in person. So they just back it autographed it Sorry graded it a 10 So saints that'll be s Sean O'Leary My house purple Yanis onto the combo. They're actually starting the game right now in about 15 minutes Milwaukee bucks Where's the bucks Anthony snow? We have a top chrome black Joey Gallo autograph Texas Rangers Rangers going to Rosario We have a Vladimir Guerrero, Jr. That is a rookie from Bowman Sterling and that's a PSA 10 Blue jays blue jays if the blue jays are not listed then that'll go to the not listed, right? Oh Sorry, it is listed. It's just started with Toronto actually Toronto blue jays Chandler There you go. Our city as a quino PSA 10 2020 tops rookie Cincinnati Reds. So some of them he put both a City in team some of them. He didn't let's see Reds. There you go. Daniel Hayes Wow, Ronald Cooney, Jr. tops chrome rookie, and it's a PSA 10. I love that guys Can't go wrong with her little Cooney rookie Tops chrome rookie Ronald Cooney, Jr. That's Braves Atlanta Braves going to Howard we have a You snail Diaz Bowman Chrome first nine five ten from 2017 Bowman Chrome Dodgers Sean earlier Tim Tebow little speckle refractor sparkle PSA 9 and A Mike Trout tops holiday PSA 10 So angels there next over And we got Mets Brandon We have a Walker beeler raider rookie for 2018 Donner's Optic a little shock PSA 9 Another one for the Dodgers there Sean We have a Bowman sir a Bowman Chrome. Sorry rookie 2010 our sorry Bowman's first Chrome card Steven Strasburg PSA 9 This dude's cards are so hot when he first came out Nationals Daniel Hayes Then we got a Sikhu Dumboya fanatics PSA 9 Raider rookie Detroit Pistons That's Matt M Terrence Mann rookie silver prism PSA 9 for the clip show Los Angeles Clippers Where's the clips Clippers Clippers there you go Steve Wright? We have a Bobby Wood, Jr. Top 100 Bowman Scouts PSA 10 for the Royals and a Louise Robert 1985 tops Chrome insert from 2020 PSA 9 so White Sox and Royals Steve Wright with the White Sox and Royals is Sean O'Leary how about a 2003 SPX Kobe PSA 9 That's a cool part of their Kobe Ryan for the Lakers Lakers Chad last spot moja. That's a cool one right there, man. We got a Walt Frazier mosaic autograph For the Knicks. That's another one there for Michael. No problem Mike. Nice a man. You got another one right now We got a Devante grand prism PSA 10 for the Charlotte Hornets Charlotte Hornets Steve Wright We have a Nate Archibald 4 to 10 distinguished autographs from flawless basketball Boston Celtics Are the Celtics on this list? All right, so it looks like Celtics are not on this list So that would go to the Where's that one? Other not listed it's going to PJ We got a NBA finalist silver prism Tim Duncan PSA 10 great insert right here Spurs San Antonio Spurs Sean O'Leary You have a nice one coming up guys a good time to get him little wander Franco Paper prospects PSA 10 first We got that is Tampa Bay Rays Anthony Snow We have a Fernando Tautis Junior rookie tops update PSA 10 Padres Greg with that one. We got showtime show hey old honey 2020 Bowman rookie PSA 10 Another one there for the Angels. We already got that one next over and it looks like two more guys two more This is Zika Elliott. Very nice. That is a rookie prism purple scopes PSA 10. That's actually really nice 45 out of 99 for the Dallas Cowboys Anthony Snow and last but not least this one's really cool. I have a Have a couple of these mild sanders and Donovan McNabb that is a black gold Mosaic out of eight eight out of eight Rob Gronkowski for the Tampa Bay Buccaneers When a Edwin hack there you go Very nice boom and done. So there you go. That was it right there So we got one two three four Five six seven eight Ten eleven twenty-four twenty-five twenty-six twenty-seven and twenty-eight so from twenty-five to thirty There you go guys that was for the collectors repack and again. We do have a hit list promo So let me quickly actually put this actually call you just back in here for now I'll grab a tub get an opportunity right now and Switched jeans and let's do this little hit list promo. So first of all before I forget I'm gonna copy and paste this list and put it in the promo list for the show notes which I believe it's right here and Bricks one through 12 and it looks like Joe has it right there So I'll paste mine there and I'll try to copy the font. What is it? Oh, yeah, you know, it's supposed to be only 37 sorry What I want to get Let me get the original list that bottom. All right, cool. That should be right there This should be the start of number nine right here whenever we saw that out All right, so everybody's in there now. Let's go with the hit list. All right, so there you go and Top two get 50 bucks. Good luck guys snake eyes one Two so there you go guys appreciate it. I'll give you guys 50 bucks each right top two Yeah, 50 bucks Appreciate it guys. Jasper's case breaks calm


So, hi, welcome everyone to the analysis seminar and today for me it's a pleasure to introduce my colleague and my friend, Eugenia, from the University of Buenos Aires, and today he's going to talk about what is a priori estimate for elliptic equations. So, thank you Andrea, thank you Manuel, I'm very comfortable here working with Presti. I'm going to talk about something that I started to study when I started my PhD like 10 years ago, and I decided to think about another product relating to this estimate. So, I will tell you about the results in the PhD and new results, okay, results that we obtained with Ricardo, Ricardo was my PhD supervisor, and we still work together. And so I will show you the results we obtained during the PhD and what we obtained now that is new and it's a generalization of that result. Okay, so let's start. You should just move also there. Click on the video. That should be okay now. Here? Yeah. The computer. Let me see. An introduction of the problem, the motivation of the problem. So, I will start with the easiest case that is considered the case of the Laplacian, we will consider the diligent left problem, or the case of the Laplacian. Okay, and we consider for our problem because it can be considered another type of domain, but we will consider a smooth domain. The regularity of the domain depends on the order of the equation. For example, for the Laplacian, we can consider a domain with a boundary C3. Okay. And so we state this problem. And it is well known that this problem has a unique solution you that can be controlled by the norm of the data. Okay, in which norm, for example, if we take a function F in LP for P between one and infinity, we can prove this weighty a theory estimate because big control the norm of the solution in terms of the data F. Okay. This was shown by any classical work by Agnus Douglas and Nidenberg. Okay. And in fact, they consider general elliptic operator, uniformly elliptic operator, I just want to introduce for the simplest case. So, this is an inequality with the constant in front. It's a constant, but I forgot to write the constant. Yes, yes, depending on the domain, of course, and the dimension. Okay, it can be depends on that constant. I'm sorry. Now, so from this equation from this problem from this estimate, we have several problems that can be solved or not, we will see. For example, I introduced the case of the Laplacian. Okay, so as I said before, it can be considered another operators that are elliptic operator, I will give you the form of that operator and what are needed for the operator to satisfy the weighty a theory estimate. But first I want to tell you in general what are the problems related to this program and then I will talk about this in more detail. Okay, okay, so this problem can be extend by for more general operators. Uniform elliptic operators, in particular cases, negative Laplacian. This was also done by Agnus Douglas and Nidenberg. In fact, they consider this operator, I just write the case of the Laplacian, but they consider this type of operators I will show you how they are later. And when we say an operator, it means a elliptic operator, we are talking about an operator involving differential operator, it means the derivative of a function of certain orders, and we multiply this operator differential operator by coefficient by functions that has certain properties. Okay, so in the case that I will show you later the coefficient are smooth. Okay, so we can ask if it can be extend for a bigger class of coefficients. Okay, I will show you in detail this later. Okay, I want just to state general ideas. Okay, so we can extend that for coefficient in a bigger class. And another program that can be considered is what happened if we consider less regularity of the domain. Okay, I say that for the case of the Laplacian because it depends on the order of the operator if I consider derivative of order two or two m. So, what happened if we want to consider less regularity. Okay, there are some results for polygonal and convex domain, for example for the case of the Laplacian. I don't know too much about this because I'm not working on this problem, but it's another possible problem to study for this type of operators. And another problem possible that it was the one I studied during my PhD is to consider another norms. I consider the last year, we say, you consider no, I will give the definition of that norms in a couple of slides. That's the problem we can solve. Okay, here are some typos here. Another problem that arise is what happened if P is equal to one because I say that the data F was in LP for P between one and infinity what happened P is equal to one. Which is a suitable subspace of L1 because in L1 it is well known that that problem does not hold. Okay, we don't have the solution and the a priori esteem. Okay, so we are already working on that problem with something else at the end of the talk. So I want to start to give more details about this problem I mentioned here. So, as I say, we will consider the problem of another norms. Okay, we consider F in LP. Now we want to consider the data F in weighted-level spaces. What we mean with weighted-level spaces? We consider this norm. Okay, we consider this norm. And what is W? W is a weight. What is a weight? A measure and positive function almost everywhere positive function. Okay, so we will consider this type of norms. Okay. Now, the special case we will consider is for Mach and Howe weights because a lot of theory of harmonic analysis is extended from levex spaces to weighted-level spaces for Mach and Howe weights. So, many results of harmonic analysis can be extended for the weighted setting. Okay, so we consider a Mach and Howe weights. So we remind for those who doesn't know the definition, we recall what is a weight in AP. If the supremium is finite and the supremium is taking over all cubes in RL. Okay, so we will consider this type of weights. And the range of the power is depends on the number P, okay, of the order P. We consider this problem. I didn't decide this because my supervisor proposed me this when I decided to start my BAT, but since he works also in numerical analysis, he knows that this kind of estimate has application to find that element method. Okay, when we want to approximate a solution of some elliptic equations, we consider a regulation of the domain and we approximate solution in each triangle of the domain. And the approximation are, for example, polynomials in each triangle, and we want to know if that approximation is good or not. What about the stability of that approximation? And the idea is that sometimes the, when you take some data or solving an equation, the data is not in LP, but it is in LPW and you can prove using that data is there. You can use it to prove that the approximate solution has a good conversion to the original solution. Okay, so it can have application. A priori estimate, in fact, the a priori estimate that are used are estimate for first order derivative. They are not need the estimate for derivative of second order, okay. So it has application. So let's state with details the problem. So I say, we consider omega a bounded domain, we consider this problem, now L is a general operator, and the operator is just for simplicity we will consider of order two, but this I will tell you today can be done for another order, any order two m, and the operator is of this form. Okay, we have, we consider derivative of order two multiplied by certain coefficients that are functions that sometimes has some regularity. For example, in the case that we study at the beginning, we consider coefficient in C3, okay. And for being an elliptic re-operator, we need some properties of the coefficient. So the properties for being uniform elliptic are this condition. Okay, in the sense of almond, Douglas, and Neyman, this is a standard. And of course, the example, typical example is negative lackluster, okay. So, as I said, the regularity of the coefficients in order to solve this problem and obtain a priori estimate depends on the order of the operator. Okay, so we will consider just the case of order two. Okay, so this problem was solved by Agmond Douglas and Neyman, they represent the solution in terms of a Calderon-Sigmund, something similar to a Calderon-Sigmund singular integrator operator, and using that theory they show this estimate. So, they represent the solution in terms of operators, and then using this representation formula, they obtain a representation formula for the second order derivative. I say second order because the operator is of order two. If the operator is two m, the most important thing to estimate is the derivative of order two m. Okay, so what we did with Ricardo, we consider this problem in the weighted setting for weights in AP. Okay, we consider the same problem and we try to extend to the weighted norms. Okay, so I will show you, I will mention what is the technique to obtain this extension. Okay. I won't mention a lot of results because I want to show you how is the connection between PDEs and harmonic analysis, how we use the techniques of harmonic analysis to obtain some results of PDEs. Okay, so I will mention the ideas of any result I will state today. Okay, so we obtain this estimate for the same problem, the same operator, the same regularity of the coefficients, the smoothe is the same, but now we consider a weight that is in the Mackenhall class AP. Okay, so what's the idea to obtain this estimate? So for the case of a smoothe, it is well known that a solution, the solution of the equation I introduced before can be written by this operator. So what is G? G is the green function associated to the equation. Okay, it depends on the domain of course. Okay, and the function has the function G and it's some derivative of the function G has a point white estimate that of course depends on the operator and depends on the domain. So we can consider a smoothe function here, know that it says F in C0 infinity, it means the function with infinite derivatives and compact support. So why we can consider this? Because these functions are dense also in LP and LPW. So then we can use an density argument to extend for any function F in LPW. So we can, if it's necessary, we consider this kind of functions. So using this representation of U, we can obtain, we can, yes, we can obtain two inequalities that are very useful to obtain weighty a purely estimate. It's not just the representation, you need to use the representation to obtain second order derivatives of the solution. Okay, so using the representation, I will show you later how is the expression for the second derivative of U. The first is in terms of the green function, the second order derivative of the green function, but using that representation, one can obtain two inequalities, one is a point weight inequality and the other is weighty norm inequality to prove the weighty a purely estimate. And it's also needed to prove the weighty a purely estimate the unweighted case. Okay. Maybe it won't be clear how we use the unweighted case because it's very technical and we have to make many calculations, but I'll give you the general idea. So the first inequality is this point was inequality between I will give you the definition now between the sharp maximum function of the operator. This is the sharp maximum function. I will write here the definition, the sharp maximum function of an operator G, okay, is a control by the maximum function, hardly readable maximum function of a function to the power s, everything to the power one over s for any s greater than zero. Okay, but what is the, what is the T is any Kaderon sigma operator, which is represent in this form. Okay, where the kernel K satisfies some properties the standard properties of Kaderon sigma, singular, the operator, the control of the size of the kernel. What is the sharp maximum function. It means that the function. Consider this sharp maximum function. Where is the here this sharp maximum function consider the supreme over the oscillations of the function f over cubes. Consider the variable maximal operator consider supreme over mean value of the model of f. Okay, so what, how is this related to what I was saying about weighty a priori estimate this estimate. This is well known that is used to prove boundedness of this type of operator in weighted norms. Okay, one, using this inequality can prove that this operator are bounded in LPW for W in AP. Okay, and how we are going to relate this to weighted a priori estimate, because it can be true. I will show you in the next ice. I think in one or two more slide that here we can substitute instead of TDF, the second order derivative of view. Okay, then we will see why this estimate is important to obtain weighty a priori estimate, but this inequality can be extended. Replacing here the data here data type in here. The second order derivative of view. Okay, the solution. So, this is the first key inequality at points where inequality. The other is a weighted norm. Sorry, not the way. Yes, we will consider the way. At first, and I'm sorry, first, I will show you the how it's written. The point was inequality in terms of the case of considering here. The second order derivative of view. And here, note that here it says omega, instead of any cube in our end, it means that when we consider the oscillation, the oscillation of our cubes, we will consider just cubes in omega. Okay. This, I don't give the proof of this estimate, but this can be proved. Okay, the extension of the inequality, the point one equality for any Calderon-Simone operator can be extended for second order derivative of the solution. Because the solution can be expressed in terms of some operator that has good point one inequality and due to this point one inequality, we can generalize the point one estimate. So, next, the second key inequality is the performance time inequality that is well known for the case of omega equal to our end. So the norm of f of a function f in the case of our end we consider just f is controlled by the sharp maximal function that is the case. I will write it here. For the case of our end, it is well known that we can control of the sharp maximal function. Okay, in the case of our end. This was proved by Pheferma and I understand that why we call Pheferma inequality. So this is the direction of that and a weighted case of that inequality because here is in our end. So it can be extended to domains, considering weights in the domain omega, and we can prove for weight in norms. And of course to prove this is based on the case of our end, but we have to make a lot of technical details, but the idea is, is similar. So we can obtain this inequality f omega it means the average of f over omega. So how do we use these two inequalities to prove weight in a purely estimate. So let's show you that if we substitute the, we need to bound the order derivatives in LPW omega but we can consider subtracting the average of f. And then this can be bound so it's no problem if we bound this difference because this can be controlled and then we can bound just this term. So how we can apply last two estimate. So we can bound by the Pheferman time we can write here the sharp maximal the local sharp maximal of the second order derivative of few estimate. So we can write here. Let's go back to the point why estimate where you see it here, then we can apply the point why estimate so I can hear right. m of the data f over s a p w. In fact, we can write, it doesn't matter because the support we can extend the function by zero upside omega and there is no problem. What happened here. The operator m that is the hardly the maximal function show you it's bounded in LP for any people. But here we don't have the norm in LP. This is like the norm. This is like the norm LP over s to the power here we have to write one over p. No, no, it's okay. This is like this norm be over s. Okay, so it is well known that the maximal operator m. It's founded in LP for the same thing we have been here, we have to write a weight in AP here. But here we have the norm LP over s. And we know that that view as hypothesis because we want to grow weight the appropriate estimate for weights in AP. We have that view in a P. What, how can we solve here and use the power less according to some properties of the market how it's here of the market how it's, it is well known that if a weight is in a P, it can be in a small AP. That is, that view is in a P over s for some s greater than one. Okay, sharp inequality the point wise sharp inequality it home for any s greater than zero any s greater than zero. So we can choose the s for which that view is in AP over s, we can choose that is, and we have the problem is here. So we have the bottleneck here, right this here over, it's always content dependent on the weight here I forgot the way I'm sorry. So here we obtain none of it. So we can follow the second order derivative of view minus the average of the second order derivative by a weight none of the data type. Okay, so in order to I won't make all the calculation just saying that this one can subtract the average. Here we write the average can't be bothered writing the definition and using the quality and using property of weight. So, in conclusion, we can obtain the weight here purely estimate using these two inequalities. So, I say that I wanted to show you how is the expression of the second order of the event that allow us to obtain the two inequalities, the point wise and the a feather man, the local feather man standing equality. So the second order derivative has this representation in terms of the second derivative of the wind function associated to the problem. And here, this is a bounded function. So this part of the operator is not problem. It is enough to bound this. Okay, in order to obtain weight the appropriate estimate. It is not trivial because this is a bounded function. And so it's enough to consider this operator and obtain the obtaining estimate for this operator. Okay, so, since this kernel, the second order derivative of the wind function is similar to a single integral operator. So this is a good estimate that can be used to prove the sharp, the point why inequality between a sharp maximal function and they have a little maximal operation. Okay, so that's the idea. As I say before, the local version of feather man is time inequality, waiting note can be extend but it is not trivial. I mean, the idea is similar but it is not trivial. Okay. And also I say that this can be extend for operator of order to end. Okay, I consider just for simplicity of the notation. I consider the case of order to. Okay. Now, what's the problem we consider now. Okay, that it's a part that we did during my PhD and next a couple of years ago, we consider the case of another type of coefficient because then we consider a smooth coefficient. What happened if we want to consider biggest class that of course contain the smooth coefficient, but all coefficient they are not necessarily smart. Okay, so which is a suitable class. So, to, to answer this question, we consider a work by the adenza for a long ago that consider the class of vanishing mean oscillation I will give the definition later. So we will consider if this class is suitable or not for our case of considering weighted norms. Okay, so we will see if this is suitable or not. And as the ideas to show this is different than the ideas I present here. Okay, because since we will consider a non necessary smooth coefficient, all the estimate for the green function. And we have the representation of in terms of this and we have estimate that depends on what estimate that depends on the question of course on the operator depends because this refunction is associated to the operator. So, the coefficients are not smooth, we don't have the same estimate. Okay, so we can use, we cannot at the moment we, we cannot use these ideas. Okay, so we will use some standard method of PDE to extend it for the vanishing mean oscillation coefficients. Okay, so we will define what is this space of vanishing mean oscillation function. So first we need to define what is a bounded mean oscillation function that is that the share maximum function is in what does it means that the oscillations are bounded. Okay, all the oscillation for all cubes in Rn are bounded. Okay. And for the case of a function in bounded bounded mean oscillation. We will consider the oscillation but for a small box. Okay, for a small box, and we will see that we will see now, I'm sorry, we will require for a bunch of TV in the vanishing mean oscillation space. The oscillation space is that the oscillations tends to zero according to the walls, the radius of the walls tends to zero. Okay. So bounded mean oscillation, the oscillation bounded and vanishing mean oscillation, the oscillation vanishing when the walls, the radius of the walls tends to zero. Okay, so we will consider the vanishing mean oscillation for the coefficient of our operator of our equation. So we will state the main result. So we consider an operator of this type where now the operators are in the class of vanishing mean oscillation. There is also a weight in AP, the domain is still smooth because as I say, we won't consider the case of non smooth domains, we just work with this class of domains. And so this problem for a data F and LP has a solution you that satisfies a the following weight is a very estimate this. Okay. So this is the solvability and the estimate and there is another result that we can state that this that the solution is unique the solution. If we consider F equal to zero, and we have the operator in a function new is equal to zero, then the solution must be zero. We will define what means zero here. Okay, and for a weight in AP and P, it is important to know that piece between a one infinity, because this theory of singular integral or similar to singular integral does not hold for P what you want. So we will consider the softness of the operators. Okay. So, we will consider this our left face is the way it's our left faces. I won't give the definition before the derivative. And when we denote zero here, it means the closure of the C infinity function with compact support and the closer it's considered in this space. Okay, according to this space. How the detail of the proof that I want to mention how we use the technique of harmonic analysis to obtain a result of this. Okay, because I, I like from these topics I like how we mix both areas. Okay, so this is very standard in PD is what I want to mention is not my idea. So, what first made the proof of the estimate for the case of a semi ball, what it is important to consider a semi ball, because the idea is one, when one consider a domain omega, a domain omega, the idea is to transform a neighborhood of the boundary in terms of a semi wall. Okay, and use that when we transform and we use the film or things to transform the neighborhood of the boundary. So the idea in general is this to consider first semi wall, because we will obtain interior estimate, estimate inside the domain and when we are close to the boundary of the domain. We consider transformation here, and if your more things in a semi ball, use the result well known in the semi ball and when we go back to the original norm, they are equivalent by this transformation. Okay, so one can make a covering of the domain, if it's inside, we can consider what's inside, if we are close to the boundary, we will transform in semi walls. Okay, so the idea is to prove in semi walls. Don't be afraid about this integral, it doesn't matter what I want to say is that the function you has a representation. Okay, it doesn't matter how it has your representation in terms of operators, okay, that has good property in here, what I want to see is that this different is important because since the function is in the functions are like vanishing in oscillation, this can be controlled, these operators are bounded, okay, it doesn't matter, this is like I'm forgetting. So we have to prove for the semi wall, okay, we have to prove for a semi wall, I don't have time to make the proof, but we have to consider the case of semi walls. So as I said, we consider to prove interior estimate, okay, and near the boundary by making a transformation of the domain into a semi wall. So this estimate, okay, the control of the second order derivative in terms of the data, and we have here this term that we have to see how can be eliminated. Okay, so one can prove first interior and near the boundary and using these two estimates, making a covering of the domain one can prove this kind of estimate. Okay, but here we still have this term that we need to see how to eliminate. Now, this part I said before that we transform the neighborhood of a boundary into a semi wall. So the operator, the original operator, we transform the domain, of course it changes the equation and the equation has it forms. The coefficients, B, Y, J are of the same type of the A, J of the original equation, we have one equation and when we transform it's appear a new equation the important here is that is of the same type. Okay, the coefficient, it can be proved that the coefficient are also in the vanishing mean oscillation. Okay, so if the coefficient are in vanishing mean oscillation, we can consider the case of the balls because here, when we prove interior estimate, here it's but say instead of semi wall, we can put here ball. So, so when we make this defiomorphins, the coefficient are still in vanishing mean oscillation, the transformant data, this is the transform data is also in LPW, but not W, the transforms way. Let me show you this. We transform the equation, the new equations are in the new coefficient are still in vanishing mean oscillation. The AP class is invariant under defiomorphins so the way transforms is in AP. The weighting norm of the solution be a new R equivalent. And for the result of the, it says one that is the case of the semi wall but the case of semi wall is similar to the case of the ball. So one can make a covering of the domain to obtain the estimate I showed you before with the terms, let me go back. We can obtain this estimate by making the covering. This is just what I said before, but I don't follow the order of the slides. So, we can obtain by using near the boundary and interior estimate and making a call. Okay, and it is important that we obtain a equivalent norms that by using that this way is also the transfer weight is also in AP. Okay, and for proving unit next is necessary to use the Pucci-Alexandrov maximum principle that is well known in PDEs. So we have this inequality for a solution of this equation, the U in this space, in this sub-left face, and of course it holds the opposite with the infimum. So we can use this maximum principle to prove the uniqueness of our problem. How? It's necessary to, it doesn't follow this calculation, but the idea is to reduce the weighted case to the unweighted case. So it can be proved that the solution, if it is here, I don't know if you notice that it says, let me show you, here it says the following. If we know that U is in some sub-left for some Q and the F is in the PW, then we have that for any P greater or equal than Q, we have this estimate. Okay, so we are going to apply that in order to obtain that we can choose a Q here greater than P. Here P plays the role of Q in the inequality I showed you before and here P plays the role of Q. So this exponent is greater than P. So according to that result, the U is in this sub-left space with exponent here Q. So by a header inequality and using property in a play way, we can show that the solution is here and we can use the Pucci-Alexandrov maximum principle. We need to show that this integral is finite, we can do by a header inequality and use it that the dual weight is in AP. This is in AP prime, so this integral is finite, so we can obtain that the solution must be zero. So the idea of this is that we reduce the weighted case to the unweighted case. The last step, no, this is yes, it's the last step, we need to prove solvability and the estimate, so we need to eliminate the term quality I showed you before. So the idea is a standard to prove solvability, the idea is to approximate our coefficients that are in the vanishing minoculation class, but see infinity coefficient. Okay, this is well known, so for the solvability is enough to use the solvability for smooth coefficient and approximate the solution that we want to find by solution in the smooth case. And to eliminate the lower order and obtain this estimate, it's necessary to use some properties such as the compactness of the ball in the weight is solvable space and the compact embedding of these spaces. Of course, it's very technical, but the idea is to suppose that this inequality does not hold, so we have the opposite inequality, it means that there is a sequence of operators of the setup of L and a sequence of function in the solvable space such that we can consider that the norm is one, and this is zero. This is equivalent to consider that this does not hold, and one can obtain a sequence here, a sub-sequence here, conversion because of the compactness of the ball, a sub-sequence conversion in the weak sense, weekly, and since the compact embedding, this compact embedding, this converges strong, U M converges to U is strong, but in W1P, W, in W1P, because that means the compact, that the embedding is compact, and the idea is to use the distance to zero to conclude that then the operator L in U must be zero, U is this function, and by uniqueness, the idea is to use uniqueness and to obtain a contradiction because the norms of this must be one, so U cannot be zero, but it's not just using that contradiction, because here is a sum of norm, so it does not contradict immediately, one has to use this inequality, we have here, maybe it's very technical, I explain it because I already studied the topic because maybe it's not easy to understand it, so we need to use this inequality because if we don't use this inequality, why we broke this inequality? It's needed to use this inequality and also the uniqueness of the problem to lead to a contradiction, right? So this is an standard method to prove the, to eliminate the lower order of the estimate using a contradiction, and I want to mention something about a future work that is, I say something about this that since the operator appearing in this representation that are singular integrals or commutator of singular integrals are not bound to the LPW, we cannot solve here this problem, but we consider a suitable sub-space of L1W that are the hardware spaces, okay? I don't have time to tell you about that, but we are still working on that problem. So that's all, any questions or comments? So you mentioned on your last slide that you're looking for potential ways of exploring the sub-space of L1, weighted, right? Yes. The original problem in the sub-space of L1, just maybe with the usual hardware space has already been considered, I assume, no? Yes. What's the status on this? So what can one say about L1W? Yes, it was proved for, at least for second order derivative, a function F in the local hardware space, because the hardware space has the good property that if you multiply by a good function, they are still in H1. And so when you consider the operator of these types, you consider the H1 of these problems, because this is not necessary in H1. So it was shown that for a function, a function because it's not being general for P equal to 1, we have function not distributions. So for this equation is you can prove the uniqueness, the solvability and the priori estimate. We're investigating some of these problems. Or some more general operators on weighted spaces. Another question or comment? I have a small question maybe, I don't know if this has sense, but at the beginning you mentioned that you work with elliptic operators and that the coefficients, the magnetic of the coefficient is symmetric. Like the Aij is symmetric, but could we also consider the non-symmetric case? If they are a standard condition for elliptic equation, that's the, when you say that something is elliptic, you consider some standard condition that is in the elliptic equation. Okay, no, I just asked because some other problems that work with kind of rough domains, sometimes they look non-symmetric. So I thought that maybe you was doing it for something special. Okay, any other question or comment? If not, let's thank Eugenia. Thank you so much for coming.


You're better off sitting here with one well-aimed shot. So my platoon was tasked with screening or conducting a screening mission at 400 Liggett essentially observing and reporting on enemy movements in the area. We were tasked to go up against an enemy mechanized armor company and essentially report back what we saw and kind of, you know, survey the area, if you will, to and pass that information along back to the battalion. So the small UAS operators being, you know, integrated into the platoon was a huge help. And, you know, I don't really see how we could have achieved our mission without the use of them or, you know, it would have made the mission success significantly harder. You know, and within minutes you can begin surveying an entire area, looking down avenues of approach for any kind of enemy movement. You can record, you know, areas of interest and upon your return, you know, give that real-time information to, you know, that next echelon leader for them to be able to make a more informed decision about, you know, flowing in, following on forces, or it really just gives you better situational awareness about what's going on within your battle space. So I can't credit them enough for, you know, the capabilities that they bring to the platoon. Yeah, so within my platoon we have three staff sergeants, very capable staff sergeants who, you know, have a significant amount of experience under their belt. So compared to a traditional rifle platoon, you know, they are able to kind of do things a little bit more decentralized. You know, I trust them to push out further and with a significant more amount of commander's intent and less detailed instruction for them to, you know, essentially own their part of the battle space. So, you know, I don't think we could have covered as much ground as we did if I didn't have those senior leaders in the positions that they were at. I think this was probably one of the most, this was definitely the first force design exercise that we were operating in such a distributed manner. You know, before we were working in a distributed manner it was a little bit smaller scale. So I think this really put to the test how well we're, you know, we'll be able to operate in the future for, you know, working so far apart from each other and kind of put the platoon commanders a greater challenge of, you know, advice what they would normally face in a normal rifle company. You know, platoon commanders and platoon sergeants and squad leaders are being tasked with things that can affect, you know, strategic level operations. So I think it was a really good opportunity to put everything that we've been doing during this work together and see where we were at and see where we need to improve upon and see the things that we did well and build upon those as well. So, yeah, but I enjoyed it. WTI is weapons and tactics instructor course and essentially they're training to create experts in air and ground tactics. This strip map iterations biggest difference is obviously the terrain, but on top of that there's more stress on the command elements because they're operating in such big distances between the companies. And at my level, the squad level, the biggest difference is probably tactics between training in the desert and training where we're used to training in the woods. The ranges here in Yuma are more enabling for larger assets, especially on squad ranges. For example, here we did a squad range and we implemented 81 millimeter mortars and 50 caliber machine guns into our squad range, whereas in Lejeune it's unlikely that we get that opportunity. It seems like WTI and strip map kind of bounce off of each other in the sense that they are getting to use us for their training and logistically picking up units and moving them. And for us it's good training in the sense that we're getting a lot of airtime and we're spreading out over larger distances, which is essentially what the commander's intent is for this force design in 2030.


I-S-S Agile Topic When teams start off using Scrum, the prospect of getting things done within a sprint can be a huge logistical and practical challenge. Gathering together all of the skills necessary to turn work from an idea into potentially deployable functionality is very difficult. Yet, if teams don't get stuff done within an iteration, then the customer isn't getting any real value, and the team doesn't really know what it is truly capable of. This retrospective assumes that teams are already achieving potentially deployable every sprint, but wish to go even further. However, the same principle of this retrospective can be used for teams who aren't quite getting done by the end of the sprint. In this retrospective, there are actually very few requirements in terms of resources, although it's always a good idea to have some index cards, some post-its, and some marker pens handy. Depending on whether you wish to embrace the multimedia aspect of this retrospective will determine whether or not you need to set up a computer and or projection capabilities in advance. This retrospective typically lasts about an hour. Hook. Welcome everyone and explain that today we're going to look at extending the team's definition of done from potentially deployable to done. And the metaphor we will be using is space. So far, by getting to potentially deployable each iteration, we are in effect reaching Earth's orbit. This is good, because this opens up new possibilities for us. We can launch satellites that help us improve communications and monitor weather, for example. While our ultimate goal may be to somehow colonize space, our next immediate challenge is to construct an international space station. Not only now do we need to reach Earth's orbit safely, but we also need to be able to navigate precisely and dock our components. We're going to play you a short video clip now to create the hook for this retrospective. You may wish to play this video to the team yourself. But why some say the moon? Why choose this as our goal? And they may well ask, why climb the highest mountain? Why 35 years ago, fly the Atlantic? Why does rice play Texas? We choose to go to the moon. We choose to go to the moon in this decade and do the other thing. Not because they are easy, but because they are hard. Because that goal will serve to organize and measure the best of our energies and skills. Because that challenge is one that we're willing to accept. One we are unwilling to postpone. And one we intend to win. To paraphrase the inspirational Kennedy, but why some say done? Why choose this as our goal? And they may well ask, why climb the highest mountain? Why fly the Atlantic? Why put a man on the moon? We choose to extend our definition of done. We choose to extend our definition of done for this iteration and every other iteration. Not because it is easy, but because it is hard. Because that goal will give our users true value and test the extent of our solutions. Because that challenge is one that we are willing to accept. One that we are unwilling to postpone. And one which we intend to win. Ask the team if working out what it would take for them to get to a state of continual delivery is something that they are keen to explore. Provided nobody objects, we can continue. But if there is an objection, listen to it and be prepared to run an alternative retrospective. However, reiterate that there is no obligation to make any changes. All we're committing to here is exploring the potential. Events. Start off by asking the team to capture all of the aspects of the product development lifecycle that they currently incorporate into their definition of done. Ask them to write them on sticky notes and place them on the wall. Acknowledge that this is already no small feat. And explain that in our space metaphor we would probably need extra components, extra training, extra protocols and enhanced communication in order to dock components to the ISS. What other considerations might we need? Write each of these on sticky notes. Ask the team to break up into small groups and take one or two of these sticky notes and explore what they would need for these categories to extend their definition of done. Give them 10 minutes, then ask them to come back to the group with their thoughts. Capture the group findings on sticky notes on the wall. Meanings. We're now moving into the meaning section. And here the aim is to help teams evaluate whether or not it's worth investing their efforts in making this happen. To go back to JFK, he realized that his project would not come for free or even cheap. That budget now stands at $5,400 million a year. A staggering sum. Though somewhat less than we pay for cigarettes and cigars every year. Explain that, of course, we're not looking at expenditure on anything like that scale, but it won't be easy nonetheless. It will take a lot of discipline and effort, so it should be worth it. Luckily, I don't think this would be as much an act of faith and vision as Kennedy was talking about. Ask the team the following questions and capture their responses. What is the value to us in extending our definition of done? What is the value to those outside of the team? How might doing this save us time and energy in the long run? Some benefits that the team might come up with include faster fixes, fewer defects, quicker feedback, shorter time to market, and the ability to scale up easier. Else. As we mentioned earlier, there is a natural variation to this retrospective where the team aren't actually getting to potentially deployable yet. And so you could focus the retrospective on how do we get out of Earth's atmosphere and into orbit. We may also wish to look at where our ultimate goal is. Of course, JFK wished to put a man on the moon. What would that look like in our version of the metaphor? What about after that? Perhaps our metaphor may extend ultimately to colonizing another planet. Explore with the team what the current understanding is about their definition of done and what it could possibly be extended to mean. Decisions. Position the next steps as a NASA press conference, scheduled to reveal the plan for the intentions and next phase of ISS agile. Ask the team to create some fictional roles that team members will be taking on. For example, the chief education officer, who's responsible for training the ops team to deploy the product. The chief communication officer, who's responsible for ensuring all necessary information, both inbound and outbound, is disseminated to appropriate parties in good time. The chief procedures officer, who's responsible for defining the necessary interfaces and protocols for smooth integration of modules. The chief robotics officer, who's responsible for increasing automation. Or perhaps the chief recruitment officer, who's responsible for acquiring new skills and knowledge into the team. Ask each of these new commanding officers, with support from other crew, to come up with a list of tasks for their area that need to be taken on in order to be confident that the definition of done can be extended in this upcoming sprint. These items will be taken on to the next sprint's backlog. You may choose to close the retrospective by replaying the Kennedy speech and asking whether this is actually a challenge that the team choose to accept. One that they are unwilling to postpone and one that they intend to win.


Who is responsible for the killing of a dozen people in Kuru District of Plateau State? Locals say headers are responsible, but the Header's Association says it's untrue. We'll be talking to people from both sides this morning. Now calls for restructuring of Nigeria continue. Different groups and people have different ideas of what they should look like. This morning, analysts will join us to discuss the kind of federalism that's right. A rattle of feet for us Nigerians. And Nigeria's representative admits Africa's Russia beauty pageant steps out to the bloodstained flag of Nigeria. She's getting praise from NSARS protesters. Glad to have you join us in the studio this morning on the breakfast. I am Annette Felix. And I'm Justin. I'm glad to have you join us. Alright, so I hope you had a great night. I hope you're feeling good this morning. I can totally understand you. You look beautiful as always. Thank you very much. And how are you? You're doing good today. It's a rainy morning. It's a rainy season again. Yes. Well, morning in Lagos. I don't know how it is across from Nigeria today. Yes, so our first top trending really just takes us all the way back to 2020, October 20. The night of the reported shooting at NSARS protesters. And we'll see Nigerians, you know, still talking about that, that very day. It seems like a day that would not be forgotten and a hit in a hurry. Because remember that a few months ago, we saw Bobby, one of Nigeria's finest comedians, step out to the head as a ward with a shirt, a blot, a shirt that is designed in a bloodstained flag. Exactly. So now there's a lady. She's representing Nigeria at the Miss Africa Russia Beauty pageant. Her name is Udecha Maka. She stepped out at that pageant with a blot-stained Nigerian flag. And this just tried to make a statement to say, remember what happened in Nigeria. It cannot be forgotten in a hurry. Exactly. Yes, indeed Nigerians are still bothered about the issues that are plaguing the country, the integrity, the issues of policing and all of that. So it was really something when I read online about how she stepped out to the blot-stained Nigerian flag. Some people are saying that is she trying to disrespect the symbolism of Nigeria. That's the green, white, green flag. Or was she just trying to remind Nigerians that just struggles and whatever they came out to do. October last year cannot just be forgotten in a hurry. I totally agree. And this also reminds me of a very popular pageant, Miss Myanmar, stepped out with... First of all, that's what you're seeing on the screen right there. She's just holding up the flag of Nigeria, stained in blood. Basically trying to call the attention of the international community to what happened on the night of October 20, 2020. And moving on now to our next top trend. And this one is something we've seen a lot of times. Regarding a very popular Nigerian figure, not even Nigerian globally, a very popular figure who has then been accused of so many things, either of intellectual property theft, sexual harassment, whatever it is. And this time around it has to do with very beloved Nigerian rights at Chimamanda. It is really very interesting when you find out that people would just want to do so many things just to get some taste of fame or just to get attached to fame. Well, the story right now is that she got some sort of insult over the social media for someone who she appreciated and she took under her wings. She quoted about two stories there. First of all, someone she thought she could relate with when it comes to feminism and all the other issues. So she came out and she took something she doesn't really do ordinarily, but she trusted this person and the person went back later and stabbed her in the back as it were and even insulted her over social media and trust Nigerians and trust the world to believe things. They see on social media even when they haven't really gotten verification or gotten the other side of the story. So Chimamanda has just gone on to say, first of all, to put that out there, Chimamanda put out a statement called violence saying that this young writer, young Nigerian feminist, she decided to mentor this person and the person took advantage of all that and is now coming out to pedal falsehoods about her. And let me read an excerpt here from the quote. Chimamanda says, no, there isn't more to the story. It's a simple story. You get close to a famous person.     So Chimamanda is basically telling her that she's been hypocritical and that she just was mediocre and tried to basically blackmail her and it just went on and on to say that she tried to... This is someone she mentored. She took special interest in during workshops but the person tried to exploit all of that and not the way the writer painted it out to be. And we've seen many things like this. We've seen many, many situations. For example, the issue of Tobore... I can't pronounce the last one. I think Tobore Ovori and Moa Budo of Eboni Live TV. So she's a journalist whom I know personally and when she reached out to me and eventually the media saying that she had written a book or she had gone on an investigation about sex trafficking in Nigeria and she published that for a site called Zim Chronicles and that Eboni Live TV basically just ripped off her story and did an adaptation of her investigation. Which was not actually true.  But she came out to say that she gave her credit. She actually appreciated the fact that it was written. It was not originally done by her. That's Moa Budo. She came out to tell the world that she actually gave her some sort of... Well, I say compensation for using her work. So most times people just come out and say half truth and you trust social media and people to just feed on it to not really get in all sides of the story and before you know it, you'll be labeled as a murderer in the Chimamanda's case and of course someone who doesn't support her people with differences as it is. To be honest, like I mentioned these stories we see every day. Yes, we do. So it all makes you want to really think about mentor-mentee relationships. When you find people who are high and up there, people in places you want to be and you're trying to approach them for that mentor-mentee relationship I'm very sure that everybody has at least one story of rejection or maybe the person you're trying to get closer to basically just gives you the cold shoulder. So when you consider things like this you can get a sense of just why or where that person is coming from. Some people just wouldn't want to get insulted at the end of the day and all sorts of lies are being peddled around the world about you. Now for instance, most people would even claim that you are their mentor when they have not even met with you they could just sort of claim all sorts of falsehood just because they admire you just want to get some sort of attachment based on your name, your face, your popularity and they could just go the extra mile just to get to that thing. What is it about fame? I don't know, people just want it don't they? Well they do but it's really really work hard, do your best somehow you just might get famous if someone wants to mentor you that is fine, if someone says you should use my biography it's fine but if you're not giving approval then don't use it even if you went to use it and someone now called you a mentee of that particular person and you are not even coming out to say that you are not after you've used the person in your own piece of work it doesn't make sense to me like she said it is obscene. Indeed, so our next up trending you know makes me ask you this question if you had dual citizenship Nigeria been one of them would you basically abandon your Nigerian citizenship for another country or would you abandon another country for Nigeria? Now this is a very important question because someone asked herself that question and gave very very daring answers and her name is Uninechi Dzog so she's a Nigerian football player she's basically switched her allegiance from Switzerland where she was born and is a citizen to play for Nigeria she's saying she wants to play for Nigeria she wants to dedicate her time to the motherland contribute to the development of sports in Nigeria and I mean that's a picture of her with her team she's on the far right they're smiling to the camera club in green exactly piece of white it is really very interesting you know what with all that we have had and read about Nigeria and how Nigeria looks at you know in the phase of things in the international in the community of nations most people would just want to dump your citizenship and gladly you know get out of the country and maybe acquire visas and look for some sort of residency there's some illegal migration by the thousands daily and people even die in such imperilous journeys in this case you have a you have a Swiss citizenship and because you're born Niger but not really most people would not want to leave you know all the comfort and the big things in quotes that come which are you know leaving abroad and being attached to those particular you know countries would you Aneta? I would need to ask myself that question but I definitely know that if you wave a foreign passport in the faces of many Nigerian youths they'll gladly jump on it about 90% of them 90% of the time 90% of the time just like this but just you know from what we know right on the streets that lots of people would grab that opportunity to leave the country most times you don't blame them because we've been plagued with a whole lot in Nigeria is it insecurity is it use unemployment is it people going to bed without even eating there's poverty in the land there's hunger there's a whole lot and most times these things are not really what you would really want to say that I am so soon Nigerian to Nigerian, I believe in the cause of Nigeria, I believe in the Nigerian dream, whatever that is. Yes, yes, definitely. So most Nigerians want to jack back. But then I'm still Nigerian, I believe in Nigeria. Are you sure you're not serving for your IELTS? You have to answer that question. So Onyichi's dog, you know, basically has picked a side and it's that of her motherland. Also, another reason why Onyichi's dog is trending is because of her beauty. I mean, take a look at the pictures we've been showing on your screen. She is, exactly, she is one beautiful woman.  Are you single? Are you married? Do you have a man in your life? So questions like that, you know, Nigerians just like to, Nigerians just like to feel good. You know, awesome, man. We always want to be attached to beautiful ladies and that. So you don't blame us when we see fine ladies and that. Of course, when there's no working on the thing that you just, you know, tempted to ask her, are you single? Do I stand the chance? You know, all of us, all of this just come to mind. So you don't blame us. It's just a Nigerian thing. You think so? It's a human thing. It's a human thing. Everybody wants to fix something. Who doesn't want to be attached to beauty? Who doesn't want to be beautiful or beautiful ladies around them? Exactly. I do. Justin for the ladies. Well, you can say that again. All right. Let's take a break here and we'll return with off the press where Mr. Ade Mola Kimbola, the publisher of the podium media is standing by to help us analyze the papers this morning.