Today I'm gonna show you how to flag any folder as a favorite. What's up YouTube? My name is Adam.  Today I'm gonna show you how to add any folder in your library as a favorite so it makes it really easy to find and this is something that you will probably use if you're someone who has a lot of folders in your library but you often frequent one particular folder quite a bit or maybe you know a few folders quite frequently or maybe you're working on a project and so you're constantly finding yourself going to a folder over and over and over again and instead of like scrolling through all of your folders in your library you want a quick and easy way to go to a folder. Another thing that's really nice about flagging a folder as a favorite is now with this new search feature we talked about the search feature in a earlier video this week I'll leave a link in the card up above if you haven't seen that video but now that you can search for folders it's really easy to search for folders that you've added as a favorite. Let's jump into Lightroom and show you how to apply favorites so that you can easily look for them. It's really simple to do so right above your film strip there's a little black section that separates the film strip from your grid view or your loop view and that is right in here and all you have to do is click in that area one time and it will open up this little dialog box and this is a dialog box you may didn't even know existed but adding folders to your favorites is something that you could have done in the past so if you're someone who is using an older version of Lightroom you can actually add folders to your favorites but with the new update for Lightroom Classic users there's an easier way so whether you're on the old version or the new version you can do this. If you are on the older version of Lightroom all you have to do is look under your recent sources and it would show you all the folders that you've recently opened in the past and if there's a folder in here in particular that you wanted to add as a favorite all you had to do was click on add to favorites and then the next time that you click in this little black section you'll see now that a folder has been added to the favorite source. If you want to remove it from your favorite source you just go down here and click on remove from favorites but with the new update to Lightroom Classic there's an easier way to do this you basically just right click on the folder that you want to mark as favorite and choose mark as favorite and you can see here that it puts a little star icon in the bottom corner of the folder that star means that your folder has been added as a favorite but once it's not a favorite anymore or you're done in a particular project and you're not going to this folder anymore just make sure you go in there and mark it not as a favorite anymore because this is meant to speed up your workflow and make it easy to get in and out of folders but as I mentioned you can search for stuff so if you go up to your search bar right above your folder section and you click on the magnifying glass this is the key you click on the little magnifying glass in the search area you'll see that you can search for either all folders or you can just search for your favorite folder so if I click on favorites it's only gonna pull up the one folder that I just favored it a second ago but if you had two or three or four folders favorited this makes pulling up those folders and those folders only really quick so you can get in there get to work hopefully you found this tip useful I know that I think this is something that I'll use quite frequently in my own workflow and hopefully you will too let me know in the comments down below if you think that this is something that you use or maybe you won't use I don't know either way thank you so much for watching my name is Adam have a great day everybody


Rickland, you cannot say that and be serious. People can cry, oh that yeah I replied that guy. Serious. They're actively genocide in Ukrainians by the letter of the law, by shipping them out and putting ethnic Russians in. The only reason they haven't done it to the rest of the country is the fact that they don't have control of it. Crimea, Crimea was only handed over in the boundary of Ukraine because of Stalin in 1954, if I recall correctly. That's 95 percent Russian, Russian ancestry for centuries. And because of the USSR for the bureaucracy of it he drew the line saying Crimea, Ukraine. We'll put it under the umbrella of Ukraine because of bureaucracy.


So while people are sort of digesting their coffee in that, one of the challenges Debbie that you alluded to and Wendy as well is that these do require very large collaborations and publication is sort of the coin of the realm here and there is great angst in terms of how one gets appropriate credit when you are the 103rd author in a 200 author paper and maybe I guess you, Debbie, you were involved in several of those from the hat map and that. Can you give some comment as to how the people who were involved in that, how that has affected their careers and... It hasn't really affected their careers that much because when they've participated, they've been full participants. So a hundred people worked with them on this and they could ask any one of a hundred individuals for a reference letter for promotion or any other thing and they could, so I think that, and people know exactly what they contributed because it's out in the open and it's clear. Also, people can come in from the outside and say, hey, I have a great idea. I want to try this and there is a feeling of, ooh, this sounds really good and people inside say, well, let's have this person come in and collaborate with us too. They don't even have to do that. They can just do it on the outside. The other thing is that the data goes in the database and anybody can touch it before a publication. What I want to say is that all the journals have also been respectful. You may not know, but there were many papers submitted on the human genome sequence before the people who sequenced the human genome actually had a paper, but they were similarly rejected from the journals. So this does happen and by forming large consortiums, the major journals are all aware that this is an issue and they make sure that they quality control what comes into them from a particular group. So I think that people should think about that. Do you want to comment on that? Well, I think that the promotions committees are hopefully understanding the changes in the way first authorship works and understanding that these are large collaborative studies that many people have contributed significantly to and that now with NIH allowing multiple PIs on a grant, we're also seeing more and more manuscripts with multiple first authors with the asterisks saying these two or three authors contributed equally. So I think we're just evolving in that arena and I'm optimistic that it will continue to evolve as we continue to have these large collaborative efforts. Hi, Vasant from Framingham. Just to add to the discussion, I think one of the strata we need to protect are the young investigators because they are evaluated on the basis of first authorship or whatever. So the consortium model works very well for the senior people. So in parallel with the progress, I think some of the evaluation criteria for the young investigators also need to be changed so that they are not just based on the number of first author publications they have. Otherwise, they will be left out of this important field, which is moving ahead. And some of our fellows have raised this issue that when we participate in this, what do we get out of this? Would we be the first author? And that really is not very practical when we are talking about the consortium model. So it would be nice if there are strategies to protect the career of young investigators who are interested in genomics and population epidemiology. One other comment is that we also realize that there are going to be so many things to write about because there's so much data that there should be something for everyone to be able to have a first author paper on even if it's not the major finding that comes out of collaborative work. So I think we find that in these, like in MESO, we have hundreds of paper proposals, and then we count the number of papers that come in. So there's a wealth of potential opportunities, and so although we all sort of worry about it, in reality there's more than enough for everybody to do. I just want to reiterate that last. There are so many paper opportunities in these types of, there's no limitation. And many of the junior people in genomics have as many first author publications as they can write. And they do participate at a level as junior investigators that, frankly, many people would kill to be a part of. So I think there are many advantages to young investigators. Yes, on the floor, please. Hello. Dr. Dickerson talked quite a bit about non-coding sequences that may regulate transcription and translation. And I was wondering, in understanding these better, what are new techniques that people need to think about to start understanding the importance of those regions? Well, I think we need to tackle some of those regions. Unfortunately, we tend to tackle those regions that have a gene first. We often can predict and don't really care about mechanism, even though we know mechanism is important for treatment. For example, in the CAD link that everybody saw, you might have promised I'm not. There's a microRNA in a couple genes, but it's mostly non-coding sequence. We don't even have a complete picture of the sequence variation in those regions. For example, we've never sequenced a complement H factor in multiple individuals with macular generation with complement H. So we need to go in and look at the variation in this region, get a more complete spectrum, and then begin to apply some of the tools that we know, like transcription profiling across it, but we can't get all the tissues, predictions of transcription profiling, multi-species comparison, and begin to actually do animal model studies on some of these if we can. And those are very costly studies, so we need to know a lot more before we do that. But I think with projects like INCODE, we go across and apply everything that we know in a very large-scale way. It won't be perfect, like the HATMAP, but it will capture a lot of what we know. But we need to know more about even the variation that we have inside the genes and regions to explore that. Debbie, could you comment a little bit on how population studies might be useful in this kind of work? For instance, in the sequencing you talked about looking at the tails of the distribution, but are there other ways? People who have one variant who then you kind of consider them to be almost a knockout and look for, you know, a human knockout and look for other sequence variations in them? So it's unclear actually how to do these sequencing right now. Even when you have a hit, do you just take your affected allele from multiple individuals and compare it to what you consider the wild type and what is the wild type? I'm not sure I know what a wild type human is. When I look around the room, boy, we're wild. I'll say that. Me too. And so I think that knowing what to choose, there are differences in if there are things, there's something called there are ways using multiple populations. If you have a phenotype that's present in multiple populations, for example, you can, we know that the association between SNPs in different populations, for example, individuals of African ancestry have less correlations among SNPs than individuals of more, of isolates from that like Europe or Asia. And so you could think about looking across populations to narrow regions. You can think about looking, but I think early on we should sequence as many people as we can afford. And I think that's one of the areas that people are getting all these hits, but they're not really thinking about how to actually go in and look at these regions. We're just going to start piling these things up and not really have a clue in terms of what the actual functional alleles are. And I think people will believe that the functional allele is the SNP that was on that chip. And most likely it's not. That's an excellent point. And I think, too, we ought to be thinking about how we can use population studies to do, in addition to sequencing, are there ways to do some transcriptional profiling? And I confess, I don't know what samples you need to even do that. So are those just DNA samples you could get, or do you need to stimulate someone? Or is it RNA? No, no, no. Most of the time what we have is B cell lines sometimes, or blood. And that's just one type of tissue. We don't often have cardiac biopsies of the same people that were studying for cardiovascular disease, for example. It'd be great if we did. I'm not sure we'd get too many donors. But we do need to think about making repositories of some of these tissues over time so that we could, for example, even if we had 100 individuals that we had a complete genome scan on, we could think about expression profiling on those 100, and that might contribute more to the knowledge database using the best technology of its time. So these could be independent studies that we could develop over time. We don't know what the appropriate tissue is, because we're not really sure when we get our hit where it's expressed in, or if it's even expressed in the genome, although we know expression is very pervasive, almost every region is expressed. And we have no idea if that's meaningful. Yes? That actually brings up my point, my question, which is what about the role of abnormal methylation and these epigenetic kinds of mechanisms? And this is something that the ENCODE project is getting at, and profiling on a large scale. They have the patterns of methylation. They have patterns, they have almost every pattern on 1% of the genome. But what they did when they were doing that was get away that they could, and they're now thinking about how they could put all this across the genome. Now, methylation also is very tissue-specific. I mean, there are many things that we need to get at in tissues that we're not getting at right now, because we don't have the archives of the tissues. But I think if we do even have a small semblance of these tissues and could develop it over time, it would be extremely important. I think you'll see eventually some time, I wouldn't be surprised if there was something that allowed people to donate their bodies to science for like an anatomy, a genome anatomy project or something like this. Yes, please. Can you comment on the portability of the HATMAP for the populations not represented? Yes. So there's a couple things I should say about the HATMAP. The HATMAP gives very good coverage of many different populations and people have looked at coverage going to different populations. Is it effective for other populations? It drops in a little bit in its ability, its transability, maybe by 10%. But what's important to know is the chips are not actually mirror images of the HATMAP. They also have a drop in how much information that they capture. So you get obviously drops in the amounts, like 10%, 20%. The coverage is not as good for all populations. The drops are about the same, though, from where you're starting from. But obviously we have significant power to detect. If we have a scan on individuals, we can only, we can add to that and get better coverage as we understand more about the immunogenome. The HATMAP is just a starting point. It's not the end point. We can just continue to add to our data sets. But people have looked at portability. It's extremely portable. And we might note that the original three HATMAP populations, which were the CEU population, I forgot what it stood for. But anyway, the European Caucasian population, the Yoruba in Nigeria and the Hanshinese and the Japanese were the original three populations. But those have been recently expanded by seven other population samples. Again, I believe this were all trios, if I'm not mistaken. They could be. But it was to get at this issue of trans-ability. Many people though have tested this independently in their populations and have seen that it's quite transferable and I believe it will be for most of those populations. But we will have actual tests of it, which is really important. And keep in mind, too, that the HATMAP was looking at common variation and so that's probably why it's so transportable and it may be that some of the rare variants will not be as... I just want to bring up something, if I can, on the CEFs. The CEFs actually, for those of you who are interested, were the original families that were used to build the linkage maps of the human genome. So they became carried over into building the association maps of the human genome because we had cell lines from these individuals and unlimited DNA resources. And the individuals had mostly consented to do the types of studies that we had already planned to do. So it seems like an unusual population, but it was the population that was used for the linkage maps that were developed. Yes, please. What about the centromeres and telomeres? Is ENCODE going to try to tackle those at all? The structural variation project may get at that because there's a lot of structural variation there. There's probably the most difficult regions for us to tackle, and they won't be great regions for these whole genome sequencing strategies that we're coming up with because they're the most repeated. Those regions are extremely important. We're getting more and more information, but those are the tough regions. What you should know about genomics is we do whatever we can do in the quickest way possible. And getting 80% of the data is always easy. We have about 70% that takes the next 20 years. It's like we have... Jim could probably tell you, what percentage of the genome do we actually have in the database now? It's not every single nucleotide, but... Yeah, right. So, because we're missing some sequences around the telomeres and centromeres, we know genes exist that are not actually on the reference sequence that we have. Where are those genes in the assemblies? We know that it's not perfect. So over time, though, we go after more and more of those regions. And actually, those regions are the starts of some great genomic careers. I can think of a young person who started out working on these regions of structural variation who actually was an assistant professor in the genome, Evan Eichler, and now is the leader of the structural variation project because that was... That's how he started. What? That's how he started. Yeah, that's how he started. And because he was looking at these regions of the genome that nobody else wanted, these segmental duplications that everybody wanted to stay away from, and he made his career around that. So, you know, there are lots of exciting and interesting findings that will lead to brand new projects in areas. And I think it's the same way for epidemiology. I think that these datasets provide such an opportunity for young people to really find an area that can be their own and that they can take forward. I would agree, and maybe even come back to a comment you made at the beginning of the discussion session. Debbie, when you said that you felt participating in the HATMAP had no effect on people's careers, I assume you meant had no adverse effect on their careers. Because I can think of several very young folks, Mark Daly, Paul DeBacher, Isaac Peer, and that who basically, you know, had questions that they sort of wanted to play with the data and late at night or whenever else, came at it and looked at it in a very different way from anyone else and really came up with some wonderful insights and perspectives. And you don't have to, you know, even come up with fantastically brilliant things to be able to contribute dramatically to just looking at associations and controlling for confounding and other things that epidemiologists do extraordinarily well and that we have the patience to do because, you know, the substance of many, many careers. And so I think rather than being concerned about our young investigators, we should find ways to fund them to do these kinds of analyses and encourage them to put their data into databases so that others behind them can do the same kind of work.


In that Morocco, where firefighters are forced to bring a fire under control in the outskirts of Casablanca, the country's most populous city. The fire started in a gas warehouse on Thursday evening in an industrial area north of the city. Five tankers, each with an eight ton load, were destroyed by the flames. An investigation has also been launched to determine the circumstances of the fire.


This is Classics of Liberty from Libertarianism.org and the Cato Institute, narrated by Caleb Brown. Today's classic is Abram D. Smith and Nullification, Part 2. In this second of two selections from Smith's decision in Rebooth, 1854, Justice Smith verbally elaborated before the court his earlier written decision. Smith argues that state judges have obligations and duties to protect the rights of citizens regardless of the unconstitutional pronouncements of the Supreme Court and the Congress of the United States. He maintains that the parties to the Constitution, the states, the people, and the United States have equal rights and duties to interpret the document and protect the party's rights as need be. Though Missouri pronounced Joshua Glover a slave and the national government unconstitutionally employed slave-catching marshals, Smith declares it his duty and right as a representative of Wisconsin and the people of Wisconsin to resist both any and all actions and legislations that violate the terms of the compact. In Rebooth, rocketed Smith to superstardom in the emerging anti-slavery community and steadily progressed to the United States Supreme Court. In Abelman v. Booth, 1859, Chief Justice Taney and a unanimous court overruled Smith. Meanwhile, a land scandal rocked Wisconsin state politics and upset the careers of many, including Smith, who stood accused of accepting $10,000 in railway bonds as a bribe. While he admitted to taking the bonds, he claimed it never influenced his opinion regarding land grants to the railroad. While Smith retained much of his anti-slavery fame, his credibility never recovered and the Democratic Party nominated a new candidate to Smith's position on the court. He was thus rendered a man without a party and though 40 Republican editors throughout the state endorsed his nomination on their own 1859 ticket, he received only 15 of 61 votes in caucus. From December 1860 to early 1862, Smith published the Milwaukee Free Democrat and co-authored the Direct Tax Act while visiting Washington in late 1861. The act levied land taxes in rebellious states. If landowners failed to pay, the national government assumed control of the property. The following year, Smith joined the Direct Tax Commission in the South Carolina Sea Islands. He arrived at his post in the Concord Islands to a gaggle of northern missionaries and a progressive atmosphere of freed slaves engaging in education, development, and training projects. Smith adjudicated disputes relating to the divisions of property and delinquent taxes, advocated full citizenship for freedmen, and he delighted in General Sherman's Special Field Order Number 15 in early 1865 which granted huge tracts of land to freed slaves throughout the South, believing that, quote, the betterment of the freedmen was inherently tied to their ability to own land in the area where they had been born and raised. He struggled with severe alcoholism and probably harder drug use that continually impacted his job performance. The man without a party, the former president of Canada and the one-time anti-slavery hero of states' rights, died in 1865 and root to New York after being relieved of his commission in disgrace. Abram D. Smith on nullification, part two. One great aim of the founders of our government, among others, was to secure beyond contingency personal liberty and to protect and preserve, as far as practicable, the independence and sovereignty of the respective states without whose agency such personal liberty could not be protected and secured. As far as was consistent with the practical efficiency of the federal government about to be organized. It should be remembered that error does not become truth by being often repeated, nor does truth lose any of its force or beauty by being seldom promulgated. Nor does vice become virtue by persistence in its practice, nor bad government grow better by acquiescence in its evils. Nor where a people have adopted a written fundamental law for the government alike of themselves and their rulers, does the infraction of that law become healed by a denial of its occurrence. I am willing that the decision of the Supreme Court of the United States in every case determined by them, within the scope of their jurisdiction, should be regarded as full and binding authority, as the law of the particular case so determined. But when it is strenuously contended that I am compelled to adopt their interpretation of the Constitution and laws of the United States and of their own powers and the powers of Congress without thought or inquiry to take what is written as the end of the law simply because it is written, that my own conscience and oath must be tamely subjected to the prescriptions of another tribunal governed by the same laws and bound by the same oath, notwithstanding the high respect approaching even to veneration, which I have for that high tribunal. I must be permitted to say that no man or body of men is made by the Constitution the keeper of my conscience, nor does it impose upon any man or body of men the fulfillment of my official oath and obligations or the power of releasing me therefrom. When duty and obligation require a steady and undeviating adherence to authority and precedent, no one will be more firm and anxious in insisting upon such adherence. But when the like duty and obligation require a departure from such precedent and authority in obedience to a paramount law, the fundamental law to which each and all are equally bound, I hope to be found just as firm in my adherence to the latter. The Constitution of the United States is, in its more essential and fundamental character, a tripartite instrument. The parties to it are the states, the people, and the United States. The latter is indeed a resulting party brought into existence by it, but when thus created, bound in all respects by its provisions. It is practically represented by its several departments, deriving their power directly and severally through its respective grants. It is derivative, not original. Previous to the operative vitality of the Constitution, this third party to the instrument was non-existent, and, of course, powerless. The other two parties, the states and the people, were pre-existent, endowed with all the essential elements of sovereignty. One great and fundamental mistake has been made in respect to the second party to the Constitution, namely the people. This party here spoken of cannot be considered as the people inhabiting the whole territory embraced within the boundaries of the original 13 states, as operating en masse as one undivided and indivisible community. Previous to the formation of the government of the United States, there was no such political existence. The people mentioned in the preamble to the Constitution, and often referred to in judicial discussions, must, it seems to me, necessarily mean the people of the United States. That is, the people of the several states united. So many uniting as were deemed a sufficient number to warrant the institution of the new government and render safe the delegation of certain powers before possessed by the respective states. The state governments pre-existent as the people of the respective states did they adopt the Constitution. By the authority of the states were the people called upon to adopt or reject the Constitution. By the people of the respective states was it adopted, and when ratified by nine states, not a majority of the people of the Union to be formed, was it to become operative. The states as such were distinctly recognized through every stage of progress, from the inception of the consummation of the plan of Union, and through the state organizations only could the first step be taken. And through those organizations only can the people of the Union now impress their will upon the measures or action of the government. Indeed, the federal Constitution provides no mode by which, in any case, can the people of the Union affect the federal government. By through the state organizations and by the instrumentalities furnished by the governments of the respective states. The states derived not one single attribute of power or sovereignty from the Constitution of the United States. On their separation from Great Britain, they were each sovereign and independent, as completely so as the government from which they had revolted. They retain all the attributes of sovereignty which they have not delegated or relinquished. Nor does the Constitution address itself, in a single instance, to the people of the whole Union as one indivisible community, but always to the people or to the constituted authorities of the respective states. But the new entity brought into existence by the Constitution does derive every jot and tittle of its power from that instrument. Without it, the state existed and performed all the functions of government. Without it, the federal government had not a shadow of existence. If that instrument ceased to operate, the states would move on, performing their present functions and probably resuming the powers before delegated, but the government of the Union would cease altogether. I make these remarks because persons in their zeal for federal supremacy seem to have lost sight of the true relations subsisting between the Confederacy and its members. The rights and sovereignty of the latter would seem to be sacrificed to the exaltation and glory of the former. But returning to elementary principles, it will not be difficult to determine the just rights and limitations of both. Test the third clause of the second section of the fourth article of the Constitution by this rule. No person held to service or labor in one state under the laws thereof, escaping into another, shall in consequence of any law or regulation therein be discharged from such service or labor, but shall be delivered upon claim of the party to whom such service or labor may be due." What power or authority did the states relinquish by this clause? At most, the right and power, if you will, to enact any law or regulation by which such escaping fugitive shall be discharged from such service or labor. They also covenanted that the fugitive should be delivered up, but did they delegate to the federal government the right to enter their territory and seize him? Did they authorize that government to organize a police establishment either permanently or temporarily, armed or unarmed, to invade their territory at will in search of fugitives from labor ranging throughout their whole extent, subject to no state law, but enjoying a defiant immunity from all state authority or process while executing their mission? Did the states relinquish the right or power to prescribe the mode by which they would execute their own solemn compact in delivering up the fugitive? Every just regard to dignity and self-respect on the part of the states forbids it. Every sentiment of delicacy, not to say justice, on the part of the national functionaries should revolt at it. In Virginia, he may be indeed a chattel, but in Wisconsin, he is a man. The laws of Virginia make him a chattel there, but the Constitution of the United States and the laws of Wisconsin regard him as a person here. Under the Constitution, the fugitive leaves the attribute of the chattel behind him in the state from which he flees and goes forth as a person. The law which makes him property in Virginia does not go with him beyond the limits of that state. On his escape from such limits, he ceases to be property, but is a person liable to be reclaimed. The person may escape, but the property cannot. The states are no more bound to recognize the fugitive slave as property than a fugitive apprentice as property. The relation of master and servant is recognized so far and so far only as the obligation of service is implied from such relation. Even such obligation is not recognized as full, complete, present and operative, but as attaching to that relation in another state. So much of the law of the state from which he fled as required of him service to his master there is to be regarded, and from that obligation of service imposed by that law, the state may not discharge him. The law of Virginia which requires of the slave service to his master is recognized as the law there, not here. We may not discharge a fugitive from the service which by law he owes in Virginia, but by that law he owes no service here. The master may capture him in Wisconsin. We must deliver him up to his master on the establishment of his claim, but his master has no right to command his service in Wisconsin. He must not beat him. He may take him back to Virginia, but he cannot command his service here. When he gets to Virginia, he will owe service by the law of that state, but not till then. By the law of that state, he owes the service and by that law only. That is the law of Virginia, but not the law of Wisconsin. If the master demands service here of his fugitive and beat him for disobedience, he is punishable by our laws. Nor could the master, having captured the fugitive in this state, sell or hire him to another. He has just the control over him requisite to his extradition and no more. If the free states are bound by the fugitive slave clause of the Constitution to recognize the full and complete rights in the owner of the fugitive slave as property to the same extent as they were recognized in the state from which he escaped, then it will soon be claimed that the free states may be made a highway for slaveholders traveling with their slaves, a thoroughfare for internal slave traders over which to transport their living chattels from state to state and state sovereignty itself must succumb to the slaveholders' authority. The simple answer to this that the Constitution does not guarantee the right. It guarantees no right. No power is granted in the Constitution to the federal government to enforce or guarantee any right in regard to fugitive slaves or any other slaves. Every day's experience ought to satisfy all that the states will never quietly submit to be disrobed of their sovereignty, submit to the humiliation of having the execution of this compact forced upon them or rather taken out of their hands by national functionaries. And that too on the avowed ground that they are so utterly wanting in integrity and good faith that it can be executed in no other way. On the contrary, if the federal government would abstain from interference, the states would adequately fulfill all their duties in the premises and peace and order would be restored. But they will never consent that a slave owner, his agent or an officer of the United States armed with process to arrest a fugitive from service is clothed with entire immunity from state authority. To commit whatever crime or outrage against the laws of the state that their own high prerogative writ of habeas corpus shall be annulled, their authority defied, their officers resisted, the process of their own courts treated with contempt, their territory invaded by federal force, the houses of their citizens searched, the sanctuary of their homes invaded, their streets and public places made the scene of tumultuous and armed violence. And state sovereignty succumb, paralyzed and aghast before the process of an officer unknown to the Constitution responsible to its sanctions. At least such shall not become the degradation of Wisconsin without meeting as stern remonstrance and resistance as I may be able to interpose so long as her people impose upon me the duty of guarding their rights and liberties and of maintaining the dignity and sovereignty of their state. That was Abram D. Smith and nullification part two. Find more classics of liberty at www.charingism.org


What is going on guys welcome back to the channel critical overlord here, so we have some news regarding Jurassic World Dominion I haven't done a Jurassic Park video. I believe ever Really on this channel, so I know this is a breath of fresh air for a lot of you Hopefully I attracted some Jurassic Park fans. Hopefully a lot of my subscribers are already Jurassic Park fans I'm gonna just try to venture out and discuss other film franchises while we were remain Waiting in in wait waiting for news regarding the primary series I've been following for the past couple of months of scream Halloween Theory videos on Jeepers Creepers, but to get into what I want to talk about We got confirmation that a specific major character from the original film Louis Dodgson who was responsible for Having Dennis Nedry give him embryos in that shaving cream can back in the original movie Who was who was a competitor working for a competitor rival company against engine Bioson? He will be returning that character will be returning in Jurassic World Dominion He will be portrayed by Campbell Scott, and I'm really excited about this because Already, we know that Jurassic World Dominion will have a lot of the returning a lot of the main Characters from the first two films and the third one Jurassic Park 3 We have Sam Neil coming back to play Dr. Grant We have Ian Malcolm back once again being portrayed by Jeff Goldblum, and we have I Can't think of her name, but the blonde the blonde character from Jurassic Park I literally just went dead in regards to what her name was She's gonna be back as well with Dr. Grant and Ian Malcolm So it'll be really interesting to see how they are introduced to the film and whether or not this Bioson returning character Dodgson whether he will be the one that brings them into this into the fold in regards to our new characters Because the Jurassic World series it's not being led by the characters from the original trilogy It's being led by Chris Pratt and Bryce Dallas Howard. They play the characters of Claire and Owen Who are they've been like the one thing I will say about Jurassic World this new trilogy the consistency of the work of the cast The original trilogy the first three Jurassic Park's We have cast members that returned Jeff Goldblum returned from the first movie to the Lost World Then he was gone from the third Sam Neil returned from the first to the third like I Just appreciate the fact that with Jurassic World we have now a consistency in regards to who our lead characters are With this duel of Bryce Dallas Howard and Chris Pratt. They've been very good in their roles For the most part, I think there's new reimagining or continuation from Jurassic Park 3 with Jurassic World and Jurassic World Fallen Kingdom have been decent Jurassic World being the better of the two Jurassic World Fallen Kingdom was kind of a setback in a It basically was a glorified Retread of the Lost World Jurassic Park, which is fine, but I just hope With Jurassic World Dominion we get some fresh new things to see and I hope it's not I hope Jurassic World Dominion somehow is not a recreation of Jurassic Park 3 because if that's the case I'll be severely disappointed just because the fact that in the the Fallen Kingdom the dinosaurs at the end they escaped they are basically set loose and We actually had a short film come out recently from Universal Showing us what's happened in between Fallen Kingdom and Jurassic World 3 where we had a family being terrorized By a dinosaur in their trailer while they were looks like they were camping So I'm really excited to see how Jurassic World Dominion will build upon that I'm hopefully getting a rise of the planet of the apes type of movie I'm hoping for it to be very action-packed I can't wait to see how this character of Dodgson will factor into the plot because he Was a lot of people included myself after seeing the demise of Dennis Nedgery in the original movie after that Velociraptor took him out in in the in the truck or in that Jeep from Jurassic Park It was very interesting how that shaving cream can was highlighted and pinpointed on after his demise when it was rolling down Into the mud and it was kind of teasing us that at some point the series would revisit that character and that concept And go back to biosyn and possibly make them Relevant again somewhere down the road now that didn't happen in the Lost World or Jurassic Park 3 It did not happen in Jurassic World It did not happen in Jurassic World the Fallen Kingdom, but it's nice to know that we will be getting Louis Dodgson back in Jurassic World Dominion what role he will play honestly I think he's gonna be one of the main villains if not the main villain will be Revisiting what they've done with those embryos if they ever managed to get other ones because to my knowledge Dennis Nedgery he never From the original movie we saw he never got to deliver those because the Velociraptor took him out What I think will be interesting is if people from biosyn maybe visited that park Undetected after it was a after it was abandoned and John Hammond decided not to go through with it after Ian Malcolm and the other experts who he had given who he gave a tour of the park to decided to not endorse it and Decided to convince him to not go ahead with the idea. I think it'll be very interesting to find out that Biosyn some people in connections with biosyn who worked for that rival company who was trying to catch up with the engine and the Recreations of dinosaurs maybe they went there and they retrieved that shaving cream can that Dennis Nedgery Unfortunately was unable to deliver with those embryos. Maybe they've done something with them and that'll factor somehow into the main Conflict with Jurassic World Dominion I can't wait to see what happens with this film. I know production is supposed to start up They're looking to start up production in the coming weeks or coming days if they haven't already I know they have their all all their new guidelines to accommodate with the pandemic. So It'll be really interesting to see once the first few set photos come out What actually we'll have a better idea an understanding of what we can expect from Jurassic World Dominion I All to the best of my knowledge. I'm expecting a rise of the planet of the apes type of movie I'm really am I'm expecting this movie to be one of the best what I'm disappointed with though is the fact that It's not going to be the final chapter if I'm not mistaken recently the director of the previous two movies He has come out and stated that Jurassic World Dominion will Hopefully they're hoping this would like bridge us into a new wave of films and I might come on now. I thought I literally thought Jurassic World Was I thought the whole point of that 2015 film was to set up a new trilogy Not to then I mean, maybe maybe I'm Underestimating or not understanding fully what he's saying. Maybe they're gonna end that trilogy With Jurassic World Dominion, but then Jurassic World Dominion will then lead us into another trilogy. That's the part I'm not really too satisfied with I don't want to go into another trilogy I think Jurassic Park and this whole idea in this concept has run its course I think it's time to let this rest and I know that coming from me Someone who's a big fan of Michael Myers someone who's a big fan of Jason Freddy all of those horror icons that have a long resume of films that far exceed four movies like What Jurassic Jurassic Park has five movies actually far exceed five movies, but at the same time I don't think those those franchises have gone on this long. I don't think that Michael Myers should have over eight movies like it does. I don't think The Nightmare Nam Street series should have gone on past Dream Warriors I don't think Friday the 13th should be as long of a series as it is because now I feel like at some point we need to favor quality over quantity and I don't get that from Jurassic World when I hear stuff like this It's great to me that you want to keep this franchise in this universe alive because I love this world and I love these characters and I really am entertained by What we now have to offer to the franchise from these last two movies But at the same time it'd be nice for someone to finally Pick up. That's why I really have an appreciation for what David Gordon Green wants to do with Halloween He wants to have his trilogy of movies and he wants to be done He doesn't want to bridge out and do all these other things I just appreciate when filmmakers have an idea They map it out and they execute it if everything goes right on the first stage They move into the second stage and then they get to finish out their plan with the third stage With a night trilogy or however many movies you have planned to plan to go With the news that they don't want to even they hope that Jurassic World Dominion will bridge into another trilogy Or just more movies that tells me they don't even know when they're going to end this series They don't know when they're going to end it. They probably have no plans to end it but I'm really excited. I can't wait to see all these returning characters come back Can't wait for production to start so we can hear more details about Dodgson and or doxin. I think that's how you pronounce him. I'm not really too sure But I'm really looking forward to seeing how biosin and that character factor into this And I can't wait to see what Jurassic World Dominion has to offer And I hope it's a level up from Fallen Kingdom because again Fallen Kingdom was a Was a downgrade downgrade compared to what we got from the first Jurassic World from 2015 But let me know what you guys think about all this down in the comments section below if you haven't already Make sure you subscribe to our post notifications. That never is a video in the description I have links on my social media accounts on my facebook twitter and instagram You can message me there to let me know if there's any movies news or views you'd like me to cover in the future We'll add it in the mind guys. I will see you in the next video


I have not been this excited for a jet season in a very long time. And rightfully so, we have a young, deep team that showed a lot of promise and we finally got our quarterback. I'll start off the season with a big Monday night win versus the Bills, but I'll give the Cowboys B2. Go one-on-one again in weeks three and four with the Patriots and Chiefs. This is a tough schedule. We'll start the year. Broncos will take a win. I'll give the Eagles one. And then a nice three-game win tree versus the Giants, Chargers and Raiders. We split with the Bills, so they take this one, then we'll take the Dolphins home game. The Falcons and the Texans are putting in the win column. We split with the Dolphins, so they can take this one. And then close out, leaving Washington going down in Cleveland and then sweeping the Pats for the first time since 2000. Over under is set at nine and a half wins. This will put the team at 11 and six on the year, so I am definitely taking the over. And we are going to end the playoff throughout this year.


Hi, I'm Ann Ruffing from Sandia National Laboratories, and in this video I'll be discussing our recent publication in biotechnology and bioengineering, entitled Physiological Effects of Pre-Patiacin Production in Genetically-Engineered Cinecococcus Elongatus 7942. Cyanobacteria like Cinecococcus are among the types of microalgae under consideration for biofuel production. Like other microalgae that can fix carbon dioxide into organic molecules like fuels and fuel precursors, using photosynthesis as the energy source. Unlike eukaryotic algae, however, cyanobacteria do not accumulate liquids in the form of triosolblissarol. However, there are some distinct advantages to using cyanobacteria. First, they're very easy to genetically manipulate. So this allows us to introduce fuel-producing pathways and use other common metabolic engineering techniques to boost fuel production. Secondly, they're mentioned to excrete fuel precursors like free fatty acids. This enables a continuous production system, which would reduce the requirements for nutrients like nitrogen and phosphates, and also reduce the time required for microalgae growth. In this work, we engineer a model cyanobacterium, Cinecococcus elongatus 7942, to produce and excrete free fatty acids. We show free fatty acid excretion for two mutants. SC01, which has gene knockout of the acyl-ACP synthetase involved in free fatty acid recycling. And SC02, which has the same gene knockout, but also expresses a thioesterase for release of the free fatty acid. While the engineered strains excrete free fatty acids, they have reduced final cell concentrations compared to the wild type, as well as reduced photosynthetic yields. Accompanying these changes, there is also a change in the photosynthetic pigments with a selective degradation of chlorophyll A. Using hyperspectral imaging and MCR analysis, we show changes in the subcellular localization of photosynthetic pigments for the free fatty acid-producing strains. Specifically, the phycabilia proteins phycocyanin and alofycocyanin are shown to aggregate at the cell poles. This is in contrast to the wild type, which shows these pigments being evenly dispersed throughout the thylakoi membrane. We identify two possible causes for these physiological effects. First, unsaturated free fatty acids, like linoleic acid, may oxidize into a variety of products which can be toxic to the cell. Exogenous linoleic acid addition was shown to inhibit cell growth and lead to the complete degradation of all photosynthetic pigments, including the phycabilia proteins and chlorophyll A. The second potential cause is that manipulating the fatty acid biosynthesis pathway may lead to changes in the chemical composition of membranes, specifically the photosynthetic membranes. In turn, this may affect the membrane structure, the activities of enzymes embedded within the membrane, or the attachment of light-harvesting pigments to the photosynthetic membrane. Changes in the degree of saturation of membrane fatty acids provides preliminary evidence to support this mechanism. The physiological changes described in the study will have a significant impact on large-scale bio-filled productivities. Therefore, further investigation is necessary to identify ways to prevent or overcome these detrimental effects of free fatty acid production.


Appalachian Wireless has some of the best data rates in wireless on our plans and not to mention an amazing deal on one of our hottest smartphones, but that's all the time I have to explain. See store for details, better service, bigger savings. That's today's Appalachian Wireless. The streets of Pocville are quiet Saturday morning just hours before the start of the final day of hillbilly days, but vendors set up for the festival are already hard at work. The hours between 6 and 10 in the morning are valuable prep times for the food vendors as they get ready for the day's sales to the hillbilly days crowd. It usually takes us about an hour to get everything ready, so we got to get everything cooked ahead so that we can get fired off in the morning when the rush hits us. We use a sourdough starter, so that's going to take a little bit to set up, a little bit to get it going, so by 7.15 to 7.30 we were mixing our first batch to have it ready by 8 o'clock. Without that prep time, the hillbilly day's fun could be a bust. Well, it's really important for us so we can get prepared for the rush that normally we'll have. You know, it starts about 10 o'clock or we're going to get busy, and so it's really important for us to get in here early in the morning and get ready to go. But even with the empty streets, one vendor was already hard at work just after 8 a.m. Auntie Ruth's donuts set up on Hambly Boulevard already had a long line some 90 minutes before booths were scheduled to open for the day. Whenever, if they know we open at 8, they're here before 8 o'clock. They're always waiting before we open. There's only so much prep work you can do beforehand, but I got here about 6 this morning. That makes early morning prep time all the more valuable for the donut vendor. We like to think we have a good product, and I think we do, and the customers think so as well, but it's just one of those things you do the best you can. I just thank these wonderful folks up here for making it a good event. It's a lot more than just our booth that makes it an event. It's all the people involved. It's very good. With Hillbilly Day's earlier wrap, the food vendors ensured that those who braved Saturday's rainy weather left Pikeville with their bellies full. From Hillbilly Days, Chris Anderson, EKB News.


What's up guys and welcome to Trendy Update Suhua SPM Buzz. Now onto the trending news today is a YouTuber content creator, Andrew Kibbe, whose account was terminated on Mandibar and unfortunately, he was not given a reason as to why his account was terminated. Now we are going to be looking at some of the reasons as to why Andrew Kibbe's channel could have been terminated. Now we all know that channel had 474,000 subscribers, quite a good number that he was ranked among first, you know, the Kenya fastest growing channels in Kenya. Of course, he creates a content or rather runs the channel from Texas where he reside. Now we all know it is not uncommon for Google to terminate an account to, you know, say to a violated company's rules and many Kenyans have suffered the same fate including NTV which lost an account channel that had at least 2.1 million subscribers. Now we are going to be looking at the reasons as to why Andrew Kibbe's YouTube channel could have been terminated. Now one of the reasons could be spamming or misleading content. Now of course these include uploading videos or comments that are repetitive, irrelevant and deceptive. For example, someone could upload a video and that is just a link to another website so they made comment on other videos with spammy links. Now that is one.    That is a very serious violation of YouTube's terms and service and of course any account that contains this type of content could be terminated immediately. But impersonation appears, this includes pretending to be someone that you're not so you could probably pretend you're a celebrity, you're another YouTube user who could create an account in the name of that person and that is account impersonation. Another one is copyright infringement and this one includes now uploading videos that contain other people's content, copyrighted content without their permission. Of course, in case you pay up a mission, you won't be able to credit the owner of the video and by then if you upload a video of another movie or a song and by then it's your yaku that your account could be terminated as well. Another and the last one is terms of service violation. This includes any activity that violates YouTube's terms of service, including uploading miscleaning content or hate speech. So if your account is terminated, you get an email and explaining the reason of termination are. But unfortunately, Andrew Keebe mentioned that he was not given a reason as to why his account was terminated. It was just told that his account is terminated. Did you can't get it up? Are these reasons are my name is a kusoma. What are your reasons or rather what are your thoughts on why YouTube content creator Andrew Keebe account was terminated? Other than that, Andrew Keebe has come back on a platform and by Niyake, a meza kwambia wachuna we have created a platform of our own and you can upload content by your platform here Andrew Keebe. Nyeza w ka fatili o kona i nasim maji. Let us know down on the comments section what are your thoughts about this conversation but in the meantime keep it trendy updates, hawa SPM buzz.


Electric vehicles, EVs, have been gaining popularity due to their independence from oil and lack of emissions. Despite this, there are still many obstacles preventing them from becoming widely adopted. These include the high cost of infrastructure, scarcity of charging stations, limited range or range anxiety, and the performance of batteries. In order to overcome these challenges, potential solutions include enhancing the infrastructure, increasing the number of charging stations, using battery swapping techniques, and improving battery technology to address range anxiety and reduce charging times. Governments should provide incentives to encourage consumers to purchase EVs, while industry stakeholders should work together with government to build a robust charging infrastructure. If these measures are taken, EVs could help reduce carbon emissions and air pollution. This article was authored by Fayez Al-Anazzi.


So, now we carry on with more examples of CW1 classes. For any sub family F of n cubes in P not necessarily finite, we have this notation WF equal to union of all the n cubes sigma which are present in F. We also have this notation F prime is all those sigma inside F which are maximal, what is the meaning of that? If sigma is contained inside tau, tau is also in F that means sigma is equal to tau, that is the meaning of maximal, only the maximal elements of F are taken in F prime. Then clearly the union of all the members is equal to union of all the maximal members, okay. So, this is where we use the fact that, you know remember I will show you that interior of sigma, intersection interior of sigma 2 is nonemptive, right? Either sigma is contained in sigma 2 or sigma 2 is contained in sigma 1. Either they are non-overlapping, okay, completely or one of them is contained in the other. So, you can take only the maximal one to take care of the union. Now, here is the extension of the earlier example for infinite case. So earlier we had this example for only on sigma. Now, I am taking it for finite members of P. Let F be a finite collection of members of P. Let F denote the collection of all k faces 0 less than k less than ornate of members of F prime. So earlier there is only one sigma. Now I am taking a finite collection. Then we can choose a function F on this large family F, okay, which is an integer value function satisfying condition 1 and 2 of example 1.7. Let me recall that. What are these conditions? This is F tau is taken to R for all tau whenever this is a member of P R, tau 1 and tau 2, one contained in the other implies F tau 1 is greater than or equal to F tau 2, okay. So this must be true for all now all members of this curly F, okay. So that is the condition 1 and 2. Such that the CW structure sigma F remember this is coming from 1.7 that example for F in F prime patch up to define a CW structure on WF so that the n cells are in 1-1 correspondence with members of F prime. There in that example let F be with this property. I did not say anything about whether there is one such and so on. There are many actually, okay. But here I say we can choose a function F. If some F is given with these two condition, I am not saying that this will happen, okay. Condition 1 and 2 are necessary even to have a CW structure on each sigma F, there is sigma F, okay. So once you have that each K phase here members of this one will have the sigma structure. Now the claim is that if F is chosen properly then these sigma structures, these sigma F structures on each of sigma will patch up to define a structure on WF itself. This is the claim not only that the second part says that the n cells of this structure are precisely the members of this P with which you started with which are maximal namely F prime, the members of F prime. So if you have understood the sigma F earlier, earlier example, then this will be an easy consequence of that, okay. So let me write down the formula. What is the F? Let F from curly F to n be the function defined as follows.  There is at least one because each tau is a phase of some F prime, some F, some member of F, therefore it is a member of F prime, okay. So it is a phase of some member of F prime. So this set is non-empty but of course it is finite also. Anyway, all those sigma prime in F prime, sigma prime intersection interior of tau is non-empty.  We can take F tau to be the maximum of all T such that P T intersection A tau is non-empty. So each member of A tau belongs to some P T. If that P T is there, put that T in this set, okay. So P T intersection A tau must be non-empty. So take those set. This is anyway finite set. It is non-empty also. So take this maximum. It is some finite number. This defines F tau. As tau varies, this F tau will be different than matter. What you have to see is that it satisfies the condition 1 and 2. That is what we want, right. The condition 1 and 2 must be satisfied, okay. Note that if sigma is in F prime, intersection say P R, one of the members, sigma is one of the members in P R as well as it is a maximum one. Then F of sigma is R because this whole set itself will be just singleton here, namely R. So it is maximum, it is actually equal to R itself by the definition of F prime because F prime is maximal. It will not intersect any other sigma prime. Sorry, a sigma is maximal. It will not intersect any other sigma prime. It is maximal, right. The interior does not intersect. So this should be only sigma prime will intersect. So that is the only member. So that will be R, okay. You have to see that this satisfies the two conditions 1 and 2 in the above example, okay. The point here is, the interior I have put here, if the interior intersects, the sigma prime intersects the closure, the whole closure also. It will perhaps contain the whole closure, sigma prime first itself, okay. So that is the trick here. So if you take sigma prime intersection interior tau non-empty and say tau prime is a phase of tau, this will be true for tau prime also. So all those t which are present in A tau will be there in A tau prime. Therefore, this will be a bigger set for tau prime than tau which means that maximum is bigger, okay. So tau 1 contain her tau 2 will imply f tau 1 is bigger than a quota of tau 2, etc. You have to verify this, okay. Hope you will be able to do it on your own now. Anyway, I have explained the second one. Condition 1 also you can see what happens, okay. So now the crucial thing in the lemma namely, we claim that the C-deb structures f sigma, as sigma varies over f prime, patch up to define a C-deb structure on w f prime which is same thing as w f, okay. So we want to do an inductive step here starting with suppose f prime has only one member. That is already covered in the example, okay. So therefore, we just want to do induction here, alright. So write f as sigma 1 sigma 2 sigma p. So this is an enumeration of members of f prime. It is a finite family. Let us check the notation w j equal to union of those sigma i for i less than equal to j. So w 1 is just sigma 1, w 2 is sigma 1 union sigma 2 and so on, okay. So we will induct on this number p. If this p is 1, we know it already. There is nothing to prove. Nothing to prove means what we have discussed this in the example 1.7. Inductively assume that the structure sigma f i have patched up to define a C-deb structure on w j. So i less than equal to j, suppose it is true for w j and that structure temporarily we should denote it by w j f. Is it coming from f, okay. Put next one a equal to w j plus 1, take the next member here. Intersect with w j, put a equal to that. If a is empty, you have two disjoint C-w structures, C-w complexes. Union is automatically C-w complex in which w j structure will be kept as it is and sigma j plus 1 structure will be kept as it is. Both of them will be sub complex. There is no problem, okay. The problem comes when a is non-empty, right. So suppose a is non-empty. The idea is to use our earlier lemma. So let us denote C-w, denote the C-w structure on a coming from w j 1 and coming from j plus 1, sigma j plus 1, okay. This intersection of these two things, right, respectively. Let us denote this one by a0, that one by a1, okay. What we wanted is that either a0 is finer than a1 or a1 is finer than a a0. Then we could have put them together. So that was our earlier lemma, right. So here we claim that a0 is actually equal to a1. So we are through. So we can patch it and then induction will take care of it, okay. So let us look at this. What is this a? a0, a1 are not structures. a is the subspace. a is the intersection of, a is the intersection of w a1. This intersection never contains any interior of sigma j plus 1. It is exactly some part of the boundary, okay. Because sigma j plus 1, these are all maximal simplex, this after all. So it is a boundary, part of boundary of j plus 1. A part of the boundary will be made up of k phases of sigma j plus 1 for k less than or equal to the dimension of sigma, namely n less than strictly less than n, okay. And hence the union of some proper phases of tau, phases tau of sigma j plus 1. On each of these tau, the CW structure coming from either side is precisely equal to that coming from p of f tau.  And that is what we have chosen, okay. Maximum, among us all the members, not only just sigma j plus 1, for all w j's, whatever sigma i's are there, if they intersect, they will give you the whatever number there. So this is the maximum. So this is finer than all these, okay, it is either side. The CW structure coming from either side, okay. This proves that A naught is A1 because it is actually equal to the p of f tau. So it just depends upon tau, it does not depend upon where this tau is, whether it is sigma j plus 1 or of w j or so on, okay. So we can put them together. That is it. So from them, it follows that w j plus 1 as a CW structure, which you denote by w j plus 1f, in which both the old one, the inductive step, w jf and sigma f are sub complexes, okay. So the structure keeps extending as you extend the the space itself, okay. So that completes the proof that wf prime has CW structure as proposed.  Where was that? We have observed that one, no? f sigma itself is r, okay. So if sigma is in pr, there is no subdivision, it is just one single n cell, okay. So it should be noted, the final structure on wf, which is, let us call it as wp, does not depend upon the enumeration we have chosen on the members of f prime. And for j less than p, each wj is a sub complex of this wf, okay. So it all depends upon my f. Once f is chosen, no matter, this is only for our convenience that the enumeration was only for our convenience, okay, for the proof. But the final structure, so wf prime is nothing but wp here, okay, when the number a number of p is hit there, that structure is the same. So let us now go for infinite case. This was our aim finally.  So we are not into completely in the sea of all infinite family, that is not, that is not the aim, that is not possible either. So we allow f to be infinite but locally finite. Let f prime be the members of all k faces, one less than or two k less than or n, of all members of f prime. You take sub phases and so on for once you start with f prime, okay. Then there exists a function f on this f prime to end such that the cw structure sigma f patch up to define a cw structure on wf prime namely it is the incident of wf. So I am not saying here that f satisfies one and two, okay. So it should satisfy otherwise I do not talk about the structure here. The structure sigma f is defined only when f satisfies the condition one and two, okay. So here the finiteness is not there, you see. So we have to be little more careful but it is not at all all that difficult. We start off as in the proof of the previous lemma. So let f denote the collection of all k faces of all members of f prime, okay. Observe that each set a tau defined as in the above lemma, a tau equal to the set of all sigma prime and f prime such that sigma prime intersection integral tau is nonempty. This a tau is the same thing, okay. This will depend upon all those sigma prime which intersect integral tau and in particular they intersect tau. By local finiteness this family is finite, f prime may not be finite. So all this sigma prime which satisfies this one is finite. So each a tau is finite, okay. Therefore as before we can take f tau to be maximum of this thing, that will be finite. So this is the replacement for family not being finite. The local finite mistakes are obviously. Therefore, okay, we have the function f and it has satisfied the same properties one and no problem. The second thing is f prime is not just infinite but it is countable. Why? Because p itself is countable, okay. If it is finite then the proof is over by the previous lemma, okay. So assume that it is infinite and write f prime as again just like in the previous lemma, enumeration, sigma 1, sigma 2, sigma j and so on. The only difference is now this is infinite family. This take an enumeration, okay. As before put wj equal to the union of all sigma j i less than equal to j, okay. As in the example for each j, okay, the Cw structure sigma fi, i less than equal to j, patch up to define a Cw structure on wfj on wj which is a sub complex of wfj plus 1. So all these is there in the previous lemma because these are all finite now, okay. It is union of finite in many of them. It follows wfj themselves patch up define a Cw structure on wfj. So each finite stage you have and what is wf prime? It is union of these finite things. So one stereo structure is sub complex of the next one. Therefore, okay, look at any n cell or any k phase whatever is in wf prime. It would be in one some at finite stage and there things are nice. You can attach it, okay. You know attaching a k cell should make sense. How does it make sense provided its boundary is inside a k minus 1, k minus 1 skeleton. That is all you have to see. So that is automatically satisfied here, okay. Also note that since interior of any two members of a primary disjoint follows that for sigma belong to Pr, we have f sigma equal to r, same thing as in the previous case. Therefore, the n cells of wf are in one-one correspondence in member of f prime. There is no subdivision of the n cells, okay. In other words, the proof of previous lemma is due to words provided this once a tau is finite for all tau. And that is ensured by local finite. That is all the difference, okay. We now come to an important result which I have that is why we I have put it as theorem but this is also an example but this will be useful. This will be used elsewhere while studying manifolds we will be using this one. This was our aim of doing all this fundamental work here, okay. Take x to be any closed subset of an open set v inside Rn. So v is some open subset of Rn, x is a closed subset of that. I want to tell you right in the beginning that v could be the whole of Rn and on the other extreme x could be empty also. So that case is also allowed here, okay. And they are of some importance also, okay. They may be trivially true sometimes but they are of importance, okay. You can take just v to be any open subset and x to be empty and that is one important case, all right. So x is a closed subset of a open subset v of Rn. Let a equal to all the members of p. p, remember p is a union of all pk's such that this sigma is inside v. This is no more open set. This is some n cell which should be contained inside v and intersection of x with sigma must be empty. It should be away from x. So that is my a. Second one is b is all those p, all those sigma and p such that again sigma is contained inside v. So v is allowing us, should not go out of v that is the first thing. But the second condition is x intersection sigma is non-empty, okay. So this is difference between these are quite different if x is non-empty then these are different. If x is empty set then this should be not there only. This is always true. So it would be just all those sigma adjacent, okay. So these two are quite different things which are both subspaces of v, okay. Then w a which is the union of all the members of a, w b which is the union of all members of v are both n dimensional Cw complexes which are countable, pure and locally finite, okay. So I am not including here when the case a and b may be empty and so on. So those things are obvious. You can just figure it out, okay. So the proof is though there are two statements here for a and b they are similar. So I am going to put them together that c denote either a or b as above. As observed before first of all w over c union of all members of c same thing as w over c prime which is c prime is what? a prime or b prime which is maximal numbers of c. In view of the lemma, okay all that we need to do is to verify that c prime is locally finite. That was our previous case if it is locally finite then we have the structure, okay. So that was the previous thing here. It is a locally finite family then there is an f then you have this sigma f's patch up to define such a structure. So in view of this lemma all that we need to do to verify c prime is locally finite. At all the points of w c this is your union this is the this is my x whatever or that whatever, okay. So at all points this family should be locally finite, okay. So here is pro take any point in w c but choose epsilon for epsilon positive let us have this notation standard notation of course u epsilon of x is the ball of radius epsilon open ball of radius epsilon centered at x norm of x minus norm of norm of x minus y is less than epsilon. So this is a ball in capital R n, right and this is an open ball strictly less than epsilon. So this is open ball centered at x and having radius epsilon. So this is notation. Now choose epsilon such that this ball is contained inside the open set v minus x if we are working namely c equal to a c equal to a x belongs to the v a which means that x is not a point of capital X and its x is a of course a point of v. So it seems a v minus x whenever v minus x is an open subset we can choose epsilon sufficiently small set u epsilon of x is contained inside that. So that is no problem. Similarly in the second case u epsilon x is just contained inside v, okay. In the second case this is easier. The first case you have to throw away x and then take that open set. Once you have taken u epsilon contains at v this u epsilon is going to ensure local finiteness. We claim that only finitely many members of c prime will intersect u epsilon fax. The emphasis is on c prime. If you ask for c members of c that is not true. Here it is definitely not true. May not be true is not correct not true at all, okay. Only for c prime it is true. Finitely many members of c prime will intersect u epsilon. Local finiteness is for c prime at every point, okay. Let us prove this one. Given any r, okay. Clearly there are only finitely many members of pr which intersect a given bounded set. Take a bounded set inside r. Members of pr will intersect it, okay. Because bounded set is inside some large square and inside the large square, large cube only finitely many members of pr will be there, okay. When r is fixed given any fixed r this is true, okay. Hence in particular how many will intersect u epsilon x, u epsilon x is the bounded set. Finitely many will intersect, okay. Therefore there are only finitely many members of if take all the s less than equal to r each of them will contribute finitely many. So how you have only finitely many for pr r union of pr where s is x quantity equal to r, okay. The problems comes with members of p s where s is bigger than r. That means these are smaller, smaller, smaller, smaller cubes. They can create problem. But this is the point is here. We shall see that none of them belong to c prime if r is chosen appropriately. U epsilon is already chosen.  I mean p s they will intersect but they are not members of c prime, sorry. They will not belong to c prime. So number of members of c prime which intersect u epsilon will be finite they will be coming from here only, okay. So once I have said this one it should be obvious to you but let me just make it more clear. To choose r let r belong to n be such that this number this is the real integer square root of n divided by 2 over r minus 1 is less than epsilon. This 2 over r minus 1 is side length. But what is square root of n? The square root of this denotes the diagonal of any n cube of side length 1 by 2 over r to 4 r minus 1. So that is why the square root of n times 2 over r minus 1 is there. So diagonal should be less than epsilon. Let f be the collection of all members of p r. Now r has been chosen correctly, okay. Which intersects u epsilon, okay.  Then this u epsilon of x will be contained in the interior of all these w f. f is all of them which intersect. It should contain inside of them, okay. Like if I have single point for example, if I take all of them it will be a big bigger cube of side length double of that one 2 over r around that point. Similarly, if I take an edge also that will be the same thing. So this maybe may not be like look like that it may be bloated up thing. But it will be some open set that open set will contain u epsilon of x, okay. It will be union of cells after all, okay. So this is what you have to understand why this is happening. It is elementary geometry whatever topology which is inside r n, okay. Suppose now tau is in p s where s is bigger than r. And tau intersection u epsilon is non-empty, okay. So this is some member n cube in p s where s is bigger than r which is very tiny member s is bigger than r here, okay. So tau intersection interior is non-empty. This intersection tau intersection epsilon is non-empty would mean that interior of tau will mean the interior of this larger one because this is open subset contained inside this open set, okay. If some tau intersect interior tau intersects this open subset here it must intersect one of the interiors of members of this one, okay. So it will follow that interior of tau will mean the interior of WF and hence will mean the interior of some member sigma of s, okay. So this tau intersection interior tau intersection sigma that is non-empty that will imply that tau is contained in that sigma, okay. Just tau is smaller size sigma is of sigma is of size r where s is bigger than r. So it cannot be that sigma is inside tau but tau must be inside sigma. Both tau and sigma are inside c. Therefore, I mean that is what I want to say that we see how if both of them are inside c only one of them can be inside c prime. Sigma is already in c prime so tau is not in c prime. So you wanted to show that none of these tau are in c prime, right. So why both of them are in c? That is what I have to say, okay. So that is easy. The case a equal to c equal to a and b have to be done separately. In case c equal to a what does this imply? Both tau and sigma belong to c. Why? Because interior of tau, you see what is the definition of, definition of c here. Go back, okay c a definition of a. Sigma intersection x is non-empty. If tau intersection x is non-empty sigma intersection is also non-empty. And both of them are sigma itself is inside b tau will be also inside b. So both of them will be inside a, right. So that is the case when similarly the other one but this one is more since a both of c and tau are inside a. Therefore tau is not inside a prime. Okay. Now consider the case when c is b, okay. Now it is something different then it may happen that tau intersection x is empty, okay. But we want members of b to b that tau intersection x is non-empty. If this is empty that would mean tau is not in b already if it is not in b it cannot be in b prime. So that case is over. Now on the other hand tau intersection x is non-empty then tau sigma intersection x is also non-empty because sigma is larger. Both of them will be inside b. Once again tau cannot be inside b prime. So that completes the proof that this both a and b have a nice CW structures and dimensional CW complexes which are countable, pure and locally finite. You can further assume which I am not going to assume which I am not going to use but you can further assume that the n cells in each of them is equal to I mean they are coming from in 1-1 correspondence with members of a prime or members of a prime respectively. That can also be assured, okay. I want to make one more remark. W a's which I have made already I want to draw your attention. Note that W a is exactly equal to V minus x. Take any point in V minus x then there is a positive distance of that point to the the complement of V minus x, okay. After that you can choose a small neighborhood U epsilon which is completely contending side V minus V minus is open subset. Then if R is large enough then square of n divided by 2 power R minus 1 will be less than epsilon and so the smaller and smaller discs will intersect, okay. Therefore what we get is W a's actually equal to V minus x. That means V minus x gets a CW structure. If x is empty what we conclude? Every open subset of R n can be given a CW structure. Every open subset, okay. So that is the conclusion of this one. Further all these CW structures are so nice they are all made up of n cubes just to star them, okay. Each face of each of these n cubes the starring will take care of that it will become a simplicial complex. Therefore what we have proved here is every open subset of R n can be given a simplicial complex structure, okay. So we will use them in studying manifolds, okay. Here are some easy exercises for you. Try them, keep trying them. If we do not get them we are here to explain it to you. Our TAs will explain it to you and so on, okay. So that is it for today. Thank you.


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