In this video, we're going to apply the remainder estimation theorem to find the minimum degree of the Taylor polynomial about x equals 0 needed to calculate sine of 1 with an error of less than 0.0005. Let's first define our function as sine of x. What do we know about a potential value for m in the theorem? We know that any derivative of sine of x is among the following. Sine of x, negative sine of x, cosine of x, and negative cosine of x. Now the maximum value of any of these for any x is 1. So we have that the absolute value of the n plus first derivative of f must be less than or equal to 1. So in our theorem, let's let m be 1. As a result, we now have that the error bound is bounded above by 1 over n plus 1 factorial times the absolute value of x minus x naught to the n plus 1. Well, what is our interval of interest? Well, we're considering the Taylor polynomials centered at 0. So x naught is 0. Since we're investigating the error involved in approximating sine of 1, the other side of our interval is 1. So this value will be 1. This value will be 0. So we're interested in the error on the interval from 0 to 1, which means that we want to find n such that the error bound is bounded above by 1 over n plus 1 factorial times the absolute value of 1 minus 0 to the n plus 1, which is equal to 1 over n plus 1 factorial. And we want that error to be less than 0.0005. Another way to think about this is to identify a value of n such that n plus 1 factorial is greater than 20,000. Since n represents a degree of a polynomial, we will test only whole numbers for n. Pause this video and test some values out. We find that since 6 plus 1 factorial is equal to 5,040 and 7 plus 1 factorial is 40,320, we need at a minimum a seventh degree Taylor polynomial to approximate the actual value of sine of 1 with less than a 0.0005 error.


Today we're going to talk about a roadmap Flux B2 through to general availability or GA and peak beyond that. I'm going to talk about roadmap weighted somewhat towards future developments then here we'll give you a demo of what you can do with Flux B2 today. First of all, who am I, we're just some engineer, let's move on. To summarize the history of flux, what's now called Flux B1 was created at Weaveworks to deploy new versions of services to a software as a service implementation called Weave Cloud. Prior to version 1.0 it concentrated mainly on upgrading, upgrading container images. But then we made a big change in June 2017, which turned it all around so that it applied everything from Git and made updates by committing to Git. This was the big bang event for GitOps. Flux was inducted into the CNCF as a sandbox project in August 2019. But by then it was creaking, it needed to be modernized. Things like custom resource definitions went around when it was created. In early 2020 what became Flux V2 was started with the same scope but using modern, modern tooling like controller runtime and supporting multiplexing, e.g. running more than one sync. The Flux project was classified as adopt in the CNCF technology radar in 2020 and promoted to incubation status in 2021. Here we are roughly 18 months after the inception of Flux V2, looking forward to it being generally available. What does that mean? What does general availability mean? Usually it's taken to mean that a piece of software is considered ready to run in production. With an open source people have different appetites for risk, so many will have already adopted a product before GA release. But here is what it means Flux V2. First of all, it means covering the bases that Flux V1 covered, roughly speaking, syncing from Git to a cluster, updating image reps in YAML files and committing that to Git, sending notifications, e.g. to Slack when things have been done, and clarity of installation of helm charts. Most of these things have been improved and generalized on Flux V2, and in particular as I said everything is now multiplexed. You can define as many sources, syncs, updates and notifications as you need. After a GA release, the rule is usually backward compatibility. Public APIs must be stable from this point on, so it's a point of no return in a way. There are a few areas that still need development before we're happy to make it generally available. To have confidence that Flux does what it says, and that the bug escape rate is under control, we need better coverage for end-to-end tests. The controls in Flux have varying unit test coverage, because some things are just hard to test, like notifications to external services, but it's mostly okay. The command line tool has end-to-end tests, but these are closer to smoke tests for the whole set of controllers together, and coverage is not that great. So one initiative in progress is to rewrite these so that it's possible to permute the inputs and thereby cover a much wider variety of scenarios. Another area still in development is standardization, and this comes in two parts. The standard for Flux types and controllers, and libraries to help controllers stick to that standard. The API for Flux goes beyond just the CRDs. There are protocols for how to interoperate with the various objects, for example. There's a protocol for requesting that a resource is processed outside of its specified schedule, and that's used by the command line tool to synchronize with operations that it's initiated. And there's a protocol for discovering and fetching a bundle from a source object, like a git repository or bucket, which other controllers need to understand. Flux uses case status for determining whether dependencies are ready, and it's important that its own types are compatible with case status for that reason. And there are standard metrics that Flux dashboards expect controllers to export. So it's important to have these standards stabilized and implement them consistently. And that will mean that people building on Flux can be sure that their software will interoperate correctly and continue to do so. A big upcoming addition to Flux is a new model for restricting syncs with Kubernetes role-based access control or RBAC. Flux is different from most controllers, because by the nature of syncing, it can affect almost anything in the system, including the rules for what it can affect. So you have to be very careful. The proposed new security model gives platform operators more power to restrict what Flux can do per namespace or tenant using RBAC. And this model also enables scenario, which often comes up, where you want to separate concerns so that, for example, a platform operator can create a secret giving access to a git repository, and an application team can sync using the git repository without learning the secret. So the picture on the slide shows the general situation, the source, the application, and the target for the application can all have different owners or namespaces, but you still want RBAC to be in effect. If you know Flux well, you might ask, isn't this already possible? Yes it is, but it's not well enforced. So at present, you can give a service account for Flux to use when syncing, but the default is to not use a specific service account, which usually means that it inherits cluster admin privileges. This is convenient for clusters in which everyone is trusted, but it needs work and extra infrastructure to undo otherwise. In the new model, it'll be possible to default to an unprivileged user, making this aspect secure by default. Another initiative underway is to orient Flux around the different, the documentation around the different reasons people have for looking at documentation. Sometimes you just want to get something up and running, or you sometimes want to learn how it all works, or sometimes you need a recipe for some specific task, like setting up on a particular cloud platform. Sometimes you're tweaking a working system, and you need to dive into a reference to find the name of the right option. The aim of reorganizing the documentation then is to better serve these different entry points and get people to the most helpful material for them directly. Another change coming up for documentation is having specialized sections for the cloud platforms. For example, you can use your GCP AWS and so on. It's useful to collect docs together this way, because typically you're using one of those at a time. And so the sections will serve as a one-stop shop for all the relevant information. These changes to documentation are not all prerequisites for GA, but I wanted to mention them here because they're nonetheless a part of Flux's growth and maturity as a product. The ability is merely the end of the beginning. There are more plans of it in no particular order. Flux v1 and Flux v2 have always included Kubectl or Kubikutl in their container images as the means of applying configurations to the Kubernetes API. And that's because Kubectl contains a lot of logic that's difficult to reuse or to reproduce, for example doing three-way merges of resource definitions. It's now possible to rely on Kubernetes API server to do the tricky bits like merging definitions and avoid the need for shelling out to Kubectl. And so the field management mechanism also lets you see fine-grained conflicts, which means Flux can be more sophisticated in how it applies configs and keep out of the way of other actors like autoscaders. One goal of Flux is to provide primitives to people to build their own GitOps flavor continuous delivery. And there's different varieties of that, hooking up your system to the inputs and outputs of the controllers in Flux, which I would call integration. Making your own types and controllers that can be used with those in Flux, which I would call extension. For the purpose of extension, the API is not as open as it could be. By open, I mean that you don't need Flux to be changed for you to be able to add to it. Right now, it is possible, as shown in the picture there, to write a controller that consumes sources. It is not possible to write a controller that provided new kind of source because the source types are hard coded. There is a design in progress illustrated here. For removing that limitation by using a type representing the output of a source. And this would open up the API to new kinds of source, like images and other artifacts from OCI repositories, for example. This exciting flagger is part of the Flux project, but doesn't yet use the Flux conventions that the other controllers do. It's its own standalone thing. It's also something of a monolith, which does all of creating shadow workload objects and controlling the version and the routing of traffic to different versions of those objects. And it's also reweighting kinds of various kinds of gate during a rollout like running a load test and sampling metrics to determine when a rollout can proceed. Some of these are pretty close to being standalone and generic functions in themselves, which could be moved into their own API types and controllers. The benefit to doing this would be extensibility again. It would be easier to build a flagger analog that caters to a scenario not in the scope of flagger. For example, flagger deals with workloads, deployments and demon sets, but you might want it to roll out another kind of object that doesn't follow the same rules. You could use the metrics, routing and gating parts of the flagger and write your own shadowing logic to have your own form of progressive delivery. One last thing I want to mention regarding future plans is other kinds of automation. So version two includes an automation controller, which fulfills the same purpose as the image update feature of flag flux V1. And that controller was developed to give some continuity for people who would be migrating from flux V1, but people have also found it to be useful in its own right. The picture in the slide shows how bits relate and image repository scans for an images tags and image policy gives a rule for choosing a tag from those. And the image update automation updates YAML files according to the policies in its namespace. The pattern value put together can be used for other kinds of automation. For example, one kind that's been requested is updating the chart version of a Houston and Helm release. So that's a possibility. I have not talked much about what flux version two does as of today. So I'm now going to hand over to Hida, who will fill that gap with a demonstration. Thank you. Thank you, Michael and a lot of folks just another engineer who's going to demo the most recent part of flux to reach this table API image update automation. If this is not a topic of interest to you, you can find other demos and how to use on the flux website.  So with my screen shared, we are ready to dive into the automation of image updates with flux. Flux comes with the CLI, which allows you to generate flux API objects and control the lifecycle of them, but also provide access to bootstrapping features. For this demo, we are going to bootstrap against GitHub. The bootstrapping against GitLab or plain Git server is possible as well. I'm now going to type out the bootstrap command and explain some of the flags as you may have already noticed by typed out the bootstrap command. Flux is built out of components and not all of them are included by default. To allow it to keep track of image tags and automate updates to Git, we need to install the image components using the components extra flag. In addition, we need to enable a read-write key so that later on we can write back to Git. Now let's run the command. Now as you can see, the bootstrap command showed that everything was actually already set up and the controllers are already running in the cluster. This is because I cheated and run the command before the demo to save us some time and show you it's idiom potent and will only make changes if things aren't up to date. Let's quickly clone our bootstrap repository and navigate into it. To showcase the image update, we first need something to update running in the cluster. Because deploying isn't the main topic of this demo, I'm going to deploy the Stefan's PodInfo application to a namespace with the same name without providing too many further details. As shown on the screen, I created a simple deployment which makes use of PodInfo 5.0.0 in the PodInfo namespace. After we get commit and a quick on-demand reconcile request using the Flux CLI, the PodInfo deployment should now show up on the cluster. Let's see if this is true. As you can see, the deployment has been applied by the customized controller and is now running on the cluster. Now, as you may have noticed, I put a template for the deployment from my storage. Stefan tends to be a busy bee and I can be quite sure the version I deployed isn't actually the latest version. Let's see what Flux has to offer to automate this. On the diagram on the right side of my screen, you can see the controllers are just bootstrapped. And that we just need a couple of custom resource guides to actually set up the automation. I will set them up one by one and explain what they do, starting with the image repository. The image repository object I just created, once applied, tells the image reflector controller to start maintaining an index of the available tags for that image. But this is just a bag of references and doesn't make Flux understand what's the latest version you want to have running on your cluster. This is where image policies come into play, which define rules for selecting the latest image from the bag of references. As PodInfo is released with proper SAM versioning, I created a policy to select the latest version within the major 5 SAM file range. Now, with these two objects, Flux knows where to look for image tags and what the latest tag is based on the defined policy. But it doesn't do anything with this information yet. To make it start looking into outdated references in a git repository, we need to create an image automation object that tells Flux where it should look and how it should write the new images back to git. Let's go ahead and create this object. Now, because the command arguments are quite verbose, I'm going to give you a quick overview of what is included in the generated YAML. The source ref at the bottom of the YAML document defines a reference to a git repository object which contains the information to connect to the git repository. While other Flux controllers use artifacts that are generated from the git repository object to apply resources to the cluster, the image automation controller performs a git clone by itself to ensure it's looking at the latest information. The git object at the top defines what git reference should be checked out and expected for outdated images.  With the current configuration options, changes are pushed directly to the branch that is synchronized to the cluster and updates should be fully automatic. The commit template ensures that multiple images are updated in a single commit, their list in the commit mail data. The update object defines the scope of the update and limits it to the part of the cluster. The strategy is something you can forget about for now as there is only a single option available at the moment. Now let's commit these files to git and see what happens. I run Flux Git GetImagesAll, which provides you an overview of all the image-related objects in the cluster. While you can see it has applied the image-update-outstimation object, it also mentions that no updates have been made to the git repository. This is because their deployment object lacks a small but important item. Markers. Flux doesn't really understand the structure of objects. It doesn't know where the image is in the deployment or other resource. You have to mark the image values in your resources so that it can identify and compare them to the value of the policy. The markers come in three flavors. Then you can mark a full image, an image tag, or an image name, which together should be sufficient to mark almost any type of resource. Let's add the marker to a resource referred to, the podinfo v5 image policy I created earlier on. Let's commit the file to git and see if we finally have our automation running. We now reconciled the last changes. So while the image automation object doesn't show that it made a commit yet, I'm quite sure this is a glitch in the distributed matrix. So let's go ahead and see if actually nothing has changed in git. It happened to be a glitch and a commit was made. Let's inspect the podinfo deployment in the cluster and see what version it is running. As you can see, the customized controller has processed the git commit by now and applied the change made by the image automation controller onto the cluster. And podinfo is now happily running version 5.2.1. Congratulations, you have witnessed image update automation driven by Flux. Now before I call this the end of the demo, I want to highlight the existence of two dock pages that will probably help you further shape a mental model of how image automation works with Flux. The first page is the one showed earlier in the demo which lists the markers. Remember this page and come back to it if you ever happen to be stuck. Also remember the markers can be put in any resource and the sky is like the limit. Second, if you create an image automation object that pushes to another branch that it reads from, you can create a gated automation approach which will force the updates through a pool request. The page shown explains how to do this for github, but the same technique can be used with many other git providers. Having made those last remarks, this concludes my demo and I wish you all a happy day. If you have any questions, don't be afraid to ask them during the Q&A or find us on the CNCS Slack in the Flux channel.


Don't laugh at me. I need to be like looking into the camera. This is the most difficult part, trust me is the intro because it's like so straight on point. Welcome to our Azure Stack Partner Solutions series. I'm here with Tiberiu Radu, also known as Tibi. He is a Principal Program Manager at the Intelligent Edge Solutions team. We are going to talk about Azure Stack Hub Solutions, delivered by our partners and service providers in a short video series. But before we actually going to talk about that, I want to give him the chance to basically explain what Azure Stack and what Azure Stack Hub actually is. Hi Thomas. Hello everyone. The Azure Stack Hub story started a few years ago, as we were looking at covering both the on-premises requirements of the world, the customers, as they were going through their journey to move to the Cloud, so to say. In the meantime, the Intelligent Cloud, Intelligent Edge story started to materialize growing from a hyperscale Cloud, which is Azure. You can look at it as a global computer, which is always available and has infinite scale. Then it goes through various devices all the way to very small microcontrollers, which are the Azure Steer devices. Now, in between these, we have the Azure Stack family of products. The Azure Stack family of products includes Azure Stack Hub, Azure Stack Edge, and Azure Stack HCI. Azure Stack Edge is a Cloud-managed appliance, meaning you can order it from the Azure portal, and it's intended to bring that compute power closer to the Edge where it's needed. So in cases like machine learning or where you have ID solutions, so that's what Azure Stack Edge is intended to do to bring that Cloud-managed appliance closer to the Edge. Azure Stack HCI is a hyper-convert solution, which is intended to provide a scalable virtualization and storage platform that's intended for scenarios like remote branch offices or where you have a high-performance workload in a traditional virtualization platform with the extension of the Cloud services towards it. So Azure services are extended towards the Azure Stack HCI and you're able to manage these resources from the Cloud as well as gain those hybrid capabilities. Azure Stack Hub is a Cloud-native integrated system. That means it brings the actual control plane of Azure down to the on-premises. So it works in fully disconnected scenarios, scenarios where you have data sovereignty or you're doing application modernization. You need to have the data on-premises, either in a certain geo or in a certain location. So Azure Stack Hub brings the services themselves to the premises of the customers. That's awesome. Thank you for explaining that. Again, I remember Ignite 2019 where Jason Sanders who is the engineering lead for all the Azure services came up and said basically in his keynote that we really believe that hybrid is an end-state for our customers and not just an in-between state until everything is moved to the Cloud. I know that there are very, very good reasons why customers want to deploy services and applications and not just in the Cloud, but also on their location in their own data center, at the edge or in their stores or wherever they are running services. So can you quickly explain a little bit why customers would actually deploy services in their locations? There are a number of reasons, but mostly we've seen three buckets out there, where they fit into. Data sovereignty or regulatory scenarios where due to some security reasons or regulations or whatever that the case might be, that data cannot leave the on-premises. Then there are the application modernization scenarios where you're looking to move certain parts of your application to something like a past services. So using app services or using some higher end service like a past service. So using the platform directly to provide that application modernization. Then you have the scenarios where you have a connected or a completely disconnected Cloud running in your premises. All of these you keep a consistent application development as well as operations across your public estate, as well as your private estate. So the on-premises would leverage the same. You would have operational consistency across your public and your private Cloud. But when you're considering the Azure Stack family of products, then the hybrid use cases also evolve with that family of products. So those three components, although you find parts of them in these use cases that we're showing now, they are building up towards a more complex hybrid use cases. You're leveraging each component of the family for specific reasons. When you're closer to the edge, you would go with Azure Stack Edge. When you're using more of a traditional virtualization platform enhanced by the Cloud in certain areas you would use Azure Stack ACI. When you need the full power of the control plane of Azure services running in your premises, either in a connected or a disconnected manner, then you would use Azure Stack Hub specifically. Then the networking aspect of things, these are solutions that need to be close application, for example. So if you have a latency issue, if you have a bandwidth issue or you have a remote location that has a long path to go through to an Azure region, and Azure regions are coming up all over the world, and more and more are being deployed. But still, there are places where connectivity is an issue and having that better latency helps. Yeah. No, absolutely. I agree with these. I heard that from many, many customers that these are, for example, the reasons why they also want to run certain workloads in their own location. I also had a customer where we were actually talking about, for example, factories or stores where they just didn't want to take the dependency on their internet connectivity to the Cloud where they just want to have something there. They're still going to use the Cloud for processing power in the long-term or in the mid-term, but they don't want to close down their store or factory just because the internet connectivity is not available. Now, with the Azure Stack portfolio, and especially with Azure Stack Hub, we get that in-between piece from where you need these Cloud skills, but also you have these traditional on-prem skills. We see a lot of people going out and the need to migrate these services, for example, to Azure, but also especially to Azure Stack Hub, you need to have certain set of skills. You need to have these modern Cloud skills, as I mentioned, but then also understand these traditional workloads and also operate, for example, the Azure Stack Hub solution. This is where we're going to talk about in this series. We have different partner solutions and partners which help us with this. Can you explain a little bit what partners these are and what help they can deliver for our customers? The operator role is something very specific to Azure Stack Hub. We try to be very conscious about staying away from the administrator name because in a traditional virtualization platform, you would have a administrator actually managing everything from the hardware all the way up to the actual application. With Azure Stack Hub, you have an operator which needs to operate the platform on top of which you have the various workloads deployed. This offers from a user perspective, from the person that consumes these services, it's going to look and feel just like Azure and it has that consistent approach of deploying from things as simple as deploying an ARM template to deploying solutions all up. Now, the thing which is different from Azure is the operator side of things. Because in Azure, obviously you have the Azure teams that are managing the platform itself. But on Azure Stack Hub, you would have these operators actually managing the platform itself. For this, there are specific things which make sense from an Azure perspective and knowing Azure and knowing Azure infrastructure, IaaS platform does help towards this, but there are specific knowledge that you also need to cover for this. The way we've approached this over the last couple of years is essentially we're seeing two types of customers. There are ones which are looking at things from a self-service perspective, meaning they have their own IT teams and they are ramping up those teams towards understanding the operator role and learning about the operator role. Then we have a more of a managed service approach where you have a partner which is a service provider, it could be a CSP, an MSP, an SI, whoever it is, that's managing that platform for you, and then you're just consuming the things which are on top of it. For the self-service side of the world, as we started the journey with the early adopter programs, we came up with a number of documents, white papers, videos, guidance on how to actually operate an Azure Stack Hub. These set of documents, we've grouped them together in what it's called the Azure Stack Foundational Series. These are a set of learning videos that help somebody go from knowing Azure IaaS all the way to understanding what an operator role actually is. These are something like 16 hours worth of content, and they are all doubled by the operator workshops where we've published a set of guides that an operator would get more hands-on experience with Azure Stack Hub. Understand what the differences are, understand how the platform itself is managed, all the way to how do you troubleshoot and how do you create a support packet and things like this. These documents help somebody have a glimpse into what an operator actually is and what the Azure Stack Hub operator actually needs to do. That's awesome. I've seen those guides and those videos, and I absolutely love the content we have there. It really helps to understand how you actually operate the Azure Stack Hub infrastructure, if you will, in a way where, as you said, you don't have to do everything, right? Azure Stack Hub is really like an appliance, but there's still certain things you need to know how to operate. This is great. I know that we also talked about when it comes to partner solutions and managed service providers. One topic very often comes up is migration, or as you also say, it's very important not just to talk about migration but also modernization. Can you highlight there a little bit more about what the offers are there and what solutions we have there? From a migration standpoint, you shouldn't be talking about migration as a lift and shift only. When if you would just move that virtual machine over on Azure Stack Hub, that's not, you're not taking advantage of the whole platform, and you're not taking advantage of what's offered from the platform. We've actually came up with a blog series that highlights how Azure Stack Hub at its core is an IaaS platform, and the things that are exposed through the platform itself, out of the box, so to say, and how it helps modernize those applications. Now, in terms of an actual migration, there are things as simple as creating an ARM template. Instead of deploying a certain application, just install the VM, deploy the application, figuring out the settings, and then giving that to your customers, you can automate the whole thing with an ARM template. In some cases, we've seen customers go from days worth of time that it took to figure out how to deploy that application, because separate teams would need to be involved down to tens of minutes, because everything would be standardized inside an ARM template, and deployed end-to-end. When one team needs to change something, they would just come in, change that respective bit, and everything would be sorted. So, from a migration perspective, it's always about migration and modernization. To highlight this, in the next couple of weeks, the Azure Stack Hub will be listed as one of the migration targets for the CAF set of documents. CAF is the Cloud Adoption Framework, and this helps enterprise customers and customers all up with their Azure journey. It's a pretty massive set of documents, and it includes a number of things. It focuses on the Azure migrations, but we will have a set of Azure Stack Hub-specific guidance materials that will help customers with their Azure Stack Hub journey as well. That is fantastic. I'm really looking forward to that, because when I speak to customers, especially when they want to set up their hybrid environment, they are looking for that guidance, and having that in the Cloud Adoption Framework really, really helps. I also want to quickly ask you, we talked a lot about the partner solutions for Azure Stack Hub. If I want to find these solutions, is there a way to get these solutions? Definitely. Everything we've mentioned so far sort of fits into the self-service bucket, if you want. This is with the appropriate aircoats, of course, but this is where you learn, you build that muscle to actually have a migration and modernization story, as well as operate the Azure Stack Hub itself. Now, if you are looking for having a managed service approach, we have came up with a list of partners that offer these services, and these are service providers that offer this managed services approach. That means they will operate the Azure Stack Hub for you. In some cases, they even go further than that, and they help you with the migrations or they help you with the application modernization pieces, meaning if you're looking to rewrite your application and move it to take advantage of the cloud, the PAD services and the things that the cloud exposes, they would help you with those as well. So, this set of partners are offering this as a managed services approach. We also have a number of ISVs listed in this website, which are the partners that are building solutions on top of Azure Stack Hub. So, these are solutions running on Azure Stack Hub and taking advantage of the platform one way or another. In some cases, they're extending the capabilities that the cloud has. In others, they're adding their own specific vertical solutions on top. That's perfect. And again, these are the two types of partners. We're actually going to have a look in this Azure Stack Partner Solutions video series. So, we'll have both service providers but also ISV solutions to make your Azure Stack Hub deployment even better. So, before we actually go out and close this, I want to ask you, for people who want to learn more about Azure Stack Hub, especially about the migration guidance as well as the partner solutions, what would you recommend? What's the next step, except for watching our video series? There are various levels of guidance out there. They start from the foundational series. They add on top the workshops and the guidance that is described in these workshops and then go all the way to things like the tech communities where we have a common space that we're publishing articles across the Azure Stack family of products. And then even on Twitter, where most of the PMs are present one way or another on Twitter, we tend to post new things on Twitter and we have various updates there as well. Now, there's also a number of MVPs, for example, that are publishing articles and they're publishing their own content that enhances this. And there are a set of guides which you've actually created that help with the migration and these are a set of videos that help with the migration and how to think about the migration all up. And these give you the set of articles that you can use as a self-service sort of an approach. If that's not the way you're looking for and you're looking for a managed service, then our service providers can help you with that as well. Now, this is awesome. Thank you very much for all these resources. Again, I really appreciate when I work with or when I work with the Azure Stack team or the Intelligent Edge solution team is like the responsiveness and the way we can actually engage, right? And there's a ton of content out there which help our customers to be successful with their deployments. So thank you very much, Tibi. Thank you very much for having you. In the next coming videos, we will talk more about the different Azure Stack Hub partner solutions. So tune in for the next one. Thank you so much. Thank you so much.


I'd like to thank the organizer for this very nice invitation. So I am very, it's a great pleasure and honor to talk to the ear. So a few memory, personal memory. So I arrived in Paris in 1983 as a student of ENSGF, JF, so for girls. And I have to get registered in some university in Paris. So I went to Paris 7, and I registered as a student of the license and the maitris at the same time, it was the rule of ENS. And I needed to choose some courses. So as a license courses among the list of possible courses, there was one that I took, which was integration by Roger, sorry, and a second course in the level of maitris, which has the funny name of Analyzes III, and also by Godement. And this course was about differential geometry. And both courses were really great, so I learned a lot from them. And so in integration, it was about the big measure. And the theory of measure. And in the second course, I still remember very well Godement on the blackboard. I explained him to us how to define cards. So in French it's cards, cards, cards. And this was very useful for his course. And also for the course on algebraic geometry, I took the semester, the next semester with Le Potit. So and then I passed my aggregation, and the year after, I tried to find an advisor to do my, at that time it was called DEA, Diplomedétude à Profondi, which corresponds now to master degree. And at ENSGF, so Cèvres in Mont-Rouge, there were some mathematicians who had offices there. And among them, there was Marie-France Vigneras. And so she agreed to be my master advisor. And she gave me a paper of Roger Howe on how correspondence for final groups. So it was my first, the first time I met the world of reductive group. So I remember the date, the preside date when she gave me that paper. It was on the 14th of July, 1984. So it was very strange to me that mathematicians are working on the 14th of July. But I was still used to some strange thing already, because Godman was giving his integration course at lunchtime, and the other courses, and that history was at dinner time. And so Marie-France told me that there was a seminar in Jussieu, and that I will need to go at that seminar, but I will understand probably nothing. But the music will be useful for my ear. And she was very strict about it. She said, it is imperative to go. And so on next fall, I went to this seminar, which the title of this conference is taken from. And I remember the third lecture, and it was really great. And the third lecture at the seminar was given by Corinne Blondel, and I was very, very lucky because it was on the paper I have to study as my master degree. And in these days, the seminar was something like a usual kind of seminar. I mean, invited people, visitors or French people to explain her work, but also doing like a running seminar, so studying some difficult subjects which change each year. And that year, it was on the data correspondence over periodic fields. And the seminar was also a good time to learn in a more deeply way one subject. And also, many times, it provides books after the seminar by some of the speakers. And sometimes we're just reading books, and I remember one summer in the beginning when I was reading the Goethe-Mont-Jacques. And I took it with me from my holidays and I was on the beach with Goethe-Mont-Jacques. And because I had to prepare a talk for the seminar on one chapter of it. And so just a few written for you. So this book, which is well known now, it's a reference about data correspondence from the, which was written after the seminar. Another book, by Mughlin and Vaspurgé, which was quoted by Jim Arthur yesterday. And a small story about that book. So as you see, there is an English version. And I met the person who was doing the tradition in English, and Les Lèche-Neps, and at that time, and she told me the following story. So she started first by doing it by automatic way. So she takes the book and just to use some automatic way to translate. And then a new English word show up. And do you, I guess you have no idea what this name, this word. So many adverbs in French, like probablement are translated in English, are probably. And the new word, whichever, was godly. So this is just a story. So it is, but it's good. It's a book which is a paraphrase of a scripture. So godly is good to pure in it. And then some other book from the time of a seminar, especially this one, which is a project started by Michael, and many other French people, not any French writer, but many articles written by French people. OK, just one thing before I go to my talk. June 1987, do you remember what, in that year, there was a special event in Paris. I wonder if someone remembers what it is. Yes, Luminé, you are right. Luminé was part of a special period, period special, which gives also very good books in hysterics. But this year is very important for me because at that time, during the special period, I met for the first time many of you, but in June, it was in Paris. And it was not really included in the special, but it was an event which was closely related, but did run. And in that event, I remember a very nice talk by Hervé Jacquet about, was about one theorem of Valls-Purget, and some of that talk. And so I give you the title of this event because nobody remembers, unfortunately. But for me, it was good because it was the first time I was attended to a conference. So it's probably why I remember. OK, good morning. Ah, yeah. OK. Too long, too long time ago. So now I'm going to my over here. So I will talk about joint work with Ahmed Mosaoui and Martin and Martin Solleveld. And so it's about N-hands, long-distance parameter. So I will take a group G. G will be the F-points of redactives. Connected at a bright group defined over F. As usual, F is a local non-alchymedian film. And the long-distance parameter, or L parameter, is a certain morphism from, so I will take this as a, this group as a version of the Vadorian group in the L group. And for the version of the L group, I will take this one where G, my notation is the following. So GF is a redactive group over C with root datum, dual to that of G. So the situation is well known for all of you. Just a small example. If G is GLN, F, this one is same group. If G is, for instance, if it is SLN, this one will be PGL. And here it is SP. And of course, if the Piardet group is SP to N, you will get SO to N plus 1. If this is SO to N, you will get SO to N over C. And this parameter is supposed to have, I assume to have some good property that the usual one that you can find in Corvallis, in the paper by Boren in Corvallis, for instance. So, what do I want to say? So this is the parameter. And I am especially interested of a notion, I would like to define a notion, first, a notion of hospitality. So as usual, this is, I will talk on objects like this as objects living on the dual side and the automorphic side, which is just the representation side, just the side of representation of G, a smooth representation of G. So, and of course the idea is that this hospitality on this object on the dual side should correspond to the usual notion of hospitality, of hospitality or super-hospitality at the level of representation. And of course we, you know the, it occurs for instance in the talk of Bertrand Le Maire, for instance. You have, it was mentioned, the Bernstein center of the Bernstein decomposition, which is a decomposition of a representation of, a smooth representation of G due to Bernstein as a product of certain, certain full subcategory and these subcategories are indexed, so Bernstein decomposition. So this is the category of smooth representation. You can do it as a product on some element of some category that I will call R as G of G. And this element S are given by, actually I think L will be a levy of G, a levy of a parabolic, of a rational levy of a rational parabolic. And sigma is a irreducible super-hospital, super-hospital representation of L. And this relation is slightly weaker than the conjugate, the G conjugation. We allow to twist by an unamplified character of a levy. So if you believe in local and non-correspondence, which is known to be true in many cases, like for G, equal to GLNF due to the work of Ari Santelor and also never proved by Enyar and thought-proof by Schultz. It's also known for classical groups, thanks to the book by James Archer. And if we add the stabilization, which was conditional to the stabilization of the local transformer, which is now done in this book. So as I said, just a few lectures, but if you come from a number of pages, you will see why there are some things like this. So if you believe in local and non-correspondence, the irreducible representation, I will call this, first this will give you a partition of, as the level of irreducible representation of the set of the smooth dual of G as it will be partitioned, the smooth dual of G in some subset. And I will call this Bernstein series. So this has nothing to do with local and non-correspondence. But now, if you believe in local and non-correspondence, this set should be in bijection in a nice way with certain objects attached to the dual side. And so it means that you should also have a decomposition similar in the dual, on the dual side. And so the idea was to try to define such a partition directly on the dual side without assuming local and non-correspondence. But here, this is just not possible for the reason that to an L parameter, local and non-correspondence should attach an L packet of irreducible representation of G. And in a given L packet, it happens many often that you have simultaneously irreducible super-cospital representation and non-super-cospital representation. Of course, all these non-super-cospital representation need to be discrete. Discrete series because the parameter to have super-cospital in its packet need to be discrete. And that kind of situation already arrived in SP4 or G2 example. So you have a packet of irreducible representation. So it's when G is this group, which contains super-cospital and non-super-cospital. And for instance, we can have a packet which contains, I will say, representation with unipotent reduction. That are with some are super-cospital and some are in the principal series. And so principal series will mean that in the Berthin decomposition, they will occur in the Berthin series attached to a tolus. So the levy L will be a tolus. And the super-cospital occurs in the Berthin series attached to a levy L which is equal to G.  In fact, you will see at the end of the talk that it is some course of the composition is in fact compatible. So you cannot define cospitality for an L-parameter. We will turn to the notion of N-amps in French Enrichie, L-parameter. Some people say that the L-parameter is the name of the object and the enhancement will be its... The L-parameter is the last name and the enhancement will be the surname. So in any case, it will be pairs like this where c-phi is an L-parameter and rho is an irreducible representation of some group that I will call s-phi. So here is a definition, so s-phi. So we use the following construction which is due to Jim Arthur. But this construction is compatible also with the one given by Tashokareta. And the service, as you can imagine, this group is the component group of some group and the group has this fancy notation which we'll explain now. So we are on the dual side, so we have the long-long dual of G and this is its adjoint group, so the quotient of the group by its center. And this notation is for the simply connected cover of this one. And we have a long-long parameter. And this group, so this, this one, yeah, this is the definition, it's the centralizer in G-add of the image of phi and the image of phi just means this. And here, this one with one is just the inverse image of this group. So this is taken for the paper by Arthur and a note on the help group. And so we take this group and, of course, the group in general, for GLN it is connected, but in general, no, even no, sorry, in general this group is not connected, as I defined it, and I take, I call S phi its group of component and so the parameter is, and by an irreducible representation of that group. And this kind of situation occurs in a special case, for instance, for representation which, which I admit E-warifix vector, which are a special case of the representation with unipotent reduction. And this occurs in many years ago in an old paper by Lustig on square integrable representations. Okay. So this could be the end of the story, but in fact, the point is for us, the key point is that this group has another interpretation. So I have considered the group, oh, sorry, I need the things to, between the two. Okay, this group, I define, this group, I define it as a centralizer in this group of the anti-image of phi. And then I can also consider not this group, but just the phi group, and it's what I will do now. So I will introduce... This phi is finite. S phi is finite, yes. And I will introduce this group, same notation, the center of Sg of phi of wf. So I just ask that this group only request that the element of the group centralize the image of wf. And so in this group, and also I have this group, and I will also introduce some unipotent element, which is the one you can imagine. So the image by phi, because this, if I do that, I will only consider the restriction of phi to wf, so I need to do something with SL2c part. So I will just consider the image of this element. And this is a unipotent element, and it can be viewed as a unipotent element of g phi. And so we have a group, and this one is not finite in general, and in general it is disconnected, not necessarily connected. And okay, I have this, and sorry, what I call it, g phi, sorry, it's not a unipotent element, it's a unipotent element of a dual group of L of g. And I will consider the group, the group I will call g phi, sorry, is just the component group of this one. And the point is that it is easy to check that, in fact, s phi is... No, no, it's supposed to be right. Sorry, I'm just confusing my notation. This is not the component group. I was right in the beginning, sorry. This is this group, g is an unipotent of a g, which can be viewed as an element of g phi. And the group of component, sorry, it's the group that I call to be more precise, I should call it, i g phi of u phi. So in g phi, I take the group which is... So this is g phi. This is not this, sorry. Excuse me. So we have g phi, so we are in g phi. I can consider the centralizer of... in g phi of u phi. So g phi maybe is possibly disconnected and if you take a unipotent element in it, it could be even more disconnected. And so you take this group and the point is that this group is isomorphic. So it's a consequence of Jacobson, Morozov, and all of them, of course, final groups. So the idea to look at this group, g phi and that group, i phi is due to my student, Ahmed Musawi, and sorry, it was part of his thesis. And the point is that now you are of a complex redactive group, g phi, possibly disconnected and the unipotent element in it and you are looking at irreduci... and rho, which was an irreducible representation of s phi. You can view it as an irreducible representation of that group. So you have a complex league group maybe disconnected, unipotent element in it and a representation of that. And when you see this, you think of springer correspondence or more precisely it's generalized version due to lipstick. Yeah, by reconstruction it was irreducible representation of s phi but when you view s phi as this group you can view it as a representation of that. And so now you can say but you can define a notion of hospitality. This is cuspidol f. Of course we want that phi is discrete. So its image is not contained in a proper levy subgroup of the L group and also a condition on rho and the condition on rho is just that this with the above notation is cuspidol per in this group g phi. So I will not have time to explain what it means. So this notion is due to lipstick in the case when this group is connected. So it is in this paper in Inventionist I think, yeah. This paper about intersection homology complexes. It is the paper in which it defines generalized springer correspondence. And in fact it is possible to look to view the representation rho as some irreducible local system and there is a notion of cuspidol f for local system which involves to look at some perverships on the unipotron variety of the group g phi and on this notion of perversive you have a parabolic induction and Jacky restriction which are just, I mean, of course it's different way of defining them but cuspidality is the same definition as you know the object which are Jacky restriction when you apply the Jacky restriction it is equal to zero if a levy is a proper levy of a group. So it's something similar which that lipstick has done in this setting and this also has a nice property that it also work for L modular representation and this is due to Acha, Anderson, Gito and Rich. Okay, anyway, you have cuspidol pass but since I will not give you the definition I decided to give you at least two examples. So first example is for this group so that you already saw in Vincent Cechard talk so same notation this is a division D is a division algebra so object that you already were familiar with thanks to the Godman-Jacky book so it's a division algebra so the dimension over with center f and so the dimension of D over f is a square and the condition m with our called md so if I impose this condition all this group are in a form of gln over f and for later use I will define so this group are not split and even more they are all non-quasi-split except in the case where D is 1 and in that case we have a quasi-split form which is gln which in fact is split and sorry this is the example now the dual of gln is maybe I should first give you this yes which is well known that all the inner form are the same long-length group which is the long-length groups of the inner it's given by the other quasi-split group in the inner form of the class and of course it's simply connected cover is sln and the group here if you compute it it's a cyclic group and it's n z mod zn now if your parameter has to be discreet so it's the example of hospital and as long as parameter because the fee discreet it has to be the following form so here this is an irreducible representation of the vague group and this one is an irreducible representation of sl2c and we need a name for this dimension we call it d' and then we can check that so it is written in our first paper which is now published in the manuscript which says that this is hospital if and only if of course we take this fee fee as above we need a condition about the objects so the left common multiple of d and n mod d' is equal to n and of course if you look so this is the condition that you get if you just compute hospital pairs and of course if you look at long nose correspondent look at long nose correspondent for g so we attach which is known to be true we attach to your parameter some representation 5p irreducible representation of your group and in general of course even if you take fee discreet long nose parameter this representation is not always supercospital but pi fee is supercospital if and only if the same condition so you can find it for in when f is zero characteristic in the book by Delene Cashdown and Vigneras on the Bernstein center where you also can find the Bernstein center so this means in particular that there is a nice compatibility between the hospitality on the dual side and the hospitality on the group side in this special case a second case that I would like to give to you is so for classical group sp and so so even you can look in the paper by Ahmed which is now in publishing representation theory so another very similar case which is the case of a unitary group given the quasi-split you should make quasi-split and E is a quadratic extension of course of f VL group is just GLN over C some indirect product with a Galois group of E now this is in a slightly nicer way so my group G5 so my group G5 what is this I was following form it's a product of M sigma M sigma E prime some orthogonal group so if you take an arbitrary long lens not M sigma an arbitrary L parameter you will get the group G5 will be a product of that kind of group and times a product of some group of GLN type but because you assume you're considering a hospital L parameter so f has to be discrete then there is no GL part and of course now we compute the group and this group is of course it's a product similar with some similar group in M sigma C of some so my unipotent element I wrote it as a product of unipotent element depending to this decomposition and of course the group is a product of group and SP and so what I need and so U is a product of U sigma and the condition to get cuspidality so in a lipstick paper you also find that if a pair is cuspidal the unipotent element needs to be distinguished which in this case gives you that so the unipotent so U sigma is a unipotent element in this orthogonal group so it corresponds to a partition of M sigma and the partition you have in this situation of cuspidality is the following it has the following form and so this is when sigma is in I plan so I take just I plan and we can present what means this index it means that the corresponding parameter is of orthogonal type or simplistic type but we don't want to enter in this and here it is the following partition 2, 4 2d sigma and so the group are just Z mod 2z Z mod 2z and now so the group I forgot about the orthogonal or simplistic group this group is a product it's a product of Z mod 2z so it is exactly like this so this sigma is the dimension in that way in that way that u sigma is an element of the corresponding orthogonal or simplistic group my notation are not good but anyway I would like to be going too fast so I need to see pair so Z to a minus 1 and this is Z mod 2z so it's a product of copies of copies of Z mod 2z and in SP is similar I would like to have this fancy things I could have just write it Z a but the reason which appears soon I would prefer to write it like this okay yeah and this is but then and my this is the point I will define rho and rho sorry rho is an irreducible representation of group A g phi of u phi so it's a product a tensor product of representation of all these groups and rho is a tensor product of representation that I would call rho sigma for sigma in i prime cup i prime prime and I need to define what are the rho sigma and so rho sigma is defined okay so now it's a non-trivial character of a two-element group in each time I mean it's a rho sigma on the z z i is minus 1 at some power of i but okay at some at some power at some okay to be i or it could be written in more properly and if you do this in a more proper way then you will see so you can explain more precisely what is i depending if I take 2a or 2a minus 1 and the point is that we have the point is we can choose we can compute and we find what we have to put here and then you will get this is the point that epsilon sigma of za minus 2 is always equal is always equal to a minus 1 and if you look at this condition and especially this one these are exactly the condition which are funded by Collette Mughlin in his work about the long-nose parameter for unitary groups and the description of a superacospital and so the condition and you will get that the condition you have on the unipotent element and the condition of this character correspond exactly to the fact that the Jordan set.5 which occurs in Collette and Marco work for instance or in other paper by Collette Mughlin this is as no all and this is the condition that for the character how to be alternated and so it gives a bijection between between cospital pairs and the superacospital representation of quasi-split unitary groups and so the conjecture in generalis is just that these two kind of objects correspond by local long-nose correspondence and then I will pass quickly to the other things so yes oh yes it's a very good question so it's because I'm okay it's it's hidden in the picture but I am in fact considering at the same time all the inner twist of a given group yes and the other inner form of it's why it's not equal to one and in fact yes we can if you want to look to a given group and not to all these inner twist you need to take not all the character of belonging to S5 so first we need to only consider the phi parameter which are relevant for the given group and there is also a notion of relevance for the and we just take the one which are relevant for G and so the yes because this is just using a codevitz we use the codevitz isomorphism and then you know the action of rho on some some group which is containing in S5 and you just take the one which and this group and the action of that group should correspond to the codevitz isomorphism to the this form of inner twist of a group I mean this is okay but it's just not have time just to really go into it but just say yeah just to answer to your question so we have two group we have this group so just to answer to that specific question so this group is just the group which occur if you look at the group which is assumed to be quasi-split then we have this group sorry the center of group and you consider have a connected component of that group and then we have a central X and then S5 is a central extension of some group of that group by S by some group Z so we have this exact short exact sequence and five and one and where this group is just the centralizer of the center sorry of that group mod its intersection with the connected component of the center of the image of phi and then when you have this you look at the restriction of your irreducible character who of S5 to that group and from that group using Godwitz isomorphism you can understand which you can get a character and then there's something to do but it's and this is the same this one so if your group is quasi-split this is enough to consider but if your group is not you need to consider this okay and then okay I will just analyze this and the conjecture is thus there is a bijection between what I will call phi of Lg which is this will be G conjugacy classes of n-n's l-parameter and here I will add cusp to mean that I will only consider the cuspidal 1 and the union of irreducible cuspidal representation of G alpha for all G alpha in such that G alpha is for all the inert all G alpha inert twist of G and if you take here of course in here you can take the one which are G relevant for some G then this one should correspond to here cusp of the same corresponding G if you consider the element here which are G relevant for given G we conjecture that we are in bijection with a super cuspidal irreducible representation of G so as I told you this conjecture is true for quasi-splitonatory group of course for GLN for SLN of the division of the trivlar for classical split classical groups and in that case is due to Ahmed and it's also true for representation with unipotent reduction of any PID group which is split which became split on a unharmified an unharmified extension of F and this is a recent paper in the archive by a student of Eric Obdam Obdam and Martin Sodeveld and of course when when G is simple of adjoint type simple adjoint type this follow follow from realistic classification of representation with unipotent reduction of such kind of group so I guess my time is finished what you say yeah so I would just just two minutes so this is the conjecture then from from that what we can do is we we write on this so then if you take all the for say for 1G the set of Anas-Langlance parameter first it is inside the theorem so Musaoui-Soleveld it's a Gs so we we of course we have this notion of hospitality now we can take as a cuspidal pair so Levy now this one is a cuspidal is an Anas-Langlance parameter for L and the relation is similar as in Bernstein so we have an action of some object which reflect the which correspond by Levy-Langlance correspondence for Chorus 2 of the Anamifile character and we can similarly to Bernstein define this kind of inertial relation on this object so this is the first thing so it's kind of avatar of a Bernstein decomposition but only at the level of irreducible object and the second thing is that this is in bijection with a simple module of a twisted because I do not have time to explain what it is twisted a fine so say Gs I would just say that the root system so when you have a fine algebra there is some root system attached to a final group so it is correspond to the root system of some group some final group and I will not have time to explain what it is and so we have this and the last word of course this decomposition is not the analog on the dual side of a Bernstein decomposition but now if you consider not as I said in the beginning an L packet is not an union of Bernstein component and the Bernstein component is not an union of L packet but from this it's very easy by considering the stable Bernstein center an union of L packet which in this terminal in this setting here it means to considering some union of of this subset and so there is a well defined union of that subset and so we can have a course decomposition of this as some I phi where this one union of this object and so this give a way to link this kind of decomposition not in L packet but in union of L packet and this union of L packet is the one which occurs in the stable Bernstein center so I will stop here Do you have any questions? Yes I have two questions one is just about notation previously the input of phi was the dual group of the complex numbers L, G Now you write G and index is about G check Are you alright? You are perfectly right It's not this It's LL Thank you It's LL It's L It's this And you said you have two questions The other question is is there a natural topology on this? I have no idea I would be very happy to have any insight on this kind of things At the moment it's just set It's just a set of So it's why it's not as nice as a Bernstein decomposition because it's not the decomposition of it's just at the moment you see a visible object but I guess there should be something but it's unclear to me I want to I want to correspond with the modules of the Hecker That's a a bijection, is there more than one way to normalize the bijection? Okay So the definition of a bijection is very complicated and it's completely based on the Lustig construction in the first graded of the Hecker algebra so it passed through that and also the Hecker algebra we get depends on indeterminate and at the moment there are I mean there are too much freedom on the indeterminate I mean we can choose several kind of we can take just one and determinate but if you do that you have the right algebra because in the Piad case we can have different and so but at the moment so it's not the definition is too as to freedom in it but when you specialize for example for classical groups we can for quasi-split symplectic you know for gonadal group we have check using the work of Volcker-Hierman and also the work of Colin Mughlin they correspond to the there is a canonical choice of indeterminate in order to get an algebra which is the right one but I think there is something to do still like I mean to input like L-functions or something like that in the picture which to make it more canonical so in this in this question there is no multiplicity for in a form which appeared all appear with only ones the inner twist appear so it means that sometimes we have two inner forms which correspond to different inner twist so this is the trick the two inner forms can appear can appear, yes


Hello everyone, I am Dr. Pankaj Bundyal, Consultant Radiologist and Director of Bundyal Radiodical Speak Center at Amravati Mahashtra. I have been practicing all imaging modalities since last 21 years, however CT and MRI are my main areas of interest. It is a wonderful feeling to be on Indian Radiology channel. Today, I am speaking on tumor mimics in brain. The condition which mimics tumor are demyelination, inflammatory, infective, that is tuberculosis, fungal or pyogenic, post-treatment, that is post-radiation, vascular malformation, ischemia, arterial and venous, traumatic and degenerative. 20-year-old lady presented with sudden onset of Diplopia with case palsy since 15 days, with history of mild ataxia. Hyper intense signal is seen in pons, almost full pons is involved, no enhancement seen. So no significant expansion, almost entire pons involvement, no enhancement. This is a classical pontine demyelination. Follow-up scan after the treatment shows significant reduction of the hyper intense lesion seen in the pons. Comparison case, a hyper intense signal seen in the pons. The lesion is eccentric in position. Expansion of pons is seen with patchy contrast enhancement. So expansion is eccentric and patchy enhancement. This is a classical pontine glioma. Then we come on case number 2, 16-year-old girl with acute onset of excessive sleepiness for 10 days and history of psychiatric disturbances. Hyper intense signal seen in the region of hypothalamus just adjacent to the third ventricle on T2 weighted images. On coronal flare it is very nicely seen, hyper intense signal in the region of hypothalamus. Post-contrast study shows poor enhancement and follow-up MR treated with IV methylprandisol for 3 days followed by oral steroid, significant improvement in all the symptoms and there is a significant reduction in the hyper intense signal seen in the region of the hypothalamus. So this was a classical hypothalamic demyelination. Then we come on case number 3, low-grade fever, URTI since 5 days with history of convulsions. Multiple hyper intense signal seen on flare and T2 which appears hyper intense on T1. On contrast study open ring like enhancement and the incomplete portion of ring is on the grey matter side of the region. No mass effect, no surrounding edema on diffusion weighted images, mildly increased ADC. MR spectroscopy shows elevation of choline and elevation of lactate. So this was classical TUM effective demyelination. Laps sided severe headache, repeated third nopalcy, single episode of convulsions. An abnormal hyper intense signal seen in the left temporal region on T2 weighted images which appears hyper intense on flare, on administration of the contrast patchy enhancement was seen. So this was classical Rosydafman. It is a idiopathic benign lymphofroliferative disorder mimics lymphoma and other inflammatory conditions on imaging. Lymphadenopathy is also seen, diagnosis based on classical histopath findings. 17 year old boy presented with altered sensorium and history of convulsions. On flare and T2 hyper intense signal seen in the bifrontal region which appears hyper intense on T1 and on administration of the contrast intense enhancement seen along the faults. So this was a classical felsine tuberculosis. Then we come on fungal lesions, history of B cell lymphoma, induction therapy and history of convulsions. MRI shows an iso intense signal on T1 and flare and patchy or punctate hyper intense CT on T2 in the right occipital region. Post contrast MRI shows irregular ring enhancement. On diffusion weighted images and ADC the wall as well as the nodular soft tissue display diffusion restriction while the content don't shows restriction. Single voxel MR spectroscopy over enhancing area demonstrates diminished and acetyl aspartate to creatinine ratio, normal colline to NNA ratio, normal colline to creatinine ratio and diminished myo-inositol to creatinine ratio. So this was a classical fungal lapses. Patient has developed febrile neutropenia, neutrophil count 0, similar lesions in lungs, chest, liver and splem. Succumb to it post-martin confirmed the finding. Case 7 20 year old boy studying in third year of college presented with left sided hemiplegia, lower limb more than upper limb. Hyper intense lesion with hyper intense rim seen on T2 weighted images and on T1 the lesion was hyper intense and rim was hyper intense. After administration of the contrast ring thick walled enhancing lesion was seen. So this was reported as lymphoma or tuberculoma operated outside. Follow-up scan post-op again shows a ring enhancing lesion. This was classical aspergilloma. Aspergillus abscess appear radiologically identical to other brain abscesses as classical ring enhancing lesions with striking high signal intensity on diffusion weighted images. However aspergillus abscess may have peripheral low signal intensity on T2 weighted images. This is better appreciated on GRE or SWI sequences. 50 year old man presented with fever since 5 days left hemiparesis. Abnormal lesion seen in the right parietal region. Mild hyper intense rim on T1 dark to rim due to free radicals, diffusion restriction, peripheral enhancement and lactate peaks on MR spectroscopy. So this was classical pyogenic abscess. Comparison case similar ring like enhancing lesion seen in the right occipital region. Nodularity of the wall is seen. Non-uniform restricted diffusion is regularity of the inner margin on post-contrast scan. In addition to lactate high choline peak was there and on perfusion hyper perfusion of wall was seen. So this was classical necrotic tumor that is GVF. Then we come in to other case. In this we are seeing a hyper intense lesion on T1 with surrounding vasogenic edema in the right thalamus with mild mass effect. Which appears hypo intense on T1 with surrounding hyper intensity. Mild diffusion restriction was seen after administration of the contrast intense enhancement was seen. So this was classical giant conglomerate tuberculoma. Giant conglomerate tuberculoma appears T2 dark, slightly T1 hyper intensity, minimal diffusion abnormality, no GRE susceptibility and intense post-contrast enhancement. This is the other case. In this we are seeing a hyper intense lesion in the right basal ganglia which appears hypo intense on T1. Diffusion restriction was seen but appears hyper intense on ADC. On administration of the contrast heterogeneous enhancement was seen. On MR perfusion it was showing low perfusion and on spectroscopy peak lipid and glutamate seen. So this is a classical intratranial tuberculomas. Comparison case abnormal hyper intense lesion seen in the left basal ganglia which appears hypo intense on T1 with surrounding vasogenic edema. Ring like enhancement seen on the contrast study with mild mass effect in the form of compression of ipsilateral lateral ventricle. MR perfusion shows relatively cerebral blood volume was 2.5. So this was classical metastasis and patient was known case of CA breast. Then case number 9 40 year old men presented with altered sensorium and right hemiplegia. Abnormal lesions were seen in the both cerebral hemispheres of varying sizes with surrounding vasogenic edema throughout the brain. Largest was in the left basal ganglia. On MR spectroscopy reduced NNA mildly increased choline and lactate peak. So this was classical toxoplasmosis. Toxoplasmosis will also show a eccentric dot sign. A ring like lesion with hyper intense rim seen within the ring. Some of the lesion of toxo may be hemorrhagic. So toxoplasmosis it is iso to hypo intense on T1 weighted images. Hettogenously hyper intense with occasionally hypo intense rim that is target sign on T2 weighted images. Variable post contrast enhancement no minimal enhancement no or minimal enhancement to ring or nodular enhancement that is eccentric dot. Hemorrhage occasionally present and MR spectro shows high lipid lactate peak. Lymphoma versus toxoplasmosis. Lymphoma usually periventricular in location while toxo is seen in deep grey structures. No hemorrhage in lymphoma often hemorrhage is seen in toxo. Solitary or multiple lesions in lymphoma always multiple toxoplasmosis. Diffusion restriction with dark ADC values are seen in lymphoma. Usually no restriction if present it is peripheral. High colon or MR spectroscopy in lymphoma only mild increase in colon peak in toxoplasmosis. Now we come on post treatment. 58 year old male operated case of GBM received radiotherapy presented with multiple episodes of seizures and six nerve pulsing. Abnormal expansion region seen in the left occipital region in the white matter extending to the cortex with Swiss pattern of the interior arqueate fibas. MR perfusion shows low cerebral blood volume that is less than one. So this was classical radiation necrosis. Now we come on vascular malformation oblique aneurysms. 45 year old male patient came with sudden onset right plover limb weakness. Well defined hyper intense signal seen in the left frontal region on CT. It appears hyper intense on flare. MR angiography was normal and cavernous malformation are angiographically occult and do not demonstrate arteriovenous shunting. Follow up MRI on T2 and GRE shows blooming. T1 SAGE re and post contrast. Some enhancement is seen on contrast study. So this was classical cavernous angioma cavernous angioma are collection of sinusoidal vascular spaces. Clotted unclotted blood in various stages of oxidation. Slow circulation without a vision ting on MR typical cavernomas looks like raspberry or popcorn appearance. Gradient ecosystem shows this region better than T1 and T2 weighted images. These are the multiple sites of the cavernomas on MRI and CT scan. Now case number 12 35 year old lady presented with headache occasionally tingling in all the limbs vertigo on and off. This is a classical brain capillary telangiectasis. HWI may have high sensitivity equal to GRE in detecting capillary telangiectasis in brain. More often solitary but sometime it may be multiple. Common locations are pons cerebellum or spinal cord. T2 and flare shows normal or slightly increased signal. T1 shows ISO or low signal. T1 contrast study shows tippled enhancement if large can show branching or linear draining veins. So this is classical brain capillary telangiectasis. Case number 13 holocranial headache on and off since three years. Three episodes of TIS in last one year. Weakness of left upper and lower limb with slurring of speech. On examination patient was conscious oriented and no focal neurological deficit. On CT well defined hyper intense signal with surrounding vasogenic edema seen in the right temporal region. On contrast study ring like enhancement was seen but inside the ring there was a intensely enhancing legion seen within the ring. In same study we have done a CT angiography and it was classically a partly thrombosed right MCA aneurysm. It is very nicely seen on CT angiography study. DSA confirms this thrombosed aneurysm. Now we come on in facts and hemorrhages. Case number 14 45 year old male presented with altered sensorium and irrelevant talk. Lower limb weakness came for neurosurgical review. Abnormal hyper intense legion on T2 weighted images seen in the region of corpus callosum. This legion were hyper intense on flare, hyper intense on T1 and diffusion restriction was seen. It was thought to be a corpus callosal lymphoma oblique glioma. But MR angiography was done and there was a narrowing seen in the both sided anterior cerebral artery. So these were bilateral ACA territory in facts. Case number 15 24 year old transplant recipient on triple immunosuppressant presented with left hemiparesis and Diplopia. Abnormal hyper intense signal seen in the right thalamus with surrounding vasogenic edema. On contrast enhancement was seen and on diffusion restriction was seen. On SAGE hyper intense sinuses were seen. So this was on MR angiography. There was a thrombosis of the right transverse sinus and right sigmoid sinus. So this was classical venous in fact. This is the other case in which we are seeing abnormal signal in both thalamus that appears hypo on T1, hyper on T2 and flare. On MR venography there was a thrombosis seen in the straight sinus. So this was a straight sinus thrombosis. Newborn 45 day old baby presented symptoms of raised ICT refused to feed examination shows sun setting. Clinical diagnosis was meningitis or encephalitis. On T2 weighted sequence mixed intensity lesions seen in the mid frontal region on T2 and T1 also. Coronal T2 also shows very nicely a mixed intensity lesion in the mid frontal region. GRE shows blooming, diffusion source restriction which appears hyper intense on ADC. So this was classical non accidental injury. 45 year old male patient presented with left sided tatexia and tremors of left upper and lower limb with dysmectria. No nystagmus history of CNS tuberculosis 3 years back for which completed course of AKT was taken. Clinical impression was relapse of CNS tuberculosis in posterior fossa, abnormal signal seen in the region of the just anterior to the fourth ventricle. On T2 weighted image these lesions were hyper intense with hyper intense valve. So this was classical hypertropic olivary degeneration. So diagnosis T2 hyper intense lesion confined to olivary nucleus with lack of enhancement or diffusion restriction associated with lesions in contralateral dented nucleus, contralateral superior cerebellar peduncle, axilateral red nucleus, axilateral pontine tegmentum. So we have covered almost all conditions which mimics tumor. Thank you.


One of the more important tools is the ability to use function notation. And this emerges as follows. We frequently express the value of a quantity using this function notation. Some examples. We might talk about the velocity of a projectile in meters per second. Note that units are very important here. T seconds after an experiment begins. Or another possibility. We might look at the population of a country in millions of persons. T years after some designated starting year. Or possibly we may look at the percentage of voters who support a candidate. T days before an election. And in every case the function notation tells us something useful about the quantity. For example, let's let v of t be the velocity of an object in meters per second. T seconds after an experiment begins. And what is it telling us? Interpret v of zero equal to 25. So one very useful thing to do here is we can instantiate the definition by writing it down. And then replacing the terms of the definition with some specific values. So our definition v of t is the velocity of the objects in meters per second. T seconds after an experiment begins. Paper is cheap. Understanding is priceless. v of t equals. We'll write down that definition. Now what do we have? Well what we know is we have this v of zero. So t and zero. We're going to replace those two. And first step this is v of zero. T has been replaced by zero. This is the velocity of an object in meters per second. Zero seconds after an experiment begins. And then v of zero is the same thing as 25. So I'll make that next replacement. And so I have the 25 is the velocity of an object in meters per second. Zero seconds after the experiment begins. And so this suggests v of zero equals 25. We can read this as at the start of the experiment the velocity of the object is 25 meters per second. Well let's take a look at another example here. Let v of t equals 150 minus 5t. Here we have an explicit formula for the velocity of the object. We want to determine how rapidly the object is moving after six seconds. Well a little bit of analysis goes a long way. In this problem our function actually gives the velocity of an object in meters per second. And what we're looking for how rapidly is the object moving. Well that sounds a lot like a velocity. So we should write down our function expression. Note that we've included the units of that function expression. Again paper is cheap and it doesn't do any good to skimp on the writing when it makes the comprehension much more difficult. Now we've been given an amount in seconds and the only thing in the problem that's measured in seconds is going to be our t value. So that suggests that t is equal to 6. So we'll let t equal 6. We'll substitute that into our expression v of t v of 6 150 minus 5 times 6 meters per second. And after all the dust clears v of 6 equals 120 meters per second. And again we should state our answer including the actual units. The velocity after six seconds was 120 meters per second. Well here's another problem. The height of an object after t seconds is going to be given by some formula. When did the object hit the ground? Again a little bit of analysis goes a long way. In this problem we're given the height of an object. And so we know something about the height of the object. And we're looking for a when. So that suggests a time. And again what that suggests is we should begin by writing down our function expression. Once again notice that we've included the units of our expression. Now notice we've been given a condition when did the object hit the ground. And what that suggests is that we have the height of zero meters. So we'll substitute that in. Our height is zero meters.  So we can now solve this equation. Our quadratic formula, simplification and our approximate values. And we finally go to our interpretation. t is the time after the object is fired from the cannon. So only the second solution makes sense. The first solution is a time before the object is fired. And that doesn't make a lot of sense. And so we can state our final answer including the units. The object hits the ground after 60.033 seconds. Well how about a graph? So I suppose f of x is the temperature of a location in degrees Fahrenheit x hours after 6 a.m. And now I have this graph showing the plot of y equals f of x. And we want to know when did the temperature drop below zero degrees Fahrenheit? Well f of x is the temperature. y equals f of x. So the question, when did the temperature drop below zero degrees? Well temperature, when did f of x, when did y drop below zero? And so there's the important connection we make. We want to look at the place where our y values dropped below zero. And if we look at our graph, that's going to occur right here at t equals 5. Not here because the temperature did not actually drop below zero there. Rather at t equals 5. And so at x equals 5, x is the number of hours after 6 a.m. So when did the temperature drop below 5 hours after 6 a.m. Or we could say 11 a.m.


Does your smartphone have these apps? Cause it should. Smartphones are great. They're an indispensable part of modern life and have almost everything you need pre-installed right out of the box. But if you want to improve the security, the functionality, the usability of your phone, you're going to want to make sure you have a few other apps installed right from the start. Here are the top five smartphone apps you need. Number five. A web browser. I know what you're saying. There's a web browser already, but the default browser pre-installed on your phone, while great, doesn't give you privacy. If you don't want anyone tracking your web surfing activity, consider a third-party web browser. Alternatives like DuckDuck, though, which has a nifty one-touch clear-all tabs and data mode, or the Brave browser with its ad-blocking engine. Mozilla even has one called Firefox Focus. Coming in at number four, a payments app. You've got to set one up. If you want to use your phone to pay for things, you need to have the right payments app installed. If you don't use Apple Pay, Google Pay, or Samsung Pay, you'll need to make sure you have Venmo, PayPal, Zell, or whatever service you use. Up to number three, a VPN app. Data privacy also means using a data network you trust. And if you're roaming or using public Wi-Fi and you want a little extra security, get a trusted VPN service and download the app. Sliding into number two, a password manager.  Password managers increase both security and convenience, plus you can use the same password manager account across different machines. So you'll have them at the ready no matter what machine you're using. At number one, an authenticator app. Two-factor authentication is crucial to secure your online accounts. And yes, you can use SMS, but it's less secure than an authenticator app.  That code typically expires 90 seconds after it's issued. So use it. Got a new smartphone? Well, smartphones are only as good as the apps you have installed on them. And remember to keep those apps updated. What's your must-have app? What did we miss? Let us know down below in the comments. And if you want more great tech news and info, follow us right here on YouTube. Hit that like and subscribe, youtube.com slash daily tech news show. Get our podcast at dailytechnewshow.com and support us at Patreon. Patreon.com slash DTNS. I'll see you there.