So, well, ladies and gentlemen, now I'd like to invite most power-packed leaders for this session. First up, Ms. Anmol Gill, head customer marketing India and neighboring countries, Bacardi, India Private Limited, Dr. Ipsitha Chatterjee, head innovation, development and brand strategy Lotus Herbal's Private Limited, Mr. Prasanna Raman, advisor, solutions lead, Snapchat India, Mr. Rajiv Jain, VP Marketing DS Group, and this session will be chaired by Mr. Akshay Mathur, CRO Tyro. Well, ladies and gentlemen, with this, I'd now like to welcome our entire panel who's joined us and given us their valuable time. And I'd now like to pass it on to our session chair, Mr. Mathur, for taking it forward with us Abel Pandu. Over to you. Thank you. Very good evening to all who've joined us for this panel discussion on building brands of tomorrow through mobile from branding to performance. We have a very rich panel today, and we have Anmol Ipsitha, Prasanna and Rajiv in the panel. And I'm very excited to kind of talk about how mobile marketing has been shaping up in the digital ecosystem and how they've been using their brands and leveraging that as a medium. We have a great mix of brands and publishers in the panel today and just excited to kind of talk through with them. I'll start with you Rajiv. Mobile marketing is like the dominating in the digital pie right now. So how significant has it become for your brand at DS Group? Rajiv, I think you're on mute. Am I audible now? Yes. Yes, absolutely. Good evening everyone. The number of mobile, when we talk about mobile, I think digital is basically becoming synonymous with mobile only now because I think that more than 90% of the access of digital is for mobile only. Even during pre-COVID also, the numbers were increasing and during COVID time, the numbers have grown up very, very high. This has been helping the market at DS brand team and all in various ways. Number one, various of our brands, they are high impulse category brands which needs sustained visibility throughout the year and needs various consumer engagement activities. In TV and other conventional medium, this becomes difficult to be present in the media throughout the year. You need a mobile factor level of budget. Number two, it's not possible to target your communication in particular consumer segments or in particular georefinal territories. Mobile is giving us a lot many other options of if I want to focus my communication in particular states, particular consumer segments on particular occasion in some particular context. So mobile is giving me various these kinds of options. Number two, as I said that various of our brands, they are high impulse category brand which needs a lot of consumer engagement activities. In TV, it's not possible at times in TV or maybe other conventional medium. But in digital, we can do a lot many consumer engagement activities which may include gaming, which may include influencer activity, which may need some consumer contest and all that. Plus apart from brand building, what I am seeing the benefit of mobile marketing, it captures the complete journey. How the consumer journey is captured that it's the journey from awareness to advertising. It's not only giving the platform for building awareness. Also, I can map the journey, whether my brand has come into consideration whether he has done some searches around my brand or not, whether he has actually purchased and after purchased what has been his feedback. That is the most important thing in fact. So this gives the complete journey of the consumer, which is perhaps not possible in any other conventional medium. Apart from that, it's playing very important role in the personality. Nowadays, the consumer is very demanding. Every consumer has a different kind of demands and needs and wants and all that. Mobile can map the consumer's motivation, barriers, the triggers, interests in a very, very focused manner. And I can do the targeting of the consumer. I can do the need satisfaction of the consumer, the consumer journey in a much more effective way. Absolutely. Sorry, absolutely. You're absolutely right. When you say that there is a focus targeting and you're able to kind of capture the right audience that you want to target and segment those users and then show creative solutions or relevant creatives rather to the right audience set. And mobile actually helps you do that much easier than you would be able to do in a traditional medium. In traditional medium, you tend to just bucket audience bases, the publishers itself. So if it is MTV, then you say, okay, anyone who watches MTV is of a certain age group. And that's how you decipher your audience. But in mobile and of course it starts with digital, there is more user targeting possible. Doctor, how are you seeing mobile marketing in Lotus Herbal and how are you using that to leverage for your brand? You guys are very offline driven and I do understand that you have an e-commerce site and you're doing things online now. But how has mobile marketing worked for you previously and now? Definitely. I think there has been a major shift for Lotus Herbal's Private Limited as a beauty conglomerate for shifting the focus towards a mobile-first brand. And I think the kind of timelines we've had to deliver it during the COVID period was phenomenal. The team worked very, very hard and we did manage to move towards a mobile-first strategy. And of course, there were a lot of changes which happened internally strategy-wise in the marketing plan also. And of course, now it has become really big for us. So I think we are up for the challenge as 95% consumers who are online interacting with us are through mobile devices. So I think Lotus, of course, our physical presence, our retail presence was our top priority before the COVID times. But with changing times, I think we've adopted better and of course mobile marketing is a very big part of our digital strategy now. And also I think given the kind of lockdown period we had, the kind of influencer marketing campaigns we took over. So all of this contributed to this entire digital first movement for Lotus Herbal's Private Limited and that's how now, of course, we are moving towards a mobile-first strategy for sure. Absolutely, right. So COVID has actually accelerated a lot of things. And we were shocked by the timelines we met. I think the team worked phenomenally. And the strategies for a digital first brand, which normally takes about a year and a half to put into place, we have managed to do all of that and in the coming future you will see a lot of changes in the brand as well. Great, great. Anul, how are you seeing Bacardi? Of course, Bacardi has been using mobile as a medium for a long time in their digital mix. Post COVID, how do you see this significantly changing or you see that it remains the same or you want to use it in a different strategy? Yeah, hi Akshay. So at Bacardi, because Bacardi always keeps the consumer at the heart of whatever we do. So with the changing consumer behavior, we've always evolved our marketing strategies accordingly. Now let's look at from a consumer lens. If we keep traditional media, ATL on one side and mobile marketing on the other, with the penetration of mobiles, with the penetration of internet across markets in India, even some markets, tier two markets, the amount of time spent on the mobile is much more than the amount of time spent on television. The second big thing is that television and radio, et cetera, they're all broadcasting mediums. They're great traditional mediums and they're broadcasting in nature. So anywhere where transactional information is concerned or you need to have an engagement with the brand, the consumer seeks mobile platforms for that. So Bacardi has very effectively in the last few years done a lot of activities on ground, which have a digital connect where you know the consumers are more active or digital for an experiential activity that they're going forward, they're going with their friends too. So that integration started happening much earlier. Hence, during this time, we again trended with how the consumer was behaving. So if the consumer is spending more time on mobile, we invest more in content, which is again very targeted sort of content which is done by co-creators. So it's not brand first, it's content first. And now increasingly content to commerce, but as some part of the parts of the country, now I also have e-commerce coming for alcohol brands like us. The second important thing is in times of pandemic, safety is utmost important. So in the times of safety, the consumer, they want to go out, but they want to engage with the brand, even the physical property, even the physical properties of the brand, very, very carefully. So this is where the second bit of mobile marketing, which is all your QR codes, scanning, et cetera, comes really handy. And this year, we have in Bacardi used that technology at the forefront of most of our activities in our channels. That's great. Yes, I've seen a couple of your mobile marketing campaigns and they are very engaging like you say. And you rightly said that TV is more broadcasting and of course digital has been an engaging medium and it's a two way medium, but kind of mobile makes it even more right there at the user level. I just want to follow up with this and more to you with another question. In the last nine months, the way content is consumed and promoted on mobile and has gone massive shifts. And is there a new playbook at work in this pandemic, as far as mobile marketing is concerned, what have you been that additions for Bacardi, being a liquor brand, being associated with lifestyle, as a lifestyle company? Actually, when we talk about the playbook, the playbook has effectively changed now, keeping the codes of pandemic for the consumer first. So there is of course, whatever we do, even in mobile marketing, we've got safety, we've got convenience, we've got availability, et cetera, which is handy. These are the three code words which I've been using for all my campaigns, for all the campaigns of Bacardi. So the playbook right now is more to do with how do you effectively target your consumer, keeping these three things in mind. So if it is about safety, if the consumer is purchasing your brand, there should be safety norms which is followed while creating engagement. So this is where the QR codes came into place and it's very effective in terms of talking to your consumer and we want to take it forward in most of our campaigns. Then comes the convenience and the availability part, which is also backend for brands like us to be available in neighborhood places where the consumer is available, geo-targeting the consumer, relevant content to the consumer, which is very personalized. In today's time, let's imagine a consumer who purchases our brands and this could be any target segment, any brand that we are talking about and the path to reach how the consumer is going to test to is to go to a 24-hour cycle of this consumer and say, where all is this consumer using the mobile and how effectively they want to use the mobile to interact with your brand? Are you available in a set of consideration? So even the content cannot be any content. It has to be relevant content, very personalized messaging for the consumer. That is where the love story with the brand starts. Absolutely, yeah. You're absolutely right when you say that the messaging and what the content is super critical. Prasanna, just moving on to you. In your view, as marketing trends move from traditional and ATL to mobile marketing, what do you think are a few of the trends and formats that will dominate? Like there is content, there is AR, native, etc. What do you think would be a dominating format in the times to come? Sure. Hi, actually, firstly, I think I've worked in a couple of media agencies before working in these platforms. So at that time, digital used to be the last section that gets presented to the client. And within that mobile used to be the absolute last. There have been times where I've been told saying, hey, it's a TV, digital and mobile will happen tomorrow. So you have to come back another day, but it's great to see all the panelists talking enthusiastically about how mobile is an integral part of the media mix. And obviously, it's equal to digital. So that's great to see. But specifically to how Snapchat is looking at it is pandemic or even otherwise, there were two trends that were emerging. People with the time that they have during this lockdown, people are sitting at home. They're able to wrap up work most of the times. They can have some free time in hand. So people either want to upskill themselves or they want to use that time to watch content that entertains them. So we have some strategy for going for each of the two. For example, in India, Snapchat has launched new original series in the Discover section. Discover section is where people consume content on Snapchat. And that launched just in 2018. In fact, in the last quarter of two years, we have seen phenomenal growth. In fact, we saw close to 60% growth on a quarterly basis in the last quarter on the Discover section. And we are also investing in games for that matter. And because people are consuming content in the native language, we have introduced nine Indian language in which people can access Snapchat. And as a result of all these efforts, daily active users grew over 150% in Q3 2020. And if you go into our interface right now, we have close to 60 million users. They're all metro, youth-centric. And contrary to popular belief, around 25% of the Snapchat users in India are below 18. Everybody else is above 18, so 75% above 18. So a lot of these trends are emerging. And we are kind of activating these and to the other section that we spoke about earlier about upscaling people. Like children are, I don't know if it's by force or by interest, a lot of people are being asked to upscale on their coding skills. Earlier, the peer pressure used to be, it's not just anymore Sharmajika Ladakh. Over and over, it's like peer pressure against Mark Zuckerberg learning coding at five years of age. And somebody else did at seven years of age. You also learned coding. So a lot of that upscaling thing is also happening. So we are in that game as well in encouraging more and more college students to become augmented reality-based lens creators. So we have our program going across different colleges in India where we are certifying people in creating these lenses, which again, a lot of brands are using. But we're also getting democratizing augmented reality in a lot of people, in a lot of these colleges as well. In fact, the lens that's there in my background, Capital Social is also something that was created as a result of this program where we are educating a lot of people that wants to be upscaled in this area of augmented reality, which is great for all of us. Absolutely. Like you rightly said, people are upscaling and things are moving more towards newer type of content, like you mentioned for SNAP. Even the phenomenon of OTTs have suddenly exploded, post-COVID also. And who had thought that movies would start getting launched on a mobile screen and big budget movies would start getting launched there. And this is like a dramatic shift that we are seeing. And this here, of course, but you never know it to be a reality going forward also. And that's definitely one trend where content coming in this small screen and long-format content coming in this small screen would also be there, of course, from the content that already exists. And of course, like you mentioned, newer technologies, mobile technologies like AR, et cetera, also. For example, I think I was seeing somewhere that trials are happening on mobile, right? So trial of a car. And I think there are trial AR. Through AR, you can see a car itself. You can touch it, but you can go inside it and actually see how it is. And that is actually changing how we are living in this society right now. That's great. Also, Ipsita, just coming back to you, Lotus Herbal is into skincare, hair care, makeup, et cetera. And like we mentioned, it is heavily offline into detail stores, et cetera. My question to you is, how local has become an effective tactic for marketers? How are you using mobile advertising for this purpose? I think you're correct. Hyperlocal has become the way forward for marketers because there's a lot of respect for our internal culture, our culture, which is very, very local. So all especially related to beauty, the services, and also a lot of user-generated content is being created with Lotus Herbal and collaborations with the influencers at a very local level and also tagging geo-conquesting and like Anmol said, geo-targeting is something we are looking at. And that's how we're working. Also, I would like to tell that in post the pandemic, the borders between beauty and wellness have actually merged together. So the kind of content consumption currently, like you said, there's a new playbook which we need to play with. People are focusing more towards holistic beauty, holistic wellness. So beauty brands are not beauty brands anymore. There's a lot of extension to it.  And of course, wellness plays a very important part. And I would also like to tell that, you know, this year has been successful for us because we have a latest acquired world's first organic Ayurvedic wellness and beauty brand, which is Soul Tree. So like we said, you know, as a strategy, we are moving forwards toward the content where beauty and wellness have merged with each other and also local and hyper local is of course a part of our strategy. Also, just to let you know, a lot of live streams have happened and a lot of, you know, performance marketing videos, which is user generated is in local languages. So we want to connect to our people from wherever they are beyond boundaries and we want to reach out to them. So like I said, hyper local is definitely the way forward for the brand. Right. In fact, Rajeev, for you also, I think as the S group, you know, hyper local would be important.  And of course, you know, India is so diverse.  So is hyper local an effective tactic for you guys and how are you using that and interesting to hear your thoughts? You are very right that in today's context, hyper local is definitely very, very important. And when the digital and mobile, they're giving you that flexibility of targeting your consumer as per the location, as per the geo targeting. So this is definitely very relevant. However, we have seen that this is working a lot in case of retail brands. Because that the consumer, when he's around their retail outlet, and when he's getting, that time he's getting the message around the brand and with particularly with some offer, the chances of his purchase intention, his purchase action enhances many folding. Apart from that, as I said in the beginning, not also that mobile and digital is giving a lot of flexibility. If I want to focus a particular colony, particular town, particular consumer, I know that because in India that even in Delhi, I see that there's a lot of difference between consumer at Chandni Chow or maybe consumer at GK1 or GK2 and all that. If I want to focus the Chandni Chow consumer and may not the GK1 consumer, so this is giving me a lot of flexibility on that. Earlier, like when we were doing the brand planning, we used to think what we will do at test marketing in a particular town. But now, when we do the planning, we don't think at town, we think at which particular area in their particular town. Because we have seen there is a lot of diversity in a particular town itself in fact, particularly in Madras in fact. So we have been using all these techniques a lot in fact. As I said that our consumer base of a particular brand we have found is much lower than the overall universe. I see that how to profile my consumers and where do they stay and what are their motivation and barriers. With all those things, this mobile is helping us a lot. Interesting. In fact, Anmol, I wanted to ask this question to you also that how is hyperlocal, does that benefit you in a particular way? I'm not sure people think liquor is liquor, right? I'm sure there are nuances to that also that it will be consumed in a particular way in Boa or Vizavi, you know, at Chennai or maybe a different place. Or that's a perception. Or you think that there is a need for hyperlocal targeting, getting messaging to the right audience of a right demographic at the right place is very essential for a brand like Bacardi. Yeah. So actually, you know how I view hyperlocaling is more like near me. What is it that is near me? So it has, there is two ways how we can work and how we usually work. There are a lot of consumers who actually attend a lot of brand experiential platforms of ours like Bacardi Weekend, our Breezer Vivid Shuffle and Bacardi Weekend of this year has gone digital. So on fifth and sixth, it is all online. So when we do that, a lot of geotargeting happens to these consumers, reminding them of the famous brand platform that they've been to and they have a huge emotional connect with. The second way of the other channel to do it is we are quite partners, strong partners with our on trade and off trade channels which means liquor stores and places where which are pubs and bars who partner with Bacardi and have products and drinks available.  You know, you suddenly say this is Gorya, this is what I want. So this is when you're near me marketing, this hyper local marketing of near me can be done by partnering with your respective on trade channels which is your restaurant. So happy hours pop up. Bacardi Weekend is happening at this outlet. Of course during pandemic times we are not concentrating as much as on trade and in the second channel but we're definitely doing a lot with the consumer database we already have. Interesting. Interesting. And coming to you Prasanna I just wanted to ask how are you hyper local getting into that snap? Hyper local actually is an integral part of how one can advertise on snap as well because what we notice is I think as a company we've realized in the long run that India is not just one country. It's like multiple countries put into one. So there are many such occasions and India is a very moment specific country. There would be different moments in different parts of India. So I would say for example certain festivals are celebrated in certain states and so on and so forth. So therefore the communication needs to be different and it needs to be tailor made to those people and because people are consuming content in these different languages and these different states we normally encourage brands to communicate in a manner that's more applicable to that specific region for example Durga Pooja in West Bengal or Achad Pooja in Bihar and so on and so forth. So we have tasted success with a lot of campaigns in fact some of the global brands that have tested success with us by engaging with us. So in terms of making it hyperlocal and specific to those audience in that specific region it's super critical and in fact we collaborate with both the creative agencies and the client to be able to tailor made content which is specific to those regions so that's how we tackle the kind of hyperlocal possibilities that exist. So you did mention that there are multiple languages that are available on Snap and there is content also available in multiple languages? Absolutely content is also available in multiple languages we have in the discover section and that's what has resulted in the significant increase in discover section consumption across different channels we have some really good curated content and everything is almost every content that goes on Snap available for people to view goes through a bit of screening so therefore we get to decide what gets prominent so therefore there is minimal hate like for example we got into trouble for a certain leader of who's not the leader anymore of a certain country when they wanted to put out a lot of hate content and then we stood up against him them so that kind of goes to show the kind of content on the platform which is absolutely curated so it's a brand safe environment on Snap as well over and about the fact that you have different languages in which people are consuming content in the discover section. Sure, we have about 10 minutes so I just wanted to go through the panel and ask one final question As per the latest E4M marketing report in the next 3 years while ad spends in India is going to expect to continue to grow like a figure of about 45% and reach about 80 to 20,000 close that is by 2020 what are the new emerging areas to watch out for in the mobile space in India that you think which will define mobile marketing in 2021 is it whether it is video it is AI AI we hear super apps being built out by Tata and Reliance etc what is the next big thing or the trend that we should look out for in 2021 if you had to sum it up in few sentences I will start with you Dr. if you can go ahead Like I said for beauty of course it's a very different ship towards the mobile first strategy and like you said most of our ad spends and our advertising is now going to be adopting a mobile first strategy so I think the biggest trend upcoming in the field of beauty would be am I audible yeah yeah so I was saying that the biggest trend upcoming in the field of beauty and beauty brands of course is AR because that's the way forward there's a lot of virtual application virtual try-ons are the future also there's a lot of omni channel associations where the retail will be a virtual retail concept in future so I'm adding to that so there's a lot of omni channel activations that can be accessed through mobiles also another thing I think social commerce and end commerce is up on the rise so that will be the future in the beauty industry as well so this is my parting thoughts thank you Dr. Raji you want to add some yeah I think 2-3 areas where I see a lot of potential in FMCG e-commerce contributes hardly 3 or 3.5% in the overall sale in the COVID scenario a lot many people they have got into habit of purchasing online that I think will continue after post-COVID period also impact so that gives us a lot of opportunities because this mobile is giving us the complete consumer journey he can see my ad there only he can click he can purchase he can give his review also impact so this I am quite excited that this will open a lot of opportunities number 2 AI is playing very very important role because in our brand as I said there are many challenges that overall universe is huge but actually my target consumer is very limited I have to target them based on their interest their lifestyle, their behaviour, psychographies and all AI is playing very important role in giving me the exact photos targeting number 3 I think wise will play very important role in the coming years that time but you can say 150 words in the same time in the consumers they have limited time they have so many things to do on mobile and digital wise will play very very important role these are the areas where I see a lot of opportunities in the coming years absolutely you are right AI and voice we see that that is going to grow in the next year all the devices which are coming up like Alexa and Google I think they are definitely something that will define the future Anmol your parting thoughts around this so pandemic has taught us everything taught us one thing that the technology barriers that existed with some group of people whether they were kids or they were senior citizens or they were people who were not comfortable using mobile devices or internet those barriers have come crashing down so by force or by choice everybody had to go online like how Rajiv said it is here to stay people will now continue using mobile technology new technology new platforms to interact with the world to entertain themselves to buy their products and even if even if they have to give feedback review you don't talk to the brand it will be all on the mobile so if we talk about alcohol specifically I think it will be all on the mobile technology these are all new platforms they are very interesting to see but my interest would lie more in how do we apply this in an effective way that leads to doing what we do with the consumer that has said that is where my interest would lie and I think it will come in time to come it will come these are brilliant technologies and we already are seeing integration of these technologies across so for a brand like Bacardi entertaining the consumer on the platforms that they do and who would not mind a bar on the cloud so yeah I think these three technologies come together would be great absolutely I think it is more from an experiential for the customer what you mentioned more from an experiential point of view that these technologies like AR or voice etc makes a lot of difference especially in a scenario where nobody can meet anybody or there is restrictions going on etc who have your aftars etc and probably what brands can do fun around those I think but yes you rightly said how a brand applies these technologies would make a lot of difference personal anything you want to add what are your thoughts around what will be the next few trends in 2021 yeah absolutely I think one thing to watch out for is just because audience users have a lot of time on hand I think brands should learn to appreciate that time as well and not abuse it by showing long form creatives to satisfy certain people internally because I see in certain places basic best practices of mobile communication is not being followed for example the brand needs to be up front it needs to be short form people have choices people do not have appointment viewing anymore even when they do they still have choices so a lot of those choices need to be respected I would say that should be because just because people have time doesn't mean you can run long format and have the suspense big reveal not for mobile I think that marketers should definitely watch out for and just to take something out of what Anmol said earlier about all these technologies like AI and AR are great but they need to deliver for us in terms of real business results I just wanted to touch upon that there was a study done as early as 2018 by mindshare which shows that when somebody is using augmented reality they are 45% more likely to pay attention to the communication that's there in augmented reality so I have seen I know that using try-ons for example for make up and stuff so that's one use case for augmented reality but very recently we have used augmented reality to be able to communicate the kind of benefits that the brand can have using a gamified lens in certain other cases using certain other lens technology where people are able to use augmented reality to be able to imbibe and understand that RTB that the brand wants to communicate so when you have augmented reality in the media mix along with hard working creatives which were story ads or snap ads a lot of those hard working creatives the combination gives great brand resonance and I can tell you from first hand knowledge augmented reality actually works like a dream when it comes to brand resonance across the funnel be it top funnel mid funnel or even bottom funnel for that matter so I would strongly encourage people to use augmented reality to deliver brand resonance rather than using it as a new shiny toy so definitely encourage brands to consider augmented reality to do the hard working tasks of building brand resonance great so what I take from it is online mobile commerce is going to be one formidable trend in 2021 given the current scenario AI voice is something that will develop and we see that AR would also be one of the trends because of specially because of having you know trials at home possible etc and these are the 3-4 trends that we see I think in the next year and time will tell how these panel thank you all for your time Mr. Mathur just to cut in short we still have a couple of questions coming in from the audience and we just have another 4 minutes in hand as for the ISD timing so Mr. Mathur I'll just quickly throw it to our esteemed panel firstly thank you so much for moderating it so wonderfully to all our esteemed panelists I've just posted in the chat the first question is in mobile first strategy are performance campaigns more in demand compared to branding campaigns so could any of you just like to give in your a perils of wisdom on the same should I Mr. Jai yes please go ahead when there is definitely there is a good opportunity of performance math and campaigns on mobile but I still feel that mobile gradually is becoming very important from branding perspective also because when the consumer spends more time in mobile and spends less time on the conventional media so definitely the strictness the involvement the engagement is more nowadays but I'm noticing that on the conventional media even if you're watching something whether be TV or newspaper our attention spend is becoming less impact when we see some ad on TV that time we start watching our facebook we start checking our facebook install that moreover lately I've seen atleast two campaigns popular campaigns which most of my colleagues my people to whom I've interacted they haven't seen anywhere on TV or newspaper anywhere they have been only on digital platform one was from Cadbury that hyper local campaign one from Facebook and many people they remember the campaign only on mobile and digital they are no way by any means the performance campaign they are pure branding campaign they feel that increase in the time spend and in the population of mobile and all so this will not work only for performance this will work a lot for branding absolutely thank you so much Mr. Dan now ladies and gentlemen with two minutes in hand I still have another audience question which has just come in it says that with digital growing dramatically is there any one key learning which you would like to pass on to the new age market who are wanting to get into digital so I think what I'll do is for this I just like a note from every panelist if you can just summarize it with this one key learning so first up Ms. Gill if you'd like to just give your one key learning on the safe over to you so I think the key learning for me has always been think like the consumer don't think like the brand person the moment you start thinking like a consumer you will automatically know what are your drivers what are your barriers and how you would want to consume information from the brand and automatically your campaigns will start resonating with consumer thoughts so there is no rocket science to digital marketing it's relatively very very easy sure thank you so much Ms. Gill Dr. Chatterjee if you could like to say your key learning for the same I think one of my key learnings and my key learnings which I would like to pass on to the new age marketer and who are making the slow transition from the field of a marketer to a digital first strategy marketer I would like to say that content is the king in digital you should have thumb stopper content so I call them thumb stoppers not show stoppers thumb stopper so you know the attention span is barely like I would say 8 to 10 seconds maybe and of course Prasanna would have a better idea in terms of your content if you are moving towards a digital you should think from a perspective of having a thumb stopper content otherwise the entire digital campaign or the digital marketing strategy doesn't work out well and that is one of my key learnings so you have to create very good thumb stopper content absolutely thank you so much Dr. Chatterjee now I would just like to go directly to Mr. Raman for the final key learning over to you yeah I think my key learning for somebody that is starting out on digital is do not settle for metrics that meets your eye many times we could make that mistake in digital saying just because there is a metric like CTR I mean we all know like people have never swiped up or clicked on a TV ad and still brands were built so I mean let's focus on metrics that really matter to you rather than focusing on metrics that meets the eye I think you know that's a longest topic but in essence I would say dig deep and find what metrics really matter and ensure that the translation from a from a campaign objective into specific outcomes that you are picking up from the campaign or outputs and ensure that they talk to each other and don't just be don't just settle for things that meets the eye absolutely so thank you so much Mr. Raman Ms. Skill, Dr. Chatterjee, Mr. Marthur and Mr. Jain thank you for joining us and the dear brand was a great and a very engaging panel discussion once again a big thank you


Hello all, I am Stalin Fidel Kumar from Sevahangai. I have secured All India Rank 75 in Indian Forest Services examination 2021. My optional is Geology and Forestry. My journey with Shankara Aisakrami started in 2016 with Prelims Come Main's batch and Forestry optionals. Regarding Forestry optionals, Karthikeyan Saal has been a great support to me regarding how to study forestry and how to tackle an examination. With the one-on-one interview with Karthikeyan Saal and interview guidance which he gave actually has been helpful to clear the examination. Apart from that I would like to thank Rajitha ma'am who has been a great support when I was down few years ago. So my journey will not be complete without a person who is Shankar Saal and he has been a great support to me and I would like to dedicate this success to him. Thank you.


Let us first see what is the present situation with this course. So, far we have introduced the following. First we introduce the notion of formal languages and then we look for the finite representation. In that context, we have introduced the notion called regular expressions and the respective languages called regular languages. So, after introducing these regular languages, given a language whether it is regular or not, it is not that easy to understand. We understood that point because giving regular expression is not that easy. So, in that context, we took the help of grammars and we specialized the two so called right linear grammars that we are calling regular grammars because our intention is to understand that these regular grammars characterizes regular languages. Of course, we have not proved that we are going to prove all these things. So, through regular expressions, whatever that we have defined so called regular languages, our intention is to show that regular grammars characterizes that means given any regular language, we can understand that through a regular grammar. Still there we have certain examples of regular languages, we are unable to give regular grammar for that. In that context, we gave a tool called finite automaton and through finite automaton, we are understanding regular languages and as I hope you understood clearly that giving regular expression is not that easy and regular grammar is relatively better, but for many of the things I hope now you could have given this finite automaton. In that we have variants of course, so called DFA and NFA and we have observed that these two are equivalent. So, here the notion of finite automaton that may be DFA or NFA is essentially to capture the information related to regular languages, this characterizes that. So, in this context when we are proving DFA is equivalent to NFA, the number of states when you are. So, the DFA is introduced and whenever you are given in NFA, if you want to convert it to your DFA, what is happening the situation here, if you have n states here in this conversion, you are getting 2 power n states that is exponential number of states. Now, in order to understand that what is a possible minimal DFA or you know when you are considering a particular language corresponding to that if you are if you are constructing a NFA and converting to DFA, is it the situation that you will always have that many number of states or whether there are any states which are useless. So, that kind of aspects now in this module we are going to discuss. So, here our concentration is to understand that what is a minimal DFA to accept a regular language that is our concentration now. Anyways, so in this cycle of course, once again when to conclude here this particular cycle, this finite automata and we are going to observe that in the following lectures that all these are equivalent. So, the present concentration of course, at present we are concentrating to understand that what is the minimal DFA to accept a regular language. So, the present lecture is essentially present module essentially concentrating on that 1 or 2 lectures in this direction. In this direction, we require one important tool called Mahill Neurot theorem that I will be discussing in this lecture and in fact, this is a very important characterization for the languages acceptable DFA. Of course, eventually what we are understanding the language acceptable DFA is regular languages. So, this is very important characterization for regular languages. So, in this lecture I will concentrate on so called Mahill Neurot theorem and to introduce that theorem I require some basic definitions in the direction. First let me introduce this an equivalence relation tilde on sigma star like strings over an alphabet sigma is said to be right invariant. If you take any two strings x y if they are related with respect to tilde then any string you take that and concatenate it on right side of both the strings then they should also be related with respect to tilde. So, if this property is satisfied there for all strings x y which are related if you concatenate any string z on the right hand side the resultant string should also be equivalent with respect to the tilde. So, such an equivalence relation we call it as right invariant equivalence relation. Let us look at an example so that you can understand this concept better. So, this example I am defining tilde l of course that I will be using throughout this lecture. So, please remember now you consider a language over sigma that means it is a subset of sigma star define the relation tilde l on sigma star with this condition that x related if you take any two strings x and y in sigma star you relate them if and only if for any string z x z is in l if and only if y z is in l. So, whenever x z is in l y z should also be in l. So, this advice ours of course. So, with this condition we are defining this tilde l and of course you can quickly understand that this is an equivalence relation because to understand a relation is an equivalence you have to understand reflexivity symmetry and transitive. So, any string x is related to itself because if you concatenate any string z the resultant string on both the situations here are same. And therefore, both are in l or not in l and thus you can quickly understand that it is reflexive symmetry follows very quickly because of this if and only if condition here and a transitive to also you can verify to understand that this tilde is an equivalence relation on sigma star. Now, our point here of course important thing you have to cross check here in this context is right invariant property. That means x related to y implies x z related to y z for all z that is what we have to understand for right invariant relation. So, take x y in sigma star and choose take x y in sigma star and assume they are related and pick up any z arbitrary in from sigma star. Now, we have to show that x z related to y z with respect to this relation that is we have to show that for all w in sigma star x z w is in l if and only if y z w is in l this is the condition we have to cross check. Now, if you write for any w take an arbitrary w in sigma star and write u to be z w and now you see that since x related to y with respect to tilde l what we have x u related x u is in l if and only if y u is in l this is the definition of tilde l. So, x u is in l if and only if y u is in l and what is u we have chosen that is z w that means what are the property that we are aiming to that is this we have got it that is this x z w is in l if and only y z w is in l and hence this tilde l is a right invariant equivalence relation. Now, I give you one more example this is also very important in this lecture that is I am defining it as tilde a for a d f a a consider a d f a q sigma delta q naught f take a d f a now define the relation tilde a on sigma star as below x and y are related with respect to tilde a if and only if you put those strings in the initial state you should essentially reach to the same state if you are reaching to the same state then you say they are related and you can understand again that this is a equivalence relation because if you take any string that is related itself because what are the state that you are reaching to with respect to x of course, that is fixed in a d f a and hence x related to x for all x and symmetry is very straight forward because here we have we are putting equal to. So, x is reaching to a particular state and you know y is reaching to a particular state and they are same then of course, vice versa and hence x related to y implies y related to x and when you are looking for a trans duty because of the equal to here you are getting trans duty also very quickly and hence this is an equivalence relation tilde a is an equivalence relation. So, here what we have to understand that it is a right invariant equivalence relation let us see take x related to y that is this property satisfied. Now, you pick up any arbitrary z in sigma star what I have to observe that delta cap q naught x z is equal to delta cap of q naught y z that is what we have to observe to show that x z is related to y z. Now, consider this delta cap q naught x z I hope by now you have proved that this is equal to this you can use induction to prove this property and now since this two are equal you can take in place of delta cap of x naught sorry q naught at x you can replace it with delta cap q naught at y and again using the property you can have this and thus x z is related to y z for all z x here that is an arbitrary thing and thus you can understand this is also a right invariant equivalence relation on sigma star of course, on sigma star tilde a is defined on sigma star. Now, let me state Mahily road theorem let I will be a language over sigma the following three statements regarding l are equivalent what are those statements statement 1 l is accepted by a DFA of course, once eventually if you prove that the finite automata is a character a character it captures the properties of regular languages that means it is it is characterizing regular languages. Then this theorem gives you a characterization for regular languages as I had mentioned what is the point here is there exist a right invariant equivalence relation tilde of finite index on sigma star such that l is the union of some of the equivalence classes of tilde. So, what we are trying to say here in the second point you can find a right invariant equivalence relation tilde such that l is and of course, it is a finite index such that l is union of some of its equivalence classes and number 3 is the equivalence relation tilde l as defined just now with respect to l because given a language l you can talk about tilde l as just we have defined the equivalence relation tilde l is a finite index. So, this theorem claims that these three are equivalent l is accepted by DFA and there is a right invariant equivalence relation tilde of finite index such that l is union of some of its equivalence classes and the equivalence relation tilde l is a finite index these three are equivalent. We prove 1 implies 2 and 2 implies 3 and 3 implies 1 to show that these three are equivalent to show 1 implies 2 assume l is accepted by DFA say a to be this q sigma delta q naught f. First what we will show that in point number 2 what we have to show there is a right invariant equivalence relation which is a finite index and l is union of some of its equivalence classes this is the point number 2 we have to prove. Here in this context what we will observe that that tilde into that is you know tilde a with respect of a will satisfy the tilde into that means what we have to observe that tilde is a finite index because we have already observed that tilde a is a right invariant equivalence relation on sigma star. So, what we have to observe the remaining two points one is it is a finite index and number 2 is l is union of some of its equivalence classes these two points we have to observe. So, tilde a you know already the claim is it is a finite index. So, to show it is a finite index first let me observe the following two points for x and sigma star delta cap of q naught x if it is equal to p then the equivalence class containing x is all those strings in sigma star if you put it if you put them in the initial state if you reach to p that is equivalent to x because if you remember the definition of tilde a two strings with respect to tilde a they are equivalent. If you put those two strings in the initial state of d f a here that is in q naught you should reach to the same state then we say those two are equivalent. That means if you take any string x what are the strings equivalent to x that is the equivalence class containing x is precisely all those strings in sigma star if you put them in the initial state q naught if you reach to the state p what is p here the state that you are reaching via x from the initial state q naught this is what is precisely the equivalence class containing x this is the point number 1 and point number 2 because here the state to which you are reaching that is characterizing the equivalence class. So, now corresponding to each state now let us look at all those strings let me call for a given q in q for a given state q consider c q the class of q I am calling is all those strings if you put them in the initial state q naught if you reach to the state q let me call it as c q this is an equivalence class of course this can be empty if no string can reach to this state q because if that is not reachable via using any string then of course this is this is empty let me consider empty also. So, in this context so what I am saying here this is an equivalence class of tilde a because what are all those strings what are all those strings if you put in the initial state to reach to q they are all equivalent to each other and this forms an equivalence class. Thus what you can understand equivalence classes of tilde a are completely determined by the states of a because from the point number 1 you understood that the state is coming into the picture. Now, take every state and set the class c q and collect all those strings this can be empty if that particular state is not reachable from the initial state, but wherever it is non empty all those strings which are reaching from the initial state to that particular state they are all equivalent and that is an equivalence class with respect to this tilde a from this 2 points what you can understand that this tilde a is completely determined by the states of a and moreover the number of equivalence class of tilde a is less than or equal to the number of states of a the reason why then every state gives an equivalence class to you of course, possibly empty in whenever the state is not reachable the equivalence class what I am calling here is empty set. So, thus the number corresponding to each state you get an equivalence class possibly an empty class some of the cases. So, thus the number of equivalence class of tilde a is less than or equal to the number of states of a and hence tilde a is a finite index because a has only finite number of states therefore, tilde a is a finite index of course, here index we mean index of an equivalence relation we mean the number of equivalence classes of that relation. Now, we want to prove that the second point l is union of some of its equivalence classes here this tilde a is equivalence classes. Now, consider this l by definition this is all those x and sigma star if you put them in the initial state you will be reaching to the to one of the final states that means, delta cap of q naught x is in f that is union of these sets because if you reach to a particular final state because for all final states I am considering these sets some of the non final some of the final states may not be reachable in which class this set is empty. So, now just we have proved that this set is an equivalence class we are calling it as C p corresponding to a state p and thus you understand that l is union of C p's for p belongs to f as desired. And thus we have observed that 1 implies 2 that means we could identify a right invariant equivalence relation here that is tilde a which is a finite index number 1 and number 2 l is the language l is union of some of its equivalence classes we have proved 1 implies 2. Now, let us consider 2 implies 3 to prove 2 implies 3 what we have to observe that tilde l is a finite index the number of equivalence classes of tilde l is finite. Suppose, tilde is an equivalence as in 2 that means tilde is a right invariant equivalence relation of finite index number 1 and number 2 is and number 2 is l is union of some of this its equivalence classes. So, that is what I am assuming assuming tilde is an equivalence as in 2 point number 2 of the theorem we show that tilde is a refinement of tilde l what is refinement. If you suppose this is the set under consideration and if you consider an equivalence relation equivalence relation you know equivalence relation partitions the set. Now, if you further refine this relation that means if you break this equivalence classes further for example, maybe you are breaking this direction or whatever. Whatever the classes here I am getting you can clearly see that what are the partition that I am getting here each equivalence class that means equivalence class here I mean the portions that are shown with the boundaries of this cross lines with the vertical lines. So, this portion is this is a new equivalence class for example here this is a new equivalence class these equivalence classes are contained in the original equivalence classes because they are subsets of the because we are partitioning further partitioning this. So, now we say this is a refinement we are further refining this equivalence classes. So, that means if you take any 2 elements in this set if they are related with respect to new equivalence relation they are related to the original with respect to original relation because new equivalence class is a subset of original equivalence is a subset of one of original equivalence class. So, what we are trying to show that this tilde is a refinement of tilde l that means once we show that this is a refinement as I have explained that the number of equivalence classes of tilde is more than the number of I mean more or equal to the number of equivalence classes of tilde l. That means a index of tilde is greater than or equal to index of tilde l as it is given that tilde is a finite index tilde l is also is a finite index. So, once you we show that tilde is a refinement of tilde l then we are through now suppose we have as I have mentioned that if x related to y with respect to tilde we have to show that x related to with respect to tilde l also to observe that this is tilde is a refinement of tilde l. Now, for that purpose for x y in sigma star suppose x related to y with respect to tilde to show x related to with y with respect to tilde l what we have to show for all that if you concatenate that from its right side this is in l if and only if y that is in l. Now, since tilde is right invariant that is what is hypothesis given into the point number 2 of the theorem since tilde is right invariant what we have x z related to y z for all that and in point number 2 we have this information also l is union of some of its equivalence classes some of the equivalence classes of tilde. That means whenever x z related to y z that means x z is in l then y z should also be in l. So, that is we have for all that x z is in l if and only if y z is in l that is what is the desired property we look for to show that x related to y with respect to tilde l and hence what we have observed x related to y with respect to tilde implies x related to y with respect to tilde l also and hence tilde is a refinement of tilde l and thus tilde l is a finite index. So, this is this gives you 2 implies 3 now we prove 3 implies 1 in 3 what we have tilde l is a finite index let us assume that and what we have to show we have to show that there is a d f a to accept l l is accepted by some d f a. Now, we construct the d f a in this portion 3 implies 1 now construct a l by taking q sigma delta q naught f of course, here I have to give you that q the state set you consider it as sigma star by tilde l that means the partition of sigma star with respect to the equivalence relation tilde l hence all those equivalence classes bracket x x and sigma star that is what we consider it as state set in a l and what is the initial state you consider the equivalence class containing epsilon the empty string and then you consider the final states are as here all those equivalence classes with which are in you know if you take any string in l the equivalence class containing that x you consider as a final state and now you have to give a transition function in case of d f a. Let me define delta from q cross sigma to q q cross sigma to q defined by delta of bracket x at a we assign it to bracket x a the equivalence class containing the string x a of course, for all states bracket x and for all a in sigma we have to understand that this is a d f a to understand this is d f a what do you require this of course, sigma is given to be alphabet that is a finite set alphabet means I have what I have to understand that q is a finite set number 1 and number 2 this delta is a function from q cross sigma to q that means we have to look at we have to prove the well defined s of delta. So, that this is a d f a so to observe these two points of course, since tilde l is a finite index since tilde l is a finite index we can quickly see that the number of equivalence classes of tilde l is finite and thus the partition that means the number of equivalence classes here is finite. So, q is a finite set very quickly that q is a finite set to understand that this delta is well defined how is delta defined this is defined as to show this delta is well defined consider bracket x equal to bracket y consider two states for bracket y bracket x in q these are the states consider and consider arbitrary a in sigma the claim is bracket x a is equal to bracket y a because what we have to show delta of bracket x at a is equal to delta of bracket y at a that is what we have to show that bracket x is equal to bracket y a. So, that means what we have to show with respect to to show these two equivalence classes are same we have to show that this x a with respect to tilde l it is related to y a that is what we have to show. And what is tilde a tilde a you know already it is a right invariant equivalence relation that means, if you concatenate any string on it is on right hand side the resultant strings are also equivalent. Here a is in sigma that is a string you can of length one if you if you concatenate a to x as well as y they should also be related with respect to tilde a because tilde l because tilde l is a right invariant equivalence relation. And hence you have this property once you have this property these two strings are in the same equivalence class that means, the equivalence classes containing x a and y a are same. And thus what do you have delta of a bracket x at a is equal to delta of bracket y at a. And thus you can understand this delta is well defined this is well defined. So, once you understand that this delta is a map from q cross sigma to q this with this assignment we have this a l is a d f a deterministic finite automaton.  For that purpose we show that delta cap of q naught w is equal to bracket w. So, the state the equivalence class the state the resultant state will be the equivalence class containing w this serves the purpose. If you show this this point this is sufficient because if you take any string w is in l if and only if bracket w is in f the reason why w is in l if and only if bracket w is in f. How this is once we have proved this thing delta cap of q naught w equal to bracket w once we prove this the initial state here of course is delta cap of we are proving this that is what we are proving. If this state is in final state then w is in f that is straight forward if this is w is in l and whenever w is in l what is the requirement whenever you put this w in the initial state you should reach to a state that is in f. So, with this criteria with this criteria delta cap of q naught w is equal to bracket w if you can prove then w is in l if and only if bracket w is in f is clear. And thus what will happen the language accepted by a l is equal to p l we prove this assertion by induction on the length of w. So, induction basis is clear for strings of length 0 because if you take string of length 0 that is epsilon if you put in this that is what is q naught by definition of delta cap of course in any d f a here q naught is bracket epsilon thus we have the basis of the induction that is for those strings of length 0 of course empty string. Moreover you can in fact observe that by definition of this delta this assertion is true for those strings of length 1 also because delta we have defined with this condition. Now, if you consider delta cap of q naught at a for the strings of length a that means that is delta cap is nothing else but delta and delta is defined as so and this string is nothing else but a epsilon a is nothing else but a that is you are getting a equivalence class containing a. And hence what are the assertion we have that is true for all those strings of length 0 and all those strings of length 1. So, induction basis from this point you can observe for inductive step you consider a string in sigma star some x and consider a in sigma what I have to observe the delta cap of q naught x a is equal to bracket x a that is what we have to prove. So, consider delta cap of q naught x a this equal to this that is the definition of delta cap by inductive hypothesis we have this delta cap of q naught at x is bracket x that is delta of bracket x at a by definition of delta you know this is bracket x a and thus what we have delta cap of q naught x a is equal bracket x a. And hence by induction what we have delta cap of q naught at w is equal to bracket w for all w in sigma star. And hence any string is in l if you put it in the initial state of this particular automaton you will be reaching to a final state precisely the equivalence class containing the string.  So, we have proved that 1 implies 2 2 implies 3 and 3 implies 1 thus the condition the statements 1 2 and 3 are equivalent. Now, let me give you a remark the proof of 2 implies 3 shows that the number of states of any d f a accepting l is greater than or equal to the index of tilde l given a language l consider any d f a accepting l in 2 implies 3 what we have proved the number of states of that d f a a is greater than or equal to the index of tilde l that is what we have proved. Because to observe that tilde l is a finite index we have proved this point in 2 implies 3. And in the proof 3 implies 1 we have provided a d f a namely a l with the number of states equal to the index of tilde l. Because in that a l the states are essentially the equivalence classes of tilde l. So, that means the states what we have considered they are equivalence classes that is the number of equivalence classes is equal to the number of states of that a l. So, from these 2 points you can understand that a l is a minimum state d f a accepting l. So, given l you know what is tilde l from tilde l you know the equivalence classes from those equivalence making those equivalence classes as states we have constructed a l. And since any d f a accepting l should have the number of states of any d f a accepting l should have more or equal states than the index of tilde l. And a l is in fact having the number of states equal to the index of tilde l we can conclude that a l is minimum state d f a accepting l. So, we have from main linear node theorem we have a minimum state d f a accepting l that is what is the conclusion at this point of time. Now, let me give 1 or 2 examples as applications of main linear node theorem, because main linear node theorem is characterizing once again I am emphasizing that main linear node theorem is characterizing the languages that are accepted by d f a eventually what we point out that main linear node theorem characterizing regular languages. So, let me consider this example you know this is a regular language you can give a regular expression for this consider the language x in consider the language with all those strings having a b as substringer what is the regular expression for this this is any string is here is of the form x a b y for any x and y in over a b.  And you know d f a accepting this language also that we have already discussed have constructed and from main linear node theorem we should understand that the tilde l should be of finite index. So, let us calculate the number of equivalence class of tilde l in this example to understand using main linear node theorem this is a regular I mean this is a language actually by d f a. First observe that the strings epsilon a and a b are not equivalent to each other let us observe this point. So, to show epsilon a are not equivalent to each other we have to find a string we have to find a string such that if you concatenate that string on the right to its right hand side of epsilon and to a one string should be in l one string should not be in other string should not be in l that is how we have to find. Now, in particular suppose if you consider b the string b epsilon b is b you know b is not in l, but if you concatenate b to a on its right hand side that is a b that is in l. So, you can quickly observe from this that a and epsilon are not equivalent because we could identify a string b here that concatenating b to the right hand side of both the strings one string we are getting in l other string we are not getting in l and thus a and epsilon are not equivalent with respect to tilde l. Similarly, we observe that this epsilon and a b are not equivalent if you consider any string which is not containing a b if you concatenate that to epsilon right hand side you will get of course, it is a same string and thus that is not in l and if you concatenate that string any string to a b on the right hand side anyway that is having a b and thus that is a string in l. So, you get strings one is in l other is not in l, so that you can understand that epsilon and a b are not equivalent. So, what I am trying to say here is to show a and a b are not equivalent with respect to tilde l you choose any string x in which a b is not a substring for which a b is not a substring. Suppose, if you consider this concatenate x to epsilon right hand side this is resultant string is x only and if you concatenate x to a b on its right hand side resultant string anyway is having a b a substring. You see this string x which is epsilon x is not having a b a substring, so that is not in l whereas, this string is always in l as a b is substring of this resultant string and hence we observe that epsilon and a b are not equivalent any string x a general string I have mentioned in particular here we are considering b if you choose b b is not having a b a substring and you can understand that epsilon and a b are not equivalent. Now, we also observe that a and a b are not equivalent here I have taken the string a if you concatenate a to its right side then of course, the resultant string is a a that is not in l if you concatenate a to a b at the right hand side that is a anyway string of l and thus a a when this little a distinguishes a and a b and the therefore, a and a b are not equivalent because to show two strings are equivalent with respect to tilde l if you pick up any string from sigma star concatenating that string right side of both the strings the resultant strings both should be in l or should not be in l. So, with that criteria we have to cross check and from these three points I hope you can you understood that these three are distinguishable each other and thus what is the conclusion here these three strings should be in different equivalence classes. So, let me say epsilon is in the equivalence class bracket epsilon the equivalence class containing epsilon let me call bracket a for the equivalence class containing a bracket a b for this. So, let me assume these three classes are there from this. Now, what we are going to show that if you take any other string in sigma star that string should be in one of these equivalence classes thus precisely we will have these three equivalence classes and hence we can understand that tilde l is a finite index let me under this point also pick up a string x and now I see the property of x in two cases whether a b is a substring of x or a b is not a substring of x. If a b is a substring of x then x clearly will be in the clearly suppose a b is a substring of x let me use this notation a b less than equal to x I mean a b is a substring of x. So, what we are trying to say that x is related to a b so that this x will be in bracket a b. This is very quick because to x if you concatenate any string z whether it has a b or not having a b since already there is a b in x in the resultant string x that you will have a b a substring and since with the same property if you concatenate the z with to a b this resultant string is also having a b a substring thus this is in l and this is also in l irrespective of whatever that z you are concatenating and hence these two are equivalent x and a b are equivalent. So, if you take any string x in a b star if it is having a b a substring that clearly will be in bracket a b now if you consider the other case that if a b is not a substring of x then what are the possibilities if a b is not a substring of x then let me see x may be of the form a power n or may be b power n in which a b is not there or when you have some a's and b's since a b should not be there you can have some number of b's followed by some number of a's but of course after that you should not have any b's after a's because if you have any b after following to this a's then you will always get a b a substring thus I can categorize those strings x for which a b is not a substring means it may be a power n may be b power n result of the form b power n or b power n a power n these three cases. Now, in each case we can discuss that these strings will fall in some of the equivalence classes among bracket epsilon bracket a of course whether it will be in bracket a b or not you will observe here for n greater than equal to 1 because if it is empty string that is one situation that I will consider in the case of b power n for n greater than equal to 0 here at least 1 a for n greater than equal to 1 and the other case I will consider. So, for n greater than equal to 1 if you consider x 2 b of the form a power n in this case x will be in bracket a as I have discussed so far you can take it as an exercise and observe that x will be in bracket a in this case and if you take x which is of the form b power n of course n greater than equal to 0 also you can put where in which case that is epsilon that will be in bracket epsilon. And if x has some a's and b's as I had mentioned x must be of the form x must be of the form this b power n a power n for m n greater than equal to 1 in this case you can understand that this will be in bracket a thus this tilde l has exactly 3 equivalence classes and hence it is a finite index and the conclusion from Mahler's theorem that this language can be accepted by a DFA of course you know already that this language a DFA accepting this language. So, let me consider one more example we know this language a power n b power n for m greater than equal to 1 so far we have not constructed any of course we have a grammar for this but not a regular grammar we have constructed a contrasted grammar for this in this example we observe that tilde l for this language l a power n b power n n greater than equal to 1 we observe that tilde l is of the index of tilde l is infinite it is not finite. And thus using Mahler's theorem you can say this is not this cannot be accepted by any DFA and as we are saying that DFA characterizes the properties of regular languages of course from this we can conclude that a power n the language a power n b power n greater than equal to 1 is not a regular language. So, let me observe this point we show that the index of tilde l is not finite for instance you consider these two strings a power n b power m these two strings from sigma star here sigma is a b with m different from n consider these two strings they are not equivalent to they are not equivalent to equivalent with respect to tilde l because if you consider b power n if you concatenate b power n to a power n on its right hand side that is an element of l whereas, since m different from n a power m b power n is not an element of l. And thus for each n there has to be one equivalence class to accommodate a power n the string a power n. And thus this tilde l is not a finite index and therefore, the conclusion from Mahler's theorem is that there cannot be any DFA accepting this language.


So here at the Port of Livorno, we're loading up 206 MATVs and 30 containers, getting ready to load up and be sailed off. There is nothing more rewarding than working with the vessel crew over here with the U.S. Naval Ship Bob Hope and also to working with our Italy Detachment. So we have a very tight relationship with our Italy Detachment and with Port authorities here at the Port of Livorno. It shows a tight partnership as far as like the cargo coming in and cargo going out, coming up with a distinct list of everything for the future movement, so here to the Port of Livorno.