se so about a hundred percent up on what it was last year Right So doesn't that seem to be across the board Okay In terms of fertilizers Right, okay, but still if if if you want results you're gonna have to dig deep for them That's just the bottom line There's no alternatives. Is there any sort of cheaper alternatives? Um, once again you can use seaweed stuff like that On a seaweed is a good, it is an element within us called just mannitol I think you've got Which actually unlocks the sort of fertilizer that's already in the ground So it can be quite good But generally seaweed you get away with for one year Half that you probably end up having to go back to that nitrogen based stuff And does the seaweed, I mean does it go on in seaweed form Or is it grounded down to a powder or what way does that work? You get it in many forms now Uh, generally you can actually buy it in liquid form That's actually made in Donegal down here And Donegal so it's a good local product Okay, because you know if I had not asked you that It would have just ended up at like there was a high tide at my house Yes But funny we actually do that quite often for people with raised beds and stuff Yeah What people would do actually is go down and say September, October Harvest seaweed just in its own form And it's less noticeable obviously in that environment Pretty much and just lay it on top of the bed And actually just leave it over the winter then chop it in All right, uh, Paul let's get to some questions here We'll talk about watering and stuff as we wear on What is a good fast growing hedge for a windy site? Pretty much our most popular hedge down there by far would be Grysalinia Which is a nevergreen hedge sort of limey green leaf And the stems on are quite flexible So on a windy site they tend to flex with the wind As opposed to standing up to it Okay, uh, I have Sweet Pea grown from seed And I planted out five and they got eight at base What's attacking my Sweet Peas? The Sweet Pea chances are slugs I would have thought it'd be my number one suspect This year has been actually because it's been mild winter And that seems to be a huge amount of aphids About the green fly and white fly Seems to be doing a huge bit If you look at the underside of whatever leaves is left Pretty much you'd see any signs of green fly or that Otherwise I would suspect slugs Okay, so this is, I suspect a little bit like some A lot of us has pulled out the pots, right? They've taken the old plants out And they're sitting there full with compost from last year They want to know can they reuse that compost? Can you just maybe give it a squirt of chicken poo Or tomato food and plant into it? Yeah, sorry That pretty much would depend the size of the pot A lot of the time you're putting in your flowers And you want to get the best out of them So you need a fair bit of nutrition there You could sort of rejuvenate the old stuff But ideally what you're doing a fair size pot Would be actually remove half of it And as you remove it once again and get back to your slugs Check it for like slug and snail eggs Or something like that Because it is enriched with nutrients presumably And the flowers eat that And they draw that food source from it So you end up with what might look the same But doesn't contain the same Pretty much, and on a big pot It's not practically good at all The compost is really new at all every year So if you do half every year What could be eating my sunflower plants? Once again, probably either your aphids or your slugs What's an aphid again? An aphid is the green flower, the white flower Now this caller says the sunflower plants Some of them are completely eaten down to the stem It's the first year this has happened Yes Once again, I'd say your mild winter is probably the root cause All of that Because the eggs and stuff didn't get killed off With the natural frost as should have been But sort of just keep an eye on it And you get organic slug pellets and stuff like that Now so you can actually put them around it And it should help If you do that once or twice it should sort of dwindle the population Discourage them, yeah, okay Right, I can't seem to keep any indoor cactus alive Should the soil be dry because they're desert plants Do you have any tips on this? Now I would have thought if you aren't really green fingered The thing you kind of go towards is a cactus Yeah, pretty much cactus should be very easily grown So it should, the arid desert plants will do like it dry So they do, and funny, some people think about that Get around what can happen when indoor plants are regular water Is like you sort of forget about them for two, three weeks or both And then pretty much you'd remember twice in two days And sort of flood them if you know what I mean So a good test of that is actually to use your fingernail If you dip your fingernail into the soil at the top of the pot Generally a plant if it's still dry down to the depth of your fingernail It needs water, but cactus would tend to sort of go a wee bit less Yeah, so how often would you water a cactus? It depends on the humidity in your house, of course And stuff like that, but is there a rule of thumb? Even time of year Probably if you get into the habit of checking every couple of weeks I would say, but generally probably three to four weeks Because the top will dry out anyway because of the environment of the house That's why you're digging in a little bit to find out what's happening in the soil Yeah, and sometimes actually if we're too close to like a north facing window as well It can be chill to get some Yeah, all right Okay, you don't want chilly cactus Trying to grow an evergreen hedging around my garden And I'm doing it from clippings What's the best method to go about this? Generally clippings, a lot of people do it just by putting Obviously it depends on the type of hedge that you're growing But a lot of people what you do is sort of August, September time Is take loads of clippings and put them into pots And actually put them, you can make like a, if you google it Even you can make like a makeshift sort of greenhouse You want to see their taking first before you come to them, yeah? Yeah, and the other thing about it is taking cutlass like that It's sort of, it's an exact science And what you want to do, if you say you want to get 50 plants We'll be putting 150 Yeah Because you're never going to get 100% Right, okay So just make sure your volume is on person If you're clipping the hedge, you're getting the cutlass for free anyway So you meant we do plenty All right I've seen videos online where potted plants are sat in a basin of water And watered this way, you leave them until the plant has stopped sucking up the water Ever heard, tell her that? Is it beneficial? Yes, it would be quite beneficial, all right If you go on holidays, I think, or away for a week or so Pretty much leaving them permanently in the dish of water Is probably not good in terms of taking up too much water And can actually rot the roots a bit So what you would do is, generally, if you have a saucer Someone set it in it for about 20 minutes That should be enough time for it Okay, so it's not a long-term watering solution And if you go away for the thing, a lot of people would do is To make a sort of a wick As they get a few bath towels into them in the bath As you wet them Use them like a wick and set your mask down So it regulates the water intake to some extent But it doesn't stop it I have never gone before And I want to put up hanging baskets at the front door Which do you recommend for first-timers And can they stay out in heavy rain? Yes, probably the easiest plant for hanging baskets Would be the Petunias or Nestertiums Either of which would be a very easy sort of mandate Pretty much all you have to do with them is just water Once you water regularly, they'll actually fly on So they won't have... Exactly, it is easy I mean, it's not complicated You know, plants are like weeds They're in the same family If they're fed and they're in a decent environment They'll grow and they'll keep growing all summer for you Absolutely And you'll get great results really quick If you do nothing else and just keep up in the water You're pretty much guaranteed results And another thing I've noticed too If you go into a garden center like yours or others The people there can't help you enough Yes, absolutely not, they're what they got We can be a bit shy to ask But once you ask, they go, well, where is it? What do you want? What colors do you like? And next thing you know, they're, you know Yeah, no, like I said People in the garden centers tend to be fairly passionate About their plants So they are only too happy to help Skimmy in pots Do I use Eric? It's almost like this is... Ericaceous Hold on, I was going to try it myself Don't spoil it I'm going to try it again Please ask for planting skimmy in pots Do I use Ericaceous compost? Ericaceous compost Did it do alright? Yes, perfect So Ericaceous compost Yes, the wood benefit probably from Ericaceous compost Alright, that's the lime-free stuff Yeah, alright We get, you know yourself all the time We get questions on roses So what should or shouldn't we be doing it This time of year? Roses pretty much Ideally by now you want to fed them They're just sort of starting to come into bloom now So they are So generally a good feed at the start of the year We're a granular sort of fertilizer And keep an eye on them For the likes of your green flyer Because roses, green flyers love them So they do Right Then just keep an eye as well You get black spot Tends to come on them at this time of year This a day like today Sort of a mildish damp day Ideal time for black spot Right, okay What can you do about that though? Generally get sprayed They're very cold They're actually They're a few sort of more organic methods you can use But generally the big thing is If your rose is well fed And a healthy plant It's able to fight off Attacks better than Well, you wouldn't be in favour of Because you know obviously they're beautiful Different colours Beautiful smells Is there any advantage to spraying them You know routinely As a deterrent to the likes of black spot Yeah, like a lot of those rose growers Would actually spray every 10 days Regardless So they would Yeah Generally what the thing is going to A fungal treatment is going to help them Keep away the black spot So generally if you get into a routine That probably is If you want So it shows specimen roses That would be the way to go And another enemy of the roses The green flies Is there anything Yes, the green flyer They just love them Yeah, what can you do Rather than give them a banquet There are a few things there One would be to actually Spray them just water and spray liquid Yeah Would actually help to jet them off that way Okay, brilliant stuff Listen Paul, thanks as always Grant, thank you very much I really appreciate it And thank you to everyone Who texted in with questions Anything we didn't get to We'll carry over Paul will be back for us With us in a couple of weeks That's Paul McLaughlin Of Balloch Dirt Garden Centre Just down the road from us Here at the mountaintop In Leather County, isn't that right? Yeah, that's the joke Okay, great to have you in Thanks very much Paul We have some live music Coming up for you before 12 Don't you know The Ninetal Noon Show With Letter Kenny Credit Union Now offering myCU current account And debit mastercard Bringing full banking features Delivered with the same Local trustworthy service Of your credit union Come along to a great night Of song and dance With Evelyn and friends This Saturday the 4th In a clanry hotel In aid of the Donegal branch Of the Alzheimer's Society Music by the Percy Robinson band And special guests include Hugh McLean The invoice show band Malcolm Fisher And Amy Mehan Also Ann Biddy Hilary Ann Hetherington Eddie Gulliver The country kickers line dancers And compared Tommy Rosney That's from nine this Saturday night In the clanry hotel Letter Kenny Tickets from hotel reception Clox news agents Or pay at the door Your mental health is important And now it's easier than ever To find the supports and services Closest to you You can free phone the Your mental health information line At 1-800-111-888 anytime Our team will tell you Which supports and services are near you Where they are located And how you can access them You can also visit Yourmentalhealth.ie For more information on mental health For you or for people you know Yourmentalhealth.ie From the HSE Come along to the Big Family Funday This Saturday the 4th At Lifford Old Courthouse Explore the sales Try out the stocks And meet your favourite characters With clowns, storytellers Face painting And much more Admission is free And everyone's welcome From 12 to 2 Or enjoy the great afternoon tea With sandwiches and treats For 15 euro per person In the comfort of the Grand Courthouse To book afternoon tea Call Lifford Old Courthouse On 9141733 Getting value on your shopping Has never been more important That's why at Supervalue We have great offers like Save 33% in our Irish Steak Sale Selected goodies like Fanta And Nestle Arrow Any 4 for 4 euro And get new weekly money off vouchers On the Real Rewards App For low prices that compete with anyone It's got to be Supervalue Okay Karen Kelly And Simon McCafferty Collectively are without Willow And they join us in the studio now Karen, good morning to you first Hello How are you keeping? Ah, not too bad Fantastic What about you, Simon? Got a laden kickin' Good stuff Listen, you used to sound very exciting Heading off on tour next month Yep What's going on there? Where are you going? What are you doing? We're going to Ottawa and Canada We're playing a few shows We're playing the Almond Keltfest And then we're playing Redbird Live And are we winery? And we're doing a wee songwriting workshop And we're over there So a few things lined up Is it exciting? Or are we nerve wracking our mixer? Yes, both Because we've been out of, obviously, for obvious reasons We've had a bit of time back now But we've been out of the groove a wee But we've all changed a little bit You know, it's not just about artistically It's about, you know, our attitudes to everything and stuff True Plus the Q's and Dublin airport's not helping Well, hopefully they'll get it sorted by July Or we're completely Banjaxed all together And are you out sort of meeting a new audience In Ottawa, Simon? Or do you have a fan base there already? Or what's the story? Hopefully We have some people I don't know if we have a few Canadian fans yet Actually, one of the messages saying we'll be there At the festival So that's cool We've never met these people But it's a class that we can connect That's through social media, presumably So those connections have been built up Anywhere else? Is it going anywhere else? Or just focusing on Canada for now? Canada with a few local shows lined up too We'll be playing actually for the Bringing it Home Festival And on Fana He on the 24th of June before we go So that'll be in the oyster And then when we come back We'll be headlining the club yard on the 19th of August So now a few things lined up Very good And what in terms of songwriting and stuff You know, I mean I presume you still did a lot of that Through lockdown Worked together Were they any good?  To be fair, like our songs were sad before But oh my god It's different level stuff now all together It's another level And we've been at both, you know And we're kind of, you know It was a nice love of the time as well Yeah I think at the start to a lockdown I think every musician was in the same way It was just like, oh, we know we've all this time to write music And then when it came to it we were just like Yeah, because you can't just pull it out of the air, can you? I mean, it has to be You're presuming you have to be in a space to sort of You know, good or bad or whatever it might be It was where I think I feel like you're putting so much pressure on that first time That you're just like, oh, we have all this time We have to write And when you're the busyest That's probably when the ideas come flying in You have to take your notes But you have least time to You've got all the You've got the momentum I don't know how long the flight is to Canada But sure, you can bang out a couple A couple of songs on the plane In terms of releasing music What's the situation there? Yeah, we just released a single there on Friday Called Nightingale So that's actually available now everywhere Spotify, Amazon Everywhere online Preferably if you can buy it Ideally, if you want to buy it, that'd be great But Spotify is great for getting We talked earlier on about reaching fans all over the place All through that And them being able to interact And feed back with you And of course there's, I presume, like other parts of Canada There's probably a decent Irish community over there as well It would be a real treat for them To see a couple of guys from Donegal landing over Yeah, well, the festival we're playing at Is kind of a Celtic festival anyway So it'll be nice, yeah Good stuff Right, so are you going to perform for us now? What are you going to do? The new singles? We'll do the new single, yeah Okay, so this is Nightingale Right, okay And it'll be heard in Leo's tavern in August and in Atua in July But live for you now here on The Lantel Noon Show Here's Karen Kelly, Simon McCaffery Collectively Without Willow Alright, that's lovely Nightingale from Karen Kelly and Simon McCaffery Which begs the question Because I think these are, you know, one of the most well known groups What's taking four years to get a song out Since your last single, it's four years, isn't it? Yep Oh, just life, you know We didn't get the word, we weren't able to meet up for a long time Yeah, I'm really Just didn't get round to it I like it, that's the right answer That's literally my answer to everything I just didn't get round to it So it's quite refreshing to hear Do you actually write songs together? Is that possible? Can two people write a song at the same time? Yeah, we have kind of sat down at the table and say we'll have a guitar idea And I'll just kind of start writing a melody and vocals award Because I would do a lot of her And then you say, well, that would sound good in there and all that kind of stuff Yeah, brilliant to suppose, yeah, it works well Are you just doing the one or are you doing another one for us? Can you do another one? Yeah, that'd be fantastic As I say, just for you tuning in, we are in the company of Without Willow, Karen and Simon That was their brand new song, Nightingale The first single released since 2018 It's released ahead of a miniature of Ottawa in July Karen's also going to be running songwriting workshops over there Which will be of huge benefit if you've just listened to what they've put together And locally in August in Leo's Tavern All right, what are we going to hear next? Do you think? So this is another new one called As Autumn Falls Lovely, okay, whenever yous are ready I will view the trees I've changed since that last song My night's grown Cobblestone and suitcase said Hoping you might stay by a stone Cold leaves with the ease fall Could let go and trust it Cobblestone and suitcase said Cobblestone and suitcase said All right, Without Willow there The new single is available for you now on all the usual platforms And you can go and hear the studio version of that Which was engineered by Ari McBrurty And featured the talents of Cello player Laura McFadden Listen, thank you both very much Safe journeys and huge success for the future as always What do you sing in the shower? Out of interest Because obviously the music you hear I don't sing in the shower A lot of you, yeah of course you do A lot of your own music is, you know, it's haunting And melodic and that kind of stuff I'm not, would you sing it in the shower? I would sing like, also like the big 80s hits in the shower Yeah, like Carlson's When I'm Fun or something Exactly All right, okay What about you, Simon? What do you sing in the shower? Actually, don't I just get Good night or just not so sure Worried like that I wash three bits and away we go Listen Listen, both of you, thank you very much indeed I really appreciate it That's without willow there Back with more on the program after the break Stay where you are It's not need to go on about this Yeah, yeah, yeah, yeah Strawberry He says he's quite fresh like But I'm a little strawberry Like come on Torn up, what did you have to say? Yeah, yeah, you're joking me He did what? Everyone loves little beef Well, almost everyone Lidl, more for you Your specialty is quality tiles, bathroom suites and wooden floors Who is the best range of tiles and dunny gold? Crawford tiles The best wood flooring? Crawford tiles The best bathroom suites? Crawford tiles Five day bathroom refits? 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It's the first point of contact For customers when entering your premises At C&M and Broadway in Letter Kenny They have a huge range of clothing Covering all areas of the workplace It's widely known The customers warm to and trust employees That present themselves well Have your company name and broiderage are printed On all your work uniforms Contact C&M and Brodery on 07491 28097 And get your staff looking their best Okay, so you might remember earlier in this week We were contacted by a lady who I believe she felt she may have been exposed to chicken pox And she was in the early stages of pregnancy About 12 weeks And went to the hospital Was directed to the hospital And had to spend an awful lot of time sitting in the ED And she said that this was nerve-racking for her Because she was going in to make sure she didn't have chicken pox But there were people getting sick with diarrhea And all the usual ailments you might see In an emergency department And she thought that it may have been the case in the past That she'd be brought in through the front door There'd be a separate seating area They're not going to be inundated with pregnant women Of course A separate seating area before she could go in to Guiney And surprisingly, not surprisingly But I wasn't necessarily expecting it Quite a few people then contacted us afterwards To talk of similar situations Now, of course, when we're talking about health It's kind of sensitive So Caroline, good morning to you Good morning, Greg, how are you? Fine, and as much as you're comfortable in explaining the situation And what happened, that's up to you But we know that someone close to you Ended up having to go to hospital So pick it up from there as much as you want to say, Caroline Yeah, well, my good friend She went to hospital eight weeks And she was sleeping in a mass carriage And she had a certain A and E for four hours While she was going through this And that's a very personal journey To go through and a very traumatising journey to go through And I couldn't understand it Because whenever I was having my children My eldest is 25 and my youngest is 8 And the thing like that was to happen to me Or with my friends We just went to the doctor and directly to Guiney And the ladies was very private And very personal there And gave us the treatment that With empathy and care and kindness And very gentle But when you're sitting in an A and E department With God knows maybe 20 or 30 people around you You're always sure to meet somebody You know, and they're saying, oh, what's wrong? You're already in for it You don't want to be disgusted And it's not about, you know, no disrespect To someone with a broken ankle There's an awful lot of emotional stuff Potentially going on What I find confusing about this, Carol Caroline, sorry Is because ourselves during lockdown For similar reasons Had to go to the hospital And went in through the front door Now officer, I wasn't allowed in It wasn't through the ED So how recently is this changed? Oh, see, I don't know And that's what I'm hoping that you shall find out For me because So then my friend went on then at 12 Because she actually did suffer a loss And it was, but she still had to go then To A and E again Even though she was talking to her doctor And talking to Guiney She still had to go through this protocol Of being tree-eyed in A and E Now we're going backwards with all our Health up there, can I? Because of 25 years ago I could have went straight to Guiney from my doctor And now you have to go through A and E That's so inhumane Now in relation to the second experience Was it a long wait in ED as well at that time? Three hours at that time And she was actually in that mess She was there in the toilet That's what I was going to say Yeah, I was going to say We spoke her baby And I'm sorry, that's why Not we shouldn't be saying this over here But that is Well, you know what? We can't hide from him So what do you mean? Yeah Then dignity of it all And the same age And my heart broke for her Because she didn't have the privacy of her own wee room Where you go to Guiney If something like that happens You get your private wee room The nurses is tough and hazy They're just so good at what they do up there And they're giving you all the empathy that you need And privacy Look, privacy is the main thing You know And her partner wasn't even with her Until she went up to Guiney You know? Yeah That's crazy I mean, so this is She's not At that point, she's not in a pre-care situation Where you're sitting waiting for care That is in any situation Where people understand the human side of things And the practicalities of what she was going through And what she has to deal with and manage Right, not just mentally but physically That should be right You're in You're in and we have a specific department for that You shouldn't be sitting there While the process is happening Waiting for that care We can't lose our empathy and our care No, we definitely can't It's a very sensitive situation again And you're going through trauma And it's not just you going through the trauma And you need your care system Your partner there with you To hold your hand and tell you, look I'm here with you It's very important for women to know That they're being heard and being listened to But you can't get that there When you're sitting in an open room With 20 or 30 other people that have broken lums Or like vomiting, bog and stuff like that And you're saying to yourself, please just get me to a private room You know, and why has this changed? Why now, can we not just go back to that? Maybe it was because of COVID I don't know, but I mean like Could we not just go back to going to the doctor And the doctor send us directly to the guy Now you're getting to that You know what's going to happen Caroline Is, and if it's not all happening already People are going to stay at home They're going to say I am not doing this I know what's happening Whether they're 100% right or not You know yourself I am not going to do this publicly Yeah, but then you have to think That they have to get checked No, I know Yes, of course I know But my point is, is unfortunately Someone might make that decision Because of this is the way they would be treated And that's what worries me They shouldn't They have to get in there and get seen straight away Because everything isn't always as it seems But I'm just worried that People go, I cannot physically go in there And go through this process Not under the care of the hospital And in public Yeah, and you know It has a very heart-breaking Thing to experience And it's very traumatising But this is an insult to injury By making ladies that are threatening To mascara or, you know, the likes To go through any That is just so There's no dignity in it at all People are, as far as I understand The way emergency departments work is They will decide who needs to be seen first Right, and I get that So someone has to make a decision That a woman clearly having a miscarriage Is down the line Three or four hours down the line I would have thought that this is one of those medical emergencies That is dealt with with compassion And through the front door Yeah, 100% Yeah, which it was It used to be like that all the time As I say, we don't need to be going backwards We have to start Keep going forward all the time And improving our maternity services I did just contact transforming births on EGAL They have now sent emails to the Minister for Health And I have contacted Thomas Pringle's office And he has sent emails to the hospital To find out why is this happening And why ladies in this position Have to sit for three and four hours Or even more There was a thread on the letter And oh my god, about this here And the amount of ladies that commented and said Yes, I had to go through an A and A This happened in that And I was sitting there thinking This is very traumatising for me to read Nevermind for them ladies to go through You know And it's heartbreaking Like in this teenage, I don't want my girls I have three girls and one of them is 22 I don't want her to ever have to experience a loss But if she does, I want her to I mean can you imagine sitting there for two hours And you feel for obvious reasons That you need to use a bathroom And you've got other people in there Maybe with vomiting and diarrhea And they're trying to get into that bathroom And you're in this public space This is happening to you And there are other people And you're thinking I know they're trying to get in How long can I stay in here You know, just even in that moment How that must feel Now we got a response from the HSE And I'm really disappointed by it To be honest with you Because firstly, they're hiding behind GDPR And secondly, they're effectively saying Sorry, this happened You can make a complaint That to me is an extension of the impassionates As I don't know if that's the right word But the impassionate treatment of this person Because not only is what she went through That was bad enough But now they're saying She can make a complaint if she likes We're not commenting on it That's ridiculous That's just And if she doesn't like the answer to the complaint She can go to the ombudsman Where is the heart there? Where is the humanity in any of this? Either in the initial treatment or in that response Sorry for the negative experience You know, make a complaint Have your say If you don't like our answer You can take it up with the ombudsman I mean, is that where we're at now? Yeah, well, that's ridiculous To be honest with you, that's a really pathetic answer But this lady did make a complaint So she did And then she was talking to a member of staff And she said to the member Why did I have to do this here? Why did I go through this? And the member of staff said Well, it's just because a senior member of staff Thought during Covid that that was the right thing to do Because you were triaged and you were swabbed And you were all out there And to stop infection But like Everything else has gone back Mostly to normal I don't know why You can't just go to your doctor To your Guineap department And that's it This is not a complaint There's enough people setting an A&E For other things without ladies that must carry something there too Yeah You know This is This also the response, okay This is not It's as if we're complaining about how long someone had to wait in A&E That's not the The issue here is not the length of wait Though that compounds the situation This is women that are clearly miscaraging Maybe have a GP referral But you know yourself and you head straight to the hospital That's what the plan is with that And you believe you go in the front door And you prepare yourself for every outcome You don't I don't ever associate If someone is having a suspected miscarriage Or is evidently having a miscarriage That that is a sit and wait in ED medical condition So make it complained They might say Well, we apologise for your long delay This should be about what is the policy here That's the conversation here Not sorry for your inconvenience This is what is your policy We can't continue to have these women And their families and partners And it's an incredibly stressful time for partners as well Have a situation whereby it's treated similarly to a broken finger Yeah As a bereavement at the end of the day Sadly, you know, a lot of them are being bereaved Like it's not been treated the way it should be treated Whereas the ampers here That's unhuman There's no dignity in it at all And what is the cut-off? Is there a different treatment for four weeks For eight weeks For 12 weeks What happens at 16 weeks? I know if you're over 16 weeks If you're over 16 weeks You go to the maternity then department But explain that to us Let us understand that What is the thought process behind that? Is it that oh well sure It's only 12 weeks I'm not trying to upset anyone okay And I'm not trying to throw fuel on the fire But we need to understand Why is what what's the the cut-off definition Has things changed since maybe Ireland's attitude to Things have changed I don't know like but we need answers All right We need thank you All right Caroline And my best wishes to your your friend Yeah, thanks Okay Genuinely all right Okay, thank you for that Listen what's your views on it Are we overstating it here 08 660 25000 What's that what's app and text to that number The 9 till noon show With letter Kenny credit union Now offering mortgages With life cover provided At no additional cost Letter Kenny credit union 9102127 A lot of the best things in life Have been a little out of reach for a while Like those belly clutching Back of the bus laughs Or having the chats with their mates As you're getting ready to go out We've all missed loads of things But luckily most of them are making a comeback And so are we at Faroaga Clubs Join thousands of adult volunteers Who make Faroaga Clubs a community Where young people can become who they want to be By giving them a space where they belong Become a volunteer today at faroaga.ie Okay, open wide there Perfect Let's have a look at these teeth A range of treatment benefits are available to employed Self-employed and retired people Who have the required number of PRSI contributions Some benefits are free Like eye tests and dental check-ups While for others you receive a contribution Towards the treatment Such as for hearing aids or wigs And now if you're 25 to 28 years old The number of contributions you need has been reduced Find out more at gov.ie Ford slash treatment benefit The treatment benefit scheme It's there for your benefit A government of Ireland initiative Brought to you by the Department of Social Protection Okay, I meant to mention actually As we were wrapping things up with Without Willow If you're out there, a musician out there You're creating or releasing new music Get in touch with us We have a number of slots over the course of the year We'll try and see what we can do now That we can have live music back in studio again Thankfully So if you are a musician or a band out there And don't prejudge what we play on Highland And it means it wouldn't fit in Get in touch with us And we'll see if we can work with you To give you the platform Which is not insignificant to hope It's the overnight staying of camper vans Staying its free parking I kind of put that point to that listener I'm going to run through some of your comments On different subjects here Greg, in January this year There was a tiny wee piece On the bottom of RTE's website Reporting that domestic violence against men Had risen 32% in the previous year I didn't hear it reported anywhere else Can you imagine if it had been a huge rise In violence against women What the outcry would have been Hopefully this Johnny Depp case Will encourage more male victims to come forward That comes in from Noel Yes, indeed Noel But I'll tell you what I don't want to be involved in Is versus versus You know women versus men You know men get We need to stamp all of that stuff out Now the 32% increase in violence against men Would be coming from a very low base It happens Men should report it And this should be backed up in doing so, okay But the reality is the overwhelming majority Of domestic violence victims are women But that does not mean We have to sort of compete one with the other I don't know where we get on with that Right, Pat O'Toole's political correspondent With the Irish Farmers Journal Good morning Pat Good morning We had Minister Charlie McConnelog earlier on I put three of the findings of your poll On farmers sentiment towards him It was a mixed bag There was some good in there, some bad And also you pull people on Their support for political parties What were your highlights that jumped out From you Pat on it? I suppose it's the Rose of Chintain They previously had polled 100 and 6% In a succession of polls over the last 15 years And that would have been in 2018 They reached 6% Now they're on 16 It's a significant increase I suppose it matches the rate of increase they're having In national opinion polls in the general population But at a lower rate So when Chintain were at 10% to 12% In the general population They were at 5% to 6% Half that level of support among farmers Now they're polling in the mid 30s And they're at half that in the farming population So they're still trending The farmers' patterns are changing more conservatively They're changing more slowly But they are changing And the impact of Chintain's emergence As a third force in electoral politics Is now manifesting in the farmer vote as well As of course are the rural independence at 20% Rural independence Or independence in rural constituencies Are a very significant force in our politics Although not in government But there's 23 independents You could say that 17 of them Are identifiably rural independents Who push very strongly on rural issues Whenever something like turf comes up Or rural transport Or farming supports Those people will come to the fore And their level of support is high and steady Yeah, and interestingly I think in fairness One of the first people to come out To counteract what Eamon Ryan was saying Was Leo Faradkar And he obviously was speaking to farmers I was not aware that really Finnegale seems to be Although they've dropped somewhat The party for the farming community Yeah, so traditionally And in page 16 In the third page of our analysis We look at a poll of polls Going back to 2007 Traditionally, and was picked up In those early polls Finnefal and Finnegale would have split The farming vote between them Polling in the low 40s Sharing that vote equally Finnefal's vote collapsed After the economy collapsed And some hard decisions Had to be made around farming And around the general economy It collapsed down in the low 20s And they've been stuck there ever since Finnegale on the other hand Surged up to over 60% And then It's only Pardon me In this recent poll That we've seen their level of support Has fallen very sharply down From the mid 40s to 37% But if you combine them 6 out of 10 farmers Still, you know, are in favour Of the government parties More than less than 50% Are dissatisfied With Charlie McConnelog It's not a bad school report For them given the environment Muyin, is it? Given the environment I think Charlie McConnelog's A little unfortunate I'm from Wexford down the far end Of the country But he would have Toured the march People wouldn't have known them That well in the southeast But especially since he became minister We've spent a lot of our time in lockdown But he toured the march To present the cap proposals Debated the issues with farmers And I think there was a level of respect for him And they think that the minister's rating In part Is held back by the rating That we're seeing for his party That Finnegale just can't gain traction They're stuck in the mid 20s They can't regain the level of support They had in the past You're right That there's absolutely no doubt That Finnegale and Finnegale have higher level of support In the farming population Than they do in the overall population But it's that historic load Combined support that Finnegale and Finnegale have At the moment is a little over 60% And that's historically low How did the grains get on If you dared ask? Negligible Yeah, they barely feature At sort of between 0 and 1% Consistently across poles The Labour Party have seen an increase in support But again off a low base So and that comes down to candidates often You know, you get regional disparities But you get disparities within individual constituencies So if you say take Sean Sherlock in Cork East He's a very popular candidate Agriculture spokesperson for the Labour Party Even though he's urban based He has a good support I think across And a lot of respect in the farming population So you pick up boats for the Labour Party In a constituency like that Whereas other rural constituencies Where the Labour candidate Or TD or Senator is less wedded into Rural issues in the rural community The Labour vote will be negligible In Ireland I suppose Because of our system of electoral politics The personality is very important And that's I suppose Why the rural independence are doing so well Even though they have no party base behind them And say for instance in your analysis of it Although Sinn Féin are showing gains Are they traditionally not seen to have strong policies on farming Or farmers don't like their policies Or they're not vocal enough on them I mean do you have in the last 30 seconds An analysis of that Pat? I think that Sinn Féin In terms of the policies we've gone for They are more popular with smaller farmers And part-time farmers They have been very strong advocates Of the flattening of payments in CAP They would have pushed for an absolutely flattening of CAP So they have targeted the constituency They have a coherent policy platform They have a very presentable Ag spokesperson and a very articulate Ag spokesperson and an invested one in Matt Carthy And I think that he has before him Martin Kenny similarly spoke very well On farming issues and I think broadening the appeal Of the party to the farmer base But I think generally the reality is That Sinn Féin are maturing as an electoral party And we're seeing the fruits of that in this poll Right and those polls and an awful lot more Are inside this week's Irish Farmers Journal Thanks Pat Appreciate it Pat, I'll tool their political correspondent With the Irish Farmers Journal Thanks to all of you for listening Or watching and taking part in the


Good. So first of all, thank you very much. A big thanks to the organizers and personal, a personal thanks for inviting me here. I could speak for four days about Wigner semi-circle laws and deeper subjects that arise there in, but it's not going to be quite that long. It's going to be only four hours, for our day lectures. All right, so let's see how to transform this blue screen into a PDF. Good. Okay, so since I'm the first lecturer of the summer school, I feel like I should say a few words about random matrices in general and then delve into the more specific subject at hand. So of course, coming in here, you can ask if you haven't seen such things before and I know that some in the audience might not have been familiarized before this lecture with random matrices. You can ask, what is a random matrix? And a random matrix by and large can be thought of as a model. So an object that can be used to model things. In general, you want to model uncertain, noisy, hard-to-quantify information. So for example, random matrices have a reason as sample covariance matrices in statistics. They have great important roles to play in signal processing and information theory, quantifying noise. There are, they can serve as testers for algorithms in numerical linear algebra. If something works, you know, what does it mean for an algorithm to work well on a typical case? Well, let me take a random matrix and see how it performs on it, right? And on and on and on and on. So they serve as models and you can extract information from from them about the system that you're working with or about the situation or about perhaps like I said, the algorithm that you want to analyze. There are roughly in classical random matrix theory, there are two paths that have been introduced to random matrices. One of them came from statistics and it came with the work of John Wishart. Well, that's the most the first prominent instance in which random matrices have been used explicitly and they were used to model covariance matrices, sample covariance matrices. And then in the 50s, Wigner and a bunch of physicists, mathematical physicists coming after him, including Dyson, have looked at random matrices in the context of particle interaction, many body problems. The two types of matrices that they worked with are similar, in many respects, but different. We're gonna focus in the course of this series of lectures on the second kind of matrix, the Wigner random matrix. So it was introduced to model the behavior of complex systems given by Hamiltonians. And Wigner himself considered two types of random matrices, the ones which have binomial entries plus or minus one and the ones that have Gaussian entries. But with that, I'm introducing the idea of a random matrix in terms of its entries, so let me say a little bit about that. So how do we define a Wigner matrix? And I'm going to talk about real random matrices. So you might see that the title of this slide is random real facts. Random just stands in for the fact that whatever I say is going to relate to random variables. A real does not want to emphasize the fact that they are facts. Although in today's world, you never know, it's perhaps good to put a qualifier on things. But it's meant to emphasize the fact that we're gonna talk about real entries. So I'm only going to talk about real entries in this lecture. Pretty much everything that I say will be extendable. Nicely and smoothly to complex entries or quaternion entries, but that's going beyond the scope of my first lecture. Okay, so random real facts. We look at a matrix, say an n by n matrix. We put a condition on in for it to be symmetric. So w i is equal to w j, i j is equal to w j i for all i j. And we sample these entries from distributions. So w i j are independent and centered. So that means expectation zero. And the diagonal can be differently distributed than the off diagonal, as long as they're all centered. And we normalize the off diagonals to have variance one. This is expected w i one, one two squared is equal to one. That's because it's already centered, so we don't need to subtract the expectation. We don't condition the variance of the diagonal. It'll turn out not to matter for our purposes. So this is what a random matrix is. You can think of it as just sampling one variable at a time from these distributions with the constraints that I emphasized. For example, if you look at the off diagonal entries as being distributed, and this is a notation that I have to explain, n of zero one, n of zero one stands in for the standard Gaussian, so centered variance one, Gaussian variable. And the diagonal ones are n of zero two. So the variance here is two instead of one. And what you get if you do this is the Gaussian orthogonal ensemble. It has beautiful properties. It has been analyzed very closely. It serves as a benchmark in a sense and is something for the rest of the ensembles, an ensemble here being defined by the kind of distributions that you choose to put on w ij and w ii. It serves as something to project yourself onto. So how different will a Wigner ensemble with some other distributions be from the one that has Gaussian distributions? And this question is going to be answered more in depth in the next week, I guess, when we're going to talk about universality and the like. But for starters, for this week, for this lecture in particular, we're not going to go quite as deeply. We're just going to mention that this ensemble is particularly nice because things can be computed for it. Exactly. Okay, so one more assumption that I'm going to make, which I will eventually remove, but for now to make things easy as a stepping stone, I will assume that the distributions that the w ij's come from have moments, have finite moments of all orders. Okay, this is what this assumption is saying. So a moment is the expected value of x to the k with respect to some distribution. That's the case order moment. So these are the assumptions that I'm going to make on my Wigner matrices this lecture. And with this assumption, so we have now, we have a model, we know what the entries look like. We have an idea of maybe what what symmetries are. What are we interested in about this matrix? Well, it turns out actually to really normalize things properly. We're going to have to work not with the matrix that I just told you, but with the scaling of it. So we're going to just scale by one over square root of n. The reason why you want to scale by this one over square root of n intuitively is that this makes the variance on each row be bounded. And therefore the spectral norm is likely to be bounded. Okay, so you want to bound the spectral norm. You want to be able to say something about the eigenvalues being with high probability in some compact interval, right off the bat. Okay, so that's why we need to to introduce this scaling by one over square root of n. So I'm going to work with the matrix W bar, which is the scaling of the matrix that I introduced. So the main quantity of interest for Wigner per specifically, but for subsequent generations and generations of mathematicians and mathematical physicists that came after him has been the spectrum. The eigenvalues of this matrix were supposed to model the energy levels of the system, of the system that he was interested in and therefore introduced this random matrix as a model for. And the question is, what's the distribution of the eigenvalues of this matrix? So what do I mean by that? What's the distribution of the eigenvalues of this matrix? Well, let's take a look. So we have, for some matrix A, which is square and symmetric. Suppose that its eigenvalues are lambda 1 of A, lambda 2 of A, lambda n of A. You can think of a distribution of one uniformly random eigenvalue chosen from this matrix as being 1 over n sum of delta lambda I of x. So this is the delta Dirac function. All this is saying is that you choose from amongst the delta, sorry, the lambda I of A's uniformly, okay? This is called the empirical spectral distribution or for short ESD. I'm going to use this shortened acronym for the matrix A. So this is the empirical spectral distribution ESD of A. This right here. In fact, let me write it down. So if A is n by n, and I'm going to look at it as being symmetric. It doesn't have to be, but it sure helps me think of the distribution with eigenvalues lambda 1 of A less than or equal to lambda n of A. Then the empirical distribution of A's fA of x is equal to 1 over n summation delta lambda I of A of x. So this is the quantity that I'll be looking at. And now we can ask what is this ESD for the matrix W bar, this random matrix that I introduced. And let's pretend that this is an actual real-time experiment that you can write on a MATLAB, on a machine that has MATLAB on it. I know we're supposed to use Mathematica and the like, but I'm a MATLAB girl, sorry about that. And let's take a look at what I'm doing here, okay? I'm picking a 200 by 200 random matrix. In MATLAB, random denotes sampling from the normal distribution, okay? So this is now a square matrix of independent standard Gaussians. It's not symmetric, so I'm going to have to symmetrize it and I do that by just adding the transpose and dividing by two. Then I scale it, as I promised, W bar is what I'm interested in, so I divide by 1 over square root of n. I take its eigenvalues, I do the histogram of the eigenvalues, and then I say hold on, and then I plot something else, which is actually going to give me a density. It's the semicircular density, so it's from minus 2 to 2 and it should be a semicircle. Well, okay, I'm going to qualify that. It's not a semicircle because you're dividing by 1 over square root 1 over 2 pi, so it's an ellipse really. But it looks like a semicircle. And if you do this experiment on MATLAB, you're going to get something that looks like this. The empirical, empirically plotted spectral distribution is going to be awfully close to the continuous semicircular distribution that I've also plotted here. You can say, okay, but this in itself is an interesting fact, of course, but you have chosen a very particular type of Wigner matrix, the one that has Gaussian entries, and you just told us that they're nice. That they're particular, they're nice, we like them, right? Well, it turns out that what you write here doesn't really matter. You can start off with brand of 200, which is uniform distribution. Well, you'd have to center it, so you have to start with something that's entered. But you can start, you can construct your own distribution to start off with. Go through this process, plot the results, and get something extremely similar to this, extremely similar. So you can see that immediately you can conjecture, okay, so it looks like the eigenvalue distribution for one of these Wigner matrices, for n large enough, n is 200, but I think that you can start actually seeing it. I did 200 because I wanted the histogram to have enough meat in it, so that you know, you can look at it more closely, but you can probably start seeing this thing at 50, 100, it's already there. So it looks like the distribution of the eigenvalues as n, the size of the matrix, increases, approaches that of this semicircular distribution. And you can conjecture this regardless of the distribution on the entries that you start from. So that's pretty interesting because it says two things. First of all, it says that even though you have a random matrix and therefore, you know, you would expect, okay, so you're going to have random eigenvalues, that's clear, right? And random matrix means random eigenvalues. But there's tremendous structure behind these random eigenvalues. They may be random, but the group of them kind of arrange themselves in a very nice pattern. So their structure behind the random matrices, in fact, there's very strong structure, and this is where the second thing comes in. It's a stable structure. It's what it's called a universal structure. It does, because it doesn't depend on the distribution of the entries that you start with. Okay? This is pretty powerful. It's a very simple. It's the first step, really, in this examination of universality for Wigner matrices. But even so, it's pretty powerful. And now, hopefully after I wad you a little bit with this, I will tell you that there are two lies on this slide. Can anyone point them out? Yes. The normalization, oops, oh, sorry, I think that this is my laptop conking out. Yes, you are completely correct. The normalization of the histogram is different. As you can see, I've plotted things pretending that things were from minus two to two, but the plot has minus one to one. I'm sorry about that. I realized this this morning. It comes, the plot comes from a time when I insisted on doing things my way and plotting things between minus one and one instead of minus two and two. So good. That's one of them. What's the second? How familiar are you with histograms? It's what? The frequencies? It's normalized, exactly. So if I just plot the histogram, it's just going to, the heights are going to be integers. It's going to show me how many eigenvalues I have in a particular bin. But what I have actually done here is I've normalized the histogram to have area one, because if I don't, there's no point in comparing it with a distribution or with a curve that has area one, right? So that's the second, let's call it like, as I knew what I was doing when I said it's good to take responsibility. But there are easily fixable lines. So you just have to introduce a little bit more code here. Instead of the histogram, you have to plot an normalized histogram so that the area is one. So this is the experiment that will tell you what we're going to be doing for the next, let's say lecture and a half. The fact that you pick a single matrix and you see such a good convergence, such a good approximation to the semicircular distribution tells you something. Suggests that the convergence of this ESD to the semicircular distribution is going to happen in some sense in probability, because you've chosen one matrix at random and you plotted it and you see some good results. So that's what we'll be proving for the next, I guess, lecture and a half, maybe less now, because I've been talking for a while, so we don't technically have the rest of the lecture, it's not a full lecture. So this here is the formalization of what I was saying. We look at the ESD of W bar and we say that FW bar converges weekly in probability to the distribution with density given by the semicircle. That's what we're going to be proving. And the way we're going to be proving this is we're going to look at the method of moments. The method of moments is something very more usually used in probability when you have to deal with countable structures and we'll see in a moment why matrices qualify. And it's based on the following idea. Under certain conditions, and I'm just going to enumerate a few, you're going to see a couple of them, you're going to see in the problem session, in the first problem session for this course. Carleman-Riss or compact support, a distribution can be defined by set of moments. So if you know the expected value of x to the k with respect to this distribution, then you know the distribution. And if you talk about distributions, you should probably get a feeling of why this is true.  You know that if you know the distribution on compact continuous functions, then you know compactly supported continuous functions. Then you know in a sense what it is. Those compact continuous functions can be approximated by polynomials, polynomials are linear combinations of moments. So that's the idea. Now, the method of moments is the following thing. If a distribution sigma is uniquely defined by its moments, and if you have a sequence of distributions fn that have the property that their moments for each k, the kth moment of the sequence of distributions converges to the kth moment of sigma, then fn converges weekly in probability to sigma. This is something that again you'll be discussing more in depth in the problem session, but we're going to rely on this fact today. So in other words, if I want to show convergence, let me go back, if I want to show convergence in probability to a certain distribution, it will be sufficient to prove for each moment convergence of the moment. Does that make sense? I hope it does because I'm going to be using that a lot. So this is the moment method or the method of moments, and like I mentioned, we'll be using it many times. What are the moments of the semicircle? Well, it turns out that the moments of the semicircle are very easily expressible. By delta k over 2 is in Z. So what do I mean by that? I mean that this variable here is going to be 1 if k is even and 0 if it is odd. So the moments of the semicircle, the odd moments are 0. The even moments are equal to the ck over 2, ck over 2 being the k over second kettlen number. The k over second kettlen number can easily be defined as such. So k choose k over 2 divided by k over 2 plus 1. These kettlen numbers are actually very important in combinatorics. They count all sorts of structures. We'll see some of the structures that they count. And they're at the core of some very interesting phenomenon. And one of them is that they are the moments of the semicircular distribution. So in other words, if I were able to show that for any k, the kth moment of the empirical spectral distribution converges to these kettlen numbers, I'd be done. It would follow that the distributions converge weekly in probability to the semicircular distribution, which is my goal. What are the moments of the SD? Well, let's take a look. So here is the SD. Suppose that I want to look at the expected value of x to the k with respect to this distribution. What is it going to be? Well, since I'm applying x to the k to delta Dirac functions, those are going to be just lambda i of h to the k. So the kth moment of fw is going to be 1 over n summation of lambda i to the k lambda fw bar, let's say, to the k. And lo and behold, that's just 1 over n trace of w bar to the k. So that's the nice thing, that the moment of the empirical spectral distribution is expressible in terms of the matrix. It's expressible in terms of, as we will see, the entries of the matrix. And that's what allows us to calculate it. So as I mentioned, this is what I want to be proving without the expectation. But I will start off by taking an additional expectation. I'll say, okay, I don't want to look at fw bar. I want to look at the expected f of w bar. And that can be achieved by averaging over w bar. How do you do that? You average over the entries. So I'm going to be looking at a slightly different object. I'm going to be looking at the expectation over w bar of this object. This is going to be the moment of the distribution that has, quote, unquote, level density. So when Wigner was looking at these matrices, he was looking at the eigenvalues as being energy levels. He was interested in the level density and therefore he was looking roughly at how this would average out over the ensemble. The nice thing about this is that even though the ESD itself is always atomic, okay, it's a sum of delta functions, when you average out over the matrix, you will get a continuous object. You'll get a continuous distribution. In fact, these have beautiful expressions in terms of orthogonal polynomials for the case of the Gaussian orthogonal ensemble. And the plots, if you were to plot the level density against the semicircle, you would see what I like to call finger plots. So essentially for any n, you would see a little bump around the semicircle around there. It would look something like maybe this. So if suppose that you were looking at n is equal to five, you would have the semicircle like this and you would see something that has five bumps, kind of like that. But that's the level density, which is the average of the empirical spectral distribution. And I'm going to be looking at the moments of that first. The reason why I'm going to be looking at the moments of that first is that it'll turn out that the empirical spectral density is actually very close to the level density. The moments are very concentrated. The moments of the ASD are very concentrated around their means, which are the moments of the level density. And that's going to be intrinsic to proving things. Okay. So let me look at this object. I'm going to look at the one over n expected value over W bar of trace of W bar to the k. Remember that W bar is scaled by one over root n. So when I raise it to the k, I get one over n to the k over two. There's an additional plus one there because of this one over n. And what's the rest? Well, what does the trace of W bar look like? Or trace of any matrix to a power. So trace of A to the k is going to be a sum over I1, I2, Ik, all being indices between one and n of AI1, I2, AI2, I3, AIk, I1. This is elementary linear algebra. This is how you multiply matrices. So if you multiply the same matrix A to itself, k minus one times, that's what you're going to get. So I'm going to be interested in taking the expectation over these variables of products such as this. Let me call this ordered k tuple, I1, I2, Ik, the indices. And I will write WI for WI1, I2, etc, etc. Okay, so this is just notation for now. I'll call this a word, although I think that I'm probably not going to use the word word again necessarily in this presentation. For each such word, we have an associated graph G. Let me give you an example. So suppose that I want to look at a term that looks like this. W12, W24, W43, W34, W42, W24, W42, W21. This is an acceptable word. What's the corresponding index set I? It's 1, 2, 4, 3, 4, 2, 4, 2. It's length is 8. That's how many terms you have in this product that constitutes a word. And if I were to represent this as a graph, I would do it as follows. Let me first draw four points, 1, 2, 3, 4, and then see if I can connect them in the order indicated by the set of indices, 1 to 2, then 2 to 4, then 4 to 3, then I come back, 3 to 4, then I come back to 2, then I advance again to 4, back to 2, and then at the end I come back to 1. So what do I do? I execute a walk on this simple graph with four vertices and three edges. I execute a closed walk on it. That's all I do. And for this graph, so if I identify this with the graph g sub i, I have three edges, just three edges, and I have four vertices. And if you look again at this word, you will see that actually you can collapse it in and of itself and use the fact that the matrix is symmetric to rewrite this. So w12 is the same as w21 because of symmetry. W43 and 32 are also the same. And then everything that's remaining, w24 and w42, are one and the same. So this is actually going to be w12 square w24 to the fourth w34 square. That's what it is. So knowing that every one of the words can be expressed as such and that each one of them has a graph associated with it, let's think a moment about what we're doing when we're summing all of these things. We're picking graphs which have to have at most k edges because each word gives you a closed walk of length k. So you have at most k edges in that. So you'll have at most k plus one vertices in the in the graph. And that means that for every k you have a finite number of such objects. Once you've picked a graph and a way to walk, all you have to choose are labels. To figure out, you know, how many terms in here will correspond to that particular graph in that particular walk, but with different labels, perhaps. You have to count the labels. And that's the interesting part that you can do very crude approximations. Okay. Now I said that you can have up to k edges, but there's one thing to note, which is that the entries w i, w ij are all independent. So that means, since I'm taking an expectation, if you have an edge that gets walked on only once, so if the power in the expression kind of like this here on the board is going to contain a w ij to power one, then what that means is that that disappears. It wants to take the expectation it disappears because all the variables are centered. So I'm not interested in those. And therefore I'm only interested in walks in which each edge is walked on at least twice. But what does that mean? Well, it means, let's see, here's, it means that this gives me a sharper bound than I mentioned before on the number of edges. Since each edge will have to be walked on at least twice, that means that k, the number of edges in the walk is going to have to be at least as large as twice the number of effective edges because each edge is walked at least, walked on at least twice. So that means that E is going to be at least, at most k over two on the one hand. On the other hand, we shall have that this graph G is connected after all I'm executing a walk. So if the graph is connected, then we know that the number of vertices it has is at least, sorry, the number of edges it has is at least the number of vertices minus one. So we have this interesting set of inequalities here. So the first one comes from the fact that each edge has to be walked on at least twice. Otherwise, that's a term that I'm not interested in the expectation. And the second one comes from the fact that the graph G sub i is going to be connected. So all right, so what does that mean? It means that V is less than or equal to k over two plus one. Why is this important? Well, as I mentioned, the sum that I'm interested in is a sum over all graphs of fixed size less than or equal to k of ways of walking on these graphs. And all of those are finite numbers because k is finite. There's no end so far. And intervenes only when I start putting labels on these graphs. Because that's where the meat is. It's how many times does this particular walk appear in the expression for the trace, given that you have n vertices? And the answer is that for each effective vertex in the graph G i, you'll have about n choices. Of course, you don't have exactly n choices. It matters. The order in which you choose the vertices matters. But overall, you have about n choices. So you'll have about n choices for that one, about n choices for the two, for the four, and for the three. So that means that effectively, each walk on each graph is going to appear about O, so big O of n to the v times. Big O is notation saying that it's on the order of. It's going to be constant with some constant, which is independent of n, times n to the v, where v is the number of vertices. So this is O of n to the v choices. Let me go back for a moment. So for each walk, and there will be a finite number of types of walks, I'll have O of n to the v choices. Those choices get scaled by this n to the k over 2 plus 1. So what do we notice? As asymptotically as n grows large, the only walks that are going to survive are those for which there are k over 2 plus 1 vertices. If you have fewer than that, the contribution from such terms is going to wash away, because you'll have n to the v over n to the k over 2 plus 1. It's order of 1 over n, and that disappears. So the only thing, the only walks that will matter, that are asymptotically relevant, are those for which v is precisely k over 2 plus 1. Now, immediately you should be able to see that this creates a problem. If k is odd, k over 2 plus 1 is not an integer. So what can I conclude? All the odd moments are going to be zero. I can't satisfy this. I can have only fewer than that vertices. Therefore, all the terms in the sum are smaller than constant, and asymptotically, odd moments are going to be zero. But what if k is even? Then I can satisfy this. I can have v is equal to k over 2 plus 1. What kind of walks do I have there? Let's think about it a moment. If v is precisely equal to k over 2 plus 1, what does that mean? Well, I obtained that inequality as a consequence of this string of two inequalities. So to satisfy this as an equality, both of these have to be equalities. So that means two things. One is that each edge is walked on exactly twice, and the second one is that the number of edges is precisely the number of vertices minus 1. What kinds of connected trees have the number of edges be the number of vertices minus 1? Oh, no. Tarnit. All kinds of connected graphs have e is equal to v minus 1. Duh, trees. Terms with this property will come from walks on trees. That's where I'm going to get this highest order contribution, walks on trees. So here's the kicker. You have a one to one correspondence between a rooted unlabeled tree. What does that mean? Well, I mean something like this. You have a structure with no labels, but you have a marked vertex, a starting marked vertex. You have a tree with a marked vertex. You can do what is called a breadth first search or a depth first search on it. And that will give you a way of walking. So it will give you a type of walk. It's going to be in a one to one correspondence with this. How do you find out what terms you have in the sum corresponding to these walks? All you have to do is just put labels on it. 100, 275, 1310, and 1. That's all. And you have o of n to the k over 2 plus 1 ways to do that. So all you have to do really is count the number of unlabeled but rooted trees. That's all. And all of those will give you the asymptotically relevant terms. Well, it turns out oh, why do we only count that? I think I have it, oops, sorry. I have it over here. So if you look at the asymptotic contribution, you will be taking the expectation over moments of these variables. Since in this case, each edge will be walked on twice, all the powers here will be 2. And because of the way we set things up, expected of WIJ squared is 1. So they come with no extra weight. All you have to do really is just count the number of rooted unlabeled trees. That's all. And that number is a Catalan number. And that's the end of it. That tells you that the moments of the expected ESD or the level density are going to indeed converge to 0 if you have odd moments and Catalan numbers if you have even moments. So we've shown that the expected ESD converges to the semicircular. That's not all there is. Let's see how much time I have. Why good. We would like to actually show not that this is true for the expected ESD, but that it's true for the moments of the ESD in general. So I want to remove that expectation. Now there's a very simple technique for removing the expectation or showing concentration for a random variable. And that is to examine the variance. If I look at a random variable, and I know that its expectation is over here and its variance is very, very small, then that will tell me that the variable converges asymptotically almost surely to the expectation. Okay, that's all. So I'm going to do precisely that. I'm going to remove the expectation by showing that the variance is small. What's the variance? It's a variance of this quantity, 1 over n trace of W bar to the k, which can be expressed as such. And when you take the variance, you will get that this is 1 over n to the k plus 2. The reason is these two guys multiply, so it squares. Covariance of I, W, J, where I and J are k tuples, and W, I, W, J are the words corresponding to these k tuples. It's a covariance, because I will look at the expected W, I, W, J minus expected W, I, minus expected W, J. Okay? That's what I'll have to compute. So it's enough to focus on one of these guys and see what happens. Okay? Each one of them will come with its own graph, and in fact you can think of the join of the two graphs. So I'll have W, I, W, J, this is G, I, this is G, J. Let's think of the two graphs happening somewhere here. So I have one over here and I have another one, let's say over here. Okay? So this is G, I, this is G, J. I'm going to look at it at the join. The variables W, I, J, which correspond to edges in these two graphs are independent. So if these two graphs don't have an edge or a vertex in common, what is the covariance of the two words, of the product of two words? What happens to covariances of independent variables? Zero, thank you. So covariance of a product of independent variables is zero. And if the two graphs are disjoint, then the variables are independent. In fact, this is true even if they have a vertex in common. Because remember, the variables are defined by edges, not by vertices. So when I look at these covariances, the only ones that will stick around are those that correspond to words with overlap. The two words have to overlap. The graphs have to overlap at least on an edge, otherwise the covariance is zero and the term disappears. Okay? So now let's take a look. So the number of edges in the join of the two graphs, G, I, and G, J, the total number of edges is at most K. Oh, by independence. So it's the same considerant as before. If I have an edge in this join of graphs that is only walked on by either I or J, once, well actually not by the sum of I and J or by the concatenation of I and J, if I only walk on an edge once, then by the fact that there's independence and centering of the variables, that will disappear. So yet again, I'm only interested in walks for which each edge is walked on at least twice. Now I have K edges that I walk on in I, I have K edges in J, so for a total of at most two K edges, but the effective edges have to be walked on at least twice. So the total number of edges in G, I joined with G, J is at most K. Therefore, the number of vertices, as we know, V minus one is less than or equal to E and since it's less than or equal to K, so I'm applying the same kind of reasoning as before. I have this, therefore I have that V necessarily is going to be at most K plus one. Again, I can wave my hands and say that for any fixed K, the number of pairs of such walks is going to be finite unless I look at the labels in the absence of the labels, the numbers of such types of walks is going, the pairs of such types of walks is going to be finite. So I'll wave my hand around and say that the only thing that matters is in how many ways can I place labels. The number of ways in which I can place labels again is going to be O of N to the V. V is less than K plus one. What do I normalize by? N to the K plus two. So that means that the total contribution from any kind of type of pairs of walks is going to go to zero. I just don't have enough choices because I have to overlap them. I just don't have enough choices. So I'm going to have that the variance just simply goes to zero because the highest order term is of the order N to the K plus one and I divide by N to the K plus two. So that means that the variance goes to zero, which means that the moments are concentrated, which is what I set out to do, but actually I can say a little bit more. The variances are indeed going to zero, but they're going to zero a little bit faster than I stated. They're going to zero as one over N squared and the reason for that is simple. It's the condition of overlapping. Let's look at why I cannot have terms for which V is K plus one. So suppose that V were K plus one, right? What does that mean? Just as before it means that these two inequalities are both equalities. So that means that in the join of graphs the number of edges is equal to the number of vertices is minus one. So that means that the graph is a tree. So I have a tree. I have a tree in which each edge is worked on precisely twice because this inequality becomes equality. If the join of the two graphs is a tree, that means that, well, okay, they're both connected. Both GI and GG are connected and since their join is a tree, it follows that both of them are trees. But since the number, okay, both of them are trees, the walks I execute on these trees have to be closed because the words represent closed walks. So that means that already in WI and WJ, I walk on the trees in such a way that each edge is walked on twice. And when I join them, each edge is still walked on only twice so I cannot have any overlap. If I had overlap, I'd have to have at least one edge walked on four times. Is that clear? Okay. So that's a contradiction because if there's no overlap, if I have WI and WJ and there's no overlap, then the covariance is what? Zero. So that means that I cannot be in this situation because it doesn't allow for overlap. So what does that mean? It means that this term doesn't appear at all. In fact, the most choices we're going to have is n to the k and when you divide by n to the k plus two, then you get that the covariance is, that the variance, the variance of the moment of the ESD is actually going to converge to zero, like one over n squared. You can do actually pretty nice close estimate on the things that I've just waved my hands around, which are the numbers of walks, the numbers of joints of graphs and so on. You can estimate those things pretty clearly and it will allow you to construct an almost sure argument by using Borrell-Cantelli, but this is a little bit beyond of the scope that I want to achieve today. So now what do we know? We know that the moments are concentrated and we know that they converge to the Catalan numbers or to zero, depending if they're odd moments. So I'm going to stop here and tomorrow I will show you how to use these two facts together with a various trust approximation to conclude that not just the level density distribution converges to the semicircle law, but that the ESD itself converges to the semicircle law. So rather than the smoothed out deterministic object converging to the semicircle, we will show that the random object also converges to the semicircle. And there will be a problem session later today in which you'll be able to kind of dig a little bit deeper and settle some of these notions before tomorrow's lecture. All right, thank you. I should have said from the beginning that I really appreciate questions and I'm trying to make this interactive. So you didn't ask questions in the body of the presentation, but if you'd like to ask some questions now, you're completely welcome to do so. Any questions? We have time for maybe one or two. What happens if you let the variance of the off diagonal elements go to zero? So in other words, the matrix, so you're looking at a sparse matrix. Yes, so you'll be talking about that in problem session, I think, tomorrow. It's an Erdos-Schraini ensemble and you can still prove that as long as the weight in which this variance goes to zero is not too slowly, then you get the same kind of semicircular distribution. Very good question.


Further item at the internally displaced persons camp in Wargo, Vietnam is the local government area over dual states following the dumping of a newborn baby by yet to be identified mother at the entrance of the camp. Speaking to journalists who visited the camp, coordinator of the camp, Pastor Solomon for Lawrence show review, the dual the camp is known for giving to the needy over the years. This was the first time they had seen such in the camp around 11 a.m. on Friday while all the children we are you know in school and everything was just going on well then the security they they were checking around and they saw a baby just close to our security post under the mango tree it is so a baby and newborn baby like few hours old or one day old you know and then they went close and so the baby sleeping with a note written and the note says that I can't take care of this baby and I don't want to kill the baby so from now henceforth the baby is your own I've come to give you the baby so we just thought around what do we do we are registered charity and NGO took care for the needy in the society and caring for children is one of our main a trust so we felt okay let's take this baby in and then we went to the police and to other relevant agencies we're still in the process so that's how we're here today with this they happen of this nature this is the first time we've not had anybody drop a baby here you know but we've had the other children why beats older than this one there are displaced or rather orphans that we've been taking care of for many years so but having to come and drop a baby just by our doorstep is the first experience we are having yes we love to keep the baby we we want to keep the baby we want to train the child once you see this child grow want to make something out of the child this woman or whoever she is had approached us and said I can take care of this child maybe this what my condition we have done it for many people what would have done is okay this child need to be breastfed and so can we just probably get you a room somewhere or stay in our center and then we draw kind of sympathy people coming to support her to train each other that's what we would have done you


These co-circuits photographers in action. On the left is the female photographer with skirt. On the right is the male photographer with tripod. This is the aero stick kit. I have glued it to a rectangle of black styrene and attached it to the yellow SUV. Styrene and SUV are not included. Press 1 is half line back and forth. Press 2 is half line back and forth quickly. Press 3 is full line blinking. Press 4 is full line strobing. Press 5 is left arrow blinking. Press 6 is right arrow blinking. Press 7 is bi-directional arrow blinking. Press 8 is off.


Hello everyone, and welcome back to theCUBE's live ongoing coverage of HPE Discover Barcelona 2023. I'm your host, Rebecca Knight, along with my co-host and analyst, Rob Stretche. This is day two, there's a lot of energy and excitement in the room, I feel it. Yeah, we're getting into the meat of things here as well and getting some great partners is part of it and I'm really excited about this next guest. Yes, yes, he was on the main stage this morning. Carl Harvard, managing director at Tiger Cloud, great name for the company. Well done, congratulations on being up there with Antonio this morning. Yeah, it was fun, it was fun. What really struck me was how you began, you began with such a note of optimism about this moment in time. Yeah, yeah, and I mentioned my gray beard at the time as well. You did, you did. Which I've related to. Yeah, I was relatable, right. So yeah, it is, isn't it? I mean, and I don't know if he caught the lady towards the end who was going to be a judge and decided not to be. She took the other side of the coin as well. I think it's absolutely right, it's a balanced thing. I think we're our moment in time with Genitive AI and AI as a whole. The way technology has evolved to support human creativity, who knows what's on the horizon, and which is great. People's imagination can now be brought to life. But then on the flip side of that, imagination can go everywhere. So I think we have to do it in absolutely the most ethical way we possibly can and support organizations and right organizations to, I guess, develop what they want to develop and qualify those, call them rogue type of organizations that want to get hold of this technology and prevent them from doing wrong things with it. Right, right, so doing AI right, doing AI ethically or responsibly is critical. Tell our viewers a little bit about Tyga Cloud. Okay, so Tyga Cloud is Europe's first, largest and cleanest Genitive AI cloud service provider. So we focus on Genitive AI. Our infrastructure is all GPU based. We don't do any app migration or any big data storage. Purely about helping organizations get access to this technology to train, tune and infer the large language models or the models that they're building for AI purposes. So the way we have gone about building our business so far and the way we'll continue to build it is, as well as being the first and largest in Europe, which is great, and that's a lot of help with HPE here. We're also doing it back to your ethical way of making sure that this compute power, which is really high-density power drawing equipment is housed in the right way. And what I mean by that is data centers that are clean energy driven, so hydroelectric powered, not necessarily nuclear, so I should say renewable energy. So hydroelectric powered predominantly, but also cooled in the same way as well through natural sources. I think I gave the example this morning that our data center in Sweden, just outside the Arctic Circle, is completely naturally cooled. Our left dial data center in Norway is in a disused mine below sea level so it uses the natural violent recalling. So that means that people can unconsciously decide how to train their generative AI because we're transparent about it. The other thing we're transparent about is not just the source of energy, it's the efficiency of the energy we use. So I don't know if you're familiar with PUEs, the power usage effectiveness, so yeah. So I think acceptable levels of 1.4, 1.45, all our data centers are 1.2 or below. Our Bowdoin data center is 1.06, which is, no, that's a good number. We know we can do better, so we want to draw that down. But so a sustainable way of creating your applications with generative AI is absolutely the heart of what we do. There's other things I can keep going on about, if you want to say. Well, I think it's, I mean, today's the first day of COP28 down in Dubai. So I think it's very timely to be talking about this. And I think that when you start to look at the gen AI and all the data and all the crunching and how they have, I mean, people are talking tens of thousands of GPUs to go and train models. What are you seeing from your customers? Because you talked about customers of all different sizes and democratizing AI and kind of get into that a little bit. Yeah, I mean, that's one of the things core values, which I think Tiger shares with HPE and vice versa. Because if you look at the market dynamics, which clearly you have, there is a big demand for this compute capability and organizations are buying it in bulk.  However, I think that limits the access to startups and enterprises who have great ideas. They've got the right skill sets. They just don't have access to the right compute power to bring it to life. So our whole ethos is to democratize access to this. So buying quite a few GPUs, I mean, the announcement we made today was an additional 330 million investment with HPE. That's on top of what we previously invested. So we're now about 800 million total.   But what the whole point of us is to say, we'll buy these GPUs, we'll make them live, we'll put them in the right environments, but what we'll do is make these open and accessible. If you're a small startup and you only need access to a couple of H100 machines, you can do that with us. If you're a bigger enterprise and you've got this great model you wish to train, but you only need it for a couple of months, we'll do that as well. Organizations that we've had knock on our door say, we'll take everything you've got for three years. We won't do those deals. We want to actually be able to have, I guess, fueling the European economy from a generative AI point of view to be that part of the ecosystem which again was mentioned this morning, allow people to roll on and roll off as and when they want to. And sorry, second part of your question I think was the types of customers. So we're not focused on specific organizations, you know, who are going to take a lot of GPUs. You know, startups that share the same values as us are absolutely at the heart of our target audience, call it that, because if we can help them succeed, great, we've done part of our job. Likewise, life science companies around drug discovery, universities on research around things like that, perfect customers as well. So yeah, we want to make sure everyone has a fair share.  So it comes out of Europe versus, hey, we're taking Lama two or we're taking one of these other foundational models. Yeah, yeah, we are, it's more so now. You know, a few months ago, we weren't necessarily seeing that. And I think there's been a few scares from a EU compliance perspective, which affects not just European organizations, but it affects, as you know, global organizations that have European customer bases. Then we've had a lot of people come to us to say, okay, we like what you're doing, you're a European organization, you can offer us true sovereignty, true compliance with its GDPR, et cetera. And we know if anything happens inadvertently, I don't know, a diplomatic disagreement between countries, either side of the Atlantic, and it won't happen. But if it did happen, we can help them tick their compliance box to say they have full control over their data in the region that they're within. So, and I think we're getting more US companies requesting this too, which is interesting. Well, it's an uncertain geopolitical environment. I understand that. One of the things you had just mentioned is having the same kind of values as the startups you work with and how that's a real priority for you. It's also, I would imagine, something that you share with HPE. This idea of a shared value, shared priorities, shared commitment to sustainability and to democratizing AI. Why is that so important for your working relationship and what does that bring to customers? What's the value that customers feel from that? I think it's like, not just what we offer with Gerontive AI, but if you've got that common purpose, that common aim of objective, and it's not a monetary objective. I mean, that is important in some instances, but you say, okay, here's this customer here. They can accelerate what they're doing in genomic research in order to define a cure for cancer or whatever it might be. Then HPE obviously are very interested in helping those. And in partnership with us, we can help that customer in a specific area, but HPE have a breadth and depth of other things they can also bring to the table. So that's great for the customer. If you take the customer-centric view, together, Tiger and HPE, we can sort of offer that customer full-rounded solution or with the intent of helping them succeed in their own objectives, ideally for the benefit of the planet or humankind or whatever it may be. But yeah, I just feel working with these guys for quite some time now, even on a personal level, you can sense those values are true. How do you see the customers, are they transient or do they come and host, they build and host there? How does that work for you? Because if you have a lot of GPUs and if you're doing inference, for instance, maybe you train the model and then do inference, maybe you don't need as many H100s, for instance. Yeah, when you move from the training and tuning to inference, absolutely, it's a slightly, I won't say different, but a config that's required to help organization move to inference. So what we do is, as Tiger, is we have infrastructure as a service. So simple terms, we rent the compute power out and host the customer, et cetera, et cetera. We do have a sister company in our group called Ardent who are managing a lot of data centers, building them out. They will offer the Kolo capabilities as well. So if organizations have access to their own compute power, we can bring them on stream as well and offer them that hybrid model should they want to. So yeah, did I ask your question? Yeah, I guess are people coming in, renting it for a month, two months, training, or is it, hey, they come and, not three-year contract, but they're building out their apps, they're training, then they're tuning the models, then they start doing some inference there and they're re-tuning the models, or what's happening. So that's the, I guess, the beauty of what we try to construct as a European cloud for GNII. So we have these islands of 2032 GPUs, all configured in the same way, located in different locations. Now, if someone wants to train a model for, say, a three- or six-month period, we can do that. When they move to inference, we can give them a slightly different config and move them to a data center probably more suited to for inference. So in our suite, we have tier three data centers, which means high levels of security, minimal downtime, I think 1.6 hours per year of downtime that you're allowed, with also round trip times of 12 milliseconds or less. So when it comes to inferencing, even though it may be less demand on the GPUs, the CPUs come into play though, but you need that balance. So that's when we probably move them to the slightly different area of our cloud to help them suit. But they can pick and mix, that's the beauty. And to your point about contract lengths, now, one, three, six months fine, 12 months, 24 months, 36 months is fine. Our longer-term contracts tend to be with organizations that serve a number of customers themselves. So they reserve a bank of compute power and they actually serve several customers within that. So it still fits within democratizing access. It's almost doing LLMs as a service or something like that. It's a two-degree, yeah, yeah. Absolutely. And we need to get better at that because at the moment in our evolution path, we are very much infrastructure service, but we know we're having to enhance our managed services on top of that and also our software skills. So we're ramping up our skill sets around people that really understand how to construct, advise, even audit LLMs in order that they are configured in the right way, et cetera. The skill sets, and we talk about this a lot, the skill set around that is still so nascent for a lot of organizations. Are you finding that, and this is why you're going down that path, is that... Yeah, it is. I mean, it's like, when any new, and I know JNAI is different, but when any new technology emerges, it merges rapidly. Well, it's Chachi BD's birthday today. Oh, is it? Oh, okay, I didn't know that, I didn't send a card. So, but yeah, the skills tend to follow, don't they? And therefore, there's always this catch up. So we know we need to be ahead of the game, and being an elite partner of NVIDIA helps us because we get future sight of what's coming down the line. So we can see, you know, H200, Grace Hopper, et cetera, advanced notices of what that entails. We get to see what they're developing on their software stacks, same with HPE. So we're sort of slightly ahead of the game, so we can attract the right talent in. But if we don't get the talent in ourselves, which is probably not our true model, we're not going to scale heavily that way, that's where the partnership with HPE comes in, because they have that talent in the world. They've got the talent, right, exactly. Yeah, yeah, yeah. Yeah, excellent. Well, Carl, thank you so much for coming on theCUBE, it was a pleasure having you here. Thank you, glad to be here. Thanks. I'm Rebecca Knight for Rob Stretch, I stay tuned for more of theCUBE's live coverage of HPE Discover Barcelona. You are watching theCUBE, the leader in high-tech enterprise coverage.