No, okay, so good morning everybody. Welcome to the second seminar of January of our statistical physics joint STP system meeting. And it's my pleasure today to introduce to you Ginestra Bianconi from Queens Mary University of London and other to the Institute of London. Ginestra is an expert in several aspects of statistical mechanics in particular she has pioneered ideas in network theory that span several fields. And she has also had a steam tier before setting in London for several years so it's also double pleasure for us to introduce a year, and today she will be telling about their recent research on the DRAC operator and dynamics of topological signals. Thank you so much for the kind. Oh, sorry. This on. Okay, so thank you so much for the kind invitation is a real pleasure to be back in Trieste and meet so many friends. Today I want to tell you about work that we have been doing during COVID. So, you will see we we took a route that is quite original probably due to the isolation that we experienced. And in particular, I want to tell you about research work on topological signals and the DRAC operator. This works parts from the interest on higher order networks and higher order networks are characterizing interaction between two or more nodes in a complex systems typically, and they come in different flavors so they can be hyper graph in which you have interaction that are not only per wise but includes many body, I get order interaction. You have simple shell complex that are a special type of hyper graph that are amenable to topological treatment, and then network with triadic So it's going to be interesting and research has been is been done here in ICTP regarding those and, but I will not speak about those today. For instance, higher order interaction are attracting a lot of interesting brain research when you can have three region of the brain that are correlated per wise, or activated at the same time and this leads to a higher order interaction, which is depicted in a triangle formed by three nodes and trillings, but a field triangle in which you have this higher order interaction. Another very intuitive example of higher order network is for instance collaboration network where you have team that collaborated together so each paper is co author by more than true author typically. And so this has sparked the interest on higher order network. And of course one of the central team in network science has been the interplane between structure and dynamics, and there has been in network science a lot of research in this direction, in particular focusing on the combinatorial and statistical property of networks for instance how the scale free distribution change the critical behavior of critical system critical phenomena such as the easy model percolation and so on. But when you look at higher order networks not only combinatorial and statistical property are important, but also network topology and network geometry. And this is the central team of my talk today. And this is kind of the message of a little book that I published by Cambridge University Press. And this is a very personal account of higher order networks, but actually if you want to see some wider, wider point of view from a more comprehensive set of people. You can look at our nature physics perspective article on higher order network, in which we highlight the importance, the importance of topological signals. The dynamical variable, not only associated to the node of the network, as it's usual, but also associated to the links to the triangles to the theater Idra. And this dynamical variable here depicted as a clock on the links can be projected up and down on triangle or on nodes using topological tools and I will speak about that. So here we will look at higher order network in the term of simplicial complex. Simplicies are set of D plus one node, it's the dimensional synthesis are set of D plus one nodes. And they encode these higher order interaction. So it's zero simplex is one node, a one simplex is a link, a two simplex is a triangle, and so on. And they encode these higher order interaction which I mentioned before, but also they can, they are amenable to to be treated in geometry and geometry. And a face of a simple shell complex is a simplex, which is built by a subset of the node. So if you have a tetrahedra, the faces, or a three simplex the faces are four nodes, six links, and for triangle. And these are important because a simple shell complex is a set of simplices, but close under the inclusion of the phases of is simplex. So if you have a triangle, then you should also include all the links and all the nodes of this triangle in your, in your set in your simple shell complex and this is a little, a little condition in order to apply all the beautiful tool of topology to this object. Here I want to mention that so although now this talk is mostly about dynamics structure and dynamics. We have also worked a lot on structure and on modeling and this is a simple shell complex since they describe this network geometry, they, they are very interesting object to characterize network topology and geometry from data.  One question is whether the network geometry of complex system is a priori prerequisite of the network evolution so where nodes know that are embedded into space or where there is the special aspect of this network is an emerging phenomenon of network dynamics. And this leads to the concept of emergent geometry. So a mathematical tool which allows to describe a network that are geometrical, but starting only for poorly combinatorial and statistical property of the of the model. So let's look on this approach that of course is a wide history. And, and here we have looked at a very simple model for instance a model in which you stuck a triangle to a link randomly with this one one to partner move. When you do that deterministic deterministically choosing this link randomly, and with the probability P you do this other move that is closing this pie which is also close to one partner move in a way that each link is in is incident to at most two triangle. And the network that result is this kind of very interesting manifold that is generated because the constraint that each link is incident at most to triangle is the, in addition to the connectivity of the of this, of this simple shell complex is a condition to build manifold is a combinatorial condition to build manifold so we can have this emergent geometry here, and then we, this is two dimensional. And so we went with model with higher or higher dimensional. So not only stuck in triangle to links, but also having to try to and stuck in to try to triangles. And we do all only this move of stacking a triangle to a link and we can generate a kind of the selection of hyperbolic plane. So these emergent hyperbolic play deep geometry, it's emergent in this way or you can have three, three D manifold. And when you add feature to to the faces or to the notes, then you can have transition also of of the dynamics from something that grows in every direction more or less uniformly to something that chooses one direction is also called a kind of kind of spine direction, or you can have also very cramped configuration. So, yeah, so for this model we have theoretical understanding of the homogeneous face. And like this, this, this, this space transition this fine thing is, is related to my previous work on both Einstein condensation on complex network so is a similar transition, but is also well also in network from the mathematical perspective that the transition to the both Einstein condensation is not fully understood. So this is still still research ongoing but okay so for instance you look at the diameter them out the diameter scale with the network science. This is a small world, so it scales like the logarithm of the network science here, like a beta lattice. And in this case is polynomial. This is finite dimensional but you know in one direction. Another approach to modeling is is is maximum entropy and so a configuration model and so we have also this kind of approach for simple shell complexes. Okay, so let me just now go back so this model you can take it as model of reference where we can run dynamical process if we want. And let's now look back at this diagram out this different aspect to shape the interaction between structure and dynamics. And let me focus on the in the role of topology in shaping higher order dynamics. And so topology is of course, the study of shapes and their invariant. So they connected component of a network or the cavity for instance the one dimensional cavity of a circle or the two dimensional cavity of a sphere. And these are accounted for instance from Betty numbers. And when you look at the network for instance this fungal network is clear that the cycle structure is important there. So I want to use topology to do that, and there are ways to do that for instance persistent homology you can, you can study the significance of the cyclic pattern in the structure. But actually what I want to tell you here is something different how to use topology to study dynamics topological signals. The topological signals are dynamical variable sitting not only on the notes, but also on the links or eventually on triangle or the simple as you, you like, also in cell complex you can have a square or a cube. So, these topological signals are attracting increasing attention in the context of a complex systems. There are data about it. So for instance, you can think of a citation that a team of older gods, you know, not the team of three older or pair of two older got on the paper could live together, or you can have fluxes in biological representation networks that are defined on the links synaptic signals between the nodes. There is all of sports that are pushing on edge signals at the level of brain region. And you can have also vector field so vector define on on the node for instance of a simple shell complex, but then as for instance speed of wind at given location or current at given location in the ocean, and you have a tessellation of the surface of the earth. So you project this vector on the tessellation so you define. You treat these vector field, especially as a co change so as a function defined on the links of the triangulation. In order to treat topological signals we need algebraic topology. So, a simple as I said is a set of the simple as a set of D plus one nodes. So, simplices have an orientation. So, because in algebraic topology because they are considered as the element of a vector space. So, for instance, you have a link i j and this has a different orientation of the link j i opposite orientation of the link. This is here is indicated with our role but it's not the direction is all in orientation and a triangle you can have a anti clockwise direction 123 and then, you know, you have the opposite orientation that is. With in the clockwise direction. So, with the synthesis with oriented system places you can construct and chain. So linear combination of these oriented synthesis with coefficient either in the integer or in the real. And from this chain, you can have a kind of interpretation of what is an unchanged so for instance you have 13 minus 23 plus 24, you go from 123 from three to two because there is a minus in front and then from 224 so you can construct can have algebraic expression for instance also cycle or things like that. So, on this and chain you can define a boundary operator and this is an important operator that maps each and say and chain to n minus one chain. And so it can be defined by his action on each simplex of the simple shell complex. And if you apply to a link. The boundary operator gives the two and points with opposite with two minus one. If you apply the boundary operator to a triangle 123 you get the three link at its boundary and for this is the name. You have 123 and minus 13 so you go from three to one along along the orientation of the triangle. And there's nothing to be very scared about it because these boundary matrices are only represented by matrices rectangular matrices that have no the links. So here, the boundary of the link one two is one is two minus one, and the boundary of the triangle is 123 minus 123. You construct this rectangular matrices, and then this boundary operator actually have also geometrical interpretation so be one can be interpreted interpreted as a discrete divergence be one transpose as the discrete gradient be to transpose as the discrete discrete curve. And you have this property that is a very important topological property that the boundary of the boundaries new. So because if you have a triangle, the boundaries the links at the boundary of the triangle but this is a close cycle. So the boundary of the boundaries. So practically only one dimension up or one dimension down with with the boundary operator. So, by concatenating the boundary operator you can construct a large Laplacian, which describe diffusion from and synthesis to and synthesis either going one dimension down and minus one, or one dimension up. So the for the node you can only go one dimension up and this is the graph Laplacian for the links you can go down and up and if you are if we have only know the links and triangles for for for the, for the triangle you can only go down to the link. And so this odd Laplacian can be written in this way so you go either one dimension up or one dimension down so you can go from no from link to link to note or from link to link through triangle. And so, certainly this odd Laplacian encode topology because the dimension of the kernel is then Betty number, and actually also the again vector of the kernel can be. There is a basis of the kernel, they can vector of the kernel in which they can vector are localized on the cavity on the n dimensional cavity. And this Ellen down and Ellen up they have very interesting property because the image of one is in the kernel of the other. And this leads to a very nice property that is called odd the composition, which tells you that if you have a seniors on the links that is a unique way to write it as a gradient flow, going away from a source to a sink and harmonic component along the cavity and a curl flow or solid on your component. Okay, so this is a very nice tool that is taking a lot of interest in in higher order networks to treat topological signals. And what we did is we use it to define a kura motto model on for topological signals. So the kura motto model is this model proposed by kura motto in 75. So you have faces associated to the nodes of the network. So the faces also laid at their own frequency, and then they are coupled in such a way that each node can try to align to the face of the nearby node. And in a fully connected network in the infinite limit you have a phase transition between an unsynchronized phase to a synchronized phase with disorder parameter. And despite each phase as his own intrinsic frequency that is drawn randomly from a Gaussian distribution for instance so they, there is no coherence if there is no interaction. Then if the strength the coupling constant is larger than a bit there is this finite fraction of oscillator that starts to oscillate together. And the question is, can we define this kura motto model for fancy places associated to the links. Okay, so, so what we did is it's quite simple so we started by an expression of the above the above equation for the kura motto model in the boundary operator this is exactly equal to the equation that kura motto road. And so, in terms of the boundary operator you see that the coupling couple face of the node if they share a link in this non linear type of coupling. And so what we did is we def. So this is a vector of the phases of the node with this is the vector of the frequencies internal frequency of the links and this is the coupling term. So what we did is we consider a phase associated to the links now intrinsic frequency associated to the links and then the coupling such that link to link try to align to each other. So if they share a triangle or if they share a node. And the linearized dynamics is. Can they be that there are sites which are not coupled by anybody. Well, if they are easily isolated. It means, not to any triangle like. Yeah, but it is, there is the other term. So they can be con links can be. Are coupled. If they are connected by triangle or they are connected by a link and if they are connected by. So the linearized dynamics is the one defined by the large Laplacian. And from this is apparent that the harmonic moment. So when the fee is aligned into the kernel of the Laplacian, the harmonic model are the one that are free to oscillate and the other modes are dumped. The synchronization that remains if the other mode freeze is actually only a synchronization along the cavity. So the dynamics select the cavity and oscillate only along the cavities. So in order to measure what happens to the other mode, we, we filter so we project the face on the links to the triangles or to the nodes, and the space. And from this projected dynamics that is now totally the couple, we can see that essentially this other mode that are not harmonic freeze with with when we raise the coupling constant sigma. And we have two other parameter, one for the faces projected on the node for instance and one for the faces projected on the links. So what we do then is what we can kind of modulate the coupling constant of one term coupling links that share triangles with the older parameter of the projected dynamics on the node and vice versa. And the two projected dynamics are not anymore the couple and we get an explosive transition discontinues. So this is a configuration model of simple shell complexes, which I mentioned before, is a kind of random graph, but we did it also on NGF, and we did it also. On the on real data set. So we can have real connect on we build a simple shell complex from in a way that is called a click complex, and we find this phenomenon so it seems to be quite independent on on the topology, although I mean we want to to go back to this so I can check it carefully. So in particular, I think the cavity distribution would maybe in on a random graph. Yes. So this is on a fully connected network on a random graph. Yes, you have a transition. The analytic tool to characterize this transition are not exact, but there are approximations so there are the annealed approximation which is called that we also use to to you for our model gives good result on random graph for instance, doesn't need to be fully connected. Yeah, yeah. Although with the network with spectral dimension, you might have some some instability of the fully synchronized state. So there is a critical spectral dimension that should be, you can have some kind of frustrated localized synchronization. So the question is, can we now have a dynamics, the couple node and links. And so for this we need, we need the data cooperator, we need to leave the ocean laplacian and treat the odd, the data cooperator. So here, I just wanted to put this slide to be sure so the data cooperator is not something we invented or I invented. I saw it fast in a paper by satellite garnet on and Zanardi in 2016, which is a very nice paper proposing quantum algorithm for topological data analysis. They don't cite anybody for the data cooperator. So I worked in my group for a long time without having other reference for that. But recently I find a very interesting reference by Olaf post in 2009, which is a mathematician, which identifies a lot of property of the data cooperator. And also he doesn't cite anybody. And but I could go back and see that Davis as in non commutative geometry as a paper in which he mentioned the data con graph. And this is as far as I could go back. Because there are also other paper that sites this paper but and and we are working quite intensively on that, and starting from my 2021 paper. And I will, I will speak about this work. So the data cooperator us on a topological spinner. So you have, for instance, if you have no links and triangles, you have a node signal, a link signals and a triangle signal. And the data cooperator is defined in terms of the boundary. And so this is a block structure between node links and triangle. This is when you have a simple shell complex of two dimension but you can go higher. And so when you applied the data cooperator on the topological spinners. The links to be projected on the nodes, the nodes to be projected on the links and so on so you can allow cross talking about these signals in different dimension using the data cooperator. What is it called the data cooperator, because it's most characteristic property as did I can use paper was looking for is that the square of the data cooperator is the laplacian so is a matrix was diagonal. And so what are the large laplacian. And. Okay, so of course they can value of the data cooperator will be the square of the again value of the laplacian, the laplacian so you have positive and negative again values not anymore positive definite. And so you can decompose in the one and the two the one only couple northern links, the two only couple links and triangle. And for this, direct the composition that we call you, you can play the same game of odd the composition so you can have that that anything, any topological spinners and not links and triangles can be decomposing a unique way into northern links links and triangles or an harmonic component. And so the DRAC operator as the spectrum that are these square plus or minus the square root of the laplacian, and then plus zero again value which has this generously the sum of the Betty number. And they can vector are they can vector either of the one or the two or harmonic and they can vector of the one are related by chirality. The, the again vector of the one of this type either you won we won or we won minus we won where you won and we won are the matrices that indicate the left and the right singular value of the boundary operator. Let's start to consider on a network on a network. This definition of the data cooperator is can be written as D plus the star where D is the serial derivative and the star is is dual. So this is a self adjoint operator. And so this is the most basic way to define the DRAC on a network. For the moment not. Okay, but. Okay, so let me, let me go. I think, I think DRAC is correct in this case so you can put a complex number in front. As long as this modulus is one, the square will be the laplacian as well so this complex number as we will see plays the role of a gamma metrics and as if you work for instance in one dimension or two dimension. This is enough. And if you want to have like this instead you need to kind of upgrade to a gamma function. You have a topological spinner that now is defined on node and links. So, the vector defined on all the nodes and all the links so this is a spinner but as a geometrical interpretation. So let me start writing the equation in DRAC equation in an Hamiltonian form. So with this DRAC operator that I defined before this D plus the star and beta is this kind of matrix familiar matrix where D and beta anticommute. And these of course is related to the chirality of of the again back store of the. Just to show you that you can do this, you can study this, this again, again value problem. And when you do that you see that key is an again back to the graph laplacian with a and the, and see is a game back to the L one down laplacian. And the dispersion relation is the relativistic one. So, the energy state are of course positive and negative. And so you have this kind of matter anti matter symmetry but the state of energy equal to M, they correspond to the harmonic again vector. So, the pattern might not be symmetric on a chain you have the the harmonic again back to the nodes and the links is the same, because for the quality a chain and no then the links are the same. But in general for a network you can have, you might have more states at value energy M or than energy minus M or vice versa. So, they get back to it can be represented on any network, you know for a certain node and links you can plot it play with this. And of course, it's also interesting you can play with weights weights of the node and weights of the need because well, of course I mentioned to you the laplacian but that is also the way that laplacian or the normalize laplacian so you can introduce some weight. And, and have a data cooperator in which now here you have the odd to all of the, this is, this is the co boundary and this is the odd to all of the co boundary. We applied this to many classical problem. So, for instance synchronization again, we find this discontinuous transition forward transition and continuous backward transition, but as a function of Z of this Z so with this non linear coupling, we can have to this continuous transition, and here the node and the links are coupled locally so the node feels only the nearby links and the link feel only the nearby nodes. It's quite different from what I show you before with this global coupling constant that that was depending on the order parameter here is the dynamics is local and you have this phenomena. The linearized dynamics is this one. The harmonic component are the one that remain unchanged, while the other component here. We treat the, the, the component along the vector lambda and minus lambda together. So they, they have a, they don't freeze with a dumping with a stable fixed point but they have a stable focus. So there is a kind of rhythm at merging and also in the non linear model. And this emergence of low frequency that we are quite fascinated with. So this is treatment that emerged from the dynamics. For the moment we don't have a theoretical understanding. In addition to to the fact that the linearized dynamics is a stable focus but for the fully non linear model. This emergence of low frequency it may just a numerical observation for the moment. Yeah. So, so if you put that equal to zero. These dynamics the couple and will be the dynamics of note, the standard model, and the standard link link connected to note. So this z, then you have this local coupling of northern links so this will start speaking together. So, yeah, in some sense you can go back to. We also consider touring patterns. Not all topology satisfy the condition to have touring pattern on their structure, but square lattice do so we can have stripes we can have more complex pattern. And we are also applying to signal processing. We have signals on the nodes and on the links and you want to do signal processing jointly. But now I want to tell you something more about three plus one dimensional like this. So, if you have a. We see that we have D plus the star, this is the data cooperate on a graph, but we can play with this complex number. And so this complex number can be used to play and distinguish between link index direction or in the wide direction so when I do the boundary on the x direction I put to be equal to one and when I do the boundary on the y direction I want to be equal to I so that I can distinguish between the two direction, but in three dimension. I need more. And also in four dimension three plus one dimension I need more. So this is what I tried to do. So I want to distinguish between four different direction x, y, z and T. I want to have weight on the links, eventually. So I now change a bit to this scenario. And I have the function on the links is now not to define on each link. So in such a way that is known localized. But each link is directed so you have I to J. And I have a function defined on the link I to J, which in some sense describe a flux going from I to J. And a function defined on J to I such that they define the opposite direction flux so then this way they, the faith function is now localized so I don't have any more this problem that the theory is not localized. And since I want to distinguish between these different direction, I have a part on the node a part on the directed link, but the part on the node will be double. So I will have to function define on the node and to function define on the links. And the, the matrix the weighted matrices here are e to the a one and e to the zero. So in algebraic topology these metrics are taken real and diagonal here I took the liberty to take it more general. Although there might be geometric constraint that that I'm not taking into account yet that that needs to apply. I found a re operator that is for each link is one and minus one one link at one and minus the link on the other land, but these are only on link in the x direction in the y direction in the z direction. And I make it weighted with this weighted matrices so G zero is a node by node matrix G one is the link by link matters. So we can have the laplacian which is be one be one dagger now and in direction x and the laplacian in the old network is the sum of the Laplacian x y z and eventually also LT. If the network is not weighted this Laplacian commute, but if it's weighted in an arbitrary metric this Laplacian might not commute. Okay. Instead of having the complex number I have now the gamma matrices that take this form and they're expressed in terms of the Pauli matrices. And the, the data cooperator the serial derivative and links to all are indicated in this way so where I practically double because I have two. And two zero crocheting so this is a matrix that is. Two by two in in this block form. And so the data cooperator now as the gamma D mu plus D mu star. So we have to contract. And the way we contract the indices is by assuming that the metrics is Euclidean so we absorb the complex number into the, the, the signature of the metric into the definition of the gamma matrices. The square of the directional data cooperator is the Laplacian for space spatial one. But for the time, time, like directional operator, the square is the Laplacian with the minus in front, as we want. So that I noticed in 2001 is that this data cooperator do not anti commute and do not commute either. So it's not like the derivative that commute is in many cases is like a derivative but they don't commute.  And then you have an operator that go from a link in the y direction to a node you find zero because you are on the X links on the X direction, but if you are from the link index you go on the node, and then you go back in the link on the why, then you have a non zero term. And so you have this non zero common anti commutator and commutator. And one can interpret the commutator as a curvature. This is non zero only in the link link sector. There are already two other indices hidden here that are the indices of the link one non zero in the X direction here and one non zero in the y direction. So, one might want to define a riches color by summing over these. These matrices the element in the link mu and the element in the link new. And so here it becomes more and more speculating my argument but I think it's a kind of program direction so you can have a anti commutator that is also non negative and from from the spatial direction I call it. magnetic magnetic field. And so you can define the data cooperator where the, the, the nodes, the node part follows practically the claim Gordon equation. And depending on the laplacian in time and in space commuting or anti commuting you can have different different discussion relation because this p square might not be the sum of px plus p y plus p z because if they don't commute is the Agen vector of their sum. And when they all commute you get the, the relativistic dispersion relation, and, and then I want to mention here, one calculation which is quite particular when you have the time and the space laplacian with not which, which commute but the space laplacian commute between each other. And then we have key that follows the claim Gordon equation psi follows the shreddinger equation with the general magnetic constant to and the game, the magnetic field is played by the by the anti commutator of the data cooperator this is why we call it magnetic fields and interesting enough. The key is non negligible is of order e to e minus m. So, we can write some action for the metrics are contractor at new new. And then we can try to minimize or contract it four times along, for instance a placate. And of course one can play the game and try to think, propose some equation, which would be an equation of motion of both the matter fields so the topological spinner and the weight. So this matrix a one a zero because these action is only determine in term of the way. So let's go back to complexity. Of course they're not only three dimensional lattices. Of course, in general in a graph you would like to have a gamma metrics the right algebra for any generic graph and this is a very challenging thing to do. If you applied the side one can have also multi layer networks in which you can have two nodes connected by two link only one link or only the other link in the other layer. So you can consider these three different type of links and treat them with the directional data cooperator. And in this case, the, the Laplacian do not commute with each other typically. So I reached my conclusion. Thank you so much for your attention. So, here we have covered a series of work on topological signals which are. I believe I hope you believe you see is an amazing objects. So they are dynamical variable not only associated to the note but also to the links. And they can undergo synchronization and this synchronization as a relation with topology and in particular with the homology of the simple shell complex. And when we want to treat topological signal of different dimension we can couple with the data cooperator. And this data cooperator as a lot of interesting use they can use in classical dynamics in a very wide set of scenario. But I think also exploring the fact that the data cooperator my propose a field theory in which the spinner is as a geometrical interpretation is quite interesting and of course, possibly this kind of idea could be maybe also simulated in in some scenario so just constructed on, I don't know, quantum quantum quantum material right to construct an artificial field theory of this type. Okay, so I need to thank all my collaborators. My group is rather small, I should admit, but these are people that have worked with me over the year. So, at each time, the team has been very small. And this is the selected recent reference. Thank you so much for your attention. Let me start. When you define the possible actions that you get for this spirit, I would have said okay one just take the trace of the black cat. And the which is kind of the square term. And the trace of that we know if you know locally on a single day which is the square term. Why don't you keep the simple choices that are also at this first time. So, yeah, so the history is that I took this, this fast to action fast. Yes, so these are along the placets. So this, this can can be a kind of curvature going along the placets. And this kind of electromagnetic like inspired expression. Because the other one. Yeah, so it's very strange because this f this commutator and anti commutator are quite interesting. And of course the anti commutator as, as the, the term in which the demo demo right that is known zero. But then, when you do a new new a new new. And when you go and do this calculation together is a bit like the electromagnetic interaction also is like you have also the magnetic part because because of this particular structure so practically what changes only the sign in between. And when you multiply this anti commutator with itself, essentially, then you have, you go from Y link to X links and then you need to contract with X links with Y links. And so you have only a sign of difference, maybe between the two, when you do the square of this one. It's very strange. Yeah, it is like they are strictly related to this to quantity so writing in term of the anti commutator is, it's not so different from writing in term of the commutator. Yes. Yeah, so. Okay, I let me repeat the question the question is that, if you have no synchronization. So you have this synchronization could amount of like, and then for a random graph, similar to kind of like, but then if the network have a geometry and a spectral dimension. Then this synchronization might be unstable and then you have this frustrated synchronization. And so we have work on this and we have seen also these uppers and spectral dimension and year for direct synchronization essentially we did not check carefully on simply shell complex with the spectral dimension it might be that when you have the complex synchronization. It might be that it is important not only the graph spectral dimension but also the simple shell complex spectral dimension. There might be an interplay because one can have spectral dimension both of graph and I wrote that I wrote the dynamics so there might be a way we are exploring that. Yeah, let me say that she will be around the week so in case you want to know. Thank you next again. Thank you.


I feel like we're doing everything we can to get better every week. Now, Friday nights with the high school kids, you never know what's going to come out of the box, so hopefully we'll be ready. It's like every week we work hard. It is Pikeville Week, so we are going to put a little more effort. Because the rivalry we have with the team down the creek. They understand last year we just weren't good enough. Pikeville's better than us, and sometimes you just got to understand that. We played hard and they won. And that's part of it. And this year it's a new year. And we're just trying to learn from every game and try to get better every week and see what happens. They're just another team, really. I mean, they're going to be good as they were last year. They're the number one team in class A, so we're just looking for revenge. There is a better taste in our mouth, and we're going to try to play really physical football and hopefully go with the win. Tiger fans come out and support us Friday 7.30 kickoff. Be there.


So, warm welcome to the 16th lecture on the subject of digital signal processing its applications. We will take a minute to recapitulate what we did in the previous lecture. We had been looking at the rational Z transform in the previous lecture and we had evolved different ways to deal with the inversion of the rational Z transform. Essentially, we had looked at partial fraction expansion and the use of the derivative. I mean the derivative of the Z transform. So, we multiply the sequence by n when you take the derivative essentially there is an association between multiplication of the sequence by the time index and taking a derivative in the Z domain. Of course, you know you do need to multiply by the factor minus Z after taking the derivative and so on, but there is a relationship between multiplying by n or the time index or a polynomial in the time index. So, polynomial in the time, so let us write this down. You see polynomial in the time index polynomial in time index n. Essentially, something of the form summation k going from 0 to say m a k n raise the power k multiplied by x n involves taking essentially a linear combination of the kth derivative. This corresponds to summation k going from 0 to m. Now, please note we should interpret this carefully what I am saying here minus Z d dz to the power k x Z. This means applying the operator first take the derivative with respect to Z and then multiply by minus Z together and this should be treated as an operator. And for example, if you are doing this twice if k is equal to 2 what you mean is take the derivative with respect to Z multiply by minus Z then take the derivative again with respect to Z and multiply by minus Z. So, apply the whole operator minus Z d dz twice that is how one should interpret it. So, this multiplied by B k some B k some constant. So, essentially there is a there is a one to one association in the Z domain between a polynomial in the time index multiplying the sequence and a polynomial in this operator minus Z d dz multiplying the Z transform operating on the Z transform. So, I have I have also asked you to put down this relationship precisely by using inductive reasoning. I have left it to you as an exercise to do that and based on this principle we could see that we could handle any Z transform here any rational Z transform based on this principle and partial fraction expansion we can handle any rational Z transform. One observation that we made last time was that you know the rational Z transform is kind of frequent you know currents that is why we said that we were spending so much of time on the rational Z transform. We shall take the next few minutes to understand why this is the case why is it that the rational Z transform is so important why does it why does it you know mean a lot to us. In fact, let us assume that we have a system described by a system function which is rational. So, for example, let us take a concrete system as an example where the transfer function or the system function is rational consider an LSI system with system function let us say 1-1 4th Z inverse divided by 1-1 3rd Z inverse into 1-1 ½ Z inverse this is capital H of Z is essentially the Z transform of the impulse response of that system. Now let us expand let us draw out HZ HZ can be rewritten as 1-1 4th Z inverse divided by 1-1 3rd plus half Z inverse plus 1 6 Z raised to power minus 2 which of course becomes 1-1 4th Z inverse 1-3 plus 2 makes it 5 by 6 Z inverse plus 1 by 6 Z raised to power minus 2. Now what is HZ HZ is the ratio of the output Z transform to the input Z transform in fact that is obvious because the input Z transform multiplied by HZ would give you the output Z transform XZ multiplied by HZ would give you YZ and therefore another way to understand the system function is the ratio of the output Z transform to the input Z transform this also tells us something about LSI systems. If you look at the ratio of the output Z transform to the input Z transform and if you find that this ratio is independent of the input then the system is LSI that is another way of looking at it. So in the Z domain the fact that the output Z transform divided by the input Z transform is independent of the input makes the system LSI alright. So in effect what we are saying is that HZ which is YZ by XZ in this case is 1-1 4th Z inverse by 1-5 by 6 Z inverse plus 1 by 6 Z raised to power minus 2 and we can of course cross multiply we can cross multiply to get YZ into 1-5 by 6 Z inverse plus 1 by 6 Z raised to power minus 2 is XZ into 1-1 4th Z inverse and we can go back to time go back to the natural domain by taking the inverse Z transform both sides. So of course it is very easy to see that you would get Yn here and here you would get minus 5 by 6 the inverse Z transform of Z inverse YZ is Y of n-1 essentially Y delayed by 1 and so on. So one can take the inverse Z transform on both sides we will denote this as the inverse Z transform you have Yn-5 by 6 Y of n-1 plus 1 by 6 Y of n-2 is Xn-1 by 4 Xn-1 and of course we can rewrite this. So you have you know what we want to do ultimately is to write Yn in terms of past values of Y and X and its past value. So we can keep Yn on one side and move all the other terms to the other side. So we have Yn is 5 by 6 Yn-1-1 by 6 Yn-2 plus Xn-1 by 4 Xn-1 and this is an example of what is called a linear constant coefficient difference equation it is abbreviated by L double CdE difference equation let us understand the terms one by one difference equation is a relation between the input and the output in the sampled natural domain. The difference equation is a relation between the discrete input and the discrete output the constant coefficient difference equation is a relationship where the constants that are involved where the multipliers that are involved are all constant. So here these are the multipliers 1-1 4 5 by 6 and – 1 by 6 all of them are constant that is why it is a constant coefficient difference equation and linear because it obeys this entire input output relationship obeys the principle of superposition. That means it obeys additivity and homogeneity and therefore this is called a linear constant coefficient difference equation. If you drop the terms one by one for example you could have a constant coefficient difference equation which is not linear for example suppose in this equation you brought in y of n-2 the whole squared then it would become a constant coefficient difference equation but not linear and so on right and there could be of course a linear equation but not constant coefficient for example some of these terms could depend on n here right. So each of these terms in the expression L double CdE is important now L double CdEs are very important you see these this kind of an equation is very fundamental to realizability. In fact let us spend a minute on seeing how you can realize this very relationship here that is how would you translate this into a hardware structure or a software structure we can see this right away. So in fact suppose you happen to have you know a circuit which constantly generates yn that means you have you know let us assume that you have a timing mechanism which times the samples the sampling points and the samples you know in every unit time one sample of the input is fed and one sample of the output is generated so you have a timing mechanism you have to have a timing mechanism. Now in that timing mechanism what you do is introduce what you call memory so you introduce a delay buffer which we will denote by D in fact introduce two of them so when you feed yn to a delay buffer what you get is y of n minus 1 that means a delay buffer stores the previous value of the output I mean stores the previous value of the input given to it so in this case it is yn but in general a delay buffer stores the previous value of the input given to it and of course here you would get y of n minus 2 and similarly you could have a delay buffer operating on the input and that would give us x of n minus 1. Now what does this equation tell me this equation tells me that I have taken y of n minus 1 multiplied it by 5 by 6 I have taken y of n minus 2 and multiplied it by minus 1 by 6 I have added these two I will show addition by a summation yeah I have added these two I have also taken the input and added to it minus 1 by 4 times the input delayed by one sample and I have added these two together and thus generated the output I have taken care here to allow only for what are called two input adders so we will assume that all the elements need to be of uniform structure so all the adds are of two inputs of course you could have additions with more than two inputs because addition is a commutative and associative operation you can add more than two at a time but we will assume for the sake of modularity and implementability that you use a two input adder. So what we have done is to realize this relationship this linear constant coefficient difference equation by using three kinds of elements delay buffers these d's multipliers constant multipliers and two input adders let us write that down we can realize realize means translate into hardware or and or software a linear constant coefficient difference equation and hence a rational system function a rational system function is realizable it is realizable because you can use a finite amount of hardware and software to translate it into an implementable structure finite it important what kind of hardware and software have we used here let us again list that down elements that are used one sample delay buffer which we denoted by D constant multipliers or scaling units two input adders these are three kinds of elements that we use of course you can very easily see that each of these elements can either be thought of as a hardware element or as a software element for example if you think about the one sample delay you could think of it as a little program statement which stores the previous value and keeps it at a memory location and of course in half way you can think of it as a register or a combination of registers similarly a constant multiplier can be thought of either as a hardware element or as a software element in software it is a one line multiplication by a constant in hardware essentially a multiplier implemented with any of the known multiplier algorithm constant multiplier of course and you know that there are very efficient ways of implementing multiplier if you know them to be constant multiplier for example one could use distributed arithmetic or any other efficient way of doing multiplication both algorithms you like right anyway so much so for the fact that and two input adders again we know how to do that the tools complement or any other structure right so we know how to implement it in hardware and of course in software again so one line statements all of these are really unit statements in software or unit elements in hardware modular elements in hardware and with a combination of them we can realize a rational system so rational system function is realizable now to contrast let us take an example of an irrational you see all this while we have been saying rational so the natural question that comes to mind is can you have an irrational system function so can a system function or can a z transform be rational and the answer of course has to be yes of course xz is e raised the power z inverse with the region of convergence mod z greater than 0 now note that here you cannot include the point z equal to 0 in the region of convergence in fact this is an example of what is called an essential singularity you know in some sense a singularity of infinite degree we will see in a minute now e raised to power z inverse is very easy to expand you know those of us all of us would have been exposed to Taylor series at some right the Taylor series expansion of e raised to power z inverse is essentially summation n going from 0 to infinity z inverse raised to power of n divided by n factorial with 0 factorial defined as 1 and n factorial defined as n into n-1 factorial this is well known this is essentially the Taylor series expansion of e raised to power of x now when we write down this Taylor series expansion it is very easy to identify what sequence corresponds to this xz that is very easy to do in fact it turns out that this sequence xn would simply be 1 by n factorial un where n factorial has been defined already and un you know un is the unit step in fact very interestingly if an LSI system were to have this impulse response suppose xn where the impulse response of an LSI system we could ask whether the system is causal and stable and it is very easy to see the system is of course causal because the impulse response is 0 for n less than 0 so of course if this were to be the impulse response of an LSI system the LSI system would be causal would it be stable how would we answer the question you would need to look at the absolute sum of the impulse response and in fact the absolute sum of xn is very easy to calculate so summation n over all integers mod xn is very easy to calculate it is essentially summation n going from 0 to infinity mod 1 by n factorial which is essentially 1 by n factorial because n factorial is positive and this is very easy to evaluate this is essentially erase the power of x evaluated at x equal to 1 this is e the standard neperian base and that is of course finite and therefore this is absolutely summable and therefore the system is stable so if a system were to have this impulse response it would be both causal and stable however erase the power z inverse can never be written as a ratio of 2 finite series in z it is impossible to write erase the power z inverse as a ratio of 2 finite series in z z or z inverse does not matter and therefore xz is irrational and concomitant with that is also that if this happen to be the system function of some LSI system which it can be nothing stops it from being the system function of an LSI system if it were that system is not realizable that means there is no finite hardware or at least no known at least today there is no known finite hardware or software structure which can realize this as of date irrational systems are unrealizable in hardware or software the rational systems are not see we must not confuse irrational with unstable or non-causal or something of the kind if rationality or irrationality for an LSI system is a property over and above the other property causality stability yes of course rationality has a meaning only for LSI system one cannot talk about rationality or irrationality when the system is not LSI in fact not only LSI it must be LSI and its impulse response must have a Z transform otherwise there is no sense in talking about rationality or irrationality so now we have a classification of LSI system you have the whole class of LSI system they are characterized by the impulse response among them you have a smaller class whose impulse response has a Z transform that means that the Z transform converges somewhere in the Z plane and among them you have Z transforms or LSI systems which have a Z transform in the Z transform is rational that is the innermost core which we are calling rational system and those in the system that we are going to design implement and build.


Sink Tech, Hawaii, Code Green. Welcome to all of you on a balmy Honolulu day, temperature 82 degrees, trade winds at 10 to 12 miles an hour who could get any more beautiful than that. And we're going to talk today about resource efficiency resource. I thought this was an energy efficiency book. No. No resource and energy efficiency go hand in hand because we're going to be talking about low income or houseless or housing for houseless people via shipping containers and think about it shipping containers are already built. You don't need to pour new steel, new concrete new anything into them you just modify the existing containers. So you don't have to chop down trees you don't have to go to the steel steel mill and build new steel structures. It's all embodied energy there. Not to mention just a wee bit of cost saving in the process, which is why we can offer homeless and low income housing. And to talk about that today is for Neema McCutcheon, whose name you doubtless recognize because she's the former president of a I a Honolulu. So welcome for Neema. Thank you. Thank you Howard. Pernima is going to talk about two projects one and award winning project in Los Angeles, and another innovative project on a small site in New York City. So by all means, take it away Pernima. Thank you. Thanks for that introduction. And yeah, I am pretty much a cut in a local architect. I was raised in India but I've lived on the islands since the 99 piece. I did spend about eight years overseas during that period in India and in Mongolia. And I feel like I might have a little bit of a global perspective to this very, you know this crisis that's looming all around us here and globally. So I thanks so much Howard and I wanted to take this opportunity really to share two projects that I've worked on. These are not projects that I worked on at work. These are projects actually design competitions that I did along with a friend. Robert McDonald R I B a he's, he's a well established residential architect in the UK. And with Lewis darn turn who helped us with our renderings and modeling all of in all of it for for the submissions for the competitions. The first one was right was one that we did in 2019 and it was in New York, and on a very small site, providing low cost housing and the one at the bottom, which I'm going to start by sharing first is for schedule in Los Angeles. If you, if you can move to the next slide please. The schedule. 2020 international design competition was sponsored by a committee on design as well as Los Angeles County. And we were actually honored to be awarded third place for our submission here. So for those that are not familiar schedule is basically a 54 block area and downtown Los Angeles that has become synonymous with homelessness and poverty. And on any given night, like several thousand people sleep on the streets out here. So for this one, the design brief was to provide transitional housing, as well as social services for the homeless community of skid row. And it was your provided this very narrow small, not very convenient lot, perhaps, and the idea was to provide a temporary structure for a lifespan of three to 10 years which can be disassembled. It was to be done sustainably cost efficiently. It was meant to be done fast because you know there's a crisis. And this was also kind of during COVID actually this competition was 2020. And so the need was very urgent so it was also supposed to be something that was replicable. And I think I mentioned that we were not actually given a program I don't think I mentioned it but we weren't given a program as to how many units to fit and how dense it should be, or even what social services were acquired we were tasked with figuring all of that out. So this is our solution. This is the street side we had the use containers and I'll go into that a little bit more. Maybe we go into the next slide. So on the lower part of the screen is what we call a site plan and on the right hand side. So that what's, you know, the boundary, the gray boundary around the adjacent properties and our properties in the center to the right is the street or the entry into the site. Now this, this and you can see the image on the top which is sort of a view looking at the facility from the street. Now on on the bottom side or on that image on the left hand side is an existing hotel it's called the Panama hotel it's a residential hotel. And then on the right is a lighthouse. I mean sorry a chapel. It's called the Susa Azusa lighthouse chapel and and to the rear of the site as a warehouse. But, but this is what we would call a zero lot line development because the properties adjacent to these are everybody's allowed to build right to the lot line, and it is zoned so that you can go up to six stories high. We could be pretty much locked within the site with no no light, you know, and ventilation. So our design. We basically ended up doing something that was inward looking and created courtyards, and we avoided long corridors we ended up having a very simple circulation between these, you know, the residences services. So everything like we allowed for natural ventilation to happen easily that way and provided people a nice pleasant area to look at rather than looking at just walls around, you know, around them. So, so as I was saying, I was going to talk a little bit more about why we chose containers. So we reviewed a variety of options. We reviewed, you know, going with traditional construction we also looked at purpose built containers that you know they build it exactly to what your specifications are. But then we finally settled on repurpose containers because partly because there are 11 million unused containers in the world. It just seems more environmentally sensitive to try to reuse that it also creates more employment if you can reuse it, you know, you're kind of trying to take care of multiple problems at the same time, but it also would be initial lower initial cost would be lower labor cost would be a little bit higher than purpose built, but, but it's good for us to have labor, and then it's durable it's strong. And so why not. So that's kind of we came up with the idea to use containers and moving on to the next slide. So this, this plan basically shows a typical plan. I'm going to have a better plan for you later but this plan basically is from levels two to six, and it actually makes it easier for me to explain the upper floor plans because they're repetitive, and then move to the lower floor plan. But so in this diagram I've got a bunch of units on the right hand side which is repeatable pod a the blue pod and then you have the pink pod. The blue pod has the units that face the the street, they're like shotgun units, you know parallel to the street with balconies overlooking the street. And then it has, it has like a central, actually both of them have a central core where you have the stairs and the gray areas is your elevator block, and all of the white areas are basically open to the elements on, at least on one side. Okay, so we decided to do have two pods and smaller, you know corridors rather than a long corridor that went from one end of the site to the other, but also because these pods can be then replicated on a variety of different sites.  So these become with some tweaking they can be reused in different places. So that's, that's like one of the key takeaways is like we can replicate something then we can really bring the costs down. So anyway so that with that I can move on to the next one. And per name and let me point out that LA has a relatively mild climate. So you're taking advantage of that mildness by opening up absolutely as much as you can. And that certainly reduces energy consumption also, and gives people access to fresh air and fresh views. Exactly. And applicable to Hawaii too. Okay, so this one again is a more detailed plan showing the, all of the, the level of the residential levels which is levels from two to six. And so again the, the gray areas are all of our units and the green areas are the courtyard, the white areas are stairs and corridors. There's little orange, or in this screen it's showing up as little pink rectangles here and there. There are little benches where people can kind of sit and engage with whoever's passing by. We also have. It's hard to see on this drawing but we have common laundry facilities on each floor. So it allows for social interactions, you know, amongst the residents to kind of help create a community. So that was one of our thinking and then all of the orange arrows are basically arrows through windows doors. It's basically showing how views, you know, how, how we can create some sort of a natural surveillance. So the residents are always looking on to some sort of a common area from their units. And so that you know we don't have these. It's to create some sort of a natural security. You know, people will never be in some sort of a dark corner or you know where they might feel threatened there's always some natural oversight by the residents. In fact, the courtyard at the, to the left of the screen. That's actually a kids play area it's kind of secured but you know you always have people looking on or the ability to look on from upper floors. So you know that if there's any any situation, you can always call security right away and take care of it. So security and you know, a place where people really feel safe and joyful and happy was really important to us. I guess we could move on to the next slide. All right, so maybe we can zoom in a little bit on this one. So we provided basically for unit types and each each unit type has its own bathroom we thought it was important for them to have the privacy of their own bathrooms. So even though this this one unit is just a single container it's unit type a. And actually before I go on to it's the specific unit type I also want to mention that all the units are acoustically internally insulated. They're not provided with air conditioning they provide the intention is to really create spaces that are well ventilated naturally and with ceiling fans and lit also with natural daylight. So in this particular unit. This is type a it's for we think it would be more for like folks that are chronically homeless so we haven't provided kitchen services and this we have common kitchen area on the lower floor which I'll show you later. And it does have a bathroom it has some storage areas a bed, little, like you know little places sit out and then a balcony, and the drawing on the right shows all of the units kind of lined up on each floor like you can have six units, and the balconies are like a little bit funky, you can you know add, you know, variety of play of in and out zigzag up and down so you can create some fun without an interesting design without really spending a lot of money on it. We don't see that as an expensive element. And then the drawing in the middle which we can't see too well but basically it's a sketch showing how a small space can be really organized well to, you know, provide you with the high functionality. So, you know, storage built in and space, like that brown area you know where we have a lower sort of ceiling where you can, you know, put some bags and things so just efficient planning can go a long way in making a small container be much more, you know, usable. Um, I think we could, let's see if there's anything else. Yeah, I think that was it for this one. And so we can move to the next slide. So like, so we basically had four unit types and that was unit type a, and then here in the blues is the ADA units which we have on level one of one of the pods. And the others are unit type B and C and just very briefly. The idea is that they kind of like Lego pieces kind of go together. So the ADA, for example, there's a unit on the left and the right but the bathrooms are central. And so they can be stacked and it's less expensive when you do something like that. There's a lot more turning around space and, you know, make very comfortable, much more comfortable for somebody who might be in a wheelchair or have other accessibility issues in the ADA unit. But it can accommodate up to two adults and a young child on a pullout bed. So I mean this, you know, that's your unit type ADA and then the one in the pink is quite similar but with more, more. It doesn't have to be as tight. So, you know, we have more dining area and things like more closet like that. But very similar and they could be converted either or depending on if you need more, more ADA units, you could convert the pink ones into more ADA units. And then the bottom one, the purple one is actually for a larger group like maybe a family or, you know, maybe four or five people living together.  All of these units, like these three, unlike the type A half kitchenette so you can cook as well if you want to. And it has a children's bedroom as well, the one, the purple one.  So I mean it allows for more. So we have these different varieties of unit types so it allows for, you know, many, many groups of users to be able to use this facility. I think, oh, I, and I think I pointed out that earlier, you know, the areas between the two units have the ability to either have like, you know, washer dryer or a bench or so. So we are using anyways, every little bit of space that we can to provide, you know, to think of as many things as we think they need amenities and facilities. So with that, I can go on to the next slide. I hope, Howard, this is fine. You've got about eight minutes per name. Oh, okay, I didn't realize. Okay, so we will move faster than so we used about 126 containers 111 were used for housing about 158 adults 59 children 15 were used for supportive services and we move on to the next slide. So I'm just going to start by zoning and I just want to quickly point out so this is level one on the right hand side of the street. It's the most complicated level because we have residents up on the upper floors but they need to be entering safely and securely, but we also wanted to provide services for others that aren't so fortunate to be able to sleep in inside. That are still on the streets. So the bottom right orange is where we have a bathroom or toilet facilities and drinking water facilities and that would be available to anybody off the street 24 seven. And the little light blue area above it is a security information reception for the sort of secure areas, but also place where you can find out where you could go for certain services. So it's a peach or light pink band above. It has a kitchenette where up in the entry like a dining area where you can pick up your trees and pick up food and then go into the, you know, into these common areas where anybody off the street anytime is able to come in at scheduled times for a meal like breakfast lunch dinner. And then in the middle the dark pink area is where you have counseling offices job training and the like. And then and supportive services offices and then the yellow area is also a place where people can get like their, you know, some podiatry, some hair care and showers, and that would be like within certain like you know 12 hours of the day scheduled so that you can accommodate a lot of people. And purple and blue areas are the secure residence areas and they have different ways to get in there so you're depending on your comfort level if you want to go through the purple one you can go through there you can go from the outside. And I think that was it in a nutshell, I mean it's very complex and we just want to make sure everybody's safe but we want to provide maximum services. And Pernima you do have, I guess 24 seven security. Yes, there is yeah. Yes. Yeah, so it would be required, I think, and, you know, at least at the early stages. Okay, and then we can go on to the next one which I just the next slide. Okay, I just want to point out that you know we use very simple methods of getting an interest in the design a little bit of perforated screen at the bottom for like light and shadow. Very balcony is some pop of color and you know inexpensive method but we still believe in the power of design to heal. We believe in like you know making a building really lovely for the users. With that I can go to the next one. So this one actually. So this, this one is in New York City. And this was actually a really good project project basically what the city of New York did is they identified 23 sites. They gave us very narrow and inconvenient lots, and they want, they gave us one site to design to but they want that design to be flexible so that it can be applied to 23 sites, and perhaps more later. So on the top right you see a little image with, you know, two buildings with a narrow sliver in the middle. Well that's 17 feet wide and about 100 feet deep. And that's your site. I don't know if you can picture 17 feet I mean out of that you lose, you know, two feet just for the walls 15 feet it's it's it's like maybe somebody's, you know, dining area or living area in the house or something very narrow. But we managed to put eight units on this lot. And we had some height restrictions and all but we have an entry level lobby and 8888 studio on the floor floor, and we have a bunch of studios upstairs but we have a little roof deck amenity with some, you know, gardening opportunities for the community. Anyhow, so I'm going to go on to the next slide and explain it a little bit more. So this is what it looks like when you cut a section through the building so you can see the lobby at the bottom. One of the strategies we used was like to go split split level. So we can keep the core that's in yellow right now. Very small and give more space to the units on the small side so it's a strategy we use and then we also wanted to get a lot of light in so we have translucency to that elevator core so you know so the light from the balconies on the outside. So we have, we have means to bring it all the way inside and you see that on the next slide. But we have photo will take some of the roof out there, I should point out but anyways, and so in this. This is a detailed plan on the lower part of the drawing and upper part of the drawing is a section through one of the units. So you can see that the light is coming in through the balcony side and bouncing all the way in because we didn't go. I mean these are studios we decided not to go with walls all the way up to the ceiling, and we also have little, you know, glazed areas between the, the, you know, in the walls, so that the light bounces all the way back to the corridor and then, or the core and then you have light coming in from there so very simple strategy to have natural daylight throughout. So I think with that I'll just move on to let me let me point out, pretty much the benefits of daylighting these are low income people. Yeah. And this is for low income. I should buy this project is for low income group. And they might have been living in very cramped quarters before with very little daylight, you're bringing daylight in, and you are automatically improving their quality of life. And just with this daylight all over. And from an energy efficiency standpoint, you can't light a space much more efficiently than with daylight. Right. That was our idea. I want to point out another thing on that same slide. In the New York project we did not use actually if you want to zoom in. I want to just point out there's a diagram on the right that shows how, how we conceived this so this one was not actually designed with using containers but rather modules that could be lifted into position. And then these modules can be, you know, you can have like the one that has the bathroom module and then intermediate modules all the way up to the street side. So the core would have to be built separately, also as modules. But so we kind of, you know, figured out what, what sort of sizes, what trucks can bring into the city and you know how they can be delivered and all of that. So this is a different setup. Moving on to the next slide. So this is what it looks like very simple, but each unit has a balcony so that the community can engage with what's going on in the street you want to make sure that people can really engage it's really important. And then we have, you know, all of these houses have old houses have stoopes, but we put a little reinterpreted stoop at the bottom, where people within the community can also hang out and again interact with with whoever's passing by so that was kind of important. Yeah, very simple. Going on and of course we use reclaimed materials and all not, you know, all the environmental. Yeah, two minutes. Okay, I'm on my last slide I think. Yeah, one more. Okay, so this one is basically. Yeah, they gave us 23 sites I've shown about six of them and you can see the dark blue on the gray areas those are the sites they're super narrow and small. Lots, but we were able to show demonstrate how we could create with like to like varying to widths, how we could replicate this solution on 23 sites. And, you know, successfully like use this idea of modularity and I think my, you know, if I have to emphasize on anything throughout this presentation that would be the key is like, if we can create a prototype and test it out it's going to take a little bit of effort and and cost to do that. But if we can then replicate it with tweaking, you know, still providing great design on a variety of sites, I think that, you know, modular and replicability is going to be what we can use to sort of solve some of our crisis. Like affordable housing housing home for the house list. So, sorry, I took all my time. No, no, I'll just jump in with a couple of comments you use the word replicability several times, and prior to 1912 automobiles were very, very expensive only a thing for the rich, then came along Henry Ford, and said I'm going to build an automobile that the workers can afford, and you have the same print Oh, and the secret there is replicability you build every car exactly the same way your cost goes way way down. And you are doing replicability in the form of modular housing where it's all built the components are built in a factory and you just have cranes. You put them all together and that really, really brings the cost down, and you are afforded a high quality material because it's these things are built in the factory. And, and it's also wonderful looking I made a, I never thought that I would dream of living in a home for for a very low income people but this looks like a little mini dream home that you created here pretty much.  Yeah, you use a wonderful phrase there, the power of design to heal. That's obviously what you've got in mind here. And I know you're just starting out. This is going to be a new career for you for Neema and you are definitely serving humanity, and that portion of humanity that most needs to get in this case off the streets are out of miserable housing, and into housing that will increase their sense of well being and hopefully their sense of productivity their health, the whole nine yards. And we all see homeless in the streets we see low income people, and you are doing your bit to make the world a better place for Neema so I thank you and appreciate you very much. And that very cherry note we must bid fond you to think take away cold green green Neema McCutcheon and Howard wig. See you next time. Mahalo.