 Angelo, are you online? Hello, yes, I am. So I think it has enough lines. Reza, you have to stop sharing your. Yeah. Yeah. Yeah. Oh, sure. Sorry. OK, so Angelo, can you share your slides? Yes. Can you see it? So we see you, not your slides yet? Uh-huh. OK. OK. So go on. Start. Yeah. OK. So hello, everyone. Good morning, good afternoon. I am Angelo Marco-Ramosa from the National Institute of Physics, UP, University of the Philippines, Wilhelmine, from the Philippines. So the talk I will give for 20 minutes is entitled First Passage Characteristics of Resetting Bias Diffusion with Mixed Absorbing and Reflecting Boundaries. So since it's only a 20-minute talk, we'll just cut some stuff. And we'll try to manage. Angelo, I'm sorry I'm interrupting you. It's a 50-minute talk, actually. Oh, 15 minutes. Uh-huh. Yeah. So after talk. OK. Go on. So from this title, since it's a diffusion, we know a diffusion has so many things to talk about. You have, it can be anything. So from the title, we'll narrate down to what I'm going to talk for 15 minutes. So this is the title. So from the title, first we have boundaries. So the boundaries, let's say, for this talk, it will be only a one-dimensional lattice. And then it's just limited from x equal to 0 to x equal to 1. And then we also have absorbing and reflecting boundaries. So here, we set x equal to 0 to be an absorbing boundary. While x equal to 1, the reflecting boundary. Also, in this talk, we also included, we're going to talk about the first class characteristics, such as the mean and the variance, since mean and variance can show the behavior of your distribution. So from here, the implications are, since we have an absorbing boundary at x equal to 0, the limit of our distribution at any time when the space approaches that boundary x equal to 0 should be 0. While the reflecting boundary at x equal to 1, the implication of this is the flux at x equal to 1 should be 0 at any time p. Also, since we included the setting, in this talk, our variable for setting is small r. And then our random walker will reset to a position, say, x0. Also, we included a bias. And then in here, the bias will be indicated by the variable v. So the plan to solve the first passage character 6 is, so first, how do we solve for the first passage or the mean first passage time? So in papers, we read that so that they can solve for this, their process is, first, they solve the probability distribution with the setting, and then compute the first passage time with the setting, and then get the mean first passage. But this, for me, it's not intuitive. And then I'm going to show a more, I say this, friendly approach. So for me, I will start with the distribution, and then solve for the last transform of it, our probability distribution, and then compute the first passage time without the setting. And then later on, we include the setting onto it, then solve for the mean first passage time so that we can see the characteristics of the first passage. So in this process, my plan, we have milestones. So first, we need to identify the evolution of the probability. So in here, we start with this. And then after we identify the evolution of our probability, we need to get the probability rate. After this, we will move on to the last transformative probability distribution. And then from here, we can get easily the last transform of the first passage. And then in this talk, the main idea in here, or my contribution, or such, is we include their setting from a non-resetting first passage. Later, I will show how it is related. And then after this, we can now get the moments, like mean and variance. So in here, in this process, in these milestones, it is much easier since we are going just to take the derivatives and not to take integrations. And we know it's easier to take integratives than integrations. So the mathematical methods for this is first, from our evolution of probability, we're going to use Taylor expansion to get the probability rate. And then after to get the, we will take the Laplace transform with the application of our conditions, boundary conditions, continuity conditions, and initial conditions. We will get the Laplace transform. After that, there's a definition of first passage that will connect the Laplace transform of probability distribution to the first passage time distribution. And then we will add the setting. Then there's a definition of moments from this. So can I interrupt you for a second? Can you just, I may have missed this, but what does resetting correspond to physically in X or S space? Sorry. So resetting means, so since we have a random walker, it means that while it traverses the region, there's a probability that our random walker will be teleported into a resetting position. In this talk, it will be X naught. Here, the setting means our random walker, the only probability will go to the X naught. I see. Okay. Yes, so it's a sudden jump. He says teleportation, but it's not the same. Okay, okay, I see. In this talk, we'll be teleportation. Well, some you can set actually a traversing time, the setting time, traversing the setting time to go to that position. So there are papers there, you can look for them and then you can read about them. So here, so let's start in the solving for this. So the evolution of the probability. So for example, at the bottom of the screen, you can see that there's a sample of a lattice and then the blue dot is our position X. So the evolution of probabilities means that from time T, what will happen to our random walker from T to time T plus DT? So in here, as you can see, all the P's are our probability at a position X and then T and for time T and then T plus DT, which is the reference for the next time step. So the things that are not common here are the case. So example, this K, K plus at location X minus HT is the probability of a random walker from X minus H, where H is the spacing size or spacing step, X minus H. So the K plus is the probability of it moving to the right to going to position X, while K minus X plus HT is the probability of a random walker at X plus H to move to position X at time T. So as you can see, at the position X, there's an increase of probability or increase of numbers of random walker going to there, since you can go from X minus H and X plus H to go to position X. Furthermore, the probability will decrease since the random walker can move from PXT, the position X, leaving it. So if the random walker, it can go from position X, it can go to the left given by the probability K minus or probability rate K minus and then probably K plus for moving to the right. So this is the, for me, this is more intuitive because they can look at a lattice and then think what you want to have or like a recipe or the ingredients you want so that if it can move two steps, you can add more terms in here. So from here, from our milestones, we tell you expand this. So we tell you expand this on DT and H and we'll have this. Well, I omitted the higher orders of DT squared and H cubed, okay? So from here, it's kind of not common. It's very uncommon. But in the next slide or next equation or there's a simpler notation such that this equation seven is the more and the equation eight are the more common equation we know, okay? So again, here, V is our variable that shows bias and D is our diffusion coefficient, okay? So we have a homogeneous media, yes? Anjalo, I'm sorry for the interruption. We are almost at the 10 minute mark. So you are five minutes flexible but leave some time for questions, thank you. Okay, so we have a homogeneous media. So V and D are just constants. Now, so our conditions I said earlier are boundary conditions which are given by nine and 10 which I said earlier, initial condition equation 11 we said that say the random variable start at the resetting position. We can put it anywhere else but for here, let's set it at the resetting position and then we have a continued condition. So since we have two different mid boundary it's wise to cut your probability distribution into two so that we're here it's P less than and P greater than so that you can apply a boundary condition on one say, which is the P at X equal to zero and then another function P greater than where you apply boundary condition at X equal to one, okay? So when you use less transfer we will get this equation, okay? So in here, we have a some, constant of, so this is a constant of X, this some value C. Next, after that we solve for the first passage. So first passage is just the flux wherein we can think of the first time the run of Walker Beach at position X. Here, we want to see at X equal to zero and we'll get this equation. Now, to get the first passage probability with the setting with this paper by Pal we obtain this equation 18 where the relationship of first passage time without the setting and first passage time with the setting we obtain this equation such that we will be able to get our the characteristics of the first passage. So the end moment is given by this one. So equation 20 is the definition and the equation 21 is using the our previous equation. So from this we will be able to get the first passage characteristics mean the first moment. It is given by this equation. So it is actually very long. So we use equation 23 to simplify our equation, okay? And then this is the second moment, the variance. We will able to have this equation. So we have three more large functions of capital C, capital gamma and capital lambda where they scale as equation 25. So example of our first passage of the mean is given by this. So where the y-axis is the first moment the mean first passage time and then the x-axis are resetting rate R, okay? So in here we can see that the difference of the colors are the different V and then the D we just set equal to one. So in here we can see that the bias increases when you increase or rather the first moment increases as to increase the bias, which makes sense because the bias when it's positive it goes to the right. So if the random worker wants to move to the right going away from our absorbing boundary it will take more time. And also in this plot it shows that the points the huge points in the plot are the minimum point at this region. We can see that it is different for a different V and also as you can see in the shape of the plot there are some resetting part, example here. So Joel we have less than one minute you have to wrap it up, okay, go on. So in here it takes more time to go to the absorbing boundary than if you increase the resetting rate it will take less time and then if you go further it will take again more time. So there's an optimal resetting rate on this such. So that's all of my talk. So thank you for listening. Lots of love from the fieldings. Thank you. Well, thank you very much. It's nice we have such a diverse range of topics for today afternoon. We don't have time for questions again but if you have a question please.