Instructor (Andrew Ng):Okay, good morning. Welcome back. So I hope all of you had a good Thanksgiving break. After the problem sets, I suspect many of us needed one. Just one quick announcement so as I announced by email a few days ago, this afternoon we'll be doing another tape ahead of lecture, so I won't physically be here on Wednesday, and so we'll be taping this Wednesday's lecture ahead of time. If you're free this afternoon, please come to that; it'll be at 3:45 p.m. in the Skilling Auditorium in Skilling 193 at 3:45. But of course, you can also just show up in class as usual at the usual time or just watch it online as usual also.
Just to recap, right, in the previous lecture, I defined the Reinforcement Learning problem and I defined MDPs, so let me just recap the notation. I said that an MDP or a Markov Decision Process, was a ? tuple, comprising those things and the running example of those using last time was this one right, adapted from the Russell and Norvig AI textbook. So in this example MDP that I was using, it had 11 states, so that's where S was. The actions were compass directions: north, south, east and west.
The state transition probability is to capture chance of your transitioning to every state when you take any action in any other given state and so in our example that captured the stochastic dynamics of our robot wondering around [inaudible], and we said if you take the action north and the south, you have a .8 chance of actually going north and .1 chance of veering off, so that .1 chance of veering off to the right so said model of the robot's noisy dynamic with a [inaudible] and the reward function was that +/-1 at the absorbing states and -0.02 elsewhere. This is an example of an MDP, and that's what these five things were. Oh, and I used a discount factor G of usually a number slightly less than one, so that's the 0.99. And so our goal was to find the policy, the control policy and that's at ?, which is a function mapping from the states of the actions that tells us what action to take in every state, and our goal was to find a policy that maximizes the expected value of our total payoff. So we want to find a policy. Well, let's see. We define value functions Vp (s) to be equal to this. We said that the value of a policy ? from State S was given by the expected value of the sum of discounted rewards, conditioned on your executing the policy ? and you're stating off your [inaudible] to say in the State S, and so our strategy for finding the policy was sort of comprised of two steps. So the goal is to find a good policy that maximizes the suspected value of the sum of discounted rewards, and so I said last time that one strategy for finding the [inaudible] of a policy is to first compute the optimal value function which I denoted V*(s) and is defined like that. It's the maximum value that any policy can obtain, and for example, the optimal value function for that MDP looks like this. So in other words, starting from any of these states, what's the expected value of the sum of discounted rewards you get, so this is V*. We also said that once you've found V*, you can compute the optimal policy using this.
And so once you've found V*, we can use this equation to find the optimal policy ?* and the last piece of this algorithm was Bellman's equations where we know that V*, the optimal sum of discounted rewards you can get for State S, is equal to the immediate reward you get just for starting off in that state +G(for the max over all the actions you could take)(your future sum of discounted rewards)(your future payoff starting from the State S(p) which is where you might transition to after 1(s). And so this gave us a value iteration algorithm, which was essentially V.I.
I'm abbreviating value iteration as V.I., so in the value iteration algorithm, in V.I., you just take Bellman's equations and you repeatedly do this. So initialize some guess of the value functions. Initialize a zero as the sum rounding guess and then repeatedly perform this update for all states, and I said last time that if you do this repeatedly, then V(s) will converge to the optimal value function, V*(s) and then having found V*(s), you can compute the optimal policy ?*.
Just one final thing I want to recap was the policy iteration algorithm in which we repeat the following two steps. So let's see, given a random initial policy, we'll solve for Vp. We'll solve for the value function for that specific policy. So this means for every state, compute the expected sum of discounted rewards for if you execute the policy ? from that state, and then the other step of policy iteration is having found the value function for your policy, you then update the policy pretending that you've already found the optimal value function, V*, and then you repeatedly perform these two steps where you solve for the value function for your current policy and then pretend that that's actually the optimal value function and solve for the policy given the value function, and you repeatedly update the value function or update the policy using that value function. And last time I said that this will also cause the estimated value function V to converge to V* and this will cause p to converge to ?*, the optimal policy.
So those are based on our last lecture [inaudible] MDPs and introduced a lot of new notation symbols and just summarize all that again. What I'm about to do now, what I'm about to do for the rest of today's lecture is actually build on these two algorithms so I guess if you have any questions about this piece, ask now since I've got to go on please. Yeah.
Student:[Inaudible] how those two algorithms are very different?
Instructor (Andrew Ng):I see, right, so yeah, do you see that they're different? Okay, how it's different. Let's see. So well here's one difference. I didn't say this 'cause no longer use it today. So value iteration and policy iteration are different algorithms. In policy iteration in this step, you're given a fixed policy, and you're going to solve for the value function for that policy and so you're given some fixed policy ?, meaning some function mapping from the state's actions. So give you some policy and whatever. That's just some policy; it's not a great policy. And in that step that I circled, we have to find the ? of S which means that for every state you need to compute your expected sum of discounted rewards or if you execute this specific policy and starting off the MDP in that state S.
So I showed this last time. I won't go into details today, so I said last time that you can actually solve for Vp by solving a linear system of equations. There was a form of Bellman's equations for Vp, and it turned out to be, if you write this out, you end up with a linear system of 11 equations of 11 unknowns and so you can actually solve for the value function for a fixed policy by solving like a system of linear equations with 11 variables and 11 constraints, and so that's policy iteration; whereas, in value iteration, going back on board, in value iteration you sort of repeatedly perform this update where you update the value of a state as the [inaudible]. So I hope that makes sense that the algorithm of these is different.
Student:[Inaudible] on the atomic kits so is the assumption that we can never get out of those states?
Instructor (Andrew Ng):Yes. There's always things that you where you solve for this [inaudible], for example, and make the numbers come up nicely, but I don't wanna spend too much time on them, but yeah, so the assumption is that once you enter the absorbing state, then the world ends or there're no more rewards after that and you can think of another way to think of the absorbing states which is sort of mathematically equivalent. You can think of the absorbing states as transitioning with probability 1 to sum 12 state, and then once you're in that 12th state, you always remain in that 12th state, and there're no further rewards from there. If you want, you can think of this as actually an MDP with 12 states rather than 11 states, and the 12th state is this zero cost absorbing state that you get stuck in forever. Other questions? Yeah, please go.
Student:Where did the Bellman's equations [inaudible] to optimal value [inaudible]?
Instructor (Andrew Ng):Boy, yeah. Okay, this Bellman's equations, this equation that I'm pointing to, I sort of tried to give it justification for this last time. I'll say it in one sentence so that's that the expected total payoff I get, I expect to get something from the state as is equal to my immediate reward which is the reward I get for starting a state. Let's see. If I sum the state, I'm gonna get some first reward and then I can transition to some other state, and then from that other state, I'll get some additional rewards from then. So Bellman's equations breaks that sum into two pieces. It says the value of a state is equal to the reward you get right away is really, well. V*(s) is really equal to +G, so this is V*(s) is, and so Bellman's equations sort of breaks V* into two terms and says that there's this first term which is the immediate reward, that, and then +G(the rewards you get in the future) which it turns out to be equal to that second row.
I spent more time justifying this in the previous lecture, although yeah, hopefully, for the purposes of this lecture, if you're not sure where this is came, if you don't remember the justification of that, why don't you just maybe take my word for that this equation holds true since I use it a little bit later as well, and then the lecture notes sort of explain a little further the justification for why this equation might hold true. But for now, yeah, just for now take my word for it that this holds true 'cause we'll use it a little bit later today as well.
Student:[Inaudible] and would it be in sort of turn back into [inaudible].
Instructor (Andrew Ng):Actually, [inaudible] right question is if in policy iteration if we represent ? implicitly, using V(s), would it become equivalent to valuation, and the answer is sort of no. Let's see. It's true that policy iteration and value iteration are closely related algorithms, and there's actually a continuum between them, but yeah, it actually turns out that, oh, no, the algorithms are not equivalent. It's just in policy iteration, there is a step where you're solving for the value function for the policy vehicle is V, solve for Vp. Usually, you can do this, for instance, by solving a linear system of equations. In value iteration, it is a different algorithm, yes. I hope it makes sense that at least cosmetically it's different.
Student:[Inaudible] you have [inaudible] representing ? implicitly, then you won't have to solve that to [inaudible] equations.
Instructor (Andrew Ng):Yeah, the problem is - let's see. To solve for Vp, this works only if you have a fixed policy, so once you change a value function, if ? changes as well, then it's sort of hard to solve this. Yeah, so later on we'll actually talk about some examples of when ? is implicitly represented but at least for now it's I think there's – yeah. Maybe there's a way to redefine something, see a mapping onto value iteration but that's not usually done. These are viewed as different algorithms.
Okay, cool, so all good questions. Let me move on and talk about how to generalize these ideas to continuous states. Everything we've done so far has been for discrete states or finite-state MDPs. Where, for example, here we had an MDP with a finite set of 11 states and so the value function or V(s) or our estimate for the value function, V(s), could then be represented using an array of 11 numbers 'cause if you have 11 states, the value function needs to assign a real number to each of the 11 states and so to represent V(s) using an array of 11 numbers. What I want to do for [inaudible] today is talk about continuous states, so for example, if you want to control any of the number of real [inaudible], so for example, if you want to control a car, a car is positioned given by XYT, as position and orientation and if you want to Markov the velocity as well, then Xdot, Ydot, Tdot, so these are so depending on whether you want to model the kinematics and so just position, or whether you want to model the dynamics, meaning the velocity as well.
Earlier I showed you video of a helicopter that was flying, using a rain forest we're learning algorithms, so the helicopter which can fly in three-dimensional space rather than just drive on the 2-D plane, the state will be given by XYZ position, FT?, which is ?[inaudible]. The FT? is sometimes used to note the P[inaudible] of the helicopter, just orientation, and if you want to control a helicopter, you pretty much have to model velocity as well which means both linear velocity as well as angular velocity, and so this would be a 12-dimensional state.
If you want an example that is kind of fun but unusual is, and I'm just gonna use this as an example and actually use this little bit example in today's lecture is the inverted pendulum problem which is sort of a long-running classic in reinforcement learning in which imagine that you have a little cart that's on a rail. The rail ends at some point and if you imagine that you have a pole attached to the cart, and this is a free hinge and so the pole here can rotate freely, and your goal is to control the cart and to move it back and forth on this rail so as to keep the pole balanced. Yeah, there's no long pole in this class but you know what I mean, so you can imagine. Oh, is there a long pole here?
Instructor (Andrew Ng):Oh, thanks. Cool. So I did not practice this but you can take a long pole and sort of hold it up, balance, so imagine that you can do it better than I can. Imagine these are [inaudible] just moving back and forth to try to keep the pole balanced, so you can actually us the reinforcement learning algorithm to do that. This is actually one of the longstanding classic problems that people [inaudible] implement and play off using reinforcement learning algorithms, and so for this, the states would be X and T, so X would be the position of the cart, and T would be the orientation of the pole and also the linear velocity and the angular velocity of the pole, so I'll actually use this example a couple times.
So to read continuous state space, how can you apply an algorithm like value iteration and policy iteration to solve the MDP to control like the car or a helicopter or something like the inverted pendulum? So one thing you can do and this is maybe the most straightforward thing is, if you have say a two-dimensional continuous state space, a S-1 and S-2 are my state variables, and in all the examples there are I guess between 4-dimensional to 12-dimensional. I'll just draw 2-D here. The most straightforward thing to do would be to take the continuous state space and discretize it into a number of discrete cells.
And I use S-bar to denote they're discretized or they're discrete states, and so you can [inaudible] with this continuous state problem with a finite or discrete set of states and then you can use policy iteration or value iteration to solve for V*(s)-bar and ?*(s)-bar. And if you're robot is then in some state given by that dot, you would then figure out what discretized state it is in. In this case it's in, this discretized dygrid cell that's called S-bar, and then you execute. You choose the policy. You choose the action given by applied to that discrete state, so discretization is maybe the most straightforward way to turn a continuous state problem into a discrete state problem.
Sometimes you can sorta make this work but a couple of reasons why this does not work very well. One reason is the following, and for this picture, let's even temporarily put aside reinforcement learning. Let's just think about doing regression for now and so suppose you have some invariable X and suppose I have some data, and I want to fill a function. Y is the function of X, so discretization is saying that I'm going to take my horizontal Xs and chop it up into a number of intervals. Sometimes I call these intervals buckets as well. We chop my horizontal Xs up into a number of buckets and then we're approximate this function using something that's piecewise constant in each of these buckets. And just look at this.
This is clearly not a very good representation, right, and when we talk about regression, you just choose some features of X and run linear regression or something. You get a much better fit to the function. And so the sense that discretization just isn't a very good source of piecewise constant functions. This just isn't a very good function for representing many things, and there's also the sense that there's no smoothing or there's no generalization across the different buckets. And in fact, back in regression, I would never have chosen to do regression using this sort of visualization. It's just really doesn't make sense.
And so in the same way, instead of X, V(s), instead of X and some hypothesis function of X, if you have the state here and you're trying to approximate the value function, then you can get discretization to work for many problems but maybe this isn't the best representation to represent a value function. The other problem with discretization and maybe the more serious problem is what's often somewhat fancifully called the curse of dimensionality. And just the observation that if the state space is in RN, and if you discretize each variable into K buckets, so if discretize each variable into K discrete values, then you get on the order of K to the power of N discrete states. In other words, the number of discrete states you end up with grows exponentially in the dimension of the problem, and so for a helicopter with 12-dimensional state space, this would be maybe like 100 to the power of 12, just huge, and it's not feasible. And so discretization doesn't scale well at all with two problems in high-dimensional state spaces, and this observation actually applies more generally than to just robotics and continuous state problems. For example, another fairly well-known applications of reinforcement learning has been to factory automations. If you imagine that you have 20 machines sitting in the factory and the machines lie in a assembly line and they all do something to a part on the assembly line, then they route the part onto a different machine. You want to use reinforcement learning algorithms, [inaudible] the order in which the different machines operate on your different things that are flowing through your assembly line and maybe different machines can do different things. So if you have N machines and each machine can be in K states, then if you do this sort of discretization, the total number of states would be K to N as well. If you have N machines and if each machine can be in K states, then again, you can get this huge number of states. Other well-known examples would be if you have a board game is another example. You'd want to use reinforcement learning to play chess. Then if you have N pieces on your board game, you have N pieces on the chessboard and if each piece can be in K positions, then this is a game sort of the curse of dimensionality thing where the number of discrete states you end up with goes exponentially with the number of pieces in your board game. So the curse of dimensionality means that discretization scales poorly to high-dimensional state spaces or at least discrete representations scale poorly to high-dimensional state spaces. In practice, discretization will usually, if you have a 2-dimensional problem, discretization will usually work great. If you have a 3-dimensional problem, you can often get discretization to work not too badly without too much trouble. With a 4-dimensional problem, you can still often get to where that they could be challenging and as you go to higher and higher dimensional state spaces, the odds and [inaudible] that you need to figure around to discretization and do things like non-uniform grids, so for example, what I've drawn for you is an example of a non-uniform discretization where I'm discretizing S-2 much more finally than S-1. If I think the value function is much more sensitive to the value of state variable S-2 than to S-1, and so as you get into higher dimensional state spaces, you may need to manually fiddle with choices like these with no uniform discretizations and so on. But the folk wisdom seems to be that if you have 2- or 3-dimensional problems, it work fine. With 4-dimensional problems, you can probably get it to work but it'd be just slightly challenging and you can sometimes by fooling around and being clever, you can often push discretization up to let's say about 6-dimensional problems but with some difficulty and problems higher than 6-dimensional would be extremely difficult to solve with discretization. So that's just rough folk wisdom order of managing problems you think about using for discretization. But what I want to spend most of today talking about is [inaudible] methods that often work much better than discretization and which we will approximate V* directly without resulting to these sort of discretizations. Before I jump to the specific representation let me just spend a few minutes talking about the problem setup then. For today's lecture, I'm going to focus on the problem of continuous states and just to keep things sort of very simple in this lecture, I want view of continuous actions, so I'm gonna see discrete actions A. So it turns out actually that is a critical fact also for many problems, it turns out that the state space is much larger than the states of actions. That just seems to have worked out that way for many problems, so for example, for driving a car the state space is 6-dimensional, so if XY T, Xdot, Ydot, Tdot. Whereas, your action has, you still have two actions. You have forward backward motion and steering the car, so you have 6-D states and 2-D actions, and so you can discretize the action much more easily than discretize the states. The only examples down for a helicopter you've 12-D states in a 4-dimensional action it turns out, and it's also often much easier to just discretize a continuous actions into a discrete sum of actions. And for the inverted pendulum, you have a 4-D state and a 1-D action. Whether you accelerate your cart to the left or the right is one D action and so for the rest of today, I'm gonna assume a continuous state but I'll assume that maybe you've already discretized your actions, just because in practice it turns out that not for all problems, with many problems large actions is just less of a difficulty than large state spaces. So I'm going to assume that we have a model or simulator of the MDP, and so this is really an assumption on how the state transition probabilities are represented. I'm gonna assume and I'm going to use the terms "model" and "simulator" pretty much synonymously, so specifically, what I'm going to assume is that we have a black box and a piece of code, so that I can input any state, input an action and it will output S prime, sample from the state transition distribution. Says that this is really my assumption on the representation I have for the state transition probabilities, so I'll assume I have a box that read take us in for the stated action and output in mixed state. And so since they're fairly common ways to get models of different MDPs you may work with, one is you might get a model from a physics simulator. So for example, if you're interested in controlling that inverted pendulum, so your action is A which is the magnitude of the force you exert on the cart to left or right, and your state is Xdot, T, Tdot. I'm just gonna write that in that order. And so I'm gonna write down a bunch of equations just for completeness but everything I'm gonna write below here is most of what I wanna write is a bit gratuitous, but so since I'll maybe flip open a textbook on physics, a textbook on mechanics, you can work out the equations of motion of a physical device like this, so you find that Sdot. The dot denotes derivative, so the derivative of the state with respect to time is given by Xdot, ?-L(B) cost B over M Tdot, B. And so on where A is the force is the action that you exert on the cart. L is the length of the pole. M is the total mass of the system and so on. So all these equations are good uses, just writing them down for completeness, but by flipping over, open like a physics textbook, you can work out these equations and notions yourself and this then gives you a model which can say that S-2+1. You're still one time step later will be equal to your previous state plus [inaudible], so in your simulator or in my model what happens to the cart every 10th of a second, so ? T would be within one second and then so plus ? T times that. And so that'd be one way to come up with a model of your MDP. And in this specific example, I've actually written down deterministic model because and by deterministic I mean that given an action in a state, the next state is not random, so would be an example of a deterministic model where I can compute the next state exactly as a function of the previous state and the previous action or it's a deterministic model because all the probability mass is on a single state given the previous stated action. You can also make this a stochastic model. A second way that is often used to attain a model is to learn one. And so again, just concretely what you do is you would imagine that you have a physical inverted pendulum system as you physically own an inverted pendulum robot. What you would do is you would then initialize your inverted pendulum robot to some state and then execute some policy, could be some random policy or some policy that you think is pretty good, or you could even try controlling yourself with a joystick or something. But so you set the system off in some state as zero. Then you take some action. Here's zero and the game could be chosen by some policy or chosen by you using a joystick tryina control your inverted pendulum or whatever. System would transition to some new state, S-1, and then you take some new action, A-1 and so on. Let's say you do this for two time steps and sometimes I call this one trajectory and you repeat this M times, so this is the first trial of the first trajectory, and then you do this again. Initialize it in some and so on. So you do this a bunch of times and then you would run the learning algorithm to estimate ST+1 as a function of ST and AT. And for sake of completeness, you should just think of this as inverted pendulum problem, so ST+1 is a 4-dimensional vector. ST, AT will be a 4-dimensional vector and that'll be a real number, and so you might run linear regression 4 times to predict each of these state variables as a function of each of these 5 real numbers and so on.
Just for example, if you say that if you want to estimate your next state ST+1 as a linear function of your previous state in your action and so A here will be a 4 by 4 matrix, and B would be a 4-dimensional vector, then you would choose the values of A and B that minimize this. So if you want your model to be that ST+1 is some linear function of the previous stated action, then you might pose this optimization objective and choose A and B to minimize the sum of squares error in your predictive value for ST+1 as the linear function of ST and AT. I should say that this is one specific example where you're using a linear function of the previous stated action to predict the next state.
Of course, you can also use other algorithms like low [inaudible] weight to linear regression or linear regression with nonlinear features or kernel linear regression or whatever to predict the next state as a nonlinear function of the current state as well, so this is just [inaudible] linear problems. And it turns out that low [inaudible] weight to linear regression is for many robots turns out to be an effective method for this learning problem as well.
And so having learned to model, having learned the parameters A and B, you then have a model where you say that ST+1 is AST plus BAT, and so that would be an example of a deterministic model or having learned the parameters A and B, you might say that ST+1 is equal to AST + BAT + ?T. And so these would be very reasonable ways to come up with either a deterministic or a stochastic model for your inverted pendulum MDP. And so just to summarize, what we have now is a model, meaning a piece of code, where you can input a state and an action and get an ST+1. And so if you have a stochastic model, then to influence this model, you would actually sample ?T from this [inaudible] distribution in order to generate ST+1.
So it actually turns out that in a preview, I guess, in the next lecture it actually turns out that in the specific case of linear dynamical systems, in the specific case where the next state is a linear function of the current stated action, it actually turns out that there're very powerful algorithms you can use. So I'm actually not gonna talk about that today. I'll talk about that in the next lecture rather than right now but turns out that for many problems of inverted pendulum go if you use low [inaudible] weights and linear regressionals and long linear algorithm 'cause many systems aren't really linear. You can build a nonlinear model.
So what I wanna do now is talk about given a model, given a simulator for your MDP, how to come up with an algorithm to approximate the alpha value function piece. Before I move on, let me check if there're questions on this. Okay, cool. So here's the idea. Back when we talked about linear regression, we said that given some inputs X in supervised learning, given the input feature is X, we may choose some features of X and then approximate the type of variable as a linear function of various features of X, and so just do exactly the same thing to approximate the optimal value function, and in particular, we'll choose some features 5-S of a state S.
And so you could actually choose 5-S equals S. That would be one reasonable choice, if you want to approximate the value function as a linear function of the states, but you can also choose other things, so for example, for the inverted pendulum example, you may choose 5-S to be equal to a vector of features that may be [inaudible]1 or you may have Xdot2, Xdot maybe some cross terms, maybe times X, maybe dot2 and so on. So you choose some vector or features and then approximate the value function as the value of the state as is equal to data transfers times the features. And I should apologize in advance; I'm overloading notation here. It's unfortunate. I use data both to denote the angle of the cart of the pole inverted pendulum. So this is known as the angle T but also using T to denote the vector of parameters in my [inaudible] algorithm. So sorry about the overloading notation.
Just like we did in linear regression, my goal is to come up with a linear combination of the features that gives me a good approximation to the value function and this is completely analogous to when we said that in linear regression our estimate, my response there but Y as a linear function a feature is at the input. That's what we have in linear regression. Let me just write down value iteration again and then I'll written down an approximation to value iteration, so for discrete states, this is the idea behind value iteration and we said that V(s) will be updated as R(s) + G [inaudible].
That was value iteration and in the case of continuous states, this would be the replaced by an [inaudible], an [inaudible] over states rather via sum over states. Let me just write this as R(s) + G([inaudible]) and then that sum over T's prime. That's really an expectation with respect to random state as prime drawn from the state transition probabilities piece SA of V(s) prime. So this is a sum of all states S prime with the probability of going to S prime (value), so that's really an expectation over the random state S prime flowing from PSA of that. And so what I'll do now is write down an algorithm called fitted value iteration that's in approximation to this but specifically for continuous states. I just wrote down the first two steps, and then I'll continue on the next board, so the first step of the algorithm is we'll sample. Choose some set of states at random. So sample S-1, S-2 through S-M randomly so choose a set of states randomly and initialize my parameter vector to be equal to zero. This is analogous to in value iteration where I might initialize the value function to be the function of all zeros. Then here's the end view for the algorithm. Got quite a lot to write actually. Let's see. And so that's the algorithm. Let me just adjust the writing. Give me a second. Give me a minute to finish and then I'll step through this. Actually, if some of my handwriting is eligible, let me know. So let me step through this and so briefly explain the rationale. So the hear of the algorithm is - let's see. In the original value iteration algorithm, we would take the value for each state, V(s)I, and we will overwrite it with this expression here. In the original, this discrete value iteration algorithm was to V(s)I and we will set V(s)I to be equal to that, I think. Now we have in the continuous state case, we have an infinite continuous set of states and so you can't discretely set the value of each of these to that. So what we'll do instead is choose the parameters T so that V(s)I is as close as possible to this thing on the right hand side instead. And this is what YI turns out to be. So completely, what I'm going to do is I'm going to construct estimates of this term, and then I'm going to choose the parameters of my function approximator. I'm gonna choose my parameter as T, so that V(s)I is as close as possible to these. That's what YI is, and specifically, what I'm going to do is I'll choose parameters data to minimize the sum of square differences between T [inaudible] plus 5SI. This thing here is just V(s)I because I'm approximating V(s)I is a linear function of 5SI and so choose the parameters data to minimize the sum of square differences. So this is last step is basically the approximation version of value iteration. What everything else above was doing was just coming up with an approximation to this term, to this thing here and which I was calling YI. And so confluently, for every state SI we want to estimate what the thing on the right hand side is and but there's an expectation here. There's an expectation over a continuous set of states, may be a very high dimensional state so I can't compute this expectation exactly. What I'll do instead is I'll use my simulator to sample a set of states from this distribution from this P substrip, SIA, from the state transition distribution of where I get to if I take the action A in the state as I, and then I'll average over that sample of states to compute this expectation. And so stepping through the algorithm just says that for each state and for each action, I'm going to sample a set of states. This S prime 1 through S prime K from that state transition distribution, still using the model, and then I'll set Q(a) to be equal to that average and so this is my estimate for R(s)I + G(this expected value for that specific action A). Then I'll take the maximum of actions A and this gives me YI, and so YI is for S for that. And finally, I'll run really run linear regression which is that last of the set [inaudible] to get V(s)I to be close to the YIs. And so this algorithm is called fitted value iteration and it actually often works quite well for continuous, for problems with anywhere from 6- to 10- to 20-dimensional state spaces if you can choose appropriate features. Can you raise a hand please if this algorithm makes sense? Some of you didn't have your hands up. Are there questions for those, yeah?
Is there a recess [inaudible] function in this setup?
Instructor (Andrew Ng):Oh, yes. An MDP comprises SA, the state transition probabilities G and R and so for continuous state spaces, S would be like R4 for the inverted pendulum or something. Actions with discretized state transitions probabilities with specifying with the model or the simulator, G is just a real number like .99 and the real function is usually a function that's given to you.
And so the reward function is some function of your 4-dimensional state, and for example, you might choose a reward function to be minus – I don't know. Just for an example of simple reward function, if we want a -1 if the pole has fallen and there it depends on you choose your reward function to be -1 if the inverted pendulum falls over and to find that its angle is greater than 30° or something and zero otherwise. So that would be an example of a reward function that you can choose for the inverted pendulum, but yes, assuming a reward function is given to you so that you can compute R(s)I for any state. Are there other questions?
Actually, let me try asking a question, so everything I did here assume that we have a stochastic simulator. So it turns out I can simply this algorithm if I have a deterministic simulator, but deterministic simulator is given a stated action, my next state is always exactly determined. So let me ask you, if I have a deterministic simulator, how would I change this algorithm? How would I simplify this algorithm?
Student:Lower your samples that you're drawing [inaudible].
Instructor (Andrew Ng):Right, so Justin's going right. If I have a deterministic simulator, all my samples from those would be exactly the same, and so if I have a deterministic simulator, I can set K to be equal to 1, so I don't need to draw K different samples. I really only need one sample if I have a deterministic simulator, so you can simplify this by setting K=1 if you have a deterministic simulator. Yeah?
Student:I guess I'm really confused about the, yeah, we sorta turned this [inaudible] into something that looks like linear state regression or some' you know the data transpose times something that we're used to but I guess I'm a little. I don't know really know what question to ask but like when we did this before we had like discrete states and everything. We were determined with finding this optimal policy and I guess it doesn't look like we haven't said the word policy in a while so kinda difficult.
Instructor (Andrew Ng):Okay, yeah, so [inaudible] matters back to policy but maybe I should just say a couple words so let me actually try to get at some of what maybe what you're saying. Our strategy for finding optimal policy has been to find some way to find V*, find some way to find the optimal value function and then use that to compute ?* and some of approximations of ?*. So far everything I've been doing has been focused on how to find V*. I just want to say one more word. It actually turns out that for linear regression it's often very easy. It's often not terribly difficult to choose some resource of the features.
Choosing features for approximating the value function is often somewhat harder, so because the value of a state is how good is starting off in this state. What is my expected sum of discounted rewards? What if I start in a certain state? And so what the feature of the state have to measure is really how good is it to start in a certain state? And so for inverted pendulum you actually have that states where the poles are vertical and when a cart that's centered on your track or something, maybe better and so you can come up with features that measure the orientation of the pole and how close you are to the center of the track and so on and those will be reasonable features to use to approximate V*. Although in general it is true that choosing features, the value function approximation, is often slightly trickier than choosing good features for linear regression.
Okay and then Justin's questions of so given V*, how do you go back to actually find a policy? In the discrete case, so we have that ?*(s) is equal to all [inaudible] over A of that. So that's again, I used to write this as a sum over states [inaudible]. I'll just write this as an expectation and so then once you find the optimal value function V*, you can then find the optimal policy ?* by computing the [inaudible]. So if you're in a continuous state MDP, then you can't actually do this in advance for every single state because there's an infinite number of states and so you can't actually perform this computation in advance to every single state.
What you do instead is whenever your robot is in some specific state S is only when your system is in some specific state S like your car is at some position orientation or your inverted pendulum is in some specific position, posed in some specific angle T. It's only when your system, be it a factor or a board game or a robot, is in some specific state S that you would then go ahead and compute this augmax, so it's only when you're in some state S that you then compute this augmax, and then you execute that action A and then as a result of your action, your robot would transition to some new state and then so it'll be given that specific new state that you compute as augmax using that specific state S that you're in.
There're a few ways to do it. One way to do this is actually the same as in the inner loop of the fitted value iteration algorithm so because of an expectation of a large number of states, you'd need to sample some set of states from the simulator and then approximate this expectation using an average over your samples, so it's actually as inner loop of the value iteration algorithm. So you could do that. That's sometimes done. Sometimes it can also be a pain to have to sample a set of states to approximate those expectations every time you want to take an action in your MDP.
Couple of special cases where this can be done, one special case is if you have a deterministic simulator. If it's a deterministic simulator, so in other words, if your similar is just some function, could be a linear or a nonlinear function. If it's a deterministic simulator then the next state, ST+1, is just some function of your previous stated action. If that's the case then this expectation, well, then this simplifies to augmax of A of V* of F of I guess S,A because this is really saying S prime=F(s),A. I switched back and forth between notation; I hope that's okay. S to denote the current state, and S prime to deterministic state versus ST and ST+1 through the current state. Both of these are sorta standing notation and don't mind my switching back and forth between them. But if it's a deterministic simulator you can then just compute what the next state S prime would be for each action you might take from the current state, and then take the augmax of actions A, basically choose the action that gets you to the highest value state.
And so that's one case where you can compute the augmax and we can compute that expectation without needing to sample an average over some sample. Another very common case actually it turns out is if you have a stochastic simulator, but if your similar happens to take on a very specific form of ST+1=F(s)T,AT+?T where this is galsie noise. The [inaudible] is a very common way to build simulators where you model the next state as a function of the current state and action plus some noise and so once specific example would be that sort of mini dynamical system that we talked about with linear function of the current state and action plus galsie noise. In this case, you can approximate augment over A, well.
In that case you take that expectation that you're trying to approximate. The expected value of V* of S prime, we can approximate that with V* of the expected value of S prime, and this is approximation. Expected value of a function is usually not equal to the value of an expectation but it is often a reasonable approximation and so that would be another way to approximate that expectation and so you choose the actions according to watch we do the same formula as I wrote just now. And so this would be a way of approximating this augmax, ignoring the noise in the simulator essentially. And this often works pretty well as well just because many simulators turn out to be the form of some linear or some nonlinear function plus zero mean galsie noise, so and just that ignore the zero mean galsie noise, so that you can compute this quickly.
And just to complete about this, what that is, right, that V* F of SA, this you down rate as data transfers Fi of S prime where S prime=F of SA. Great, so this V* you would compute using the parameters data that you just learned using the fitted value iteration algorithm. Questions about this?
Student:[Inaudible] case, for real-time application is it possible to use that [inaudible], for example for [inaudible].
Instructor (Andrew Ng):Yes, in real-time applications is it possible to sample case phases use [inaudible] expectation. Computers today actually amazingly fast. I'm actually often surprised by how much you can do in real time so the helicopter we actually flying a helicopter using an algorithm different than this? I can't say. But my intuition is that you could actually do this with a helicopter. A helicopter would control at somewhere between 10hz and 50hz. You need to do this 10 times a second to 50 times a second, and that's actually plenty of time to sample 1,000 states and compute this expectation.
They're real difficult, helicopters because helicopters are mission critical, and you do something it's like fast. You can do serious damage and so maybe not for good reasons. We've actually tended to avoid tossing coins when we're in the air, so the ideal of letting our actions be some up with some random process is slightly scary and just tend not to do that. I should say that's prob'ly not a great reason because you average a large number of things here very well fine but just as a maybe overly conservative design choice, we actually don't, tend not to find anything randomized on which is prob'ly being over conservative. It's the choice we made 'cause other things are slightly safer. I think you can actually often do this.
So long as I see a model can be evaluated fast enough where you can sample 100 state transitions or 1,000 state transitions, and then do that at 10hz. They haven't said that. This is often attained which is why we often use the other approximations that don't require your drawing a large sample. Anything else? No, okay, cool. So now you know one algorithm [inaudible] reinforcement learning on continuous state spaces. Then we'll pick up with some more ideas on some even more powerful algorithms, the solving MDPs of continuous state spaces. Thanks. Let's close for today.


After only 8 months of blogging and less than 2 years of marriage, My Marriage in Motion has made it onto the Top 10 Marriage Blogs list! I'd like to send a HUGE thank you to everyone that voted our blog into the #8 spot on this year's top 10 list! I couldn't believe I was nominated. I'm even more shocked that we made the top 10!
A special thank you to Stu Gray and the Stupendous Marriage blog for hosting the top 10 list. Stu and his wife are also Nashvillians. Small world!
It's been a crazy few months, but things are finally starting to settle down. Although it won't be calm around here for long, we enjoyed our time together last weekend and are really looking forward to finding our new normal. How was your Memorial Day weekend?


I have to admit that this week's menu was planned pretty hurriedly. We went out of town this weekend with some friends, and got back yesterday evening. It was a great trip. Our friends also have two small children – girls, though – and J and their older daughter had a blast playing together. J has decided recently that it's fun to declare, "I want to play ROCK BAND!" over and over again (we don't know where this came from; Brian plays Rock Band but J hasn't "played" with him in months). When you picture this, you have to understand the way it's said. He's not whining (this time) – he bellows it out and really amps up the last two words of the statement. He apparently passed his love of saying this on to our friends' daughter, because yesterday morning as we sat enjoying our continental breakfast they took turns shouting it back and forth to each other with huge grins on their faces. Ahh, the fun of toddlers.
Back to my point, though… we got back last night, and you know what comes with returning from a trip: a) tiredness; and b) unloading the car. Oh, and feeding two hungry little boys since we arrived at dinner time. By the time we got all settled in and had the boys ready for bed, I really didn't feel like planning our meals for the week, let alone making a trip to the grocery store. If I waited until today to go, I'd have to take both boys with me, and that mental image was enough to get me motivated. I kept it simple this week. The good news is I also kept our grocery bill down – $50 again this week!
Monday: Chicken stir fry with broccoli, carrots, onion, and rice. I use teriyaki sauce, soy sauce, chinese five spice, and a little bit of red pepper flakes to season our stir fry. My brother and his girlfriend are coming over again, and he requested something Asian.
Tuesday: Chick-Fil-A Family Night. Definitely looking forward to not cooking! Brian won't be able to eat with us tomorrow night, so for the cost of just my meal the boys and I will all get dinner (free kids' meals!!). Plus, that means no clean-up. LOVE that.
Wednesday: Mexican food, of course, for Cinco de Mayo! I am going to make my mom's recipe for Chicken Enchilada Casserole. It's a favorite of mine – very easy, and makes a ton of leftovers too.
Thursday: We're grilling out hot dogs and will bake some sweet potato fries to go with them. Honestly, I don't really love hot dogs, but Brian and J do, so I'll go for them every once in a while.
Friday: Margarita chicken with black beans & rice – I got the chicken recipe from the Copykat cookbook I wrote about a while back. Brian's going to cook the chicken on the grill – I love spring and summer because that means we grill out all the time. So good!
Saturday: Burritos. This is a nice go-to meal for us. I cook some lean ground beef with fajita seasoning, and then we add black beans, rice, shredded cheddar, lettuce, cilantro, and salsa. It's our version of Qdoba or Chipotle burritos at home – sometimes we'll change up the filling, but this is our stand-by combination.
I realize that yet again I've procrastinated in trying a fish recipe… I think I might actually be a little nervous about it! I'm resolved to actually follow through and make seafood next week. I'm kind of glad I have my Menu Plan Monday post to hold me accountable!
I hope you all have a wonderful week! As always, if you have any recipes to suggest I try next week, share them in the comments section and I'll include them in an upcoming post.
P.S. – Don't forget to enter my giveaway that runs through this week! Who doesn't want a free wallet?
We love Chick-Fil-A and that really is a great deal to pay for only one meal and get two free. I did that a while back when my hubby was out of town, but I didn't realize that I would get both my kid's meals free. It was a nice surprise. Enjoy your night off from cooking and cleaning.
.-= Amanda @ Mommy's Idea Book´s last blog ..Menu Plan Monday (5/3 – 5/9) =-.
Thanks, Amanda! It's a crazy week at work, so a night off this week is especially appreciated.
All sounds good. Great you only kept the bill down to $50 again. Gotta love it. I always struggle to keep the bill down to budget. Looks like a good mexican meal week. You are brave I think i would have skipped the meal plan monday post, but that's just me. Anyway thanks for sharing. I'm needing some meal inspirations this week! Off to start the grocery list!
.-= Rebecca´s last blog ..looking at motherhood together =-.
Thanks Rebecca – we LOVE Mexican food around here!
Yum… I am hungry now! I'm going to have to do better with my menu planning so we can save $$ too.
.-= Kristen´s last blog ..Far Above Rubies: Proverbs 31:12 =-.
Yummy. How do you make Chicken Enchilada Casserole?
Jacquie, I added the link to the recipe – I had meant to do that originally so thanks for reminding me! :) Hope you like it… it's a little spicy (not too much) so be careful of it giving you heartburn!
Thanks for posting the recipe. I have been really bored on bedrest and in desperate need of a project so I decided to redo my recipe book. I am including our favorites and then I am including recipes that are budget friendly, recipes that are healthy, etc. I can't be in the kitchen cooking yet but once this baby is born I can't wait to get back into the kitchen. :-) I saw a delicious recipe today on Rachael Ray and I can't wait to make it. It is for the Adirondack Red Wing Burger (it is a healthy chicken or turkey burger on an English Muffin). So now I have her recipe and your recipe to add to my book. Thanks again!
.-= Charla´s last blog ..Starbuck's Frappuccinos Half-Price this Friday, 3-5 PM =-.


Move, delete, and rename indexes.
Open multiple indexes at the same time.
Share indexes on the network.
Specify the location of new indexes.
A command-line tool for searching and indexing.
Moving indexes from one place to another (eg. from one drive to another).
Creating secondary indexes that aren't tied to a solution.
1.6.3 also introduces the idea of a "secondary index" – each solution gets (or shares) a primary index as before, but you can open several secondary indexes as well. These are really useful for library code – for instance, Source Search is built using Lucene.NET; sometimes I want to include the Lucene sources in my searches, and sometimes not. I can build a secondary index for Lucene, and only open it when I want to.
Often library code doesn't change, so secondary indexes can be read-only – you don't need to waste your PC's time scanning for changes if you don't want to, because you can open secondary indexes read-only.
Taking the idea of a secondary index even further, they can live on a network drive. Some organisations keep library code on a network drive, and Source Search now supports this. One network server maintains the index for the shared code, updating it in real time if it changes, and all the developers open that index as a read-only secondary index.
Update an index to pick up changes to the code (the command-line tool doesn't watch the filesystem; it's more for batch operation).
It's early days for ess.exe – there's lots it could do but can't do yet. Let me know what you want from it!
As always, this new release is available from the download page or the gallery.


Looking for great deals on Lubeck hotels? A hotel in Lubeck is easy to find with check-rates.com. Choose from a great selection of Lubeck hotels and make your reservation today. Whether you are looking for your hotel by price or by star rating, we can provide all the information you need to select the best Lubeck hotel for your needs. It has never been so easy to find and reserve Lubeck hotels.
We recently found 51 hotels in Lubeck. Find the perfect accommodation for your stay and make your Lubeck hotel reservation today. Find hotels in Lubeck compare hotel rates and book the best hotel deals online. We will search hundreds of travel websites and return a listing of accommodation options including: inns, resorts, motels, hostels, bed and breakfast and more.
Use the filters to view hotels & accommodation in a specific area of Lubeck. Select price range, hotel themes, accommodation type, facilities, brand, or hotel class from basic (1 Star) to luxury hotels (5 Stars) in Lubeck. Enter your travel dates to view the best hotel deals for your Lubeck accommodation needs.


Welcome back to the 2U Showcase! This is where we'll regularly share cool collectibles & pop culture paraphernalia we've come across during our adventures across time, space, and the internet. Click each image for pricing and order information!
Entertainment Earth Exclusive! This logically designed Bif Bang Pow! The Big Bang Theory / Star Trek: The Original Series 3 3/4-inch scale action figure of Sheldon Cooper captures Sheldon dressed in the same blue and black uniform as the legendary Starfleet science officer. This detailed rendition of Sheldon was inspired by The Big Bang Theory's second season episode titled "The Codpiece Topology". Also included are his very own Communicator, Phaser and Tricorder accessories.
Dr. Sheldon Cooper is smarter than you – and he's not afraid to say so. A theoretical physicist possessing a master's degree, two Ph.D.'s, and an IQ of 187, Sheldon is originally from East Texas, where he was a child prodigy, starting college at the age of 11. He has zero interest in romance, as his only real love is science (although his "girlfriend" Amy is constantly finding loopholes in their "Relationship Agreement" to disprove this theorem).
In The Big Bang Theory, Leonard and Sheldon are brilliant physicists – geniuses in the laboratory, but socially challenged in the real world. Enter beautiful, street-smart neighbor, Penny who aims to teach them a thing or two about life. Despite their on-again, off-again relationship in the past, Leonard found true love with fiancé, Penny. Even Sheldon has found a female companion – entering into a "Relationship Agreement" with neurobiologist Amy Farrah Fowler. In their free time, Leonard and Sheldon continue to enjoy frequent trips to the comic book store and fantasy role-playing games with their ever-expanding universe of friends, including fellow scientists Raj and Howard, and Howard's adorable microbiologist wife, Bernadette.
Princess Diana, mighty amazon warrior.
Exclusive 11-ounce mug transforms from black to a full-color image!
Heat activation reveals Wonder Woman's hidden image.
Relax in Galactic Peace in this amazingly comfortable distressed Star Wars vintage tee.


Our Mission is to provide Temporary Shelter to OFW's and Seafarers who come to Manila before or after going to work abroad. The SCPM promotes human dignity and respects the rights of the people on the move. The SCPM pretends to be a home for those OFW and Seafarers by creating an environment of friendship and welcoming.
If you are an OFW or a Seafarer, you are very welcomed to the SCPM, if during your staying in Manila becomes exhausting remember that in the SCPM you can find relief.
Welcome to the SCPM, we share with you the hope that moves you and the faith that keeps you walking.


text:
Hans Hürlimann, né le  et mort le  à Bâle, est un botaniste, chercheur et enseignant suisse.

Biographie

Jeunesse
Hans Hürlimann naît le . Il fréquente l'école primaire et le lycée à Winterthour. Il étudiée ensuite la biologie, avec une spécialisation en botanique, à l'université de Zurich. Parmi ses professeurs figurent A.U. Däniker et E. Schmid. Il obtient son doctorat en  pour sa thèse . Ce travail primé est toujours d'actualité en ce qui concerne les problèmes liés aux roseaux. Il montre, entre autres, que l'espèce alors appelée Phragmites communis ne peut germer que sur la terre ferme. Déjà pendant ses études, mais aussi par la suite, Hans Hürlimann enseigne la biologie, la chimie et la géographie dans différentes écoles secondaires.

Travaux
La même année que son doctorat, il est chargé d'entreprendre un voyage d'étude en Nouvelle-Calédonie avec son camarade Marcel Baumann et le professeur français André Guillaumin. Ce voyage dure deux ans et porte sur l'étude de la flore (plantes vasculaires et mousses). Il visite également d'autres îles du Pacifique, notamment le Royaume de Tonga. Le traitement des collections de ces domaines aura occupé le chercheur jusqu'à sa retraite. Ses travaux scientifiques comprennent la description de deux genres de la famille des Santalaceae avec un collègue, en plus de plusieurs nouvelles espèces de mousses et de plantes à fleurs.

En , Hans Hürlimann prend un poste au département pharmaceutique de la CIBA à Bâle. Ses activités professionnelles le mènent à effectuer des voyages de recherche et des visites de congrès dans les régions tropicales du monde entier. Au cours de ces voyages, il profite de ses grandes compétences linguistiques. En plus de sa connaissance du grec et du latin, il parle le français, l'anglais, l'espagnol, le néerlandais et le sursilvan.

Hans Hürlimann étudie également la mousse indigène en Suisse. Il est membre de la commission de cartographie pour l'inventaire de la flore bryophyte suisse. Il entretient ses propres collections, qui proviennent principalement du Val Lumnezia, des environs de Bâle et de Winterthour.

Tout au long de sa vie, il s'intéresse en particulier pour les hépatiques. Il s'engage dans la conservation de la nature, même en Nouvelle-Calédonie.

Vie privée
En , il se marie avec Margrit (Gritli) Högger. Il l'avait rencontrée dans l'ancien jardin botanique de l'université de Zurich. Ils vivent à Bâle et élèvent deux enfants, Vreni et Hans Caspar (Peuss). Il a également par la suite des petits-enfants, et a également le privilège de compter deux arrière-petits-enfants.

En tant qu'ancien scout nommé « Juan », il entretient une amitié de longue date avec ses camarades scouts ; également avec des personnes de la « Compagnie de wagons-restaurants » (en allemand ) qu'il avait rencontrées lors de ses fréquents voyages en train entre Bâle et Zurich. Il est également en bons termes avec de nombreux collègues dans son pays et à l'étranger.

Hans Hürlimann avait une relation particulière avec l'Association suisse de bryologie et de lichénologie. Il en a été membre depuis sa fondation, et a assisté à ses réunions aussi longtemps que possible. Plus d'une fois, il a conduit les membres lors d'excursions autour de Lumbrein (Val Lumnezia, GR), où il possédait une maison de vacances. En , il a été fait membre honoraire de la Société, comme d'ailleurs il l'avait été un an plus tôt avec la Société botanique suisse. Il était également membre honoraire de la Société botanique de Bâle, où il avait siégé au conseil d'administration.

Fin de vie
Il prend une retraite anticipée afin d'achever un ouvrage de grande envergure. Ce supplément en deux parties à l'ouvrage classique de Karrer sur les substances végétales organiques a été rédigé en collaboration avec un chimiste (Hürlimann & Cherbuliez 1981-1985).

Sa femme meurt deux ans et demi avant lui, ce qui est une très grande perte pour lui. Il meurt le  dans une maison de retraite à Bâle. Il aura vécu jusqu'à l'âge de presque .

Publications

Ouvrages

Taxons
Il a décrit et nommé  différents :

 Citronella macrocarpa Hürl., Mem. Mus. Hist. Nat., Paris, Ser. Bot. xv. 60 (1964).
 Daenikera Hürl. & Stauffer, Vierteljahrsschr. Naturf. Ges. Zurich 102: 332 (1957).
 Dicarpellum paucisepalum Hürl. ex M.P.Simmons, Fl. N. Caléd. 25: 14 (-17; fig. 2A-B) (2004).
 Elaeodendron baas-beckingii Hürl., Candollea xv. 74 (1955).
 Elaeodendron pininsulare Hürl., Candollea xv. 75 (1955).
 Salaciopsis glomerata Hürl., Candollea xv. 71 (1955).
 Salaciopsis sparsiflora Hürl., Candollea xv. 72 (1955).
 Salaciopsis tapeinospermophylla Hürl., Candollea xv. 73 (1955).
 Sphenostemon oppositifolius Hürl., Mém. Mus. Nation. Hist. Nat. Nouv. Sér., B (Paris) 23: 12 (1974) (1974).

Éponymes
Plusieurs espèces sont nommées en son honneur :
 Aktinokentia huerlimannii H.E.Moore (Palmae)
 Alyxia hurlimannii Guillaumin (Apocynaceae)
 Austrobuxus huerlimannii Airy Shaw (Picrodendraceae)
 Cheilolejeunea huerlimannii Tixier (Marchantiophyta)
 Cololejeunea huerlimannii Tixier (Marchantiophyta)
 Dysoxylum hurlimannii Guillaumin (Meliaceae)
 Elaphoglossum hurlimannii Guillaumin (Filices)
 Eugenia hurlimannii Guillaumin (Myrtaceae)
 Ficus hurlimannii Guillaumin (Moraceae)
 Frullania huerlimannii S.Hatt. (Marchantiophyta)
 Herbertus huerlimannii H.A.Mill. (Marchantiophyta)
 Lasiochlamys hurlimannii (Guillaumin) Sleumer (Salicaceae)
 Litsea hurlimannii Guillaumin (Lauraceae)
 Aponardia huerlimannii (Váňa & Grolle) Váňa (Marchantiophyta)
 Phyllanthus hurlimannii Guillaumin (Euphorbiaceae s.l.)
 Plagiochila huerlimannii Inoue (Marchantiophyta)

Notes et références

Liens externes

Botaniste suisse du XXe siècle
Naissance en décembre 1921
Décès en octobre 2014
Décès à Bâle
Décès à 92 ans

redpajama_set_name: RedPajamaWikipedia


Please e-mail us using this form, or you can use this e-mail address: advising@highline.edu to send e-mail using your own e-mail program.
To be assigned a faculty advisor, please complete the form listed below and submit. We will make every attempt to get you information on your new HC faculty advisor within 7 to 10 business days from when you submit your request.
I plan to transfer to a four-year college or university.
*If you select this option, please select an institution and major.
Every effort will be made to assign you to the requested faculty advisor.
Thank you for your response. You will receive an email within 1-2 weeks with your advisor assignment.
Please contact us using this form, or you can use this e-mail address: advising@highline.edu to send e-mail using your own e-mail program.


text:
Aloft Minneapolis
900 Washington Avenue, Minneapolis, Minnesota 55415
Featured, Festivals & Fairs
November 05, 2022 to November 07, 2022
map-Aloft Minneapolis Northrup King Building, 1500 Jackson St. NE, Minneapolis, MN, 55413
Add to my calendar - Art Attack
Enjoy a weekend packed with the visual artworks of over 300 visual artists and the sounds of a variety of singer/songwriters and composers. Put on your walking shoes and allow plenty of time to explore the 4 floors of the Northrup King Building, talk with the artists, listen to the music and try your hand at an interactive art activity.
call-Art AttackN/A
map-Aloft Minneapolis Getting here Click on Getting here button
The Great Northern
Dec 26, 2022 - Feb 5, 2023 Dec 26, 2022
- Minneapolis, MN
2023 Art Shanty
Jan 21, 2023 - Feb 12, 2023 Jan 21, 2023
St. Paul Winter Carnival
Jan 26, 2023 - Feb 5, 2023 Jan 26, 2023
- St--Paul,

redpajama_set_name: RedPajamaCommonCrawl


text:
account-security

A simple account security helper.

Assumes the 'account' object has a password field to store the hashed password.

Require the module

    var Security = require('account-security');

Instansiate it, plugging in your db calls.

The constructor takes three functions:

    new Security(
        function getAccount(accountName, callback){
            // get the account that matches this email
            // call callback with (error, email)
        },
        function createAccount(accountName, password, accountDetails, callback){
            // accountDetails will be the object passed into security.register
            // but with the password field now hashed
        },
        function saveAccount(account, callback){
            // account will be an object that was returned from getAccount or createAccount.
        }
    );

Mongo with Mongoose example:

    var security = new Security(

        // Get account
        function(accountName, callback){
            db.Account.findOne({
                accountName: accountName
            }, callback);
        },

        // Create account
        function(accountName, hash, accountDetails, callback){
            var newAccount = new db.Account({
                firstName: accountDetails.firstName,
                surname: accountDetails.surname,
                accountName: accountName,
                password: hash
            });

            callback(null, newAccount);
        },

        // Save account
        function(account, callback){
            account.save(callback);
        }
    );

Use the object:

    // Register
    function(request, response){
        security.register(
            accountName: // retrieved from the request somehow,
            password: // retrieved from the request somehow,
            {
                // extra account data retrieved from the request somehow
            },
            function(error, account){
                if(error){
                    // handle error
                }

                // Wooo it worked!
            }
        );
    }

    // Sign In
    function(request, response){
        security.authenticate(
            /* accountName from request */,
            /* password from request */,
            function(error, account){
                if(error){
                    // handle error
                }

                // User authenticated. Set cookies or whatever..
            }
        );
    },

    // Change Password
    function(request, response){
        security.changePassword(
            /* accountName from request */,
            /* old password from request */,
            /* new password from request */,
            function(error){
                if(error){
                    // handle error
                }

                // password changed.
            }
        );
    }

redpajama_set_name: RedPajamaGithub


Inspiration please, bent nails, wood and beauty. The art of Lee Jae-Hyo.
We came across the art of Lee Jae-Hyo and felt the need to share it. The Korean born artist is, perhaps, best known for his more recent pieces which feature hundreds of bent nails hammered into wood.
The wood is burnt, the nails are filed down and the results are stunning. His work crops up in a number of galleries or you can contact the artist through his own website. In any event, we wanted to share some images of his work simply because we enjoyed them so much.
Glittering metallic nails on black charcoaled wood. A strange contrast between what was and what is. Sharp, now smooth. Once something to avoid touching becomes irresistible tactile.
Would you want to sit on a bench covered with bent nails? Could you resist it?
This piece rolls like the waves on the ocean, with the play of light creating ever different patterns.
Close up the effect is fascinating. Each nail is, of course, unique. The creative transformation through striking and then rendering smooth becomes apparent.
Earlier pieces have the same quality of making the material an integral part of the piece's shape and texture.
The reference to the natural world is as apparent as is the use of the qualities of the raw materials as the essence of the piece.
These are quite beautiful. Thank you for sharing them.


Perry Ellis 360 perfume, presented to Perry Ellis by perfumer nose, Sophia Grojsman introduces the top notes of melon, mandarin, osmanthus and blue rose. A heart of lemon, green notes of lily-of-the-valley, lotus and sage are introduced into the composition, with a warm base of sandalwood, powdery vanilla, vetiver, amber and sensuous musk.
Citrus and juniper berries are the top notes of the aromatic fragrance of Perry Ellis 360 cologne. Middle notes include lavender, sage and cardamom. This is a scent that evolves from a clean scent that is taken over by spicy notes with a dry down that mellow to a warm woody and spicy combination.
Pefume.net is your one–stop-shop for all your favorite women's perfumes and men's colognes. Every fragrance we sell is authentic, we guarantee it. And not only do you save on each and every purchase, your shipping is free on U.S. orders of $70 or more. Discover Scentsational savings on your favorite fragrances.
For winter dry skin Put bath oil in a spray bottle. After a shower or bath, while still wet, spray your skin and then rub. Blot yourself with a towel. Now you are not only moisturized, but your tub is not slick with oil!


text:
You are at:Home»COUPONS»DEALS»Grab the Elephone PX Christmas Bundle with Elephone Watch 3 For Just $279 + Free Global Shipping with Up To 55% OFF
Grab the Elephone PX Christmas Bundle with Elephone Watch 3 For Just $279 + Free Global Shipping with Up To 55% OFF
By Reyan Siddiqui on	 December 24, 2018 DEALS, LATEST NEWS, PHONES, Elephone
The Elephone brand, known for inexpensive smartphones and fitness bracelets, launched a fundraising campaign for the launch of the Elephone PX smartphone on the Indiegogo platform. For the Christmas holidays, Elephone offers to order a kit from a smartphone and an Elephone Watch 3 for $279. At the same time, the smartphone costs $ 249.99, which means the watch will cost $30.
It is possible to buy watches separately, within the framework of the same action, for $30, at their full cost of $129. The proposal is interesting, but, for some reason, it seems that we are being deceived somewhere.
Elephone Watch 3 is an advanced waterproof smartwatch designed to pair and optimize the smartphone experience. It features a 1.33-inch color display with a 2.5D arc curved glass on top to add beauty and to protect the screen. In addition, the W3 incorporates a 180 mAh battery with magnetic charging. The battery takes only 3 hours to fully charge and offers the smartwatch up to 25 days in standby.
Elephone PX offers a 6.34-inch screen with minimal bezels thanks to its dual pop-up camera. It also has a dual rear camera, which is 16+2MP (to match the front-facing camera). However, its true-full-screen appearance is countered by compromises elsewhere in its specs. The PX's display resolution is 720×1480 (for a 19:9 aspect ratio), and it has 4GB of RAM. It has 64GB of internal storage, although it can be expanded with up to 400GB of microSD card support. Elephone also claims that the PX will come with a 3500mAh battery, USB type C and Android Pie (9.0) out of the box. On the other hand, it does not have a headphone jack.
The PX is to be powered by the MediaTek P22 processor, which has 4 Cortex-A53 (2.0GHz) and 4 more A53 cores at 1.5GHz. It is also portrayed as having a raised accent around the rear of the pop-up module, as well as a fetching aluminum strip along the bottom half of the rear panel. However, the rest of the body is specified as plastic.
ELEPHONE Watch 3, which is an IP67 rated waterproof smartwatch with plenty of features like heart rate monitor and others, then you have a very solid bundle, which is priced for Xmas only at $279. It's available on the Indiegogo crowdfunding platform in limited quantities and the delivery is estimated roughly in February 2019.
Previous ArticleXiaomi Mi Play Measured Running Score: Performance Level Of The Helio P35
Next Article LEVETOP T1 Review: GPS Brushless RC Drone with Camera 1080P For just $156 at TOMTOP in Flash Sale


text:
Scientists discover fish have been jumping out of bay because it smells bad
By John Jacobs | February 25, 2019
Bayshore – Hillsborough County – Monday, 2.25.2019
If you've ever walked down Bayshore Boulevard, you may have noticed that fish sometimes unexplainably, jump out of the water. Many speculated reasons why this may be, but up until recently, nobody knew the official answer.
Now, scientists have proven that the reason the fish have been jumping out of the water is because, "it smells bad."
"The bay smells bad," says senior marine biologist Sheila Ziper. "We studied many different variables of the bay itself," Ziper says as she motions her hands together like that ancient aliens meme guy. "We studied water height, water content, water taste, touch, feel, wetness," she pauses, "but then we noticed something." She points towards her nose. "The water smells," she smiles, "bad."
"The water smells bad, and the fish have to spend all day in it. So when they get into the most polluted parts of the bay, they jump out, to get a breathe of 'fresh' air for brief moment, then it's back into the sewage." Ziper concluded her quote by staring blankly at me.
The bay has been known to fill up with gas station polar foam cups, leftover beads from Gasparilla, and good-old-fashioned garbage.
"For years I wondered why they were jumping out of the water!" Says Clem Hanley as he speeds around us on his recumbent bicycle. "I live right on the water! My house cost millions of dollars! But every time I open my front door, I get a huge whiff of s**t, right in my face! It hits me like a brick wall!" He's having more fun than he should for some reason. "I mean, you'd think, 'house right on the water, this is perfect!' But in reality, it's only almost perfect. Because until the government," he points at a stop sign, "gets their s**t together, it's always going to smell like s**t around here!" His own cyclical seemingly-mistaken semi-joke immediately registers in his head, and he tries to act like he made that statement on purpose.
What this information means for the future of Bayshore and the cleanliness of the bay is still up in the air.
"Most people didn't even realize fish could smell," says Ziper. "I didn't even know until last week!" She covers her head to keep her hat from blowing off in the wind. "I got this gig, and I studied the stuff they wanted to, but I put a magnifying glass over the fish's face, and I noticed it has what looks to be a nose." She defeatedly shrugs as if she thinks she's just been exposed as a fraud, however we are not judging her. "You can tell my parents I'm not going to be coming home." She says to us. "You can tell them… I'm a Tampanian now." We stare at her. "Oh yea, I'm from Boston by the way, I forgot to mention that. Go Sox!"
The city has encouraged residents to do anything they can to "stop the bay from smelling bad," however no official measures have been released.
← Ferris Wheel Considered for Future Method of TransportationMan missing since Gasparilla found alive in Port-O-John →
About John Jacobs
MTV Reality TV Star and Award-Winning Tampa News Force Correspondent. Subscribe to YouTube Channel, Follow on Twitter: @MaybachDiamonds Instagram: @MaybachDiamonds
View all posts by John Jacobs →


text:
Victoria Fleischer Victoria Fleischer
https://www.pbs.org/newshour/arts/soul-musician-curtis-harding-keeping-tradition
Soul musician Curtis Harding on keeping with tradition and staying current
Arts Jul 30, 2014 3:33 PM EST
Curtis Harding spoke to Art Beat about his debut album, "Soul Power."
Soul musician Curtis Harding originally wanted to be an oceanographer. He loves the ocean, but growing up in a musical family left its mark.
"My mother's a gospel singer. My sister plays piano. We all sang in the church, sang on the road, traveled around a lot to different cities," Harding told Art Beat during a recent tour stop at DC9 in Washington, D.C.
"I've always loved music, so I just kind of fell into it… Just like if your mother's a doctor, your father's a doctor, you might become a doctor. Keeping with tradition."
Harding released his first solo album, "Soul Power," in May, which he describes as "slop 'n' soul." "Slop" is a southern term for "leftovers," and alludes to the way he incorporates different musical elements into his songs.
He says his style is essentially gospel, but his mother didn't always see how his music was connected to that genre. "Gospel music is soul music; blues, all that, it comes from gospel," said Harding. "It takes a while for, especially, someone who comes from a religious background to understand certain things about secular music… She likes it now."
While Harding is working on his solo endeavors, touring the East Coast through the fall, he also makes music as part of the band Night Sun, along with The Black Lips' Cole Alexander and others. But Night Sun is a very different project, unrelated to what many have described as the retro sound of "Soul Power" — a label that Harding doesn't necessarily embrace.
"I don't think of myself in any terms. If someone wants to relate my music to the past, that's cool, too, but I think you should remain relevant to the times. It's 2014 — unless I had a time machine — you know what I'm saying."
Watch Curtis Harding's performance of his single "Keep on Shining" from his concert at DC9:
Videos by Victoria Fleischer and Justin Scuiletti. Live concert audio mix and recording by Randy Lancelot/DC9.
@vlfleischer
Visit CANVAS, PBS NewsHour's art hub.
Visit Now →


Lionsgate has picked up a pitch for a low-budget horror film from Mike Flanagan and Jeff Howard, and the story has kind of a strange Silent Hill meets The Purge sorta slant. It's based on a short story by Irish novelist Kealan Patrick Burke, and finds a couple characters trapped in a town in which the entire population has been replaced by grinning dopplegangers who like to hide around corners. That's where the title Peekers comes from, and while it sounds a bit sill, the original story has enough of a creepy tone that, adapted and expanded properly, it might make for a good film.
A terrified man walks into a police station to report that the residents on his entire street are missing, insisting the victims were done in by eerie, grinning doppelgangers. When the man disappears from the locked interrogation room, the young female officer who took his statement must venture out into the rapidly vanishing community to find her son and get him to safety as the community is infested by these deadly grinning idiots.
In fact, the short story has already been adapted into a short film, and the eight minute film runs longer than it'll probably take you to read the story. The actual story is almost completely summarized by that paragraph above, and the pitch for this film is essentially the continuation of the tale after it concludes on the page. Here's the short film, which is basically a microbudget version of the tale.


What an honor to have these two amazing children's ministers from Bogatanga, Ghana join the growing number of international SSCM graduates.
On the left is Paul Forderer, a great help and support to Pastor Francis Opoku (right) who had been up to his eyebrows ministering to the children of his nation through his own group called Action Children Mobilization. He had formed dozens of groups ministering to children called Royal Cadets, but had dreamed for a long time of joining his ministry to PowerClubs. As of 2012 there were 47 Royal Cadet PowerClubs serving 10,000 children.
This happened just a few short years ago, and his groups are now called Royal Cadets PowerClubs. Our premiere PowerClub trainer and KIMI Ambassador at Large, Pamela Ayres, went to train his existing leaders on how to bring children into the supernatural lifestyle of God. The leaders began teaching their 5,000+ children how to hear the voice of God, pray for the sick, and just take them into the deeper truths of God's word. Some of our most intriguing testimonies come from his PowerClubs.
Francis has told us that while not all of his leaders have fully caught the vision, others are doing remarkably well. Several are teaching on Spirit Gifts and experiencing spiritual encounters. Word of wisdom and Word of Knowledge are paramount among them. Others are walking in a higher level of Discernment than myself.
"One of the leaders, Felix, operates in the deliverance ministry and especially against witchcraft spirits," Francis continued. "He is teaching his children to exercise their power over evil spirits.
Pastor Francis has one of the biggest heart and visions of anyone we have met in Africa that go so far beyond his work in PowerClubs alone. He is active in feeding the hungry, and obtaining Bibles and books for the children of Ghana. He is a leader in the Right to Play program in Ghana, and for years housed as many as twenty street children in his home with his own six children.
New SSCM graduate Andrea Hall was serious about receiving her certificate in person. She was not able to attend the entire conference, but was determined to receive her SSCM certificate in person.
She rode a bus all night over 27 hours one way to make it to the ceremonies in time. Because she was not even going to be in town a full 24 hours, she didn't even purchase a hotel room, but slept on Becky's couch until the next day.
John Tasch, a true spiritual father, joined our team in giving Andrea a prophetic word.
Our KIMI leaders gathered around Andrea to pray over her as we do with all of the graduates in attendance.
Andrea is leaving soon to do missionary work in Africa and hopes to establish PowerClubs there. We salute you, Andrea!
2013 Our Biggest SSCM Graduating Class Ever!
In October 2013, we held our 4th Annual KIMI Leadership Conference "Pressing for the Prize," in which John Tasch, was our keynote speaker. John was a breath of fresh air of encouragement and inspiration. During this conference we also had our Annual School of Supernatural Children's Ministry graduation ceremonies for all the new SSCM grads from literally all over the world.
While only ten were able to actually fly into North Dakota to join us, we actually had the largest graduation class ever in one year. There were forty-two new grads from about fifteen different nations! We were thrilled. As always, we laid hands on and prayed over the ones who came in person, but we also prayed over the remaining certificates and those they represented before mailing them out.
One graduate told us: "This training has given me a new paradigm for the call of God on my life. During the 'casting vision' part of the training course all I could say after every teaching was "wow." I was awakened to the reality of the need and potential of our kids to be taught and equipped in their callings as kids of the kingdom." Lyn C.
We look forward to seeing these graduates get further involved in KIMI and helping spread the message of boys and girls walking in the supernatual power of God!
If you are a new student of SSCM, please plan to join us for Graduation in 2014. Click here for more information.
I wish to express my personal heartfelt thanks to Laura for the many years of faithful dedication and devotion she has given to Kids in Ministry, and to me as her mentor and friend. Laura and I have worked very closely together over the years, especially when she volunteered to take the responsibilities that came with overseeing the School of Supernatural Children's Ministry. Laura was also the director of KIMI Wisconsin and established a network of leaders and PowerClubs in Uganda.
"I have been so very blessed and honored to mentor under Becky Fischer and to be a part of Kids in Ministry International since the first day I was trained to be a KIMI Missionary Representative in November of 2004. At that time you didn't know what to do with any of us! That was the day God started me on the path to my destiny of sending me to children in the nations.
Laura has been with us from the beginning. She's a part of the spiritual "furniture" of KIMI. But I have known in my heart that if I did my job right, we would be a launching pad for a number of leaders and that not all will stay. There are others God will eventually call out as well. That is one of the ways we will spread this vision all over the world!
I remember what it was like when I realized I could no longer be satisfied with being a children's pastor because the vision in my heart was "too big for the house (local church)". This is really as much of a time to rejoice as it is to mourn, because Laura will be taking a piece of us with her. I know beyond a doubt that whatever she does she will be helping to "redefine children's ministry in the 21st century!
You can reach Laura at crownofglory.laura@yahoo.com.


text:
Q: What's different between gstreamer0.10-fluendo-mp3 and gstreamer0.10-fluendo-plugins-mp3-partner? I was installing some MP3 codecs when i noticed these two, why do we have two of these? what's different?

A: Raw package information:
Package: gstreamer0.10-fluendo-plugins-mp3-partner
Source: gstreamer0.10-fluendo-plugins-partner
Priority: extra
Section: sound
Installed-Size: 256
Maintainer: Brian Thomason
Architecture: i386
Version: 7.0.20100316-3
Replaces: gstreamer0.10-fluendo-mp3, gstreamer0.10-fluendo-plugins,
gstreamer0.10-fluendo-plugins-mp3, gstreamer0.10-fluendo-plugins-offline
Provides: gstreamer0.10-fluendo-mp3, gstreamer0.10-fluendo-plugins-mp3
Depends: libc6 (>= 2.4), libglib2.0-0 (>= 2.12.0), libgstreamer0.10-0 (>= 0.10.0),
libxml2 (>= 2.6.27), zlib1g (>= 1:1.1.4), debconf (>= 0.5) | debconf-2.0,
gstreamer0.10-plugins-base (>= 0.10.14)
Conflicts: gstreamer0.10-fluendo-mp3, gstreamer0.10-fluendo-plugins,
gstreamer0.10-fluendo-plugins-mp3, gstreamer0.10-fluendo-plugins-offline
Pre-Depends: debconf
Filename: pool/partner/g/gstreamer0.10-fluendo-plugins-partner/gstreamer0.10-
fluendo-plugins-mp3-partner_7.0.20100316-3_i386.deb
Size: 92048
MD5sum: 478ec57fd9233c5400a8c9784a27659d
SHA1: 2282a621166a18b1565c306c961a8ada2346ad99
Description: MP3 codec support for GStreamer

The GStreamer multimedia framework is being used in more and more systems,
ranging from server media processing systems, end-user desktops and various
embedded and mobile devices. The Fluendo Plugins are a set of plugins for
GStreamer which implement support for various media formats and hardware.
These plugins give you an opportunity to quickly and easily add proprietary
support to their GStreamer-based products.

This package contains an enhanced mp3 codec.

*

*gstreamer0.10-fluendo-plugins-mp3-partner replaces gstreamer0.10-fluendo-mp3. Most likely the last one will someday disappear or be replaced by the 1st one.

*gstreamer0.10-fluendo-plugins-mp3-partner contains an enhanced mp3 codec.

A: From what I could gather, the new -partner lib doesn't depend on liboil, which is GPL (or LGPL). The non suffixed lib does depend on that. Liboil is a library to handle optimizations (like using MMX, SSE or the like when it can).
Not depending on a (L)GPL library allows Fluendo to distribute their library more easily.
It is explained in better detail here: http://www.fluendo.com/shop/product/fluendo-mp3-decoder/
I quote:
"The fully licensed binary GStreamer plug-in
Fluendo has paid the license of Fraunhofer and Thomson to be able to distribute a binary MP3 decoder. This means that people who want MP3 support for the desktop music players can get a fully licensed plug-in directly from this site.
"
"Issues to be aware of
If you are living in a country where the MP3 patents don't apply, you are entitled to use the source code provided by Fluendo (or anyone else) to get legal MP3 support onto your Unix/GNU/Linux desktop.
In contrast, if you live in a country where patents do apply, or if you are a distribution maker working in countries where the patents apply, you need the licensed binary from Fluendo. If this is the case, please be aware that, even if our binary is made from MIT licensed source code, the resulting binary, combined with our license, is not free software, at least not GPL-compatible. This means that if you ship GStreamer with our binary MP3 plug-in, you need to be sure that you don't ship any GPL-licensed plug-ins that could end up being used together with the MP3 plug-in, as this would be a violation of the GPL. You also need to make sure you don't ship any GPL-licensed players which would use this plug-in.
"

redpajama_set_name: RedPajamaStackExchange


text:
Q: Sequence of unbounded continuous functions I'm having trouble with an exercice from Rudin's Real and Complex Analysis:
Does there exist a sequence $(f_n)$ of continuous positive functions on $\Bbb R$ such that $f_n(x)$ is unbounded if and only if $x \in\Bbb Q$ ?
What if $x \in\Bbb R \setminus\Bbb Q$?
Now, replace unbounded by $f_n(x) \rightarrow \infty $ as $ n \rightarrow \infty$ and answer the analogues to the previous questions.
I don't know very well how to construct such functions, as they would have to be continuous. My guess is that you can do it for $x \in\Bbb Q$ , as their measure is zero, but I don't know how to prove that you can or cannot do it.
Edit:
I have found the answers for $f_n(x) \rightarrow \infty $, but I don't understand yet how to prove the first cases. I think I have to use Baire's theorem but I don't know how to get there.

A: In the first case such a sequence can't exist. Note that $\mathbb{Q}$ is $G_\delta$ set, since if $\mathbb{Q} = \cap_n V_n$ with $V_n$ open, then $\mathbb{R} = \{ r_m \} \cup (\cup_n V_n^c)$ would be of first category (this is discussed more detailed here). On the other hand, the set $\cap_m \cup_n \{x : f_n(x) > m \}$ of points at which a sequence of positive continuous functions is unbounded is a $G_\delta$ set.
In the second case there is such a sequence. Let $\mathbb{Q} = \{q_k\}$ and consider
$$f_n(x) = \min_{1 \leq k \leq n} \{k + n \vert x - q_k \vert \} \geq 1.$$
Then for each $q_m \in \mathbb{Q}$ and $n \geq m$, one has $f_n(q_m) \leq m + n \vert q_m - q_m \vert = m$, hence $\{f_n(q_m)\}$ is bounded. On the other hand, if $x \in \mathbb{Q}^c$, then given $M> 0$, there is some $N = N(M) > M$ such that $n \vert x - q_i \vert > M$ for all $1 \leq i \leq M$ provided $n > N$. Therefore,
$$f_n(x) =  \min_{1 \leq k \leq n} \{k + n \vert x - q_k \vert \} > M,$$
that is, $f_n(x) \longrightarrow \infty$.


text:
Il doppio del Grand Prix SAR La Princesse Lalla Meryem 2012 è stato un torneo di tennis facente parte del WTA Tour 2012.

Le detentrici del titolo erano Andrea Hlaváčková e Renata Voráčová, ma hanno deciso di non partecipare.

Petra Cetkovská e Aleksandra Panova hanno superato in finale Irina-Camelia Begu e Alexandra Cadanțu per 3-6, 7-65, [11-9].

Teste di serie

  Anastasija Rodionova /  Arina Rodionova (ritirate, primo turno)
  Eléni Daniilídou /  Klára Zakopalová (ritirate)

  Nina Bratčikova /  Darija Jurak (quarti di finale)
  Ol'ga Govorcova /  Ol'ga Savčuk (quarti di finale)

Tabellone

Note

Grand Prix SAR La Princesse Lalla Meryem 2012


Teysha Technologies unveils 'breakthrough' in bio-degradable bio-plastics.
Teysha Technologies, a UK-based bioplastics startup, has developed a new polymer technology that it claims could help to reduce plastic pollution. According to the company, the natural polycarbonate platform it has developed can create fully biodegradable substitutes for existing petroleum-based plastics.
AggiePol, the bioplastic, is derived from sustainable feedstocks and can be physically, mechanically and chemically tuned to suit the needs of its intended application, according to Teysha.
Teyshas technology uses a plug-and-play system that takes monomers and co-monomers, the natural building blocks of plastics, to create an eco-friendly alternative to traditional polymers.
Instead of using hydrocarbon-based petrochemicals, which are sourced from fossil fuels and generate various pollutants in the manufacturing of the material, Teyshas platform uses natural products such as starches and agricultural waste products.
By controlling the chemistry, formulation and polymerisation conditions, the polycarbonate materials created by Teyshas technology can be precisely tuned.
Unlike existing bioplastics such as PLA and PHA, the degradation rate of Teyshas AggiePol can also be tuned, minimising the environmental impact of plastic products after the end of their useful lifetime, according to Teysha.
Until now, strategies to reduce plastic pollution have relied on changing human behaviour, such as encouraging recycling and banning certain plastic items like carrier bags, said Dr. Ashlee Jahnke, Co-inventor and Head of Research at Teysha Technologies.
By considering the ultimate fate of an applications material in the initial design stage, we can reduce the environmental impacts that occur from plastics.
Read:Thyssenkrupp-commissioned bio-plastic plant starts production in China.
Read:DuPont announces bio-plastic breakthrough with pilot plant world first.
Industry heavyweights back scheme to help end plastic waste in the environment.
5 minutes with… Patrick Krieger, Assistant Director of the Plastics Industry Association.
Supermarket giants gain compostable bag boost with the aid of expanded Novamont plant.
Video: Making edible food packaging made from milk proteins.


The Nixon 2 light wall bracket is a beautiful piece from Endon Lighting. This wall light is the perfect addition for those who are looking to cultivate a classy, traditional look within their home. The Nixon 2 light features a gorgeous sconce design and a sleek antique brass finish.
It also comes complete with a white half shade included.
The length, height and width of the Nixon 2 light wall bracket are 183, 260 and 350mm respectively. It also requires 2x SES (E14 cap) light bulbs with a maximum wattage of 40w.


text:
Fluajul este variația lentă, pe durate mari, a eforturilor unitare la întindere și a deformațiilor materialelor supuse unor solicitări continue. Este o creștere dependentă de timp a deformației rezultate din aplicarea în mod continuu a forței de întindere. Testele de fluaj se realizează la solicitări și temperaturi constante.

Fenomenul de fluaj se poate identifica la metalele solicitate la o sarcină constantă, la o temperatura înaltă. Aceste metale vor căpăta deformații de lungime din ce în ce mai mari odată cu creșterea timpului.

Mecanica mediilor continue
Deformare (mecanică)