We will begin discussion on a new topic today, this is on time integration of equation of motion, so at the end of our previous discussions we have seen that the governing equation of motion typically is formulated in this form MU double dot plus U dot plus KU, for sake of generality I am now introducing a non-linear term, we have not discussed how this originates but we can believe that this type of terms would arise if we include non-linear behavior either in strain displacement relations or in stress strain relations, so this is the equation of motion and these are the specified initial condition, so this equation constitutes a set of coupled second order non-linear ordinary differential equations, we say that these equations are semi discretized equations because in arriving at this equation we have discretized space but the time is still a continuous parameter, so they also constitute a set of initial value problem, so these are a set of ordinary differential equations, semi discretized ordinary differential equations which constitute an initial value problem and these equations are coupled. Now we consider the problem of discretizing in time, so what we aim to do is to we consider solution of this equation at a set of discrete time instance ordered as T0, T1, T2, Tn with delta Tn being the step size Tn plus 1 minus Tn, the basic idea here is to replace the derivatives appearing in the equation of motion by finite difference approximations and then solve the resulting algebraic equations, that is the main idea. Now to be able to do that what we will do is we will reorganize the equation of motion in a slightly different form, so this is the equation of motion that we have MU double dot plus CU dot plus KU plus non-linear terms is F of T and specified initial conditions, now I pre-multiply by M inverse, so this equation now becomes, takes this form, I will introduce now a new set of variables X1 and X2, X1 is U of T, X2 is U dot of T, I can define another 2N cross 1 vector which in which I assemble X1 and X2 as shown here, we call this as system state vector, so this is a state vector consisting of displacements and velocities at the N degrees of freedom. Now what is X1 dot? X1 dot is U dot, U dot is X2, so this is X1 dot is X2, X2 dot is U double dot, so that I obtain from the governing equation and we write in this form. Now I will assemble this equation in the matrix form, I will have X1 dot X2 dot is equal to this matrix into X1 X2 plus the non-linear term plus the excitation term. Now I can rewrite this set of equations in a general form as X dot is some function A X of T, T where A, this vector A is 2N cross 1, we say that this equation is in state space form, by state I mean displacement and velocity vector together that constitute the state, and this equation is said to be written in the configuration space where we have only displacements and we get second order differential equations. In state space we have a set of 2N first order coupled ordinary differential equations which are initial value problems and we say that the equation is in the state space form. So what we have done is the set of N coupled second order ordinary differential equations given by this in the configuration space have been thus recast as a following set of 2N first order coupled ordinary differential equation. Now what I do is I introduce a notation X of T, X naught, T naught I denote the solution of this differential equation with X of T naught equal to X naught, so there is T naught, X naught and T of course is the independent variable, in discrete time approximation time will be discretized as in a increasing sequence T naught T1, T2, TN with steps as delta TN as I mentioned before, and I denote the value of the system state at T equal to TN by YN, so YN is X of TN where initial conditions at T equal to T naught is X naught. Now the main question is how to approximate X dot in terms of YN, okay, that is the problem. Now we could use finite difference approximation, for example X dot of T can be written using forward difference scheme X dot of T is approximately equal to X of T plus delta T minus X of T by delta T, so consequently I can write this as YN dot is equal to YN plus 1 minus YN by delta TN, so from this I can derive now the equation for YN plus 1 which will be YN plus delta TN into YN dot, YN dot is nothing but A of YN comma TN, where N runs from 0, 1, 2, 3 with Y naught is equal to X naught at T equal to T naught. Now delta TN are the algorithmic parameters, delta T1, delta T2, delta T3 are the algorithmic parameters, if all of them are equal then delta T is the only algorithmic parameter. Now the accuracy of this solution depends upon this choice of these algorithmic parameters, delta TN, if delta TN are all equal the accuracy depends basically on the choice of delta T. Now we talk about errors, we call something on a local discretization error this is X of TN plus 1 comma XN TN minus YN plus 1, that means this is I have moved from T equal to TN to T equal to TN plus 1 to get X of TN plus 1, and that is approximated as YN plus 1, so this error is called local discretization scheme that means we are basically moving from TN to TN plus 1. On the other hand if you move from T naught to TN plus 1 that means all the preceding steps up to N plus 1 step if you cover then we call this error as the global discretization error, there is other error called round off error which is due to finite precision calculations, so you do either double precision or single precision calculation typically we do double precision calculations, so on a computer the digits get terminated so it is inevitable, so there will be errors that we call as RN plus 1, of course there are other errors which we call as dynamical system modeling errors, for example does algorithm correctly capture dynamical properties of system like natural frequencies, frequency response functions, free vibration amplitudes is the energy conserved, suppose I am dealing with an undamped free vibration problem is the energy conserved, so if there is any compromise on any of these issues these can be grouped as dynamical system modeling errors. Now the basic question that we are looking for is how to formulate the integration algorithms to achieve acceptable accuracy, there are so many errors of different kinds so associated with that there is a definition of an accuracy, so how to understand the nature of different errors, okay, and does this error as N tends to infinity norm of EN does it go to 0 or does it remain finite or does it become unbounded, so at every time step certain error is made either because of truncating of we are approximating the derivative by a finite difference approximation and there is a round of error, so consequently at every time step there will be an error, so how does this error committed at one step how does it propagate to the future steps, so does it go to 0 or it remains bounded or does it become unbounded, so now how to select algorithmic parameters to achieve acceptable performance of the integrators, so what are the expectations from a good integrator, so these are the type of questions we want to now address. Now let's spend some time on dynamical system modeling errors so that we understand what exactly is meant, so now let us consider the scalar equation of motion U double dot plus omega square U equal to 0, so it's a single degree freedom system natural frequency is omega and it starts with initial condition U naught and U naught dot, we can solve this problem exactly and we know that this is the solution, we have an amplitude and a phase that they are given by this they are functions of initial conditions and system natural frequency, now the question is suppose if I solve this equation using the discrete approximation, do we get amplitude of response to be constant, for example do we get the solution in this form where R is a constant, otherwise we get numerical dissipation or growth, if amplitude is not constant either it will decay or it will grow, so then that would mean that the algorithm has introduced some artificial dissipation mechanism into the system which is not there in the original equation of motion, then do we get the impulse response function correctly, is the distortion acceptable, what is acceptable distortion, now is the vibration energy conserved, some of these questions are important, so answering these questions would enable us to discuss the dynamical system modeling errors, we could also consider for example a damped harmonically driven oscillator and we know that in steady state this type of systems have certain well known features, there is a resonance and where the resonance occurs, what is the resonance amplitude, what is the shape of the frequency response function and so on and so forth, there are well known qualitative features associated with the response of this system, now the question we can ask is does the discretized version of the solution possess these well known features or not, okay, so that is one, similarly if there is a nonlinear term, we are not discussed about nonlinear term but we can think of adding say for example alpha U cube here, now such systems display a pattern of bifurcations, we will come to that later but at this stage it is suffice to observe that nonlinear systems have certain qualitative features in the associated with their behavior and we can ask the question does the discretized version of the equation display the same type of qualitative behavior as the continuous version, okay, these are, this type of issues can be grouped under the heading of dynamical system modeling errors. So what we are going to do is there will be a few mathematical preliminaries that are needed to understand the questions and the answers that we will be discussing related to the properties of these integration schemes, and there are some set of terminologies, we talk about implicit schemes, explicit schemes, single step method, multi-step method, self-starting and not self-starting, we talk about consistency of the stability of these methods and we discuss energy accuracy, energy conservation properties and so on and so forth, so what we will do is we will try to discuss these issues as we go along, so we will start with some simple mathematical preliminaries, we discussed this earlier in one of the lectures, we will be using this so called gauge notations, the O and capital O and lowercase O notations, we say that F of X is of order G of X as X tends to A, if as X tends to A this ratio F of X by G of X remains finite, okay, and if this goes to 0 then we say that it is lowercase order G of X, okay, the mathematical description is given here I will not get into these details, I have provided it for sake of completion but I will list right with few examples. Now let's consider this function A into X to the power of 7 BXQ plus EX plus D, we make a statement that this function is order X to the power of 7 as X tends to infinity, so how do you verify? You would divide this by X to the power of 7, okay, and you can see that this goes to, as X tends to infinity this goes to A, right, the first term will be A, second term will be B by X to the power of 4, C by X to the power of 6, D by X to the power of 7, so in the denominator as X tends to infinity all the terms go to 0 except the first term which is A, so we say that this statement is verified by checking this, similarly this function is order X to the power of 0 as X goes to 0, how do I check? You take X to the power 0, as X goes to 0 only D will remain, so this is finite, right, so this polynomial is order X to the power of 0, now A into X to the power of 7 is order X to the power of 7 as X goes to 0, how do you verify? You divide by X to the power of 7 and go take X to the power of 0 I get A which is finite, so there are a few more examples you can verify, you know, get a feel for what these terminologies mean. So I have a few more examples, let's quickly run through this, sine X is order X as X goes to 0, why?  There is one, another mathematical preliminary that we would be needing is the, we should quickly recall what is the mean value theorem. Now consider a function F of X to be continuous for X lying between A and B and differentiable for, you know, this is the closed interval, this is open interval, it is differentiable in the open interval. Then according to the mean value theorem there exists at least one C satisfying the condition that C lies between A and B such that DF by DX at C is exactly given by this, F of B minus F of A divided by B minus A, that you can see here this is my F of X and this is A and this is B and you can see that at this point and at this point if you draw the curve DF by DX which is given by this, this will be parallel to the true, you know, this is approximation so this will be parallel to this, which is a tangent to F of X at those points. So at C1 and C2 this is an exact representation, but of course for a given F of X we would not know where is that C, how many of them are there whether there is, you know, where it is located we would not know. Now Taylor series is something that we are all familiar with, so let's quickly see the statement of some of the results associated with Taylor series expansions, so again let F be a function of X with derivatives of all orders throughout an interval containing point A, the Taylor series generated by F at X equal to A is given by F of A plus F prime A into X minus A, F double prime A by 2 factorial X minus A whole square and so on and so forth, this is a well known Taylor's series. Now if you truncate the Taylor series at say the nth term we get what is known as the Taylor's polynomial, so that is other conditions remaining the same I write PN of X as this, so this is known as Taylor's polynomial, okay. Now the corollary to the Taylor's theorem, if F has derivatives of all orders in an open interval I containing A then for each positive integer n and for each X in the interval I, F of X can be written as F of A plus F prime A X minus A so on and so forth, after nth term there is a reminder term and this reminder is given by this for some C between A and X, this again we are using mean value theorem in writing this, now if this limit of this reminder term goes to 0, right for all X in I we say that Taylor series generated by F at X equal to A converges to F on I and we write F of X as this, okay. Now we talked about forward difference, backward difference and central difference, we talked about finite difference, so some of the examples are forward difference, backward difference, central difference, so let's see quickly what it means. So let's consider a function F of X and I write the Taylor's expansion F of X plus H is F of X plus H F prime X plus H square by 2 factorial F double prime F X and so on and so forth. Now from this I can write the solve for F prime of X, F prime of X will be given by F of X plus H minus F of X divided by H, so you are taking this other term to the other side you are dividing by H therefore this becomes H, H by 2 factorial plus H square by 3 factorial and so on and so on. So this term is clearly of the order H because if you divide by H and take H to 0 this will be a finite quantity, right. So this we say that this is a forward difference approximation to F of X at X. The backward difference approximation to derive that we consider instead of F of X plus H I consider F of X minus H, so this will be F of X minus H F prime of X and so on and so on. Again if I solve for F prime of X I get this expression and divide by, we have divided by H and we can see that these terms are of order H as H goes to 0, so this approximation is known as backward difference approximation to F prime of X. To derive the central difference approximation what we do is we consider F of X plus H which is this expansion and also F of X minus H which is this expansion. Now I subtract these two, so if moment I subtract these two these two get cancelled these two add up I get 2 H F prime of X and similarly this H square term will get cancelled and I will get the next term will be of the 2 H cube by 3 factorial so on and so on. Now if I solve for F prime of X from this I get F prime of X is F of X plus H minus F of X minus H divided by 2 H plus this term will be divided by H, so this will be order H square so as H goes to 0, okay. So this is the central difference approximation for derivative of F at X. Intuitively we can see that if error is of the order H and if we half H that is a step size H the error gets half, similarly if the error is of the order H square and if we half the step size the error will get quartered, okay, so that is the advantage of order H square, okay. Now let's return to the equilibrium equation in the standard form we have already seen that this equation can be written in this standard form in the state space. Now what I will do is I will replace this X dot by finite difference approximation and see what we get, so we will consider a sequence of increasing time instant 0, T1, T2, T capital N, capital TN is TF which is a final time instant up to which we want to integrate this equation and we denote YN as value of the system state at T equal to TN by starting at T equal to 0 with an initial condition X naught. Now for X dot of T if I use forward difference approximation X dot of T will be written as X of T plus D delta T minus X of T by delta T and this must be equal to A of X of T, T. So now if I solve for X of T plus delta T which is YN plus 1 will be equal to YN that is X of T plus delta T is on the other side delta T YN dot, YN dot is A of YN, so I will get YN plus 1 as YN plus delta T A YN and this order of approximation is order delta T, okay. Now if we want now backward difference approximation we consider time instant T plus delta T and go back in time and I get X dot of T plus delta T is X of T plus delta T minus X of T by delta T, so this looks similar to this but here you must notice that I am writing this at time instant T, okay T plus delta T is ahead of the time at which I am writing this, whereas here the T is lagging, okay T plus delta T is the current time and whereas T is the previous time instant. Now here again I can write YN plus 1 as YN plus delta T this is important this is Y dot of N plus 1 whereas here it is Y dot of N, so this will be YN plus delta T A of YN plus 1 whereas this is A of YN and the order of accuracy is still order of delta T, the central difference scheme I can write this so again at time T I write X dot of T is X of T plus delta T minus X of T minus delta T divided by 2 delta T this must be equal to A of X of T comma T, so I want to write for X of T plus delta T I will get YN plus 1 this is YN minus 1 2 delta T into YN dot this is YN dot so that is YN minus 1 plus 2 delta T A of YN and this approximation is order delta T square, okay. Now we can also develop another scheme which is slightly different from logic is slightly different, suppose X of T I write it as 0 to T X dot of SDS, that's the definition of a derivative, so if you consider T equal to TN plus 1 this will be 0 to TN plus 1 X dot of SDS, this 0 to TN plus 1 I can write it as 0 to TN plus TN to TN plus 1, so this first term is nothing but X of TN and this is an increment, for this second term I will use a trapezoidal rule of integration, so I will therefore I will get this as YN plus 1 as YN plus half of delta T YN dot N plus 1 plus YN dot, okay, so from this I get this as the approximation, now for YN plus 1, Y dot N plus 1 I will write A of YN plus 1 and YN dot I will write it as YN plus delta T A YN, so this approximation is known as the approximation based on trapezoidal rule. Now we can summarize all this, so in the forward difference scheme I had this representation, in backward difference scheme I had this representation, in central difference I had this, in trapezoidal rule I had this. Now a generic form based on this we can see that a generic form can be written as YN plus 1 is alpha 1 YN plus alpha 2 YN minus 1, so on and so forth alpha M YN plus 1 minus M plus delta T this is on Y of T and these are derivatives, some beta naught Y dot N plus 1, beta 1 Y dot N so on and so forth up to this kth term, this is a general form, all these schemes will fit into this by selecting alpha 1, alpha 2, etc, beta naught, beta 1, etc in a suitable manner I can recover these schemes. Let's consider the generic form, again you look at the term beta naught, so I am writing YN plus Y at T equal to TN plus 1 and this involves the derivative of the state at N plus 1 if B naught is not 0, okay, if B naught is not 0 therefore YN plus depends upon derivatives of the state at TN plus 1 that is upon Y dot N plus 1, so such schemes are known as implicit schemes, okay, otherwise if beta naught is 0 then the scheme is said to be explicit. Now if we now consider alpha 2, alpha 3 and alpha M as 0 and beta 2 up to beta M are 0 then we say that the scheme is a single step scheme, otherwise it is called multi-step scheme. Now the scheme is said to be self-starting if YN for N less than 0 does not enter the calculations, okay, right, so if I write this for N this term is a previous step, okay, suppose if N equal to 0 this will be YN and this is alpha 1 Y naught there is no problem but in certain schemes you will soon see that I want to start with N plus 1 equal to 0 and this I will be needing Y of minus N and such schemes are said to be not self-starting, okay, we need to see that. So let's quickly see in this scheme of things how does this different classification schemes can be viewed as, suppose if you now consider forward difference approximation which is given by this, now you can see that this is a single step explicit self-starting scheme, okay, explicit because YN plus, we have YN plus 1 here and there is no derivative of YN plus on the other side.  Similarly central difference approximation is it's a multi-step method, it is explicit and it is not a self-starting scheme, because here YN minus 1 you will see that in the due course, for example for N equal to 0 this will be YN and I will be needing Y of minus 1, and Y of minus 1 is a hypothetical quantity which is not there as a part of definition of the original problems of mechanics, problem of mechanics, so this is not a self-starting scheme and we need to devise some special methods to start the solution. Similarly trapezoidal rule is single step implicit and it is self-starting, okay, why implicit because I have A of YN plus 1 here which is YN plus 1 dot, okay, so if system is non-linear you can see that in implicit schemes you need to solve every time step algebraic non-linear equation, okay, so the implicit schemes that way demand computational efforts especially if system is non-linear, if system is linear we can take these terms to the left side and rearrange the terms which is straight forward, so the difference between implicit scheme and explicit scheme will be strongly felt in computational effort for non-linear systems, okay. Let's make some observations and remarks, so we have discretized time T0 to TF in terms of 0, T1, T2, TN which is a non-decreasing sequence of, actually increasing sequence of time instance. If we now take TN to be N delta T we say that we have a constant step size, okay, now system states are U of T, U dot of T and U double dot of T in our calculation. Now if we assume that system states at T equal to TN are known and that is what we assume then the problem on hand is to find system states at T equal to TN plus 1, the integration algorithms essentially achieve this, so these are known as time marching techniques, so we move from TN to TN plus 1 and we want to take state from TN to TN plus 1, so these are called time marching techniques. Now the computation effort is proportional to the number of time steps, okay, to advance the solution from T to 0 to T to TF, therefore the step size need not be smaller than what is essential, what is the meaning of something being essential it is something to do with accuracy basically, so what we need to be concerned about? We need to be concerned about growth of errors, does error committed at a time step say TN grow as time advances, if it grows we say that the solutions are unstable, the scheme is unstable, otherwise it is if it remains constant we say it is stable, if it goes to 0 we say that it is asymptotically stable. The accuracy of the solution, if the errors don't grow it doesn't mean that we get accurate answers, okay, how to judge accuracy, that's a different issue. Now for example for undamped free vibration of a single degree freedom system we can check for amplitude, amplitude should remain constant and frequency distortions must not be there, we know that frequencies say for example square root K by M, do we get that square root K by M in numerical simulations or not? The time integration method is the most generally applicable method to solve equilibrium equation, we have talked about frequency response function based methods or dynamic stiffness and so on and so forth, they are valid only for linear systems and of course there are additional requirements that system should reach harmonic steady state, but here the system can be nonlinear, the excitation could be transient, this method remains applicable. Now for linear systems the solution actually does not require transformation to natural coordinates, see for linear system we can do, we can find natural frequencies, mode shapes and uncouple the equation, we have that option, but if you are trying to integrate the equation numerically it is not needed to actually perform the uncoupling of equations of motion, if you can do it it helps but it is not essential, so this would mean that we have greater flexibility in modeling damping, see we have seen that if damping is classical there are many advantages in doing analysis based on mode superposition, but if you are doing direct integration there is no special requirement on what should be the damping matrix because you can directly integrate. Now but in practice we prefer to specify damping in terms of modal ratios, in that case damping will be specified in terms of formulation which employs normal modes, natural coordinates and normal modes, so if you intend to use direct integration in such situations you have to formulate the C matrix from the known information about modal damping ratios, so we have discussed how to do that in the previous class. As I already said the method remains applicable even when equations of motions are nonlinear. Now let's look at modeling of linear multi degree freedom systems, so how many degrees of freedom should be included in our model, we will discuss this in again at a later stage but we can start asking these questions at this stage. Now the size of the model that is number of degrees of freedom actually depend on details of mesh used in the spatial discretization, now these details need to be chosen such that the system behavior is represented with acceptable accuracy over a given frequency range. What is the frequency range that we should select? We should look at excitation, suppose you are dealing with say earthquake like excitation the frequency range can be up to say about 20 to 30 hertz from low frequency to up to that frequency, wind it could be 0 to 2 hertz or 3 hertz wave also in the same region, so if you are performing say seismic response analysis of engineering structure you should ensure that say within the frequency range of say 0 to 20 hertz whatever are the natural modes that the structure possess all of them should be modeled accurately. So we have seen that if you want to capture first say 5 modes in a model you should have about 50 degrees of freedom in your model that means the spatial discretization should be fine enough so that you get a model with 50 degrees of freedom so that at least 1 tenth of the natural frequencies will be computationally trustworthy. So that would mean if omega max is the highest frequency we will say that the discretization scheme should be such that all the modes that lie in frequency range up to 1.25 omega max need to be captured well, so the mesh details should be such that the model has at least 10R degrees of freedom where R is a number of degrees of freedom, R is a number of modes that you expect the structure to have in frequency range up to 1.25 omega max. A consequence of this is that when you are using direct integration you will always be dealing with models which has spurious higher order modes, suppose you are making a model with 100 degrees of freedom we know that the first 10 modes are likely to contribute to the response and modes from 10 to 100 are the higher order spurious modes. Now it is desirable that the discretization scheme that is the time discretization scheme should have some inherent capability to numerically dissipate the higher order spurious modes but not affecting the genuine lower order modes which contribute significantly to the response. So that would mean that we look for certain numerical dissipation characteristics in our integration schemes when we formulate the time integration schemes. So what is desirable is that the lower modes should not be numerically dissipated but the higher spurious modes should be numerically dissipated, okay, we will see more about that but this is one of the concern that we need to appreciate at the outset. So what we will do now is we will take up discussions on few methods, I propose to discuss the following method, the forward Euler method, the backward Euler method, the central difference method, and there are methods known as Newmark's family of methods and HST alpha method and so on and so forth. So we will see the development of these methods and as we go along we will see what motivates, what are the motivations to develop different schemes. After getting for example backward Euler scheme what is the need to go for central difference method, and what is the need to go to Newmark's family of methods when you already have done central difference method and so on and so on. So the questions will be on stability, accuracy, dissipation of higher spurious modes and so on and so forth, methods being implicit or explicit these issues also will come up. So what we, the strategy we will take is we will develop the basic formulary for each of these schemes and I will provide a pseudo code for implementation then we will analyze the method for its, how the errors behave in each, the specific method, and then we will illustrate each of these method with a few examples and we will conclude by discussing the relative merits of different methods. So this is a scheme of things that we will follow. So let me start with the discussion on forward Euler method, so this is a time axis T naught is here, TN, TF is the final time instant which is here. So let's the equation of equilibrium is MU double dot plus U dot plus KU equal to F of T at time T, these are the initial conditions. Now at time T I will approximate the velocity U dot of T using forward difference scheme which is U of T plus delta T minus U of T delta T, so from this I can get U N plus 1 which is U of T plus delta T as delta T into U dot that is U dot N plus U N, a similar approximation I can make for acceleration also, so that will be U dot of T plus delta T minus U dot of T by delta T where U dot of N plus 1 is obtained as delta T U N double dot plus U N dot, so you can see here this is an explicit scheme because when I am writing the state at N plus 1 I don't need acceleration at N plus 1, okay. Now so let us consider the condition for equilibrium at TN plus 1, so this is written as MU double dot N plus 1 plus for U N dot, U N plus 1 dot I will use this, okay, this is this, then for U N plus 1 I will use this, so this is this, this is equal to FN where F1 is F of TN, now I will rearrange these terms what is not known is U N plus 1 double dot and that is obtained in terms of the remaining terms like this. So how do we implement the method? You can see here the scheme is very clear I have to bank on this equation, this equation and the final equation, so I will start with T equal to 0, N equal to 0 we will input the initial condition U naught and U naught dot, I will also need the initial acceleration I will obtain it from the equation of motion.  Then I will find the velocity and the displacement, then I will increment N and if N delta T is more than TF I will like stop, otherwise I will go to 2, so this is the simple minded outline of how to implement a forward difference scheme, but if you carefully look at this it is not necessary to invert M matrix at every time T, you need not have to multiply M inverse and C at every time T and so on and so forth, so we can refine this a bit, so what I will do is I will calculate these matrices inverse and these products outside the loop, they need to be done only once, we need not do every time T, so if you do that I get a more usable implementation scheme, so I will input KMC matrices and delta T there is some algorithmic parameter and we will also store all the excitation and then the initial conditions, so I will define A as M inverse, B as A into C, C is damping, D is A into K, now we will start with N equal to 0 and we will accept this U naught and U naught dot, form U naught double dot, so I am not inverting any of these matrices nor I am multiplying any matrix now, so U N plus 1 double dot is given by this, I get similarly velocity and displacement and I will increment time and if I cross the final time instant I will stop, otherwise I will restart with this step, so most of the calculation gets done in the steps 4 to 6 and I am not repeating any calculation, if I can avoid I have avoided all that, okay, so this is a forward, you will learn forward difference scheme, okay, it's logically simple, so what we can do now is we can think of asking how does errors behave when I apply this scheme to analysis of simple systems, now for purpose of illustration I will consider a single degree freedom system under some excitation F of T, now this is the scheme I will get using forward difference scheme for this problem I will get this is the scheme, okay, now this can be written as capital Y N plus 1 as A Y N plus L N, okay, where A is this matrix, okay, now suppose at the nth step the state that is U N plus 1 and U N plus 1 dot is contaminated by noise gamma N, now the question is how does gamma N behave, so I will substitute this into this equation, so Y N plus 1 will be Y N plus 1 plus gamma N plus 1 is equal to A Y N plus A gamma N plus L N, so now Y N plus 1 is solution to this equation therefore this and this cancel out they are equal, the sum of these two is equal to this so that gets out of reckoning, so the error is gamma N plus 1 is A gamma N, so this is the equation for evolution of the error gamma, so we will digress a bit now, we will return to that shortly but let us examine the nature of this kind of finite difference equations, suppose you consider a scalar equation X N plus 1 is A X N and we start with some nonzero initial conditions, so X 1 will be A X naught, X 2 will be A X 1 which is A square X naught and X N will be A to the power of N X naught, now if I am interested in knowing what happens to X N as N tends to infinity, we can easily see here that this function will go to 0 if modulus of A is less than 1, suppose A is 0.1, A square will be 0.01, A to the power of 3 will be 0.01 and so on and so forth, A to the power of 10 will be 1 into 10 to the power of minus 10 so on and so on, so this is the condition. Now if A is 1, plus 1 or minus 1, X N will stay at X naught as N tends to infinity, okay, and similarly if A is greater than 1, suppose A is 2, initially it will be 2 X naught then 4 X naught then 16 X naught and so on and so forth, it goes to infinity as N tends to infinity, so the behavior of this X N as N tends to infinity is crucially governed by the value of A, if the absolute value of A is less than 1, X N goes to 0 as N tends to infinity, this is a scalar equation.  Now let's do the following, let us introduce the transformation X N is equal to phi Z N, so this equation for X N plus 1 will be phi Z N plus 1 will be equal to A phi Z N, now let phi be such that phi transpose phi is identity and phi transpose A phi is a diagonal matrix, say if A is such that this is possible, how do we select? We can select phi by solving the eigenvalue problem associated with A that we have seen in few occasions how to do that, now suppose A be such that this is possible, then I can write phi Z N plus 1 equal to A phi Z N and pre-multiply by phi transpose I get this equation, now since phi transpose phi is I, I get Z N plus 1 and phi transpose A phi is a diagonal matrix of lambda I I get this, okay, so I can, Z is a S cross 1 vector therefore I can write the kth element of that Z N plus 1 k as lambda k Z N k where k runs from 1 to S, so this is the equation I get Z N plus 1 k, now the behavior of this individual Z as k tends to infinity depends on absolute value of lambda k, the kth eigenvalue, right, now if you are interested in jth component of your vector X that is given by this summation, okay, so the behavior of this as N tends to infinity crucially depends on the highest eigenvalue, if you rank order this lambda k depending on the absolute value of its, that is the absolute value of lambda k we are interested in, if you are interested in knowing what happens to this modulus of XJ, XNJ as N tends to infinity for this to go to 0 we require that the maximum value of this lambda k must be less than 1, okay, because each one is a scalar equation, for each scalar equation to go to 0 the each of the eigenvalue absolute value should be less than 1, so if the highest eigenvalue is modulus is less than 1 then XNJ will go to 0 as N tends to infinity, so therefore if you are interested in behavior of XNJ as N tends to infinity that is controlled by the highest modulus of the eigenvalues of A, okay, now alternatively we can also consider XN as A XN plus 1, A into XN minus 1 I can write it as A to the power of N X naught, now if we seek the solution now of the form XN is alpha N into X naught suppose if we seek this solution, for some value of alpha such a solution may be possible for which value of alpha this is possible we have to see, this alpha could be scalar which can in general be complex value, so now you substitute this into this equation I get XN is alpha N X naught therefore XN plus 1 is alpha N plus 1 X naught, now if you consider this equation XN plus 1 is A XN A to the power of N X naught and make these substitutions I get alpha N plus 1 X naught is A alpha N X naught, so from this I get A X naught is alpha X naught, so this would mean for non-trivial solutions the determinant of A minus alpha I must be equal to 0 and if we represent the roots of this equation which are the eigenvalues of A, suppose Ith root is written as alpha 0 I exponential I theta I, this is a complex number so I can always represent like this. The condition for XN J modulus of that as N tends to infinity to go to 0 is given by the condition that the maximum value of absolute value of alpha I must be less than 1, so that would mean actually this quantity which is a maximum value of modulus of the eigenvalue for I running from 1 to S is known as spectral radius of A, okay, so what we are asking is the spectral radius of A must be less than 1, so that would mean if you look at the eigenvalue in the complex plane we have real part here imaginary part here and this is the so called unit circle, okay it has radius equal to 1, what we want is the root should be inside this unit circle for stability, asymptotic stability, if it is right on the unit circle the errors don't grow but it is stable but if it is outside the unit circle the errors would grow, okay, so this is what we need to verify if you are interested in studying the growth of errors. One more thing that we should notice here is the only question we need to answer is whether roots lie within unit circle or not, we are not so much interested in knowing the absolute value of the root, we are simply interested in knowing a qualitative feature associated with the roots, so we do not need the value of the eigenvalues, we need to only ascertain if all the roots of the characteristic equation lie within the unit circle in the complex plane, given that we are asking a simple question then actually we are not interested in determining the value of all the eigenvalues, we are simply interested in knowing whether all eigenvalues lie within the unit circle or not, to answer this simpler question we can develop an alternative method which is computationally simpler than finding all the roots and finding out whether they lie within unit circle or not. So that takes us to discussion on what is known as Jury's criterion where given a polynomial by using Jury's criterion we can verify whether roots of a polynomial lie within a unit circle or not, so that has bearing on discussion on stability of the finite difference time integration schemes, so what we will do is we will take up that issue in the next lecture and we will conclude this lecture at this stage.


Welcome to Spotlight. I'm Ryan Gertzma. And I'm Robin Basilen. Spotlight uses a special English method of broadcasting. It is easier for people to understand, no matter where, in the world they live. Every year in May, actors, filmmakers, and experts gather together in Cannes, France. They gather for the International Cannes Film Festival. There, they watch new films from all over the world. And groups of experts judge the films. In 2010, one of those new films was called Benda Bellilli. Filmmakers Renault Ballet and Florent de la Toulée created this film. In it, they tell a true story of an unusual music group from the city of Kinshasa. Today's Spotlight is on this film and the staff Benda Bellilli music group. Kinshasa is in the Democratic Republic of the Congo. Life in this city can be difficult. Most people suffer from poverty. Often, they do not have enough food to eat. And large groups of people live together in shelters. In these conditions, diseases like polio spread very easily. Papa Ricky Lakabu and Coco Gambali grew up in Kinshasa. They both had the disease polio as children. Polio left Papa Ricky and Coco disabled or handicapped. The two men could not use their legs. However, both men could play music. In fact, they were extremely skilled. Still, many people could not look beyond their handicaps. Papa Ricky told BBC News that he could not find work as a musician. When I went to see my musician friends to look for work, they did not want to work with me, even though I knew how to sing and play the guitar. They did not want to work with me at all, because I am handicapped. They said, you are always late and you cannot dance. Working with us just will not work. So Papa Ricky and Coco started their own music group called Staff Benda Belili. Soon, other musicians joined their group. They were also handicapped from polio. Some played instruments, some sang, and some of them even danced by using their hands like feet. All of the men worked other jobs during the day, but they played their music as often as they could. The men of Staff Benda Belili believed they could do anything they wanted, even with their physical handicaps. Benda Belili means look beyond appearances. And this is what they wanted people to do. Look beyond their physical handicaps. The members of Staff Benda Belili dreamed of being famous for their music, not for their handicaps. They hoped that one day their music would even be popular in Europe. Staff Benda Belili started by playing their music on street corners. They played their music outside bars and restaurants where people ate. People liked their music and gave them money. But the musicians did not earn enough money to stop working their other jobs. Then, in 2004, filmmakers Renaud Berrette and Florent de la Toulée heard Staff Benda Belili performing in the streets. In their film Benda Belili, Berrette describes how they felt when they first met these musicians. When we met Staff Benda Belili on the corner of a street, we immediately loved these undiscovered skilled musicians. Their songs were about living on the streets of a city that had no mercy. Their music went straight to our hearts. After that, they decided they wanted to work with this music group. They wanted to help them make a music album. And they wanted to film the process. Staff Benda Belili and the filmmakers began to work together. The filmmakers recorded everything. They filmed the musicians as they practiced their music at the Kinshasa Zoo. They filmed the musicians when they played in the streets. They even filmed the musicians when they were not playing music. And they were filming when Staff Benda Belili met Roger Landoux, a 13-year-old musician. Roger worked in Kinshasa to support his family. He created his own instrument from a bent wood bow, a tin can, and one metal wire. He called his instrument a satongue. With it, he played popular songs to make money. Papa Ricky and Staff Benda Belili heard Roger play and were very amazed. Papa Ricky knew that Roger could be a very skilled musician if he trained more. So Papa Ricky invited Roger to join Staff Benda Belili. Roger practiced with the group. He learned from the more experienced musicians. Soon he began to perform with them. After a while, the group went to a music studio to record their first music album. Recording was not easy. The musicians made many mistakes. And during the recording, Papa Ricky's wife called. She told him there was a fire. The fire destroyed the shelter where many of them lived. As a result, the men went home without finishing their album. A year later, Beret and Dula Tulei came back to Kinshasa. They had found a music recording company that would help Staff Benda Belili create their first music album. In their film, Beret and Dula Tulei show Papa Ricky encouraging the rest of the group before recording. Let us put our problems away until we have an album in our hands. Finally, in March 2009, after five years and many difficulties, Staff Benda Belili finished recording their album. They called it Tre Tre Fort, which means very, very strong. A few months later, they traveled to Europe to play for a music celebration event in France. Staff Benda Belili continues to travel and play their music, sometimes for crowds with thousands of people. The men of Staff Benda Belili have seen their dream become real. Their music became famous in many countries around the world, and they traveled to Europe. They achieved this because they never stopped hoping and working hard. Staff Benda Belili is also thinking about the future. Music gave Papa Ricky and the other men hope. It helped them think positively about the future. They want to share that hope and positive thinking with future generations. Dula Tulei explained, with the money they earn from the film, they want to start an organization. They want to build a place to teach young musicians and continue the Staff Benda Belili experience. Staff Benda Belili should have no trouble achieving their new dream. Roger is a good example. He learned a great deal from the men of Staff Benda Belili. He said, I am the youngest member of the group. I grew up with the others, so I cannot let them down. When I met Ricky, he told me that someday I would become the leader of the Staff Benda Belili. So I have a lot of responsibility and a lot of work left to do. The writer of this program was Courtney Scott. The producer was Ryan Gertzma. The voices you heard were from the United States and the United Kingdom. All quotes were adapted and voiced by Spotlight. You can find our programs on the internet at www.radioenglish.net. This program is called Benda Belili Achieving Dreams Through Music.


What is going on guys, welcome back to the channel and I appreciate you being here. Today I am back with another Destiny 2 video and today I bring you more information on the season of the Drifter which drops next week as last night I brought you basically a recap of the Viaduct they dropped explaining a lot of what we will get with the Joker's World DLC, but the Trab last night covered much more information on this next DLC for Destiny 2 which I want to dive deep into today guys and to be honest last night I wasn't too optimistic about the DLC I mean some parts of it looked promising but I was still underwhelmed a little with what they seemed to offer but after reading the Trab I mean like my hopes are high now they really are but before we get into that guys if you do enjoy the video and would like to show you support you can by hitting that like button if you are new around here and enjoy daily Destiny 2 videos be sure to subscribe okay so yesterday we got the Viaduct which explained a lot of what we will get with the Joker's World with the Gambit Prime new game mode with the Wrakenin new game mode a few other things two new Exhaust requests so many other things to be honest if you missed that video you'll find it linked within a video description it covered basically what you needed to know about the Viaduct I mean the Viaduct if you ain't seen also you'll find that linked within a video description but I'm pretty sure going through this Trab today we'll go through everything that Viaduct offered so yes let's get into it people so firstly within the Trab what I want to cover they showcase this new Roadmap which you guys can see on screen now which actually shows us what we will be getting over the next month or so now we see March 5th we get Gambit Prime and Wrakenin Tier 1 March 5th is when the Joker's World will drop which is next Tuesday we've got Gambit Prime we get a new Gambit map too called New Arcadia we also get Gambit Private matches on March 8th we get Wrakenin Tier 2 this is that other game mode which looks like some weird push forward horde mode but to me looks absolutely amazing I love this sort of thing the only problem I have with it though is within the Viaduct it seems as though it's tied to Gambit so you have to go back and forth between Gambit Prime and Wrakenin I mean I am a massive Gambit fan I mean Gambit Prime could change my mind and that I love the fact that it's only one round based as well and you unlock perks which make it progress over time that sounds cool hopefully as we get into this Trab today they explain a little more about the actual loop you have to take between these two modes so we will see Gambit Prime Deep 6th is a new map which will come on March 12th also on March 12th we will get a new quest for the form called the Allegiance Quest March 15th we get Invitation of the Nine and Wrakenin Tier 3 I wonder what this is I mean moving into Viaduct we did see the Nine talking to Judge Drifter it seems like they're cooking something up so we will see about that March 19th we'll get Gambit Prime Legions Foley not sure what this is March 26th we'll get Gambit Prime Emerald Coast I'm guessing these are just maps to be honest they look like maps on April 2nd we'll get Gambit Prime all maps available private matches Gambit Prime New Arcadia and Deep 6th I'm guessing that will all come into one that's what's going to happen there so progress towards April 2nd each week we'll get a new map by an April 2nd we'll get it all at the same time and April 9th we'll get Art Week 3 to all players no idea what that is an April 6th to May 6th we'll get the Reverie which is a new event and this is 3-2 all players so that's a road map going forward with the season of the Drifter so I'm moving on with a twab and we see state's prime accessibility and this talks about colorblind mode because obviously if you didn't know with Gambit Prime there will be select roles with each role you'll get a certain armor each armor is represented by a certain color so basically the four roles are Reaper Invader Collector and Sentry the Reaper is the person who kills all the combatants the Invader is the person who invades and kills the other team the Collector is the guy who collects all the malts and the Sentry is the guy who takes out the taken within that prime evil battle and we can see what these four colors for these four represented armors will look like within colorblind modes on stream now now also with the joke as well we will get new pinnacle weapons there's three in total is a vanguard one called Oxygen SR3 which is a precision scout rifle which will offer that solar burn and the Oxygen's unique perk Dragonfly deals more damage based on a number of positions it's made or beforehand which sounds pretty cool for the crucible we get the recluse a lightweight machine gun and it's void this thing's unique perk is called Master of Arms kills with any weapon improve these weapons damage for a short time and then there is a Gambit one which is called Delirium I believe that's pronounced rapid fire machine gun and this will offer arc the delirium's unique perk is called killing tally kills increase the weapons damage until it is stalled or reloaded damn that sounds pretty cool with their reintroduction of bullet hole star machine guns we figured a perk that played with large magazine sizes and deal with waves of enemies would be appropriate unlike the other two weapons this one has one of the new weapon perks we're introducing in season of the drifter Overflow to kick the magazine size out even further or true spray and play action while we're on the subject of weapons here's another preview of some changes made for the season of the drifter Exotic power weapons in destiny have been the goal to for quite some time due to getting the most bang for your book from raw damage these changes along with others that will be shown when the full patch notes for 2.2.0 are revealed are intended to make legendary power weapons a good option for those who enjoy their exotic kinetic and energy weapons we believe that the exotic you choose to run should be an expression of your play style rather than just the math working out in your favor exotics will always have their unique properties but a well-gold legendary power weapons should be able to compete in raw damage with an exotic power weapon although the legendary weapon mainly conditional setups and mods to outright match nonetheless the gap should not be so wide that you the player feel it necessary to run an exotic in any specific slot grenade launchers pve damage increased by 25 percent reserve ammo increased on all grenade launchers in most cases grenade launchers gain three rounds in reserves but this amount may vary based on the perks you have good prep etc magazine perks and mods no longer affects ammo reserves grenade launchers had a bit of a reserve issue due to the magazine having an impact on reserves and as a result the reserve totally widely changed typically not in the favor of the player this was addressed and they get a pve damage bonus on top of it too to make them more competitive in that slot rocket launchers pve damage increased by 60 to 65 percent exotic rocket launchers have had their damage tuned separately cluster bomb damage reduced by 80 percent like what this lost damage was moved to the rocket launchers main projector and noted above in its damage increase okay then i get that rocket launchers were entirely too dependent on cluster bomb to be effective like the full autopilot construct guns cluster bombs would nearly double the damage output of any given rocket under optimal conditions the result was that on any target that you couldn't get all the cluster bombs to impact a detonate rocket launchers fell by the wayside pretty harshly by moving most of the damage of cluster bombs into the rocket itself rocket launchers will now be more reliable and more effective around especially against aerial targets that cluster bombs would almost never damage okay so a few changes to weapons there nothing major to be honest i just wonder how that grenade launcher change is going to affect them in pvp i mean on older pve damage is increased by 25 percent but i'm guessing the increased reserve ammo is counting for both pve and pvp they weren't really clear on that to be honest okay so we're going to move on to eververse with a seasonal update with each season we have an opportunity to update our goals surrounding eververse and the ways players engage with it in season of the drifter we've put more focus on giving players control in the ways they acquire the items they wish to equip the following is from the dev team last year we talked about our efforts to give you more control over how you purchase eververse items we released the prismatic matrix as an experiment to partially address this but we believe we can do it even better for season of the drifter we want to try something new we will be removing the prismatic matrix instead every week there will be unique bundles available that can be directly purchased for silver allowing you to directly buy exactly the items you want all unique bundles will also contain an exclusive vanity item available only through that weekly bundle if you currently have any prismatic facets you can still use them up until March 5th after the beginning of the new season they will turn into expired prismatic facets and will dismantle into 150 bright dust bright dust off and will also continue to offer a direct path to acquiring items found within bright engrams as always we will continue to months feedback and work to improve the eververse experience each season and when we are ready to try something new we will share those plans directly with you for now here are a few items you can expect to see in next season's bright engrams and we see some duty in chicken we see a new armor set and a new i believe ornament for the malfeasance which is pretty cool and a few other bits and bobs right there now i've seen a lot of feedback on this already about them removing the prismatic matrix and giving us the option to actually buy direct items from eververse with our money basically because that's like silverage you have to purchase silver in game for real life money now this is a natural thing i think is a good choice if they're giving us options to actually buy what we want instead of getting things at random because that's how it was before people would purchase silver and you just buy engrams and what you've got from the engrams is completely random they are giving us the opportunity to buy what we want for our money there's no wasting your money now there's no rng there's no gambling and this in my opinion is a great great thing i mean a lot of people will not agree with that a lot of people think this should be taken out of the game altogether there should be no loot boxes whatsoever i mean i understand that point too but at the end of the day people for them to continue maintenance on the game going forward brings all this new shit they need to make their money somehow and giving us direct options to purchase what we want from eververse is a good thing in my opinion like i said there's no more gambling anymore there's no more purchasing silver buying these engrams and not getting anything you want i mean you can probably still do that but now each week there will be bundles which are for exclusive things also which you can purchase direct for silver that we rely for money money and i think this is a step in the right direction if you ask me i mean yes a happier world would be no eververse no loot boxes at all but at the end of the day people they are a part of games now they ain't going anywhere no matter how much we bitch and complain about them this in my opinion is just a better system than what we had before and it's as simple as that now the trap just goes on to talk about fixes and bugs and all sorts of things come in within the next update 2.2.0 which will be resolving a number of issues we've been having which you guys can see on the screen now if you do want to check them out but me i just want to recap on why i believe this season the drift that could be exciting times to go forward the real could now after the V-Duck i like i said i wasn't too optimistic about it i mean it looked okay but i wasn't sure how long content was going to last and let's face it people there are other games out there to play there's apex legends there's anthem then you are playing that the division is not far away either and there's many other games people are playing but what we see the roadmap offering with the season the drift it looks quite promising and it seems as though they are dropping content more or less every single week for the next month or two leading up into the reverie event whatever this is so although this isn't a massive dlc this isn't a taking king this isn't a forsaken it seems as though at least we will get steady content coming over the coming months and this flow of content this is what destiny truly needs right now because at the minute let's not be around a bush people it's pretty damn boring there's nothing to do whatsoever but as long as they get the rewards right as long as they get the end game right it could be quite good i mean it's always exciting times for a destiny player when new dlc is about to drop but until it drops people we don't truly know how it's going to be i mean everybody looked forward to the black armory and it was good for the first couple of weeks maybe even first month but then it just died down relatively quickly the roadmap right here in my opinion is about 100 levels higher than what we got with the black armory roadmap but still a game we won't know until we actually get our hands on it but with them introducing these all new armor sets a new game mode in the reckoning a new game mode in gambit prime it could be quite exciting times a head it really could the one thing i still do have an issue with though is a the announcement of none new exotic armors no new faction rallies no weapon factions at all to be honest no new crucible or vanguard weapons beside their pinnacle ones and only two new exotics one of them i believe tied to uh basically season pass holders but hey there is more to a game than just chasing exotics i mean they are a massive thing for many many people but as long as those loot to chase and reasons for us to continue to play then it should be quite good the gambit prime armors in my opinion look quite interesting i'm guessing the way these will be unlocked is you have to play a certain role for a certain period to get these if you want to get the collector armor set you will have to collect a certain amount of moats no doubt and then push forward into the reckoning game mode and do something there and like i said if they get this right people it could be that grind we all love but we will see we will see but guys let me know what you think about the season of the drifter are you excited to come back to destiny did you never leave destiny let me know down below in that comment section guys i hope you enjoyed the video and hopefully people i will see you on that next one


Hello, this is Watson. In this video, we are reviewing the Watson 0609 Mini CNC Router. Now we are going to get a handle on its design and features, and we'll find out what is great about this machine. This is a tabletop milling machine with a 600x900mm working area. For this reason, it is perfectly suitable for machining small sized workpieces. The machine will easily cope with a fast startup of production. Despite belonging to the junior type of CNC routers, it can be used for performing demanding production tasks at the highest level. The machine is equipped with a 1.5 kilowatt spindle and a collet chuck, ER11, with a rotation speed up to 24,000 rpm. The height of spindle stroke at z-axis is 100mm. If necessary, a 2.2 kilowatt spindle can be installed. The Watson 0609 Mini is perfectly suited for woodworking and furniture production, light industry, interior and exterior decoration, as well as for souvenir production. The solid cast machine bed provides the required rigidity level and vibration resistance. In addition, it guarantees reliability and accuracy of machining. In order to provide durability of the frame construction, it is fired in a special furnace to relieve stress. The 20mm high wind guide rails are installed. The Watson 0609 Mini comes with two phase stepper motors that provide movements along Y, X and Z axes. Also, the machine can be equipped with hybrid steppers or servo motors. The machine can be used for making knife handles, milling butts of rifles, engraving on flasks and thermoses. In addition, you can make three-dimension images and bass reliefs, wooden boards for serving steaks and burgers, micro-circuit carriers and much more on this equipment. Its dimensions are 1080 x 1080 x 1340 mm, and its weight is 175 kg. It is compact and is easy to transport. The positioning accuracy of the machine is 0.05 mm, and its maximum working speed is 5,000 mm per minute. The gantry is moved by a stepper motor controlled via NC Studio. The system has an intuitive graphical interface and a clever contour calculation system. The connection type is D-sub-USB. An advanced DSP controller, rich Auto-A11, can be installed for ease of use. The Watson 0609 Mini is perfect for performing express orders. With this machine, you can upgrade your hobby and turn it into a business, from making your own products to workshops and training at schools. We hope you found this video useful and interesting. Please like it and subscribe to the channel. If you still have questions, feel free to share them in comments. We will be glad to answer them. This is Watson. See you in the next video.


I'm so excited to be here. The event's going great. We just finished up mentorship and we're moving on to some more programming. One of the great things about Patriot Boot Camp is the infusion of ideas, creativity, and solutions that we get from our participants, the veterans and spouses who are part of the program, the entrepreneurs that are so creative, they have so many solutions. One of those solutions is Aliamati's T-Care program. Aliamati's a Patriot Boot Camp alumnus does great things but because of what he did in his connection to DAV and we helped him and now he's helping us with caregivers. If you're a caregiver, please tune in to DAVcaregiver.org for resources, the program's free but it really helps caregivers get respite. Just last month we did a recon raider remembrance ride. Some of you might have tuned in for that. Great program, honoring DAV, honoring recon raiders who are fallen and we're here today with a special army veteran who does great things himself. Doug Reed, tell us about your company and what you guys do. What I actually do, I'm a motivational speaker but more around African-American community, empowering men and women to actually understand being in the marginalized community but understanding your role and empowerment around your wellness, understanding how it is to just be transparent in your healing. And in addition to outreach, you're actually, your goal is to put on events, correct? Yes sir, yes sir. So what that events is bringing them into a space, a branch of discussion and one thing I've actually learned is once you bring men and women to a free space, we call it, they're actually able to identify with it and open up about some of their problems and that problem comes in their healing and able to actually be better workers, be more like a DAV employee. Yeah. Well, tell us, what have you learned from the event? Oh my gosh, it's so much, I didn't really understand as far as DAV and what they do, but the lobbying that you guys actually doing, then coming to the actual boot camp and finding the resources that actually avail and the camaraderie as far as what the military offers, bring coming into that space, getting the knowledge, the mentors, the mentees, so much knowledge is actually being offered to us to make a difference in our communities and of course within the world itself. And you and I have something in common when it comes to the PACDAC. Why don't you speak to that just a little bit? Well, one thing about the PACDAC, I mean, the tremendous work being a whistleblower that you did. I have a community of reservists, we're actually, we're talking about the PACDAC, talking about the burn pits, but one of the things that we learned was it was more in regards to the burn pits, it was a toxic exposure. We started to learn more about the different bases in the toxic exposure. So actually to meet you in person, the actual whistleblower is like heartfelt and actually will make a difference because now I'm able to attract and tell people about the DAV and what the services that you guys offer, but actually for you to be here on site with so many other veterans and the impact of the organization. Well, that's very humbling. There are other things too, veterans like this, entrepreneurs like this are gonna be hiring people and then we connect them with the employment department. He's going to be helping DAV by getting more people who need their benefits, getting that thing squared away. So it's just a great event. Thank you, Doug, so much for being here. It's an honor to meet you, sir. Thank you for your service. CDAV will be back.


Question is from Phoebes Krake, I live in France and gyms may be forced to close due to the coronavirus. What are some tips to maintain muscle during this period of time? This is going to be a much more of a common question, I think, because of this problem. Well, obviously, if you have home gym equipment, you're totally fine, right? You can work out at home. If you have a barbell dumbbells, adjustable bench, you're pretty much set. You could do almost any exercise and train every body part with that type of basic equipment. But let's say you don't. Let's say you don't have any equipment whatsoever. What are some good, I guess, body weight, you know, no gym-required type of exercise and workouts? Now, we created a program specifically around this called Maps Anywhere. So it's a full workout program that doesn't need or require equipment. It just requires bands and your body weight. There's a couple things you could do to make body weight exercises more challenging. Because the thing for me with body weight exercises, like a body weight squat sometimes, is it gets too easy. How many of these do I need to do before I start to feel these or whatever? Because I don't have 200 to 300 pounds on my back. Just move to one leg.  And you can get great workouts with some of these. Also, this is a great time to do tempo and isolation exercises, too. Or isometric exercise. Isometric. Yeah, isometric and tempo stuff is great. Like, man, I can take somebody and do 10 body weight squats and make it extremely intense. Slow. Oh, yeah, slow it down, do an isometric hold at the bottom of it. And yeah, watch you get, and I mean, of course, another great option is if you can do a single leg squat. But that's normally a really big jump, right? A lot of people. You can assist. I guess you could hold on to something while doing it. Yeah, but even then, it's still really hard for a lot of people mobility-wise, strength-wise, to make the jump from, okay, body weight squats are really easy for me. Let me go to a single leg pistol. Like, even with the assistance, it's typically pretty challenging for most people. So I, my first way to progress body weight squats is just simply slowing down the tempo and doing isometric holds at the bottom. And you can make those really, really challenging. So, and then if you have the ankle mobility and hip mobility to do and strength to do a single leg squat, I think that's an excellent advice. I think to rubber bands, now that we have different versions, like they have some now that are really, you know, applied a lot of like thickness to it. So it's got like more weight resistance that you can really like challenge yourself with. Like before it used to be like, if I worked out with bands, it was basically like, I mean, it's pretty light. It wasn't any much better than like working out with just your body weight. But I think there's a lot more options now with these, these thick bands that you can really get some serious resistance that, that, you know, you can work against and do pretty much just about every exercise you can think of, as long as you have something to wrap it around. And like our, I know as a rubber band, it's we have has handles for these as well as two. So you can still do like all kinds of stuff. Dude, seriously you could do, you know, and I did this years ago, traveled to Southern Italy with my, with my family and the small town that we were staying in, they have a gym that just so happened to be closed for the entire month that we were there. So I had no gym to go to unless I went to the next town, but I didn't have a car. So all I had access to was a pull up bar, resistance bands, and my body. And I had phenomenal workouts. Obviously with the pull up bar, you can do all the best back exercises, no demand, which is pull ups. You could do all kinds of pressing exercises for the upper body by elevating your feet, which increases the resistance, you could balance with your feet up against the wall. If you want to do upside down push ups, those tend to be very difficult. And then of course, you can use bands, but here's something else that I found because of the lack of high resistance for a lot of exercises, frequency become became more important. So normally when I'd go to a gym, for example, I'd hit full body, maybe three days a week with heavy weights, but now let's say I'm at home, I don't have access to heavy weights. Well now increase the frequency. So instead of doing three days a week, do six days a week, six days a week, I'm training the whole body with a 30 to 45 minute body weight band based workout.  In fact, what you may actually notice is if you switch out of the gym and do that for let's say two or three months or even one month and then go back to the gym, you may actually find that you have better stability, better fitness. It's a new novel stimulus.  And at the very least, it'll prevent lots of fitness loss and muscle loss and that kind of stuff. So frequency becomes more important when you don't have the weight that will create the tension that you want.


There was a route into the east, traveled by merchants who brought back oriental luxuries. But this journey was long and difficult. Some sea captains began to seek a more convenient sea route around Africa, while others, certain that the earth was round, thought they could reach their goal by sailing west. Christopher Columbus, the first sea captain to look for India and China across the Atlantic Ocean, was an Italian serving the king and queen of Spain. In the year 1492, he found islands which he claimed for Spain. Thinking they were the East Indies, he called their people Indians. Now all the kings of Europe want land in this new world. In 1497, John Cabot, another Italian, crosses the ocean and claims the territory he discovers for the king of England. In 1500, Cabral, a Portuguese serving his own king, is blown off his course and reaches a southern shore, the coast of what is now Brazil. Sailing from the islands discovered by Columbus, Balboa reaches land which he claims for Spain. He crosses it and looks down upon a great new ocean, the Pacific. Magellan, a Portuguese, outlines the limits of the continent as he pushes south, then west and north again. The Italian Verrazano, traces the southern coastline of North America. He serves the king of France, as does Cartier, who plants the French flag farther north at the mouth of a great river. It is the 16th century and all Europe is discovering America, named for Amerigo Vespucci, the Italian who was one of the first to write of the new world. The northern continent is a great wilderness of plains and mountains, of forests, lakes and rivers, stretching from the Arctic to the tropics, between the shores of two great oceans. In the 20th century, 155 million people will make their homes in the United States. But now fewer than a million Indians live in this area, hunting the bison or the deer and cultivating their fields of maize and other vegetables. The people of this land make their home in the forests. Leave the wigwams of their villages to hunt or to fight rival tribes for control of the best hunting grounds. Their clothing is made from the skins of the animals they kill for food. They peel the bark of trees to make canoes that take them swiftly over lakes and rivers. Their tools are primitive, but they have many skills now lost to man. The skills they need for survival in the wilderness. Canoes are well suited to the Indians' needs, for they can move in very shallow water, and their shell-like hulls can be maneuvered skillfully between the sharp rocks of the rapids. While these men know nothing of the world beyond their shores, they know every turn in their rivers and every pathway through their forests. Spaniards are the first to bring European customs to America. They colonize the island discovered by Columbus, then spread over Central and South America. Churches replace Indian temples. First, the Indians think the strangers must be gods and offer precious gifts. But the Spaniards are only men, men who find wealth beyond their dreams. They see the jewels worn by Indian chiefs. The vases hammered out of precious metals by craftsmen and royal workshops. They see gold and silver dug from the earth by Indian miners. Before long, Spaniards are gathering new world riches for themselves. Fortunes in gold and silver, in sugar harvested on great plantations, are sent to Spain, while shiploads of colonists come from Spain. They build towns and settlements and bring old world architecture to the new continent. They teach Spanish laws and customs to the Indians and spread the Christian faith. But European nations grow envious of the wealth that flows from America to Spain. Gallions loaded with rich cargoes are anchored in colonial harbors. They face sudden raids by rival fleets. Toreus English captains are received in triumph at London's Royal Palace. Great honours are bestowed upon them for their courage and good fortune. Their stories of adventure in a land of endless wonders kindle dreams in many heads. As she listens, Queen Elizabeth begins to understand that the new world can add to the power and prosperity of the kingdom over which she rules. For now in England, the times are hard for many people. It was once a country where all who tilled the soil could find a simple livelihood. But the lords of great estates have turned farmlands into grazing lands. Farmers are homeless and dispossessed in their own land. Despairing of finding a living in the country, they plowed their way along the road that lead to London, already overcrowded. Few newcomers can find work in London's bustling streets. And soon the starving take to begging. The beggars and petty thieves are harshly punished in an age of cruel law. If men disobey the rules of the state church, they may be put to death or thrown into prison. A man of influence at the English court, Sir Walter Rolly, sees hope for his suffering countrymen in these lands across the sea. Can Englishmen live in the new world? He sends explorers to North America to make detailed reports. They find that Northern Indians have no precious metals. However, they raise plentiful crops around their peaceful villages. All waving stalks with ears of golden grain called maize. And tobacco plants whose broad green leaves the Indians roll and smoke for pleasure. Most plentiful, sweet, fruitful and wholesome of all the world are the words used to describe the new land of promise. In 1607, a party of Englishmen comes to establish the first settlement. Indians approach them with suspicion, but the strangers offer tools and trinkets in exchange for a fresh-killed game. This simple trade is the beginning of a cautious friendship. John Smith, an English captain, takes command when men start off to hunt for gold. He sets them to building shelters and defenses. Soon the food they've brought will be exhausted. They must clear land to harvest their own crops. They must learn from their Indian friends how to raise maize and tobacco. When new settlers, both men and women, arrive to work the fields and a way is found to cure tobacco, it becomes the colony's most valuable crop for export. Soon Virginia ports are visited by the European trading ships, which have been sailing to Africa, taking on cargoes of captured Negroes and selling them in the Spanish colonies. Now the traders bring the Negroes to the new English colony, and the small farms grow into great tobacco plantations worked by slaves. In 1620, a second-list settlement was started far to the north in Massachusetts, where a hundred people landed on a bare and windy shore. Seeking freedom from the English church, their purpose was to live according to God's will as they understood it. For this, they were ready to confront the grim and grisly face of poverty. They copied the Indian way of raising crude shelters, but the New England winters soon forced them to build wooden huts. They covered them with thatch, like the homes they left in England. But for their governor, they built a better house, and they met under his roof to worship God in their own way. At first, there is little but hard work and faithful duty in their lives. Their Indian friends have taught them to grow maize and pound the grain to meal. They have shown them where to cast their nets, to find fish in abundance in the waters off the shore. To survive in the wilderness, the pilgrim family has adopted Indian ways. But in the evening, in the new world as in the English home they have abandoned, the mother spins while the father reads the Bible and the children sleep nearby. Their bedding is hung up by day to make more space in the crowded home. A second group of nearly a thousand Englishmen soon settles near the pilgrims. They are Puritans who also oppose the ceremonies of the English church. Their leaders are men of property who have come with servants, laborers, and craftsmen to supply the comforts they could not bring with them. In this colony, religion dictates government. The ministers of the church proclaim the laws and rule the conduct of the citizens. Since they preach that hard work is a virtue, the communities grow rapidly and soon begin to prosper. But to make sure of obedience, the judges impose many of the punishments that drove the Puritans from England. Toiling to win a livelihood from the thin and rocky soil tightly bound by the rules of their religion, the colonists have few diversions. Their society is centered in their meeting house. Here ministers teach the religion that shapes town laws passed by the congregation. But the Puritans still surrounded by the wilderness have many things to fear. They cannot relax their vigilance or their laborers. Within the next hundred years, these little New England settlements turn into bustling towns. New settlers, laborers, and craftsmen pour in from Europe to the new land where there is work for all willing hands. Many New Englanders work in the shipping industry which grows as the towns turn into ports. The rich forests provide wood for the fishing boats and for the ships of commerce that will sail across the ocean back to England and southeast to the West Indies and Africa. Many skills are needed for the making and handling of new vessels. Depended on a lifeline between the New World and the Dips carry their produce, lumber, grain and salted fish to other parts of the British Empire and back to the land from which they came so ill provided barely a hundred years before. The ocean which separates them from the rest of the world becomes their high road to prosperity. By 1730, a New England merchant can find time from his work to sit for his portrait. His eye proudly surveys the many comforts that he can afford. His wife has become the mistress of a house where luxury is joined to the simplicity of Puritan tradition. His son was born here and will grow up thinking of this new world as his home. He is native land, but it's not only in New England that colonists have prospered. Thousands of European families from many nations have found relative peace and security on this distant continent. For in a hundred years, 13 separate colonies have brought civilization to the wilderness. Settlements line the coast from New Hampshire in the north to Georgia and its broad tobacco fields in the south. Virginia, tobacco, and the work of Negro slaves have created vast plantations and, for their wealthy owners, the basis for a new kind of aristocracy. From his study, the planter manages the broad and varied industry of his estate. By 1750, the ships that carry his tobacco crops to England also carry his orders for European goods. The furniture of his mansion has been shaped by the skilled hands of old world craftsmen. But the food and almost all the clothing on the plantation and the tools needed to produce them come out of local workshops. The planter's family leads a life of ease but not of idleness. For the making of a colony requires industry, vigilance, and the constant search for ways to meet new problems. The planter's son finds the latest theories of European thinkers in the library of his home. For in Virginia, as in the other colonies, new principles of government are being studied. At the same time, the elegance of English aristocracy is carefully cultivated. The colonies growing Atlantic trade and the arrival of new European settlers by the thousands is developing great harbor towns such as Charleston, Boston, Philadelphia, and New York. New York, first settled by the Dutch in 1624 and taken over by the English in 1664, now receives the people of many lands who bring their different customs, their skills, their arts along with their dreams of opportunity and freedom. Building houses that recall ancestral homes in Holland, the New Yorkers create a snug harbor town on the banks of the Hudson and East Rivers. They live in comfort true to their old customs of neatness and cleanliness, surrounded by the things they love. This home in New York looks Dutch and is furnished like homes along the Zyder Zee.  Just as Dutch furniture produced in America becomes American furniture, American culture has been born of this and many other cultures. Southwest of New York, land had been granted by the English king to William Penn, a leader of the Society of Friends. The Quakers, as they were called, practicing their creed of universal brotherhood, made a pact of friendship with the Indians. Here through the years, settlers dealt justly with the Indians and the Indians kept faith with them. The Quakers invited people of other faiths and other nations to join them. Many Germans came to Pennsylvania, pious and hardworking farmers who reaped the harvest of comfort and security.  In the new land, old German crafts can flourish. In time they will be called American, like the German settlers themselves, with their solid virtues and their devotion to religion. The Bible, printed in German and in many other languages, is the book most widely read in all the colonies, just as the church is always the most important meeting place. A man's right to worship, as he chooses, will be the heritage of coming generations. Religious freedom and the promise of peace and prosperity had brought the first colonists from England to New England and Virginia. The Dutch had found new prosperity in their new Amsterdam. Germans of many persecuted faiths were flourishing on the rich soil of Pennsylvania. Protestants had come from Catholic France and brought their culture to Carolina, while hardy Scots moved beyond the coastal settlements, pushing up the rivers to make homes in the wilderness. By 1750, more and more colonists are moving inland. In distant hamlets and on isolated farms, they must rely on their own efforts for all their needs. There are skilled tasks and simple manual tasks to fill long days of work for all. Frontier families have no servants, and there are many mouths to feed. So, resourcefulness becomes part of their character. Gears and pulleys can roast a chicken. Devices like this are the first of many which Americans will invent to meet the shortage of working hands. The table of pioneers like their customs and their language shows the mixture of their traditions. For even the Indian's maze has become part of their daily meals, along with dishes that have English, Scottish or German names. In the wilderness, the family is self-sufficient and self-taught. Their school is the farm kitchen. By candlelight, after days spent in the fields or in the forest, parents hand down their knowledge and children answer with their own new thoughts, born of their new life on the frontier. As they pushed westward, the colonists had pushed the Indians before them. Now the settlers who are living on the far edge of civilization face the constant menace of attack. At the warning of a raid, homes are abandoned and people flee from Indian vengeance. Most settlers have traded fairly with the Indians, exchanging firearms for land and furs. But some have cheated them and killed them. Enraged, the Indians rise and turn their guns on both the innocent and the guilty. The settlers have come to know the wilderness as hostile, and the Indians as a mortal enemy who can suddenly destroy in a few hours the work of many months. How still another enemy appears, for the Indians are often armed and led by Frenchmen. The French, strong rivals of England in Europe, also contend with the English for control of the new continent. While the English were first settling in Virginia, the French were building an outpost at Quebec. In time, the French province reached the Gulf of Mexico. Explorers spread a network of small forts to mark their claims to land. By the middle of the 18th century, the French and English are face to face. But danger does not stop the forward surge of pioneers in search of homes. It is 250 years after the new world was discovered by Columbus. From its coastal towns and cities, from Europe across the ocean, England's colonists are pushing into the heart of the American continent. Cutting their way through forests, clearing ground, building log houses, raising stockades and forts, spreading their roots in the new land of promise. Food migration that begins in far off Europe. Men and women of many faiths in many countries are joined in a common adventure for freedom and security. Out of their rich and varied qualities, the continent that lies before them will forge a character. The character of a new people, the character of a new country they are making in the wilderness.


The study aimed to investigate the prevalence of TSHIB in a pediatric cohort with AITD and healthy controls, using cell-based bioassays to measure TSHR stimulating, TSI, and blocking, TBI, immunoglobulins. The results showed that children with GD were positive for TSI while those with HT had increased TSI levels in the presence of eye disease. TBI was negative in all tested samples of children with GD, but positive in one HT sample. Overall, the study suggests that TSI is prevalent in children with GD and AITD with orbitopathy, while TBI may be a useful diagnostic tool for differentiating between GD and other autoimmune diseases. This article was authored by Carolina Stosek, Otto Basowski, Katazena Ziora, and others.