tag:blogger.com,1999:blog-5908830827135060852.post3405384193953794718..comments2024-10-04T12:06:07.606-04:00Comments on Bond Economics: Techniques For Finding SFC Model SolutionsBrian Romanchukhttp://www.blogger.com/profile/02699198289421951151noreply@blogger.comBlogger44125tag:blogger.com,1999:blog-5908830827135060852.post-15050636700525734782020-08-04T01:30:21.113-04:002020-08-04T01:30:21.113-04:00This comment has been removed by a blog administrator.saivenkathttps://www.blogger.com/profile/07265662654934474648noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-59198284363838014392019-03-02T07:24:22.525-05:002019-03-02T07:24:22.525-05:00This comment has been removed by a blog administrator.seoexperthttps://www.blogger.com/profile/06801410501138696016noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-90656306637484967582018-10-24T06:11:12.424-04:002018-10-24T06:11:12.424-04:00This comment has been removed by a blog administrator.Anonymoushttps://www.blogger.com/profile/01016866745731972164noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-76477891012903645972016-03-30T03:52:14.426-04:002016-03-30T03:52:14.426-04:00Lol... yes, I went way overboard. I'm more hav...Lol... yes, I went way overboard. I'm more having fun learning to use MathJax than anything else (to do the equation formatting).<br /><br />"Are you saving Excel as a web page and then posting the html file on your blog? Or are you using the Google Doc spread sheet? Or another way?"<br /><br />Back when I did the interactive version of Nick Edmond's model (that Brian links to in his post), I found that Microsoft was offering free access to an online version of Excel if you signed up for a hotmail account. It was a case in which MS actually had something that worked a lot better than Google Docs (and it was just as free). Maybe that has changed since, but the MS version of an embedded spreadsheet is truly interactive without exposing the original to modification by the unwashed masses (which was the big disadvantage of using a Google Docs version). All I did was develop my spreadsheet online (storing it on a free "OneDrive" server), and then choosing the "embed" option directly from MS's OneDrive (cloud) interface. It produces an HTML line that you simply copy and paste into the HTML editor side of blogger, and the spreadsheet appears there, and people can use it just like you've established they can (you have control over what they can do when you create the embedded link). But since everyone who sees it essentially has their own copy, I generally set it up to let people edit them right there on the blog page, so they can see the results w/o having to download the sheet.<br /><br />Since that time I've purchased a copy of Excel which comes with access to OneDrive as well. I did this to gain access to the advanced features not present in the free online version (I don't even know if the free version is offered anymore). So I can either edit a copy on my laptop and then save it to OneDrive, or just keep it on OneDrive all the time. <br /><br />To gain access to the "embed" feature go out to OneDrive (assuming you have an account) through your normal browser, find your file, put a check mark on it, and select "embed" from the menu along the top. I didn't find a way to do that from inside Excel. It works for Word documents as well (see my SIM4 post... that was my solution for putting fancy formatted equations on my blog before I learned MathJax).<br /><br />It's very easy to do. Go to my SIM6 post with your browser, and select "view source" to see the HTML. Then search for "iframe" to find what the line looks like that embeds the sheet. You can edit that line directly to change things like the width and height, and the cells it displays (which is handy... so you don't have to go back to OneDrive to do that stuff).<br /><br />Then anytime I change the original, it's automatically updated on the blog. No need to go back to OneDrive and make a new "iframe" HTML line. It works great!Tom Brownhttps://www.blogger.com/profile/17654184190478330946noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-47965380217539776492016-03-29T23:41:40.036-04:002016-03-29T23:41:40.036-04:00Tom, I think this is indeed a short course on line...Tom, I think this is indeed a short course on linear systems! You have a great amount of content here. I confess that it is past my present understanding but it is a goal to gain understanding of the basics (and more) that you are describing. <br /><br />I notice that you present SIM6 as interactive spreadsheet. Are you saving Excel as a web page and then posting the html file on your blog? Or are you using the Google Doc spread sheet? Or another way?<br /><br />This evening has been spent using my equations to build a new Excel worksheet. So far, I am following a blend of your SIM6 and the original SIM screen layout. It works on the SIM parameters and makes a nice updating chart. Next, add more complexity.<br /><br />So far, I have used the " 1-α1(1-θ) = X " in the expanded form. Then I can vary theta and alpha to move the GDP curve. These parameters could be calibrated if desired but that would be a challenge in itself.Roger Sparkshttps://www.blogger.com/profile/01734503500078064208noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-14123100605904943412016-03-29T18:27:13.196-04:002016-03-29T18:27:13.196-04:00Roger, what I wrote above about modes is illustrat...Roger, what I wrote above about modes is illustrated here as a <a href="https://3.bp.blogspot.com/-_D2zy8aC_js/VvCT4qGpDaI/AAAAAAAAJQM/kEAdimGDgLUvG-9rR0c91YoYjPxq3jFzg/s1600/RLC%2Bcircuit.png" rel="nofollow">collection of different single mode homogeneous solutions (AKA "impulse responses")</a>. Note that they all appear to be stable (real parts < $0$) and the oscillatory ones have an unplotted imaginary component, 90 degrees out of phase (AKA "orthogonal" to them). Strictly real valued oscillations arise from equal magnitude conjugate pair modes (and a strictly real initial value or impulse, of course). $A$ being strictly real ensures that any complex modes present will be paired in this fashion with their conjugates. It's possible that a single large $A$ matrix could produce a superposition of all the modes pictured. The (possibly complex) value associated with each mode is an eigenvalue of $A$. Sometimes those are referred to as the spectral components of $A$. Thus concludes my short course on linear systems. Lol... (C:Tom Brownhttps://www.blogger.com/profile/17654184190478330946noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-23237135685997004762016-03-29T15:29:08.281-04:002016-03-29T15:29:08.281-04:00In particular, for an LTI system $A_d = e^{A_c T_s...In particular, for an LTI system $A_d = e^{A_c T_s} \approx I + A_c T_s$ (the approximation holding when the matrix norm $\lVert A_c T_s \rVert = \lVert A_c \rVert T_s \ll 1$, ... an generalization of the approximation you may recall from <a href="http://ramblingsofanamateureconomist.blogspot.com/2016/03/low-interest-rates-are-contractionary.html?showComment=1458842876394#c5013595352362210487" rel="nofollow">this comment I left you on John Handley's blog</a>). The subscripts $d$ and $c$ denote discrete and continuous time, respectively. Note that $A_d$ is different for every sample period $T_s$ used. $B_d$ is generally more complicated, but can be approximated as $B_d \approx A^{-1}_c (A_d - I) B_c$. This approximation is exact when $\mathbf{u}$ is a "zero order hold" (ZOH) function, meaning it stays constant over each sample period. More generally, for LTI systems with an arbitrary input $\mathbf{u}$ you must integrate:<br />$$\mathbf{x}_{n+1} = e^{A_c T_s} \mathbf{x}_{n} + \int_{0}^{T_s} e^{A_c (T_s - \tau)} B_c\, \mathbf{u}(\tau + n T_s) d \tau \tag 9$$<br />or more generally still, for a time-varying linear system<br />$$\mathbf{x}_{n+1} = \Phi_{(n+1) T_s, n T_s} \mathbf{x}_{n} + \int_{n T_s}^{(n+1) T_s} \Phi_{(n+1) T_s, \tau} B_c (\tau) \mathbf{u}(\tau) d \tau \tag {10}$$<br />Where $\Phi_{t_2, t_1}$ is the state transition matrix from time $t_1$ to time $t_2$. Or you can do a <a href="http://banking-discussion.blogspot.com/2016/03/sim4.html" rel="nofollow">Taylor expansion on $u$ and find a separate $B_{di}$ for each $i^{th}$ term</a>. As Brian has previously pointed out, a numerical integration accomplishes much the same thing, only instead of Taylor terms sampled at one time, it uses $\mathbf{u}$ sampled at a several times over a single nominal sample period.<br /><br />Finally, as I mentioned in a comment above, some non-linear dynamic systems can be approximated over each sample period as a linear system. For example, you may have a non-linear system such as $\dot{\mathbf{x}} = \mathbf{f}(\mathbf{x},\mathbf{u} ,t$), where $\mathbf{f}, \mathbf{x}$ and $\mathbf{u}$ are generally vectors. One way to accomplish this is to approximate $\mathbf{f}$ as $A_c$ and $B_c$ where these matrices are the <a href="https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant" rel="nofollow">Jacobians</a> of $\mathbf{f}$ wrt $\mathbf{x}$ (i.e. $A_c = \partial \mathbf{f}/\partial\mathbf{x}$) and $\mathbf{u}$ (i.e. $B_c = \partial \mathbf{f}/\partial\mathbf{u}$) respectively, and to rewrite the differential equation in terms of a small perturbation of $\mathbf{x}$ (call it $\delta{\mathbf{x}}$) about a nominal solution $\mathbf{x}*$: $\delta{\mathbf{x}} \equiv \mathbf{x} - \mathbf{x}*$. Similarly define $\delta{\mathbf{u}} \equiv \mathbf{u} - \mathbf{u}*$. Then we can write for an approximate perturbation to this non-linear differential equation $\delta{\dot{\mathbf{x}}} = A_c \delta\mathbf{x} + B_c \delta\mathbf{u}$. The discrete time approximation I gave above ($A_d \approx I + A_c T_s$) can be seen in this light as a two term vector Taylor expansion of $\mathbf{f}$ wrt $\mathbf{x}$, i.e. $A_d \approx I + (\partial\mathbf{f}/\partial\mathbf{x}) T_s$.Tom Brownhttps://www.blogger.com/profile/17654184190478330946noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-20866443540201281042016-03-29T12:37:21.455-04:002016-03-29T12:37:21.455-04:00Hi Roger,
I'm glad you found the comments inte...Hi Roger,<br />I'm glad you found the comments interesting. That Wikipedia page link will have a more coherent presentation I'm sure. I'm definitely no expert on SFC modeling though. In the case of SIM it overlaps with my comments above. Like Brian says that's not always true. I don't even know if what you're postulating about X carries over to other models.<br /><br />A couple of final thoughts: A and B for continuous time and discrete time "equivalents" are not the same. The transition matrix solution in continuous time is the A matrix for discrete time, for example.Tom Brownhttp://www.google.comnoreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-40190648013163891892016-03-29T11:37:01.800-04:002016-03-29T11:37:01.800-04:00Tom, I find your comments very interesting. They a...Tom, I find your comments very interesting. They are opening a view and providing a link into a vastly expanded world of geared relationships.<br /><br />It seems to me like the relation "1-α1(1-θ) = X" is the key to making spreadsheet models into sequential flow displays of the economy as it flows through time. SIM begins this, but SIM describes only the initiation of an ongoing economic system. We need the ability to change the theta and alpha parameters "on the fly" to see how GDP and wealth react to changes.<br /><br />Brian certainly initiated an interesting discussion!<br /><br />Roger Sparkshttps://www.blogger.com/profile/01734503500078064208noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-53491247951429184552016-03-29T10:34:18.415-04:002016-03-29T10:34:18.415-04:00Yes, it loads the processing code from the MathJax...Yes, it loads the processing code from the MathJax site. I find it loads much slower on my underpowered computer that uses Chrome than on my iPad.<br /><br />The only thing with Python is that you would probably need to track down the various math packages. I have been using Python as a scripting language to glue my database data acquisition together, and not for time series analysis, so I have not actually used those packages. I hacked together something simple that did what I wanted.Brian Romanchukhttps://www.blogger.com/profile/02699198289421951151noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-36254890620214785322016-03-29T09:51:52.844-04:002016-03-29T09:51:52.844-04:00Brian,
Thanks for your answer. I find Python to b...Brian,<br /><br />Thanks for your answer. I find Python to be flexible and powerful language so I assume it would work for fairly sophisticated models. I have thought about automating graphs of Fed flow of funds data using Python and Bokeh (a fairly simple process of reading and plotting data from a large file).<br /><br />The math is appearing today. I did notice a file loading to support mathjax -- I think that file must download from the central server every time one lands on the page or refreshes the page. Odds are the support file did not download into chrome the other day.Joe Leotehttps://www.blogger.com/profile/01292763300917387201noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-83697101808529639192016-03-29T06:05:57.433-04:002016-03-29T06:05:57.433-04:00When $u$ is a step function (i.e. $0$ when $t <...When $u$ is a step function (i.e. $0$ when $t < 0$, and stepping up to a constant $u_0$ for $t \geq 0$), the heterogeneous (i.e. $x_0 = 0$) solution to $\dot{x} = a x + b u$ is $(1 - e^{a t}) \bar{x}$ for $t \geq 0$ (and $0$ for $t < 0$), where $\bar{x} = (-b/a) u_0$ is (in the case of a stable system (i.e. $a < 0$)) the steady state solution for $x$. So the time constant $Tc$, in this case, is the time for the system to rise to $(1 - e^{-1}) \approx 0.63$ of $\bar{x}$, which is the situation with the example input G (a step) in SIM. More generally, the solution is the sum of the heterogeneous and homogeneous solutions.Tom Brownhttps://www.blogger.com/profile/17654184190478330946noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-7095788963315983432016-03-29T05:15:07.787-04:002016-03-29T05:15:07.787-04:00"It is not nearly as sophisticated as your (a..."It is not nearly as sophisticated as your (and Brian's) method"<br /><br />I wouldn't say that. IMO, you're trying to gain insight into what the equations mean. I was more interested in putting them into a form I was familiar with. It's no accident that Brian and I named the "matrices" (really just both scalars in this case) $A$ and $B$ in the "state update equation" or why I named the two "matrices" (both 4x1 vectors in this case) in the measurement equation $C$ and $D$ (really $C_Z$ and $D_Z$ in my case since $C$ is unfortunately already the name of a measurement). It's because those four: $A, B, C$ and $D$ are commonly used in a <a href="https://en.wikipedia.org/wiki/State-space_representation" rel="nofollow">"state space"</a> representation of a system, both for continuous time and discrete time. It really only applies to linear systems. If $A, B, C$ and $D$ are functions of time, then it's a time varying linear system, and if they are not, then it's a <a href="https://en.wikipedia.org/wiki/LTI_system_theory" rel="nofollow">linear time-invariant (LTI) system</a>. Of the two, LTI systems are a bit easier to deal with. Once a system can be modeled in that way (linear or LTI), there's a huge variety of tools that you can use on it that have been developed over the decades. Even if a system is non-linear, sometimes it can be approximated as a linear system between time steps (where multiple iterations are sometimes required per time step for it to converge). Linear systems have the huge advantage of superposition, meaning that different solutions can be added together to produce the final output. This means you can look at the response of the system to an initial state $x_0 = x(0)$ and to the exogenous input $u$ (both of which can be further divided into sub-components) as additive. In discrete time the state update equation implements what's known as a 1st order difference equation, and in continuous time, this is referred to as a 1st order linear differential equation. Both are generalizations of the scalar 1st order differential equations you probably learned about somewhere along the line: $\dot{x} = a x + b u$, the "homogeneous" (i.e. $u = 0$) solution of which is $x(t) = e^{a t} x_0$. Which is where the term "time constant" I use comes from... it's the time constant $T_c = 1/|a|$ of that exponential. When $a < 0$ (i.e. the system is stable), $T_c$ is the time required for the system to decay to $e^{-1} \approx 0.37$ of $x_0$. This solution is known as a "mode" of the system. In the more general case where $x$ is n-dimensional, there will be up to n modes in the solution, which again, are all additive, with some modes potentially being complex (i.e. having real and imaginary components, and producing various kinds of oscillatory behavior). The notation $e^{A t}$ (the <a href="https://en.wikipedia.org/wiki/Matrix_exponential" rel="nofollow">"matrix exponential"</a> AKA the "state transition matrix") is used to denote this more general multi-mode solution of an LTI system. (note: time varying linear systems also have a transition matrix, but it's not a matrix exponential in general).<br /><br /><a href="http://wolfweb.unr.edu/~fadali/ee472/DTState-Space.pdf" rel="nofollow">This</a> is a good summary about converting between discrete and continuous time.Tom Brownhttps://www.blogger.com/profile/17654184190478330946noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-38128980433164813232016-03-29T00:47:31.284-04:002016-03-29T00:47:31.284-04:00[What follows is the derivation of the equations f...[What follows is the derivation of the equations found in my earlier comment]<br /><br /><b> Goal </b><br /><br />Find the annual GDP (AGDP) increase from a single Original InJection (OIJ).<br /><br /><b> Background </b><br /><br />We know that a single money injection by government can be recovered by government with taxation. [We begin with zero and end with zero. In between zero and zero, we have additional GDP]. We are asking "How much activity can the money generate if every transaction is taxed?" <br /><br />This is easy to answer if we ask the question in the correct way. Ask "If government injects money and then takes that money away with a tax on each transaction, what is the value of the total transactions required to recover the initial injection?"<br /><br />GDP*Tr = OIJ<br />or<br />GDP = OIJ/ Tr<br /><br />where monetary activity is GDP and tax rate is Tr.<br /><br />This value of GDP is a limit. It says nothing about the timing of GDP events.<br /><br /><b> Convert into a series of annual events </b><br /><br />We can convert GDP to annual events (AGDP) by considering every step is a division between two taxing authorities. The primary authority will receive the Tr share and the second authority will receive the remainder. Write this in two equations.<br /><br />(1) AGDP*Tr = AT<br /><br />where AT is the Annual Tax, and<br /><br />(2) AGDP*(1-Tr)*Rr = AR<br /><br />where Rr is the Remainder rate and AR is the Annual Remainder.<br /><br />Notice that the sequence of events is important here. Tax is removed before a remainder can be calculated.<br /><br />We know that the sum of the two tax divisions is the original injection:<br /><br />(3) AT + AR = OIJ<br /><br />Substitute 1 into 3 to get<br /><br />(4) AGDP*Tr + AR = OIJ<br /><br />Substitute 2 into 3 to get<br /><br />(5) AGDP*Tr + AGDP*(1-Tr)*Rr = OIJ<br /><br />Simplify 5 to get<br /><br />(6) AGDP*( Tr + (1-Tr)*Rr) = OIJ<br /><br />[ Now we have<br /><br />Tr + (1-Tr)*Rr = 1-α1(1-θ) = X <br /><br />which brings congruence with Tom Brown's comment.]<br /><br />This explains the derivation of the first equation contained in my earlier comment<br /><br />“(Period National Income) * (θ + (1-θ)(1-α1)) = Government”<br /><br />wherein I try to align my terms with the SIM model parameters.<br /><br />[Tom: That is how I came up with the equations. It is not nearly as sophisticated as your (and Brian's) method.]<br /><br />Roger Sparkshttps://www.blogger.com/profile/01734503500078064208noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-91457801614536273972016-03-28T21:04:56.743-04:002016-03-28T21:04:56.743-04:00I am still wrapping my mind around this. It sounds...I am still wrapping my mind around this. It sounds like three different approaches came to the same basic conclusion. That would be good.<br /><br />I missed that <br /><br />(θ + (1-θ)(1-α1)) = 1-α1(1-θ) = X<br /><br />I agree now that you pointed it out.<br /><br />You are certainly demonstrating excellent math skills. It will take me a lot of study to understand your more general approach.<br /><br />My derivation was inverted from yours. I will try to post the derivation either here or on my blog. That may not happen until Wednesday but will try for sooner.<br /><br />Thanks for taking this detailed look at the three (four) equations.Roger Sparkshttps://www.blogger.com/profile/01734503500078064208noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-17545272443825386552016-03-28T19:54:43.215-04:002016-03-28T19:54:43.215-04:00And in terms of what Brian wrote in his post, my X...And in terms of what Brian wrote in his post, my X = 1 - A1, so A1 = α1(1-θ), and while we're at it, his A2 = α2θ and his B1 = θ, which causes his A and B to equal mine.Tom Brownhttps://www.blogger.com/profile/17654184190478330946noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-50256138741720157502016-03-28T19:52:03.641-04:002016-03-28T19:52:03.641-04:00And yes, if you set α1 = α2 = 1, then my equation ...And yes, if you set α1 = α2 = 1, then my equation (8A) at SIM7 shows that Tc/Ts = -1/log(0) -> 0 <br />I.e. the time constant Tc compared the sample period Ts goes to zero. However, G&L write that you must have 0 < α2 < α1 < 1 in their equation 3.7 <a href="https://4.bp.blogspot.com/-KKkQf2zscr4/VtjXISV_4wI/AAAAAAAAJEE/iI9TIv_8oBQ/s1600/godley%2B2.png" rel="nofollow">here</a>.Tom Brownhttps://www.blogger.com/profile/17654184190478330946noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-34794958646943115302016-03-28T19:44:44.157-04:002016-03-28T19:44:44.157-04:00This comment has been removed by the author.Tom Brownhttps://www.blogger.com/profile/17654184190478330946noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-82619442782349129662016-03-28T19:35:42.139-04:002016-03-28T19:35:42.139-04:00Roger, you write:
(A) "(Period National Inco...Roger, you write:<br /><br />(A) "(Period National Income) * (θ + (1-θ)(1-α1)) = Government"<br /><br />First of all<br /><br />(θ + (1-θ)(1-α1)) = 1-α1(1-θ) = X<br /><br />It shows up a lot, so I label it X in my comment above and on SIM7.<br /><br />By "Period National Income" do you mean Y? Well, T = θY, so that's right.<br /><br />Y[2] = G[2]/X (from my equations above, because H[1] = 0, so your (A) is correct.<br /><br />(B) "Period Wealth = (Last Period Wealth) + Government - Tax"<br /><br />That's H[n+1] = H[n] + G[n+1] - T[n+1], so yes, your (B) is correct.<br /><br />(C) "(Period National Income)*(θ + (1-θ)(1-α1)) =<br />Government + (Last Period Wealth) * α2"<br /><br />That's Y[n+1] = α2*H[n]/X + G[n+1]/X<br /><br />Which matches my equation for Y[n+1] when n > 1 (i.e. when H[n] > 0, so that means (C) is good too). Or at least I agree with your equations (A), (B) and (C). All seems OK to me.Tom Brownhttps://www.blogger.com/profile/17654184190478330946noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-61640580155694219432016-03-28T19:05:19.643-04:002016-03-28T19:05:19.643-04:00Mathjax: back when I first encountered that on Jas...Mathjax: back when I first encountered that on Jason's blog, I saw the LateX code, not the formulas. <a href="http://informationtransfereconomics.blogspot.com/2014/03/the-islm-model-again.html?showComment=1396510569711#c5498535460855094305" rel="nofollow">Here's my little dialog with him</a>... I haven't had trouble since.Tom Brownhttps://www.blogger.com/profile/17654184190478330946noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-25794259672180156742016-03-28T17:24:50.062-04:002016-03-28T17:24:50.062-04:00The limitations of my framework are peculiar to th...The limitations of my framework are peculiar to the way I built it, and not the issue with the language. I built it an away so that I could simulate fairly complex systems, but I gave up on tracking the algebraic equations. This would rule it out for some academic uses, and I could not directly simulate the Godley & Lavoie models. The way my system works, each decision relies on information that is "currently available" (which includes some information from the current period), so there is no need to simultaneously solve for the "in-period equilibrium". It would have been possible to emulate the iterative techniques used by Matlab/R in Python.<br /><br />As for the post - I use Chrome, and it works for me. I have no idea how to make it display, other than checking to see whether your browser is out of date. This sort of thing is why I avoiding using MathJax in the first place. Although it's clunky, I kind of prefer pasting in a screen shot of the equations, since any browser can support that. I am supposed to be finishing off my report, but I expect that I might do a PDF with the formulae sooner or later. (One of my next projects will be a report on SFC models, but I will do it as a PDF, which I hope can be distributed on Amazon.) Brian Romanchukhttps://www.blogger.com/profile/02699198289421951151noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-58173038823013997952016-03-28T16:28:29.051-04:002016-03-28T16:28:29.051-04:00Brian,
Could you comment on the limitations you h...Brian,<br /><br />Could you comment on the limitations you have encountered for using Python for simulating SFC models? <br /><br />Today this post is filled with red fields that say [Math Processing Error]. I am using chrome browser.Joe Leotehttps://www.blogger.com/profile/01292763300917387201noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-19013954072339601172016-03-28T13:00:24.544-04:002016-03-28T13:00:24.544-04:00BTW, I erased my comment about what I think Jason ...BTW, I erased my comment about what I think Jason is doing... too much speculation on my part.Tom Brownhttps://www.blogger.com/profile/17654184190478330946noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-69137474439143571082016-03-28T12:48:47.488-04:002016-03-28T12:48:47.488-04:00Equation 8A is the ratio of the system time consta...Equation 8A is the ratio of the system time constant to the sample period: Tc/Ts = -1/(log(1-V)) which is approximately 1/V for |V|<<1 (V is a stand in for the ratio of parameters there).Tom Brownhttps://www.blogger.com/profile/17654184190478330946noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-27424452674826862162016-03-28T12:43:14.566-04:002016-03-28T12:43:14.566-04:00In particular, equations 5 (KT = 1), 3A, 4A, 6A an...In particular, equations 5 (KT = 1), 3A, 4A, 6A and 8A.Tom Brownhttps://www.blogger.com/profile/17654184190478330946noreply@blogger.com