text:
Integrated visual analysis of patterns in time series and text data : Workflow and application to financial data analysis

2016
Journal article
Published in
Information Visualization ; 15 (2016), 1. - pp. 75-90. - ISSN 1473-8716. - eISSN 1473-8724
Abstract
In this article, we describe a workflow and tool that allows a flexible formation of hypotheses about text features and their combinations, which are significantly connected in time to quantitative phenomena observed in stock data. To support such an analysis, we combine the analysis steps of frequent quantitative and text-oriented data using an existing a priori method. First, based on heuristics, we extract interesting intervals and patterns in large time series data. The visual analysis supports the analyst in exploring parameter combinations and their results. The identified time series patterns are then input for the second analysis step, in which all identified intervals of interest are analyzed for frequent patterns co-occurring with financial news. An a priori method supports the discovery of such sequential temporal patterns. Then, various text features such as the degree of sentence nesting, noun phrase complexity, and the vocabulary richness, are extracted from the news items to obtain meta-patterns. Meta-patterns are defined by a specific combination of text features which significantly differ from the text features of the remaining news data. Our approach combines a portfolio of visualization and analysis techniques, including time, cluster, and sequence visualization and analysis functionality. We provide a case study and an evaluation on financial data where we identify important future work. The workflow could be generalized to other application domains such as data analysis of smart grids, cyber physical systems, or the security of critical infrastructure, where the data consist of a combination of quantitative and textual time series data.
Subject (DDC)
004 Computer Science
Keywords
Heterogeneous data, time series analysis, frequent financial data analysis, text document analysis, interest point detection, interesting interval patterns, hybrid temporal pattern mining (HTPM), hypothesis generation
Cite This
ISO 690WANNER, Franz, Wolfgang JENTNER, Tobias SCHRECK, Andreas STOFFEL, Lyubka SHARALIEVA, Daniel A. KEIM, 2016. Integrated visual analysis of patterns in time series and text data : Workflow and application to financial data analysis. In: Information Visualization. 15(1), pp. 75-90. ISSN 1473-8716. eISSN 1473-8724. Available under: doi: 10.1177/1473871615576925
BibTex
@article{Wanner2016-01-01Integ-31189,
year={2016},
doi={10.1177/1473871615576925},
title={Integrated visual analysis of patterns in time series and text data : Workflow and application to financial data analysis},
number={1},
volume={15},
issn={1473-8716},
journal={Information Visualization},
pages={75--90},
author={Wanner, Franz and Jentner, Wolfgang and Schreck, Tobias and Stoffel, Andreas and Sharalieva, Lyubka and Keim, Daniel A.}
}

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Yes


text:
### Retucl's blog

By Retucl, history, 4 months ago,

I solved 1621I - Two Sequences in this way but a bit different.

I calculated the maximum number of copies of the minimum suffix of each prefix of a string in a simple way:

int A[N];

bool cmp(int l1,int r1,int l2,int r2){
if(r1-l1!=r2-l2) return false;
for(int i=0;i<=r1-l1;++i)
if(A[l1+i]!=A[l2+i]) return false;
return true;
}

// mSuf1[i] means that A[mSuf1[i]:i] is the minimum suffix of A[1:i]
// mSuf2[i] means that A[mSuf2[i]:i] is the maximum number of copies of the minimum suffix of A[1:i]

for(int t=L;t<=R;++t){
mSuf2[t]=mSuf1[t];
if(cmp(mSuf1[mSuf1[t]-1],mSuf1[t]-1,mSuf1[t],t))
mSuf2[t]=mSuf2[mSuf1[t]-1];
}

// The loop above looks so slow but it was accepted.

My submission: 143043686.

So I wanna know the upper bound of the time complexity of this loop and how to construct the string to get this bound. The size of the character set can be $O(|S|)$.

• +21

format: plain


text:
# How to obtain digital IIR filter coefficients for known z-plane zero-pole plot?

My clue is just to expand brackets and obtain coefficients of numerator and denominator of my transfer function. Am I right?

• Yes, you are right, only if your filter structure is a direct form. – Juancho Dec 23 '16 at 15:33
• going from poles and zeros to Direct Form coefficients is easy. going in the other direction is a little harder because it involves factoring polynomials. – robert bristow-johnson Feb 24 '17 at 4:00

That is, all transfer functions $(Kb,a)$, where $K$ is a constant, have the same pole-zero representation. For this reason, we usually use pole-zero-gain terminology.


text:
How to share your video-conference window among attendees – or, the many ways of splitting a rectangle in many

These days, in the covid world we live in we have multiple videoconferences every day, with colleagues, family, friends… The average number of people attending has grown, and when I was in a call some time ago, I wondered, what would be the “optimal” way of splitting the videconference application window among the people in the call? What would be the definition of “optimal”? And, in how many ways could the space be split?, as sometimes you might prefer, for instance, one where space is shared equally or one where the current speaker has predominance.

This intrigued me, and I started some research to try to find answers to these questions. I started with the assumption that the videoconference application owns a window on the screen, which is a rectangle (we will call it the bounding rectangle), and we want to divide it in $$N$$ different rectangles, being $$N$$ the number of people in the call. We can define different criteria for optimizing like, say, dividing equally the window area among the callers. However, for that we need to define mathematically the dividing rectangles, and here is where the first difficulty arises, as the mathematical description changes depending on how you split the window. For instance, two of the ways of dividing the bounding rectangle (let’s call that “rectangulation”) for $$N=3$$ are as displayed:

Here we assign a width and a height to each rectangle. Note that optimizing means assigning values to these $$w_i$$ and $$h_i$$ to fulfill the optimizations criteria, which is like moving the “walls” between rectangles. The height $$h$$ and width $$w$$ of the bounding rectangle are considered fixed. In each rectangulation, it is possible to expose the implicit restrictions by equations that relate widths and heights of the different rectangles. For the first one in the image, the equations are:

$$\begin{array}{lcl} w_1+w_2 & = & w \\ w_1 & = & w_3 \\ h_1+h_3 & = & h \\ h_1+h_3 & = & h_2 \\ \end{array}$$

For the second one:

$$\begin{array}{lcl} w_1+w_2+w_3 & = & w \\ h_1 & = & h \\ h_1 & = & h_2 \\ h_2 & = & h_3 \end{array}$$

When we optimize we need to take into account these restrictions, which implies that for each rectangulation we have a different optimization problem with different solutions due to having different restrictions/equations in each case. The conclusion is that we need to solve the optimization problem for all possible rectangulations for a given $$N$$, and then compare the different optimal values to select the best one. In summary, the steps we need to follow are

1. Find all possible rectangulations for $$N$$ callers
2. Decide an optimization criteria
3. Solve the optimization problem for all rectangulations, finding the best sizes for each way of splitting the bounding rectangle
4. Choose the best rectangulation by finding out which of them has the optimal solution that fulfills better the optimization criteria

In the following sections we will go through each of these steps. I have developed some python code that implements the algorithms and that includes some unit tests that drive them and will be mentioned in different parts of the article. You can find it in github. If on an Ubuntu system, you can install dependencies and clone the repository from the command line with:

$sudo apt install python3-pip $ pip3 install numpy matplotlib scipy sympy
$git clone https://github.com/alfonsosanchezbeato/optimal-rectangulations N.B. In the mathematical expressions we use one-based indexes but, for practical reasons, the code uses zero-based indexes. Rectangulating a rectangle There is quite a bit of mathematical literature on the topic of rectangulations, motivated by applications like placing elements in integrated circuits. Depending on how you define them there can be more or less ways to “rectangulate” a rectangle. We want to take into account all possible rectangulations, but at the same time we need to be careful to avoid duplications. In our framework, a single rectangulation is one that can represent all rectangle sets that can fulfill a given set of equations as those shown at the beginning. Practically, that means that we can slide segments in the rectangulation (without changing the vertical or horizontal orientation) along the limiting segments, as that will not change the relationship that the equations describe. Each of these rectangulations is a class of equivalence of all possible rectangle sets described by a set of equations, and we need to find each of them and then find the optimal rectangles sizes for them, so we cover all possibilities. It turns out that each of these equivalence classes can be represented by what is called in the literature mosaic floorplans or diagonal rectangulations. How to obtain these rectangulations? I will follow (mostly) the method explained in 1. First, we create a matrix of size $$N$$x$$N$$, which conceptually splits the bounding rectangle in cells. We will call it the rectangulation matrix. We number the rectangles consecutively and put the numbers in the matrix diagonal. Then, we choose a permutation (any can do) for the numbers. This permutation determines how the rectangulation will look like. Finally, we follow the order in the permutation to create the rectangles, one by one. To build each rectangle, the rules are 1. Start from the free cell as left and bottom as possible that can include the cell with the rectangle number 2. Do not include cells with numbers for other rectangles 3. Make it as big as possible, with the restriction that it must not surpass the right of the rectangle below it and it must not surpass the top of the rectangle to its left. The bottom and left walls of the bounding rectangle are considered bottom and left rectangles for rectangles touching them. This defines a map between permutations and diagonal rectangulations that we will call $$\rho$$. Figure 2 illustrates this procedure for the rectangulation $$\rho(4,1,3,5,2)$$. This is implemented in the do_diagonal_rectangulation() function. There is a test to create random rectangulations that can be run with $ ./rect_test.py TestRectangles.test_diagonal_rectangulation_random

Figure 3 shows an example of a run, with numbered rectangles. The sizes are as returned by do_diagonal_rectangulation(), so we can see that all rectangles have part of them in the top-left to bottom-right diagonal.

It turns out that this map between permutations and diagonal rectangulations is surjective, so it would be interesting to filter out the permutations that produce duplicated rectangulations to reduce the number of calculations. Fortunately, it has been proved that it is possible to establish a bijection between a subset of permutations, called Baxter permutations, and diagonal rectangulations2.

Baxter permutations

Baxter permutations can be defined by using the generalized pattern avoidance framework3. The way it works is that a sequence is Baxter if there is no subsequence of 4 elements that you can find in it that matches one of the generalized patterns 3-14-2 and 2-41-3. This is an interesting topic but I do not want to digress too much: I have based my implementation on the definition of section 2 of 4. The function is_baxter_permutation() checks whether a sequence is Baxter or not by generating all possible subsequences of 4 elements and checking if they avoid the mentioned patterns. An example using it can be run with

./rect_test.py TestRectangles.test_baxter_permutation Using Baxter permutations to filter out permutations can save lots of time for bigger $$N$$. For instance, for $$N=10$$, only around 10% of the permutations are Baxter. There is a closed form formula for the number of Baxter permutations as a function of $$N$$5, so we can compare easily with the total number of permutations, $$N!$$. Rectangle equations Once we have a rectangulation, we need to extract the equations that define the relations between widths and heights across the rectangles. To do this, we can loop through the rectangulation matrix and detect the limits between rectangles, which define horizontal and vertical segments. When found, we will create an equation that defines that the sum of the widths/heights of the rectangles at each side of the segment must be equal. There will be also two equations for the bounding rectangle that make the sum of width/height of rectangles at the side be equal to the total width/height. Note that these are 2 and not 4 equations, as 2 of the 4 we can extract would be linearly dependent on the full set of equations (this can be seen because if we take the equation on one side and follow the parallel segments equations while moving to the opposite side we can deduce the rectangles that touch the bounding rectangle at the opposite limit). Example of the equation sets that would be extracted are those we have already seen for the rectangulations in figure 1. In the slightly more complex case of figure 2, the equations would be $$\begin{array}{lcl} w_1+w_2 & = & w \\ w_2 & = & w_3+w_5 \\ w_1+w_3 & = & w_4 \\ h_1+h_4 & = & h \\ h_1 & = & h_2+h_3 \\ h_3+h_4 & = & h_5 \end{array}$$ The function that implements this algorithm is build_rectangulation_equations(). It returns a matrix with the equation coefficients that will be used when optimizing the rectangulations. The final number of linearly independent equations we will get will be always $$N+1$$. This is a corollary of Lemma 2.3 from 6. This lemma defines two operations, “Flip” and “Rotate”, that allow transforming any rectangulation on another one by a finite number of these operations. These operations do not modify the number of segments in the rectangulation. Therefore, we could transform any rectangulation in one having only, say, vertical segments. This rectangulation would have $$N-1$$ segments splitting the bounding rectangle, which implies that the original rectangulation had also $$N-1$$ segments, because flips and rotations keep the number of segments the same. Finally, after considering two additional equations for the bounding rectangle we find out that the total number of equations for any rectangulation would be $$N+1$$. In each rectangulation, some of the segments will be horizontal, defining $$J$$ equations, and others will be vertical, defining $$K$$ equations, with $$J+K=N+1$$ if we consider now two of the sides of the bounding rectangle segments. We can then define a set of coefficients $$a_{j,i}$$ and $$b_{k,i}$$ with $$j \in \{1,\ldots,J\}$$, $$k \in \{1,\ldots,K\}$$ and $$i \in \{1,\ldots,N\}$$ to have finally the equations $$\begin{array}{lcl} a_{1,i}w_i & = & w \\ a_{j,i}w_i & = & 0 & \textrm{for } j \in \{2,\ldots,J\}\\ b_{1,i}h_i & = & h \\ b_{k,i}h_i & = & 0& \textrm{for } k \in \{2,\ldots,K\} \end{array}$$ where the two of them with independent term will be the equations to account for the limits imposed by the bounding rectangle. Note that here we are passing all non-independent terms to the left, so the values for the coefficients can be $$0$$, $$1$$ or $$-1$$. Optimization problem Now that have defined arithmetically a rectangulation, we need to decide, what do we want to optimize for? Depending on the circumstances, we might want to optimize with different targets. For instance, we might want to split a window equally among callers, but depending on the rectangulation that could produce very narrow rectangles either horizontally or vertically. So, this cannot be the only thing to consider. In the end, I decided to adopt two criteria: 1. Make the area the same for all callers as far as possible 2. Preserve the camera aspect ratio so we can display camera output without cropping These two criteria overdetermine the system (although this would not be the case if we had selected only criterion 1, as in this case we would have $$2N$$ equations in total), so it is not possible to achieve both fully at the same time. So, I used a ponderated optimization function with a coefficient $$c$$ to balance between the two parts of the function. With that and taking into account the previously calculated rectangulation equations, we can pose the optimization problem as \begin{aligned} \min_{w_1,\ldots,w_N,h_1,\ldots,h_N \in \mathbb{R}^{2N}} F(w_1,\ldots,w_N,h_1,\ldots,h_N) = \\ = \sum_{i=1}^N{\left(w_i h_i – \frac{w h}{N}\right)}^2 + c\,h^2 \sum_{i=1}^N{\left(w_i – k h_i\right)}^2 \\ \textrm{subject to} \quad \begin{aligned} a_{1,i}w_i = w \\ a_{j,i}w_i = 0 \\ b_{1,i}h_i = h \\ b_{k,i}h_i = 0 \end{aligned} \end{aligned} where $$w$$ and $$h$$ are the width and height of the bounding rectangle and $$w_i$$ and $$h_i$$ are the width and height of the $$i$$-th rectangle of the rectangulation. The coefficient $$k$$ is the desirable width over height ratio for the rectangles, which we will want usually to be the same as the usual camera ratio between $$x$$ and $$y$$ resolution. In the first term of the objective function $$F$$ the difference between the rectangles areas and the area that we would have if the total area was equally split is calculated, while the second term measures how far we are from the target aspect ratio in each rectangle. Note that in the equation we multiply $$c$$ by the the squared total height. This is needed so resulting sizes are scaled when $$w$$ and $$h$$ are scaled. That is, when multiplying the size of the bounding rectangle by a constant $$q > 0$$ so $$w \rightarrow qw$$ and $$h \rightarrow qh$$, we would like the sizes of the solution to scale by the same value. We can see that is actually the case if we pose a new optimization problem for the scaled bounding rectangle (with new variables $$w_i^\prime$$ and $$h_i^\prime$$) and then substitute in the new target function $$F^\prime$$: \begin{aligned} F^\prime = \sum_{i=1}^N{\left(w_i^\prime h_i^\prime – \frac{q^2 w h}{N}\right)}^2 + c\,q^2 h^2 \sum_{i=1}^N{\left(w_i^\prime – k h_i^\prime\right)}^2 \\ = q^4 \sum_{i=1}^N{\left(\frac{w_i^\prime h_i^\prime}{q^2} – \frac{w h}{N}\right)}^2 + c\,q^4 h^2 \sum_{i=1}^N{\left(\frac{w_i^\prime}{q} – k \frac{h_i^\prime}{q}\right)}^2 \end{aligned} The constant $$q^4$$ does not affect the location of the minima of the function, so we can ignore it. Then, if we do the change of variable $$w_i^\prime = q w_i$$ and $$h_i^\prime = q h_i$$ we end up with the original objective function $$F$$, which proves that the minima of $$F^\prime$$ are $$q$$ times the minima of $$F$$. The constraints also behave well under the change of variables: the equations without independent terms are unaffected by scaling, and for the other two, when we replace $$w$$ by $$qw$$ and $$h$$ by $$qh$$ we see that $$q$$ can go to the left side of the equations and we would have $$w_i^\prime/q$$ and $$h_i^\prime/q$$ in all terms, showing that with the change of variables we would have the original minimization problem. What this is telling us is that solutions to this problem depend only on $$N$$, $$k$$, $$c$$ and the ratio $$w/h$$, minus scaling. If we have a solution for a set of these variables, we know the solution if we scale proportionally the bounding rectangle. Something similar happens if we scale only one of the dimensions, with the difference that this would happen only if we scale $$k$$ too, so $$w \rightarrow qw$$ and $$k \rightarrow qk$$ would scale widths by $$q$$, while $$h \rightarrow qh$$ and $$k \rightarrow k/q$$ would scale heights by $$k$$. Finally, this is defining the solution to one rectangulation, determined by a given Baxter permutation that we will call $$r$$, that defines a set of optimal values $$\{w_1^r,\ldots,w_N^r,h_1^r,\ldots,h_N^r\}$$. The final solution would be the one that minimizes $$F$$ across the optimal values for all possible rectangulations. If we create a set $$S_{B(N)}$$ where each element is a set with the optimal solution to a rectangulation, being $$B(N)$$ the total number of rectangulations, the global optimal values would be $$\min_{\{w_1^r,\ldots,w_N^r,h_1^r,\ldots,h_N^r\} \in S_{B(N)}} F(w_1^r,\ldots,w_N^r,h_1^r,\ldots,h_N^r)$$ Solving the optimization problem It is possible to solve the optimization problem of the last section in a fully analytical way. For that, we can use Lagrange multipliers to include the restrictions in a new optimization function, and then find the derivatives to be able to find the extrema. The python sympy module comes in handy for solving analytically this problem: this is implemented by the solve_rectangle_eqs() function. There is a test that uses it:  ./rect_test.py TestRectangles.test_lagrange_method_analitycal

However, it turned out that when we have enough rectangles, sympy can take a very long time to get a solution, and we have many problems to solve, one for each possible rectangulation. Therefore, I used in the end the minimize function from scipy library to solve numerically the problem. We can calculate analytically the Jacobian for the objective function and provide good initial estimates by using the proportions from the diagonal rectangulation. The minimize function can also incorporate constraints, so we can easily add the rectangulation equations to the mix. Scipy will choose the SLSQP algorithm to perform the minimization, due to these constraints. The results are consistently fast and accurate. The function that implements this is minimize_rectangulation(). Some of the tests that use it can be run with:

$./rect_test.py TestRectangles.test_scipy_minimize $ ./rect_test.py TestRectangles.test_diagonal_rectangulation_15rect
$./rect_test.py TestRectangles.test_diagonal_rectangulation_random The solution for the rectangulation in figure 3 for a window of size 320×180 for $$c=0.05$$ and $$k=1.5$$ can be seen in figure 4. The area element of $$F$$ is predominant in this case. Results Now that we have everything in place we can answer the questions we posed at the beginning of the post. For instance, in a videoconference with 5 callers, what would be the ideal window split? For this, the function get_best_rect_for_window() calculates all possible rectangulations for a given $$N$$, making sure we filter non-Baxter permutations, then minimizes according to the target function we defined previously (by using minimize_rectangulation()), and finally selects the rectangulation that produces the global smallest value for that target. A test that calculates the optimum for $$N=5$$, $$c=0.05$$, $$k=1.33$$, $$w=320$$ and $$h=180$$ can be run with: $ ./rect_test.py TestRectangles.test_best_5rect

Visually, we get the rectangulation shown in figure 5.

How things change as we start to resize the window? One of the tests shows what happens when the ratio $$w/h$$ changes (remember that solutions simply scale if we scale width and height with the same constant, so this ratio interests us more than real resolutions sizes in a monitor). For $$c=0.05$$, $$k=1.33$$ and area $$wh=320 \cdot 200 = 57600$$ pixels, this test finds the optima for different ratios:

\$ ./rect_test.py TestRectangles.test_best_rect_for_w_h_ratio

We can see the optimal rectangulations for different ratios calculated by this test in figure 6. It is also interesting to see how things change if we change the balance between the members of the objective function. In figure 7 we show the solutions for same test but with $$c=0.5$$.

Outcome is in general lines as you would expect. Having a prime number of callers forces rectangles to not be the same all the time, but the algorithm follows the optimization criteria as far as possible and makes a reasonable job. With a bigger $$c$$ we see that having rectangles with a width/height ratio near to $$k$$ starts to predominate and difference between areas starts to grow.

Conclusion

We have seen in this post the ways in which a rectangle can be split in another rectangles and how we can optimize the layout with different criteria. This of course has more applications than splitting the window of a video-conference application in different camera views, being the most well known one optimizing the placing of components on an integrated circuit. In any case, I had not expected to go this far when I started to work on this problem, but it was a fun ride that hopefully you have enjoyed as well, especially if you have gone this far in the reading!

Analysis and Plots of Solutions to Complex Powers

In chapter 5 of his mind-blowing “The Road to Reality”, Penrose devotes a section to complex powers, that is, to the solutions to

$$w^z~~~\text{with}~~~w,z \in \mathbb{C}$$

In this post I develop a bit more what he exposes and explore what the solutions look like with the help of some simple Python scripts. The scripts can be found in this github repo, and all the figures in this post can be replicated by running

git clone https://github.com/alfonsosanchezbeato/exponential-spiral.git
cd exponential-spiral; ./spiral_examples.py

The scripts make use of numpy and matplotlib, so make sure those are installed before running them.

Now, let’s develop the math behind this. The values for $$w^z$$ can be found by using the exponential function as

$$w^z=e^{z\log{w}}=e^{z~\text{Log}~w}e^{2\pi nzi}$$

In this equation, “log” is the complex natural logarithm multi-valued function, while “Log” is one of its branches, concretely the principal value, whose imaginary part lies in the interval $$(−\pi, \pi]$$. In the equation we reflect the fact that $$\log{w}=\text{Log}~w + 2\pi ni$$ with $$n \in \mathbb{Z}$$. This shows the remarkable fact that, in the general case, we have infinite solutions for the equation. For the rest of the discussion we will separate $$w^z$$ as follows:

$$w^z=e^{z~\text{Log}~w}e^{2\pi nzi}=C \cdot F_n$$

with constant $$C=e^{z~\text{Log}~w}$$ and the rest being the sequence $$F_n=e^{2\pi nzi}$$. Being $$C$$ a complex constant that multiplies $$F_n$$, the only influence it has is to rotate and scale equally all solutions. Noticeably, $$w$$ appears only in this constant, which shows us that the $$z$$ values are what is really determinant for the number and general shape of the solutions. Therefore, we will concentrate in analyzing the behavior of $$F_n$$, by seeing what solutions we can find when we restrict $$z$$ to different domains.

Starting by restricting $$z$$ to integers ($$z \in \mathbb{Z}$$), it is easy to see that there is only one resulting solution in this case, as the factor $$F_n=e^{2\pi nzi}=1$$ in this case (it just rotates the solution $$2\pi$$ radians an integer number of times, leaving it unmodified). As expected, a complex number to an integer power has only one solution.

If we let $$z$$ be a rational number ($$z=p/q$$, being $$p$$ and $$q$$ integers chosen so we have the canonical form), we obtain

$$F_n=e^{2\pi\frac{pn}{q} i}$$

which makes the sequence $$F_n$$ periodic with period $$q$$, that is, there are $$q$$ solutions for the equation. So we have two solutions for $$w^{1/2}$$, three for $$w^{1/3}$$, etc., as expected as that is the number of solutions for square roots, cube roots and so on. The values will be the vertex of a regular polygon in the complex plane. For instance, in figure 1 the solutions for $$2^{1/5}$$ are displayed.

If $$z$$ is real, $$e^{2\pi nzi}$$ is not periodic anymore has infinite solutions in the unit circle, and therefore $$w^z$$ has infinite values that lie on a circle of radius $$|C|$$.

In the more general case, $$z \in \mathbb{C}$$, that is, $$z=a+bi$$ being $$a$$ and $$b$$ real numbers, and we have

$$F_n=e^{-2\pi bn}e^{2\pi ani}.$$

There is now a scaling factor, $$e^{-2\pi bn}$$ that makes the module of the solutions vary with $$n$$, scattering them across the complex plane, while $$e^{2\pi ani}$$ rotates them as $$n$$ changes. The result is an infinite number of solutions for $$w^z$$ that lie in an equiangular spiral in the complex plane. The spiral can be seen if we change the domain of $$F$$ to $$\mathbb{R}$$, this is

$$F(t)=e^{-2\pi bt}e^{2\pi ati}~~~\text{with}~~~t \in \mathbb{R}.$$

In figure 2 we can see one example which shows some solutions to $$2^{0.4-0.1i}$$, plus the spiral that passes over them.

In fact, in Penrose’s book it is stated that these values are found in the intersection of two equiangular spirals, although he leaves finding them as an exercise for the reader (problem 5.9).

Let’s see then if we can find more spirals that cross these points. We are searching for functions that have the same value as $$F(t)$$ when $$t$$ is an integer. We can easily verify that the family of functions

$$F_k'(t)=F(t)e^{2\pi kti}~~~\text{with}~~~k \in \mathbb{Z}$$

are compatible with this restriction, as $$e^{2\pi kti}=1$$ in that case (integer $$t$$). Figures 3 and 4 represent again some solutions to $$2^{0.4-0.1i}$$, $$F(t)$$ (which is the same as the spiral for $$k=0$$), plus the spirals for $$k=-1$$ and $$k=1$$ respectively. We can see there that the solutions lie in the intersection of two spirals indeed.

If we superpose these 3 spirals, the ones for $$k=1$$ and $$k=-1$$ cross also in places different to the complex powers, as can be seen in figure 5. But, if we choose two consecutive numbers for $$k$$, the two spirals will cross only in the solutions to $$w^z$$. See, for instance, figure 6 where the spirals for $$k=\{-2,-1\}$$ are plotted. We see that any pair of such spirals fulfills Penrose’s description.

In general, the number of places at which two spirals cross depends on the difference between their $$k$$-number. If we have, say, $$F_k’$$ and $$F_l’$$ with $$k>l$$, they will cross when

$$t=…,0,\frac{1}{k-l},\frac{2}{k-l},…,\frac{k-l-1}{k},1,1+\frac{1}{k-l},…$$

That is, they will cross when $$t$$ is an integer (at the solutions to $$w^z$$) and also at $$k-l-1$$ points between consecutive solutions.

Let’s see now another interesting special case: when $$z=bi$$, that is, it is pure imaginary. In this case, $$e^{2\pi ati}$$ is $$1$$, and there is no turn in the complex plane when $$t$$ grows. We end up with the spiral $$F(t)$$ degenerating to a half-line that starts at the origin (which is reached when $$t=\infty$$ if $$b>0$$). This can be appreciated in figure 7, where the line and the spirals for $$k=-1$$ and $$k=1$$ are plotted for $$20^{0.1i}$$. The two spirals are mirrored around the half-line.

Digging more into this case, it turns out that a pure imaginary number to a pure imaginary power can produce a real result. For instance, for $$i^{0.1i}$$, we see in figure 8 that the roots are in the half-positive real line.

That something like this can produce real numbers is a curiosity that has intrigued historically mathematicians ($$i^i$$ has real values too!). And with this I finish the post. It is really amusing to start playing with the values of $$w$$ and $$z$$, if you want to do so you can use the python scripts I pointed to in the beginning of the post. I hope you enjoyed the post as much as I did writing it.


text:
# Dealing with stator magnetic saturation

In my previous experiments demonstrating torque feedback (full rate inverse dynamics, ground truth torque testing), I’ve glossed over the fact that as the stator approaches magnetic saturation, the linear relationship between torque and current breaks down. Now finally I’ll take at least one step towards allowing moteus to accurately work in the torque domain as motors reach saturation.

## Background

The stator in a rotor consists of windings wrapped around usually an iron core. The iron in the core consists of lots of little sub-domains of magnetized material, that normally are randomly oriented resulting in a net zero magnetic field. As current is applied to the windings, those domains line up, greatly magnifying the resulting magnetic field. Eventually most of the sub-domains are aligned, at which point you don’t get any more magnifying effect from the iron core. In this region, the stator is said to be “saturated”. You can read about it in much more depth on wikipedia or with even more detail here. The end result is a curve of magnetic field versus applied current that looks something like this:

To date, moteus assumes that you are operating completely in the “Linear” region, where the torque and current are linearly related.

## Operating in the Rotation Region

To operate in the “rotation” region I ended up using the following formula:

$\tau = K_T * I_c + ts * log2(1 + (I - I_c) * is)$

Where $I$ is the input current, $K_T$ is the motor torque constant, $ts$, $I_c$ and $is$ are three constants that I fit to measured torque data. With some approximations, this can be calculated relatively efficiently on the STM32G4 that drives the moteus controller, adding only a microsecond to the overall loop time to go in both directions.

I then ran a torque sweep with my load-cell fixture from before, and sure enough, the input and output torque match much better now across the entire range of operation, despite the fact that the phase current needs to start growing very rapidly near the top end:

# Testing qdd100 stator windings

My initial design torque for the qdd100 was a little over 17 Nm. However, when I did my first ground truth torque testing, I found that some servos had a lower maximum torque than I had specified. While working to diagnose those, I built a qdd100 that used an alternate stator winding of 105Kv instead of the 135Kv that are in all the beta units. The Kv rating of a stator describes how fast the motor will spin for a given applied voltage. If you assume the same amount of copper mass of wiring, a lower Kv will mean that there are thinner wires that wrap around the stator more turns (or fewer wires in parallel). A higher Kv will have thicker wires with fewer overall turns.

On paper, if you assume a perfect controller, this shouldn’t make much of a difference. The same input power should be required for the same output torque. The only differences should come into play once you have a controller with either a limited maximum voltage or a limited maximum current. The higher Kv motor will be able to go faster given a fixed maximum voltage, and the lower Kv motor will have more torque for a given maximum current.

I wanted to verify that this was true as part of my evaluation to identify the cause of my decreased torque, so I used a slightly upgraded torque testing fixture:

For now, I rigged up the world’s cheapest load cell from amazon to a Nucleo configured to report the load in grams over the serial port. I also wired up my Chroma power supply over USB using the linux USBTMC driver. With those two things hooked up, I was able to run tests that sweeped across torque commands, while recording output torque, phase current, and input power.

At higher torques, the input power was pretty sensitive to the temperature of the windings — hotter windings increased the resistance, which increased the power required to achieve a given phase current, thus my plot isn’t perfect as it was grabbed over several different runs. For the highest power samples I couldn’t use my Chroma, as it is limited to around 600W. Thus those samples don’t record the input power.

Plotting the input power vs output torque on the same chart shows that indeed, modulo some measurement error, they are the same for the two stators:

So, this experiment reaffirmed my understanding of stator magnetics and confirmed that the stator winding was not the cause of my decreased torque.

# micro-BOM management

I’ve now built 3 or 4 complete quad A1 style robots depending upon how you look at it. Each was somewhat of a one-off, incrementally modified over time as I discovered failure modes and improved the design. Before starting to serially build quad A1 style robots, I wanted to get a better understanding of how much actually goes into making one. The quad A1 has a fair number of sub-assemblies, custom PCBs, harnesses, and assembly steps that go into its production. During previous builds, I kept running into problems where I would run out of some component, fastener, or raw material unexpectedly, then have to wait for its lead time before I could continue.

I’m currently using a simplified Kanban inventory tracking system where I re-order parts and components when their inventory level reaches a critical threshold. However, given the discontinuous nature of my production, setting those levels is really hard. I didn’t know, for instance, just how many M3x8 bolts were necessary between the chassis, legs, and all the other sub-assemblies, since I had never really built multiple of all of them at one time before.

## Incremental improvement

To make my life a little easier, I’ve started on a “micro-BOM” management solution. It currently is just a simple C++ application that reads a tree of JSON5 (mostly so I can use the obvious comment style) files on disk. Each JSON5 file describes a single stock item, what sub-components it consists of, and what it takes to acquire it in terms of cash, assembly time, 3d printer time, ordering information, and assembly procedures.

{
"version" : "20200701.0",
"components" : [
{ "name" : "quada1_leg_foot", "qty" : 1 },  // the squash ball
],
"resources" : [
{ "mk3s" : 7.5, "petg_black" : 71.54, "hours_tech1" : 0.1 },
],
"procure" : {
"print" : "20200625-lower_leg_and_bracket.3mf",
},
}

That tool can then follow the hierarchy to count up how many stock units of each type are necessary for any top level item, and how many “resources” are necessary to make that happen.

Thus after entering in ~85 separate “stock units”, I now know that the current version of a full quad A1 requires approximately 300 hours of MK3s printer time between all of its assemblies and sub-assemblies, 132 heat set inserts, and 310 fasteners of various types (assuming the qdd100s are pre-assembled)!

## Next steps

Given that I have relatively deep sub-assembly trees, and that I expect to be keeping some stock of all of them, I want to make this tool inventory-aware. That way it knows I already have N leg sub-assemblies in stock or Y qdd100s already built, and thus can tell me how many more bolts and such are necessary to build a full robot.

I’d also like to be able to export the results from this to replace the human-readable bom.txt entries sprinkled throughout git. That will make them both more accurate, and easier to maintain.

# High speeds with the moteus controller

Someone contacted me not too long ago who wanted to use the moteus controller, but wasn’t sure if it would be able to hit their target mechanical velocity of 6000rpm. I honestly wasn’t either, so I tested it. After a quick firmware fix, the devkit motor when run at 34V seems to be able to do it no problem.

It should be noted that the current firmware assumes you are within a thousand or so revolutions of 0. You can exceed that pretty quickly running at 120 revolutions per second!

# mjbots Monday: New lower prices

One of my goals with mjbots is to make building dynamic robots more accessible to researchers and enthusiasts everywhere. To make that more of a reality, I’m lowering the prices in a big way on the foundational components of brushless robotic systems, the moteus controller and qdd100 servo.

Don’t worry, if you purchased any of these in the last month, you should be getting a coupon in your email equivalent to the difference.

Happy building!

# Cartesian leg PD controller

As I am working to improve the gaits of the mjbots quad A1, one aspect I’ve wanted to tackle for a long time is improving the compliance characteristics of the whole robot. Here’s a small step in that direction.

## Existing compliance strategy

The quad A1 uses qdd100 servos for each of its joints. The “qdd” in qdd100 stands for “quasi direct drive”. In a quasi direct drive actuator, a low gearing ratio is used, typically less than 10 to 1, which minimizes the amount of backlash and reflected inertia as observed at the output. Then, high rate electronic control of torque in the servo based on current and position feedback allows for dynamic manipulation of the spring and dampening of the resulting system.

Another option is a series elastic actuator, which uses a traditional high gear reduction servo with a mechanical spring or elastic mechanism inline with the load. Sometimes a separate motorized actuation mechanism can be used to vary the damping properties of the elastic element. This is in principle similar to the quasi direct drive approach, but suffers from a limited overall control bandwidth. Despite being “springy”, QDD servos are still able to have a very high effective mechanical control bandwidth, on the order of hundreds of hertz.

For the quad A1 to date, the compliance it exhibits is largely due to the qdd100’s internal control algorithms, and to a very minor extent, flexing in the mechanical structures of the quad A1 itself. This does work, and gives decent results.

## Limitations

The biggest limitation of solely using this approach, is that since the compliance is performed at the joint level, it has no knowledge of the current 3d configuration of the leg. The resulting compliance in 3D space is highly non-linear and depends upon where in configuration space the leg is at that point in time. For instance, if the back legs are configured to have the knee very bent, but the front legs are not, then the back knee needs a much larger restorative torque per unit rotation to have the same linear restorative force at the tip of the leg.

That results in artifacts like shown in the video at the bottom. When the robot falls with the legs not in an identical configuration, the robot ends up pitching or rolling depending upon how the compliance interacts with the current leg geometry.

## A “fix”

In my original designs for the moteus controller, I had left a high rate “inter-leg” bus option in the design, where each controller could exchange IK information at the full control rate, so that all compliance could be performed in the 3D space, rather than in joint space. However, as the design progressed, and I failed to implement it, I dropped that capability to simplify and reduce costs.

Here, I ended up implementing something purely in software which doesn’t have the same level of performance as that system would have, but also doesn’t require additional dedicated high rate communication transceivers on every servo control board. The 3D PD controller is just run on the raspberry pi at the regular control update rate (400Hz currently). That makes the control flow look like this:

## Results

While this solution isn’t perfect, it does give better results in many scenarios. I applied some disturbances to the robot with either solely joint level controllers, or joint plus XYZ controllers. For the two cases, I tried to tune the controllers to a similar level of stiffness and damping to make the comparison as fair as possible. Walking is generally improved as well, even with just a constant compliance throughout the gait cycle.

# New mjbots.com

https://shop.mjbots.com is now https://mjbots.com (don’t worry, the old site redirects)! The functionality is largely the same, you can still get your qdd100 actuators or moteus controllers. The biggest differences are 1) it looks slightly nicer, and 2) shipping rates are improved, and international shipping rates drastically so. For instance, DHL “Express” 2 day shipping to some points in Europe is now under $35 USD, whereas previously 2 day shipping was over$300. That is often cheaper than even USPS International Priority — which is typically 2-4 weeks.

I’ll be adding some more products over the next couple of weeks, and I wanted to make them as accessible to a worldwide audience as possible!

# Updated serialization library (diagnostics part 1)

Now that I have the qdd100 servo in beta phase, the IMU working at full rate, and the quad A1 is moving around I’m getting closer to actually working to improve the gaits that the machine can execute.  To date, the gaits I have used completely ignore the IMU and only use the feedback from the joints in order to maintain force in 3D.  With tuning and on controlled surfaces this can work well, but if you go outside the happy regime, then it can undergo significant pitch and roll movements during the leg swing phase, which at best results in a janky walk, and at worst results in oscillation or outright instability.

There are also a number of as-yet-unidentified problems that seemingly cause the feet to not track the ground position properly, resulting in the feet slipping on the floor despite being nearly fully loaded.

To tackle all these new domains requires some improvements to my diagnostics infrastructure and tools.  I’ll cover the improvements I’ve made in a few posts, since the work that has gone into it has covered a fair amount of ground.  I’ll start with something I mostly completed back in the summer of 2019 and has the least direct impact, but gives at least a background for some of the other upcoming changes.

## Telemetry format

Super Mega Microbot since its inception in 2014 used a self-describing serialization and telemetry format that was loosely based on work I had done professionally previously at Bluefin Robotics and then Jaybridge Robotics.  This format was then the basis for later work at Jaybridge and Toyota Research Institute.  The basic idea breaks down like this:

• The schema which describes the data and the data are separate entities
• The schema is recorded alongside the data whenever it is written to persistent storage
• The schema contains sufficient information to reconstruct a CSV or JSON like representation of the data with no additional meta-data
• Structure tools can map a given on disk-schema to a possibly different in-memory one using a schema evolution algorithm
• The data is serialized and stored in a manner which is very efficient to write at high rates from realtime processes

Compared to other serialization mechanisms, this has different trade-offs.

• Formats like JSON, XML, either completely include the schema in each data instance, or include a large amount of self-describing information in each data instance that is not strictly necessary to represent it
• Formats like protobuf, capnproto, flatbuffers, and SBE have a different tradeoff.  They are geared towards performance, but largely also assume a single canonical source of schema data that is shared through an independent side channel and has a single linear revision history.  This makes sense for server RPC, where client and server are each distributed (possibly different) versions of the schema and want to communicate without having to exchange it.  They also include more metadata in the data stream than is strictly required many of them are more expensive to serialize or deserialize.
• The closest to this work is Apache AVRO.  It uses the same principle of separate schema and data, and expects the schema to be stored alongside the data.  It also requires no code generation, which many of the above tools do require.

The unique pieces in this work over AVRO are that:

• The data format is such that many common in-memory structures can simply be bit copied as serialized data with no further effort.  Those that do require some manipulation still require no additional in-memory structures associated with serialization.  This combines the properties of protobuf in that the serialization objects can be used as mutable state, with those of capnproto that allows zero cost serialization.
• No recursion or pointers are supported, which renders the necessary code very simple.  The entirety of the C++ serialization and deserialization library is only a few hundred lines of code and took less than a week overall to write, unit test, and debug over the 6 years I’ve been using it.  It also functions perfectly fine in microcontroller-based embedded environments like the moteus controller.
• The on-disk format is designed for rapid random seek access in time, assuming that small-ish records are written regularly.

The downsides are that it isn’t widely supported, isn’t optimized to handle single structures which have very large serialized representations, and the only language bindings aside from C++ are read only ones for python and TypeScript.

In future articles, I’ll describe a bit of the detail of the recently revised design, then go into the tools that use it.

# More MLCC learning

It seems that I’m learning much about PCB design the very hard way.  Back last year I wrote up my discovery of MLCC bias derating.  Now I’ll share some of my experiences with MLCC cracking on the first production moteus controllers.

When I was first putting the production moteus controllers through their test and programming sequence, I observed a failure mode that I had yet to have observe in my career (which admittedly doesn’t include much board manufacturing).  When applying voltage, I got a spark and puff of magic smoke from near one of the DC link capacitors on the left hand side.  In the first batch of 40 I programmed, a full 20% failed in this way, some at 24V, and a few more at a 38V test.  I initially thought the problem might have been an etching issue resulting in voltage breakdown between a via and an internal ground plane, but after examining the results under the microscope and conferring with MacroFab determined the most likely cause was cracking of the MLCCs during PCB depanelization.

Here’s a video describing the problem and potential solutions in way more detail than I’ll go into:

Needless to say, I hadn’t managed to see this failure in the 100 or so previous moteus controllers I’ve built, or I would have figured this out and resolved it!

For this first round of production controllers, I went and replaced every single capacitor near the edge of the board with a TDK variant that has internal soft termination, then tested them all at max voltage and a little beyond.  Future revisions will use that variant of capacitor everywhere, as well as relocating the capacitors to reduce the mechanical stress they experience during manufacturing, handling, and installation.

# Programming and testing moteus controllers

Like with the fdcanusb, I built a programming and test fixture for the moteus controllers.  The basic setup is similar to the fdcanusb.  I have a raspberry pi with a touchscreen connected via USB to a number of peripherals.  In this case, there is a STM32 programmer, a fdcanusb, and a label printer.  Here though, unlike with the fdcanusb fixture, I wanted to be able to test the drive stage of the controllers and the encoders too.

My solution was to create a mechanical fixture that each board slots onto, with pogo pins that connect to test points for the phase outputs.

While it doesn’t make as good a connection as the solder through holes normally used to connect a motor, it is good enough to verify that the controller works.  As a side-bonus, it also makes it trivial to test that the absolute magnetic encoder works properly.

This video shows how the programming and testing process works, and walks through testing a few boards.


text:
main-content

## Swipe to navigate through the articles of this issue

01-10-2020 | Original Article | Issue 19/2020

# A multistage hybrid model for landslide risk mapping: tested in and around Mussoorie in Uttarakhand state of India

Journal:
Environmental Earth Sciences > Issue 19/2020
Authors:
Mukunda Mishra, Tanmoy Sarkar
Important notes

## Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Abstract

The study aims to develop a hybrid model approach for the assessment of the landslide (LS) risk qualitatively. It involves multiple consecutive stages of statistical prediction, machine learning, and mapping in the GIS environment. At the first stage, a landslide susceptibility map has been developed using the analytic hierarchy process (AHP) algorithm, coupled with the binary logistic regression (BLR) technique. The AHP model incorporates 11 geo-hydrological and environmental variables as predictors sourced from remote-sensing datasets to generate the LS susceptibility as output. Twenty-three field-based validation locations validate the test result. Pearson's correlation coefficient (r) between the observed ($${{\mathrm{\L}}}_{{{\text{COMPUTED}}}}$$) and predicted ($${{\mathrm{\L}}}_{{{\text{PREDICTED}}}}$$) values of LS susceptibility is 0.928 at 0.01 level of significance. At the next stage, the LS risk is evaluated considering the ‘risk trio,’ i.e., the combination of the hazard, exposure, and vulnerability. This stage involves the transformation of a range of qualitative datasets to the virtual workspace of machine learning. The landslide risk output has been predicted with an initial fuzzy model, incorporating a set of 32 rules for membership functions (MF). This initial model uses randomly selected 20% datasets to tailor the fuzzy rules through the adaptive neuro-fuzzy interface (ANFIS). The training to ANFIS results in framing 120 fuzzy rules for the best possible prediction of the outcome. The final LS risk map from the ANFIS output shows that more than 70% area is under high-to-very high LS risk. The model is tested in a 5′ × 5′ grid around the famous hill station Mussoorie in the state of Uttarakhand, India. The model exhibits a satisfactory level of accuracy for the present-study area, which has made us confident to recommend it. The multistage model is worthy of being applied for landslide risk mapping for the similar kinds of study areas, and also for other areas of landslide with necessary customization as deemed necessary.

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Literature


text:
- Current Issue - Ahead of Print Articles - All Issues - Search - Open Access - Information for Authors - Downloads - Guideline - Regulations ㆍPaper Submission ㆍPaper Reviewing ㆍPublication and Distribution - Code of Ethics - For Authors ㆍOnline Submission ㆍMy Manuscript - For Reviewers - For Editors
 << Previous Issue Communications of the Korean Mathematical Society (Vol. 22, No. 2) Next Issue >>
 Commun. Korean Math. Soc. 2007 Vol. 22, No. 2, 157—321
 Isomorphism of modular group algebras of abelian groups with semi-complete $p$-primary components Peter Danchev MSC numbers : 20C07, 16S34, 16U60, 20K10, 20K15, 20K20, 20K21 Commun. Korean Math. Soc. 2007 Vol. 22, No. 2, 157—161
 Determination of all subfields of cyclotomic function fields with divisor class number two Jaehyun Ahn, Hwanyup Jung MSC numbers : 11R58 Commun. Korean Math. Soc. 2007 Vol. 22, No. 2, 163—171
 Fuzzy subalgebras with thresholds in BCK/BCI-algebras Young Bae Jun MSC numbers : 03G10, 03B05, 03B52, 06F35 Commun. Korean Math. Soc. 2007 Vol. 22, No. 2, 173—181
 Composition of Binomial Polynomial Eunmi Choi MSC numbers : 12E05, 11D41 Commun. Korean Math. Soc. 2007 Vol. 22, No. 2, 183—194
 Maximal column rank preservers of integer matrices Seok-Zun Song, Kyung-Tae Kang MSC numbers : Primary 15A03, 15A04, 15A36 Commun. Korean Math. Soc. 2007 Vol. 22, No. 2, 195—205
 Corrigendum on "The Number of Points on Elliptic Curves $E~:y^{2}=x^{3}+cx$ over $mathbb{F}_{p}$ mod 8" Ilker Inam, Gokhan Soydan, Musa Demirici, Osman Bizim, Ismail Naci Cangul MSC numbers : 11G20, 14G05 Commun. Korean Math. Soc. 2007 Vol. 22, No. 2, 207—208
 Complemented sublattice of the Banach envelope of $operatorname{Weak}L_1$ isomorphic to $ell^p$ JeongHeung Kang MSC numbers : 46B03 Commun. Korean Math. Soc. 2007 Vol. 22, No. 2, 209—218
 Some Identities Involving the Legendre's Chi-Function Junesang Choi MSC numbers : Primary 33E20; Secondary 11M41 Commun. Korean Math. Soc. 2007 Vol. 22, No. 2, 219—225
 A note on the monotone interval-valued set function defined by the interval-valued Choquet integral Lee-Chae Jang MSC numbers : 28A45, 28A48, 28E10 Commun. Korean Math. Soc. 2007 Vol. 22, No. 2, 227—234
 Exponent-quasiadditive properties and application Gendi Wang, Xiaohui Zhang, Yuming Chu MSC numbers : 26D07, 33C05 Commun. Korean Math. Soc. 2007 Vol. 22, No. 2, 235—240
 Relation between fractal measures and Cantor measures In-Soo Baek MSC numbers : 28A78 Commun. Korean Math. Soc. 2007 Vol. 22, No. 2, 241—246
 On a system of generalized nonlinear variational inequalities Jingchang Li, Zhenyu Guo, Zeqing Liu, Shin Min Kang MSC numbers : 47J20, 49J40 Commun. Korean Math. Soc. 2007 Vol. 22, No. 2, 247—258
 Continuity for multilinear integral operators on Besov spaces Liu Lanzhe MSC numbers : 42B20, 42B25 Commun. Korean Math. Soc. 2007 Vol. 22, No. 2, 259—275
 Energy finite $p$-harmonic functions on graphs and rough isometries Seok Woo Kim, Yong Hah Lee MSC numbers : Primary 31C20 Commun. Korean Math. Soc. 2007 Vol. 22, No. 2, 277—287
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 Upper bounds for bivariate Bonferroni-type inequalities using consecutive events Min-Young Lee MSC numbers : Primary 60E15; Secondary 60C50 Commun. Korean Math. Soc. 2007 Vol. 22, No. 2, 305—313
 On the exponential inequality for negative dependent sequence Tae-Sung Kim, Hyun-Chull Kim MSC numbers : 60F15 Commun. Korean Math. Soc. 2007 Vol. 22, No. 2, 315—321


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# Publications 32

Here we present a list of publications that were a result of projects funded by EuHIT or were published by members of EuHIT consortium.

Page 2 of 4

• Article
Jackson, M. J.; Kolosov, O.; Schmoranzer, D.; Skrbek, L.; Tsepelin, V.; Woods, A. J.
Journal of Low Temperature Physics. 2016, Vol. 183, Issue 3, p. 208-214. DOI: 10.1007/s10909-015-1397-4
• Publication URL http://dx.doi.org/10.1007/s10909-015-1397-4
• Abstract We present proof-of-concept measurements of the vortex line density generated by a quartz tuning fork resonator probed by the attenuation of second sound in superfluid $$^{4}$$He at 1.6 K. The force–velocity response of a quartz tuning fork operating at a frequency of 31 kHz exhibited the onset of extra damping at a velocity of 0.5 ms$$^{-1}$$. Attenuation of the 5th resonant mode of second sound was observed at the same velocity, indicating the production of vortex lines. Our measurements demonstrate that an increase of the drag coefficient corresponds to the development of quantum turbulence.
• Conference Proceedings
Usama, S.; Kopec, J.; Tellez, J.; Kwiatkowski, K.; Redondo, J.M.; Nadeem, M.
Geophysical Research Abstracts. EGU2017-65903-1. 2017, Vol. 19, p. 1.
• Keywords Fractal grids, Turbulence, Sparce Flow, Intermittency
• Abstract Flat 2D fractal grids are known to alter turbulence characteristics downstream of the grid as compared to the regular grids with the same blockage ratio and the same mass inflow rates. This has excited interest in the turbulence community for possible exploitation for enhanced mixing and related applications. Recently, a new 3D multi-scale grid design has been proposed, such that each generation of length scale of turbulence grid elements is held in its own frame, the overall effect is a 3D co-planar arrangement of grid elements. This produces a ‘sparse’ grid system whereby each generation of grid elements produces a turbulent wake pattern that interacts with the other wake patterns downstream. A critical motivation here is that the effective blockage ratio in the 3D Sparse Grid Turbulence (3DSGT) design is significantly lower than in the flat 2D counterpart – typically the blockage ratio could be reduced from say 20 in 2D down to 4 in the 3DSGT. If this idea can be realized in practice, it could potentially greatly enhance the efficiency of turbulent mixing and transfer processes clearly having many possible applications. Work has begun on the 3DSGT experimentally using Surface Flow Image Velocimetry (SFIV) at the European facility in the Max Planck Institute for Dynamics and Self-Organization located in Gottingen, Germany and also at the Technical University of Catalonia (UPC) in Spain, and numerically using Direct Numerical Simulation (DNS) at King Fahd University of Petroleum Minerals (KFUPM) in Saudi Arabia and in University of Warsaw in Poland. DNS is the most useful method to compare the experimental results with, and we are studying different types of codes such as Imcompact3d, and OpenFoam. Many variables will eventually be investigated for optimal mixing conditions. For example, the number of scale generations, the spacing between frames, the size ratio of grid elements, inflow conditions, etc. We will report upon the first set of findings from the 3DSGT.
• Article
Jackson, M. J.; Tsepelin, V.; Poole, M.; Woods, A. J.; Človečko, M.; Skrbek, L.; Schmoranzer, D.
Physical Review B. 2016, Vol. 94, Issue 21, p. 214503. DOI: 10.1103/PhysRevB.94.214503
• Abstract We report recent investigations into the transition to turbulence in superfluid $$^{4}$$He, realized experimentally by measuring the drag forces acting on two custom-made quartz tuning forks with fundamental resonances at 6.5 kHz and 55.5 kHz, in the temperature range 10 mK to 2.17 K. In pure superfluid in the zero temperature limit, three distinct critical velocities were observed with both tuning forks. We discuss the significance of all critical velocities and associate the third critical velocity reported here with the development of large vortical structures in the flow, which thus starts to mimic turbulence in classical fluids. The interpretation of our results is directly linked to previous experimental work with oscillators such as tuning forks, grids, and vibrating wires, focusing on the behavior of purely superfluid $$^{4}$$He at very low temperatures.
• Conference Paper
Tuliszka-Sznitko, Ewa; Kielczewski, Kamil
"Computational Methods in Science and Technology". 2015, Vol. 21, p. 211.
• Conference Paper
Tuliszka-Sznitko, Ewa; Kiełczewski, Kamil
Contributions to the Foundations of Multidisciplinary Research in Mechanics, edited by J.M. Floryan. 2017, Vol. 2, p. 1015.
• Publication URL http://iutam.org/publications/ictam-proceedings/ictam_2016/
• Keywords Taylor-Couette flow, DNS/SVV, PIV
• Abstract The authors present the results obtained during a numerical investigations of the Tylor-Couette flow in configurations of aspect ratios 11.75 and 3.76, different radii ratios and end-wall boundary conditions. The main purpose of the research is to investigate the influence of geometrical parameters and end-walls boundary conditions on the flow structure, the torque fluctuations and its total values. The results are discussed in the light of Brauckmann, Eckhardt [1] and Dong [2] data obtained with periodicity condition in axial direction. The results are compared also with Wendt [3] experimental results. For the cavity of aspect ratio 3.76 the authors study the laminar-turbulent transition for the radii ratio from the range: 0.375 - 0.821 (the comparison with the EuHIT PIV results are presented). The authors have found that the transition from the three-cell structure to the one-cell structure occurs only for the lower ratios Computations are performed for Re up to 3100. All presented results have been obtained with the use of DNS/SVV method in which an artificial viscous operator is added to Laplace operator to stabilize the computational process for higher Reynolds numbers.
• Conference Proceedings
Tellez, Jackson; Redondo, Jose Manuel; Malik, Nadeem; Kopec, Jasec; Kwiatokowski, Kamil; Usama, Syed
International Workshop on Complex Turbulent Flows (IWCTF2017). 2017.
• Abstract The experimental campaign performed during 2017 in the Low Turbulence Wind Tunnel at the Max-Planck-Institute for Dynamics and Self-Organization is described and some of the preliminary results presented. The results compare several types of fractal and multifractal wakes in new configurations, these are based upon previous numerical and experimental turbulence data obtained by Vassilicos et al. [1,2], but we also include different types of grids (2D and 3D). Additional long velocity time series were obtained in order to also evaluate higher order structure functions in the scope of the Euhit Funded project,ˆaTurbulence Generated by Sparse 3D Multi-Scale Grid” in Gottinguen, Germany. The presented turbulence velocity data has been collected by means of a single hot-wire anemometry, other PIV and flow visualizations will be discussed elsewhere. We compare the properties of turbulent flow generated by plane grids such as the configuration shown in Fig. 1 as well as Sierpinski type of configurations. the concept of a sparse three-dimensional multi-scale grid turbulence generator (3DSG) is expected to reduce the blockage of the grids, here the layers of grids of different size (Fig. 2) in a multi-plane configuration is similar to the co-planar arrangement [2], but in the case of sparce grids [3, 4] producing an overall blockage ratio which is substantially less than in a comparable 2D arrangement. The experimental results will be compared with data generated from traditional fractal grids, and also with theoretical and numerical calculations where posible. In this case in order to compare the same 2D grate located in the wind tunnel, the developmment of a procedure capable of removing part of the effect of the 2D grid, so is posible to convert the structure of a grid in either fractal type grid or as simple grid. This code is written in MATALB, and may also be used for Sierpinski grids [5]. The comparison of PDF’s, Spectra and Higher order Structure Functions [6,7] is being used in the different facilities describing non-homogeneous turbulence, because it is important to study and compare the role of non-uniform turbulence and intermittency in the different grids, Fig. 3 shows some results of the spectra and structure functions of Sierpinski grids of order 1-2-4. The 3D sparse grid turbulence generators(3DSGT) which reduces the effective blockage ratio, seem to possess a much greater parameter space [8,9], which could allow further optimisation of the turbulence, when compared to the similar structure 2DF grids [1,2]. Here, we report some results. The blockage ratio (solidity) is greatly reduced in the 3D sparse (3DS) grid system, while the mass flow rate is greatly enhanced and is sensitive to the largest scale in the grid system. (Fig. 4 ) Within the range of parameters examined, the enhancement in the mass fluxes was found to be between 21 and 38.

References

[1] S. Laizet J. C. Vassilicos. “DNS of Fractal-Generated Turbulence”. In Flow Turbulence Combustion 87 (2011) 673-705.

[2] Queiros-Conde J. C. Vassilicos. “Turbulent Wakes in 3-D Fractal Grids”.In Intermittency in Turbulent Flows. (ed. J. C. Vassilicos.) Cambridge University Press 1(2001).

[3] N. A. Malik. “Sparse 3D Multi-Scale Grid Turbulence Generator”. In US Patent No. US 9,599,269 B2 1 (2017).

[4] N. A. Malik. “Sparse 3D Multi-Scale Grid Turbulence Generator”. In EPO Patent no. 2965805 1 (2017).

[5] C. E. Leyton-Pavez, J. M. Redondo, A. M. Tarquis-Alfonso, J. C. Gil-Marti‘In, J. D. Tellez-Alvarez. “Fractal analysis of growing cities and its relationship with health centre distribution”.In Proceedings of ISP RAS 29 2 (2017), 201-214.

[6] Mahjoub, O. B., Babiano A. and Redondo, J. M. “Structure functions in complex flows”. In Flow, Turbulence and Combustion 59 (1998) 299-313.

[7] Tarquis, A. M., Platonov, A., Matulka, A., Grau, J., Sekula, E., Diez, M., Redondo, J. M.. “ Application of multifractal analysis to the study of SAR features and oil spills on the ocean surface”. In Nonlinear Processes in Geophysics 21 (2014) 439-450.

[8] J. M. Redondo, A. M. Tarquis, J. D. Tellez. “Fractal analysis of Sierpinski Flows”.In Proceedings of ISP RAS 30 1 (2018),

[9] S. S. Strijhak, J.M. Redondo, J. Tellez. “Multifractal Analysis of a Wake for a Single Wind Turbine”.In Proceedings Topical Problems of Fluid Mechanics 2017, Prague, 2017 Edited by David Simurda and Tom´as Bodn´ar 360 pp. 275-284. ISBN 978-80-87012-61-1, ISSN 2336-5781.
• Article
Ferrero, E.; Mortarini, L.; Manfrin, M.; Solari, M.; Forza, R.
Journal of Geophysical Research: Atmospheres. 2014, Vol. 119, Issue 11, p. 6292-6305. DOI: 10.1002/2013JD021243
• Keywords physical simulation, rotating tank, microburst
• Abstract A laboratory simulation of atmospheric microbursts is presented. The physical model of the atmospheric phenomenon is reproduced at a reduced scale in a rotating tank (TURLab, Italy), the similitude is based on the Froude number. Different experiments were carried out varying the Rossby number, and the analysis of four significant cases is presented. The velocity, vorticity, and turbulent kinetic energy fields are evaluated together with the swirling strength analysis. The comparison with the natural prototype is eventually shown and discussed.
• Conference Paper
Toffoli, A.; Proment, D.; Salman, H.; Monbaliu, J.; Stramignoni, E.; Forza, R.; Manfrin, M.; Onorato, M.
Proc. 36th International Conference on Ocean, Offshore and Arctic Engineering (OMAE). 2017, p. OMAE2017-61156.
• Abstract The probability of occurrence of rogue waves in wind- generated fields is investigated experimentally in an annular wind-wave flume. Unlike many experiments on rogue waves, where waves are mechanically generated, here the wave field is forced naturally by wind as it is in the ocean. The peculiar geometry of the flume makes waves propagating circularly in an unlimited-fetch condition. Water surface elevation was measured at specific cross-sections under the effect of different wind speeds to monitor the temporal evolution of the wave field. Results show that the kurtosis of the surface elevation, the fourth order mo- ment of the probability density function and a measure of the percentage of extremes in a wave record, gradually increases in time with the evolution of the wave field. Deviations from Nor- mal statistics are observed to be a function of wind speed. The maximum departure from Normality resembled the one induced by quasi-resonant wave-wave interactions and it is observed at the final stage of wave growth and immediately before reaching the fully development regime.
• Conference Proceedings
Malik, N.
Geophysical Research Abstracts. 2017, Vol. 1.
• Keywords Diffusion, Turbulence, 3DSGT
• Abstract Presentationin the Short course, Experimental and Field Turbulence in Geophysical and Environmental Flows. SC2/AS2.8/NP9.2
• Article
Poulain, Cedric; Mazellier, Nicolas; Gervais, Philippe; Gagne, Yves; Baudet, Christophe
Flow, Turbulence and Combustion. 2004, Vol. 72, Issue 2, p. 245-271. DOI: 10.1023/B:APPL.0000044414.48888.25
• Abstract In this paper we report an experimental investigation of various statistical properties of the spatial Fourier modes of the vorticity field in turbulent jets for a large range of Reynolds numbers (530 ≤R$$\lambda$$≤ 6100). The continuous time evolution of a spatial Fourier mode of the vorticity distribution, characterized by a well-defined wavevector, is obtained from acoustic scattering measurements. The spatial enstrophy spectrum, as a function of the spatial wave-vector, is determined by scanning the incoming sound frequencies. Time-frequency analysis of the turbulent vorticity fluctuations is also performed for different length scales of the flows. Vorticity time-correlations show that the characteristic time of a Fourier mode behaves as the sweeping time. Finally, we report preliminary Lagrangian velocity measurements obtained using acoustic scattering by soap bubbles inflated with helium. Gathering a large number of passages of isolated bubbles in the scattering volume, one is able to compute the Lagrangian velocity PDF and velocity spectrum. Despite the spatial filtering due to the finite size of the bubble, the latter exhibits a power law, with the -2 exponent predicted by the Kolmogorov theory, over one decade of frequencies.


text:
# NMTC Final Inter 2016

Q1) A, B, C are three points on a circle. The distance of C from the tangents at A and B to the circle are a and b respectively. If the distance of C from the chord AB is c, show that c is the geometric mean of a and b.

Q2)Find all integer solutions to the equation x³+(x+4)²=y².

Q3)Two right angled triangles are such that the incircle of one triangle is equal in size to the circumcircle of the other. If P is the area of the first triangle and Q is the area of the second triangle, then show that P/Q≥3+2√2.

Q4) (a)Find the maximum value k for which one can choose k integers from 1,2,.....,2n so that none of the chosen integers is divisible by any other chosen integer.

(b) F(x) is a polynomial of degree 2016 such that all the coefficients are non negative and non exceed F(0).Show that the coefficient of x^2017 in (F(x))² is at most F(1)²/2.

Q5) (a) n>=3 and a1, a2, a3,....., an are different positive integers. Given that, except the first and the last, each one is a harmonic mean of its immediate neighbors. Show that none of the given integers is less than n-1. (b)Show that the shortest side of a cyclic quadrilateral with circumradius 1 is at most √2.

Q6)C1,C2,C3 are circles with radii 1,2,3 respectively, touching each other as shown in Figure. Two circles can be drawn touching all these circles. Find the radii of these two circles.

Note by Naitik Sanghavi
3 years, 1 month ago

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- 3 years, 1 month ago

Thanks for mentioning. Do you want me post the solutions, or you were just kind enough to help me have a glimpse of the paper?

- 3 years, 1 month ago

As you wish! 😊

- 3 years, 1 month ago

For question 6, use Descartes' Theorem.

- 3 years, 1 month ago

I have posted Junior Level Paper here.

- 3 years, 1 month ago

You have not mentioned question 5 b

- 3 years, 1 month ago

I have some questions about NMTC, please care to answer:

• Is NMTC objective or subjective?
• Is it IITJEE or Olympiad Mathematics?

- 3 years, 1 month ago

Well NMTC screening test is objective and NMTC final test is subjective.It is Olympiad Mathematics

- 3 years, 1 month ago

Done!BTW I am getting only two solutions for (Q2)!Am I correct?

- 3 years, 1 month ago


text:
Outlook: MAGELLAN GLOBAL FUND. is assigned short-term Ba1 & long-term Ba1 estimated rating.
Dominant Strategy : Sell
Time series to forecast n: 05 Feb 2023 for (n+4 weeks)
Methodology : Modular Neural Network (Social Media Sentiment Analysis)

## Abstract

MAGELLAN GLOBAL FUND. prediction model is evaluated with Modular Neural Network (Social Media Sentiment Analysis) and Polynomial Regression1,2,3,4 and it is concluded that the MGF stock is predictable in the short/long term. According to price forecasts for (n+4 weeks) period, the dominant strategy among neural network is: Sell

## Key Points

1. Technical Analysis with Algorithmic Trading
2. Stock Rating
3. What is the use of Markov decision process?

## MGF Target Price Prediction Modeling Methodology

We consider MAGELLAN GLOBAL FUND. Decision Process with Modular Neural Network (Social Media Sentiment Analysis) where A is the set of discrete actions of MGF stock holders, F is the set of discrete states, P : S × F × S → R is the transition probability distribution, R : S × F → R is the reaction function, and γ ∈ [0, 1] is a move factor for expectation.1,2,3,4

F(Polynomial Regression)5,6,7= $\begin{array}{cccc}{p}_{a1}& {p}_{a2}& \dots & {p}_{1n}\\ & ⋮\\ {p}_{j1}& {p}_{j2}& \dots & {p}_{jn}\\ & ⋮\\ {p}_{k1}& {p}_{k2}& \dots & {p}_{kn}\\ & ⋮\\ {p}_{n1}& {p}_{n2}& \dots & {p}_{nn}\end{array}$ X R(Modular Neural Network (Social Media Sentiment Analysis)) X S(n):→ (n+4 weeks) $\stackrel{\to }{S}=\left({s}_{1},{s}_{2},{s}_{3}\right)$

n:Time series to forecast

p:Price signals of MGF stock

j:Nash equilibria (Neural Network)

k:Dominated move

a:Best response for target price

For further technical information as per how our model work we invite you to visit the article below:

How do AC Investment Research machine learning (predictive) algorithms actually work?

## MGF Stock Forecast (Buy or Sell) for (n+4 weeks)

Sample Set: Neural Network
Stock/Index: MGF MAGELLAN GLOBAL FUND.
Time series to forecast n: 05 Feb 2023 for (n+4 weeks)

According to price forecasts for (n+4 weeks) period, the dominant strategy among neural network is: Sell

X axis: *Likelihood% (The higher the percentage value, the more likely the event will occur.)

Y axis: *Potential Impact% (The higher the percentage value, the more likely the price will deviate.)

Z axis (Grey to Black): *Technical Analysis%

## IFRS Reconciliation Adjustments for MAGELLAN GLOBAL FUND.

1. For example, an entity may use this condition to designate financial liabilities as at fair value through profit or loss if it meets the principle in paragraph 4.2.2(b) and the entity has financial assets and financial liabilities that share one or more risks and those risks are managed and evaluated on a fair value basis in accordance with a documented policy of asset and liability management. An example could be an entity that has issued 'structured products' containing multiple embedded derivatives and manages the resulting risks on a fair value basis using a mix of derivative and non-derivative financial instruments
2. If changes are made in addition to those changes required by interest rate benchmark reform to the financial asset or financial liability designated in a hedging relationship (as described in paragraphs 5.4.6–5.4.8) or to the designation of the hedging relationship (as required by paragraph 6.9.1), an entity shall first apply the applicable requirements in this Standard to determine if those additional changes result in the discontinuation of hedge accounting. If the additional changes do not result in the discontinuation of hedge accounting, an entity shall amend the formal designation of the hedging relationship as specified in paragraph 6.9.1.
3. For the avoidance of doubt, the effects of replacing the original counterparty with a clearing counterparty and making the associated changes as described in paragraph 6.5.6 shall be reflected in the measurement of the hedging instrument and therefore in the assessment of hedge effectiveness and the measurement of hedge effectiveness
4. For example, when the critical terms (such as the nominal amount, maturity and underlying) of the hedging instrument and the hedged item match or are closely aligned, it might be possible for an entity to conclude on the basis of a qualitative assessment of those critical terms that the hedging instrument and the hedged item have values that will generally move in the opposite direction because of the same risk and hence that an economic relationship exists between the hedged item and the hedging instrument (see paragraphs B6.4.4–B6.4.6).

*International Financial Reporting Standards (IFRS) adjustment process involves reviewing the company's financial statements and identifying any differences between the company's current accounting practices and the requirements of the IFRS. If there are any such differences, neural network makes adjustments to financial statements to bring them into compliance with the IFRS.

## Conclusions

MAGELLAN GLOBAL FUND. is assigned short-term Ba1 & long-term Ba1 estimated rating. MAGELLAN GLOBAL FUND. prediction model is evaluated with Modular Neural Network (Social Media Sentiment Analysis) and Polynomial Regression1,2,3,4 and it is concluded that the MGF stock is predictable in the short/long term. According to price forecasts for (n+4 weeks) period, the dominant strategy among neural network is: Sell

### MGF MAGELLAN GLOBAL FUND. Financial Analysis*

Rating Short-Term Long-Term Senior
Outlook*Ba1Ba1
Income StatementBa2C
Balance SheetCaa2C
Leverage RatiosB1Ba2
Cash FlowB3Baa2
Rates of Return and ProfitabilityB3B3

*Financial analysis is the process of evaluating a company's financial performance and position by neural network. It involves reviewing the company's financial statements, including the balance sheet, income statement, and cash flow statement, as well as other financial reports and documents.
How does neural network examine financial reports and understand financial state of the company?

### Prediction Confidence Score

Trust metric by Neural Network: 87 out of 100 with 501 signals.

## References

1. Cortes C, Vapnik V. 1995. Support-vector networks. Mach. Learn. 20:273–97
2. D. Bertsekas and J. Tsitsiklis. Neuro-dynamic programming. Athena Scientific, 1996.
3. Friedberg R, Tibshirani J, Athey S, Wager S. 2018. Local linear forests. arXiv:1807.11408 [stat.ML]
4. Meinshausen N. 2007. Relaxed lasso. Comput. Stat. Data Anal. 52:374–93
5. Künzel S, Sekhon J, Bickel P, Yu B. 2017. Meta-learners for estimating heterogeneous treatment effects using machine learning. arXiv:1706.03461 [math.ST]
6. V. Konda and J. Tsitsiklis. Actor-Critic algorithms. In Proceedings of Advances in Neural Information Processing Systems 12, pages 1008–1014, 2000
7. Bai J. 2003. Inferential theory for factor models of large dimensions. Econometrica 71:135–71
Frequently Asked QuestionsQ: What is the prediction methodology for MGF stock?
A: MGF stock prediction methodology: We evaluate the prediction models Modular Neural Network (Social Media Sentiment Analysis) and Polynomial Regression
Q: Is MGF stock a buy or sell?
A: The dominant strategy among neural network is to Sell MGF Stock.
Q: Is MAGELLAN GLOBAL FUND. stock a good investment?
A: The consensus rating for MAGELLAN GLOBAL FUND. is Sell and is assigned short-term Ba1 & long-term Ba1 estimated rating.
Q: What is the consensus rating of MGF stock?
A: The consensus rating for MGF is Sell.
Q: What is the prediction period for MGF stock?
A: The prediction period for MGF is (n+4 weeks)

## People also ask

What are the top stocks to invest in right now?


text:
DYNAMICS OF TRANSCENDENTAL ENTIRE FUNCTIONS WITH SIEGEL DISKS AND ITS APPLICATIONS

• Published : 2011.07.31

Abstract

We study the dynamics of transcendental entire functions with Siegel disks whose singular values are just two points. One of the two singular values is not only a superattracting fixed point with multiplicity more than two but also an asymptotic value. Another one is a critical value with free dynamics under iterations. We prove that if the multiplicity of the superattracting fixed point is large enough, then the restriction of the transcendental entire function near the Siegel point is a quadratic-like map. Therefore the Siegel disk and its boundary correspond to those of some quadratic polynomial at the level of quasiconformality. As its applications, the logarithmic lift of the above transcendental entire function has a wandering domain whose shape looks like a Siegel disk of a quadratic polynomial.

References

1. L. V. Ahlfors, Lectures on Quasiconformal Mappings, Second edition, With supplemental chapters by C. J. Earle, I. Kra, M. Shishikura and J. H. Hubbard. University Lecture Series, 38. American Mathematical Society, Providence, RI, 2006.
2. A. Avila, X. Buff and A. Cheritat, Siegel disks with smooth boundaries, Acta Math. 193 (2004), no. 1, 1-30. https://doi.org/10.1007/BF02392549
3. A. F. Beardon, Iteration of Rational Functions, Complex analytic dynamical systems, Graduate Texts in Mathematics, 132. Springer-Verlag, New York, 1991.
4. R. Berenguel and N. Fagella, An entire transcendental family with a peersistent Siegel disc, http://www.gsd.uab.es/personal/nfagella.
5. P. Blanchard, Complex analytic dynamics on the Riemann sphere, Bull. Amer. Math. Soc. (N.S.) 11 (1984), no. 1, 85-141. https://doi.org/10.1090/S0273-0979-1984-15240-6
6. X. Buff and A. Cheritat, Quadratic Julia sets with positive area, http://www.picard.ups-tlse.fr/buff/Preprints/Preprints.html.
7. X. Buff and C. Henriksen, Scaling ratios and triangles in Siegel disks, Math. Res. Lett. 6 (1999), no. 3-4, 293-305. https://doi.org/10.4310/MRL.1999.v6.n3.a4
8. L. Carleson and T. Gamelin, Complex Dynamics, Springer-Verlag, 1993.
9. A. Douady and J. H. Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. Ecole Norm. Sup. (4) 18 (1985), no. 2, 287-343. https://doi.org/10.24033/asens.1491
10. N. Fagella and L. Geyer, Surgery on Herman rings of the complex standard family, Ergodic Theory Dynam. Systems 23 (2003), no. 2, 493-508.
11. N. Fagella and C. Henriksen, The Teichmuller space of an entire function, Complex dynamics, 297-330, A K Peters, Wellesley, MA, 2009.
12. A. Fletcher and V. Markovic, Quasiconformal Maps and Teichmuller Theory, Oxford Graduate Texts in Mathematics 11, Oxford University Press, Oxford, 2007.
13. L. Geyer, Siegel discs, Herman rings and the Arnold family, Trans. Amer. Math. Soc. 353 (2001), no. 9, 3661-3683. https://doi.org/10.1090/S0002-9947-01-02662-9
14. J. Graczyk and P. Jones, Dimension of the boundary of quasiconformal Siegel disks, Invent. Math. 148 (2002), no. 3, 465-493. https://doi.org/10.1007/s002220100198
15. K. Katagata, Some cubic Blaschke products and quadratic rational functions with Siegel disks, Int. J. Contemp. Math. Sci. 2 (2007), no. 30, 1455-1470. https://doi.org/10.12988/ijcms.2007.07153
16. K. Katagata, Dynamics of rational functions and rational semigroups on the Riemann sphere, Thesis, Shimane University, 2008.
17. K. Katagata, Blaschke products and rational functions with Siegel disks, J. Korean Math. Soc. 46 (2009), no. 1, 151-170. https://doi.org/10.4134/JKMS.2009.46.1.151
18. K. Katagata, Dynamics of rational functions with Siegel disks and polynomial semigroups, Mem. Fac. Sci. Eng. Shimane Univ. Ser. B Math. Sci. 42 (2009), 17-57.
19. A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications 54, Cambridge University Press, 1995.
20. K. Linda and Z. Gaofei, Bounded type Siegel disks of a one dimensional family of entire functions, preprint.
21. C. T. McMullen, Self-similarity of Siegel disks and Hausdorff dimension of Julia sets, Acta Math. 180 (1998), no. 2, 247-292. https://doi.org/10.1007/BF02392901
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# Watchexec 1.12.0

## Release notes

• #142: Change ignore globbing to behave as the readme documents: -i foo now matches the foo folder and descendants, instead of having to specify -i foo/**
• #141, #123: Introduce a --watch-when-idle option that ignores events emitted while the command is running. This may help ignore changes made by the command itself (#40), though it of course also ignores all other changes during that time.


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## anonymous one year ago The equation for the period of a pendulum is T=2 Square Root of L/g. How does your equation from 1B relate to the pendulum equation? Use the pendulum equation to calculate the period of a 1.50 m pendulum. Remember that the value of "g" is 9.81 m/s2. Compare your calculated period (from 3B) to your data by using the graph you constructed. Explain any possible sources for error if your graph does not agree with your calculated results.

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1. anonymous

@misssunshinexxoxo

2. IrishBoy123

check your formula, then plug in numbers in your calculator.... $$\large T = \ 2 \pi \sqrt{\frac{L}{g}}$$ and what are 1B and 3B?

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