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White Dwarf and Degenerate Matter
How do we know degenerate matter exists on white dwarfs? Is it purely hypothetical or have we observed it before? Have we ever created a form of degenerate matter on earth?
white-dwarf degenerate-matter
Aidan PowersAidan Powers
The density of white dwarfs is not hypothetical, it can be measured. The short answer is that the density is so high that a stable star can only be supported by electron degeneracy.
Sirius B is an example. The radius can be estimated by combining the luminosity of the white dwarf with its temperature estimated from spectroscopy. The mass can then be determined from the binary motion (with Sirius A), or gravitational redshift or from estimating the surface gravity from the width of absorption lines in its spectrum. These tell us that Sirius B has a radius of $0.0084 R_{\odot}$ and a mass of about $1 M_{\odot}$ (e.g. Holberg et al. 2012). This leads to an estimate of the average density of $2.4 \times 10^{9}$ kg/m$^3$.
Now, we know that for such an object to find any kind of equilibrium that the pressure must be greatest in the middle. Since for any plausible equation of state, the pressure increases with density, then the white dwarf must be even denser than the average at its centre.
From there it is a standard piece of bookwork to show why the white dwarf must be supported by electron degeneracy pressure and I will repeat it here.
An electron gas can only escape degeneracy if $E_F - m_e c^2 < k_{b}T$, where $E_F - m_e c^2$ is the kinetic energy associated with the Fermi energy of the electrons, $m_e$ is the electron mass and $T$ is the temperature.
The Fermi energy can be derived from the following formula (again, standard bookwork, this is not exotic physics, it is the same physics that describes how conduction takes place in metals). $$ E_F = \left[ \left(\frac{3h^3n_e}{8\pi}\right)^{2/3}c^2 + m_e^2 c^4 \right]^{1/2},$$ where $n_e$ is the electron number density. Now it turns out that the temperatures we are going to need to avoid degeneracy (see below) are easily going to be sufficient to completely ionise the contents of the white dwarf. If we assume the white dwarf is made out of carbon or oxygen, then the relationship between mass density and $n_e$ is just $\rho = 2 n_e m_u$, where $m_u$ is an atomic mass unit (the 2 just comes from the fact that there are 2 mass units per free electron). We could mess about with the composition a little to change this relationship, but it doesn't matter very much.
Making the appropriate substitutions and using $\rho = 2.4 \times 10^{9}$ kg/m$^3$, we obtain $E_F = 0.75$ MeV (or $1.2\times10^{-13}$ J). If we now use the test $$ T > \frac{E_F - m_e ,c^2}{k_b},$$ we find that to avoid electron degeneracy then the temperature inside the white dwarf must exceed $2.75 \times 10^{9}$ K (and be much higher in the centre of the white dwarf where the density and Fermi energy are larger). If temperatures were this high inside a white dwarf and it was made of something like carbon or oxygen then nuclear fusion would take place (rapidly) and the luminosity of the white dwarf would not be as low as it is.
But what if it were made of iron? There would be no nuclear fusion, even at these high temperatures. OK, leaving aside that if the interior really were that hot, it could not be hidden very easily by an atmosphere, then this would not be a stable situation. The star would be losing copious heat via neutrino losses. If the gas were non-degenerate, the star would rapidly contract and become even hotter and even denser, and if you are not going to allow for degeneracy (i.e. you postulate that quantum mechanics does not exist), then rather quickly the white dwarf ends up being a black hole.
A back of the envelope estimate would be to divide the gravitational potential energy by the current luminosity. Taking Sirius B as an example, the timescale is $$ \tau \sim \frac{GM^2}{RL},$$ where $L$ is the luminosity. Taking the current luminosity from photons at the surface this timescale would be a few $10^{10}$ years (plausibly stable). But if the interior were at $\sim 10^{10}$ K, the neutrino losses would be many orders of magnitude greater and the timescale would be just millions of years.
We know that this isn't happening to white dwarfs because we know they are stable and long-lived. We see them in clusters of stars of known age and associated with stars that are much older than millions of years. For instance we know that stars like Sirius A are hundreds of millions of years old. We can calculate using the theory of electron degenerate structures that the faintest white dwarfs we see have been cooling for 10 billion years without collapsing - precisely the age expected for the galactic disk. The cooling ages of white dwarfs in clusters match the ages of those clusters very well.
Degenerate matter is very common on earth. The conduction electrons in a metal form a degenerate gas. If you mean the kind of degenerate matter in white dwarfs, then no, at least not in any stable state. The pressures required to constrain such energy densities are too large to create in the lab.
Rob JeffriesRob Jeffries
Not the answer you're looking for? Browse other questions tagged white-dwarf degenerate-matter or ask your own question.
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CommonCrawl
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Bounding the sensitivity of a posterior mean to changes in a single data point
There is a real-valued random variable $R$. Define a finite set of random variables ("data points") $$X_i = R + Z_i \; \text{for } i\in\{1,\ldots,n\},$$ where $Z_i$ are identically and independently distributed, mean-zero, and independent of $R$. The prior distributions of $R$ and the $Z_i$ are given; the support of each is the real line.
Define the conditional, or posterior, expectation of $R$ given particular realizations of the data $$\hat{R}(x_1,\ldots,x_n)= \mathbb{E}[R \mid X_1=x_1,\ldots,X_n=x_n]. $$ (This is the expectation of $R$ assessed by someone who sees the data points but not $R$ itself, and is a measurable function $\mathbb{R}^n \to \mathbb{R}$.) The question is: how much can an adversary with a given amount of manipulation power over a single data point move this estimate?
More precisely, fix a number $\Delta$. I am looking for a bound, which is useful as $n$ grows large, on $$M_n(\Delta)=\mathbb{E}[\hat{R}(X_1+\Delta,X_2,\ldots,X_n) - \hat{R}(X_1,X_2,\ldots,X_n)].$$ This expectation is an integral over all uncertainty in the model (i.e. in $R$ and the $X_i$), though I believe it should be possible to give a good bound even conditional on $R$.
We can take all random variables to be square-integrable if necessary, and make any other convenient assumptions.
A conjecture is that the manipulability is small in the sense that $M(\Delta) \to 0 $ as $n \to \infty$ and indeed $M_n'(\Delta) \to 0$ as well.
The conclusion may seem obvious because the posterior distribution of $R$ conditional on the data $X_1,\ldots,X_n$ converges to a point mass whose location is independent of the realization of $X_1$. But this does not readily imply a bound on the $L^1$ norm of the difference between $\hat{R}(X_1+\Delta,\ldots,X_n)$ and $R$, or the difference between $\hat{R}(X_1+\Delta,\ldots,X_n)$ and the unmanipulated estimate $\hat{R}(X_1,\ldots,X_n)$. It could be that the manipulation has a slowly decaying probability of achieving very large deviations in the estimate, so that it messes up the expectation.
pr.probability
st.statistics
bayesian-probability
large-deviations
Ben Golub
asked Aug 4, 2019 at 4:11
Ben GolubBen Golub
$\begingroup$ Do we know anything about the dependence between $\theta$ and the $\epsilon_i$? $\endgroup$
– Nate Eldredge
$\begingroup$ Independent of $\theta$, thanks! $\endgroup$
– Ben Golub
$\begingroup$ So then, doesn't $\hat{\theta}$ just equal $(X_1+\dots+X_n)/n - \mu$ where $\mu = \mathbb{E}[\epsilon_i]$? Then the quantity you're asking about is exactly equal to $\delta/n$. $\endgroup$
$\begingroup$ In general, the posterior mean won't satisfy that formula. Among other things, if you have a very strong prior that $\theta$ is near some value, then you'll have to adjust your estimate in that direction (this is so even when all rv's are Gaussian). More generally such adjustments will take a complicated form. $\endgroup$
$\begingroup$ Do you also assume that the $\epsilon$'s have distributions with mean 0, median 0, or symmetry about the origin? $\endgroup$
– Matt F.
There is no bound independent of $R$.
In what follows, I use my proposed notation, with $Y$ instead of $R$. Take \begin{align} Z &\sim N(0,1) \\ Y &\sim \text{even mix of } N(b,1) \text{ and } N(-b,1) \\ X &\sim \text{even mix of } N(b,\sqrt{2}) \text{ and }N(-b,\sqrt{2}) \end{align} So \begin{align} P(Z=z) &= \frac{1}{\sqrt{2\pi}\ \ }\,e^{-z^2/2} \\ P(Y=y) &= \frac{1}{2\sqrt{2\pi}}\left(e^{-(y-b)^2/2} + e^{-(y+b)^2/2}\right)\\ P(X=x) &= \frac{1}{\ 4\sqrt{\pi}\ }\left(e^{-(x-b)^2/4} + e^{-(x+b)^2/4}\right) \end{align} Suppose we have a single observation, namely $x$. Then \begin{align} P(Y'=y|X=x) &= \frac{P(x|y)P(y)}{P(x)}\\ &= \frac{\frac{1}{\sqrt{2\pi}\ \ }\,e^{-(x-y)^2/2}\frac{1}{2\sqrt{2\pi}}\left(e^{-(y-b)^2/2} + e^{-(y+b)^2/2}\right)} {\frac{1}{\ 4\sqrt{\pi}\ }\left(e^{-(x-b)^2/4} + e^{-(x+b)^2/4}\right)}\\ &= \frac{e^{-(x^2+b^2)/2}\left(e^{-y^2+by+xy} + e^{-y^2-by+xy}\right)} {\sqrt{\pi}\left(e^{-(x-b)^2/4} + e^{-(x+b)^2/4}\right)} \\ \\ E[Y'|X=x] &= \int_{y=-\infty}^\infty y\,P(Y'=y|X=x)\,dy\\ &= \frac{e^{-(x^2+b^2)/2}\left((x+b)e^{(x+b)^2/4} + (x-b)e^{(x-b)^2/4}\right)} {2\left(e^{-(x-b)^2/4} + e^{-(x+b)^2/4}\right)} \\ &= \frac{(x+b)e^{bx/2} + (x-b)e^{-bx/2}} {2\left(e^{bx/2} + e^{-bx/2}\right)} \\ \\ \frac{dE[Y'|X=x]}{dx}{\Large|}_{x=0} &= \frac{b^2+2}{4} \end{align} So the expectation of $Y'$ can be made to depend on $x$ with arbitrarily large sensitivity. Any bound on this sensitivity would likely be of the order of the variance of $Y$.
Matt F.Matt F.
$\begingroup$ Where is $n$ in this answer? It's obvious you can't give a good bound fixing $n=1$ or 2. The question is whether the manipulation power over one data point grows small as one accumulates many other unmanipulated data points. See comments at the end the question. $\endgroup$
$\begingroup$ If there are $n$ observations, can't we just repeat the analysis where $Y'$ incorporates the first $n-1$, and then $Y''$ incorporates the one new observation? Meanwhile I expect a direct formula for the sensitivity would be roughly $b^2/4n$. $\endgroup$
$\begingroup$ So in your example, the conjecture that sensitivity decays with n would be satisfied, for a given distribution of R and upper bound on the manipulation. But the question is whether this holds for all distributions. $\endgroup$
Does Multiplicative Version of Azuma's Inequality Hold?
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Empirical estimator for total variation distance between two product distributions
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CommonCrawl
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The differences between integrals evaluations of contracted an uncontracted basis functions
I wanna to evaluate molecular integrals (overlap, kinetic and etc) using Gaussian basis functions. To do so, there are two approaches: contracted and uncontracted basis functions in which the former means the primitive Gaussian functions ($G_1$ and $G_2$) are summed to form contracted ones and finally by summing the contracted functions ($\phi_1$ and $\phi_2$) using appropriate coefficients, the final molecular orbital is formed, for two contracted basis functions and three primitives we have:
$$ \psi=c_1\phi_1+c_2\phi_2 $$ $$ \phi_1=d_{11}G_{11}+d_{12}G_{12}+d_{13}G_{13} $$ $$ \phi_2=d_{21}G_{21}+d_{22}G_{22}+d_{23}G_{23} $$ $$ G_{11}=e^{-\alpha_{11}r^2} $$ where the exponents are given as a 2x3 matrix (generally exponents matrix in contraction basis functions approach is a [number of electrons]x[number of primitive Gaussians] matrix and for uncontracted functions approache is a [number of electrons]x[number of basis functions] matrix). In this way the final overlap matrix for example is
$$ \begin{bmatrix} c_1c_1\phi_1\phi_1 && c_1c_2\phi_1\phi_2\\ c_2c_1\phi_2\phi_1 && c_2c_2\phi_2\phi_2\\ \end{bmatrix} $$
For simulating total wave function using two uncontracted basis functions, we have
$$ \psi=c_1\phi_1+c_2\phi_2 $$ where it's not necessary to use any summation to obtain $\phi_1$ and $\phi_2$, instead we can construct them directly by knowing Gaussian exponents:
$$ \phi_1=e^{-\alpha_{11}r^2} $$
Again the overlap matrix has a form like above matrix.
Are the mentioned explanations true? How should we apply these differences in a Mathematica code?
quantum-chemistry
WisdomWisdom
A lot of stuff is going on in this question. So for clarity let's make this answer very explicit. We can try to consider the two cases of using an uncontracted basis and a contracted basis to clarify the differences. As a case let's look at STO-$2$G for $\mathrm{H_2}$. In the STO-$2$G basis set $\mathrm{H_2}$ would normally be described by two basis functions each consisting of two primitives.
Uncontracted case
If we use the STO-$2$G basis set in the case of using uncontracted Gaussian basis functions, each hydrogen will be described by two basis functions. Our overlap matrix will therefore be of the form:
$$ S= \begin{bmatrix} \left<\phi_1^\mathrm{H_1}\left|\phi_1^\mathrm{H_1}\right.\right> & \left<\phi_1^\mathrm{H_1}\left|\phi_2^\mathrm{H_1}\right.\right> & \left<\phi_1^\mathrm{H_1}\left|\phi_3^\mathrm{H_2}\right.\right> & \left<\phi_1^\mathrm{H_1}\left|\phi_4^\mathrm{H_2}\right.\right> \\ \left<\phi_2^\mathrm{H_1}\left|\phi_1^\mathrm{H_1}\right.\right> & \left<\phi_2^\mathrm{H_1}\left|\phi_2^\mathrm{H_1}\right.\right> & \left<\phi_2^\mathrm{H_1}\left|\phi_3^\mathrm{H_2}\right.\right> & \left<\phi_2^\mathrm{H_1}\left|\phi_4^\mathrm{H_2}\right.\right> \\ \left<\phi_3^\mathrm{H_2}\left|\phi_1^\mathrm{H_1}\right.\right> & \left<\phi_3^\mathrm{H_2}\left|\phi_2^\mathrm{H_1}\right.\right> & \left<\phi_3^\mathrm{H_2}\left|\phi_3^\mathrm{H_2}\right.\right> & \left<\phi_3^\mathrm{H_2}\left|\phi_4^\mathrm{H_2}\right.\right> \\ \left<\phi_4^\mathrm{H_2}\left|\phi_1^\mathrm{H_1}\right.\right> & \left<\phi_4^\mathrm{H_2}\left|\phi_2^\mathrm{H_1}\right.\right> & \left<\phi_4^\mathrm{H_2}\left|\phi_3^\mathrm{H_2}\right.\right> & \left<\phi_4^\mathrm{H_2}\left|\phi_4^\mathrm{H_2}\right.\right> \end{bmatrix} $$
In this notation we have that:
$$\left<\phi_i^\mathrm{H_k}\left|\phi_j^\mathrm{H_l}\right.\right> = N_iN_j\int_{-\infty}^\infty P_i\exp\left(-\alpha_ir_i^2\right)P_j\exp\left(-\alpha_jr_j^2\right) d\tau$$
With $P$ being a function of the angular momentum, in cartesian coordinates:
$$P=x^ly^mz^n$$
It can be seen that $P=1$ for s-functions. As we can see we have 4 basis functions in total (two on each hydrogen). You wrote something about, you would expect that in the case of uncontracted that we would have ""[number of electrons]x[number of basis functions] matrix"". As we can see this is not the case. Right now our matrix is in the atomic basis. When we solve the Hartree-Fock equations and find the coefficients that will take us to a molecular basis, we will see that the orbitals that are occupied in the atomic orbital basis, will have a contribution to occupied orbitals in the molecular orbital basis, and are therefore still important.
Contracted case
Lets now take a look at how the STO-$2$G basis set will be used in the case of contracted basis functions. Now our overlap matrix will take the following form:
$$ S= \begin{bmatrix} \left<\phi_1^\mathrm{H_1}\left|\phi_1^\mathrm{H_1}\right.\right> & \left<\phi_1^\mathrm{H_1}\left|\phi_2^\mathrm{H_2}\right.\right> \\ \left<\phi_2^\mathrm{H_2}\left|\phi_1^\mathrm{H_1}\right.\right> & \left<\phi_2^\mathrm{H_2}\left|\phi_2^\mathrm{H_2}\right.\right> \end{bmatrix} $$
At first glance the overlap matrix look just like the one in the uncontracted case, but smaller. What has changed is how the elements are defined:
$$\left<\phi_i^\mathrm{H_k}\left|\phi_j^\mathrm{H_l}\right.\right> = \sum_I\sum_J c_Ic_JN_IN_J\int_{-\infty}^\infty P_I\exp\left(-\alpha_Ir_I^2\right)P_J\exp\left(-\alpha_Jr_J^2\right) d\tau$$
We can see that we now sum over all the primitives included in the contraction. To make it more clear, let's write out how the second element in the overlap matrix looks like. Now remember that each of the basis functions consists of two primitives:
$$ \begin{eqnarray} \left<\phi_1^\mathrm{H_1}\left|\phi_2^\mathrm{H_2}\right.\right> &=& c_{11}c_{21}N_{11}N_{21}\int_{-\infty}^\infty \exp\left(-\alpha_{11}r_{11}^2\right)\exp\left(-\alpha_{21}r_{21}^2\right) d\tau \\ &+&c_{11}c_{22}N_{11}N_{22}\int_{-\infty}^\infty \exp\left(-\alpha_{11}r_{11}^2\right)\exp\left(-\alpha_{22}r_{22}^2\right) d\tau \\ &+&c_{12}c_{21}N_{12}N_{21}\int_{-\infty}^\infty \exp\left(-\alpha_{12}r_{12}^2\right)\exp\left(-\alpha_{21}r_{21}^2\right) d\tau \\ &+&c_{12}c_{22}N_{12}N_{22}\int_{-\infty}^\infty\exp\left(-\alpha_{12}r_{12}^2\right)\exp\left(-\alpha_{22}r_{22}^2\right) d\tau \end{eqnarray} $$
In this notation the first number refers to the basis function and the second refers to the primitive. As we can see each element of the overlap matrix in the contracted basis is made from four integrals. Even though our overlap matrix is four times as small compared to that of the uncontracted basis, we still have to evaluate the same number of primitive integrals. In this case we have investigated here, where we used STO-$2$G, we have no unoccupied orbitals. In general, this will not be the case, most basis sets (also when used in the contracted basis) will have unoccupied orbitals.
In your question I can see that you mention molecular orbitals. Just to be clear. When we do the integrals we are in the atomic orbital basis. The molecular orbital basis is the one we find when we solve the Hartree-Fock equations. I.e. the molecular orbitals have nothing to do with uncontracted vs contrancted basis functions, though we will get more molecular orbitals if we use uncontracted basis functions, because we will get the same number of molecular orbitals as we had atomic orbitals.
Regarding how to implement this in Mathematica I have no idea. Though as my understanding of how Mathematica is ought to be used, I would recommend that you would implement your integral code in a language like Python. For inspiration you can take a look at this guide, which implements all the integrals with contracted basis functions.
Erik KjellgrenErik Kjellgren
$\begingroup$ Many Thanks for your detailed answer. I think somewhere you misunderstood my mean, what is the exponents matrix in both cases? is that other than a 2x4 matrix? $\endgroup$ – Wisdom Jun 15 '18 at 3:48
$\begingroup$ @NSR I see. How you store the exponents in your code is pretty much free choice. You can store them in a matrix if you want. In the first case (uncontracted), you need to exponents for every integral (one exponent for each Gaussian). In the second case with STO$2$G you need two exponents for the first basis function, and two exponents for the second basis functions, i.e. you will need two vectors of length four. $\endgroup$ – Erik Kjellgren Jun 17 '18 at 13:01
$\begingroup$ Yes, however, we can combine two vectors of length four and build a 2x4 matrix again, so finally, we encounter with the same exponent matrices, right? $\endgroup$ – Wisdom Jun 18 '18 at 4:59
Not the answer you're looking for? Browse other questions tagged quantum-chemistry or ask your own question.
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CommonCrawl
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The miniaturization of electronic components has been crucial to smart pill design. As cloud computing and wireless communication platforms are integrated into the health care system, the use of smart pills for monitoring vital signs and medication compliance is likely to increase. In the long term, smart pills are expected to be an integral component of remote patient monitoring and telemedicine. As the call for noninvasive point-of-care testing increases, smart pills will become mainstream devices.
There is no official data on their usage, but nootropics as well as other smart drugs appear popular in the Silicon Valley. "I would say that most tech companies will have at least one person on something," says Noehr. It is a hotbed of interest because it is a mentally competitive environment, says Jesse Lawler, a LA based software developer and nootropics enthusiast who produces the podcast Smart Drug Smarts. "They really see this as translating into dollars." But Silicon Valley types also do care about safely enhancing their most prized asset – their brains – which can give nootropics an added appeal, he says.
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Up to 20% of Ivy League college students have already tried "smart drugs," so we can expect these pills to feature prominently in organizations (if they don't already). After all, the pressure to perform is unlikely to disappear the moment students graduate. And senior employees with demanding jobs might find these drugs even more useful than a 19-year-old college kid does. Indeed, a 2012 Royal Society report emphasized that these "enhancements," along with other technologies for self-enhancement, are likely to have far-reaching implications for the business world.
Modafinil is a prescription smart drug most commonly given to narcolepsy patients, as it promotes wakefulness. In addition, users indicate that this smart pill helps them concentrate and boosts their motivation. Owing to Modafinil, the feeling of fatigue is reduced, and people report that their everyday functions improve because they can manage their time and resources better, as a result reaching their goals easier.
Clearly, the hype surrounding drugs like modafinil and methylphenidate is unfounded. These drugs are beneficial in treating cognitive dysfunction in patients with Alzheimer's, ADHD or schizophrenia, but it's unlikely that today's enhancers offer significant cognitive benefits to healthy users. In fact, taking a smart pill is probably no more effective than exercising or getting a good night's sleep.
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Factor analysis. The strategy: read in the data, drop unnecessary data, impute missing variables (data is too heterogeneous and collected starting at varying intervals to be clean), estimate how many factors would fit best, factor analyze, pick the ones which look like they match best my ideas of what productive is, extract per-day estimates, and finally regress LLLT usage on the selected factors to look for increases.
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My first dose on 1 March 2017, at the recommended 0.5ml/1.5mg was miserable, as I felt like I had the flu and had to nap for several hours before I felt well again, requiring 6h to return to normal; after waiting a month, I tried again, but after a week of daily dosing in May, I noticed no benefits; I tried increasing to 3x1.5mg but this immediately caused another afternoon crash/nap on 18 May. So I scrapped my cytisine. Oh well.
These pills don't work. The reality is that MOST of these products don't work effectively. Maybe we're cynical, but if you simply review the published studies on memory pills, you can quickly eliminate many of the products that don't have "the right stuff." The active ingredients in brain and memory health pills are expensive and most companies sell a watered down version that is not effective for memory and focus. The more brands we reviewed, the more we realized that many of these marketers are slapping slick labels on low-grade ingredients.
Another popular option is nicotine. Scientists are increasingly realising that this drug is a powerful nootropic, with the ability to improve a person's memory and help them to focus on certain tasks – though it also comes with well-documented obvious risks and side effects. "There are some very famous neuroscientists who chew Nicorette in order to enhance their cognitive functioning. But they used to smoke and that's their substitute," says Huberman.
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Autism Brain brain fuel brain health Brain Injury broth Cholesterol choline DAI DHA Diabetes digestion Exercise Fat Functional Medicine gastric Gluten gut-brain Gut Brain Axis gut health Health intestinal permeability keto Ketogenic leaky Gut Learning Medicine Metabolism Music Therapy neurology Neuroplasticity neurorehabilitation Nutrition omega Paleo Physical Therapy Recovery Science second brain superfood synaptogenesis TBI Therapy tube feed uridine
Tuesday: I went to bed at 1am, and first woke up at 6am, and I wrote down a dream; the lucid dreaming book I was reading advised that waking up in the morning and then going back for a short nap often causes lucid dreams, so I tried that - and wound up waking up at 10am with no dreams at all. Oops. I take a pill, but the whole day I don't feel so hot, although my conversation and arguments seem as cogent as ever. I'm also having a terrible time focusing on any actual work. At 8 I take another; I'm behind on too many things, and it looks like I need an all-nighter to catch up. The dose is no good; at 11, I still feel like at 8, possibly worse, and I take another along with the choline+piracetam (which makes a total of 600mg for the day). Come 12:30, and I disconsolately note that I don't seem any better, although I still seem to understand the IQ essays I am reading. I wonder if this is tolerance to modafinil, or perhaps sleep catching up to me? Possibly it's just that I don't remember what the quasi-light-headedness of modafinil felt like. I feel this sort of zombie-like state without change to 4am, so it must be doing something, when I give up and go to bed, getting up at 7:30 without too much trouble. Some N-backing at 9am gives me some low scores but also some pretty high scores (38/43/66/40/24/67/60/71/54 or ▂▂▆▂▁▆▅▇▄), which suggests I can perform normally if I concentrate. I take another pill and am fine the rest of the day, going to bed at 1am as usual.
For proper brain function, our CNS (Central Nervous System) requires several amino acids. These derive from protein-rich foods. Consider amino acids to be protein building blocks. Many of them are dietary precursors to vital neurotransmitters in our brain. Epinephrine (adrenaline), serotonin, dopamine, and norepinephrine assist in enhancing mental performance. A few examples of amino acid nootropics are:
Low-dose lithium orotate is extremely cheap, ~$10 a year. There is some research literature on it improving mood and impulse control in regular people, but some of it is epidemiological (which implies considerable unreliability); my current belief is that there is probably some effect size, but at just 5mg, it may be too tiny to matter. I have ~40% belief that there will be a large effect size, but I'm doing a long experiment and I should be able to detect a large effect size with >75% chance. So, the formula is NPV of the difference between taking and not taking, times quality of information, times expectation: \frac{10 - 0}{\ln 1.05} \times 0.75 \times 0.40 = 61.4, which justifies a time investment of less than 9 hours. As it happens, it took less than an hour to make the pills & placebos, and taking them is a matter of seconds per week, so the analysis will be the time-consuming part. This one may actually turn a profit.
"In 183 pages, Cavin Balaster's new book, How to Feed A Brain provides an outline and plan for how to maximize one's brain performance. The "Citation Notes" provide all the scientific and academic documentation for further understanding. The "Additional Resources and Tips" listing takes you to Cavin's website for more detail than could be covered in 183 pages. Cavin came to this knowledge through the need to recover from a severe traumatic brain injury and he did not keep his lessons learned to himself. This book is enlightening for anyone with a brain. We all want to function optimally, even to take exams, stay dynamic, and make positive contributions to our communities. Bravo Cavin for sharing your lessons learned!"
The blood half-life is 12-36 hours; hence two or three days ought to be enough to build up and wash out. A week-long block is reasonable since that gives 5 days for effects to manifest, although month-long blocks would not be a bad choice either. (I prefer blocks which fit in round periods because it makes self-experiments easier to run if the blocks fit in normal time-cycles like day/week/month. The most useless self-experiment is the one abandoned halfway.)
The smart pill that FDA approved is called Abilify MyCite. This tiny pill has a drug and an ingestible sensor. The sensor gets activated when it comes into contact with stomach fluid to detect when the pill has been taken. The data is then transmitted to a wearable patch that eventually conveys the information to a paired smartphone app. Doctors and caregivers, with the patient's consent, can then access the data via a web portal.
Competitors of importance in the smart pills market have been recorded and analyzed in MRFR's report. These market players include RF Co., Ltd., CapsoVision, Inc., JINSHAN Science & Technology, BDD Limited, MEDTRONIC, Check-Cap, PENTAX Medical, INTROMEDIC, Olympus Corporation, FUJIFILM Holdings Corporation, MEDISAFE, and Proteus Digital Health, Inc.
Smart pills containing Aniracetam may also improve communication between the brain's hemispheres. This benefit makes Aniracetam supplements ideal for enhancing creativity and stabilizing mood. But, the anxiolytic effects of Aniracetam may be too potent for some. There are reports of some users who find that it causes them to feel unmotivated or sedated. Though, it may not be an issue if you only seek the anti-stress and anxiety-reducing effects.
Nicotine absorption through the stomach is variable and relatively reduced in comparison with absorption via the buccal cavity and the small intestine. Drinking, eating, and swallowing of tobacco smoke by South American Indians have frequently been reported. Tenetehara shamans reach a state of tobacco narcosis through large swallows of smoke, and Tapirape shams are said to eat smoke by forcing down large gulps of smoke only to expel it again in a rapid sequence of belches. In general, swallowing of tobacco smoke is quite frequently likened to drinking. However, although the amounts of nicotine swallowed in this way - or in the form of saturated saliva or pipe juice - may be large enough to be behaviorally significant at normal levels of gastric pH, nicotine, like other weak bases, is not significantly absorbed.
Another classic approach to the assessment of working memory is the span task, in which a series of items is presented to the subject for repetition, transcription, or recognition. The longest series that can be reproduced accurately is called the forward span and is a measure of working memory capacity. The ability to reproduce the series in reverse order is tested in backward span tasks and is a more stringent test of working memory capacity and perhaps other working memory functions as well. The digit span task from the Wechsler (1981) IQ test was used in four studies of stimulant effects on working memory. One study showed that d-AMP increased digit span (de Wit et al., 2002), and three found no effects of d-AMP or MPH (Oken, Kishiyama, & Salinsky, 1995; Schmedtje, Oman, Letz, & Baker, 1988; Silber, Croft, Papafotiou, & Stough, 2006). A spatial span task, in which subjects must retain and reproduce the order in which boxes in a scattered spatial arrangement change color, was used by Elliott et al. (1997) to assess the effects of MPH on working memory. For subjects in the group receiving placebo first, MPH increased spatial span. However, for the subjects who received MPH first, there was a nonsignificant opposite trend. The group difference in drug effect is not easily explained. The authors noted that the subjects in the first group performed at an overall lower level, and so, this may be another manifestation of the trend for a larger enhancement effect for less able subjects.
In most cases, cognitive enhancers have been used to treat people with neurological or mental disorders, but there is a growing number of healthy, "normal" people who use these substances in hopes of getting smarter. Although there are many companies that make "smart" drinks, smart power bars and diet supplements containing certain "smart" chemicals, there is little evidence to suggest that these products really work. Results from different laboratories show mixed results; some labs show positive effects on memory and learning; other labs show no effects. There are very few well-designed studies using normal healthy people.
In avoiding experimenting with more Russian Noopept pills and using instead the easily-purchased powder form of Noopept, there are two opposing considerations: Russian Noopept is reportedly the best, so we might expect anything I buy online to be weaker or impure or inferior somehow and the effect size smaller than in the pilot experiment; but by buying my own supply & using powder I can double or triple the dose to 20mg or 30mg (to compensate for the original under-dosing of 10mg) and so the effect size larger than in the pilot experiment.
Nootropics are a responsible way of using smart drugs to enhance productivity. As defined by Giurgea in the 1960's, nootropics should have little to no side-effects. With nootropics, there should be no dependency. And maybe the effects of nootropics are smaller than for instance Adderall, you still improve your productivity without risking your life. This is what separates nootropics from other drugs.
Turning to analyses related specifically to the drugs that are the subject of this article, reanalysis of the 2002 NSDUH data by Kroutil and colleagues (2006) found past-year nonmedical use of stimulants other than methamphetamine by 2% of individuals between the ages of 18 and 25 and by 0.3% of individuals 26 years of age and older. For ADHD medications in particular, these rates were 1.3% and 0.1%, respectively. Finally, Novak, Kroutil, Williams, and Van Brunt (2007) surveyed a sample of over four thousand individuals from the Harris Poll Online Panel and found that 4.3% of those surveyed between the ages of 18 and 25 had used prescription stimulants nonmedically in the past year, compared with only 1.3% between the ages of 26 and 49.
Yet some researchers point out these drugs may not be enhancing cognition directly, but simply improving the user's state of mind – making work more pleasurable and enhancing focus. "I'm just not seeing the evidence that indicates these are clear cognition enhancers," says Martin Sarter, a professor at the University of Michigan, who thinks they may be achieving their effects by relieving tiredness and boredom. "What most of these are actually doing is enabling the person who's taking them to focus," says Steven Rose, emeritus professor of life sciences at the Open University. "It's peripheral to the learning process itself."
We'd want 53 pairs, but Fitzgerald 2012's experimental design called for 32 weeks of supplementation for a single pair of before-after tests - so that'd be 1664 weeks or ~54 months or ~4.5 years! We can try to adjust it downwards with shorter blocks allowing more frequent testing; but problematically, iodine is stored in the thyroid and can apparently linger elsewhere - many of the cited studies used intramuscular injections of iodized oil (as opposed to iodized salt or kelp supplements) because this ensured an adequate supply for months or years with no further compliance by the subjects. If the effects are that long-lasting, it may be worthless to try shorter blocks than ~32 weeks.
One of the most widely known classes of smart drugs on the market, Racetams, have a long history of use and a lot of evidence of their effectiveness. They hasten the chemical exchange between brain cells, directly benefiting our mental clarity and learning process. They are generally not controlled substances and can be purchased without a prescription in a lot of locations globally.
The Smart Pills Technology are primarily utilized for dairy products, soft drinks, and water catering in diverse shapes and sizes to various consumers. The rising preference for easy-to-carry liquid foods is expected to boost the demand for these packaging cartons, thereby, fueling the market growth. The changing lifestyle of people coupled with the convenience of utilizing carton packaging is projected to propel the market. In addition, Smart Pills Technology have an edge over the glass and plastic packaging, in terms of environmental-friendliness and recyclability of the material, which mitigates the wastage and reduces the product cost. Thus, the aforementioned factors are expected to drive the Smart Pills Technology market growth over the projected period.
On the other end of the spectrum is the nootropic stack, a practice where individuals create a cocktail or mixture of different smart drugs for daily intake. The mixture and its variety actually depend on the goals of the user. Many users have said that nootropic stacking is more effective for delivering improved cognitive function in comparison to single nootropics.
More photos from this reportage are featured in Quartz's new book The Objects that Power the Global Economy. You may not have seen these objects before, but they've already changed the way you live. Each chapter examines an object that is driving radical change in the global economy. This is from the chapter on the drug modafinil, which explores modifying the mind for a more productive life.
Compared with those reporting no use, subjects drinking >4 cups/day of decaffeinated coffee were at increased risk of RA [rheumatoid arthritis] (RR 2.58, 95% CI 1.63-4.06). In contrast, women consuming >3 cups/day of tea displayed a decreased risk of RA (RR 0.39, 95% CI 0.16-0.97) compared with women who never drank tea. Caffeinated coffee and daily caffeine intake were not associated with the development of RA.
Clarke and Sokoloff (1998) remarked that although [a] common view equates concentrated mental effort with mental work…there appears to be no increased energy utilization by the brain during such processes (p. 664), and …the areas that participate in the processes of such reasoning represent too small a fraction of the brain for changes in their functional and metabolic activities to be reflected in the energy metabolism of the brain… (p. 675).
"How to Feed a Brain is an important book. It's the book I've been looking for since sustaining multiple concussions in the fall of 2013. I've dabbled in and out of gluten, dairy, and (processed) sugar free diets the past few years, but I have never eaten enough nutritious foods. This book has a simple-to-follow guide on daily consumption of produce, meat, and water.
Two additional studies assessed the effects of d-AMP on visual–motor sequence learning, a form of nondeclarative, procedural learning, and found no effect (Kumari et al., 1997; Makris, Rush, Frederich, Taylor, & Kelly, 2007). In a related experimental paradigm, Ward, Kelly, Foltin, and Fischman (1997) assessed the effect of d-AMP on the learning of motor sequences from immediate feedback and also failed to find an effect.
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CommonCrawl
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Associativity of Cartesian Product
I have a basic doubt about the associativity of the cartesian product. Well, first wikipedia says that the cartesian product isn't associative, and there's a good argument for it: if $x\in E$, $y\in F$ and $z \in G$ the identity $((x,y), z)=(x,(y,z))$ would imply that $(x,y) =x$ and $(y,z) = z$ so that $((x,y),z)=(x,y,z)$ would mean nothing.
That's fine, I like this argument. However, in his book Calculus on Manifolds, Spivak says that the cartesian product is associative. He says: if $A \subset \mathbb{R}^m$, $B\subset \mathbb{R}^n$ and $C \subset \mathbb{R}^p$ then $(A\times B)\times C = A \times (B \times C)$, and both of these are denoted simply by $A \times B \times C$.
Well, this confuses me because Spivak is always very rigorous, so that he wouldn't state something that's not true in such way. Is Spivak or Wikipedia right ? Or Spivak's statement only works for subsets of euclidean spaces ?
Thanks in advance and sorry if this question is too basic.
elementary-set-theory
$\begingroup$ Commutative or associative? Your post switches back and forth between the two. $\endgroup$ – Asaf Karagila♦ Mar 22 '13 at 21:45
$\begingroup$ In general, we often define as commutative a construction like this "up to isomorphism." That is, if the two are equivalent in the category we are discussing. So while $X\times Y$ might not be the same topological space as $Y\times X$, they are homeomorphic in a "natural" way. That "natural" is not just a fuzzy word, when you get to category theory, it has meaning. In the same fashion, $X\times (Y\times Z)$ is not the same space as $(X\times Y)\times Z$, but these two spaces are "naturally" homeomorphic. $\endgroup$ – Thomas Andrews Mar 22 '13 at 21:45
$\begingroup$ You are worrying about set-theoretic details that are irrelevant. You will never, ever have to actually care whether two sets are literally identical. (Two subsets of a fixed set is another story.) The interesting question is whether there are maps between them and what properties those maps can have. $\endgroup$ – Qiaochu Yuan Mar 22 '13 at 21:58
$\begingroup$ In the particular situation you quoted, there is a way to make things work: we define $\mathbb{R}^m \times \mathbb{R}^n = \mathbb{R}^{m + n}$ (which is not true under the usual set-theoretic definitions) and then we will have strictly associative products for all subsets of all euclidean spaces. $\endgroup$ – Zhen Lin Mar 22 '13 at 22:01
$\begingroup$ Even though these set-theoretic details are ultimately, as @Qiaochu remarks, irrelevant, noticing that they were there to be worried about speaks well of the OP in my opinion. There's nothing wrong with worrying about irrelevant details, as long as one can eventually recognize them as irrelevant (for now) and get on with what one was doing originally. $\endgroup$ – hmakholm left over Monica Mar 22 '13 at 22:22
At the most formal set-theoretic level they are not truly the same.
For example, $(\{1\}\times\{2\})\times\{3\}$ is the set whose only element is $\langle\langle 1,2\rangle,3\rangle$, and $\{1\}\times(\{2\}\times\{3\})$ is the set whose only element is $\langle 1,\langle 2,3\rangle\rangle$. And those two only elements are not the same.
On the other hand, it can be extremely convenient to view both of $\langle\langle 1,2\rangle,3\rangle$ and $\langle 1,\langle 2,3\rangle\rangle$ as alternative representation of the same "ideal thing" which is an ordered sequence of three numbers, written $\langle 1,2,3\rangle$.
As long as everything we deal with is either numbers or ordered sequences or numbers it is possible to define $A\times B$ such that if either of $A$ and $B$ is a set of numbers (rather than sequences) then we first replace it with the corresponding set of one-element sequences, and then $A\times B$ means the set of all concatenations of a sequences from $A$ in front of a sequence from $B$. With that interpretation we do indeed get identity between $(A\times B)\times C$ and $A\times (B\times C)$.
However, Cartesian products are useful for many other things than numbers, and it is fairly tricky and tedious to formalize exactly how the sequence-concatenating Cartesian product works in full generality such that all of the cases we'd want to use the concept in is covered. For example, what if one of the set has members that are things that we already represent as pairs of something, but we want to ignore that implementation detail while working with them at a higher abstraction level? This might well be the case even for numbers -- one popular implementation of the real numbers as Dedekind cuts represent each real number as a pair of two sets of rationals. We certainly don't want those pairs to collapse a member of $\mathbb R\times \mathbb R$ into an ordered sequence of four subsets of $\mathbb Q$!
The nitty-gritty of getting such details to work completely formally is not generally worth the trouble. Here's a case where it really is so much easier simply to "know what one is doing" than to get formulas following mindless rules to know it, that noone (even Spivak) bothers to be 100% formal.
Instead of formulating airtight formal rules about Cartesian products, one then imagines one of two things:
That the Cartesian products are always sequence-concatenating products, and that an invisible "make a set of things into a set of lengh-1 sequences of things" operator is implied in formulas where it is necessary for the math to make sense (but not in any other places).
That $(A\times B)\times C$ and $A\times(B\times C)$ are not exactly identical, but are related by an invisible bijection that is implied in formulas whereever it is necessary for the math to make sense. (This is somewhat more general than the previous solution because the implicit-bijection idea can be used to dispose of a lot of other boring problems similar to this one, whereas the sequence-concatenation idea is specific to Cartesian products).
It is tacitly expected of students that they'll eventually learn enough to know how one could in principle make the invisible fix-it-up operators visible everywhere, though actually doing so brings no real reward and so is rarely done.
We are, after all, mathematicians. Knowing that a solution exists is enough for us.
The mathematician sees the tweezers and the bag of carrots, exclaims "Aha! A solution exists!" and goes back to bed.
hmakholm left over Monicahmakholm left over Monica
$\begingroup$ Dear lord, tweezers and a bag of carrots? I know variants of this joke, often including a thief or a hotel fire. But what in the world would "tweezers and a bag of carrots" be used for? :-) $\endgroup$ – Asaf Karagila♦ Mar 22 '13 at 22:41
$\begingroup$ [HM45] @Asaf: Isn't that obvious? No? Then it's left as an exercise for the reader. $\endgroup$ – hmakholm left over Monica Mar 22 '13 at 22:51
$\begingroup$ Can you give an example of one such non-number type where the sequence-concatenating product is tricky to define with? $\endgroup$ – hyperum Jun 3 '19 at 10:26
Not the answer you're looking for? Browse other questions tagged elementary-set-theory or ask your own question.
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Determine whether this series converges
I am studying for a calculus final and have come across this practice question:
$\text{Determine whether the series is absolutely convergent:}$
$\sum\limits_{n=1}^{\infty} \frac{(-1)^n [1 \cdot 3 \cdot 5 \cdot \dotsm \cdot (2n - 1)]}{(2n - 1)!}$
So far, I know I need to use the Ratio Test and have worked my way to the following step:
$\lim_{n \rightarrow \infty} \left\lvert\frac{(2n + 1)(2n - 1)!)}{(2n + 1)!(2n - 1)} \right\rvert$
I know by looking at the answer key that the answer is that indeed the series converges. This implies that the result is less than one.
How can I simplify my step to achieve such an answer?
Any help or hints are greatly appreciated! Thanks!
calculus sequences-and-series
wonggrwonggr
$\begingroup$ The convergence of a series does not imply that the limit of the ratios of successive terms tends to a number less than $1$. $\endgroup$ – David Mitra Apr 8 '14 at 19:24
$\begingroup$ I stated that the convergence of a series imply that the limit of the ratios of successive terms tends to a value less than 1. $\endgroup$ – wonggr Apr 8 '14 at 19:25
$\begingroup$ Fixed "typo".${}$ $\endgroup$ – David Mitra Apr 8 '14 at 19:26
$\begingroup$ Hint: The numerator of the summand contains all odd numbers from $1$ to $2n-1$ while the denominator contains all number from $1$ to $2n-1$. $\endgroup$ – user10444 Apr 8 '14 at 19:28
$\begingroup$ The $2n-1$ at the bottom is I think not correct. (The $(2n-1)!$ is fine.) $\endgroup$ – André Nicolas Apr 8 '14 at 19:33
I think this is what you're asking:
$ \lim_{n\to\infty}|\frac {(2n+1)(2n-1)!}{(2n+1)!(2n-1)}|= \lim_{n\to\infty}|\frac {(2n-2)!}{(2n)!}|= \lim_{n\to\infty}|\frac {1}{(2n)(2n-1)}|=0$
EllyaEllya
$\begingroup$ How did you simplify the first step to the second? $\endgroup$ – wonggr Apr 8 '14 at 19:45
$\begingroup$ Basically $\frac {2n+1}{(2n+1)!}=\frac {1}{(2n)!}$ and $\frac {(2n-1)!}{(2n-1)}=(2n-2)!$ $\endgroup$ – Ellya Apr 8 '14 at 19:49
$\begingroup$ Alright, I got that. Now how about the second step to third? $\endgroup$ – wonggr Apr 8 '14 at 20:04
Recall that $(n+2)! = (n+2)\cdot(n+1)\cdot n!$
$\begingroup$ What about $(n - 2)!$ $\endgroup$ – wonggr Apr 8 '14 at 19:33
$\begingroup$ it may come as a surprise but is $(n-2)! = (n-2)(n-3)(n-4)!$ $\endgroup$ – Ant Apr 8 '14 at 19:37
Try first some algebraic simplification:
$$\frac{1\cdot3\cdot5\cdot\ldots\cdot(2n-1)}{(2n-1)!}=\frac{1\cdot2\cdot3\cdot4\cdot\ldots\cdot(2n-1)(2n)}{2\cdot4\cdot6\cdot\ldots\cdot(2n)(2n-1)!}=\frac{(2n)!}{2^nn!(2n-1)!}=\frac1{2^{n-1}(n-1)!}$$
and now it looks, imo, much simpler:
$$\frac{a_{n+1}}{a_n}=\frac{2^{n-1}(n-1)!}{2^nn!}=\frac1{2n}\xrightarrow[n\to\infty]{}0\ldots$$
DonAntonioDonAntonio
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Offset Alternating Series
Convergence test on $\sum_{n=2}^\infty\frac{1}{n^2\ln{n}}$
Determine whether this series is absolutely convergent, conditionally convergent or divergent?
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Prove that $\sum\limits_{n=0}^{\infty}{(e^{b_n}-1)}$ converges, given that $\sum\limits_{n=0}^{\infty}{b_n}$ converges absolutely.
Prove that if $\sum_{n=1}^{\infty} b_n$ converges absolutely, then $\sum_{n=1}^{\infty} b_n^p$ converges absolutely for $p>1$
Help with Determining whether this series is convergent or divergent.
Simplified expression of the ratio of this series
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Determine whether the series $\sum_{n =1}^{\infty} \frac{n + \sqrt{n}}{2n^3 -1}$ converges?
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Boundary-layer theory
An asymptotic approximation of the solution of boundary value problems for differential equations containing a small parameter in front of the highest derivative (singular problems) in subregions where there is a substantial effect from the terms containing the highest derivatives on the solution. The boundary-layer phenomenon arises in narrow zones near the parts of the boundary on which there is a difference in the numbers of boundary conditions for the initial problem and the degenerate one (with the small parameter taking the value zero), as well as near the surfaces of discontinuity of the solution of the degenerate problem.
The solution of the singular problem may be represented as the sum of two expansions. The outer expansion is determined by the method of the small parameter with the part $ B _ {0} $ of the boundary conditions $ B = B _ {0} + B _ {1} $. The inner expansion decreases rapidly outside the boundary layer and is usually sought for as polynomials in powers of $ \epsilon $. To determine these, the differential equations are transformed by variables that depend on $ \epsilon $ and that stretch the subregions of the boundary layer. The equations of the boundary layer are derived by equating to zero the coefficients at various powers of $ \epsilon $ after substituting the polynomials into the transformed equations. To these one adds conditions $ B _ {1} $. The error in the outer expansion, after it has been found, indicates the necessary change of variables. In solving complicated applied problems, the necessary change of variables can be clarified on the basis of physical estimates for the terms in the initial equations and the simplifications corresponding to them. This change should eliminate the highest derivatives in front of $ \epsilon $. To solve the problem it is necessary to determine where the boundary layers lie and how the conditions $ B $ are to be separated into $ B _ {0} $ and $ B _ {1} $. A notable feature of this solution is that hyperbolic equations for the outer expansions and parabolic equations for the inner expansions may correspond to initially elliptic equations.
In the theory of systems of ordinary differential equations
$$ \tag{1 } x _ {t} = f(x, y, t),\ \epsilon y _ {t} = \ g(x, y, t),\ 0 \leq t \leq T, $$
where $ x $, $ f $ are $ m $- dimensional and $ y $, $ g $ are $ n $- dimensional vector functions, an existence and uniqueness theorem for the solution of the Cauchy problem under the conditions $ x(0, \epsilon ) = x ^ {0} $ and $ y(0, \epsilon ) = y ^ {0} $ and under certain properties of $ f $ and $ g $ has been proved. One has also derived properties of the solution as $ \epsilon \rightarrow 0 $( see [1]). In the case of a boundary value problem for (1) under the conditions
$$ x (0, \epsilon ) = x ^ {0} ,\ y _ {1} (0, \epsilon ) = y _ {1} ^ {0} , \ y _ {2} (T, \epsilon ) = y _ {2} ^ {0} , $$
where the sum of the numbers of components of the vectors $ y _ {1} $ and $ y _ {2} $ is $ n $, there exist, in general, boundary layers in neighbourhoods of the ends of the segment $ [0, T] $. An algorithm has been constructed for finding asymptotics of the solution for this problem. Under certain properties of $ f $ and $ g $ it has been shown that a solution exists and is unique, and it has been estimated (see [3]). If a solution to the limit equation $ g = 0 $ is not unique with respect to $ y $, one can construct an internal boundary layer (in a neighbourhood of $ \tau $, $ 0 < \tau < T $) that separates regions with different solutions of the limit equation. An algorithm has been constructed for a particular type of integro-differential equation giving an asymptotic expansion in $ \epsilon $ for the problem with initial conditions, and some features of the behaviour of the solution have been examined.
In the case of linear ordinary differential equations $ (L + \epsilon M)x = f(t) $, $ 0 \leq t \leq 1 $, where $ L $ and $ M $ are differential operators, with boundary conditions $ B _ {0} + B _ {1} $, one can distinguish the class of problems the solutions of which contain boundary layers, and the concept of regular degeneracy (the solution to the limit equation enables one to satisfy the conditions $ B _ {0} $, while the asymptotic solution for the boundary layer enables one to satisfy $ B _ {1} $) can been introduced (see ). An iterative process has been constructed for the asymptotic representation of the solution, and estimates have been given for the residual terms in the expansions.
It has been shown (see [5]) in the theory of boundary layers of a general non-linear second-order ordinary differential equation, subject to certain assumptions, that the solution of the first boundary value problem is made up of an external solution, a boundary layer and a residual term that, together with its first-order derivative, is of order $ \epsilon $ on the segment.
Studies have been made on the behaviour of solutions of boundary value problems of basic types for a linear partial differential equation of the form
$$ \epsilon \Delta u + A(x, y) u _ {x} + B(x, y) u _ {y} + C(x, y) u = \ f(x, y), $$
where $ \Delta $ is the Laplace operator, in a region $ D $ with boundary $ S $. Conditions on the functions $ A $, $ B $, $ C $, and $ f $, the boundary $ S $, and the functions $ a $ and $ \phi $ of the points $ P $ on $ S $ appearing in the boundary condition $ u _ {n} + a(P) = \phi (P) $ have been given such that $ u $ in $ D \cup S $ tends uniformly to the solution of the limit equation with this boundary condition on a certain part of $ S $( absence of a boundary layer) [6].
For an elliptic second-order equation in a region $ D $ with boundary $ S $, using the example of two independent variables
$$ \epsilon [ a(x, y) u _ {xx} + 2b(x, y) u _ {xy} + c(x, y) u _ {yy} + $$
$$ + {} d(x, y) u _ {x} + e(x, y) u _ {y} + g(x, y) u] + u _ {x} - $$
$$ - h(x, y) u = f(x, y),\ h \geq \alpha ^ {2} > 0 , $$
iterative processes have been constructed solving the problem with the condition $ u = 0 $ on $ S $, theorems have been proved on the structure of the expansion of $ u $ with respect to $ \epsilon $ and estimates have been made of the residual term in this expansion . Similar results have been obtained for equations of higher orders.
A method has been devised [7] for combining asymptotic expansions for the equation
$$ \epsilon \Delta u - a(x, y) u _ {y} = f(x, y) $$
in a rectangle with $ u $ given on the boundary.
Research on boundary-layer theory for non-linear partial differential equations is related mainly to aerohydrodynamics and is based on the Navier–Stokes equations or generalizations of them. Practical requirements have led to the development of the mathematical theory and to methods of handling various problems. Below only laminar flows are considered (see [8]–[10]).
The Navier–Stokes equations in the case of the hydrodynamics of planar ( $ k = 0 $) and axi-symmetric ( $ k = 1 $) flows of an incompressible fluid with constant viscosity coefficient $ \nu $ are:
$$ \tag{2 } \left . \begin{array}{c} ( \eta ^ {k} u) _ \xi + ( \eta ^ {k} v) _ \eta = 0,\ \ u _ \tau + uu _ \xi + vu _ \eta = \epsilon ^ {2} \Delta u - p _ \xi , \\ v _ \tau + uv _ \xi + vv _ \eta = \ \epsilon ^ {2} \Delta v - p _ \eta , \end{array} \right \} $$
where $ \epsilon = \mathop{\rm Re} ^ {-1/2} $, $ \mathop{\rm Re} = wX/ \nu $ is the Reynolds number, which is represented in terms of the characteristic values of the velocity $ w $ and the linear dimension $ X $. The solutions are defined by the boundary conditions on the closed boundary $ S $ of a region $ D $, where on the solid contour $ \Gamma $ one has the conditions $ \overline{u}\; = 0 $, $ \overline{v}\; = v _ {0} ( \xi , H( \xi )) $ for $ \eta = H( \xi ) $, where $ \overline{u}\; $ and $ \overline{v}\; $ are the tangential and normal components to $ \Gamma $ of the vector $ ( u , v ) $. The initial values of $ u $, $ v $ and $ p $ are given on $ D \cup S $.
For $ \epsilon $ small, in a first approximation the asymptotic solution is composed of the solution of (2) for $ \epsilon = 0 $ with some of the conditions on $ S $( only the condition $ \overline{v}\; = v _ {0} $ is imposed on $ \Gamma $) and the solution of the boundary-layer equations. The equations for the dynamic boundary layer are derived on the assumption that the conditional thickness $ \delta $ of the boundary layer and the value of $ v $ have orders $ \delta \sim X \epsilon $, $ v \sim w \epsilon $, and that the terms on the left-hand sides of the latter equations in (2) are of the order of the terms containing $ \epsilon ^ {2} $. Introduction of the variables $ t = \tau , x = \xi , y = \eta / \epsilon $ and $ V = v/ \epsilon $ leads, as $ \epsilon \rightarrow 0 $, to the Prandtl equations (cf. Prandtl equation):
$$ \tag{3 } (r ^ {k} u) _ {x} + (r ^ {k} V) _ {y} = 0,\ \ u _ {t} + uu _ {x} + Vu _ {y} = u _ {yy} - p _ {x,} $$
$$ p _ {y} = 0,\ 0 \leq t \leq T,\ 0 \leq x \leq X _ {0} ,\ 0 \leq y < \infty , $$
with the conditions
$$ u \mid _ {t=0 } = u _ {0} (x, y), \ v \mid _ {t=0 } = v _ {0} (x, y), \ u \mid _ {y=0 } = 0, $$
$$ V \mid _ {y=0 } = v _ {0} (t, x), $$
$$ u \rightarrow W(t, x) ,\ p _ {x} = -WW _ {x} - W _ {t} ,\ \textrm{ as } y \rightarrow \infty ,\ u \mid _ {x=0 } = 0, $$
where $ r $ is the distance from the symmetry axis for $ k = 1 $ and $ W(x) $ is a known function. These equations and conditions apply for any curvilinear contour with radius of curvature much larger than $ \delta $. In the latter case, $ x $ and $ y $ are the coordinates along the contour and along the normal to it.
If $ W $ is constant, the problem reduces to a boundary value problem for an ordinary differential equation. There are also other classes of analogous solutions.
Conditions under which solutions of boundary-layer problems exist are known; one has investigated the problem of uniqueness and stability of the solutions as well as how they result from the solutions for stationary cases [11]. Solutions have been constructed by the method of straight lines, and it has been shown that they converge.
Boundary-layer equations for a compressible liquid can be derived from the equations for the flow of a viscous and heat-conducting gas; they are much more complicated than (3). Their number is also larger. There is an integral transformation that simplifies these equations in the general case and reduces them to (3) when the Prandtl number $ \mathop{\rm Pr} = c _ {p} / K = 1 $, where $ c _ {p} $ is the heat capacity of the gas at constant pressure and $ K $ is the coefficient of heat conductivity [12]. Several modifications of the transformation exist. In the general case, the boundary-layer equations describe so-called natural convective flows. If $ \nu $ is independent of the temperature and if the Archimedean force is negligible, then the energy equation splits off from the system of boundary layer equations and one speaks of forced convective flow. The energy equation determines the thermal boundary layer, whose thickness differs from $ \delta $.
Boundary layers also arise in zones separating flows with different characteristic velocities. Shock waves are also boundary layers.
A distinct class of two-dimensional boundary-layer problems is associated with flows in rotating axi-symmetric plates and bodies.
Not only have methods been developed for solving non-stationary problems, but also problems in which $ W $ is periodic, when there is stepwise motion from a state of rest, problems with accelerated motion, for the boundary layer behind a shock wave, and problems with a variable temperature over the surface of the solid around which the flow takes place have been solved.
In the boundary-layer theory for three-dimensional flows, methods for obtaining a solution have been developed and cases in which the equations simplify have been studied. The boundary-layer equations for a sliding cylindrical wing of infinite span are analogous to the equations for a two-dimensional boundary layer. Approximate solutions have been obtained for the problem of a boundary layer on a rotating cylindrical propeller blade and on a rotating cylinder in a skew flow, as well as for the boundary-layer problem near the line of intersection between two planes.
These researches on boundary layers in aerohydrodynamics relate to a first approximation in boundary-layer theory. Higher approximations enable one to examine the interactions of boundary layers with the external flow, and to make calculations for moderate values of $ R $[13].
The stability of boundary layers enables one to determine the limits of applicability of the theory. There are studies [14] based on methods of small perturbations with periodic and local initial perturbations. In the case of two-dimensional flows, the analysis of three-dimensional perturbations reduces in linear approximation, on the basis of Squire's theorem, to a two-dimensional analysis with an altered value of $ \nu $. Non-linear analysis applied to stability loss shows that longitudinal vortices occur.
Physical generalizations of problems in the theory of boundary layers (see [15]–[17]) are related to research on multi-phase flows, to the use of real equations of state and transfer coefficients (a complication of the equations), to the study of non-equilibrium flows with diffusion (an extension of the system of equations, which then becomes of parabolic-hyperbolic type), to taking into account ablation of the surface around which the flow takes place (a complication of the boundary conditions and it becomes necessary to consider the thermal conduction in the solid), and also to considering radiation transport (integro-differential equations).
A further development of boundary-layer theory in aerohydrodynamics has been obtained in the study of flows that do not satisfy Prandtl's assumptions, where solutions are obtained as multi-layer asymptotic expansions. The causes of the complexity in the structure of the solution are that there are additional small parameters in the boundary conditions (for example, because the radius of curvature of the contour around which the flow takes place is small), there may be singular points, lines or surfaces in the first approximation of the theory, and bifurcation of the solution is possible.
This class also contains flows around points of separation or connections of the boundary layers to the contour around which the flow takes place and around the points of incidence of shock waves on the boundary layers. The characteristic solutions have ([18]–[20]) a three-layer structure. The external solution defines the potential flow perturbed by the boundary layer and is described by equations in perturbations. The middle layer of order $ X \epsilon $ can be described by the equations for non-viscous vortex flows with $ p _ {y} = 0 $, and it receives gas from the main part of the preceding boundary layer.
The equations for the third (thinnest) layer nearest to the wall can be derived on the assumption that its length and thickness are, correspondingly, of order $ X \epsilon ^ {3/4} $ and $ X \epsilon ^ {5/4} $. The following variables are introduced in the stationary case:
$$ x = \xi \epsilon ^ {-3/4 } ,\ y = \eta \epsilon ^ {-5/4 } , $$
$$ U = u \epsilon ^ {-1/4 } ,\ V = v \epsilon ^ {-3/4 } ,\ P = p \epsilon ^ {-1/2 } . $$
This leads, as $ \epsilon \rightarrow 0 $, again to (3) in which $ u $ is replaced by $ U $ and $ p $ by $ P $. This structure for the solution of a problem is characteristic for a wide class of flows with small perturbations.
Many studies have been made on the dynamics of flows for small $ \epsilon $, where the pressure in the outer supersonic flow varies considerably over short distances ( $ M > 1 $, $ M = w/c $, where $ w $ is the velocity of the gas and $ c $ is the velocity of sound). This includes problems on calculating flows around contours of large local curvature and flow attachment to the surface of the body. In these cases $ p _ {y} \neq 0 $ in the middle layer of the three-layer scheme for the solution.
The class of problems in which the perturbed solution occupies a finite region has also been studied. These solutions occur when there is moderate or strong interaction of the boundary layer with an external hypersonic flow ( $ M \rightarrow \infty $) or in supersonic flow around a body of finite length in which there are intense gas injections through the surface ( $ v _ {0} > 0 $). In these cases, the pressure on the outer boundary of the boundary layer is determined from the solution to the complete problem. The perturbation from the trailing edge of the body propagates upstream, which is caused by the non-uniqueness of the solution in a neighbourhood of the leading edge.
[1] A.N. Tikhonov, "Systems of differential equations containing small parameters in front of the derivatives" Mat. Sb. , 31 : 3 (1952) pp. 575–586 (In Russian)
[2] W. Wasov, "Asymptotic expansions for ordinary differential equations" , Interscience (1965)
[3] A.B. Vasil'eva, V.F. Butuzov, "Asymptotic expansions of solutions of singularly perturbed equations" , Moscow (1973) (In Russian)
[4a] M.I. Vishik, L.A. Lyusternik, "Regular degeneracy and boundary layers for linear differential equations with a small parameter" Uspekhi Mat. Nauk , 12 : 5 (1957) pp. 3–122 (In Russian)
[4b] M.I. Vishik, L.A. Lyusternik, "Solutions to some problems on perturbations in the case of matrices and non-self-adjoint differential equations" Uspekhi Mat. Nauk , 15 : 3 (1960) pp. 3–80 (In Russian)
[5] J.D. Cole, "Perturbation methods in applied mathematics" , Blaisdell (1968)
[6] O.A. Oleinik, "on equations of elliptic type with a small parameter in front of the highest derivatives" Mat. Sb. , 31 : 1 (1952) pp. 104–117 (In Russian)
[7] A.M. Il'in, E.F. Lelikova, "The method of matched asymptotic expansions for the equation in a rectangle" Mat. Sb. , 96 : 4 (1975) pp. 568–583 (In Russian)
[8] N.E. Kochin, I.A. Kibel', N.V. Roze, "Theoretical hydrodynamics" , 2 , Interscience (1964) (Translated from Russian)
[9] L.G. Loitsyanskii, "Laminar boundary layers" , Moscow (1962) (In Russian)
[10] H. von Schlichting, "Grenzschicht-Theorie" , Karlsruhe (1951)
[11] O.A. Oleinik, "Mathematical problems of boundary-layer theory" Uspekhi Mat. Nauk , 23 : 3 (1968) pp. 3–65 (In Russian)
[12] A.A. Dorodnitsyn, "Boundary layers in an incompressible gas" Prikl. Mat. i Mekh. , 6 : 6 (1942) pp. 449–486 (In Russian)
[13] M. van Dyke, "Perturbation methods in fluid mechanics" , Parabolic Press (1975)
[14] R. Betchov, V.O. Criminale, "Stability of parallel flows" , Acad. Press (1967)
[15] S. Soo, "Fluid dynamics of multiphase systems" , Blaisdell (1971)
[16] W.H. Dorrance, "Viscous hypersonic flow" , McGraw-Hill (1962)
[17] W.M. Kays, "Convective heat and mass transfer" , McGraw-Hill (1971)
[18] V.Ya. Neiland, "On the theory of separation of laminar boundary layers in supersonic flow" Izv. Akad. Nauk SSSR Mekh. Zhidk. i Gaza : 4 (1969) pp. 53–57 (In Russian)
[19] V.Ya. Neiland, "Asymptotic problems in the theory of viscous flows" Trudy TsAGI : 1529 (1974) (In Russian)
[20] K. Stewartson, "Multistructured boundary layers on flat plates and related bodies" Adv. App. Mech. , 14 (1974) pp. 145–239
The article above presents a one-sided view of the subject and is in many aspects not up-to-date. Furthermore, very substantial contributions in the Western literature are neglected. Mathematically boundary layers arise in problems of singular perturbations and are treated by methods of asymptotic analysis. In linear and non-linear, ordinary and partial, differential equations much more has been accomplished than described above in the first part. Overviews that are partially complementary can be found in [a1]–[a3].
From the point of view of applications boundary-layer theory is described above as a branch of fluid- and gasdynamics, where it originated. However, boundary layers and transition layers also occur in many other fields, such as for example combustion, geophysics, free boundary problems, or epidemology. Samples and further literature can be found in [a4], [a5].
Systems of (ordinary) differential equations of the type (1) above are often referred to as singular perturbations (cf. also the editorial comments to Perturbation theory).
For an idea of the role of singular perturbations, boundary layers and multiple time scales in control and system theory, cf. [a6].
[a1] J.L. Lions, "Perturbation singulières dans les problèmes aux limits et en contrôle optimal" , Lect. notes in math. , 323 , Springer (1973)
[a2] W. Eckhaus, "Asymptotic analysis of singular perturbations" , North-Holland (1979)
[a3] K.W. Chang, F.A. Howes, "Nonlinear singular perturbation phenomena: theory and application" , Springer (1984)
[a4] C.M. Brauner (ed.) B. Gay (ed.) J. Mathieu (ed.) , Singular perturbations and boundary layer theory , Lect. notes in math. , 594 , Springer (1977)
[a5] F. Verhulst (ed.) , Asymptotic analysis II , Lect. notes in math. , 985 , Springer (1983)
[a6] P. Kokotovic (ed.) , Singular perturbations and time scales in modelling and control of dynamical systems , Univ. Illinois at Urbana-Champaign (1980)
Boundary-layer theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boundary-layer_theory&oldid=46125
This article was adapted from an original article by Yu.D. Shmyglevskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
Retrieved from "https://encyclopediaofmath.org/index.php?title=Boundary-layer_theory&oldid=46125"
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Integrating the error term of a Taylor polynomial: example
Prototypes: More serious questions about Taylor polynomials
Integrating Taylor polynomials: first example
Being a little more careful than for the previous example, let's keep track of the error term in the example we've been doing: we have $${1\over 1-x}=1+x+x^2+\ldots+x^n+ {1\over (n+1)}{ 1 \over (1-c)^{n+1 }}x^{n+1}$$ for some $c$ between $0$ and $x$, and also depending upon $x$ and $n$. One way to avoid having the ${ 1 \over (1-c)^{n+1 }}$ 'blow up' on us, is to keep $x$ itself in the range $[0,1)$ so that $c$ is in the range $[0,x)$ which is inside $[0,1)$, keeping $c$ away from $1$. To do this we might demand that we integrate over the interval $[0,T]$ with $0\le T <1$.
For simplicity, and to illustrate the point, let's just take $0\le T\le {1\over 2}$. Then in the worst-case scenario $$\left|{ 1 \over (1-c)^{n+1 }}\right|\le { 1 \over (1-{1\over 2 })^{n+1}}= 2^{n+1}$$
Thus, integrating the error term, we have \begin{align*} \left|\int_0^T{1\over n+1}{ 1 \over (1-c)^{n+1 }}x^{n+1}\;dx\right|&\le \int {1\over n+1}2^{n+1}x^{n+1}\;dx ={ 2^{n+1} \over n+1}\int_0^Tx^{n+1 }\;dx\\ &=\frac{ 2^{n+1}}{n+1}\left[{x^{n+2} \over n+2}\right]_0^T ={2^{n+1}T^{n+2} \over (n+1)(n+2) } \end{align*} Since we have cleverly required $0\le T\le {1\over 2}$, we actually have \begin{align*} \left|\int_0^T{1\over n+1}{ 1 \over (1-c)^{n+1 }}x^{n+1}\;dx\right|&\le { 2^{n+1}T^{n+2} \over (n+1)(n+2) }\\ &\le { 2^{n+1}({1\over 2})^{n+2} \over (n+1)(n+2)}={1\over 2(n+1)(n+2) } \end{align*}
That is, we have $$\left|-\log(1-T)-\left[T+{T^2\over 2}+\ldots+{T^n\over n}\right]\right|\le {1\over 2(n+1)(n+2)}$$ for all $T$ in the interval $[0,{1\over 2}]$. Actually, we had obtained $$\left|-\log(1-T)-\left[T+{T^2\over 2}+\ldots+{T^n\over n}\right]\right|\le { 2^{n+1}T^{n+2} \over 2(n+1)(n+2) }$$ and the latter expression shrinks rapidly as $T$ approaches $0$.
Previous: Integrating Taylor polynomials: first example
Classic examples of Taylor polynomials
Computational tricks regarding Taylor polynomials
Determining tolerance/error in Taylor polynomials.
How large an interval with given tolerance for a Taylor polynomial?
Achieving desired tolerance of a Taylor polynomial on desired interval
The idea of the derivative of a function
Derivatives of polynomials
Garrett P, "Integrating the error term of a Taylor polynomial: example." From Math Insight. http://mathinsight.org/integrating_error_term_taylor_polynomial_example_refresher
Keywords: ordinary derivative, single integral, Taylor polynomial
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Section 3.2: Homotopy Groups
Subsection 3.2.7: Closure Properties of Homotopy Equivalences (cite)
3.2.7 Closure Properties of Homotopy Equivalences
We now apply Whitehead's theorem (Theorem 3.2.6.1) to establish some stability properties for the collection of homotopy equivalences between Kan complexes (and weak homotopy equivalences between arbitrary simplicial sets).
Proposition 3.2.7.1. Suppose we are given a commutative diagram of Kan complexes
\[ \xymatrix { X \ar [r]^-{g} \ar [d]^{f} & X' \ar [d]^{f'} \\ S \ar [r]^-{h} & S', } \]
where $f$ and $f'$ are Kan fibrations and $h$ is a homotopy equivalence. Then the following conditions are equivalent:
The morphism $g$ is a homotopy equivalence.
For each vertex $s \in S$ having image $s' = h(s)$ in $S'$, the map of fibers $g_{s}: X_{s} \rightarrow X'_{s'}$ is a homotopy equivalence.
Remark 3.2.7.2. In the situation of Proposition 3.2.7.1, the assumption that $S$ and $S'$ are Kan complexes can be eliminated at the cost of working with weak homotopy equivalences in place of homotopy equivalences: see Proposition 3.3.7.1.
Proof of Proposition 3.2.7.1. Assume first that $(1)$ is satisfied. Let $s$ be a vertex of $S$ having image $s' = h(s)$ in $S'$; we wish to show that the induced map $g_{s}: X_{s} \rightarrow X'_{s'}$ is a homotopy equivalence. By virtue of Remark 3.1.5.5, it will suffice to show that for every simplicial set $W$, the induced map $\operatorname{Fun}(W,X_{s} ) \rightarrow \operatorname{Fun}( W, X'_{h(s)} )$ is bijective on connected components. Replacing $X$ by $\operatorname{Fun}(W,X)$ (and making similar replacements for $X'$, $S$, and $S'$), we may reduce to the problem of showing that $g_{s}$ induces a bijection $\pi _0( X_ s) \rightarrow \pi _{0}( X'_{s'} )$. Let us regard $\pi _0(X_ s)$ and $\pi _0( X'_{s'} )$ as endowed with actions of the fundamental groups $\pi _{1}(S,s)$ and $\pi _{1}(S',s')$, respectively (Variant 3.2.4.5). Using our assumption that $g$ and $h$ are homotopy equivalences, we conclude that the induced maps
\[ \pi _0(X) \rightarrow \pi _0(X') \quad \quad \pi _0(S) \rightarrow \pi _0(S') \quad \quad \pi _{1}(S,s) \rightarrow \pi _{1}(S',s') \]
are bijective. Applying Corollaries 3.2.5.3 and 3.2.5.5, we conclude that $g_{s}$ induces a bijection $G \backslash \pi _0(X_ s) \rightarrow G \backslash \pi _0( X'_{s'} )$. It will therefore suffice to show that, for every vertex $x \in X_{s}$, the stabilizer in $G$ of the connected component $[x] \in \pi _0(X_ s)$ is equal to the stabilizer of the connected component $[g(x)] \in \pi _0(X'_{s'} )$. This follows from Corollary 3.2.5.7, since $g$ induces an isomorphism $\pi _{1}(X,x) \rightarrow \pi _{1}(X', g(x) )$.
We now show that $(2) \Rightarrow (1)$. Assume that, for each vertex $s \in S$ having image $s' = h(s)$ in $S'$, the induced map $g_{s}: X_{s} \rightarrow X'_{s'}$ is a homotopy equivalence. We wish to show that $g$ is a homotopy equivalence. We first show that the map $\pi _0(g): \pi _0(X) \rightarrow \pi _0(X')$ is bijective. Our assumption that $h$ is a homotopy equivalence guarantees that the map $\pi _0(h): \pi _0(S) \rightarrow \pi _0(S')$ is bijective. It will therefore suffice to show that, for each vertex $s \in S$ having image $s' = h(s)$, the induced map $\pi _0(X) \times _{ \pi _0(S)} \{ [s] \} \rightarrow \pi _0( X' ) \times _{ \pi _0(S') } \{ [s'] \} $ is bijective. Using Corollaries 3.2.5.3 and 3.2.5.5, we can identify this with the map of quotients $(\pi _{1}(S,s) \backslash \pi _0(X_ s)) \rightarrow ( \pi _1(S',s') \backslash \pi _0(X'_{s'} ) )$. The desired result now follows from the bijectivity of the map $\pi _0(g_ s): \pi _0( X_ s ) \rightarrow \pi _0( X'_{s'} )$ and of the group homomorphism $\pi _{1}(S,s) \rightarrow \pi _{1}(S',s')$.
To complete the proof that $g$ is a homotopy equivalence, it will suffice (by virtue of Theorem 3.2.6.1) to show that for every vertex $x \in X$ having image $x' = g(x)$ and every positive integer $n$, the group homomorphism $\pi _{n}(X,x) \rightarrow \pi _{n}(X',x')$ is an isomorphism. Setting $s = f(x)$ and $s' = f(x')$, we have a commutative diagram of exact sequences
\[ \xymatrix { \pi _{n+1}(S,s) \ar [r] \ar [d]^{\sim } & \pi _{n}(X_ s, x) \ar [r] \ar [d]^{\sim } & \pi _{n}(X,x) \ar [r] \ar [d] & \pi _{n}(S,s) \ar [r] \ar [d]^{\sim } & \pi _{n-1}(X_ s,x) \ar [d]^{\sim } \\ \pi _{n+1}(S',s') \ar [r] & \pi _{n}(X'_{s'}, x') \ar [r] & \pi _{n}(X',x') \ar [r] & \pi _{n}(S',s') \ar [r] & \pi _{n-1}( X'_{s'}, x'). } \]
Our assumptions that $g_{s}$ and $h$ are homotopy equivalences guarantee that the outer vertical maps are bijective, and elementary diagram chase shows that that the middle vertical map is an isomorphism. $\square$
Proposition 3.2.7.3. Let $\operatorname{\mathcal{W}}$ denote the full subcategory of $\operatorname{Fun}( [1], \operatorname{Set_{\Delta }})$ spanned by those morphisms of simplicial sets $f: X \rightarrow Y$ which are weak homotopy equivalences. Then $\operatorname{\mathcal{W}}$ is closed under the formation of filtered colimits in $\operatorname{Fun}( [1], \operatorname{Set_{\Delta }})$.
Proof. Suppose we are given a filtered diagram $\{ f_{\alpha }: X_{\alpha } \rightarrow Y_{\alpha } \} $ in $\operatorname{\mathcal{W}}$, so that each $f_{\alpha }$ is a weak homotopy equivalence of simplicial sets. We wish to show that the induced map $f: (\varinjlim _{\alpha } X_{\alpha }) \rightarrow (\varinjlim _{\alpha } Y_{\alpha } )$ is also a weak homotopy equivalence. Using Proposition 3.1.6.1, we can choose a diagram of morphisms $\{ u_{\alpha }: Y_{\alpha } \hookrightarrow Y'_{\alpha } \} $ with the following properties:
Each of the maps $u_{\alpha }$ is anodyne, and the induced map $u: (\varinjlim _{\alpha } Y_{\alpha }) \rightarrow (\varinjlim _{\alpha } Y'_{\alpha } )$ is anodyne.
Each of the simplicial sets $Y'_{\alpha }$ is a Kan complex, and (therefore) the colimit $\varinjlim _{\alpha } Y'_{\alpha }$ is also a Kan complex.
Since every anodyne morphism is a weak homotopy equivalence (Proposition 3.1.5.12), we can replace $\{ f_{\alpha }: X_{\alpha } \rightarrow Y_{\alpha } \} $ by the diagram of composite maps $\{ (u_{\alpha } \circ f_{\alpha }): X_{\alpha } \rightarrow Y'_{\alpha } \} $, and therefore reduce to the case where each $Y_{\alpha }$ is a Kan complex.
Let us regard the system of morphisms $\{ f_{\alpha } \} $ as a morphism from the filtered diagram of simplicial sets $\{ X_{\alpha } \} $ to the filtered diagram $\{ Y_{\alpha } \} $. Applying Proposition 3.1.6.1 again, we see that this diagram admits a factorization $\{ X_{\alpha } \} \xrightarrow { \{ g_{\alpha } \} } \{ X'_{\alpha } \} \xrightarrow { \{ h_{\alpha } \} } \{ Y_{\alpha } \} $ with the following properties:
Each of the morphisms $g_{\alpha }$ is anodyne, and the induced map $g: (\varinjlim _{\alpha } X_{\alpha }) \rightarrow (\varinjlim _{\alpha } X'_{\alpha } )$ is anodyne.
Each of the morphisms $h_{\alpha }$ is a Kan fibration, and (therefore) the induced map $( \varinjlim _{\alpha } X'_{\alpha }) \rightarrow (\varinjlim _{\alpha } Y_{\alpha })$ is also a Kan fibration.
Arguing as before, we can replace $\{ f_{\alpha }: X_{\alpha } \rightarrow Y_{\alpha } \} $ by the diagram of morphisms $\{ h_{\alpha }: X'_{\alpha } \rightarrow Y_{\alpha } \} $, and thereby reduce to the case where each $f_{\alpha }$ is a Kan fibration. In this case, Proposition 3.2.6.8 guarantees that each $f_{\alpha }$ is a trivial Kan fibration. It follows that the colimit map $f: (\varinjlim _{\alpha } X_{\alpha }) \rightarrow (\varinjlim _{\alpha } Y_{\alpha } )$ is also a trivial Kan fibration, and therefore a (weak) homotopy equivalence by virtue of Proposition 3.1.5.9. $\square$
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Tag: Sebbar
Everything's wrappable to a sphere
Published January 11, 2018 by lievenlb
One of the better opening quotes of a paper:
"Even quite ungainly objects, like chairs and tables, will become almost spherical if you wrap them in enough newspaper."
The paper in question is The orbifold notation for surface groups by John Conway.
Here's Conway talking leisurely about Thurston's idea to capture the acting group via the topology of the orbifold space and his own notation for such orbifolds.
Here's another version of the paper, with illustrations: The orbifold notation for two-dimensional groups, by Conway and Daniel H. Huson.
A very accessible account are these lecture notes:
A field guide to the orbifolds, notes from class on "Geometry and the Imagination" in Minneapolis, with John Conway, Peter Doyle, Jane Gilman and Bill Thurston, on June 17–28, 1991.
And, here are notes by Thurston on The Geometry and Topology of Three-Manifolds, including stuff about orbifolds.
I came across these papers struggling my way through On the discrete groups of moonshine by Conway, McKay and Sebbar.
On the genus $0$ property of moonshine groups they have this to say:
"As for groups of the form $(n|h)+e,f,\dots$, the genus can be determined from the fundamental regions using the Riemann-Hurwitz formula. Since most of the groups are not subgroups of the modular group, the calculations of the genus, which cannot be produced here because of their length, are carried out by finding the elliptic fixed points and the cone points in the orbifolds attached to the fundamental regions. The Euler characteristic of the orbifold determines the genus of the group. See [paper] for more details on orbifold techniques."
the 171 moonshine groups
Monstrous moonshine associates to every element of order $n$ of the monster group $\mathbb{M}$ an arithmetic group of the form
(n|h)+e,f,\dots \]
where $h$ is a divisor of $24$ and of $n$ and where $e,f,\dots$ are divisors of $\frac{n}{h}$ coprime with its quotient.
In snakes, spines, and all that we've constructed the arithmetic group
\Gamma_0(n|h)+e,f,\dots \]
which normalizes $\Gamma_0(N)$ for $N=h.n$. If $h=1$ then this group is the moonshine group $(n|h)+e,f,\dots$, but for $h > 1$ the moonshine group is a specific subgroup of index $h$ in $\Gamma_0(n|h)+e,f,\dots$.
I'm sure one can describe this subgroup explicitly in each case by analysing the action of the finite group $(\Gamma_0(n|h)+e,f,\dots)/\Gamma_0(N)$ on the $(N|1)$-snake. Some examples were worked out by John Duncan in his paper Arithmetic groups and the affine E8 Dynkin diagram.
But at the moment I don't understand the general construction given by Conway, McKay and Sebbar in On the discrete groups of moonshine. I'm stuck at the last sentence of (2) in section 3. Nothing a copy of Charles Ferenbaugh Ph. D. thesis cannot fix.
The correspondence between the conjugacy classes of the Monster and these arithmetic groups takes up 3 pages in Conway & Norton's Monstrous Moonshine. Here's the beginning of it.
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Relationship between overfitting and robustness to outliers
What's the relationship between overfitting and sensitivity to outliers? For example:
Does robustness to outliers make necessarily models less prone to overfitting?
What about the other way around? Are models that are less prone to overfitting usually more robust to outliers?
Or do these concepts bear no relationship at all?
Noise driving overfitting and outliers
Consider for example this definition in Wikipedia:
"The essence of overfitting is to have unknowingly extracted some of the residual variation (i.e. the noise) as if that variation represented underlying model structure", that suggests a deeper connection between noise and overfitting.
So clearly some form of noise plays a role in overfitting. Similarly, one often models outliers as noise that the model may generate, i.e. it's something you can protect yourself against by using a noise model that would explain outliers with e.g. fat tail distributions.
So maybe the relationship here comes down to what type of noise we are fighting against in overfitting vs outliers? And if so, what is a good definition of these types of noise, and what's their relationship?
outliers overfitting robust noise
JoshJosh
$\begingroup$ If the current answers do not satisfy you, could you comment on what they are lacking in your view? $\endgroup$
– jhin
$\begingroup$ For instance, would you prefer if I extended my answer with a few more formal equations? I intentionally left them out because I felt the question was more about intuition than about precise maths, but of course, one could also formalize things. $\endgroup$
$\begingroup$ Thanks - @jhin I left a comment seeking clarification on your answer $\endgroup$
– Josh
How does a model become "robust to outliers"? It does so by acknowledging their presence in the specification of the model, by using a noise model that contains outliers. In probabilistic modeling, this may be achieved by assuming some kind of fat-tailed noise distribution. From an optimization perspective, the same thing can be achieved by using an "outlier-robust cost function" (such as the Huber loss function). Note that there is an equivalence between these two worlds, e.g., whereas L2 norm error minimization corresponds to the assumption of Gaussian noise, L1 norm error minimization (which is more robust to outliers) corresponds to the assumption of Laplacian noise. To summarize, robustness to outliers has nothing to do with the model of the process itself; it depends only on the correctness of the noise model.
How does a model become "robust to overfitting"? Overfitting is a symptom of model mismatch: the process model is too flexible and the noise model is incorrect. If we knew exactly what level of measurement noise to expect, even a very flexible model would not overfit. In practice, robustness to overfitting is achieved by using a flexible model class but biasing the model towards simpler explanations by means of regularization (using a prior over the parameters or, equivalently, an L1/L2 regularization term).
What's the relation of the two properties? Use a flexible model class without appropriate parameter priors or regularization and assume a fat-tailed noise distribution or a robust loss function, and you have an inference procedure that is robust to outliers but not to overfitting. Use an appropriate regularization term but usual L2 error minimization, and you have a method that is robust to overfitting but not to outliers. The two properties are orthogonal to each other, since they relate to different components of the assumed statistical model: robustness to outliers depends on the correctness of the noise model / error loss function, whereas robustness to overfitting depends on the correctness of the parameter priors / regularization term.
jhinjhin
$\begingroup$ Thanks, maybe it's a matter of semantics, but e.g. consider this: "The essence of overfitting is to have unknowingly extracted some of the residual variation (i.e. the noise) as if that variation represented underlying model structure". That suggests a deeper connection between noise and overfitting. Maybe this comes down to types of noise, and what we mean by "outlier", but clearly "noise" can influence both overfitting and whether or not we fit to outliers. So what's your take on these types of noise, their definition and relationship? $\endgroup$
$\begingroup$ @Josh those are interesting remarks. Unfortunately, I'm a bit short on time right now but will update my answer once I have time again! $\endgroup$
Interesting questions posed. I will address the two questions for the use case of statistical classifiers in order to demarcate the analysis to a model domain we can oversee.
Before embarking onto an elaborate answer I do want to discuss the definition of Robustness. Different definitions have been given for the concept of robustness. One can discuss model robustness - as opposed to outcome robustness. Model robustness means that your general model outcome - and hence the distribution of its predictions - that they are less sensitive or even insensitive to an increasing amount of extreme values in the training set. Outcome robustness, on the other hand, refers to the (in)sensitivity to increasing noise levels in the input variables with respect to one specific predicted outcome. I assume that you address model robustness in your questions.
To address the first question, we need to make a distinction between classifiers that use a global or local distance measure to model (probability of) class dependency, and distribution-free classifiers.
Discriminant analysis, k-nearest neighbor classifier, neural networks, support vector machines - they all calculate some sort of distance between parameter vectors and the input vector provided. They all use some sort of distance measure. It should be added that nonlinear neural networks and SVMs use nonlinearity to globally bend and stretch the concept of distance (neural networks are universal approximators, as proved and published by Hornik in 1989).
'Distribution-free' classifiers
ID3/C4.5 decision trees, CART, the histogram classifier, the multinomial classifier - these classifiers do not apply any distance measure. They are so-called nonparametric in their way of working. This having said, they are based on count distributions - hence the binomial distribution and the multinomial distribution, and nonparametric classifiers are governed by the statistics of these distributions. However, as the only thing that matters is whether the observed value of an input variable occurs in a specific bin/interval or not, they are by nature insensitive to extreme observations. This holds when the intervals of input variable bins to the leftmost and rightmost side are open. So these classifiers are certainly model robust.
Noise characteristics and outliers
Extreme values are one kind of noise. A scatter around a zero mean is the most common kind of noise that occurs in practice.
This image illustrates scatter noise (left) and salt-and-pepper noise (right). Your robustness questions relate to the right-hand kind of noise.
We can combine the true value of classifier input $i$, $z(i)$ with scatter noise $\epsilon$, and an outlier offset $e$ as
$ x(i) = z(i) + \epsilon + e \cdot \delta(\alpha) $
with $\delta(\alpha)$ the Kronecker delta function governed by the parameter $\alpha$. The parametrized delta-function determines whether the outlier offset is being added, or not. The probability $P(\delta(\alpha)=1) \ll 1$, whereas the zero-mean scatter is always present. If for example $P(\delta(\alpha)=1) = \frac{1}{2}$, we do not speak of outliers anymore - they become common noise additive offsets. Note also that distance is intrinsic to the definition of the concept outlier. The observed class labels themselves in a training set cannot be subject to outliers, as follows from the required notion of distance.
Distance based classifiers generally use the L2-norm $\mid \mid {\bf x} \mid \mid_2$ to calculate degree of fit. This norm is well-chosen for scatter noise. When it comes to extreme values (outliers), their influence increases with the power of $2$, and of course with $P(\delta(\alpha)=1)$. As nonparametric classifiers use different criteria to select the optimal set of parameters, they are insensitive to extreme value noise like salt-and-pepper.
Again, the type of classifier determines the robustness to outliers.
Overfitting
The issue with overfitting occurs when classifiers become 'too rich' in parameters. In that situation learning triggers that all kinds of small loops around wrongly labeled cases in the training set are being made. Once the classifier is applied to a (new) test set, a poor model performance is seen. Such overgeneralization loops tend to include points pushed just across class boundaries by scatter noise $\epsilon$. It is highly unlikely that an outlier value, which has no similar neighboring points, is included in such a loop. This because of the locally rigid nature of (distance-based) classifiers - and because closely grouped points can push or pull a decision boundary, which one observation in its own cannot do.
Overfitting generally happens between classes because the decision boundaries of any given classifier become too flexible. Decision boundaries are generally drawn in more crowded parts of the input variable space - not in the vicinity of lonely outliers per se.
Having analyzed robustness for distance based and nonparametric classifiers, a relation can be made with the possibility of overfitting. Model robustness to extreme observations is expected to be better for nonparametric classifiers than for distance-based classifiers. There is a risk of overfitting because of extreme observations in distance-based classifiers, whereas that is hardly the case for (robust) nonparametric classifiers.
For distance-based classifiers, outliers will either pull or push the decision boundaries, see the discussion of noise characteristics above. Discriminant analysis, for example, is prone to non-normally distributed data - to data with extreme observations. Neural networks can just end up in saturation, close to $0$ or $1$ (for sigmoid activation functions). Also support vector machines with sigmoid functions are less sensitive to extreme values, but they still employ a (local) distance measure.
The most robust classifiers with respect to outliers are the nonparametric ones - decision trees, the histogram classifier and the multinomial classifier.
A final note on overfitting
Applying ID3 for building a decision tree will overgeneralize model building if there is no stopping criterion. The deeper subtrees from ID3 will begin fitting the training data - the fewer the observations in a subtree the higher the chance of overfitting. Restricting the parameter space prevents overgeneralization.
Overgeneralization is in distance based classifiers also prevented by restricting the parameter space, i.e. the number of hidden nodes/layers or the regularization parameter $C$ in an SVM.
So the answer to your first question is generally no. Robustness to outliers is orthogonal to whether a type of classifier is prone to overfitting. The exception to this conclusion is if an outlier lies 'lightyears' away and it completely dominates the distance function. In that really rare case, robustness will deteriorate by that extreme observation.
As to your second question. Classifiers with well-restricted parameter spaces tend to generalize better from their training set to a test set. The fraction of extreme observations in the training set determines whether the distance based classifiers be led astray during training. For non-parametric classifiers, the fraction of extreme observations can be much larger before model performance begins to decay. Hence, nonparametric classifiers are much more robust to outliers.
Also for your second question, it's the underlying assumptions of a classifier that determine whether it's sensitive to outliers - not how strongly its parameter space is regularized. It remains a power-struggle between classifier flexibility whether one lonely outlier 'lightyears away' can chiefly determine the distance function used during training. Hence, I argue a generally 'no' to your second question.
Match Maker EEMatch Maker EE
$\begingroup$ Thanks, I like much about this answer, but there is an aspect of overfitting that I think still went unanswered. You mention "Overfitting generally happens between classes" but one could also consider outliers as examples of data points that come from the wrong class. Moreover, you discuss noise in the context of robustness to outliers, but noise can also drive overfitting: "[From Wikipedia] The essence of overfitting is to have unknowingly extracted some of the residual variation (i.e. the noise) as if that variation represented underlying model structure" (see more context on the OP) $\endgroup$
$\begingroup$ See extensions in text and my modified conclusions. I enjoy such an analytic discussion. $\endgroup$
– Match Maker EE
$\begingroup$ Thanks. "Hence, nonparametric classifiers are much more robust to outliers." I'm not sure if you can make this type of broad generalization. A Gaussian process is often considered non-parametric (its ability to interpolate the data grows with the data), and I think if I use, say, a linear GP kernel (linear covariance function), you are still fitting data to a Normal distribution on a linear model that is more sensitive to outliers than, say, a parametric linear regression with a Laplace (fatter tails) distribution. No? $\endgroup$
$\begingroup$ A with regards to nonparametric classifiers, the salt-and-pepper noise is commonly removed by a 3 x 3 median filter by convolution. Hence, there the (nonparametric) histogram based technique has found its way. It is true, however, that using the L1-norm as a fit criterion does not suffer from increasing influence of an extreme value. $\endgroup$
There is a lot of things to influence the outliers, if the model is overfitting then it will learn specific details of data including noise data points like outliers. But it's not necessarily that if model not robust to outliers then it's overfitting, there is models is sensitive to outliers.
Ali MostafaAli Mostafa
Per Wikipedia on contraposition to quote:
In logic and mathematics, contraposition refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as proof by contraposition.[1] The contrapositive of a statement has its antecedent and consequent inverted and flipped. For instance, the contrapositive of the conditional statement "If it is raining, then I wear my coat" is the statement "If I don't wear my coat, then it isn't raining."...The law of contraposition says that a conditional statement is true if, and only if, its contrapositive is true.[3]
So, on the slightly reworded question: Is a model that does not overfit easily than one that does, necessarily implied more robustness to outliers, the contraposition is, as 'not more' is 'equal or less': Does equal or less robustness necessarily follow from a model that overfits easily than one that does not?
To assist in the answer, take the case of Least Absolute Deviation regression which is known for its robustness. It also curious in the case of estimation of single parameter, it reduces to a median estimate as opposed to the mean (which is highly susceptible to outliers as it incorporates all the data). So, the mean can be viewed as 'overfitting' but in samples, the mean and median can be close due to a balancing of large positives and negative values.
Per the 'if and only if standard' placed on the veracity of the councontrapositive, necessarily less robustness does not follow from a model that overfits easily than one that does not, so my answer is no.
AJKOERAJKOER
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Thermogravimetric and kinetic analysis to discern synergy during the co-pyrolysis of microalgae and swine manure digestate
Arun K. Vuppaladadiyam1,
Hao Liu2,
Ming Zhao ORCID: orcid.org/0000-0002-5801-55931,
Abdul F. Soomro1,
Muhammad Zaki Memon1 &
Valerie Dupont3
Co-pyrolysis of wastes with other feedstock can synergistically improve the rate of biomass decomposition and also help to resolve the issues related to limited availability feedstock. In this regards, synergistic interaction between feedstock during co-pyrolysis is an important aspect of research. As the constituents of aquatic and lignocellulosic biomass are different, and the decomposition pattern of aquatic biomass is dissimilar when compared to lignocellulosic biomass, it is important to understand whether these two biomasses interact during co-pyrolysis.
Synergism in the co-pyrolysis of microalgae (MA), swine manure digestate (SWD), and their blends (MA/SWD) (w/w %), 2.5/7.5 (MD-1), 5/5 (MD-2), and 7.5/2.5 (MD-3), was evaluated based on decomposition behavior, gas yields, extent of thermal degradation, and kinetics. Extractives and volatiles in biomass enhanced the reaction kinetics and products yields, as indicated by the reduction in apparent activation energy of the blends, accompanied by an increase in H2, total gas yield, and extent in degradation. Thermogravimetric data, via isoconversional methods, were interpreted to achieve the apparent activation energies for the thermal degradation of the MA, SWD, and their blends. The best fit reaction models were identified using compensation effect and generalized master plots methods. Semi-quantitative method was used to quantify the evolved gas species. H2, CO, and CO2 were noted to be the dominant gases, implying that tar cracking and reforming reactions were predominant.
Overall, synergy was noticed with respect to the pyrolysis of SWD biomass to gas products in the presence of MA biomass, whereas synergy was witnessed up to 50 w/w % MA in view of kinetic parameters as evaluation criteria.
The alternative energy sources, particularly biomass derived biofuels, have been highlighted as substitutes for conventional petroleum fuels. Though, there are numerous advantages associated with biofuels, the first-generation biofuels (produced from edible crops) could result in diverse side effects such as water shortage, increase in the price of edible crops, etc. The negative effects associated with the first-generation biofuels, to a major extent, can be eliminated by shifting to the second generation (produced form non-edible biomass) or third generation (produced from aquatic biomass such as macro-/microalgae) as they do not present any threat to food chain or ecosystem [1]. Moreover, microalgae can grow in fresh water, brackish, and wastewater, making them suitable for simultaneous wastewater treatment and biomass generation [2]. In addition, microalgae cultivation can be integrated with large CO2 point sources, such as power or cement plants, and wastewater treatment facilities, making it a sustainable pathway for biomass generation coupled with carbon capture [3].
The livestock sector has been reported to be one of the principal contributors to grave environmental issues around the world. In China, it is reported that animal manure is one of the major sources of water pollution, mainly through the transfer of nitrogen (N) and phosphorus (P) [4]. Furthermore, it is estimated that 837 million tons of animal manure are produced in China, 208 million of which contributed by swine manure alone [5]. These considerable amounts of wastes present a serious threat to the environment if they are not managed properly. Furthermore, swine manure is more recognized as a pollutant and is often a derelict bioenergy resource. However, use of manure as a bioenergy feedstock could reduce waste disposal problems and alleviate pressure on environment by providing clean energy [6]. In particular, generation of energy from livestock manure eliminates the most common problems, such as unwanted transfer of pathogens into ecosystem, eutrophication caused by the leaching of nutrients into nearby water bodies, etc., associated with the conventional means of waste management [7].
A wide range of technologies are available for converting biomass to bioenergy, which includes biochemical such as anaerobic digestion, transesterification, fermentation, etc., and thermochemical techniques, such as gasification, liquefaction, pyrolysis, etc. [2]. Till date, anaerobic digestion (AD) has been widely acknowledged as a means to add value and stabilize solid wastes. However, the high CO2 content of biogas generated from AD process impairs the fuel quality and necessitates several purification steps. In addition, the carbon deposited in the microorganisms lowers the carbon conversion from manure to biofuels. Among the thermochemical conversion techniques, pyrolysis has been the most widely accepted technology to convert biomass, especially agricultural and aquatic biomass, into bioenergy owing to its potential advantages such as less pollution emission, reasonable cost, and simple operation [8]. However, co-pyrolysis and catalytic pyrolysis help to improve the nature, quantity, and quality of the end products [9]. Furthermore, it is reported that co-pyrolysis of wastes with other feedstock can synergistically improve the rate of biomass decomposition and also help to resolve the issues related to limited availability feedstock [10]. In this regards, synergistic interaction between feedstock during co-pyrolysis is an important aspect of research. As the constituents of aquatic and lignocellulosic biomass are different, and the decomposition pattern of aquatic biomass is dissimilar when compared to lignocellulosic biomass, it is important to understand whether these two biomasses interact during co-pyrolysis. In the former, because of its nutritional value, microalgae decomposition has been mainly related to its lipids, proteins, and carbohydrates content [11], while the latter, intended as biofuel, the decomposition has been mainly studied in terms of its cellulose, hemicellulose, and lignin content [12].
The knowledge of pyrolysis kinetics is essential for many reasons such as predicting the behavior of biomass during pyrolysis which forms the basis for reactor design. A number of successive and/or parallel reactions occur during the thermal breakdown of biomass making it a complicated process. For a thermal decomposition process, the International Confederation for Thermal Analysis and Calorimetry (ICTAC) highly recommends isoconversional methods to identify the 'kinetic triplet' that include apparent activation energy, pre-exponential factor, and reaction mechanism [13, 14]. In the present study, an attempt has been made to evaluate the synergistic influence of microalgae and swine manure digestate during co-pyrolysis as compared to individual pyrolysis. Thermal decomposition behavior of two different category of biomass, namely microalgae and swine manure digestate, and their blends in different ratios have been studied using thermogravimetric analyser coupled with mass spectrometer. Furthermore, the kinetic parameters, which include apparent activation energy (Eα), and pre-exponential or frequency factor (A) were identified using isoconversional methods (Kissinger–Akahira–Sunose (KAS) [15] and Flynn–Wall–Ozawa (FWO) [16, 17] methods) and compensation effect respectively. The reaction model, f(α), was identified using compensation effect [18] and generalized master plots method [19]. Furthermore, the gases evolved during pyrolysis were analyzed and reported. The most commonly used reaction mechanisms, along with their differential f(α) and integral expressions g(α), are presented in Table 1.
Table 1 Common solid-state reaction mechanisms [44, 45]
Biomass preparation and characterization
The microalgae, Spirulina platensis (MA), sample was collected from Phycospectrum Environmental Research Centre (PERC), Chennai, Tamil Nadu, India. The isolated culture was then inoculated and grown in a 1 L Erlenmeyer flask with working volume of 500 mL using CFTRI (developed by Central Food and Technology Research Institute, Mysore, India) [20] medium for 30 days. The composition of CFTRI media is presented in Table 2. The cultivation was carried out in an incubator, maintaining a temperature of 30 °C and a light intensity of 500 lx throughout the cultivation period. After the cultivation phase, the microalgae cultures were harvested by phase separation in a centrifuge at 6500 rpm for 15 min to obtain the microalgae biomass. The solid phase biomass was washed with deionized water multiple times and was maintained at 80 °C in a ventilated oven to procure dried algal biomass.
Table 2 Composition of CFTRI media [46, 47]
Partially digested manure was collected from swine manure anaerobic digestion plant Donghua, Beijing. The Automatic Methane Potential Test System II (Bioprocess Control, Sweden) was used to completely digest the sample. The digestion process was run in triplicates, in a 0.6 L reactors, to ensure the complete digestion. The system consisted of three units: Unit A is a water bath containing 15 glass bottles for anaerobic digestion (AD) and is maintained at mesophilic temperature (35 °C); Unit B, CO2 adsorption using 3 M sodium hydroxide (NaOH); and Unit C, in which the volume of CH4 released from Unit A was automatically recorded. A mixing rod with slow mechanical rotation was used in each bottle in Unit A. The glass bottles containing manure samples were placed in the water bath in Unit A and only gas generated during the digestion process is allowed to pass to Unit B, where bromothymol blue indicator is used to monitor the change in the pH. However, there is no possibility of the manure samples in Unit A to react with 3 M NaOH in Unit B. The run was stopped after ensuring the methane generation was less than 10 mL per day. The growth and digestion characteristics of MA and swine manure digestate (SWD), respectively, as they are out of the scope of current study, are not discussed. Elemental analysis was undertaken using a carbon–hydrogen–nitrogen analyzer (model CE 440; EAI, Oakland, NJ, USA), biocompounds (lipid, protein, and carbohydrate) analysis [21], and structural component analysis (lignin, cellulose, and hemicellulose) [22] were carried out as per standard procedures and are presented in Table 3.
Table 3 Proximate analysis, elemental composition, and chemical composition of biomass samples
TG–DTG–MS analysis
Thermal analyser (Q600) was used to perform the thermogravimetric (TG) and derivative thermogravimetric (DTG) analyses of individual and blended biomasses. Sample, ca. 3 mg weight, was heated to 800 °C from ambient temperature. Initially, a constant heating rate of 15 °C min−1 was used, and later, the experiments were run at different heating rates of 10 and 20 °C min−1, from room temperature to 800 °C. For all the experiments, argon (Ar) gas was used as a purge gas and the flow was set at 500 mL min−1 to ensure an inert environment. The results reported are the average of the data obtained by conducting the experiments in triplicate.
The pyrolysis gas was collected and delivered to the mass spectrometer (MS) by heated capillary, wherein the gas molecules are ionized and differentiated based on their mass-to-charge ratio (m/z). The ions that were scanned and their respective evolved gases are presented in Table 4.
Table 4 Ion fragments and their representative gas species
With respect to the purge flow rate of Ar and weight of biomass, normalization of the raw signals from MS was done based on the following equation:
$$ {\text{Normalized signal for key molecule fragments }}`i '= \text{ }{{\left( {{\text{IC}}_{i} *500} \right)} \mathord{\left/ {\vphantom {{\left( {{\text{IC}}_{i} *500} \right)} {\left( {{\text{IC}}_{\text{Ar}} *{\text{wt}}_{\text{sample}} } \right)}}} \right. \kern-0pt} {\left( {{\text{IC}}_{\text{Ar}} *{\text{wt}}_{\text{sample}} } \right)}}, $$
where, ICi and ICAr indicate molecular m/z signals for molecular ion fragments and Ar, 'i' (arbitrary unit), and wtsample indicate the weight of biomass sample (g). The detailed procedure for the analysis of evolved gas species is discussed in our previous study [23].
Kinetic analyses
Biomass pyrolysis varies for different biomass, mainly because of differences in their chemical structure. However, the overall path of biomass pyrolysis can be defined as follows: \( {\text{Biomass}}\, \to {\text{Char}}\, + \,{\text{Volatiles}}\, + \,{\text{Gases}}. \) The rate constant k(T), according to the Arrhenius equation, can be expressed as follows:
$$ k(T) = {\text{Ae}}^{{\left( {\frac{{ - E_{\upalpha} }}{RT}} \right)}} , $$
where A (s−1) and Eα (J mol−1) are pre-exponential and apparent activation energy of the reaction, respectively, and R and T (°K) are universal gas constant (8.314 J mol−1 K−1) and absolute temperature, respectively. The kinetics of solid-state thermal degradation can be defined as follows:
$$ \frac{{{\text{d}}\alpha }}{{{\text{d}}t}} = k(T)f(\alpha ) = {\text{Ae}}^{{\left( {\frac{{ - E_{\upalpha} }}{RT}} \right)}} f(\alpha ), $$
where α is the degree of conversion at time t and f (α) indicate the reaction mechanism function.
The thermal decomposition is reflected by the conversion degree α which could be defined as follows:
$$ \alpha = \frac{{m_{0} - m_{t} }}{{m_{0} - m_{\infty } }}, $$
where m0, mt, and m∞ indicate the initial, instantaneous, and final masses during thermal degradation, respectively.
By understanding that temperature increases with respect to time under constant heating rate (β), β can be expressed as follows:
$$ \beta = \frac{{{\text{d}}T}}{{{\text{d}}t}} = \frac{{{\text{d}}T}}{{{\text{d}}\alpha }}{ \times }\frac{{{\text{d}}\alpha }}{{{\text{d}}t}}. $$
From Eqs. (4) and (2):
$$ \frac{{{\text{d}}\alpha }}{{{\text{d}}T}} = \frac{A}{\beta }e^{{\left( {\frac{{ - E_{\alpha } }}{RT}} \right)}} f(\alpha ). $$
The integrated form of f (α) can be expressed as follows:
$$ g(\alpha ) = \int_{0}^{\alpha } {\frac{{{\text{d}}\alpha }}{f(\alpha )}} = \frac{A}{\beta }\int_{{T_{0} }}^{T} e^{{\left( {\frac{{ - E_{a} }}{RT}} \right)}} {\text{d}}T. $$
An exact solution for the above integral cannot be obtained, and thus, Eq. (7) needs to be solved by employing approximations or numerical methods. The isoconversional methods, in view of their good adaptability and validity, are known to provide a viable method to identify the apparent activation energy. Thus, in the present study, two isoconversional methods Kissinger–Akahira–Sunose (KAS) and Flynn–Wall–Ozawa (FWO) methods are applied to determine the apparent activation energy.
The FWO method can be expressed as follows:
$$ \ln (\beta ) = \ln \left[ {\frac{{AE_{\alpha } }}{Rg(\alpha )}} \right]\, - 5.331 - 1.0516\frac{{E_{\alpha } }}{RT}. $$
The KAS method can be expressed as follows:
$$ \ln \left( {\frac{\beta }{{T^{2} }}} \right) = \ln \left[ {\frac{AR}{{E_{\alpha } g(\alpha )}}} \right] - \frac{{E_{\alpha } }}{RT}. $$
At constant conversion rate (α) and multiple heating rates, the plots ln (β/T2) vs. 1/T (KAS method) and ln (β) vs. 1/T (FWO method) result in straight lines, whose slope can be used to calculate the apparent activation energy. Generalized master plots' methods was used to identify the reaction mechanism function f(α). Theoretical master plots, listed in Table 1, are considered as a reference and were compared against the experimental master plots. The underlying concept related to generalized master plots method is, at infinite temperature, the generalized time (θ) required to achieve a certain degree of conversion (α) [24, 25] can be given by the following equation:
$$ \theta = \int_{0}^{t} {e^{{\left( {\frac{ - E\alpha }{RT}} \right)}} } \,{\text{d}}t. $$
By differentiating the above equation:
$$ \frac{{{\text{d}}\theta }}{{{\text{d}}t}}\, = \,e^{{\left( {\frac{ - E\alpha }{RT}} \right)}} . $$
By combining (1), (2), and (10), we get the following:
$$ \frac{{{\text{d}}\alpha }}{{{\text{d}}\theta }}\, = \,\frac{{{\text{d}}\alpha }}{{{\text{d}}t}}\,e^{{\left( {\frac{ - E\alpha }{RT}} \right)}} . $$
We can relate a proper kinetic model, by selecting α = 0.5 as a reference, to reduced generalized rate of reaction as given by the following:
$$ \frac{{\frac{{{\text{d}}\alpha }}{{{\text{d}}\theta }}}}{{\left( {\frac{{{\text{d}}\alpha }}{{{\text{d}}\theta }}} \right)0.5}}\, = \,\frac{f(\alpha )}{f\left( \alpha \right)0.5}\, = \,\left( {\frac{{\frac{{{\text{d}}\alpha }}{{{\text{d}}\theta }}}}{{\left( {\frac{{{\text{d}}\alpha }}{{{\text{d}}\theta }}} \right)0.5}}} \right)\,\frac{{e^{{\left( {\frac{ - E\alpha }{RT}} \right)}} }}{{e^{{\left( {\frac{ - E\alpha }{RT0.5}} \right)}} }}. $$
Therefore, by plotting the generalized reaction rate from the right-hand side of Eq. (13), and the theoretical plots from the left-hand side of Eq. (13) against conversion (α), appropriate reaction mechanism function can be inferred on comparison.
Compensation effect
While the model free methods are sufficient to determine apparent activation energy, compensation effect allows to accurately determine pre-exponential factor and reaction model. A strong correlation in the form of linear relationship between the Arrhenius parameters, ln Aj and Ej estimated using single heating rate method, is known as compensation effect. Taking logarithm on both sides and rearranging terms, Eq. (3) results in Eq. (14):
$$ \ln \left[ {\frac{1}{fi(\alpha )}\frac{{{\text{d}}\alpha }}{{{\text{d}}t}}} \right]\, = \,\ln Ai(\alpha ) - \,\frac{E\alpha (\alpha )}{RT}, $$
where i indicate the reaction model in Table 1.
Selecting any reaction model fi(α) mentioned in Table 1, and by plotting the left-hand side of Eq. (13) against the inverse of temperature, a pair of ln Ai and Eαi can be generated from the intercept and slope of straight line. Then the compensation equation is given as follows:
$$ \ln A_{i} \, = \,a^{*} E_{\alpha i} \, + \,b^{*} , $$
where a* and b* are generated from linear fitting of these pairs of ln Ai and Eα,i. By substituting the activation energy obtained from model free isoconversional methods (Eq. 9) and a*, b* values obtained from Eq. (14) in the following equation, the pre-exponential factor at given α can be calculated using the following equation:
$$ \ln A_{\alpha } = \, a^{*} E_{\alpha } + \, b^{*} , $$
where Eα was obtained from isoconversional methods. By rearranging Eq. (3), the reaction model f(α) can be defined as follows:
$$ f(\alpha )\, = \,\left( {\frac{{{\text{d}}\alpha }}{{{\text{d}}t}}} \right)_{\alpha } \,\left[ {A\alpha \exp \left( { - \,\frac{E\alpha }{RT}} \right)} \right]^{ - 1} . $$
By substituting the experimental values for (dα/dt) and Tα and using the values of activation energy and pre-exponential factor derived from isoconversional methods and compensation effect, respectively, Eq. (16) yields numerical values for f(α), which can be matched against theoretical f(α) models to identify the most accurate reaction model.
The thermodynamic parameters such as, free Gibbs energy (ΔG), enthalpy (ΔH), and change in entropy (ΔS) were evaluated using Eqs. (18) to (20) [26, 27]:
$$ \Delta G\,\, = \,\,E_{\alpha } + RT_{m} \ln \left[ {\frac{{k_{B} T_{m} }}{h\,A}} \right] $$
$$ \Delta H\,\, = \,\,E_{\alpha } - RT $$
$$ \Delta S\,\, = \frac{\Delta H - \Delta G}{{T_{m} }}, $$
where Tm, kB, and h indicate DTG peak temperature, Boltzmann constant (1.381 × 10−23 J K−1), and Plank constant (6.626 × 10−34 J s), respectively.
Thermal behavior of microalgae, digestate, and their blends
The TG and DTG curves for the (co-)pyrolysis process for selected biomass samples, which include MA, SWD, and their blends MD-1 (25% MA + 75% SWD w/w) MD-2 (50% MA + 50% SWD w/w) and MD-3 (75% MA + 25% SWD w/w), generated at 15 °C min−1 heating rate, are presented in Fig. 1.
TG and DTG curves of biomass samples and their blends: a microalgae, b digestate, c MD-1, d MD-2, and e MD-3 at heating rate 15 °C min−1
During pyrolysis, the organic components of biomass are decomposed into different vapor phases and gas compounds leaving a carbon-rich solid residue (pyrolysis char). The mass loss in samples during pyrolysis process is mainly because of evolution of vapor and gas from the biomass. The pyrolysis process of MA, SWD, and their blends (MD-1, 2, and 3) can be divided into three zones. In the first zone, the pyrolysis took place at low temperatures (< 200 °C), and this can be attributed to the evaporation of physically absorbed moisture [28]. The main pyrolysis of the plain samples, MA and SWD, took place between 200 and 500 °C, and are in good agreement with the other studies reported in the literature [29, 30]. The second zone could be considered as the main pyrolysis stage as maximum weight loss was noticed in this zone for the individual samples and their blends. For MA sample (Fig. 1a), the second zone occurred in the temperature range 200–600 °C with a characteristic peak at 315 °C, followed by a shoulder in between 400 and 500 °C. The main peak can be attributed to the decomposition of proteins. The maximum degree of weight loss was noticed at ca. 300–340 °C. As microalgae do not contain cellulose and hemicellulose, weight loss can be attributed to the decomposition of structural components such as lipids, lignin, proteins, and carbohydrates. In this temperature range, these compounds are reported to undergo a set of reaction mechanisms, which include decarboxylation, depolymerization, and cracking of primarily carbohydrates, lipids, and proteins [31]. The small shoulder at the end of the second zone could possibly be the result of lipids and proteins present in microalgae (Table 2).
On the other hand, the DTG profile for the pyrolysis of SWD showed three distinct peaks (Fig. 1b). The main degeneration stage was identified to be in the temperature range 200–500 °C. The main peak was noticed at ca. 330 °C and can be ascribed to the degradation of hemicellulose and glucoside linkage depolymerisation [30]. Swine manure cannot be considered as common lignocellulosic biomass as it not only contains cellulose, hemicellulose, and lignin, but also contains a large amount of extractives (such as sugars, proteins, lipids, starches, etc.). These extractives cannot be overlooked as they contribute significantly to the total weight of biomass and have a decomposition range matching closely that of hemicellulose [8]. A small shoulder at the right of the major peak can be ascribed to the decomposition either of lipids or of other N-containing compounds. In addition, a smaller peak close to 700 °C was witnessed and this could be because of the dehydration or calcination of mineral components [32]. The DTG profiles of MD-1 (Fig. 1c), MD-2 (Fig. 1d), and MD-3 (Fig. 1e) were similar to the biomass that contributed to the major portion of blend. For instance, the DTG profile of MD-1 is similar to that of SWD, while that for MD-3 the DTG profile was close to MAs. However, the peak temperatures of the blends decreased when the proportion of MA was increased. The peak temperatures for MD-1, 2, and 3 were 320, 316, and 314 °C, respectively, with the major weight loss for all the three blends happening in the temperature range of 270–370 °C. When the TGA data of the individual biomass (SWD) are compared against the blends of MA and SWD, no remarkable synergistic effect can be seen during co-pyrolysis, as the amount of solid residue left at the end of co-pyrolysis (at 800 °C) is an intermediate value between the residues during the pyrolysis individual biomasses. However, the residues obtained from the pyrolysis of blends were much lower than the SWD pyrolysis residue. This could be because of evolution of hot gaseous species as a result of thermal decomposition of volatile matter or could be because of thermal decomposition of biomass enhanced by catalytic activity of metal contents in the MA and SWD ash [33].
Analysis of evolved gas species
The gas evolution trends for all the samples are depicted in Fig. 2. There were two distinct phases of gas evolution from pyrolysis. The first gas release coincided with the main pyrolysis weight loss, and the released gases, primarily CO and CO2, evolved within the temperature range of 250–400 °C. At temperatures 450 °C and above, H2 released due to tar cracking and reforming reactions was noted and it continued up to 800 °C. There was very little CH4 released during the pyrolysis of all the samples. These appeared to be the general trends of pyrolysis which were shared across all the samples, however, to intensities of gas releases seemed to differ across samples. For MA (Fig. 2a), two CO2 evolution peaks, one at 220 °C and the other at 324 °C, were noticed. These peaks can be associated with the decomposition of saccharides and main decomposition of proteins and lipids [34]. For SWD (Fig. 2b), two distinct CO2 evolution peaks were noticed at 333 °C and 688 °C. These can be attributed to the decomposition of volatiles and solid residue respectively, as observed in thermogravimetric analysis of SWD. For the blends (Fig. 2c-MD-1-, Fig. 2d-MD-2, Fig. 2e-MD-3), with the increase in MA content, the second peak value of CO2 decreased gradually. Similar behavior was observed during the co-pyrolysis of microalgae with textile dying sludge [35]. It can be noticed from Fig. 2 that the CO evolution trend appeared to be similar for all the samples. The evolution of CO started from temperatures ca. 200 °C and continued thorough out the run. The possible reasons could be decomposition of volatiles and Boudouard reactions occurring in temperatures < 500 °C and 500–800 °C, respectively. For blends, the second evolution peak for CO followed similar trend as seen for CO2 and the second peak gradually decreased as the proportion of MA grew. The trends from MA and SWD, Fig. 2a, b, showed a similar H2 release, but their CO and CO2 releases differed. However, there was a mutually synergistic effect on total gas yield between the two biomass types. It can be noticed from Fig. 2f that the gas yields increased with the rise in the composition of MA in the blends. In addition, the total gas yield was obtained in the order MD-3 > MD-2 > MD-1 > SWD > MA. The plausible reasons for an increase in individual gas compositions as well as total gas yield could be char gasification reactions and reforming reactions. In addition, the alkali metals present in the ash of MA and SWD biomasses could have improved the yield rates and consequently also the final yields.
Gas yield rates for samples: a MA b SWD, c MD-1, d MD-2, e MD-3, and f cumulative gas yields for all the samples
Evidence of synergy during pyrolysis of biomass blends
To evaluate quantitatively the possible synergistic effects during the co-pyrolysis of biomass blends, the experimental results (TGA data) were compared with the calculated results, i.e., weighted residual mass values (additive model), if the biomasses had been pyrolysed independently. This is equivalent to a complete lack of synergistic interaction between the two biomass samples during pyrolysis, so that the calculated values are the sum of individual values proportional to their mass ratio. More detailed explanation regarding the procedure can be obtained from the work done by Mallick et al. [33], The weighted residual mass values as functions of time can be obtained from the following equation:
$$ \alpha_{\text{cal}} \, = \,(1 - f_{\text{MA}} )\;\alpha_{\text{SWD}} \, + \,f_{\text{MA}} \,\,\alpha_{\text{MA}} , $$
where fMA is the fraction of microalgae in the mixture, and αMA and αSWD are the conversions of individual biomasses at a given time. Hence, if individual biomasses are pyrolysed independently, αcal can be defined as theoretical conversion of a given biomass blend with zero synergy.
Using Eq. (21), the theoretical conversions of biomass blends, calculated by using the TGA data of individual biomasses, are compared with the experimental data and the results are presented in Fig. 3a–c. A significant discrepancy can be noticed between the experimental and calculated profiles of blends MD-1 and MD-2 and MD-3, especially in the temperature range of 250–650 °C. The experimental thermal degradations of all the blends were more extensive than the calculated additive degradations of individual biomasses of same mass ratio, and this essentially is the evidence of mutual synergy for thermal degradation between the two biomasses and is discussed in greater details in the subsequent sections.
Comparison of experimental and calculated TGA profiles of biomass for heating rate of 15 °C min−1: a MD-1, b MD-2, and c MD-3
Kinetic analysis
Estimation of apparent activation energy of thermal degradation
Thermogravimetric data was used to analyze the kinetic parameters of the samples and blends that are considered in the present study. The description of transport phenomena together with chemical kinetics is crucial in the design and optimization of thermochemical conversion systems [36]. Considering the TGA data obtained at heating rates 10, 15, and 20 °C min−1 and using the isoconversional methods, namely KAS and FWO methods, the apparent activation energies were determined for the materials used in the study. According to Eqs. (7) and (8), the apparent activation energies were calculated within a selected conversion range of 0.1–0.8 with an interval of 0.05, and are presented in Table 5. Activation energy is defined as the minimum energy required to initiate a reaction, which imply that a reaction with higher activation energy either need higher reaction temperature or longer duration to gain adequate energy to initiate the reaction [37]. When many reactions are present, an apparent activation energy is derived from the data, representative of the ensemble of reactions taking place, as is the case for pyrolysis. The change in the apparent activation energy with respect to conversion is presented in Fig. 4. From Table 5 and Fig. 4, it can be noticed that apparent activation energy (Eα) is highly dependent on conversion which indicates that pyrolysis of biomass is, as expected, a complex process involving many reactions occurring simultaneous at the same stage [38]. The overall pyrolysis process can be characterized as a multi-stage reaction in which every single stage contributes to global mechanism to some extent depending on the decomposition. The average activation energies for samples and their blends using KAS and FWO methods were in the range of 153.17–157.55 for MA, 206.45–209.51 for SWD, 176.07–176.25 for MD-1, 177.13–178.99 for MD-2, and 172.29–172.94 for MD-3. The activation energies obtained in this study are close to the values reported in the literature for similar feedstock. For instance, Shuping et al. [39] and Fernandez-Lopez et al. [40] reported an average range for Eα of 145.71–146.42 and 210–213 for microalgae Dunaliella tertiolecta and swine manure, respectively, using KAS and FWO methods.
Table 5 Activation energies (Ea) and correlation factors (R2) for different conversion values using KAS and FWO models
Variation in activation energy with progressing conversion for a using KAS and b using FWO models
The Eα were noticed to increase from α = 0.15 which indicate the pyrolysis of main components of biomass (lipid, protein, and carbohydrate in MA and cellulose, hemicellulose, and lignin in SWD). With the increase in the conversion, the pyrolysis of biomasses and their blends can be roughly categorized into different stages (depending on the biomass and their blending ratio). For MA, the pyrolysis with respect to conversion can be divided into two stages: α = 0.1–0.25 and α = 0.25–0.8. The temperatures corresponding to stages I and II are 250–300 and 300–500 °C, respectively. During stage I, the activation energy increased from 160 to 172 kJ mol−1. As per the previous literature and DTG curves, discussed above in this study, carbohydrates and proteins of microalgae biomass decomposed in this stage. In the stage II, the activation energy decreased continuously from 164.8 kJ mol−1 at α = 0.3 to 133.1 kJ mol−1 at α = 0.8. For SWD biomass and blends, MD-1 and MD-2, the pyrolysis process, with respect to conversion, can be divided into three stages. Based on the DTG analysis in this study, it can be inferred that most of the extractives (lipids, proteins, and carbohydrates) and hemicellulose decomposed in stage I, which varied from α = 0.1–0.25 for SWD and MD-1 and α = 0.1–0.35 for MD-2. The difference in the range for MD-1 and MD-2 could be the variation in the proportion of biomasses. In stage I, the Eα values increased from 207.9 to 218.3, 139.5 to 180.3, and 174 to 190 kJ mol−1 for SWD, MD-1, and MD-2, respectively. In stage II, for SWD, the activation energy almost kept a constant value for α varying from 0.25 to 0.55, while the activation energy decreased continuously from 171 to 164 and 172–153 kJ mol−1 for blends MD-1 and 2, respectively. Finally, the activation energies climbed up in stage III as shown in Fig. 4. It should be noted that the behavior of blends is similar to the individual mass-dominant biomass. For MD-3 the trend of activation energy against conversion is similar to MAs. The overall pyrolysis for MD-3 can be divided into two stages, where, in stage I, the activation energy increases continuously to 192.7 kJ mol−1 at α = 0.55 and then decreases to 146.5 kJ mol−1 at α = 0.8. The synergistic effect in co-pyrolysis of biomass is evident in the terms of activation energy. It can be witnessed from Table 4 that average activation energies of blends MD-1 and 2 have a synergy of SWD with respect to MA in up to 50 w/w %, as the calculated weighted blend activation energies (0 synergy) were higher than those found experimentally. All the blends displayed a higher activation energy than MAs, revealing lack of mutual synergy. The plausible reason could be that the extractives present in MA biomass may not have enhanced the degradation of structural components in SWD biomass, majorly lignin, cellulose, and hemicellulose. Furthermore, the synergistic effect of additional heating due to volatiles content is also limited as the SWD biomass contain lower volatiles when compared with other forms of biomass such as lignocellulosic biomass [33]. As a result, the activation energy of blends is marginally higher than that of MA biomass. However, a synergy can be noticed as the average activation energy of blends is significantly lower than pure SWD biomass.
Evaluation of pre-exponential factor and other thermodynamic parameters
The compensation effect was used to estimate the pre-exponential factor by using Eqs. (15) and (16), while the other thermodynamic parameters, such as activation enthalpy, activation entropy, and Gibbs free energy, were identified using Eqs. (18)–(20). Using the activation energy obtained using FWO method, pre-exponential factor can be obtained from the compensation line. Considering each fi(α) from Table 1 into Eq. 14, 15 pairs of ln Ai and Eα,i are obtained and are plotted in Fig. 5a–e. The variation of ln (Aα) and A (s−1) with respect to Eα is presented in Fig. 5f–j for MA, SWD, MD-1, MD-2, and MD-3, respectively. The values of pre-exponential factor and other thermodynamic parameters for biomasses MA, SWD, and for blends MD-1, 2, and 3 are listed in Table 6. The pre-exponential factor (A) showed variation in a wide range from 1010 to 1020 (depending on material and blending ratio) over the conversion range 0.1–0.8. This reflects the complex nature of biomasses and of the reactions that occur during the process of pyrolysis. The A values ≤ 109 s−1 indicate surface reactions, but, if the reactions are not dependent on the surface area, a low A may also indicate a closed complex (tight junctional complex), while values above 109 s−1 indicate a loose junctional complex [36, 41]. For A ranging in between 1010 and 1012 s−1, when compared to the initial reagent, the activated complex was restricted in rotation [42]. In case of unimolecular materials, the complex is further expected to interact more intensely with its neighbours by expanding its size. From Table 5, within the decomposition range of lignin component (α = 0.5–0.8), the values of A of more than 1014 s−1 indicate a slower and more difficult degradation effect and imply the need of higher molecular collision. In such case, the reaction demands more energy and this scenario is in agreement with the activation energy characteristics (Table 5).
The compensation line of Arrhenius parameters for a MA, b SWD, c MD-1, d MD-2, and e MD-3; lnAα vs. Eα dependencies and Aα vs. Eα dependencies for f MA, g SWD, h MD-1, i MD-2, and j MD-3
Table 6 Kinetic and thermodynamic parameters of thermal degradation of MA, SWD, MD-1, MD-2, and MD-3 under the heating rate (β) 15 °C min−1
From Table 6, it is evident that the change in activation enthalpy is in good agreement with the activation energy. It should be noted that, when the difference between Eα and ∆H is minimum, it indicates favorable conditions for the formation of activated complex [43]. From Table 5, when the Eα and ∆H values are compared against each other, a small energy barrier (~ 5 kJ mol−1) indicates the ease of reaction happening under the mentioned conditions. The changes in the Gibbs free energy (ΔG) indicate the increase in the total energy of the reaction system during the formation of activated complex. The positive values of ΔG indicate unfavorable conditions and requirement of excess energy as input [33]. The ΔG values were positive for all the materials and the blends considered in this study. Furthermore, the negative values of ΔS, as well as ΔG values higher than ΔH indicate that a significant amount of heat supplied to the system is unused or free. Entropy (ΔS) is commonly interpreted as the degree of disorder of the system. A small activation entropy indicates that the material is brought to a new state near its own thermal equilibrium after it has been through some kinds of physical or chemical aging phenomena. On the other hand, high values of activation entropy indicate that the material is far from its own thermal equilibrium. In the former case, the material shows little reactivity, which necessitates an increase in reaction time to form an activated complex. In the latter case, the material shows high reactivity and requires shorter reaction time to form an activated complex [42]. As can be seen in Table 6, the ΔS values for MD-1, 2, and 3 increased from − 33.9 to 98.8, − 26.1 to 88.3, and − 16.1 to 51.7 J mol−1, indicating an increase in the reactivity of the system.
Evaluation of reaction model
The generalized master plots are strictly influenced by the kinetic model used to fit the reaction but not by the heating rates. Therefore, in principle, the experimental master plots should take similar shapes for any heating rate. Using Eq. (13), experimental and theoretical master plots for different kinetic models mentioned in Table 1 were compared against each other. In addition, after identifying the pre-exponential factor (A) and apparent activation energy (Eα), the f(α) function is numerically evaluated by using (17). The f(α) functions deduced from compensation effect and generalized master plots method for the materials studied are presented in Fig. 6, MA (Fig. 6a, b), SWD (Fig. 6c, d), MD-1 (Fig. 6e, f), MD-2 (Fig. 6g, h), and MD-3 (Fig. 6i, j). Though there are a number of reaction models available (Table 1), they can be categorized into three major groups describing the rate of change of conversion with temperature; accelerating, decelerating, and sigmoidal. The f(α) deduced from compensation effect can be used to categorize the reaction model into any of these three categories, while generalized master plots methods can be used to match experimental f(α) curves to theoretically available models. For sample MA, the experimental reaction model deduced using compensation effect indicates f(α) as a monotone function decreasing continuously with conversion. Based on the progression of f(α) with respect to conversion, f(α) can be classified into order-based models. This finding is in accordance with the curve obtained using the master plots method. The experimental curve closely matches with random nucleation with three nuclei on the individual particle (F3) reaction model (Fig. 6a, b). Similar trend was witnessed for SWD biomass sample, where the f(α) generated by using compensation effect was found to decrease continuously with the extent of conversion. This type of behavior can be related to reaction-order models or diffusion models.
Reaction mechanisms for samples obtained (a), (c), (e), (g) and (i) using compensation effect (on the left) and (b), (d), (f), (h) and (j) using generalized master plots method (on the right side)
It can be understood from Fig. 6d, that the f(α) function followed three-dimensional diffusion (Jander equation) (D3) in the entire conversion range. For blends MD-1 and MD-2, the experimental curves closely matched to D3 mechanism at lower conversions (α < 0.5) and then shifted to F3 mechanism at higher conversions. The reaction model of MD-3 was similar to that of MA biomass for conversions > 0.2. However, at lower conversions (α < 0.2), the blend MD-3 was in between D3 and three-dimensional diffusion (Ginstling–Brounshtein equation) (D4) mechanisms. Intuitive reasons why the 3D nucleation and nuclei growth model provided a better fit with MA samples, while the 3D diffusion model fitted best the SWD samples, can be supported by their different compositions and structures. Lack of lignin/cellulose/hemicellulose in MA samples resulted in a faster decomposition concurrent with low temperature evolution of CO and CO2. In contrast, SWD with cellulose (potentially crystalline), hemicellulose, and lignin which are harder to degrade, would need to overcome solid diffusion barriers, resulting in a wider spread evolution of the CO and CO2 gas products towards the higher temperatures. Decarboxylation and decarbonylation would occur more readily via nuclei and nuclei growth mechanism on the easily accessed solid reagents in the absence of strong fibre components, such as cellulose, hemicellulose, and lignin. The synergy would be reflected in a larger influence of 3D nucleation and nuclei growth than expected, and lowered diffusion barriers consistent with ash catalyzed reactions.
The present study attempts to investigate the synergistic impact of microalgae biomass during its co-pyrolysis with swine manure digestate. TG and DTG profiles of plain and blended samples showed three zones of devolatilization. The volatiles and extractives in the biomass samples enhanced the kinetics of thermal decomposition of biomass blends. The mineral content of the ash in the blends enhanced their kinetics, which is evident based on gas yields and low activation energy with blends when compared to SWD biomass. If heat flow and disorder change are comprehensively evaluated, the higher activation ∆G values indicated favorability for the reactions to happen. The synergy between biomasses was evident in the gas evolution trends, as the total gas yield was noticed to increase with increase in the proportion of MA in the blends. In addition, the second evolution peaks of CO2 and CO were found to decrease with a rise in the proportion of MA.
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Maia AAD, de Morais LC. Kinetic parameters of red pepper waste as biomass to solid biofuel. Bioresour Technol. 2016;204:157–63.
Chen C, Ma X, He Y. Co-pyrolysis characteristics of microalgae Chlorella vulgaris and coal through TGA. Bioresour Technol. 2012;117:264–73.
Garcia-Maraver A, Perez-Jimenez JA, Serrano-Bernardo F, Zamorano M. Determination and comparison of combustion kinetics parameters of agricultural biomass from olive trees. Renewable Energy. 2015;83:897–904.
Shuping Z, Yulong W, Mingde Y, Chun L, Junmao T. Pyrolysis characteristics and kinetics of the marine microalgae Dunaliella tertiolecta using thermogravimetric analyzer. Bioresour Technol. 2010;101(1):359–65.
Fernandez-Lopez M, Pedrosa-Castro GJ, Valverde JL, Sanchez-Silva L. Kinetic analysis of manure pyrolysis and combustion processes. Waste Manage. 2016;58:230–40.
Turmanova SC, Genieva S, Dimitrova A, Vlaev L. Non-isothermal degradation kinetics of filled with rise husk ash polypropene composites. Express Polymer Letters. 2008;2(2):133–46.
Xu Y, Chen B. Investigation of thermodynamic parameters in the pyrolysis conversion of biomass and manure to biochars using thermogravimetric analysis. Bioresour Technol. 2013;146:485–93.
Mehmood MA, Ye G, Luo H, Liu C, Malik S, Afzal I, et al. Pyrolysis and kinetic analyses of Camel grass (Cymbopogon schoenanthus) for bioenergy. Bioresour Technol. 2017;228:18–24.
Poletto M, Zattera AJ, Santana RMC. Thermal decomposition of wood: kinetics and degradation mechanisms. Bioresour Technol. 2012;126:7–12.
Ali I, Naqvi SR, Bahadar A. Kinetic analysis of Botryococcus braunii pyrolysis using model-free and model fitting methods. Fuel. 2018;214:369–80.
Nyabuto DK, Kewei C, Mariga AM, Kibue GW, Meilin H, Changhai W. Growth performance and biochemical analysis of the genus Spirulina under different physical and chemical environmental factors. Afr J Agric Res. 2015;10(36):3614–24.
Pandey J, Tiwari A, Mishra R. Evaluation of biomass production of Spirulina maxima on different reported media. J Algal Biomass Utln. 2010;1(3):70–81.
School of Environment, Tsinghua University, Beijing, 100084, China
Arun K. Vuppaladadiyam
, Ming Zhao
, Abdul F. Soomro
& Muhammad Zaki Memon
Beijing Guohuan Tsinghua Environmental Engineering Design & Research Institute Co., Ltd., Beijing, China
Hao Liu
School of Chemical and Process Engineering, The University of Leeds, Leeds, LS2 9JT, UK
Valerie Dupont
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The manuscript was written through contributions of all authors. All authors read and approved the final manuscript.
Correspondence to Hao Liu or Ming Zhao.
Authors agreed to publish this article.
Vuppaladadiyam, A.K., Liu, H., Zhao, M. et al. Thermogravimetric and kinetic analysis to discern synergy during the co-pyrolysis of microalgae and swine manure digestate. Biotechnol Biofuels 12, 170 (2019) doi:10.1186/s13068-019-1488-6
Co-pyrolysis
Isoconversional
Thermogravimetric
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CommonCrawl
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1-SEP: Scientific Method & Design Vocabulary - Mrs. Kim (ECMHS)
kim-black-ecmhs
scientific method vocab
Terms in this set (118)
the number of times an experiment is performed; a reliable experiment consists of at least 3 trials
step-by step instructions used in a lab investigation that include how to perform the experiment, the type of data to collect, and how to collect the data
(accepted value - experimental value) / accepted value x 100
the group in an experiment whose conditions are being changed
manipulated variable
the variable that is changed or controlled by the scientist during an experiment, also known as the independent variable
responding variable
factor in an experiment that a scientist wants to observe, which may change in response to the manipulated variable; also known as a dependent variable
operational definitions
describes the actions or operations that will be used to measure or control a variable
observations and measurements recorded during an experiment
a diagram that uses vertical or horizontal bars to show comparisons among two or more separate but related items
DRY-MIX
a way to remember dependent and independent variables and how to plot them on a line graph
Dependent-Responding-Y axis
Manipulated-Independent-X axis
linear graph
a line graph in which the data points yield a straight line
non-linear graph
a graph in which the data points do not fall along a straight line
circle graph
a data display that uses pieces of a circle to show parts of a whole
A statement that sums up the experimental results or what you think the data means and how it relates to your hypothesis
This is any object or system that is specifically designed for the purpose of increasing or maintaining the capabilities of people. This technology belongs to the broader assistive-technology category.
This is a process of interpreting experimental results.
This means to observe carefully and in detail so as to identify causes, key factors, or possible results.
This type of science is more concerned with practical uses for scientific discoveries.
This type of science is concerned mainly with formulating and testing new hypotheses and theories.
any object or system that increases or maintains the capabilities of people with or without disabilities
the sum of each number in a set added together and then divided by the total number in the set
an unwanted influence on a sample
the use of biological processes, organisms, or systems to manufacture products intended to improve the quality of human life
determining the value or amount of something by mathematical or logical methods
the condition or set of conditions which bring about an effect
refers to two or more people working together towards a common goal or researching a common topic
computer model
a computer program, or network of computers, that attempts to simulate an abstract model of a particular system
control group
an experimental group, used as a baseline comparison, that receives all the control variables, but does not receive the independent variable
control variable
variables which remain constant, or unchanged, in an experiment
the process of gathering, organizing, and displaying data with the goals of finding useful information, drawing conclusions, and helping with decision making
the act of writing down the required information you get from a laboratory experiment
a compilation of information, usually quantitative, that represents measures of a variable or set of variables
the correct method of recording numerical information from an experiment in labeled rows and columns
an organized collection of data that has one or more uses
the observed variable in an experiment or study whose changes are determined by the effect of independent variables. When graphed, this variable is usually on the vertical axis.
the process of giving form to an idea, could be a plan of action or a description of a physical thing
dimensional anaylsis
a technique of problem-solving that uses the units that are part of a measurement to help solve the problem
the state or object brought about by a cause
the field that deals with applying science, mathematics and technology to develop safe and economic solutions to problems
equipment error
an error due to some type of equipment malfunction, often related to the calibration of the instrument not being done or being done incorrectly
data from an experiment, which is used to verify or reject the original hypothesis in the conclusion
scientific procedure of conducting tests to find results or to test a hypothesis
experimental variable
the condition which is changed within the experiment
extrapolation
the process of using a set of known data points on a graph to predict where an unknown data point outside of that range would be
to be unjustifiably credited with a particular accomplishment of quality; based on deceit
the image obtained by plotting dots, bars or lines to show scientific data
a type of calculator that lets you quickly visualize math; you type an equation and the calculator draws the graph for you
a prediction of the outcome of scientific processes based on analysis; an educated testable guess
also called the manipulated variable in an experiment; determines the change in the dependent variable; usually graphed on the horizontal axis
the process of making a prediction based on the results of prior knowledge, observations or similar events
the act of trying to find the answer to an unknown question by the use of experimentation and research
the process of predicting the value of an unknown data point within the range of a set of known data points
the method of researching or studying a topic, examination, inquiry
a graph that is represented by points connected by segments that shows how one variable (responding/dependent) is affected by another variable (manipulated/independent)
the science of correct reasoning; the basic components are statements that can be true or false, but never both
a measure of central tendency; the average
a collection of data made by comparing objects in standard units; metric units are used in science
the difference between the accepted value and the experimental value
a measure of central tendency; the number in the middle of a data set
the decimal system of weights and measures based on the meter (length), gram (mass), liter (volume)
the value that occurs the most frequently in a data set
this describes how science should be practiced and how scientific truths are created; the idea that scientists must try to eliminate personal bias and emotional involvement
a process of watching an experiment and noting what occurs
Occam\'s Razor
the principle that states when choosing a scientific explanation, the simplest one is to be preferred
the evaluation of scientific research, creative work, or performance by other people in the same field
how well the results of an experiment agree with the accepted value
personal error
when the observer makes inaccurate or biased observations; can be overcome by taking an average based on several measurements
an issue related to an individual or a relationship that can cause concern or conflict
physical model
a physical construction that accurately represents something that is either unseen, too small or too large to view under normal circumstances; could also represent an idea
uses pictures or symbols to represent an assigned amount of data
uses a circle divided into sections to represent percentage data
position-time graph
a line graph used to describe the motion of an object
the ability to consistently repeat a measurement
a statement that a particular event will occur, based on reasonable observations of prior events
a question that is proposed for solution, discussion or experimentation
the process of finding a solution to a problem
standards designed to ensure the safety of products; an important consideration of the design process
an equation which states that two ratios are equal
an original, full-scale, and sometimes working model of a new product or new version of an existing product
qualitiative
information from an experiment that does not have a numerical value
information from an experiment that can be assigned a number or a quantity and can often be used to construct a graph
random error
error caused by unpredictable variation in equipment readings or in an experimenter's interpretation of the readings
the difference between the smallest and the largest numbers in a data set
a way of showing the relative size of two numbers; usually expressed as a fraction
an important principle of the scientific method in which an experiment can accurately be reproduced by different scientists and with different lab equipment
the systematic investigation into and study of materials and sources in order to establish facts and reach new conclusions.
one of the first steps in scientific research; starts with how, what, when, who, which, why, where
the outcome of an experiment
the number of individuals in an experimental group in an experiment; the larger the number the more accurate the experiment
scientific law
A statement that describes what scientists expect to happen every time under a particular set of conditions
a series of steps that scientists use to answer questions and solve problems
a shorthand way of writing very large or very small numbers; expressed as a number between 1 and 10, times 10 raised to an exponent
a person hired to do research in any of the science fields
all the digits in a measurement that have been measured exactly, plus one digit whose value has been estimated
a set of requirements that gives an exact description of an object or a process; often includes measurements
systematic error
an error in measurement caused by an imperfection in the equipment, persistent mistakes made by the experimenter or some other consistent problem/issue
the use of scientific tools to help make life easier and better
A measure of how hot or cold something is or the average kinetic energy of a substance
used to describe a characteristic of a hypothesis where two variables may be related; cause and effect
a scientifc conclusion based on supporting evidence
time-series graph
a type of line graph in which the independent variable is plotted on the X-axis and the dependent variable is plotted on the Y-axis
a test or rehearsal of something new in order to evaluate the product
a standard quantity (such as a centimeter, second, square foot, or gallon) that is used to measure and describe an object; included on the X & Y axes on a line graph
the extent to which a test measures or predicts what it is supposed to
a factor that can be changed in an experiment; important when testing a hypothesis
velocity-time graph
a type of line graph that represents acceleration; the slope of the line tells us the rate of change in velocity
water displacement
method used to measure the volume of an irregularly shaped object
X-axis
in a line graph comparing two variables, the independent variable is plotted along this horizontal axis.
X-Y graph
this is a type of graph where the dependent variable is plotted vs. the independent variable as unique points.
in a line graph comparing two variables, the dependent variable is plotted along this vertical axis
An approximation of a number based on reasonable assumptions; not a guess
anomalous data
Data that do not fit with the rest of a data set.
the number of times an event occurs in a given period
similar to a bar graph but it is used to display and analyze continuous numerical data that are grouped
a point that is not part of the trend
a graph that shows data plotted at points, like a line graph
a creative process in which a team works together to solve a problem by using each other's suggestions to spark other ideas
The giving up of one benefit or advantage in order to gain another which is regarded as more favorable.
Poetry Terms (25)
gracroo18
Observations, Inferences, and Predictions
andreasummey16
Stems 19
mals5mp
Stem List 18
R_Maxey8
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In each sentence below, place a caret $(\wedge)$ to indicate where an apostrophe should be inserted. **Example 1.** Write $0_{\,\, \hat{}} \mathrm{~s}$ with a slash through them to distinguish them from the letter $O$. 7. Does the phrase "mind your $p s$ and $q s^{\prime \prime}$ have an equivalent in Finnish?
For each of the following sentences, add end marks where needed. Also, underline any errors in the use of abbreviations and write the correction above the error. **Example** 1. Will $\underline{\text{Mister}}$ (Mr.) Benson be our teacher next year? 14. What a wonderful speech Sen. Brown gave
What were the main features of the African slave trade, and what effects did it have on Africa?
How did the Inca emperor ensure that newly conquered territories were loyal to him?
Physics Exam
ChristinaStudent22
Series 66 Poor Man's Flash Cards
dlcaaron
Business Policy, Holt, Chpt 1
krstedman
Sara_Renzi4
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CommonCrawl
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A vessel containing a weak acid/water mixture has a calibrated level range of 0 to 10 feet, measured by a pressure transmitter at the bottom. Suppose that the specific gravity of the normal mixture is 1.12 (112
Now, suppose some pure acid is added to the vessel, increasing the actual liquid level {\it and} increasing the liquid density at the same time. The new liquid level is 7 feet high and the specific gravity has increased to 1.35. Assuming the transmitter has been calibrated for the original specific gravity of 1.12, what will its output signal correspond to in feet of liquid level? In other words, if this transmitter's output signal were driving a level indicator device for an operator to read, how many feet of level would the indicator device register?
Identify an alternative method for liquid level measurement that would not be affected by changes in liquid density.
If the liquid level in the vessel rises to 7 feet, the transmitter should (ideally) output a final signal corresponding to 7 feet of level. However, since the transmitter in question is a hydrostatic pressure type, and the liquid density has increased as well, it will not register accurately anymore.
The old specific gravity was 1.12, and the new specific gravity is 1.35. The error in span will be the ratio of the new specific gravity to the old, or 1.35/1.12 = 1.2054. In other words, the new acid/water mixture is 1.2054 times denser than it is supposed to be, according to how the transmitter was calibrated. Thus, the transmitter output will correspond to a liquid level of 7 feet times the error factor of 1.2054, or 8.4375 feet.
A good way to solve this problem is to apply the problem-solving technique of {\it simplification}. Imagine the two specific gravities being much simpler numbers: 1 and 2 instead of 1.12 and 1.35, respectively. It should be obvious now that a doubling of specific gravity will result in a doubling of level indication (i.e. the transmitter will register twice as much liquid level as there actually is inside the vessel). It should also be obvious that the error factor is 2:1, which is precisely the ratio of new:old specific gravity.
Alternatively, liquid level could be measured with a float, a capacitive sensor, ultrasonic or radar gauge, or some other instrument functioning on the detection of the liquid/vapor interface.
An open vessel contains water at 60$^{o}$ F. A pressure transmitter located at the bottom of the vessel measures the hydrostatic pressure (``head'') generated by the water and outputs a signal corresponding to level. Suppose that the temperature of this vessel were to increase over time to 110$^{o}$ F due to exposure to very hot outside air (the vessel is located in Death Valley, California during the summer). Knowing that an increase in water temperature will result in a decrease in density, what will happen to the level transmitter's output as the vessel heats up? Will the transmitter output increase, decrease, or stay the same? Why?? Assume that no water enters or exits the vessel during the period of heating from 60$^{o}$ F to 110$^{o}$ F.
The transmitter output will stay the same.
If you thought that the transmitter output would decrease due to the water becoming less dense, I recommend you explore the concept of density a little deeper with the following ``thought experiment:''
{\bullet} Imagine the vessel filled half-way with liquid.
{\bullet} Imagine that liquid heating up and expanding until it is only {\it half} as dense as it was at the beginning of the experiment.
{\bullet} Calculate the new hydrostatic pressure with the expanded, less-dense liquid.
In a way, this is a ``trick'' question. As any given mass of water heats up, its volume will increase. This is what makes it less dense than before: increased volume with the same mass (weight). Knowing how to calculate hydrostatic pressure from height and specific gravity, you might have approached this problem by assuming the water column height would have stayed the same while its density decreased, resulting in a lesser hydrostatic pressure and decreased transmitter output. However, this answer is incorrect.
If the water volume expands as a result of a temperature increase, the level inside the vessel {\it must rise}, because it will require a higher vessel level to contain a greater volume of water. The level will rise by the same percentage that the density decreases, resulting in a cancellation of level increase with density decrease. As a result, the hydrostatic pressure at any point in the vessel remains constant.
If you have difficulty picturing this effect, consider an exaggerated example to make things simpler. Suppose that a vessel has been filled to a height of 10 feet with cold liquid:
Now, imagine that same vessel being heated until the liquid expands to exactly {\it twice} its former volume:
The liquid level must double to accommodate twice the volume in the same vessel, assuming a vessel of constant diameter, and its density will be cut in half (twice the volume with the same mass). Consequently, the two factors of liquid level change and liquid density change cancel each other out to give the exact same hydrostatic pressure as before. In other words, the transmitter will register the same liquid level as it did when the liquid was cold, and will output the exact same signal, even though the actual vessel level is quite a bit more than it was when cold. Pretty tricky, huh?
A vessel holding some process liquid needs to have its level monitored. The range of level in this vessel is 0 to 20 feet, and the process liquid has a specific gravity of 1.0 (like water). Someone decides to attach a pressure transmitter to the bottom of the vessel to infer level from hydrostatic pressure like this:
Later, a top is added to this vessel to keep rain from entering in. Unfortunately, though, this process liquid tends to emit vapor which will be trapped by the closed vessel and create a pressure inside of it. A small vent is added to the top of the vessel to permit the vapor to escape, but it is a {\it small} vent, not big enough to ensure a total absence of vapor pressure buildup at all times:
What problem in level measurement will result from there being an occasional vapor pressure buildup inside this vessel? How may this problem be corrected so that the liquid level will be accurately measured at all times?
Any vapor pressure will be sensed by the transmitter and interpreted as increased liquid level!
Two solutions to this problem:
(1) Use two transmitters (one at top of vessel, one at bottom) and electronically subtract their output signals.
\par}
(2) Connect the ``Low'' side of the one $\Delta$P transmitter to the top of the vessel to naturally compensate for vapor pressure.
Since the transmitter infers level from the amount of pressure sensed at the bottom of the vessel, any vapor pressure buildup inside the vessel will be falsely interpreted as additional liquid level, since the transmitter senses the {\it sum total} of hydrostatic pressure plus any vapor pressure trapped inside the vessel.
For example, if the process liquid level were at 10 feet (50
In many applications, the gas pressure inside a vessel far exceeds the hydrostatic pressure generated by the column of liquid inside of it, so this problem can be a very serious one in level measurement. We need to fully understand the nature of the problem and how to solve it in order to successfully measure liquid level in many industrial applications.
The solution to this dilemma is to measure the {\it difference} in pressure between the top and bottom of the vessel. We could do this by using two pressure transmitters, one at the top and one at the bottom, and detect hydrostatic pressure by subtracting the top transmitter's measurement from the bottom transmitter's measurement. A computer may be used to perform the mathematical subtraction of signals:
Another, more elegant method, incorporates a {\it differential} pressure transmitter with pipe connections to both ends of the vessel. This solution performs the subtraction mechanically, by directly exposing the transmitter's sensing element to the {\it difference} of two applied pressures:
Because the differential pressure transmitter solution requires only one sensing instrument rather than two, and results in better accuracy because we are only dealing with the inaccuracies of a single instrument rather than the compounded inaccuracies of two instruments (three, if you include the subtraction unit), the differential, or ``d/p'' solution is the one more widely used.
In this particular level measurement application, a differential pressure instrument would have the exact same calibration points (lower and upper range-values) as a ``normal'' pressure transmitter connected to an open vessel.
The principle of buoyancy may be used to create a level transmitter instrument, generating an output signal proportional to the change in weight of a ``displacer'' rod suspended in a liquid:
Often, the displacer is housed inside its own ``cage'' for easy removal from the process, as shown above, or it may be inserted directly into the process vessel like this:
If the vessel is open and the liquid inside is turbulent due to mixing or other agitation, a {\it stilling well} serves the same purpose as a cage:
To calibrate such an instrument, it must be isolated from the process liquid. Sometimes this means simply closing block valves and draining the cage. Other times it means removing the displacer mechanism from the vessel entirely. But once the displacer is hanging dry, there is the problem of simulating a 100
Describe a way to make the displacer ``think'' it is fully submerged in process liquid when it in fact is hanging freely in the air. Explain how you would be able to {\it precisely} and {\it accurately} simulate this condition, as well as any given condition of partial displacer submersion for that matter.
One way to simulate a full condition is to use a {\it hand scale}:
The following storage vessel holds water. The hydrostatic-pressure level transmitter is located 5 feet below the bottom of the vessel, and the desired level measurement range is 8 feet to 12 feet:
Assuming a pneumatic transmitter with an output range of 3 PSI to 15 PSI, and a calibration accuracy of +/- 1
$$\begin{array} {|l|l|} \hline Process & Percent of & $\Delta$ pressure & Output signal & Output signal & Output signal \\ \hline level (ft) & span ( \\ \hline \ & 0 & & & & \\ \hline & 10 & & & & \\ \hline & 25 & & & & \\ \hline & 50 & & & & \\ \hline & 75 & & & & \\ \hline & 90 & & & & \\ \hline & 100 & & & & \\ \hline \end{array}$$
$$\begin{array} {|l|l|} \hline Process & Percent of & $\Delta$ pressure & Output signal & Output signal & Output signal \\ \hline level (ft) & span ( \\ \hline \8 & 0 & 156 & 3 & 2.88 & 3.12 \\ \hline 8.4 & 10 & 160.8 & 4.2 & 4.08 & 4.32 \\ \hline 9 & 25 & 168 & 6 & 5.88 & 6.12 \\ \hline 10 & 50 & 180 & 9 & 8.88 & 9.12 \\ \hline 11 & 75 & 192 & 12 & 11.88 & 12.12 \\ \hline 11.6 & 90 & 199.2 & 13.8 & 13.68 & 13.92 \\ \hline 12 & 100 & 204 & 15 & 14.88 & 15.12 \\ \hline \end{array}$$
Determine the LRV and URV settings for the water seal drum lever transmitter (LT-21), assuming the LRV point is at the lower nozzle and the URV point is at the upper nozzle (the two nozzles being 3 feet 8 inches apart from each other), and that the remote seal fill fluid has a specific gravity of 0.934:
The elevation for this transmitter (i.e. the total differential pressure applied by the height of fill fluid on both sides) is equal to the total height difference between the remote seal diaphragms multiplied by the specific gravity of the fill fluid:
$$P_{elevation} = (44 \hbox{ in})(0.934) = 41.1 \hbox{ "WC}$$
In the LRV condition, this is the only pressure seen by the transmitter. Therefore, 41.1 "WC is the appropriate LRV setting for this transmitter. If we assume that the ``H'' port of this DP transmitter connects to the lower nozzle, the LRV will be -44.1 "WC. If we assume the ``H'' port connects to the upper nozzle, the LRV will be +41.4 "WC.
In the URV condition, we have the exact same amount of elevation (the fill fluid inside the capillary tubes) but on the lower nozzle we have the hydrostatic pressure of 44 vertical inches of water (i.e. the water inside the seal drum). Thus, in the URV condition the transmitter sees a differential pressure of:
$$P_{differential} = 44 \hbox{" WC} - 41.1 \hbox{ "WC} = 2.9 \hbox{ "WC}$$
If we assume the ``H'' port of this DP transmitter connects to the lower nozzle, the URV will be +2.9 "WC. If we assume the ``H'' port of this DP transmitter connects to the upper nozzle, the URV will be -2.9 "WC.
This question is a good candidate for a ``Virtual Troubleshooting'' exercise. Presenting the diagram to students, you first imagine in your own mind a particular fault in the system. Then, you present one or more symptoms of that fault (something noticeable by an operator or other user of the system). Students then propose various diagnostic tests to perform on this system to identify the nature and location of the fault, as though they were technicians trying to troubleshoot the problem. Your job is to tell them what the result(s) would be for each of the proposed diagnostic tests, documenting those results where all the students can see.
During and after the exercise, it is good to ask students follow-up questions such as:
{\bullet} What does the result of the last diagnostic test tell you about the fault?
{\bullet} Suppose the results of the last diagnostic test were different. What then would that result tell you about the fault?
{\bullet} Is the last diagnostic test the best one we could do?
{\bullet} What would be the ideal order of tests, to diagnose the problem in as few steps as possible?
Calculate the hydrostatic pressure generated at the bottom of this vessel (in units of PSI) when it is completely filled with water:
Now calculate the hydrostatic pressure at the bottom of this vessel (in units of PSI) when it is completely filled with gasoline (density = 42 lb/ft$^{3}$):
What do you think the pressure will be at the bottom of the vessel if it is exactly half-full of gasoline and half-full of water, with a gasoline-water {\it interface} at the 5.5 foot mark? Explain your reasoning.
Hydrostatic pressure when completely full of water = 4.769 PSI
Hydrostatic pressure when completely full of gasoline = 3.208 PSI
Hydrostatic pressure when water-gasoline interface is at the 50
{\bf Summary Quiz:}
Calculate the hydrostatic pressure at the bottom of this open vessel, holding 7 feet of gasoline over 6 feet of water:
{\bullet} 190.9 inches W.C.
Calculate the differential pressure sensed by the level transmitter at three different water levels in this boiler steam-drum level measurement system: 0
Assume a density for (hot) boiler drum water of 36 lb/ft$^{3}$, a density for steam in the drum of 7 lb/ft$^{3}$, and a density for (warm) water in the ``wet leg'' of 61.8 lb/ft$^{3}$. If the pressure at the ``low'' (L) side of the transmitter is greater than the pressure at the ``high'' (H) side, be sure to express the differential pressure quantity as a negative number.
Credit will be given for correctly calculating each of the differential pressures:
{\bullet} {\bf (6 points)} Transmitter $\Delta$P at 0
{\bullet} {\bf (6 points)} Transmitter $\Delta$P at 50
{\bullet} {\bf (6 points)} Transmitter $\Delta$P at 100
A displacer-type density transmitter registers a displacer weight of 6.3 pounds with the cage completely full of sample liquid. The displacer has a dry weight of 14 pounds, a density of 120 lb/ft$^{3}$, and is cylindrical in shape. Calculate the density of the liquid in units of pounds per cubic foot.
The liquid has a density of 66 pounds per cubic foot.
Determine a basic 5-point (0
The cylindrical displacer weighs 15 pounds (dry) and has a diameter of 2 inches. The process liquid is water. The 0
$$\begin{array} {|l|l|} \hline Process & Percent of & Buoyant & Output signal \\ \hline level (in) & span ( \\ \hline \ & 0 & & \\ \hline & 25 & & \\ \hline & 50 & & \\ \hline & 75 & & \\ \hline & 100 & & \\ \hline \end{array}$$
$$\begin{array} {|l|l|} \hline Process & Percent of & Buoyant & Output signal \\ \hline level (in) & span ( \\ \hline \ 0 & 0 & 0 & 4 \\ \hline 10 & 25 & 1.135 & 8 \\ \hline 20 & 50 & 2.270 & 12 \\ \hline 30 & 75 & 3.405 & 16 \\ \hline 40 & 100 & 4.540 & 20 \\ \hline \end{array}$$
The cylindrical displacer weighs 9 pounds (dry) and has a diameter of 2 inches. The process liquid is water. The 0
$$\begin{array} {|l|l|} \hline Process & Percent of & Buoyant & Output signal \\ \hline level (in) & span ( \\ \hline \ 0 & 0 & 0 & 4 \\ \hline 5 & 25 & 0.567 & 8 \\ \hline 10 & 50 & 1.135 & 12 \\ \hline 15 & 75 & 1.702 & 16 \\ \hline 20 & 100 & 2.270 & 20 \\ \hline \end{array}$$
An Overview of Sequential Function Chart (SFC) Programming
Understanding Series Circuits from an Industrial Perspective
Phoenix Contact Offers New High-Voltage DC/DC Converter
New Actuator Control Panel Designed for Hazardous Conditions
by Jacob Ramirez
An Introduction to Parallel Circuits for Industrial Engineers
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CommonCrawl
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Advances in Continuous and Discrete Models
Theory and Modern Applications
On some wavelet solutions of singular differential equations arising in the modeling of chemical and biochemical phenomena
Mo Faheem1,
Arshad Khan1 &
E. R. El-Zahar2,3
Advances in Difference Equations volume 2020, Article number: 526 (2020) Cite this article
This paper is concerned with the Lane–Emden boundary value problems arising in many real-life problems. Here, we discuss two numerical schemes based on Jacobi and Bernoulli wavelets for the solution of the governing equation of electrohydrodynamic flow in a circular cylindrical conduit, nonlinear heat conduction model in the human head, and non-isothermal reaction–diffusion model equations in a spherical catalyst and a spherical biocatalyst. These methods convert each problem into a system of nonlinear algebraic equations, and on solving them by Newton's method, we get the approximate analytical solution. We also provide the error bounds of our schemes. Furthermore, we also compare our results with the results in the literature. Numerical experiments show the accuracy and reliability of the proposed methods.
The solution of Emden–Fowler type equation is vital because of its numerous applications in engineering and technical problems. There are several phenomena like astrophysics, aerodynamics, stellar structure, chemistry, biochemistry, and many others (see [19, 38, 40, 41]) which can be modeled by the Lane–Emden equation of shape operator w given by [18]
$$\begin{aligned} u''(z)+\frac{\mathtt{w}}{z}u'(z)+f(u,z)=0, \quad \mathtt{w} >0. \end{aligned}$$
A number of research papers are inclined toward the numerical solution of such type of differential equations. The numerical methods for the solution of Lane–Emden equation based on B-spline have been studied in [24, 30–32]. Homotopy analysis methods and iterative schemes for fast convergence and accuracy of solutions of singular and doubly singular BVPs have been developed in [21, 22, 26, 33]. Roul et al. have dealt with the solution of a class of two-point nonlinear singular boundary value problems with Neumann and Robin boundary conditions by deploying a high order compact finite difference method [25]. A least square recursive approach together with convergence analysis for solving Lane–Emden type initial value problems has been developed in [27], in which they simply reduce the solution of the original initial value problem to the solution of an integral equation. The B-spline method fails to provide a satisfactory approximation in the presence of singularity; on the other hand, the Adomian decomposition methods (ADM) fail to establish a convergent series solution to strongly nonlinear BVPs. To overcome these shortcomings, Roul came up with the combination of ADM and B-spline collocation methods for accurate solution, see [23]. Madduri and Roul developed a fast converging iterative scheme for the solution of a system of Lane–Emden equations converting them into equivalent Fredholm integral equations and treating them with homotopy analysis method [14]. In this paper, we discuss and solve some mathematical models of the chemical and biochemical phenomena using wavelet methods.
Model of electrohydrodynamic (EHD) flow in a circular cylindrical conduit
The effect of the electric and magnetic field on fluid has been studied by many researchers. Phenomena involving the conversion of electrical and magnetic energy into kinetic energy are known as electrohydrodynamics (EHD) and magnetohydrodynamics (MHD). The effect of the electric field on fluids gives extra means of controlling flow conditions and has various technical applications such as EHD thruster, EHD flow, heat transfer enhancement, EHD drying and evaporation, and functional electrostatic bowler (EHD pump). EHD pump has been designed for semiconductor cooling [5], electrospray mass spectrometry, and electrospray nanotechnology [45]. The MHD flow has a wide range of applications in the fields of chemistry and biology, for instance, the fabrication in cancer tumor therapy resulting hypothermia, decreasing bleeding in the state of acute injuries, magnetic resonance visualizing, and various other diagnostic experiments [3]. Magneto-hybrid nanofluids flow via mixed convection past a radiative circular cylinder was studied in [4]. The EHD flow of a fluid is modeled by a set of partial differential equations, which can be reduced to an ordinary differential equation as in [16], and results in the following Emden–Fowler type of equation:
$$\begin{aligned} u''(z)+\frac{1}{z}u'(z)+H^{2} \biggl(1-\frac{u}{1-\alpha u} \biggr)=0, \end{aligned}$$
subject to the boundary conditions
$$\begin{aligned} u'(0)=0,\qquad u(1)=0, \end{aligned}$$
where \(u=-\frac{\overline{u}}{KE_{0}\alpha }\), \(\alpha = \frac{K}{j_{0}}\frac{\partial p}{\partial z}-1\).
Here, the pressure gradient \(\frac{\partial p}{\partial z}\) is a constant that measures the nonlinearity and \(H=\sqrt{\frac{j_{0}a^{2}}{\mu K^{2}E_{0}}}\) is the Hartmann number [16]. A schematic diagram of EHD flow is given in Fig. 1.
Schematic diagram of EHD flow in a circular cylindrical conduit
Equation (1.2) is a strong nonlinear differential equation having a singularity at \(z=0\). Finding the exact solution to this problem is quite complicated, and therefore the development and use of numerical techniques for the solution of this problem play an important role. Only few numerical methods are available for the solution of (1.2). For instance, Mastroberardino developed homotopy analysis method [15], Ghasemi et al. used least square method [6], Mosayebidorcheh applied Taylor series [17], and Roul et al. gave a new iterative algorithm [28] for the solution of strongly nonlinear singular boundary value problems.
Nonlinear heat conduction model in human head
Biomechanics is the area of science in which mechanics laws and formulae are used to study the behavior of the human body. The heat flow in the human body is quivering and vital field that helps to analyze the human heat stress at various temperatures. The human head is the only organ in the human body that controls different parts and functions in the body. The authors in [37] and [13] studied the effect of digital mobile phone emission on the human brain and concluded that the cellular phone waves can cause several brain problems, like exciting the brain cell, weakening the neural behavior, and possible disruption in the functionality of the nervous system. Ketley [11] points out the neuropsychological squeal of digital mobile phone exposure in humans. Similarly, the thermal effect of wave and radiation from digital phones on the human nervous system and brain is studied in [7, 12, 39, 44].
The following Emden-type equation is used to model the distribution of heat source in the human head [2]:
$$\begin{aligned} u''(z)+\frac{2}{z}u'(z)+ \frac{p(u)}{\gamma }=0, \quad 0< z< 1, \end{aligned}$$
$$\begin{aligned} u'(0)=0, \qquad -\nu u'(1)=\mu (u-u_{k}), \end{aligned}$$
where \(p(u)\) is the heat production rate per unit volume, u is the absolute temperature, z is the radial distance from the center. Figure 2 shows the schematic diagram of human heat conduction model.
Schematic diagram of human heat conduction model
Many researchers have shown their interest in solving this model numerically. For example, Wessapan et al. [43] derived a numerical algorithm of specific absorption rate and heat transfer in the human body to leakage electromagnetic field. Keangin et al. [9] gave an analysis of heat transfer in liver tissue during microwave ablation using single and two double slot antennae. Wessapan and Rattanadecho [42] used a three-dimensional human head model for simulating the heat distribution by applying 3-D finite element mesh (see Fig. 3).
Human head exposed to mobile phone radiation [42]
Mathematical model of spherical catalyst equation
The following Lane–Emden equation is used to model the dimensionless concentration of chemical species which occur in a spherical catalyst [19]:
$$\begin{aligned} u''(z)+\frac{2}{z}u'(z)-{\rho }^{2}u(z)e^{ ( \frac{\sigma \beta (1-u(z))}{1+\beta (1-u(z))} )}=0, \end{aligned}$$
$$\begin{aligned} u'(0)=0, \quad u(1)=1, \end{aligned}$$
where \({\rho }^{2}\), σ, and β denote the Thiele modulus, dimensionless activation energy, and dimensionless heat of reaction, respectively, and are given by
$$\begin{aligned}& {\rho }^{2}=\frac{\kappa _{ref}\mathtt{R}^{2}}{\mathtt{D}},\qquad \sigma =\frac{\mathtt{E}}{\mathtt{R_{g}}\mathtt{T_{s}}}, \qquad \beta = \frac{(-\Delta \mathtt{H})\mathtt{D}\mathtt{C}_{\mathtt{As}}}{\mathtt{KT_{s}}},\qquad z=\mathtt{\frac{R}{r}},\quad \text{and}\quad u=\mathtt{\frac{C_{A}}{C_{As}}}\quad \text{[19]}. \end{aligned}$$
The effectiveness factor of spherical pellet is defined as [34]
$$\eta =\frac{3}{{\rho }^{2}}u'(z)\quad \text{at }z=1. $$
Mathematical model of spherical biocatalyst equation
The following Lane–Emden equation is used for modeling the spherical biocatalyst equation [34]:
$$\begin{aligned} u''(z)+\frac{2}{z}u'(z)-{\rho }^{2} \frac{(1+\beta )u(z)}{1+\beta u(z)}=0, \end{aligned}$$
$$\begin{aligned} u'(0)=0, \qquad u(1)=1, \end{aligned}$$
$$\begin{aligned}& {\rho }^{2}= \frac{-\mathtt{r_{As}}\mathtt{R}^{2}}{\mathtt{DD_{As}}},\qquad \beta =\frac{\mathtt{C_{As}}}{\mathtt{K_{m}}}, \qquad z=\mathtt{\frac{R}{r}},\quad \text{and}\quad u=\frac{\mathtt{C_{A}}}{\mathtt{C_{As}}} \quad \text{[34]}. \end{aligned}$$
$$\begin{aligned}& \eta =\frac{3}{{\rho }^{2}}u'(z)\quad \text{at }z=1. \end{aligned}$$
The schematic diagram of spherical biocatalyst is shown in Fig. 4.
Schematic diagram of spherical biocatalyst
Several numerical techniques have been adopted for solving non-isothermal reaction–diffusion model equations. For instance, Singh [34] applied optimal homotopy analysis method, and Jamal and Khuri [8] used Green's function and fixed point iteration approach for solving such type of equations. Rach et al. [19] reduced this model equation into an equivalent Volterra integral equation and then solved it by coupling the modified Adomian decomposition method and the Volterra integral technique.
Jacobi wavelet is the family of wavelets reduced into Legendre wavelet, Chebyshev wavelet, and Gegenbauer wavelet for the specific value of κ and ω. There are a lot of research papers available for the solution of ordinary and partial differential equations using Jacobi and Bernoulli wavelets, for instance, see [1, 10, 20, 46]. In this study, we introduce two methods based on Jacobi and Bernoulli wavelets for solving models of electrohydrodynamic flow in a circular cylindrical conduit, nonlinear heat conduction model in the human head, spherical catalyst equation, and spherical biocatalyst equation. These wavelets transform these model equations into a system of nonlinear algebraic equations, and on solving them, we get the unknown wavelet coefficients. With the help of these coefficients, we get the approximate analytical solution that is valid over all the problem domain, not only at grid points. The outline of this paper is as follows: The second section describes the Jacobi wavelet, function approximation by Jacobi wavelet, and integration of Jacobi wavelet. Similarly, the third section describes the Bernoulli wavelet, function approximation by Bernoulli wavelet, and integration of Bernoulli wavelet. In the fourth section, the wavelet approximation method for all the above models is given. In the fifth section, we state some theoretical proof for error bounds of our methods. In the sixth section, the numerical experiments confirm that our methods converge fast.
Jacobi wavelet
Jacobi polynomials
Jacobi polynomials, which are often called hypergeometric polynomials, are denoted by \(\mathtt{J}_{m}^{\kappa ,\omega }(z)\) and can be defined by the following explicit formula:
$$\begin{aligned}& \mathtt{J}_{m}^{\kappa ,\omega }(z)= \frac{\Gamma (\kappa +m+1)}{m!\Gamma (\kappa +\omega +m+1)}\sum _{i=0}^{m} \binom{m}{i} \frac{\Gamma (\kappa +\omega +i+m+1)}{\Gamma (\kappa +i+1)} \biggl( \frac{z-1}{2} \biggr)^{i}. \end{aligned}$$
Some first few Jacobi polynomials are given by
$$\begin{aligned}& \mathtt{J}_{0}^{\kappa ,\omega }(z)=1, \\& \mathtt{J}_{1}^{\kappa ,\omega }(z)=\kappa +1+(\kappa +\omega +2) \frac{z-1}{2}, \\& \begin{aligned} \mathtt{J}_{2}^{\kappa ,\omega }(z)&=\frac{(\kappa +1)(\kappa +2)}{2}+( \kappa +2) ( \kappa +\omega +3)\frac{z-1}{2}\\ &\quad {}+ \frac{(\kappa +\omega +3)(\kappa +\omega +4)}{2} \biggl(\frac{z-1}{2} \biggr)^{2},\qquad \ldots. \end{aligned} \end{aligned}$$
These polynomials are orthogonal on [−1, 1] with respect to the weight \((1-z)^{\kappa }(1-z)^{\omega }\) and satisfy the following properties:
$$\begin{aligned}& \mathtt{J}_{m}^{\kappa ,\omega }(-1)=(-1)^{m}\binom{m+\omega }{m}, \\& \mathtt{J}_{m}^{\kappa ,\omega }(-z)=(-1)^{m}\mathtt{J}_{m}^{\omega ,\kappa }(z), \\& \mathtt{J}_{m}^{\kappa ,\kappa }(z)= \textstyle\begin{cases} \frac{\Gamma (m+\kappa +1)\Gamma (\frac{m}{2}+1)}{\Gamma (\frac{m}{2}+\kappa +1)\Gamma (m+1)} \mathtt{J}_{\frac{m}{2}}^{\kappa ,\frac{-1}{2}}(2z^{2}-1), & \text{if } m \text{ is even}, \\ \frac{\Gamma (m+\kappa +2)\Gamma (\frac{m}{2}+1)}{\Gamma (\frac{m}{2}+\kappa +1)\Gamma (m+2)}z \mathtt{J}_{\frac{m}{2}}^{\kappa ,\frac{1}{2}}(2z^{2}-1), & \text{if } m \text{ is odd}. \end{cases}\displaystyle \\& \int _{-1}^{1}(1-z)^{\kappa }(1+z)^{\omega } \mathtt{J}_{m}^{\kappa , \omega }(z)\mathtt{J}_{n}^{\kappa ,\omega }(z)\,dz= \frac{2^{\kappa +\omega +1}}{2m+\kappa +\omega +1} \frac{\Gamma (m+\kappa +1)\Gamma (m+\omega +1)}{\Gamma (m+\kappa +\omega +1)m!} \delta _{nm}, \end{aligned}$$
where \(\delta _{nm}\) is Kronecker delta.
$$\begin{aligned} &2m(m+\kappa +\omega ) (2m+\kappa +\omega -2)J_{m}^{\kappa ,\omega }(z)\\ &\quad =(2m+ \kappa +\omega -1) \bigl((2m+\kappa +\omega ) (2m+\kappa +\omega -2)z \\ &\qquad {}+{\kappa }^{2}-{\omega }^{2}\bigr)\mathtt{J}_{m-1}^{\kappa ,\omega }(z)-2(m+ \alpha -1) (m+\omega -1) (2m+\kappa +\omega )\mathtt{J}_{m-2}^{\kappa , \omega }(z). \end{aligned}$$
Jacobi wavelet of shifted Jacobi polynomial
Jacobi wavelet of the shifted Jacobi polynomial defined on six arguments k, n, κ, ω, m, z is denoted by \(\mathscr{J}(k,n,\kappa ,\omega ,m,z)=\mathscr{J}_{n,m}^{\kappa , \omega }(z)\), and can be defined on [0, 1) as follows [1]:
$$\begin{aligned} \mathscr{J}_{n,m}^{\kappa ,\omega }(z)= \textstyle\begin{cases} 2^{\frac{k}{2}}\mu _{m}^{\kappa ,\omega }\mathtt{J}_{m}^{\kappa , \omega }(2^{k}z-2n+1), & \text{if } z \in [\xi _{1}, \xi _{2}), \\ 0, & \text{otherwise}, \end{cases}\displaystyle \end{aligned}$$
where \(\xi _{1}=\frac{n-1}{2^{k-1}}\), \(\xi _{2}=\frac{n}{2^{k-1}}\), and \(\mu _{m}^{\kappa ,\omega }=\sqrt{ \frac{(2m+\kappa +\omega +1)\Gamma (2m+\kappa +\omega +1)m!}{2^{\kappa +\omega +1}\Gamma (m+\kappa +1)\Gamma (m+\omega +1)}}\).
Equivalently, for any positive integer k, Jacobi wavelet can also be defined as follows:
$$\begin{aligned} \mathscr{J}_{i}^{\kappa ,\omega }(z)= \textstyle\begin{cases} 2^{\frac{k}{2}}\mu _{m}^{\kappa ,\omega }\mathtt{J}_{m}^{\kappa , \omega }(2^{k}z-2n+1), & \text{if } z \in [\xi _{1}, \xi _{2}), \\ 0, & \text{otherwise}, \end{cases}\displaystyle \end{aligned}$$
where i is wavelet number determined by \(i=n+2^{k-1}m\), where \(n=0,1,2,\ldots \) and \(m=0,1,2,\ldots,M-1\), where m is degree of polynomial. M can be determined by \(M=\frac{N}{2^{k-1}}\), where \(k=1,2,\ldots \) .
Function approximation by Jacobi wavelet
Let \(\{\mathscr{J}_{1,0}^{\kappa ,\omega },\ldots,\mathscr{J}_{1,M-1}^{ \kappa ,\omega },\mathscr{J}_{2,0}^{\kappa ,\omega },\ldots,\mathscr{J}_{2,M-1}^{ \kappa ,\omega },\mathscr{J}_{2^{k-1},0}^{\kappa ,\omega },\ldots, \mathscr{J}_{2^{k-1},M-1}^{\kappa ,\omega }\}\) be a set of Jacobi wavelets.
Any function \({f}(z) \in L^{2}[0,1)\) can be expressed in terms of Jacobi wavelet as follows [1]:
$$\begin{aligned}& f(z) = \sum_{n=1}^{\infty }\sum _{m=0}^{\infty }a_{n,m}\mathscr{J}_{n,m}^{ \kappa ,\omega }(z) = \sum_{i=1}^{\infty }a_{i} \mathscr{J}_{i}^{ \kappa ,\omega }(z). \end{aligned}$$
For approximation, we truncate this series for a natural number N, and we get
$$\begin{aligned} f(z) \approx &\sum_{n=1}^{2^{k-1}}\sum _{m=0}^{M-1}a_{n,m}\mathscr{J}_{n,m}^{ \kappa ,\omega }(z)= \sum_{i=1}^{N}a_{i} \mathscr{J}_{i}^{\kappa , \omega }(z) \end{aligned}$$
$$\begin{aligned} =&a^{T}\mathscr{J}(z), \end{aligned}$$
where a and \(\mathscr{J}(z)\) are matrices of order \(N\times 1\) given by
$$\begin{aligned}& \begin{aligned}[b] a&=[a_{1,0},a_{1,1},\ldots, a_{1,M-1},a_{2,0},a_{2,1},\ldots,a_{2,M-1}, \ldots,a_{2^{k-1},0},\ldots,a_{2^{k-1},M-1}]^{T} \\ &= [a_{1},a_{2},\ldots,a_{N}]^{T}, \end{aligned} \end{aligned}$$
$$\begin{aligned}& \begin{aligned}[b] \mathscr{J}(z) &=\bigl[\mathscr{J}_{1,0}^{\kappa ,\omega }(z), \ldots, \mathscr{J}_{1,M-1}^{\kappa ,\omega }(z),\mathscr{J}_{2,0}^{\kappa , \omega }(z), \ldots,\mathscr{J}_{2,M-1}^{\kappa ,\omega }(z),\mathscr{J}_{2^{k-1},0}^{ \kappa ,\omega }(z), \ldots,\mathscr{J}_{2^{k-1},M-1}^{\kappa ,\omega }(z)\bigr] \\ &= \bigl[\mathscr{J}_{1}^{\kappa ,\omega }(z),\ldots, \mathscr{J}_{N}^{\kappa , \omega }(z)\bigr],\end{aligned} \end{aligned}$$
where the coefficient \(a_{i}\) can be determined by \(a_{i} = \langle f(z),\mathscr{J}_{i}^{\kappa ,\omega }(z)\rangle = \int _{0}^{1}f(z)\overline{\mathscr{J}_{i}^{\kappa ,\omega }(z)}\,dz\).
Integration of Jacobi wavelet
Let \(\mathscr{J}^{1}_{i}(z)\), \(\mathscr{J}^{2}_{i}(z)\), and \(\mathscr{J}^{3}_{i}(z)\) be the first, second, and third integration of Jacobi wavelet from 0 to z respectively. These integrations can be determined as follows:
$$\begin{aligned}& \mathscr{J}_{1,i}^{\kappa ,\omega }(z) = \textstyle\begin{cases} 2^{\frac{-k}{2}}\mu _{m}^{\kappa ,\omega } ( \frac{1}{(m+\kappa +\omega )} ) \{{\mathtt{J}}_{m+1}^{\kappa -1, \omega -1}(\hat{z})-{\mathtt{J}}_{m+1}^{\kappa -1,\omega -1}(-1) \}, & \xi _{1}\leq z < \xi _{2}, \\ 2^{\frac{-k}{2}}\mu _{m}^{\kappa ,\omega } ( \frac{1}{(m+\kappa +\omega )} ) \{{\mathtt{J}}_{m+1}^{\kappa -1, \omega -1}(1)-{\mathtt{J}}_{m+1}^{\kappa -1,\omega -1}(-1) \}, & \xi _{2}\leq z \leq 1, \end{cases}\displaystyle \end{aligned}$$
$$\begin{aligned}& \mathscr{J}_{2,i}^{\kappa ,\omega }(z) = \textstyle\begin{cases} 2^{\frac{-3k}{2}}\mu _{m}^{\kappa ,\omega } ( \frac{1}{(m+\kappa +\omega )} ) \{ ( \frac{1}{(m-2+\kappa +\omega )} ) \{{\mathtt{J}}_{m+2}^{\kappa -2, \omega -2}(\hat{z}) \\ \quad {}-{\mathtt{J}}_{m+2}^{\kappa -2,\omega -2}(-1) \}-(1+\hat{z}){ \mathtt{J}}_{m+1}^{\kappa -1,\omega -1}(-1) \}, & \xi _{1}\leq z < \xi _{2}, \\ 2^{\frac{-3k}{2}}\mu _{m}^{\kappa ,\omega } ( \frac{1}{(m+\kappa +\omega )} ) \{ ( \frac{1}{(m-2+\kappa +\omega )} )\{{\mathtt{J}}_{m+2}^{\kappa -2, \omega -2}(1) \\ \quad {}-{\mathtt{ J}}_{m+2}^{\kappa -2,\omega -2}(-1)\}-2{\mathtt{J}}_{m+1}^{ \kappa -1,\omega -1}(-1) \\ \quad {}+(\hat{z}-1)\{{\mathtt{J}}_{m+1}^{\kappa -1,\omega -1}(1)-{\mathtt{J}}_{m+1}^{ \kappa -1,\omega -1}(-1)\} \}, & \xi _{2}\leq z \leq 1, \end{cases}\displaystyle \end{aligned}$$
where \(\hat{z}=2^{k}z-2n+1\).
Bernoulli wavelet
Bernoulli polynomials
Bernoulli polynomials are denoted by \(\mbox{\ss} _{m}(z)\), where m is the degree of polynomials and can be defined by the following explicit formula:
$$\begin{aligned}& \mbox{\ss} _{m}(z)=\sum_{i=0}^{m} \binom{m}{i} \mbox{\ss} _{m-i}z^{i}, \end{aligned}$$
where \(\mbox{\ss} _{k}, k=0,1,2,\ldots m\), are the Bernoulli numbers. Another explicit formula for these polynomials is given by
$$\begin{aligned}& \mbox{\ss} _{m}(z)=\sum_{i=0}^{m} \frac{1}{i+1}\sum_{j=0}^{i}(-1)^{j} \binom{i}{j}(z+j)^{m}. \end{aligned}$$
The first few Bernoulli polynomials are given by
$$\begin{aligned}& \mbox{\ss} _{0}(z)=1,\qquad \mbox{\ss} _{1}(z)=z-\frac{1}{2}, \qquad \mbox{\ss} _{2}(z)= z^{2}-z+ \frac{1}{6},\qquad \mbox{\ss} _{3}(z)=z^{3}-\frac{3}{2}z^{2}+ \frac{1}{2}z. \end{aligned}$$
Bernoulli polynomial satisfies the following properties:
$$\begin{aligned}& \mbox{\ss} _{m}(1)=(-1)^{m} \mbox{\ss} _{m}(0),\\& \mbox{\ss} _{2m+1}(0)=0,\qquad \mbox{\ss} _{2m-1}\biggl(\frac{1}{2} \biggr)=0,\\& \mbox{\ss} _{m}(1-z)=(-1)^{m} \mbox{\ss} _{m}(z),\\& \mbox{\ss} _{m}(z+1)- \mbox{\ss} _{m}(z)=mz^{m-1},\\& \int _{0}^{1} \mbox{\ss} _{n}(z) \mbox{\ss} _{m}(z)\,dz=(-1)^{n-1}\frac{n!m!}{(m+n)!} \mbox{\ss} _{m+n}. \end{aligned}$$
Bernoulli polynomials can be calculated by the following recursive formula: \(\mbox{\ss} _{m}'(z)=m \mbox{\ss} _{m-1}(z)\).
Bernoulli wavelet defined on four arguments k, n, m, z is denoted by \(\mathscr{B}(k,n,m,z)=\mathscr{B}_{n,m}(z)\) and can be defined on [0,1) as follows [10]:
$$\begin{aligned} \mathscr{B}_{n,m}(z) = \textstyle\begin{cases} 2^{\frac{k-1}{2}}\overline{ \mbox{\ss} }_{m}(2^{k-1}z-n+1), & \xi _{1}\leq z \leq \xi _{2}, \\ 0, & \text{elsewhere}, \end{cases}\displaystyle \end{aligned}$$
where \(\xi _{1}=\frac{n-1}{2^{k-1}}\) and \(\xi _{2}=\frac{n}{2^{k-1}}\) and
$$\begin{aligned} \overline{ \mbox{\ss} _{m}}(z) = \textstyle\begin{cases} 1, & m = 0, \\ \frac{1}{\sqrt{\frac{(-1)^{m-1}(m!)^{2}}{(2m)!} \mbox{\ss} _{2m}}} \mbox{\ss} _{m}(z), & m > 0, \end{cases}\displaystyle \end{aligned}$$
where \(\mbox{\ss} _{2m}\) is the Bernoulli number.
On the interval \([0, 1)\), for any positive integer k, Bernoulli wavelet can also be defined as follows:
$$\begin{aligned} \mathscr{B}_{i}(z) = \textstyle\begin{cases} 2^{\frac{k-1}{2}}\overline{ \mbox{\ss} }_{m}(2^{k-1}z-n+1), & \xi _{1}\leq z \leq \xi _{2}, \\ 0, & \text{elsewhere}. \end{cases}\displaystyle \end{aligned}$$
Here, i is wavelet number and can calculated by the relation \(i = n+2^{k-1}m\), where \(n=0,1,2,\ldots \) and \(m=0,1,2,\ldots,M-1\), m is degree of polynomials. For \(k=1,2,\ldots \) , M can be found by \(N=2^{k-1}M\).
Function approximation by Bernoulli wavelet
Let \(\{\mathscr{B}_{1,0},\ldots,\mathscr{B}_{1,M-1},\mathscr{B}_{2,0},\ldots, \mathscr{B}_{2,M-1},\mathscr{B}_{2^{k-1},0},\ldots,\mathscr{B}_{2^{k-1},M-1} \}\) be a set of Bernoulli wavelets.
Any function \({f}(z) \in L^{2}[0,1)\) can be expressed in terms of Bernoulli wavelet as follows [46]:
$$\begin{aligned}& f(z) = \sum_{n=1}^{\infty }\sum _{m=0}^{\infty }b_{n,m}\mathscr{B}_{n,m}(z) = \sum_{i=1}^{\infty }b_{i} \mathscr{B}_{i}(z). \end{aligned}$$
$$\begin{aligned} f(z) \approx& \sum_{n=1}^{2^{k-1}}\sum _{m=0}^{M-1}b_{n,m}\mathscr{B}_{n,m}(z)= \sum_{i=1}^{N}b_{i} \mathscr{B}_{i}(z) \end{aligned}$$
$$\begin{aligned} =&b^{T}\mathscr{B}(z), \end{aligned}$$
where b and \(\mathscr{B}(z)\) are matrices of order \(N\times 1\) given by
$$\begin{aligned}& \begin{aligned}[b] b&=[b_{1,0},b_{1,1},\ldots, b_{1,M-1},b_{2,0},b_{2,1},\ldots,b_{2,M-1}, \ldots,b_{2^{k-1},0},\ldots,b_{2^{k-1},M-1}]^{T} \\ &= [b_{1},b_{2},\ldots,b_{N}]^{T}, \end{aligned} \end{aligned}$$
$$\begin{aligned}& \begin{aligned}[b] \mathscr{B}(z) &=\bigl[\mathscr{B}_{1,0}(z),\ldots, \mathscr{B}_{1,M-1}(z), \mathscr{B}_{2,0}(z),\ldots, \mathscr{B}_{2,M-1}(z),\mathscr{B}_{2^{k-1},0}(z),\ldots, \mathscr{B}_{2^{k-1},M-1}(z)\bigr] \\ &= \bigl[\mathscr{B}_{1}(z),\ldots,\mathscr{B}_{N}(z) \bigr].\end{aligned} \end{aligned}$$
The coefficient \(b_{i}\) is calculated by \(b_{i} = \langle f(z),\mathscr{B}_{i}(z)\rangle = \int _{0}^{1}f(z) \overline{\mathscr{B}_{i}(z)}\,dz\).
Integration of Bernoulli wavelet
Let \(\mathscr{B}_{1,i}(z)\) and \(\mathscr{B}_{2,i}(z)\) be the first and second integration of Bernoulli wavelet from 0 to z, respectively. These integration can be determined as follows:
$$\begin{aligned}& \mathscr{B}_{1,i}(z) = \textstyle\begin{cases} 2^{\frac{-k+1}{2}}\zeta (\frac{1}{m+1} ) \{{ \mbox{\ss} }_{m+1}( \hat{z})-{ \mbox{\ss} }_{m+1}(0) \}, & \xi _{1}\leq z < \xi _{2}, \\ 2^{\frac{-k+1}{2}}\zeta (\frac{1}{m+1} ) \{{ \mbox{\ss} }_{m+1}(1)-{ \mbox{\ss} }_{m+1}(0) \}, & \xi _{2}\leq z \leq 1, \end{cases}\displaystyle \end{aligned}$$
$$\begin{aligned}& \mathscr{B}_{2,i}(z) = \textstyle\begin{cases} 2^{\frac{-3k+3}{2}}\zeta (\frac{1}{m+1} ) \{ ( \frac{1}{m+2} ) \{{ \mbox{\ss} }_{m+2}(\hat{z})-{ \mbox{\ss} }_{m+2}(0) \}-( \hat{z}){ \mbox{\ss} }_{m+1}(0) \}, & \xi _{1}\leq z < \xi _{2}, \\ 2^{\frac{-3k+3}{2}}\zeta (\frac{1}{m+1} ) \{ ( \frac{1}{m+2} )\{{ \mbox{\ss} }_{m+2}(1)-{ \mbox{\ss} }_{m+2}(0)\}-2{ \mbox{\ss} }_{m+1}(0) \\ \quad {}+(\hat{z}-1)\{{ \mbox{\ss} }_{m+1}(1)-{ \mbox{\ss} }_{m+1}(0)\} \}, & \xi _{2}\leq z \leq 1, \end{cases}\displaystyle \end{aligned}$$
where \(\hat{z}=2^{k-1}z-n+1\) and \(\zeta =\frac{1}{\sqrt{\frac{(-1)^{m-1}(m!)^{2}}{(2m)!} \mbox{\ss} _{2m}}}\).
Methods for solution
In this section, we discuss the methods for the solution of the models described above. The following notations have been introduced:
$$\begin{aligned} &\phi _{1,i}(z)= \int _{0}^{z}\phi _{i}(z)\,dz, \end{aligned}$$
$$\begin{aligned} &\phi _{2,i}(z)= \int _{0}^{z}\phi _{1,i}(z)\,dz, \end{aligned}$$
$$\begin{aligned} &\Phi _{1,i}= \int _{0}^{1}\phi _{1,i}(z)\,dz, \end{aligned}$$
$$\begin{aligned} &\Phi _{2,i}= \int _{0}^{1}\phi _{2,i}(z)\,dz. \end{aligned}$$
Method for solution of model of electrohydrodynamic flow in a circular cylindrical conduit
We can express the second derivative of (1.2) in terms of wavelet series as follows:
$$\begin{aligned} u''(z)=\sum_{i=1}^{N}c_{i} \phi _{i}(z). \end{aligned}$$
Integrating (4.5) twice from 0 to z, we get
$$\begin{aligned}& u'(z)=\sum_{i=1}^{N}c_{i} \phi _{1,i}(z)+u'(0), \end{aligned}$$
$$\begin{aligned}& u(z)=\sum_{i=1}^{N}c_{i}\phi _{2,i}(z)+zu'(0)+u(0). \end{aligned}$$
Using boundary conditions (1.3) in (4.6)–(4.7), we get
$$\begin{aligned}& u'(z)=\sum_{i=1}^{N}c_{i} \phi _{1,i}(z), \end{aligned}$$
$$\begin{aligned}& u(z)=\sum_{i=1}^{N}c_{i}\phi _{2,i}(z)+u(0). \end{aligned}$$
Putting \(z=1\) in (4.9) and after simplifying, we get
$$\begin{aligned} u(0)=-\sum_{i=1}^{N}c_{i}\Phi _{2,i}. \end{aligned}$$
Therefore equation (4.9) becomes
$$\begin{aligned} u(z)=\sum_{i=1}^{N}c_{i}\bigl( \phi _{2,i}(z)-\Phi _{2,i}\bigr). \end{aligned}$$
Putting the values of \(u(z)\), \(u'(z)\), and \(u''(z)\) from equations (4.5), (4.8), (4.11) in equation (1.2) and collocating at \(z=z_{l}=\frac{l-0.5}{N}\), where \(l=1,2,\ldots,N\), yields the following system of nonlinear equations:
$$\begin{aligned} \sum_{i=1}^{N}c_{i}\phi _{i}(z_{l})+\frac{1}{z_{l}}\sum _{i=1}^{N}c_{i} \phi _{1,i}(z_{l})+ \mathtt{H^{2}} \biggl(1- \frac{\sum_{i=1}^{N}c_{i}(\phi _{2,i}(z_{l})-\Phi _{2,i})}{1-\alpha \sum_{i=1}^{N}c_{i}(\phi _{2,i}(z_{l})-\Phi _{2,i})} \biggr)=0. \end{aligned}$$
On solving this system of nonlinear equations by Newton's method, we get the unknown wavelet coefficients \(c_{i}\)'s. After putting these \(c_{i}\)'s in equation (4.11), we get the approximate solution.
Method for solution of nonlinear heat conduction model in the human head
We can approximate the second derivative of equation (1.4) in terms of wavelet series as follows:
Integrating (4.13) twice from 0 to z, we get
Using boundary conditions (1.5) in (4.14)–(4.15), we get
Putting \(z=1\) in (4.16)–(4.17) and multiplying (4.16) by ν and (4.17) by μ and after solving these equations for \(u(0)\), we get
$$\begin{aligned} u(0)=u_{\kappa }-\frac{1}{\mu }\sum_{i=1}^{N}c_{i}( \nu \Phi _{1,i}+ \mu \Phi _{2,i}). \end{aligned}$$
Therefore equation (4.17) becomes
$$\begin{aligned} u(z)=\sum_{i=1}^{N}c_{i}\biggl( \phi _{2,i}(z)-\frac{1}{\mu }(\nu \Phi _{1,i}+ \mu \Phi _{2,i})\biggr)+u_{\kappa }. \end{aligned}$$
Putting the values of \(u''(z)\), \(u'(z)\), and \(u(z)\) from equations (4.13), (4.16), (4.19) in equation (1.4) and collocating at \(z=z_{l}=\frac{l-0.5}{N}\), where \(l=1,2,\ldots,N\), yields the following system of nonlinear equations:
$$\begin{aligned} \sum_{i=1}^{N}c_{i}\phi _{i}(z_{l})+\frac{2}{z_{l}}\sum _{i=1}^{N}c_{i} \phi _{1,i}(z_{l})+ \frac{p (\sum_{i=1}^{N}c_{i}(\phi _{2,i}(z_{l})-\frac{1}{\mu }(\nu \Phi _{1,i}+\mu \Phi _{2,i}))+u_{\kappa } )}{\gamma }=0. \end{aligned}$$
After solving this system of nonlinear equations by Newton's method, we get the unknown wavelet coefficients. On putting these coefficients in equation (4.19), we get the approximate wavelet solutions of nonlinear heat conduction model in the human head.
Method for solution of spherical catalyst equation
Putting \(z=1\) in (4.25) and after simplification, we get
$$\begin{aligned} u(0)=1-\sum_{i=1}^{N}c_{i}\Phi _{2,i}. \end{aligned}$$
$$\begin{aligned} u(z)=\sum_{i=1}^{N}c_{i}\bigl( \phi _{2,i}(z)-\Phi _{2,i}\bigr)+1. \end{aligned}$$
Putting the values of \(u''(z)\), \(u'(z)\), and \(u(z)\) from equations (4.21,4.24,4.27) in equation (1.6) and collocating at \(z=z_{l}=\frac{l-0.5}{N}\), where \(l=1,2,\ldots,N\), yields the following system of nonlinear equations:
$$\begin{aligned} \begin{aligned}[b] &\sum_{i=1}^{N}c_{i}\phi _{i}(z_{l})+\frac{2}{z_{l}}\sum _{i=1}^{N}c_{i} \phi _{1,i}(z_{l})\\ &\quad {}-{ \rho }^{2} \Biggl(\sum_{i=1}^{N}c_{i} \bigl(\phi _{2,i}(z_{l})- \Phi _{2,i}\bigr)+1 \Biggr)e^{ \{ \frac{\gamma \beta (1- (\sum _{i=1}^{N}c_{i}(\phi _{2,i}(z_{l})-\Phi _{2,i})+1 ) )}{1+\beta (1-(\sum _{i=1}^{N}c_{i}(\phi _{2,i}(z_{l})-\Phi _{2,i})+1) )} \}}=0. \end{aligned} \end{aligned}$$
After solving this system of nonlinear equations by Newton's method, we get the unknown wavelet coefficients \(c_{i}\)'s. On putting these \(c_{i}\)'s in equation (4.27), we get the approximate wavelet solutions of spherical catalyst equation.
Method for solution of spherical biocatalyst equation
The same procedure has been implemented as in case of spherical catalyst equation. Substituting the values of \(u''(z)\), \(u'(z)\), and \(u(z)\) from equations (4.21), (4.24), (4.27) in equation (1.8) and collocating at \(z=z_{l}=\frac{l-0.5}{N}\), where \(l=1,2,\ldots,N\), yields the following system of nonlinear equations:
$$\begin{aligned} \sum_{i=1}^{N}c_{i}\phi _{i}(z_{l})+\frac{2}{z_{l}}\sum _{i=1}^{N}c_{i} \phi _{1,i}(z_{l})-{ \rho }^{2} \frac{(1+\beta ) (\sum_{i=1}^{N}c_{i}(\phi _{2,i}(z_{l})-\Phi _{2,i})+1 )}{1+\beta (\sum_{i=1}^{N}c_{i}(\phi _{2,i}(z_{l})-\Phi _{2,i})+1) }=0. \end{aligned}$$
Solving this system of nonlinear equations, we get the unknown wavelet coefficients \(c_{i}\)'s. After putting the values of \(c_{i}\)'s in equation (4.27), we get the approximate wavelet solutions of spherical biocatalyst equation.
Error bounds
Lemma 5.1
Let \(u(z)\in C^{M}[0, 1]\)with \(|u^{M}(z)|\leq \lambda \), \(\forall z \in (0,1); \alpha >0\)and \(u(z)\simeq [4] \sum_{n=1}^{2^{k-1}}\sum_{m=0}^{M-1}a_{n,m}\phi _{n,m}(z)= \sum_{i=0}^{N}a_{i}\phi _{i}(z) \), where \(\phi _{n,m}(z)\)is Jacobi or Bernoulli wavelet. Then \(|a_{i}|\leq \alpha _{M} {2^{-m(M+\frac{1}{2})}}\lambda \).
Let \(u(z)\in C^{M}[0, 1]\)and \(u(z)\simeq \sum_{n=1}^{2^{k-1}}\sum_{m=0}^{M-1}a_{n,m}\phi _{n,m}(z)= \sum_{i=0}^{N}a_{i}\phi _{i}(z) \). Let \(\varepsilon _{m}(z)\)be the error of approximation. Then \(|\varepsilon _{m}(z)|\leq \alpha _{M} \frac{2^{-mM}}{1-2^{-m}}( \frac{M}{2}-1)\lambda \alpha _{\Phi }\).
\(\varepsilon _{m}(z)=u(z)-P_{V_{m}}u(z)=\sum_{m=M}^{\infty }\sum_{n=2^{k-1}+1}^{ \infty }a_{n,m}\phi _{n,m}(z)\), \(z\in R\) and \(|\varepsilon _{m}(z)| \leq [4] \sum_{m=M}^{\infty }\sum_{n=2^{k-1}+1}^{ \infty }|a_{n,m}\phi _{n,m}(z)|\), \(z\in R\).
Using Lemma 5.1, we get
$$\begin{aligned}& \bigl\vert \varepsilon _{m}(z) \bigr\vert \leq \sum _{m=M}^{\infty }\sum_{n=2^{k-1}+1}^{ \infty } \alpha _{M} {2^{-m(M+\frac{1}{2})}}\max_{z \in I_{n}^{m}} \bigl\vert u^{M}(z) \bigr\vert 2^{ \frac{m}{2}}\alpha _{\phi }, \end{aligned}$$
where \(\alpha _{\phi }= \max_{z \in I_{n}^{m}}|\phi (2^{m}z-n)|\).
$$\begin{aligned}& \bigl\vert \varepsilon _{m}(z) \bigr\vert \leq \sum _{m=M}^{\infty }\alpha _{M} {2^{-mM}}(2M-1) \max_{z \in I_{n}^{m}} \bigl\vert g^{M}(z) \bigr\vert \alpha _{\phi {G}}. \end{aligned}$$
For very large m, \(|\varepsilon _{m}(z)|\leq \alpha _{M} \frac{2^{-mM}}{1-2^{-m}}(2M-1) \lambda \alpha _{\phi }\), where \(\lambda = \max_{z \in I_{n}^{m}}|g^{M}(z)|\). □
Theorem 5.3
Let \(u(z)\)be the exact solution of (1.2), (1.4), (1.6), and (1.8) and \(u_{N}(z)\)be the approximate solution, and let \(\varepsilon _{m}(z)\)be the error of approximation. Then \(|\varepsilon _{m}(z)|=O(2^{-mM})\).
Here, we calculate the error bounds for solution of (1.2). The same procedure can be applied for equations (1.4), (1.6), and (1.8).
The error is given by
$$\begin{aligned} \bigl\vert \varepsilon _{m}(z) \bigr\vert = \bigl\vert u(z)-u_{N}(z) \bigr\vert = \Biggl\vert \sum _{m=M}^{\infty }\sum_{n=2^{k-1}+1}^{ \infty }a_{n,m} \bigl(\phi _{n,m}^{2}(z)-\Phi _{n,m}^{1} \bigr) \Biggr\vert , \end{aligned}$$
where \(\phi _{n,m}^{2}(z)\) is the second integration of \(\phi _{n,m}(z)\) from 0 to z and \(\Phi _{n,m}^{1}\) denotes the second integration of \(\phi _{n,m}(z)\) from 0 to 1. Therefore,
$$\begin{aligned} \bigl\vert \varepsilon _{m}(z) \bigr\vert \leq \sum _{m=M}^{\infty }\sum_{n=2^{k-1}+1}^{ \infty } \bigl\vert a_{n,m}\bigl(\phi _{n,m}^{2}(z)-\Phi _{n,m}^{1}\bigr) \bigr\vert . \end{aligned}$$
$$\begin{aligned} \bigl\vert \varepsilon _{m}(z) \bigr\vert &\leq \sum _{m=M}^{\infty }\sum_{n=2^{k-1}+1}^{ \infty } \alpha _{M}{2^{-m(M+\frac{1}{2})}} \bigl\vert \bigl(\phi _{n,m}^{2}(z)-\Phi _{n,m}^{1}\bigr) \bigr\vert \end{aligned}$$
$$\begin{aligned} &=\sum_{m=M}^{\infty }\sum _{n=2^{k-1}+1}^{\infty }\alpha _{M}2^{-m(M+ \frac{1}{2})}2^{\frac{m}{2}} \bigl\vert \bigl(\phi ^{2}\bigl(2^{m}z-n\bigr)-\Phi _{n,m}^{1}\bigr) \bigr\vert \end{aligned}$$
$$\begin{aligned} &\leq \sum_{m=M}^{\infty }\alpha _{M}2^{-mM}(2M-1)\lambda \gamma _{ \phi }, \end{aligned}$$
where \(\gamma _{\phi }=\max_{z\in I_{n}^{m}}|(\phi ^{2}(2^{m}z-n)-\Phi _{n,m}^{1})|\). Hence \(|\varepsilon _{m}(z)|=O(2^{-mM})\). □
It is clear that each of the Jacobi and Bernoulli wavelet methods has an exponential rate of convergence/spectral accuracy.
Numerical simulation
In this section, we solve the examples of electrohydrodynamic model flow in a circular cylindrical conduit, nonlinear heat conduction model in the human head, spherical catalyst equation, and spherical biocatalyst equation. For the sake of comparison, the resultant approximate analytical solution has been used to find the solution at any point in the interval \([0, 1]\). We have chosen the initial guess as a zero vector of length N. We have used optimality tolerance =10−06 and function tolerance =10−03 in stopping criteria for Newton's method.
Numerical treatment of EHM equation
We applied Bernoulli wavelet series method (BWSM) and Jacobi wavelet series method (JWSM) \((\kappa =-\frac{3}{7},\omega =-\frac{1}{8})\) for the solution of (1.2). First we study the effect of nonlinearity \((\alpha )\) on the velocity profile at small value of the Hartmann number \((H)\) and observe that as we increase H, the velocity profile becomes flatter near to the center, see Fig. 5. For small value of H, the velocity profile almost remains parabolic with change in α, see Fig. 6. We also study the influence of different H with fixed α and see that a strong boundary layer is build up in velocity for a large value of H, see Figs. 7 and 8. We see that BWSM result for fixed values \(H^{2}=2, 100\) with different value of \(\alpha =0.1,0.5,1\) and for fixed values of \(\alpha =0.1, 1\) with different value of \(H^{2}=0.5, 2, 16, 36, 49, 64\) agrees with the result of SSNM (sixth-order spline numerical method), see Figs. 10, 12, 3, and 4 of [29]. The numerical solution by BWSM and JWSM for different values of \(H^{2}\) is given in Tables 1 and 2, respectively. The absolute residual errors and CPU time for different values of J are shown in Table 3.
Graph of BWSM solution of EHD equation for \(J=3\), \(M=8\) and \(k=1\), \(H^{2}=100\)
Graph of JWSM solution of EHD equation for \(J=3\), \(M=8\) and \(k=1\), \(H^{2}=2\)
Graph of BWSM solution of EHD equation for \(J=3\), \(M=8\) and \(k=1\), \(\alpha =0.1\)
Graph of JWSM solution of EHD equation for \(J=3\), \(M=8\) and \(k=1\), \(\alpha =1\)
Table 1 BWSM solution of EHM equation for \(\alpha =0.1\) with \(J=3\), \(M=8\), \(k=1\)
Table 2 JWSM solution of EHD equation for \(\alpha =1\) with \(J=3\), \(M=8\), \(k=1\)
Table 3 Maximum absolute residual errors of EHD equation for \(\alpha =1\) and \(H^{2}=0.5\)
Numerical treatment of nonlinear heat conduction model in the human head
Consider equation (1.4) along with boundary conditions (1.5) and \(p(u)=e^{-u}\), \(\gamma =1\), \(\nu =1\), \(\mu =2\), and \(u_{k}=0\) and get the following Emden–Fowler type equation:
$$\begin{aligned} u''(z)+\frac{2}{z}u'(z)+e^{-u}=0, \end{aligned}$$
and the boundary conditions become
$$\begin{aligned} u'(0)=0;\qquad u'(1)+2u(1)=0. \end{aligned}$$
We used Bernoulli and Jacobi wavelets for solving this problem. The calculation has been done by taking \(\kappa =-\frac{1}{4}\) and \(\omega =-\frac{1}{3}\) in Jacobi wavelet. A comparison of our results with the results of Haar solution [35] and ADM [36] is given in Table 4. We show the absolute residual errors and CPU time for different J in Table 5. Figures 9 and 10 show the BWSM and JWSM solution at \(J=3\) for different values of γ, respectively.
Graph of BWSM solution of nonlinear heat conduction model in human head equation for different γ with \(J=3\), \(M=8\), and \(k=1\)
Graph of JWSM solution of nonlinear heat conduction model in human head equation for different γ with \(J=3\), \(M=8\), and \(k=1\)
Table 4 Numerical solution of nonlinear heat conduction model in the human head for \(\gamma =1\) with \(J=3\), \(M=8\), \(k=1\)
Table 5 Maximum absolute residual errors of nonlinear heat conduction model for \(\gamma =1\)
Numerical treatment of spherical catalyst equation
Consider equation (1.6) with (1.7) by taking \(\beta =1\), \(\rho =1\). We have performed BWSM and JWSM \((\kappa =\frac{1}{5}, \omega =-\frac{1}{6})\) for the solution of (1.6). The influence of different values of activation energy is shown in Figs. 11 and 12. The numerical solution of (1.6) for \(\sigma =0.5, 1, 1.5\) is given in Table 6. We compare our results with the results of OHAM [29] in Table 7.
Graph of BWSM solution of spherical catalyst equation for \(J=3\), \(M=8\) and \(k=1\), \(\sigma =0.5, 1, 1.5\)
Graph of JWSM solution of spherical catalyst equation for \(J=3\), \(M=8\) and \(k=1\), \(\sigma =0.5, 1, 1.5\)
Table 6 Numerical solution of spherical catalyst equation for \(\beta =1\), \(\rho =1\) with \(J=3\), \(M=8\), \(k=1\)
Table 7 Maximum absolute residual errors of spherical catalyst equation for \(\beta =\rho =1\) with \(J=2\)
Numerical treatment of spherical biocatalyst equation
Consider equation (1.8) with (1.9) by fixing \(\beta =2\). We have performed BWSM and JWSM \((\kappa =\frac{-3}{5}, \omega =-\frac{1}{8})\) for the solution of (1.8). The influence of different values of Thiele modulus is shown in Figs. 13 and 14. The numerical solution of (1.6) for \(\rho =1, 1.5, 2\) is given in Table 8. We compare our results with the results of OHAM [34] in Table 9.
Graph of BWSM solution of spherical biocatalyst equation for \(J=3\), \(M=8\) and \(k=1\), \(\beta =2\)
Graph of JWSM solution of spherical biocatalyst equation for \(J=3\), \(M=8\) and \(k=1\), \(\beta =2\)
Table 8 Numerical solution of spherical biocatalyst equation for \(\beta =2\) with \(J=3\), \(M=8\), \(k=1\)
Table 9 Maximum absolute residual errors of spherical biocatalyst equation for \(\beta =2\) with \(J=2\)
In this paper, we have studied EHD flow in a charged circular cylinder conduit, nonlinear heat conduction model in the human head, non-isothermal reaction–diffusion model equations in a spherical catalyst, and non-isothermal reaction–diffusion model equations in a spherical biocatalyst which are modeled by Lane–Emden type equations having strong nonlinearity. We have solved these models by two numerical methods based on Jacobi and Bernoulli wavelets. These wavelet methods solved Lane–Emden type equations by converting them into a system of nonlinear equations. In the study of EHD flow, we observed that the effects of Hartmann number and nonlinearity have an important impact. Further we also compare our results with the results of SSNM [29], Haar [35], ADM [36], and OHAM [34]. The graphs show the efficiency of our methods. Moreover, the present semi-analytical numerical methods have lower computational cost than ADM, Haar, and OHAM, since in our methods there is no need for symbolic successive integration which is computationally higher than numerical methods.
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The authors are indebted to the anonymous reviewer for his helpful, valuable comments and suggestions in the improvement of the manuscript. First author is thankful to Council of Scientific and Industrial Research (CSIR), Govt. of India, for providing Junior Research Fellowship.
Department of Mathematics, Jamia Millia Islamia, New Delhi, 110025, India
Mo Faheem & Arshad Khan
Department of Mathematics, College of Sciences and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Alkharj, 11942, Saudi Arabia
E. R. El-Zahar
Department of Basic Engineering Science, Faculty of Engineering, Menoufia University, Shebin El-Kom, 32511, Egypt
Mo Faheem
All authors drafted the manuscript, and they read and approved the final version.
Correspondence to Arshad Khan.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Faheem, M., Khan, A. & El-Zahar, E.R. On some wavelet solutions of singular differential equations arising in the modeling of chemical and biochemical phenomena. Adv Differ Equ 2020, 526 (2020). https://doi.org/10.1186/s13662-020-02965-7
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Modelling the effect of a dengue vaccine on reducing the evolution of resistance against antibiotic due to misuse in dengue cases
Ana Kurauchi1,
Claudio Jose Struchiner2,
Annelies Wilder-Smith3,4 &
Eduardo Massad ORCID: orcid.org/0000-0002-7200-29161,2
Theoretical Biology and Medical Modelling volume 17, Article number: 7 (2020) Cite this article
This paper intends to check whether and how a hypothetical dengue vaccine could contribute to issue of evolution of bacteria resistance against antibiotics by reducing the number of patients that would inappropriately being treated with antibiotics.
We use a new mathematical model that combines, in a novel way, two previously published papers, one on the evolution of resistance against antibiotics and one classical Ross-Macdonald model for dengue transmission.
The model is simulated numerically and reproduces a real case of evolution of resistance against antibiotics. In addition the model shows that the use of a hypothetical dengue vaccine could help to curb the evolution of resistance against an antibiotic inappropriately used in dengue patients. Both the increase in the proportion of resistant bacteria due to the misuse of antibiotics in dengue cases as a function of the fraction of treated patients and the reduction of that proportion as a function of vaccination coverage occur in a highly non-linear fashion.
The use of a dengue vaccine is helpful in reducing the rate of evolution of antibiotic resistance in a scenario of misuse of the antibiotics in dengue patients.
Antibiotics are one of the major breakthroughs in the history of medicine and have saved millions of lives [1]. However, its misuse or overuse can also have disastrous consequences [2]. Overuse is frequent: approximately two-third (68%) of antibiotics are prescribed for upper respiratory infections (URTIs) [3, 4] but > 80% of such prescriptions have been found to be unnecessary and inappropriate with adverse outcome including the menace of antibiotic resistance.
Antibiotic resistance (the ability of microbes to evolve and withstand the effects of antibiotics) is a significant cause of morbidity and mortality globally [5,6,7], and antibiotic over-consumption is the main driver of antibiotic resistance [8]. The association between antibiotic consumption and resistance is well documented across spatial and temporal scales at individual hospitals [9], nursing homes [10], primary care facilities [11], and communities [12], as well as across countries [13].
Bacteria have become resistant to antimicrobial agents as a result of chromosomal changes or the exchange of genetical material via plasmids and transposons [14].
Many countries have adopted national action plans on antimicrobial resistance (AMR) that aim to reduce per capita antibiotic consumption. The Global Action Plan on Antimicrobial Resistance endorsed by the member states of the World Health Organization (WHO) and affirmed at the high-level meeting on antimicrobial resistance during the 71st General Assembly of the United Nations [15], recommends that all countries collect and report antibiotic consumption data [16]. Surveillance data on country-level antibiotic use are needed to monitor national and global trends over time; compare antibiotic use among countries; provide a baseline for the evaluation of future efforts to reduce antibiotic use; enable epidemiological analysis of the association between antibiotic use and resistance over time [17, 18]; and support policies that aim to reduce antibiotic resistance.
Making better use of existing vaccines and developing new vaccines are important ways to tackle AMR and reduce preventable illness and deaths.
Dengue is a mosquito-borne infection, transmitted by the Aedes aegypti mosquitoes, characterized by sudden onset of fever and severe headache; resulting in shock and hemorrhage leading to death [19] in many of such patients [20, 21]. The clinical manifestations are often nonspecific, with signs and symptoms that overlap with many other febrile illnesses including bacterial infections. As a result, health care practitioners often prescribe antibiotics empirically, e.g. without confirming the diagnosis, which leads to unnecessary use of antibiotics.
In one particular study in India [22], of 370 confirmed dengue cases, 267 (74.6%)cases were prescribed antibiotics. A single antibiotic was prescribed to 225 cases (60.8% of all cases), 2 antibiotics to 33 dengue cases (8.9%), and 3 antibiotics to 9 (2.4%). Triple therapy antibiotics included cefotaxime in all prescriptions with cefixime, azithromycin, amoxyclav, doxycycline, and ceftriaxone in different combinations. Antibiotics given as dual therapy were ceftriaxone with doxyxycline, cefotaxime, or amoxyclav, and cefotaxime with doxycycline, cefixime, or metronidazole.
In another study in West Java, Indonesia [23], showed that there were 547 (17.8%) out of 3078 dengue patients that received antibiotics.
One of the authors (AK) found in the Clinics Hospital in São Paulo, Brazil, that among 103 confirmed dengue cases, 35 (34%) inappropriately received antibiotics.
Of course antibiotics are indicated when secondary bacterial infections are present in some dengue cases [24]. However, the problem of antibiotics misuse in dengue cases is related to misdiagnosing dengue with URTI [21]. To distinguish between the appropriate and inappropriate antibiotic use in dengue cases it is necessary to demonstrate the presence of bacterial infection, for instance, by collecting material for bacterial culture procedures [21].
Therefore, is seems that there is enough evidence of misuse of antibiotics in dengue patients. Such misuse could worsen the selective pressure that leads to the evolution of resistance against those antibiotics.
This paper intends to investigate whether and how a hypothetical dengue vaccine could contribute to issue of evolution of bacteria resistance against antibiotics by reducing the number of patients that would inappropriately being treated with antibiotics. We do so by using a new mathematical model that combines, in a novel way, two previously published papers, one on the evolution of resistance against antibiotics, illustrated by the case of resistance of K. pneumoniae resistance against Amikacine (this was one clear example of rapide evolution of resistance studied by one of the authors of the present paper) and one classical Ross-Macdonald model for dengue transmission.
The model combines two distinct models, one proposed previously in Massad, Yang and Lundberg [25] for the study of the evolution of antibiotic resistance, and one Ross-Macdonald type of dengue model [26], including the possibility of vaccination.
The composite model is described in Fig. 1:
Composite model combining an antibiotic resistance model and a Ross-Macdonald dengue model with vaccination
In Fig. 1, the left-hand side picture describes the model by Massad, Yang and Lundberg [25], designed to study the evolution of resistance against antibiotics. This model considered a population in a hospital environment, in which X(t) represents individuals who have been hospitalized by diverse causes with rate Λ, and are susceptible to a given infectious agent. These individuals may acquire an hospital infection by a strain of the pathogen which is sensitive to a specific antibiotic against that pathogen with a rate β1. Once infected with the sensitive strain these individuals are denoted Y(t). A fraction of p of those Y(t) individuals are treated with the specific antibiotic and recovers to the susceptible state again with rate γ.The fraction (1 − p) of non-treated individuals die from the infection with rate α. However, Y(t) individuals may be discharged from the hospital with rate μ. Alternatively, the susceptible individuals X(t) may acquire the infection by a strain of the pathogen which is resistant to the specific antibiotic against that pathogen with a rate β2. Once infected with the resistant strain these individuals are denoted Z(t). These individuals may either be discharged from the hospital with rate μ (like everyone else in the model), or die from the infection with rate α. The model consider the evolution of antibiotic resistance by two alternative mechanisms, one consisting in mutation, and one by plasmid transfer from the sensitive to resistant strains. These two mechanisms are represented in the figure by the composite rate F. The mutational component of rate F is described by the expression (θ + pξ)Y(t) + (ω + pδ)(Y(t)Z(t)). In this expression, θ is the treatment-independent mutation rate and pξ is the mutation rate induced by antibiotic treatment. The second component, comprises the term ω, which is the treatment-independent plasmid transfer and the term pδ, which is the plasmid transfer rate induced by antibiotic treatment (note the cross-infection term (Y(t)Z(t)). The back-mutational component of rate G is described by the expression σ Z(t). In this expression, σ is the back-mutation rate.
The right-hand side picture shown in Fig. 1 describes the Ross-Macdonald model to be used to represent dengue infection with a vaccination component. The model considers that people born with rate μH (assumed equal to the natural mortality rate) and who are susceptible to dengue are denoted SH(t). These individuals may acquire dengue infection with incidence \( ab\left(\frac{I_M}{N_H}\right) \), where a is the mosquitoes' biting rate, b is the probability of infection from mosquitoes to humans, IM(t) is the number of infected mosquitoes and NH is the total human population (assumed constant by equating the birth and death rates as μH). Dengue infected individuals are denoted \( {I}_H^T(t) \), a fraction h of whom are interned in the same hospital as in the first model. Non-hospitalized dengue individuals are denoted \( {I}_H^G(t) \). Alternatively the susceptible individuals may be vaccinated with rate υH and are then denoted VH(t). Hospitalized dengue-infected individuals, denoted IH(t), may either be infected with plasmids they acquire from the resistant strain infected individuals in the same hospital, ZH(t), provided that a fraction p' of them is mistreated with the antibiotics (note that the rate of plasmid transfer δ is the same), or are discharged from the hospital with rate μH (like everyone else in the model), or die from dengue infection with rate αH, or are infected with the sensitive strain and treated and recovered from dengue with rate γH. Therefore, the term p ' δIH(t)Z(t) represents the plasmid transfer from ZH(t) to IH(t), by a cross-infection mechanism. Like in the classical Ross-Macdonald model, susceptible mosquitoes, denoted SM(t) may acquire dengue infection with incidence \( ac\left(\frac{I_M}{N_H}\right) \), where again a is the mosquitoes' biting rate, c is the probability of infection from humans to mosquitoes, IM(t) is the number of infected mosquitoes and NH is the total human population. Once infected, these mosquitoes get into a latent state, denoted LM(t) and then either die or evolve to the infective state IM(t). Note that mosquitoes are born and die with the same rate μM, which implies that the total mosquito population is assumed constant.
The model's variables and parameters are described in Table 1.
Table 1 The Model's variables, parameters, biological meaning and values
The model is described by the following system of equations:
$$ {\displaystyle \begin{array}{l}\mathrm{Dengue}\ \mathrm{Model}\\ {}\\ {}\frac{dS_H(t)}{dt}=- ab\frac{I_M(t)}{N_H}{S}_H(t)+{\mu}_H\left[{N}_H-{S}_H(t)\right]-{\upsilon}_H{S}_H(t)\\ {}\frac{dI_H^G}{dt}= ab\frac{I_M(t)}{N_H}{S}_H(t)\left(1-h\right)-\left({\mu}_H+{\alpha}_H\right){I}_H^G(t)-k{\gamma}_H{I}_H^G(t)\\ {}\frac{dI_H(t)}{dt}= ab\frac{I_M(t)}{N_H}{S}_H(t)h-\left({\mu}_H+{\alpha}_H+d\right){I}_H(t)-\left(1-p\hbox{'}\right){\gamma}_H{I}_H(t)-p\hbox{'}\delta {I}_H(t)Z(t)\\ {}\frac{dR_H(t)}{dt}=\left(1-p\hbox{'}\right){\gamma}_H{I}_H(t)+k{\gamma}_H{I}_H^G(t)-{\mu}_H{R}_H(t)\\ {}\frac{dV_H(t)}{dt}={\upsilon}_H{S}_H(t)-{\mu}_H{V}_H(t)\\ {}\\ {}{I}_H^T(t)={I}_H^G(t)+{I}_H(t)\\ {}{N}_H={S}_H+{I}_H+{R}_H+{V}_H\\ {}\\ {}{Vac}_{\mathrm{cov}}(t)=\frac{1}{N_H}\underset{0}{\overset{t}{\int }}{\upsilon}_H{S}_H(s) ds\\ {}\\ {}\frac{dS_M(t)}{dt}=- ac\frac{I_H^T(t)}{N_H}{S}_M(t)+{\mu}_M\left({N}_M-{S}_M(t)\right)\\ {}\frac{dL_M(t)}{dt}= ac\frac{I_H^T(t)}{N_H}{S}_M(t)-\left({\gamma}_M+{\mu}_M\right){L}_M(t)\\ {}\frac{dI_M(t)}{dt}={\gamma}_M{L}_M(t)-{\mu}_M{I}_M(t)\\ {}\\ {}{N}_M={S}_M+{L}_M+{I}_M\\ {}\\ {}\mathrm{Antiobtic}\ \mathrm{Resistance}\ \mathrm{Model}\\ {}\\ {}\frac{dX(t)}{dt}=-{\beta}_1X(t)Y(t)-{\beta}_2X(t)Z(t)-\mu X(t)+ p\gamma Y(t)+\Lambda \\ {}\frac{dY(t)}{dt}={\beta}_1X(t)Y(t)-\left[\left(1-p\right)\alpha + p\gamma \right]Y(t)-\mu Y(t)-F\left[Y(t),Z(t)\right]+G\left[Z(t)\right]\\ {}\frac{dZ(t)}{dt}={\beta}_2X(t)Z(t)-\alpha Z(t)+F\left[Y(t),Z(t)\right]-\mu Z(t)-G\left[Z(t)\right]+p\hbox{'}\delta {I}_H(t)Z(t)\\ {}\\ {}N=X+Y+Z\\ {}F\left[Y(t),Z(t)\right]=\left(\theta + p\xi \right)Y(t)+\left(\omega + p\delta \right)Y(t)Z(t)\\ {}G\left[Z(t)\right]=\sigma Z(t)\\ {}\\ {} resis(t)=\frac{1}{N}\underset{0}{\overset{t}{\int }}\left\{{\beta}_2X(s)Z(s)+p\hbox{'}\delta {I}_H(s)Z(s)+F\left[Y(s),Z(s)\right]\right\} ds\end{array}} $$
Model (1) was simulated, first with the antibiotic resistant component only in order to reproduce the results obtained by Massad, Yang and Lundberg [25] with the data from Klebsiella pneumoniae strains resistant against the antibiotic Amikacin in the Clinics Hospital of the School of Medicine of the University of São Paulo, Brazil. Results are shown in Fig. 2.
Performance of the model of antibiotic resistance (black line) simulated with parameters as in Table 1, and actual evolution of Klebsiella pneumoniae resistance against Amikacin (red line). Real data from Massad, Yang and Lundberg [24]
Note that the model tallies the actual data with good accuracy for a fraction of antibiotic treated individuals of 70%. In just 5 years resistance evolved from less than 10% to more than 70%.
The complete model (1) was then simulated with variables and parameters as in Table 1 in order to estimate the impact of inappropriately treating dengue patients with the antibiotics and the impact of vaccination against dengue on the evolution of antibiotic resistance of dengue-infected individuals mistreated with the same antibiotic (Amikacin) against the same pathogen (Klebsiella pneumoniae). The result can be seen in Fig. 3.
Performance of the complete model of antibiotic resistance and dengue simulated with parameters as in Table 1. Continuous purple line represents the equilibrium after 60 months of treatment in the absence of dengue, that is, the base line evolution of resistance against antibiotics for that specific community. Other lines represent effect of vaccination with several proportions of antibiotic misuse against dengue, varying from 10% (lower light green line) to 50% (upper blue line)
Note that the final proportion of resistant bacteria varies in a non-linear fashion with the increase in the proportion of dengue patients inappropriately treated with the antibiotic.
Note that the vaccination coverage necessary to reduce the resistance against the antibiotic in this extreme situation is very high.
In this paper we propose a composite model to test the hypothesis that a hypothetical vaccine against dengue could help to hamper the evolution of resistance against antibiotics due to their misuse in dengue patients. This hypothesis was tested with a composite model combining a previously published model for studying the evolution of antibiotic resistance, with a classical Ross-Macdonald dengue model [26]. The simulation of a real setting involving the overuse of amikacin in patients infected with K. pneumoniae in a large hospital in Sao Paulo, Brazil [25] and the inclusion of dengue patients (see Fig. 1) into the bacterial dynamics part of the composite model. The inappropriate use of antibiotics in dengue patients increased the evolution of resistance against these antibiotics in a non-linear fashion. Hence, if 10% of dengue patients were treated with antibiotics, the proportion of bacteria resistant to the drugs would increase from the baseline of 70% to almost 89% and so on as seen in Fig. 3.
The result of the simulation of the impact of the theoretical dengue vaccine also resulted in a highly non-linear decrease in the proportion of resistant bacteria with the increase in the vaccination coverage (Fig. 3). Although it should be expected a reduction in the proportion of resistant bacteria with the reduction of susceptible individuals due to the vaccine, the simulations show that the necessary coverage to result in a significant reduction in the proportion of resistant bacteria is very high. In addition, the higher the proportion of dengue patients mistreated with antibiotics, the higher the necessary vaccination coverage to reduce the antibiotic resistance to base level (Fig. 3).
Our model has several oversimplifications and limitations. Firstly it assumes a homogeneously mixing transmission, both to the bacterial infection and to the dengue infection. The model is deterministic, ignoring eventual stochastic fluctuations in the compartments dynamics. Many of the parameters used in the simulations are not based on empirical observations, although the antibiotic resistant part of the model reproduces a real scenario with a reasonable accuracy. The dengue model is not stratified by serotypes but considers dengue as an all-or-nothing infection. And finally, the theoretical vaccine is assumed to be 100% efficient to all dengue serotypes. Therefore if such a vaccine would be available and if 100% of susceptible people were vaccinated, then no dengue case would occur and no misuse of antibiotic would occur. Considering 50 million dengue cases per year worldwide, considering that between 20 and 40% are mistreated with antibiotics and considering the average cost of one antibiotic course of US$9.91 [27] for each episode of wrongly diagnosed upper respiratory infection (the main cause of antibiotic misuse in dengue patients [2], then it should be expected an economic gain of between US$99,100,000.00 and US$198,200,000.00 per year.
Notwithstanding the above oversimplifications, we think that the composite model served its purposes since it was designed to qualitatively investigate how a hypothetical vaccine could curb the evolution of resistance against antibiotics that is caused by the inappropriate use of these drugs in dengue patients.
It is possible, therefore, based on the results of the simulation of our model that a dengue vaccine would reduce the rate of evolution of antibiotic resistance in a scenario in which dengue patients are inappropriately treated with the drug.
Please contact author for data request.
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This work was partially supported by the project ZikaPLAN, funded by the European Union's Horizon 2020 research and innovation programme under Grant Agreement No. 734584, by LIM01-HFMUSP, CNPq and FAPESP.
School of Medicine, University of Sao Paulo, Sao Paulo, Brazil
Ana Kurauchi & Eduardo Massad
School of Applied Mathematics, Fundacao Getulio Vargas, Rua Praia de Botafogo 190, Rio de Janeiro, CEP - 22250-900, Brazil
Claudio Jose Struchiner & Eduardo Massad
Department of Disease Control, London School of Hygiene and Tropical Medicine, London, UK
Annelies Wilder-Smith
Heidelberg Institute of Global Health, University of Heidelberg, Heidelber, Germany
Ana Kurauchi
Claudio Jose Struchiner
Eduardo Massad
All authors contributed equally for the designing, analysis and writing of this paper. The author(s) read and approved the final manuscript.
Correspondence to Eduardo Massad.
This study was approved by the Ethics Review Board of the "Hospital das Clinicas da Faculdade de Medicina da USP" under protocol number 3.764.549.
We declare that we do not have any conflict of interest.
Kurauchi, A., Struchiner, C.J., Wilder-Smith, A. et al. Modelling the effect of a dengue vaccine on reducing the evolution of resistance against antibiotic due to misuse in dengue cases. Theor Biol Med Model 17, 7 (2020). https://doi.org/10.1186/s12976-020-00125-8
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The $\ell ^{1}$-indices of Tsirelson type spaces
by Denny H. Leung and Wee-Kee Tang PDF
Proc. Amer. Math. Soc. 131 (2003), 511-521 Request permission
If $\alpha$ and $\beta$ are countable ordinals such that $\beta \neq 0$, denote by $tilde{T}_{\alpha ,\beta }$ the completion of $c_{00}$ with respect to the implicitly defined norm \[ \Vert x\Vert = \max \{\Vert x\Vert _{\mathcal {S}_{\alpha }},\frac {1}{2}\sup \sum _{i=1}^{j}\Vert E_{i}x\Vert \}, \] where the supremum is taken over all finite subsets $E_{1}$, …, $E_{j}$ of $\mathbb {N}$ such that $E_{1}<\dots <E_{j}$ and $\{\min E_{1}, \dots , \min E_{j} \} \in \mathcal {S}$_{�}$. It is shown that the Bourgain$ℓ^{1}$-index of$T̃_{�,�}$is$�^{�+�⋅�}$. In particular, if$�_{1}>�=�^{�_{1}}⋅m_{1}+…+�^{�_{n}}⋅m_{n}$in Cantor normal form and$�_{n}$is not a limit ordinal, then there exists a Banach space whose$ℓ^{1}$-index is$�^{�}$.$
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D. H. Leung and W.-K. Tang, The Bourgain $\ell ^{1}$-indices of mixed Tsirelson spaces, in preparation.
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Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 46B20, 05C05
Retrieve articles in all journals with MSC (2000): 46B20, 05C05
Denny H. Leung
Affiliation: Department of Mathematics, National University of Singapore, Singapore 117543
Email: [email protected]
Wee-Kee Tang
Affiliation: Mathematics and Mathematics Education, National Institute of Education, Nanyang Technological University, 1 Nanyang Walk, Singapore 637616
Email: [email protected]
Received by editor(s) in revised form: July 10, 2001, and September 20, 2001
Communicated by: N. Tomczak-Jaegermann
Journal: Proc. Amer. Math. Soc. 131 (2003), 511-521
MSC (2000): Primary 46B20; Secondary 05C05
DOI: https://doi.org/10.1090/S0002-9939-02-06586-3
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Quantitative Finance Stack Exchange is a question and answer site for finance professionals and academics. It only takes a minute to sign up.
Exponential Moving Average Data Set Average Age
Asked 1 month ago
Why is the smoothing coefficient of the EMA (exponential moving average) calculated as:
$${\displaystyle \alpha =2/(N+1)}?$$
Brown R.G, on page 107 of "Smoothing, forecasting and prediction of discrete time series (1963)" goes about giving an explanation using the following folowing derivation for what he calls the average age of the data set:
(1) $$avg=0\alpha+1\alpha\beta+2\alpha\beta^2+...$$ (2) $$avg=\alpha\sum_{k=0}^{\infty}k\beta^k$$ (3) $$avg=\frac{\beta}{\alpha}$$
He then goes about saying that if want the exponential average age of the data set (3) to be the same as the simple (N-datapoints) moving average you just need to solve the equation:
$${\displaystyle \beta/\alpha =(N-1)/2}?$$
Or in plain english:
The exponential average age of the data = simple N average age of the data.
It's all fine but I cannot bridge the gap between equations 2 and 3, how did he solve the power serie into that fraction?
"The average age is the age of each piece of data used in the average, weighted as the data of that age would be weighted. In the exponential smoothing process, the weight given data k periods ago is is:
$$\alpha\beta^k, \beta=(1-\alpha)$$
time-series moving-average
MBdrMBdr
Bridging the gap between equation 2 and 3:
Let us use the fact that the infinite series $\sum_{k=0}^\infty k \cdot \beta^k$ has a closed-form solution given by (see this post for proof):
$$\sum_{k=0}^\infty k \cdot \beta^k = \frac{\beta}{(1-\beta)^2},$$
when $|\beta| = |1-\alpha|<1$ or $\alpha < 1$, for the infinite series to converge to the above result.
Now, let us calculate the average in a few more steps:
\begin{align} \bar{k} &= \alpha \sum_{k=0}^\infty k \cdot \beta^k \\ &= \alpha \cdot \frac{\beta}{(1-\beta)^2} \\ &= \alpha \cdot \frac{\beta}{\left(1-(1-\alpha)\right)^2}\\ &= \alpha \cdot \frac{\beta}{\alpha^2}\\ &= \frac{\beta}{\alpha} \end{align}
I hope this clarifies some things.
PlebPleb
$\begingroup$ That's perfect. Thank you! $\endgroup$
– MBdr
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The association between internal migration and pulmonary tuberculosis in China, 2005–2015: a spatial analysis
Wei-Bin Liao1,
Ke Ju1,
Ya-Min Gao2 &
Jay Pan1,3
Internal migration places individuals at high risk of contracting tuberculosis (TB). However, there is a scarcity of national-level spatial analyses regarding the association between TB and internal migration in China. In our research, we aimed to explore the spatial variation in cases of sputum smear-positive pulmonary TB (SS + PTB) in China; and the associations between SS + PTB, internal migration, socioeconomic factors, and demographic factors in the country between 2005 and 2015.
Reported cases of SS + PTB were obtained from the national PTB surveillance system database; cases were obtained at the provincial level. Internal migration data were extracted from the national population sampling survey and the census. Spatial autocorrelations were explored using the global Moran's statistic and local indicators of spatial association. The spatial temporal analysis was performed using Kulldorff's scan statistic. Fixed effects regression was used to explore the association between SS + PTB and internal migration.
A total of 4 708 563 SS + PTB cases were reported in China between 2005 and 2015, of which 3 376 011 (71.7%) were male and 1 332 552 (28.3%) were female. There was a trend towards decreasing rates of SS + PTB notifications between 2005 and 2015. The result of global spatial autocorrelation indicated that there were significant spatial correlations between SS + PTB rate and internal migration each year (2005–2015). Spatial clustering of SS + PTB cases was mainly located in central and southern China and overlapped with the clusters of emigration. The proportions of emigrants and immigrants were significantly associated with SS + PTB. Per capita GDP and education level were negatively associated with SS + PTB. The internal migration flow maps indicated that migrants preferred neighboring provinces, with most migrating for work or business.
This study found a significant spatial autocorrelation between SS + PTB and internal migration. Both emigration and immigration were statistically associated with SS + PTB, and the association with emigration was stronger than that for immigration. Further, we found that SS + PTB clusters overlapped with emigration clusters, and the internal migration flow maps suggested that migrants from SS + PTB clusters may influence the TB epidemic characteristics of neighboring provinces. These findings can help stakeholders to implement effective PTB control strategies for areas at high risk of PTB and those with high rates of internal migrants.
Over the past two decades, despite considerable effort directed towards tuberculosis (TB) prevention and control, China has continued to experience one of the most significant TB endemics worldwide, with an estimated annual incidence of 67 per 100 000 population in 2015 [1]. In China, among the class A and B infectious diseases, TB has one of the highest notification rates and is a leading cause of death and poverty [2, 3]. The prevalence of TB slowly decreased between 2000 and 2010 [4]. However, due to population growth, the overall number of TB cases continues to increase and there are differences in prevalence between the east, central, and western regions of China [5].
With global economic development, migration flows have become increasingly intense; these play an important role in TB epidemiology, especially in China. With the reform of the household register system (hukou system), the number of internal migrants within China is growing rapidly [6]. In 2015, there were an estimated 247 million internal migrants in China (18% of the total population), and the growth trends are likely to continue [7]. Most internal migrants in China have traveled from rural areas to prosperous economic regions to pursue higher incomes and better lifestyle opportunities [8, 9]. Compared with permanent residents, internal migrants face substantial problems in accessing medical care, insurance, and social security [10]. According to previous studies, migrants are more vulnerable due to low income, poor working and living conditions, less education, and poor awareness of health protection [11,12,13]. Therefore, internal migration poses a challenge for TB control and prevention in China.
Spatial and spatiotemporal analysis of TB notification cases could provide crucial epidemiological information to guide interventions. In recent years, the geographical information system (GIS) and spatial statistics were used to detect the spatial characteristics of TB in China [14,15,16,17,18]. Several studies have demonstrated that TB is not randomly distributed. Furthermore, significant differences in cluster patterns were found between local residents and migrants, indicating that migrants may have been infected before their arrival [19, 20]. However, few studies have simultaneously focused on the spatial and spatiotemporal cluster patterns of TB, and the association between TB and internal migration in China.
This study aims to investigate trends in spatial clusters of sputum smear-positive pulmonary TB (SS + PTB) incidence between 2005 and 2015. Additionally, this study aims to identify the spatial distribution of internal migration flows and to evaluate the relationship between SS + PTB and internal migration at the provincial level. We hope that this study can contribute to informing effective medical resource allocation, guide the design of internal migration interventions, and identify factors underlying the spread of SS + PTB in high-risk areas.
Provincial incidence of notified SS + PTB cases was obtained from the national, web-based Notifiable Infectious Diseases Reporting Information System (NIDRIS), which includes sputum smear-positive, sputum smear-negative, sputum not done, and sputum culture-positive PTB data. Due to its high risk of transmission among the population, SS + PTB is of greatest concern; therefore, we focused on SS + PTB cases from 2005 to 2015 within China. Hong Kong, Macau, and Taiwan of China were not included in our analysis because of data accessibility. The classification of Eastern areas, Central areas and Western areas is based on the standard of the National Statistics Bureau.
The internal migration data were extracted from the national population sampling surveys in 2005 and 2015 (these surveys were conducted nation-wide and covered 16.99 million people (1.325% of the total population) and 21.31 million people (1.55% of the total population), respectively) and from the census in 2010. In the current study, internal migration was defined as a move from one province to another province within China. Internal migration was divided into emigration and immigration. We then calculated the proportion of emigrants (POE) and immigrants (POI) in the total population for each province. Other variables included gross domestic product (GDP) per capita (RMB 10 000; PCGDP), the proportion of people in the population with a college degree or higher (EDU), human immunodeficiency virus (HIV) incidence rate, urbanization rate (UR), and population density (persons per square kilometers; PD). Detailed information for these variables is shown in Tables 1 and 2.
Table 1 Specification of the variables
Table 2 Descriptive statistics of independent variables
Spatial autocorrelation analysis
Global Moran's I is a measure of spatial autocorrelation developed by Moran [21]. It is widely used in public health to investigate spatial clusters of infectious diseases [22, 23]. We calculated global Moran's I statistics in GeoDa (version 1.6.7, GeoDa Center for Geospatial Analysis and Computation, Arizona State University, AZ, USA) in order to examine the spatial autocorrelations of SS + PTB, POE, and POI in the study area. The equation for Moran's I statistic is:
$$ I=\frac{n}{S_0}\frac{\sum_{i=1}^n{\sum}_{j=1}^n{w}_{ij}\left({x}_i-\overline{x}\right)\left({x}_j-\overline{x}\right)}{\sum_{i=1}^n{\left({x}_i-\overline{x}\right)}^2} $$
where n is the number of spatial units, xi or xj is the incidence of SS + PTB and the proportion of emigration or immigration in province i or j, wij is a matrix of spatial weight between province i and j, and S0 is the sum of all \( {w}_{ij}:{S}_0={\sum}_{i=1}^n{\sum}_{j=1}^n{w}_{ij} \). The value of Moran's I usually ranges from − 1 to 1, with positive values indicating a positive association and negative values indicating a negative association. A value approaching − 1 or 1 indicates a stronger association. The Z-statistic is used to test the significance of Moran's I: Z = I − E[I]/STD[I], where E[I] = − 1/(n − 1), STD = E[I2] − E[I]2.
The spatial relationships among the provinces were characterized by the spatial weight matrix. In our study, a first-order queen continuity weights matrix, which defines neighbors as those with either a shared border or vertex, was used for spatial weights. A queen weights matrix was constructed in GeoDa using the province level polygon-shaped file.
Global Moran's I reveals the overall relationship of all the research units in the area. Local Moran's I is a method to explore the local spatial distribution characteristics [24]. In this study, local Moran's I was used to describe the local spatial autocorrelation; it is an indication of the extent of significant spatial clustering of similar values. The sum of local Moran's I is proportional to global Moran's I. The following equation was used to calculate the local Moran's I statistic:
$$ I=\frac{\left({x}_i-\overline{x}\right)}{\frac{1}{n}\sum \limits_{i=1}^n{\left({x}_i-\overline{x}\right)}^2}\sum \limits_{j=1}^n{w}_{ij}\left({x}_j-\overline{x}\right) $$
where n is the number of spatial units, xi or xj is the proportion of emigrants or immigrants in province i or j, and wij is a matrix of spatial weight between province i and j. Like global Moran's I, the value of local Moran's I also ranges from -1 to 1, with a positive value indicating clustering of similar values and a negative value indicating the opposite. In this study, the local Moran's I was used to make the LISA clusters maps. The cluster maps were presented in ArcGIS (version 10.2, ESRI Inc., Redlands, CA, USA).
Analysis of spatial variation in temporal trends
The spatial variation in temporal trends was based on Kulldorff's scan statistics, which are used for the identification of areas with exceptionally different temporal trends [25]. This method assumes that the average annual percentage change of SS + PTB within the scanning window is the same as that outside the window. A circular window is imposed on each location in turn; then, a number of circular windows that are flexible in both size and location are constructed. For each window, a likelihood is calculated, and the most likely cluster is defined as the window with the maximum likelihood, that is, the cluster least likely to be due to chance. Under the null hypothesis, the P-value is obtained from Monte Carlo hypothesis testing. In this study, the Poisson probability model was used, in which the number of cases in each location is under a Poisson distribution. The maximum number of replications for the Monte Carlo simulation was set to 999, and P < 0.05 was considered to be statistically significant [26].
It is reported that only 12% of immune-sensitized individuals actually develop clinical symptoms of TB, and the total number of PTB notification cases in China was approximately 11 million between 2005 and 2015 [27]. Therefore, this suggests that 100 million (8% of the total population in China) people in this time may have been sensitized and were at risk of developing TB. Many studies utilizing spatial temporal mapping suggest that the main guideline for selecting an optimal scanning window is reducing the overlapping areas, or that a single cluster should make up no more than 15% of the whole study area [28, 29]. Moreover, previous research in China at the prefecture level found that 11% was the optimal parameter for spatial cluster sizes, so we analyzed the notification rate of SS + PTB setting the maximum sizes from 8 to 11% of the total population at risk by increments of 1% [18]. When the maximum size is set at 8 to 11%, there are fewer overlaps, and the biggest cluster covered no more than 15% of all the provinces. As such, we choose 8% as the maximum spatial cluster size. The spatial variation in temporal trends of SS + PTB was examined using SaTScan (version 9.4.2, Kulldorff and Information Management Services, Inc., Boston, USA).
Panel data analysis
A fixed-effects model was used to estimate the effects of internal migration, demographic factors, and socio-economic factors on SS + PTB incidence. The internal migration factors included emigrants and immigrants. Emigrant refers to those people who left their household registration place for more than half a year. Immigrant refers to those people who were settled in the current resident area for more than half a year.
We used GDP per capita (RMB 10 000), population density, education level, and urbanization level to reflect the social-economic situation. GDP represents the level of economic development of a region. Education and urbanization level can indirectly affect SS + PTB incidence via the effects of income or health education on TB prevention [30, 31]. The natural logarithm of each variable was used in the construction of the model. The basic descriptive statistics for these variables are presented in Table 2. Due to internal migration data accessibility, only three years (2005, 2010, and 2015) were included in the model. The model can be expressed as:
$$ {Y}_{i,t}=\alpha +{X}_{i,t}\beta +{\varepsilon}_{i,t} $$
where Yi, t is the incidence of SS-PTB, i and t are the province and year, respectively, α is the intercept term, Xi, t is a vector of independent variables, β is the coefficients of the independent variables, and εi, t is the error term. The descriptive analysis and the fixed effects model were performed in Stata (version 12.0; StataCorp, TX, USA).
Descriptive analysis of SS + PTB cases
A total of 4 708 563 SS + PTB cases were reported in China between 2005 and 2015, of which 3 376 011 (71.7%) were male and 1 332 552 (28.3%) were female. The number of male cases was twice that of female cases. The notification rates of SS + PTB decreased from 41.90 cases per 100 000 population in 2005 to 17.93 cases per 100 000 population in 2015, with an annual average rate of 29.84 per 100 000 population. In addition, a significant proportion of the SS + PTB infections were aged > 60 years old (around 30%) and 45 to 60 years old (around 26%). Among the reported cases, around two-thirds were peasants; the percentage of SS + PTB cases that were classified as a householder or unemployed increased over the years of the study (see Table 3).
Table 3 The demographic characteristics of SS + PTB cases in China from 2005 to 2015
Figure 1 shows the spatial distribution of the annual average notification rate of SS + PTB, and the proportions of internal emigrants and immigrants in China at the provincial level from 2005 to 2015. There were obvious spatial variations in the annual average notification rate of SS + PTB, with rates ranging from 9.87 to 54.48 per 100 000 population. The highest SS + PTB notification rates were found in the provinces of Xinjiang, Qinghai, Hubei, Jiangxi, and Hainan, primarily in the northwest, southeast, and south of China.
The annual average notification rate of SS + PTB and the proportion of emigrants/immigrants at province level in China, 2005–2015. a illustrates the notification rate of SS + PTB. b and c illustrate the proportion of immigrants and emigrants, respectively. TB: Tuberculosis; SS + PTB: Sputum smear-positive pulmonary TB
Anhui (14.01%), Jiangxi (11.69%) provinces and Chongqing (10.62%) Municipality had the highest levels of internal emigrants. Further, provinces in the Central South had higher levels of emigrants (around 9.25% for each province). Provinces with the highest levels of internal immigrants were located in the eastern regions, such as in Beijing (31.15%), Tianjin (19.6%), Shanghai (34.86%), Zhejiang (18.51%), and Guangdong (20.19%) provinces. Provinces with lower levels of immigrants were also located in southeast areas close to Guangdong or Zhejiang. Interestingly, those provinces also had lower levels of internal emigrants.
Global and local spatial autocorrelation
The global Moran's I statistics showed positive spatial autocorrelations in SS + PTB in China each year (as presented in Table 4). Further, there was an increasing trend in global Moran's I, which can be divided into three periods: 2005–2007, 2008–2009, and 2010–2015. The highest spatial autocorrelations were observed in 2011–2015, ranging from 0.319 to 0.388. Furthermore, the proportion of internal emigrants and immigrants also exhibited significant positive spatial autocorrelations each year (see Table 5).
Table 4 Globe Moran's I statistics of SS + PTB in China, 2005–2015
Table 5 Globe Moran's I statistics of emigrant and immigrant in China, 2005, 2010 and 2015
Figure 2 shows the local Moran's I statistics. We observed the stability of spatial clusters each year during the study period, and the clusters were stable within most provinces. Provinces such as Shaanxi, Henan, Chongqing, Guizhou, Inner Mongolia, and Hubei (mostly located in central China) showed a low-low type of relationship, indicating that these provinces had a low proportion of internal immigrants and that the surrounding provinces also had low proportions of immigrants. Jiangsu Province, which is located on the southeast coast of China, had a high-high type of relationship, meaning that a high proportion of immigrants were found in Jiangsu and that the surrounding provinces also had high proportions of immigrants. Chongqing, Guizhou, Hunan, and Hubei exhibited high-high types of relationships for the proportion of internal emigrants. On the other hand, Hebei Province exhibited a low-low relationship and Guangdong a low-high relationship.
The LISA cluster map of the internal emigrant and immigrant in China. a, b and c for immigrant, d, e and f for emigrant. a, b and c show the spatial clustering of immigrants in 2005, 2010 and 2015, respectively. d, e and f show the spatial clustering of emigrants in 2005, 2010 and 2015, respectively. LISA: Local indicator of spatial association
Spatial variation in temporal trends
From 2005 to 2015, there was a 6.96% annual average decrease in the notification rate of SS + PTB. We identified one most likely cluster and ten secondary clusters; two provinces/municipalities showed increasing annual trends; and nine provinces showed a slower decreasing annual trend compared to the outside time trend (Table 6). Guizhou and Beijing showed increasing annual average trends of 0.207 and 0.222%, respectively. Guangdong, Hunan, Jiangxi, Zhejiang, Liaoning, Qinghai, Hainan, Guangxi, Xizang, and Hubei showed decreasing annual average trends of 3.098, 3.256, 2.734, 3.581, 4.286, 4.082, 6.037, 4.358, and 6.483%, respectively. Figure 3 showed the spatial distribution of the most likely and secondary clusters. Most clusters were located in central and southern provinces of China; although Qinghai and Tibet are in west China, while Beijing and Liaoning are in northeast China.
Table 6 Spatial clusters of temporal trends of smear positive PTB in China, 2005–2015
The spatial variation in temporal trends of smear positive PTB in China, 2005–2015. PTB: Pulmonary TB
The association between internal migration and SS + PTB
Three fixed-effect models were examined: one with the proportion of internal emigrants (model 1), one with the proportion of internal immigrants (model 2), and another with both the proportion of internal emigrants and the proportion of internal immigrants (model 3); panel regression results are presented in Table 7. The results showed that the proportion of emigrants and immigrants, per capita GDP, and the urbanization rate were found to be significantly associated with the SS + PTB rate. From model 1 and model 2, the coefficients for emigrants and immigrants were 0.632 and 0.536, respectively. Furthermore, we found that the proportion of emigrants was significantly positively related to SS + PTB, while the proportion of immigrants was not significantly related to SS + PTB in model 3. Further, model 1 had the highest R-square value; the model with only emigrants was able to explain 40% of the variation in SS + PTB rate.
Table 7 The result of fixed effect model
Internal migration flow maps and reasons
Based on the results of SS + PTB spatial cluster, the most likely cluster and the five secondary clusters were chosen to visualize the internal migration flow maps. Among these clusters, Guangdong, Beijing and Zhejiang are developed and prosperous provinces, and Guizhou, Hubei and Hunan are located in southern China, near Guangdong and Zhejiang provinces with a large immigrant population. The proportion of emigrants in these two provinces was significantly higher than the proportion of immigrants in Guizhou (POE: 10.22% vs POI: 2.08%), Hubei (POE: 9.01% vs POI: 1.94%), and Hunan (POE: 10.15% vs POI: 1.13%). In contrast, the proportion of immigrants was obviously higher than the proportion of emigrants in Beijing (POI: 31.15% vs POE: 1.66%), Zhejiang (POI: 18.51% vs POE: 3.45%), and Guangdong (POI: 20.19% vs POE: 1.03%).
Figure 4 shows the flow of internal migrants for the six spatial clusters. The highest proportion of immigrants from Hebei (21.60%) flowed into Beijing, with immigrants from other spatial clusters accounting for 17.07%. Similarly, the highest proportion of immigrants from Anhui (19.36%) flowed into Zhejiang, and other spatial clusters accounted for 41.08%. The highest portion of immigrants from Hunan (21.48%) flowed into Guangdong, with immigrants from other spatial clusters accounting for 42.96%. In contrast, 33.45 and 26.56% of the emigrants in Guizhou flowed into Zhejiang and Guangdong, respectively. Further, 43.87 and 13.58% of the emigrants in Hubei flowed into Guangdong and Zhejiang, respectively. We also found that the highest proportion of emigrants from Hunan flowed into Guangdong (67.18%) and Zhejiang (8.84%).
The internal migration flow of Beijing, Zhejiang, Guangdong, Guizhou, Hubei and Hunan province. a, b and c for immigrant flow, d, e and f for emigrant flow. a, b and c present the immigrant flow of Beijing, Zhejiang, Guangdong over 2005–2015, respectively. d, e and f present the emigrant of Guizhou, Hubei and Hunan over 2005–2015, respectively. TB: Tuberculosis; SS + PTB: Sputum smear-positive pulmonary TB
Based on these results, it appears that internal migrants are more likely to flow into neighboring provinces or southern provinces. In Table 8, over 70% of internal migrants were leaving their place of household registration for work and business. Other reasons, such as study and training, only accounted for around 30% of internal migration.
Table 8 The reasons of internal migration in 2005, 2010 and 2015
In the current study, we identified spatial clusters and spatial variations in temporal trends in the distribution of SS + PTB cases in China between 2005 and 2015. There was a decreasing trend in the notification rate of SS + PTB; averaging an annual decrease of 6.96% in notifications. We also observed spatial variations in the distribution of internal migration. Further, these clusters were stable across the year in each year examined: 2005, 2010, and 2015. Compared with the flow of immigration, the flow of emigration was more consistent with the distribution of SS + PTB.
The global Moran's statistic results indicated that although there was a decreasing trend in the SS + PTB notification rate, the distribution of the SS + PTB notification rate became increasingly clustered over time. One potential reason is the flow of patients from prefecture cities or counties to the provincial capital for better diagnosis and treatment; this could impact on the clustering of SS + PTB [8]. It should be noted that due to the time dimension limitation, local Moran's test was not used to identify clusters of SS + PTB [32, 33]. Instead, we used the spatial variation of temporal trend method to evaluate the space-time distribution of SS + PTB based on Kulldorff's scan statistical methodology.
The results showed that the most likely cluster and the ten secondary clusters were located in South, Northeast, and West China. Among these clusters, Guizhou, Hubei, Hunan, Guangxi, and Jiangxi provinces are in central-southern China. The main reasons for the high SS + PTB risk in these five provinces include low levels of socioeconomic development, the large proportion of the population living in poverty, and poor medical care [34]. Guangdong, Hainan, and Zhejiang are developed and prosperous provinces; internal migrants in these provinces accounted for a large part of the population, especially in Guangzhou and Hangzhou, the capital cities of Guangdong and Zhejiang, respectively. Research in Zhejiang has indicated that nearly one-third of reported TB cases are migrants [35]. Beijing was another cluster that exhibited an increasing trend in SS + PTB cases. It is believed that internal migration plays an important role in promoting growth in the TB epidemic in Beijing [6, 20]. Similarly, in Liaoning Province, the number of migrant TB cases increased by 30.55% since 2006, with the majority of cases observed in Dalian and Shenyang [36]. Further, we identified two clusters in Qinghai and Tibet, in the Northwest of China. Qinghai and Tibet are the largest political subdivisions in China, accounting for one-fifth of China's total territory. Poor traffic conditions, uneven allocation of public health resources, and limited knowledge of TB are possible reasons for the high rates of TB in Qinghai and Tibet [32, 37]. With respect to traffic conditions, the Qinghai-Tibet railway was not operational until 2006; and the disparity of traffic infrastructure in such regions likely kept people from seeking medical help.
We also found significant positive spatial autocorrelations in both the proportion of internal emigrants and immigrants, and the clusters where this was noted were concentrated in central-southern China. In contrast, some high-high clusters for the proportion of emigrants overlapped with the low-low clusters for the proportion of immigrants. This may suggest that people in those areas are more likely emigrate to other provinces, or that these areas are less attractive for internal immigrants.
We observed that the proportions of emigrants and immigrants were statistically significant in the models 1, 2 and 3, while emigrants were more significant than immigrants. Previous studies have demonstrated that the number of rural-to-urban migrants has been increasing steadily, and this has a significant impact on urban TB epidemics [6, 19, 20]. In this study, internal migration was assessed in terms of emigration and immigration. The results of the fixed effects model suggested that the model with the emigration variable was able to explain more variation of SS + PTB. Further, we compared the spatial temporal clusters of SS + PTB and high-high clusters of emigration. Some emigration clusters overlapped with the SS + PTB clusters, primarily in central-southern China. Therefore, we found that the migrant population in those clusters was at an increased risk of SS + PTB infection and transmission.
The results of the emigration flow maps indicate that migrants from central-southern clusters would likely prefer to emigrate to Guangdong and Zhejiang. On the contrary, the results of the immigration flow maps indicate that migrants in Guangdong, Zhejiang, and Beijing primarily come from neighboring provinces. These results are consistent with the provincial distribution of TB cases in the migrant population in China [38]. Therefore, strategies to control TB in these areas must consider the characteristics of internal migration flow. Since the reform of the Chinese economy, southeast coastal provinces have become the most economically developed areas in China. When people are aware that local health services cannot meet their needs, they will seek better health services in different provinces. By the end of 2015, it was estimated that there were around 8.22, 32.01, and 23 million internal migrants in Beijing, Guangdong, and Zhejiang, respectively. As shown in Table 8, the main reasons for internal migration were the pursuit of work and business. The high-speed China railway also makes it convenient for people to migrate to neighboring provinces within a few hours.
Per capita GDP and education level were found to be statistically significant in the models 1, 2 and 3, indicating that economic development, and improved awareness of TB could help to alleviate the SS + PTB epidemic. Our results are consistent with other studies [15, 39, 40]. Evidence indicates that TB is a poverty-related disease, with an average treatment period of 6 months [41]. Due to different medical insurance systems, internal migrants who come from country areas are usually covered by the new rural cooperative medical system, and their health insurance is generally not transferable to a new location. Further, migrants are required to return to their hometowns (houkou registered places) to seek medical treatment under their insurance. However, due to low level medical and transportation costs, thus, they are less likely to go back to their hometown to benefit from health insurance [42]. Despite the available strategies for free diagnosis and treatment of TB, such as the Direct Observed Treatment Short Course (DOTS) strategy, there remains a lack of knowledge about these among the migrant population. This is a critical problem in China and is a barrier to accessing TB care [43, 44]. The national plan to control TB 2011–2015 required 85% of the public to possess some knowledge of TB by the end of 2015. However, national and local surveys show that poor awareness of TB remains in most areas.
Urbanization, the HIV incidence rate, and population density were predictors that not statistically significant in the models 1, 2 and 3. We found that HIV incidence rate was positively associated with SS + PTB. This may be due to a disparity in the distribution of SS + PTB and HIV; for example, there may be a high prevalence of TB/HIV co-infection in poor areas [45]. Due to data accessibility limitations, we did not have data on the PTB/HIV coinfection incidence rate. Population density was also positively correlated with SS + PTB. Similar results have been reported in other studies [46,47,48]. Due to poor financial conditions, migrants are more likely to be living in crowded places that favor the spread of Mycobacterium tuberculosis; this explains the association between population density and SS + PTB. In addition, we observed that urbanization was negatively associated with SS + PTB. This means that urban areas that have well-developed public health infrastructure, better-qualified health care workers, and where most of the residents are covered by medical insurance (such as urban resident basic health insurance or urban employee basic health insurance), are less at risk of SS + PTB [44, 49]. In contrast, the prevalence of TB in rural areas is consistently higher than in urban areas in China [4, 50, 51]. Due to their slower developing economies, these areas have limited healthcare resources and continuing government input is required to improve TB prevention and control efforts in these areas.
Several limitations of this research should be noted. First, this study did not include migrant SS + PTB cases, and this may affect the distribution of PTB. This is because the national health and family planning commission began to conduct dynamic surveys of migrants in 2009; thus reliable data on the migrant population was not available in 2006–2009. Second, the spatial level analysis was conducted by province, and the observed cluster patterns may depend on the spatial scale chosen. It is preferable to identify spatial distribution by smaller geographical units such as a county. Third, the selection of the maximum circle size of the scanning window may influence the results of the spatial-temporal scan statistics. In this study, the maximum size of the scanning window was set as 8%; further investigation is needed to test the sensitivity of the scanning window. Finally, this was an ecological study examining the association between SS + PTB rate and risk factors; the potential ecological fallacy is inevitable.
In short, there was a decreasing trend in the notification rate of SS + PTB between 2005 and 2015. We found spatial-temporal clustering of SS + PTB and spatial variation in internal migration in China. The SS + PTB clusters were mainly located in central-southern China, and the internal migration clusters were mainly located in central inland China. The proportions of emigrants and immigrants were positively correlated with SS + PTB, while per capita GDP and education level were negatively correlated with SS + PTB. The proportion of emigrants was a more significant predictor of SS + PTB and could explain more variation in SS + PTB compared to the proportion of immigrants. Further, we found that the SS + PTB clusters overlapped with emigration clusters, and the internal migration flow maps suggested that migrants from SS + PTB clusters may influence the TB epidemic characteristics of neighboring provinces. Therefore, we recommend that policymakers acknowledge that migrants are a vulnerable population group. Cooperative efforts should be strengthened between provinces where there are high proportions of emigration and immigration in order to enable effective TB control. Further research is needed to explore the TB epidemic characteristics associated with internal migration based individual migrant data, particularly in central-southern China.
The monthly reported all-forms PTB cases from January 2005 to December 2015 in each of 31 provinces of mainland China were obtained from the web-based national Notifiable Infectious Diseases Reporting Information System (NIDRIS). We would like to share statistical results of this study. If anyone needs these data, please contact the corresponding author for a soft copy.
EDU:
College degree or higher
GDP:
GIS:
Geographical information system
Local indicator of spatial association
NIDRIS:
Notifiable Infectious Diseases Reporting Information System
PCGDP:
Gross domestic product per capita
Proportion of emigrants
POI:
Proportion of immigrants
SS + PTB:
Sputum smear-positive pulmonary TB
TB:
UR:
Urbanization rate
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This study was supported by the National Science Foundation of China (Grants No.: 71303165 and 71874116).
West China School of Public Health and West China Fourth Hospital, Sichuan University, No. 17, Section 3, Ren Min Nan Road, Chengdu, 610041, Sichuan, China
Wei-Bin Liao, Ke Ju & Jay Pan
Medical College, Northwest Minzu University, Lanzhou, China
Ya-Min Gao
West China Research Center for Rural Health Development, Sichuan University, Chengdu, China
Jay Pan
Wei-Bin Liao
Ke Ju
WBL conceived this study. WBL and YG designed the study. WBL and YG collected documents and extracted data. WBL and KJ analyzed data and produced the figures and tables. WBL wrote the first draft of the paper. WBL, YG, KJ and JP revised the paper. All authors approved the final submitted version.
Correspondence to Jay Pan.
In this study, TB data were collected by web based PTB surveillance system. Therefore ethical consent was not essential.
Liao, WB., Ju, K., Gao, YM. et al. The association between internal migration and pulmonary tuberculosis in China, 2005–2015: a spatial analysis. Infect Dis Poverty 9, 5 (2020). https://doi.org/10.1186/s40249-020-0621-x
DOI: https://doi.org/10.1186/s40249-020-0621-x
Internal migration
Spatial inequality, infectious diseases and disease control
Human migration, conflict and Infectious Diseases
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CommonCrawl
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Application of response surface methodology for color removing from dyeing effluent using de-oiled activated algal biomass
Samar A. El-Mekkawi ORCID: orcid.org/0000-0003-0669-24501,
Rehab A. Abdelghaffar2,
Fatma Abdelghaffar2 &
S. A. Abo El-Enin1
Conservation of the ecosystem is a prime concern of human communities. Industrial development should adopt this concern. Unfortunately, various related activities release lots of noxious materials concurrently with significant leakage of renewable resources. This work presents a new biosorbent activated de-oiled microalgae, Chlorella vulgaris, (AC) for biosorption of Acid Red 1 (AR1) from aqueous solution simulated to textile dyeing effluent. The biosorption characteristics of AC were explored as a function of the process parameters, namely pH, time, and initial dye concentration using response surface methodology (RSM).
Optimization is carried out using the desirability approach of the process parameters for maximum dye removal%. The ANOVA analysis of the predicted quadratic model elucidated significant model terms with a regression coefficient value of 0.97, F value of 109.66, and adequate precision of 34.32 that emphasizes the applicability of the model to navigate the design space. The optimization depends on the priority of minimizing the time of the process to save energy and treating high concentrated effluent resulted in removal % up to 83.5%. The chemical structure and surface morphology of AC, and the dye-loaded biomass (AB) were characterized by Fourier transform infrared spectroscopy (FTIR), scanning electron microscopy (SEM), energy-dispersive X-ray analysis (EDX), and transmission electron microscope (TEM). The activation process transforms the biomass surface into a regular and small homogeneous size that increases the surface area and ultimately enhances its adsorption capacity
The optimization of the process parameters simultaneously using RSM performs a high-accurate model which describes the relationship between the parameters and the response through minimum number of experiments. This study performed a step towards an integrated sustainable solution applicable for treating industrial effluents through a zero-waste process. Using the overloaded biomass is going into further studies as micronutrients for agricultural soil.
The demand for industrial products is accelerating because of population growth (Heffron et al. 2020). Dyes are necessary for coloring and finishing the final products' packages of various industries such as textiles, leather, cosmetics, and food (Ribeiro and Veloso 2021; Tang et al. 2021). As a consequence of the variety of industrial products, enormous amounts of effluent discharge contaminated with dyes require treatment (Abdulrazzaq et al. 2021, Wang et al. 2021). The textile industry uses around 30% of synthetic dyes. This industry is developed globally, growing 1 trillion dollars, employing 35 million employees, and contributing 7% of the total world exports (Lellis et al. 2019; de Oliveira Neto 2021). Despite its high attention, this industry is an extremely global polluter as a consequence of the large amounts of the drained dyes in the textile effluents (da Silva et al. 2021).
The discharge of the untreated effluents overloaded with dyes into the water bodies is eco-destructive (Kishor et al. 2021). It prevents light penetration through water, causing a reduction in the photosynthesis rate of phytoplankton which affects the aquatic biota (Berradi et al. 2019; Watari et al. 2020). The dyes also act as carcinogenic, toxic, and mutagenic agents (Liang et al. 2020). It is rather difficult to treat dye effluent for such dyes of aromatic molecular structure as acid, basic, direct, and reactive dyes. The complex aromatic structure enhances molecular stability and tolerates biological degradability (Zhang et al. 2020).
Environmental regulations have grown more stringent all over the world. Therefore, the dye effluent should be treated carefully before discharge. There are several methods for color removal as coagulation and flocculation (Abdulrazzaq et al. 2021), membrane filtration (Mansor et al. 2020), foam separation (Soylu et al. 2020), and chemical precipitation (Shen et al. 2019). A novel approach based on activated carbon is the potential for dye removal. Some limitations are mentioned for this approach, such as the shortage of renewable resources and the difficulties in the regeneration of the substrate. The global demand for the production of activated carbon using alternative environmentally friendly resources has annually increased (Liu et al. 2020; Suhas et al. 2021).
Lately, the use of renewable raw materials, plenteously available and inexpensive to produce valuable activated carbon has attracted a lot of attention (Taer et al. 2021). Thermal activation can be done in two stages including carbonization and subsequently activation. Dehydration of the samples enhances the yield of carbonization as a consequence of the sufficient removal of the volatile compounds (Suganya et al. 2020).
Several microorganisms like yeasts, bacteria, fungi, and algae are used to develop cheaper alternatives (Cheng et al. 2020; Semião et al. 2020). In this respect, microalgae have high sorption capacity and high binding affinity due to their functional groups, such as amino, sulfate, phosphate, carboxyl, and hydroxyl groups (Khorasani and Shojaosadati 2019). The other merits of using microalgae as a feedstock are the rapid growth rate due to its high photosynthetic efficiency, and less land area required without competing with the food crops (Chew et al. 2021).
The efficiency of the biosorption process depends on several parameters like the concentration of dyes as the pollutant, pH solution, and the processing time (Adeogun et al. 2019). Optimization of these parameters requires a large number of experiments to evaluate the effect of each one. Therefore, response surface methodology (RSM) is the best route to optimize the three parameters simultaneously and investigate the interaction between the parameters (Khalifa et al. 2019). The RSM based on the central composite design (CCD) ensures the best fitting of the predicted model in case of significant curvature of the relation between the model parameters and the resulted response (Asfaram et al. 2016).
The previously reported researches tackled micro-organisms as biosorber in their live form or dried biomass. Asfaram et al. (2016) studied the potentiality of isolated Yarrowia lipolytica isf7 to remove malachite green from culture. The optimum dye removal % was 80% at pH 7, temperature 25 °C, 24 h, and 35 mg/L initial dye concentration with a sharp decrease in dye removal for alkaline pH above 7.5 and acidic pH below 5. Yeddou-Mezenner (2010) performed a kinetic study about the effect of waste P. mutilus dry biomass (50–215 µm) on removing Basic Blue 41 dye from aqueous solutions. The maximum percent removal was 65% at initial concentration 50 mg/L, pH 9–8, 30 °C, and 60 min.
To the best of our knowledge, there are no published reports on the application of the de-oiled activated carbon from algae in biosorption. Based on the integrated process to utilize the extracted oil out of Chlorella vulgaris in different industries, the residual biomass is used as an effective biosorbent via activation. The potentiality of the overloaded biosorbent as a nano-nutrient for agriculture is suggested for further research.
In this study, the biosorption of Acid red1 (AR1) dye from aqueous solutions using activated de-oiled microalgae dried biomass was studied. The pH of the effluent stream, the contact time, and the dye concentration as three parameters affecting the dye removal from a simulated dye effluent were studied and analyzed using CCD in RSM. Finally, a prediction model for the optimum dye removal percentage was derived considering the priority of minimizing the contact time at maximum concentration. Fourier transform infrared spectroscopy (FTIR) analysis was performed to elucidate the functional groups involved in the adsorption process. The change of the amorphous structure of biosorbent was to be investigated using scanning electron microscopy (SEM) analysis, and the elemental analysis was to be detected by energy-dispersive X-ray analysis (EDX).
Dry algal biomass of Chlorella vulgaris was acquired by the Algal Biotechnology Unit, National Research Center, Egypt. The microalga was defatted using ethanol in a ratio of 1:6 W/V. The extracted oil was used in other research work for biodiesel production, while the residual biomass (RB) was collected to examine its potentiality as a dye biosorbent. The used dye Acid Red 1 (AR1) of the structure illustrated in Fig. 1 was purchased from Sigma-Aldrich Company.
Dye structure
Preparation of biosorbent
Activation took place by adding 1 mL of 97% H2SO4 drop by drop to 10 g of RB and kept at room temperature for 24 h. The excess solution was clarified off, while the residue was air-dried and carbonized at 400 °C for 1 h in a muffle furnace. The dye solutions were prepared by dissolving a known amount of dye according to the requirements of each experiment (15–50) mg in a 1000-mL measuring flask with deionized double distilled water.
Batch adsorption study
Batch adsorption experiments were carried out in 1000-mL Erlenmeyer flasks containing 25 mL of dye solution with scheduled experimental design, and 2 g of AC. The flasks were agitated using a magnetic stirrer for the required time. After centrifugation at 3000 rpm for 5 min, the remaining dye concentration was measured using UV–Vis spectrophotometer at 600 nm. The adsorption efficiency as a reflection of the dye removal percentage and the adsorption capacity was calculated by Eqs. (1) and (2), respectively (Asfaram et al. 2016):
$$\hbox{Dye Removal} \%=\frac{C_{{\rm o}} - C_{{\rm f}}}{C_{{\rm f}}} \times 100$$
$$q=\frac{C_{{\rm o}} - C_{{\rm f}}}{X} \times V$$
where Co is the initial dye concentrations (mg/L), Cf (mg/L) is the final dye concentrations (mg/L), q is the amount of a dye adsorbed (mg/g), V is the solution volume (L), and X is the adsorbent dosage (g/L).
The process parameters, namely initial dye concentration (D), pH (P), and contact time (T), were optimized using response surface method (RSM). The relationship between the parameters and the response was determined using central composite design (CCD) in RSM using Design Expert-6.0.8 software during a trial period. The examined range of the pH was 3–10, while the range of dye concentration was 15–50 mg/L, and the range of contact time was 30–110 min. The selected ranges as shown in Table 1 have been chosen based on preliminary experiments. The parameters presented in Table 1 were five levels coded as − α, − 1, 0, + 1, + α, respectively. The total number of experiments was 20 with eight factorial points (23), six axial points (2 × 3), and six replicate points according to CCD.
Table 1 Matrix for the central composite design (CCD)
The mathematical relationship between the parameters and the response was to be approximating by the following second-order polynomial model (El-Mekkawi et al. 2019)
$$Y={a}_{0}+{a}_{1}{X}_{1}+{a}_{2}{X}_{2}+{a}_{3}{X}_{3}+{a}_{11}{X}_{1}^{2}+{a}_{22}{X}_{2}^{2}+{a}_{33}{X}_{3}^{2}+{a}_{12}{X}_{1}{X}_{2}+{a}_{13}{X}_{1}{X}_{3}+{a}_{23}{X}_{2}{X}_{3}$$
where Y is the predicted response (dye removal %); a0 is the intercept regression constant; a1, a2, …. a23 are regression coefficients; and X1, X2 and X3 are the independent variables investigated. The analysis of variance (ANOVA) was to be performed to justify the significance and fitting of the developed regression model.
Measurements for characterization of the algal biosorbent
Fourier transform infrared (FTIR) spectroscopy analysis
The functional groups of RB, the activated de-oiled algae (AC), and those overloaded with dye (AB) were detected using FTIR spectrophotometer Model-JASCO FT/IR-4700 in the infrared spectral range of 400–4000 cm−1 wavelength (Brangule et al. 2020). FTIR figures the information specific to the groups that explain the interactions of the group with the dye molecule.
Scanning electron microscope (SEM) & energy-dispersive X-ray spectroscopy (EDX)
Morphological analysis was performed using SEM (Model, Joel JAX-480A) with an accelerating voltage of 30 kV and magnification between 10.000X and 400.000X (Ahmed et al. 2019). The EDX unit was used to evaluate the carbon w% in AC and detect the elements in AB (Sastre de Vicente et al. 2020).
High-resolution transmission electron microscope (TEM)
The particle sizes of algal biomass samples RB and AC were examined under a high-resolution transmission electron microscope [(JEM-1230) JEOL] (Zohoorian et al. 2020).
The results revealed that the dye removal percent of AR1 is 3.4% using RB, and it achieved 96% using AC at the same conditions of operation. These conditions are 15 mg/L initial dye concentration and 3 pH for 1 h. Therefore, the model describes the relationship between the parameters affecting the dye removal process is applied for AC as biosorbent.
Description and analysis of the model
The model results for the design points show a significant curvature; thus the axial points are used to fit the model. The quadratic model is selected depending on the statistical results of the suggested model as summarized in Table 2. The probability F-value revealed that the suggested model is significant, while the error function is not significant. The ANOVA analyses of the quadratic model in Table 3 show that the model terms are the linear factors (D, P, T) and the quadratic factor P2. The statistical analyses revealed that the regression coefficient R2 is 0.97 and that the adjusted R2 is in reasonable agreement with the predicted R2. The value of Prob > F is less than 0.05, whereas the lack of fit is not significant.
Table 2 Suggested model summary statistics
Table 3 ANOVA results for the CCD
In addition to the linear relationship of the normal plot of the residuals illustrated in Fig. 2a, the actual values of the response are represented versus the predicted values in Fig. 2b. The data points are spread out from the mean referring to the high variance as the coefficient of variation (C.V.) is greater than one as elucidated in Table 4. Therefore, the model can accurately predict the relationships between the model terms within the selected range. The regression equation in terms of the actual variables is as follows:
a Normal plot of the residuals, b actual values of the response versus the predicted values: c simultaneous effect of initial dye concentration and effluent pH on Dye removal % at the center value of time, and the model contour, d interaction relationships between the model parameters
Table 4 Statistical analysis for the response
$$Removal \%=111.1621-0.45771 D-5.38829 P+0.075686 T+0.39671 {P}^{2}$$
The model graph and its contour are exhibited in Fig. 2c. It represents the effect of pH on the model in a parabolic shape that reflects on the contour graph while Fig. 2d elucidates the nonexistence of any interaction in the design boundaries.
Optimization of the model solutions
The actual values of the experimental results are very close to the predicted values of the model; this congruency cause a linear relationship as illustrated in Fig. 2b. Meanwhile, the maximum dye removal percent 98% is achieved at 15 mg/L dye concentration, 110 min, and 3 pH; 97% dye removal is achieved at 10 pH. Remarkably, 96% dye removal is progressed at 32.5 mg/L initial dye concentration, 12.39 pH, and 70 min. The optimization of the three parameters and their response simultaneously is such difficult; therefore, the derived model is the best route for optimization.
The optimization of the model parameters is achieved by defining the priority of each parameter using the Design-Expert software to achieve maximum response. The results of the predicted model revealed that the optimum conditions to achieve 83.5% dye removal are 37.87 mg/L initial dye concentration and 3 pH for 30 min. Remarkably, the same results are achieved at 10 pH at a confidence interval of 95% and a prediction interval of 95%. The adsorption capacity calculated using Eq. 2 is 15.8 mg/g. Contrastingly, for 100% desirability, the operating conditions are 15 mg/L initial dye concentration, pH 3, and 110 min are required to achieve 100% dye removal. The optimum conditions should consider the priority of treating high concentrations of pollutants at minimum time as discussed in detail in Discussion section.
Characterizations of de-oiled algal biosorbent
Fourier transform infrared spectroscopy (FTIR)
FTIR analysis in Fig. 3 illustrates the functional groups of RB, AC, and AB. The presence of weak secondary amines -NH and primary amines –NH2 is proved by the peaks ranging from 3758.58 to 3830.9 cm−1 (Tan et al. 2018; Déniel et al. 2020; Silva et al. 2020). The border absorption peaks at wavelength 3156.90–3275.50 cm−1 elucidate the existence of stretching hydroxyl group –OH (Kumar et al. 2019) that found in all samples with a higher-intensity value for biosorbent RB.
FTIR bands for RB, AC, and AB
The stretching –CH group and aliphatic –CH2 group have a leading role in the adsorption process (Asemani et al. 2020). The peaks at 2924.52–2965.02 cm−1 are exhibiting the presence of –CH2 group with asymmetric stretching in AC more than RB. The methyl symmetric stretching –CH group exists in both AC and AB at wavelength 2880.17 cm−1.
Sulfur exists as –SH group at the wavelength 2372.01 cm−1 for sample AC. The –NO as symmetric stretching vibration group appears at wavelength 1542.77 cm−1 for RB. The nitro group is strongly electron-withdrawing; this elucidates the lower efficiency of RB than AC as biosorbent for AR1. The IR spectrum at 1094.4–1092.48 cm−1 is illustrating the Si–O group (Asemani et al. 2020).
Scanning electron microscope (SEM)/(EDX)
Figure 4 illustrates the microstructural and morphological changes in each of RB, AC, and AB algal biomass that was analyzed using scanning electron microscopy (SEM). The shape of RB shows a smooth and irregular surface with random heterogeneous sizes. The surface of AC has a regular and small homogeneous size that increases the surface area and ultimately enhanced its adsorption capacity. On the other hand, the AB has elongated regular fiber bundles with a smooth distinct surface. The carbon w% is measured and examined using EDX analysis. The most significant result is the increase of carbon w% in AC by 29%.
SEM images of RB, AC, and AB
Transmission electron microscope (TEM)
The results of TEM illustrated that the particle size of the used RB is over 200 nm as illustrated in Fig. 5 while the particle size of AC is in the range of 12–31 nm. This analysis is used to figure the effect of activation process that enhances reduction of the particle size. This contributes in explanation of the wide difference between RB and AC performance as biosorbent.
TEM images of RB, and AC
Eco-friendly environmental disposal of wastes:
EDX analysis detected several elements in AB as illustrated in Fig. 6. These elements as C, O, Na, Si, P, S, and Ca are mixed to agricultural soil as shown in Fig. 6. The plant growing in soil mixed with AB samples is significant. This prospective function of over-loaded biomass as micro-nutrients for agricultural soil needs more study for a long period with further analyses.
Elemental analysis for AB using EDX and its reflection on plant
The quadratic model is selected depending on the model summary statistics in Table 2. The cubic model is aliased, which prevents the proper navigation and transformation for the cubic model. Meanwhile, the quadratic model has the maximum values of R2, Adj. R2, and Pred. R2; they are 0.97, 0.96, and 0.94, respectively. It also has a minimum PRESS value that indicates the predictive ability of the model. The probability function of the model elucidated Prob > F smaller than 0.05 for both linear and polynomial models. However, the sequential sum of squares favors the highest order polynomial, where the additional terms are significant and the model is not aliased while the probability function of the lack of fit is significant for both linear and factorial models that elucidate high error for both.
In Table 3, the model F value of 109.66 implies the model is significant. The model terms D, T, P2 are significant for Prob > F smaller than 0.05. Despite the term P is not significant, it is required to support hierarchy. In Table 4, the adequate precision ratio of 34.32 indicates an adequate signal-to-noise ratio that elucidates that this model can be used to navigate the design space. The coefficients of the actual factors in the model equation are to be displayed in six decimal places without any approximation; this action is necessary to justify the required accuracy as much as possible.
The normal probability plot of the studentized residuals is illustrated in Fig. 2a elucidates the normality of residuals distribution. The actual values of experiments versus the predicted values through the model calculations in Fig. 2b are very close using linear regression. The model graph and its contour in Fig. 2c represents the effect of the term P2 on the response. Either low or high pH reflects increasing the percent removal, while neutrality exists at the vertex point. The absence of interaction between the parameters as illustrated in Fig. 2d emphasizes the ability of the model to navigate the design space.
The main target of developing the predicting model is to optimize the process parameters to achieve maximum the percent removal. Thus, the highest importance has been given to the response, followed by the process parameters. The logical goal is to treat maximum dye concentration during the minimum time, whereas neutral solutions have a lower response than acidic and ones alkaline, i.e., pH has lower importance. The lower response area, as shown in the 3D model graph Fig. 2c and its contour, is located at the bottom of the convex while the response increases through navigation the curve from neutral pH towards acidic or alkaline region. Contrastingly, the desirability function to approach 100% dye removal results in having to change operating conditions to be the lowest boundary of the model limits for initial dye concentration and pH, and the highest limits for time. This approach decreased the adsorption capacity by half while increasing the water purification by 17%. Particularly, selecting a specific approach should follow the industrial requirements and the environmental aspects.
Characteristics of de-oiled biosorbent
FTIR analysis contributes to the explanation of biosorption mechanism through the description of their functional group. The presence of secondary amines –NH and primary amines -NH2 in RB, and AC clarified the ability of algal biosorbent to form the positive cations performing ionic pond with AR1 ions (Tan et al. 2018; Déniel et al. 2020; Silva et al. 2020). The stretching ponds in the –CH group and aliphatic -CH2 group have a central role in the adsorption process as they possess a high ability to form free radicals that attract the AR1 ions (Asemani et al. 2020). The increasing of carbon w% in AC detected by EDX emphasizes the presence of an aliphatic CH2 group and stretching CH group in AC.
SEM analysis illustrates the change in the microstructure of AC and AB. The regular and small homogeneous size of AC increases the surface area and ultimately enhanced its adsorption capacity while the elongated regular fiber bundles with a smooth distinct surface of AB elucidate the adsorption process for the surface and the core absorption inside the biosorbent particle.
The reduction of nanoparticle size of AC, as figured by TEM analysis, increases the active sites and encourages rapid contact between the dye molecules and the binding sites (Zohoorian et al. 2020). This analysis explains the wide variation in RB and AC performance as the dye removal percent using RB and AC are 3.4%, and 96%, respectively. These results are integrated with that detected by SEM analysis, where the morphology and the homogeneous shape of AC facilitate the dye removal.
The desorption process is necessary to complete the process cycle. However, several obstacles contradict this process; some of them are related to the cost and the other is related to the environment (Farooq et al. 2018). The minor elements included in such material enhance its potentiality to be used as a micro-nutrient for agricultural soil. Further research has to be performing to cover the plant requirements and human health.
The recent research is concerned with applying micro-organisms in dye removal from textile wastewater. Microalgae are a potential source for several industries. The extracted oil from C. vulgaris is suitable for biofuel and biomedical. Thus; using de-oiled microalgae in integrated industries enhance its potentiality. The activation process transforms the biomass surface into a regular and small homogeneous size that increases the surface area and ultimately enhances its adsorption capacity. The simultaneous optimization of the process parameters is achieved by CCD in RSM. This method decreases the number of essential experiments and provides a predicting model with a CI of 95%. Further experiments are encouraged to be performing to study more process parameters as the AC amount and process temperature. The reusability of dye-loaded bio sorbent as micro-nutrient is an important economical aspect to avoid the obstacles of the desorption process and provide a useful by-product.
All data generated or analyzed during this study are included in this published article [and its supplementary information files].
Dye-loaded biomass
Activated de-oiled microalgae
AR1:
Acid Red 1 dye
ANOVA:
Analysis of variance—statistics
CCD:
Central composite design
Coefficient of variance
FTIR:
Fourier transform infrared spectroscopy
RB:
De-oiled microalgae
RSM:
Response surface method
TEM:
Transmission electron microscope
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Kishor R, Purchase D, Saratale GD, Saratale RG, Romanholo Ferreira LF, Bilal M, Chandra R, Bharagava RN (2021) Ecotoxicological and health concerns of persistent coloring pollutants of textile industry wastewater and treatment approaches for environmental safety. J Environ Chem Eng 9(2):105012. https://doi.org/10.1016/j.jece.2020.105012
Kumar S, Ahluwalia AS, Charaya MU (2019) Adsorption of Orange-G dye by the dried powdered biomass of Chlorella vulgaris Beijerinck. Curr Sci 116(4):604–611. https://doi.org/10.18520/cs/v116/i4/604-611
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Liang Z, Wang J, Zhang Y, Han C, Ma S, Chen J et al (2020) Removal of volatile organic compounds (VOCs) emitted from a textile dyeing wastewater treatment plant and the attenuation of respiratory health risks using a pilot-scale biofilter. J Clean Prod 253:120019. https://doi.org/10.1016/j.jclepro.2020.120019
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Sastre de Vicente ME, Rodriguez-Barro P, Herrero R, Vilariño T, Lodeiro P, Barriada JL (2020) Biosorption of chemical species by Sargassum algal biomass: equilibrium data. In: Konur O (ed) Handbook of algal science, technology and medicine. Academic Press, pp 675–696. https://doi.org/10.1016/B978-0-12-818305-2.00042-5
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This paper was supported by the National Research Centre in Egypt; we highly appreciate the central laboratories for conducting the tests in this research.
The National Research Centre, Dokki, Egypt, has provided this research a financial support Grant Number "12010203".
Chemical Engineering and Pilot Plant Department, Engineering Research Division, National Research Centre, P.O. 12622, Giza, Egypt
Samar A. El-Mekkawi & S. A. Abo El-Enin
Dyeing, Printing and Textile Auxiliaries Department, Textile Industries Research Division, National Research Centre, P.O. 12622, Giza, Egypt
Rehab A. Abdelghaffar & Fatma Abdelghaffar
Samar A. El-Mekkawi
Rehab A. Abdelghaffar
Fatma Abdelghaffar
S. A. Abo El-Enin
"SA Abo El-Enin and RA conceptualized the main idea of this work. SA Abo El-Enin and SA El- Mekkawi prepared the de-oiled activated carbon. RA and FA performed the characterization analyses of the biosorbent and the dye adsorption experiments. SA El- Mekkawi performed the statistical analysis of the CCD; she is also the corresponding author. All authors contributed to writing the manuscript. All authors read and approved the final manuscript.
Correspondence to Samar A. El-Mekkawi.
El-Mekkawi, S.A., Abdelghaffar, R.A., Abdelghaffar, F. et al. Application of response surface methodology for color removing from dyeing effluent using de-oiled activated algal biomass. Bull Natl Res Cent 45, 80 (2021). https://doi.org/10.1186/s42269-021-00542-w
DOI: https://doi.org/10.1186/s42269-021-00542-w
De-fated microalgae
Activated biosorbent
Dye removal
Desirability function
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CommonCrawl
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Determine the Values of $a$ such that the 2 by 2 Matrix is Diagonalizable
1-a & a\\
-a& 1+a
\end{bmatrix}\] be a $2\times 2$ matrix, where $a$ is a complex number.
Determine the values of $a$ such that the matrix $A$ is diagonalizable.
(Nagoya University, Linear Algebra Exam Problem)
To find eigenvalues of the matrix $A$, we determine the characteristic polynomial $p(t)$ of $A$ as follows.
p(t)&=\det(A-tI)=\begin{vmatrix}
1-a-t & a\\
-a& 1+a-t
\end{vmatrix}\\[6pt] &=(1-a-t)(1+a-t)+a^2\\
&=(1-t)^2-a^2+a^2=(1-t)^2.
If you put $b=1-t$, then $(1-a-t)(1+a-t)=(b-a)(b+a)=b^2-a^2=(1-t)^2-a^2$.
Thus, the eigenvalue of $A$ is $1$ with algebraic multiplicity $2$.
Let us determine the geometric multiplicity (the dimension of the eigenspace $E_1$).
A-I=\begin{bmatrix}
-a & a\\
-a& a
\end{bmatrix}
\xrightarrow{R_2-R_1}
\begin{bmatrix}
\end{bmatrix} .
If $a\neq 0$, then we further reduce it and get
\end{bmatrix}\xrightarrow{\frac{-1}{a}R_1}
\end{bmatrix}.
Hence the eigenspace corresponding to the eigenvalue $1$ is
E_1=\calN(A-I)=\Span \left\{\, \begin{bmatrix}
\end{bmatrix} \,\right\},
and its dimension is $1$, which is less than the algebraic multiplicity.
Thus, when $a\neq 0$, the matrix $A$ is not diagonalizable.
If $a=0$, then the matrix $A=\begin{bmatrix}
\end{bmatrix}$ is already diagonal, hence it is diagonalizable.
In conclusion, the matrix $A$ is diagonalizable if and only if $a=0$.
Quiz 12. Find Eigenvalues and their Algebraic and Geometric Multiplicities (a) Let \[A=\begin{bmatrix} 0 & 0 & 0 & 0 \\ 1 &1 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 \end{bmatrix}.\] Find the eigenvalues of the matrix $A$. Also give the algebraic multiplicity of each eigenvalue. (b) Let \[A=\begin{bmatrix} 0 & 0 & 0 & 0 […]
How to Diagonalize a Matrix. Step by Step Explanation. In this post, we explain how to diagonalize a matrix if it is diagonalizable. As an example, we solve the following problem. Diagonalize the matrix \[A=\begin{bmatrix} 4 & -3 & -3 \\ 3 &-2 &-3 \\ -1 & 1 & 2 \end{bmatrix}\] by finding a nonsingular […]
Given the Characteristic Polynomial of a Diagonalizable Matrix, Find the Size of the Matrix, Dimension of Eigenspace Suppose that $A$ is a diagonalizable matrix with characteristic polynomial \[f_A(\lambda)=\lambda^2(\lambda-3)(\lambda+2)^3(\lambda-4)^3.\] (a) Find the size of the matrix $A$. (b) Find the dimension of $E_4$, the eigenspace corresponding to the eigenvalue […]
Eigenvalues and Algebraic/Geometric Multiplicities of Matrix $A+cI$ Let $A$ be an $n \times n$ matrix and let $c$ be a complex number. (a) For each eigenvalue $\lambda$ of $A$, prove that $\lambda+c$ is an eigenvalue of the matrix $A+cI$, where $I$ is the identity matrix. What can you say about the eigenvectors corresponding to […]
Find the Limit of a Matrix Let \[A=\begin{bmatrix} \frac{1}{7} & \frac{3}{7} & \frac{3}{7} \\ \frac{3}{7} &\frac{1}{7} &\frac{3}{7} \\ \frac{3}{7} & \frac{3}{7} & \frac{1}{7} \end{bmatrix}\] be $3 \times 3$ matrix. Find \[\lim_{n \to \infty} A^n.\] (Nagoya University Linear […]
Find All the Eigenvalues and Eigenvectors of the 6 by 6 Matrix Find all the eigenvalues and eigenvectors of the matrix \[A=\begin{bmatrix} 10001 & 3 & 5 & 7 &9 & 11 \\ 1 & 10003 & 5 & 7 & 9 & 11 \\ 1 & 3 & 10005 & 7 & 9 & 11 \\ 1 & 3 & 5 & 10007 & 9 & 11 \\ 1 &3 & 5 & 7 & 10009 & 11 \\ 1 &3 & 5 & 7 & 9 & […]
Eigenvalues and Eigenvectors of Matrix Whose Diagonal Entries are 3 and 9 Elsewhere Find all the eigenvalues and eigenvectors of the matrix \[A=\begin{bmatrix} 3 & 9 & 9 & 9 \\ 9 &3 & 9 & 9 \\ 9 & 9 & 3 & 9 \\ 9 & 9 & 9 & 3 \end{bmatrix}.\] (Harvard University, Linear Algebra Final Exam Problem) Hint. Instead of […]
Find the Eigenvalues and Eigenvectors of the Matrix $A^4-3A^3+3A^2-2A+8E$. Let \[A=\begin{bmatrix} 1 & -1\\ 2& 3 \end{bmatrix}.\] Find the eigenvalues and the eigenvectors of the matrix \[B=A^4-3A^3+3A^2-2A+8E.\] (Nagoya University Linear Algebra Exam Problem) Hint. Apply the Cayley-Hamilton theorem. That is if $p_A(t)$ is the […]
Tags: algebraic multiplicitycharacteristic polynomialdimensioneigenspaceeigenvaluegeometric multiplicitylinear algebraNagoyaNagoya.LA
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Characteristic Polynomials of $AB$ and $BA$ are the Same
Are Coefficient Matrices of the Systems of Linear Equations Nonsingular?
If the Sum of Entries in Each Row of a Matrix is Zero, then the Matrix is Singular
Introduction to Matrices
Elementary Row Operations
Gaussian-Jordan Elimination
Solutions of Systems of Linear Equations
Linear Combination and Linear Independence
Nonsingular Matrices
Inverse Matrices
Subspaces in $\R^n$
Bases and Dimension of Subspaces in $\R^n$
General Vector Spaces
Subspaces in General Vector Spaces
Linearly Independency of General Vectors
Bases and Coordinate Vectors
Dimensions of General Vector Spaces
Linear Transformation from $\R^n$ to $\R^m$
Linear Transformation Between Vector Spaces
Orthogonal Bases
Determinants of Matrices
Computations of Determinants
Introduction to Eigenvalues and Eigenvectors
Eigenvectors and Eigenspaces
Diagonalization of Matrices
The Cayley-Hamilton Theorem
Dot Products and Length of Vectors
Eigenvalues and Eigenvectors of Linear Transformations
Jordan Canonical Form
Cosine and Sine Functions are Linearly Independent
Are these vectors in the Nullspace of the Matrix?
Isomorphism Criterion of Semidirect Product of Groups
More in Linear Algebra
A Matrix Equation of a Symmetric Matrix and the Limit of its Solution
Let $A$ be a real symmetric $n\times n$ matrix with $0$ as a simple eigenvalue (that is, the algebraic multiplicity...
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CommonCrawl
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Problem J
Mr. Turtle loves drawing on his whiteboard at home. One day when he was drawing, his marker dried out! Mr. Turtle then noticed that the marker behaved like an eraser for the remainder of his drawing.
Mr. Turtle has a picture in his head of how he wants his final drawing to appear. He plans out his entire drawing ahead of time, step by step. Mr. Turtle's plan is a sequence of commands: up, down, left or right, with a distance. He starts drawing in the bottom left corner of his whiteboard. Consider the $6 \times 8$ whiteboard and sequence of commands in the first diagram. If the marker runs dry at timestep $17$, the board will look like the second diagram (the numbers indicate the timestep when the marker is at each cell). Note that it will make a mark at timestep $17$, but not at timestep $18$.
Mr. Turtle wants to know the earliest and latest time his marker can dry out, and he'll still obtain the drawing in his head. Can you help him? Note that timestep $0$ is the moment before the marker touches the board. It is valid for a marker to dry out at timestep $0$.
Each input will consist of a single test case. Note that your program may be run multiple times on different inputs. The input will start with a line with $3$ space-separated integers $h$, $w$ and $n$ ($1 \le h,w,n \le 1\, 000\, 000, w \cdot h \le 1\, 000\, 000$) where $h$ and $w$ are the height and width of the whiteboard respectively, and $n$ is the number of commands in Mr. Turtle's plan.
The next $h$ lines will each consist of exactly $w$ characters, with each character being either '#' or '.' . This is the pattern in Mr. Turtle's head, where '#' is a marked cell, and '.' is a blank cell.
The next $n$ lines will each consist of a command, of the form "direction distance", with a single space between the direction and the distance and no other spaces on the line. The direction will be exactly one of the set $\{ \texttt{up}, \texttt{down}, \texttt{left}, \texttt{right} \} $, guaranteed to be all lower case. The distance will be between $1$ and $1\, 000\, 000$ inclusive. The commands must be executed in order. It is guaranteed that no command will take the marker off of the whiteboard.
Output two integers, first the minimum, then the maximum time that can pass before the marker dries out, and Mr. Turtle can still end up with the target drawing. Neither number should be larger than the last timestep that the marker is on the board, so if the marker can run to the end and still draw the target drawing, use the last timestep that the marker is on the board. If it's not possible to end up with the target drawing, output -1 -1.
Sample Input 1
Sample Output 1
...#....
########
#..#...#
#..#####
#.......
up 3
down 2
###.####
#......#
.#.
Problem ID: whiteboard
CPU Time limit: 4 seconds
Memory limit: 1024 MB
Sample data files
Source: North American Invitational Programming Contest (NAIPC) 2016
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CommonCrawl
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Shor's Algorithm
02-11-2021 02-11-2021 blog 21 minutes read (About 3160 words) 0 visits
Shor's algorithm is a quantum computer integer factorization algorithm that runs in asymptotically polynomial time. Concretely, given an integer $N$, it finds it prime factors relative efficiently. Shor's algorithm consists of two parts, a classical part and a quantum part. The classical part is self-contained and has been extensively discussed in my previous article "Composite Number Factorization Using Modular Exponentiation Period".
In this blog post, I would like to discuss the Shor's algorithm, and mainly focus on the quantum part.
It is highly recommended that the readers read the following articles before reading this article on Shor's algorithm.
Euclidean Algorithm
Composite Number Factorization Using Modular Exponentiation Period
Discrete Fourier Transform
Discrete Fourier Transform for 0/1 Periodic Sequences
Hadamard Operator
Most of the important properties of Hadamard operator have been derived in the prerequisites section of my previous blog post on Deutsch-Jozsa algorithm. Unlike Deutsch-Jozsa algorithm, Shor's algorithm is only going to use a small fraction of the Hadamard operator properties that Deutsch-Jozsa algorithm has used. I would just copy the properties useful for Shor's algorithm algorithm. For the derivation, proof, and other properties of Hadamard operator, the reader should refer to my previous blog post.
To extract an arbitrary column $j$ from $H^{\otimes {n}}$, we prepared a one-hot quantum system basic state vector $| \mathbf{y} \rangle = [y_0, y_1, \cdots, y_{2^n-1}]^{\top}$, where $y_j = 1$ and $y_k = 0$ for $k \neq j$.
H^{\otimes {n}}_{:,j} &= H^{\otimes {n}} | \mathbf{y} \rangle \\
&= H^{\otimes n}[\mathbf{0}, \mathbf{j}] | \mathbf{x}_0 \rangle + H^{\otimes n}[\mathbf{1}, \mathbf{j}] | \mathbf{x}_1 \rangle + \cdots + H^{\otimes n}[\mathbf{2^n-1}, \mathbf{j}] | \mathbf{x}_{2^{n}-1} \rangle \\
&= \frac{1}{\sqrt{2^n}} (-1)^{\langle \mathbf{0}, \mathbf{j} \rangle} | \mathbf{x}_0 \rangle + \frac{1}{\sqrt{2^n}} (-1)^{\langle \mathbf{1}, \mathbf{j} \rangle} | \mathbf{x}_1 \rangle + \cdots + \frac{1}{\sqrt{2^n}} (-1)^{\langle \mathbf{2^n-1}, \mathbf{j} \rangle} | \mathbf{x}_{2^{n}-1} \rangle \\
&= \frac{1}{\sqrt{2^n}} \sum_{i=0}^{2^n-1} (-1)^{\langle \mathbf{i}, \mathbf{j} \rangle} | \mathbf{x}_i \rangle\\
&= \frac{1}{\sqrt{2^n}} \sum_{ \mathbf{x} \in {0,1}^n } (-1)^{\langle \mathbf{x}, \mathbf{j} \rangle} | \mathbf{x} \rangle\\
&= \frac{1}{\sqrt{2^n}} \sum_{ \mathbf{x} \in {0,1}^n } (-1)^{\langle \mathbf{x}, \mathbf{y} \rangle} | \mathbf{x} \rangle\\
where $| \mathbf{x}_i \rangle$ is a quantum system one-hot basic state vector, $|\mathbf{x}_i\rangle = [x_0, x_1, \cdots, x_{2^{n}-1}]^{\top}$, where $x_i = 1$ and $x_k = 0$ for $k \neq i$.
Specifically, if $j = 0$, $| \mathbf{y} \rangle = [\underbrace{1, 0, 0, \cdots, 0}_{2^n} ]^{\top} = | \mathbf{0} \rangle$,
H^{\otimes {n}} | \mathbf{0} \rangle
&= \frac{1}{\sqrt{2^n}} \sum_{i=0}^{2^n-1} (-1)^{\langle \mathbf{i}, \mathbf{0} \rangle} | \mathbf{x}_i \rangle \\
&= \frac{1}{\sqrt{2^n}} \sum_{i=0}^{2^n-1} (-1)^{0} | \mathbf{x}_i \rangle \\
&= \frac{1}{\sqrt{2^n}} \sum_{i=0}^{2^n-1} | \mathbf{x}_i \rangle \\
&= \frac{1}{\sqrt{2^n}} \sum_{ \mathbf{x} \in {0,1}^n } | \mathbf{x} \rangle \\
In the "Discrete Fourier Transform", we have discussed how to perform discrete Fourier transform and inverse discrete Fourier transform using unitary matrix multiplication.
We define the inverse discrete Fourier transform Vandermonde matrix $\mathbf{DFT}^{\ast\dagger}$ as
\mathbf{DFT}^{\ast\dagger} &=
\begin{bmatrix}
\omega_N^{0 \cdot 0} & \omega_N^{0 \cdot 1} & \cdots & \omega_N^{0 \cdot (N-1)} \\
\vdots & \vdots & \ddots & \vdots \\
\omega_N^{(N-1) \cdot 0} & \omega_N^{(N-1) \cdot 1} & \cdots & \omega_N^{(N-1) \cdot (N-1)} \\
\end{bmatrix}
and of course the discrete Fourier transform Vandermonde matrix $\mathbf{DFT}^{\ast}$ is
\mathbf{DFT}^{\ast} &=
\omega_N^{-0 \cdot 0} & \omega_N^{-0 \cdot 1} & \cdots & \omega_N^{-0 \cdot (N-1)} \\
\omega_N^{-(N-1) \cdot 0} & \omega_N^{-(N-1) \cdot 1} & \cdots & \omega_N^{-(N-1) \cdot (N-1)} \\
where $\omega_N^{i \cdot j} = (\omega_N^{i})^j = \omega_N^{ij}$, and $\omega_N^{0}, \omega_N^{1}, \cdots, \omega_N^{N-1}$ are the $N$th root of unity, i.e., $\omega_N^{k} = e^{i\frac{2\pi k}{N}}$.
The unitary inverse discrete Fourier transform matrix and unitary discrete Fourier transform matrix are
U_{\text{iDFT}} = \frac{1}{\sqrt{N}} \mathbf{DFT}^{\ast\dagger} \\
U_{\text{DFT}} =\frac{1}{\sqrt{N}} \mathbf{DFT}^{\ast} \\
Using some math tricks, we could also rewrite the unitary inverse discrete Fourier transform matrix and unitary discrete Fourier transform matrix as
U_{\text{iDFT}} = \frac{1}{\sqrt{N}} \sum_{x=0}^{N-1} \sum_{y=0}^{N-1} \omega_{N}^{x \cdot y} | y \rangle \langle x | \\
U_{\text{DFT}} = \frac{1}{\sqrt{N}} \sum_{x=0}^{N-1} \sum_{y=0}^{N-1} \omega_{N}^{- x \cdot y} | y \rangle \langle x | \\
where $|x \rangle$ and $|y \rangle$ are one-hot encoded state vector for scalar $x$ and $y$.
Note that $| y \rangle \langle x |$ is an outer product resulting in a $N \times N$ matrix.
Quantum Fourier Transform
Quantum Fourier transform and the inverse quantum Fourier transform are just the discrete Fourier transform and inverse discrete Fourier transform analogues in quantum systems. In stead of applying the discrete Fourier transform Vandermonde matrix and the inverse discrete Fourier transform Vandermonde matrix to a sequence of scalars, quantum Fourier transform and the inverse quantum Fourier transform applies the same Vandermonde matrices to the superposition, i.e., the linear combination, of a sequence of quantum states. We will see this more concretely in the Shor's algorithm.
The unitary inverse quantum Fourier transform matrix and unitary quantum Fourier transform matrix are the same to the ones used for inverse discrete Fourier transform and discrete Fourier transform.
U_{\text{iQFT}} = \frac{1}{\sqrt{N}} \sum_{x=0}^{N-1} \sum_{y=0}^{N-1} \omega_{N}^{x \cdot y} | y \rangle \langle x | \\
U_{\text{QFT}} = \frac{1}{\sqrt{N}} \sum_{x=0}^{N-1} \sum_{y=0}^{N-1} \omega_{N}^{- x \cdot y} | y \rangle \langle x | \\
where $|x \rangle$ and $|y \rangle$ are quantum state vector for basic states, which are also represented as one-hot encoded vectors.
Shor's Algorithm
Classical Part
From the "Composite Number Factorization Using Modular Exponentiation Period", we learned that given a composite integer $N$, and integers, such as an $a$, that are relatively prime to $N$, if somehow we could compute the period $r$ of the modular exponentiation remainder sequence $f_{a,N}$, we will be able to factorize the composite integer $N$ efficiently using classical algorithms.
We have also shown in the "Composite Number Factorization Using Modular Exponentiation Period" that the minimal period $r$ must be equal or smaller than $N$. When $N$ is small, we could the generate the modular exponentiation remainder sequence $f_{a,N}$ for a sequence length of at least $2N$ using modular algorithmic I described, and find out the period. However, when $N$ is extremely large, such as a number that is hundreds or even thousands of digits long, generating the modular exponentiation remainder sequence $f_{a,N}$ of length $2N$ using modular algorithmic takes exponential asymptotic complexity $O(10^n)$ which is intractable, where $n$ is the number of digits in the number $N$ using base of $10$.
To find out the period of the modular exponentiation remainder sequence $f_{a,N}$ when $N$ is extremely large, it turns out that quantum computer with quantum Fourier transform could do it in polynomial asymptotic complexity which is tractable. Let's look at how to find out the period using quantum circus and quantum algorithms.
Quantum Part
Similar to all the quantum oracle circuits used for other quantum algorithm, we have a quantum oracle circuit specific for the function $f_{a,N}(x)$ for Shor's algorithm.
We explicitly define $f_{a,N}(x)$ as the $x$th value in the $0$-indexed $f_{a,N}$ sequence. Mathematically,
f_{a,N}(x) = a^{x} \bmod {N}
Note that in the quantum oracle circuit, $\mathbf{x} \in \{0, 1\}^m$ and $\mathbf{y} \in \{0, 1\}^n$ are just the binary representations for $x$ and $y$ and $\oplus$ is (bit-wise) $\text{XOR}$ (binary addition modulo 2).
With the quantum oracle circuit $U_{f_{a,N}}$, we could further design the quantum algorithm to find out the period of $f_{a,N}$.
To compute the quantum states $|\varphi_0\rangle$, $|\varphi_1\rangle$, and $|\varphi_2\rangle$, we have
|\varphi_0\rangle &= |\mathbf{0}\rangle \otimes |\mathbf{0}\rangle\\
&= |\mathbf{0}, \mathbf{0}\rangle \\
|\varphi_1\rangle &= (H^{\otimes m} \otimes I) |\varphi_0\rangle \\
&= (H^{\otimes m} \otimes I) (|\mathbf{0}\rangle \otimes |\mathbf{0}\rangle) \\
&= H^{\otimes m}|\mathbf{0}\rangle \otimes I | \mathbf{0} \rangle \\
&= \bigg( \frac{1}{\sqrt{2^m}} \sum_{ \mathbf{x} \in {0,1}^m } | \mathbf{x} \rangle \bigg) \otimes | \mathbf{0} \rangle \\
&= \frac{1}{\sqrt{2^m}} \sum_{ \mathbf{x} \in {0,1}^m } | \mathbf{x} \rangle \otimes | \mathbf{0} \rangle \\
&= \frac{1}{\sqrt{2^m}} \sum_{ \mathbf{x} \in {0,1}^m } | \mathbf{x}, \mathbf{0} \rangle \\
|\varphi_2\rangle &= U_{f_{a,N}} |\varphi_1\rangle \\
&= U_{f_{a,N}} \bigg( \frac{1}{\sqrt{2^m}} \sum_{ \mathbf{x} \in {0,1}^m } | \mathbf{x}, \mathbf{0} \rangle \bigg) \\
&= \frac{1}{\sqrt{2^m}} \sum_{ \mathbf{x} \in {0,1}^m } U_{f_{a,N}} | \mathbf{x}, \mathbf{0} \rangle \\
&= \frac{1}{\sqrt{2^m}} \sum_{ \mathbf{x} \in {0,1}^m } | \mathbf{x}, \mathbf{0} \oplus f_{a,N}(\mathbf{x}) \rangle \\
&= \frac{1}{\sqrt{2^m}} \sum_{ \mathbf{x} \in {0,1}^m } | \mathbf{x}, f_{a,N}(\mathbf{x}) \rangle \\
&= \frac{1}{\sqrt{2^m}} \sum_{ \mathbf{x} \in {0,1}^m } | \mathbf{x}, a^{\mathbf{x}} \bmod {N} \rangle \\
We could see that the periodic $f_{a,N}$ sequence of length $2^m$ from $f_{a,N}(0)$ to $f_{a,N}(2^{m}-1)$ has been encoded in the $|\varphi_2\rangle$ qubits. The bottom qubits store the remainder information. Because the remainder is less than $N$, $n = \log_2 N$ qubits are just sufficient. Regarding the value of $m$, although personally I think $m = \log_2 (2N) = n + 1$ qubits should be sufficient to find out the period in theory, in practice, we used $m = \log_2 N^2 = 2 \log_2 N = 2n$ qubits.
After measuring the bottom qubits, the state of the bottom qubits becomes determined. Suppose the bottom qubits we found after measurement is
a^{\mathbf{\overline{x}}} \bmod {N}
Because $f_{a,N}$ is periodic with a period of $r$, for $s \in \mathbb{Z}$ and $\mathbf{\overline{x}} + sr \geq 0$, we have
a^{\mathbf{\overline{x}}} \equiv a^{\mathbf{\overline{x}} + sr} \pmod {N}
Because of the quantum entanglement, many possibilities for the top qubits have been eliminated. The remaining possibilities are the $\mathbf{x}$ whose $a^{\mathbf{x}} \equiv a^{\mathbf{\overline{x}}} \pmod {N}$, and there are $\lfloor \frac{2^m}{r} \rfloor$ different possibilities. So we have
|\varphi_3\rangle &= \frac{1}{\lfloor \frac{2^m}{r} \rfloor} \sum_{a^{\mathbf{x}} \equiv a^{\mathbf{\overline{x}}} \pmod {N}}^{} | \mathbf{x}, a^{\mathbf{\overline{x}}} \bmod N \rangle \\
We could further rewrite $|\varphi_3\rangle$ as
|\varphi_3\rangle &= \frac{1}{\lfloor \frac{2^m}{r} \rfloor} \sum_{j = 0}^{\lfloor \frac{2^m}{r} \rfloor - 1} | t_0 + j r, a^{\mathbf{\overline{x}}} \bmod N \rangle \\
where $t_0$ is the smallest $\mathbf{x}$ that $a^{\mathbf{x}} \equiv a^{\mathbf{\overline{x}}} \pmod {N}$, and is referred as the offset.
We expand $|\varphi_3\rangle$ as
&= \frac{1}{\lfloor \frac{2^m}{r} \rfloor} \sum_{j = 0}^{\lfloor \frac{2^m}{r} \rfloor - 1} | t_0 + j r \rangle \otimes | a^{\mathbf{\overline{x}}} \bmod N \rangle \\
&= \bigg( \frac{1}{\lfloor \frac{2^m}{r} \rfloor} \sum_{j = 0}^{\lfloor \frac{2^m}{r} \rfloor - 1} | t_0 + j r \rangle \bigg) \otimes | a^{\mathbf{\overline{x}}} \bmod N \rangle \\
We apply the inverse quantum Fourier transform unitary matrix $U_{\text{iQFT}}$ for the top qubits.
U_{\text{iQFT}} \bigg( \frac{1}{\lfloor \frac{2^m}{r} \rfloor} \sum_{j = 0}^{\lfloor \frac{2^m}{r} \rfloor - 1} | t_0 + j r \rangle \bigg)
&= \bigg( \frac{1}{\sqrt{2^m}} \sum_{x=0}^{2^m-1} \sum_{y=0}^{2^m-1} \omega_{2^m}^{x \cdot y} | y \rangle \langle x | \bigg) \bigg( \frac{1}{\lfloor \frac{2^m}{r} \rfloor} \sum_{j = 0}^{\lfloor \frac{2^m}{r} \rfloor - 1} | t_0 + j r \rangle \bigg) \\
&= \frac{1}{\sqrt{2^m} \lfloor \frac{2^m}{r} \rfloor} \bigg( \sum_{x=0}^{2^m-1} \sum_{y=0}^{2^m-1} \omega_{2^m}^{x \cdot y} | y \rangle \langle x | \bigg) \bigg( \sum_{j = 0}^{\lfloor \frac{2^m}{r} \rfloor - 1} | t_0 + j r \rangle \bigg) \\
&= \frac{1}{\sqrt{2^m} \lfloor \frac{2^m}{r} \rfloor} \bigg( \sum_{j = 0}^{\lfloor \frac{2^m}{r} \rfloor - 1} \sum_{y=0}^{2^m-1} \omega_{2^m}^{(t_0 + j r) \cdot y} | y \rangle \bigg) \\
&= \frac{1}{\sqrt{2^m} \lfloor \frac{2^m}{r} \rfloor} \bigg( \sum_{y=0}^{2^m-1} \sum_{j = 0}^{\lfloor \frac{2^m}{r} \rfloor - 1} \omega_{2^m}^{(t_0 + j r) \cdot y} | y \rangle \bigg) \\
&= \frac{1}{\sqrt{2^m} \lfloor \frac{2^m}{r} \rfloor} \bigg( \sum_{y=0}^{2^m-1} \Big( \sum_{k = 0}^{2^m-1} \omega_{2^m}^{k \cdot y} F[k] \Big) | y \rangle \bigg) \\
&= \frac{1}{\sqrt{2^m} \lfloor \frac{2^m}{r} \rfloor} \bigg( \sum_{y=0}^{2^m-1} \Big( \sum_{k = 0}^{2^m-1} e^{i\frac{2\pi ky}{2^m}} F[k] \Big) | y \rangle \bigg) \\
&= \frac{1}{\sqrt{2^m} \lfloor \frac{2^m}{r} \rfloor} \bigg( \sum_{y=0}^{2^m-1} \Big( \sum_{k = 0}^{2^m-1} F[k] e^{i\frac{2\pi ky}{2^m}} \Big) | y \rangle \bigg) \\
&= \frac{1}{\sqrt{2^m} \lfloor \frac{2^m}{r} \rfloor} \bigg( \sum_{y=0}^{2^m-1} f[y] | y \rangle \bigg) \\
F[k]
&=
\begin{cases}
1 & \text{if $k = t_0 + j r$}\\
0 & \text{else}\\
\end{cases} \\
f[y] = \sum_{k = 0}^{2^m-1} F[k] e^{i\frac{2\pi ky}{2^m}}
Note that in the above derivation, we used the fact that for one-hot encoded state vector $| x \rangle$ and $| y \rangle$,
\langle x, y \rangle
1 & \text{if $x = y$}\\
This is exactly the inverse discrete Fourier transform for $0/1$ periodic sequence. In "Discrete Fourier Transform for 0/1 Periodic Sequences", we have shown and proved that the discrete Fourier transform and the inverse discrete Fourier transform for $0/1$ periodic sequence will change the sequence period from $r$ to $\frac{N}{r}$. More importantly, the leading zeros, i.e., the offset, will be eliminated after the transform. In our case, the original sequence period is $r$ for sequence $F$ and the sequence period for $f$ after inverse discrete Fourier transformation is $\frac{2^m}{r}$, and $f[0] \neq 0$.
In "Discrete Fourier Transform for 0/1 Periodic Sequences", we have shown and proved that if $r$ divides $2^m$, the sequence after inverse discrete Fourier transform will be a perfect $0/v$ periodic sequence with a period of exactly $\frac{2^m}{r}$. However, if $r$ does not divide $2^m$, the sequence after inverse discrete Fourier transform will not be a perfect $0/v$ periodic sequence.
Let's discuss what Shor's algorithm will do in these two scenarios.
When $r$ divides $2^m$, after inverse quantum Fourier transform, the quantum states for the top qubits are the superposition of the basic states $\{|y_0\rangle, |y_1\rangle, |y_2\rangle, \cdots, |y_k\rangle, \cdots \}$ with equal probabilities where $y_k = k\frac{2^m}{r}$. So when we measure the top qubits, the value we observed must be $x = k\frac{2^m}{r}$, multiples of $\frac{2^m}{r}$. Because $2^m$ is known, so we could know the value of $\frac{k}{r}$.
\frac{k}{r} = \frac{x}{2^m}
Does knowing the value of $\frac{k}{r}$ tells us the exact value of $r$? Not necessarily. For example, if $r = 2^6$ and $k = 3$, $\frac{k}{r} = \frac{3}{2^5}$, the denominator is irreducible, so there is no ambiguity about the value of $r$. However, if $r = 2^6$ and $k = 2\times3 = 6$, $\frac{k}{r} = \frac{3}{2^5}$, the denominator is reduced so the value of $r$ cannot be unambiguously determined. So in general, if $k$ is odd, we could immediately determine the value of $r$ without any ambiguity. In practice, we got $50%$ chance that $k$ is odd. So we can run the quantum circuits $s$ times, and get the largest denominator as the value for $r$. The probability of sampling at least one odd $k$ in $s$ runs is $1 - \frac{1}{2^s}$. When $s = 20$, the probability is already greater than $0.999$. So this approach to determine the value of $r$ is feasible in practice.
When $r$ does not divide $2^m$, which is more common in practice, after inverse quantum Fourier transform, the value we observed from the top qubits, in addition to $x = k\lfloor\frac{2^m}{r}\rfloor$ and $x = k\lceil\frac{2^m}{r}\rceil$ with relatively high probabilities, might also be $x = k\lfloor\frac{2^m}{r}\rfloor - 1$ and $x = k\lceil\frac{2^m}{r}\rceil + 1$ with relatively low probabilities. In this scenario, we would like to again run the quantum circuits $s$ times, and try to analyze the observations to extract the most likely value for $r$. Since the random observation generation process has been explicitly modeled in my description, it is just a statistical inference for the random variable $r$ given the observation from running the quantum circuits $s$ times. I have written several articles on statistical inference and I am not going to elaborate it here for inferring the most likely value for $r$.
Once we got the period $r$, we used the classical part of the Shor's algorithm to factorize the integer $N$.
I have been writing the article on Shor's algorithm intermittently during the past year and I found it really difficult to make it mathematically friendly to the readers. However, if we don't know the math, we will not understand why Shor's algorithm will work.
Quantum Computing for Computer Scientists
https://leimao.github.io/blog/Shor-Algorithm/
PyTorch Pruning
Bootstrap Methods
2Prerequisite
2.1Hadamard Operator
2.2Discrete Fourier Transform
2.3Quantum Fourier Transform
3Shor's Algorithm
3.1Classical Part
3.2Quantum Part
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<!-- w0973701.png $#A+1 = 35 n = 0 $#C+1 = 35 : ~/encyclopedia/old_files/data/W097/W.0907370 Weibull distribution Automatically converted into TeX, above some diagnostics. Please remove this comment and the {{TEX|auto}} line below, if TeX found to be correct. --> {{TEX|auto}} {{TEX|done}} A particular [[Probability distribution|probability distribution]] of random variables $ X _ {w} $, characterized by the distribution function $$ \tag{* } F _ {w} ( t, p, \sigma , \mu ) = \ \left \{ \begin{array}{ll} 1- \mathop{\rm exp} \left \{ - \left ( \frac{t- \mu } \sigma \right ) ^ {p} \right \} , & t > \mu , \\ 0, & t \leq \mu , \\ \end{array} \right. $$ where $ p $ is a parameter of the shape of the distribution curve, $ \sigma $ is a scale parameter and $ \mu $ is a shift parameter. The family of distributions (*) was named after W. Weibull [[#References|[1]]], who was the first to use it in the approximation of extremal data on the tensile strength of steel during fatigue testing and to propose methods for estimating the parameters of the distribution (*). Weibull's distribution belongs to the limit distributions of the third kind for the extremal terms of a series of order statistics. It is extensively used to describe the laws governing the breakdown of such things as ball bearings, vacuum instruments and electronic components. The [[Exponential distribution|exponential distribution]] ( $ p = 1 $) and the [[Rayleigh distribution|Rayleigh distribution]] ( $ p = 2 $) are special cases of the Weibull distribution. The distribution functions (*) do not belong to the Pearson family. There are auxiliary tables, from which the Weibull distribution functions may be calculated ([[#References|[2]]]). If $ \mu = 0 $, the $ q $-level quantile is $ \sigma [ - \mathop{\rm ln} ( 1 - q)] ^ {1/p} $, $$ {\mathsf E} X _ {w} ^ {k} = \sigma ^ {k} \Gamma \left ( 1+ \frac{k}{p} \right ) ,\ \ k = 1, 2 \dots $$ $$ {\mathsf D} X _ {w} = \sigma ^ {2} \left [ \Gamma \left ( 1+ \frac{2}{p} \right ) - \Gamma ^ {2} \left ( 1+ \frac{1}{p} \right ) \right ] , $$ where $ \Gamma ( x) $ is the gamma-function; the coefficients of variation, skewness and excess are independent of $ \sigma $, which facilitates their tabulation and the compilation of auxiliary tables which can be used to obtain parameter estimates. If $ p \geq 1 $, the Weibull distribution is unimodal with mode equal to $ \sigma ( p - 1) ^ {1/p} $, while the risk function $ \lambda ( t) = pt ^ {p- 1 } / \sigma ^ {p} $ is non-decreasing. If $ p < 1 $, the function $ \lambda ( t) $ monotonically decreases. It is possible to construct so-called Weibull probability paper ([[#References|[3]]]). On this paper, $ {F _ {w} } ( t, p, \sigma , 0) $ becomes a straight line; if $ \mu > 0 $, the graph of $ {F _ {w} } ( t, p, \sigma , \mu ) $ is concave, and if $ \mu < 0 $ it is convex. Estimates of the parameters of the Weibull distribution by the quantile method yields equations which are much simpler than those obtained by the maximum-likelihood method. The simultaneous asymptotic efficiency of the estimates of the parameters $ p $ and $ \sigma $ (for $ \mu = 0 $) by the quantile method is maximal (and equal to 0.64) if level $ - 0.24 $ and level $ - 0.93 $ quantiles are employed. The distribution function (*) is well-approximated by the distribution function of the log-normal distribution, $$ \Phi \left ( \frac{ \mathop{\rm ln} ( t- b)- a }{c} \right ) $$ (where $ \Phi ( x) $ is the distribution function of the standardized normal distribution and $ - \infty < b < \infty $, $ - \infty < a < \infty $, $ c > 0 $): $$ \inf _ {p, \sigma ,a,c } \ \sup _ { t } \ \left | F _ {w} ( t, p, \sigma , 0)- \Phi \left ( \frac{ \mathop{\rm ln} t- a }{c} \right ) \right | = $$ $$ = \ \inf _ { a,c } \sup _ { t } \left | F _ {w} ( t, 1, 1, 0) - \Phi \left ( \frac{ \mathop{\rm ln} t- a }{c} \right ) \right | = 0.038 . $$ ====References==== <table><TR><TD valign="top">[1]</TD> <TD valign="top"> W. Weibull, "A statistical theory of the strength of materials" , Stockholm (1939)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> B.V. Gnedenko, Yu.K. Belyaev, A.D. Solov'ev, "Mathematical methods of reliability theory" , Acad. Press (1969) (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> L. Johnson, "The statistical treatment of fatigue experiments" , Amsterdam (1964)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946)</TD></TR></table> ====Comments==== ====References==== <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> N.L. Johnson, S. Kotz, "Distributions in statistics. Continuous univariate distributions" , Wiley (1970)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J. Galambos, "The theory of extreme order statistics" , Wiley (1987)</TD></TR></table>
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Weibull distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weibull_distribution&oldid=52396
This article was adapted from an original article by Yu.K. BelyaevE.V. Chepurin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
Retrieved from "https://encyclopediaofmath.org/wiki/Weibull_distribution"
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Double zero singularity and spatiotemporal patterns in a diffusive predator-prey model with nonlocal prey competition
Global and exponential attractors for the 3D Kelvin-Voigt-Brinkman-Forchheimer equations
September 2020, 25(9): 3437-3460. doi: 10.3934/dcdsb.2020068
Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system modeling coral fertilization
Ling Liu 1, , Jiashan Zheng 2,, and Gui Bao 2,
Department of Basic Science, Jilin Jianzhu University, Changchun 130118, China
School of Mathematics and Statistics Science, Ludong University, Yantai 264025, China
Received June 2019 Revised October 2019 Published September 2020 Early access April 2020
We consider an initial-boundary value problem for the incompressible four-component Keller-Segel-Navier-Stokes system with rotational flux
$ \begin{align} \left\{ \begin{array}{l} n_t+u\cdot\nabla n = \Delta n-\nabla\cdot(nS(x,n,c)\nabla c)-nm,\quad x\in \Omega, t>0,\\ c_t+u\cdot\nabla c = \Delta c-c+m,\quad x\in \Omega, t>0,\\ m_t+u\cdot\nabla m = \Delta m-nm,\quad x\in \Omega, t>0, \qquad \qquad \qquad \qquad \qquad \qquad (*)\\ u_t+\kappa(u \cdot \nabla)u+\nabla P = \Delta u+(n+m)\nabla \phi,\quad x\in \Omega, t>0,\\ \nabla\cdot u = 0,\quad x\in \Omega, t>0\\ \end{array}\right. \end{align} $
in a bounded domain
$ \Omega\subset \mathbb{R}^3 $
with smooth boundary, where
$ \kappa\in \mathbb{R} $
is given constant,
$ S $
is a matrix-valued sensitivity satisfying
$ |S(x,n,c)|\leq C_S(1+n)^{-\alpha} $
with some
$ C_S> 0 $
$ \alpha\geq 0 $
. As the case
$ \kappa = 0 $
(with
$ \alpha\geq\frac{1}{3} $
or the initial data satisfy a certain smallness condition) has been considered in [18], based on new gradient-like functional inequality, it is shown in the present paper that the corresponding initial-boundary problem with
$ \kappa \neq 0 $
admits at least one global weak solution if
$ \alpha>0 $
. To the best of our knowledge, this is the first analytical work for the full three-dimensional four-component chemotaxis-Navier-Stokes system.
Keywords: Navier-Stokes system, Keller-Segel model, global existence, tensor-valued sensitivity.
Mathematics Subject Classification: 35K55, 35Q92, 35Q35, 92C17.
Citation: Ling Liu, Jiashan Zheng, Gui Bao. Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system modeling coral fertilization. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3437-3460. doi: 10.3934/dcdsb.2020068
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export.arXiv.org > nucl-ex
Nuclear Experiment
Title: New $α$-Emitting Isotope $^{214}$U and Abnormal Enhancement of $α$-Particle Clustering in Lightest Uranium Isotopes
Authors: Z. Y. Zhang, H. B. Yang, M. H. Huang, Z. G. Gan, C. X. Yuan, C. Qi, A. N. Andreyev, M. L. Liu, L. Ma, M. M. Zhang, Y. L. Tian, Y. S. Wang, J. G. Wang, C. L. Yang, G. S. Li, Y. H. Qiang, W. Q. Yang, R. F. Chen, H. B. Zhang, Z. W. Lu, X. X. Xu, L. M. Duan, H. R. Yang, W. X. Huang, Z. Liu, X. H. Zhou, Y. H. Zhang, H. S. Xu, N. Wang, H. B. Zhou, X. J. Wen, S. Huang, W. Hua, L. Zhu, X. Wang, Y. C. Mao, X. T. He, S. Y. Wang, W. Z. Xu, H. W. Li, Z. Z. Ren, S. G. Zhou
Subjects: Nuclear Experiment (nucl-ex)
A new $\alpha$-emitting isotope $^{214}$U, produced by fusion-evaporation reaction $^{182}$W($^{36}$Ar, 4n)$^{214}$U, was identified by employing the gas-filled recoil separator SHANS and recoil-$\alpha$ correlation technique. More precise $\alpha$-decay properties of even-even nuclei $^{216,218}$U were also measured in reactions of $^{40}$Ar, $^{40}$Ca with $^{180, 182, 184}$W targets. By combining the experimental data, improved $\alpha$-decay reduced widths $\delta^2$ for the even-even Po--Pu nuclei in the vicinity of magic neutron number $N=126$ were deduced. Their systematic trends are discussed in terms of $N_{p}N_{n}$ scheme in order to study the influence of proton-neutron interaction on $\alpha$ decay in this region of nuclei. It is strikingly found that the reduced widths of $^{214,216}$U are significantly enhanced by a factor of two as compared with the $N_{p}N_{n}$ systematics for the $84 \leq Z \leq 90$ and $N<126$ even-even nuclei. The abnormal enhancement is interpreted by the strong monopole interaction between the valence protons and neutrons occupying the $\pi 1f_{7/2}$ and $\nu 1f_{5/2}$ spin-orbit partner orbits, which is supported by a large-scale shell model calculation.
Title: Isospin diffusion measurement from the direct detection of a Quasi-Projectile remnant
Authors: A. Camaiani, G. Casini, S. Piantelli, A. Ono, E. Bonnet, R. Alba, S. Barlini, B. Borderie, R. Bougault, C. Ciampi, A. Chbihi, M. Cicerchia, M. Cinausero, J.A. Dueñas, D. DellAquila, Q. Fable, D. Fabris, C. Frosin, J. D. Frankland, F. Gramegna, D. Gruyer, K. I. Hahn, M. Henri, B. Hong, S. Kim, A. Kordyasz, M. J. Kweon, H. J. Lee, J. Lemarié, N. LeNeindre, I. Lombardo, O. Lopez, T. Marchi, S. H. Nam, P. Ottanelli, M. Parlog, G. Pasquali, G. Poggi, J. Quicray, A. A. Stefanini, S. Upadhyaya, S. Valdré, E. Vient
Journal-ref: Phys. Rev. C 103, 014605 (2021)
The neutron-proton equilibration process in 48 Ca+ 40 Ca at 35 MeV/nucleon bombarding energy has been experimentally estimated by means of the isospin transport ratio. Experimental data have been collected with a subset of the FAZIA telescope array, which permitted to determine Z and N of detected fragments. For the first time, the QP evaporative channel has been compared with the QP break-up one in a homogeneous and consistent way, pointing out to a comparable n-p equilibration which suggests close interaction time between projectile and target independently of the exit channel. Moreover, in the QP evaporative channel n-p equilibration has been compared with the prediction of the Antisymmetrized Molecular Dynamics (AMD) model coupled to the GEMINI statistical model as an afterburner, showing a larger probability of proton and neutron transfers in the simulation with respect to the experimental data.
Title: Some remarks on the discovery of Md-244
Authors: F.P.Hessberger, M.Block, Ch.E.Duellmann, A.Yakushev, M.Leino, J.Uusitalo
In two recent papers by Pore et al. and Khuyagbaatar et al. discovery of the new isotope Md-244 was reported. The decay data, however, are conflicting. While Pore et al. report two isomeric states decaying by alpha emission with E(1)=8.66(2) MeV, T_1/2=0.4+0.4/-0.1s and E(2)=8.31(2) MeV, T_1/2 approx 6 s, Khuyagbaatar et al. report only a single transition with a broad energy distribution of E=(8.73-8.86) MeV and T_1/2=0.30/-0.09 s. The data published by Pore et al. very similar to those published for Md-245 (E=8.64(2), 8.68(2) MeV T_1/2=0.35+0.23/-0.16}$ s ). Therefore, we compare the data presented for Md-244 by Pore et al. with those reported for Md-245 by Ninov et al. and also by Khuyagbaatar et al.. We conclude that the data presented by Pore et al. shall be attributed to Md-245 with small contributions (one event each) from Fm-245 and probably Md-246.
Title: Persistence of the ${Z=28}$ shell gap in ${A=75}$ isobars: Identification of a possible ${(1/2^-)}$ $μ$s isomer in ${^{75}}$Co and $β$ decay to ${^{75}}$Ni
Authors: S. Escrig, A. I. Morales, S. Nishimura, M. Niikura, A. Poves, Z. Y. Xu, G. Lorusso, F. Browne, P. Doornenbal, G. Gey, H.-S. Jung, Z. Li, P.-A. Söderström, T. Sumikama, J. Taprogge, Zs. Vajta, H. Watanabe, J. Wu, A. Yagi, K. Yoshinaga, H. Baba, S. Franchoo, T. Isobe, P. R. John, I. Kojouharov, S. Kubono, N. Kurz, I. Matea, K. Matsui, D. Mengoni, P. Morfouace, D. R. Napoli, F. Naqvi, H. Nishibata, A. Odahara, E. Şahin, H. Sakurai, H. Schaffner, I. G. Stefan, D. Suzuki, R. Taniuchi, V. Werner, D. Sohler
Comments: 15 pages, 6 figures, 3 tables. Physical Review C
Background: The evolution of shell structure around doubly-magic exotic nuclei is of great interest in nuclear physics and astrophysics. In the `south-west' region of $^{78}$Ni, the development of deformation might trigger a major shift in our understanding of explosive nucleosynthesis. To this end, new spectroscopic information on key close-lying nuclei is very valuable.
Purpose: We intend to measure the isomeric and $\beta$ decay of $^{75}$Co, with one proton- and two neutron-holes relative to $^{78}$Ni, to access new nuclear structure information in $^{75}$Co and its $\beta$-decay daughters $^{75}$Ni and $^{74}$Ni.
Methods: The nucleus $^{75}$Co is produced in relativistic in-flight fission reactions of $^{238}$U at the Radioactive Ion Beam Factory (RIBF) in the RIKEN Nishina Center. Its isomeric and $\beta$ decay are studied exploiting the BigRIPS and EURICA setups.
Results: We obtain partial $\beta$-decay spectra for $^{75}$Ni and $^{74}$Ni, and report a new isomeric transition in $^{75}$Co. The energy ($E_{\gamma}=1914(2)$ keV) and half-life ($t_{1/2}=13(6)$ $\mu$s) of the delayed $\gamma$ ray lend support for the existence of a $J^{\pi}=(1/2^-)$ isomeric state at 1914(2) keV. A comparison with PFSDG-U shell-model calculations provides good account for the observed states in $^{75}$Ni, but the first calculated $1/2^-$ level in $^{75}$Co, a prolate $K=1/2$ state, is predicted about 1 MeV below the observed $(1/2^-)$ level.
Conclusions: The spherical-like structure of the lowest-lying excited states in $^{75}$Ni is proved. In the case of $^{75}$Co, the results suggest that the dominance of the spherical configurations over the deformed ones might be stronger than expected below $^{78}$Ni. Further experimental efforts to discern the nature of the $J^{\pi}=(1/2^-)$ isomer are necessary.
[5] arXiv:2101.06036 (cross-list from nucl-th) [pdf, ps, other]
Title: Statistical and dynamical bimodality in multifragmentation reactions
Authors: S. Mallik, G. Chaudhuri, F. Gulminelli, S. Das Gupta
Comments: arXiv admin note: substantial text overlap with arXiv:1802.05047
Subjects: Nuclear Theory (nucl-th); Nuclear Experiment (nucl-ex)
The bimodal behavior of the order parameter is studied in the framework of Boltzmann-Uehling-Uhlenbeck (BUU) transport model. In order to do that, simplified yet accurate method of BUU model is used which allow calculation of fluctuations in systems much larger than what was considered feasible in a well-known and already existing model. It is observed that depending on the projectile energy and centrality of the reaction, both entrance channel and exit channel effects can be at the origin of the experimentally observed bimodal behavior. Both dynamical and statistical bimodality mechanisms are associated in the theoretical model to different time scales of the reaction, and to different energy regimes.
Title: Production of super-heavy nuclei in cold fusion reactions
Authors: V. Yu. Denisov, I. Yu. Sedykh
Comments: 23 pages, 10 figures, 5 tables. To be published in Chinese Physics C
The model for the cold-fusion reactions related to the synthesis of super-heavy nuclei in collisions of heavy projectile-nuclei with $^{208}$Pb target nucleus is discussed. In the framework of this model the production of the compound nucleus by two paths through, the di-nuclear system and the fusion way, are taken into account simultaneously. The formation of the compound nucleus in the framework of the di-nuclear system is related to the transfer of nucleons from the light nucleus to the heavy one. The fusion way is linked to the sequential evolution of the nuclear shape from the system of contacting nuclei to the compound nucleus. It is shown that the compound nucleus is mainly formed by the fusion way in the cold-fusion reactions. The landscape of the potential energy related to the fusion path is discussed in detail. This landscape for very heavy nucleus-nucleus systems has the intermediate state, which is linked to the formation of both the compound nucleus and the quasi-fission fragments. The decay of the intermediate state is taken into account in the calculation of the compound nucleus production cross sections and the quasi-fission cross sections. The values of the cold-fusion cross sections obtained in the model are well agreed with the experimental data.
[7] arXiv:2101.06062 (cross-list from nucl-th) [pdf, other]
Title: New signatures of phase transition from Statistical Models of Nuclear multifragmentation
Authors: G. Chaudhuri, S. Mallik, P. Das, S. Das Gupta
The study of liquid-gas phase transition in heavy ion collisions has generated a lot of interest amongst the nuclear physicists in the recent years. In heavy ion collisions, there is no direct way of measuring the state variables like entropy, pressure, energy and hence unambiguous characterization of phase transition becomes difficult. This work proposes new signatures of phase transition that can be extracted from the observables which are easily accessible in experiments. It is observed that the temperature dependence of the first order derivative of the order parameters in nuclear liquid gas phase transition exhibit similar behavior as that of the variation of specific heat at constant volume Cv which is an established signature of first order phase transition. This motivates us to propose these derivatives as confirmatory signals of liquid-gas phase transition. The measurement of these signals in easily feasible in most experiments as compared to the other signatures like specific heat, caloric curve or bimodality. Total multiplicity, size of largest cluster are some of the order parameters which have been studied. Statistical Models based on canonical ensemble and lattice gas model has been used for the study. This temperature where the peak appears is designated to be the transition temperature and the effect of certain parameters on this has also been examined. The multiplicity derivative signature proposed in this work has been further confirmed by other theoretical models as well as in experimental study.
[8] arXiv:2101.06122 (cross-list from physics.ins-det) [pdf, other]
Title: Electron beam studies of light collection in a scintillating counter with embedded wavelength-shifting fibers
Authors: M. Lauß, P. Achenbach, S. Aulenbacher, M. Ball, I. Beltschikow, M. Biroth, P. Brand, S. Caiazza, M. Christmann, O. Corell, A. Denig, L. Doria, P. Drexler, I. Friščić, J. Geimer, P. Gülker, M. Kohl, T. Kolar, W. Lauth, M. Littich, M. Lupberger, S. Lunkenheimer, D. Markus, M. Mauch, H. Merkel, M. Mihovilovič, J. Müller, B. S. Schlimme, C. Sfienti, S. Širca, E. Stephan, S. Stengel, C. Szyszka, S. Vestrick, A. Wilczek (for the MAGIX Collaboration)
Subjects: Instrumentation and Detectors (physics.ins-det); Nuclear Experiment (nucl-ex)
The light collection properties of several wavelength-shifting fiber configurations embedded in a box-shaped plastic scintillating counter were studied by scanning with minimum ionizing electrons. The light was read out by silicon photomultipliers at both ends. The light yield produced by the 855-MeV beam of the Mainz Microtron showed a strong dependence on the transverse distance from its position to the fibers. The observations were appropriately modeled by attributing the total light yield to the collection of diffuse light inside the counter and of direct light reaching a fiber. This was compared to the light collection properties of a scintillating counter without fibers. These studies were carried out within the development of plastic scintillating detectors as an active veto system for the DarkMESA electron beam-dump experiment that will search for light dark matter particles in the MeV mass range.
[9] arXiv:1907.03658 (replaced) [pdf, other]
Title: Scale and Scheme Independence and Position-Momentum Equivalence of Nuclear Short-Range Correlations
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Speed at aphelion is decreasing for Parker Solar Probe with each new orbit despite being closer to the Sun
Sorry for the really bad picture
I am looking at some data from the Parker Solar Probe mission and I was looking at the speeds of PSP at perihelion and aphelion at each new, different, orbit (the orbits caused by a gravity assist of which there are 8) and I found something which I can't explain. Firstly the speeds at perihelion increase after each orbit which makes sense because PSP is getting closer and closer to the Sun (shown in the picture v3>v2>v1). Then I noticed that the speeds at aphelion actually decrease after each new orbit (v6>v5>v4) which I don't understand because the aphelion positions are also getting closer and closer to the Sun. I was thinking that the only way for these speeds to be decreasing would be because of the gravity assist. If this is the case I still don't understand why these speeds would be decreasing since the Venus assist should get less effective as PSP gets faster (not more effective which is what is suggested by the fact that the speeds are decreasing each new orbit). In fact the assist wouldn't just have to be getting slightly more effective but quite a bit more effective in order to not only decrease the speed more than the previous orbit, but do so with PSP moving much faster than before (each new orbit is closer to the Sun). As far as I know the factors that effect the magnitude of the delta-v from a gravity assist are the hyperbolic excess speed (where, generally, lower speed is more desirable), periapsis altitude (which stays fairly constant for each gravity assist - there are a couple outliers but I see no trend with those outliers and the speed of aphelion data), and gravitational parameter (constant). On top of this, even if the gravity assist was capable of getting more effective and decreasing the speeds more and more, this would mean that Venus would always have to be positioned before aphelion in order to decrease its speed by the time it reaches aphelion which is also something I think is unlikely. Does anyone know what may be happening here? *** all my knowledge on this subject is from the internet so I might be missing something really simple ***
v_peri = 84.362, 95.293, 109.0, 129.348, 148.007, 162.669, 176.786, 190.492
v_apo = 17.239, 16.865, 16.213, 15.04, 13.988, 13.253, 12.577, 11.987
This is the velocity at perihelion and aphelion data where the first row is orbit 1, the second is orbit 2, third is orbit 3, ..., and the last is orbit 8. There appears to be similarities with the 2nd and 3rd orbits and the 5th and 6th which is also possibly important. I know that the pairs of gravity assists 1st and 2nd, 3rd and 4th, 5th and 6th occur at roughly the same position of Venus in its orbit but even this doesn't seem to be what is reflected by the similarities between orbits 2 and 3, 5 and 6 in the data below.
This is a table of all the data I calculated (the perihelion and aphelion distances I found from NASA).
I also found a graph of PSP's speed over time and it displays the same thing that I found: despite the speed at perihelion increasing, the aphelion speed is slightly decreasing.
I have the idea that this decreasing in speed might be the reason why PSP is able to keep its velocity going into the gravity assist fairly constant (I also calculated this) despite its speed at perihelion increasing significantly. I calculated the incoming v_inf 's of PSP for each gravity assist and they are all approximately 22.9 km/s despite the fact that PSP would be getting a lot more speed from the Sun after each assist (I have a weird exception in the speeds for the 6th flyby which I found is 43.7 km/s but I suspect something is wrong with this data since it also has an eccentricity of 140 000). I was wondering how PSP is able to keep such a constant v_inf going into the assist and now I guess it has something to do with it decreasing speed at aphelion.
orbital-mechanics gravity-assist astrodynamics astronomy parker-solar-probe
Alexander Ivanov
Alexander IvanovAlexander Ivanov
$\begingroup$ I wonder if it would be possible to calculate the change in orbital energy and figure out whether there is a discrepancy larger than the change rendered by the gravitational assists. $\endgroup$ – BMF Jul 11 '20 at 0:27
$\begingroup$ @BMF So your suggesting to calculate the difference in orbital energies (E = -GM/2a) of 2 orbits, say the 1st and 2nd, and then the difference in orbital energies of the two corresponding hyperbolic orbits cause by the gravity assists? (which should be the same formula except without the negative sign I think). I don't really understand why this would be useful to do. Could you clarify how these energies might help explain the decrease in aphelion speeds? $\endgroup$ – Alexander Ivanov Jul 11 '20 at 1:18
Another way to express the results of the vis-viva equation at apoapsis is $${v_a}^2 = \frac{2\mu}{r_p+r_a}\frac{r_p}{r_a} = \frac{2\mu}{r_a}\frac{r_p}{r_p+r_a}\tag{1}$$ where $v_a$ is the velocity at apoapsis, $\mu$ is the standard gravitational parameter $\mu\equiv GM$, and $r_a$ and $r_p$ are the apoapsis and periapsis distances.
On holding the apoapsis distance $r_a$ constant, the first term on the right hand side of equation (1), $\frac{2\mu}{r_a}$, is constant, making the apoapsis velocity solely vary according to latter term on the right hand side of equation (1), $\frac{r_p}{r_p+r_a}$. This latter term monotonically increases as the periapsis distance $r_p$ increases from zero to $r_a$.
Alternatively, this latter term monotonically decreases as $r_p$ decreases from $r_a$ to zero. In other words, decreasing periapsis distance while holding apoapsis distance constant results in a decrease in apoapsis velocity.
David HammenDavid Hammen
$\begingroup$ Yet another way to look at it is to solve for $r_p$ given $v_a$, $r_a$, and $\mu$:$$r_p = \frac{{v_a}^2 r_a}{\frac{2\mu}{r_a}-{v_a}^2}$$Reducing the apoapsis velocity reduces the periapsis distance. Yet another way to look at it: The delta v needed to make an object dive into the Sun is a good deal more than is the delta v needed to make an object escape the solar system. $\endgroup$ – David Hammen Jul 11 '20 at 7:25
$\begingroup$ Oh wow this is a nice way of understanding it. The data actually shows that the apoapsis distance is also decreasing (I added in a table of all my data). However, the apoapsis distances is decreasing at a much slower rate than the periapsis distance and so it makes sense that the overall apoapsis velocity should be decreasing (and not by much, which is exactly what is seen). Thank you! $\endgroup$ – Alexander Ivanov Jul 11 '20 at 7:51
Just a short supplementary to try and add a more "intuitive" understanding to the two excellent "equation-based" answers.
For me the easiest way to think of this is that you have cause and effect reversed in your description of the problem. Consider PSP at aphelion, which is always more or less at Venus distance from the Sun, so that it can use Venus' gravity to shape its orbit. It is moving "around" the Sun (since it's at aphelion, it is momentarily getting neither closer or further away). Where it goes next is determined by how fast it is moving. If it was moving fast enough it would stay in a circular orbit, with the Sun's gravity pulling it around just enough to stop it getting further away, but not enough to bring it closer. (this is what Venus does, near enough). If it is going less fast than that, then the Sun's gravity bends its trajectory in more sharply, so it ends up curving "in" towards the Sun. As it moves closer, gravity both bends and accelerates its trajectory, until in the end it speeds up enough that it stops getting closer to the Sun (perihelion) and then starts to move away again. The slower it was moving at aphelion, the more "falling in" it needs to do to reach perihelion, so the closer to the Sun it gets, but, it turns out, the faster it will be going at perihelion. So this is what the equations are showing -- for a fixed aphelion distance, the slower you are going at aphelion, the lower you perihelion will be, but, in fact, the faster you will be going at perihelion.
Steve LintonSteve Linton
$\begingroup$ Thank you for this explanation! It seems so obvious now. In order to keep bending the orbit enough to get PSP to be closer and closer to the Sun, it must be moving slower at aphelion as otherwise it would simply have enough speed to make a more circular orbit than before (and the goal is to be more eccentric to get closer to the Sun). $\endgroup$ – Alexander Ivanov Jul 11 '20 at 15:21
When thinking about the speeds and distances in a Keplerian orbit we turn to our friend the vis-viva equation:
$$v^2 = GM \left( \frac{2}{r} - \frac{1}{a} \right)$$
where $v$ is the speed at distance $r$ for an object with a semi-major axis $a$ and $GM$ is the gravitational constant $G$ times the Sun's mass M. We can call that product the standard gravitational parameter of the Sun and it's about 1.327E+20 m^3/s^2.
The distances at periapsis and apoapsis $r_p, r_a$ are given by
$$r_p = a(1-\epsilon)$$ $$r_a = a(1+\epsilon)$$
where $\epsilon$ is the eccentricity of the orbit. Put these together and the speeds at periapsis and apoapsis $v_p, v_a$ are
$$v_p^2 = \frac{GM}{a} \frac{1+\epsilon}{1-\epsilon}$$
$$v_a^2 = \frac{GM}{a} \frac{1-\epsilon}{1+\epsilon}$$
Since you have two equations and two unknowns for each orbit, in principle you can solve for the semimajor axis and the eccentricity for each row in your table. You can imagine that as long as both parameters can vary, you can make the speed at aphelion either increase or decrease even though the speed at perhihelion decreases.
You can get a big hint from the fact that the orbit starts circular and ends very elliptical and suppose that eccentricity generally increases, and remember that Parker needs to regularly access Venus's orbit each time it moves closer to the Sun since it uses Venus' gravity rather than propulsion to lower its perihelion, so it's probably a good first guess to assume that the orbit keeps a roughly constant aphelion and its perihelion and eccentricity both decrease over time.
An orbit with a fixed apoapsis and decreasing periapsis will have increasingly slower speed at apoapsis so that it "falls closer" to the Sun.
uhohuhoh
$\begingroup$ Yes I actually used that formula to get the speeds at perihelion and aphelion. I only had the values for the distance to perihelion and aphelion for each orbit so I used that to get a and e and solve for the velocities. And when you mentioned that the apoapsis would remain at the same position, the data I have (from NASA) shows that it doesn't. The aphelion distance definitely decreases for each orbit and so does the velocity at aphelion, I will put in the whole table of data that I made into the question. $\endgroup$ – Alexander Ivanov Jul 11 '20 at 6:20
$\begingroup$ So I added in the table and you can see that r_a decreases while v_a also decreases which is what I am confused about. Since this point is getting closer to the Sun how could it be possible that PSP is moving slower at it. $\endgroup$ – Alexander Ivanov Jul 11 '20 at 6:24
$\begingroup$ Right, I suppose my question is not specifically about PSP. Thank you for thinking about it. Another idea I had (I'm kinda just guessing though) is that maybe it is to do with the fact that the closer PSP is to the Sun, the farther down it is in the Sun's gravitational well. And in that case I guess it could be possible that the farther down you go, the "steeper" the travel back up is and so the harder it is to go back up. So I was thinking maybe thats why it would be slower for an aphelion that is actually closer (it has to climb out of a steeper part of the well). $\endgroup$ – Alexander Ivanov Jul 11 '20 at 7:25
$\begingroup$ That idea came when I was thinking about those pictures of how relativity describes gravity (same principle as a trampoline). But again I have no idea how accurate that model is or really anything about it, it's just a thought. $\endgroup$ – Alexander Ivanov Jul 11 '20 at 7:27
$\begingroup$ @AlexanderIvanov If you keep apoapsis near Venus for gravity assists for orbit lowering and the potential energy at apoapsis therefore stays the same, then the only way to get closer to the Sun is to reduce total energy, which means kinetic energy at apoapsis must be lower which means velocity at apoapsis must be lower. $\endgroup$ – uhoh Jul 11 '20 at 8:24
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Complex earthquakes and Teichmüller theory
Author: Curtis T. McMullen
Journal: J. Amer. Math. Soc. 11 (1998), 283-320
MSC (1991): Primary 30F10, 30F40, 32G15
DOI: https://doi.org/10.1090/S0894-0347-98-00259-8
MathSciNet review: 1478844
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: It is known that any two points in Teichmüller space are joined by an earthquake path. In this paper we show any earthquake path $\mathbb R \rightarrow T(S)$ extends to a proper holomorphic mapping of a simply-connected domain $D$ into Teichmüller space, where $\mathbb R \subset D \subset \mathbb C$. These complex earthquakes relate Weil-Petersson geometry, projective structures, pleated surfaces and quasifuchsian groups. Using complex earthquakes, we prove grafting is a homeomorphism for all 1-dimensional Teichmüller spaces, and we construct bending coordinates on Bers slices and their generalizations. In the appendix we use projective surfaces to show the closure of quasifuchsian space is not a topological manifold.
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F. Bonahon. The geometry of Teichmüller space via geodesic currents. Invent. math. 92(1988), 139–162.
C. J. Earle. On variation of projective structures. In I. Kra and B. Maskit, editors, Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference, pages 87–99. Annals of Math. Studies 97, Princeton, 1981.
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L. Keen and C. Series. Pleating coordinates for the Maskit embedding of the Teichmüller space of a punctured torus. Topology 32(1993), 719–749.
S. Kerckhoff. The Nielsen realization problem. Ann. of Math. 177(1983), 235–265.
S. Kerckhoff. Earthquakes are analytic. Comment. Math. Helvetici 60(1985), 17–30.
S. Kerckhoff and W. Thurston. Non-continuity of the action of the modular group at Bers' boundary of Teichmüller space. Invent. math. 100(1990), 25–48.
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O. Lehto. Univalent functions and Teichmüller spaces. Springer-Verlag, 1987.
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S. Nag. The Complex Analytic Theory of Teichmüller Space. Wiley, 1988.
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Retrieve articles in Journal of the American Mathematical Society with MSC (1991): 30F10, 30F40, 32G15
Retrieve articles in all journals with MSC (1991): 30F10, 30F40, 32G15
Received by editor(s): March 8, 1996
Received by editor(s) in revised form: October 21, 1997
Additional Notes: The author's research was partially supported by the NSF
Article copyright: © Copyright 1998 American Mathematical Society
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Divergent Series
Why is it that divergent series make sense?
Specifically, by basic calculus a sum such as $1 - 1 + 1 ...$ describes a divergent series (where divergent := non-convergent sequence of partial sums) but, as described in these videos, one can use Euler, Borel or generic summation to arrive at a value of $\tfrac{1}{2}$ for this sum.
The first apparent indication that this makes any sense is the claim that the summation machine 'really' works in the complex plane, so that, for a sum like $1 + 2 + 4 + 8 + ... = -1$ there is some process like this:
going on on a unit circle in the complex plane, where those lines will go all the way back to $-1$ explaining why it gives that sum.
The claim seems to be that when we have a divergent series it is a non-convergent way of representing a function, thus we just need to change the way we express it, e.g. in the way one analytically continues the Gamma function to a larger domain. This claim doesn't make any sense out of the picture above, & I don't see (or can't find or think of) any justification for believing it.
Futhermore there is this notion of Cesaro summation one uses in Fourier theory. For some reason one can construct these Cesaro sums to get convergence when you have a divergent Fourier series & prove some form of convergence, where in the world does such a notion come from? It just seems as though you are defining something to work when it doesn't, obviously I'm missing something.
I've really tried to find some answers to these questions but I can't. Typical of the explanations is this summary of Hardy's divergent series book, just plowing right ahead without explaining or justifying the concepts.
I really need some general intuition for these things for beginning to work with perturbation series expansions in quantum mechanics & quantum field theory, finding 'the real' expanation for WKB theory etc. It would be so great if somebody could just say something that links all these threads together.
quantum-field-theory mathematical-physics renormalization regularization perturbation-theory
bolbteppabolbteppa
$\begingroup$ Isn't the technique usually just: 1) Find an expression of the series that is convergent and has an adjustable parameter, and 2) sum the series 3) take the limit of the sum as the parameter approaches something that would make the sum superficially divergent? What's so complicated? $\endgroup$ – Jerry Schirmer Jan 10 '14 at 14:32
$\begingroup$ 3 years of mathematical education I guess, worrying so much about convergence of series that I was prepared to learn all the weird tests by Kummer, Schomilch, Bertrand, De Morgan etc... I have a ton of analysis books implying divergent series are bad, against that I have only what I've written above telling me divergent series are not bad. I don't know why, say, one can start with $\sum (-1)^n e^n = \tfrac{1}{1+e}, |e| < 1$, let $e = 3$ and say that $\tfrac{1}{1+3}=\tfrac{1}{4}$ should make any sense when the series doesn't converge for $1 \leq e$, because people 'assert' it has some meaning. $\endgroup$ – bolbteppa Jan 10 '14 at 14:44
$\begingroup$ A good place to start: terrytao.wordpress.com/2010/04/10/… $\endgroup$ – Yvan Velenik Jan 10 '14 at 14:49
$\begingroup$ Yvan just provided the best text about these strange sums (also called Ramanujan summations). This topic seems to show up quite regularly... yesterday on the physics SE and also on the math SE here. $\endgroup$ – Tom-Tom Jan 10 '14 at 15:28
$\begingroup$ @bolbteppa. How is this related to the WKB approximation ? $\endgroup$ – Tom-Tom Jan 10 '14 at 15:30
I've been thinking about divergent series on and off, so maybe I could chip in.
Consider a sequence of numbers (in an arbitrary field, e.g. real numbers) $\{a_n\}$. You may ask about the sum of terms of this sequence, i. e. $\sum a_n$. If the limit $\lim_{N\rightarrow\infty} \sum^N |a_n|$ exists then the series is absolutely convergent and you may talk about the sum $\sum a_n$. In case the limit does not exist but $\lim_{N\rightarrow\infty} \sum^N a_n$ exists then the sequence is conditionally convergent, and as (I assume) Carl Witthoft commented above there is a theorem stating that you may sum the sequence in a different order and get a different result for the limit. In fact by judiciously rearranging you may get any number desired. I included this just to mention that although divergent series may seem most bizarre, in the sense of summing terms and that by each term it gets nearer a limit, only the absolutely convergent series make connection with our intuiton. So we may ask about making sense of series in general.
As G. H. Hardy's "Divergent Series" explain in page 6, the trick is to understand that our usual notion of sum of a serie is a way to define something we call a "sum". In other words given a sequence we have a map that atributes to this sequence a number. The "sum map" being the trivial operation of summing the terms if the series is absolutely convergent. The idea behind divergent series is to realize that this map althogh in a sense canonical is not unique.
To be more specific, consider the space $V$ of all sequences together with operations of addition and scalar multiplication (given two sequences $\{a_n\}$ and $\{b_n\}$ and a number $\lambda$ we define addition by $\{a_n\}+\{b_n\}=\{a_n+b_n\}$ and scalar multiplication by $\lambda\cdot\{a_n\}=\{\lambda a_n\}$). Now the space of sequences with these operations is a (infinite-dimensional) vector space (there is a good question about coordinates, since I am assuming one specific basis here, but let's not worry about this now). The absolutely convergent series can be seen to form a subspace $U$ of $V$ and the "sum" $S$ is just a linear functional on this subspace, $S:U\rightarrow\mathbb{R}$. The problem is that this functional is not defined anywhere else.
So to make sense of divergent series one asks if is there another map $S'$ defined on a subspace $W$ that contains $U\subset W$ such that when restricted to this subspace is just the usual sum, i.e. is there $S'$ such that $S'|_U=S$?
And in fact many such functionals do exist. And each of these we call a different "summation method" in the sense that it attributes a value to a sequence and that when such sequence corresponds to a convergent series it gives the usual values.
For instance, Cesaro Summation says that maybe the series does not converge because it keeps oscillating (like the $1-1+1\cdots$ you mentioned). Then we could take the arithmetic mean of the partial sums $s_n=a_1+\cdots a_n$ and define the Cesaro sum of the sequence $\{a_n\}$ as $\lim_{N\rightarrow\infty}\frac{1}{N}\sum^N s_n$. Is not hard to see that this gives the usual result for convergent series (although a bit obscure that it is a linear map), but it also gives $1/2$ to the alternating series of $1-1+1\cdots$. So one must give a new meaning to the word "sum", and then you can get new results. For instance Fejer's theorem roughly states that (given mild conditions) the Fourier series of a function may not be stricty convergent, but it is always summable in the sense of Cesaro. So it tells you that the worst divergence that appears in Fourier series is of oscillating type, i. e. the series never diverges to $\pm\infty$. Furthermore by Cesaro summing you can tell around which value the series oscillates about. But this does not "sum" the series in the sense of making it convergent in the usual sense.
Other ideas for functionals is by analytic continuation. The most obvious is the geometric series $\sum x^n=\frac{1}{1-x}$. Is is only convergent in the $|x|\leq 1$ radius, but one may use analytic continuation to turn around the problem at $x=1$ and say that $1+2+4+\cdots=\frac{1}{1-2}=-1$. In this case is easier to picture the linear algebra idea. The space $U$ is of all geometric series with $|x|\leq 1$ and $S$ is the usual sum. Now we introduce a functional $S'$ by $S':W\rightarrow\mathbb{R},x\mapsto\frac{1}{1-x}$ so that $W$ is now every geometric series except the one with $x=1$. A functional which reduces to this case for the geometric series but does it also for other power series is Abelian Summation.
So the idea is not to "sum" the series really (only absolutely convergent series sum in usual sense) but to redefine the notion of sum by generalizing the concept and then using this different notion to attribute a finite value to the serie through the corresponding sequence. This finite value should tell you something about the series, like the Cesaro sum tells you around which value the series oscillates or like Abel sum is able to reconstruct the function that generated the series. So summation methods are able to extract information from divergent series, and this is how to make sense of them.
With respect to the physics, is important to stress that perturbative series in quantum field theory are (generally) divergent but neither renormalization nor regularization have to do (fundamentally) with "summing" divergent series (zeta regularization being one technique, not mandatory, although useful). Rather what does occur is that it sometimes one gets a asymptotic serie in perturbation theory. In general there are different functions with the same asymptotic serie, but with supplementary information it may, or may not, occur that one can uniquely find the function with that specific asymptotic series. In this case one can use a summation method known as Borel Summation to fully reconstruct the entire function. When such thing happens in QFT is normally associated with the presence of some sort of instanton. You can take a further look in S. Weinbergs "Quantum Field Theory Vol. 2", page 283. So the idea is to get non-perturbative information out of the perturbation series, and not to tame some sort of infinity. Renormalization is something completely different (and much worse since in fact it is highly non-linear for starters).
For further information try finding a copy of Hardy's book (it's a gem), or for the linear algebra babble J. Boos, F. P. Cass "Classical and Modern Methods In Summability".
cesarulianacesaruliana
We can compute finite value of the sum of a divergent series as general limit of their partial sum. For answer, see this link (there are method and way to sum all divergent series and examples of 24 sums of divergent series): https://m4t3m4t1k4.wordpress.com/2015/11/25/general-method-for-summing-divergent-series-using-mathematica-and-a-comparison-to-other-summation-methods/
We can use that general limit to compute finite value of divergent integrals, here is 3 examples: https://m4t3m4t1k4.wordpress.com/2015/02/14/general-method-for-summing-divergent-series-determination-of-limits-of-divergent-sequences-and-functions-in-singular-points-v2/
Sinisa BubonjaSinisa Bubonja
$\begingroup$ m4t3m4t1k4.wordpress.com/2016/12/30/… $\endgroup$ – Sinisa Bubonja Jan 2 '17 at 21:51
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A Bayesian nonparametric test for conditional independence
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Topological reconstruction of sub-cellular motion with Ensemble Kalman velocimetry
June 2020, 2(2): 123-154. doi: 10.3934/fods.2020008
Hierarchical approximations for data reduction and learning at multiple scales
Prashant Shekhar , and Abani Patra
Data Intensive Studies Center, Tufts University, Medford, MA, 02155, USA
* Corresponding author: Prashant Shekhar
Fund Project: This work was funded by the grants NSF1821311, NSF1645053, NSF1621853
Figure(14) / Table(2)
This paper describes a hierarchical learning strategy for generating sparse representations of multivariate datasets. The hierarchy arises from approximation spaces considered at successively finer scales. A detailed analysis of stability, convergence and behavior of error functionals associated with the approximations are presented, along with a well chosen set of applications. Results show the performance of the approach as a data reduction mechanism for both synthetic (univariate and multivariate) and a real dataset (geo-spatial). The sparse representation generated is shown to efficiently reconstruct data and minimize error in prediction. The approach is also shown to generalize well to unseen samples, extending its prospective application to statistical learning problems.
Keywords: Sparse representation and learning, data reduction, approximation error analysis.
Mathematics Subject Classification: Primary:68W25, 65D15;Secondary:65D12.
Citation: Prashant Shekhar, Abani Patra. Hierarchical approximations for data reduction and learning at multiple scales. Foundations of Data Science, 2020, 2 (2) : 123-154. doi: 10.3934/fods.2020008
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Algorithm 1 on TF2. The plots in the left column((a), (c), (e) and (g)) show the density of basis functions at each scale, selected while identifying the sparse representation. Rank here refers to the cardinality of basis function set $ B^s $. The right column ((b), (d), (f) and (h)) shows the corresponding scalewise reconstruction of the underlying function. The small dots show the samples from true function while star markers represent the sparse representation $ D^s_{sparse} $. The solid curve is the reconstruction from the sparse representation. The fraction in bracket (for example (11/200) for (b)) shows the number of data points chosen out of 200 as $ D^s_{sparse} $">Figure 4. Scalewise performance of Algorithm 1 on TF2. The plots in the left column((a), (c), (e) and (g)) show the density of basis functions at each scale, selected while identifying the sparse representation. Rank here refers to the cardinality of basis function set $ B^s $. The right column ((b), (d), (f) and (h)) shows the corresponding scalewise reconstruction of the underlying function. The small dots show the samples from true function while star markers represent the sparse representation $ D^s_{sparse} $. The solid curve is the reconstruction from the sparse representation. The fraction in bracket (for example (11/200) for (b)) shows the number of data points chosen out of 200 as $ D^s_{sparse} $
Algorithm 1">Figure 1. Univariate {(a): Gramancy and Lee function - Test function 1 (TF1); (b): 1-D Schwefel function - TF2} and multivariate {(c): Dropwave function - TF3; (d): 2-D Schwefel function - TF4} test functions considered for studying the performance of Algorithm 1
Figure 2. Convergence behavior on the test functions measured in 2-norm prediction error on the observed data. Top row ((a) and (b)) shows the performance on univariate functions with bottom row for multivariate functions ((c) and (d)). Each of the plots also show the Critical scale ($ S_c $) and Convergence scale ($ S_a $) along with the $ \% $ of data sampled as the sparse representation ($ D^{s}_{sparse} $) at each scale
Figure 3. Convergence measured as the decay of upper bound (equation (45)) to the inner product in the native Hilbert space between approximation $ A_sf $ and the approximation error $ E^s $ for considered univariate ((a): TF1 and (b): TF2) and multivariate ((c): TF3 and (d): TF4) test functions
Algorithm 1 on TF3. Left column ((a), (c), (e) and (g)) shows the distribution of the sparse representation selected at multiple scales. Here we have shown the projection of the sparse representation (star markers) on the X-Y plain for ease of presentation. The right column ((b), (d), (f) and (h)) shows the corresponding reconstruction for the dropwave test function from the respective $ D^s_{sparse} $. Again the fraction in the header of plots (right column) shows the proportion of dataset chosen as $ D^s_{sparse} $">Figure 5. Scalewise performance of Algorithm 1 on TF3. Left column ((a), (c), (e) and (g)) shows the distribution of the sparse representation selected at multiple scales. Here we have shown the projection of the sparse representation (star markers) on the X-Y plain for ease of presentation. The right column ((b), (d), (f) and (h)) shows the corresponding reconstruction for the dropwave test function from the respective $ D^s_{sparse} $. Again the fraction in the header of plots (right column) shows the proportion of dataset chosen as $ D^s_{sparse} $
Figure 6. Confidence and Prediction Intervals for reconstruction from $ D^s_{sparse} $ from scale 0 to scale 5 for TF2. Along with the approximation produced at these scales (solid curve), the plots also show the sparse representation selected (star marked points) with the $ 95\% $ t-confidence interval (thinner dark shaded region) and $ 95\% $t-prediction interval (broader light shaded region)
Figure 7. (Ⅰ): Importance ranking for the sparse representation (important points) sampled at scale 0 for TF2 based on ordering governed by pivoted QR decomposition; (Ⅱ) Histogram of the top 3 most important points selected for TF2 (shown as a solid curve in each subplot). Here along the columns we have the increment in scale (abbreviated as S) and along the rows we have shown the histogram of the first, second and third most important point respectively (represented with abbreviation I for Importance)
5] (represented as 'Berm') and Algorithm 1 (represented as 'Shek') for the four test function TF1, TF2, TF3 and TF4 in (a), (b), (c) and (d) respectively">Figure 8. Comparative behavior of the decay of $ 2 $-$ norm $ error for the Multiscale extension algorithm from [5] (represented as 'Berm') and Algorithm 1 (represented as 'Shek') for the four test function TF1, TF2, TF3 and TF4 in (a), (b), (c) and (d) respectively
Algorithm 1 ($ Shek $). Here plots (a), (c), (e) and (g) ((e) and (g) just show the projection on XY plane) show the distribution of the training and testing data (50/50 % split) with corresponding plots on the right hand side ((b), (d), (f) and (g)) showing the reconstructions obtained just using the training data. The header on ((b), (d), (f) and (h)) also show the corresponding testing error">Figure 9. Performance on unseen samples for TF1, TF2, TF3 and TF4 by Algorithm 1 ($ Shek $). Here plots (a), (c), (e) and (g) ((e) and (g) just show the projection on XY plane) show the distribution of the training and testing data (50/50 % split) with corresponding plots on the right hand side ((b), (d), (f) and (g)) showing the reconstructions obtained just using the training data. The header on ((b), (d), (f) and (h)) also show the corresponding testing error
5] ($ Berm $) on TF1, TF2, TF3 and TF4 with same experimental setup (same training and testing data as (a), (c), (e) and (g) in figure 9). Here again the corresponding testing error is displayed as a header with corresponding reconstructions from training data">Figure 10. Performance of Algorithm 4 from [5] ($ Berm $) on TF1, TF2, TF3 and TF4 with same experimental setup (same training and testing data as (a), (c), (e) and (g) in figure 9). Here again the corresponding testing error is displayed as a header with corresponding reconstructions from training data
5] ($ Berm $) and our Algorithm 1 ($ Shek $)">Figure 11. Generalization analysis with repeated (100 times) random splitting of data into training and testing (again 50/50 split). Here we compare the mean and standard deviation of the Root Mean Square Error (RMSE) for prediction on testing data ($ D_{test} $) by Algorithm 4 from [5] ($ Berm $) and our Algorithm 1 ($ Shek $)
Figure 12. (a): Contour plot for Digital Elevation Model (DEM) data with 60m resolution; (b): Training DEM obtained by sampling the nodes at even indices (from DEM in (a)) resulting in a 120m resolution DEM
Figure 13. Scale dependent reconstructions for the DEM dataset in figure 12 (a). Here (a), (b) and (c) show the reconstructed contour plot for scales 0, 2 and 6 respectively. The header on these plots also show the the proportion of dataset selected as the sparse representation along with the $ \infty-norm $ of the reconstruction error
Figure 14. Comparison of 60m DEM reconstruction by $ Shek $ (a) and $ Berm $ (b) using the 120m DEM training data shown in figure 12 (b). The header shows the RMSE for prediction of height at the testing locations (testing RMSE for $ Shek $: 2.74 and testing RMSE for $ Berm $: 10.7)
Algorithm1
Algorithm 1 Hierarchical approach
1: INPUT:
Hyperparameters: $ [(TOL \geq 0, T>0, P > 1) \in \mathbb{R}^3] $
Dataset: $ [(X \in \mathbb{R}^{n \times d}) \subset \Omega $, $ (f|_{X} \in \mathbb{R}^n)] $
Prediction points: $ [(X_{*} \in \mathbb{R}^{n^{*} \times d}) \subset \Omega] $
2: OUTPUT:
Convergence Scale: $ [S_a \in \mathbb{N}] $
Sparse Representation ($ D_{sparse}^{S_a} $): $ [(X_{S_a} \in \mathbb{R}^{\{l_{S_a} \times d \}}),(C_{S_a} \in \mathbb{R}^{l_{S_a}})] $
Predictions: $ [P^* \in \mathbb{R}^{n^{*}}] $
3: Initialize: $ s =0 $
4: while $ TRUE $ do
5: Compute covariance kernel: [$ G_s $ on $ X $ with $ (\epsilon_s = T/P^s $)]
6: Compute numerical Rank: [$ l_s = rank(G_s) $]
7: Remove sampling Bias: [$ (W = AG_{s}) $ with $ A \in \mathbb{R}^{k \times n} $ and $ (a_{i,j} \sim N(0,1)) $]
8: Generate permutation information: [$ WP_R = QR $]
9: Produce bases at scale s: $ [B^{s} = (G_{s}P_R)[:,1:l_s] $, \quad \text{with}\ $ B^{s} \in \mathbb{R}^{n \times l_s}] $
10: Subset the sparse representation in $ X_s $
11: Compute projection coordinate: [$ C_{s} = {(B^{s})}^{\dagger} f|_X $; $ {B^s}^{\dagger} = ({B^s}^TB^s)^{-1}{B^s}^T $]
12: Generate approximation at s: [$ (A_sf)|_{X} = B^{s}C_s $]
13: If $ (||f|_X- (A_sf)|_{X}||_2 \leq TOL) $ : $ S_a = s $; $ Break $
14: Update scale: [$ s=s+1 $]
15: end while
16: Compute bases for prediction at $ X_{*} $: [$ G^{*}_{S_a} $ centered at $ X_{S_a} $with $ (\epsilon_{S_a} = T/P^{S_a} $)]
17: Predict: $ P^* = G^{*}_{S_a} C_{S_a} $
Table Options
Download as excel
Table 1. Comparison of training and prediction time for Algorithm 1 ($ Shek $) and Algorithm 4 from [5] ($ Berm $). For training analysis, we have considered the case of different training data sizes ($ |X| $) for all test functions. Correspondingly, for testing prediction latency, we again have sets $ |X_{pred}| $ of different cardinality. For each case, the approach taking shorter time has been boldfaced
Training time (s) Prediction time (s)
$ |X| $ $ Shek $ $ Berm $ $ |X_{pred}| $ $ Shek $ $ Berm $
$ TF1 $ 50 4.61e-03 3.25e-02 100 1.47e-02 7.22e-02
100 9.25e-03 1.40e-02 200 4.15e-02 3.09e-01
200 2.55e-02 5.63e-02 400 8.18e-02 1.08
400 2.12e-01 2.17e-01 800 1.08 4.10
$ TF3 $ 900 8.39e-01 7.98e-01 1225 3.13 8.67
1600 4.19 6.40 2025 5.29 2.14e+01
2500 2.76e+01 2.67e+01 3025 7.91 3.47e+01
3600 7.19e+01 8.24e+01 4225 1.16e+01 4.95e+01
10000 1.44e+03 1.88e+03 11025 3.27e+01 2.32e+02
$ TF4 $ 900 1.42 8.70e-01 1225 4.10 1.53e+01
1600 6.94 8.68 2025 1.05e+01 3.21e+01
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Prashant Shekhar Abani Patra
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CommonCrawl
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Home gardens as a predictor of enhanced dietary diversity and food security in rural Myanmar
Anu Rammohan1,
Bill Pritchard2 &
Michael Dibley3
Home gardens have been found to improve food security and dietary diversity in a wide range of settings. However, there is a need to place home gardens within the larger food and nutrition system landscapes that shape the construction of household diets. Myanmar offers a unique opportunity to study these research questions, given the decades of political isolation, high levels of food insecurity and poor nutrition levels.
The aim of our paper is to use household survey data from three distinctive agro-ecological settings in rural Myanmar to empirically analyse the role of home gardens in influencing household food insecurity and dietary diversity. Our analysis is based on unique survey data conducted in rural Myanmar. The sample includes 3230 rural households from three States/Districts (Magway, Ayeyarwady and Chin). Using information on two dimensions of food security, a series of variables capturing a household's self-reported food security status and coping strategies when food is not available; and a measure of household's dietary diversity based on 24-h recall data, we empirically estimate a household's probability of being food insecure and the diversity of their diets.
There are statistically significant associations between access to home gardens and measures of food security and improved dietary diversity. In particular, for landless households, the ownership of home gardens/ fruits and vines is statistically significant and is associated with a 6.6 percentage points lower probability of a household having to change their diet, and a 7.9 percentage points lower probability of being in hunger.
From a policy perspective, our results show that promoting home gardens among vulnerable households can improve food security and dietary diversity among vulnerable rural households in Myanmar.
Home gardens or kitchen gardens, a common form of food production in many rural communities in developing countries offer great potential for improving household food security and alleviating micronutrient deficiencies. Studies from a wide range of settings have found home gardens to be positively associated with better food security and nutritional diversity [1,2,3,4,5,6]. Home gardening can directly enhance household food security through providing access to a diversity of nutritionally-rich foods, increased purchasing power from savings on food bills and income from sales of garden products, and fall-back food provision during seasonal lean periods [7]. However, the intricate detail of how home gardens alleviate food security and improve dietary diversity remains empirically and theoretically under-researched, and is likely to differ across settings. It is critical to place home gardens within the larger food and nutrition system landscapes that shape the construction of household diets. This implies investigating the association between socio-economic factors and home gardens, and whether home gardens influence households' dietary and food security.
Furthermore despite the dominance of agriculture, in many developing countries, rural populations experience poor nutritional outcomes, and lag in measures of social and economic progress. Dietary quality remains poor in many developing countries [8], and food consumption is dominated by cheap, starchy foods and there is limited consumption of energy rich nutrient-dense foods (fruits, vegetables and animal protein) [9]. Increasingly, there is a growing recognition of the need to also account for dietary diversity, by taking into consideration the number of different food groups consumed in a household over a given reference period [10, 11]. Previous research shows that diversity indices reflect overall dietary quality, and is positively associated with measures of food security [12,13,14]. It is in this context that access to home gardens can play a critical role in improving household food security and dietary diversity.
However, the diversity of livelihood and agro-environmental contexts among communities across different contexts means that the role of home gardens varies across settings.
The aim of this paper is to examine the role of home gardens in alleviating household food insecurity and improving dietary diversity among rural households in Myanmar. Myanmar is a mainly agrarian country with severe problems of rural poverty and malnutrition. To the best of our knowledge the present study is the first detailed quantitative assessment of home gardens in the Myanmar context, given its decades of political isolation. Since 2011 several donor-funded, national surveys have been undertaken, but none specifically asked questions about home gardens. Household respondents were not asked about this topic in the 2011 and 2013 LIFT ("Livelihoods and Food Security Trust Fund") surveys, which are typically regarded as providing the most extensive and comprehensive databases on the rural conditions of the country.
The data used in this analyses are drawn from a survey of 3320 households from three distinctive agro-ecological settings of rural Myanmar that the authors conducted between February and April 2016. These include the dry zone in the central plains; the fecund delta of the Ayeyarwady River, and the mountainous region of southern Chin State.
Our analysis finds statistically significant associations between access to home gardens and measures of food security and improved dietary diversity, particularly among households without agricultural land holdings.
In the next section, we review previous research on home gardens, followed by Methods section which contains a description of our data and the methods used in the empirical analysis. The main findings from the empirical analyses are presented in Results section, followed by the Conclusions section.
Home gardens can be loosely defined as a "traditional land use system around a homestead, where several species of plants are grown and maintained by the household members and their products are primarily intended for the family consumption" [15].Footnote 1 They exist "in backyards, farmyards, kitchens, containers, small patches of available land, vacant lots, on rooftops and tabletops, and along roadsides and the edges of fields. They are generally close to a house and source of water, and are managed by family members using low-cost inputs" [16]. Given the complexity of home gardens within wider household food production systems, including field-based cropping, the collection of semi-cultivated and wild foods, and the rearing of livestock, it is difficult to precisely articulate the concept beyond general terms [17]. Nevertheless, although the forms and functions of home gardens differ widely across the developing world, they tend to exercise similar social, economic and nutritional roles whatever their settings.
Our conceptual framework for linking home gardens to dietary diversity and food security indicators is developed from recent research into patterns of rural livelihood change in the global South. Smaller proportions of rural populations in the global south today depend on own agriculture as their sole means of sustenance. It is increasingly the norm that households are involved in diverse livelihood activities across subsistence and wage-labour in the agricultural and non-agricultural sectors [18, 19]. This has dramatic impacts on the pathways through which households procure their food, with implications for dietary diversity and quality [20]. As households shift from own-production to market exchange, patterns of nutrition become dependent on what is physically present in local shops and markets. This may widen food choices however also raise consumption of highly-processed, calorie-dense foods hence undermining initiatives to promote healthy diets [21].
These contexts frame contemporary scholarly attention to home gardens. They contribute to household food availability and generate economic and nutritional benefits through direct and indirect pathways. By directly increasing overall food supply, home gardens reduce the need for households to rely on the market to meet their food requirements, thereby freeing up cash for other uses. Furthermore, many of the fruits and vegetables that are typically cultivated in home gardens tend to be both rich in micronutrients and relatively expensive in shops and market, vis-à-vis more economically accessible staples and, increasingly, cheap processed snack foods. The considerable species diversity in home gardens means that although they may not be primary sources of household sustenance, they add important variety to household diets [17]. In Nepal for example, in the wetter, middle hill regions of the country, more than 75% of home gardens had 21 to 50 diverse species per household [15].
A large body of literature has analysed the links between agriculture and agricultural interventions on household nutrition [22]. A 2008 review of 23 studies [23], found home gardens to positively associated with intakes of fruits and vegetables in 14 cases,Footnote 2 improvements in anthropometric measures in six cases, improvements in serum retinol levels (a biomarker of Vitamin A deficiency) in one case, and mixed results or no effects in two cases.Footnote 3 A meta-analysis of 11 interventions promoting home gardens from 1993 to 2000 found evidence of increased intake of fruits and vegetables in eight cases, improvements in anthropometric measures in one case, and indeterminate or negative associations in two cases [24].Footnote 4 A third study reviewing 23 home garden interventions 1995 and 2009 found mixed results, with unclear evidence of the influence of home gardens on diets other health indicators (stunting, wasting, etc) [25].
The overall tendency in the literature is to associate home gardens with superior food intake, diets and nutrition outcomes. With regards to home gardens research, a project initiated by Helen Keller International in rural Bangladesh integrating home gardens, livestock and nutrition education programs [3, 26,27,28], and a South African initiative to address Vitamin A deficiencies through home gardens [29] have been influential.
Even if home gardens are empirically found to improve food intake, diets or nutrition, questions remain about over how these dynamics play out within different rural communities. In particular, there is a paucity of knowledge on the role of home gardens in addressing food security and improving dietary diversity in rural Myanmar, particularly among vulnerable households.
Around 10% of Myanmar's population of 60 million is estimated to be below the official food poverty line, with many pockets of high levels of food and nutrition insecurity across various states/regions and villages [30].
The data for this study comes from a unique survey of 3230 rural households from three States/Districts each representing distinctive agro-climatic zone of Myanmar. These include: Magway (the Dry Zone, with agriculture dominated by pulses, maize and, in areas adjacent to watercourses, rice), Ayeyarwady (in the fertile Delta region, the traditional rice bowl of the country, and with important fishery resources), and Chin (in the western hilly zone, and has traditionally been regarded as the most food insecure area of Myanmar). In each State/District, two adjacent townships were selected.Footnote 5 These include the townships of Yesagyo and Pakokku (in Magway), Kyaiklet and Maubin townships (in Ayeyarwady), and Mindat and Kanpetlet townships (in Chin). To establish a representative sample for each township, population counts for each township were obtained from the 2014 Myanmar Census, and a Probability Proportional to Size (PPS) method was applied to randomly select 20 villages. In each village, household lists were obtained from local authorities and a random sample of 30 households selected for survey. This method provided a target sample of 600 households per township, or 3600 for the entire survey.
Selected households were interviewed face-to-face by a team of enumerators employed by two local partner institutions, the University of Community Health Magway, and the University of Public Health Yangon, under supervision from the research team. The survey took place between in the lean season February–April 2016, which is in the lean season. Information on home gardens was asked as part of a wider set of questions on household demographics, assets, livelihoods, food security and dietary diversity. It is important to note that all our data is at the household level, as we are unable to account for intra-household differences in allocation of food.
Empirical strategy
The empirical aim is to analyse the role of access to home gardens on household-level food security and dietary diversity in rural Myanmar. To this end, the first step is to identify measures of household food security and dietary diversity.
Measures of food security
We measure food security using a slightly varied application of the Household Food Insecurity Access Scale (HFIAS) methodology, developed by USAID [28]. With a recall period of 4 weeks, this methodology was originally developed for the FANTA (USAID) initiative with the aim of providing a holistic methodology to capture the experience of food insecurity [31]. The senior-most knowledgeable female of the household was asked a series of questions on household member's experiences of food insecurity. In particular, questions were asked of household's food intake and coping strategies in the event of non-availability of food, over the past 4 weeks.
Each respondent's self-reported assessment of their household's food security was classified into the following five discrete categories: (i) 'Shortage of food'= 1 if in the last 4 weeks: respondent worried that the household would not have enough food; or any household member had to eat a limited variety of foods; or any household member had to eat a smaller meal than needed; or any household member had to eat fewer meals in a day; or there was ever no food to eat of any kind in the household; or any household member went to sleep hungry at night because of lack of food; or any household member had to go 24 h without eating anything because of lack of food, 0 otherwise. (ii) 'Hunger' = 1 if at any point in the last 4 weeks, there was no food of any kind in the household; or any household member went to sleep hungry at night because of lack of food; or any household member had to go 24 h without eating anything because of lack of food, 0 otherwise; (iii) 'change food' = 1 if in the last 4 weeks: any household member had to change their diet to cheaper; or less preferred foods, or were not able to eat the kinds of foods their prefer, because of a lack of resources, 0 otherwise; (iv) 'reduce food' = 1 if in the last 4 weeks: any household member had to eat a limited variety of foods because of a lack of resources; or eat a smaller meal than they felt was needed, or eat fewer meals in a day, because there wasn't enough food, 0 otherwise; and (v) 'Borrow' = 1 if in the past 4 weeks: the household took food on credit from a local shop; or had to borrow food from relatives or neighbours, 0 otherwise.
Responses to (i) and (ii) directly assess levels of household food insecurity, whereas (iii)- (iv) relate to household's coping strategies in the event of food shortages.
Given that these responses capture different elements of food insecurity, the dependent variable food security is measured separately for each of the five potential food security indicators. Accordingly, we estimate binary choice reduced form univariate Probit models for each of the five food insecurity indicators. Formally, the model can be written in the following general form:
$$ Food\ securityi={\propto}_0+{\propto}_1 socio-{econ}_i+\kern0.5em {\propto}_2 land-{ownership\ status}_i+\kern0.5em {\propto}_3 homegarden+{\propto}_4{geographical}_i+{\varnothing}_i $$
Where the dependent variable Food securityi captures the food security in household i, the vector socio-econ refers to the socio-economic and demographic characteristics of the household measured using household size and the gender of the household head; household's economic characteristics are captured using wealth quintiles (based on household assets and calculated using principal components analysis). The household's land-owning status is measured using a dummy variable that takes on a value of one if the household owned land, 0 otherwise, and for land-owning households, we include categorical variables for land size. We include a dummy variable for whether or not the household has a home garden and the vector geographical includes indicator variables for the six townships in the sample. Further details on the explanatory variables and descriptive statistics are provided below.
Measure of dietary diversity
Our next dependent variable is a measure of dietary diversity. As previously discussed, dietary diversity is commonly used as a proxy measure for the quality of human diets [13, 14]. Previous research has found dietary diversity to be positively associated with measures of food security [32, 33]. Using a 24-h food recall methodology [10], self-reported consumption of food items were grouped into ten food groups in accordance with the Minimum Dietary Diversity – Women (MDD-W) methodology [7]. The same household member answering the food security questions also provided information on the food intake of household members. The diversity of household diets is measured as the intake of food from the ten discrete food groups among household members over the previous 24 h, and takes on a value of 1 if anyone in the household consumed those foods in the previous 24 h, 0 otherwise.
The dietary diversity methodology has previously been used to construct the variable Dietary Diversity Score (DDS), which measures the number of unique food groups (rather than number of different foods) consumed by members of the household over the last 24 h [33,34,35]. We are interested in the diversity of food groups rather than the number of foods because it is possible that a household has consumed a large number of foods, but they may all be from the same food group, thus not providing any diversity in diet.
The ten food groups considered are those with the most density of nutrients, and therefore those most important in diet: starchy staples, beans and peas, dairy, flesh foods, eggs, nuts and seeds, dark, leafy greens (Vitamin A rich), Other fruits and vegetables (Vitamin A rich), Other fruits and other vegetables.
To construct the DDS variable, binary response variables are defined for each of the values taken by the DDS variable. A household is classified as being in the category DDS2 if household members consumed at least one food from two different food groups, and zero otherwise; a household is defined as having a DDS3 if it consumed at least three of the food items and zero otherwise and so on. In our sample, we observe that on average households in Chin ate from 3.5 food groups; households in Ayeyarwady from 3.9 food groups and households from Magway from 4.9 food groups. The DDS is clustered between 2 and 5 food groups, with DDS2 being the worst outcome, and DDS5 being the best outcome. Since nearly all the households in our sample consumed from at least two food groups, and given the natural ordering of the DDS variable, we use the Ordered Probit model for our empirical analyses.
Specifically, following previous research [34], we categorise the DDS into four categories: DDS2, DDS3, DDS4 and DDS5, where DDS2 is the lowest category of dietary diversity. The food consumption categories are represented by an ordered variable V that assumes the discrete ordered values of 0, 1,.. .and j. The ordered probit model for V (conditional on explanatory variables x) can be derived from a latent variable model.
Assume that the latent variable D* is determined by D* = x0b + e, where x is a vector of household's socioeconomic and community-level characteristics entering the equation and e refers to the error term, which we assume is normally distributed across observations.
However, D*, the propensity to consume from a particular food group, is unobserved. Given that we observe D, the household's dietary diversity status, the observed aspects of a household's dietary diversity status can formally be written as:
$$ D=\left\{\begin{array}{c}0\ if\ only\ 2\ food\ groups\ are\ consumed\\ {}1\ if\ 3\ food\ groups\ are\ consumed\\ {}2\ if\ 4\ food\ groups\ are\ consumed\\ {}3\ if\ 5\ or\ more\ food\ groups\ are\ consumed\end{array}\right. $$
and each of these categories is a discrete category of the dependent variable, which can be explained by the same set of explanatory variables.
Explanatory variables
The key explanatory variable used in this paper is a measure of whether the household had access to a home garden.
Questions about home gardens were incorporated into the questionnaire as part of a larger group of questions about household food production. We sought to elicit as full a description as possible about households' food production activities by asking respondents if they: (i) had land (owned or leased) upon which they grew crops or grazed livestock; (ii) had fruit trees or vines; (iii) owned livestock for food purposes in or around their homestead; (iv) had a 'home garden' (defined as growing crops, fruits, or vegetables) in or around their homestead; (v) caught or collected wild animals, fruits or other foods from forests, common land or rivers, lakes, etc. This wider context is relevant, because in some agro-ecological settings, distinctions between home gardens and households' other food production activities can be blurred [36].
Specifically, respondents were asked if they owned a home garden, their access to fruit trees and vines, and their ownership of livestock. Using these responses, we combine home gardens & fruits and vines into one variable, which is categorised as follows: = 1 if yes, and 0 if no. This is because they both represent responses on non-agricultural production. However, we create a separate categorical variable for livestock ownership. This is because during the qualitative survey which followed the household data survey we realised that there was some confusion in respondents' answers to these questions. For example, fruit trees were sometimes counted in home gardens, so in the empirical analyses we combined the trees and vines variable with the home gardens variable.
Although livestock ownership may be complementary to home gardens if animals provide a source of manure and natural weeding, domestic animals eat and trample produce, in which case households may face a trade-off between keeping livestock and maintaining home gardens.Footnote 6
Furthermore, for those households that had access to home gardens/ fruits and vines, we asked respondents to nominate the crops grown in their home gardens/ fruits & vines. We used these responses to construct indicator measures of home garden crop diversity- ranging from home garden crop diversity 0 (no diversity in home garden crops grown) to home garden diversity 3 (indicating that the household grew three or more diverse crops in their home gardens).
From Table 4 we note that on average our households grow over two home garden crops, except in Maubin township where they grow just under two home garden crops.
We also include a set of variables to capture the household's demographic and socio-economic characteristics. These include household socio-economic and demographic characteristics including variables such as the household head's sex and household size; economic status variables measured where using data on asset ownership for each household we create a wealth index using principle components analysis. The households in the sample are then categorized into one of five wealth quintiles (ranging from poorest to richest households). These wealth quintiles provide a more permanent measure of household wealth that is less affected by transitory income changes, and is less likely to be subject to measurement error compared to income.
Additionally, we include information on household land ownership, which is defined as aggregate total land operated by all household members. In these questions, 'land ownership' is defined as being under the direct control of any household member, without the obligation to pay rent or a share of production. In this sense, households may either have formal title to the land through government documentation, or land could be held without formal title but via de facto possession. Our interest is not in the legality of tenure, but in the economic principle of a household having access to land. Therefore, for landholding households (i.e., those answering 'yes' to the question of whether they own land), we ask the respondent to indicate the total area of land that is owned by all members of the household. We include four indicator variables, no land, land area < 2 acres, land area between 2 and 5 acres, and land area ≥ 5.
Note that all land in Myanmar is owned by the state and cultivators only retain the tilling rights, and land-use rights can now be sold, mortgaged or inherited. This pattern of intergenerational land transfer has increased land fragmentation of holdings and small farming households. Although the new land laws of "Vacant, Fallow and Virgin Land Management Law" and "Farmland Law" were both enacted in March 2012 to solve land related problems, land reform continues to be a major challenge [37].
Finally, we include variables to capture geographical differences across our study sites, by including indicator variables for the six study townships: Yesagyo and Pakokku (in Magway), (Kyaiklet and Maubin townships (in Ayeyarwady), and Mindat and Kanpetlet townships (in Chin).
The descriptive statistics for the variables included in the empirical analysis are presented in Tables 1 and 2. Table 1 presents the descriptive statistics classified by whether or not the household had access to home gardens; and in Table 2 we present descriptive statistics disaggregated by DDS categories. The main point to note from Table 1 is that 959 households have access to home gardens out of our sample of 3239 households. Furthermore, 38% of households with home gardens do not own agricultural land, and 16% of households with home gardens are in the poorest wealth quintile.
Table 1 Descriptive statistics- comparison of households with home gardens and those without
Table 2 Summary statistics by food security indicators
In general food insecurity is high in our sample, and there is a strong association between food insecurity measures and DDS. In particular, over three-quarters of the households that report having to change their diet, reduce their food intake, or face food shortage are also categorized as being in the lowest DDS category (DDS2). Furthermore, 87% of the households in DDS2 category report a shortage of food. Indeed this is 61% even among households in DDS5 category 5, where over half the sample reporting having to change their diet due to a lack of resources, suggesting a high degree of seasonality. Surprisingly on average, only 23% of the households report being in hunger in DDS2 category (5% among the households in the highest dietary diversity category- DDS5). However, it is likely that households are taking measures to address their hunger when they face food shortages. For example, a high proportion of the households report having to change their diets, borrow or reduce their food intake due to lack of resources.
Not surprisingly, dietary diversity is highest among the wealthiest households, with approximately 51% of the households in the wealthiest quintile being in the two highest DDS categories.
There are also regional differences across our sample, with households in the townships of Pakokku and Yesagyo (Magway state) having the most diverse diets (DDS5). Together these two townships account for over half the households represented in the DDS5 category. On the other hand, the townships of Maubin and Kyaiklet (Ayeyarwaddy) have the highest proportion of households in the lowest DDS category, and together they represent approximately 48% of the households in DDS2 category. Households in Mindat, Kanpetlet, Maubin and Kyaiklet have at least 20% of the households surveyed reporting DDS = 2.
The unconditional means also show that access to home gardens is associated with better dietary diversity, with 22% of the households in the highest DDS category (DDS5) owning a home garden compared to 17% in the lowest DDS category (DDS2). In terms of the links between land ownership and home gardening, we note that only 12% of landless households have access to a home garden, compared to 31% among landed households.
In Table 3 we present land ownership patterns by township. It is unsurprising that land sizes are the largest in the two townships in the fertile Ayeyarwaddy delta, where on average a household in Kyaiklet township owns approximately 7.34 acres of land. However, it is noteworthy that while the average land size is large in these townships, nearly 63% of households in our sample do not own land in the township of Kyaiklet, and 75% do not own land in Maubin in the Ayeyarwaddy delta. Together with Table 2 they indicate high levels of food insecurity in these Ayeyarwaddy townships which also have high levels of landless households, and large inequities in land ownership.
Table 3 Land holding patterns by Township
On the other hand, in the townships of Mindat and Kanpetlet in the mountainous Chin state, land sizes are small with households owning on average just 3.12 and 2.80 acres of land, respectively. Moreover, the proportion of households reporting no land ownership is relatively low (approximately 25%). It is important to point out that land ownership patterns in these two townships are a bit complicated as they are not a major agricultural growing region in Myanmar, and land is sometimes communally owned, and it is unclear whether we are observing land ownership or cultivation rights.
The situation in the townships of Yesago and Pakokku (in Magway) are similar to the Ayeyarwaddy townships, albeit with lower levels of landlessness.
To get a better sense of home garden ownership, in Table 4 we report ownership patterns for home gardens, fruits and vines, and livestock across our townships. We observe that on average there are just over 2 home garden crops grown in township, but the average number of crops grown differ across townships, with 2.6 different crops grown in Yesagyo (in Magway) and just 1.9 crops in Maubin (in the Ayeyarwady). Ownership of home gardens ranges between a high of 34.8% in Kanpetlet township (Chin state) to a low of 10% in Pakokku. However, home gardens are often complementary to livestock ownership and are substitutes with fruits and vines, as together they account for informal food growing patterns. We observe that across our sample, approximately 83% of the households in Mindat and Kanpetlet (in Chin state) have access to home gardens, fruits and vines and livestock, and below 50% in the two Magway townships (34.2% in Pakokku and 48.5% in Yesagyo). The townships in Ayeyarwaddy exhibit similar patterns to the Magway townships with regards to home garden access. Our results show that 21 and 15% of the households in Kyaiklet and Maubin, respectively have access to just home gardens, and approximately 74 and 66% have access to either of home gardens/ fruits and vines or livestock.
Table 4 Ownership of Kitchen Garden, Fruits and Vines, and Livestock by Township
However, these unconditional means do not provide a full picture of the links between food security, dietary diversity and access to home gardens. In the next section, we present the main estimation results of our analyses (Tables 5, 6 and 7).
Table 5 Univariate Probit Estimations results for landless households- Food Security
Table 6 Ordered Probit Estimation results for landless households- Dietary Diversity
Table 7 Estimation results – full sample
Role of home gardens on food security and dietary diversity of non-agricultural land-owning households
We are interested in explaining if access to home gardens improves food security and dietary diversity among vulnerable households. This is likely to be particularly critical for the sample of non-land owning households, who are likely to be among the most vulnerable. Therefore, we begin by presenting the estimation results for the Probit and Ordered Probit models for the sample of non-agricultural land owning households, in Tables 5 and 6 respectively. The Univariate Probit estimations are presented separately for each of our five food security indicators, for the sample of non-agricultural land owning households. For ease of interpretation we present marginal effects, which show the percentage change in the probability of the outcome variable when the value of a regressor changes, holding all other variables constant.
From Table 5 we observe that for landless households, the ownership of home gardens/ fruits and vines is statistically significant and is associated with a 6.6 percentage point lower probability of a household having to change their diet, and a 7.9 percentage point lower probability of being in hunger, with no statistically significant influence on the other food security indicators.
Household wealth is found to play a statistically significant and positive role in mitigating food insecurity. Relative to households in the poorest wealth quintile, households from each of the higher wealth quintiles have a lower probability of being food insecure, across each of our food security indicators, with the size of the association increasing monotonically with each higher wealth quintile, with the largest effects observed for the wealthiest households. In particular, relative to a household in the lowest wealth quintile, a household in the highest wealth quintile has a 38.2 percentage points lower probability of reporting 'change', a 17.7 percentage points lower probability of reporting 'hunger', and a 44.4 percentage point lower probability of reporting a shortage of food.
There are also regional differences across our sample. Relative to Maubin township, where 75% of the households had no land, we observe that households living in the other townships had a statistically significant and negative association with reporting needing to 'change their diet', having to reduce their food consumption, face 'shortages' or 'hunger'.
In Table 6, we present the Ordered Probit estimation results for dietary diversity, using the DDS categories as our dependent variable and the same set of explanatory variables as with the food security regressions. Since our data has information on the diversity of fruits and vegetables grown in home gardens, we additionally include among our explanatory variables indicator variables for the diversity of home garden crops (according to FAO classification). These variables capture information on whether the household's home garden includes cultivation across one food group, two, three or more.
From Table 6 we observe that in the sample of non-agricultural land owning households, access to home gardens/ fruits and vines is strongly associated with better DDS. More specifically, access to home garden garden/ fruits & vines is associated with a 7.7 percentage points lower probability of being in the lowest DDS category (DDS 2) and a 13.1 percentage points higher probability of being in the highest DDS category, relative to a household with no home gardens.
Not surprisingly, having greater diversity in home garden crops grown also improves dietary diversity, with growing three or more diverse crops in the home garden being associated with a 7.9 percentage points lower probability of being in DDS category 2 and a 17.3 greater probability of being in the highest dietary diverse category (DDS 4). There is no statistically significant correlation between home garden crop diversity and dietary diversity for three or less groups.
We observe that after controlling for all other characteristics, there is a statistically significant and monotonically increasing relationship between greater dietary diversity and household wealth. In particular, relative to a household in the lowest quintile, a household in the second quintile has a 5.7 percentage points higher probability of being in DDS4 and a 4.4 percentage points lower probability of being in the lowest DDS category. This positive association between household wealth and dietary diversity is particularly large for households in the highest wealth quintile, who have a 30.9 percentage points higher probability of being in DDS5 relative to a household in the lowest wealth quintile.
Does access to home gardens improve food security and dietary diversity for agricultural land-owning households?
The above section found a positive association between home gardens and measures of food security and dietary diversity. To understand if home gardens play a similarly important role in the full sample (including both land owning and non-land owning households), we additionally control for the influence of land size using four discrete land ownership categories- no agricultural land, land below 2 acres, land between 2 and 5 acres, land over 5 acres. These results are presented in Table 7 for Probit and Ordered Probit models reporting marginal effects for food security indicators and dietary diversity score (DDS), in Panel A and B respectively.
From Panel A, it is noteworthy that despite controlling for household wealth, relative to households with no land, for the largest land-owners is significantly associated with a lower probability of reporting food insecurity across all our five measures. Specifically, relative to a household with no land, ownership of over 5 acres of agricultural land reduces the probability of a household reporting the need to 'change' their diet by 7.7 percentage points, with a 6.7 percentage points lower probability of 'hunger', a 7.4 percentage points lower probability of experiencing food shortages and a 7.8 percentage points lower probability of needing to reduce food intake. These results are in keeping with previous research from Myanmar which used the LIFT dataset, and found that agricultural land ownership plays a key role in reducing food insecurity [30]. However, their study did not take into account the role of home gardens.
However, home garden access is only statistically significant and negatively associated with the 'hunger' indicator for food insecurity, where we observe that a household with home gardens has a 3 percentage points lower probability of reporting hunger. Surprisingly, we find a positive and statistically significant association between access to home gardens and food shortage. Since our dataset is a cross-section, it is difficult to infer causality, and it is possible that households participate in home garden cultivation when they face food shortages.
Notably, from Panel B we observe that land size has no statistically significant influence on dietary diversity. However, home gardens are positively and significantly associated with greater dietary diversity. In particular, we observe that a household with access to home gardens has a 4.9 percentage points lower probability of being in the lowest dietary diversity category (DDS2), and there is a 4.7 percentage point positive association between home garden access and being in the second highest dietary diverse category.
Robustness tests
In the previous sections, we demonstrated the positive associations between measures of food security and dietary diversity and access to home gardens. Since home gardens may have a differential influence on households depending on the size of their land holdings, we include some additional interaction terms in our regressions. In particular, we interacted the variable home gardens access with different categories of land size, both for the food security and dietary diversity regressions. These interaction terms are statistically significant and have the expected signs, both in models for food security and dietary diversity. In other words, home gardens reduce food insecurity and improve dietary diversity, and the higher the size of agricultural land, the greater the role that home gardens play in reducing food insecurity and improving dietary diversity.
The results for the other variables are as expected, with higher wealth quintiles being associated with better dietary diversity and food security.
There is wide acknowledgement of the critical role played by home gardens in addressing the food security and dietary needs to rural populations in developing country settings. The aim of this paper was to analyse the role of access to home gardens on household-level food security and dietary diversity in rural Myanmar. Myanmar offers a unique opportunity to study these research questions, given the decades of political isolation, high levels of food insecurity and poor nutrition levels. To the best of our knowledge the present study is the first detailed assessment of home gardens in the Myanmar context.
Using information on two dimensions of food security, a series of variables capturing a household's self-reported food security status and coping strategies when food is not available; and a measure of household dietary diversity based on 24-h recall data, we empirically estimate a household's probability of being food insecure and the diversity of their diets. Our analysis has three key findings. Firstly, in the sample of landless households, access to home gardens is statistically significant and negatively associated with food insecurity (measured using change and hunger); and positively associated with higher dietary diversity. Secondly, an improvement in the diversity of crops grown in the home garden improves dietary diversity. Finally, access to home gardens is also significantly associated with lower food insecurity and better dietary diversity, however, the magnitude of the effects is larger for the largest landowners.
Recent research has pointed to home gardens as a playing vital role for households in the context of far-reaching food system transformations [21]. They help maintain consumption of diverse, healthy foods in situations where emergent pathways of food availability favour unhealthy contributors to diets and in contexts where poor households have limited access to agricultural land, non-farm livelihoods to support a diverse diet. Our results show that promoting home gardens among vulnerable households can improve food security and dietary diversity among rural households in traditional food systems such as Myanmar.
Home gardens can also go by the names homestead garden, backyard garden, or kitchen garden. We treat these as synonymous. It also needs noting that our usage of the term 'home gardens' is different to that used in some African contexts where it denotes small-scale agroforestry and livestock practices at the village level.
Some of these studies focused only on Vitamin A-rich fruits and vegetables.
3 There are 29 studies in their meta-analysis, but four assessed the effects of livestock interventions only, and three assessed nutrition education only. Of the remaining 23 studies, all involved a home garden intervention, either singly or in combination with nutrition education or livestock.
This study includes reference to a total of 27 intervention studies, of which 11 specifically refer to home gardens.
'Townships' are the third tier of government administration in Myanmar, and outside of the larger cities they typically comprise a region characterized by a rural population dispersed in villages and hamlets and an adjacent service town.
Note that our survey question on home gardens did not explicitly ask whether output was for own-consumption or sale. However, a prior question in the survey asked respondents to list any crop, fruit or vegetable they produced for sale, so we can infer that all the data we collected on home garden foods were for own-consumption.
DDS:
Dietary diversity score
FANTA:
Food and Nutrition Technical Assistance
HFIAS:
Household Food Insecurity Access Scale
Livelihoods and Food Security Trust Fund
MDD-W:
Minimum Dietary Diversity – Women
Brownrigg L. Home gardening in international development: what the literature shows. League for International Food Education: Washington, DC; 1985.
Cabalda AB, Rayco-Solon P, Solon JA, et al. Home gardening is associated with Filipino preschool children's dietary diversity. J Am Diet Assoc. 2011;111:711–5. https://doi.org/10.1016/j.jada.2011.02.005.
Bushamuka VN, de Pee S, Talukder A, et al. Impact of a homestead gardening program on household food security and empowerment of women in Bangladesh. Food Nutr Bull. 2005;26:17–25.
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Galhena DH, Freed R, Maredia KM. Home gardens: a promising approach to enhance household food security and wellbeing. Agric Food Sec. 2013;2. https://doi.org/10.1186/2048-7010-2-8.
Food and Agriculture Organization (FAO) & FHI 360. Minimum dietary diversity for women: a guide for measurement. Rome: FAO; 2016. http://www.fao.org/3/a-i5486e.pdf
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Gautam R, Sthapit BR and Shrestha PK (eds.). Home gardens in Nepal: proceedings of a workshop on "enhancing the contribution of home garden to on-farm management of plant genetic resources and to improve the livelihoods of Nepalese farmers: lessons learned and policy implications", 6–7 August 2004, Pokhara, Nepal. LI-BIRD, Bioversity International and SDC. 2006.
Arimond M, Hawkes C, Ruel MT, et al. Agricultural interventions and nutrition: lessons from the past and new evidence. In: Thompson B, Amoroso L, editors. Combating micronutrient deficiencies : food-based approaches. Oxford: CABI; 2011. p. 41–75.
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Pritchard B, Vicol M, Jones R. How does the ownership of land affect household livelihood pathways under conditions of deagrarianization? 'Hanging in', 'stepping up' and 'stepping out' in two north Indian villages. Singap J Trop Geogr. 2017;38(1):41–57.
Gillespie, S., Harris, J. & Kadiyala, S. The agriculture-nutrition disconnect in India: What do we know?. IFPRI Discussion Paper 001187. 2012. online. http://www.ifpri.org/sites/default/files/publications/ifpridp01187.pdf.
HLPE. Nutrition and food systems. A report by the high level panel of experts on food security and nutrition of the committee on world food security. Rome: HLPE; 2017.
Webb P. Impact pathways from agricultural research to improved nutrition and health: literature analysis and research priorities. Paper for ICN2 better nutrition, better lives, Rome: Food and Agriculture Organization and Geneva: World Health Organization. 2013. http://www.fao.org/3/a-as573e.pdf.
Haider BA, Bhutta ZA. Web appendix 16 – dietary diversification strategies including home gardening, livestock farming and dietary modifications. Lancet. 2008;371:417–40.
Hawkes C, Ruel MT, Arimond M, et al. From agriculture to nutrition: pathways, synergies, and outcomes. Washington, DC: IBRD and the World Bank; 2010.
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Talukder A, Kiess L, Huq N, et al. Increasing the production and consumption of vitamin A-rich fruits and vegetables: lessons learned in taking the Bangladesh homestead gardening programme to a national scale. Food Nutr Bull. 2000;21:165–72.
Karim R, Desplats G, Schaetzel T, et al. Seeking optimal means to address micronutrient deficiencies in food supplements: a case study from the Bangladesh integrated nutrition project. J Health Popul Nutr. 2005;23:369–76.
Iannotti L, Cunningham K, & Ruel MT. Improving diet quality and micronutrient nutrition: homestead food production in Bangladesh. International food policy research institute, IFPRI discussion paper 00928. 2009.
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Faber M, van Jaarsveld PJ. The production of provitamin A-rich vegetables in home-gardens as a means of addressing vitamin a deficiency in rural African communities. J Sci Food Agric. 2007;87:366–77.
UNDP. IHLCA: health and nutrition profile report. Yangon: UNDP; 2011.
Coates J, Swindale A, Bilinsky P. Household food insecurity access scale (HFIAS) for measurement of household food access: Indicator guide (v. 3). Washington, D.C: Food and Nutrition Technical Assistance Project, Academy for Educational Development; 2007. http://www.fao.org/fileadmin/user_upload/eufao-fsi4dm/doc-training/hfias.pdf
Hillbruner C, Egan R. Seasonality, household food security, and nutritional status in Dinajpur, Bangladesh. Food Nutr Bull. 2008;29:221–31. https://doi.org/10.1177/156482650802900308.
Rammohan A, Pritchard B. The role of landholding as a determinant of food and nutrition security in rural Myanmar. World Dev. 2014;64:597–608.
Swindale A, Bilinsky P. Household food insecurity across scale (HFIAS) for measurement of food access: Indicator guide (Ver. 2). Washington, DC: Food and Nutrition Technical Assistance/Academy for Educational Development; 2006.
Hoddinott J & Yohannes Y. Dietary diversity as a food security indicator. International food policy research institute, food consumption and nutrition division discussion paper, 136. 2002.
Fanzo J, Remans R, Termote C. Smallholders, agro-biodiversity and mixed cropping and livestock systems. In: Pritchard B, Ortiz R, Shekar M, editors. The handbook of food and nutrition security. London: Routledge; 2016. p. 299–318.
The authors acknowledge the contributions of Min Kyaw Htet and Mark Vicol who oversaw the data collection. We also wish to thank the University of Community Health, Magway, who assisted with the collection of field data, and our team of enumerators, from the University of Community Health, Magway, and the University of Public Health, Yangon.
The authors acknowledge funding from the Australian Research Council (ARC) Discovery Project grants. The funds from the grant were used to facilitate data collection and employ researchers to help in the data analyses.
Discipline of Economics, University of Western Australia, Perth, 6007, Australia
Anu Rammohan
School of Geosciences, University of Sydney, Sydney, Australia
School of Public Health, University of Sydney, Sydney, Australia
Michael Dibley
AR, BP and MD conceptualized the study and were involved in the data collection. AR conducted data analysis. BP conducted literature review and worked with AR to interpret data. All authors read and approved the final manuscript.
Correspondence to Anu Rammohan.
This study was conducted according to the guidelines laid down in the Declaration of Helsinki and all procedures involving human subjects/patients were approved by the Ethics Review Committee on Medical Research, Department of Medical Research, Union of Myanmar (117/Ethics 2015), and by the Ethics Committees of the University of Sydney and the University of Western Australia. Written [or Verbal] informed consent was obtained from all participants. Where verbal consent was obtained, it was witnessed and formally recorded.
Rammohan, A., Pritchard, B. & Dibley, M. Home gardens as a predictor of enhanced dietary diversity and food security in rural Myanmar. BMC Public Health 19, 1145 (2019). https://doi.org/10.1186/s12889-019-7440-7
Dietary diversity
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CommonCrawl
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Trivial and non-trivial zeros
I am new to DSP, and I'm self studying. Could someone please explain to me what do we mean by trivial and non-trivial zeros?
z-transform transfer-function poles-zeros zeros
ShadyShady
$\begingroup$ Could you give a reference that uses these terms? $\endgroup$ – Matt L. Nov 22 '18 at 11:49
I wasn't familiar with that term in the context of signal processing. (Instead, I've seen the term being used in the context of the Riemann zeta function.) But I've found a document and this book where the term is used in a DSP context. The (obvious) definition is that trivial poles and zeros are the ones at the origin $z=0$ and at infinity $|z|=\infty$. They're called trivial because they don't affect the magnitude of the corresponding frequency response. Multiplying a given $\mathcal{Z}$-transform with $z$ (i.e. adding a pole at infinity and a zero at the origin) just advances the corresponding sequence by one sample, and multiplying by $z^{-1}$ (i.e., adding a pole at the origin and a zero at infinity) delays the corresponding sequence by one sample.
As an example, note that a causal FIR filter has as many poles as zeros, but all of them are at the origin $z=0$, i.e., they can't be used to shape the magnitude response. The same is true for an all-pole filter: all its zeros are at the origin, so they don't help to create a stop band (which would be one possible function of non-trivial zeros).
I think it's important to point out that the term trivial zero (or pole) is not a standard term used in DSP - as far as I know - but it appears to be an idiosyncratic use of a few authors.
$\begingroup$ the typical way in the literature to refer to something idiosyncratic , "so called trivial zeros" $\endgroup$ – Stanley Pawlukiewicz Nov 22 '18 at 15:52
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CommonCrawl
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Stacks in the Zariski topology?
I have two naive questions about stacks.
1) Is it possible to define stacks in the Zariski topology?
Presuming you can:
2) If a stack has a coarse moduli, and the coarse moduli space is a scheme, then does that mean that your stack is a stack in the Zariski topology?
In general, I am trying to understand why a new notion of open cover is necessary if all I am interested in is remembering stabilizers. Certainly this is too simple a mind-set, so feel free to enlighten me.
ag.algebraic-geometry stacks
David SteinbergDavid Steinberg
1.) It's possible to define stacks on ANY category equipped with a Grothendieck topology (such a category with a topology is called a site). In particular, this holds true for the Zariski site. Moreover, there is always a way to define an "Artin stack"- these are those stacks which arise as torsors for a groupoid object in your site. Outside of algebraic geometry, these give rise to notions of topological and differentiable stacks, for instance.
EDIT: As long as groupoid objects exist in your category.
2.) As in Harry's post, any stack which is a stack in a site which is finer than the Zariski topology is also a stack in the Zariski topology.
To address your general question as to "why a new notion of open cover is necessary if all I am interested in is remembering stabilizers", you should learn a bit about Grothendieck topologies. I'll make a couple remarks:
i) If all you cared about were stabilizers, then you wouldn't need to use any covers at all; ordinary fibered categories would do the trick!
Indeed, take a group object in your site acting an object, and take the action groupoid- it is a groupoid object. Look at the pseudo-functor which assigns each object of your site the groupoid of maps into this groupoid object (considering the object as a groupoid with all identity arrows). This remembers the stabilizers for this action.
ii) (subcanonical) Grothendieck topologies are a choice of a type of cover for your objects, in such a way that this object is the colimit of these covers, AND "this is important to remember". This is a little imprecise, so, allow me to elaborate via an example from topology:
Let $U_i$, $i\in I$ be an open cover of a space X. Then, continuous maps from X to another space Y are in bijection with with continuous maps $f_i:U_i \to Y$ which agree on their intersection. This is just saying that X is the colimit of this open cover. Instead, we can view this a property of the presheaf $Hom(blank,Y)$ represented by $Y$ on the category of topological spaces (for you set theorists, choose a Grothendieck universe).
For any $X$ and any open cover $U_i$, $i\in I$ of $X$, (let $Hom(blank,Y)=F$)
the natural map $F(X) \to \varprojlim \left[{\prod{F(U_i)}} \rightrightarrows {\prod{F(U_{ij})}}\right]$
is a bijection.
If $F$ is any presheaf, this is just saying $F$ is a sheaf. Since this is NOT true for an ARBITRARY presheaf $F$, X is no longer the colimit of its open covers in the full category of all presheaves. The same argument holds for all fibred categories- it's only true if we restrict to STACKS (and $X$ then becomes the weak colimit of this cover, but, never mind).
The reason you add the condition for descent for covers, is so that "all maps into your stack from a space are continuous". More precisely, and more generally, it's so that maps from a space, scheme, whatever you site is, into a stack can be determined by mapping out of elements of some covering of your object in a way that glues (for stacks, rather than sheaves, they don't need to AGREE on the intersection, but, agree up to an invertible 2-cell, plus some coherency conditions).
Combining these ideas, if you have a group acting on an object, the pseudo-functor produced by the action groupoid is rarely a stack with respect to your topology, but you can stackify it, and then it will become one and still remember all the stabilizers. I hope this helps!
David CarchediDavid Carchedi
$\begingroup$ What you said about Artin stacks being definable over any site is not true. You need a "geometric context" in the sense of Toën. That is, you need a distinguished class of morphisms that commutes with finite sums, is squarable, has finite pullbacks, is compatible with the topology etc. Then an Artin stack is a a stack representable by a "geometric space" in a suitable sense. Your examples are true, but the total statement is not. It's in Cours 2. $\endgroup$ – Harry Gindi Apr 16 '10 at 2:27
$\begingroup$ Well, I suppose that yes, I do need a class of groupoid objects to exist such that when I take their enriched simplicial nerve, each structure map admits setions with respect to the Grothendieck topology. $\endgroup$ – David Carchedi Apr 16 '10 at 10:02
$\begingroup$ I don't know what you're talking about, but it sounds very cool. Where can I read about it? $\endgroup$ – Harry Gindi Apr 16 '10 at 10:10
$\begingroup$ If you have groupoid objects (which means that certain pullbacks need to exist), you can look at those stacks arising as torsors of groupoid objects (these will be the stackification of pseudo-functors of the form Hom( ,G)). Whether or not these deserve to be called Artin stacks in general may indeed be contentious, but this is why I used quotation marks. I agree that you will need additional axioms for these "Artin stacks" to be equivalent to a bicategory of groupoids and torsors, so maybe this is what you mean. What is "Cours 2"? I would be interested in looking at what these conditions are. $\endgroup$ – David Carchedi Apr 16 '10 at 11:45
$\begingroup$ @Harry, my other comment (about simplicial objects) is not quite what I meant to say. But, what I should have said is that for these stacks to behave nicely (other than what I said above), sometimes the structure maps of the groupoid are required to be some sort of "local fibration". To see this in the topological context, see: Noohi's Foundations of Topological Stacks I. $\endgroup$ – David Carchedi Apr 16 '10 at 11:52
1.) Yes, you can define a stack in on any site.
A stack is merely a fibered category satisfying descent along all covers (with respect to a specific Grothendieck topology). There's a particularly beautiful form of the descent condition given in Vistoli's notes. That is, a fibered category $F$ on $\mathcal{C}$ is a stack if given any object $S$ of $\mathcal{C}$, and any covering sieve $T$ of $S$, the induced functor $Hom(S,F)\to Hom(T,F)$ is an equivalence of categories. Here, of course, we replace $S$ and $T$ with their representable pseudofunctors/fibered categories.
If you meant an algebraic stack, they are étale stacks, so they are in particular Zariski stacks. There are also representability conditions (the diagonal is representable by an algebraic space), but those are not particularly important for this discussion.
2.) Since every stack in a finer Grothendieck topology is automatically a stack in the Zariski topology, then it follows that these two things are unrelated (in particular, the finer the topology, the more covering sieves you have, which makes it harder to meet the descent condition.)
I can't recommend the notes by Angelo Vistoli on Descent, Grothendieck Topologies, Fibered Categories, and Stacks enough. They're really top-notch (and avaiable on his website).
For your underlying question:
I'm sure that you've probably heard things said like "The Zariski Topology is the 'Wrong' topology for Algebraic Geometry". This is true for a number of reasons involving cohomology that I don't really understand. However, it turns out that if you live in the étale topology, a lot of things become much easier to work with in terms of geometry and definitions. In particular, a scheme is merely an étale sheaf of sets on $Aff$ satisfying a covering condition by smooth monomorphisms of representable functors. I don't know if you have ever seen the definition of the global Zariski topology, but it's really quite nasty. (It's in Demazure-Gabriel, if you're dead set on reading about it).
Anyway, I'm not the person to tell you about the wonderful geometric properties of the étale topology, but the rough idea is that the étale topology is similar to the analytic topology for complex schemes.
However, the nice geometric properties of the étale topology come at the price of having to prove a lot of foundational results from commutative algebra. In particular, many of the proofs characterizing (formally) smooth, unramified, and étale morphisms require Zariski's main theorem, which is actually very deep and notoriously hard to prove.
Edit: Toën's notes on stacks are wonderful as well, but they assume that you've got a pretty good grasp of descent. In particular, schemes are simply sheaves, while the classical presentation of a scheme as a locally ringed space is simply given as the "locally ringed space associated with the scheme". This viewpoint is quite useful to make the transition to thinking about (particularly) the small étale topos associated with a scheme, which looks almost like a locally ringed space but in fact gives us a lot of useful information not available in the classical presentation. The small étale topos is the perfect illustration of why a topos is a "generalization of the notion of a space".
Vistoli's Notes on Descent
Toën's Notes on Stacks
Edit 2: To address Brian's concerns, I'd like to note that the Zariski topology is an extremely important tool (especially for commutative algebra in my experience, however little that may be). I have problems with the global Zariski topology, which I maintain is "more trouble than it's worth".
Harry GindiHarry Gindi
$\begingroup$ Harry, do not parrot a claim about erroneous nature of "most" published proofs of ZMT unless you verified it for yourself. Proofs in EGA and Hartshorne are fine. ZMT in no deeper than other things (e.g., Artin's work) needed to make algebraic spaces useful. Don't besmirch value of Zariski topology, lest you find yourself caught in circular reasoning (e.g., dubious you could get anywhere without having to prove that formally etale lfp maps are open for the Zariski topology; real work isn't avoided by making definitions). You lack the experience to pass judgement on "best" foundations. $\endgroup$ – BCnrd Apr 16 '10 at 3:07
$\begingroup$ Brian, my emphasis in this post was on avoiding the definition of the global Zariski topology on the category of affines. The Zariski topology on a scheme is an invaluable tool for proving things in commutative algebra and algebraic geometry. I guess you could say that I was besmirching the global Zariski topology, but I have never seen it used as more than a proof of concept for the Grothendieck topologies. With respect to ZMT, I've removed the note because you're right, I have not checked them myself. That's not to say I don't trust Mel, but you make a good point that I should check myself. $\endgroup$ – Harry Gindi Apr 16 '10 at 9:04
$\begingroup$ The global Zariski topology has its uses, see: A. Hirschowitz: Cohérence et dualité sur le gros site de Zariski. Proceedings Trento 1988, Lect. Notes in Math.1389 (1989) 91-102. $\endgroup$ – Leo Alonso Jan 20 '11 at 11:33
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Investigating the physiology of normothermic ex vivo heart perfusion in an isolated slaughterhouse porcine model used for device testing and training
Benjamin Kappler ORCID: orcid.org/0000-0001-6816-31621,2 na1,
Carlos A. Ledezma3 na1,
Sjoerd van Tuijl2,
Veronique Meijborg4,
Bastiaan J. Boukens4,
Bülent Ergin5,
P. J. Tan3,
Marco Stijnen2,
Can Ince5,
Vanessa Díaz-Zuccarini3,6 &
Bas A. J. M. de Mol1,2
BMC Cardiovascular Disorders volume 19, Article number: 254 (2019) Cite this article
The PhysioHeart™ is a mature acute platform, based isolated slaughterhouse hearts and able to validate cardiac devices and techniques in working mode. Despite perfusion, myocardial edema and time-dependent function degradation are reported. Therefore, monitoring several variables is necessary to identify which of these should be controlled to preserve the heart function. This study presents biochemical, electrophysiological and hemodynamic changes in the PhysioHeart™ to understand the pitfalls of ex vivo slaughterhouse heart hemoperfusion.
Seven porcine hearts were harvested, arrested and revived using the PhysioHeart™. Cardiac output, SaO2, glucose and pH were maintained at physiological levels. Blood analyses were performed hourly and unipolar epicardial electrograms (UEG), pressures and flows were recorded to assess the physiological performance.
Normal cardiac performance was attained in terms of mean cardiac output (5.1 ± 1.7 l/min) and pressures but deteriorated over time. Across the experiments, homeostasis was maintained for 171.4 ± 54 min, osmolarity and blood electrolytes increased significantly between 10 and 80%, heart weight increased by 144 ± 41 g, free fatty acids (− 60%), glucose and lactate diminished, ammonia increased by 273 ± 76% and myocardial necrosis and UEG alterations appeared and aggravated. Progressively deteriorating electrophysiological and hemodynamic functions can be explained by reperfusion injury, waste product intoxication (i.e. hyperammonemia), lack of essential nutrients, ion imbalances and cardiac necrosis as a consequence of hepatological and nephrological plasma clearance absence.
The PhysioHeart™ is an acute model, suitable for cardiac device and therapy assessment, which can precede conventional animal studies. However, observations indicate that ex vivo slaughterhouse hearts resemble cardiac physiology of deteriorating hearts in a multi-organ failure situation and signalize the need for plasma clearance during perfusion to attenuate time-dependent function degradation. The presented study therefore provides an in-dept understanding of the sources and reasons causing the cardiac function loss, as a first step for future effort to prolong cardiac perfusion in the PhysioHeart™. These findings could be also of potential interest for other cardiac platforms.
Isolated perfused hearts have been used for cardiac research since the groundbreaking work of Langendorff [1] in 1895. Hearts isolated from the body and perfused ex vivo offer results with higher reproducibility, when compared to in vivo counterparts, because they are not affected by systemic influences, such as neurohumoral control and systemic circulation. Extensive work has been done to customize ex vivo heart platforms for precise research purposes and to improve and accelerate the development of cardiac prototypes and interventions. The optimal perfusion with warm oxygenated blood enables realistic device validation, while these setups can be also medical devices themselves (e.g. donor heart transportation) [2, 3]. Nowadays, ex vivo models are available that offer more in-dept research possibilities such as electrophysiological studies [4, 5] and working heart studies [6, 7], in which blood is pumped in a natural way (i.e. the blood enters the heart through the left atrium, it is then pumped to the left ventricle and it is finally ejected through the aorta). This "working mode" allows measurements of pump function, cardiac pressures (i.e. ventricular, aortic, pulmonary) and flows (i.e. aortic, coronary, etc.) and was first described by Neely, Liebermeister [6] in 1967. As a result of these setup developments, isolated heart preparations are used for a variety of investigations in cardiology, cardiac surgery, physiology and pharmacology to investigate physiological, biochemical, pharmacological and morphological characteristics as well as cardiac function [8,9,10,11,12,13].
Pig hearts are appreciated for investigations, which specifically require physiological conditions similar to patient application, as these hearts are a good match to the morphology and physiology of human hearts [10]. However, potential significant differences (i.e. shape, opening of superior and inferior caval veins into the atrium, prominent left azygous vein drainage, number of pulmonary veins, etc.) are known between porcine and human hearts [14].
Considering the increase of cardiac investigations while bearing in mind the costs and ethical issues related to laboratory animal experiments, isolated slaughterhouse hearts could under carefully chosen circumstances be used for feasibility studies that can precede the conventional and contentious animal testing. Slaughterhouse heart experiments are less cost intense, as experimentation protocols do not need ethical approval, while uncertified teams can perform several experiments a day, due to the abundance of slaughterhouse hearts without sacrificing additional animals for the research conducted. This results in an improved learning curve as investigations can originate faster [15].
One customized model based on slaughterhouse hearts is the PhysioHeart™, developed by LifeTec Group B.V. (Eindhoven, The Netherlands). Slaughterhouse pig hearts revived in this commercially available isolated heart model have previously shown cardiac output, stroke volume, pressures, valve interactions and dynamic changes that are comparable to those observed in humans [15]. In the last decade, this model has been used to successfully visualize transcatheter aortic valve implantations [16], to assess computer tomographic myocardial perfusions [17], to evaluate magnetic resonance imaging-based 4D flow analysis and to study left ventricular assist devices [18, 19], intra-aortic balloon pump support [20] and coronary autoregulation [21].
In the face of, extensive work, novelties and decades of experience, isolated heart perfusion remains demanding, in particular in use of slaughterhouse hearts. The unavoidable warm ischemia (between exsanguination of the pig and the cardiac arrest) and the cold transport to the laboratory influence the experimentation outcome negatively. These shortcomings encountered within these experiments, specifically the time-dependent contractility degradation and edema, are under specific aspects similar to those observed within human DCD (donation after circulatory determined death) heart preservation, which are considered for transplantations [22].
Although the PhysioHeart™ model is well-established for device and therapy testing, less is known about its cardiac metabolic, biochemical and electrical physiology as the experiments progress in time, specifically regarding the source of the abovementioned shortcomings. Therefore, in this study, we report a comprehensive recording of time-dependent metabolic, biochemical, electrical and hemodynamic variables acquired from isolated normothermic, hemoperfused, slaughterhouse porcine hearts. The goal of this study was to create a much as possible complete inventory of the changes over time to identify the causes of the progressive deterioration of cardiac function and development of edema in the PhysioHeart™. This study provides the basis for further investigations and improvements to extend the cardiac function of the slaughterhouse hearts revived in the PhysioHeart™ platform. We envision that this study with its comprehensive recordings could be of potential interest for cardiac normothermic perfusion in other platforms.
Seven hearts were obtained from Dutch Landrace pigs slaughtered for human consumption. Each animal had a weight of approximately 110 Kg. All protocols followed by the slaughterhouse and laboratory were consistent with EC regulations 1069/2009 regarding the use of slaughterhouse animal material for diagnosis and research, supervised by the Dutch Government (Dutch Ministry of Agriculture, Nature and Food Quality) and were approved by the associated legal authorities of animal welfare (Food and Consumer Product Safety Authority).
Isolation and administration of cardioplegia
The procedure for harvesting the hearts was equivalent in all the animals and is summarized in this section. Before heart harvesting, the pig was electrically stunned, hung and exsanguinated, but not heparinized. Afterwards a parasternal incision was made in the thorax and the heart and lungs were removed en-bloc. The heart was immediately topologically cooled. Subsequently, the pericardial sac was opened, the pulmonary artery was transected under the bifurcation and the aorta was transected under the first supra-aortic vessel. The heart was then isolated and prepared as described in a previous study [21]. Immediately after removal, the aorta was cannulated and 2 L of heparinized modified St. Thomas 2 crystalloid cardioplegic solution (Table 1) was administered through the coronary arteries at a mean pressure of 80–100 mmHg and a temperature of 4 °C. Warm ischemic time was measured and never exceeded 4 minutes. During the harvesting, 10 L blood to be used for reperfusion were collected, by exsanguination, from subsequently slaughtered pigs; this blood was also heparinized at 5000 IU/L. The heart was transported to the laboratory submerged in the St. Thomas solution at 4 °C and the blood was transported in a Jerry can at room temperature with no additional treatments. The heart and blood were stored for 2 hours (transport and preparation) and, after 1 hour in storage, an additional liter of cold cardioplegic solution was administered to the heart and the blood was filtered with a 200 μm filter.
Table 1 Modified St. Thomas solution for crystalloid cardioplegia
Mounting isolated hearts onto the circuit
The ex-vivo perfusion of the slaughterhouse hearts was performed using the PhysioHeart™ platform, which has been described previously [15]. The left atrium and the aorta were connected to the platform, the pulmonary artery was cannulated in order to return the venous blood to the reservoir and measure coronary flow. Temporary pacing leads (Medtronic Inc., Minneapolis, MN, USA) were placed on the right ventricular outflow tract (RVOT) to monitor electrical activity and paced when needed to maintain a regular rhythm. A porcine heart mounted in the platform in shown in Fig. 1. The perfusion circuit was primed with normothermic (38 °C) heparinized oxygenated blood (6 L, hematocrit 18–25%; pH 7.40 ± 0.05) and supplemented with insulin (0.32 units/L). The hearts were then perfused in Langendorff mode to keep the coronary perfusion pressure at 80 mmHg. Steady contractile myocardial activity was restored within 5 minutes, providing defibrillation when needed.
Porcine heart mounted in the PhysioHeart™ platform
After a supplementary stabilization time of about 15 min, preload and afterload were opened, so the platform was switched from Langendorff to working mode. During the working mode, the left ventricle (LV) ejects the perfusate into the afterload. The atrial pressure (ATP) and aortic pressure (AP) were adjusted to create a mean load of 10–20 mmHg and of 60–100 mmHg, respectively. All related pressures and flows were monitored and kept at physiological values. The blood glucose level was maintained, manually, between 5 and 7 mmol/L by the addition of a mixture of glucose and insulin. The pH was maintained with sodium bicarbonate. The mean cardiac output and coronary flow rate were measured using two ultrasound flow probes (SonoTT™ Clamp-On Transducer, em-tec GmbH, Finning, Germany), placed after the afterload and the pulmonary artery, respectively. The hemodynamic parameters were continuously monitored and adapted according to the pump function of the heart to fit the optimal clinical scenario as reported in Schampaert, van Nunen [20], shown in Fig. 2.
Physiological clinical scenario followed during the PhysioHeart™ experiments
Blood analysis and control
Arterial blood samples were taken from the oxygenator before the heart was connected to the loop, this was called the baseline measurement, and then every 60 min after reperfusion. Blood gas values, temperature, and electrolytes were measured using a VetScan i-STAT 1 (Abaxis, Union City, CA, USA). Based on the i-STAT 1 measurements, the pH, glucose, and ionized calcium were maintained at physiological levels by adding sodium bicarbonate, a glucose/insulin mixture (2 mmol/U) and calcium chloride. For further detailed analysis, full blood was collected in blood collection tubes. Tubes for plasma and serum analysis were centrifuged at 500 g for 10 min. Plasma and full blood samples were stored at 4 °C and serum samples were stored at − 80 °C overnight. The following day, samples were transported for analysis to a clinical laboratory (Máxima Medisch Centrum, Veldhoven, the Netherlands) and were examined with a C8000 analyser (Roche).
Pacing protocols, electrical acquisitions and signal processing and analysis
After the first four experiments, it was hypothesized that the analysis of the hearts' electrophysiology could give further insight on how to prolong normothermic perfusion. Hence, as a proof of feasibility, an electrophysiological analysis was performed in the last three experiments as follows. The hearts were paced, when necessary, with a pacing lead placed at the RVOT. To avoid breakthrough beats, pacing was provided at a higher frequency than the exhibited sinus rhythm of around 100 bpm. In order to investigate their electrical restitution (i.e. the physiological change in wave propagation velocity with respect to the change in heart rate), the hearts were paced at 100, 120 and 150 bpm during working mode. Unipolar epicardial electrograms (UEG) were acquired, on the left ventricle, with two different custom-made acquisition systems. Using two different systems allowed to compare which would be a better option in cardiac transplant applications. First, the UEG were measured using a square grid containing 11 × 11 electrodes, with 5 mm inter-electrode spacing. When using this grid, the UEG were recorded, simultaneously from all electrodes, using a BioSemi ActiveTwo acquisition and preprocessing system at a sampling frequency of 2048 Hz. The second acquisition system was formed by a rectangular grid of 6 × 8 electrodes (AD-TECH FG48G-SP05X-0E2). The signals from the latter grid were recorded simultaneously from all electrodes by a National Instruments (NI) 6031E acquisition card at 500 Hz. The acquisition card was connected to the grid using an NI SCB-100 shielded connector block followed by a NI-SH100100 connector cable. The configuration of the acquisition card and the recording of the signals was made using a custom-made virtual instrument developed in NI Labview 2013. Both grids are shown, as placed during the experiments, in Fig. 3.
Electrode grids used to measure the UEG during the PhysioHeart™ experiments. (left) 11 × 11 electrode grid used with BioSemi ActiveTwo acquisition system (right) 6 × 8 electrode grid used with the National Instruments acquisition system
After their acquisition, the UEG were pre-processed using a digital, Butterworth, band-pass filter with cutoff frequencies fc1 = 0.5 Hz and fc2 = 40 Hz. This filter removed undesirable DC components, baseline wander and high frequency, low-amplitude, noise. Afterwards, the activation time of the tissue under each electrode was measured as the point when the signal's derivative was minimal during the QRS complex of each beat. The propagation of the activation wave was assessed using activation maps, these were constructed, for each beat, by measuring the activation time of the tissue under each electrode and drawing isochrones that joined the areas of the LV that activated simultaneously. Finally, the velocity at which the activation wave propagated, called wave propagation velocity (WPV), was measured from the activation maps using the following equation:
$$ WPV=\frac{d\left({p}_2,{p}_1\right)}{AT\left({p}_2\right)- AT\left({p}_1\right)} $$
where p2 and p1 are two points in the direction of propagation, d(p2, p1) is the Euclidean distance between the two points and AT(p) is the activation time at point p. The UEG post-processing was done using Matlab (R2017b).
The seven hearts selected for this study were those that showed initial physiological cardiac hemodynamics which we considered to be at least CO 3 L/min, ATP 10–20 mmHg and AP 60 mmHg at the beginning of the working mode. Statistically significant differences in the mean values among treatment groups were determined by one-way analysis of variance. A paired t-test was used to confirm differences between heart weights before and after the experiment. A p-value ≤0.05 was set as a criterion for significance. All values are presented as the mean ± standard deviation. All the statistical tests were performed with Sigmaplot 11.0.
Cardiac hemodynamics and weight
The porcine hearts used in the experiments had a mean weight of 513 ± 104 g and showed hypertrophic cardiomyopathy before they were connected to the experimental platform, a general observation on the use of slaughterhouse hearts. After the experiments, the hearts had increased their weight to 657 ± 173 g, which is a statistically significant (p = 0.018) increase of 28%. This result can be observed in Fig. 4.
Weight of the hearts before connecting them to the platform and after the end of the experiments, * indicates a statistically significant difference
All seven hearts were revived and produced initial normal cardiac flows under the relevant pressures (ATP, 13 ± 2 mmHg; LVP, 71 ± 18 mmHg; AP,75 ± 8 mmHg) during the working mode. At baseline, the initial cardiac output (5.1 ± 1.7 L/min) was physiological in all hearts but decreased over time with a deterioration rate of 12.5 ± 2.7% per hour, while pressures were kept at physiological levels. All hearts preserved physiological hemodynamics for at least 2 hours. Only two of the seven hearts were able to maintain physiological hemodynamics for up to 4 hours. The progression of the cardiac output during the experiments can be observed in Fig. 5; in the figure, the values are shown as a percentage of their baseline values. The figure indicates that the degradation of the cardiac output was similar across all the experiments. The coronary flows and aortic pressures degraded with a similar trend.
Progression of the cardiac output across all the experiments. The values are presented as a percentage of their baseline value. * indicates a statistically significant difference
Blood values
The electrolyte concentrations that were documented across all seven experiments are presented in Fig. 6. At baseline, before connecting the hearts into the circuit, hyper-chloremia, −kalemia, −phosphatemia and -osmolarity were observed. Potassium values remained at its baseline level (7 ± 0.3 mmol/L) during the experiments. Sodium, total calcium, chloride, phosphate and magnesium increased steadily each hour by 3.9 ± 1.2 mmol/L, 0.3 ± 0.2 mmol/L, 4.7 ± 2.4 mmol/L, 0.2 ± 0.08 mmol/L and 0.2 ± 0.1 mol/L respectively, which resulted in hyper-natremia, −calcemia and -magnesemia during reperfusion. In all experiments, the electrolyte concentrations in blood were unphysiologically high before reperfusion or rose to unphysiological concentrations as the experiment progressed.
Time dependent changes in electrolyte concentrations shown as scatter and column plots. The dashed lines indicate upper and lower reference limit. a. Sodium, b. Potassium, c. Phosphate, d. Osmolarity, e. Chloride, f. Total calcium, g. Magnesium. * denotes a statistically significant difference in mean value
The metabolic panel, and its evolution in time, is presented in Fig. 7. The hearts revived in the platform showed symptoms of an aerobic metabolism during the working mode. As can be observed in Fig. 7b, this was evidenced by the consumption of approximately 1 mmol/L of glucose and 1 mmol/L of lactate per hour. However, lactate levels stabilized or even rose in the last hour, this is an indication of an anaerobic metabolism towards the end of the experiment. The manual adjustments in glucose helped maintaining it within physiological ranges but resulted in a wide variation during the third hour with an increase in the mean value, as can be observed in Fig. 7a. Additionally, as can be observed in Fig. 7c the mean values of free fatty acids decreased with an exponential trend from 0.62 ± 0.33 mmol/L to 0.22 ± 0.06 mmol/L in the first hour and dropped below the lower reference limit in the second hour; this is also an indication of an aerobic metabolism. Concentrations of triglycerides were stable at 0.7 ± 0.1 mmol/L during the experiment; however, an increase in the mean value after 1 hour of reperfusion can be observed in Fig. 7g. Urea and creatinine were stable at 3.5 ± 0.1 mmol/L and 140.2 ± 6.4 μmol/L respectively with signs of elevated creatinine levels. In turn, hyperammonemia could already be observed at baseline (305 ± 76 μmol/L) and increased at a rate of 132.5 ± 34.2 μmol/L per hour during the experiment.
Scatter and column plots showing the changes of metabolic biomarker concentrations during the ex-vivo beating heart experiments. Dashed lines indicate upper and lower reference limits. a. Glucose, b. Lactate, c. Free fatty acids, d. Ammonia, e. Urea, f. Creatinine, g. Triglycerides. * denotes a statistically significant difference in mean value
Cell damage biomarkers
Before connecting the hearts to the circuit, the mean values of the cell damage markers were above the reference limits, as can be observed in Fig. 8. Within the first hour after cardiac reperfusion, all cell damage markers showed the highest increase compared to the later measurements. Aspartat-Aminotransferase (ASAT), Creatine Kinase (CK), Troponin I, L-Lactatdehydrogenase (LDH) and Myoglobin were rising throughout the experiment with 337 ± 175 U/L, 4099 ± 1699 U/L, 29717 ± 6954 ng/L, 464 ± 1699 U/L, 1289 ± 1026 μg/L respectively.
Scatter and column plots showing the cardiac necrosis markers values during hemoperfusion in the PhysioHeart™ platform. a. Aspartat-Aminotransferase (ASAT), b. Creatine Kinase (CK), c. Troponin I, d. L-Lactatdehydrogenase (LDH) and e. Myoglobin. * denotes a statistically significant difference mean value
Additional plasma values
Manual pH balancing ensured the stability of the mean values of pH (7.40 ± 0.02) and base excess (− 1.32 ± 0.43 mm/L) but resulted in high variance in these markers, see Fig. 9d and e. Although albumin concentrations (28 ± 3 g/L) were also stable, their levels were low in relation to the reference range and identified the pathological nature of hypoalbuminemia. Similarly, hypervitaminosis D was identified due to high and not changing calcitriol levels (315 ± 15 pmol/L) above the reference range, see Fig. 9c. Finally, free hemoglobin increased during the experiment by approximately 0.02 ± 0.01 mmol/L per hour.
Scatter and column plots of blood related biomarkers during PhysioHeart™ experiments. a. Albumin, b. Free hemoglobin, c. Calcitriol, d. Base Excess, e. pH, f. Arterial oxygen saturation (SaO2). * denotes a statistically significant difference in mean values
Activation maps and wave propagation velocity measurements
Activation maps have been recorded on the epicardial surface of the hearts with two different electrode grids (11 × 11: Fig. 10, 6x8: Fig. 11). Figure 10 shows representative examples, at three different instants in time, of the activation maps obtained during one of the experiments, in which the heart was paced at 100 bpm during working mode. The activation patterns do not show signs of conduction block or arrhythmic nodes at any moment of the working mode on the area covered by the grid. However, the three activation patterns show a delay in the arrival of the depolarizing wave over time. The moment the wave arrives to the grid in Fig 10 Fig 11a is at around 130 ms after the pacing signal, this is increased to around 140 ms in Fig. 10b and is over 150 ms at the endpoint of the experiment (see Fig. 10c).
Activation maps observed during the working mode of the baseline PhysioHeart™ experiment. The heart was paced at 100 bpm. Electrode (2,6) malfunctioned, so the data from that channel was ignored. a. Was measured at the beginning of the working mode, b was measured 30 min after a. and c. was made 30 min after b. at the end-point of the experiment
Activation maps observed during the working mode of a PhysioHeart™ at three frequencies. Three different frequencies (sinus rhythm at 100 bpm, 120 bpm and 150 bpm) have been included in the measurement. In a. the heart was beating at sinus rhythm, whereas in b. and c. a pacemaker drove the beating of the heart
The activation maps measured in another experiment included a pacing protocol, in which three different frequencies (at sinus rhythm at 100 bpm, 120 bpm and 150 bpm) have been considered, which can be observed in Fig. 11. These measurements were made using the 6 × 8 electrode grid configuration. The direction of the depolarizing wave during pacing (Fig. 11b-c) was different compared to sinus rhythm (see Fig. 11a), since the natural heart rhythm starts in the sinus node and the pacing is provided at the RVOT, this phenomenon is justified. As before, the region that was monitored by the grid showed no arrhythmic nodes or areas of conduction block during the working mode. Furthermore, as can be observed in Fig. 11b-c changing pacing frequencies had no observable effect on the propagation pattern.
Differences between the activation maps in Fig. 10 and 11 are a consequence of the grid configurations. Activation maps in Fig. 10a-c look more uniform due to the larger grid (11 × 11 electrodes, 55x55mm) compared to Fig. 11a-c (6 × 8 electrodes, 30x48mm), which is also reflected in the shorter travelling time in Fig. 11. Differences in direction are attributable to the different positions of the grids (see Fig. 3).
The wave propagation velocities measured across all PhysioHeart™ experiments at the beginning of the working mode are presented in Table 2, they indicate normal physiological behavior. Indeed, the mean value of the wave propagation velocity is within the boundaries observed in healthy hearts [23] and it exhibits a physiological restitution effect [24], (i.e. a decrease in both action potential duration and wave propagation velocity as the stimulation frequency increases). However, the standard deviation of the WPV shows a beat-to-beat variability, of around 10 cm/s, which is unphysiological when compared to the relatively small variability observed in healthy measurements [23]. Also, towards the end of the experiment, the measurements of wave propagation velocity indicated a value of around 65 cm/s at 100 bpm (not shown in the table). This decrease in velocity indicates an impairment of the electrical conduction as the experiment progresses.
Table 2 Mean value (μWPV) and standard deviation (σWPV) of the wave propagation velocities measured during working mode at different pacing frequencies. The table summarizes the acquisitions made across all PhysioHeart™ experiments
The main objective of this study was to present an in-depth biochemical, hemodynamic and electrophysiological characterization of our isolated ex-vivo slaughterhouse heart experiments in the PhysioHeart™ platform. Initially, the resuscitated porcine hearts showed physiological metabolic, electrical and hemodynamic activities. However, electrophysiological and hemodynamic cardiac functions gradually diminished due to the initiation of waste product intoxication, reduction of essential nutrients, ion imbalances, cardiac necrosis and, most likely lastly, reperfusion injury and inflammation. On one hand, we conclude that the variability observed in the baseline pump function is a consequence of the 'random' selection of the slaughterhouse animals and the harvesting techniques. On the other hand, the superimposed progressive diminishment in cardiac function is concluded to be a result of the isolated slaughterhouse heart pathophysiology. Namely, the observed loss of function is associated with an increased level of metabolites and electrolytes, declining nutrients, a gradual loss of tissue integrity with edema and cell death which we believe is a result of the lack of hepatic and nephrological plasma clearance in the isolated heart setting. Figure 12 and Table 3 summarize these results, which resemble the deterioration of the heart function in a multi-organ failure situation. These observations support the use of plasma clearance interventions and support the working hypothesis that isolated hearts should be treated, as far as possible, as heart-and-organ failure environment.
Timeline of process-dependent physiological changes during PhysioHeart™ experiments
Table 3 Result summary
The limited duration of acceptable performance during the isolated heart experiments highlights that the isolated working heart needs to be in an environment that resembles the in-vivo physiology to avoid loss of its morphological and functional integrity. The slaughterhouse pigs used in this study were in general good health and were examined by a veterinarian prior to slaughtering. However, previous research has shown that domestication, selective breeding, scarce physical activity and improved feeding efficiency lead to morphological abnormalities in slaughterhouse-derived porcine hearts [25]; this was observed in our specimens as hypertrophic cardiomyopathy. Also, the baseline blood measurements revealed elevated levels in damage markers (i.e. CK, ASAT, LDH, troponin and myoglobin). As previous research has shown, elevated damage markers were most likely caused by the limited heart capacity observed in farm animals due to an intensive selection pressure and high stress during regrouping, transport and slaughtering [26]. The baseline measurements also revealed high ion concentrations and hyperosmolarity in blood; as previously reported by Heinze and Mitchell [27], this was probably a consequence of water accumulation in the intra- and inter-cellular space caused by the electrical stunning. Moreover, it is believed that the hyperkalemia observed at baseline was a consequence of the rapid drop in pH produced by the slaughtering; this drop in pH is known to lead to a cellular intake of H+ and release of K+ as a physiological process of pH balancing [28]. Electrical stunning also produces muscle contraction, which leads to hypoglycemia, hyperlactatemia, elevated creatinine levels and hyperammonemia [29]. These contractions can further lead to acidosis (low pH) and hyperlactatemia, an effect that has previously been reported during epileptic seizures when the muscles suffer from hypoxia [30, 31].
For each heart experiment, blood from different pigs was collected immediately after exsanguination and stored for about 2 h until preparation for reperfusion. Generally, pooling blood leads to transfusion reaction in humans, but the particular characteristics of the porcine hematopoietic system make porcine blood pooling less harmful as it causes transfusion reaction only in very rare cases [32]. However, the storage lesion of erythrocytes, during which glucose is consumed, levels of 2,3-diphosphoglycerate (DPG) and ATP decrease, and ammonia and potassium levels increase [33,34,35], is most likely contributing to the pathological blood values observed already at baseline.
Despite the quick harvesting process, warm cardiac ischemia is still expected to occur and to cause cardiac nutrient deficiency, hypoxia, acidosis and necrosis. It is expected that these processes will continue to damage the tissue during the cold storage period. Finally, these already stressed, hypertrophic hearts, were stored in a St. Thomas solution 2, a hypooncotic solution that promotes the influx of water through the endothelial layer into the intracellular space; this causes a further risk of cardiac edema [36, 37]. A more complex composed cardioplegic solutions like Custodiol [38], Somah [39], Celsior [40, 41] or UWS [41], could be of favor during hypothermic storage of slaughterhouse hearts. However, the use of a more complex solution also requires a careful consideration of price and advantages, which are currently under evaluation.
Therefore, the here above described 'slaughterhouse-associated' adverse effects should not be ignored when comparing the results with the carefully removed heart. These effects result in an increased chance for a reduced preservation, loss of cardiac tissue and function of the slaughterhouse porcine hearts. Despites these limitations one can learn from this pig heart the following:
Biomarkers and electrolytes
Immediately after cardiac resuscitation, an increase in potassium and magnesium in blood is observed. This is probably due to the washout of the cardioplegic solution from the coronary system. This solution, which is administered during harvesting, contains potassium and magnesium at 16 mmol/L to ensure cardiac arrest during storage. Figure 6g illustrates this wash out on the example of magnesium which experiences its largest increase in the first hour.
Throughout the experiment, we observe a rise in cardiac injury markers caused by reperfusion injury [42] and inflammatory responses of leukocytes and platelets. It remains speculative, but possible causes for the increasing markers may be heterogeneous cardioplegia delivery to the myocardium, harvest-related thrombosis, air emboli, and/or hypertrophic myocardium. These circumstances vary amongst hearts and therefore result in the observed fluctuating necrosis marker concentrations [43], initial cardiac outputs and pump functions of slaughterhouse-based hearts.
Hypertrophic hearts are known to be more vulnerable to ischemia and reperfusion injury [44] due to dilated epicardial coronaries, reduced capillary density and vascular dilatation reserve, which reduces the diffusion of nutrients and oxygen [45] and could potentially negatively influence the cardiac arrest. The presence of dilated, hyperemic coronaries in beating pig hearts revived in the PhysioHeart™ platform has been recently confirmed by Schampaert, van 't Veer [21] who associated the hyperemia to an endothelial response to the organ harvest and preparation. However, whether the hyperemic circulation is related to these preparation processes or a hypertrophy-related impairment to pharmacological and physiological stimulation, as other works suggest [46, 47], is still not fully understood.
The acidic environment during cardiac storage reduced the pH of the blood pool after cardiac resuscitation. The pH balancing with sodium bicarbonate led to an increase of sodium and reduction of ionized calcium [48] which was then counter balanced with calcium chloride administration resulting in constant rise of sodium and chloride in the blood. Besides these processes, the revived hearts showed a physiological aerobic cardiac metabolism, supported by free fatty acid uptake and lactate as well as glucose metabolism similar to previous reports [40, 49,50,51]. The constant rise of ammonia also confirms an amino acid catabolism.
However, as the cardiac hemoperfusion progresses, essential cardiac nutrients like free fatty acids decrease and toxic waste products like ammonia increase; this is known to cause edema and to disturb oxidative phosphorylation in the mitochondria [52]. This could explain the increasing lactate values and gain of heart weight of more than 20% at the end of the experiments.
The rise of plasma free hemoglobin in our study was not significant. However, in only one experiment free hemoglobin passed 0.08 mmol/L, which occurred already from the beginning of the experiment. That could have resulted from pre-experimental blood handling. We identified the centrifugal pump as the source with the highest risk to induce hemolysis. Finally, the static concentrations of albumin, triglycerides, urea, creatinine, calcitriol but also potassium exclude the possibility that the rise of electrolytes could arise from evaporation of free water in our system.
Epicardial electrical activity during the working mode
Electrical measurements showed physiological electrical activities of hearts revived in the PhysioHeart™ platform and during the working mode. This can be appreciated in the activation patterns presented in Fig. 10 and Fig. 11, which show unaltered electrical conduction pathways with no observable conduction block or ischemic effects in the areas of interest. Also, WPV restitution (i.e. a decrease in wave propagation velocity as the pacing frequency is increased) was observed and presented in Table 2. The analysis of restitution effects is central in the early detection of arrhythmia and in testing anti-arrhythmic drugs and devices; consequently, observing restitution effects in the PhysioHeart™ platform enables its use to investigate these phenomena within the scope of normothermic perfusion.
Although normal physiological behavior was observed during the working mode, all PhysioHeart™ experiments showed abnormally high sodium, potassium and ionized calcium concentrations in blood. These concentrations increased as the experiment progressed, this is evident from Fig. 6. The abnormally high sodium concentration translated, as observed in Table 2 and as supported by previous research [53], in high wave propagation velocity. Abnormally fast depolarization waves could induce arrhythmias because they may cause re-entrant waves or conduction block. High ionized calcium concentration in blood has also been shown to be related to longer action potentials [54] and abnormal membrane excitability [53]. Also, the observed hyperkalemia is known to cause elevated resting membrane potentials and reduced cellular excitability [55] and, consequently, arrhythmia such as atrial fibrillation or ventricular tachycardia. The use of insulin in our experiments may have support these effects as insulin leads to a dose-dependent influx of potassium into the cells [56]. This last fact was also evident because, in some experiments, stimulation protocols induced arrhythmias when pacing higher than sinus rhythm.
These observations put in evidence the importance of, simultaneously, monitoring the ion concentrations in blood and the electrophysiological activity of the cardiac tissue. In particular, the use of electrode grids within normothermic perfusion platforms could enable the detection of localized ischemia and abnormal conduction patterns that could result in arrhythmia during transport. Moreover, the monitoring of the ion concentrations in blood would also enable to determine the causes of any unphysiological electrical behavior, which can result in fast action to prevent the decreased performance of the heart.
Achieving normal cardiac physiology during ex-vivo slaughterhouse heart perfusion
The PhysioHeart™ platform, with its starling resistor as preload and a standard four-element Windkessel model as afterload, generates flow patterns and pressure curves in the revived slaughterhouse hearts that are similar to those measured in humans [15]. For an average of 3 h, physiological and morphological cardiac characteristics, with normal electrical and metabolic activities, can be obtained without any corrective measures. Although not all blood values are physiological prior and during reperfusion, isolated beating slaughterhouse porcine hearts seem to tolerate these pathologies for a limited period. Therefore, it is inferred that the isolated working normothermic heart can be used as a baseline model to study cardiac intervention methods (LVADs, TAVI valve replacements, etc.).
These interventions may be unloading, moderate hypothermia, filtration of plasma for inflammatory components and metabolic waste, addition of nutrients and protective drugs. In view of ethical constraints regarding use of animals for short-term and uncertain-outcome experiments, the platform provides several benefits including availability, low cost, and no ethical objections.
In view of improved and prolonged preservation of the PhysioHeart™ model is the mitigation of the immune response of the pig blood. This can be achieved by separating lymphocytes and platelets, to obtain platelet and lymphocyte-poor blood in combination with administering inflammatory and autoimmune depressing drugs (i.e. dexamethasone, prednisone). The use of antibiotics and fungistatic medication would further serve to avoid infections. In addition, including anti-arrhythmic drugs in the platform will help mitigate the effects of high electrolyte concentrations in blood and, consequently, extend the time in the working mode.
We have identified that the PhyioHeart™ lacks hepatological and nephrological clearances and substance supplementations. In the upcoming future, the blood perfusate should be kept physiological and renewed constantly to maintain metabolomics and proteomic profiles and to remove toxins either with dialysis, new platelet-poor plasma [52] or a similar complex medium (i.e. HCO3, HEPES, inorganic salts, amino acids, carbohydrates, fatty acids, lipids, vitamins (Cernevit), trace elements, colloids and hormones for vasodilation like Milrinone). Dialysis and hemofiltration may be helpful by removing excess water, toxins and stabilizing the electrolyte and ion balance as the periods of arrhythmia observed in the PhysioHeart™ experiments were associated with high ion concentrations. Finally, these attempts are assumed to attenuate the loss of cardiac function in the platform and would lead to more standardized and improved experimentation.
Limitations of the current PhysioHeart™ experiments
Despite low animal costs, big animal isolated heart experiments remain costly. Therefore, the scope of this study was rather exploratory in order to determine the boundary conditions and attempts needed to prolong physiological cardiac perfusion in our model.
All the hearts had different initial left ventricular function and electrical activity, which result in a 'normal variation' that may affect small-number experiments. Therefore, it is important to standardize and optimize the harvest and mounting procedures to avoid outlying negative performance at the start of the experiment. An additional limitation is that the blood pool volumes between experiments were not equal could have influenced blood marker concentrations; a standard initial volume is required to ensure consistency across experimental acquisitions. Additionally, the priming volume for deairing the circuit was not equal between experiments and, most likely, biased the baseline measurements; in future experiments, the priming volume will be considered when analyzing baseline values. Finally, one may consider non-invasive epicardial monitoring such as echocardiography, stress-strain imaging and speckle imaging to detect early signs of ischemia that may result in corrective or supporting actions. Moreover, the use of square grids limits the electrophysiological study to a small region of the heart; the use of more sophisticated electrode arrays, capable of making whole-heart measurements, would enable a more comprehensive electrophysiological study. Finally, micro-puncture histology during and at the end of the experiments may be useful as a hard outcome parameter.
The isolated working slaughterhouse heart is a practical (e.g. abundant numbers, no need to specifically sacrificed laboratory animals, no ethical approval of study protocols, etc.) and cost-efficient model to perform investigative and therapeutic experiments. This study was meant to identify factors limiting these experiments of isolated slaughterhouse porcine heart in the PhysioHeart™ perfusion model. Our findings confirm the viability and function loss of the isolated slaughterhouse hearts are best described by the phenomenon of "time-dependent multi-organ" failure. The blood parameters, biochemical and electrophysiological changes observed in the PhysioHeart™ platform provide a better understanding of the necessary effort to overcome the challenges. Diligent and strictly protocolized harvesting and installation will reduce variation at the start of the PhysioHeart™ platform experiment. This study identified several mechanisms and provided explanations of the potential sources which limit the ex vivo cardiac viability and perfusion time which in turn could be corrected. Perfusate renewal and clearance has been recognized as crucial for prolonged cardiac perfusion in the PhysioHeart™ model which could be of potential interest for other heart platforms.
Aortic pressure
ASAT:
Aspartat-aminotransferase
ATP:
CK:
DPG:
2,3-diphosphoglycerate
FFP:
Fresh frozen plasma
LDH:
L-Lactatdehydrogenase
LV:
Left ventricle
NI:
RVOT:
Right ventricular outflow tract
UEG:
Unipolar epicardial electrograms
WPV:
Wave propagation velocity
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We thank Dr. Dirk Bakkeren, Anita Groos, Riejean Kuylaars and the rest of the team from the clinical laboratory at the Máxima Medical Centre Veldhoven for the biochemical analysis.
This project has received financial funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska Curie grant agreement No 642612. Bastiaan J. Boukens received financial funding from the Dutch Heart Foundation (2016 T047).
Benjamin Kappler and Carlos A. Ledezma contributed equally to this work.
Department Cardiothoracic Surgery, Amsterdam University Medical Center, Meibergdreef 9, Amsterdam, The Netherlands
Benjamin Kappler & Bas A. J. M. de Mol
LifeTec Group B.V, Eindhoven, The Netherlands
Benjamin Kappler, Sjoerd van Tuijl, Marco Stijnen & Bas A. J. M. de Mol
Department of Mechanical Engineering, University College London, Torrington Place, London, UK
Carlos A. Ledezma, P. J. Tan & Vanessa Díaz-Zuccarini
Department of Medical Biology, Amsterdam University Medical Center, Meibergdreef 9, Amsterdam, The Netherlands
Veronique Meijborg & Bastiaan J. Boukens
Department of Translational Physiology, Amsterdam University Medical Center, Meibergdreef 9, Amsterdam, The Netherlands
Bülent Ergin & Can Ince
WEISS Centre for Surgical and Interventional Sciences, UCL, Gower Street 10, London, UK
Vanessa Díaz-Zuccarini
Benjamin Kappler
Carlos A. Ledezma
Sjoerd van Tuijl
Veronique Meijborg
Bastiaan J. Boukens
Bülent Ergin
P. J. Tan
Marco Stijnen
Can Ince
Bas A. J. M. de Mol
BK, CAL, SvT and MS participated in study design. BK, CAL and SvT carried out the study and prepared the manuscript which was revised and approved by BE, CI, PJT, VD-Z and BAJMdM. BJB and VM assisted with the assessment of the epicardial ECGs. All authors read and approved the final manuscript.
Correspondence to Benjamin Kappler or Vanessa Díaz-Zuccarini.
All protocols followed by the slaughterhouse and laboratory were consistent with EC regulations 1069/2009 regarding the use of slaughterhouse animal material for diagnosis and research, supervised by the Dutch Government (Dutch Ministry of Agriculture, Nature and Food Quality) and were approved by the associated legal authorities of animal welfare (Food and Consumer Product Safety Authority).
B. Kappler worked with LifeTecGroup B.V. before his employment as PhD researcher at the department of Cardiothoracic Surgery of the Academic Medical Center at the University of Amsterdam. S. van Tuijl. is an employee of the LifeTec Group B.V. M. Stijnen and B.A.J.M. de Mol are founders of LifeTec Group BV and are respectively Head of Medtech Innovation and director of Medical Affairs. Professor B.A.J.M. de Mol was chief of the department of Cardio-thoracic Surgery at the Amsterdam University Medical Center. LifeTec Group B.V. developed the PhysioHeart™ platform and has the proprietary rights. LifeTecGroup BV supported this study with professional laboratory services and provision of materials.
Kappler, B., Ledezma, C.A., van Tuijl, S. et al. Investigating the physiology of normothermic ex vivo heart perfusion in an isolated slaughterhouse porcine model used for device testing and training. BMC Cardiovasc Disord 19, 254 (2019). https://doi.org/10.1186/s12872-019-1242-9
Ex vivo
Normothermic perfusion
Cardiac physiology
Cardiac electrophysiology
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CommonCrawl
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Journal of Dynamics and Differential Equations
Instability of Equilibrium Solutions of Hamiltonian Systems with n-Degrees of Freedom Under the Existence of Multiple Resonances and an Application to the Spatial Satellite Problem
Daniela Cárcamo-Díaz
Claudio Vidal
In this paper, we prove the instability of one equilibrium point in a Hamiltonian system with n-degrees of freedom under two assumptions: the first is the existence of multiple resonance of odd order s (without resonance of lower order) but with the possible existence of resonance of higher order; and the second is the existence of an invariant ray solution of the truncated Hamiltonian system up to order s. It is shown that in the case of resonance without interaction, the necessary conditions for instability have important simplifications with respect to the general case. Examples in three, four and six degrees of freedom are given. An application of our main result to the spatial satellite problem is considered.
Hamiltonian system Equilibrium solution Stability Normal form Resonance Invariant ray solution Chetaev's theorem
Mathematics Subject Classification
37C75 34D20 34A25
The authors would like to thank the referee for valuable comments, which improved an earlier version of this paper. Claudio Vidal was partially supported by project Fondecyt 1180288. This paper is part of the Daniela Cárcamo-Díaz Ph.D. thesis in the Program Doctorado en Matemática Aplicada, Universidad del Bío-Bío. Daniela Cárcamo-Díaz acknowledges funding from CONICYT PhD/2016-21161143.
Consider the system of ordinary differential equations
$$\begin{aligned} \dot{u}=f(u), \, u\in {\mathbb {R}}^n, \,f(0)=0. \end{aligned}$$
Theorem 6.1
Assume that the cone K with vertex in 0 (see Fig. 4) has in cross-section the form of a cube (of dimension \(n-1\)) and that the following conditions are fulfilled:
The trajectories may only enter on each pair of opposite side faces of K.
For the points of Int(K) there exists a differentiable function of the type of the "Lyapunov function" (\(L(0)=0, L(u)>0\) for \(u\ne 0\), \(u\in Int(K)\)) and \(\dot{L}(u)>0\) by virtue of the system (42) on Int(K).
Then the equilibrium state \(u=0\) of (42) is unstable in the Lyapunov sense.
The proof of this theorem can be found in [10].
Representation of the cone K in Chetaev's Theorem 6.1
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© Springer Science+Business Media, LLC, part of Springer Nature 2018
1.Departamento de Matemática, Facultad de CienciasUniversidad del Bío-BíoConcepción, VIII-RegiónChile
2.Grupo de Investigación en Sistemas Dinámicos y Aplicaciones-GISDA, Departamento de Matemática, Facultad de CienciasUniversidad del Bío-BíoConcepción, VIII-RegiónChile
Cárcamo-Díaz, D. & Vidal, C. J Dyn Diff Equat (2019) 31: 853. https://doi.org/10.1007/s10884-018-9679-6
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CommonCrawl
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How to find the length of an indefinitely repeated sequence in a list
Here, I am not looking for simply a repeated element, but an indefinitely repeated sequence. For example, if I have the list
{1,3,4,3,4,2,3,3,2,3,3,2,3,3,2,3,3,2,3,3}
I would like Mathematica to output $3$, since the sequence $(2,3,3)$ (with length 3) repeats until the end of the sequence after a certain point. Note, the repeated sequence needs to continue until the end of the list. For example, the sequence $(3,4)$ does repeat itself near the beginning of the list, however I would not want an output of $2$ due to this repetition since the sequence $(3,4)$ does not continue after a certain point.
More formally: If I input a list for which the $i^{th}$ element is $x_i$, I want Mathematica to output the smallest number $n$ for which $x_i=x_{i+n}$ for all sufficiently large $i$.
What function/code would allow me to input a list and then would output the desired result?
list-manipulation sequence
volcanrbvolcanrb
$\begingroup$ duplicate?: Find cycle length in a list $\endgroup$ – kglr Nov 11 '18 at 23:27
Perhaps this?:
Length@Last@FindTransientRepeat[
{1, 3, 4, 3, 4, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3}, 2]
(* 3 *)
For whether the second argument 2 is sufficient for all your inputs, see FindTransientRepeat.
Michael E2Michael E2
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CommonCrawl
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Validation of the brief international cognitive assessment for multiple sclerosis (BICAMS) in the Portuguese population with multiple sclerosis
Cláudia Sousa ORCID: orcid.org/0000-0002-9070-95441 na1,
Mariana Rigueiro-Neves2 na1,
Telma Miranda1,
Paulo Alegria3,
José Vale3,
Ana Margarida Passos2,
Dawn Langdon4 &
Maria José Sá1,5
The validation of international cognitive batteries in different multiple sclerosis (MS) populations is essential. Our objective was to obtain normative data for the Portuguese population of the Brief International Cognitive Assessment for Multiple Sclerosis (BICAMS) and assess its reliability.
The BICAMS was applied to 105 MS patients and 60 age, gender and education matched healthy controls (HC). In order to test its reliability, BICAMS was re-administered in a subset of 25 patients after a 7-month interval.
Most participants were women, with a mean age of 37, 21 years and a mean of 14,08 years of education. The vast majority of the MS patients (92.4%) had the relapsing remitting type, 58.1% were professionally active, mean disease duration was 6.52 years, median EDSS score was 1.5 (range: 0–6.0) and the median MSSS score was 2.01 (IQR range: 3.83). The MS group presented significantly higher scores of anxiety and depression than HC and 47,4% had fatigue. The MS group performed significantly worse than the control group across the three neuropsychological tests, yielding the following values: SDMT: t(165) = 3.77, p = .000; CVLT-II: t(165) = 2.98, p = .003; and BVMT-R: t(165) = 2.94, p = .004. The mean raw scores for Portuguese normative data were as follows: SDMT: 58.68 ± 10.02; CVLT-II: 60.47 ± 10.12; and BVMT-R: 24.68 ± 5.52. Finally, test–retest reliability coefficients for each test were as follows: SDMT: r = .90; CVLT-II: r = .71; and BVMT-R: r = .84.
The Portuguese version of BICAMS here in described is a reliable monitoring instrument for identifying MS patients with cognitive impairment.
Multiple sclerosis (MS) is a chronic inflammatory demyelinating disease of the central nervous system that can impair any body function, including cognition [1]. Cognitive dysfunction affects 40 to 70% patients [2, 3]. Irrespective of age and gender [3, 4], may occur at all stages of the disease, even at the very early beginning [5, 6] and definitely impacts the lives of MS patients and their families [3, 7, 8].
The characteristic pattern of cognitive impairment in MS has been described early on to include memory, information processing efficiency, executive functioning, attention and processing speed [1]. However, the cognitive domains most likely to be affected in MS are information processing speed and memory, whilst visual processing and executive function are less likely to be impaired and language is largely intact [1, 9,10,11,12].
The most frequently used neuropsychological batteries for patients with MS such us, the Brief Repeatable Battery of Neuropsychological tests and the Minimal Assessment of Cognitive Function in MS, require specialized technical and human resources and take a considerable time for evaluation in the daily clinical setting [1, 13]. Recently, the Brief International Cognitive Assessment for MS (BICAMS) was developed and recommended as a validated and standardized international screening test, because it is an easier assessment tool that can be administered by a technician who is not a specialist in neuropsychology and lasts only about 15 min to apply [14, 15]. Besides, the three instruments that compose BICAMS – Symbol Digit Modalities Test (SDMT) [16], California Verbal Learning Test (CVLT-II) [17] and Brief Visuo-spatial Memory Test Revised (BVMT-R) [18] – have previously been shown to have good psychometric properties.
The aims of this study are to describe the normative values of the Portuguese version of the BICAMS with gender, age and education corrections and to test the validity of this battery in a sample of Portuguese patients with MS.
A group of 105 patients with MS diagnosed according to the McDonald criteria [19] and a control group of 60 age, gender and education matched healthy subjects (HC), entered this study, and conducted in the period 2015–2016.
The MS patients were consecutively recruited at the MS Clinics from two hospitals located in separate regions of the country, Hospital de São João (Oporto; North) and Hospital Beatriz Ângelo (Loures; South), whereas the HC group was recruited from the community and among relatives and friends of MS patients. All participants were aged between 17 and 69 years and they were fluent in Portuguese as first language.
Exclusion criteria were current or past neurological disorder other than MS, presence of major psychiatric illness, history of learning disability, history of serious head trauma, presence of alcohol or drug abuse, relapse and/or corticosteroid use within 4 weeks preceding the neuropsychological assessment. HC were also required to present scores > 21 on Montreal Cognitive Assessment Portuguese version (MoCA) [20, 21].
The study was approved by the ethical committees of both hospitals. All the participants, from MS group and HC, volunteered to participate in this study, giving written informed consent.
An initial demographic interview was conducted. This was based on a common script that included a demographic questionnaire, medical history, drinking and drug habits and present health status. The MS data, such as type, duration, and degree of disability and severity, as assessed by the Expanded Disability Status Scale (EDSS) [22] score and the Multiple Severity Status Score (MSSS) [23], respectively, were obtained in the clinical protocols.
Then, participants underwent the BICAMS battery [14], which included the oral version of Symbol Digit Modalities Test (SDMT) [16], the learning trials from the California Verbal Learning Test-II (CVLT-II) [17] and the Brief Visuo-spatial Memory Test-Revised learning trials (BVMT-R) [18].
The SDMT [16] examines sustained attention, concentration and processing speed. In the oral version, the participant examines a series of nine meaningless geometric symbols, which are labeled from 1 to 9. Then, during 90 s the participant is instructed to say the corresponding number to each symbol, as rapidly as possible. The test score corresponds to the number of correct responses.
The CVLT-II [17] is a measure of verbal learning and memory. The test begins with the examiner reading a list of 16 words to the patient and then he/she is asked to report as many of the items as possible, in any order. After recall is recorded, the entire list is read again followed by a second attempt at recall. Altogether, there are five learning trials. The outcome measure is the total number of recalled items over the five learning trials.
The BVMT-R [18] is a measure of visuo-spatial learning and memory. The participant is exposed to a matrix of six simple abstract designs for 10 s followed by an unaided recall; we used the form 1 of the original test. After that, the participant is asked to render the designs using paper and pencil, taking as much time as needed for reproduction. The scoring criterion is based on location and accuracy of each design (from 0 to 2, maximum total score for each array 12). The outcome measure of this test corresponds to the total recall score across the three trials.
The validation was conducted per the international standards given by the expert consensus committee [15]. As the first step, the CVLT-II list of words were translated and re-translated from English to Portuguese and vice versa respectively; the other two tests did not require translation due to their nature. In the second step, the test instructions were translated into Portuguese.
In both groups, anxiety and depression symptoms were also measured using the Portuguese version of Hospital Anxiety and Depression Scale (HADS) [24]. In the MS group the level of fatigue was measured with the Modified Fatigue Impact Scale (MFIS) [25,26,27].
"The participants of both groups were asked to return for a follow-up session to allow for test–retest reliability analyses. A subgroup of 26 patients and 13 HC returned after a mean time of 7 months and all the tests administered in the first session were repeated in the same manner and in the same order."
Well-trained clinical psychologists conducted all sessions and the tests were applied in a standardized way and in a fixed order. The mean time for BICAMS application was 15 min, as described [14, 15].
Statistical analysis was performed using the Statistical Package for the Social Sciences (IBM SPSS), version 23.0. Descriptive statistics (e.g., mean, standard deviation, median, interquartile range and percentages) were used for demographic characterization of both groups. Student's t-test for independent samples was used to analyze the differences between groups, at the level of p < .05. The values shown in the tables are bilateral p-values. The effect sizes of those differences were calculated using Cohen's d. Spearman's correlations (Ρ) were used to analyze reliability measures and the relationship between BICAMS, HADS and MFIS results. Raw scores were analyzed for the full sample and Z-scores were calculated. Multiple regression analysis was used to produce normative data.
Demographics and MS characteristics
The groups were similar with regard to age (MS group: M = 38.26 years±11.03; HC: M = 36.17 years ±12.01, p = .63), gender (MS group: %Female = 66.7; HC: %Female = 58.3, p = .28) or number of educational years (MS group: M = 13.55 ± 3.71; HC: M = 14.62 ± 3.47, p = .42). With respect to professional status, the majority of subjects were employed, with a much higher proportion of HC than MS, as is usually reported (n = 56, 94.9%; n = 61, 58.1%, respectively). In the MS group, 92,4% (n = 97) of patients had the relapsing remitting type and 3,8% (n = 4) secondary progressive type and 3,8% (n = 4) clinically isolated syndrome. The average disease duration was 6.52 years (SD = 5.95) and the median EDSS score was 1.5 (range: 0–6.0). The MSSS score, calculated in patients from 1 to 30 years of disease duration (n = 95), had a median value of 2.01 (IQR range: 3.83).
Criterion-related validity: Group differences
Means, standard deviations and t test's for independent samples from the three tests are presented in Table 1. The results showed that MS group performed significantly worse than the HC group on all measures. Cohen's d was analyzed for each neuropsychological test and were satisfactory: SDMT - 0.65 (large); CVLT-II - 0.49 (medium); BVMT-R - 0.45 (medium) [28].
Table 1 Group differences on BICAMS measures
Reliability: Test-retest
The test–retest reliability data obtained in a subgroup of MS patients are presented in Table 2. The test-retest reliability coefficients showed a strong to a very strong and significant effect for all BICAMS tests.
Table 2 Test–retest means and correlations for MS group (n = 26)
The test-retest results in the HC were not considered in view of the low number of cases.
Regression based-norms
To obtain a regression-based normative model for BICAMS, the distribution of the SDMT, CVLT-II and BVMT-R raw scores was analyzed for the complete sample and the Z scores were calculated. The raw scores were then converted into scaled scores (M = 10 and SD = 3), as presented in Table 3. For each test a multiple regression analysis with a stepwise method using the scaled scores as dependent variable and age, gender and education as predictors was performed. Education was introduced as the number of regular academic school years that the participant successfully completed. As some studies suggest that there is a curvilinear relationship between demographic variables and cognitive function [29], the quadratic term of age and education were also introduced as predictors. These results allow us to detect which variables contributed significantly to explain each of the scaled neuropsychological test scores.
Table 3 Raw score to scaled score conversions for the BICAMSs tests
The T-scores corrected for education, age and gender were generated through a procedure suggested by Diehr and colleagues [30]. Therefore, another multiple regression (enter method) with each of the BICAMS test scaled scores as the dependent variable and the significant predictors of each test was performed. The non-standardized predicted values of this equation were saved and a new variable was calculated corresponding to the difference between an individual's actual and predicted scale score (i.e., the residual) divided by the standard deviation of those residuals. These values were then rescaled for a T-score (M = 50 and SD = 10).
Finally, another multiple regression analysis with corrected T-score as the dependent variable was performed to generate each test normative formula for the Portuguese population. The final formula to calculate the T-scores for each of BICAMS's test are presented below:
$$ \mathbf{SDMT}\ \mathrm{T}\ \mathbf{score}=10.511+\left({0.007}^{\ast }\ {\mathrm{age}}^2\right)+\left(-{0.966}^{\ast }\ \mathrm{years}\ \mathrm{of}\ \mathrm{education}\right)+\left({4.138}^{\ast }\ \mathrm{scaled}\ \mathrm{score}\right) $$
$$ \mathbf{CVLT}-\mathbf{II}\ \mathrm{T}\ \mathbf{score}=3.195+\left({0.006}^{\ast }\ {\mathrm{age}}^2\right)+\left({3.761}^{\ast }\ \mathrm{scaled}\ \mathrm{score}\right) $$
$$ \mathbf{BVMT}-\mathbf{R}\ \mathrm{T}\ \mathbf{score}=-8.004+\left({0.514}^{\ast }\ \mathrm{age}\right)+\left({\mathrm{3,833}}^{\ast }\ \mathrm{scaled}\ \mathrm{score}\right) $$
In determining impairment, the 5th percentile value based on the performance of healthy control sample was calculated for each test. Participants were considered impaired if their score was equal of below the percentile 5thof the control group (results are presented on the Table 4) [31]. Then, using the previously reported criteria of impairment defined by "one or more abnormal tests" [32, 33], it was found that 24.8% of the MS sample was impaired at baseline.
Table 4 The prevalence of cognitive impairment in MS patients according to the 5th percentile value of HC on BICAMS tests
Analysing the degree of disability assessed by EDSS and cognitive performance, we found significant correlations with all cognitive tests (SDMT: −.497, p = .000; CVLT: −.334, p = .000; BVMT: −.275, p = .005).
Regarding anxiety and depression symptoms, it was found that MS group presented higher scores on these measures than HC, and that these differences were statistically significant: anxiety (MS group: M = 7.85 ± 4.51; HC: M = 6.32 ± 3.00, t = − 2.348; p = .20) and depression (MS group: M = 5.14 ± 3.95; HC: M = 3.18 ± 2.57). Anxiety symptoms were found to be more frequent (n = 56; 53.3%) than depression symptoms (n = 29; 27.6%) in MS patients. In the MS group depression symptoms had a modest significantly negative effect only on CVLT-II results (R – 0.196; p = .45), whereas anxiety was not significantly correlated with any BICAMS test. The assessment with the MFIS scale (n = 95) showed that fatigue was present in 50 MS patients (47,4%) and was significantly correlated with the EDSS score (R – .279; p = .006), and with anxiety (R – .631; p = .0001) and depression symptoms (R – .754; p = .0001). Conversely, fatigue was negatively correlated with SDMT score (R – .266; p = .009); similar results were observed in both MFIS subscales, physical (M = 18.04 ± 9.66; R-.289; p = .005) and cognitive (M = 17.84 ± 10.21; R-.203; p = .049).
An international consensus committee of experts recently recommended a short battery of tests for cognitive assessment in MS that allows monitoring of cognition over time and is a fast and reliable instrument that may be administered by healthcare professionals with no specific experience in neuropsychological testing. According to the international standards for validation [15], several validation studies of BICAMS have been carried out in different cultures and languages, with the aim of making this psychometric tool more solid and internationally applicable. Up to now, there exists normative data for populations of several countries, such as Czech Republic [32], Italy [34], Hungary [35], Ireland [36], Brazil [37], Lithuania [38] Argentina [39], Canada [33], Greece [31], Belgium [40], Japan [41] and Turkish [42].
The current study followed the recommendations and standards of the BICAMS consensus committee [14, 15] and is the first to publish the Portuguese normative data for SDMT, CVLT-II and BVMT-R. Our results showed that MS group performed significantly worse than HC group on all measures (SDMT, CVLT-II and BVMT-R), a finding that is in agreement with the other recently published validations. These differences were more marked in the SDMT and CVLT-II than the BVMT-R, and similar results were found by O'Connell and colleagues (2015), Spedo and colleagues (2015) and Vanotti and colleagues (2016). Test-retest reliability in our population fits the recommended international standards for BICAMS validation [14]. Test–retest reliability for raw scores was adequate to excellent for all the three tests in this validation; more than .80 in SDMT and BVMT, replicating prior finds [33, 34]. Yet our results are lower than those of Vanotti and colleagues (2016). In addition we confirmed that the SDMT has particularly high test-retest reliability. We used a wider time span than other authors [37, 40] in order to avoid the learning effect, since at both evaluation times the same forms were applied.
The BICAMS tasks were able to identify cognitive impairment in 24.8% of MS patients using the criteria of impairment defined by one or more abnormal tests. This is a lower value than those found in other studies, which ranged from 47.3 to 58% [31,32,33, 35, 36]. This result may reflect the characteristics of our MS sample, which were mainly RRMS and rather early cases (mean disease duration 6.5 years) and a correspondingly low level of physical disability, median EDSS 1.5 [31,32,33, 36]. The lower level of disability in our sample is further supported by our MSSS data [23].
We found a significant correlation between EDSS and cognitive performance in the three tests used, that is, the higher the EDSS score the worse the cognitive test performance.
Regarding anxiety and depression symptoms, we found that the MS group also presented with higher scores on these measures then the HC, fitting the results of other BICAMS validation studies [32, 33, 37]. The Hungarian BICAMS validation reported a negative correlation of fatigue with all BICAMS tests [31]. In our study an association with fatigue was only seen in the SDMT test, possibly reflecting the lower fatigue in our patients as well as the lower physical disability.
This study was some limitations. First, follow-up assessments were done in a low number of cases, especially in the HC group, which is due to the fact that some individuals live far from the Hospital and incur additional personal costs. Another limitation is the fact that effect size for CVLT and BVMT-R although satisfactory, is on the threshold of the effect size classified as medium.
In conclusion, our study provides the Portuguese BICAMS standards for use with MS patients and evidences the strong psychometric properties of the Portuguese BICAMS version. The normative data of the BICAMS for the Portuguese population enables the use of the battery in clinical practice, for longitudinal patient assessments and as an outcome measure of cognitive functioning in clinical trials. Future prospective studies with larger samples of MS patients, with different types of disease evolution, will certainly add valuable information concerning the clinical applicability of the Portuguese BICAMS version.
BICAMS:
Brief International cognitive assessment for multiple sclerosis
BRB-N:
Brief repeatable battery of neuropsychological tests
BVMT-R:
Brief visuo-spatial memory test – revised
CVLT-II:
California verbal learning test – II
HADS:
Hospital anxiety and depression scale
HC:
Healthy subjects
IBM SPSS:
Statistical package for the social sciences
MACFIMS:
Minimal assessment of cognitive function in multiple sclerosis
MFIS:
Modified fatigue impact scale
MoCa:
Montreal cognitive assessment
MSSS:
Multiple severity status score
SDMT:
Symbol digit modalities test
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Silva AM, Vilhena E, Lopes A, Santos E, Gonçalves MA, Pinto C, et al. Depression and anxiety in a Portuguese MS population: association with physical disability and severity of disease. J Neurol Sci. 2011;306:66–70.
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Larson RD. Psychometric properties of the modified fatigue impact scale. Int J MS Care. 2013;15(1):15–20.
Gomes LR. Validação da versão portuguesa da Escala de Impacto da Fadiga Modificada e da Escala de Severidade da Fadiga na Esclerose Múltipla (Validation of the Portuguese version of the Modified Fatigue Impact Scale and the Fatigue Severity Scale in Multiple Sclerosis). Thesis.Minho University. 2011.
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Polychroniadou E, Bakirtzis C, Langdon D, Lagoudaki E, Kesidou E, Theotokis P, et al. Validation of the brief international cognitive assessment for multiple sclerosis (BICAMS) in Greek population with multiple sclerosis. Multiple Sclerosis and Related Disorders. 2016;9:68–72.
Dusankova JB, Kalincik T, Havrdova E, Benedict RH. Cross cultural validation of the minimal assessment of cognitive function in multiple sclerosis (MACFIMS) and the brief international cognitive assessment for multiple sclerosis (BICAMS). Clin Neuropsychol. 2012;26(7):1186–200.
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The authors would like to thank all participants of this study.
An earlier version of this paper was presented at the ECTRIMS-ACTRIMS Meeting, Paris, France, 25–28 October 2017. The abstract of the e-Poster was published in the Multiple Sclerosis Journal (2017): 23: (S3): 680 – 975. http://0-journals.sagepub.com.brum.beds.ac.uk/doi/10.1177/1352458517731285.
Availability of data materials
The datasets used and analysed during the current study are available from the corresponding author on reasonable request.
This work was funded by an unrestricted educational grant from Bayer, which had any role in the study, namely in its design, sample collection, analyses and interpretation of data and in the writing of the manuscript.
Cláudia Sousa and Mariana Rigueiro-Neves contributed equally to this work.
MS Clinic, Department of Neurology, Centro Hospitalar São João Porto, Alameda Prof. Hernâni Monteiro, 4200 – 319, Porto, Portugal
Cláudia Sousa, Telma Miranda & Maria José Sá
BRU-IUL, Instituto Universitário de Lisboa (ISCTE-IUL), Lisbon, Portugal
Mariana Rigueiro-Neves & Ana Margarida Passos
Department of Neurology, Hospital Beatriz Ângelo, Loures, Portugal
Paulo Alegria & José Vale
Department of Psychology, Royal Holloway, University of London, London, UK
Dawn Langdon
Faculty of Health Sciences, University Fernando Pessoa, Porto, Portugal
Maria José Sá
Cláudia Sousa
Mariana Rigueiro-Neves
Telma Miranda
Paulo Alegria
José Vale
Ana Margarida Passos
CS contributed in study concept and design, drafting and revising the manuscript and in the acquisition and interpretation of data. MRN contributed in study concept and design, drafting and revising the manuscript, in the acquisition and interpretation of data and statistical analysis. TM contributed in the acquisition of data and statistical analysis. PA and JV contributed in patient recruitment, acquisition of clinical data and revising the manuscript. AMP contributed in study concept and design, revising the manuscript, in the interpretation of data and statistical analysis. DL contributed in study concept and design and in revising the manuscript. MJS contributed in study concept and design, drafting and revising the manuscript, in the analysis and interpretation of data and study supervision. All authors read and approved the final manuscript.
Correspondence to Cláudia Sousa.
All patients and all healthy control subjects provided written informed consent to participation in the study. The Ethics Committees of Centro Hospital S. João, Oporto and Ethics Committees Hospital Beatriz Ângelo, Loures, granted approval for the research project.
MJS has received consulting/speaker fees from Bayer, Biogen, CSL Behring, Merck, Novartis, Roche, Sanofi and Teva.
DL has participated in speaker bureau for Bayer, Merck, Almirall, Execemed, TEVA, Roche, Novartis, Biogen, Sanofi; has had consultancy from Novartis, Bayer, Merck, Biogen, TEVA, Sanofi; has had research grants from Bayer, Merck, Novartis, Biogen. All are paid into DL's institution.
The other authors have nothing to disclose regarding this study.
Sousa, C., Rigueiro-Neves, M., Miranda, T. et al. Validation of the brief international cognitive assessment for multiple sclerosis (BICAMS) in the Portuguese population with multiple sclerosis. BMC Neurol 18, 172 (2018). https://0-doi-org.brum.beds.ac.uk/10.1186/s12883-018-1175-4
BICAMS
Normative values for Portugal
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The formation of bilirubin and p-nitrophenyl glucuronides by rat liver
Biochemical Journal 99(2):507-12
DOI:10.1042/bj0990507
Geraldine A. Tomlinson
Sumner J. Yaffe
1. Glucuronide formation of bilirubin and p-nitrophenol in vitro with excess of UDP-glucuronic acid by UDP-glucuronyltransferase from livers of young and adult rabbits was studied. 2. The development of UDP-glucuronyltransferase for the two substrates followed a markedly different pattern during maturation of young rabbits, p-nitrophenol-conjugation ability being much higher at birth than that for bilirubin. 3. Mg(2+) increased bilirubin conjugation, but inhibited p-nitrophenyl glucuronide formation. 4. p-Nitrophenol acted as a potent non-competitive inhibitor for bilirubin conjugation but bilirubin did not affect p-nitrophenyl glucuronidation. 5. The enzyme for bilirubin conjugation was inactivated at pH9 during treatment with snake venom, whereas in the same preparation the activity of the corresponding enzyme for p-nitrophenol was enhanced. In addition, some solubilization of the latter enzyme could be achieved by this method. 6. The possibility of the existence of more than one enzyme system for the formation of O-glucuronides is discussed.
... Because of its clinical implications the enzyme activity with bilirubin as substrate has attracted particular attention. Initial reports (Isselbacher, Chrabas & Quinn, 1962;Tomlinson & Yaffe, 1966) indicated that attempts to solubilize liver microsomes by a variety of procedures resulted in partial or complete loss of bilirubin glucuronyltransferase activity. Halac & Reff (1967) reported, however, that a relatively stable, solubilized enzyme activity was obtained by dialysis of rat liver microsomes against alkaline EDTA followed by treatment with deoxycholate. ...
... However, glucuronyltransferase activity was inhibited by the greater affinity of ATP for Mg2+ than Ca2+ concentrations of bilirubin above about 0.3mM. suggests that there is another factor too, possibly This has also been observed with rabbit liver the relative affinities of the enzyme for the two microsomes (Tomlinson & Yaffe, 1966). cations. ...
The effect of steroids and nucleotides on solubilized bilirubin uridine diphosphate glucuronyltransferase
B.P.F. Adlard
G H LATHE
1. It was confirmed that bilirubin glucuronyltransferase can be obtained in solubilized form from rat liver microsomes. 2. Michaelis-Menten kinetics were not followed by the enzyme with bilirubin as substrate when the bilirubin/albumin ratio was varied. High concentrations of bilirubin were inhibitory. 3. The K(m) for UDP-glucuronic acid at the optimum bilirubin concentration was 0.46mm. 4. Low concentrations of Ca(2+) were inhibitory in the absence of Mg(2+) but stimulatory in its presence; the converse applied for EDTA. 5. UDP-N-acetylglucosamine and UDP-glucose enhanced conjugation by untreated, but not by solubilized microsomes. 6. The apparent 9.5-fold increase in activity after solubilization was probably due to the absence of UDP-glucuronic acid pyrophosphatase activity in the solubilized preparation. 7. The activation of solubilized enzyme activity by ATP was considered to be a result of chelation of inhibitory metal ions. 8. The solubilized enzyme activity was inhibited by UMP and UDP. The effect of UMP was not competitive with respect to UDP-glucuronic acid. 9. A number of steroids inhibited the solubilized enzyme activity. The competitive effects of stilboestrol, oestrone sulphate and 3beta-hydroxyandrost-5-en-17-one, with respect to UDP-glucuronic acid, may be explained on an allosteric basis.
... This leads to the phenomenon of latency of UGT activity in preparations of ER (i.e., microsomes). The phenomenon of latency derives from experimental observations that in isolated microsomal preparations, UGTs generally show increased activities after the membranes are physically or chemically disrupted [20][21][22][23][24][25][26][27][28]. Other enzymes present in the lumen of the ER, such as glucose-6-phosphatase, also display latency [29]. ...
Revisiting the Latency of Uridine Diphosphate-Glucuronosyltransferases (UGTs)-How Does the Endoplasmic Reticulum Membrane Influence Their Function?
Yuejian Liu
Michael W H Coughtrie
Uridine diphosphate-glucuronosyltransferases (UGTs) are phase 2 conjugation enzymes mainly located in the endoplasmic reticulum (ER) of the liver and many other tissues, and can be recovered in artificial ER membrane preparations (microsomes). They catalyze glucuronidation reactions in various aglycone substrates, contributing significantly to the body's chemical defense mechanism. There has been controversy over the last 50 years in the UGT field with respect to the explanation for the phenomenon of latency: full UGT activity revealed by chemical or physical disruption of the microsomal membrane. Because latency can lead to inaccurate measurements of UGT activity in vitro, and subsequent underprediction of drug clearance in vivo, it is important to understand the mechanisms behind this phenomenon. Three major hypotheses have been advanced to explain UGT latency: compartmentation, conformation, and adenine nucleotide inhibition. In this review, we discuss the evidence behind each hypothesis in depth, and suggest some additional studies that may reveal more information on this intriguing phenomenon.
... occurs at different rates. These rate differences are consistent with the existence of either separate transferase enzymes or a single enzyme with different substrate affinities [1,4,13,14,33]. Thus, the homozygous jaundiced Gunn rat conjugates p-nhrophenol normally but has only a partial capacity to conjugate o-aminophenol and is unable to conjugate bilirubin [6,28,34]. Consequently, measurements of UDP-glucuronyltransferase activity for bilirubin should employ bilirubin as the substrate in an assay system that is optimal for conjugation of that substrate and permits accurate determination of the conjugated products. ...
Bilirubin Uridine Diphospho-glucuronyltransferase in Rat Liver Microsomes: Genetic Variation and Maturation
LEON STREBEL
Gerard B. Odell
Extract: Optimal conditions for the in vitro assay of bilirubin uridine diphospho- (UDP) glu-curonyltransferase activity in rat liver microsomes are described. Solvent partitioning was used to separate the conjugated from nonconjugated bilirubin, thus avoiding dependency on the rate of coupling with diazotized sulfanilic acid for the distinction between bilirubin and its conjugated form. The inclusion of uridine diphospho-N-ace-tylglucosamine (UDPNAG) in the reaction mixture permitted the rate of conjugation of bilirubin by fresh rat liver homogenates and microsomes to occur at greater saturation of the available enzyme with the substrates bilirubin and UDP-glucuronic acid. Liver microsomes, isolated in 0.15 M KC1, increased their activity for bilirubin conjugation and decreased their dependency on UDPNAG during the first 10 days of storage at — 15°. Chromatographic separation of the azo pigments of the conjugated bilirubin gave evidence to suggest that bilirubin monoglucuronide was the initial product and bilirubin diglucuronide appeared in increasing amounts in more prolonged incubations. These results suggested that bilirubin monoglucuronide can be intermediate to the formation of bilirubin diglucuronide. Bilirubin UDP-glucuronyltrans-ferase activity in hepatic microsomes of adult homozygous Gunn rats was not demonstrable. In microsomes of heterozygous Gunn rats and normal Wistar and Sprague-Dawley rats bilirubin UDP-glucuronyltransferase activity was found to be 31.0 and 58.0 μg bilirubin conjugated/mg microsomal N/30 min, respectively. Measurements in developing rats indicated that the maturation in enzyme activity occurred by at least two distinct means: increase of specific activity of the microsomes, and an increase in the content of microsomes per gram of liver (Table IV).
Regulation of Microsomal Enzymes by Phospholipids
Donald A. Vessey
David Zakim
Treatment of bovine liver microsomes with a partially purified preparation of phospholipase A from Naja naja venom leads to activation of UDP-glucuronyltransferase with p-nitrophenol as glucuronyl acceptor. This stimulation of activity is due to a 6-fold increase in activity at Vmax. As activity at Vmax increases, there is a progressive decrease in binding affinity of the enzyme for both substrates, and although the enzyme remains stable at 23°, it becomes unstable at 37°. This unstable form of UDP-glucuronyltransferase decays to another stable form with a maximum activity 2.5-fold greater than that of untreated enzyme. As with phospholipase A, treatment with phospholipase C also activates UDP-glucuronyltransferase, but to a lesser extent. In addition to phospholipases other agents which can alter microsomal lipids also activate UDP-glucuronyltransferase. Triton X-100, sonication, and exposure to pH 9.8 increased activity at Vmax and had variable effects on the binding constants for UDP-glucuronic acid and p-nitrophenol. Maximal activation by these treatments was less than that obtained with phospholipase A; no two treatments had similar effects on all kinetic parameters of the enzyme. Nevertheless, additive effects could not be demonstrated. Although Triton stimulated glucuronidation of p-nitrophenol by the enzyme, it was without effect on the reverse reaction. This fact plus the other data indicate that the activation of UDP-glucuronyltransferase in these experiments cannot be attributed to compartmentation of the enzyme but is due to phospholipid-induced alterations of enzyme conformation.
Bilirubin Metabolism
Thomas R. C. Sisson
It was pointed out by With(122) in his exhaustive monograph on bile pigments that we are indebted to the work of Hans Fischer and his colleagues for much of our knowledge of the bile pigments. Their metabolism, principally that of bilirubin IXa, is of importance to the pediatrician because hyperbilirubinemia is singular among the common symptoms appearing in the newborn in that it in itself, regardless of cause, can produce severe and irrevocable damage to the developing CNS.
UDP-Glucuronyltransferase
Jörg Frei
Ella Schmid
Heidi Birchmeier
This chapter discusses UDP-glucuronyltransferase (GT). Important biological compounds, such as bilirubin, steroid hormones, thyroid hormones, phenols produced by the intestinal flora, drugs, and other exogenous substances or their metabolic products, can be conjugated in the organism by GT to glucuronides and excreted in this form in the bile and urine. GT is not present in muscle, spleen, and leucocyte. To determine the activity of this enzyme in very small amounts of tissue, such as that obtained from needle biopsy material, only microanalytical, chemical, or radiochemical methods are suitable. UDPGA solution is stable for only a few days, even in the deep freeze. In duplicate experiments, the accuracy is better than 5%; no statistical evaluation is yet available. The GT activity in human liver is significantly decreased in Gilbert's disease, in cases of Crigler–Najjar disease, in viral hepatitis, and in drug-induced hepatitis, while the activity is only moderately decreased in cirrhosis and steatosis.
Hepatic Microsomal Drug Metabolism in the Perinatal Period
James R. Fouts
Drug metabolism refers to any chemical change which a drug undergoes during its sojourn in the body. These chemical changes can involve oxidation, reduction, cleavage (most often hydrolysis) and conjugation (with glucuronic acid, etc.). These chemical changes can have quite different effects on the action of any given drug, and the point I wish to emphasize is that drug metabolism and detoxication are not synonymous. Drug metabolism can cause little or no change in drug action, increase in drug action, or decrease in drug action. Besides these quantitative considerations, metabolism can affect drug action qualitatively—making a stimulant of the central nervous system from an inactive parent molecule or a molecule having a CNS depressant action.
Hereditäre hepatische Hyperbilirubinämien
M. Vest
Wie der Titel angibt, sollen diejenigen Hyperbilirubinämien behandelt werden, bei denen, auf Grund eines familiären Vorkommens und anderer Kriterien, eine vererbte Störung angenommen werden kann. Außerdem muß die Störung in der Leber lokalisiert sein. Eine Abgrenzung ist heute noch schwierig, weil es erst bei wenigen der in Frage kommenden Syndrome gelungen ist, die eigentliche Ursache mit Sicherheit aufzuklären, z. B. den Defekt des Enzyms Glucuronyltransferase beim Crigler-Najjar-Syndrom. Bei den meisten anderen hereditären hepatischen Ikterusformen ist man nicht in der Lage, etwas Sicheres über die Pathogenese, geschweige denn über die Ursache der Störung auszusagen. Dadurch ist es nicht möglich zu entscheiden, ob klinisch ähnliche, aber nicht ganz identische Krankheitsbilder durch einen einheitlichen Defekt bedingt sind, oder ob sie auf unter-schiedlichen pathogenetischen Mechanismen beruhen (z. B. Dubin-Johnson- und Rotor-Syndrom).
Effects of Drugs on Bilirubin Metabolism
H. L. Rayner
Brent Schacter
L G Israels
Type I: A hereditary disorder of bilirubin metabolism in which bilirubin glucuronyl transferase activity is lacking abd the bile is free of bilirubin glucuronide. Severe unconjugated hyperbilirubinemia usually leads to kernicterus in infancy.
The Biosynthesis of Glucuronides
G.J. Dutton
Glucuronide-forming enzymes
Handbook Exp Pharmacol
G. Dutton
Perinatal Bilirubin Metabolism
Adv Pediatr
M. Michael Thaler
The clearance of bilirubin in the newborn infant presents special problems. Bilirubin is produced more actively than at later ages; maternal inhibitors (presumably progestational hormones) may interfere with hepatic bilirubin metabolism, and deconjugation in the intestine may allow reabsorption of unconjugated pigment. In addition, deficiencies in hepatic uptake, conjugation and excretion may contribute to neonatal hyperbilirubinemia. Hemolytic conditions accelerate the accumulation of bilirubin after birth. Despite the frequent coincidence of these factors in the newborn period, severe hyperbilirubinemia is relatively rare in normal newborn infants. Hepatic processes which are normally inoperative or peripheral in adults may minimize the tendency to bilirubin retention in the immature organism. Alternate pathways of heme catabolism may produce pyrrolic compounds which are less toxic or easier to excrete than bilirubin. During the first few days after birth hepatic glucuronide stores may be conserved by conjugation of bilirubin to its monoglucuronide rather than diglucuronide derivative, or by formation of nonglucuronide conjugates. The development and regulation of bilirubin metabolism has been investigated in the newborn rat. The enzyme system responsible for bilirubin production is fully developed in utero, whereas conjugating activity is poorly developed. After birth heme catabolism is stimulated by heme and by hormones responsive to hypoglycemia, i.e., glucagon and epinephrine. Thus, hemolysis and starvation enhance the capacity for bilirubin production. In contrast, the postnatal development of conjugating activity is retarded by hemolysis and starvation. In addition to hemolysis, hypoglycemia is frequently present in erythroblastosis fetalis. Hypoglycemia may develop in infants of diabetic mothers and in babies whose feedings are delayed after birth. Under these circumstances bilirubin accumulates rapidly in newborn human infants, compared with adults. The metabolic relationships observed in newborn rats may clarify the factors responsible for severe neonatal hyperbilirubinemia in these situations.
The Effects of Lipid-Protein Interactions on the Kinetic Parameters of Microsomal UDP-Glucuronyltransferase
The study of glucuronidation reactions began 100 years ago with the isolation of a conjugate of o-nitrotoluene from the urine of dogs. Acid hydrolysis of this conjugate yielded an acidic sugar as one of the products. An identical sugar was obtained on acid hydrolysis of a conjugate of chloral hydrate exreted in human urine. The formula of the reducing sugar was given correctly as $$ {\left( {{\rm{CHOH}}} \right)_4}\left\{ {\begin{array}{*{20}{c}} {{\rm{CHO}}}\\ {{\rm{COOH}}} \end{array}} \right. $$by Schmiedeberg and Meyer in 1879 (Smith and Williams, 1970). A wide variety of organic compounds are now known to be conjugated with glucuronic acid. Predominant among these are aromatic molecules containing phenolic and carboxylic groups which form O-ether and O-ester glucuronides, respectively. Several aromatic thiols are glucuronidated, and N-glucuronides also have been reported (Dutton, 1966). The donor of the glucuronic acid moiety in these reactions was identified as UDP-glucuronic acid (Dutton and Storey, 1953; Dutton, 1966).
Postnatal developmental changes in hepatic bilirubin UDP glucuronyl transferase. Studies on the solubilized enzyme
Joseph Krasner
Mont R. Juchau
S J Yaffe
Activity of bilirubin glucuronyl transferase in mouse liver homogenates is low at birth, reaches a peak at 14 days of age, 2 to 3 times greater than adult values. Kinetic studies showed marked differences between microsomes derived from adult and 14 day old animals. Michaelis Menten constants [K(m)] were 5.9 x 10-5 M and 4 x 10-4 M respectively at these 2 ages. K(m) values increased from 5.9 x 10-5 M to 6.7 x 10-4 M when enzymes derived from adult animals were solubilized with deoxycholate. In contrast, solubilization did not modify the K(m) values in preparations from 14 day old mice. The newborn K(m) value for unsolubilized enzyme was 8.8 x 10-5 M and 6.7 x 10-4 M for the solubilized preparation. Pretreatment of pregnant animals with barbiturates to induce enzymic activity was necessary for the kinetic studies in the newborn. These findings indicate that different forms of the enzyme exist at the several stages of development studied. Alterations produced by solubilization procedures suggest a common structural form which undergoes alteration during maturation.
Reduction of serum-unconjugated-bilirubin with phenobarbitone in adult congenital non-haemolytic unconjugated hyperbilirubinemia
MaryJeanne Kreek
MarvinH. Sleisenger
Two adult siblings with congenital non-hæmolytic unconjugated hyperbilirubinæmia and glucuronyl-transferase deficiency were treated with sodium phenobarbitone. Bilirubin levels began to fall 12-36 hours after therapy was begun. The brother was treated for eleven days with concomitant, striking reduction in bilirubin level and with a return to former levels after therapy was discontinued. The sister was maintained on therapy with normal bilirubin levels for 5 months; bilirubin levels slowly rose to pretreatment levels during a 5-month tapering of phenobarbitone, but again fell when the dose of phenobarbitone was increased. The mechanism of this phenobarbitone effect upon bilirubin metabolism is unknown. The most intriguing hypothesis is that glucuronyl transferase is enhanced; a second possibility is augmentation of pre-existing alternative pathways of bilirubin metabolism. Changes in sulphobromophthalein-sodium metabolism were also noted during phenobarbitone therapy, possibly indicating a drug effect on the hepatic excretory transport system.
Phenobarbital-induced protection against toxicity of toluene and benzene in the rat
Masayuki Ikeda
Hatsue Ohtsuji
Treatment of rats with phenobarbital stimulates the in vitro activities of hepatic side-chain hydroxylase and aromatic hydroxylase. This effect is paralleled by enhanced in vivo metabolism of toluene and benzene and results in increased tolerance of the rats to the narcotic action of toluene and the leukcopenic action of benzene. No comparable effect of the pretreatment is observed on the rates of oxidation of aromatic alcohol to the corresponding acid, phenolic sulfation, phenolic glucuronidation, or glycine conjugation of benzoic acid. Rapid disappearance of toluene from blood due to enhanced hepatic metabolism together with reduced sensitivity of central nervous system offer an explanation for the shortened sleeping time after toluene injection. The protective effect of phenobarbital against the leukopenic action of benezene is discussed in relation to the etiology of benzene intoxication.
Studies on the activation in vitro of glucuronyltranferase
Biochim Biophys Acta Enzymol
Arnt Winsnes
1.1. The relationship between the different activating principles acting of glucuronyltransferase (UDP-glucuronate glucuronyltransferase EC 2.4.1.17) in vitro has been studied using five acceptor substrates (bilirium o-aminophenol, 4-methylumbelliferone, p-nitrophenol and phenolphthalein) in assay of enzyme activity2.2. Preincubation of mouse and rat-liver suspensions in vitro resulted in an increased activity of glucuronyltransferase which was highly variable depending on the acceptor substrate used. While 4-methylumbelliferone glucuronyltransferase exhibited an activity that was 45 times the initial one at maximum, the o-amninophenol enzyme was not activated by this procedure3.3. Detergents (Triton X-roo and digitonin) and UDP-N-acetylglucosamine were also found to activate glucuronyltransferase; only bilirubin glucuronyltransferase was not activated by UDP-N-acetylglucosamine4.4. Activation of glucuronyltransferase could not be increased by combining preincubation, detergents or UDP-N-acetylglucosamine, whereas activation of ratliver o-aminophenol gluconyltransferase by detergents and UDP-N-acetylglucosamine was strongly potentiated by diethylnitrosamine5.5. The slightly different kinetic of nonactivated and activated enzyme that were found at varying substrate (UDP-glucuronic acid, p-nitrophenol concentrations could not explain the activation observed6.6. The results are compatible with an activation by preincubation, detergents and UDP-N-acetylglucosamine due to exposition of active sites of glucuronyltransferases that have been nonfunctioning in unactivated tissue homogenates7.7. The degree of activation at constant detergent concentration was the same over a wide range of enzyme-protein concentrations8.8. The pH optimum for detergent-activated p-nitrophenol glucuronyltransferase was found at 6.2 6.6 while the activity towards bilirubin and o-aminophenol was maximum at pH 7.6. Detergent-activated 4-methylumbelliferone and phenolphthalein glucuronyltransferases, on the other hand, revealed about the same activity in the range between pH 6.2 and 7.6
Bilirubin Glucuronide Formation in Developing Guinea Pig Liver
H Flodgaard
R Brodersen
Flodgaard, H. J. & Brodersen, R. Bilirubin Glucuronide Formation in Developing Guinea Pig Liver. Scand. J. din. Lab. Invest. 19, 149–155, 1967. Formation of bilirubin diglucuronide by guinea pig liver tissue slices and microsomes was studied, using a radioisotope derivative method for the quantitative determinations, at various stages of perinatal development. O-aminophenol inhibits competitively the conjugation of bilirubin and vice versa. From the Km and Ki values it is concluded that bilirubin and OAP are conjugated by two different enzymes. The liver slices rapidly lose UDPGA to the medium, and the rate of bilirubin conjugation decreases. From the course of these processes an approximate value of the Michaelis constant for UDPGA was calculated. The possible role of the low UDPGA content of the liver of the foetus and newborn in the pathogenesis of neonatal jaundice is discussed.
Solubilization of udp glucuronosyltransferase by digitonin from trypsin-digested rat liver microsomes
Int J Biochem
Raija Puukka
Mariitta Laaksonen
1.1. Solubilization of UDP glucuronosyltransferase (pi-nitrophenol) from rat liver was achieved by digesting microsomes first with trypsin and then with digitonin. Sixty to sixty-five per cent of the enzyme activity was recovered in the solubilized fraction obtained by centrifugation at 105,000 g for 60 minutes. The remaining activity was resedimented.2.2. The frozen solubilized preparation maintained 60–70 % of its activity after two weeks.3.3. The enzyme was further purified by the elution of the solubilized fraction from a column of Sepharose 6B. After gel filtration the increase in the specific activity was 40–fold in the best preparations. This increase in the specific activity was, however, partially due to the activation of the latent enzyme during trypsin-digitonin treatments.4.4. The partially-purified preparation was shown to contain other proteins and membrane phospholipids, which suggests that UDP glucuronosyltransferase may still be part of a complex structure derived from the microsomal membrane.
Nitro and Nitroso Groups: Volume 2 (1970)
Jan Venulet
Robert L. Van Etten
IntroductionBiological Oxidation–Reduction Processes and Oxidative PhosphorylationBiochemistry and Pharmacology of Naturally Occurring Compounds Containing the Nitro or Nitroso GroupBiochemistry and Pharmacology of Synthetic Compounds Containing the Nitro or Nitroso GroupThe Role of Nitro and Nitroso Compounds in the Formation of MethemoglobinAddendum and Final RemarksReferences
VARIATIONS IN DETOXICATION ENZYMES DURING MAMMALIAN DEVELOPMENT*
ANN NY ACAD SCI
Charlotte S. Catz
A Study on the Enzymatic Mechanism of Guinea‐pig Hepatic‐Microsomal Bilirubin Glucuronyl Transferase
Eur J Biochem
James L. Spratt
Robert F. Potrepka
1Initial velocity studies for the two-substrate system of bilirubin glucuronyl transferase were performed with Triton X-100 as an activator. These studies were performed by varying the concentration of bilirubin in the presence of different concentrations of uridine 5′-diphosphate glucuronic acid (UDP-GlcUA) and gave an intersecting pattern (characteristic of a sequential mechanism) when the data were graphed as double-reciprocal plots.2Competitive inhibition was observed for UDP-GlcUA and noncompetitive inhibition for bilirubin when UDP (product) or UMP (dead end) were used as inhibitors. Alternate product inhibition with phenolphthalein glucuronide produced noncompetitive inhibition for both UDP-GlcUA and bilirubin.3It was concluded that the reaction under study probably has an ordered mechanism although the iso-Theorell-Chance mechanism is still a possibility.4Kinetic constants (mean for 3 animals) for a sequential mechanism yielded 52.3 μM as the apparent Km for bilirubin, 2.3 mM as the apparent Km for UDP-GlcUA and 20.2 nmol bilirubin conjugated per mg of microsomal protein per 30 min as the V for this reaction.
Competition between p‐Aminophenol, p‐Nitrophenol, and Bilirubin for Glucuronidation in Cultures of Rat Hepatoma Cells and Homogenates of the same Cells
Pharmacol Toxicol
Hans Erik Rugstad
Erik Dybing
Glucuronidation of p-aminophenol (PAP), p-nitrophenol (PNP), and bilirubin has been studied in cultures of a clonal cell line with liver-like functions and homogenates from the same cells. The purpose was to study the effect of simultaneous addition of substrates to cultures of living cells and to homogenates with an excess of UDPGA, and to compare the glucuronidation rates in living cells and in homogenates from the same cells, fortified with UDPGA. The rates of glucuronidation of PAP, PNP, and bilirubin by cells in culture were about 50, 70, and 15 nmol/mg protein/hour respectively. In fortified cell homogenates the respective glucuronidation rates were about 500, 1800, and 15 nmol/mg protein/hour. PNP inhibited the glucuronidation of PAP and bilirubin both in cell cultures and cell homogenates. PAP inhibited the glucuronidation of bilirubin in cell cultures. Bilirubin did not inhibit the glucuronidation of the other substrates. PNP glucuronide did not inhibit the glucuronidation of bilirubin or PAP in cell cultures or homogenates.
Steigerung der Glucuronidierungsrate in der Neugeborenenperiode durch therapeutische Phenobarbitaldosen
Windorfer, A., Jr
Neugeborene Kaninchen erhielten Phenobarbital oral in einer Konzentration von 5–7 mg/kg; durch mehrtgige Verabfolgung lie sich eine Aktivittssteigerung der Bilirubin-Glucuronyltransferase erreichen. Eine deutlichere Aktivittszunahme sahen wir nach zustzlicher Phenobarbitalgabe an die Muttertiere einige Tage vor der Geburt. Bei menschlichen Neugeborenen prften wir die Glucuronidierungsfhigkeit fr Paracetamol mit und ohne Phenobarbitalvorbehandlung. Nach mehrmaliger Gabe von 3–5 mg/kg Phenobarbital erfolgte ein Anstieg der Glucuronidierungsrate fr Paracetamol. Es wre vorstellbar, da sich bei menschlichen Neugeborenen auch die Bilirubin-Glucuronyltransferase, wie im Tierversuch, durch therapeutische Phenobarbitaldosen in ihrer Aktivitt vermehren lt.Newborn rabbits were given Phenobarbital in doses of 5–7 mg/kg. Application over several days induced activation of glucuronyl transferase. A further increase in activation could be achieved by treatment of the mother animals for several days before birth. In human newborns we tested the glucuronidation of Paracetamol with and without induction by Phenobarbital. After several doses of 3–5 mg/kg Phenobarbital an increased rate of glucuronidation of Paracetamol was seen. It may be assumed that the glucuronyl transferase for bilirubin can be induced by therapeutic doses of Phenobarbital in human newborn infants in a similar manner to that demonstrated in animal experiments.
Differentiation of homologous forms of hepatic microsomal UDP-glucuronyltransferase. II. Characterization of the bilirubin conjugating form
Jovita Goldenberg
The glucuronidation of bilirubin by UDPglucuronyltransferase (EC 2.4.1.17) was investigated using a kinetic assay. The role of albumin in the assay was studied at concentrations of bilirubin both above and below its limit of solubility. In assays saturated with respect to bilirubin, albumin was almost without affect on initial rates. At concentrations of bilirubin below its limit of solubility, albumin was inhibitory. Thus, the bilirubin-albumin complex is not a substrate for the enzyme. In assays containing saturating concentrations of bilirubin, the rate of glucuronidation was influenced by conditions which "salt in" bilirubin. Using a bisubstrate kinetic analysis, the dissociation constant for the enzyme bilirubin complex was determined to be 12–18 μM. The enzyme was shown to be stimulated by the allosteric effector UDP-N-acetylglucosamine which caused an increase in the apparent affinity of the enzyme for UPDglucuronic acid. The bilirubin conjugating form of UDPglucuronyl-transferase showed the same general behavior toward sulfhydryl reagents and perturbers of the lipid environment as previously studied forms. However, because of differences in the response of the rate of conjugation of bilirubin to metals, UDP-N-acetylglucosamine, and sulfhydryl reagents, it has been concluded that the bilirubin conjugating enzyme differs from the p-nitrophenol, o-aminophenol and o-aminobenzoate forms of UDPglucuronyltransferase.
Partial purification of hepatic UDP-glucuronyltransferase: Studies of some of its properties
A.P. Mowat
Irwin M Arias
1.1. Purification of UDP-glucuronyltransferase (UDP-glucuronate glucuronyltransferase, EC 2.4.1.17) from guinea pig liver, achieved by ultrasonication, sucrose gradient fractionation and Sephadex G-200 elution, increased the capacity to conjugate bilirubin in vitro 20-fold and 50-fold. The stability of enzymatic activity in this preparation was greater than previously described for this enzyme in preparations with a comparable increase in initial activity and was strikingly similar with both substrates. The purification procedure did not separate conjugating ability for any of the substrates tested.2.2. The inadequacy of gravitational criteria alone in indicating solubilization of a membrane-bound enzyme was demonstrated by electron microscopy. Methods used unsuccessfully in attempting true solubilization are listed.3.3. The purified enzyme preparation was shown to contain a number of proteins including microsomal enzymes and lipids. The possible effect of these on kinetic studies is critically discussed.4.4. Application of the purification procedure to livers of cat, Gunn rat and hypophysectomized or thyroidectomized rats also yielded a stable purified enzyme preparation, without separating conjugating ability for the substrates tested.5.5. The results of these investigations are discussed and their relationship to possible multiplicity of UDP-glucuronyltransferase considered.
Bilirubin and the heterogeneity of microsomal uridine diphosphate glucoronyltransferase from rat liver
G.J. Mulder
1.1. The glucuronide conjugation of bilirubin by rat liver microsoma UDP-glucuronyltransferase (UDPglucuronate glucuronyltransferase (acceptor unspecific), EC 2.4.1.17) was studied; the enzyme preparation was activated by addition of Triton X-100.2.2. when bilirubin was dissolved in NaOH solution or solubilized in an albumin solution, equal initial velocities could be obtained at pH 7.3, but much higher concentrations of albumin-solubilized bilirubin were required for this.3.3. Bilirubin glucuronide conjugation could be inhibited by other substrates such as , 4-methylumbelliferone, phenolphthalein and .4.4. glucuronide conjugation was inhibited by very low concentrations of bilirubin (10 μM) only if bilirubin was added in the absence of albumin. With increasing concentrations of albumin this inhibition disappeared. Increasing concentrations of microsomal enzyme protein also reversed the inhibitory effect of bilirubin. The inhibition appeared to be competitive.5.5. Contrary to most earlier findings, the present results suggest that bilirubin and are conjugated at the same active site.
Partial purification and kinetics of oestriol 16α-glucuronyltransferase from the cytosol fraction of human liver
Govind S. Rao
Marie Luise Rao
H Breuer
An enzyme that conjugates the 16alpha-hydroxyl group of oestriol with glucuronic acid was found in the cytosol fraction of human liver. The enzymic activity could not be sedimented when the cytosol fraction was centrifuged at 158000g(av.) for 120min. The oestriol 16alpha-glucuronyltransferase was purified 100-fold by 0-30% saturation of the cytosol fraction with ammonium sulphate followed by filtration of the precipitate through Sephadex G-200. The activity was eluted at the void volume. The product of the reaction, oestriol 16alpha-monoglucuronide, was identified by paper chromatography and by crystallization of radioactive product to constant specific radioactivity. The optimum temperature was 37 degrees C, and the activation energy was calculated to be 11.1kcal/mol. The apparent Michaelis-Menten constants for oestriol and UDP-glucuronic acid were 13.3 and 100mum respectively. Cu(2+), Zn(2+) and Hg(2+) inhibited, whereas Mg(2+), Mn(2+) and Fe(2+) stimulated the enzyme. Substrate-specificity studies indicated that the amount of oestradiol-17beta, oestradiol-17alpha and oestrone conjugated was not more than about 5% of that found for oestriol. Oestriol 16alpha-monoglucuronide, a product of the reaction, did not inhibit the 16alpha-oestriol glucuronyltransferase; in contrast, UDP, another product of the reaction, inhibited the enzyme competitively with respect to UDP-glucuronic acid as the substrate, and non-competitively with respect to oestriol as the substrate. ATP and UDP-N-acetylglucosamine did not affect the oestriol 16alpha-glucuronyltransferase. 17-Epioestriol acted as a competitive inhibitor and 16-epioestriol as a non-competitive inhibitor of the glucuronidation of oestriol. 5alpha-Pregnane-3alpha,20alpha-diol also inhibited the enzyme non-competitively. It is most likely that the oestriol 16alpha-glucuronyltransferase described here is bound to the membranes of the endoplasmic reticulum.
Salycylamide glucuronide formation in liver disease and its change by drugs
Gastroenterol Jpn
Yukihiko Adachi
G. Wakisaka
Tatsuo Yamamoto
Salicylamide glucuronide (SAMG) in 0-6 and 6-12 hours-urine specimens was determined after oral administration of salicylamide in 7 normal volunteers (NV), in 51 cases of various liver diseases and hyperbilirubinemias, and in 19 cases after drug administration, to predict the in vivo drug metabolism in man and its change by drugs. Maximal glucuronide formation was obtained by 1.0 g of salicylamide administered to NV; thus, this dosage was used in the present study. SAMG as percent of total salicylamide, the percent of SAMG, from 0-6 hours-urine specimens was high and constant in NV (71.3 +/- 8.3 (Mean +/- S.D.)). 0-0.08% of the total salicylamide was confirmed as free salicylamide in 0-12 hours-urine specimens of NV. The percent of SAMG of 0-6 hours-urine specimens was 57.2 +/- 8.6 in acute hepatitis, 66.6 +/- 10.9 in chronic hepatitis, and 48.6 +/- 10.7 in liver cirrhosis (mean +/- S.D.). Free salicylamide increased slightly in liver diseases. Serum bilirubin levels tended to be inversely correlated with the percent of SAMG. In most cases of Gilbert's syndrome, the percent of SAMG remained at a normal level. The percent of SAMG in cases with unconjugated hyperbilirubinemias of other geneses were almost within normal limits. Bucolome and phenobarbital increased the percent of SAMG in patients with various liver diseases. After rifampicin or phenytoin administration, the percent of SAMG of the patients with lung tuberculosis or epilepsy did not surpass that of NV.
Perinatal Development of Bilirubin UDP-Glycosyltransferase Activities in Rat Liver
C De Wolf-Peeters
Ronald de vos
J Fevery
K. P. M. Heirwegh
Bilirubin UDP-glucuronyltransferase and UDP-xylosyltransferase activity could already be demonstrated in rat liver from day 19 of fetal life onwards (4 days before birth). Bilirubin-glucuronide was present in hepatocytes of 21-day-old fetal liver strongly suggesting that the enzyme detected in vitro was really active in vivo. This further supports the theory that secretion from the cell is also immature. The known deficiency of UDP-glucose dehydrogenase in fetal and neonatal rat liver could lead to decreased intracellular concentrations of UDP-glucuronic acid and possibly to increased concentrations of UDP-glucose. However, no glucosides were present in the fetal hepatocytes.
Physiologic disposition of [3H]epinephrine in the rabbit fetus. Effect of promethazine
BIOCHEM PHARMACOL
Kathleen T. Shiverick
Samuel Solomon
[3H]epinephrine was injected into fetal rabbits through the uterine wall on day 26 of gestation in order to study placental transfer, and the uptake, retention and metabolism of the hormone in fetal tissues. Less than 0.1 per cent of the administered [3H] was found in maternal blood or within uninjected littermates. Accumulation of [3H]/g of tissue was greatest in the fetal kidney, liver and intestine, and less in the heart and lung. Levels of [3H] in the fetal serum, heart, lung and placenta were comparable in value at 5 and 30 min, but all fell by 60 min. In contrast, the fetal kidney and intestine show a 2- to 4-fold accumulation of [3H] in 30 min as compared to 5 min, while the fetal liver exhibits a 10-fold accumulation over 30 and 60 min. [3H] metabolites present in fetal serum at 30 min, expressed as a per cent of the total radioactivity, are 45 per cent for metanephrine and 27 per cent for O-methylated deaminated products, while 25 per cent is present as unmetabolized [3H]epinephrine. O-methylated metabolites are predominant in the liver, where 85 per cent of the radioactivity comprises metanephrine in conjugated and unconjugated forms. Administration of promethazine to the doe prior to injection of [3H]epinephrine into rabbit fetuses resulted in a 40–64 per cent decrease in accumulation of [3H] in the fetal brain, liver, lung, kidney, intestine and heart. The pattern of metabolites in the serum of promethazine-treated fetuses shows an increase in the per cent of [3H] in the O-methylated, deaminated fraction from 27 to 37 per cent. While the total per cent of O-methylated metabolite is not altered in the liver of treated fetuses, metanephrine is predominantly present in the conjugated form. The liver of the rabbit fetus is an important site for accumulation of catecholamine metabolites and is pharmacologically sensitive to promethazine administered to the mother.
The radiometric assay of microsomal UDP-glucuronyltransferase using 125I-labelled phenolphthalein
CLIN CHIM ACTA
S.E. Aw
A rapid and sensitive method is described for assaying UDP-glucuronyl-transferase (EC 2.4.1.17) activity. The substrate is the relatively apolar phenolphthalein, labelled with 125I on one of its two phenolic rings leaving a free hydroxyl group for glucuronidation. Extraction with ethyl acetate leaves the glucuronide in the aqueous reaction medium which is then counted. Less than 10 mug microsomal protein is required and the glucuronidation of 10 ng 125I-labelled phenolphthalein (0.03 nmoles) is easily detected.
Novobiocin-inhibition and magnesium-interaction of rat liver microsomal bilirubin UDP-glucuronosyltransferase
Philippe Duvaldestin
Jean-Louis Mahu
Anne-Marie Preaux
P Berthelot
Novobiocin inhibited bilirubin UDP-glucuronosyltransferase (EC 2.4.1.17) from rat liver both in vitro and in vivo, in a dose-dependent fashion. This inhibition was immediate, and was fully reversed when novobiocin was removed by dialysis or by ultracentrifugation through 0.6 M sucrose. The inhibition could not be explained by an alteration in the membrane conformation of this enzyme, since the same kinetic changes were observed in digitonin-activated and in non-activated microsomes. Novobiocin exerted a non-competitive inhibition of bilirubin UDP-glucuronosyltransferase with either bilirubin or UDP-glucuronic acid as the substrate. Kinetic studies demonstrated uncompetitive inhibition of novobiocin or bilirubin UDP-glucuronosyltransferase as a function of Mg2+ concentration, whether the assays were EDTA-free or not. Thus, similarities seem to exist between the known effect of novobiocin on membrane-bound enzymes of the bacterial wall and its inhibitory effect on bilirubin UDP-glucuronosyltransferase: both these enzymic systems require metal divalent cations for maximal activity. The uncompetitive inhibition pattern observed with novobiocin with regard to Mg2+ suggests that this antibiotic acts on bilirubin conjugation by affecting Mg2+-enzyme complexes.
Studies on bilirubin and steroid glucuronidation by rat liver microsomes
Martin Jacobson
W Levin
Allan H. Conney
The addition of digitonin or other detergents to liver microsomes from control and phenobarbital-treated rats increased the rate of glucuronidation of bilirubin, testosterone, estradiol, estrone and estriol. The extent of activation by digitonin as well as the concentration of detergent causing maximal activation was substrate dependent. Treatment of rats with sodium phenobarbital (75 mg/kg/day, i.p.) for 3 days significantly increased the glucuronidation of testosterone and estradiol per mg of protein but had no effect on estrone glucuronidation. Administration of 0.1% sodium phenobarbital in the drinking water for 20 days increased bilirubin, testosterone and estrone glucuronidation per mg of protein, but had little or no effect on estradiol and estriol glucuronidation. When UDP-glucuronyl transferase activity was calculated on a per g liver (wet wt) basis, however, an increase in the glucuronidation of estradiol and estriol was seen; this increase could be accounted for by a large stimulatory effect of phenobarbital on the synthesis of total microsomal protein. Liver microsomes from homozygous Gunn rats, which are deficient in the ability to conjugate bilirubin, did conjugate testosterone, estradiol. estrone and estriol, but some substrates were conjugated more rapidly than others. The addition ofdigitonin to liver microsomes from the homozygous Gunn rat increased the glucuronidation of testosterone but not that of estrone. Microsomal steroid glucuronidation in this animal was stimulated to varying degrees by the administration of sodium phenobarbital. The data suggest the presence of several glucuronyl transferases in liver microsomes.
Drug Metabolizing Function of Isolated Perfused Liver
Tsutomu Noda
Kazunori Araki
Masao Koida
Hiroshi Kaneto
How closely the isolated liver of the rat would simulate the in vivo function of the organ in terms of the metabolic pattern of the compounds such as bromosulphophthalein, p-nitrophenol, hexobarbital, and indocyanine green was investigated. In order to produce tissue with the stimulated function, the animal was pretreated with phenobarbital and, for the reverse purpose, with ethionine. Some of the indices of the function employed herein, such as the appearance pattern of the compound, the rate of biochemical transformation or the biliary excretion, showed that the perfused liver would generally well reflect the in vivo situation. The method with isolated and perfused liver could exclude the participation of other organs and also the influence of the factors unavoidable in an in vivo experiment. Thus, it is suggested that the isolated perfused liver is useful for studying directly the functional level of the organ as drug metabolizing tissue.
Regulation of UDP-glucuronosyltransferase by lipid-protein interactions. Comparison of the thermotropic properties of pure reconstituted enzyme with microsomal enzyme
Andrew J. Dannenberg
Michal Rotenberg
The temperature dependence of two kinetic properties of the GT2P isoform of microsomal UDP-glucuronosyltransferase was studied for enzyme in intact microsomes and for pure enzyme reconstituted into different types of lipid bilayers. The properties studied were the non-Michaelis-Menten kinetics of the enzyme and activity at Vmax(app). For enzyme in intact microsomes, the pattern of non-Michaelis-Menten kinetics was seen at all temperatures in the range tested (23 to 48 degrees C), and the slopes of the Hill plots of the data were constant across this range of temperatures. Although non-Michaelis-Menten kinetics were present for pure enzyme in bilayers of 1,2-dimyristoylphosphatidylcholine or 1,2-dipalmitoylphosphatidylcholine only in the gel phase (Hockman, Y., Kelley, M., and Zakim, D. (1983) J. Biol. Chem. 258, 6509-6519), it was not possible to reconstitute this pattern of kinetics for enzyme at T greater than 40 degrees C. For example, GT2P displayed Michaelis-Menten kinetics in bilayers of 1,2-distearoylphosphatidylcholine at 44 degrees C. For enzyme in microsomes, activities at Vmax(app) increased with increasing temperature in the range 23 to 48 degrees C, with a discontinuity in the slope of the Arrhenius plot at 34 degrees C. This thermotropic property also could not be reconstituted with pure GT2P. Instead, activities at Vmax(app) for GT2P reconstituted in 1,2-dioleoylphosphatidylcholine, 1,2-distearoylphosphatidylcholine, or 1,2-stearoyl oleoylphosphatidylcholine increased in the range 23 to 37 degrees C, but then decreased at T greater than 37 degrees C. The fall in activity at T greater than 37 degrees C was reversible, indicating that GT2P undergoes a reversible change at 37 degrees C to a less active form of the enzyme. The differences between pure, reconstituted GT2P and microsomal GT2P indicate that the thermotropic properties of the microsomal enzyme are not properties of the enzyme per se but depend on interactions between it and other components in the microsome. The data suggest, therefore, that the properties of GT2P in microsomes results in part from an organization of components in the microsomal membrane.
Effects of metals on the properties of hepatic microsomal uridine diphosphate glucuronyltransferase
Divalent metal ions have two important effects on the properties of UDP-glucuronyltransferases. They alter the properties of the active sites of these enzymes and modify the effect of UDP-N-acetylglucosamine on their activities. The metal ion responsible for effects on the active site is different from the metal ion which modifies the properties of the allosteric site (UDP-N-acetylglucosamine site); that is, there are at least two metal binding sites in some forms of UDP-glucuronyltransferase. The effects of metal ions also differ, according to the compound used as glucuronyl acceptor. By studying the effects of metal ions on the rates of synthesis of p-nitrophenyl, o-aminobenzoyl, and o-aminophenyl glucuronides, three types of UDP-glucuronyltransferase reactions can be distinguished. In the type I reaction, for which p-nitrophenol serves as the prototype aglycone, divalent metal jons increase activity at Vmax, and are essential for stimulation by UDP-N-acetylglucosamine. In contrast to the type I reaction, divalent metal ions are not essential for stimulation of UDP-glucuronyltransferase by UDP-N-acetylglucosamine with o-aminophenol as aglycone (type Ia reaction). Divalent metal ions, however, increase activity at Vmax for the type Ia reaction. In the type II reaction, for which o-aminobenzoate is the prototype aglycone, metal ions decrease the concentrations of UDP-glucuronic acid needed for half-maximal rates of glucuronidation and are essential for stimulation by UDP-N-acetylglucosamine. Metal ions do not increase activity at Vmax. for the type II reaction. Interactions between metal ions and UDP-N-acetylglucosamine seem to be essential components of the regulatory apparatus of UDP-glucuronyltransferase catalyzed reactions since these interactions ensure rapid rates of glucuronidation under conditions in vivo.
Critique of the Assay and Significance of Bilirubin Conjugation
ADV CLIN CHEM
J.A.T.P. Meuwissen
This chapter describes the methodology and properties of uridine diphosphate (UDP)-glucuronyltransferase and of the related UDP-glycosyltransferase activities. It also delineates the applications to human disease. Bilirubin derives largely from senescent erythrocyte hemoglobin. In most normal adult animals, conjugation of bilirubin with various sugars represents the major mechanism for its transformation. The reaction products are eliminated from the organism by excretion into the bile. Glucuronyl transfer is of predominant importance in the biotransformation of bilirubin in man and rat. Glucuronyl transference to bilirubin and to many other acceptor substrates (G5, L14, M14, P4, W4), with the exception of phenolphthalein (G5), is higher in the rough endoplasmic reticulum than it is in the smooth endoplasmic reticulum. The main field of application of the in vitro assays of bilirubin UDP glycosyltransferase activities is the evaluation of stimulation or inhibition—by drugs or endogenous substances—of the enzyme activities or of their maturation. Development of unconjugated hyper-bilirubinemia in some neonates is related to breast feeding.
Solubilization and characterization of hepatic bilirubin UDP-glucuronyltransferase
Biochim Biophys Acta
Daniel H. Gregory
Richard D. Strickland
The hepatic enzyme, bilirubin UDP glucuronyltransferase (UDPglucuronate glucuronyltransferase, EC 2.4.1.17) has been solubilized in stable form by treating rat liver microsomes with 10% (by weight) dissolved digitonin at pH 7.8. All enzyme activity was associated with the rough membranes of the endoplasmic reticulum as shown by ultracentrifugation of selected density gradients in combination with electron microscopy. Proof of solubilization was determined by analytic ultracentrifugation, gel filtration, and electron microscopy which indicated that the enzyme was an homogeneous protein with a calculated molecular weight in the range of 150 000. Studies to characterize enzyme kinetics and assay requirements were performed and differ from those reported for liver hemogenates.
Activation of hepatic microsomal glucuronyl Transferase by bilirubin
LIFE SCI
Emilio Sanchez
T R Tephly
In rat liver microsomal preparations, bilirubin markedly stimulated the glucuronidation of morphine and p-nitrophenol catalyzed by UDPglucuronyltransferase (UDPGT, EC 2.4.1.17). The activation was not due to contamination of bilirubin with bile acids. At equimolar concentrations, the activating effect of bilirubin was greater than that produced by deoxycholate, a detergent well known as an activator of UDPGT. Other results suggest that bilirubin activation of UDPGT is similar to that produced by detergents. In experiments, the rate of urinary excretion of morphine glucuronide in rats treated with bilirubin was twice that of control animals. These results suggest that bilirubin may be a physiologic activator of UDPGT activity.
UDP-glucuronyltransferase activity towards oestriol in fresh and cultured foetal tissues from man and other species
Brian Burchell
1.1. UDP-glucuronyltransferase activity towards oestriol has been demonstrated in liver preparations from foetal and adult human and mouse, and from chick embryo and chick. It was also found in foetal human kidney and gut, but appeared absent from placenta.2.2. In human foetal liver and kidney levels of UDP-glucuronyltransferase activity towards oestriol were as low as those towards o-aminophenol, and developed with age.3.3. No UDP-glucosyltransferase activity towards oestriol was detected in human foetal preparations.4.4. UDP-glucuronyltransferase activity towards oestriol develops precociously in cultures of chick embryo liver; in cultures of human foetal liver the transferase activity towards oestriol declines whilst increasing towards o-aminophenol.5.5. Pre-treatment with phenobarbital increased UDP-glucuronyltransferase activity towards oestriol in chick embryo liver, both in ovo and in culture. Pre-treatment with phenobarbital increased this activity in maternal mouse liver but not in foetal liver, in utero or in culture, nor in cultured human foetal liver.6.6. UDP-glucuronyltransferase activities towards oestriol and towards o-amino phenol differ markedly in development, in culture and in response to phenobarbital pre-treatment and activation procedures; heterogeneity of the enzyme is probably responsible.
Spontaneous and activation of a glucuronyltransferase in vitro
Kira K. Lueders
E L Kuff
The microsomal glucuronyltransferase catalyzing formation of p-nitrophenyl glucuronide in rat, guinea pig, and mouse livers, and transplantable rat hepatomas, was activated spontaneously during storage at 0 ° and by low concentrations (less than 0.05%) of deoxycholate and Triton X-100 added in vitro. Activity in liver microsomes was increased 5- to 10-fold, in hepatoma microsomes 20-fold, and in homogenates 2-to 5-fold. Higher concentrations of detergent were inhibitory to activity in fresh preparations, and preparations which had been fully activated spontaneously were inhibited by all concentrations of detergents. The activation was irreversible and not accompanied by solubilization of the enzyme. The Michaelis constants (for uridine diphosphate glucuronic acid) for detergent- and spontaneously activated liver and hepatoma microsomes appeared to be the same. Attempts to determine the mechanism of activation by kinetic analysis were unsuccessful, but the data were compatible with an increase in the relative site number per milligram microsomal protein during activation.
Biosynthesis of β-glucuronides of retinol and retinoic acid in vivo and in vitro
Kenneth Lippel
James Allen Olson
After the intraportal injection of retinol-6,7-¹⁴C to rats, the 0-ether derivative of retinol, retinyl β-glucosiduronate, appears in the bile. Both retinoyl β-glucuronide and retinyl β-glucosiduronate are also synthesized in vitro when washed rat liver microsomes are incubated with uridine diphosphoglucuronic acid (UDPGA) and either retinoic acid or retinol, respectively. The synthesis of retinoyl β-glucuronide was also demonstrated in microsomes of the kidney and in particulate fractions of the intestinal mucosa. The glucuronides were characterized by their UV absorption spectra, by their quenching of UV light or fluorescence under it, by their thinlayer chromatographic behavior in two solvent systems, and by the identification of products released during their hydrolysis by β-glucuronidase. With retinoic acid as the substrate, the UDP glucuronyl transferase of rat liver microsomes had a pH optimum of 7.0, a temperature optimum of 38°C, and a marked dependence on the concentrations of both retinoic acid and UDPGA, but was unaffected by a number of possible inhibitors, protective agents, and competitive substrates. The conversion of retinal to retinoic acid and the synthesis of retinoyl β-glucuronide from retinoic acid could not be detected in whole homogenates, cell fractions, or outer segments of the bovine retina.
Influence of phenobarbital treatment on and bilirubin glucuronidation in Wistar rat, Gunn rat and cat
Peter Jansen
P.Th. Henderson
The influence of phenobarbital pretreatment on the glucuronidation of and bilirubin has been compared in liver microsomes from Wistar rat, homozygous and heterozygous Gunn rat and cat. Both for Wistar rat and cat it appeared that phenobarbital has a different effect on and bilirubin glucuronidation. No bilirubin glucuronidating activity could be detected in homozygous Gunn rats both before and after exposure to phenobarbital. The glucuronidating capacity of Gunn rats, however, was strongly enhanced after phenobarbital treatment. These results are arguments in favour of the involvement of different enzymes in the glucuronidation of and bilirubin.
Assay and Properties of Digitonin-Activated Billrubin Uridine Diphosphate Glucuronyltransferase From Rat Liver
M Van de Vijver
1. The bilirubin UDP-glucuronyltransferase assay described by Van Roy & Heirwegh (1968) has been improved. 2. Extraction of final azo-derivatives is rendered more simple and efficient by thorough emulsification and by cooling. 3. Pretreatment of homogenates and cell fractions with digitonin increases the sensitivity of the assays and gives less variable results than those with untreated preparations. The activation procedure is flexible. 4. Blank values (obtained from incubation mixtures from which activating bivalent metal ion and UDP-glucuronic acid were omitted) are low. No endogenous conjugate formation could be detected except with untreated, fresh liver homogenates. Control incubation mixtures containing the latter preparations are preferably kept at 0 degrees C. 5. With activated microsomal preparations, rates of breakdown of UDP-glucuronic acid (as monitored by release of P(i)) were low. Little if any increase in enzyme activity was found when UDP-N-acetylglucosamine was included in the incubation mixtures. 6. Slight deviation from Michaelis-Menten kinetics with respect to bilirubin observed at low substrate concentrations is probably related to the use of binding protein in the assay mixtures. Michaelis-Menten kinetics were followed with respect to UDP-glucuronic acid. Part of the enzyme in microsomal preparations from rat liver functioned independently of added bivalent metal ions. Mn(2+) was slightly more, and Ca(2+) somewhat less, stimulatory than Mg(2+). The Mg(2+)-dependent fraction showed Michaelis-Menten kinetics with respect to the added Mg(2+). 7. The enzyme activities found were higher than values reported in the literature for untreated or purified preparations from rat liver. They were above reported values of the maximal biliary excretion rate of bilirubin.
Some Aspects of Vitamin A Metabolism
VITAM HORM
This chapter discusses the physiology and metabolism of vitamin A. In modern biology, vitamin A possesses a feminine-like charm. It is unique structurally, is essential nutritionally, is elegant in its photoreceptor role, and yet is tantalizingly mysterious in its other biological involvements. The demonstration that β-carotene is cleaved into two molecules of retinal by a cytoplasmic enzyme of the intestine and liver lays to rest the old and frustrating controversy concerning the mechanism and mode of provitamin A transformation. The rapid hydrolysis and reesterification of vitamin A ester has been demonstrated quantitatively in which the specificity of the esterification reaction for long-chain saturated fatty acid has been reaffirmed. Retinol is mainly transported as an ester in the chylomicra of the lymph, but a significant, minor portion is apparently oxidized and transported via the portal blood. The puzzling rapid disappearance of vitamin A acid after its injection into animals has been resolved by the demonstration that retinoyl β-glucuronide is rapidly formed and excreted in the bile. As retinyl β-glucuronide, the O-ether derivative of retinol, is formed after the administration of retinol, retinoic acid is not an obligatory intermediate in retinol metabolism.
Studies on bilirubin UDP-glucuronyltransferase
Elias Halac
Albert Reff
The capacity of liver fractions to conjugate bilirubin and p-nitrophenol with glucuronic acid is increased several fold by dialysis against alkaline EDTA. UDPglucuronyltransferase (UDP-glucuronate glucuronyltransferase (acceptor-unspecific), EC 2.4.1.17), with activity towards bilirubin and p-nitrophenol, has been solubilized by treating microsomes activated with EDTA and deoxycholate.Kinetic data obtained with activated microsomes, and fractionation and inactivation experiments performed with deoxycholate-solubilized enzyme have yielded evidence in favor of the existence of at least two UDP-glucuronyltransferases.Microsomal pellets were examined by electron microscopy before EDTA, after EDTA, and after EDTA and deoxycholate. The most apparent change that is concomitant with the activation by EDTA is the transformation of rough membranes into smooth membranes.
Effect of phenobarbitone on hepatic microsomal enzymes of the male rat
C.W.T. Pilcher
R.P.H. Thompson
Studies on a solubilized oestriol 16 -glucuronyltransferase from human liver microsomes
Treatment of lyophilized human liver microsomes with deoxycholate led to "solubilization" of oestriol 16α-glucuronyltransferase. The enzymic activity was stable in the presence of dithio-threitol and EDTA. A five fold enrichment of enzyme activity was obtained after filtration of the "solubilized" enzyme through Sepharose 4B. Phospholipase C reduced the enzyme activity by 70%, lysolecithin restored the activity to 75%. Phosphatidylcholine and sphingomyelin were found to be the major phospholipids in the partially purified oestriol 16α-glucuronyltransferase. From the ratio of void volume to elution volume, the apparent molecular weight of the active fraction was found to be approximately two million. These results suggest that the oestriol 16α-glucuronyltransferase "solubilized" from human liver microsomes may be a lipid-protein complex.
A study on the enzymatic mechanism of guinea-pig hepatic-microsomal bilirubin glucuronyl transferase
The synthesis of bilirubin glucuronide by tissue homogenates
Gerold M. Grodsky
John V. Carbone
Nucleotide Activation of Liver Microsomal Glucuronidation
Burton M. Pogell
Luis F. Leloir
The solubilization and partial purification of glucuronyl transferase fromrabbit liver microsomes
Kurt J. Isselbacher
Marilyn F. Chrabas
Rita C. Quinn
Protein estimation with Folin phenol reagent
Oliver H. Lowry
N.J. Rosebrough
A.L. Farr
R.J. Randall
Uridine compounds in glucuronic acid metabolism. I. The formation of glucuronides in liver suspensions
G J DUTTON
I. D. E. Storey
A Biochemical Lesion in Congenital, Non-obstructive, Non-h??molytic Jaundice
Julius Axelrod
Rudi Schmid
Lydia Hammaker
CONGENITAL, non-obstructive, non-hæmolytic jaundice is a syndrome characterized by marked elevation of `indirect-reacting' (free) bilirubin in the blood1. Recent studies have shown that bilirubin forms a polar glucuronide which is readily excreted in the bile1. The synthesis of bilirubin glucuronide has been found to be catalysed by an enzyme system in the microsomes of the liver that transfers glucuronic acid from uridine diphosphate glucuronic acid to a Suitable aglycone2. We wish to describe experiments showing that the accumulation of free bilirubin in this syndrome is the result of a defect in the glucuronide-forming mechanism.
Studies in the neonatal development of the glucuronid conjugating system
Audrey K. Brown
Wolf W. Zuelzer
The synthesis of bilirubin glucuronide in animal and human liver
Marjorie Walker
The Effect of Dietary Cirrhosis and CCl14 Poisoning on Glucuronyl Transferase Activity of Rat Liver
Zygmunt Chojecki
Fred Kern
Hepatic injury was produced in rats by a protein-deficient diet and by acute and chronic CCI 4 poisoning. Glucuronyl transferase activity in liver homogenates was measured and compared with that in a control group of animals. The homogenates were incubated in a suitable medium with uridine diphosphoglucuronic acid and with both bilirubin and phenolphthalein as acceptors of glucuronide. The glucuronide conjugation of bilirubin was significantly decreased by both nutritional cirrhosis and by acute and chronic CCI 4 toxicity. The conjugation of phenolphthalein was not affected by nutritional liver injury, but was markedly diminished by CCI 4 damage. The recovery after acute CCI 4 poisoning was more rapid for bilirubin conjugation than for phenolphthalein conjugation. The differences between bilirubin and phenolphthalein conjugation suggest that in the presence of liver damage the rate of bilirubin glucuronide synthesis should not be inferred from studies using other glucuronide acceptors, particularly phenolphthalein.
Solubilization of aromatic hydroxylase system of liver microsomes and requirement of lipid-like factor
Yoshio Imai
Ryo Sato
Jaundice of the newborn due to novobiocin
T HARGREAVES
J B HOLTON
Comparison of glucuronide synthesis in developing mammalian and avian liver
Glucuronyl Transferase Activity in Experimental Neonatal Hypothyroidism
Edmond A Werder
Some differences in the conjugation of o-aminophenol and p-nitrophenol by the uridine diphosphate transglucuronylase of mouse-liver homogenates
1. Glucuronide synthesis from uridine diphosphate glucuronate and o-aminophenol or p-nitrophenol in the presence of uridine diphosphate transglucuronylase of mouse-liver homogenates has been studied with respect to inhibition by compounds known to be conjugated under the experimental conditions, and also by thiophenol. 2. Raising the o-aminophenol concentration decreased the inhibition of o-aminophenyl glucuronide synthesis by the alternative glucuronyl acceptors phenol, menthol and benzoic acid, but was without effect on that caused by p-nitrophenol and thiophenol. 3. Raising the p-nitrophenol concentration decreased or abolished the inhibition of p-nitrophenyl glucuronide synthesis due to phenol, menthol, benzoic acid, anthranilic acid, o-aminophenol and thiophenol. 4. The o-aminophenol system was much more readily inhibited by all compounds than the p-nitrophenol system. 5. In tris buffer, pH7.4, over 30% activation of the o-aminophenol system was achieved by 2mm-Mg(2+), but 10mm-Mg(2+) was inhibitory. The p-nitrophenol system showed only inhibition from 2mm-Mg(2+) upwards. 6. The results are discussed as suggesting that there are at least two uridine diphosphate transglucuronylases.
The inhibition of the uridine diphosphate-transglucuronylase activity of mouse-liver homogenates by thiol reagents
1. A study of the catalysis of the formation of the glucuronides of o-aminophenol and p-nitrophenol by the uridine diphosphate transglucuronylase of homogenates of female mouse liver has been made, with reference to the effect of reagents reacting with thiol groups. 2. The synthesis of both glucuronides was completely inhibited by organic mercurials and N-ethylmaleimide. The inhibition was only partial with arsenite and the arsenoxides, iodoacetamide and o-iodosobenzoate. 3. The o-aminophenol system was much more sensitive than that for p-nitrophenol to all the thiol reagents, except N-ethylmaleimide, which was equally active in both systems. 4. At very low concentrations of the organic mercurials, the o-aminophenol system was activated. 5. With o-aminophenyl glucuronide formation, complete protection was given by glutathione and cysteine against the organic mercurials, N-ethylmaleimide and iodoacetamide, and partial protection against the arsenicals. Reversal was complete against the mercurials, and very limited against the arsenicals and iodoacetamide. The effects of N-ethylmaleimide and o-iodosobenzoate were irreversible. Results with p-nitrophenol were very similar. 6. Uridine diphosphate transglucuronylase was partially protected against p-chloromercuribenzoate and lewisite oxide by uridine diphosphate glucuronate, but not by o-aminophenol. 7. Glutathione did not prevent the decline in the rate of conjugation of o-aminophenol when homogenates were aged by incubation at 30 degrees . Cysteine was unable to prevent or reverse inactivation by ultrasonic radiation.
p-Nitrophenylglucuronide formation by homozygous adult Gunn rats
H van LEUSDEN
J.A.J.M. Bakkeren
ZILLIKENF
STOLTELA
Protein Measurement With Folin Fenol Reagent
O.H.N.G. Lowry
Since 1922 when Wu proposed the use of the Folin phenol reagent for the measurement of proteins (l), a number of modified analytical pro- cedures ut.ilizing this reagent have been reported for the determination of proteins in serum (2-G), in antigen-antibody precipitates (7-9), and in insulin (10). Although the reagent would seem to be recommended by its great sen- sitivity and the simplicity of procedure possible with its use, it has not found great favor for general biochemical purposes. In the belief that this reagent, nevertheless, has considerable merit for certain application, but that its peculiarities and limitations need to be understood for its fullest exploitation, it has been studied with regard t.o effects of variations in pH, time of reaction, and concentration of react- ants, permissible levels of reagents commonly used in handling proteins, and interfering subst.ances. Procedures are described for measuring pro- tein in solution or after precipitation wit,h acids or other agents, and for the determination of as little as 0.2 y of protein.
J Axelrod
R Schmid
L Hammaker
Axelrod, J., Schmid, R. & Hammaker, L. (1957). Nature, Lond., 180, 1426.
BIOCHEM J
D E Langelaan
P E Ross
Dutton, G. J., Langelaan, D. E. & Ross, P. E. (1964). Biochem. J. 93, 4P.
M Walker
Lathe, G. H. & Walker, M. (1958). Biochem. J. 70, 705.
H Lokietz
R M Dowben
D Y Y Hsia
Lokietz, H., Dowben, R. M. & Hsia, D. Y. Y. (1963). Pediatrics, Springfield, 32, 47.
Gastroenterology, 40, 521
Z Chojecki
F Kern
Chojecki, Z. & Kern, F. (1961). Gastroenterology, 40, 521. Dutton, G. J. (1963). Ann. N.Y. Acad. Sci. 111, 259.
H Lineweaver
D Burk
Lineweaver, H. & Burk, D. (1934). J. Amer. chem. Soc. 56, 658.
Y Imai
R Sato
Imai, Y. & Sato, R. (1960). Biochim. biophys. Acta, 42, 164. Isselbacher, K. J., Chrabas, M. F. & Quinn, R. C. (1962).
J BIOL CHEM
H T Malloy
I C A Evelyn
Malloy, H. T. & Evelyn, I. C. A. (1937). J. biol. Chem. 119, 481.
B M Pogell
L F Leloir
I D E Storey
Pogell, B. M. & Leloir, L. F. (1961). J. biol. Chem. 236, 293. Storey, I. D. E. (1965a). Biochem. J. 95, 201. Storey, I. D. E. (1965b). Biochem. J. 95, 209. van Leusden, H. A. I. M., Bakkeren, J. A. J. M., Zilliken, F. & Stolte, L. A. M. (1962). Biochem. biophys. Res. Commun. 7, 67.
E A Werder
Werder, E. A. & Yaffe, S. J. (1964). Biol. Neonat. 6, 8.
Neutralizing interaction between heparins and myotoxin II, A lysine 49 phospholipase A2 from Bothrop...
November 1994 · Journal of Biological Chemistry
Bruno Lomonte
Ernesto Moreno Frias
A Tarkowski
Marco Maccarana
Heparin binds to phospholipase A2 myotoxins from Bothrops asper snake venom, inhibiting their toxic activities. This interaction was investigated using purified myotoxin II, a Lys-49 phospholipase A2 of this venom, and a series of heparin variants, fragments, and other glycosaminoglycans. The binding was correlated to toxin neutralization, using endothelial cells as a target. Myotoxin II binds ... [Show full abstract] radiolabeled heparin in solution unselectively, and forms macromolecular complexes with an optimum at a heparin:toxin molar ratio of 1:5. Both O-sulfates and N-sulfates play a role in heparin binding, in the order of importance 2-O-sulfates > 6-Osulfates > N-sulfates. The shortest heparin oligosaccharides interacting with myotoxin II are hexasaccharides. The binding of a neutralizing monoclonal antibody (MAb-3) to myotoxin II was not inhibited by heparin, indicating that the two molecules interact with different sites on the toxin. A synthetic peptide (residues 115-129 in the numbering system of Renetseder et al. (Renetseder, R., Brunie, S., Dijkstra, B. W., Drenth, J., and Sigler, P. B. (1985) J. Biol. Chem. 260, 11627-11634) of myotoxin II displays both heparin-binding and cytolytic activities. It is concluded that heparin neutralizes myotoxin II by binding to a strongly cationic site in the region of residues 115-129, a possible contribution of lysines 36 and 38 suggested by molecular modeling studies. As this cationic region appears to be responsible for the cytolytic activity of the toxin, the present report constitutes the first identification of a cytotoxic region on a phospholipase A2 myotoxin.
Vanillic acid as a novel specific inhibitor of snake venom 5′-nucleotidase: A pharmacological tool i...
December 2009 · Biochemistry (Moscow)
Dhananjaya Bhadrapura Lakkappa
Nataraju Angaswamy
Raghavendra Gowda
Cletus D'Souza
Vanillic acid has been investigated for its inhibitory effect on Naja naja, Daboia russellii, and Trimeresurus malabaricus venom 5′-nucleotidase activity. Trimeresurus malabaricus venom 5′-nucleotidase activity was 1.3- and 8.0-fold higher than that of N. naja and D. russellii venoms, respectively. Substrate specificity studies showed that for all the venoms tested, 5′-AMP was the preferred ... [Show full abstract] substrate for 5′-nucleotidase. This indicates the central role of adenosine in snake envenomation. Vanillic acid selectively and specifically inhibited 5′-nucleotidase activity among several enzymes present in the three venoms tested. The inhibitor was competitive, as the inhibition was relieved by increased substrate concentration. It appears that the COOH group in vanillic acid is the determining factor for inhibition as vanillin, a structurally similar compound with respect to vanillic acid, had no inhibitory activity. This study for the first time exemplifies vanillic acid as a pharmacological tool in evaluating the role of 5′-nucleotidase in snake envenomation.
Serpentine Science: Charles Kellaway and the Fluctuating Fortunes of Venom Research in Interwar Aust...
June 2010 · Historical Records of Australian Science
Peter Hobbins
Australian medical research before the Second World War is predominantly viewed as an anodyne precursor to its conspicuous postwar successes. However, the expanding intellectual appeal and state support for local research after 1945 built upon scientific practices, networks, facilities and finances established between 1919 and 1939. Arguably the most prominent medical scientist working in ... [Show full abstract] Australia during this period was Charles Kellaway (1889–1952), director of Melbourne's Walter and Eliza Hall Institute from 1923 until 1944. Facing both financial challenges and a profoundly unsupportive intellectual climate, Kellaway instigated a major research programme into Australian snake venoms. These investigations garnered local and international acclaim, allowing Kellaway to speak as a significant scientific actor while fostering productive laboratory collaborations. The venom work spurred basic research in tissue injury, anaphylaxis and leukotriene pharmacology, yet delivered pragmatic clinical outcomes, particularly an effective antivenene. By selecting a problem of continuing public interest, Kellaway also stimulated wider engagement with science and initiated a pioneering ad hoc Commonwealth grant for medical research. In tracing his training, mentors and practices within the interwar milieu, this article argues that Kellaway's venom studies contributed materially to global biomedical developments and to the broader viability of medical research in Australia.
Monoclonal antibodies to Mojave toxin and use for isolation of cross-reacting proteins in Crotalus v...
February 1986 · Toxicon
Eppie D. Rael
Richard J. Salo
Hector Zepeda
Hybridomas secreting monoclonal antibodies against Mojave toxin were established. The antibodies were used for identifying cross-reacting proteins in individual C. s. scutulatus and other Crotalus venoms and to isolate Mojave toxin. The antibodies recognized five bands with a pI range from 5.1 to 6.1 in immunoblots of electrofocused crude venom and Mojave toxin purified by immunoaffinity ... [Show full abstract] chromatography. The specificity of the antibodies was for the basic subunit of the toxin, which resolved into four bands of pI between 9.3 and 9.6. Individual C. s. scutulatus venoms of snakes from Texas and southern Arizona had multiple bands with pI's ranging from 4.9 to 6.3. Cross-reacting proteins were also recognized by the antibodies in the electrophoresed venoms of C. basiliscus, C. d. durissus, C. d. terrificus, C. h. horridus and C. v. concolor, and may be isolated by immunoaffinity chromatography with the monoclonal antibodies.
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Location: NDSU / NDSU Mathematics / Seminars and Colloquium / Colloquium / Fall 2018
Fall 2018 Mathematics Colloquium
Time and Location:
Minard 214 at 3:00 PM (refreshments at 2:30 PM in the Math Conference Room (Minard 404))
Special Colloquia or Tri-College Colloquia venues and times may vary, please consult the individual listing.
Tuesday, September 25 Jeremiah Bartz (UND)
Multinets in the complex projective plane
Abstract: Multinets are certain configurations of lines and points with multiplicities in the complex projective plane. They appear in the study of resonance and characteristic varieties of complex hyperplane arrangement complements and cohomology of Milnor fibers. Although the notion of multinets has been around since 2006, very few examples or general properties of multinets with non-trivial multiplicities were known until much more recently.
In this talk, we will describe the motivation behind studying multinets, give numerous examples with pictures and recipes for constructing them, and discuss recent advances along with open problems in this area.
Tuesday, October 9 Kaisa Taipale (UMN)
The shape of data: applied topology
Abstract: Topological data analysis looks at the shape of data -- the underlying structure -- by adapting traditional methods from topology to modern computation and applications. Using simplicial complexes and homological algebra, we can analyze structures that persist, visualize the topological structures of datasets, and approach questions in manifold learning. Anyone can carry out data analysis with Python and R, and you might be surprised what insights mathematicians can bring to the world of applications. We'll look at real-life problems from health to finance to politics, and I hope to challenge your idea of what "applied mathematics" can mean.
Tuesday, October 16 Tri-College Colloquium
Doğan Çömez at Concordia
Quantization for probability measures on Cantor Dust
TIME and ROOM : Integrated Science Center (ISC) 301 at 3:00 Pm. Refreshments at 2:30 IN ISC 354
Abstract: Quantization for a probability measure is the process of estimating it by a discrete probability that assumes only a finite number of levels in its support. The focus of this talk is the quantization of continuous singular self similar probability measures on Cantor Dusts. For such a probability measure, we determine the optimal sets of points that ensures optimality and their quantization errors. In addition, we will show that the quantization dimension of the underlying measure exists, whereas its quantization coefficient, which determines the efficient rate of convergence of errors, does not exist. However, the interval in which accumulation points of the quantization coefficient lie is determined.
Tuesday, October 23 Semyon Litvinov (Penn State-Hazelton)
On convergence of ergodic averages for Dunford-Schwartz operators
in fully symmetric function spaces.
Abstract: Let $(\Omega,\mu)$ be a sigma-finite measure space, and let $X\subset L^1(\Omega)+L^\infty(\Omega)$ be a fully symmetric space. If $\mu(\Omega)=\infty$, we give necessary and sufficient conditions for almost uniform convergence in $X$ (in Egorov's sense) of Cesàro averages $M_n(T)(f)=\frac{1}{n}\sum_{k=0}^{n-1}T^k(F)$ for all Dunford-Schwartz operators $T$ in $L^1(\Omega)+L^\infty(\Omega)$ and any $f\in X$. Besides, if $(\Omega,\mu)$ is quasi-non-atomic, we show that the averages $M_n(T)$ converge strongly in $X$ for each Dunford-Schwartz operator $T$ in $L^1(\Omega)+L^\infty(\Omega)$ if and only if $X$ has order continuous norm and $L^1(\Omega)$ is not contained in $X$.
Tuesday, October 30 William Martin (NDSU)
PRIUM: Promoting Reasoning in Undergraduate Mathematics
PRIUM is a collaborative NSF-funded project (NSF DUE 1624906) involving mathematics department leaders and research mathematicians at North Dakota State University and Georgia State University. Our cyclic, iterative assessment model seeks insights about the development of undergraduates' abilities to read, interpret, critique and write proofs. Research-based approaches to teach mathematical proof or assessing student comprehension of proof have been scantly documented in the literature in spite of its attention in mathematics education research for over 30 years (Mejia-Ramos, Fuller, Weber, Rhoads, & Samkoff, 2012). PRIUM is designed to support mathematics educators and mathematicians to work closely to utilize insights gained from research-based proof assessments. This presentation will include a brief overview of the project and preliminary results from the assessments implemented during 2016-18. We will present several proof assessments implemented at the two universities and will engage with participants about how these assessments differ from the traditional "memorize and reproduce" techniques of proof assessment.
Mejia-Ramos, J.P., Fuller, E., Weber, K., Rhoads, K., & Samkoff, A. (2012). An assessment model for proof comprehension in undergraduate mathematics. Educational Studies in Mathematics, 79: 3–18.
Tuesday, November 6 Dan Florentin (Kent State)
Caustic duality in Minkowski billiards
TIME CHANGE: The talk starts at 3:15 pm.
Abstract: Mathematical billiards are a classical and well-studied class of dynamical systems,"a mathematician's playground". Convex caustics, which are curves to which billiard trajectories remain forever tangent, play an important role in the study of billiards. In this talk we will discuss convex caustic in Minkowski billiards, which is the generalization of classical billiards in non-Euclidean normed planes. In this case a natural duality arises from, roughly speaking, interchanging the roles of the billiard table and the unit ball of the (dual) norm. This leads to duality of caustics in Minkowski billiards. Such a pair of caustics is dual in a strong sense, and in particular they have equal perimeters and other classical parameters. We will show that, whenthe norm is Euclidean, every caustic possesses a dual caustic, but in general this phenomenon fails. Based on joint work with S. Artstein-Avidan, Y. Ostrover, and D. Rosen.
Tuesday, November 13 Ivan Yegorov (NDSU)
Optimal feedback strategies for bacterial growth.
Abstract: Mechanisms of bacterial adaptation to environmental changes are of great interest for both fundamental biology and engineering applications. This talk presents a continuous-time dynamic problem of resource allocation between metabolic and gene expression machineries for a self-replicating bacterial population. In compliance with evolutionary principles, the criterion is to maximize the accumulated structural biomass. In the model, we include both the degradation of proteins into amino acids and the recycling of the latter (i.e., using as precursors again). On the basis of the analytical investigation of our problem by Pontryagin's principle, we develop a numerical method to approximate the switching curve of the optimal feedback control strategy. The obtained field of extremal state trajectories consists of chattering arcs and 1 steady-state singular arc. The constructed feedback control law can serve as a benchmark for comparing actual bacterial strategies of resource allocation. We also study the influence of temperature, whose increase intensifies protein degradation. While the growth rate suddenly decreases with the increase in temperature in a certain range, the optimal control synthesis appears to be essentially less sensitive. Besides, we present an extended model that shows the potential of optimal control frameworks for better understanding and improving biotechnological production processes.
Tuesday, November 27 Tri-College Colloquium
Doug Anderson at NDSU
Equilibrium Stability vs. Hyers-Ulam Stability
Abstract: We compare the equilibrium stability of a simple first-order differential equation with its Hyers-Ulam stability, appropriately defined. Then we analyze these stabilities when the problem is discretized, either with difference equations or with more general time scales. Whether a given equation has Hyers-Ulam stability, and if so, whether there is a minimum Hyers-Ulam constant and unique accompanying solution is also discussed.
Tuesday, December 4 Mohamed Baghzali (NDSU)
The Effectiveness of the NDSU Math Emporium
Abstract: In the Fall 2015 semester, the NDSU Math Department created a learning center and has continuously tried to achieve three overarching goals:
1.) challenge students and assist them to become active learners;
2.) provide multiple ways to accommodate student learning; and
3.) provide appropriate and adequate resources.
Six semesters later, the true effectiveness of the NDSU Math Emporium courses redesign in introductory mathematics courses (College Algebra and Trigonometry) was measured and the results as well as the logistics will be presented during this presentation.
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Stable homotopy category and the moduli space of formal groups
The usual disclaimer applies: I'm new to all this stuff, so be gentle.
It seems like the spectrum, as defined by Balmer, of the stable homotopy category of finite complexes is something like $M_{FG}$, the stack of formal groups (that is, $Spec L/ G$ where $L$ is the Lazard ring and $G$ acts by coordinate changes). I'm not actually sure if that's true, I don't think I've seen it written quite like that, but the picture of the spectrum in Balmer's paper looks an awful lot like how I'd imagine $M_{FG}$ looking.
If the above is right, then there's another tensor triangulated category with the same spectrum, namely the derived category of perfect complexes on $M_{FG}$ (whatever that means for stacks...).
Just how far away is the stable homotopy category from actually being equivalent to this derived category? Is there a theorem to the effect that it can't be equivalent to such a thing? Do we even know that it's not equivalent?
I've heard that chromatic homotopy theory is about setting up a rough dictionary between algebro-geometric terminology regarding $M_{FG}$ and the stable homotopy category, so I guess the question is about whether or not we can make the dictionary into a proper functor.
stable-homotopy triangulated-categories ag.algebraic-geometry
Dylan WilsonDylan Wilson
$\begingroup$ Uh oh... I'm realizing that this $M_{FG}$ doesn't make much sense unless I pick a field... should I rephrase the question by localizing the stable homotopy category at a prime and picking a field of characteristic $p$? Or am I being paranoid? $\endgroup$ – Dylan Wilson Aug 19 '11 at 7:13
$\begingroup$ Section 2.4.2 of Morava's Complex cobordism and algebraic topology says something about this and provides references for sub-questions. Here's an arXiv link: arxiv.org/abs/0707.3216 . $\endgroup$ – Eric Peterson Aug 19 '11 at 8:18
$\begingroup$ The Hopf map is an obstruction for the existence of an algebraic model for the stable homotopy category $\endgroup$ – Fernando Muro Aug 19 '11 at 9:16
$\begingroup$ (You don't need to pick a field, the moduli of formal groups makes sense over any base.) $\endgroup$ – Tyler Lawson Aug 19 '11 at 13:05
$\begingroup$ @Fernando: I was thinking about this... I know there are no Hopf maps in things like $D^{perf}(X)$ when $X$ is a scheme, but I wasn't sure if this still held for stacks $\endgroup$ – Dylan Wilson Aug 19 '11 at 15:54
One useful thing to keep in mind is that the cohomological functor from the stable homotopy category to the category of quasi-coherent sheaves on the moduli stack $\mathcal{M}$ is not essentially surjective. For example, if you fix a prime $p$ and a height $n \geq 1$, then there is a closed substack $\mathcal{M}^{\geq n}$ consisting of formal groups over $\mathbb{F}_p$ having height $\geq n$. A standard problem in stable homotopy theory is to try to cook up finite spectra which map to the structure sheaf of $\mathcal{M}^{\geq n}$. You can generally only do this when $p$ is large compared with $n$. For small values of $p$ you generally have to make do with finite spectra whose image is the structure sheaf of some nilpotent thickening of $\mathcal{M}^{\geq n}$. These can always be found (a deep result of Devinatz-Hopkins-Smith) and this is what gives you such a strong connection between the topology of $\mathcal{M}$ and the "spectrum" of the stable homotopy category. But you have to work hard for it, and the connection is much weaker (closed subsets of $\mathcal{M}$ have an interpretation in the stable homotopy category, rather than closed substacks) than what you would expect if Adams-Novikov spectral sequences were to degenerate.
Jacob LurieJacob Lurie
$\begingroup$ Ah, thank you! Just to make sure I know what you're talking about: The finite spectra mapping to the structure sheaf of $\mathcal{M}^{\ge n}$ would be these $V(n)$-type spectra, yeah? And a finite spectrum with image the structure sheaf of a nilpotent thickening of $\mathcal{M}^{\ge n}$ would be the ones constructed in the proof of the periodicity theorem? This language is wonderful, conceptually! $\endgroup$ – Dylan Wilson Aug 19 '11 at 16:01
The relationship between the derived category of $M_{FG}$ and the stable homotopy category $\mathcal{S}$ is somewhat like the relationship between $D(gr_I(R))$ and $D(R)$, where $R$ is a commutative ring, and $I$ is an ideal in $R$, and $gr_I(R)=\bigoplus_nI^n/I^{n+1}$ is the associated graded ring. Complex cobordism gives a homological functor from $\mathcal{S}$ to the abelian category of quasicoherent sheaves on $M_{FG}$, but there is no useful functor from $\mathcal{S}$ to the associated derived category $D(M_{FG})$. Computationally, the morphism groups in $\mathcal{S}$ are the target of an Adams-Novikov spectral sequence whose $E^2$ page can be described as morphism groups in $D(M_{FG})$.
Neil StricklandNeil Strickland
$\begingroup$ This answer is great! I was torn between all of them... so I went with the popular vote. $\endgroup$ – Dylan Wilson Aug 19 '11 at 16:05
It's definitely known that the derived category of ${\cal M}_{FG}$ and the stable homotopy category are not equivalent. This is an instance of
The Mahowald Uncertainty Principle: Any spectral sequence converging to the homotopy groups of spheres with an $E_2$-term that can be named using homological algebra will be infinitely far from the actual answer.
(The naming is due to Ravenel; this quote is from Paul Goerss' "The Adams-Novikov Spectral Sequence and the Homotopy Groups of Spheres".) There is often a feeling that stable homotopy theory always deviates from algebra as soon as is possible.
As Neil said, the Adams-Novikov spectral sequence starts with morphisms in the derived category and computes stable homotopy groups of spheres. Every place where this spectral sequence does not degenerate indicates a point where the stable homotopy category deviates from the algebraic approximation. This includes the following phenomena.
Hidden additive extensions, such as the hidden additive extension making $\pi_3^s$ into $\mathbb{Z}/24$ rather than $\mathbb{Z}/12 \times \mathbb{Z}/2$.
Hidden multiplicative extensions. In the (2-local) stable homotopy groups there are elements $\eta \in \pi_1^s$, $\nu \in \pi_3^s$, and $\sigma \in \pi_7^s$. My recollection is that such that $\eta^2 \sigma = \nu^3$ on the $E_2$-term, but Toda showed that this relationship doesn't hold on-the-nose in stable homotopy groups of spheres.
Differentials. For any prime $p$, there is always a nontrivial differential in the Adams-Novikov spectral sequence, and the first differential is called the Toda differential.
Tyler LawsonTyler Lawson
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TECHNICAL ADVANCE
A general framework for comparative Bayesian meta-analysis of diagnostic studies
Joris Menten1,2 &
Emmanuel Lesaffre2
Selecting the most effective diagnostic method is essential for patient management and public health interventions. This requires evidence of the relative performance of alternative tests or diagnostic algorithms. Consequently, there is a need for diagnostic test accuracy meta-analyses allowing the comparison of the accuracy of two or more competing tests. The meta-analyses are however complicated by the paucity of studies that directly compare the performance of diagnostic tests. A second complication is that the diagnostic accuracy of the tests is usually determined through the comparison of the index test results with those of a reference standard. These reference standards are presumed to be perfect, i.e. allowing the classification of diseased and non-diseased subjects without error. In practice, this assumption is however rarely valid and most reference standards show false positive or false negative results. When an imperfect reference standard is used, the estimated accuracy of the tests of interest may be biased, as well as the comparisons between these tests.
We propose a model that allows for the comparison of the accuracy of two diagnostic tests using direct (head-to-head) comparisons as well as indirect comparisons through a third test. In addition, the model allows and corrects for imperfect reference tests. The model is inspired by mixed-treatment comparison meta-analyses that have been developed for the meta-analysis of randomized controlled trials. As the model is estimated using Bayesian methods, it can incorporate prior knowledge on the diagnostic accuracy of the reference tests used.
We show the bias that can result from using inappropriate methods in the meta-analysis of diagnostic tests and how our method provides more correct estimates of the difference in diagnostic accuracy between two tests. As an illustration, we apply this model to a dataset on visceral leishmaniasis diagnostic tests, comparing the accuracy of the RK39 dipstick with that of the direct agglutination test.
Our proposed meta-analytic model can improve the comparison of the diagnostic accuracy of competing tests in a systematic review. This is however only true if the studies and especially information on the reference tests used are sufficiently detailed. More specifically, the type and exact procedures used as reference tests are needed, including any cut-offs used and the number of subjects excluded from full reference test assessment. If this information is lacking, it may be better to limit the meta-analysis to direct comparisons.
There is a growing interest in diagnostic test accuracy (DTA) reviews to select the best diagnostic test procedure [1] for a given setting. Most meta-analyses of diagnostic tests, however, estimate the diagnostic accuracy of a single test [2, 3]. Selection of the best test is usually done by undertaking separate meta-analyses for each test and then comparing the results [3]. Even when formally comparing diagnostic tests in a single systematic review, the analysis may ignore study effects. Such an approach can lead to biased comparisons due to confounding by study effects, as shown in a recent review [3]. Takwoingi and colleagues showed that results from comparative studies, where two tests were directly compared and which provide the most robust comparisons, differed from those of non-comparative studies. However, only 31 % of available studies were comparative. This indicates that there is a need for meta-analytical methods via direct and indirect comparisons. Data from direct comparisons may be inconclusive while a combined analysis of direct and indirect comparisons may be conclusive and can result in more accurate estimates [4, 5].
A particular aspect of comparisons between diagnostic tests is that the diagnostic performance of the index test is nearly always determined by comparison with a second test, the reference standard. Such a reference standard is presumed to 100 % correctly classify subjects as diseased or not. However, for many diseases it is impossible to determine the true disease status with certainty [6] and reference standards are imperfect. It is well known that the use of imperfect reference standards may bias estimates of the accuracy of the index test [7]. This consideration leads to a second requirement for comparative meta-analyses of diagnostic studies: the meta-analytic methods should adjust for the use of imperfect reference standards.
The aim of this manuscript is to develop a model that can be used for the comparative meta-analysis of two diagnostic tests that conforms to the two requirements sketched above. First, we assess possible biases in the estimation of the relative accuracy of two index tests due to the use of imperfect reference tests. We describe the different parameters that can be used to estimate the relative accuracy of two tests and assess the bias resulting from the use of imperfect reference standards. This allows us to select the most appropriate summary measure to use in the comparative meta-analysis of two diagnostic tests. Subsequently, we describe and develop models that can be used in the meta-analysis of diagnostic studies to compare the relative accuracy of two tests. We start with models that presume a perfect reference test is used in each primary study and extend these models allowing for imperfect reference tests. We estimate these models using Bayesian methods, specifically using Markov-Chain Monte-Carlo (MCMC) methods through Gibbs sampling [8]. For each model we provide the model specification and offer suggestions for appropriate informative or vague priors. In addition, we assess in a simulation study the value of these newly developed models but also the bias induced by the use of incorrect methods. Finally, we apply the methods to a real data example in the field of leishmaniasis.
Our aim is to estimate and test the difference in diagnostic accuracy of two or more index tests in a meta-analysis, combining data across all available studies. The studies included in a DTA review typically test each subject with one or more index tests and with one reference test. This reference test may differ between studies. To set the scene, data from a hypothetical meta-analysis are presented in Table 1. In this example, there are three index tests (T1,T2,T3) and two possible reference tests (T4,T5). For example, in Study 1 index tests T1 and T2 are performed on all subjects as well as reference test T4. There are 30 subjects with positive results on all three tests, one subject shows positive results on T1 and T2 and a negative result on T4, etc. Studies 1 and 2 allow direct estimation of the relative accuracy of T1 and T2. Studies 3, 4 and 5 allow the estimation of the accuracy of T1 (studies 3 and 4) or T2 (study 5), but allow no direct comparison of T1 and T2. For these studies, the relative accuracy of T1 and T2 can only be estimated by estimating the diagnostic accuracy of each test separately and then comparing these estimates. This is complicated by the fact that the reference test is not the same for each study. Studies 6 and 7 do not allow direct comparison of the accuracy T1 and T2, but offer the possibility of an indirect comparison through the third index test T3. The information from this third test may help to eliminate differences among the studies.
Table 1 Tabulation of an hypothetical diagnostic test accuracy meta-analysis. Columns T1,T2,T3 indicate results for the 3 possible index tests. Columns T4,T5 indicate results for the 2 possible reference tests. + indicates a positive test result, - a negative test result. NA indicates that the test was not performed in that particular study. The observed frequency column report the number of subjects with a specific test result pattern in each study
As a first step in a comparative DTA meta-analysis, we have to select an appropriate statistic to compare the two tests. The best statistic would be one which is readily interpretable by users of the meta-analysis and which is least prone to bias. We describe the possible choices below together with the results from a small simulation study. Subsequently, we need to develop a model which allows the incorporation of all available data while ensuring that results are valid and are not biased by differences in study characteristic, such as the selection of the reference standard used. Some possible models are described below. We assessed the value of these models in a simulation study and in a practical application.
Measures of relative value of diagnostic tests
Diagnostic accuracy is characterised by sensitivity S and specificity C. These two quantities are related and comparisons between tests need to take both S and C into account. Comparisons between two tests can be summarized using the difference or relative risk in S and C for the two tests. An alternative parameterization uses the diagnostic odds ratio DOR=(S×C)/[(1−S)×(1−C)] which summarizes the accuracy of a test in a single number [9]. This parameter could be used as a summary in a meta-analysis, for example by calculating the relative DOR of two tests [3]. However, the use of imperfect reference standards can bias all above measures of relative accuracy of two index tests. We assessed the direction and magnitude of bias on these measures in a small simulation study. A description of the simulation study setup is given in Additional file 1.
Models for the comparative meta-analysis of diagnostic tests
In this section we develop models to compare J tests, by combining data across I studies in a comparative meta-analysis. All these models are hierarchical in nature. At the first level of the hierarchy, the models describe the observed data of the individual studies. The observed test outcomes depend on the disease prevalence and the accuracy of the tests in each study, and possible covariation among the test results. We describe the accuracy of the tests in terms of the study-specific sensitivity S ij and specificity C ij of test j in study i. At the second level, we specify a model for these study-specific sensitivity-specificity pair {S ij ,C ij }. Five possible models are described; they are listed in Table 2.
Table 2 Description of the different models. Example code for the models is given in Additional file 2. S ij and C ij represent the sensitivity and specificity of test j in study i
Meta-analytic models when a perfect reference standard is available
If a perfect reference test is available, the number of diseased N Di and non-diseased N NDi subjects in study i is known, as are the numbers of true positives N TPij and true negatives N TNij for each test j. In the standard bivariate model for the meta-analysis of a diagnostic test [9], the observed numbers of true positives and true negatives for each index test are assumed to be drawn from two independent binomial distributions N TPij ∼Bin(N Di ,S ij ) and N TNij ∼Bin(N NDi ,C ij ). The transformed values g(S ij ) = θ Sij and g(C ij ) = θ Cij are modeled at the next level, where g(.) is a link function to allow the use of the normal distribution. Common choices for g(.) are the logit, complementary log-log or probit functions. Several models are possible to incorporate comparisons of the diagnostic accuracy of different tests in this framework. We discuss three models below. These models can be further expanded to allow for covariates, other dependence structures or alternative parameterizations.
Model 1: Standard bivariate model for the meta-analysis of diagnostic tests
A basic approach is to estimate the average diagnostic accuracy of each test separately and subsequently compare the estimates of the average S j and C j across the different studies. In this approach, the standard bivariate model for the meta-analysis of diagnostic tests [2] can be used for each test separately. All g(S ij )=θ Sij and g(C ij )=θ Cij pairs are assumed to follow independent bivariate normal distributions:
$$\begin{array}{*{20}l} \left(\begin{aligned} \theta_{Sij} \\ \theta_{Cij} \end{aligned} \right) \sim N \left(\left[ \begin{aligned} \mu_{S_{j}} \\ \mu_{C_{j}} \end{aligned} \right], \Sigma_{j} \right), \\ \text{with} \ \Sigma_{j} = \left(\begin{aligned} & \sigma^{2}_{S_{j}} & \sigma_{S_{j}C_{j}} \\ & \sigma_{S_{j}C_{j}} & \sigma^{2}_{C_{j}} \end{aligned} \right), \end{array} $$
where \(\rho _{S_{j}C_{j}}=\sigma _{S_{j}C_{j}}/(\sigma _{S_{j}}\times \sigma _{C_{j}})\) is the correlation between \(\theta _{S_{\textit {ij}}}\) and \(\theta _{C_{\textit {ij}}}\). Estimates of the relative accuracy of the tests are obtained from the estimated \(g^{-1}(\mu _{S_{j}})\) and \(g^{-1}(\mu _{C_{j}})\). For example, the average difference in S between T1 and T2 is estimated as \(\hat {S}_{D21} = g^{-1}(\hat {\mu }_{S_{2}}) - g^{-1}(\hat {\mu }_{S_{1}})\). The advantage of the standard bivariate model is that it is relatively easy to fit using both Bayesian or frequentist techniques, with SAS [9] and WinBUGS [10, 11] example code available. However, as this model is not based on the comparisons between the index tests, but on the pooling of results for each test across all available studies, the results may be biased by study characteristics. This is equivalent to pooling findings from the active treatment arms of RCTs and comparing these estimates, an approach which is considered not to be appropriate for the meta-analysis of RCTs [12].
Model 2: Meta-Analysis Based on Direct Comparisons
To take study effects into account, the overall probability of testing positive in diseased subjects μ Si or in non-diseased subjects μ Ci for each study i could be modeled and S ij and C ij of the individual tests described as contrasts from this overall probability.
If we limit the data to studies which compare the two tests directly, we can write the study specific, transformed sensitivities g(S ij ) = θ Sij and specificities g(C ij ) = θ Cij as follows:
$$ \begin{aligned} \theta_{Si1} = \mu_{Si} + \delta_{Si}/2, \\ \theta_{Si2} = \mu_{Si} - \delta_{Si}/2, \\ \theta_{Ci1} = \mu_{Ci} - \delta_{Ci}/2, \\ \theta_{Ci2} = \mu_{Ci} + \delta_{Ci}/2. \end{aligned} $$
In case g is the logit function, δ Si = log(SOR12) and δ Ci = log(COR12), i.e. the log of the ORs of testing positive in diseased subjects for T1 compared to T2 and the log of the ORs of testing negative in non-diseased subjects in study i, respectively. To obtain average estimates of the difference in diagnostic accuracy between the two tests, δ Si and δ Ci are modeled using a bivariate normal distribution:
$$\begin{array}{*{20}l} \left(\begin{aligned} \delta_{Si} \\ \delta_{Ci} \end{aligned} \right) \sim N \left(\left[ \begin{aligned} \nu_{\delta_{S}} \\ \nu_{\delta_{C}} \end{aligned} \right], \Sigma \right) \\ \text{with} \ \Sigma = \left(\begin{aligned} & \sigma^{2}_{\delta_{S}} & \sigma_{\delta_{S}\delta_{C}} \\ & \sigma_{\delta_{S}\delta_{C}} & \sigma^{2}_{\delta_{C}} \end{aligned} \right). \end{array} $$
The \(\nu _{\delta _{S}}\) and \(\nu _{\delta _{C}}\) are the average log OR of the S and C between tests T1 and T2, respectively. The μ Si and μ Ci account for the dependence of test results obtained from the same study and can be estimated as fixed effects of in their turn modeled using bivariate normal distributions. This model is equivalent to the Smith −Spiegelhalter−Thomas model for two-treatment comparisons of RCTs [13, 14]. A similar model, but assuming a fixed, rather than random, relative accuracy between the different index tests is described in the Cochrane Handbook for Systematic Reviews of DTA studies [9].
Model 3: Meta-Analysis Based on Direct and Indirect Comparisons
As shown in Lu et al. [14] in the case of meta-analysis of RCTs, the Smith −Spiegelhalter− Thomas model can be expanded to a mixed treatment-comparison meta-analysis of more than two treatments. Similarly, we can expand Model 2 to J diagnostic tests. By taking diagnostic test T J as baseline, we can rewrite eqs. 2 and 3 as:
$$\begin{aligned} \theta_{Si1} = \mu_{Si} + (\,J-1) \times \delta_{Si1}/J - \delta_{Si2}/J - \ldots - \delta_{Si(\,J-1)}/J, \\ \theta_{Si2} = \mu_{Si} - \delta_{Si1}/J + (\,J-1) \times \delta_{Si2}/J - \ldots - \delta_{Si(\,J-1)}/J, \\ \vdots \\ \theta_{SiJ} = \mu_{Si} - \delta_{Si1}/J - \delta_{Si2}/J - \ldots - \delta_{Si(\,J-1)}/J, \end{aligned} $$
$$\begin{aligned} \theta_{Ci1} = \mu_{Ci} + (\,J-1) \times \delta_{Ci1}/J - \delta_{Ci2}/J - \ldots - \delta_{Ci(\,J-1)}/J, \\ \theta_{Ci2} = \mu_{Ci} - \delta_{Ci1}/J + (\,J-1) \times \delta_{Ci2}/J - \ldots - \delta_{Ci(\,J-1)}/J, \\ \vdots \\ \theta_{CiJ} = \mu_{Ci} - \delta_{Ci1}/J - \delta_{Ci2}/J - \ldots - \delta_{Ci(\,J-1)}/J, \end{aligned} $$
$$ \left(\delta_{Si1}, \delta_{Si2}, \ldots, \delta_{Si(\!\,J-1)}, \delta_{Ci1}, \delta_{Ci2}, \ldots, \delta_{Ci(\,J-1)} \right) \sim \!N(\boldsymbol{\nu_{\delta}},\Sigma). $$
and \(\phantom {\dot {i}\!}\boldsymbol {\nu }_{\boldsymbol {\delta }}=\left (\nu _{\delta _{S1}},\ldots,\nu _{\delta _{S(\,J-1)}}, \nu _{\delta _{C1}},\ldots,\nu _{\delta _{C(\,J-1)}}\right)\) represents the average log ORs for S and C of the J−1 tests compared to the baseline test T J . The differences in S and C between T1 and T2 on the logit scale are estimated by \(\nu _{\delta _{S1}} - \nu _{\delta _{S2}}\) and \(\nu _{\delta _{C1}} - \nu _{\delta _{C2}}\), respectively. This method allows indirect comparisons of T1 and T2 through comparison with a third test, similar to mixed treatment comparisons meta-analysis of RCTs. One complication of this model, is the specification and estimation of the variance-covariance matrix Σ. Specifying a structured variance-covariance matrix is in general complex and difficult to handle in MCMC estimation since each sampled variance-covariance matrix should be positive-definite [15]. In addition, model identification of the model with a general variance-covariance matrix will be difficult, especially when number of tests of interest is large. As an initial exploration of this model we can use a simplified variance-covariance structure, for example a diagonal or block diagonal matrix, and subsequently assess the effects of relaxing the simplifying assumptions. We describe some possible simplified variance-covariance structures in Additional file 2: Section 2.6.
Meta-analytic models when no perfect reference standard is available
The models described above presume that the disease status of all subjects in all studies is known, and consequently that the N Di ,N NDi ,N TPij and N TNij for each study i and test j is available. However, if only imperfect reference standards are available, the reported estimates of these quantities may be biased. The models described above can be expanded through latent class analysis (LCA) [16] to allow for the use of imperfect reference standards. In LCA, the true disease status of the participants of the basic studies is an unobserved, or latent, variable with two mutually exclusive categories, "diseased" and "non-diseased". This unobserved variable determines the probability to test positive or negative to a number of diagnostic tests which may include one or more imperfect reference tests. LCA models have been described for a variety of situations ranging from when a single imperfect test is observed in each study to more complex designs involving multiple tests. When multiple tests are involved, they may be treated as independent conditional on the disease status or the conditional dependence between them may be modeled using a variety of approaches [17–20]. An important underlying assumption of the latent class model is that the tests included in the model all correspond to the same underlying disease state [21]. Especially in a meta-analysis, where each study may use a different set of tests, this assumption is critical. If this assumption is not met, the underlying latent variable may differ among studies.
Description of the conditional independence latent class model
In this section, we describe the basic latent class model at the level of the individual study i in the meta-analysis. To simplify notation, we temporarily suppress the i-subscript for the study level. For latent class analysis, the basic data is not the number of true positives and true negatives for each test j, but rather the number of subjects that show a certain pattern of outcomes across the J tests performed in a study. The number of subjects with pattern y=(y1,y2,…,y j ) can be denoted as N y and is assumed to follow a multinomial distribution N y ∼Mult[N,P(y)], with y j the observed binary outcome (0 = negative, 1 = positive) for test T j , N the total sample size and P(y) the probability that y occurs.
Denoting the unobserved disease status as D (not diseased D=0, diseased D=1) and under the conditional independence assumption \(P(\mathbf {y}|D=k) = \prod _{j=1}^{J} P\left (y_{j}|D=k\right)\), the class probabilities P(y) can be described in terms of the S j and C j of the J tests. That is:
$$\begin{array}{@{}rcl@{}} P\left(\mathbf{y}\right)= \sum_{k=0}^{1} P\left(D=k\right) P\left(\mathbf{y}|D=k\right) = \\ {}\pi \prod_{j=1}^{J} S^{y_{j}}_{j} \left(1-S_{j}\right)^{\left(1-y_{j}\right)}\,+\, \left(1-\pi\right) \prod_{j=1}^{J} C^{\left(1-y_{j}\right)}_{j} \left(1\,-\,C_{j}\right)^{y_{j}}, \end{array} $$
with π the disease prevalence.
Thus LCA provides estimates for the study specific prevalence of disease π i and the S ij and C ij of the J i tests used in study i, which is a subset of the J different tests used across the I studies of the meta-analysis.
Model 4: Hierarchical Latent Class Model
In essence, the most basic hierarchical latent class model (Model 4) is constructed through a combination of equations 1 and 5. While previously the reference test was presumed to be 100 % sensitive and specific, in Model 4 all S ij and C ij , including those of the reference tests, are modeled using separate bivariate normal distributions as in Equation 1. The observed data is assumed to come from the multinomial distribution described in Equation 5. Again, like Model 1, this model ignores the correlation among test results from the same study. The prevalences π i can be assumed to be different for each study or to have a common normal distribution, \(\pi _{i} \sim N\left (\mu _{\pi },\sigma ^{2}_{\pi }\right)\).
Model 5: Network-based Hierarchical Latent Class Model
By rewriting the θ Sij and θ Cij in terms of μ Si ,μ Ci ,δ Sij , and δ Cij as in Eq 4, we can again take into account study level effects. The hierarchical modeling is equal to Model 3, the only difference is at the study level as described in Eq 5. This model thus adjusts the meta-analysis for the use of imperfect reference tests. By using the expanded Smith −Spiegelhalter− Thomas model of Lu et al. [14] at the second level of the hierarchy, study level effects are eliminated without the need to limit the analysis to direct comparisons only.
Model estimation and prior specification
Models are estimated in a Bayesian framework using Markov Chain Monte Carlo (MCMC) methods with OpenBUGS 3.0.3 called from within R 3.0.1 using the BRugs library. The Bayesian approach allows the estimation of complex, joint models and the combination of prior information, e.g. on the value of the reference test used, in the meta-analysis of new diagnostic tests. To complete the Bayesian model, priors need to be provided for all model parameters. OpenBUGS code for the models and full specifications of the priors are in Additional file 2. Convergence was checked using visual inspection of trace plots of the Markov chains and the Gelman-Rubin diagnostic statistic [22].
For parameters related to the index tests of interest, we consider it generally most appropriate to use uninformative priors. Specifically, we used normal priors with mean μ equal to zero and standard deviation σ equal to 1.69 for logit-transformed probabilities. This prior matches a uniform prior over the interval [0,1] in the first two moments on the probability scale [23]. When appropriate, these priors were bounded to avoid label switching [20]. Label switching is a problem arising in MCMC estimation of latent class models when two equivalent solutions are possible which give rise to identical observed data [24, 25]. The problem can be avoided by constraining S or C of one or more test to be ≥0.5. For the contrast in S and C, expressed as log ORs, normal priors with μ=0 and a large standard deviation, e.g. σ=10 can be used. For the variance-covariance matrices, we construct non-informative priors using uniform priors for standard deviations and correlations.
The model was specified using a logit link function and results are estimated on the log-odds scale. The MCMC approach as implemented in OpenBUGS allows to obtain posterior distributions of all functions of the estimated parameters, as the average S and C of the index tests and differences between S and C of the different tests. We illustrate this in the OpenBUGS code in Additional file 2. We used the 2.5 and 97.5 th percentiles of the sampled posterior distribution of the statistics of interest as bounds for the 95 % credible intervals.
If we want to use information from previous phases of the research, we can use informative priors. It may for example be appropriate to use information obtained from a previous meta-analysis of case-control studies when performing a meta-analysis of phase IV studies, i.e. studies recruiting clinically suspect patients consecutively in a representative clinical setting [7]. However, given that the phase IV design ensures the most realistic assessment of the performance of a test when used as a diagnostic tool in the target population [6], we may want to reduce the influence from these prior phases by using a prior which is more diffuse than the actual results from the prior meta-analysis. In the latent class model Model 4, we can use informative priors for the diagnostic accuracy of the reference test. It is likely that some information on the accuracy of the reference tests is available. In fact, standard analysis assumes S and C of the reference test to be 100 %, which can be considered to be very strong deterministic prior from a Bayesian viewpoint [26]. Priors for the accuracy of reference test can be obtained from the literature or expert opinion [10].
Simulation study
To assess the performance of the different models and to uncover possible bias of combining data without proper control for study specific effect or adjustment for the use of imperfect reference standards, we performed a simulation study using two different scenarios. For each scenario, we generated 250 sets of 20 diagnostic studies. We analyzed each simulated data set using the models described above using the logit for the link function g(.). We evaluated the models using coverage probabilities (the proportion of replications in which the 95 % credible interval contained the true value) and power (the proportion of replications in which a difference in S and C between the two tests of interest was detected). In Scenario 1, we simulated a setting without systematic bias but where a common imperfect reference test is used to assess the diagnostic accuracy of the index tests in all primary studies. In Scenario 2, we simulated the situation of two index tests which are assessed in primary studies that tend to use different reference standards. This situation may rarely occur in practice, but was selected to assess how the model performed in an extreme situation with systematic bias due to imperfect reference tests. A full description of the simulation study setup is in Additional file 3.
Real data example
We applied the models to data obtained in a meta-analysis of rapid diagnostic tests for visceral leishmaniasis, which we described earlier [10, 27]. In the published meta-analysis, the focus was on estimating the diagnostic accuracy of individual tests. We extracted the data relevant for the comparison of one rapid diagnostic test, the RK39 dipstick, with that of the direct agglutination test (DAT) as a test case for the application of the methods developed in the current paper. We limited the data for this test case to primary studies that included the RK39 dipstick or DAT with at least one other index test and microscopical examination as a reference test for which full data was available in the published primary study. We selected all index tests which were used in more than one study. In total, we included 10 primary studies, four index tests (DAT, RK39-dipstick, IFAT, KAtex) and two reference tests (parasitology including spleen aspirate, parasitology not including spleen aspirate) (Table 3). All tests are specific to VL and consequently can be expected to related to the same underlying latent variable. The data are shown in Fig. 1 and Appendix 4. Note that the current study is used as "proof-of-concept" of the statistical modeling approach and not as a complete meta-analytic comparison of the two tests which would require a more extended search strategy.
Forest plot for real data example. Estimated sensitivity and specificity of the RK39 dipstick (open circles) and DAT (closed squares) with 95 % confidence interval, using parasitology as gold standard
Table 3 Overview of the real data example: a comparative meta-analysis of the RK39 dipstick and direct agglutination test (DAT) for the diagnosis of visceral leishmaniasis. The total sample size (N) and availability of test results (X) is given for all 10 studies. Other tests: IFAT=indirect fluorescent antibody test, KAtex =latex agglutination test, spleen =parasitological examination of tissue aspirates including spleen sample, no spleen: parasitological examination of tissue aspirates not including spleen sample
The aim of this modeling exercise was to estimate the differences in S and C between the RK39-dipstick and DAT. A previous meta-analysis indicated that the diagnostic accuracy of the RK39, and possibly also of the DAT, may be lower in East-Africa compared to other geographic regions [28]. To correct for these differences, we included a fixed region effect (East-Africa vs. rest of the world) for S. We fitted the five models listed in Table 2. In the previous study [10], we obtained expert opinion on the diagnostic accuracy of the two reference tests. Expert opinion on the diagnostic accuracy of parasitology including spleen aspirate varied between 88 and 95 % for S and between 95 and 100 % for C. For parasitology without spleen aspirate, expert opinion varied for S between 70 and 80 % and between 95 and 100 % for C. We used this information to determine the priors in estimation of the models allowing for imperfect reference standards.
The results of our simulation study indicated that in a realistic setting, bias in estimating the difference in S and C between two index tests due to the use of an imperfect reference standard can be relatively limited (Additional file 1). Strong bias only occurred if the errors of one index test were strongly correlated with those of the reference test while the errors of the second index test were uncorrelated with those of the reference test. Similar observations can be made for the relative S and C. When the comparisons were expressed as odds-ratios or when using the relative Diagnostic Odds Ratio as a summary statistics, bias was more substantial and occurred even with uncorrelated errors. This corresponds to the findings from Zhang et al. who report that also in the meta-analysis of RCTs the odds-ratio is not always a suitable summary statistic [5].
Model performance: simulation study
Results of the simulation study of the model performance are described in detail in Additional file 3. The bias in estimating the contrasts in S and C between T1 and T2, expressed as a difference, relative risk or odds-ratio is summarized in Fig. 2.
Summary of simulation results. Bias in estimates of the contrasts in diagnostic accuracy from the proposed meta-analytical models applied in the simulation study. The boxplots present the bias in \(\hat {S}_{D12}\) and \(\hat {C}_{D12}\) (first row), \(\hat {S}_{RR12}\) and \(\hat {C}_{RR12}\) (second row), \(\hat {S}_{OR12}\) and \(\hat {C}_{OR12}\) (third row). The first column presents Scenario 1 where a common imperfect reference standard with moderate S and high C was used, the second scenario 2 where systematic bias is induced by differing reference standards. Full explanation of the model is in the text; full explanation of the simulation setup and results in Additional file 3. Note for Scenario 1: For models 1 to 3 disease status was estimated from the results of T4. Note for Scenario 2: Models 4 and 5 were applied both assuming it is known that the reference tests differ across studies (4a and 5a) and ignoring the difference in reference tests (4b and 5b)
In Scenario 1, where a common imperfect reference standard with moderate S and high C was used, a naive analysis assuming the reference test was perfect (Models 1–3), resulted in bias in estimating the odds-ratios (Fig. 2c) and to a lesser extent also the relative risks (Fig. 2b). If the contrast of interest was expressed as a difference (Fig. 2a), a true gold standard was available (Fig. 2a–c, Models 1–3), or if a latent class model was used to allow for imperfect reference tests (Fig. 2a–c, Models 4 and 5), no bias was apparent. Allowing for imperfect reference tests resulted in a lower power compared to the situation that a perfect reference test was available (Additional file 3, Table 3).
In Scenario 2, the reference standard of the simulated studies varied according to the index tests studies, which could result in systematic bias. In the analysis of Scenario 2, Models 4.i and 5.i correspond to the situation where the researchers knew of the differences in reference standard used across studies and that the variation in reference standard was thus a known source of bias. Models 4.ii and 5.ii correspond to the situation that researchers are unaware of the differences in reference standards among studies and that consequently the variation in reference standard was an unknown source of bias. This may occur if researchers do not provide sufficient detail in the primary publications on the exact modalities of the reference test procedures. For example, in the diagnosis of VL microscopical examination of spleen aspirates is the preferred reference test, while bone marrow aspirates show a limited S. Often researchers indicate their reference test to be based on spleen aspiration. However in closer assessment of these publications, it can become apparent that some researchers perform spleen aspiration on nearly all subjects while others may preform spleen aspiration on only a minority of subjects. Ignoring these difference in reference tests may lead to bias. Incorrectly assuming the reference test were perfect resulted in substantial bias, especially when ignore study level effects (Fig. 2d–f, Model 1). When correcting for the use of imperfect reference tests using LCA (Models 4.i and 5.i), unbiased estimates for the differences in diagnostic accuracy between T1 and T2 were obtained (Fig. 2d–f, Models 4.i and 5.i). If the data were however analyzed ignoring the differences between the reference tests, the differences in diagnostic accuracy between T1 and T2 were overestimated (Fig. 2d–f, Models 4.ii and 5.ii).
Real data example: diagnostic tests for visceral leishmaniasis
Results of modeling of the VL data are in Table 4. All models indicated that S of DAT (S2) was 8 to 11 % higher, compared to the S of RK39 (S1), in East-Africa, but this difference did not reach statistical significance. In the rest of the world, estimates of S1 and S2 were similar. Differing modeling strategies or allowing for imperfect reference standards did not impact estimates of S or comparisons of S between the two index tests of interest. This is as expected as the parasitological reference tests show a similar and high C.
Table 4 Results of the meta-analysis of the diagnostic tests for visceral leishmaniasis
In contrast, allowing for imperfect reference tests (models 4 and 5) resulted in considerably higher estimates for C of both the RK39 dipstick and DAT compared to models assuming perfect reference tests were used (models 1–3). False negative results for the reference tests may have resulted in reduced estimates of C1 (75.5–78.6 %) and of C2 (80.1–81.5 %). Allowing for imperfect reference standards resulted in considerable higher estimates for C1 (90.2–91.0 %) and C2 (93.0–94.1 %).
In the analyses that used parasitology as a, presumed perfect, reference test, a substantial difference between C1 and C2 (\(\hat {C_{2}} - \hat {C_{1}}\) = 5.3 %) was observed when limiting the analysis to direct comparisons only (Model 2). On the other hand, Model 1, based on independent estimation of C1 and C2, showed a much smaller difference (\(\hat {C_{2}} - \hat {C_{1}}\) = 1.5 %). The model using direct and indirect comparisons (Model 3) showed intermediate results (3.2 %). This can be explained by the fact that the studies in which no direct comparison was possible between the RK39 dipstick and DAT showed contradictory results to the studies with direct comparisons. These studies also used the least sensitive reference standard which may explain that results of Models 4 and 5, both allowing for imperfect reference standards, were similar.
In this paper, we developed a novel model to perform a comparative meta-analysis of the accuracy of two or more diagnostic tests when a perfect reference standard is unavailable. In a first step, we assessed the bias of comparative measures of the diagnostic accuracy of two tests induced by the use of an imperfect reference test. We observed that the difference in S and C may be the least subject to bias while at the same time being easily understandable to users of the meta-analysis results. In our modeling approach, we combined LCA with models developed for the mixed treatment comparisons meta-analysis of RCTs. The modeling framework accommodates a broad range of studies, including "Multiple Test Comparison", "Randomized Test Comparison", and "Between-Study Test Comparison" studies according to the terminology of Takwoingi et al., with the first two designs offering the most robust comparative data [3]. In a simulation study, the resulting model showed adequate performance, even if some aspects of the data generating mechanism were ignored. The simulation study also stressed the importance of accurate and complete extraction of the data from the primary studies when performing a DTA review. When differences in reference tests were ignored, biased estimates of the relative accuracy of the competing tests were unavoidable. This highlights the importance of complete and transparent reporting of DTA studies as promoted by the STARD initiative [29]. For a correct analysis of the data, the index and reference tests should be accurately described. Any cut-offs used to classify test results as positive or negative should also be reported and results for all subjects should be given, including subjects with incomplete or equivocal test results. The cross-classification of all test results should be presented in a format similar to that of the motivating example dataset in Additional file 4. The fact that meta-analysis is possible using imperfect reference tests suggests it may be more efficient to design future studies with multiple imperfect tests rather than using a single "as-accurate-as-possible" reference test, as has been shown in the analysis of epidemiological studies with imperfect measures of exposure [30, 31]. When applied to a dataset on visceral leishmaniasis diagnostic tests, the model indicated that C of the two tests of interest may have been underestimated due to the use of imperfect reference test. Our novel modeling approach, combining latent class analysis with hierarchical meta-analysis modeling, allowed the estimation of the difference in accuracy of the two index tests without making strong assumptions on the performance of the reference tests used. However, as in all meta-analyses, care should be taken that the studies combined are in fact comparable. While our approach corrects for bias and heterogeneity induced by the use of imperfect reference tests, other types of bias as publication and spectrum bias, can result in incorrect meta-analysis results. The approach can be combined with meta-regression techniques to reduce heterogeneity.
As limitations of our approach the following points can be given, which can indicate future avenues for further progress in this field. As a first limitation, we chose to compare diagnostic tests based on the sensitivity and specificity, and in particular based on the difference in these quantities among competing tests. Focusing on differences in S and C leads to results which are easily understandable for potential users. However, a test can be superior to another with respect to S while inferior with respect to C. In this case, selecting the optimal test can be difficult. Using a single summary measure of diagnostic accuracy, as the relative diagnostic odds-ratio (rDOR) can make comparisons among tests easier [32]. Theoretically, the test with the highest DOR may be preferred. However, this may not always be the case as the potential risk of a false positive result may be different from the risk of a false negative result. It may be easier for users to balance an increase in S versus decreases in C. In addition, the rDOR may be more prone to bias as we have shown for the OR difference in S and C. In our model formulation, the rDOR can be easily obtained. If the primary parameter of interest is however the rDOR, an alternative model formulation, for example an extension of the hierarchical summary ROC model [9, 33], may be more appropriate.
To allow estimation of the model, we made considerable simplifications to the variance-covariance structure of our parameter space. Not all these simplification may be warranted and a more general variance-covariance structure may refine estimates from this model. Fitting a general variance-covariance matrix however results in important computational difficulties. Our simulation study indicated that these limitations do not necessarily invalidate analysis results, but further research is needed to assess when this may no longer be the case. Modeling the variance-covariance matrix via partial autocorrelations [15] may allow the fitting of more complex model. We accommodated study effects using contrast-based (CB) approaches. However, in RCT arm-based approaches that correctly incorporate correlations have been shown to be superior to CB methods [5]. Further development of the equivalent models for DTA reviews, Models 1 and 4 in our setting, incorporating the correlations induced by study levels, is needed. Model 4 which corrects for imperfect reference tests, but ignores study effects, performed well in our simulation study. However, it is vulnerable to bias from study-specific effects. the model would need extensions to incorporate dependencies between test results from the same study, as for example was done for the meta-analysis of RCTs in Zhang et al. 2014 [5], before it is recommended as a general method for the meta-analysis of DTA studies above Model 5. However, if the accuracy of the different index tests is not strongly correlated across studies, Model 4 may perform equally well as Model 5 and may offer advantages in terms of identifiability and computational feasibility.
We showed how prior information, e.g. on the diagnostic accuracy of the reference test, can be used to aid model estimation in the case of the hierarchical latent class model. This is in line with the methods we have earlier developed for the meta-analysis of the diagnostic accuracy of a single test when using an imperfect reference standard [10]. In the case of the network based latent class model, it is however much less clear how this information can be used. The diagnostic accuracy of each test is in this model a linear combination of an overall, study-specific, probability of testing positive on all tests and a number of contrasts in diagnostic accuracy among these tests. More research is needed on how priors can be constructed for this model, e.g. using the priors for conditional probabilities rather than for S and C directly [26].
DTA studies can be expected to exhibit considerable heterogeneity and may be more prone to bias and inconsistency between direct and indirect comparisons compared to RCTs. Applications of network meta-analytic model to DTA studies must be performed with care and further development of statistical methods are needed. The literature on network-based meta-analysis of RCTs contains many additional tools, for example to assess consistency of estimates obtained from direct versus indirect comparisons [34–36], assess heterogeneity among studies [37], detect outlying studies [38] and correct for bias [39]. We only performed a limited application of techniques developed in this context. Expanding these techniques to DTA meta-analyses may be a valuable direction of research. In particular, it is important to expand the concept of consistency of comparisons across networks to the context of DTA reviews [40, 41].
Alternative approaches to the comparative meta-analysis of diagnostic tests are proposed. The regression approach of Macaskill et al. [9] can be seen as a variation of our model 2 in which the relative S and C, expressed as an odds-ratio, between tests is constant across studies. For the case all tests are applied to all subjects, Trikalinos at al. [42] describe a model which fully accounts for the within-study correlation between the tests' subject-specific S and C. This approach can be more efficient than the methods proposed in the current manuscript. However, both approaches need further empirical and simulation studies to assess their relative merits. Different models may be most appropriate depending on the application. In case it is suspected that reference tests may show only limited S or C, the analysis method should allow for the use of imperfect reference test. If important study-level effects are expected, proper control for confounding by these effects is needed. If there is important uncertainty on the value of the reference test or the presence of study level effects, it will be preferable to fit several models and assess the robustness of the results to the assumptions. At this stage of research, it is not possible to provide a general recommendation on the optimal modeling approach for the meta-analysis of comparative DTA reviews.
The models developed in this paper are promising and can improve the comparison of the diagnostic accuracy of competing tests in DTA systematic review. This is however only true if the studies and especially information on the reference tests used are described in sufficient detail. If the reporting of the studies does not provide sufficient detail, it may be better to limit the meta-analysis to direct comparisons. Further work refining the modeling approach, especially with respect to the specification of more general covariance structures and the use of measures of consistency of direct versus indirect comparisons, can further improve these methods.
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The works was supported by the Department of Economy, Science and Innovation of the Flemish Government. JM thanks Marleen Boelaert for her support and advice.
Clinical Trials Unit, Institute of Tropical Medicine, Nationalestraat 155, Antwerp, B-2000, Belgium
Joris Menten
L-Biostat, KULeuven University of Leuven, Kapucijnenvoer 35, Leuven, B-3000, Belgium
Joris Menten & Emmanuel Lesaffre
Emmanuel Lesaffre
Correspondence to Joris Menten.
JM conceived research questions, developed study design and methods, carried out statistical analysis, interpreted results and drafted the manuscript. EL advised on study design, methods, statistical analysis and commented on successive drafts. Both authors read and approved the final manuscript.
Additional file 1
Simulation Study for the Selection of Appropriate Statistics for a Comparative DTA Review. (PDF 1208 KB)
Software Code, Prior Specification, and Possible Structures for Variance-Covariance Matrices. (PDF 149 KB)
Simulation Study of the Modeling Approach. (PDF 491 KB)
Visceral Leishmaniasis Data. (PDF 44 KB)
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver(http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Menten, J., Lesaffre, E. A general framework for comparative Bayesian meta-analysis of diagnostic studies. BMC Med Res Methodol 15, 70 (2015). https://doi.org/10.1186/s12874-015-0061-7
Meta-analyses
Diagnostic test accuracy
Latent class model
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CommonCrawl
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Matrix multiplication that includes a tensor
How would I best express the following in Mathematica: $\begin{pmatrix}2 & 4\end{pmatrix} \begin{pmatrix}r_1 & r_2\\r_3 & r_4\end{pmatrix} \begin{pmatrix}6 \\ 8\end{pmatrix}$, where $r_i$ are 3-element vectors?
Is there a more elegant way than this?
r1 := {1, 2, 3}
res := {2, 4} . {{r1//Hold, r2//Hold}, {r3//Hold, r4//Hold}} . {{6}, {8}} // ReleaseHold
res[[1]]
Out= {172, 160, 172}
matrix tensors
ste12ste12
$\begingroup$ {2, 4}.Transpose[{{{1, 2, 3}, {3, 2, 1}}, {{2, 3, 1}, {2, 1, 3}}}, {1, 3, 2}].{6, 8} $\endgroup$ – J. M. is in limbo♦ Apr 25 '13 at 15:20
$\begingroup$ @J.M.: Nice, thanks. Shall I delete this question or do you want to add it as the answer? $\endgroup$ – ste12 Apr 25 '13 at 15:34
$\begingroup$ I can write an answer with some elaboration, if you don't mind a bit of a wait, or unless some other enterprising user beats me to it. $\endgroup$ – J. M. is in limbo♦ Apr 25 '13 at 15:41
$\begingroup$ @J.M. I have something minor to post but I certainly don't intend to "beat you to it." Shall I wait to post? $\endgroup$ – Mr.Wizard♦ Apr 25 '13 at 15:43
$\begingroup$ @Mr. Wizard, you know I'm a slow typist. :) Nevertheless, I've long been used to getting scooped by the quicker young-ins... so I was already half-expecting somebody had written an answer in the interim. $\endgroup$ – J. M. is in limbo♦ Apr 25 '13 at 16:36
As it stands, the dimensions of the tensor {{{1, 2, 3}, {3, 2, 1}}, {{2, 3, 1}, {2, 1, 3}}} are compatible with the vector on the left, but not on the right. We thus have
{3, 4}.{{{1, 2, 3}, {3, 2, 1}}, {{2, 3, 1}, {2, 1, 3}}}
{{11, 18, 13}, {17, 10, 15}}
and we now want to form the product of this with {6, 8} in some way. One could certainly do Transpose[{{11, 18, 13}, {17, 10, 15}}].{6, 8}, but the more appropriate way would be to match the dimensions of the original tensor with not only the left vector, but with the right vector as well.
Thus, one might initially do
Transpose /@ {{{1, 2, 3}, {3, 2, 1}}, {{2, 3, 1}, {2, 1, 3}}}
{{{1, 3}, {2, 2}, {3, 1}}, {{2, 2}, {3, 1}, {1, 3}}}
and now it is clear one can take left- and right-products of this tensor with $2$-vectors. A better route, though, is to note that the previous operation was exactly equivalent to swapping out levels $2$ and $3$, and thus we have the more compact
Transpose[{{{1, 2, 3}, {3, 2, 1}}, {{2, 3, 1}, {2, 1, 3}}}, {1, 3, 2}]
With this, we finally have
{2, 4}.Transpose[{{{1, 2, 3}, {3, 2, 1}}, {{2, 3, 1}, {2, 1, 3}}}, {1, 3, 2}].{6, 8}
{172, 160, 172}
J. M. is in limbo♦J. M. is in limbo
This seems simpler than your method with Hold:
Block[{r1, r2, r3, r4},
{2, 4}.{{r1, r2}, {r3, r4}}.{{6}, {8}}
{{172, 160, 172}}
Ways to compute inner products of tensors
Mr.Wizard♦Mr.Wizard
This really is an inner product, and can be written straightforwardly using the Inner command:
Inner[Times, Transpose[Inner[Times,{2, 4},{{r1, r2}, {r3, r4}}, Plus]], {6, 8}, Plus]
Not the answer you're looking for? Browse other questions tagged matrix tensors or ask your own question.
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Residual transpiration as a component of salinity stress tolerance mechanism: a case study for barley
Md. Hasanuzzaman1, 4,
Noel W. Davies2,
Lana Shabala1,
Meixue Zhou1,
Tim J. Brodribb3 and
Sergey Shabala1Email author
BMC Plant BiologyBMC series – open, inclusive and trusted201717:107
https://doi.org/10.1186/s12870-017-1054-y
Accepted: 6 June 2017
While most water loss from leaf surfaces occurs via stomata, part of this loss also occurs through the leaf cuticle, even when the stomata are fully closed. This component, termed residual transpiration, dominates during the night and also becomes critical under stress conditions such as drought or salinity. Reducing residual transpiration might therefore be a potentially useful mechanism for improving plant performance when water availability is reduced (e.g. under saline or drought stress conditions). One way of reducing residual transpiration may be via increased accumulation of waxes on the surface of leaf. Residual transpiration and wax constituents may vary with leaf age and position as well as between genotypes. This study used barley genotypes contrasting in salinity stress tolerance to evaluate the contribution of residual transpiration to the overall salt tolerance, and also investigated what role cuticular waxes play in this process. Leaves of three different positions (old, intermediate and young) were used.
Our results show that residual transpiration was higher in old leaves than the young flag leaves, correlated negatively with the osmolality, and was positively associated with the osmotic and leaf water potentials. Salt tolerant varieties transpired more water than the sensitive variety under normal growth conditions. Cuticular waxes on barley leaves were dominated by primary alcohols (84.7–86.9%) and also included aldehydes (8.90–10.1%), n-alkanes (1.31–1.77%), benzoate esters (0.44–0.52%), phytol related compounds (0.22–0.53%), fatty acid methyl esters (0.14–0.33%), β-diketones (0.07–0.23%) and alkylresorcinols (1.65–3.58%). A significant negative correlation was found between residual transpiration and total wax content, and residual transpiration correlated significantly with the amount of primary alcohols.
Both leaf osmolality and the amount of total cuticular wax are involved in controlling cuticular water loss from barley leaves under well irrigated conditions. A significant and negative relationship between the amount of primary alcohols and a residual transpiration implies that some cuticular wax constituents act as a water barrier on plant leaf surface and thus contribute to salinity stress tolerance. It is suggested that residual transpiration could be a fundamental mechanism by which plants optimize water use efficiency under stress conditions.
Residual transpiration
Osmolality
Osmotic potential
Leaf water potential
Cuticular waxes
Under optimal conditions plants lose typically 95–98% water from the leaf surface via stomatal pores in a process termed stomatal transpiration. However, under some environmental conditions, a relatively large portion of evaporated water may bypass the stomata and occur through the cuticle. Depending on the species and conditions, water loss through the cuticle can be as high as 28% of the water transpired through stomata [1, 2]. Moreover, some water can escape the leaf via stomata even when they are fully closed [3, 4]. Because of this, using the term "cuticular transpiration" is not always appropriate, and this process is best described as "residual transpiration". It has been estimated that leaf cuticular water permeability varies extensively among species and ranges from 10−7 to 10−4 m s−1 [2, 5]. Residual transpiration is usually localized to the area surrounding stomata, where there are more and larger cuticular pores [6]. While stomatal conductance is a dynamic process that can be rapidly controlled by ion fluxes into/out of guard cells, residual transpiration depends almost entirely on the existing (passive) lipophilic cuticular pathway of the leaf surface, and, hence cannot rapidly be adjusted to changing conditions [7, 8]. However, when stomata are closed under salinity or drought conditions, the balance between stomatal and non-stomatal transpiration is shifted. Under severe stress conditions, when stomata are closed and stomatal transpiration is reduced to nearly zero, the difference in residual transpiration becomes a significant factor determining water use efficiency. Thus, reducing non-stomatal (residual) transpiration is a potentially useful mechanism for improving plant performance under stress conditions. Genotypes having lower residual transpiration can conserve relatively more water under water stress conditions, and it has therefore been suggested as a selection trait in the breeding of cereals genotypes adapted to a dry environment [9, 10].
Cuticular wax is the outermost hydrophobic layer of the aerial plant tissues, and plays an important role in protecting plants against biotic and abiotic environmental stresses, and acts as a barrier to excessive non-stomatal transpiration [11]. The main functions of cuticular waxes include maintaining equilibrium between the transpirational water loss and root water uptake by transpiration control, defending against attack by insects and pathogens, reducing water retention on plant surfaces by controlling surface wettability, controlling loss and uptake of polar solutes, and regulating the exchange of gases and vapour [12]. Extraction of cuticular waxes from plant parts with organic solvent increases the cuticular water permeability indicating that the wax layer is a fundamental water transport-limiting barrier of the cuticle, especially when stomata are closed [13]. Some reports suggested that plants that have a thicker cuticle or a cuticle containing a larger amount of waxes are more efficient in reducing non-stomatal transpiration and thus better adapted to water stress conditions [14], and in some species total wax loads increased by 30 to 70% under water stress conditions [15]. However, the correlation between residual transpiration and the thickness of cuticle and/or amount of total cuticular waxes is still not clear-cut. Some researchers found that the total amount of cuticular waxes and cuticular thickness are negatively correlated with residual transpiration in different plants [16–19]. However, some authors reported no correlation between residual transpiration and waxes [2, 20, 21].
Residual transpiration could be influenced by the characteristics of the leaf surface and morphological structure of the plant. Some studies argued [2] that residual transpiration did not relate to the amount of wax coverage and thickness of the cuticle but could be dependant on physical properties, orientation of wax crystal structure and wax composition. It is not clear however if this conclusion can be extrapolated to all species. The cuticle layer is a cutin-rich domain with embedded polysaccharides and an overlying layer that is less abundant in polysaccharides but enriched in waxes referred to as the cuticle proper [11]. The waxes are either deposited within the cutin matrix known as intracuticular wax or accumulate on its surface known as epicuticular wax crystals, or films. Cuticular waxes is a general term for the complex mixture of homologous series of very-long-chain fatty acids, primary n-alcohols, secondary n-alcohols, n-aldehydes, n-alkanes, n-alkyl esters, and cyclic organic compounds like pentacyclic triterpenoids, flavonoids, tocopherols and hydroxycinnamic acids derivatives [22]. Specific chemical compounds of the cuticle may be related to the water barrier. Higher levels of nonpolar long chain aliphatic wax compounds of cuticular wax such as hydrophobic alcohols, n-alkanes, and aldehydes tend to be associated with a barrier against cuticular water loss while alicyclic wax components including triterpenoids and sterol derivatives are less effective as a water barrier [23–26].
It is also not clear whether residual transpiration is only related to the cuticular wax on the leaf surface or it is also associated with the plant water relations. It was suggested that residual transpiration is correlated with leaf water status such as leaf water content, osmotic potential and leaf water potential [9]. Other evidence however shown that residual transpiration is not related to relative water content or osmotic potential [27].
The objectives of this study were to investigate the effect of residual transpiration on salinity tolerance and the relationship of residual transpiration to plant water relations, and cuticular wax load at three different leaf positions under irrigated conditions of two salt tolerant and two salt sensitive barley genotypes.
Four barley (Hordeum vulgare L.) genotypes contrasting in their salt tolerance were used in this study. Cultivars Franklin and Gairdner were salt sensitive and failed to produce any grain when grown under highly saline (300 mM NaCl) conditions in the glasshouse [28], while cultivars TX9425 and ZUG293 were salt tolerant and managed to produce ~30% increased grain yield (compared with control) under same conditions. Seeds were obtained from the Australian Winter Cereal Collection and multiplied in the field at Tasmanian Institute of Agriculture facilities in Launceston. Seeds were surface sterilized with 10% commercial bleach and thoroughly rinsed with tap water, and sown in 2 L plastic pots using standard potting mixture containing 70% composted pine bark; 20% coarse sand; 10% sphagnum peat; Limil at 1.8 kg m−3, dolomite at 1.8 kg m−3. The plant nutrient balance was maintained by adding the slow release Osmocote Plus™ fertilizer (at 6 kg m−3), plus ferrous sulphate (at 500 g m−3). Plants were grown under controlled glasshouse condition (day length 14 h; day/night temperatures 25/15 °C; relative humidity 65%) at the University of Tasmania (Hobart, Australia) in January 2015. The plants were irrigated automatically twice per day.
Residual transpiration measurement
Two different methods were used for the determination of residual transpiration from the excised leaf under dark conditions as follows:
Method-1
Residual transpiration was determined following Clarke and McCaig [29] with modification. Three fully expanded leaves from each genotype at three positions (old leaf, intermediate leaf and young flag leaf) were selected for sampling (Fig. 1a). The leaves were excised and sealed with vacuum grease on the cut end immediately. Then collected leaves were immediately transported to the laboratory. Fresh weights (W0) were determined by an electronic balance. The leaves were then placed in a controlled dark room at 20–21 °C and 50% relative humidity (RH). The leaves were weighed at 2, 4 and 6 h (W2, W4 and W6 respectively) intervals and then placed in dry oven at 60 °C for 24 h and reweighed (Wd). Residual transpiration was measured per dry weight basis by using the following formula
$$ \mathrm{Residual}\ \mathrm{transpiration}=\frac{\left({\mathrm{W}}_0-{\mathrm{W}}_2\right)+\left({\mathrm{W}}_2-{\mathrm{W}}_4\right)+\left({\mathrm{W}}_4-{\mathrm{W}}_6\right)}{3\times {\mathrm{W}}_{\mathrm{d}}\left({\mathrm{T}}_2-{\mathrm{T}}_1\right)} $$
Quantifying the residual transpiration (RT) from leaves of three different positions in barley. a sampled leaves; b, c RT values measured from leaves of three different positions from 4 barley varieties contrasting in salinity stress tolerance by Method-1 and Method-2, respectively. Data is mean ± SE (n = 6). d mean RT values for plants in salt-tolerant (ZUG293, TX9425) and salt-sensitive (Gairdner, Franklin) groups estimated by two different methods. Data labelled with different lower case letters in panels (b) and (c) are significantly different at P < 0.05
where T1-T2 = time interval between two subsequent measurements (2 h).
The measured residual transpiration was then recalculated per projected leaf area basis and expressed in mg H2O cm−2 h−1.
Residual transpiration was measured according to Clarke and co-authors [9] with modification. Leaf sampling was the same as for Method-1. Initial weights were determined immediately after excision of leaves. The leaves were maintained in darkness for stomatal closure under ambient room conditions at 20–21 °C and 50% RH. The leaves were weighed again after 24 h. The leaves were dried at 60 °C for 24 h and then dry weight was determined. Residual water loss was determined per dry weight basis by using the following formula
$$ \mathrm{Residual}\ \mathrm{transpiration}=\frac{\left({\mathrm{W}}_{\mathrm{i}}-{\mathrm{W}}_{\mathrm{d}}\right)-\left({\mathrm{W}}_{24}-{\mathrm{W}}_{\mathrm{d}}\right)}{{\mathrm{W}}_{\mathrm{d}}} $$
where Wi = Initial fresh weight; W24 = Fresh weight after 24 h; Wd = Dry weight
The measured residual water loss was then recalculated per leaf area basis and expressed in mg H2O cm−2.
Measurement of leaf osmolality and osmotic potential
Three leaves at three leaf position e.g. old, intermediate and young flag leaves were taken from each genotype. Representative leaf samples were taken in centrifuge tubes and frozen at −20 °C overnight and then squeezed to extract sap. An amount of 10 μl sap was taken from each sample for measuring leaf osmolality (c) using a vapour pressure osmometer (Vapro model 5520, Wescor Inc., Logan, Utah). The osmotic potential was calculated by Van't Hoff's equation from the osmolality (mmol kg−1): osmotic potential (MPa) = −c (mmol kg−1) × 2.4789 × 10−3 at 25 °C.
Measurement of leaf water potential
Two leaves were excised from each genotype from three positions of the stem for leaf water potential determinations. The leaf blades were cut with a sharp blade and immediately sealed in an elliptical grass compression gland gasket. The leaf blades were sealed in a pressure chamber (Model 615; PMS Instruments, Albany, OR, USA), and the chamber was pressurized using compressed air at a rate of 0.1 MPa s−1 until water first appeared at the cut surface of the leaf. The total elapsed time from when the leaf was cut from the plant to the initial pressurisation of the chamber was 5–10 s. The leaf water potential data were reported in MPa.
Scanning electron microscopy (SEM)
After sampling the leaves were stored at −20 °C overnight and then lyophylised in a pre-cooled freeze drier (Mini-ultra cold, Dynavac, Aus, Techno lab). The dried samples (3–5 mm long) were mounted on SEM specimen stubs with double-sided carbon tape (one half with adaxial and the other with abaxial surface uppermost) and then coated with a thin film (2–3 nm) of Pt for 20 min using a sputter coater (BalTec SCD 050) in an atmosphere of argon to improve the electrically conducting properties of leaf and high resolution of images. Three replicates of coated samples were examined with a Hitachi SU-70 UHR field emission scanning electron microscope setting with 1.5 kV, 17.2 mm × 2.00 k SE (M). The imaging was performed in the Central Science Laboratory, University of Tasmania.
Wax extraction and analysis
Three fresh leaves at three positions of the plant from each genotype were excised and ten 0.64 cm2 disks were sampled from each by leaf punch.. The leaf segments were soaked in 5 mL of solvent (dichloromethane with n-docosane (C22 alkane, 20 mg/L) as an internal standard) for 5 min with gentle stirring [30]. The extract contained waxes from both abaxial and adaxial leaf surfaces. The extracts were evaporated to dryness under a nitrogen stream for 30 min at 58 °C. The samples were redissolved in 0.5 ml dichloromethane for analysis by combined gas chromatography-mass spectrometry (GC-MS) on a Varian 3800 gas chromatograph coupled to a Bruker-300 triple quadrupole mass spectrometer. One microlitre injections in splitless mode were made with an injector temperature of 275 °C. The column was a 30 m × 0.25 mm DB5 (0.25 μm film thickness) (Agilent, Australia) and the oven temperature program was 60 °C (held for 1 min) to 220 °C at 30 °C per minute, then to 310 °C at 10 °C per minute with a final hold time of 5 min. The carrier gas was helium at a constant flow of 3.5 ml min−1. Mass spectra were collected over the range m/z 40 to 600 every 0.3 s. Compounds were identified through a combination of MS reference databases (NIST MS database and an in-house database of relevant compounds), and Kovats' retention indices. The individual components and total wax were expressed in terms of μg equivalents of n-docosane cm−2. All subsequent μg cm−2 values are in terms of n-docosane equivalents in the text and figures.
All data were analyzed by using SPSS 20.0 for Windows (SPSS Inc.). Significant differences between different genotypes were determined by one-way analysis of variance based on Duncan's multiple range tests. Different lower case letters in the figures represent significant differences. The significance of correlations between different parameters was determined by bivariate correlations based on Pearson's correlation (two-tailed).
As both stomata density and amount of cuticular waxes depends on the leaf age, we hypothesised that a significant variation in residual transpiration should exist between leaves of different positions. A significant variation was seen in the different leaf positions for all varieties (P < 0.05; Fig. 1a and b). Old leaves transpired more water than the intermediate and flag leaves for all varieties using both methods. In Method-1, significant variation was observed between old leaves and intermediate leaves but not in intermediate and flag leaves in most genotypes. Old leaves of TX9425 (0.74 ± 0.04 mg H2O cm−2 h−1) genotype transpired the highest amount of water and Franklin transpired the lowest amount of water (0.36 ± 0.02 mg H2O cm−2 h−1). In Method-2, significant differences were seen between the three leaf position in all genotypes. Old leaves of TX9425 (10.24 ± 0.53 mg H2O cm−2) transpired the highest amount of water followed by old leaves of ZUG293 (8.01 ± 0.48 mg H2O cm−2), Gairdner (6.88 ± 0.52 mg H2O cm−2) and Franklin (6.02 ± 0.28 mg H2O cm−2), respectively. Young flag leaves of TX9425 (5.73 ± 0.25 mg H2O cm−2) transpired the highest amount of water followed by ZUG293 (3.68 ± 0.14 mg H2O cm−2), Gairdner (3.02 ± 0.17 mg H2O cm−2) and Franklin (2.86 ± 0.12 mg H2O cm−2), respectively. Salt tolerant varieties transpired more water through the cuticle than that of sensitive varieties under normal growth conditions (Fig. 1c). The cumulative loss of water of the three leaf positions of two tolerant genotypes (TX9425 and ZUG293) was higher than two sensitive genotypes (Gairdner and Franklin) in both methods. The two tolerant genotypes transpired 43% and 32% more water respectively than the two sensitive genotypes in the two methods under normal growth condition.
Leaf sap osmolality correlates negatively with residual transpiration
A significant difference of leaf sap osmolality was observed among different leaf positions (P < 0.05; Fig. 2a). Leaf sap osmolality decreased with increasing leaf age for all genotypes. The osmotic potential was highest in old leaf and lowest in flag leaf in all genotypes (P < 0.05; Fig. 3a). The highest decrease (60%) was observed in TX9425 followed by ZUG293 (43%), whereas the lowest decrease (20%) was measured in Franklin followed by Gairdner (28%), in old and young leaves respectively. A strong negative correlation (R2 = −0.86 for Method-1 and -0.92 for Method-2; significant at P < 0.01) was found between the overall leaf sap osmolality in plants grown under normal growth conditions and residual transpiration.
a genetic variability in osmolality of barley leaves at three positions in plants grown under normal (no salt) growth condition. Mean ± SE (n = 6). b correlations (Pearson's R2 values) between leaf sap osmolality and residual transpiration measured by two different methods. Data labelled with different lower case letters are significantly different at P < 0.05 and asterisks are significant at P < 0.01
a genetic variability in osmotic potential of barley leaves at three positions in plants grown under normal (no salt) condition. Mean ± SE (n = 6). b correlations (Pearson's R2 values) between leaf osmotic potential and residual transpiration measured by two different methods. Data labelled with different lower case letters are significantly different at P < 0.05 and asterisks are significant at P < 0.01
Osmotic potential and leaf water potential correlate positively with residual transpiration
The osmotic potential was highest in old leaves and lowest in flag leaves in all genotypes (P < 0.05; Fig. 3a). ZUG293 and TX9425 followed the order old > intermediate > young flag leaf, whereas Franklin and Gairdner followed old > intermediate = young flag leaf. A strong positive correlation (R2 = 0.86 for Method-1 and 0.92 for Method-2; significant at P < 0.01) was found between the overall leaf osmotic potential in plants grown under normal growth conditions and residual transpiration. A significant variation of leaf water potential was found among the three leaf positions in all four genotypes (P < 0.05; Fig. 4a). Leaf water potential increased with increasing the plant leaf age, the highest and lowest leaf water potential was found at old leaf and young flag leaf, respectively. A positive correlation (R2 = 0.59; significant at P < 0.01) was found (in Method 2) between the overall leaf water potential in plants grown under normal growth condition and residual transpiration.
a genetic variability in water potential of barley leaves at three positions in plants grown under normal (no salt) growth condition. Mean ± SE (n = 6). b correlations (Pearson's R2 values) between leaf water potential and residual transpiration measured by two different methods. Data labelled with different lower case letters are significantly different at P < 0.05 and asterisks are significant at P < 0.05
Structure and distribution of cuticular waxes on leaf epidermis
SEM analysis showed similar cuticular waxes structure in three different leaf positions of four barley genotypes. The cuticular waxes formed combined coatings of different arrangement of minute crystallised plates about 1–2 μm in size, relatively vertically oriented to the leaf epidermal surface (Fig. 5; Additional file 1: Figure S1). Cuticular wax structures were a less dense covering of adaxial surface of old leaves compared to the intermediate and young flag leaves for all genotypes. The epidermis of three different leaf positions of four genotypes was covered with waxy plates, but not fully over the guard cell of all genotypes (Fig. 6). In the case of TX9425 and ZUG293 genotypes, the guard cells of stomata were not fully covered with waxy plates, whereas the guard cells of Franklin and Gairdner were fully covered with waxy plates. No differences were found for adaxial and abaxial surface of leaves in all genotypes regarding to cuticular wax structure and density (data not shown).
Representative SEM images showing cuticular wax on the adaxial surface in leaves of three different positions in variety Franklin grown under control condition. One (of six) typical images is shown for each position
Representative SEM images showing cuticular wax on the adaxial surface of the flag leaf in barley varieties ZUG293 (1), TX9425 (2), Franklin (3) and Gairdner (4) grown under control conditions. One (of six) typical images is shown for each genotype
Total wax content of leaves correlates negatively with residual transpiration
A significant negative correlation (R2 = −0.41 for Method-1 and -0.34 for Method-2; significant at P < 0.05) was found between the total cuticular wax content of leaves and residual transpiration measured by two different methods in plants grown under normal growth conditions (Fig. 7a).
a correlations (Pearson's R2 values) between total cuticular wax and residual transpiration measured by Method-1 (mg H2O cm−2 h−1) and Method-2 (mg H2O cm−2). b correlations (Pearson's R2 values) between alcohols and residual transpiration measured by Method-1 (mg H2O cm−2 h−1) and Method-2 (mg H2O cm−2). Data labelled with asterisks are significant at P < 0.05
Cuticular wax constituents, contents and effect on residual transpiration
Across all four barley varieties the average of total leaf cuticular wax was found to be 5.37 μg cm−2 under normal growth condition. The averages of total cuticular wax of old leaves, intermediate leaves and flag leaves of all genotypes studied were 5.06 μg cm−2, 5.06 μg cm−2 and 5.98 μg cm−2 respectively. Cuticular waxes on barley leaves were dominated by primary alcohols (84.7–86.9%), aldehydes (8.90–10.1%), n-alkanes (1.31–1.77%), benzoate esters (0.44–0.52%), a phytol related compound (0.22–0.53%), fatty acid methyl esters (0.14–0.33%), β-diketones (0.07–0.23%) and alkylresorcinols constituents (1.65–3.58%). Primary alcohols consisted of odd and even numbers of carbon from C22 to C29, particularly n-docosanol (C22), n-tetracosanol (C24), n-hexacosanol (C26), and n-octasonanol (C28), and much smaller amount of odd numbered carbons. The higher n-alkane component on barley leaf consisted mainly of n-hentriacontane (C31) and n-tritriacontane (C33). The main aldehydes were n-hexacosanal (C26), n-octacosanal (C28) and n-triacontanal (C30). Benzoate esters included n-docosyl benzoate (C22), n-tetracosyl benzoate (C24) and n-hexacosyl benzoate (C26). Major fatty acid methyl esters were methyl n-octacosanoate (C28), methyl n-triacontanoate (C30) and methyl n-dotriacontanoate (C32).
Old leaves for all genotypes studied showed the average highest absolute amount of alcohols (4.39 μg cm−2) followed by aldehydes (0.45 μg cm−2) and the lowest β-diketones (Table 1). Similar results were found at intermediate and flag leaves for all genotypes (Tables 2 and 3; Additional file 2: Table S1). Among the genotypes, ZUG293 old leaves contained the highest amount of alcohols followed by Franklin. The same results were found for intermediate leaf for all genotypes (Table 2; Additional file 2: Table S1). For flag leaves of all genotypes the average highest alcohols were measured from Franklin followed by ZUG293 (Table 3).
Absolute amount (μg cm−2) of different compounds of cuticular wax on old leaf position of four barley genotypes grown under normal growth conditions (n = 4)
ZUG293
Gairdner
5.48 ± 0.16
Benzoate esters
Phytol related
Methyl esters
Diketones
0.00 ± 00
Alkylresorcinols
0.00 ± .00
Absolute amount (μg cm−2) of different compounds of cuticular wax on intermediate leaf position of four barley genotypes grown under normal growth conditions (n = 4)
Absolute amount (μg cm−2) of different compounds of cuticular wax on flag leaf position of four barley genotypes grown under normal growth condition (n = 4)
A negative significant correlation (R2 = −0.44 for Method-1; P < 0.05 and R2 = −0.36 for Method-2; significant at P < 0.05) was found between residual transpiration and primary alcohols of cuticular wax component of barley genotypes (Fig. 7b). No significant correlations were found between residual transpiration measured by two different methods and other cuticular wax components (Table 4).
Correlations (Pearson's R2 values) between residual transpiration measured by two different methods and different cuticular wax compounds of three different leaf positions of four barley genotypes grown under normal growth condition. Values labelled with asterisk are significant at P < 0.05
R2 values with residual transpiration
R2 Value
Benzoates
Phytols
Residual transpiration and plant water relations
To maintain proper growth and leaf expansion, the growing shoot needs to maintain positive turgor which can be achieved by maintaining osmotic cellular adjustment by either increasing the production of compatible solutes or inorganic ions. As plants accumulate more organic osmolytes in young leaves than old leaves to maintain turgor pressure [31], it was hypothesised that residual transpiration should be less in young leaves due to the fact that they have higher osmolality and hence better water retention, and this was found to be the case. As shown in Fig. 2a and b, young flag leaves had a higher osmolality than the older leaves, and increased osmolality had a strong negative correlation with the residual transpiration under normal growth conditions indicating that the increase of leaf sap osmolality might decrease the water transpiration through plant cuticle. An effective osmotic adjustment mechanism may maintain water status in the leaf tissue by decreasing in the cell sap osmotic potential resulting from a net increase of intracellular solutes [32].
A leaf can increase its resistance to dehydration through a reduction in cellular osmotic potential by a net accumulation of cellular solutes. In this study, young flag leaves possessed significantly lower osmotic potential than the intermediate and older leaves; a trend that was correlated positively with residual transpiration (Fig. 3a and b). This indicated that a leaf with lower osmotic potential had more turgor pressure to spend and could resist greater loss of water through the cuticle. Lower negative leaf water potential was measured with increasing leaf age for all varieties, which was negatively correlated with residual transpiration (Fig. 4a and b). Young leaves maintained less turgor at more negative leaf water potentials and tended to have less residual transpiration. Increased turgor in the epidermis stretches cuticles and causes a change in gas exchange of the cuticle. A leaf with less turgor would have a tighter cuticle, thus inhibiting gas exchange [33]. Burghardt and Riederer [14] observed that cuticle gas exchange was affected when leaf water potentials decreased. Thus, leaf water potential affects the diffusion of water vapour through the cuticular barrier, and residual transpiration is negatively correlated with lower leaf water potential [33].
Change in residual transpiration to improve water use efficiency
Salinity stress is often referred to as a "physiological drought", so some correlation between salinity and drought stress tolerance is expected. The most salinity tolerant varieties showed the highest residual transpiration under unstressed conditions (Fig. 1d). Being somewhat counterintuitive, this is in a good agreement with Bengston et al. [34] who showed that drought stress resistant oat genotypes generally transpired the highest amount of water through the cuticle under unstressed conditions, whereas it was strongly reduced under stress conditions. In addition, higher (33 to 38%) residual transpiration in wheat and cotton leaves was reported from irrigated than rainfed field-grown wheat plants [9]. On the other hand, deposition of cuticular waxes increased in tolerant genotypes during prolonged drought stress, leading to a reduced rate of residual transpiration [16, 35].
Water use efficiency can be expressed as the ratio of leaf net carbon assimilation to total transportation water loss. Plants exhibit higher water use efficiency with higher CO2 assimilation than the stomatal conductance, when non-stomatal water loss is negligible [36]. Salt tolerant genotypes transpired more water through cuticle under well irrigated condition that reveals their water use efficiency is lower than sensitive genotypes. Generally stress tolerant barley genotypes have a lower biomass and yield performance under control conditions [37]. This could be due to their higher non-stomatal transpiration under irrigated conditions resulting in lower water use efficiency. Conversely, tolerant genotypes could reduce residual water loss under water deficit conditions when stomata are closed and/or partially closed, and this increased water use efficiency could be a significant factor determining their survival capacity to hostile environmental conditions compared to the standard cultivated genotypes. It has been documented that wheat genotypes having lower residual transpiration adapted and performed better under water stress conditions [38]. Genotypes with normally low residual transpiration are at a functional advantage in water-limited environments since they make more efficient use of the water available. Thus, under conditions of water deficit, residual conductance to water vapour may be an important determinant of plant water balance and stress reactivity.
On the other hand, transpiration is the most effective way of leaf cooling of well-irrigated plants. In plants with adequate water supply stomata may regulate leaf temperature close to the optimum for metabolic processes, including photosynthesis or to prevent tissue heat damage under excessive radiation or temperature [39]. Moreover, under water limited conditions, stomatal closure and decreased transpiration, associated with high water use efficiency, may lead to a dramatic increase in leaf temperature (up to 7 °C above air temperature) [40]. At this condition, high temperatures may disrupt the photosynthetic-related enzymes and produce reactive oxygen species which would challenge the plant cell [41].
Relationship between residual transpiration and amount of cuticular waxes
Our working hypothesis in this study was that reduced residual transpiration should be positively correlated with hydrophobicity of the leaf surface (hence, amount of cuticular waxes deposited). A significant negative correlation (Fig. 7a) between the total amount of cuticular wax and residual transpiration was found in the present investigation, which indicated that amount of cuticular wax may create a protecting barrier to reduce the loss of water through the cuticle. Previous studies have reported a weak but significant negative correlation between the cuticular wax and residual transpiration in sorghum [18], wheat [17], and barley [42]. This weak correlation may be due to the protecting barrier to the diffusion of water through the cuticle depends on the structure, orientation of wax plates on epidermis, variation of epicuticular and intracuticular wax compositions and distribution of wax plates. Both intracuticular [43] and epicuticular [44] wax layer may contribute to the formation of residual transpiration barrier depending on the plant species and specific cuticle constituents. Plants generally exhibited a significant increase in the amount of cuticular wax amount per unit area of leaves under different stress condition such as water deficit and salinity [20]. The quantity of cuticular wax, however, is not the sole contributor to residual transpiration due to the complexity of water flow through the cuticle [45].
Cuticular waxes have different types of structural morphology including granules, filaments, plates and tubes [12]. According to the SEM images analysis, plate type cuticular wax observed on the leaf surface consisted of aliphatic compounds in which the primary alcohols n-hexacosanol and n-octacosanol were predominant in different leaf positions for all the barley genotypes.
Cuticular waxes on barley leaves consisted of alcohols, aldehydes, alkanes, benzoate esters, phytol related compounds, fatty acid methyl esters, β-diketones and alkylresorcinols (Tables 1, 2 and 3). Generally, plate type primary alcohol based cuticular waxes always dominate on the leaf surface in the Fabaceae and Poaceae (wheat, barley) [42] and constitute the major barrier to water loss. This was also the case in our study reported here (Fig. 7b) [45]. However, such findings could be not generalized to all species. The hydrophobic long chain alcohol, hydrocarbon and aldehyde fractions are the active components of cuticle in controlling residual transpiration in different plant species [44]. The main portion of the transpiration barrier in tomato fruits and Rhazya stricta leaves is located in the intracuticular wax layer containing large amount of pentacyclic triterpenoids whereas cuticular very long chain aliphatics play a minor role [46, 47]. Plant species containing fatty acid with very long aliphatic chain (alcohols, aldehydes and alkanes) in the epicuticular wax, together with high amount of alicyclic compounds such as triterpenoids, steroids, or tocopherols in the intracuticular wax contribute equally to the formation of residual transpiration barrier (44). In general, it is accepted that higher levels of long chain aliphatic components in the wax can lead to a higher hydrophobicity of the residual transpiration barrier and thus decrease cuticular water loss [26]. This should be kept in mind while targeting this trait in the breeding programs.
Both leaf osmotic potential and the amount of cuticular waxes are involved in controlling water loss from barley leaves under well irrigated conditions. A significant and negative relationship between the amount of primary alcohols and cuticular transpiration implies that primary alcohols may influence the water barrier more than other constituents on plant leaf surface and thus contribute to salinity stress tolerance, at least in barley.
RH:
We cordially thank Dr. Sandrin T. Feig and Dr. Karsten Goemann from the Central Science Laboratory, University of Tasmania, for the assistance with SEM. We also thank Norhawa Puniran-Hartley, Joseph Hartley, Koushik Chakraborty and Akhikun Nahar for their help with freeze drying samples and wax extraction.
This work was funded by the Grain Research and Development Corporation (GRDC) grants to Sergey Shabala and Meixue Zhou. The funders had no role in study design, data collection, analysis and interpretation of data and in writing the manuscript.
All the data pertaining to the present study has been included in table and/or figure form in the present manuscript and authors are pleased to share analyzed/raw data and plant materials upon reasonable request.
MH conducted the bulk of experiments and wrote the paper draft. SS was responsible for experimental design and data interpretation, and took the leading role in writing. NWD was leading the cuticular wax analysis, critically assessed all the data and commented on the manuscript. LS, MZ and TJB provided a logistical support for this work and contributed to data interpretation and writing. All authors read and approved the final manuscript.
Additional file 1: Figure S1. SEM images showing cuticular wax on the adaxial surface in three different positions of leaf in varieties ZUG293 (A), TX9425 (B) and Gairdner (C) grown under control conditions (PPTX 4472 kb).
Additional file 2: Table S1. Amount (μg cm−2) of different components of cuticular wax in three different positions of leaf of four barley genotypes (n = 4) (XLSX 12 kb).
School of Land and Food, University of Tasmania, Private Bag 54, Hobart, Tas, 7001, Australia
Central Science Laboratory, University of Tasmania, Hobart, Tas, 7001, Australia
School of Biological Science, University of Tasmania, Private Bag 55, Hobart, Tas, 7001, Australia
Department of Agronomy, Faculty of Agriculture, Sher-e-Bangla Agricultural University, Sher-e-Bangla Nagar, Dhaka, -1207, Bangladesh
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Plant-abiotic interactions
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CommonCrawl
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Find and prove the limit of $X_n=$ $\frac {n^{100}}{1.01^n}$
I have to find and prove the limit of the sequence $X_n=$ $\frac {n^{100}}{1.01^n}$
What is the easier way?
I tried to use Bernoulli's inequality to say lim$\frac {n^{100}}{1.01^n}$ = lim$\frac {n^{100}}{(1+1/10)^n}$ and $ (1+1/10)^n \geq 1+n(1/10).$ But I could get anything. I think another way is to use the Squeeze Theorem but I have not could find the correct sequences.
I only can use the definition, Squeeze Theorem, the Bernoulli's inequality or using operations to reduce the sequence.
calculus sequences-and-series limits
nurun nesha
Alex TurnerAlex Turner
$\begingroup$ You now have five people who have answered your question. Do any of the answers provided successfully answer your question? If so, (this goes for all the other questions you've asked on MSE as well) you should click the check mark next to the answer that you feel best answered your question. This will let other MSE users know that your question has an answer, and it will give some points to the user who provided the answer. $\endgroup$ – graydad Oct 21 '15 at 5:16
$\begingroup$ You accepted the only "wrong" answer :) Well, it's not wrong but it doesn't give you the limit of the sequence. $\endgroup$ – lcv Nov 13 '15 at 6:23
We have $$\frac {n^{100}}{1.01^{n}} = \left(\frac{n}{1.01^{n/100}}\right)^{100} $$ By Bernoulli's inequality on the inside quantity, $$\begin{align}\left(\frac{n}{\left(1+\frac{1}{100}\right)^{n/100}}\right)^{100} &\leq \left(\frac{n}{1 + \frac{1}{100}\cdot\frac{n}{100}}\right)^{100} \\ &< \left(\frac{n}{\frac{n}{10000}}\right)^{100} \\ &= 10000^{100} \end{align}$$
graydadgraydad
Theorem. If $a,b > 0$, then $(\log x)^{a}/x^{b} \to 0$ as $x \to \infty$.
Corollary. If $a,b > 0$, then $x^{a}/e^{xb} \to 0$ as $x \to \infty$.
The corollary follows immediately from the fact that formally we have $x^{a}/e^{xb} = (\log t)^{a}/t^{b}$ and the theorem.
Take $a := 100$ and $b:= \log (1.01)$; then the statement that you want to prove follows directly.
To prove the theorem, just note that, given any $a,b > 0$, we have $x \geq 1$ only if for all $c > 0$ we have $$ \frac{(\log x)^{a}}{x^{b}} = \frac{\big( \int_{t=1}^{x}\frac{1}{t} \big)^{a}}{x^{b}} \leq \frac{\big( \int_{t=1}^{x} t^{c-1} \big)^{a}}{x^{b}} = \frac{(x^{c} - 1)^{a}}{c^{a}x^{b}} < \frac{x^{ac}}{c^{a}x^{b}}; $$ then take $c := \frac{b-1}{a}$.
MegadethMegadeth
When you are dealing with sequences it is better to use a theorem related to sequences. We will use the following result:
Theorem: If $0 < r < 1$ then $r^{n} \to 0$ as $n \to \infty$ and if $r > 1$ then $r^{n} \to \infty$ as $n \to \infty$.
Also it is best to analyze the sequence $a_{n} = n^{k}/a^{n}$ in general where $k, a$ are positive real numbers. For the current question we have $a = 1.01$ and $k = 100$.
We have $$\frac{a_{n + 1}}{a_{n}} = \frac{1}{a}\left(\frac{n + 1}{n}\right)^{k} \to a\cdot 1^{k} = \frac{1}{a}$$ as $n \to \infty$. We will show that if $0 < a \leq 1$ then $a_{n} \to \infty$ and if $a > 1$ then $a_{n} \to 0$ as $n \to \infty$. Note that when $a = 1$ then $a_{n} = n^{k}$ so that $a_{n} \to \infty$ as $n \to \infty$.
Let's start first with the case $a > 1$. Then clearly $b = 1/a < 1$ and therefore there is a number $r$ such that $b < r < 1$. Since $a_{n + 1}/a_{n} \to b$ as $n \to \infty$, it follows that there is a positive integer $m$ such that $a_{n + 1}/a_{n} < r$ for all $n \geq m$. Hence we can see that $$0 < \frac{a_{m + n}}{a_{m}} = \frac{a_{m + 1}}{a_{m}}\cdot\frac{a_{m + 2}}{a_{m + 1}}\cdots\frac{a_{m + 1n}}{a_{m + n - 1}} < r^{n}$$ Since $0 < r < 1$ it follows that $r^{n} \to 0$ as $n \to \infty$ and hence applying Squeeze theorem to the above equation we see that $$a_{m + n}/a_{m} \to 0$$ and therefore $a_{m + n} \to 0$. It follows that $a_{n} \to 0$. For the current question we have $a = 1.01$ so that $a > 1$ and hence it is covered by this case and the desired limit is $0$.
In exactly the same way we can show that if $0 < a < 1$ then $a_{n} \to \infty$. For this case we have $b = 1/a > 1$ and we need to choose $1 < r < b$ and use the fact that $a_{n + 1}/a_{n} > r$ for all large values of $n$. The result will follow because in this case $r^{n} \to \infty$.
The above proof is pretty standard (and should be available in any textbook which deals with limit of sequences) and its beauty is that it does not make use of exponential and logarithm functions (and no differentiation/L'Hospital's Rule)
Paramanand SinghParamanand Singh
Since $$ e^x> \frac{x^{101}}{101!} \quad \forall x \ge 0, $$ we have $$ 1.01^n=e^{n\ln1.01} >\frac{n^{101}\ln1.01}{101!} \quad \forall n\ge 1. $$ Therefore $$ 0<\frac{n^{100}}{1.01^n}<\frac{101!}{\ln1.01}\cdot\frac1n \quad \forall n\ge 1. $$ It follows that $$ 0\le\lim_{n\to\infty}\frac{n^{100}}{1.01^n}\le \lim_{n\to\infty}\frac{101!}{\ln1.01}\cdot\frac1n=0, $$ i.e. $$ \lim_{n\to\infty}X_n=0. $$
Mercy KingMercy King
We will show that $\displaystyle \lim_{n\to\infty}\frac{n^k}{1.01^n}=0$ for all $k$. In particular, $k=100$ is the sequence you are asking about.
When $k=0$, we have $\lim_{n\to\infty}\frac{1}{1.01^n}$, which is $0$ since $1.01^n$ increases without bound.
Suppose that $k>0$ and that $\displaystyle \lim_{n\to\infty}\frac{n^{k-1}}{1.01^n}=0$. Since $\displaystyle \lim_{n\to\infty}\frac{n^k}{1.01^n}$ is an $\frac{\infty}{\infty}$ indeterminate form, we may apply l'Hopital's rule to see that the limit is equal to $\displaystyle \lim_{n\to\infty}\frac{kn^{k-1}}{\ln(1.01)1.01^n}=\frac{k}{\ln 1.01}\lim_{n\to\infty}\frac{n^{k-1}}{1.01^n}$, which equals zero by hypothesis.
By induction, the limit is zero for all $k\geq 0$.
Kyle MillerKyle Miller
$\begingroup$ L'Hospital's is not allowed for this problem $\endgroup$ – graydad Oct 21 '15 at 3:23
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$x_n$ is the smallest number for which $\sum_{i=1}^{x_n}\frac{1}{i}\geq n$ .Find $\lim_{n \to \infty}\frac{x_{n+1}}{x_n}$.
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CommonCrawl
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The pupil responds spontaneously to perceived numerosity
The size of the attentional window when measured by the pupillary response to light
Shira Tkacz-Domb & Yaffa Yeshurun
Numerosity tuning in human association cortices and local image contrast representations in early visual cortex
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Pupil dilation but not microsaccade rate robustly reveals decision formation
Christoph Strauch, Lukas Greiter & Anke Huckauf
Elisa Castaldi1,2,
Antonella Pomè2,
Guido Marco Cicchini3,
David Burr ORCID: orcid.org/0000-0003-1541-88322,4 &
Paola Binda ORCID: orcid.org/0000-0002-7200-353X1
158 Altmetric
Object vision
Although luminance is the main determinant of pupil size, the amplitude of the pupillary light response is also modulated by stimulus appearance and attention. Here we ask whether perceived numerosity modulates the pupillary light response. Participants passively observed arrays of black or white dots of matched physical luminance but different physical or illusory numerosity. In half the patterns, pairs of dots were connected by lines to create dumbbell-like shapes, inducing an illusory underestimation of perceived numerosity; in the other half, connectors were either displaced or removed. Constriction to white arrays and dilation to black were stronger for patterns with higher perceived numerosity, either physical or illusory, with the strength of the pupillary light response scaling with the perceived numerosity of the arrays. Our results show that even without an explicit task, numerosity modulates a simple automatic reflex, suggesting that numerosity is a spontaneously encoded visual feature.
Being able to efficiently estimate the number of enemies or prey is essential for survival: all animals1—from insects to humans—are capable of some form of numerosity discrimination. Much evidence suggests that numerosity discrimination is a basic and spontaneous sense, often referred to as the number sense2,3. For example, in monkeys and crows, single neurons are tuned to numerosity, before any training on numerosity tasks4,5,6; in humans, crude number discrimination has been documented as early as a few hours after birth7,8. Neural representation of numerosity is organized within a topographic principle, such as that for most sensory attributes9,10. Visual processing of numerosity is rapid, with numerosity-specific signals emerging in the occipital cortex only 75 ms after stimulus onset11, and humans can saccade rapidly to the more numerous target, as quickly as 190 ms12, implicating primitive, possibly subcortical, circuitry that quickly transforms numerosity information into an oculomotor response. This and other literature point to numerosity being a salient perceptual feature, eliciting a spontaneous perceptual response.
The pupillary light response is one of the most basic sensory responses, serving primarily to regulate light entry and aid dark adaptation13,14. However, even when luminance is kept constant, pupil size can vary predictably and systematically with the effective strength of stimulation. For example, attending to a bright or dark patch will enhance the pupillary response evoked by the patch15,16. Moreover, brightness17 and size18 illusions, and even the implied brightness of images of the sun and moon19,20, can elicit strong and reliable pupillary constriction. That pupil size is susceptible to attention, visual illusions, and to the semantic content of images suggests that the subcortical structures that control pupil size receive modulatory signals from higher-level areas21,22.
Here we exploit the pupillary light response to study the spontaneous nature of numerosity perception and show that it scales with perceived numerosity. Pupils constrict more to passively observed white arrays and dilate more to black arrays for more numerous (luminance matched) stimuli, whether perceived numerosity was manipulated by dot number or by exploiting a grouping-based illusion. This suggests an implicit association between numerosity and perceptual strength, which can be read out—objectively and quantitatively—from the pupil.
The connectedness effect
We measured pupil size, while participants passively viewed arrays of dots of different numerosities, real or perceived, leveraging on a strong numerosity illusion. Figure 1A, B show examples of the white versions of the stimuli: the experiments used both dark and bright stimuli, black and white dots and lines on a gray background. Using two versions of the connectedness illusion (panels A and B) helps to dismiss various potential artifacts (discussed later).
Fig. 1: Stimuli.
A Examples of white stimuli used for experiment 1: stimuli comprised either 18 or 24 dots, which were either connected by lines or isolated, with the lines displaced to random positions. The stimuli with black dots were identical. Participants simply maintained gaze on a central fixation point without performing any task, while the stimuli were displayed for 6 s. B Stimuli for experiment 2. The lines in the connected condition are thinner and in the isolated condition they are absorbed into the dots, which become slightly larger (about 40%). See "Methods" for further details. C Fourier amplitude (arbitrary units) of the stimuli of Fig. 1A, as a function of spatial frequency, over the range 0.3–10 cycles per degree. D Same as C, for the stimuli of Fig. 1B. Source data are provided as a Source Data file.
The stimuli varied both in physical and perceived numerosity: they comprised either 18 or 24 dots, and in half of the stimuli perceived numerosity was reduced by connecting pairs of dots to form dumbbells, a well-known illusion23,24,25, obvious on inspection of Fig. 1. In the other half, the connecting lines were either displaced to random positions between the dots (experiment 1) or removed entirely (experiment 2). In both experiments, all stimuli had the same total number of white or black pixels, irrespective of numerosity and connectedness, covering 92.7 deg2 for experiment 1 and 82.3 deg2 for experiment 2. The area, defined as convex hull, was 513 deg2 for all stimuli. Figure 1C, D show the Fourier transforms of the stimuli, discussed later.
After completing the first pupillometry experiment (where they were completely naive of the goals of the experiment), participants were asked to judge which of the two sequentially presented arrays appeared to comprise more dots. One was the standard (18 isolated dots), the other the probe (connected or isolated: see "Methods" for experimental details). Figure 2 shows example psychophysical functions for the two experimental conditions (displaced or removed dots) in a typical participant. The median of the functions (0.5 response) yields the point of subjective equality (PSE), the numerosity of the probe that matched the reference. For the displaced-lines condition, the PSE was around 13, 28% less than for measurements with the isolated dots. For the removed-lines condition, the PSE changed a little less, to 14, about 23%. Figure 2B, D show the results averaged over all participants. The average bias in the connected patterns was about 30% for displaced-line stimuli and 22% for removed-line stimuli, similar to that reported by previous studies23,24,25. Given the magnitude of the effect, we would expect the perceived numerosity of a 24 connected-dot pattern to be around 17 in experiment 1 and 19 in experiment 2, similar to that of the unconnected 18-dot pattern.
Fig. 2: Psychophysics results.
A Typical psychometric curves for one participant, measuring the connectedness effect on perceived numerosity for experiment 1. The proportion of trials in which the probe pattern was reported to be more numerous than the standard (which comprised 18 isolated dots) is plotted as a function of the number of dots in the probe pattern. Black and gray lines refer to the connected and isolated conditions, respectively. Arrows indicate the PSE measured in the two conditions. Leftward shifts of the psychometric curve imply underestimation of the test arrays. B Perceptual bias expressed as percentage of PSE difference from the reference numerosity for the two connectedness conditions. Bars represent average across participants (N = 14 participants, individual data reported as black and gray circles for the connected and isolated conditions, respectively), error bars show 1 SEM. C, D Example psychometric curve and average perceptual bias measuring the connectedness effect on perceived numerosity for experiment 2 (N = 13 participants, open circles). Bars represent average across participants and error bars show 1 SEM. The results from the first experiment (A, B) were replicated in the second one (C, D). Source data are provided as a Source Data file.
Pupillary light and dark responses
We recorded pupil size while participants passively observed the stimuli of Fig. 1, which were repeatedly displayed for 6 s. Trials with isolated and connected stimuli were intermingled in pseudo-random order within the same session, with separate sessions for different colors and numerosities. We ran two separate experiments, using the two types of isolated controls, with connecting lines displaced to random positions, or removed.
Figure 3A shows the average baseline-corrected pupillary responses for experiment 1 (displaced-lines stimuli), to all stimulus types, both white and black. As expected, the pupil constricted for white-dot stimuli and dilated for black-dot stimuli. The light-evoked constriction was predictably faster than the dark-evoked dilation21, but combining the two (by subtracting the light from the dark response) yielded a strong and sustained luminance response over the 6 s stimulus presentation (Fig. 3B). Importantly, although the total number of pixels (hence luminance) was always the same in all four conditions, the amplitude of the pupil-size modulation clearly varied across conditions.
Fig. 3: Pupil responses to different numerosities and connectedness levels.
A Pupil dilation or constriction in response to black or white stimuli for the displaced-lines condition (experiment 1). Color-coded lines refer to the different numerosity and connectedness levels defined in the legend to C. The gray shaded area on the abscissa between the two vertical dashed lines show time of stimulus presentation. Thick lines plot the time courses of the pupil size averaged across participants (N = 16 participants) and the shaded area around them the SEM. Thin color-matched lines show the average fit obtained by convolving stimulus predictors with the pupil response function (defined by parameters listed in Supplementary Table 1). B Time course of the pupil difference (dark minus light responses) estimating the net pupillary response to luminance. C Average pupil difference in the interval 1–6 s after stimulus onset (the stimulus duration minus the first second, which we excluded to discard the fast and transient pupillary onset response). Dots represent averages across participants (N = 16 participants) and error bars are SEM. D–F Same as A–C, for the removed-lines condition (experiment 2). Open circles represent averages across participants (N = 13 participants) and error bars are SEM. Repeated-measures ANOVAs tested for pupil differences between numerosity and connectedness levels. Significance of main effects and interactions are reported in text. The results from the first experiment (A–C) were replicated in the second one (D–F). Source data are provided as a Source Data file.
We used the mean pupil difference (Fig. 3B) over the 1–6 s interval to index the pupil response strength. These values are shown in Fig. 3C (for individual data, see Supplementary Fig. 1). The stimulus with the highest perceived numerosity (24 isolated dots) elicited the strongest response, and that with the lowest perceived numerosity (18 connected dots) elicited the weakest response (paired t-test comparing 24 isolated vs. 18 connected: t(15) = 4.5, p < 0.001, Cohen's d = 1.1, log10BF = 1.9); 18 isolated and 24 connected dots, which had similar apparent numerosity, elicited an intermediate response. Thus, the strength of the pupillary luminance response depended on numerosity, both the physical numerosity and the perceived numerosity of an illusory pattern. A two-way analysis of variance (ANOVA) for repeated measures (two numerosities, two connectedness levels) confirmed that the pupil response was modulated by both factors (main effect of connectedness: F(1,15) = 20.5, p < 0.001, ηp2 = 0.6, log10BF = 0.8; main effect of numerosity: F(1,15) = 6.2, p = 0.025, ηp2 = 0.3, log10BF = 0.8), which did not interact (F(1,15) = 0.13, p = 0.72, log10BF = −0.5).
Figure 3D, E show the pupillary time courses for experiment 2 (using the alternative form of isolated-dot stimuli, with lines removed), which were similar to those of experiment 1. The responses to bright and dark (Fig. 3D), their difference (Fig. 3E) for the four classes of stimuli, and the average difference over the fixed window (1–6 s, Fig. 3F) were all similar to those above them. Again, the difference in pupillary response was strongest for 24 isolated and weakest for the 18 connected (t(12) = 2.9, p = 0.014, Cohen's d = 0.8, log10BF = 0.6), and intermediate for 24 connected and 18 isolated. Results of the two-way ANOVA for repeated measures were as follows: main effect of connectedness F(1,12) = 6.7, p = 0.024, ηp2 = 0.4, log10BF = 0.5; main effect of numerosity F(1,12) = 2.6, p = 0.13, log10BF = 0; and connectedness by numerosity interaction: F(1,12) = 0.005, p = 0.95, log10BF = −0.5. The reduced effect sizes compared to experiment 1 may be related to the pupillary responses being generally weaker (possibly due to the lower total number of white or dark pixels) and to the illusion being marginally reduced (compare Fig. 2B, D).
Averaging pupil size over a fixed time window (here 1–6 s) is a standard technique of pupillometry. However, it raises the possibility that the results could depend on the length of the time window. We therefore implemented an additional analysis, modeling the pupil time courses, assuming that they resulted from the linear combination of three predictors convolved with the pupil response function26. The pupil response function was estimated for individual participants using their average response time course across conditions (for the parameters of the pupil response function, see Supplementary Table 1). The three predictors were stimulus appearance and disappearance (two impulse-like functions), and time on screen (a boxcar function); their weights represent the strength of the transient onset/offset responses and the sustained pupillary response. Given the estimated pupil response function, we fitted pupil traces from each condition and participant, yielding β-weights representing the contribution of each predictor to the observed pupillary response. Average best fit curves are shown in Fig. 3 (thin lines); their goodness of fit was generally excellent (83% variance explained averaged across participants and experiments, see Supplementary Table 1).
We focused on the sustained response, considering pupil difference traces (difference between black and white stimuli) to estimate the impact of numerosity on the net pupillary response to luminance. β-Weights for the sustained predictor varied across conditions, reinforcing the results obtained by taking the simple mean of the response over the stimulus window (see Supplementary Tables 2 and 3). β-Weights were highest for 24 isolated dots, lowest for 18 connected dots (experiment 1: t(15) = 3.8, p = 0.002, Cohen's d = 0.9, log10BF = 1.4; experiment 2: t(12) = 2.2, p = 0.047, Cohen's d = 0.6, log10BF = 0.2) and intermediate for 18 isolated or 24 connected dots. For displaced-line stimuli (experiment 1): main effect of connectedness F(1,15) = 17.1, p < 0.001, ηp2 = 0.5, log10BF = 0.5; main effect of numerosity F(1,15) = 3.1, p = 0.09, log10BF = 0.3; connectedness by numerosity interaction: F(1,15) = 0.1, p = 0.79, log10BF = −0.5). For removed-line stimuli (experiment 2): main effect of connectedness F(1,12) = 3.8, p = 0.07, log10BF = 0.2; main effect of numerosity F(1,12) = 2.4, p = 0.15, log10BF = −0.2; connectedness by numerosity interaction: F(1,12) = 0.17, p = 0.69, log10BF = −0.4.
Potential artifacts
Displays with higher perceived numerosity show stronger pupil responses, both dilation to black stimuli and constriction to white stimuli. Thus, any potential artifact that predicts unidirectional pupil changes, constriction or dilation, is unlikely to confound our results.
Although the total number of pixels, and therefore mean luminance, did not vary between conditions in each experiment, the manipulations obviously caused small variations in spatial frequency content, which could have affected the pupillary response21. Figure 1C, D plot the spatial frequency amplitude spectra for the four conditions, separately for the two experiments. Although there are no large differences in amplitude, there are some subtle differences at certain frequencies that could potentially confound the results. However, these small differences are not consistently in the direction needed to explain the effects in both experiments. For example, for experiment 1 (displaced-lines), the amplitude is higher for the isolated conditions, those that elicit stronger pupillary responses. However, the situation reverses for the stimuli of experiment 2 (removed lines), where the amplitude is higher for the connected conditions, associated with a smaller pupillary response. The Fourier amplitude can therefore not explain the results of both experiments, which was a major motivation for using both types of stimulus manipulations.
It is also possible that the eye-movement patterns could have been different for the different classes of stimuli, which would have affected pupillary responses, possibly driving the main effects. We therefore calculated the bivariate confidence interval area for each participant and condition, and analyzed these values with a repeated-measure ANOVA, with connectedness and numerosity levels as factors. None of the main effects or interactions were significant in either of the two pupillometry experiments, suggesting that unstable fixation did not contribute to the results (see Supplementary Information, Supplementary Fig. 2A, B).
A further possible artifact is that stimuli with higher perceived numerosity have higher perceived brightness, which in turn drives the pupillary response, as has been reported for images of the sun and moon19. This seems unlikely, as there are no obvious differences in apparent brightness on inspection of the stimuli of Fig. 1. Nevertheless, we measured perceived brightness in our participants with a forced choice psychophysical technique (see Supplementary Information). The results (Supplementary Fig. 2C) reveal no large, or even significant differences in apparent brightness between the four conditions, excluding the possibility that it is driving the pupillary response.
In this study, we investigated whether pupil size is spontaneously modulated by numerosity, for stimuli of identical luminance. Although participants were not required to judge numerosity, or any other aspect of the stimuli, the magnitude of pupil responses evoked by white or black dots systematically scaled with perceived numerosity: the higher the perceived numerosity, the stronger the pupil response to the stimulus luminance. The pupillary response scaled down when either the physical numerosity was decreased (from 24 to 18) or when the dots were joined by lines to form dumbbells, reducing the perceived numerosity. When the lines joining the dots were displaced or removed, the pupillary response increased. We used two different analyses: quantifying pupil size within a fixed time window (delimited by the stimulus presentation) and fitting the pupil traces on the basis of a physiologically motivated general linear model26. Both analyses produced similar results, with both types of stimuli.
The numerosity-driven pupil-size modulation observed in this experiment could not be explained by luminance differences across the stimuli, as the total number of white or black pixels was always exactly matched across all conditions, of both experiments. Nor could the pupillary response be driven by differences in perceived brightness, as has been reported for images of the sun and moon19. Psychophysical measurements revealed no statistically significant differences in apparent brightness between the four conditions of our experiment. Although we cannot rule out the possibility that numerosity may affect apparent brightness in some very subtle ways, we can conclude that any potential effects were not robust enough to be driving the responses reported here.
The pupillary response was not driven by differences in the Fourier spectrum, as the spectra were very similar over most of the visible range (Fig. 1C, D) and varied in opposite ways for the stimuli of experiments 1 and 2. Other potential artifacts may include differences in mental effort or memory load. Large differences in these factors are unlikely under the passive viewing conditions of the experiment. In any case, neither of these factors (Fourier power or mental effort) could explain the pattern of the data, as they predict a unidirectional pupil size change: constriction or dilation, respectively27,28. This could not account for enhanced constrictor and dilatator pupillary response to light and dark stimuli, suggesting multiplicative scaling of the effective stimulus strength.
The results of our study suggest that patterns perceived as more numerous are spontaneously represented as perceptually stronger and consequently evoke stronger pupillary responses. This fits well with previous evidence suggesting that humans spontaneously encode numerosity. Humans are far more sensitive to numerosity changes than to changes in area or density when asked to identify the odd-one-out of three dots arrays without instructions on the aspect the stimuli may differ29,30. Similarly, numerate adults, children, innumerate adults, and monkeys used numerosity rather than other non-numerical dimensions to classify arrays of dots as "little" or "a lot"31. The current results are also in line with other studies suggesting that numerosity is a salient visual feature that is difficult to ignore32, and can drive automatic oculomotor orientation responses12.
The mechanisms by which perceived numerosity enhances the pupillary response are far from clear. One possibility is that the effect is mediated by attention, given that pupillary responses are known to be enhanced when attention is directed towards stimuli of different luminance15,16. This is certainly feasible, although it is not obvious why participants should spontaneously attend to higher perceived numerosities and maintain attention for the whole 6 s period. Whatever the underlying mechanism, the results show that stimuli of higher perceived numerosity have a higher salience, or stimulus strength, and this is reflected in the gain of the pupillary response.
The pupillary response to luminance is one of the simplest sensory responses. Pupil control is completely involuntary33, with luminance regulation mediated by a simple subcortical circuit that starts from the retina and sends light flux information to the pupillomotor Edinger-Westphal (EW) nuclei of the tectum via the olivary pretectal nucleus (OPN)14. Clearly, the modulation of luminance responses by perceptual and cognitive factors, such as brightness and size illusions, or attention shifts15,16,17,18, implies cortical modulation of this circuit. A recent neurophysiological study in monkeys showed that electrical micro-stimulation of the prefrontal cortex (specifically of the frontal eye field), which is implicated in attentional control, modulated the pupillary light reflex in a spatial- and temporal-specific manner34. This result suggests that the prefrontal cortex may exert control over the mesencephalic pupil light reflex circuit (OPN and EW), either through direct feedback signals or indirectly via relay stations in the occipital cortex and/or in the superior colliculus22. It is well documented that prefrontal and occipito-parietal structures support numerosity processing35,36,37, even in passive viewing paradigms38,39, and there is also evidence that numerosity may be already coded at subcortical levels12,40. Moreover, electroencephalogram and functional magnetic resonance imaging studies have shown that the effect of connectedness on numerosity perception emerges as early as 150 ms after stimulus onset, at the level of V341, and continues in the parietal cortex42. These cortical and subcortical connections could support the spontaneous modulation of the pupillary response to luminance with perceived numerosity.
In conclusion, our results reinforce previous work suggesting that numerosity is a primary visual attribute, which spontaneously modulates one of the more basic sensory responses, the pupil light response. Pupillometry may prove to be an effective tool to study numerical cognition, providing a quantitative and objective index that tracks this perceptual process. The paradigm is simple, requires no specific training, is task free, and responses are recorded automatically without the need for invasive experimenter intervention. These features make it a potential candidate for future studies on populations and species for which psychophysical testing has proved difficult or unfeasible.
Sixteen participants (six males, mean age: 30 ± 3 years) with normal or corrected to normal vision participated in the study. All participants took part to the first pupillometry experiment, 14 of these participated in the numerosity discrimination experiment of Fig. 2, and 13 in the second pupillometry and psychophysics experiment. The research was approved by the local ethics committee (Commissione per l'Etica della Ricerca, University of Florence, n. 111 dated 7 July 2020) and was in accordance with the Declaration of Helsinki. All participants gave written informed consent prior to the study.
Stimuli and apparatus
For each experiment, 8 types of stimuli were used, illustrated in Fig. 1: 2 numerosities (18 and 24), connected into dumbbell-like shapes or isolated (either by displacing or removing the connecting lines). All stimulus types could be either black or white (maximum and minimum luminances on the screen, about 12.6 and 256 cd/m2), presented on a gray background of 129.3 cd/m2. We ran two separate experiments, with slightly different stimuli and hence different Fourier spectra (Fig. 1), to confront several potential artifacts. In both experiments, all stimuli had the same number of pixels, covering 92.7 deg2 for experiment 1 and 82.3 deg2 for experiment 2. For experiment 1 (Fig. 1A), the connecting lines were displaced to random positions on the screen in the isolated-dot condition: dot diameter, 2.2° for N18, 1.9° for N24; line width, 1.03°; line length 2°–3°. For experiment 2 (Fig. 1B), the connecting lines were removed in the isolated condition and their pixels absorbed into the dots: dot diameter, 2.2° for N18, 1.9° for N24 in connected condition; 2.44°–2.64° (N18) and 2.10°–2.28° (N24) in isolated condition; line width, 0.6°; line length, 1.6°–2.8°.
To match the total number of pixels in the arrays (and hence luminance), as well as the covered area (convex hull), stimuli were precalculated offline through multiple steps. In the first step, the isolated stimuli were created: coordinates were randomly generated for twice the number of the desired dot positions, half for the dots and the other half for the lines, with the constraint that neither dots nor lines could be closer than 0.5°. In a second step, the connected stimuli were created by connecting randomly chosen couples of dots, with line length as detailed above. In the third step, the total number of displayed pixels and the convex hull of the isolated dots was modified to match those of the connected stimuli. For the stimuli of experiment 2, dots were enlarged so that the overall ink of the isolated stimulus matched those of the dumbbells (which contained the connectors). For stimuli of experiment 1, the length of the isolated lines was iteratively adjusted until it matched the total number of pixels of the connected stimuli (which would otherwise contain less pixels due to the over imposition of the lines with the dots at the point of connection between the two). Then the position of the isolated dot pairs was iteratively modified until the convex hull matched that of the connected stimuli (which would otherwise have had higher probability of appearing sparser than the isolated stimuli). The final convex hull was 513 deg2.
Connected and isolated dots were intermixed within a session, whereas arrays of different numerosity or color (white or black) were presented in separate sessions. Order of trials and sessions was varied pseudo-randomly across participants. Each participant performed 4 sessions of 60 trials each. Trials started with a fixation point shown for 1 s, followed by the presentation of one of the numerosity arrays, which remained visible for 6 s. Trials ended 1 s after stimulus disappearance, giving a 2 s interstimulus interval. Importantly, participants were instructed to keep their gaze on the fixation point and simply observe the stimuli, without performing any task.
Measurements were made in a quiet dark room with participants sitting in an experimental booth surrounded by thick black curtains so that the only light source was the stimulus display (Liquid Crystal Display monitor screen, 1280 × 720 pixels, refresh rate 60 Hz). Participants sat at 57 cm from the screen, with head stabilized by chin rest. Pupil diameter was monitored at 500 Hz with an EyeLink 1000 system (SR research) with infrared camera mounted below the screen, recording from the left eye. Before each session, eye position was linearized by a standard nine-point calibration routine. Stimuli were generated and presented under Matlab using Psychtoolbox-3 routines43.
Psychophysics
A subset of participants took part in psychophysical experiments to establish the strength of the illusion (after completing the first set of pupillometry measurements and usually on a different day) and a brightness comparison task as well (see Supplementary Information). For the numerosity measurements, two arrays of dots (test and reference stimuli) were presented sequentially in a randomized order for 500 ms in central fixation and participants judged which was the more numerous. In separate sessions, stimuli were either all black or all white, in analogy with the pupillometry experiment. The test stimulus of experiment 1 comprised isolated dots and lines, of variable numerosity (8, 10, 12, 14, 16, 20, 22, or 24 dots, with half the number of lines), whereas the test stimulus of experiment 2 comprised only isolated dots, of variable numerosity (spanning from 5 to 12 dots). The reference stimulus (randomly presented first or second) had a fixed number of 18 dots and 9 lines, which could be either isolated or connected for experiment 1, whereas only the connected condition was tested for experiment 2. For each condition (color and connectedness), participants were tested with 3 sessions of 40 trials each. The participant was asked to report whether the first or second array had more dots, by pressing the corresponding key.
Pupillometry data were preprocessed to exclude blinks or signal losses. Specifically, we excluded time points where pupil size was unrealistically small (< 0.1 mm), where pupil size changes where unrealistically large (> 1 mm from the median of the trial), or too quick (any pupil size changes that were faster than 25 mm/s were treated as artifacts and the surrounding 20 ms window were excluded from the analyses). Data points that passed this quality check were down-sampled at 20 Hz and then high-pass filtered by convolving the trace with a 500 ms square window. The resulting pupil size time courses were baseline-corrected by subtracting the average pupil diameter in the 200 ms preceding the stimulus onset. To statistically compare pupil size changes across conditions, we averaged the pupil size over a period of 1–6 s after the stimulus onset, i.e., over the stimulus presentation window excluding the first second (which included the pupil size fast response triggered by the stimulus appearance). These values were analyzed with two-way repeated-measures ANOVA, with numerosity and connectedness level as factors.
We additionally modeled the pupil time courses with a General Linear Model approach, assuming that pupil responses resulted from the linear combination of three predictors (two impulse-like and one boxcar functions, each normalized to its integral) convolved by the pupil response function. The three predictors corresponded to three components known to contribute to pupil size21: a transient pupil modulation at the onset and offset of the stimulus and a sustained response lasting for the entire stimulus presentation time window. It has been proposed that these components are mediated by different mechanisms and have different characteristics: the transient pathway is characterized by poor spatial summation, bandpass temporal response, and high-contrast gain, whereas the sustained pathway exhibits large spatial summation, low-pass temporal response, and low-contrast gain21. The transient predictors (impulse-like functions at stimulus onset and offset) were to capture the grating response, which has previously been reported to be generated by visual transients21,28. The sustained predictor (a boxcar function representing stimulus duration) captures a steady-state luminance response, which is primarily driven by illumination but can be affected by perceptual and cognitive factors44.
The pupil response function was modeled by a Gamma function h(t):
$$h\left(t\right)=\frac{{(\frac{t-\delta }{\tau })}^{(n-1)}{{{{{{\rm{e}}}}}}}^{-(\frac{t-\delta }{\tau })}}{\tau \left(n-1\right)!}$$
where n was the number of filters, \(\tau\) the change rate, and \(\delta\) the delay in pupil response after stimulus onset. For each participant we found the best-fitting parameters n, \(\tau\), and \(\delta\) (each constrained in an appropriate interval: from 1 to 8, 10 to 800, and 0 to 200, respectively, see Supplementary Table 1). Once we defined the pupil response function that best fitted the average pupil trace across conditions, we estimated the β-weights for each condition separately and computed the goodness of fit of each of them. We then statistically compared the β-weights across conditions by entering them in two-way repeated-measure ANOVA with numerosity and connectedness level as factors.
To analyze the results of the numerosity discrimination task, for each condition (black or white arrays, connected or isolated), the responses were plotted as function of the probe numerosity and fitted with a cumulative Gaussian distribution, whose median defines the PSE (see Fig. 2C). We then computed bias index as:
$${{{{{\mathrm{Bias}}}}}}=100* \left(\frac{{{{{\mathrm{PSE}}}}}}{N}-1\right)$$
Statistical tests (ANOVA, t-tests, and Bayesian analyses) were conducted with Matlab or Jasp 0.14.145. For Bayesian ANOVA, models including the connectedness and numerosity factors, both these factors, and both factors plus the interaction were ordered by their predictive performance relative to the best model. The reported Bayes factors correspond to inclusion Bayes factors resulting from the analysis of the effects across all matched models46. Bayes factors are reported in logarithmic base 10 units (log10BF) and their absolute values should be interpreted as providing anecdotal (0–0.5), substantial (0.5–1), strong (1–1.5), or very strong (>1.5) evidence, in favor of the alternative hypothesis if positive or the null hypothesis if negative.
Reporting summary
Further information on research design is available in the Nature Research Reporting Summary linked to this article.
Data are available at Zenodo (https://doi.org/10.5281/zenodo.5168707)47. Source data are provided with this paper.
Codes and scripts will be provided upon reasonable request from the corresponding author.
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This research has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie (grant agreement number 885672–DYSC-EYE-7T "The neural substrate of numerical cognition in dyscalculia revealed by eye tracking and ultra-high field 7T functional magnetic imaging") and from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programs (grant agreement number 801715–PUPILTRAITS and number 832813–GenPercept).
Department of Translational Research and New technologies in Medicine and Surgery, University of Pisa, Pisa, Italy
Elisa Castaldi & Paola Binda
Department of Neuroscience, Psychology, Pharmacology and Child Health, University of Florence, Florence, Italy
Elisa Castaldi, Antonella Pomè & David Burr
Institute of Neuroscience, National Research Council, Pisa, Italy
Guido Marco Cicchini
School of Psychology, University of Sydney, Camperdown, NSW, Australia
David Burr
Elisa Castaldi
Antonella Pomè
Paola Binda
All authors contributed to the study concept and to the design. Stimuli were designed by G.M.C., E.C., and A.P. Testing and data collection were performed by E.C. and A.P. E.C., A.P., and P.B. performed the data analysis. All authors contributed to the interpretation of results. E.C. drafted the manuscript, and D.B. and P.B. provided critical revisions. All authors approved the final version of the manuscript for submission.
Correspondence to David Burr.
Peer review information Nature Communications thanks Bruno Laeng and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.
Peer Review File
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\(\epsilon \kappa \)-Curves: controlled local curvature extrema
Kenjiro T. Miura ORCID: orcid.org/0000-0001-9326-31301,
R. U. Gobithaasan ORCID: orcid.org/0000-0003-3077-87722,
Péter Salvi ORCID: orcid.org/0000-0003-2456-20513,
Dan Wang ORCID: orcid.org/0000-0003-1906-49981,
Tadatoshi Sekine ORCID: orcid.org/0000-0003-1813-822X1,
Shin Usuki ORCID: orcid.org/0000-0002-4363-63461,
Jun-ichi Inoguchi ORCID: orcid.org/0000-0002-6584-57394 &
Kenji Kajiwara ORCID: orcid.org/0000-0002-0543-93845
The Visual Computer volume 38, pages 2723–2738 (2022)Cite this article
The \(\kappa \)-curve is a recently published interpolating spline which consists of quadratic Bézier segments passing through input points at the loci of local curvature extrema. We extend this representation to control the magnitudes of local maximum curvature in a new scheme called extended- or \(\epsilon \kappa \)-curves.\(\kappa \)-curves have been implemented as the curvature tool in Adobe Illustrator® and Photoshop® and are highly valued by professional designers. However, because of the limited degrees of freedom of quadratic Bézier curves, it provides no control over the curvature distribution. We propose new methods that enable the modification of local curvature at the interpolation points by degree elevation of the Bernstein basis as well as application of generalized trigonometric basis functions. By using \(\epsilon \kappa \)-curves, designers acquire much more ability to produce a variety of expressions, as illustrated by our examples.
Avoid the most common mistakes and prepare your manuscript for journal editors.
The \(\kappa \)-curve, proposed recently by [28], is an interpolating spline which is curvature-continuous almost everywhere and passes through input points at the local curvature extrema. It has been implemented as the curvature tool in Adobe Illustrator® and Photoshop® and is accepted as a favored curve design tool by many designers (see, e.g., [4, 6]).
We consider the reasons for the success of \(\kappa \)-curve to be:
Information along contours is concentrated at local maxima of curvature.
Curves of low degree have smooth distribution of curvature.
\(G^2\)-continuous curves tend to look fairer than only \(G^1\)-continuous ones.
Attneave [1] suggested, based on his empirical study, that information along contours is concentrated in regions of high magnitude of curvature, as opposed to being distributed uniformly along the contour, and it is further concentrated at local maxima of curvature (see also [33]). Although Attneave never published the details of his methods, [20] conducted a similar experiment and obtained the same results. Levien and Séquin [12] argue similarly and assert that points of maximal curvature are salient features.
The curvature of a polynomial curve is given by a relatively complicated rational function [7], and its distribution might not be globally smooth. However, if the curve is of a low degree, the curvature distribution is more uniform and the curve is fairer, thus more suitable for illustration. The quadratic polynomial curve has the nice property that its curvature has only one local maximum, and its location is easily computable [28], which makes the handling of curvature extrema much easier.
Graphic designers often accept \(G^1\) continuity as good enough for illustration. However, discontinuity remains; for example, if you join a straight line and a circular arc with \(G^1\) continuity, the rhythm of the curve will be broken at the joint. For this reason, we give preference to \(G^2\)-continuous curves.
Nonetheless, \(\kappa \)-curves are not perfect, and their further investigation is necessary [29]. The following are two important shortcomings of \(\kappa \)-curves:
They are not curvature-continuous everywhere: at inflection points only \(G^1\) continuity is guaranteed.
Since the degree of freedom (DoF) of the quadratic segments is limited, it is impossible to control the magnitudes of local maximum curvature at the input points.
For the first shortcoming, Wang et al. [26] provided a solution through the use of log-aesthetic curves [15, 17] instead of polynomial curves. Log-aesthetic curves have a shape parameter (\(\alpha \)), which can be utilized to control the curvature distribution as shown in Fig. 1. Their method guarantees \(G^2\) continuity everywhere, including inflection points, since log-aesthetic curves with negative \(\alpha \) values can represent S-shaped curves with \(G^2\) continuity. (Note that these cannot be represented by quadratic Bézier curves.)
These curves, however, are defined by a Cesàro equation and thus take extra time to evaluate, making the interpolation method impractical for real-time design purposes.
Because of the second shortcoming, if the designer wants to increase or decrease the magnitude of the curvature extremum, she needs to add extra input points, as shown in Fig. 2. Yan et al. [27] proposed a piecewise rational, quadratic, interpolatory curve that is able to reproduce circles and other elliptical or hyperbolic shapes. Although their main intention was to reproduce circles, their method could also control the magnitude of local maximum curvature. However, only rational quadratic curves are applicable, and it is not possible to extend their method for other types.
In this paper, we propose a new method to solve the second shortcoming by degree elevation of the Bernstein basis functions, giving an extra DoF to each quadratic curve segment, providing control over the magnitudes of local maximum curvature at input points as shown in Fig. 3. In order to increase the designers' possible choices, we also introduce a new trigonometric basis for which we can perform degree elevation. In addition, we propose a general method for bases with extra shape parameters. By adding one more parameter to each of the curve segments, the designers obtain more expressive power for their illustration. The family of this new curve is denoted as \(\epsilon \kappa \)-curves.
Log-aesthetic curves with various \(\alpha \) values [26]. Input points are depicted by black boxes, and green points correspond to positions of local maximum curvature. The blue curves show the normal curvature. These curves are \(G^2\)-continuous everywhere
Addition of extra input points to control the magnitude of curvature extrema: b is the original \(\kappa \)-curve, a, c shows deformed curves with large and small curvature extrema by adding two extra input points
The left- and rightmost curves are \(\kappa \)-curves; the others are \(\epsilon \kappa \)-curves with gradual changes in the global shape parameter a. The face part (for \(a=0.95\) and \(a=2/3\)) is zoomed in for better comparison. When a=2/3, \(\kappa \)- and \(\epsilon \kappa \)-curves are identical
\(\epsilon \kappa \)-curves preserve all of the appealing properties of \(\kappa \)-curves, i.e., point interpolation, \(G^2\) continuity (except at inflection points), continuous modification (changes smoothly when the input points move), local influence, and real-time generation. With a small processing overhead, \(\epsilon \kappa \)-curves offer the ability to control the magnitude of local maximum curvature.
We have implemented \(\epsilon \kappa \)-curves in MATLAB® and Julia [11]. The source of the Julia code is available online [22].
The rest of this paper is organized as follows. Section 2 reviews the related work. Section 3 presents a method to control the magnitudes of local maximum curvature by degree elevation of the Bernstein basis functions. Section 4 introduces a new trigonometric basis and proposes a method with degree elevation similar to the one proposed in the previous section. Finally, we end with conclusions and discussion of future work.
In this section, we first review [28] and their underlying strategy developing \(\kappa \)-curves. Next, we discuss related researches on various kinds of basis function formulations for curve design.
\(\kappa \)-Curve
The basic framework of our method is adopted from [28], generating curves controlled by interpolation points. \(\kappa \)-curves have stimulated the field of interpolatory curve generation, resulting in works such as [5], which proposes a method for good control over the location and type of geometric feature points (e.g., cusps and loops).
Yan et al. [28] create a sequence of quadratic curves with \(G^2\) continuity almost everywhere. They derived explicit formulae for the point between two quadratic segments to guarantee \(G^2\) continuity and for the additional condition that input points should be interpolated at maximum curvature magnitude positions. \(\kappa \)-curves are determined by the locations of the middle control points of quadratic Bézier segments—the rest is easily derived from the continuity constraints. Hence, the variables are the locations of these middle control points.
Their basic strategy to determine these locations is to adopt a local/global approach [13, 23]: \(G^2\)-continuous connection is performed locally, while the interpolation at maximum curvature magnitude positions is done globally.
Regarding control of the magnitude of local curvature, the most common technique is to change the weight of a control point of a rational curve [7]. A larger weight attracts the curve to its control point, which makes local curvature larger. However, this technique is not applicable for interpolatory curves. Another technique is to introduce extra parameters called bias and tension to the B-spline formulation for controlling local curvature [2], but this is also not applicable to interpolatory curves.
Basis functions
As mentioned in [28], curve modeling has a long history, especially in computer-aided geometric design, as well as computer graphics. In CAGD and applied mathematics, to extend the expressive power of curves, many researchers have been trying to develop new bases with extra shape parameters. The following is a (nonexhaustive) list of such bases:
C-Bézier curve [31]
Cubic alternative curve [10]
Cubic trigonometric Bézier curve (T-Bézier basis) [8]
\(\alpha \beta \)-Bernstein-like basis [32]
Quasi-cubic trigonometric Bernstein basis [30]
Trigonometric cubic Bernstein-like basis [24]
Our method proposed in the next section can be applied for curves based not only on polynomials, but also other bases such as a trigonometric basis with degree elevation property which we will introduce in Sect. 4. All of the representations listed above have extra parameters for shape control, and we can utilize these parameters to control the magnitudes of local maximum curvature. (Not all curve types are applicable, however, as explained in Appendix C.)
It is interesting to note that most researchers to date have attempted to develop new bases using four control points, based on cubic polynomials or quadratic trigonometric functions. They prefer using four control points out of concern for connections at both ends of the curve. In order to control the magnitude of curvature at the two ends independently, at least two control points are necessary at each end. This differs fundamentally from Yan et al.'s (and our) approach, which uses only three control points.
The importance of [28] is their proposed paradigm shift for curve generation, by considering the local maximum curvature in the middle, instead of focusing on the endpoints. If we can assume that the curvature has just one local maximum in each segment, then only one extra parameter per segment is adequate to control the magnitude of its local maximum curvature.
To our best knowledge, no trigonometric basis family for arbitrary degree has been published yet. Our novel generalized trigonometric basis functions range from linear, using three control points, to any higher degree n, using \(2n+1\) control points. The curve can be evaluated by a recursive method , similar to de Casteljau's algorithm, as explained in Appendix B. Since the curve uses trigonometric functions as blending functions, it can represent a circular arc exactly, without using a rational form.
Cubic Bernstein polynomials
In this section, we extend \(\kappa \)-curves in a direct manner, by elevating the degree of quadratic Bézier segments to cubic. \(\epsilon \kappa \)-curves retain the following properties of \(\kappa \)-curves:
Interpolate all input points (control points).
All local maximum curvature points are the same as the input points.
\(G^2\) continuity is guaranteed almost everywhere (except for inflection points).
In the following, we discuss only closed curves, but it is straightforward to extend our methods to open curves as has been demonstrated for \(\kappa \)-curves.
If we elevate the degree of a planar Bézier curve, we obtain an additional control point, which has two DoFs (the x and y coordinates). To reduce these to one, we add a geometric constraint on the location of the second and third control points of the cubic Bézier curve, as shown in Fig. 4. Here, a is an internal division ratio, where the larger a is, the closer the control points \(P_1\) and \(P_2\) are to the control point \(Q_1\). We make the restriction \(2/3 \le a < 1\) because the curve should not have a complicated curvature distribution. Using \(Q_i\), the curve \(C(t;a)\) is expressed by
$$\begin{aligned} C(t;a)&= (1-t)^3 Q_0 + 3(1-t)^2t \left[ (1-a)Q_0+a Q_1 \right] \nonumber \\&\qquad + 3(1-t)t^2 \left[ a Q_1+(1-a)Q_2 \right] + t^3 Q_2. \end{aligned}$$
Note that if \(a=2/3\), the curve degenerates to quadratic.
Constrained cubic Bézier curve. If \(a=2/3\), the curve becomes quadratic
We have proved that by constraining the construction of the cubic polynomial curve as in Eq. (1), using only three control points instead of four, the curvature in one curve segment has at most one local maximum for \(2/3 \le a < 1\); see details in [16], as well as Appendix A for a general discussion on the curvature extrema of cubic polynomial curves, and a high-level summary of the proof. In addition, we have made MATHEMATICA simulation available in [9] to compute the number of curvature extrema for this curve using both classical approach and Sturm's theorem. Hence, we can safely assume that the curvature in one curve segment has at most one local maximum, and a single extra parameter for each segment is enough to control the magnitudes of local maximum curvature.
Geometric constraints
We assume that \(\epsilon \kappa \)-curves consist of a sequence of constrained cubic polynomial curves
$$\begin{aligned} c_i(t;a_i)&= (1-t)^3c_{i,0} + 3(1-t)^2t \left[ (1-a_i)c_{i,0}+a_ic_{i,1}\right] \nonumber \\&\quad + 3(1-t)t^2 \left[ a_ic_{i,1}+(1-a_i)c_{i,2}\right] + t^3c_{i,2}, \end{aligned}$$
parameterized by t and also \(a_i\), which is an extra shape parameter. The control points are given by \(c_{i,0}\), \(c_{i,1}\) and \(c_{i,2} \in {\mathbb {R}}^2\), corresponding to \(Q_i\), \(i=0,1,2\) in Fig. 4. The \(a_i\)'s are reserved for designers and can be manipulated independently.
The curvature of this curve \(c_i(t;a_i)\) is given by
$$\begin{aligned} \begin{aligned}&\kappa _i(t; a_i) = {\det \left( \frac{\partial c_{i}(t;a_i)}{\partial t}, \frac{\partial ^2 c_{i}(t;a_i)}{\partial t^2} \right) }\ \big /\ {\left\| \frac{\partial \,c_{i}(t;a_i)}{\partial t } \right\| ^3} \\&\quad = \frac{4}{3}\cdot \frac{\triangle (c_{i,0},c_{i,1},c_{i,2})\left[ a_i^2(1-t)t + a_i(1-a_i)\left( (1-t)^2+t^2\right) \right] }{\Vert (1-t)^2 a_i r_i +2(1-t)t(1-a_i)(r_i+s_i)+ t^2 a_i s_i \Vert ^3 },\nonumber \end{aligned}\\ \end{aligned}$$
where \(\triangle \) indicates the area of the triangle specified by its arguments, and \(r_i = c_{i,1}-c_{i,0}\), \( s_i = c_{i,2}-c_{i,1}\).
In the quadratic case (i.e., \(\kappa \)-curves), as the curvature has such a simple formula, we can express the parameter \(t_i\) at the point of maximal curvature explicitly, in terms of the Bézier coefficients of the \(i^\mathrm {th}\) quadratic Bézier curve as
$$\begin{aligned} t_i = \frac{\langle r_i,r_i-s_i\rangle }{\Vert r_i-s_i \Vert ^2}, \end{aligned}$$
where \(\langle a,b\rangle \) means the scalar product of vectors a and b. Then, we add the condition
$$\begin{aligned} c_i(t_i)=p_i, \end{aligned}$$
where \(p_i\) is the \(i^\mathrm {th}\) input point. Solving for \(c_{i,1}\) and substituting into Eq. (4), we get a cubic equation in \(t_i\) that depends only on the endpoints \(c_{i,0}\) and \(c_{i,2}\), and the input points \(p_i\).
Unfortunately, we cannot obtain an explicit formula like Eq. (4) for the parameter \(t_i\) in the cubic case, because of its high degree (see details in Appendix A.1), but this is not a problem. Solving Eq. (5) for \(c_{i,1}\), we arrive at
$$\begin{aligned} \begin{aligned} c_{i,1}&= \left[ p_i-(1-t_i)^3 c_{i,0}\right. \\&\quad -3(1-t_i)t_i(1-a_i)((1-t_i)c_{i,0}+t_ic_{i,2}) \\&\quad \left. -t_i^3 c_{i,2}\right] \big / (3 a_i (1-t_i) t_i). \end{aligned} \end{aligned}$$
Substituting this into the derivative of Eq. (3), and letting it equal 0, we obtain (after some simplification) a polynomial equation of degree 9 in \(t_i\). This equation can be derived by the Maxima [14] code in Fig. 13 (Appendix A). We solve this equation and select a real root in [0, 1]. Note that we have proved that there is one and only one solution for the polynomial equation of degree 9 in \(t_i \in [0,1]\) as in the case of \(\kappa \)-curves; see details in [16], as well as Appendix A.2.
However, when we use other types of curves with more complicated representations (see examples in Appendix C), this kind of formula may be hard to derive. In these cases, we can use the relaxed Newton's method to compute the maximum curvature. For this, we need to be able to compute the curvature and its derivative; we do this using a quadratic Taylor series approximation around the last value of \(t_i\).
The leftmost curve is a \(\kappa \)-curve. The other curves are \(\epsilon \kappa \)-curves: the \(a_i\) values are equal to 2/3 except for one, two and three input points, respectively, where \(a_i=0.85\)
The a values of the two wing tips in the second figure from the left are 0.95, and those of the other input points are 2/3; note the sharpening of the wings. In the third figure, the roles are reversed. The left- and rightmost curves in red are \(\kappa \)-curves with the same input points
We introduce the constant \(\lambda _i\) (\(0< \lambda _i < 1\)) according to the construction method of \(\kappa \)-curves and set
$$\begin{aligned} c_{i,2} = c_{i+1,0} = (1-\lambda _i) c_{i,1} + \lambda _i c_{i+1,1}. \end{aligned}$$
Let the curvatures at the endpoints of the curve segment be denoted by \(\kappa _i(0;a_i)\) and \(\kappa _i(1;a_i)\), then from Eq. (3)
$$\begin{aligned} \kappa _i(1;a_i)&= \frac{4}{3}\cdot \frac{(1-a_i) \triangle ^+_i}{ a_i^2 \lambda _{i}^2\Vert c_{i+1,1} - c_{i,1} \Vert ^3},\nonumber \\ \kappa _{i+1}(0;a_{i+1})&= \frac{4}{3}\cdot \frac{(1-a_{i+1}) \triangle ^-_{i+1}}{ a_{i+1}^2 (1-\lambda _{i})^2\Vert c_{i+1,1} - c_{i,1} \Vert ^3}, \end{aligned}$$
introducing the notations \(\triangle ^+_i=\triangle (c_{i,0}, c_{i,1}, c_{i+1,1})\) and \(\triangle ^-_{i}=\triangle (c_{i-1,1}, c_{i,1}, c_{i,2})\).
The leftmost curve is a \(\kappa \)-curve; the other curves are \(\epsilon \kappa \)-curves: a is equal to 0.75, 0.85 and 0.95, respectively. Note the curvature at the inflection points
By adopting the local/global approach, we treat \(c_{i,0}\) as fixed for the computation of \(\kappa _i(1;a_i)\), although it depends on \(\lambda _{i-1}\) (similarly for \(c_{i+1,2}\)).
In order to guarantee \(G^2\) continuity at the joint of two consecutive segments, the following equations should be satisfied:
$$\begin{aligned} \kappa _i(1;a_i) = \kappa _{i+1}(0;a_{i+1}). \end{aligned}$$
Hence,
$$\begin{aligned} \lambda _i = \frac{\sqrt{(1-a_i)\triangle ^+_i}}{\sqrt{(1-a_i)\triangle ^+_i} + \frac{a_i}{a_{i+1}}\sqrt{(1-a_{i+1})\triangle ^-_{i+1}}}. \end{aligned}$$
Since \(0< a_i, a_{i+1} <1\), \(\lambda _i\) is real and \(0< \lambda _i <1\).
In the global phase, we calculate the positions of the middle control points \(c_{i,1}\) by solving a linear system of equations. We treat the current values of \(\lambda _i\) (internal division ratios of \(c_{i,1}\) and \(c_{i+1,1}\)) and \(t_i\) (parameters of local maximum curvature) as fixed.
Substituting Eq. (7) into Eq. (5), we get
$$\begin{aligned} p_i&= (1-t_i)^3 \left[ (1-\lambda _{i-1}) c_{i-1,1} + \lambda _{i-1} c_{i,1} \right] \nonumber \\&\quad + 3 (1-t_i)^2 t_i \left[ (1-a_i) \left[ (1-\lambda _{i-1}) c_{i-1,1} + \lambda _{i-1} c_{i,1} \right] \right. \nonumber \\&\quad \left. + a_i c_{i,1} \right] \nonumber \\&\quad + 3 (1-t_i) t_i^2 \left[ (1-a_i) \left[ (1-\lambda _{i}) c_{i,1} + \lambda _i c_{i+1,1} \right] \right. \nonumber \\&\quad \left. + a_i c_{i,1}\right] \nonumber \\&\quad + t_i^3 \left[ (1-\lambda _{i}) c_{i,1} + \lambda _i c_{i+1,1} \right] , \end{aligned}$$
which can be solved for \(c_{i,1}\).
The optimization process is summarized in Algorithm 1.
Figure 5 shows examples of closed \(\epsilon \kappa \)-curves along with the original \(\kappa \)-curve. The input points are located at the same positions. The \(a_i\) values of these curves are equal to 2/3, except for one, two and three input points, respectively, where \(a_i\) is set to 0.85. If the \(a_i\) of all input points are 2/3, the \(\kappa \)-curve on the left is generated. Since we specify a larger value for some input points, the magnitudes of the corresponding local maximum curvature increase, as we expected.
Figure 6 shows another example of local curvature control. From the left to right, the first drawing shows a bird using \(\kappa \)-curves. In the second, we set a at the wing tips to 0.95, while leaving all others at the default 2/3. This has the effect of sharpening the wing tips. In the third figure, we reversed the role of the input points, giving \(a=2/3\) to those at the wing tips and 0.95 to all other points. Here, the wings are rounded, while other parts of the bird get sharper.
Notice that the bird's beak resembles a cusp, but is actually the start and end points of an open curve located at the same position. In our implementation, we limit \(2/3 \le a \le 1\) to make a curve with smooth curvature distribution, which disallows the generation of a cusp even at \(a=1\). In cases where the designer wants to use a cusp, the curve should be cut in two, or the input points should be relocated to form a cusp as explained in [28].
Figure 7 shows examples of global curvature control. There are three \(\epsilon \kappa \)-curves, with a set to 0.75, 0.85 and 0.95, respectively, along with the original \(\kappa \)-curve (\(a=2/3\)) for comparison. By increasing a, the magnitudes of local maximum curvature increase. As the close-up windows indicate, at the inflection point \(G^2\) continuity is violated for \(\kappa \)-curves, and only \(G^1\) continuity is guaranteed. However, for \(\epsilon \kappa \)-curves with a larger a, the magnitude of curvature at inflection points, and consequently the \(G^2\) error, becomes smaller. Note that although these curves are almost \(G^2\)-continuous everywhere, they are quite different from those in Fig. 1 (generated using the same input points).
Figures 3, 8 and 9 show the effect of changes of the global shape parameter a on various designs. As discussed above, larger a values generally induce larger local curvature extrema and steeper curvature variation. The resulting curves look more sharp at the input points and more flat between them.
Changing the global shape parameter in the bear model
Changing the global shape parameter in the elephant model
Generalized trigonometric basis
In this section, we describe our new generalized trigonometric basis. This is based on the trigonometric cubic Bernstein-like basis [24], which we are going to review first.
The trigonometric cubic Bernstein-like basis functions have an extra shape parameter \(\alpha \) and are defined by
$$\begin{aligned} f_0&= \alpha S^2 - \alpha S + C^2 = 1 + (\alpha -1) S^2 - \alpha S, \nonumber \\ f_1&= \alpha S(1-S), \nonumber \\ f_2&= \alpha (S^2+C-1) = \alpha C(1 - C), \nonumber \\ f_3&= (1-\alpha ) S^2 - \alpha C + \alpha = 1 +(\alpha -1) C^2-\alpha C, \end{aligned}$$
where \(S=\sin \frac{\pi t}{2}\), \(C=\cos \frac{\pi t}{2}\), for \(\alpha \in (0,2)\), \(t \in [0,1]\). Note that these functions satisfy partition of unity, i.e., \(\sum _{i=0}^3 f_i(t) = 1\) for any \(\alpha \). When \(\alpha =1\), the above functions are simplified to
$$\begin{aligned} f_0&= 1 - S, \nonumber \\ f_1&= S(1-S), \nonumber \\ f_2&= C(1-C), \nonumber \\ f_3&= 1 - C. \end{aligned}$$
If we add the second and third functions together and rename them to u, v and w, we obtain blending functions \(\{ u, v, w \}\) as follows:
$$\begin{aligned} \begin{aligned}&u = 1-S, \\&v = S(1-S) + C(1-C) = S+C-1, \\&w = 1 - C. \end{aligned} \end{aligned}$$
It is straightforward to define a curve by these blending functions with three control points, which we can regard as a "linear" trigonometric curve since the highest degree the trigonometric functions are in is one.
One interesting relationship among these functions is
$$\begin{aligned} v^2 = 2 u w, \end{aligned}$$
which enables
$$\begin{aligned} (u+v+w)^2 = u^2 + 2 u v + 4 u w + 2 v w + w^2, \end{aligned}$$
and yields the five blending functions \(\{u^2\), 2uv, 4uw, 2vw, \(w^2 \}\), associated with five control points. We can define a curve using these blending functions and regard it as a "quadratic" trigonometric curve since the highest power of each blending function is now degree two.
In a similar way, we can extend blending functions of "degree" n with \(2n+1\) control points. As explained in Appendix B, we can perform a recursive procedure to evaluate a curve of any degree similar to de Casteljau's algorithm avoiding the overhead of trigonometric function evaluation. This means that it is not necessary to calculate the coefficients of blending functions, or keep a coefficient table.
We formulate the \(\epsilon \kappa \)-curve in this basis using a strategy similar to that in the previous section, i.e., using a sequence of quadratic trigonometric curves with a constraint on the positions of their control points, as shown in Fig. 10. Note the location of the control point \(P_2=\left[ (1-a)Q_0+2 a Q_1+(1-a) Q_2\right] /2\). The curve \(c(t;a)\) is defined by
$$\begin{aligned} \begin{aligned} c(t;a) =&\;u^2 Q_0 + 2 u v \left[ (1-a) Q_0+a Q_1 \right] \\&+ 2 u w \left[ (1-a)(Q_0+Q_2)+2 a Q_1 \right] \\&+ 2 v w \left[ a Q_1+ (1-a) Q_2 \right] + w^2 Q_2. \end{aligned} \end{aligned}$$
When a is equal to 1/2, the curve degenerates to a linear trigonometric curve.
Constrained quadratic trigonometric curve. When \(a=1/2\) (bottom), the curve becomes linear
Geometric constraints and optimization
First, we analyze the linear trigonometric curve since it corresponds to the original \(\kappa \)-curve. Let \(c_i(t)\) be a linear trigonometric curve with control points \(c_{i,0}\), \(c_{i,1}\) and \(c_{i,2}\) and defined by
$$\begin{aligned} c_i (t) = (1 - S) c_{i,0} + (S+C-1) c_{i,1} + (1-C) c_{i,2}, \end{aligned}$$
where \(S=\sin \frac{\pi t}{2}\), \(C=\cos \frac{\pi t}{2}\) and \(t \in [0,1]\). Its curvature is given by
$$\begin{aligned} \kappa _i(t) = \frac{2 \Delta (c_{i,0}, c_{i,1}, c_{i,2})}{ \left( C^2 \Vert r_i \Vert ^2 + 2 C S \langle r_i, s_i\rangle + S^2 \Vert s_i \Vert ^2 \right) ^\frac{3}{2} }, \end{aligned}$$
where \(r_i = c_{i,1}-c_{i,0}\) and \( s_i = c_{i,2}-c_{i,1}\). The numerator of the above formula does not depend on t, so the extrema of the following \(f_i(t)\) corresponds to those of \(\kappa _i(t)\):
$$\begin{aligned} f_i(t) = C^2 \Vert r_i \Vert ^2 + 2 C S \langle r_i, s_i\rangle + S^2 \Vert s_i \Vert ^2, \end{aligned}$$
and its derivative with respect to t is given by
$$\begin{aligned} \begin{aligned} \frac{d f_i(t)}{d t} =&\;{\pi }(-CS \Vert r_i \Vert ^2+(-S^2+C^2)\langle r_i, s_i\rangle \\&+ SC \Vert s_i \Vert ^2 ). \end{aligned} \end{aligned}$$
By assuming \({d f_i(t)}/{d t}=0\) with \(S, C \ne 0\), we obtain
$$\begin{aligned} S^2 - \gamma SC - C^2=0, \end{aligned}$$
where \(\gamma =( \Vert s_i \Vert ^2-\Vert r_i \Vert ^2)/\langle r_i, s_i\rangle \). We can solve the above equation and obtain
$$\begin{aligned} C = \frac{-\gamma + \sqrt{\gamma ^2+4}}{2} S = \beta S. \end{aligned}$$
Since \(0 \le S, C \le 1\), we have the unique solution
$$\begin{aligned} S = \frac{1}{\sqrt{\beta ^2+1}}. \end{aligned}$$
$$\begin{aligned} t = \frac{2}{\pi } \arcsin \frac{1}{\beta ^2+1}. \end{aligned}$$
Note that when \(r_i\) and \(s_i\) are perpendicular to each other, if \(\Vert r_i\Vert = \Vert s_i \Vert \), then the curve becomes a circular arc, and no local maximum curvature exists. If \(\Vert r_i \Vert > \Vert s_i \Vert \), then the curvature at \(t=1\) will be maximum, and if \(\Vert r_i \Vert < \Vert s_i \Vert \), the curvature at \(t=0\) will be maximum in this curve segment.
The two curves on the top left (red) are \(\kappa \)-curves; the two on the top right (brown) are \(\epsilon \kappa \)-curves using cubic Bernstein basis functions with \(a=0.75\) and 0.9. The bottom row shows \(\epsilon \kappa \)-curves using quadratic trigonometric basis functions with \(a=0.55\), 0.6, 0.75 and 0.9
A Christmas tree drawn with \(\epsilon \kappa \)-curves using the cubic Bernstein basis functions (left) and the quadratic trigonometric basis functions (right). Note that the latter has more rounded forms and—in this case—preferable
For a quadratic trigonometric curve, the curvatures \(\kappa _i(1;a_i)\) and \(\kappa _{i+1}(0;a_i)\) at the endpoints of the constrained quadratic trigonometric curve \(c_i\) are given by
$$\begin{aligned} \kappa _i (1;a_i)&= \frac{1-a_i}{a_i^2}\cdot \frac{ \triangle ^+_i}{ \lambda _i^2 \Vert c_{i+1,1}-c_{i,1} \Vert ^3}, \nonumber \\ \kappa _{i+1} (0;a_{i+1})&= \frac{1-a_{i+1}}{a_{i+1}^2}\cdot \frac{\triangle ^-_{i+1}}{(1-\lambda _i)^2 \Vert c_{i+1,1}-c_{i,1} \Vert ^3}. \end{aligned}$$
We can calculate \(\lambda _i\) by guaranteeing \(G^2\) continuity at the joint of \(c_i(1;a_i)\) and \(c_{i+1}(0;a_{i+1})\):
$$\begin{aligned} \lambda _i = \frac{\sqrt{(1-a_i)\triangle ^-_i}}{\sqrt{(1-a_i)\triangle ^-_i} + \frac{a_i}{a_{i+1}}\sqrt{(1-a_{i+1})\triangle ^+_{i+1}}}. \end{aligned}$$
As before, we get a linear system of equations for \(c_{i,1}\):
$$\begin{aligned} p_i&= u_i^2 ( (1-\lambda _{i-1}) c_{i-1,1} + \lambda _{i-1} c_{i,1} ) \nonumber \\&\quad + 2 u_i v_i ( (1-a_i)(1-\lambda _{i-1}) c_{i-1,1}\nonumber \\&\quad + ((1-a_i)\lambda _{i-1} + a_i )c_{i,1} ) \nonumber \\&\quad +2 u_i w_i ( (1-a_i) (1-\lambda _{i-1}) c_{i-1,1} +((1-a_i)\lambda _{i-1} \nonumber \\&\quad +2 a_i+(1-a_i)(1-\lambda _{i}) ) c_{i,1} + (1-a_i) \lambda _{i} c_{i+1,1} ) \nonumber \\&\quad + 2 v_i w_i ( (a_i+(1-a_i)(1-\lambda _{i}) ) c_{i,1}\nonumber \\&\quad + (1-a_i) \lambda _{i} c_{i+1,1} ) \nonumber \\&\quad + w_i^2 ( (1-\lambda _{i}) c_{i,1} + \lambda _{i} c_{i+1,1}), \end{aligned}$$
where \(u_i=1-\sin \frac{\pi t_i}{2}\), \(v_i=\sin \frac{\pi t_i}{2} + \cos \frac{\pi t_i}{2} -1\) and \(w_i = 1-\cos \frac{\pi t_i}{2}\).
Figures 11 and 12 show examples of \(\epsilon \kappa \)-curves using the quadratic trigonometric basis functions explained in this section. In the first figure, the top left two curves (red) are \(\kappa \)-curves. The top right two curves (brown) are \(\epsilon \kappa \)-curves, using cubic Bernstein basis functions with \(a=0.75\) and 0.9. The bottom row (green) shows \(\epsilon \kappa \)-curves using quadratic trigonometric basis functions with \(a=0.55\), 0.6, 0.75 and 0.9. The curves in the bottom row are more rounded than those of the Bernstein basis. By increasing a, the differences between the two types of curves become smaller as they approach a polyline generated by connecting the input points.
The second figure shows a case where these more rounded forms are clearly preferable.
Use of built-in shape parameters
So far our strategy made use of degree elevation—from quadratic to cubic in the polynomial case, and from linear to quadratic in the trigonometric case. Yet another strategy is to use extra shape parameters built into the basis functions.
As an illustrative example, take the trigonometric cubic Bernstein-like basis functions reviewed in the previous section. In the framework of \(\epsilon \kappa \)-curves, we have to degenerate the curve by relocating the positions of its control points, essentially reducing its degree.
The trigonometric cubic Bernstein-like basis functions need four control points to define a curve. To construct "quadratic" curves corresponding to the quadratic Bézier segments of \(\kappa \)-curves, we make the second and third control points collocate. Hence, the blending functions become
$$\begin{aligned} b_0(t;\alpha )&= 1+(\alpha -1)S^2-\alpha S,\nonumber \\ b_1(t;\alpha )&= \alpha (S+C-1),\nonumber \\ b_2(t;\alpha )&= 1+(\alpha -1)C^2-\alpha C, \end{aligned}$$
where \(S=\sin \frac{\pi t}{2}\), \(C=\cos \frac{\pi t}{2}\), for \(\alpha \in (0,2)\), \(t \in [0,1]\). Incidentally, this will result in the same curve as Eq. (17), if we substitute 2a for \(\alpha \).
Note that several curve types—such as the cubic alternative curve and the \(\alpha \beta \)-Bernstein-like basis functions, for specific extra parameters—have zero curvature at the endpoints, so we cannot obtain the internal division ratio \(\lambda _i\). For these curves, the method shown in this section cannot be applied. See details in Appendix C.
We have proposed two types of \(\epsilon \kappa \)-curves as extensions of \(\kappa \)-curves, for controlling the magnitudes of local maximum curvature. Our methods use degree elevation of the Bernstein basis functions and a new family of trigonometric basis functions.
In line with Yan et al.'s paradigm shift for curve generation, we consider the local maximum curvature in the middle, instead of focusing on the endpoints. The new curves preserve all of the nice properties of \(\kappa \)-curves, i.e., point interpolation, \(G^2\) continuity (except at inflection points), continuous modification (changes smoothly when the input points move) and local influence. Computing \(\epsilon \kappa \)-curves is quite fast, and designers can manipulate them interactively, acquiring much more expressive power for curve design, as illustrated by our examples. The processing time for generating \(\epsilon \kappa \)-curves is similar to that of \(\kappa \)-curves, especially for \(\epsilon \kappa \)-curves with cubic Bernstein basis. For example, \(\kappa \)-curves consume 0.07 sec to draw Fig. 8 (bear). Under similar conditions, cubic Bézier \(\epsilon \kappa \)-curves consume 0.08 sec and generalized trigonometric \(\epsilon \kappa \)-curves takes 0.37 sec on average.
Future work includes the development of \(\epsilon \kappa \)-curve plug-ins for Adobe Illustrator® and Photoshop®. Another possible research direction is to apply the proposed method to different types of aesthetic curves, e.g., log-aesthetic curves [26], \(\sigma \)-curves [18] or \(\tau \)-curves [19].
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This work was supported by JST CREST (No. JPMJCR1911); JSPS Grant-in-Aid for Scientific Research (B, No. 19H02048); JSPS Grant-in-Aid for Challenging Exploratory Research (No. 26630038); Solutions and Foundation Integrated Research Program; ImPACT Program of the Council for Science, Technology and Innovation; and the Hungarian Scientific Research Fund (OTKA, No. 124727). The authors acknowledge the support by 2016, 2018 and 2019 IMI Joint Use Program Short-term Joint Research "Differential Geometry and Discrete Differential Geometry for Industrial Design" (September 2016, September 2018 and September 2019). The second author acknowledges University Malaysia Terengganu for approving sabbatical leave which was utilized to work on emerging researches, including this work.
Graduate School of Science and Technology, Shizuoka University, Hamamatsu, Shizuoka, Japan
Kenjiro T. Miura, Dan Wang, Tadatoshi Sekine & Shin Usuki
Faculty of Ocean Engineering Technology & Informatics, University Malaysia Terengganu, Kuala Nerus, Terengganu, Malaysia
R. U. Gobithaasan
Department of Control Engineering and Information Technology, Budapest University of Technology and Economics, Budapest, Hungary
Péter Salvi
Institute of Mathematics, University of Tsukuba, Tennodai, Tsukuba, Japan
Jun-ichi Inoguchi
Institute of Mathematics for Industry, Kyushu University, Fukuoka, Fukuoka, Japan
Kenji Kajiwara
Kenjiro T. Miura
Dan Wang
Tadatoshi Sekine
Shin Usuki
Correspondence to Kenjiro T. Miura.
Below is the link to the electronic supplementary material.
Supplementary material 1 (mp4 6667 KB)
Local maximum curvature of cubic polynomial curves
General case
The signed curvature \(\kappa (t)\) of a cubic polynomial curve \(c(t) = ( x(t), y(t) )\) is given by [3] as
$$\begin{aligned} \kappa (t) = \frac{x' y''-x'' y' }{\left( x'^2 + y'^2\right) ^{\frac{3}{2}}}, \end{aligned}$$
where \(x' = {\mathrm dx(t)}/{\mathrm dt}\), \(x'' = {\mathrm d^2x(t)}/{\mathrm dt^2}\), and higher derivatives are expressed in a similar way.
At the extremum \({\mathrm d\kappa (t)}/{\mathrm dt} = 0\) and \({\mathrm d\kappa (t)^2}/{\mathrm dt} = 2 \kappa (t) \cdot {\mathrm d\kappa (t)}/{\mathrm dt}\). Hence, if we exclude points where \(\kappa (t)=0\), we can obtain t values at the extrema.
By differentiating
$$\begin{aligned} \kappa (t)^2 = \frac{(x'(t) y''(t)-x''(t) y'(t) )^2}{( x'(t)^2 + y'(t)^2 )^3} \end{aligned}$$
with respect to t, we obtain
$$\begin{aligned} \left( x'^2 + y'^2\right) ^4 \kappa \frac{\mathrm d\kappa }{\mathrm dt}&= (x''y' - x'y'')\bigl ((x'''y' - x'y''')(x'^2 + y'^2) \nonumber \\&\quad -3(x'x'' + y'y'')(x''y' - x'y'')\bigr ), \end{aligned}$$
where \(x'y''-x''y'=0\) means the curvature is equal to 0, and it corresponds to inflection points. Consequently,
$$\begin{aligned}&(x'''y' - x'y''')(x'^2 + y'^2)\nonumber \\&\qquad -3(x'x'' + y'y'')(x''y' - x'y'')= 0 \end{aligned}$$
corresponds to curvature extrema. The above equation is regarded as (quadratic \(\times \) quartic) − (cubic \(\times \) cubic), but the coefficient of sextic terms vanishes and it becomes quintic. Therefore, a cubic polynomial curve can have at most 5 curvature extrema. Please refer to [25] for more details. Since Eq. (33) is quintic, it is generally not possible to have analytical solutions, so we must use numerical approach to obtain the points on the curve where curvature extrema occurs.
Here, we give a high-level summary of the proof [16] that the curvature of a cubic curve of the form shown in Eq. (1) has at most one local extremum in the (0, 1) parameter interval. As a consequence, the degree-9 polynomial in Sect. 3.1, computed by the Maxima [14] program in Fig. 13, has at most one real solution in the (0, 1) interval. When it has none, this means that no local extremum is present in the curvature, so the extremum occurs at
$$\begin{aligned} t_0=\mathop {{{\,\mathrm{arg\,max}\,}}}\limits _{t=0,1}|\kappa (t)|. \end{aligned}$$
Maxima [14] code to calculate the degree-9 polynomial in Sect. 3.1
Without loss of generality, place the control points as follows:
$$\begin{aligned} Q_0&=(-1,0),&Q_1&=(b,h),&Q_2&=(1,0), \end{aligned}$$
where \(b\ge 0\) and \(h>0\). (The special cases where \(Q_0=Q_2\) or \(h=0\) are also handled in [16].)
Let N(t, a) denote the left side of Eq. (33) applied to this curve. When \(b\le 3-2/a\), it is easy to see that \(\partial N(t,a)/\partial t<0\) for any \(t\in (0,1)\) and \(a\in [2/3,1]\), so N is decreasing. Since N(0, a) is always positive, this means N has at most one 0-crossing.
In the following, let us also assume that \(b>3-2/a\). Then, the following statements can also be proven:
$$\begin{aligned} N(1,a)&<0&\text { when }\partial N(1,a)/\partial t>0, \end{aligned}$$
$$\begin{aligned} \partial N(0,a)/\partial t&<0,\end{aligned}$$
$$\begin{aligned} \partial N(t,a)/\partial t&<0&\text { when }\partial N(1,a)/\partial t<0,\end{aligned}$$
$$\begin{aligned} \partial ^2N(0,a)/\partial t^2&>0,\end{aligned}$$
$$\begin{aligned} \partial ^3N(t,a)/\partial t^3&<0. \end{aligned}$$
From the above,, it is easy to prove that \(N(t,a)=0\) has exactly one solution—and thus, the curvature has at most one local extremum—in (0, 1).
Generalized trigonometric basis functions
Recursive evaluation
For our new trigonometric basis, we can derive a recursive algorithm similar to de Casteljau's algorithm . For simplicity we explain only the quadratic case, but it can be extended to a general degree n by induction. To shorten expressions, we use \(u=1-S(t)\), \(v=S(t)+C(t)-1\) and \(w=1-C(t)\), where \(S(t) = \sin \frac{\pi t}{2}\) and \(C(t) = \cos \frac{\pi t}{2}\). Note that \(v^2 = 2 u w\), and
$$\begin{aligned} \begin{aligned}&(u+v+w)^2 = \\&\qquad u(u+v+w) + v(u+v+w)+w(u+v+w). \end{aligned} \end{aligned}$$
For a quadratic curve with this basis, five control points \(P_i\) (\(i=0\dots 4\)) are used, and the curve point at t is evaluated as
$$\begin{aligned} \begin{bmatrix} u&v&w \end{bmatrix} \begin{bmatrix} P_0 &{} \quad P_1 &{} \quad P_2 \\ P_1 &{} \quad P_2 &{} \quad P_3 \\ P_2 &{} \quad P_3 &{} \quad P_4 \end{bmatrix} \begin{bmatrix} u &{} \quad &{} \\ &{} v \quad &{} \\ &{} &{} \quad w \end{bmatrix}. \end{aligned}$$
Hence, the algorithm repeats a simple blending of three points \(u P_{i-1} +v P_{i} +w P_{i+1}\) to generate a point on the given curve.
Triangle method
We can also construct a triangle using the coefficients of trigonometric basis functions, similarly to Pascal's triangle. Below is a table of degree elevation, from the first row representing degree 1 to the sixth row representing degree 6:
$$\begin{aligned} \begin{array}{ccccccccccccc} &{} \quad &{} \quad &{} \quad &{} \quad &{} \quad 1 &{} \quad 1 &{} \quad 1&{} \quad &{} \quad \quad &{} \quad &{} \quad &{}\\ &{} \quad \quad &{} \quad &{} \quad &{} \quad 1 &{} \quad 2 &{} \quad 4 &{} \quad 2&{} \quad 1&{} \quad &{} \quad &{} \quad &{} \\ &{} \quad &{} \quad &{} \quad 1 &{} \quad 3&{} \quad 9 &{} \quad 8 &{} \quad 9 &{} \quad 3&{} \quad 1 &{} \quad &{} \quad &{}\\ &{} \quad &{} \quad 1&{} \quad 4 &{} \quad 16 &{} \quad 20 &{} \quad 34 &{} \quad 20 &{} \quad 16 &{} \quad 4 &{} \quad 1 &{} \quad &{} \\ &{} \quad 1&{} \quad 5&{} \quad 25 &{} \quad 40 &{} \quad 90 &{} \quad 74 &{} \quad 90 &{} \quad 40 &{} \quad 25 &{} \quad 5 &{} \quad \quad 1 &{} \\ 1&{} \quad 6&{} \quad 36 &{} \quad 70 &{} \quad 195 &{} \quad 204 &{} \quad \quad 328 &{} \quad 204 &{} \quad 195 &{} \quad 70 &{} \quad 36 &{} \quad 6 &{} \quad 1 \end{array} \end{aligned}$$
Various basis functions
Here, we check the applicability of the bases listed in Sect. 2, except for the trigonometric cubic Bernstein-like basis functions [24], since that was already discussed in the paper.
The basis of the C-Bézier curve is \(\{ \sin t, \cos t, t, 1 \}\), and the curve is defined by the following formula:
$$\begin{aligned} B_\alpha (t)&= Z_0(t) q_0 + Z_1(t) q_1 + Z_2(t) q_2 + Z_3(t) q_3 \\&= \frac{1}{\alpha -S} \left[ \begin{array}{c} \sin t \\ \cos t \\ t \\ 1 \end{array} \right] ^\top \left[ \begin{array}{cccc} C &{} \quad 1-C -M &{} \quad M &{} \quad -1 \\ -S &{} \quad (\alpha -K)M &{} \quad -KM &{} 0 \\ -1 &{} \quad M &{} \quad -M &{} \quad 1 \\ \alpha &{} \quad -(\alpha - K)M &{} \quad KM &{} \quad 0 \end{array} \right] \\&\quad \times \left[ \begin{array}{c} q_0 \\ q_1 \\ q_2 \\ q_3 \end{array} \right] . \end{aligned}$$
Here, \(\alpha \) is a built-in shape parameter satisfying \(0 < \alpha \le \pi \), and \(S=\sin \alpha \), \(C = \cos \alpha \). The parameter of the curve is \(t \in [0,\alpha ]\), and
$$\begin{aligned} K&= \frac{\alpha - S}{1-C}, \\ M&= {\left\{ \begin{array}{ll} 1 &{} \quad \text{ if } \alpha = \pi , \\ \frac{S}{\alpha - 2K} = \frac{S(1-C)}{2 S - \alpha -\alpha C} &{} \quad \text{ if } 0< \alpha < \pi . \end{array}\right. } \end{aligned}$$
Blending functions of the degenerated (quadratic) C-Bézier curve
We degenerate the curve by adding its second and third basis functions, i.e., placing the second and third control points at the same position. The curve is then defined by three control points. However, even if we vary \(\alpha \) from 0 to \(\pi \), the blending functions do not vary much, as shown in Fig. 14. Therefore, this type of curve is not suitable for changing the magnitude of local maximum curvature.
The cubic alternative curve is similar to the cubic Bézier curve, and is defined by
$$\begin{aligned}&Z(t) = F_0(t) P_0 + F_1(t) P_1 + F_2(t) P_2 + F_3(t) P_3, \\&\quad 0 \le t \le 1, \end{aligned}$$
where the basis functions \(F_i(t)\), \(t=0\dots 3\) are
$$\begin{aligned} F_0(t)&= (1-t)^2 (1+(2-\alpha )t), \\ F_1(t)&= \alpha (1-t)^2 t, \\ F_2(t)&= \beta t^2 (1-t), \\ F_3(t)&= t^2 (1+(2-\beta )(1-t)). \end{aligned}$$
When \(\alpha =\beta =2\), the curve becomes the cubic Ball curve; for \(\alpha =\beta =3\), the classical cubic Bézier curve; and for \(\alpha =\beta =4\), the cubic Timmer curve. The basis functions are nonnegative when \(0 \le \alpha \le 3\).
We assume that \(\beta =\alpha \), and the curve is degenerated by adding the second and third blending functions. Hence,
$$\begin{aligned} A_0(t;\alpha )&= (1-t)^2(1+(2-\alpha )t), \\ A_1(t;\alpha )&= \alpha (1-t)t \\&= 1- (1-t)^2(1+(2-\alpha )t) -t^2(1+(2-\alpha )(1-t)), \\ A_2(t;\alpha )&= t^2(1+(2-\alpha )(1-t)). \end{aligned}$$
We define the curve \(c(t;\alpha )\) by
$$\begin{aligned} c(t;\alpha )&= A_0(t;\alpha ) P_0 + A_1(t;\alpha ) P_1 + A_2(t;\alpha ) P_2 \\&= A_0(t;\alpha ) (P_0-P_1) + P_1 + A_2(t;\alpha ) (P_2-P_1). \end{aligned}$$
$$\begin{aligned} \frac{d c(t;\alpha )}{d t}&= (1-t)(-3\alpha t+6t+\alpha )(P_1-P_0) \\&\quad + t(3\alpha t - 6 t-2\alpha +6) (P_2-P_1), \\ \frac{d^2 c(t;\alpha )}{d t^2}&= 2 \big [ (3 \alpha t-6t-2 \alpha +3) (P_1-P_0) \\&\quad + (3 \alpha t-6t-\alpha +3) (P_2-P_1) \big ]. \end{aligned}$$
Hence, the curvature \(\kappa _i(t;\alpha )\) for each curve segment is given by
$$\begin{aligned}&\kappa _i(t;\alpha ) =\\&\quad \ \ \frac{2 \alpha _i(3(\alpha -2)(1-t)t-\alpha _i+3) (P_1-P_0)\times (P_2-P_1) }{\Vert (1-t)(-3\alpha _it+6t+\alpha _i)(P_1-P_0)+t(3 \alpha _i t-6 t-2 \alpha _i+6)(P_2-P_1) \Vert ^3}. \end{aligned}$$
When \(\alpha =3\), this degenerates to the classical cubic Bézier curve with collocated second and third control points, and at the endpoints we get \(\kappa _i(0,3)=\kappa _i(1,3)=0\), since the directions of the first and second derivatives are the same. Hence, the proposed method is not applicable in this case.
When \(\alpha \ne 0\),
$$\begin{aligned} \kappa _i(1;\alpha _i)&= \frac{4(3-\alpha _i)\triangle ^+_i}{ \alpha _i^2 \lambda _i^2 \Vert c_{i+1,1}-c_{i,1} \Vert ^3}, \\ \kappa _{i+1}(0;\alpha _i)&= \frac{4(3-\alpha _{i+1})\triangle ^-_{i+1}}{ \alpha _{i+1}^2 (1-\lambda _i)^2 \Vert c_{i+1,1}-c_{i,1} \Vert ^3}. \end{aligned}$$
To guarantee \(G^2\) continuity at the joint of the segments, when \(0< \alpha _i, \alpha _{i+1} < 3\), we can compute \(\lambda _i\) (\(0< \lambda _i < 1\) ) by
$$\begin{aligned} \lambda _i = \frac{\sqrt{ (3-\alpha _i) \triangle ^+_i}}{\sqrt{(3-\alpha _i)\triangle ^+_i} + \frac{\alpha _i}{\alpha _{i+1}}\sqrt{ (3-\alpha _{i+1}) \triangle ^-_{i+1}}}. \end{aligned}$$
When \(\alpha _i, \alpha _{i+1}>3\),
$$\begin{aligned} \lambda _i = \frac{\sqrt{ (\alpha _i-3) \triangle ^+_i}}{\sqrt{(\alpha _i-3)\triangle ^+_i} + \frac{\alpha _i}{\alpha _{i+1}}\sqrt{ (\alpha _{i+1}-3) \triangle ^-_{i+1}}}. \end{aligned}$$
In other cases, such as \(\alpha _i>3\), \(0< \alpha _{i+1} < 3\) or \(0< \alpha _i < 3\), \(\alpha _{i+1} > 3\), \(\lambda _i\) can be determined by careful handling of the signs of \(3-\alpha _i\) and \(3-\alpha _{i+1}\).
By assuming a local maximum curvature at \(t_i\), we can express the input point \(p_i\) as
$$\begin{aligned} p_i&= (1-t_i)^2(1+(2-\alpha _i)t_i) \left[ (1-\lambda _{i-1}) c_{i-1,1} + \lambda _{i-1} c_{i,1} \right] \\&\quad + \alpha _i (1-t_i)t_i c_{i,1} \\&\quad + t_i^2(1+(2-\alpha _i)(1-t_i)) \left[ (1-\lambda _{i}) c_{i,1} + \lambda _i c_{i+1,1} \right] . \end{aligned}$$
Cubic trigonometric Bézier curve [8]
For \(\lambda \), \(\mu \in [-2,1]\), \(t \in [0, 1]\), the cubic trigonometric Bézier denoted as T-Bézier basis functions are defined by
$$\begin{aligned} b_0(t)&= (1 - S)^2 (1 - \lambda S), \\ b_1(t)&= S (1 - S)( 2+ \lambda - \lambda S ), \\ b_2(t)&= C (1 - C )( 2+ \mu - \mu C ), \\ b_3(t)&= (1 - C)^2 (1 - \mu C), \end{aligned}$$
where \(S=\sin \frac{\pi t}{2}\) and \(C=\cos \frac{\pi t}{2}\).
We assume \(\lambda =\mu =\alpha \), and the curve is degenerated by adding the second and third blending functions. Hence,
$$\begin{aligned} A_0(t;\alpha )&= (1-S)^2(1-\alpha S ), \\ A_1(t;\alpha )&= 1-(1-S)^2(1-\alpha S )- (1-C)^2(1-\alpha C), \\ A_2(t;\alpha )&= (1-C )^2(1-\alpha C). \end{aligned}$$
$$\begin{aligned} c(t;\alpha )&= (1-S)^2(1-\alpha S ) P_0 \\&\quad +(1-(1-S)^2(1-\alpha S)- (1-C )^2(1-\alpha C) )P_1 \\&\quad + (1-C)^2(1-\alpha C) P_2 \\&= (1-S)^2(1-\alpha S ) (P_0-P_1) + P_1 \\&\quad + (1-C)^2(1-\alpha C) (P_2-P_1). \end{aligned}$$
$$\begin{aligned} \frac{d c(t;\alpha )}{d t}&= \frac{\pi }{2} \big [ (1-S)C (2+\alpha -3\alpha S)(P_1-P_0) \\&\quad +(1-C) S (2+\alpha -3\alpha C) (P_2-P_1) \big ], \\ \frac{d^2 c(t;\alpha )}{d t^2}&=\frac{\pi ^2}{4} \big [ (S-1)( (2S+1)( 2+\alpha - 3 \alpha S) + 3 \alpha C^2) (P_0-P_1) \\&\quad + (1- C)( (2C+1)( 2+\alpha - 3 \alpha C) + 3 \alpha S^2) (P_2-P_1) \big ]. \end{aligned}$$
Hence, \(d c/dt \times d^2 c/dt^2\) is given by
$$\begin{aligned}&\frac{d c}{dt} \times \frac{d^2 c}{dt^2}= (1-S)(1-C) \\&\quad \big [ C (2+\alpha -3 \alpha S) ((2C +1)(2+\alpha -3\alpha C)+3 \alpha S^2) \\&\quad + S(2+\alpha -3\alpha C) ((2 S +1)(2+\alpha -3\alpha S+3 \alpha C^2 ) \big ] \\&\quad (P_1-P_0)\times (P_1-P_0). \end{aligned}$$
When \(\alpha =0\), this curve is the same as the trigonometric cubic Bernstein-like curve with \(\alpha =2\), and the curvatures at the start and end points \(\kappa (0)\) and \(\kappa (1)\) are generally not equal to zero. For other \(\alpha \) values, \(\kappa (0)=\kappa (1)=0\), so our method is not applicable. This is because the directions of the first and second derivatives are the same.
\(\alpha \beta \)-Bernstein-like basis functions [32] and quasi-cubic trigonometric Bernstein basis curves [30]
For arbitrary \(\alpha \), \(\beta \in [2,+\infty ]\), and \(t \in [0,1]\), the \(\alpha \beta \)-Bernstein-like basis functions are defined by
$$\begin{aligned} A_0(t;\alpha )&= (1-t)^\alpha , \\ A_1(t;\alpha )&= 1-3t^2 + 2 t^3 -(1-t)^\alpha , \\ A_2(t;\beta )&= 3 t^2 - 2t^3 - t^\beta , \\ A_3(t;\beta )&= t^\beta . \end{aligned}$$
When \(\alpha =\beta =2\), these degenerate to cubic Said-Ball basis functions [21]. When \(\alpha =\beta =3\), these become cubic Bernstein basis functions.
Let \(\beta =\alpha \) and we degenerate the curve by adding the second and third blending functions. Hence,
$$\begin{aligned} A_0(t;\alpha )&= (1-t)^\alpha , \\ A_1(t;\alpha )&= 1-(1-t)^\alpha -t^\alpha , \\ A_2(t;\alpha )&= t^\alpha . \end{aligned}$$
We define the curve \(c(t)\) by
$$\begin{aligned} c(t;\alpha )&= (1-t)^\alpha P_0 + (1-(1-t)^\alpha -t^\alpha ) P_1 + t^\alpha P_2 \\&= (1-t)^\alpha (P_0-P_1) + P_1 + t^\alpha (P_2-P_1). \end{aligned}$$
$$\begin{aligned} \frac{d c(t;\alpha )}{d t}&= \alpha \left[ (1-t)^{\alpha -1}(P_1-P_0) + t^{\alpha -1} (P_2-P_1) \right] , \\ \frac{d^2 c(t;\alpha )}{d t^2}&= \alpha (\alpha -1) \left[ (1-t)^{\alpha -2}(P_0-P_1) + t^{\alpha -2} (P_2-P_1) \right] . \end{aligned}$$
Hence, its curvature \(\kappa (t;\alpha )\) is given by
$$\begin{aligned} \kappa (t;\alpha ) = \frac{\alpha -1}{\alpha }\cdot \frac{(1-t)^{\alpha -2} t^{\alpha -2} (P_1-P_0)\times (P_2-P_1) }{\Vert (1-t)^{\alpha -1} (P_1-P_0)+t^{\alpha -1} (P_2-P_1) \Vert ^3}. \end{aligned}$$
When \(\alpha =2\) (cubic Said-Ball curve), the curvatures at the start and end points \(\kappa (0;\alpha )\) and \(\kappa (1;\alpha )\) are generally not zero. However, when \(\alpha > 2\), we get \(\kappa (0;\alpha )=\kappa (1;\alpha )=0\). Therefore, in this case our method is not applicable.
In the case of quasi-cubic trigonometric Bernstein basis curves, the directions of the first and second derivatives at the ends are the same; thus, our method is not applicable.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Miura, K.T., Gobithaasan, R.U., Salvi, P. et al. \(\epsilon \kappa \)-Curves: controlled local curvature extrema. Vis Comput 38, 2723–2738 (2022). https://doi.org/10.1007/s00371-021-02149-8
Issue Date: August 2022
Interpolatory curves
Curvature continuity
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Multirate PWM balance method for the efficient field-circuit coupled simulation of power converters
Andreas Pels ORCID: orcid.org/0000-0003-1517-484X1,2,3,
Herbert De Gersem ORCID: orcid.org/0000-0003-2709-25181,2,
Ruth V. Sabariego ORCID: orcid.org/0000-0001-7515-39233 &
Sebastian Schöps ORCID: orcid.org/0000-0001-9150-02191,2
Journal of Mathematics in Industry volume 9, Article number: 8 (2019) Cite this article
The field-circuit coupled simulation of switch-mode power converters with conventional time discretization is computationally expensive since very small time steps are needed to appropriately account for steep transients occurring inside the converter, not only for the degrees of freedom (DOFs) in the circuit, but also for the large number of DOFs in the field model part. An efficient simulation technique for converters with idealized switches is obtained using multirate partial differential equations, which allow for a natural separation into components of different time scales. This paper introduces a set of new PWM eigenfunctions which decouple the systems of equations and thus yield an efficient simulation of the field-circuit coupled problem. The resulting method is called the multirate PWM balance method.
Switch-mode power converters are used in various devices from small-scale applications like mobile phone chargers to industrial large-scale applications like welding devices [7]. These converters use transistors to switch on and off the input voltage to produce an output voltage, which, in average, has the desired amplitude. A filter circuit is used to smoothen the output. The simulation of these devices is computationally expensive since, through the transistor switching, steep transients occur in the converter. Furthermore often a switch-event detection is necessary to avoid step size rejection or even solver failures [14]. A multirate method has been developed in [9, 10] which uses the concept of Multirate Partial Differential Equations (MPDEs) [3, 12] and a combination of a Galerkin ansatz and conventional time discretization to efficiently solve problems with pulsed excitation. The method is applicable to power converters in which the switching behavior is idealized and known a-priori. It is particularly efficient in the case of linear elements. Some circuit elements may only be accurately represented by field models. For example the induced currents in the conducting materials of an inductor usually cause eddy current losses, which can easily be accounted for in a field model but not in a circuit model. In this paper the multirate method from [9, 10] is applied to a linear buck converter circuit (see Fig. 1) in which the inductor is represented by a 2D finite element model. This substantially increases the size of the strongly coupled system of equations. To still ensure an efficient simulation, a basis transformation is applied to the pulse-width modulation (PWM) basis functions [5] leading to decoupled systems of equations which can be solved efficiently in parallel. The resulting method is called the multirate PWM balance method in analogy with the harmonic balance method where harmonic functions take the place of the PWM basis functions. Numerical results on the buck converter show the efficiency and accuracy of the proposed method in field-circuit coupled problems.
Simplified circuit of the buck converter in continuous conduction mode with \(C=10~\text{$\mu $F}\) and \(R=30~\varOmega \). The field model of the pot inductor (axisymmetric around z-axis) is designed to have an inductance of \(L=65~\text{mH}\) and a series resistance of \(R_{\mathrm {L}}=800~\text{m$\varOmega $}\) at DC. The figure shows the equipotential lines of the magnetic vector potential
The paper is structured as follows. Section 2 introduces the concept of MPDEs and explains the solving procedure using Galerkin approach and conventional time discretization. Subsequently Sect. 3 presents the original PWM basis functions as described in [5]. In Sect. 4 the PWM eigenfunctions are developed and their advantageous properties for the solving process are highlighted. Finally Sect. 5 summarizes numerical results and compares the three different solution approaches, i.e., conventional time discretization and the MPDE approach with PWM basis functions on the one hand and PWM eigenfunctions on the other hand. The paper is concluded by summarizing its content in Sect. 6.
Multirate formulation
Let the field-circuit coupled model [13] of the converter be described by the system of ordinary differential or differential-algebraic equations
$$\begin{aligned} \mathbf {A}\, \frac {{\mathrm {d}}}{{\mathrm {d}}t}\mathbf {x}(t)+\mathbf {B}\, \mathbf {x}(t) = \mathbf {c}(t), \end{aligned}$$
where \(\mathbf {A}\in \mathbb{R}^{N_{\mathrm {s}}\times N_{\mathrm {s}}}\) is a possibly singular matrix, \(\mathbf {B}\in \mathbb{R}^{N_{\mathrm {s}}\times N_{\mathrm {s}}}\) is assumed to be a regular matrix, \(\mathbf {x}(t)\in \mathbb{R}^{N_{\mathrm {s}}}\) is the unknown solution, \(\mathbf {c}(t)\in \mathbb{R}^{N_{\mathrm {s}}}\) is the excitation, and \(t\in (0, \mathcal {T}]\) is the simulation interval. The initial state of the model is given by consistent [6] initial values \(\mathbf {x}(0)=\mathbf {x}_{0}\). The ideal pulsed excitation
$$ v_{\mathrm {i}}(t)=\textstyle\begin{cases} V_{0} & \text{for all } \tau (t) \leq D, \\ 0 & \text{otherwise}, \end{cases} $$
is used as input of the power converter circuit. We denote by \(\tau (t) = \frac{t}{T_{\mathrm {s}}} \text{ modulo } 1\) the relative time, \(T_{\mathrm {s}}\) is the switching cycle and D is the duty cycle.
The system of Multirate Partial Differential(-Algebraic) Equations (MPDEs or MPDAEs) with two time scales corresponding to (1) is given by [3, 11, 12]
$$ \mathbf {A}\biggl(\frac{\partial \mathbf {\widehat {x}}}{\partial t_{1}} + \frac{ \partial \mathbf {\widehat {x}}}{\partial t_{2}} \biggr) + \mathbf {B}\, \mathbf {\widehat {x}}(t_{1},t _{2}) = \mathbf {\widehat {c}}(t_{1},t_{2}) , $$
where \(\mathbf {\widehat {x}}(t_{1}, t_{2})\) is the unknown multivariate solution and \(\mathbf {\widehat {c}}(t_{1},t_{2})\) is the multivariate excitation. Choosing the multivariate excitation such that \(\mathbf {\widehat {c}}(t,t)=\mathbf {c}(t)\), the solution of (1) and (3) are related by \(\mathbf {\widehat {x}}(t,t)=\mathbf {x}(t)\). Thus, the solution of (1) can be calculated solving the MPDEs and extracting the solution along a diagonal through the computation domain. To solve the MPDEs, additional conditions need to be specified. For the present application, a combination of initial and boundary values is applied. Initial values are supplied by \(\mathbf {\widehat {x}}(0,t_{2})=\mathbf {h}(t _{2})\), i.e., at \(t_{1}=0\), where h with \(\mathbf {h}(0)=\mathbf {x}_{0}\) is a function defining the initial values for all \(t_{2}\). The solution along the fast time scale \(t_{2}\) is periodic, i.e., \(\mathbf {\widehat {x}}(t_{1}, t_{2}+T_{\mathrm {s}}) = \mathbf {\widehat {x}}(t_{1}, t_{2})\). The multivariate right-hand side is chosen as \(\mathbf {\widehat {c}}(t_{1}, t_{2})=\mathbf {c}(t_{2})\), i.e., the pulses of the excitation occur along the fast time scale. It is possible to use MPDEs with more than two time scales. However, in the applications of this paper, it is not necessary and furthermore often not feasible since the dimension of the computation domain increases and thus also the computational effort to calculate the solution.
To solve the MPDEs (3), a Galerkin approach and time discretization is applied [2, 10]. The j-th solution component \({\widehat {x}}_{j}(t_{1}, t_{2})\) is approximated by expanding it into periodic basis functions \(p_{k}\) depending on the fast time scale \(t_{2}\) and coefficients \(w_{j,k}\) depending on the slow time scale \(t_{1}\)
$$ {\widehat {x}}_{j}^{\,h}(t_{1}, t_{2}) := \sum_{k=0}^{N_{\mathrm {p}}} w_{j,k}(t_{1})\, p _{k}\bigl(\tau (t_{2})\bigr) , $$
where the periodicity of the basis functions is accounted for by using the relative time \(\tau (t_{2}) = \frac{t_{2}}{T_{{\mathrm {s}}}} \text{ modulo } 1\). Applying the Galerkin approach with respect to \(t_{2}\) and over one period of the excitation \([0,T_{\mathrm {s}}]\) leads to
$$ \boldsymbol {\mathcal {A}}\frac{{\mathrm {d}}\mathbf {w}}{{\mathrm {d}}t_{1}} + \boldsymbol {\mathcal {B}}\,\mathbf {w}(t_{1}) = \boldsymbol {\mathcal {C}}(t _{1}) , $$
with block matrices \(\boldsymbol {\mathcal {A}}=\boldsymbol {\mathcal {J}}\otimes \mathbf {A}, \, \boldsymbol {\mathcal {B}}=\boldsymbol {\mathcal {J}}\otimes \mathbf {B}+\boldsymbol {\mathcal {Q}}\otimes \mathbf {A}\), where
$$\begin{aligned} \boldsymbol {\mathcal {J}}= T_{\mathrm {s}}\int _{0}^{1} \bar{\mathbf{p}}(\tau ) \, \mathbf {p}^{ \top }\! (\tau ) \,{\mathrm {d}}\tau ,\qquad \boldsymbol {\mathcal {Q}}= - \int _{0}^{1} \frac{{\mathrm {d}}\bar{\mathbf{p}}(\tau )}{{\mathrm {d}}\tau } \, \mathbf {p}^{\top }\! (\tau ) \,{\mathrm {d}}\tau , \end{aligned}$$
and right-hand side
$$ \boldsymbol {\mathcal {C}}(t_{1})= \int _{0}^{T_{{\mathrm {s}}}} \bar{\mathbf{p}}\bigl(\tau (t_{2})\bigr) \otimes \mathbf {\widehat {c}}(t_{1},t_{2}) \,{\mathrm {d}}t_{2} . $$
\(\bar{\mathbf{p}}\) denotes the complex conjugate of p and ⊗ denotes the Kronecker product.
PWM basis functions
The PWM basis functions developed in [5] are built up starting from the zero-th constant basis function \(p_{0}(\tau )=1\) and the piecewise linear basis function
$$ p_{1}(\tau ) = \textstyle\begin{cases} \sqrt{3} \, \frac{2\tau -D}{D} & \text{if } 0 \leq \tau < D, \\ \sqrt{3} \, \frac{1+D-2\tau }{1-D} & \text{if } D \leq \tau \leq 1, \end{cases} $$
which includes the duty cycle D of the excitation by construction. The higher-order basis functions \(p_{k}(\tau )\), \(2\leq k \leq N_{\mathrm {p}}\) are recursively obtained by integrating the basis functions of lower order \(p_{k-1}(\tau )\) and orthonormalizing them using the Gram–Schmidt algorithm. The generated basis functions are depicted in Fig. 2.
Original PWM basis functions \(p_{k}(\tau )\), \(k\in \{0,1,2,3,4 \}\)
For the PWM basis functions, the matrices \(\boldsymbol {\mathcal {J}}\) and \(\boldsymbol {\mathcal {Q}}\) from (6) are given by \(T_{\mathrm {s}}\) multiplied by the identity matrix (due to the orthonormality of the basis functions) and a square matrix with around 25% of non-zero entries, respectively. Solving the problem requires time stepping of the entire system (5).
PWM eigenfunctions
To enable an easy parallelization of the method, the equations (5) can be decoupled, for example by diagonalizing \(\boldsymbol {\mathcal {Q}}\), i.e., a basis transformation. We define new basis functions \(g_{k}(\tau )\) as linear combinations of the PWM basis functions, i.e.,
$$ g_{k}(\tau ):=\sum_{l=0}^{N_{\mathrm {p}}} v_{k,l} \,\, p_{l}(\tau ), $$
where \(v_{k,l}\) are unknown coefficients with \(k\in \{0, \dots , N_{\mathrm {p}}\}\), and \(g_{k}(\tau )\) are eigenfunctions of the time derivative operator
$$ \frac{{\mathrm {d}}}{{\mathrm {d}}\tau } g_{k}(\tau ) = \lambda _{k} \,\,g_{k}(\tau ). $$
We enforce this property in a weak sense by a Galerkin approach, i.e.,
$$ - \int _{0}^{1} g_{k}(\tau ) \frac{{\mathrm {d}}p_{m}(\tau )}{{\mathrm {d}}\tau } \,{\mathrm {d}}\tau = \lambda _{k} \int _{0}^{1} g_{k}(\tau ) \, p_{m}(\tau ) \,{\mathrm {d}}\tau , $$
where integration by parts and the periodicity of the basis functions is used. Inserting the expansion of the basis functions into (11) gives
$$ T_{\mathrm {s}}\boldsymbol {\mathcal {Q}}\,\mathbf {v}_{k} = \lambda _{k} \boldsymbol {\mathcal {J}}\,\mathbf {v}_{k}. $$
Since \(\boldsymbol {\mathcal {J}}\) is \(T_{\mathrm {s}}\) multiplied by the identity matrix (thanks to the orthonormality of the PWM basis functions), the \(\lambda _{k}\) and \(\mathbf {v}_{k}\) are the eigenvalues and eigenvectors of the matrix \(\boldsymbol {\mathcal {Q}}\), respectively. Furthermore since \(\boldsymbol {\mathcal {Q}}\) is real-valued and skew symmetric, and therefore a normal matrix, the eigenvectors \(\mathbf {v}_{k}\) are orthonormal. The new basis functions (complex-valued) are depicted in Fig. 3 for \(N_{\mathrm {p}}=4\). Note that the basis consists of pairs of conjugate complex basis functions.
PWM eigenfunctions \(g_{k}(\tau )\), \(k\in \{0,1,2,3,4\}\), i.e., \(N_{\mathrm {p}}=4\). (top) real part. (bottom) imaginary part
Inserting the transformed basis functions instead of the PWM basis functions into (6) and (7) leads, using the orthonormality of the eigenvectors, to
$$ \boldsymbol {\mathcal {A}}\frac{{\mathrm {d}}\mathbf {w}}{{\mathrm {d}}t_{1}} + {\widetilde {\boldsymbol {\mathcal {B}}}}\, \mathbf {w}(t_{1}) = {\widetilde {\boldsymbol {\mathcal {C}}}}(t _{1}) , $$
where \(\boldsymbol {\mathcal {A}}\) as in (5),
$$\begin{aligned} &{\widetilde {\boldsymbol {\mathcal {B}}}}=\boldsymbol {\mathcal {J}}\otimes \mathbf {B}+\boldsymbol{\varLambda }\otimes \mathbf {A}, \end{aligned}$$
$$\begin{aligned} &{\widetilde {\boldsymbol {\mathcal {C}}}}(t_{1})= \int _{0}^{T_{{\mathrm {s}}}} \bar{\mathbf{g}}\bigl(\tau (t _{2})\bigr) \otimes \mathbf {\widehat {c}}(t_{1},t_{2}) \,{\mathrm {d}}t_{2} , \end{aligned}$$
and Λ is a diagonal matrix with diagonal entries \({\lambda _{0}, \lambda _{1}, \dots , \lambda _{N_{\mathrm {p}}}}\). Thus the resulting matrices in (13) are block-diagonal and the degrees of freedom can be block-wisely decoupled. This leads to \(N_{\mathrm {p}}+1\) independent systems of equations given by
$$ T_{\mathrm {s}}\,\mathbf {A}\frac{{\mathrm {d}}\mathbf {w}_{k}}{{\mathrm {d}}t_{1}} + (T_{\mathrm {s}}\,\mathbf {B}+ \lambda _{k}\, \mathbf {A})\, \mathbf {w}_{k} = \int _{0}^{T_{{\mathrm {s}}}} \bar{g}_{k}\bigl(\tau (t_{2})\bigr) \,\mathbf {\widehat {c}}(t_{1},t_{2}) \,{\mathrm {d}}t_{2} \quad\text{for } k = 0,\dots ,N_{\mathrm {p}}, $$
where \(\mathbf {w}=[\mathbf {w}_{0}^{\top }, \mathbf {w}_{1}^{\top }, \dots , \mathbf {w}_{N_{\mathrm {p}}} ^{\top }]^{\top }\). Note that if a diagonal entry in Λ is complex, there is also a complex conjugate counterpart. The solutions of the decoupled system of equations corresponding to this complex eigenvalue and its conjugate complex counterpart, are, as a result, complex conjugate to each other. Therefore it is sufficient to solve one of them. This is similar to harmonic balance methods in which the harmonic basis functions are given by pairs of complex conjugates leading to similar systems of equations. In analogy to "harmonic balance method", we call the developed method the "multirate PWM balance method".
Test case and numerical results
The method is applied to the buck converter from Fig. 1, where the pot inductor is represented by a 2D field model with conducting core material (ferrite, \(\sigma _{\mathrm{fe}}=250 \) S/m). The coils are modeled as stranded conductors. The simulation interval is given by \(\varPsi =[0,10] \) ms. The switching frequency is \(f_{\mathrm{s}}=\frac{1}{T_{\mathrm {s}}}=1000 \) Hz. For the pulsed excitation (2) we use \(V_{0}=24 \) V. All calculations are performed in MATLAB. The partial differential equations governing the magnetoquasistatic problem are given by
$$ \sigma (\mathbf {r}) \frac{\partial \mathbf {A}_{\mathrm{m}}(\mathbf {r}, t)}{\partial t} + \nabla \times \bigl(\bigl(\mu (\mathbf {r})^{-1}\bigr) \, \nabla \times \mathbf {A}_{ \mathrm{m}}(\mathbf {r}, t) \bigr) = \mathbf {J}_{\mathrm {s}}(\mathbf {r}, t), $$
where r is the position vector, t is the time, \(\mathbf {A}_{ \mathrm{m}}\) is the modified magnetic vector potential [4], \(\mathbf {J}_{\mathrm {s}}\) are the imposed currents, \(\mu =4 \pi \times 10^{-7} \) H/m is the magnetic permeability and σ is the conductivity which is only non-zero in the ferrite core (\(\sigma _{\mathrm{fe}}\)). The problem is considered on a 2D planar domain with homogeneous Dirichlet boundary conditions.
Correspondingly, the Finite Element magnetoquasistatic [13] discretization of the magnetoquasistatic inductor model is given by the differential-algebraic system of equations [8]
$$ \mathbf {M}_{\sigma } \frac {{\mathrm {d}}}{{\mathrm {d}}t}\mathbf {a}(t) +\mathbf {K}\,\mathbf {a}(t) = \mathbf {P}\,i_{\mathrm{L}}(t), $$
where \(\mathbf {M}_{\sigma }\) is the singular conductivity matrix, K is the stiffness matrix, \(\mathbf {a}(t)\) gathers the degrees of freedom (DOFs) related to the magnetic vector potential, P is the discretization of the winding function [13] and \(i_{\mathrm{L}}(t)\) is the current through the inductor. The field-circuit coupling is expressed as follows. An additional variable is introduced for the magnetic flux linkage \(\varPhi (t)=\mathbf {P}^{\top }\mathbf {a}(t)\). All equations are coupled monolithically into the index-1 differential-algebraic system of equations [1]
$$\begin{aligned} &\mathbf {M}_{\sigma } \frac {{\mathrm {d}}}{{\mathrm {d}}t}\mathbf {a}- \mathbf {P}\,i_{\mathrm{L}}+ \mathbf {K}\,\mathbf {a}= \mathbf{0}, \end{aligned}$$
$$\begin{aligned} &\mathbf {P}^{\top }\mathbf {a}- \varPhi= 0, \end{aligned}$$
$$\begin{aligned} &\frac {{\mathrm {d}}}{{\mathrm {d}}t}\varPhi + R_{\mathrm {L}}\,i_{\mathrm{L}}+ v_{\mathrm {C}}= v_{\mathrm {i}}(t), \end{aligned}$$
$$\begin{aligned} &C \,\frac {{\mathrm {d}}}{{\mathrm {d}}t}v_{\mathrm {C}}- i_{\mathrm{L}}+ \frac{1}{R} \,v_{\mathrm {C}}= 0, \end{aligned}$$
which for the example in Fig. 1 contains a total of \(11\text{,}053\) DOFs. The initial conditions are given by \(v_{\mathrm {C}}(0)=0\), \(i_{\mathrm{L}}(0)=0\) and \(\mathbf {a}(0)=\mathbf{0}\).
The initial condition for the MPDEs (3) can be written as
$$ h_{j}(t_{2}) \approx \sum _{k=0}^{N_{\mathrm {p}}} w_{j,k}(0) \, p_{k}\bigl(\tau (t _{2})\bigr), $$
where \(h_{j}\) is the j-th element of h. It only has to satisfy the condition \(\mathbf {h}(0) = \mathbf {\widehat {x}}(0,0)=\mathbf {x}_{0}\). Consequently there is a high degree of freedom in choosing the initial values \(\mathbf {w}(0)\) for the system of equations (5). However not all choices lead to an efficient simulation, i.e., low dynamics in the slow time scale. The following choice of initial values has proven advantageous. First, the steady-state solution is calculated, i.e.,
$$ \mathbf {w}^{\,s}= \boldsymbol {\mathcal {B}}^{-1}\boldsymbol {\mathcal {C}}(0). $$
Secondly, the initial coefficients for \(k=1,\dots , N_{\mathrm {p}}\) are extracted from the steady-state solution \(w_{j,k}(0)=w^{\,s}_{j,k}\) for all j. The remaining coefficients are calculated by solving the solution expansion (4) for \(w_{j,0}(0)\) and using the condition \(\mathbf {\widehat {x}}(0,0)=\mathbf {x}_{0}\). In summary the initial coefficients are given by
$$ w_{j,k}(0)= \textstyle\begin{cases} w^{\,s}_{j,k} & \text{for }k=1,\dots ,N_{\mathrm {p}}\text{ and for all } j=1, \dots ,N_{\mathrm {s}}, \\ x_{j}(0) - \sum_{l=1}^{N_{\mathrm {p}}} w^{\,s}_{j,l} \,\, p_{l}(0) & \text{for }k=0 \text{ and for all } j=1,\dots ,N_{\mathrm {s}}. \end{cases} $$
The initial conditions for the system of equations (13) are computed similarly using the PWM eigenfunctions. Other choices of initial values may still lead to the correct solution but might impair the efficiency of the method.
To calculate the reference solution with a conventional adaptive time discretization, the MATLAB solver ode15s is used. It is modified to restart the simulation at the known switching instances. Consistent initial values for the restart of the solver are calculated by using a Newton–Raphson algorithm to solve the set of algebraic equations. The required differential variables are taken from the solution at the end of the prior solution interval. After finding the new set of initial values, the initial slopes of the differential variables are calculated by solving the subsystem of ordinary differential equations for the slope.
The multivariate solution \(\mathbf {\widehat {x}}(t_{1},t_{2})\) calculated using the multirate PWM balance method, i.e., solving (13) with ode15s, is reconstruced using (4) and the multivariate voltage at the capacitor is depicted in Fig. 4. The corresponding solution component of the original system of equations (1) is extracted along a diagonal through the computation domain. Figure 5 shows the current through the inductor along with the reference solution. The agreement between the multirate PWM balance method solution and the reference solution is excellent. The Joule losses in the core material due to eddy currents are calculated by
$$ P_{\mathrm{eddy}}(t)= \int _{\varOmega } \mathbf{E}(\mathbf {r}, t) \cdot \sigma (\mathbf {r}) \mathbf{E}(\mathbf {r}, t) \,{\mathrm {d}}\mathbf {r}= \bigl(\mathbf {e}(t)^{ \mathrm{H}} \bigr) \,\mathbf {M}_{\sigma } \,\mathbf {e}(t), $$
where E is the electric field strength, Ω is the spatial computation domain, the superscript H denotes the Hermitian, i.e., the complex conjugate transposed, and \(\mathbf {e}(t)=-\frac {{\mathrm {d}}}{{\mathrm {d}}t}\mathbf {a}(t)\) is the line-integrated discrete electric field. The Joule losses are plotted as well in Fig. 5. Figure 6 depicts the solution of (13), i.e., the coefficients \(\mathbf {w}(t_{1})\), exemplary for the current through the inductor \(i_{\mathrm{L}}\). As one can see, using the initial values (25), only the coefficient \(w_{j,0}\) corresponding to the zero-th basis function varies and the others stay constant. To quantify the accuracy and efficiency of the multirate PWM balance method, it is compared to conventional time discretization and to the MPDE approach with the original PWM basis functions. Different settings are considered: To analyze the performance of the conventional time discretization, the relative and absolute tolerance setting of the solver is changed, i.e., \(\mathrm{abstol}=\mathrm{reltol} \in [10^{-6}, 10^{-1}]\); For the case of the multirate PWM balance method and the MPDE approach with the original PWM basis functions, relative and absolute tolerances are fixed at \(\mathrm{abstol}=\mathrm{reltol}=10^{-7}\) and the number of basis functions \(N_{\mathrm {p}}\in \{1,\dots,10\}\) is changed. The accuracy is measured for the voltage output of the converter, i.e., the voltage at the capacitor. The relative \({\mathrm {L}^{2}}\) error is given by
$$ \epsilon (\mathrm{tol}, N_{\mathrm {p}}) = \frac{ \Vert v_{\mathrm{C,ref}}(t)-v_{ \mathrm{C}}^{\,h}(\mathrm{tol},N_{\mathrm {p}},t) \Vert _{{\mathrm {L}^{2}}(\varPsi )}}{ \Vert v_{ \mathrm{C,ref}}(t) \Vert _{{\mathrm {L}^{2}}(\varPsi )}}, $$
where \(v_{\mathrm{C,ref}}\) is the reference solution and \(v_{ \mathrm{C}}^{\,h}\) is the solution using the multirate PWM balance method and the MPDE approach with the original PWM basis functions. The norm is approximated using mid-point quadrature. Figure 7 shows the error plotted as a function of the solution time, i.e., the time that ode15s needs. For conventional time discretization the time for solving consists of the time that is needed to calculate consistent initial values and slopes after switching, and the actual time that ode15s needs. The time to calculate consistent initial values and slopes depends on the number of switching instants and is thus constant if the switching frequency or simulation interval does not change. It is given by approximately 16 seconds. The total time displayed in Fig. 7 is the sum of both contributions.
Multivariate voltage at the capacitor calculated using the multirate PWM balance method. The solution component corresponding to the original system of equations (1) is extracted along a diagonal and marked as a black curve
(top) Reference solution calculated using conventional adaptive time discretization compared to the solution obtained by the MPDE approach with \(N_{\mathrm {p},\mathrm {pwmbal}}=4\) PWM eigenfunctions. The relative \({\mathrm {L}^{2}}\) error of the current through the inductor similar to (27) is approximately \(3 \times 10^{-5}\). (bottom) Joule losses in the core material due to eddy currents
Coefficients \(w_{1,k}\) for the inductor current calculated by solving (13) with \(N_{\mathrm {p},\mathrm {pwmbal}}=4\). (top) real part. The coefficients \(w_{1,1}, \dots , w _{1,4}\) are approximately the same therefore they are hard to distinguish visually. (bottom) imaginary part
Error ϵ as defined in (27) over time for solving the systems of equations. The MPDE approach with PWM eigenfunctions (multirate PWM balance method) is considerably faster than the MPDE approach with the original PWM basis functions and the conventional time discretization
As one can see the MPDE approach with the original PWM basis functions is considerably slower than conventional time stepping. This is due to the fact that the already large systems of equations (1) (due to field-circuit coupling) are even further increased in size through the Galerkin approach. The stagnation of the error at 10−6 in Fig. 7 for values larger than \(N_{\mathrm {p}}=7\) is caused by the chosen accuracy of ode15s. Furthermore one can see that when adding another basis function the error does decrease with every second basis function. This was already observed in [5, 10]. For this reason the error for the PWM eigenfunctions is only plotted for \(N_{\mathrm {p},\mathrm {pwmbal}}\in\{1,2,4,6,8,10\}\). Since the systems of equations resulting from the multirate PWM balance method are decoupled, they can be solved efficiently in parallel. For each basis function \(g_{k}\) with \(k=0,\dots ,N_{\mathrm {p}}\), a complex-valued initial value problem of the form (16) has to be solved. The size of these systems of equations is the same as that of the original system of equations (1). However, the time for solving is considerably smaller since less time steps are necessary for the same solution accuracy. Note that due to the choice of the initial values (25) most coefficients in (13) for this numerical example do not change and only those corresponding to the zero-th basis function vary. This means that only the decoupled system of equations which corresponds to the zero-th basis function takes considerable computational effort to solve. In a parallel computing environment one would choose as many processor cores as basis functions (\(N_{\mathrm {p}}+1\)). The overall runtime is then determined only by the initial value problem that takes the longest to integrate. For this numerical example it is \(k=0\). The communication overhead between processors is not taken into account since it is highly implementation and machine dependent. The slightly decreasing time to solution when \(N_{\mathrm {p},\mathrm {pwmbal}}>1\) is owed to the fact that initial values according to (25) take more a-priori information into account which leads to smaller number of time steps and faster simulation. The overall accuracy of the method is problem-specific and always depends on both the tolerance for the solver and the number of basis functions. An a-priori determination of the number of basis functions and the solver tolerance is not yet available. An a-posteriori estimator can be constructed by increasing the number of basis functions and comparing the solutions. The resulting error is also related to the time stepping error.
The MPDE approach works also for nonlinear problems. However, similarly as for the harmonic balance case, the decoupling is not straightforward anymore. Furthermore the PWM basis functions and thus also the PWM eigenfunctions might not be able to represent the solution of problems with nonlinear elements [10]. If the amplitude of the ripples is small compared to the amplitude of the envelope, the particular efficient approach described in [9] can be applied. It uses only the slowly varying envelope to evaluate the nonlinearities. Although the assembly of the field model matrices for a new envelope cannot be parallelized, the matrices in (13) can still be decoupled and calculations to obtain the following time step can be run in parallel.
A new efficient technique was presented for field-circuit coupled models of DC-DC power converters, in which the switches are idealized and the filtering circuit is linear. The already existing MPDE technique with PWM basis functions splits the solution into fast varying and slowly varying parts. In this paper this method has been improved by introducing a new set of PMW basis functions which decouple the systems of equations similar as in the harmonic balance method. The new method, now called multirate PWM balance method, enables a parallel solution of all PWM modes resulting in a speed-up amounting to a factor 4 for the test example.
DOFs:
PWM:
pulse-width modulation
MPDE:
multirate partial differential equation
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This work is supported by the "Excellence Initiative" of German Federal and State Governments and the Graduate School CE at TU Darmstadt. The authors thank Johan Gyselinck for fruitful discussions. Further thanks go to Jonas Bundschuh and Erik Skär for their contribution to the first implementation of the PWM eigenfunctions.
Not yet available publicly. It is, however, planned to make the code, which is used to generate the results, publicly available in the near future.
No funding to report.
Graduate School of Computational Engineering, Technische Universität Darmstadt, Darmstadt, Germany
Andreas Pels, Herbert De Gersem & Sebastian Schöps
Institut für Teilchenbeschleunigung und Elektromagnetische Felder, Technische Universität Darmstadt, Darmstadt, Germany
Department of Electrical Engineering, EnergyVille, KU Leuven, Leuven, Belgium
Andreas Pels & Ruth V. Sabariego
Andreas Pels
Herbert De Gersem
Ruth V. Sabariego
Sebastian Schöps
All authors have jointly carried out the research and worked together on the manuscript. The numerical tests have been conducted by the first author. All authors read and approved the final manuscript.
Correspondence to Andreas Pels.
There are no competing interests to report.
Pels, A., De Gersem, H., Sabariego, R.V. et al. Multirate PWM balance method for the efficient field-circuit coupled simulation of power converters. J.Math.Industry 9, 8 (2019). https://doi.org/10.1186/s13362-019-0065-8
Linear circuits
DC-DC power conversion
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CommonCrawl
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Optimization by Self-Organized Criticality
Generalized entropies, density of states, and non-extensivity
Sámuel G. Balogh, Gergely Palla, … Dániel Czégel
Connecting complex networks to nonadditive entropies
R. M. de Oliveira, Samuraí Brito, … Constantino Tsallis
The Polynomial Volume Law of Complex Networks in the Context of Local and Global Optimization
Franz-Benjamin Mocnik
How driving rates determine the statistics of driven non-equilibrium systems with stationary distributions
Bernat Corominas-Murtra, Rudolf Hanel, … Stefan Thurner
Enabling Controlling Complex Networks with Local Topological Information
Guoqi Li, Lei Deng, … H. Eugene Stanley
Minimal fatal shocks in multistable complex networks
Lukas Halekotte & Ulrike Feudel
Fragility Limits Performance in Complex Networks
Fabio Pasqualetti, Shiyu Zhao, … Sandro Zampieri
Aggregation-fragmentation and individual dynamics of active clusters
F. Ginot, I. Theurkauff, … C. Cottin-Bizonne
Thermodynamics of structure-forming systems
Jan Korbel, Simon David Lindner, … Stefan Thurner
Heiko Hoffmann ORCID: orcid.org/0000-0003-3817-56261 &
David W. Payton1
Power law
Self-organized criticality (SOC) is a phenomenon observed in certain complex systems of multiple interacting components, e.g., neural networks, forest fires, and power grids, that produce power-law distributed avalanche sizes. Here, we report the surprising result that the avalanches from an SOC process can be used to solve non-convex optimization problems. To generate avalanches, we use the Abelian sandpile model on a graph that mirrors the graph of the optimization problem. For optimization, we map the avalanche areas onto search patterns for optimization, while the SOC process receives no feedback from the optimization itself. The resulting method can be applied without parameter tuning to a wide range of optimization problems, as demonstrated on three problems: finding the ground-state of an Ising spin glass, graph coloring, and image segmentation. We find that SOC search is more efficient compared to other random search methods, including simulated annealing, and unlike annealing, it is parameter free, thereby eliminating the time-consuming requirement to tune an annealing temperature schedule.
Evidence suggests that the healthy mammalian brain operates in a critical state1,2,3,4. Critical systems are dynamical systems with many interacting particles or components that display scale-invariant fluctuations. Scale invariance has long fascinated the human mind. The laws of physics are not invariant to scale: a 10-times wider and longer wooden cylindrical beam cannot support a 1000-times larger weight. Yet, in nature, we find many examples of approximate scale-invariance, e.g., the fractal shape of flowers and trees with repeating structures at different scales.
In physics, we usually achieve criticality by fine-tuning a control parameter, e.g., the temperature, to a specific value at which we observe the scale-invariant fluctuations (see, e.g., the liquid-vapor critical point). Yet, in many natural systems, criticality occurs even without such fine-tuning, and, apparently, the brain is one example of such a system. This self-tuning to criticality has been called self-organized criticality (SOC)5.
Criticality has been further referred to as a special state between order and chaos2: systems are ordered at low temperatures and random at high temperatures, but appear the most complex at the critical temperature. So, it is appealing to think that the brain operates at criticality. Interestingly, experiments show that when the brain malfunctions, e.g., during epileptic seizures, the brain loses the characteristics of criticality4.
In the brain, criticality is often characterized by neural avalanches. Neural avalanches are bursts of neural activity, e.g., as recorded with multi-electrode arrays measuring local field potentials (LFPs)1. Here, synchronous LFP spikes across multiple electrode locations constitute an avalanche. Experiments show that the sizes of neural avalanches follow power-law distributions1, which are generally a consequence of scale invariance and thus evidence for criticality, and neural models have been developed that automatically tune networks towards the critical state6.
It is still debated, however, if the brain is actually in a critical state and what the benefits of such a property are3,7,8. As benefits, optimal dynamic range, memory, and computational power have been suggested2,3,9. For example, Bertschinger and Natschläger10 show that recurrent neural networks perform best at the critical state; there, however, the task was simple (3-bit parity), and a tuning parameter was required to achieve the critical state.
In neural network models, the ability to reach an SOC state is typically studied in isolation, without having the network at the same time carry out a "useful" computational task (see, e.g., Stepp et al.11). On the other hand, the useful deep-learning networks are not critical. The difficulty to design a network that is critical and useful at the same time motivated this present work: we isolate the SOC dynamics in one network and then use these dynamics to drive a separate system. So, we create a novel optimization algorithm that uses the SOC process to create test patterns for optimization.
To solve an optimization problem, we initially identify a graph of linked/coupled optimization variables (which, e.g., are linked through an equation). Then, we create a new graph for the SOC process that mirrors the optimization graph such that there is a one-to-one correspondence between the optimization variables and the nodes in the SOC graph. The SOC process will continuously create avalanches on its graph, and the correspondence allows us to map these avalanches onto test patterns of the optimization variables.
Surprisingly, this method, which we call SOC search, can efficiently find approximate solutions to non-convex optimization problems that have many local minima. We demonstrate the performance of SOC search on a variety of optimization problems and compare against simulated annealing12 and random search. In contrast to simulated annealing, SOC search needs no parameter tuning so long the graph structure on which the SOC process is computed matches the optimization problem.
The reminder of this article is organized as follows. Section 2 introduces a popular model of SOC, the Abelian sandpile model5, which is also the basis for our optimization process. Section 3 describes the SOC optimization algorithm. Section 4 describes our experiments, introducing each optimization problem: Ising spin glass, graph coloring, and image segmentation; the latter optimizes the energy of a Markov random field. Finally, Section 5 discusses the benefits and limits of SOC search and summarizes the work.
Self-Organized Criticality
Self-organized criticality has often been illustrated with a sand pile5,13. Grains of sand are added one at a time, and eventually, the slope of the pile converges to a specific angle. At this angle, small and large avalanches occur, and the system is balanced just right such that the avalanche sizes follow a power law (this example is illustrative; not all real sand piles are actually self-organized critical13). The system self-organizes because the slope self-adapts: grains are added to a shallow slope until avalanches occur, and a slope too steep will collapse in a large avalanche.
Bak et al. developed a model that mimics the sandpile dynamics, and this model is actually self-organized critical5. This model, called Bak-Tang-Wiesenfeld model or Abelian sandpile model14, has been originally defined on square and cubic lattices, but it can be generalized to arbitrary graphs14. Here, we consider this generalized version.
Assume a graph of nodes i that each contain x i amount of grains. The Abelian sandpile model consists of a slow and a fast process. The slow process adds one grain to a random node i, x i → x i + 1 (slow external driving force). The fast process computes the propagation of an avalanche, as described in the following. If a node is above threshold x i > d i , where d i is the degree of node i, then it sheds one grain to each of its d i neighbors,
$${x}_{i}(t+\mathrm{1)}={x}_{i}(t)-{d}_{i}$$
$$\forall j\in {{\mathscr{N}}}_{i}\,:{x}_{j}(t+\mathrm{1)}={x}_{j}(t)+1,$$
where \({{\mathscr{N}}}_{i}\) is the set of nodes that share an edge with node i (here, we consider only undirected graphs). This shedding may result in a neighbor to be above threshold, making this neighbor to also shed its grains according to the above equations. This repeated shedding can result in a cascade of events, an avalanche of grain topplings. Once an avalanche is complete, i.e., all nodes are at or below threshold, the slow process proceeds by adding another grain.
After an initial phase of self-organization, avalanches occur with sizes that follow a power law distribution (this distribution has an exponential cutoff because the number of nodes is limited). For a square lattice, Fig. 1 shows a typical sequence of avalanches. They occur at different locations, have different sizes, and appear to have fractal-like boundaries.
Sample sequence of avalanches from an SOC model on a 40 × 40 square lattice.
When we continuously add grains, they also have to disappear eventually. In the original model, the square lattice has edges and the grains just fall over the edges and disappear. In the generalized version, we need to add either a sink node or dissipation. All grains that arrive at a sink node disappear. With dissipation, grains disappear with a small probability when moving between nodes. Strictly, this dissipation breaks the SOC characteristics, but if it is sufficiently small it is not noticeable, because the exponential cutoff is dominated by the limited system size. When we increase the system size, we also need to reduce the dissipation accordingly to eliminate its impact. In our experiments, we used either a square lattice with edges or random graphs with dissipation (with a probability of 0.05).
SOC for Optimization
We use the above SOC process for non-convex optimization. To do so, we need an optimization problem with a graph structure; this article contains three examples. For example, in graph coloring, the optimization variables are the color values at each node of a graph. In an Ising spin glass, the optimization variables are the orientations of the spins that are energetically coupled to their neighbors. Moreover, we assume that the optimization variables are discrete, ∈{a1, ..., a k } with k different values.
Initially, we copy the graph structure, the set of vertices and edges, of the optimization problem and compute on this graph the Abelian sandpile model. Thus, there is a one-to-one correspondence between the nodes of the sandpile model and the optimization variables. In the sandpile model, the nodes are initialized with random grain numbers drawn uniformly from {1, ..., d i } for each node i. Grains are added and avalanches are computed until the SOC state is reached. Then, the actual optimization process begins (Algorithm 1).
Algorithm 1
SOC Search.
The avalanches act as delta patterns that change the corresponding optimization variables, computing test patterns z′. For k = 2, this change simply flips the variables (e.g., spins) in a test pattern. The resulting test patterns are used in a greedy optimization scheme. This process is iterated for a fixed number of iteration steps or until a stop criterion is reached, e.g, the change in energy E is below a threshold. Interestingly, the variables x i in Equations (1) and (2) are independent of the optimization variables z; there is no feedback onto the SOC process.
Other work before has combined the concepts of self-organized criticality and optimization, see extremal optimization15. Different from ours, however, that work uses an optimization algorithm inspired by SOC, but uses neither an SOC process to generate delta or test patterns nor SOC on a graph.
We chose three optimization problems to demonstrate the functionality and versatility of SOC search. Apart from copying the graph structure, no adjustment was required for each problem. The following three sections show results and describe our experiments for 1) optimizing the ground state of an Ising spin glass, 2) graph coloring, and 3) image segmentation.
Ising Spin Glass
We tested SOC search for finding the ground state of an Ising spin glass, which is a common benchmark test for optimization methods - see, e.g., Santoro et al.16. An Ising spin glass is an arrangement of spins, particles with a magnetic moment, in a lattice. The spins, s i , point either up or down and are coupled to their neighbors such that they prefer to have either aligned or opposing magnetic movements, depending on a coupling matrix, J ij . In the ground state, i.e., without any thermal fluctuations, the spins orient themselves to minimize the energy
$$E=\sum _{i,j\in {{\mathscr{N}}}_{i}}{J}_{ij}{s}_{i}{s}_{j}\mathrm{.}$$
The spins can have two states \(\frac{1}{2}\) or \(-\frac{1}{2}\). Here, we chose a 80 × 80 lattice, and the J ij were chosen randomly uniformly from the interval (−2; 2). On such a lattice, polynomial algorithms exist to find the ground state (on generic graphs the problem is NP hard); still for approximate solvers like ours, the difficulty is in principle the same, and therefore, the lattice is suitable for benchmarking16. We chose the square lattice since methods exist to compute the exact energy minimum, which we obtained from the Spin Glass Server17 and subtracted from Equation (3); so, zero energy is the optimum (Fig. 2).
Results for optimizing the energy of an Ising spin glass on a 80 × 80 lattice (a,c) and probability of improving the energy in a greedy search (b), showing mean ± std (n = 10). Search-pattern samples are shown in (d). In (b), the results for Random Clusters and Square Shapes have been omitted for clarity because Random Clusters produced similar probabilities as SOC Search and Square Shapes similar probabilities as Random Dots.
We compared SOC search against three methods: random search flipping one spin, random search flipping as many spins as the size of an SOC avalanche, and simulated annealing12. The first two serve as baselines and the third to illustrate the competitiveness of SOC search. One-spin random search is doing a greedy optimization flipping one random spin at a time.
To demonstrate that the shape of the SOC avalanche matters and not just the size distribution, we split the random search with avalanche sizes into three variants: random dots, square shapes, and random clusters. For each, the operation is similar to Algorithm 1, but instead of using the avalanche nodes \({\mathscr{A}}\), \(|{\mathscr{A}}|\) nodes are chosen randomly according to one of the following three procedures: 1) Random dots: A set of \(|{\mathscr{A}}|\) dots/nodes is uniformly randomly chosen from the spin lattice. 2) Square shapes: A square with an area matching \(|{\mathscr{A}}|\) as close as possible is placed at a random location (uniformly distributed). 3) Random clusters: A random cluster is grown starting from a uniformly randomly placed seed location. The cluster grows one spin at a time until it reaches the corresponding SOC-avalanche size. In each growth step, one spin is placed at a random location uniformly distributed among all sites that neighbor spins in the current cluster. These random clusters were most similar to the SOC patterns compared to the other random pattern variants (Fig. 2d).
Simulated annealing12 is often the method of choice when dealing with problems that have many local minima. The annealing process helps escape local minima by allowing moves to states with higher energy. These moves happen with probability p = exp(−ΔE/T), where T is a temperature-like variable. At high temperature, moves to a higher energy are likely, but when the temperature approaches zero, the algorithm becomes greedy. Theoretically, simulated annealing is guaranteed to find a global optimal, but only in infinite time. In practice, simulated annealing depends on the proper choice of the initial temperature and temperature schedule, also called annealing schedule.
Here, we used the same annealing schedule as in Santoro et al.16, a linearly decreasing temperature with a starting value of T = 3. Like for one-spin random search, we tested the energy change of flipping one spin at a time and flipped the spin according to above probability. Different from random search, we did not choose the spins randomly, but chose the nodes in the lattice in sequence and repeated the sequence over and over again. This strategy produced better results than the random choice.
For a fair comparison of the computational cost, we count the number of spin flips for each method because the cost of evaluating a change in energy is linear in the number of nodes flipped. For SOC search, we flip all spins in an avalanche; so, we require more flips per iteration than simulated annealing. This evaluation actually puts our method at a disadvantage because, optimally, we would need to evaluate the energy at only the boundary of an avalanche (if two neighboring spins flip, the contribution of their edge to the energy remains unchanged, see Equation 3). Still, we used this evaluation for simplicity. In the computational cost, we omitted the cost of computing the Abelian sandpile model itself because this process could be either computed in parallel or beforehand: since the sandpile operates independently without feedback, we could compute the avalanches in advance and replay from memory.
As a result of the optimization, SOC search reached the lowest energy values with the fewest number of spin flips (Fig. 2a). Simulated annealing required about 8× more flips for reaching the same energy levels. The random searches with one spin, random dots, and random squares got stuck in local minima. The random clusters performed almost as well as SOC search, but reached significantly worse energy values for sufficiently many iteration steps.
We averaged all results over 10 optimization runs, each starting with a different random spin configuration, with the same starting configuration used across all methods. As in Santoro et al.16, each data point on the energy curves is from a separate set of 10 simulation runs because simulated annealing requires a unique annealing schedule for each data point. Once annealing reaches zero temperature, it gets stuck in local minima like one-spin random search; so, the annealing schedule has to be adapted for each run.
To gain a greater insight into the workings of SOC search, we computed the probability of energy improvements in the greedy search (Fig. 2b). After many iterations, i.e., 104 flips per spin, the probability to improve the energy with SOC avalanches becomes much larger than with just avalanche-sized random-dot patterns. Instead of dropping off abruptly, the probability decayed more slowly according to a power law, having a slope of −1.5.
Graph Coloring
Graph coloring has many applications such as scheduling18, register allocation19, and timetabling. The objective in graph coloring is to color nodes in a graph such that no neighboring nodes are of the same color. On generic graphs, coloring is NP hard. The optimization problem is to find the smallest number of colors possible without edge violations.
To apply simulated annealing and SOC search, we need a suitable cost function. Here, we choose one that has been shown to work with simulated annealing:20
$$E=-\sum _{i=1}^{k}|{c}_{i}{|}^{2}+\sum _{i=1}^{k}\mathrm{2|}{c}_{i}||{e}_{i}|,$$
where |c i | is the number of nodes in color class i, |e i | the number of bad edges attached to nodes of color i, and k an upper bound on the number of colors. Using this cost function, legal colorings are local minima20, and the number of colors is implicitly minimized and is obtained as the number of nonzero |c i | values.
We compared SOC search against one-node random search, random search with random dots, and simulated annealing. For one-node random search and simulated annealing, the search selected one random node at a time. We used an exponentially decaying annealing schedule (similar to Johnson et al.20), starting at T = 10. Preliminary tests found that this schedule worked better for graph coloring.
We tested two random graphs: a small world21 and a random geometric graph, both with 1,000 nodes (Fig. 3 Left). In the small world graph, each node had 4 neighbors, and the probability of random re-connections was p = 0.05. The random geometric graph had an average degree of 6.7. For coloring, we chose k = 6 for the first graph and k = 10 for the second. Similar to above, we evaluated the computational cost in terms of the number of color changes per node, and as above, for simulated annealing, we re-ran the optimization for each displayed number of color changes.
Results of graph coloring on a small-world graph (Top) and a random geometric graph (Bottom), showing a legal coloring produced by SOC search (Left) and cost functions (Right), mean ± std (n = 5).
On both graphs, SOC search produced the smallest cost values (Fig. 3). The results were averaged over 5 optimization runs. For the small-world graph, only SOC and simulated annealing found a legal coloring with 4 colors (other methods needed 5 colors), and for the random geometric graph, only SOC and simulated annealing found a coloring with 9 colors (other methods needed 10 colors).
Surprisingly, despite the random color choice for each node in an avalanche (see Algorithm 1), the path-connectedness of an avalanche still mattered. However, the SOC benefit is smaller the more colors we need to choose from. Simulated annealing does outperform SOC for a graph that requires many different colors (100) and is closer to a fully connected graph, e.g., an Erdos-Renyi graph with 1,000 nodes and probability p = 0.5 for connecting two nodes (data not shown).
Image Segmentation
Image segmentation separates images into distinct regions and is commonly used to separate foreground objects from background. Here, for demonstration, we focus on feature-based image segmentation, which considers local features like color and texture. A different approach is semantic segmentation, which considers object ID - see, e.g., Long et al.22. Feature-based segmentation is useful for automatic separation of objects from background, e.g., for image editing. We chose image segmentation as our final test because it allows us to visually evaluate the optimization results.
A powerful method for feature-based image segmentation are Markov random fields23,24. Using a Markov random field, the set of pixel labels, \({\mathscr{S}}\), is a random field, i.e., each pixel label is modeled as a random variable. The goal is to find a segmentation \({\mathscr{S}}\) that maximizes the posterior \(P({\mathscr{S}}|{\mathscr{I}})\) for a given image \({\mathscr{I}}\). The posterior is computed according to Bayes rule, \(P({\mathscr{S}}|{\mathscr{I}})\propto P({\mathscr{I}}|{\mathscr{S}})P({\mathscr{S}})\).
The two required factors are the probability for the image model, \(P({\mathscr{I}}|{\mathscr{S}})\), and the prior probability of the image labels, \(P({\mathscr{S}})\). Here, for the image model, we use a Gaussian distribution as in Kato and Pong24,
$$P({\mathscr{I}}|{\mathscr{S}})=\prod _{i}\frac{1}{\sqrt{{(2\pi )}^{n}|{{\rm{\Sigma }}}_{{s}_{i}}|}}\exp (-\frac{1}{2}({f}_{i}-{\mu }_{{s}_{i}}){{\rm{\Sigma }}}_{{s}_{i}}^{-1}{({f}_{i}-{\mu }_{{s}_{i}})}^{T}),$$
where f i is the feature vector for pixel i, \({\mu }_{{s}_{i}}\) the feature center for label s i , and \({{\rm{\Sigma }}}_{{s}_{i}}\) the covariance matrix for label s i . For the prior, it is common to use a Gibbs distribution24,25,
$$P({\mathscr{S}})=\frac{1}{Z}\prod _{i,j}\exp (-{K}_{ij}),$$
here, with K ij = γexp(−β||f i − f j ||2) if i and j are neighbors and belong to different label classes, otherwise K ij is zero. So, if neighboring features are different, K ij is approximately zero and does not penalize the probability \(P({\mathscr{S}})\), but if the neighbors have similar features even though they belong to different classes, we get a penalty: the prior probability is reduced. Our K function is a slightly modified and simplified version of the function used in Kohli and Torr25, because our K produced better results in our experiments.
Instead of maximizing the posterior directly, it is common to minimize the exponent and define it as an energy23,24; so, \(P({\mathscr{S}}|{\mathscr{I}})\propto \exp (-E)\) with \(E=-ln(P({\mathscr{I}}|{\mathscr{S}}))-ln(P({\mathscr{S}}))\). Numerically, computing with the exponents is more robust. In the following, we use this energy for optimization, setting the constants γ = 5.0 and β = 1.0.
To obtain the image features, we combined color and local texture values24. First, images were converted to LUV color space, and the resulting three colors are the first three dimensions of the feature vector. Then, the LUV images were converted to grayscale and filtered with Gabor and Gauss filters. We repeated the filtering for 4 orientations (0°, 45°, 90°, 135°) of Gabor filters and stored the resulting values after Gaussian smoothing into the feature vector, so each pixel has a 7-dimensional feature vector (3 colors + 4 textures).
For optimization, we segmented images into two label classes and compared the same four methods as above for graph coloring. SOC search produced the most consistent and meaningful segmentations (Fig. 4). For the energy curves, results were averaged over 7 optimization runs. Here, instead of choosing the number of flips on the x-axis, we used the number of iterations (increments of t in Algorithm 1) because \(P({\mathscr{S}}|{\mathscr{I}})\) is computationally expensive and has to be re-evalated after each label change. Figure 4a shows the worst-case segmentations over 7 runs for each test image. Here, SOC produced almost the same result in all 7 simulation runs, while the other methods produced occasional bad results. For the best-case segmentations (not shown), the results of SOC search and simulated annealing were the same; the other methods did slightly worse.
Result of unsupervised image segmentation: worst-case segmentations out of 7 optimization runs (a) and energy curves for the tulip segmentation (b), mean ± stderr (n = 7).
For the first time, we used the avalanche patterns of an SOC process to alter test patterns for optimization. Surprisingly, this simple method performs remarkably well on non-convex optimization problems. On the same problems, other greedy methods get stuck in local minima, but SOC search overcomes these minima without using any annealing, without even using any parameter tuning.
A disadvantage of simulated annealing is its dependence on the annealing schedule and parameters. Its performance critically depends on the starting temperature and the number of annealing steps. Moreover, before an optimization problem is solved, it is unknown how many annealing steps are required. This limitation is a big disadvantage because once the temperature reaches near zero, the optimization ends in a local minimum. So, simulated annealing has to be restarted with a new pre-defined number of annealing steps. In contrast, the SOC process is invariant in time (over sufficiently long time scales), and a given solution can be further improved with additional SOC iterations.
Our experiments further demonstrate that the shape of an avalanche matters: the size distribution itself was insufficient for a good optimization performance. With random dot and square patterns, the optimization got stuck in local minima. In our experiments with different shaped patterns, the only difference was the shape; the temporal sequence of sizes was exactly the same between methods since the optimizations were carried out in parallel using the same SOC process to generate the size distribution. The closer the shape of the random patterns resembled the SOC avalanches the better their performance. Our random clusters resembled avalanches and almost reached the performance of SOC, but there are still differences between these types of patterns: the random clusters are generally more concentric while SOC avalanches are more varied and occasionally elongated (see, e.g., Fig. 2d).
The image-segmentation experiment further provides an illustration of how SOC overcomes local minima (Fig. 5). Since the avalanches come in a wide variety of sizes and shapes (Figs 1 and 2d), it is possible that a pattern occurs that complements a current segmentation sufficiently well to overcome a local minimum. Apparently, the distribution and variability of shapes is well (maybe optimally?) balanced to allow for efficient optimization. This topic is subject to further theoretic investigation. Future work will investigate if SOC patterns are indeed optimal and which characteristics of the avalanche shape are responsible for an optimal performance.
The SOC process results in delta patterns of a wide variety of sizes and shapes, increasing the probability that a delta pattern brings a segmentation out of a local minimum.
SOC search cannot solve all NP-hard optimization problems. The optimization variables have to be linked in a graph; i.e., neighboring variables have to have some correlation (positive or negative) with each other. Moreover, there are limits on the graph structure. A fully connected graph is unsuitable because there is no locality and the path-connectedness of an avalanche would be irrelevant, eliminating the benefit over random search with random dots. For example, as we have seen for graph coloring, if we get closer to a fully connected graph, SOC search suffers. In addition, the graph structure has to be able to support SOC, e.g., the Abelian sandpile model fails to converge to SOC on line or ring graphs.
To conclude, SOC search appears remarkably adept at recovering from local minima - without any annealing scheme. Maybe this ability helps the brain from getting stuck in a thought process. Interestingly, during tonic-clonic seizures, the brain can get stuck in a repetitive loop and, as mentioned above, during seizures, criticality is apparently lost.
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This work was funded by HRL Laboratories, LLC. The authors thank Dr. Eric P. Tressler for providing code of the Abelian sandpile model that was used in the graph-coloring experiments.
HRL Laboratories, LLC, 3011 Malibu Canyon Rd, Malibu, CA, 90265, USA
Heiko Hoffmann & David W. Payton
Heiko Hoffmann
David W. Payton
H.H. conceived the idea, performed the experiments, analyzed the data, prepared all figures, and wrote the paper. D.W.P. contributed ideas and edited the paper. All authors reviewed the manuscript.
Correspondence to Heiko Hoffmann.
Hoffmann, H., Payton, D.W. Optimization by Self-Organized Criticality. Sci Rep 8, 2358 (2018). https://doi.org/10.1038/s41598-018-20275-7
Statistical analysis of acoustic emission avalanches generated during the compressive fracture process, and Mode I fracture process in cementitious composites
Indrashish Saha
R. Vidya Sagar
International Journal of Fracture (2022)
Radiolysis generates a complex organosynthetic chemical network
Zachary R. Adam
Albert C. Fahrenbach
Dmitry Yu. Zubarev
Collective and synchronous dynamics of photonic spiking neurons
Takahiro Inagaki
Kensuke Inaba
Hiroki Takesue
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CommonCrawl
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Measured wideband characteristics of indoor channels at centimetric and millimetric bands
Wei Fan1,
Ines Carton1,
Jesper Ø. Nielsen1,
Kim Olesen1 &
Gert F. Pedersen1
Accurate characterization of spatial multipath channels at millimeter wave bands has gained significant interest both in industry and academia. A channel measurement was conducted at three different frequency bands, i.e., 2−4, 14−16, and 28−30 GHz in a line-of-sight (LOS) and an obstructed-LOS (O-LOS) scenarios in an empty room environment. A vector network analyzer connected to a virtual uniform circular array and to a rotational directional horn antenna was used in the measurements, respectively. Angle-of-arrivals and delay-of-arrivals of the multipath components were obtained from the measurements for the three frequency bands. Room electromagnetic properties for the three different frequencies at different propagation scenarios were investigated as well.
The increasing demand for higher data rates has motivated research in millimeter wave frequency bands for 5G cellular systems. Millimeter wave bands offer huge free spectrum and allow the implementation of massive antenna arrays due to the small wavelength [1, 2]. Though broadband wireless access technology local multipoint distribution service (LMDS) already exists at 28 GHz, it is still unused. This band has been considered as a potential candidate for 5G cellular frequency due to its low atmospheric absorption, and availability of high gain adaptive antennas [3–6]. Accurate characterization of their spatial multipath channel at millimeter wave bands has gained significant interest both in industry and academia, as it is important for system design and performance analysis of future millimeter wave communication systems [7, 8].
While significant studies of channel characteristics were carried out at 60 GHz for indoor and short-range scenarios, only a few measurement campaigns were conducted at 28 GHz [9, 10]. Various measurement results have been reported for the 28 GHz band in the literature for urban scenarios, e.g., 28 GHz penetration and reflection measurement in [1, 3], 28 GHz path loss and signal outage analysis in [1, 11], 28 GHz angle of arrival and angle of departure analysis in [1, 5], etc. Measurement results in an indoor building-scale environment based on a home-made synchronous channel sounder were reported in [6]. To the best knowledge of the authors, very few papers have experimentally investigated the channel characteristics at 30 GHz. Measured results of indoor propagation channels at 30 GHz were reported in [12, 13], where the focus was only on path loss and small-scale fading characteristics.
Extensive work has been done for channel characterization in the literature at current cellular frequencies [14, 15] and other frequency bands, e.g., 5.8 GHz in [16, 17]. The existing work in the literature, however, are lacking in the analysis on the frequency-dependent channel behaviors. The frequency dependency factor is important, as the larger bandwidth, e.g., several GHz or subbands separated by GHz, might be utilized for future wireless systems. It would be highly interesting to compare the channel characteristics at current cellular frequency bands and at millimeter wave bands in the same propagation scenarios.
Room electromagnetics is a simple model that considers a line-of-sight (LOS) component (if present) and an exponential power-delay profile (PDP) decaying with the so-called reverberation time parameter, depending only on the wall area, the volume of the room, and an absorption coefficient. Room electromagnetics theory is interesting, as it does not require complete knowledge of the propagation environment, and it offers an alternative description of diffuse scattering in a room using only simple parameters. Validation of room electromagnetic theory was investigated for cellular frequencies [18–20]. However, very few results are available in the analysis on frequency dependency of room electromagnetics.
In this paper, we investigate the spatial-temporal characteristics of LOS and obstructed-LOS (O-LOS) scenarios in an indoor environment at three different frequency bands, namely, 2−4, 14−16, and 28−30 GHz. Angle-of-arrival and delay-of-arrival of the main paths and room electromagnetics are investigated for different frequency bands. Through the extensive measurements, we would like to investigate: (1) whether channel characteristics at different frequency bands are different in the same propagation scenario; (2) whether room electromagnetics theory holds at different frequency bands; and (3) whether diffuse scattering is important to model at millimeter wave frequency bands.
The main contributions of the paper are summarized as follows:
A sounding system using a VNA coupled to a virtual uniform circular array (UCA) and to a rotational directional antenna to resolve multipath components in delay and spatial domains at different frequencies is described.
Spatial-temporal characteristics of the indoor channels at different frequencies are compared in LOS and obstructed-LOS scenarios. The trajectory of multipath components in a room are identified by relating the multipath angle and delay information to the room geometry.
Room electromagnetic parameters at different frequencies are compared in LOS and O-LOS scenarios.
Spatial multipath components estimated with a virtual UCA are compared with results obtained with rotational directional antenna measurements.
Virtual arrays (via mechanical displacement of a single Biconical antenna) and rotation of this paper is organized as follows. Section 2 describes the indoorpropagation channels and the frequency beamforming technique used to estimate the spatial and temporal properties of multipath channels. Section 3 discusses the measurement setup. Section 4 summarizes the measurement results. Finally, section 5 concludes the paper.
Indoor propagation channels
To resolve the multipath components in space, two measurement systems are generally used. One is to steer a highly directional antenna to scan the channel impulse response (CIR) from each angular direction. This gives the angle information of the channel directly. However, the disadvantage is that the antenna pattern is embedded into the results. The other method is to use an (virtual) array. The angle information of the multipath components can be obtained via beamforming or high-resolution algorithms [21]. A uniform circular array (UCA) is preferable to a uniform linear array (ULA), as the ULA cannot distinguish the paths symmetric with respect to the array line and the beam pattern of the ULA array is not uniform around the azimuth angle.
For static indoor measurements, the measured CIR is essentially only one snapshot of an environment, i.e., only one temporal observation is available. The spatial method discussed in [22], e.g., the classical beamforming and high resolution algorithms like MUSIC cannot be directly applied, since the covariance matrix of the array outputs would be rank-deficient due to the limited temporal samples. Different algorithms were proposed to address this issue, e.g., a spatial smoothing technique in [21] and an iterative 2D Unitary ESPRIT method in [23] for virtual planar arrays. In this paper, we are interested in both delay and angle information of the multipath components. For the sake of simplicity, a conventional frequency beamforming algorithm is used [24]. Note that frequency beamforming algorithms were only used to estimate the angle and delay of the direct path between the transmitter (Tx) and receiver (Rx) for indoor ranging and localization purpose in [24].
Assume a UCA with P array elements arranged uniformly around its perimeter of radius r, each with angle θ i =2π·i/P with i∈[0,P−1] as illustrated in Fig. 1. Note that the distance between two consecutive array elements should not be larger than λ/2 to avoid spatial aliasing effects [22]. Assume that the CIR between the Tx and the center of the UCA is composed by K multipath plane waves. The channel frequency response is:
$$ H(f)=\sum\limits_{k=0}^{K-1}\alpha_{k}\exp\left(-j2\pi f\tau_{k}\right), $$
An illustration of the UCA and the multipath channel
where α k and τ k represent the complex amplitude and delay of the kth wave with k∈[0,K−1]. The k-th wave impinging with angle of arrival φ k arrives at the i-th element with a delay \(\tau _{k_{i}}\) with respect to the UCA center:
$$ \tau_{k_{i}}=-\frac{r\cdot\cos\left(\theta_{i}-\varphi_{k}\right)}{c}, $$
where c is the speed of light. The frequency response H i (f) at the ith element can be written as:
$$ H_{i}(f)=\sum\limits_{k=0}^{K-1}\alpha_{k}\cdot\exp\left[-j2\pi f(\tau_{k}+\tau_{k_{i}})\right] $$
The array frequency response H(f,θ) is:
$$ H(f,\theta)=\frac{1}{P}\sum\limits_{i=0}^{P-1} w_{i} H_{i}(f), $$
where w i ,i∈[0,P−1] is the complex weight assigned to the ith antenna element. The basic principle of classical beamforming is to shift the phase of each frequency response into alignment. Therefore, the weighting vector can be expressed as:
$$ w_{i} = exp\left(-j 2 \pi \cdot (r/c) \cdot cos(\theta - \theta_{i})\right), $$
A power peak occurs in the array beam pattern for θ=φ k . The spatial-temporal channel impulse response h(t,θ) can be recovered from H(f,θ) via inverse discrete Fourier transform:
$$ h(t,\theta)=\sum\limits_{n=0}^{N-1}H\left(f_{n},\theta\right)\cdot\exp\left(j2\pi f_{n}t\right), $$
where f n is the frequency at the n-th frequency index.
Experimental setup
Figure 2 illustrates the block diagram of the measurement system. A wideband biconical antenna was mounted on a table as the transmitter (Tx), while two types of antennas were used as the receiver (Rx). Note that both the Tx and Rx antenna were mounted at a height of 1.1 m above the ground. With the two different Rx antennas, two sets of measurements were performed. In the first measurements, a uniform circular array (UCA) was realized virtually by mounting a biconical antenna on a positioning turntable. We obtained P=720 elements of the UCA array by automatically repositioning the biconical antenna at uniform angles around a circle perimeter with r=0.5 m. At each element position, a frequency sweep was performed for three different frequency bands 2−4, 14−16, and 28−30 GHz. N=750 samples were set for each frequency band. In the second measurement series, a horn antenna was mounted on the rotation center of the turntable. To make meaningful comparison, the same measurement settings as in the first measurement series (i.e., the same frequency sweep and orientation sweep) were adopted for the second measurement series. A complete channel measurement, i.e., frequency sweep over the three bands for all element positions, took about 15 min. Three types of antennas were used in the measurement campaign, as detailed in Table 1. The biconical antenna is omnidirectional in the azimuth plane for 2−30 GHz, while the beamwidth gets narrower in the elevation plane as the operating frequency increases. The horn antenna operating at 750 MHz – 18 GHz frequency band is less directional at the low band both in azimuth and elevation plane, and becomes more directional as the frequency increases. The horn antenna operating at 26.4−40 GHz has approximately the same beamwidth in the elevation and azimuth plane. All the antennas used in the measurements are vertically polarized.
Block diagram of the measurement system. The measurement system consists of a VNA, a LO/IF distribution unit, two reference mixer modular. The mixer modular is used to down-convert the received signals at the Rx to a lower frequency to reduce the cable loss and phase variation at the long cable (9 m). Calibration was performed prior to measurements to de-embed the Tx and Rx chain in the measurement results. Channel frequency responses at three frequency bands were recorded over one sweep
Table 1 Specifications of the antennas used in the measurement campaign
Given that the bandwidth of measurement at each frequency band is 2 GHz, the delay resolution for each band is 0.5 ns, resulting in a spatial resolution is 0.15 m. The delay range of the measurement is 350 ns, which limits the maximum measurement range to approximately 105 m path length.
The measured \(S_{i}^{21}(f)\) parameter can be expressed as a product of the Tx branch, Tx antenna, the channel, the Rx branch, and the Rx antenna. The Tx branch and the Rx branch could be removed via calibration, as illustrated in Fig. 2. The antenna gains of the Tx and Rx antenna at different frequency bands are given in Table 1, and can be de-embedded in post-processing to extract the channel frequency response.
Measurement scenario
The measurements were performed in an empty room, as depicted in Fig. 3. All walls are made of concrete except the bottom wall which is a wooden board as indicated in the figure. Two scenarios were considered in the measurement: LOS and O-LOS, as shown in Figs. 4 and 5, respectively. For the O-LOS measurement, a blackboard of dimensions 1.19×1.19 m2 with one side covered by aluminum was placed between the antennas to block the paths in the LOS direction.
An illustration of the dimensions of the empty room and locations for the Tx and Rx antennas for the measurement campaign
Photograph of the line-of-sight (LOS) scenario
Photograph of the O-LOS scenario
Measurement objectives
Five objectives were targeted in the measurement campaign:
In this paper, virtual arrays (via mechanical displacement of a single Biconical antenna) and rotation of high directive horn antennas are used to investigate the power-angle-delay profiles of the indoor propagation channels. To achieve this goal, the propagation environment should be maintained static during measurement. System stability investigation was carried out in Section 4.1.
Accurate calibration of antenna element positions is required, which becomes even more challenging at millimeter wave bands. For this reason, the measurements were repeated two to three times to investigate the repeatability of the results.
Wideband channel characteristics in delay and spatial domains and room electromagnetic in the LOS and O-LOS scenarios are analyzed and compared.
One interesting research topic in millimeter wave channel modeling is the differences and similarities in channel characteristics at different frequency bands. In this paper, measured channel characteristics at 2−4, 14−16, and 28−30 GHz are compared.
Measurement results with a virtual UCA and steering horn antennas are analyzed and compared. Note that for the measurements with horns, two different horn antennas were used to cover the three frequency bands (one for 2−4 and 14−16 GHz, and the other one for 28−30 GHz), as detailed in Table 1.
Table 2 summarizes the measurement items performed in the measurement campaign.
Table 2 Measured items during the measurement campaign
Measurement results
To investigate the channel spatial profile by using a virtual array or rotating a high-directive horn antenna, it is important to ensure a static environment [16]. In order to investigate the stability of the measurement system, wideband measurements with a biconical antenna at the Rx side are performed, without rotating the turntable. That is, the channel frequency responses were repeated P=720 times for the same position of the Rx. CIRs can be calculated via inverse Fourier transform of the measured frequency responses for each frequency band. One simple way to check stability is to investigate the phase variation of the narrowband received signal, which is obtained by summing up the CIRs coherently over the delay domain. The rms phase variation of the received signal is shown in Table 3. The rms phase variation at 2−4 GHz is up to 2.9°, which is higher compared to phase variation at 14−16 and 28−30-GHz bands both for the LOS and O-LOS scenarios. This might be caused by higher instabilities in the measurements system at lower frequencies. The rms phase variations are comparable in LOS and O-LOS scenarios at all frequency bands.
Table 3 rms phase variations of the received field unit [°]
Measurements with the horn antennas and the UCA were repeated to investigate the repeatability of the measurements. Note that in this case, the turntable is rotating as shown in Table 2. To investigate the repeatability of the results, the cross correlations of the narowband received signals are calculated for each scenario among different repetitions. The cross correlation is equal to 1 when two measurement results are identical. The worst results among the repetitions are shown in Fig. 6. The measurements are generally highly repeatable, with cross correlations between repetitions above 0.98 for all scenarios. The measurements with the horn antennas are less repeatable due to a stronger cable effect (e.g., cable bending) observed in practical measurements.
Cross correlations of the received fields in two repeated measurements for different scenarios
Horn antenna results
Main path identification
The measured power-angle-delay profiles with horn antennas for the LOS and O-LOS scenarios for different frequency bands are shown in Figs. 7 and 8, respectively. The trajectories of the multipath components in the measurement room can be obtained by relating the angle and delay information of the multipaths (Figs. 7 and 8), to the room geometry (Fig. 3), as shown in Fig. 9.
Power-angle-delay profile measured with horn antennas for LOS scenarios at different frequency bands. The power dynamic range (i.e., color bar range) is limited to 35 dB. Note that the horn antenna gain is embedded in the results for all the bands
Power-angle-delay profile measured with horn antennas for O-LOS scenarios at different frequency bands. The power dynamic range (i.e. color bar range) is limited to 35 dB. Note that the horn antenna gain is embedded in the results for all the bands
Paths identified (top view) based on the results reported at 14−16 GHz and 28−30 GHz in Figs. 7 and 8 for the LOS and O-LOS scenarios. Note that path 2, which corresponds to a ground reflection, and path 3, which corresponds to a ceiling reflection, are not present in the figure
Note that the delay range is limited to 60 ns (corresponding to 18 m in distance) in Figs. 7 and 8 for illustration of the main paths. The spatial resolution is limited by the beamwidth of the horn antennas. As detailed in Table 1, the horn antenna operated at 2−4 GHz suffers from low spatial resolution both in azimuth and elevation domains, while the horn antennas operated at 14−16 and 28−30 GHz are much more directional, with a beamwidth of 20° at the 28−30 GHz band in azimuth domains. The delay resolution, however, is the same, since the same bandwidth is set for the three frequency bands.
LOS measurements at different frequency bands are dominated by the LOS paths, while the O-LOS measurement is more spatially rich, with quite a few dominant paths in various specular directions. As seen in Figs. 7 and 8, multipath components identified in the LOS and O-LOS scenarios are quite similar, except the paths impinging the Rx with AoAs around 0o and 180o, which are blocked by the blackboard. It is worth noting that these paths have a slightly different interpretation in the LOS and O-LOS cases. For example, the path identified as 1 has the same AoA for both cases, yet it corresponds to the LOS in Fig. 7, whereas in Fig. 8 it corresponds to a diffracted path from the top edge of the obstructor. The radius of the nth Fresnel zone can be calculated as:
$$ F_{n} = \sqrt{\frac{n \lambda d_{1} d_{2}}{d_{1} + d_{2}}}, $$
where d 1, d 2 is the distance from the obstructor to the Tx and Rx, respectively. In our measurement, the radius of the first Fresnel zone at 30 GHz is 0.11 m, whereas the height difference between the blackboard and the antennas is only 0.09 m; thus the first Fresnel zone was not completely obstructed by the blackboard, resulting in the diffraction identified as path 1 in Fig. 8. The power angle spectra measured for the LOS and O-LOS scenarios at 14−16 and 28−30-GHz frequency bands are shown in Fig. 10. The power angle spectra in the LOS scenario match quite well with those in the O-LOS scenario in non-LOS directions for both frequency bands, as expected.
Power angle spectra measured for the LOS and O-LOS scenarios at 14−16 and 28−30-GHz frequency bands. Note the power values at two different bands are normalized to the LOS component at 14−16 and 28−30 GHz, respectively
Contrary to the fact that only a few specular paths are identified at 14−16 and 28−30 GHz, measured channels at 2−4 GHz present quite different results. This observation is consistent with results published in the literature, i.e., channel profiles at millimeter wave bands are characterized by fewer and sparser dominant paths compared to typical cellular bands [9, 10]. A richer multipath environment can be observed for the results at 2−4 GHz, with many more paths present in the delay domain. This might be due to a stronger diffuse scattering from objects at the low frequency band, as the wavelength is more comparable to object dimensions at the low frequency. At the higher bands, the wavelength is much smaller, and hence specular reflections are dominant. Different dominant paths can be identified at 2−4 GHz. Although some specular paths can be identified at all frequencies (e.g., path 1, 2, 3, 4, 5, and 6), some of the paths present at 2−4 GHz can not be identified at higher frequency bands (e.g., path 11 and 12). Also, some specular paths present at high bands (e.g., path 7 and 8) cannot be clearly identified at the low frequency band.
The main propagation paths from the Tx to the Rx can be identified, as shown in Fig. 9. The angles and delays of the identified paths at 14−16 and 28−30 GHz are quite similar both for the LOS and O-LOS scenarios. However, the normalized power values of the identified paths with same delay and angle at different frequency bands might be different. As shown in Fig. 10, although main paths impinge the Rx at same angles, the power values are different.
In [2], scattering effects are shown in the measurements, whereas only specular components can be seen in our measurements at 28–30 GHz. This might be due to the fact that a small furnished office, where small scatterers exist, was investigated in [2], whereas an empty room with a few scatterers was selected in this paper. Moreover, a wide dynamic range was used in [2], whereas in this paper, the dynamic range in this paper is limited to 35 dB in order to investigate only the dominant components. A richer multipath environment could be expected in a furnished room as shown in [16].
Room electromagnetic
Discussions in section 4.3.1 were focused on the accurate characterizations of the main propagation paths, i.e., the LOS path and main reflected paths from the walls and ground. We are also interested in the remaining part of the CIRs that correspond to the non-dominant paths. The rms delay spread τ rms and mean excess delay \(\bar {\tau }\) are often calculated with respect to a threshold relative to the strongest path. This ensures full compatibility of the results, since noise is excluded from the calculation. The τ rms and \(\bar {\tau }\) using a 30 dB threshold are shown in Table 4 for the LOS and O-LOS scenarios at different frequency bands. τ rms and \(\bar {\tau }\) are larger in the O-LOS scenarios compared with the LOS scenarios for all frequency bands as expected, with an exception that τ rms in the O-LOS scenario is slightly smaller at 2−4 GHz, compared to the higher frequency bands. τ rms and \(\bar {\tau }\) at 2−4 GHz in the LOS scenario are larger compared to higher frequencies due to the richer multipath environments at the low band. τ rms and \(\bar {\tau }\) at the O-LOS scenarios are, however, comparable at the three different frequency bands.
Table 4 τ rms and \(\bar {\tau }\) for the LOS and O-LOS scenarios at different frequency bands
The tail of the power delay profiles (PDPs) generally have exponential decays with approximately the same decay rates for different scenarios at different frequency bands, as shown in Table 4, with a deviation up to 0.02 dB/ns around −0.26 dB/ns decay rate. The decay rate of the delay tail is governed by the overall dimensions of the room and an average absorption coefficient, and it is irrelevant to the propagation scenarios (i.e., LOS or O-LOS) [18]. This is due to the fact that the room will be filled with energy, resulting in a constant energy density for the same delay [18]. It can be concluded based on the measurement results that the average absorption coefficient does not depend on frequency. Note that the decay rates were calculated based on CIRs ranged from 60 to 200 ns to avoid including the dominant paths. If the CIRs before 60 ns, where dominant paths are present, are considered, the calculated decay rate will be different. The PDPs for the LOS and O-LOS scenarios at 28−30 GHz and the fitted lines are shown in Fig. 11. Note that different scenarios at different frequencies not only have the delay tails with the same decay rates, but also the power values in tails are the same as well, as shown in Fig. 11.
Average power delay profiles for the LOS and O-LOS scenarios at 28−30 GHz. Kaiser window with β=6 was applied to suppress the sidelobes in the inverse Fourier transform
UCA measurement results
Path loss
For the virtual UCA measurements, the path loss at each element position i at each frequency band can be expressed as:
$$ PL(i)=\frac{1}{N}\sum\limits_{n=1}^{N}|H_{i}(f_{n})|^{2}. $$
The path loss results for the LOS and O-LOS scenarios for the three frequency bands are shown in Fig. 12. In the LOS scenario, the measured path loss results match well with the calculated path loss from Friis equation, as the LOS path is dominant. The deviation is mainly due to the inaccurate antenna gains provided in Table 1 for different frequency bands. The ripples in the measured results are caused by the paths from non-LOS directions. The path loss in the LOS scenario varies along element positions as expected, e.g., minimum path loss present at element position 360 (i.e., orientation angle 180°), which corresponds well to the element positions illustrated in Fig. 3. The path loss in O-LOS scenario is around 10 dB higher compared to LOS scenario for 14−16 and 28−30-GHz bands, while only around 4 dB higher for 2−4 GHz.
Path loss over different array element positions for three frequency bands for the LOS and O-LOS scenarios. Note that the transmit and receive antenna gains at different frequency bands are de-embeded from the results
Power-angle-delay profile
The CIRs received at different UCA element positions for the LOS scenario at different frequency bands are shown in Fig. 13. Similar to horn antenna measurement results, a richer multipath environment can be observed at 2−4 GHz, compared with higher frequency bands. Only a few specular paths can be identified from the delay variation along array element positions at 14−16 and 28−30 GHz.
CIRs recorded at different UCA element positions for the LOS scenario at different frequency bands
The power-angle-delay profiles obtained with frequency beamforming techniques discussed in Section 2 for the LOS and O-LOS scenarios at different frequency bands are shown in Figs. 14 and 15, respectively. Similar to the horn antenna measurement results, richer multipath environments can be observed at 2−4 GHz, compared to results in high frequency bands, both for the LOS and O-LOS scenarios. Similar power-angle-delay profiles are present at 14 – 16 and 28 – 30-GHz frequency bands. A few dominant specular paths can be identified for all the frequency bands. Compared with horn antenna results in Figs. 7 and 8, paths with same impinging angles and delays are observed both for the LOS and O-LOS scenarios at different the frequency bands. The measured power-angle-delay profiles with the UCA, although with higher spatial resolution, suffers from high side lobes as well. As explained in [24], the joint sidelobes in delay and angle domains are caused by the joint frequency and angle dependance inherent in phase in Eq. (4) with the conventional beamforming technique. The focus of future work will be on resolving the multipath components from the array measurement results with other techniques, e.g., frequency invariant beamforming technique [24].
Power-angle-delay profile measured with UCA for LOS scenarios at different frequency bands. The power dynamic range is limited to 35 dB
Power-angle-delay profile measured with UCA for O-LOS scenarios at different frequency bands. The power dynamic range is limited to 30 dB
This work presents an extensive measurement campaign and a detailed analysis of spatial-temporal channels at centimetric and millimetric bands in an indoor environment. Multipath components were resolved in space and delay via steering a wideband horn antenna and via measurements with a virtual wideband UCA in different frequency bands, respectively. Measurement results include stability levels, repetition levels, power-angle-delay profiles, path loss, room electromagnetic, etc. Results show that:
The measurement system is quite stable, with a rms phase drift up to 1.5° at 28−30 GHz band.
The measurement results are highly repeatable, with a correlation between two repeated measurements up to 0.98.
Richer multipath components are present at lower frequency bands due to diffuse scattering effects, while specular reflections are dominant at millimeter wave bands. The trajectory of the specular components in a room are identified by relating the multipath angle and delay information to the room geometry. Investigations in a furnished room, where more scatterers exists, will be performed in future work.
Room electromagnetics are not frequency dependent in an empty indoor scenario. The tails of the PDPs can be modeled with a exponential decay with decay rate around −0.25 dB/ns.
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This work has been supported by the Danish High Technology Foundation via the VIRTUOSO project. The authors appreciate the assistance from Kristian Bank and Yi Tan with the practical measurements. The authors would like to thank Dr. Lishuai Jing and Dr. Ming Shen for the active discussions.
Department of Electronic Systems, Faculty of Engineering and Science, Aalborg University, Aalborg, Denmark
Wei Fan
, Ines Carton
, Jesper Ø. Nielsen
, Kim Olesen
& Gert F. Pedersen
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Correspondence to Wei Fan.
Fan, W., Carton, I., Nielsen, J.Ø. et al. Measured wideband characteristics of indoor channels at centimetric and millimetric bands. J Wireless Com Network 2016, 58 (2016). https://doi.org/10.1186/s13638-016-0548-x
DOI: https://doi.org/10.1186/s13638-016-0548-x
Millimeter wave channel measurements
Spatio-temporal channel modeling
Room electromagnetics
Angle of arrival estimation
Radio Channel models for higher frequency bands
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CommonCrawl
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Single-target networks
DCDS-B Home
Weak pullback attractors for stochastic Ginzburg-Landau equations in Bochner spaces
February 2022, 27(2): 769-797. doi: 10.3934/dcdsb.2021064
On the Lorenz '96 model and some generalizations
John Kerin and Hans Engler ,
Georgetown University, Washington, DC 20057, USA
* Corresponding author: Hans Engler
Received November 2020 Revised December 2020 Published February 2022 Early access February 2021
In 1996, Edward Lorenz introduced a system of ordinary differential equations that describes a scalar quantity evolving on a circular array of sites, undergoing forcing, dissipation, and rotation invariant advection. Lorenz constructed the system as a test problem for numerical weather prediction. Since then, the system has also found use as a test case in data assimilation. Mathematically, this is a dynamical system with a single bifurcation parameter (rescaled forcing) that undergoes multiple bifurcations and exhibits chaotic behavior for large forcing. In this paper, the main characteristics of the advection term in the model are identified and used to describe and classify possible generalizations of the system. A graphical method to study the bifurcation behavior of constant solutions is introduced, and it is shown how to use the rotation invariance to compute normal forms of the system analytically. Problems with site-dependent forcing, dissipation, or advection are considered and basic existence and stability results are proved for these extensions. We address some related topics in the appendices, wherein the Lorenz '96 system in Fourier space is considered, explicit solutions for some advection-only systems are found, and it is demonstrated how to use advection-only systems to assess numerical schemes.
Keywords: Lorenz-96 model, equivariant dynamical system, Hopf bifurcation, normal form, Neimark-Sacker bifurcation.
Mathematics Subject Classification: Primary: 34C23, 86A10; Secondary: 37C81, 37G05.
Citation: John Kerin, Hans Engler. On the Lorenz '96 model and some generalizations. Discrete & Continuous Dynamical Systems - B, 2022, 27 (2) : 769-797. doi: 10.3934/dcdsb.2021064
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Figure 1. A solution of the L96 system with $ F = 2 $ starting from random initial data. (a) All 36 sites at $ t = 500 $. (b) First site for $ 500 \leq t \leq 510 $. (c) Hovmoeller plot for $ 500 \leq t \leq 510 $
Figure 3. Left: Eigenvalue curve of the linearization of the L96 system about the constant vector $ \mathbb{e} $ and eigenvalues on this curve for $ N = 36 $ sites. Right: Eigenvalue curves of $ FA - I $ for the L96 system. The curve in the left panel is stretched by $ F $ and shifted to the left by 1 unit. Black: $ F = -0.8 $. For even $ N $, a pitchfork bifurcation has occurred for $ F = -1/2 $. For odd $ N $, a Hopf bifurcation has occurred for some $ F < -1/2 $. Red: $ F = 0.5 $. The constant solution is stable. Blue: $ F = 1.1 $. A Hopf bifurcation has occurred for $ F \approx 8/9 $
Figure 4. Images of the complex unit circle under the Laurent polynomials given in Table 1. (a): Ellipse ($ G_1, \, G_2, \, G_4 $). (b): Trefoil ($ G_3 $, L96 system). (c): Butterfly ($ G_5 $). (d): Kidney ($ G_6 $). (e): Vertical line ($ G_7 $). (f): Bee ($ G_8 $)
Figure 5. Eigenvalue curves of advection terms in $ \mathcal{G}_2 $ for various values of $ F $. (a) $ G_1 $, no bifurcation occurs for positive $ F $. (b) $ G_5 $, a supercritical Hopf bifurcation occurs for positive $ F $. (c) $ -G_1 + \frac{1}{2} G_5 $, a supercritical Hopf bifurcation occurs for positive $ F $
Figure 6. Left: Eigenvalues of $ F_1A - I $ (black and red) and of $ F_1A - I + \alpha_0 C_\ell $ (black and green), for $ N = 14 $. Right: Bifurcation diagram of system (38) in the $ (F,\alpha) $ plane. Blue line: Hopf bifurcation ($ \Re \, \lambda_k = 0 $). Blue stipples: A stable limit cycle exists. Green line: Hopf bifurcation ($ \Re \, \tilde \lambda_\ell = 0 $). Green stipples: A second periodic orbit exists (unstable). Red curves: Neimark-Sacker (N-S) bifurcation. Red stipples: Two stable limit cycles coexist. Magenta line: Linear approximation of N-S bifurcation curve. Red triangle: Hopf-Hopf bifurcation at $ (F_1, \, \alpha_0) $
Figure 7. Stationary solutions of Eq. (41) with $ G = G_L, \, C = B = I $ and inhomogeneous forcing $ {\bf{F}} $ for $ N = 120 $ sites. Here $ F_i = 1 $ for $ 0 \le i < N/2 $ and $ F_i = M $ for $ i \ge N/2) $
Figure 8. Hovmoeller plots showing inhomogeneous advection and dissipation as described in Eq. (46), for $ N = 100 $. Both panels use the same parameters $ (1,1,2) $ in the left half, but solutions have very different behavior. Left: Sites in the right half have parameters $ (0.5,1,1) $. Smaller advection in the right half leads to smaller spatial amplitudes. Perturbations are seen to travel to the right. Right: Sites in the right half have parameters $ (1,1.5,1) $. Larger dissipation in the right half leads to nearly constant solutions over a substantial range of sites
Figure 9. Relative energy loss $ \Delta E(t)/E(0) $ for RK4 and scaled relative energy loss $ 10^3 \times \Delta E(t)/E(0) $ for lsoda, for $ N = 36 $, $ \Delta t = 0.05 $, and $ E(0) = 400 $
Table 1. Description of the eigenvalue curves of the eight simplest 3-localized $ \mathcal{G} $-maps identified in Section 2.2. The two rightmost columns give the types of the first expected bifurcation for $ F>0 $ and $ F<0 $ as the magnitude of $ F $ increases. Asterisks indicate exceptions for certain site numbers
$ \mathcal{G} $-map Laurent polynomial $ p_A(z) $ Shape of $ p_A(\mathbb{S}^1) $ $ F>0 $ $ F<0 $
$ G_1 $ $ - z^{-1} - 1 + 2z $ ellipse none pitchfork/Hopf
$ G_2 $ $ -z^{-2} - 1 + 2z^2 $ ellipse none pitchfork/Hopf $ ^\ast $
$ G_3 $ $ - z^{-2} + z $ trefoil Hopf pitchfork/Hopf
$ G_4 $ $ - z^{-3} -1 + 2 z^3 $ ellipse none pitchfork/Hopf $ ^{\ast} $
$ G_5 $ $ -z^{-2} - z + z^2 + z^3 $ butterfly Hopf Hopf
$ G_6 $ $ -z^{-1} + z - z^2 + z^3 $ kidney Hopf pitchfork/Hopf
$ G_7 $ $ -z^{-2} - z^{-1} + z + z^2 $ vertical line none none
$ G_8 $ $ -z^{-3} - z^{-1} + z^2 + z^3 $ bee pitchfork/Hopf Hopf
Table 2. Multiple stable limit cycles are expected if approximately $ F \ge F_3^\ast $ and are found numerically for $ \tilde F_3 \le F \le \tilde F_4 $. Limit cycles may be characterized by their spatial periods $ m_1, \, m_2 $
$ N $ $ F_1 $ $ m_1 $ $ F_2 $ $ m_2 $ $ F_3^\ast $ $ \tilde F_3 $ $ \tilde F_4 $
12 1 4 1 6 1 1 $>2 $
14 .8901 7 1.1820 14 1.5206 not observed not observed
18 .8982 9 1 6 1.1892 not observed not observed
22 .9076 22 .9343 11 .9915 .996 $>4 $
28 .8901 14 .9457 28 1.0293 1.072 $>3 $
36 .8982 9 .9025 36 .9094 .904 $>2 $
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John Kerin Hans Engler
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Path optimization in a DAG: maximizing number of least cost arcs
I've got the following problem.
I've a graph $G=(V,E)$ as in the picture and I have to calculate the optimal path from $R$ to $S$.
The optimal path has to maximize the number of least cost arcs. In the example:
R->B->S is the least cost path but it's not the optimal. R->-A->D->S costs less than R->A->C->E->S , but the latter is the optimum as it goes through more, least cost arcs.
My current problem, which constitute the object this question, is to find a way of defining formally how to compare two paths according to my intuitive understanding, as comparing them comes before choosing the optimal one.
Of two paths, the better one is the one that traverses the highest number of least cost arcs.
I have been considering three ways, some that seem not fully adequate for my purpose, but would welcome other suggestions.
My three ideas are the following:
Minimizing the average of the cost of arcs on the path.
Minimizing the average is not fully ok: imagine to have a path with cost [3,12], and a path with cost [3,10,10]. The average of the first path is less than the average of the second, but the second is the optimal as it traverses more paths.
Comparing individually the arcs of the paths
Given two paths $A$ and $B$:
Let $A$ = [$a_1,\ldots,a_n$] and $B$ = [$b_1,\ldots,b_m$] where $a_i$ and $b_i$ are the arc costs of $A$ and $B$ respectively.
Then $A$ is better than $B$ if $\sum_{i=0}^n card(B < a_i)$ < $\sum_{i=0}^m card(A < b_i)$ where $card$ represents the cardinality of the set, and $B < a_i$ is defined as $\{b_j \in B \mid b_j < a_i\}$, the comparison being on the weight of the concerned arcs.
However, with this second approach I get many ties between different paths, increasing the risk of getting a suboptimal path.
Comparing the min-max of the arcs weights for each path.
Another possibility is to compute the min-max of the arcs of each path, and choose the smaller one. It is intrinsically different from the other approaches.
But it is not fully what I seek to achieve. For example, min-max over [1,1,1,3] and [2,2,2,2] chooses the latter one, while the best one is the first.
To motivation behind my intuition is the following: each node is a "checkpoint" that the solution may traverse on its way from the node $R$ to node $S$. The goal is to output a sequence $Q$ with an associated "precision"
if the chosen path is R->A->D->S, then $\;Q\;$ is $\;\;$A $\pm$ 10, D $\pm$ 5, S $\pm$ 7
Clearly, the optimal solution has to be as precise as possible, so the optimal path has to emit the maximum number of least cost arcs.
Once the comparison of paths has been formalized precisely, my problem will be to compute the optimal path on a given graph. But right now the question is only to find a proper way of comparing paths. The role of the graph structure is only, possibly, to help convey the intuition of my problem.
algorithms graph-traversal weighted-graphs partial-order order-theory
babou
Charles G.Charles G.
$\begingroup$ Note that, in the special case where all edges have the same cost, you're trying to find the longest path between two vertices, which is NP-hard. As such, you shouldn't expect there to be an efficient algorithm for the general problem. $\endgroup$
$\begingroup$ @DavidRicherby : $\;\;\;$ How is that shown? $\:$ (Is it by using a definition of "path" that either doesn't allowing revisiting vertices or doesn't allow retracing edges?) $\;\;\;\;\;\;\;\;$ $\endgroup$
$\begingroup$ @DavidRicherby according to the definition (with ∑ and unusual notations $B<a_i$), if all edges have the same cost, all paths are equivalent since their associated conmparative values are 0. If he had used "≤" rather than "<", then these values would be |B|×|A|. and all paths would still be equivalent. So your conclusion may be right, but it requires more to be established,. $\endgroup$
$\begingroup$ Am I correct in interpreting $B<a_i$ as $\{b_j\in B\mid b_j<a_i\}$ ? $\endgroup$
$\begingroup$ I edited your question so as to clarify as much as I can, and remove the second part (only one problem in a question). I also tried to remain compatible with existing answers, - - - It is not yet clear what you are after. You should give us a brief example of the real problem you are working on, because we will never get your intuition otherwise. Where do these approximations come from? What do they mean? What really happens on an arc of the graph (some kind of unprecise processing step?) What is the process you want to optimize? Give us the real problem, not your vision. $\endgroup$
I've not thought completely through it, but some thoughts on 1) which might be useful:
If you add a constant value k to each edge traversed, you could take into account the number of nodes traversed. In your case, this constant value k should be negative (e.g. the negative value of the maximum edge weight, in your example 11).
Then you might be able to state the following:
The path maximizes the number of least links if $\sum_{i=0}^n (a_i + k)$ < $\sum_{i=0}^m (b_i + k)$.
For your example: (let $P_i$ denote the $i$-th path and $C_i$ denote the costs of path $P_i$)
$P_A=[R, B, S], C_A = -2$
$P_B=[R, A, D, S], C_B = -11$
$P_C=[R, A, C, E, S], C_C = -20$
So $C_C < C_B < C_A$ and hence $P_C$ is your optimal path.
As mentioned, I've not spent too much time to look if there is an example where this approach does not work, but maybe it's a good point to start with.
Of course you get some ties with this approach as well but I think all of these ties are equivalent in the sense that the least cost arcs are traversed and the number of traversed vertices are equivalent. Correct me, if I'm wrong.
SX.SX.
The question is proposing 3 definitions for comparing paths. The first and third are apparently not satisfactory for the author of the question. We show that the second definition is not usable either.
The second formalization seems more sophisticated, and is precisely given. It is not a metric but the definition of a "better" relation between paths, that is noted here "$\prec$".
$$A\prec B\;\;\;\mbox{ iff }\;\;\;\sum_{i=0}^n |B < w(a_i)|$ < $\sum_{i=0}^m |A < w(b_i)|$$ where $w(a)$ is the weight of an arc $a$,
and $B < w(a_i)$ is defined as $\{b_j \in B \mid w(b_j) < w(a_i0\}$
Actually, this can be redefined, I think more intuitively, as it helped me find the example below, as:
$\begin{align} A\prec B\;\;\text{ iff }\;\;|\{(a_i,b_j)\mid a_i\in A, b_j\in& B, w(b_j)<w(a_i)\}|\;<\; \\ &|\{(a_i,b_j)\mid a_i\in A, b_j\in B, w(b_j)>w(a_i)\}| \end{align}$
However, the problen is that this relation is not an order on paths, not even a preorder, since it is not transitive:
A counter example is given by the following three paths, here just given as sets of weights:
$$A=\{9,5,4\},\;\;\; B= \{6\},\;\;\; C= \{8,7,3\}$$
It is easily verified that
$$A\prec B,\;\;\; B\prec C,\;\;\; C\prec A$$
Thus, despite the notation chosen here, the relation $\prec$ cannot be used to order solutions. Hence this relation cannot be used to define the choice of an optmal path, independently of how paths are defined or the graph explored.
None of the three modes of comparison seems to meet the need of the author of the question. The motivation given by the author for his intuition is too vague to help. A full example of the real problem he is adressing should be given to have a chance for further progress.
baboubabou
$\begingroup$ Thank for your time and effort. You pointed out correctly the problem: I'm still trying to define what makes a path better than another one. I edited the question in the final part of the question $\endgroup$
– Charles G.
$\begingroup$ Thank you. So I should just ask for a proper definition of "optimal"? $\endgroup$
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doi: 10.3934/dcdsb.2020191
Abstract similarity, fractals and chaos
Marat Akhmet 1,, and Ejaily Milad Alejaily 2,
Department of Mathematics, Middle East Technical University, Ankara, Turkey
Department of Fundamental Sciences, College of Engineering Technology, Houn, Lybya
* Corresponding author: Tel.: +90 312 210 5355, Fax: +90 312 210 2972
The art of doing mathematics consists in finding that special case which contains all the germs of generalit David Hilbert
Received August 2019 Revised March 2020 Published June 2020
Fund Project: The first author has been supported by a grant (118F161) from TÜBİTAK, the Scientific and Technological Research Council of Turkey
A new mathematical concept of abstract similarity is introduced and is illustrated in the space of infinite strings on a finite number of symbols. The problem of chaos presence for the Sierpinski fractals, Koch curve, as well as Cantor set is solved by considering a natural similarity map. This is accomplished for Poincaré, Li-Yorke and Devaney chaos, including multi-dimensional cases. Original numerical simulations illustrating the results are presented.
Keywords: Abstract self-similarity, self-similar space, similarity map, fractals, chaos, multi-dimensional chaotic maps.
Mathematics Subject Classification: Primary: 28A80, 65P20; Secondary: 37B10.
Citation: Marat Akhmet, Ejaily Milad Alejaily. Abstract similarity, fractals and chaos. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020191
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Figure 1. (a) The first step of abstract self-similar set construction. (b) The illustration of the boundary agreement
Figure 2. Examples of the $ 2^{nd} $ and the $ 3^{rd} $ order subsets of the Sierpinski carpet
Figure 3. A trajectory of the point under the similarity map
Figure 4. The construction of abstract self-similar set corresponding to the Sierpinski gasket
Figure 5. The construction of abstract self-similar set corresponding to the Koch curve
Figure 6. The $ 1^{st} $ and the $ 2^{nd} $ order subsets for the Cantor set
Figure 7. The construction of abstract self-similar set using the map (12)
Figure 8. The first three iterations of DASS construction using the map (13)
Figure 9. The two trajectories of the system (13)
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Marcos C. Mota, Regilene D. S. Oliveira. Dynamic aspects of Sprott BC chaotic system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1653-1673. doi: 10.3934/dcdsb.2020177
2019 Impact Factor: 1.27
Marat Akhmet Ejaily Milad Alejaily
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Structure of finite wavelet frames over prime fields
A. Ghaani Farashahi
Numerical Harmonic Analysis Group (NuHAG), Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria.
This article presents a systematic study for structure of finite wavelet frames over prime fields. Let $p$ be a positive prime integer and $\mathbb{W}_p$ be the finite wavelet group over the prime field $\mathbb{Z}_p$. We study theoretical frame aspects of finite wavelet systems generated by subgroups of the finite wavelet group $\mathbb{W}_p$.
Finite wavelet frames
finite wavelet group
prime fields
41-XX Approximations and expansions
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A. Ghaani Farashahi, Wave packet transform over finite fields, Electron. J. Linear Algebra 30 (2015) 507--529.
A. Ghaani Farashahi, Cyclic wavelet systems in prime dimensional linear vector spaces, Wavelets and Linear Algebra 2 (2015), no. 1, 11--24.
A. Ghaani Farashahi, Cyclic wave packet transform on finite Abelian groups of prime order, Int. J. Wavelets Multiresolut. Inf. Process. 12 (2014), no. 6, Article ID 1450041, 14 pages.
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Receive Date: 12 July 2015
Revise Date: 19 October 2015
Accept Date: 19 October 2015
First Publish Date: 22 February 2017
Ghaani Farashahi, A. (2017). Structure of finite wavelet frames over prime fields. Bulletin of the Iranian Mathematical Society, 43(1), 109-120.
A. Ghaani Farashahi. "Structure of finite wavelet frames over prime fields". Bulletin of the Iranian Mathematical Society, 43, 1, 2017, 109-120.
Ghaani Farashahi, A. (2017). 'Structure of finite wavelet frames over prime fields', Bulletin of the Iranian Mathematical Society, 43(1), pp. 109-120.
Ghaani Farashahi, A. Structure of finite wavelet frames over prime fields. Bulletin of the Iranian Mathematical Society, 2017; 43(1): 109-120.
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Chapters in Books
Formally Reviewed Conference Proceedings
Weakly Refereed Conference Proceedings and Workshop Abstracts
Agglomerative Clustering of Growing Squares
Thom Castermans, Bettina Speckmann, Frank Staals, and Kevin Verbeek.
Algorithmica, 84(1):216—233, 2022.
- Abstract
<p>We study an agglomerative clustering problem motivated by interactive glyphs in geo-visualization. Consider a set of disjoint square glyphs on an interactive map. When the user zooms out, the glyphs grow in size relative to the map, possibly with different speeds. When two glyphs intersect, we wish to replace them by a new glyph that captures the information of the intersecting glyphs. We present a fully dynamic kinetic data structure that maintains a set of n disjoint growing squares. Our data structure uses O(nlog nlog log n) space, supports queries in worst case O(log <sup>2</sup>n) time, and updates in O(log <sup>5</sup>n) amortized time. This leads to an O(nα(n)log5n) time algorithm to solve the agglomerative clustering problem. This is a significant improvement over the current best O(n<sup>2</sup>) time algorithms.</p>
- BibTeX
@article{agglomerative-clustering-of-growing-squares:2022,
title = {Agglomerative Clustering of Growing Squares},
author = {Thom Castermans and Bettina Speckmann and Frank Staals and Kevin Verbeek},
bookTitle = {Algorithmica},
[ PDF ]
Alluvial Connectivity in Multi-Channel Networks in Rivers and Estuaries
Willem Sonke, Maarten G. Kleinhans, Bettina Speckmann, Wout M. van Dijk, and Matthew Hiatt.
Earth Surface Processes and Landforms, 47(2):477—490, 2022.
<p>Channels in rivers and estuaries are the main paths of fluvial and tidal currents that transport sediment through the system. While network representations of multi-channel systems and their connectivity are quite useful for characterisation of braiding patterns and dynamics, the recognition of channels and their properties is complicated because of the large bed elevation variations, such as shallow shoals and bed steps that render channels visually disconnected. We present and analyse two mathematically rigorous methods to identify channel networks from a terrain model of the river bed. Both methods construct a dense network of locally steepest-descent channels from saddle points on the terrain, and select a subset of channels with a certain minimum sediment volume between them. This is closely linked to the main mechanism of channel formation and change by displacement of sediment volume. The two methods differ in how they compute these sediment volumes: either globally through the entire length of the river, or locally. We compare the methods for the measured bathymetry of the Western Scheldt estuary, The Netherlands, over the past decades. The global method is overly sensitive to small changes elsewhere in the network compared to the local method. We conclude that the local method works best conceptually and for stability reasons. The associated concept of alluvial connectivity between channels in a network is thus the inverse of the volume of sediment that must be displaced to merge the channels. Our method opens up possibilities for new analyses as shown in two examples. First, it shows a clear pattern of scale dependence on volume of the total network length and of the number of nodes by a power law relation, showing that the smaller channels are relatively much shorter. Second, channel bifurcations were found to be predominantly mildly asymmetrical, which is unexpected from fluvial bifurcation theory.</p>
@article{alluvial-connectivity-in-multi-channel-networks-in-rivers-and-estuaries:2022,
title = {Alluvial Connectivity in Multi-Channel Networks in Rivers and Estuaries},
author = {Willem Sonke and Maarten G. Kleinhans and Bettina Speckmann and Wout M. van Dijk and Matthew Hiatt},
bookTitle = {Earth Surface Processes and Landforms},
Simultaneous Matrix Orderings for Graph Collections
Nathan van Beusekom, Wouter Meulemans, and Bettina Speckmann.
IEEE Transactions on Visualization and Computer Graphics, 28(1):1—10, 2022.
Undirected graphs are frequently used to model phenomena that deal with interacting objects, such as social networks, brain activity and communication networks. The topology of an undirected graph G can be captured by an adjacency matrix; this matrix in turn can be visualized directly to give insight into the graph structure. Which visual patterns appear in such a matrix visualization crucially depends on the ordering of its rows and columns. Formally defining the quality of an ordering and then automatically computing a high-quality ordering are both challenging problems; however, effective heuristics exist and are used in practice.<br/><br/>Often, graphs do not exist in isolation but as part of a collection of graphs on the same set of vertices, for example, brain scans over time or of different people.<br/>To visualize such graph collections, we need a single ordering that works well for all matrices simultaneously.<br/>The current state-of-the-art solves this problem by taking a (weighted) union over all graphs and applying existing heuristics. However, this union leads to a loss of information, specifically in those parts of the graphs which are different. <br/>We propose a collection-aware approach to avoid this loss of information and apply it to two popular heuristic methods: leaf order and barycenter.<br/><br/>The de-facto standard computational quality metrics for matrix ordering capture only block-diagonal patterns (cliques). Instead, we propose to use Moran's I, a spatial auto-correlation metric, which captures the full range of established patterns. Moran's I refines previously proposed stress measures. Furthermore, the popular leaf order method heuristically optimizes a similar measure which further supports the use of Moran's I in this context. An ordering that maximizes Moran's I can be computed via solutions to the Traveling Salesperson Problem (TSP); orderings that approximate the optimal ordering can be computed more efficiently, using any of the approximation algorithms for metric TSP.<br/><br/>We evaluated our methods for simultaneous orderings on real-world datasets using Moran's I as the quality metric. Our results show that our collection-aware approach matches or improves performance<br/>compared to the union approach, depending on the similarity of the graphs in the collection. <br/>Specifically, our Moran's I-based collection-aware leaf order implementation consistently outperforms other implementations.<br/>Our collection-aware implementations carry no significant additional computational costs.
@article{simultaneous-matrix-orderings-for-graph-collections:2022,
title = {Simultaneous Matrix Orderings for Graph Collections},
author = {Nathan van Beusekom and Wouter Meulemans and Bettina Speckmann},
bookTitle = {IEEE Transactions on Visualization and Computer Graphics},
Compacting Squares: Input-Sensitive In-Place Reconfiguration of Sliding Squares
Hugo A. Akitaya, Erik D. Demaine, Matias Korman, Irina Kostitsyna, Irene Parada, Willem Sonke, Bettina Speckmann, Ryuhei Uehara, and Jules Wulms.
18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022), pp. 4:1—4:19, 2022.
<p>Edge-connected configurations of square modules, which can reconfigure through so-called sliding moves, are a well-established theoretical model for modular robots in two dimensions. Dumitrescu and Pach [Graphs and Combinatorics, 2006] proved that it is always possible to reconfigure one edge-connected configuration of n squares into any other using at most O(n2) sliding moves, while keeping the configuration connected at all times. For certain pairs of configurations, reconfiguration may require ω(n2) sliding moves. However, significantly fewer moves may be sufficient. We prove that it is NP-hard to minimize the number of sliding moves for a given pair of edge-connected configurations. On the positive side we present Gather&Compact, an input-sensitive in-place algorithm that requires only O( Pn) sliding moves to transform one configuration into the other, where P is the maximum perimeter of the two bounding boxes. The squares move within the bounding boxes only, with the exception of at most one square at a time which may move through the positions adjacent to the bounding boxes. The O( Pn) bound never exceeds O(n2), and is optimal (up to constant factors) among all bounds parameterized by just n and P. Our algorithm is built on the basic principle that well-connected components of modular robots can be transformed efficiently. Hence we iteratively increase the connectivity within a configuration, to finally arrive at a single solid xy-monotone component. We implemented Gather&Compact and compared it experimentally to the in-place modification by Moreno and Sacristán [EuroCG 2020] of the Dumitrescu and Pach algorithm (MSDP). Our experiments show that Gather&Compact consistently outperforms MSDP by a significant margin, on all types of square configurations.</p>
@article{compacting-squares-input-sensitive-in-place-reconfiguration-of-sliding-squares:2022,
title = {Compacting Squares: Input-Sensitive In-Place Reconfiguration of Sliding Squares},
author = {Hugo A. Akitaya and Erik D. Demaine and Matias Korman and Irina Kostitsyna and Irene Parada and Willem Sonke and Bettina Speckmann and Ryuhei Uehara and Jules Wulms},
bookTitle = {18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)},
Preprocessing Imprecise Points for the Pareto Front.
Ivor van der Hoog, Irina Kostitsyna, Maarten Löffler, and Bettina Speckmann.
ACM-SIAM Symposium on Discrete Algorithms (SODA22), pp. 3144—3167, 2022.
@article{preprocessing-imprecise-points-for-the-pareto-front-:2022,
title = {Preprocessing Imprecise Points for the Pareto Front.},
author = {Ivor van der Hoog and Irina Kostitsyna and Maarten Löffler and Bettina Speckmann},
bookTitle = {ACM-SIAM Symposium on Discrete Algorithms (SODA22)},
Fast Reconfiguration for Programmable Matter
Irina Kostitsyna, Tom Peters, and Bettina Speckmann.
pp. 365—371, 2022.
@article{fast-reconfiguration-for-programmable-matter:2022,
title = {Fast Reconfiguration for Programmable Matter},
author = {Irina Kostitsyna and Tom Peters and Bettina Speckmann},
bookTitle = {},
Kinetic Group Density in 1D
Kevin A. Buchin, Max J.M. van Mulken, Bettina Speckmann, and Kevin A.B. Verbeek.
pp. 47:1—47:6, 2022.
We are interested in tracking the overall structure of a single group of entities (e.g., people, wildlife, or vehicles) over time. As a first step, we investigate how to characterize and kinetically maintain the density of a one-dimensional group, represented by a point set P of size n. We achieve this with a kinetic data structure that maintains the critical points of an estimate of the density function of P, which is obtained using kernel density estimation. Our KDS is local, compact, responsive, and weakly efficient. However, the total number of events can be Θ(n^2). Therefore, we also show how to maintain an ɛ-approximation of the critical points using a coreset of the trajectories of the points in P. The coreset has size O(1/ɛ^2 log(1/ɛ)), for ɛ > 0, and approximates the critical points well in a topological sense.
@article{kinetic-group-density-in-1d:2022,
title = {Kinetic Group Density in 1D},
author = {Kevin A. Buchin and Max J.M. van Mulken and Bettina Speckmann and Kevin A.B. Verbeek},
A Simple Pipeline for Coherent Grid Maps
Wouter Meulemans, Max F.M. Sondag, and Bettina Speckmann.
IEEE Transactions on Visualization and Computer Graphics, 27(2):1236—1246, 2021.
Grid maps are spatial arrangements of simple tiles (often squares or hexagons), each of which represents a spatial element. They are an established, effective way to show complex data per spatial element, using visual encodings within each tile ranging from simple coloring to nested small-multiples visualizations. An effective grid map is coherent with the underlying geographic space: the tiles maintain the contiguity, neighborhoods and identifiability of the corresponding spatial elements, while the grid map as a whole maintains the global shape of the input. Of particular importance are salient local features of the global shape which need to be represented by tiles assigned to the appropriate spatial elements. State-of-the-art techniques can adequately deal only with simple cases, such as close-to-uniform spatial distributions or global shapes that have few characteristic features. We introduce a simple fully-automated 3-step pipeline for computing coherent grid maps. Each step is a well-studied problem: shape decomposition based on salient features, tile-based Mosaic Cartograms, and point-set matching. Our pipeline is a seamless composition of existing techniques for these problems and results in high-quality grid maps. We provide an implementation, demonstrate the efficacy of our approach on various complex datasets, and compare it to the state-of-the-art.
@article{a-simple-pipeline-for-coherent-grid-maps:2021,
title = {A Simple Pipeline for Coherent Grid Maps},
author = {Wouter Meulemans and Max F.M. Sondag and Bettina Speckmann},
Maximum Physically Consistent Trajectories
Bram A. Custers, Frank Staals, Bettina Speckmann, Wouter Meulemans, and Mees van de Kerkhof.
ACM Transactions on Spatial Algorithms and Systems , 7(4):, 2021.
Trajectories are usually collected with physical sensors, which are prone to errors and cause outliers in the data. We aim to identify such outliers via the physical properties of the tracked entity, that is, we consider its physical possibility to visit combinations of measurements. We describe optimal algorithms to compute maximum subsequences of measurements that are consistent with (simplified) physics models. Our results are output-sensitive with respect to the number k of outliers in a trajectory of n measurements. Specifically, we describe an O(n log n log2 k)-time algorithm for 2D trajectories using a model with unbounded acceleration but bounded velocity, and an O(nk)-time algorithm for any model where consistency is "concatenable": a consistent subsequence that ends where another begins together form a consistent sequence. We also consider acceleration-bounded models that are not concatenable. We show how to compute the maximum subsequence for such models in O(n k2 log k) time, under appropriate realism conditions. Finally, we experimentally explore the performance of our algorithms on several large real-world sets of trajectories. Our experiments show that we are generally able to retain larger fractions of noisy trajectories than previous work and simpler greedy approaches. We also observe that the speed-bounded model may in practice approximate the acceleration-bounded model quite well, though we observed some variation between datasets.
@article{maximum-physically-consistent-trajectories:2021,
title = {Maximum Physically Consistent Trajectories},
author = {Bram A. Custers and Frank Staals and Bettina Speckmann and Wouter Meulemans and Mees van de Kerkhof},
bookTitle = {ACM Transactions on Spatial Algorithms and Systems },
The Vulnerability of Tidal Flats and Multi-Channel Estuaries to Dredging and Disposal
Wout van Dijk, Jana Cox, Jasper Leuven, Jelmer Cleveringa, Marcel Taal, Matthew Hiatt, Willem M. Sonke, Kevin A.B. Verbeek, Bettina Speckmann, and Maarten G. Kleinhans.
Anthropocene Coasts, 4:36—60, 2021.
Shipping fairways in estuaries are continuously dredged to maintain access for large vessels to major ports. However, several estuaries worldwide show adverse side effects to dredging activities, in particular affecting morphology and ecologically valuable habitats. We used physical scale experiments, field assessments of the Western Scheldt estuary (the Netherlands), and morphodynamic model runs to analyse the effects of dredging and future stresses (climate and sediment management) on a multi-channel system and its ecologically valuable intertidal flats. All methods indicate that dredging and disposal strategies are unfavourable to long-term morphology because dredging creates and propagates the imbalance between shallow and deeper parts of the estuary, causing a loss of valuable connecting channels and fixation of the tidal flats and main channel positions, while countering adverse effects by disposal strategy has limited effectiveness. Changing the disposal strategy towards main channel scour disposal can be economically and ecologically beneficial for the preservation of the multi-channel system. Further channel deepening will accelerate the adverse side effects, whereas future sea-level rise may revive the multi-channel system.
@article{the-vulnerability-of-tidal-flats-and-multi-channel-estuaries-to-dredging-and-disposal:2021,
title = {The Vulnerability of Tidal Flats and Multi-Channel Estuaries to Dredging and Disposal},
author = {Wout van Dijk and Jana Cox and Jasper Leuven and Jelmer Cleveringa and Marcel Taal and Matthew Hiatt and Willem M. Sonke and Kevin A.B. Verbeek and Bettina Speckmann and Maarten G. Kleinhans},
bookTitle = {Anthropocene Coasts},
Coordinated Schematization for Visualizing Mobility Patterns on Networks
Bram A. Custers, Wouter Meulemans, Bettina Speckmann, and Kevin Verbeek.
11th International Conference on Geographic Information Science - Part II, GIScience 2021, pp. VII, 2021.
<p>GPS trajectories of vehicles moving on a road network are a valuable source of traffic information. However, the sheer volume of available data makes it challenging to identify and visualize salient patterns. Meaningful visual summaries of trajectory collections require that both the trajectories and the underlying network are aggregated and simplified in a coherent manner. In this paper we propose a coordinated fully-automated pipeline for computing a schematic overview of mobility patterns from a collection of trajectories on a street network. Our pipeline utilizes well-known building blocks from GIS, automated cartography, and trajectory analysis: map matching, road selection, schematization, movement patterns, and metro-map style rendering. We showcase the results of our pipeline on two real-world trajectory collections around The Hague and Beijing.</p>
@article{coordinated-schematization-for-visualizing-mobility-patterns-on-networks:2021,
title = {Coordinated Schematization for Visualizing Mobility Patterns on Networks},
author = {Bram A. Custers and Wouter Meulemans and Bettina Speckmann and Kevin Verbeek},
bookTitle = {11th International Conference on Geographic Information Science - Part II, GIScience 2021},
Near-Delaunay Metrics
Nathan van Beusekom, Kevin A. Buchin, Hidde O. Koerts, Wouter Meulemans, Benjamin Rodatz, and Bettina Speckmann.
Proceedings of the 33rd Canadian Conference on Computational Geometry (CCCG 2021), pp. 1—11, 2021.
We study metrics that assess how close a triangulation is to being a Delaunay triangulation, for use in contexts where a good triangulation is desired but constraints (e.g., maximum degree) prevent the use of the Delaunay triangulation itself. Our near-Delaunay metrics derive from common Delaunay properties and satisfy a basic set of design criteria, such as being invariant under similarity transformations. We compare the metrics, showing that each can make different judgments as to which triangulation is closer to Delaunay. We also present a preliminary experiment, showing how optimizing for these metrics under different constraints gives similar, but not necessarily identical results, on random and constructed small point sets.
@article{near-delaunay-metrics:2021,
title = {Near-Delaunay Metrics},
author = {Nathan van Beusekom and Kevin A. Buchin and Hidde O. Koerts and Wouter Meulemans and Benjamin Rodatz and Bettina Speckmann},
bookTitle = {Proceedings of the 33rd Canadian Conference on Computational Geometry (CCCG 2021)},
Obstructing Classification Via Projection
Pantea Haghighatkhah, Wouter Meulemans, Bettina Speckmann, Jérôme Urhausen, and Kevin Verbeek.
46th International Symposium on Mathematical Foundations of Computer Science, MFCS 2021, 2021.
<p>Machine learning and data mining techniques are effective tools to classify large amounts of data. But they tend to preserve any inherent bias in the data, for example, with regards to gender or race. Removing such bias from data or the learned representations is quite challenging. In this paper we study a geometric problem which models a possible approach for bias removal. Our input is a set of points P in Euclidean space Rd and each point is labeled with k binary-valued properties. A priori we assume that it is "easy"to classify the data according to each property. Our goal is to obstruct the classification according to one property by a suitable projection to a lower-dimensional Euclidean space Rm (m < d), while classification according to all other properties remains easy. What it means for classification to be easy depends on the classification model used. We first consider classification by linear separability as employed by support vector machines. We use Kirchberger's Theorem to show that, under certain conditions, a simple projection to Rd1 suffices to eliminate the linear separability of one of the properties whilst maintaining the linear separability of the other properties. We also study the problem of maximizing the linear "inseparability"of the chosen property. Second, we consider more complex forms of separability and prove a connection between the number of projections required to obstruct classification and the Helly-type properties of such separabilities. </p>
@article{obstructing-classification-via-projection:2021,
title = {Obstructing Classification Via Projection},
author = {Pantea Haghighatkhah and Wouter Meulemans and Bettina Speckmann and Jérôme Urhausen and Kevin Verbeek},
bookTitle = {46th International Symposium on Mathematical Foundations of Computer Science, MFCS 2021},
Polygon-Universal Graphs
Tim Ophelders, Ignaz Rutter, Bettina Speckmann, and Kevin Verbeek.
37th International Symposium on Computational Geometry, SoCG 2021, 2021.
<p>We study a fundamental question from graph drawing: given a pair (G,C) of a graph G and a cycle C in G together with a simple polygon P, is there a straight-line drawing of G inside P which maps C to P? We say that such a drawing of (G,C) respects P. We fully characterize those instances (G,C) which are polygon-universal, that is, they have a drawing that respects P for any simple (not necessarily convex) polygon P. Specifically, we identify two necessary conditions for an instance to be polygon-universal. Both conditions are based purely on graph and cycle distances and are easy to check. We show that these two conditions are also sufficient. Furthermore, if an instance (G,C) is planar, that is, if there exists a planar drawing of G with C on the outer face, we show that the same conditions guarantee for every simple polygon P the existence of a planar drawing of (G,C) that respects P. If (G,C) is polygon-universal, then our proofs directly imply a linear-time algorithm to construct a drawing that respects a given polygon P.</p>
@article{polygon-universal-graphs:2021,
title = {Polygon-Universal Graphs},
author = {Tim Ophelders and Ignaz Rutter and Bettina Speckmann and Kevin Verbeek},
bookTitle = {37th International Symposium on Computational Geometry, SoCG 2021},
Route Reconstruction from Traffic Flow Via Representative Trajectories
Bram A. Custers, Kevin A.B. Verbeek, Wouter Meulemans, and Bettina Speckmann.
Proc. 29th International Conference on Advances in Geographic Information (SIGSPATIAL), pp. 41—52, 2021.
Understanding human mobility patterns is an important aspect of traffic analysis and urban planning. Trajectory data provide detailed views on specific routes, but typically do not capture all traffic. On the other hand, loop detectors built into the road network capture all traffic flow at specific locations, but provide no information on the individual routes. Given a set of loop-detector measurements as well as a (small) set of representative trajectories, our goal is to investigate how one can effectively combine these two partial data sources to create a more complete picture of the underlying mobility patterns. Specifically, we want to reconstruct a realistic set of routes from the loop-detector data, using the given trajectories as representatives of typical behavior.<br/><br/>We model the loop-detector data as a network flow field that needs to be covered by the reconstructed routes and we capture the realism of the routes via the strong Fréchet distance to the representative trajectories. We prove that several forms of the resulting algorithmic problem are NP-hard. Hence we explore heuristic approaches which decompose the flow well while following the representative trajectories to varying degrees. We propose an iterative Fréchet Routes (FR) heuristic which generates candidates routes with bounded Fréchet distance to the representative trajectories. We also describe a variant of multi-commodity min-cost flow (MCMCF) which is only loosely coupled to the trajectories.<br/><br/>We perform an extensive experimental evaluation of our two proposed approaches in comparison to a global min-cost flow (GMCF), which is essentially agnostic to the representative trajectories. To make meaningful claims in terms of quality, we derive a ground truth by map-matching real-world trajectories. We find that GMCF explains the flow best, but produces a large number of often nonsensical routes (significantly more than the ground truth). MCMCF produces a large number of mostly realistic routes which explain the flow reasonably well. In contrast, FR produces much smaller sets of realistic routes which still explain the flow well, at the cost of a higher running time. Finally, we report on the results of a case study which combines real-world loop detector data and representative trajectories for the region around The Hague, the Netherlands.<br/>
@article{route-reconstruction-from-traffic-flow-via-representative-trajectories:2021,
title = {Route Reconstruction from Traffic Flow Via Representative Trajectories},
author = {Bram A. Custers and Kevin A.B. Verbeek and Wouter Meulemans and Bettina Speckmann},
bookTitle = {Proc. 29th International Conference on Advances in Geographic Information (SIGSPATIAL)},
[ PDF lock ]
Stable Visual Summaries for Trajectory Collections
Jules Wulms, Juri Buchmuller, Wouter Meulemans, Kevin Verbeek, and Bettina Speckmann.
Proceedings - 2021 IEEE 14th Pacific Visualization Symposium, PacificVis 2021, pp. 61—70, 2021.
<p>The availability of devices that track moving objects has led to an explosive growth in trajectory data. When exploring the resulting large trajectory collections, visual summaries are a useful tool to identify time intervals of interest. A typical approach is to represent the spatial positions of the tracked objects at each time step via a one-dimensional ordering; visualizations of such orderings can then be placed in temporal order along a time line. There are two main criteria to assess the quality of the resulting visual summary: spatial quality - how well does the ordering capture the structure of the data at each time step, and stability - how coherent are the orderings over consecutive time steps or temporal ranges?In this paper we introduce a new Stable Principal Component (SPC) method to compute such orderings, which is explicitly parameterized for stability, allowing a trade-off between the spatial quality and stability. We conduct extensive computational experiments that quantitatively compare the orderings produced by ours and other stable dimensionality-reduction methods to various state-of-the-art approaches using a set of well-established quality metrics that capture spatial quality and stability. We conclude that stable dimensionality reduction outperforms existing methods on stability, without sacrificing spatial quality or efficiency; in particular, our new SPC method does so at a fraction of the computational costs.</p>
@article{stable-visual-summaries-for-trajectory-collections:2021,
title = {Stable Visual Summaries for Trajectory Collections},
author = {Jules Wulms and Juri Buchmuller and Wouter Meulemans and Kevin Verbeek and Bettina Speckmann},
bookTitle = {Proceedings - 2021 IEEE 14th Pacific Visualization Symposium, PacificVis 2021},
Coordinating Programmable Matter Via Shortest Path Trees
@article{coordinating-programmable-matter-via-shortest-path-trees:2021,
title = {Coordinating Programmable Matter Via Shortest Path Trees},
Crossing Numbers of Beyond-Planar Graphs Revisited
Nathan van Beusekom, Irene Parada, and Bettina Speckmann.
Graph drawing beyond planarity focuses on drawings of high visual quality for non-planar graphs<br/>which are characterized by certain forbidden edge configurations. A natural criterion for the quality<br/>of a drawing is the number of edge crossings. The question then arises whether beyond-planar<br/>drawings have a significantly larger crossing number than unrestricted drawings. Chimani et al.<br/>[GD'19] gave bounds for the ratio between the crossing number of three classes of beyond-planar<br/>graphs and the unrestricted crossing number. In this paper we extend their results to the main<br/>currently known classes of beyond-planar graphs characterized by forbidden edge configurations and<br/>answer several of their open questions.
@article{crossing-numbers-of-beyond-planar-graphs-revisited:2021,
title = {Crossing Numbers of Beyond-Planar Graphs Revisited},
author = {Nathan van Beusekom and Irene Parada and Bettina Speckmann},
Bram A. Custers, Bettina Speckmann, Wouter Meulemans, and Kevin A.B. Verbeek.
author = {Bram A. Custers and Bettina Speckmann and Wouter Meulemans and Kevin A.B. Verbeek},
Can Real Social Epistemic Networks Deliver the Wisdom of Crowds?
Emily Sullivan, Max Sondag, Ignaz Rutter, Wouter Meulemans, Scott Cunningham, Bettina Speckmann, and Mark Alfano.
Oxford Studies in Experimental Philosophy, Volume 3, pp. 29—63, 2020.
@article{can-real-social-epistemic-networks-deliver-the-wisdom-of-crowds-:2020,
title = {Can Real Social Epistemic Networks Deliver the Wisdom of Crowds?},
author = {Emily Sullivan and Max Sondag and Ignaz Rutter and Wouter Meulemans and Scott Cunningham and Bettina Speckmann and Mark Alfano},
bookTitle = {Oxford Studies in Experimental Philosophy, Volume 3},
Geometry and Topology of Estuary and Braided River Channel Networks Automatically Extracted from Topographic Data
Matthew Hiatt, Willem M. Sonke, Elisabeth Addink, Wout van Dijk, Marc J. van Kreveld, Tim A.E. Ophelders, Kevin A.B. Verbeek, Joyce Vlaming, Bettina Speckmann, and Maarten G. Kleinhans.
Journal of Geophysical Research: Earth Surface, 125(1):, 2020.
Automatic extraction of channel networks from topography in systems with multiple interconnected channels, like braided rivers and estuaries, remains a major challenge in hydrology and geomorphology. Representing channelized systems as networks provides a mathematical framework for analyzing transport and geomorphology. In this paper, we introduce a mathematically rigorous methodology and software for extracting channel network topology and geometry from digital elevation models (DEMs) and analyze such channel networks in estuaries and braided rivers. Channels are represented as network links, while channel confluences and bifurcations are represented as network nodes. We analyze and compare DEMs from the field and those generated by numerical modeling. We use a metric called the volume parameter that characterizes the volume of deposited material separating channels to quantify the volume of reworkable sediment deposited between links, which is a measure for the spatial scale associated with each network link. Scale asymmetry is observed in most links downstream of bifurcations, indicating geometric asymmetry and bifurcation stability. The length of links relative to system size scales with volume parameter value to the power of 0.24–0.35, while the number of links decreases and does not exhibit power law behavior. Link depth distributions indicate that the estuaries studied tend to organize around a deep main channel that exists at the largest scale while braided rivers have channel depths that are more evenly distributed across scales. The methods and results presented establish a benchmark for quantifying the topology and geometry of multichannel networks from DEMs with a new automatic extraction tool.
@article{geometry-and-topology-of-estuary-and-braided-river-channel-networks-automatically-extracted-from-topographic-data:2020,
title = {Geometry and Topology of Estuary and Braided River Channel Networks Automatically Extracted from Topographic Data},
author = {Matthew Hiatt and Willem M. Sonke and Elisabeth Addink and Wout van Dijk and Marc J. van Kreveld and Tim A.E. Ophelders and Kevin A.B. Verbeek and Joyce Vlaming and Bettina Speckmann and Maarten G. Kleinhans},
bookTitle = {Journal of Geophysical Research: Earth Surface},
Quantitative Comparison of Time-Dependent Treemaps
E. Vernier, M. Sondag, J. Comba, B. Speckmann, A. Telea, and K. Verbeek.
Computer Graphics Forum, 39(3):393—404, 2020.
<p>Rectangular treemaps are often the method of choice to visualize large hierarchical datasets. Nowadays such datasets are available over time, hence there is a need for (a) treemaps that can handle time-dependent data, and (b) corresponding quality criteria that cover both a treemap's visual quality and its stability over time. In recent years a wide variety of (stable) treemapping algorithms has been proposed, with various advantages and limitations. We aim to provide insights to researchers and practitioners to allow them to make an informed choice when selecting a treemapping algorithm for specific applications and data. To this end, we perform an extensive quantitative evaluation of rectangular treemaps for time-dependent data. As part of this evaluation we propose a novel classification scheme for time-dependent datasets. Specifically, we observe that the performance of treemapping algorithms depends on the characteristics of the datasets used. We identify four potential representative features that characterize time-dependent hierarchical datasets and classify all datasets used in our experiments accordingly. We experimentally test the validity of this classification on more than 2000 datasets, and analyze the relative performance of 14 state-of-the-art rectangular treemapping algorithms across varying features. Finally, we visually summarize our results with respect to both visual quality and stability to aid users in making an informed choice among treemapping algorithms. All datasets, metrics, and algorithms are openly available to facilitate reuse and further comparative studies.</p>
@article{quantitative-comparison-of-time-dependent-treemaps:2020,
title = {Quantitative Comparison of Time-Dependent Treemaps},
author = {E. Vernier and M. Sondag and J. Comba and B. Speckmann and A. Telea and K. Verbeek},
bookTitle = {Computer Graphics Forum},
Vulnerability in Social Epistemic Networks
International Journal of Philosophical Studies, 28(5):731—753, 2020.
<p>Social epistemologists should be well-equipped to explain and evaluate the growing vulnerabilities associated with filter bubbles, echo chambers, and group polarization in social media. However, almost all social epistemology has been built for social contexts that involve merely a speaker-hearer dyad. Filter bubbles, echo chambers, and group polarization all presuppose much larger and more complex network structures. In this paper, we lay the groundwork for a properly social epistemology that gives the role and structure of networks their due. In particular, we formally define epistemic constructs that quantify the structural epistemic position of each node within an interconnected network. We argue for the epistemic value of a structure that we call the (m,k)-observer. We then present empirical evidence that (m,k)-observers are rare in social media discussions of controversial topics, which suggests that people suffer from serious problems of epistemic vulnerability. We conclude by arguing that social epistemologists and computer scientists should work together to develop minimal interventions that improve the structure of epistemic networks.</p>
@article{vulnerability-in-social-epistemic-networks:2020,
title = {Vulnerability in Social Epistemic Networks},
bookTitle = {International Journal of Philosophical Studies},
Hiding Sliding Cubes - Why Reconfiguring Modular Robots is Not Easy
Tillmann Miltzow, Irene Parada, Willem Sonke, Bettina Speckmann, and Jules Wulms.
36th International Symposium on Computational Geometry (SoCG 2020), pp. 78:1—78:5, 2020.
<p>Face-connected configurations of cubes are a common model for modular robots in three dimensions. In this abstract and the accompanying video we study reconfigurations of such modular robots using so-called sliding moves. Using sliding moves, it is always possible to reconfigure one face-connected configuration of n cubes into any other, while keeping the robot connected at all stages of the reconfiguration. For certain configurations Ω(n <sup>2</sup>) sliding moves are necessary. In contrast, the best current upper bound is O(n <sup>3</sup>). It has been conjectured that there is always a cube on the outside of any face-connected configuration of cubes which can be moved without breaking connectivity. The existence of such a cube would immediately imply a straight-forward O(n <sup>2</sup>) reconfiguration algorithm. However, we present a configuration of cubes such that no cube on the outside can move without breaking connectivity. In other words, we show that this particular avenue towards an O(n <sup>2</sup>) reconfiguration algorithm for face-connected cubes is blocked. </p>
@article{hiding-sliding-cubes-why-reconfiguring-modular-robots-is-not-easy:2020,
title = {Hiding Sliding Cubes - Why Reconfiguring Modular Robots is Not Easy},
author = {Tillmann Miltzow and Irene Parada and Willem Sonke and Bettina Speckmann and Jules Wulms},
bookTitle = {36th International Symposium on Computational Geometry (SoCG 2020)},
Ordered Strip Packing
K. Buchin, D. Kosolobov, W. Sonke, B. Speckmann, and K. Verbeek.
LATIN 2020, pp. 258—270, 2020.
<p>We study an ordered variant of the well-known strip packing problem, which is motivated by applications in visualization and typography. Our input consists of a maximum width W and an ordered list of n blocks (rectangles). The goal is to pack the blocks into rows (not exceeding W) while obeying the given order and minimizing either the number of rows or the total height of the drawing. We consider two variants: (1) non-overlapping row drawing (NORD), where distinct rows cannot share y-coordinates, and (2) overlapping row drawing (ORD), where consecutive rows may overlap vertically. We present an algorithm that computes the minimum-height NORD in O(n) time. Further, we study the worst-case tradeoffs between the two optimization criteria—number of rows and total height—for both NORD and ORD. Surprisingly, we show that the minimum-height ORD may require Ω(log n/ log log n) times as many rows as the minimum-row ORD. The proof of the matching upper bound employs a novel application of information entropy.</p>
@article{ordered-strip-packing:2020,
title = {Ordered Strip Packing},
author = {K. Buchin and D. Kosolobov and W. Sonke and B. Speckmann and K. Verbeek},
bookTitle = {LATIN 2020},
Simplification with Parallelism
Arthur I. van Goethem, Wouter Meulemans, Andreas Reimer, and Bettina Speckmann.
Abstr. 23rd ICA Workshop on Generalisation and Multiple Representation, 2020.
@article{simplification-with-parallelism:2020,
title = {Simplification with Parallelism},
author = {Arthur I. van Goethem and Wouter Meulemans and Andreas Reimer and Bettina Speckmann},
bookTitle = {Abstr. 23rd ICA Workshop on Generalisation and Multiple Representation},
Uncertainty Treemaps
Max Sondag, Wouter Meulemans, Christoph Schulz, Kevin Verbeek, Daniel Weiskopf, and Bettina Speckmann.
2020 IEEE Pacific Visualization Symposium, PacificVis 2020 - Proceedings, pp. 111—120, 2020.
<p>Rectangular treemaps visualize hierarchical numerical data by recursively partitioning an input rectangle into smaller rectangles whose areas match the data. Numerical data often has uncertainty associated with it. To visualize uncertainty in a rectangular treemap, we identify two conflicting key requirements: (i) to assess the data value of a node in the hierarchy, the area of its rectangle should directly match its data value, and (ii) to facilitate comparison between data and uncertainty, uncertainty should be encoded using the same visual variable as the data, that is, area. We present Uncertainty Treemaps, which meet both requirements simultaneously by introducing the concept of hierarchical uncertainty masks. First, we define a new cost function that measures the quality of Uncertainty Treemaps. Then, we show how to adapt existing treemapping algorithms to support uncertainty masks. Finally, we demonstrate the usefulness and quality of our technique through an expert review and a computational experiment on real-world datasets.</p>
@article{uncertainty-treemaps:2020,
title = {Uncertainty Treemaps},
author = {Max Sondag and Wouter Meulemans and Christoph Schulz and Kevin Verbeek and Daniel Weiskopf and Bettina Speckmann},
bookTitle = {2020 IEEE Pacific Visualization Symposium, PacificVis 2020 - Proceedings},
Volume from Outlines on Terrains
Marc Van Kreveld, Tim Ophelders, Willem Sonke, Bettina Speckmann, and Kevin Verbeek.
11th International Conference on Geographic Information Science, GIScience 2021, pp. 16:1—16:15, 2020.
<p>Outlines (closed loops) delineate areas of interest on terrains, such as regions with a heightened risk of landslides. For various analysis tasks it is necessary to define and compute a volume of earth (soil) based on such an outline, capturing, for example, the possible volume of a landslide in a high-risk region. In this paper we discuss several options to define meaningful 2D surfaces induced by a 1D outline, which allow us to compute such volumes. We experimentally compare the proposed surface options for two applications: Similarity of paths on terrains and landslide susceptibility analysis.</p>
@article{volume-from-outlines-on-terrains:2020,
title = {Volume from Outlines on Terrains},
author = {Marc Van Kreveld and Tim Ophelders and Willem Sonke and Bettina Speckmann and Kevin Verbeek},
bookTitle = {11th International Conference on Geographic Information Science, GIScience 2021},
Orthogonal Schematization with Minimum Homotopy Area
Bram A. Custers, Kevin A.B. Verbeek, Bettina Speckmann, Wouter Meulemans, Irina Kostitsyna, and Jeff Erickson.
@article{orthogonal-schematization-with-minimum-homotopy-area:2020,
title = {Orthogonal Schematization with Minimum Homotopy Area},
author = {Bram A. Custers and Kevin A.B. Verbeek and Bettina Speckmann and Wouter Meulemans and Irina Kostitsyna and Jeff Erickson},
Computing Representative Networks for Braided Rivers
Maarten G. Kleinhans, Marc J. van Kreveld, Tim A.E. Ophelders, Willem M. Sonke, Bettina Speckmann, and Kevin A.B. Verbeek.
Journal of Computational Geometry, 10(1):423—443, 2019.
Drainage networks on terrains have been studied extensively from an algorithmic perspective. However, in drainage networks water flow cannot bifurcate and hence they do not model <i>braided rivers</i> (multiple channels which split and join, separated by sediment bars). We initiate the algorithmic study of braided rivers by employing the descending quasi Morse-Smale complex on the river bed (a polyhedral terrain), and extending it with a certain ordering of bars from one river bank to the other. This allows us to compute a graph that models a representative channel network, consisting of lowest paths. To ensure that channels in this network are sufficiently different we define a <i>sand function</i> that represents the volume of sediment separating them. We show that in general the problem of computing a maximum network of non-crossing channels which are δ-different from each other (as measured by the sand function) is NP-hard. However, using our ordering between the river banks, we can compute a maximum δ-different network that respects this order in polynomial time. We implemented our approach and applied it to simulated and real-world braided rivers.
@article{computing-representative-networks-for-braided-rivers:2019,
title = {Computing Representative Networks for Braided Rivers},
author = {Maarten G. Kleinhans and Marc J. van Kreveld and Tim A.E. Ophelders and Willem M. Sonke and Bettina Speckmann and Kevin A.B. Verbeek},
bookTitle = {Journal of Computational Geometry},
Locally Correct Fréchet Matchings
Kevin Buchin, Maike Buchin, Wouter Meulemans, and Bettina Speckmann.
Computational Geometry, 76:1—18, 2019.
<p>The Fréchet distance is a metric to compare two curves, which is based on monotone matchings between these curves. We call a matching that results in the Fréchet distance a Fréchet matching. There are often many different Fréchet matchings and not all of these capture the similarity between the curves well. We propose to restrict the set of Fréchet matchings to "natural" matchings and to this end introduce locally correct Fréchet matchings. We prove that at least one such matching exists for two polygonal curves and give an O(N<sup>3</sup>logN) algorithm to compute it, where N is the number of edges in both curves. We also present an O(N<sup>2</sup>) algorithm to compute a locally correct discrete Fréchet matching.</p>
@article{locally-correct-fr-chet-matchings:2019,
title = {Locally Correct Fréchet Matchings},
author = {Kevin Buchin and Maike Buchin and Wouter Meulemans and Bettina Speckmann},
bookTitle = {Computational Geometry},
Non-Crossing Paths with Geographic Constraints
Rodrigo I. Silveira, Bettina Speckmann, and Kevin Verbeek.
Discrete Mathematics and Theoretical Computer Science, 21(3):, 2019.
<p>A geographic network is a graph whose vertices are restricted to lie in a prescribed region in the plane. In this paper we begin to study the following fundamental problem for geographic networks: can a given geographic network be drawn without crossings? We focus on the seemingly simple setting where each region is a vertical segment, and one wants to connect pairs of segments with a path that lies inside the convex hull of the two segments. We prove that when paths must be drawn as straight line segments, it is NP-complete to determine if a crossing-free solution exists, even if all vertical segments have unit length. In contrast, we show that when paths must be monotone curves, the question can be answered in polynomial time. In the more general case of paths that can have any shape, we show that the problem is polynomial under certain assumptions.</p>
@article{non-crossing-paths-with-geographic-constraints:2019,
title = {Non-Crossing Paths with Geographic Constraints},
author = {Rodrigo I. Silveira and Bettina Speckmann and Kevin Verbeek},
bookTitle = {Discrete Mathematics and Theoretical Computer Science},
Optimal Morphs of Planar Orthogonal Drawings II
Arthur I. van Goethem, Kevin A.B. Verbeek, and Bettina Speckmann.
arXiv, 2019.
Van Goethem and Verbeek recently described an algorithm that morphs between two planar orthogonal drawings $\Gamma_I$ and $\Gamma_O$ of a connected graph $G$ of complexity $n$, while preserving planarity, orthogonality, and linear complexity of the drawing during the morph. Necessarily drawings $\Gamma_I$ and $\Gamma_O$ must be equivalent; there exists a homeomorphism of the plane that transforms $\Gamma_I$ into $\Gamma_O$. The algorithm of van Goethem and Verbeek uses a linear number of linear morphs, however, if the graph $G$ is disconnected, then their method requires $O(n^{1.5})$ linear morphs. In this paper we present a refined version of their approach that allows us to also morph between two planar orthogonal drawings of a disconnected graph with a linear number of linear morphs while preserving planarity, orthogonality, and linear complexity. <br/><br/>Van Goethem and Verbeek measure the structural difference between the two drawings in terms of the so-called \emph{spirality} $s = O(n)$ of $\Gamma_I$ relative to $\Gamma_O$ and describe a morph from $\Gamma_I$ to $\Gamma_O$ using $O(s)$ linear morphs. We show how to combine all intermittent morphs that act on links of spirality $s$ into one single linear morph and prove that $s+1$ linear morphs are always sufficient to morph between two planar orthogonal drawings, even for a disconnected graph. The resulting morphs are quite natural and visually pleasing.
@article{optimal-morphs-of-planar-orthogonal-drawings-ii:2019,
title = {Optimal Morphs of Planar Orthogonal Drawings II},
author = {Arthur I. van Goethem and Kevin A.B. Verbeek and Bettina Speckmann},
bookTitle = {arXiv},
SolarView
Thom Castermans, Kevin Verbeek, Bettina Speckmann, Michel A. Westenberg, Rob Koopman, Shenghui Wang, Hein van den Berg, and Arianna Betti.
IEEE Transactions on Visualization and Computer Graphics, 25(10):2969—2982, 2019.
<p>We propose a novel type of low distortion radial embedding which focuses on one specific entity and its closest neighbors. Our embedding preserves near-exact distances to the focus entity and aims to minimize distortion between the other entities. We present an interactive exploration tool SolarView which places the focus entity at the center of a "solar system" and embeds its neighbors guided by concentric circles. SolarView provides an implementation of our novel embedding and several state-of-the-art dimensionality reduction and embedding techniques, which we adapted to our setting in various ways. We experimentally evaluated our embedding and compared it to these state-of-the-art techniques. The results show that our embedding competes with these techniques and achieves low distortion in practice. Our method performs particularly well when the visualization, and hence the embedding, adheres to the solar system design principle of our application. Nonetheless - as with all dimensionality reduction techniques - the distortion may be high. We leverage interaction techniques to give clear visual cues that allow users to accurately judge distortion. We illustrate the use of SolarView by exploring the high-dimensional metric space of bibliographic entity similarities.</p>
@article{solarview:2019,
title = {SolarView},
author = {Thom Castermans and Kevin Verbeek and Bettina Speckmann and Michel A. Westenberg and Rob Koopman and Shenghui Wang and Hein van den Berg and Arianna Betti},
Spatially and Temporally Coherent Visual Summaries
Jules J.H.M. Wulms, Juri Buchmüller, Wouter Meulemans, Kevin A.B. Verbeek, and Bettina Speckmann.
When exploring large time-varying data sets, visual summaries are a useful tool to identify time intervals of interest for further consideration. A typical approach is to represent the data elements at each time step in a compact one-dimensional form or via a one-dimensional ordering. Such 1D representations can then be placed in temporal order along a time line. There are two main criteria to assess the quality of the resulting visual summary: spatial quality -- how well does the 1D representation capture the structure of the data at each time step, and stability -- how coherent are the 1D representations over consecutive time steps or temporal ranges? We focus on techniques that create such visual summaries using 1D orderings for entities moving in 2D. We introduce stable techniques based on well-established dimensionality-reduction techniques: Principle Component Analysis, Sammon mapping, and t-SNE. Our Stable Principal Component method is explicitly parametrized for stability, allowing a trade-off between the two quality criteria. We conduct computational experiments that compare our stable methods to various state-of-the-art approaches using a set of well-established quality metrics that capture the two main criteria. These experiments demonstrate that our stable algorithms outperform existing methods on stability, without sacrificing spatial quality or efficiency.
@article{spatially-and-temporally-coherent-visual-summaries:2019,
title = {Spatially and Temporally Coherent Visual Summaries},
author = {Jules J.H.M. Wulms and Juri Buchmüller and Wouter Meulemans and Kevin A.B. Verbeek and Bettina Speckmann},
A Practical Algorithm for Spatial Agglomerative Clustering
Thom Castermans, Bettina Speckmann, and Kevin Verbeek.
Proceedings of the Twenty-First Workshop on Algorithm Engineering and Experiments, pp. 174—185, 2019.
<p> We study an agglomerative clustering problem motivated by visualizing disjoint glyphs (represented by geometric shapes) centered at specific locations on a geographic map. As we zoom out, the glyphs grow and start to overlap. We replace overlapping glyphs by one larger merged glyph to maintain disjointness. Our goal is to compute the resulting hierarchical clustering efficiently in practice. A straightforward algorithm for such spatial agglomerative clustering runs in On <sup>2</sup> log n time, where n is the number of glyphs. This is not efficient enough for many real-world datasets which contain up to tens or hundreds of thousands of glyphs. Recently the theoretical upper bound was improved to Onα(n) log <sup>7</sup> n time [10], where α(n) is the extremely slow growing inverse Ackermann function. Although this new algorithm is asymptotically much faster than the naive algorithm, from a practical point of view, it does not perform better for n ≤ 10 <sup>6</sup> . In this paper we present a new agglomerative clustering algorithm which works efficiently in practice. Our algorithm relies on the use of quadtrees to speed up spatial computations. Interestingly, even in non-pathological datasets we can encounter large glyphs that intersect many quadtree cells and that are involved in many clustering events. We therefore devise a special strategy to handle such large glyphs. We test our algorithm on several synthetic and real-world datasets and show that it performs well in practice. </p>
@article{a-practical-algorithm-for-spatial-agglomerative-clustering:2019,
title = {A Practical Algorithm for Spatial Agglomerative Clustering},
author = {Thom Castermans and Bettina Speckmann and Kevin Verbeek},
bookTitle = {Proceedings of the Twenty-First Workshop on Algorithm Engineering and Experiments},
Bram Custers, Mees van de Kerkhof, Wouter Meulemans, Bettina Speckmann, and Frank Staals.
SIGSPATIAL '19: Proceedings of the 27th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems, pp. 79—88, 2019.
<p>Trajectories are usually collected with physical sensors, which are prone to errors and cause outliers in the data. We aim to identify such outliers via the physical properties of the tracked entity, that is, we consider its physical possibility to visit combinations of measurements. We describe optimal algorithms to compute maximum subsequences of measurements that are consistent with (simplified) physics models. Our results are output-sensitive with respect to the number k of outliers in a trajectory of n measurements. Specifically, we describe an O(n logn log <sup>2</sup>k) time algorithm for 2D trajectories using a model with unbounded acceleration but bounded velocity, and an O(nk) time algorithm for any model where consistency is "concatenable": a consistent subsequence that ends where another begins together form a consistent sequence. We also consider acceleration-bounded models which are not concatenable. We show how to compute the maximum subsequence for such models in O(nk <sup>2</sup>logk) time, under appropriate realism conditions. Finally, we experimentally explore the performance of our algorithms on several large real-world sets of trajectories. Our experiments show that we are generally able to retain larger fractions of noisy trajectories than previous work and simpler greedy approaches. We also observe that the speed-bounded model may in practice approximate the acceleration-bounded model quite well, though we observed some variation between datasets. </p>
author = {Bram Custers and Mees van de Kerkhof and Wouter Meulemans and Bettina Speckmann and Frank Staals},
bookTitle = {SIGSPATIAL '19: Proceedings of the 27th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems},
Arthur van Goethem, Bettina Speckmann, and Kevin Verbeek.
Graph Drawing and Network Visualization - 27th International Symposium, GD 2019, Proceedings, pp. 33—45, 2019.
<p>Van Goethem and Verbeek [12] recently showed how to morph between two planar orthogonal drawings (Forumala Presented). of a connected graph G while preserving planarity, orthogonality, and the complexity of the drawing during the morph. Necessarily drawings (Forumala Presented). must be equivalent, that is, there exists a homeomorphism of the plane that transforms (Forumala Presented). Van Goethem and Verbeek use O(n) linear morphs, where n is the maximum complexity of the input drawings. However, if the graph is disconnected their method requires (Forumala Presented). linear morphs. In this paper we present a refined version of their approach that allows us to also morph between two planar orthogonal drawings of a disconnected graph with O(n) linear morphs while preserving planarity, orthogonality, and linear complexity of the intermediate drawings. Van Goethem and Verbeek measure the structural difference between the two drawings in terms of the so-called spirality (Forumala Presented). and describe a morph from (Forumala Presented). using O(s) linear morphs. We prove that linear morphs are always sufficient to morph between two planar orthogonal drawings, even for disconnected graphs. The resulting morphs are quite natural and visually pleasing.</p>
author = {Arthur van Goethem and Bettina Speckmann and Kevin Verbeek},
bookTitle = {Graph Drawing and Network Visualization - 27th International Symposium, GD 2019, Proceedings},
Preprocessing Ambiguous Imprecise Points
35th International Symposium on Computational Geometry (SoCG 2019), 129:42:1—42:16, 2019.
<p>Let R = {R <sub>1</sub>,R <sub>2</sub>, . . . ,R <sub>n</sub>} be a set of regions and let X = {x <sub>1</sub>, x <sub>2</sub>, . . . , x <sub>n</sub>} be an (unknown) point set with x <sub>i</sub> ϵ R <sub>i</sub>. Region Ri represents the uncertainty region of x <sub>i</sub>. We consider the following question: how fast can we establish order if we are allowed to preprocess the regions in R? The preprocessing model of uncertainty uses two consecutive phases: a preprocessing phase which has access only to R followed by a reconstruction phase during which a desired structure on X is computed. Recent results in this model parametrize the reconstruction time by the ply of R, which is the maximum overlap between the regions in R. We introduce the ambiguity A(R) as a more fine-grained measure of the degree of overlap in R. We show how to preprocess a set of d-dimensional disks in O(n log n) time such that we can sort X (if d = 1) and reconstruct a quadtree on X (if d ≥ 1 but constant) in O(A(R)) time. If A(R) is sub-linear, then reporting the result dominates the running time of the reconstruction phase. However, we can still return a suitable data structure representing the result in O(A(R)) time. In one dimension, R is a set of intervals and the ambiguity is linked to interval entropy, which in turn relates to the well-studied problem of sorting under partial information. The number of comparisons necessary to find the linear order underlying a poset P is lower-bounded by the graph entropy of P. We show that if P is an interval order, then the ambiguity provides a constant-factor approximation of the graph entropy. This gives a lower bound of (A(R)) in all dimensions for the reconstruction phase (sorting or any proximity structure), independent of any preprocessing; hence our result is tight. Finally, our results imply that one can approximate the entropy of interval graphs in O(n log n) time, improving the O(n <sup>2.5</sup>) bound by Cardinal et al. </p>
@article{preprocessing-ambiguous-imprecise-points:2019,
title = {Preprocessing Ambiguous Imprecise Points},
SETH Says: Weak Fréchet Distance is Faster, but Only if It is Continuous and in One Dimension
Kevin Buchin, Tim Ophelders, and Bettina Speckmann.
Proc. 30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 2887—2899, 2019.
<p>We show by reduction from the Orthogonal Vectors problem that algorithms with strongly subquadratic running time cannot approximate the Fréchet distance between curves better than a factor 3 unless SETH fails. We show that similar reductions cannot achieve a lower bound with a factor better than 3. Our lower bound holds for the continuous, the discrete, and the weak discrete Fréchet distance even for curves in one dimension. Interestingly, the continuous weak Fréchet distance behaves differently. Our lower bound still holds for curves in two dimensions and higher. However, for curves in one dimension, we provide an exact algorithm to compute the weak Fréchet distance in linear time.</p>
@article{seth-says-weak-fr-chet-distance-is-faster-but-only-if-it-is-continuous-and-in-one-dimension:2019,
title = {SETH Says: Weak Fréchet Distance is Faster, but Only if It is Continuous and in One Dimension},
author = {Kevin Buchin and Tim Ophelders and Bettina Speckmann},
bookTitle = {Proc. 30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)},
Kinetic Volume-Based Persistence for 1D Terrains
Tim Ophelders, Willem Sonke, Bettina Speckmann, and Kevin Verbeek.
Persistence is the method of choice to simplify terrains by removing insignificant features while retaining topologically important ones. Motivated by applications in geomorphology, we study volume-persistence, a variant of persistence which is based on the volume underneath the terrain (instead of the usual vertex heights). Specifically, we want to kinetically maintain a volume-simplified time-varying terrain. In this paper we describe a kinetic data structure (KDS) that maintains the pruned split tree of an area-persistent 1D terrain under linear vertex motion. The main ingredient of this KDS is an algorithm that detects those combinatorial events when a pruned part of the terrain attains a certain threshold volume.
@article{kinetic-volume-based-persistence-for-1d-terrains:2019,
title = {Kinetic Volume-Based Persistence for 1D Terrains},
author = {Tim Ophelders and Willem Sonke and Bettina Speckmann and Kevin Verbeek},
We study the problem of detecting outlying measurements in a GPS trajectory. Our method considers the physical possibility for the tracked object to visit combinations of measurements,using simplified physics models. We aim to compute the maximum subsequence of the measurements that is consistent with a given physics model. We give an O(n log³ n) time algorithm for 2D-trajectories in a model with unbounded acceleration but bounded velocity, and an output-sensitive algorithm for any model where consistency checks can be done in O(1) time and consistency is transitive.
Colored Spanning Graphs for Set Visualization
F. Hurtado, M. Korman, M.J. van Kreveld, M. Löffler, V. Sacristán, A. Shioura, R.I. Silveira, B. Speckmann, and T. Tokuyama.
Computational Geometry, 68:262—276, 2018.
<p>We study an algorithmic problem that is motivated by ink minimization for sparse set visualizations. Our input is a set of points in the plane which are either blue, red, or purple. Blue points belong exclusively to the blue set, red points belong exclusively to the red set, and purple points belong to both sets. A red-blue-purple spanning graph (RBP spanning graph) is a set of edges connecting the points such that the subgraph induced by the red and purple points is connected, and the subgraph induced by the blue and purple points is connected. We study the geometric properties of minimum RBP spanning graphs and the algorithmic problems associated with computing them. Specifically, we show that the general problem can be solved in polynomial time using matroid techniques. In addition, we discuss more efficient algorithms for the case in which points are located on a line or a circle, and also describe a fast ([Formula presented]ρ+1)-approximation algorithm, where ρ is the Steiner ratio.</p>
@article{colored-spanning-graphs-for-set-visualization:2018,
title = {Colored Spanning Graphs for Set Visualization},
author = {F. Hurtado and M. Korman and M.J. van Kreveld and M. Löffler and V. Sacristán and A. Shioura and R.I. Silveira and B. Speckmann and T. Tokuyama},
Computing the Similarity Between Moving Curves
K. Buchin, T. Ophelders, and B. Speckmann.
<p>In this paper we study similarity measures for moving curves which can, for example, model changing coastlines or retreating glacier termini. Points on a moving curve have two parameters, namely the position along the curve as well as time. We therefore focus on similarity measures for surfaces, specifically the Fréchet distance between surfaces. While the Fréchet distance between surfaces is generally NP-hard, we show for variants arising in the context of moving curves that they are polynomial-time solvable or NP-complete depending on the restrictions imposed on how the moving curves are matched. We achieve the polynomial-time solutions by a novel approach for computing a surface in the so-called free-space diagram based on a relation between obstacles.</p>
@article{computing-the-similarity-between-moving-curves:2018,
title = {Computing the Similarity Between Moving Curves},
author = {K. Buchin and T. Ophelders and B. Speckmann},
Homotopic c-Oriented Routing with Few Links and Thick Edges
B. Speckmann and K.A.B. Verbeek.
Computational Geometry, 67:11—28, 2018.
<p>We study the NP-hard optimization problem of finding non-crossing thick C-oriented paths that are homotopic to a set of input paths in an environment with C-oriented obstacles, with the goal to minimize the total number of links of the paths. We introduce a special type of C-oriented paths—smooth paths—and present a 2-approximation algorithm for smooth paths that runs in O(n<sup>3</sup>logκ+k<sub>in</sub>logn+k<sub>out</sub>) time, where n is the total number of paths and obstacle vertices, k<sub>in</sub> and k<sub>out</sub> are the total complexities of the input and output paths, and κ=|C|. The algorithm also computes an O(κ)-approximation for general C-oriented paths. In particular we give a 2-approximation algorithm for rectilinear paths. Our algorithm not only approximates the minimum number of links, but also simultaneously minimizes the total length of the paths. As a related result we show that, given a set of (possibly crossing) C-oriented paths with a total of L links, non-crossing C-oriented paths homotopic to the input paths can require a total of Ω(Llogκ) links.</p>
@article{homotopic-c-oriented-routing-with-few-links-and-thick-edges:2018,
title = {Homotopic c-Oriented Routing with Few Links and Thick Edges},
author = {B. Speckmann and K.A.B. Verbeek},
Simultaneous Visualization of Language Endangerment and Language Description
Harald Hammarström, T.H.A. Castermans, Robert Forkel, Kevin Verbeek, Michel A. Westenberg, and Bettina Speckmann.
Language Documentation & Conservation, 12:359—392, 2018.
The world harbors a diversity of some 6,500 mutually unintelligible languages. As has been increasingly observed by linguists, many minority languages are becoming endangered and will be lost forever if not documented. Urgently indeed, many efforts are being launched to document and describe languages. This undertaking naturally has the priority toward the most endangered and least described languages. For the first time, we combine world-wide databases on language description (Glottolog) and language endangerment (ElCat, Ethnologue, UNESCO) and provide two online interfaces, GlottoScope and GlottoVis, to visualize these together. The interfaces are capable of browsing, filtering, zooming, basic statistics, and different ways of combining the two measures on a world map background. GlottoVis provides advanced techniques for combining cluttered dots on a map. With the tools and databases described we seek to increase the overall knowledge of the actual state language endangerment and description worldwide.
@article{simultaneous-visualization-of-language-endangerment-and-language-description:2018,
title = {Simultaneous Visualization of Language Endangerment and Language Description},
author = {Harald Hammarström and T.H.A. Castermans and Robert Forkel and Kevin Verbeek and Michel A. Westenberg and Bettina Speckmann},
bookTitle = {Language Documentation & Conservation},
Stable Treemaps Via Local Moves
M. Sondag, B. Speckmann, and K.A.B. Verbeek.
IEEE Transactions on Visualization and Computer Graphics, 24(1):729—738, 2018.
<p>Treemaps are a popular tool to visualize hierarchical data: items are represented by nested rectangles and the area of each rectangle corresponds to the data being visualized for this item. The visual quality of a treemap is commonly measured via the aspect ratio of the rectangles. If the data changes, then a second important quality criterion is the stability of the treemap: how much does the treemap change as the data changes. We present a novel stable treemapping algorithm that has very high visual quality. Whereas existing treemapping algorithms generally recompute the treemap every time the input changes, our algorithm changes the layout of the treemap using only local modifications. This approach not only gives us direct control over stability, but it also allows us to use a larger set of possible layouts, thus provably resulting in treemaps of higher visual quality compared to existing algorithms. We further prove that we can reach all possible treemap layouts using only our local modifications. Furthermore, we introduce a new measure for stability that better captures the relative positions of rectangles. We finally show via experiments on real-world data that our algorithm outperforms existing treemapping algorithms also in practice on either visual quality and/or stability. Our algorithm scores high on stability regardless of whether we use an existing stability measure or our new measure.</p>
@article{stable-treemaps-via-local-moves:2018,
title = {Stable Treemaps Via Local Moves},
author = {M. Sondag and B. Speckmann and K.A.B. Verbeek},
A Framework for Algorithm Stability and Its Application to Kinetic Euclidean MSTs
Wouter Meulemans, Bettina Speckmann, Kevin Verbeek, and Jules Wulms.
LATIN 2018: Theoretical Informatics, pp. 805—819, 2018.
<p>We say that an algorithm is stable if small changes in the input result in small changes in the output. This kind of algorithm stability is particularly relevant when analyzing and visualizing time-varying data. Stability in general plays an important role in a wide variety of areas, such as numerical analysis, machine learning, and topology, but is poorly understood in the context of (combinatorial) algorithms. In this paper we present a framework for analyzing the stability of algorithms. We focus in particular on the tradeoff between the stability of an algorithm and the quality of the solution it computes. Our framework allows for three types of stability analysis with increasing degrees of complexity: event stability, topological stability, and Lipschitz stability. We demonstrate the use of our stability framework by applying it to kinetic Euclidean minimum spanning trees.</p>
@article{a-framework-for-algorithm-stability-and-its-application-to-kinetic-euclidean-msts:2018,
title = {A Framework for Algorithm Stability and Its Application to Kinetic Euclidean MSTs},
author = {Wouter Meulemans and Bettina Speckmann and Kevin Verbeek and Jules Wulms},
bookTitle = {LATIN 2018: Theoretical Informatics},
A Philosophical Perspective on Visualization for Digital Humanities
Hein van den Berg, Arianna Betti, Thom Castermans, Rob Koopman, Bettina Speckmann, Kevin Verbeek, Titia van der Werf, Shenghui Wang, and Michel A. Westenberg.
Proc. 3rd Workshop on Visualization for the Digital Humanities (VIS4DH), 2018.
In this position paper, we describe a number of methodological and philosophical challenges that arose within our interdisciplinary Digital Humanities project \catvis, which is a collaboration between applied geometric algorithms and visualization researchers, data scientists working at OCLC, and philosophers who have a strong interest in the methodological foundations of visualization research. The challenges we describe concern aspects of one single epistemic need: that of methodologically securing (an increase in) trust in visualizations. We discuss the lack of ground truths in the (digital) humanities and argue that trust in visualizations requires that we evaluate visualizations on the basis of ground truths that humanities scholars themselves create. We further argue that trust in visualizations requires that a visualization provides provable guarantees on the faithfulness of the visual representation and that we must clearly communicate to the users which part of the visualization can be trusted and how much. Finally, we discuss transparency and accessibility in visualization research and provide measures for securing transparency and accessibility.
@article{a-philosophical-perspective-on-visualization-for-digital-humanities:2018,
title = {A Philosophical Perspective on Visualization for Digital Humanities},
author = {Hein van den Berg and Arianna Betti and Thom Castermans and Rob Koopman and Bettina Speckmann and Kevin Verbeek and Titia van der Werf and Shenghui Wang and Michel A. Westenberg},
bookTitle = {Proc. 3rd Workshop on Visualization for the Digital Humanities (VIS4DH)},
13th Latin American Theoretical INformatics Symposium (LATIN), pp. 260—274, 2018.
<p>We study an agglomerative clustering problem motivated by interactive glyphs in geo-visualization. Consider a set of disjoint square glyphs on an interactive map. When the user zooms out, the glyphs grow in size relative to the map, possibly with different speeds. When two glyphs intersect, we wish to replace them by a new glyph that captures the information of the intersecting glyphs. We present a fully dynamic kinetic data structure that maintains a set of n disjoint growing squares. Our data structure uses O(n(log n log log n)<sup>2</sup>) space, supports queries in worst case O(log<sup>3</sup> n) time, and updates in O(log<sup>7 </sup>n) amortized time. This leads to an O(n α(n) log<sup>7 </sup>n) time algorithm (where α is the inverse Ackermann function) to solve the agglomerative clustering problem, which is a significant improvement over the straightforward O(n<sup>2</sup> log n) time algorithm.</p>
bookTitle = {13th Latin American Theoretical INformatics Symposium (LATIN)},
Bettina Speckmann, Csaba D. Tóth, and Xavier Goaoc.
34th International Symposium on Computational Geometry, SoCG 2018; Budapest; Hungary; 11 June 2018 through 14 June 2018, 99:xi, 2018.
@article{foreword:2018,
title = {Foreword},
author = {Bettina Speckmann and Csaba D. Tóth and Xavier Goaoc},
bookTitle = {34th International Symposium on Computational Geometry, SoCG 2018; Budapest; Hungary; 11 June 2018 through 14 June 2018},
R.I. Silveira, B. Speckmann, and K.A.B. Verbeek.
Graph drawing and network visualization - 25th International Symposium, GD 2017, Revised Selected Papers, pp. 454—461, 2018.
<p>A geographic network is a graph whose vertices are restricted to lie in a prescribed region in the plane. In this paper we begin to study the following fundamental problem for geographic networks: can a given geographic network be drawn without crossings? We focus on the seemingly simple setting where each region is a unit length vertical segment, and one wants to connect pairs of segments with a path that lies inside the convex hull of the two segments. We prove that when paths must be drawn as straight line segments, it is NP-complete to determine if a crossing-free solution exists. In contrast, we show that when paths must be monotone curves, the question can be answered in polynomial time. In the more general case of paths that can have any shape, we show that the problem is polynomial under certain assumptions.</p>
author = {R.I. Silveira and B. Speckmann and K.A.B. Verbeek},
bookTitle = {Graph drawing and network visualization - 25th International Symposium, GD 2017, Revised Selected Papers},
Optimal Algorithms for Compact Linear Layouts
Willem Sonke, Kevin Verbeek, Wouter Meulemans, Eric Verbeek, and Bettina Speckmann.
2018 IEEE Pacific Visualization Symposium, PacificVis 2018, pp. 1—10, 2018.
<p>Linear layouts are a simple and natural way to draw a graph: all vertices are placed on a single line and edges are drawn as arcs between the vertices. Despite its simplicity, a linear layout can be a very meaningful visualization if there is a particular order defined on the vertices. Common examples of such ordered - and often also directed - graphs are event sequences and processes. A main drawback of linear layouts are the usually (very) large aspect ratios of the resulting drawings, which prevent users from obtaining a good overview of the whole graph. In this paper we present a novel and versatile algorithm to optimally fold a linear layout of a graph such that it can be drawn nicely in a specified aspect ratio, while still clearly communicating the linearity of the layout. Our algorithm allows vertices to be drawn as blocks or rectangles of specified sizes to incorporate different drawing styles, label sizes, and even recursive structures. For reasonably-sized drawings the folded layout can be computed interactively. We demonstrate the applicability of our algorithm on graphs that represent process trees, a particular type of process model. Our algorithm arguably produces much more readable layouts than existing methods.</p>
@article{optimal-algorithms-for-compact-linear-layouts:2018,
title = {Optimal Algorithms for Compact Linear Layouts},
author = {Willem Sonke and Kevin Verbeek and Wouter Meulemans and Eric Verbeek and Bettina Speckmann},
bookTitle = {2018 IEEE Pacific Visualization Symposium, PacificVis 2018},
Social Network-Epistemology
M. Alfano, S. Cunningham, W. Meulemans, I. Rutter, M. Sondag, B. Speckmann, and E. Sullivan.
Proceedings - IEEE 14th International Conference on eScience, e-Science 2018, pp. 320—321, 2018.
@article{social-network-epistemology:2018,
title = {Social Network-Epistemology},
author = {M. Alfano and S. Cunningham and W. Meulemans and I. Rutter and M. Sondag and B. Speckmann and E. Sullivan},
bookTitle = {Proceedings - IEEE 14th International Conference on eScience, e-Science 2018},
Volume-Based Similarity of Linear Features on Terrains
Willem Sonke, Marc van Kreveld, Tim Ophelders, Bettina Speckmann, and Kevin Verbeek.
26th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems, ACM SIGSPATIAL GIS 2018, pp. 444—447, 2018.
<p>Linear features on terrains model the boundaries of ground cover regions, delineate glaciers, or form the boundary of rivers and lakes. When computing the similarity between such linear features, it is important to also take their context into account: the terrain. We hence explore the possibilities of volume-based distance measures for linear features on a terrain. Our measures construct suitable base surfaces between the linear features, which can slice through the input terrain and also hover above. The similarity between two linear features is then captured by the volume of "earth" above the base surface and below the terrain, and possibly also by the volume of "air" below the base surface and above the terrain. We suggest six ways of choosing a suitable base surface. These choices give rise to different measured volumes and can be useful in different application scenarios.</p>
@article{volume-based-similarity-of-linear-features-on-terrains:2018,
title = {Volume-Based Similarity of Linear Features on Terrains},
author = {Willem Sonke and Marc van Kreveld and Tim Ophelders and Bettina Speckmann and Kevin Verbeek},
bookTitle = {26th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems, ACM SIGSPATIAL GIS 2018},
W. Meulemans, B. Speckmann, K.A.B. Verbeek, and J.J.H.M. Wulms.
We say that an algorithm is stable if small changes in the input result in small changes in the output. This kind of algorithm stability is particularly relevant when analyzing and visualizing time-varying data. Stability in general plays an important role in a wide variety of areas, such as numerical analysis, machine learning, and topology, but is poorly understood in the context of (combinatorial) algorithms. <br/><br/>In this paper we present a framework for analyzing the stability of algorithms. We focus in particular on the tradeoff between the stability of an algorithm and the quality of the solution it computes. <br/>Our framework allows for three types of stability analysis with increasing degrees of complexity: event stability, topological stability, and Lipschitz stability. <br/>We demonstrate the use of topological stability by applying it to kinetic Euclidean minimum spanning trees.
author = {W. Meulemans and B. Speckmann and K.A.B. Verbeek and J.J.H.M. Wulms},
A KDS for Discrete Morse-Smale Complexes
pp. 3:1—3:2, 2018.
The Morse-Smale complex of a terrain is a topological complex that provides information about the features of the terrain. It consists of the critical points (minima, saddles and maxima), together with steepest-descent paths from saddles to minima and steepest-ascent paths from saddles to maxima. We describe a kinetic data structure to maintain the Morse-Smale-complex for a triangulated terrain whose vertex heights change continuously. This can be used to efficiently analyze time-varying data.
@article{a-kds-for-discrete-morse-smale-complexes:2018,
title = {A KDS for Discrete Morse-Smale Complexes},
Geometry and Topology of Estuary and Braided River Channel Networks Extracted from Topographic Data
Matthew Hiatt, Willem Sonke, Elisabeth Addink, Wout van Dijk, Marc J. van Kreveld, Tim Ophelders, Kevin Verbeek, Joyce Vlaming, Bettina Speckmann, and Maarten G. Kleinhans.
Channels are ubiquitous features of Earth's surface that are important pathways for the transport of water, solids, and solutes across landscapes, provide a range of ecosystem services, and support economic activity. Networks are mathematical representations of the connections among a set of objects and are useful representations of topology, geometry, and connectivity in channelized environments. However, objective and automatic extraction of channel networks from topography in multi-channel systems like braided river and estuaries has remained elusive. We present a mathematically-rigorous framework from extracting network topology and geometry from digital elevation models (DEMs) of braided rivers and estuaries. The concept of the "sand function" is introduced, which quantifies the volume of material separating channels and is a useful metric for identifying the relative scales of channels in the network. Four case studies are included: DEMs from the Western Scheldt estuary (Netherlands) and the Waimakariri River (New Zealand), as well as DEMs generated by numerical models of the morphodynamics in a braided river and an estuary. We show that larger scale channels (with higher sand function values) in the estuaries tend to be significantly deeper than smaller scale channels in the network. The results suggest that the main channel in an estuary is significantly deeper than the rest of the network, while the braided rivers tend to have channel depths that are evenly-distributed across channel scales. In all cases, the length of channels relative to system size scales with sand function scale to the power of 0.24-0.35, while the number of nodes against system scale does not exhibit a power-law relationship. The methods and results presented in this study provide a benchmark for evaluating both geometric and topologic characteristics of multi-threaded channel networks across scales.
@article{geometry-and-topology-of-estuary-and-braided-river-channel-networks-extracted-from-topographic-data:2018,
title = {Geometry and Topology of Estuary and Braided River Channel Networks Extracted from Topographic Data},
author = {Matthew Hiatt and Willem Sonke and Elisabeth Addink and Wout van Dijk and Marc J. van Kreveld and Tim Ophelders and Kevin Verbeek and Joyce Vlaming and Bettina Speckmann and Maarten G. Kleinhans},
Wouter Meulemans, Willem Sonke, Bettina Speckmann, Eric Verbeek, and Kevin Verbeek.
Linear layouts are a simple and natural way to draw a graph: all vertices are placed on a single line and edges are drawn as arcs between the vertices. Despite its simplicity, a linear layout can be a very meaningful visualization if there is a particular order defined on the vertices. Common examples of such ordered - and often also directed - graphs are event sequences and processes: public transport systems tracking passenger check-in and check-out, banks checking online transactions, or hospitals recording the paths of patients through their system, to name a few. A main drawback of linear layouts are the usually (very) large aspect ratios of the resulting drawings, which prevent users from obtaining a good overview of the whole graph. In this paper we present a novel and versatile algorithm to optimally fold a linear layout of a graph such that it can be drawn effectively in a specified aspect ratio, while still clearly communicating the linearity of the layout.
author = {Wouter Meulemans and Willem Sonke and Bettina Speckmann and Eric Verbeek and Kevin Verbeek},
The Effects of Dredging and Disposal Activity on the Resilience of Estuary Morphodynamics
Wout van Dijk, Jasper Leuven, Jana Cox, Jelmer Cleveringa, Marcel Taal, Matthew Hiatt, Willem Sonke, Kevin Verbeek, Bettina Speckmann, and Maarten G. Kleinhans.
Shipping fairways in estuaries are continuously dredged to maintain access to major ports for large ships. However, various estuaries worldwide show adverse side effects to dredging activities, including shifts from a multi-channel system to a single channel, loss of ecologically-valuable intertidal areas, and increased muddiness. The Western Scheldt (Netherlands) is an example of a system where several million m3 sediments are dredged annually and disposed of within the estuary. Here, the effects of dredging and disposal strategies on the channel-shoal morphodynamics, including the evolution of the multi-channel system and its ecological impact are studied using field observations, numerical and physical models. Time series of bathymetry, morphodynamic model runs and physical experiments in the Metronome tilting flume show for the first time that the deliberate dredging and disposal strategy increases the surface area of inter-tidal habitats, which decreases the connectivity between channels and destabilizes the multi-channel system. We quantify these changes in scale and topology using a novel and mathematically-rigorous network extraction, which identifies the partial switching of the main channel with the side channel in the past and that continuous disposal of sediment in the side channel results in siltation. The physical experiments indicate that the estuarine network is dramatically changed by dredging even for a long period following the halting of any dredging and dumping works. The morphodynamic model similarly shows that dredging of the sills is essential to maintain the current shipping fairway. A new disposal strategy targeting the deepest scours of the main channel appears to be the optimal solution for ecological value and sustains dynamic channel-shoal interactions and the multi-channel pattern. Sea level rise scenarios demonstrate the importance of keeping the sediment in the system for sustainable management, but present dredging and disposal techniques put current ecosystems under pressure.
@article{the-effects-of-dredging-and-disposal-activity-on-the-resilience-of-estuary-morphodynamics:2018,
title = {The Effects of Dredging and Disposal Activity on the Resilience of Estuary Morphodynamics},
author = {Wout van Dijk and Jasper Leuven and Jana Cox and Jelmer Cleveringa and Marcel Taal and Matthew Hiatt and Willem Sonke and Kevin Verbeek and Bettina Speckmann and Maarten G. Kleinhans},
Magnús M. Halldórsson, Naoki Kobayashi, and Bettina Speckmann.
Information and Computation, 261(2):159, 2018.
@article{preface:2018,
title = {Preface},
author = {Magnús M. Halldórsson and Naoki Kobayashi and Bettina Speckmann},
bookTitle = {Information and Computation},
A Framework for Algorithm Stability
W. Meulemans, B. Speckmann, K.A.B. Verbeek, and J. Wulms.
We say that an algorithm is stable if small changes in the input result in small changes in the output. Algorithm stability plays an important role when analyzing and visualizing time-varying data. However, so far, there are only few theoretical results on the stability of algorithms, possibly due to a lack of theoretical analysis tools. In this paper we present a framework for analyzing the stability of algorithms. We focus in particular on the tradeoff between the stability of an algorithm and the quality of the solution it computes. Our framework allows for three types of stability analysis with increasing degrees of complexity: event stability, topological stability, and Lipschitz stability. We demonstrate the use of our stability framework by applying it to kinetic Euclidean minimum spanning trees.
@article{a-framework-for-algorithm-stability:2017,
title = {A Framework for Algorithm Stability},
author = {W. Meulemans and B. Speckmann and K.A.B. Verbeek and J. Wulms},
Column Planarity and Partially-Simultaneous Geometric Embedding
L. Barba, W.S. Evans, M.H.W. Hoffmann, V. Kusters, M. Saumell, and B. Speckmann.
Journal of Graph Algorithms and Applications, 21(6):983—1002, 2017.
<p>We introduce the notion of column planarity of a subset R of the vertices of a graph G. Informally, we say that R is column planar in G if we can assign x-coordinates to the vertices in R such that any assignment of y-coordinates to them produces a partial embedding that can be completed to a plane straight-line drawing of G. Column planarity is both a relaxation and a strengthening of unlabeled level planarity. We prove near tight bounds for the maximum size of column planar subsets of trees: every tree on n vertices contains a column planar set of size at least 14n/17 and for any ɛ > 0 and any sufficiently large n, there exists an n-vertex tree in which every column planar subset has size at most (5/6 + ɛ)n. In addition, we show that every outerplanar graph has a column planar set of size at least n/2. We also consider a relaxation of simultaneous geometric embedding (SGE), which we call partially-simultaneous geometric embedding (PSGE). A PSGE of two graphs G<sub>1</sub> and G<sub>2</sub> allows some of their vertices to map to two different points in the plane. We show how to use column planar subsets to construct k-PSGEs, which are PSGEs in which at least k vertices are mapped to the same point for both graphs. In particular, we show that every two trees on n vertices admit an 11n/17-PSGE and every two outerplanar graphs admit an n/4-PSGE.</p>
@article{column-planarity-and-partially-simultaneous-geometric-embedding:2017,
title = {Column Planarity and Partially-Simultaneous Geometric Embedding},
author = {L. Barba and W.S. Evans and M.H.W. Hoffmann and V. Kusters and M. Saumell and B. Speckmann},
bookTitle = {Journal of Graph Algorithms and Applications},
Multi-Granular Trend Detection for Time-Series Analysis
A.I. van Goethem, F. Staals, M. Löffler, J. Dykes, and B. Speckmann.
Time series (such as stock prices) and ensembles (such as model runs for weather forecasts) are two important types of one-dimensional time-varying data. Such data is readily available in large quantities but visual analysis of the raw data quickly becomes infeasible, even for moderately sized data sets. Trend detection is an effective way to simplify time-varying data and to summarize salient information for visual display and interactive analysis. We propose a geometric model for trend-detection in one-dimensional time-varying data, inspired by topological grouping structures for moving objects in two- or higher-dimensional space. Our model gives provable guarantees on the trends detected and uses three natural parameters: granularity, support-size, and duration. These parameters can be changed on-demand. Our system also supports a variety of selection brushes and a time-sweep to facilitate refined searches and interactive visualization of (sub-)trends. We explore different visual styles and interactions through which trends, their persistence, and evolution can be explored.
@article{multi-granular-trend-detection-for-time-series-analysis:2017,
title = {Multi-Granular Trend Detection for Time-Series Analysis},
author = {A.I. van Goethem and F. Staals and M. Löffler and J. Dykes and B. Speckmann},
Packing Plane Spanning Trees and Paths in Complete Geometric Graphs
O. Aichholzer, T. Hackl, M. Korman, M.J. van Kreveld, M. Löffler, A. Pilz, B. Speckmann, and E. Welzl.
Information Processing Letters, 124:35—41, 2017.
<p>We consider the following question: How many edge-disjoint plane spanning trees are contained in a complete geometric graph GK<sub>n</sub> on any set S of n points in general position in the plane? We show that this number is in Ω(n). Further, we consider variants of this problem by bounding the diameter and the degree of the trees (in particular considering spanning paths).</p>
@article{packing-plane-spanning-trees-and-paths-in-complete-geometric-graphs:2017,
title = {Packing Plane Spanning Trees and Paths in Complete Geometric Graphs},
author = {O. Aichholzer and T. Hackl and M. Korman and M.J. van Kreveld and M. Löffler and A. Pilz and B. Speckmann and E. Welzl},
bookTitle = {Information Processing Letters},
Complexity Measures for Mosaic Drawings
Q.W. Bouts, B. Speckmann, and K.A.B. Verbeek.
WALCOM: Algorithms and Computation - 11th International Conference and Workshops, WALCOM 2017, Proceedings, pp. 149—160, 2017.
<p>Graph Drawing uses a well established set of complexity measures to determine the quality of a drawing, most notably the area of the drawing and the complexity of the edges. For contact representations the complexity of the shapes representing vertices also clearly contributes to the complexity of the drawing. Furthermore, if a contact representation does not fill its bounding shape completely, then also the complexity of its complement is visually salient. We study the complexity of contact representations with variable shapes, specifically mosaic drawings.Mosaic drawings are drawn on a tiling of the plane and represent vertices by configurations: simply-connected sets of tiles. The complement of a mosaic drawing with respect to its bounding rectangle is also a set of simply-connected tiles, the channels. We prove that simple mosaic drawings without channels may require Ω(n<sup>2</sup>) area. This bound is tight. If we use only straight channels, then outerplanar graphs with k ears may require Ω(min(nk, n<sup>2</sup>/k)) area. This bound is partially tight: we show how to draw outerplanar graphs with k ears in O(nk) area with L-shaped vertex configurations and straight channels. Finally, we argue that L-shaped channels are strictly more powerful than straight channels, but may still require Ω(n<sup>7/6</sup>) area.</p>
@article{complexity-measures-for-mosaic-drawings:2017,
title = {Complexity Measures for Mosaic Drawings},
author = {Q.W. Bouts and B. Speckmann and K.A.B. Verbeek},
bookTitle = {WALCOM: Algorithms and Computation - 11th International Conference and Workshops, WALCOM 2017, Proceedings},
Computing Optimal Homotopies over a Spiked Plane with Polygonal Boundary
B. Burton, E.W. Chambers, M.J. Van Kreveld, W. Meulemans, T.A.E. Ophelders, and B. Speckmann.
Proc. 25th European Symposium on Algorithms (ESA), pp. 1—14, 2017.
<p>Computing optimal deformations between two curves is a fundamental question with various applications, and has recently received much attention in both computational topology and in mathematics in the form of homotopies of disks and annular regions. In this paper, we examine this problem in a geometric setting, where we consider the boundary of a polygonal domain with spikes, point obstacles that can be crossed at an additive cost. We aim to continuously morph from one part of the boundary to another, necessarily passing over all spikes, such that the most expensive intermediate curve is minimized, where the cost of a curve is its geometric length plus the cost of any spikes it crosses. We first investigate the general setting where each spike may have a different cost. For the number of inflection points in an intermediate curve, we present a lower bound that is linear in the number of spikes, even if the domain is convex and the two boundaries for which we seek a morph share an endpoint. We describe a 2-Approximation algorithm for the general case, and an optimal algorithm for the case that the two boundaries for which we seek a morph share both endpoints, thereby representing the entire boundary of the domain. We then consider the setting where all spikes have the same unit cost and we describe a polynomial-Time exact algorithm. The algorithm combines structural properties of homotopies arising from the geometry with methodology for computing Fréchet distances.</p>
@article{computing-optimal-homotopies-over-a-spiked-plane-with-polygonal-boundary:2017,
title = {Computing Optimal Homotopies over a Spiked Plane with Polygonal Boundary},
author = {B. Burton and E.W. Chambers and M.J. Van Kreveld and W. Meulemans and T.A.E. Ophelders and B. Speckmann},
bookTitle = {Proc. 25th European Symposium on Algorithms (ESA)},
M. Kleinhans, M.J. van Kreveld, T.A.E. Ophelders, W.M. Sonke, B. Speckmann, and K.A.B. Verbeek.
33rd International Symposium on Computational Geometry, SoCG 2017, pp. 1—16, 2017.
Drainage networks on terrains have been studied extensively from an algorithmic perspective. However, in drainage networks water flow cannot bifurcate and hence they do not model braided rivers (multiple channels which split and join, separated by sediment bars). We initiate the algorithmic study of braided rivers by employing the descending quasi Morse-Smale complex on the river bed (a polyhedral terrain), and extending it with a certain ordering of bars from the one river bank to the other. This allows us to compute a graph that models a representative channel network, consisting of lowest paths. To ensure that channels in this network are sufficiently different we define a sand function that represents the volume of sediment separating them. We show that in general the problem of computing a maximum network of non-crossing channels which are δ-different from each other (as measured by the sand function) is NP-hard. However, using our ordering between the river banks, we can compute a maximum δ-different network that respects this order in polynomial time. We implemented our approach and applied it to simulated and real-world braided rivers.
author = {M. Kleinhans and M.J. van Kreveld and T.A.E. Ophelders and W.M. Sonke and B. Speckmann and K.A.B. Verbeek},
bookTitle = {33rd International Symposium on Computational Geometry, SoCG 2017},
Computing the Fréchet Distance Between Real-Valued Surfaces
Proc. 28th Annual Symposium on Discrete Algorithms (SODA), pp. 2443—2455, 2017.
<p>The Fréchet distance is a well-studied measure for the similarity of shapes. While efficient algorithms for computing the Fréchet distance between curves exist, there are only few results on the Fréchet distance between surfaces. Recent work has shown that the Fréchet distance is computable between piecewise linear functions f and g : M → Rk with M a triangulated surface of genus zero. We focus on the case k = 1 and M being a topological sphere or disk with constant boundary. Intuitively, we measure the distance between terrains based solely on the height function. Our main result is that in this case computing the Fréchet distance between f and g is in NP. We additionally show that already for k = 1, computing a factor 2 - ϵ approximation of the Fréchet distance is NP-hard, showing that this problem is in fact NP-complete. We also define an intermediate distance, between contour trees, which we also show to be NP-complete to compute. Finally, we discuss how our and other distance measures between contour trees relate to each other.</p>
@article{computing-the-fr-chet-distance-between-real-valued-surfaces:2017,
title = {Computing the Fréchet Distance Between Real-Valued Surfaces},
bookTitle = {Proc. 28th Annual Symposium on Discrete Algorithms (SODA)},
GlottoVis: Visualizing Language Endangerment and Documentation
T.H.A. Castermans, B. Speckmann, K.A.B. Verbeek, M.A. Westenberg, and H. Hammarström.
Proc. 2nd Workshop on Visualization for the Digital Humanities (VIS4DH), pp. 1—5, 2017.
We present GlottoVis, a system designed to visualize language endangerment as collected by UNESCO and descriptive status as collected by the Glottolog project. Glottolog records bibliographic data for the world's (lesser known) languages. Languages are documented with increasing detail, but the number of native speakers of minority languages dwindles as their population shifts to other more dominant languages. Hence one needs to visualize documentation level and endangerment status at the same time, to browse for the most urgent cases (little documentation and high endangerment) and to direct funding aimed at describing endangered languages. GlottoVis visualizes these two properties of languages and provides an interface to search and filter. Our tool is web-based and is comprised of glyphs on a zoomable geographic map. Clustering and (visual) data aggregation are performed at each zoom level to avoid overlap of glyphs. This reduces clutter and improves readability of the visualization. Preliminary tests with expert users confirm that our tool supports their desired workflow well.
@article{glottovis-visualizing-language-endangerment-and-documentation:2017,
title = {GlottoVis: Visualizing Language Endangerment and Documentation},
author = {T.H.A. Castermans and B. Speckmann and K.A.B. Verbeek and M.A. Westenberg and H. Hammarström},
bookTitle = {Proc. 2nd Workshop on Visualization for the Digital Humanities (VIS4DH)},
Non-Crossing Geometric Steiner Arborescences
I. Kostitsyna, B. Speckmann, and K.A.B. Verbeek.
28th International Symposium on Algorithms and Computation, ISAAC 2017, pp. 1—13, 2017.
Motivated by the question of simultaneous embedding of several flow maps, we consider the problem of drawing multiple geometric Steiner arborescences with no crossings in the rectilinear and in the angle-restricted setting. When terminal-to-root paths are allowed to turn freely, we show that two rectilinear Steiner arborescences have a non-crossing drawing if neither tree necessarily completely disconnects the other tree and if the roots of both trees are "free". If the roots are not free, then we can reduce the decision problem to 2SAT. If terminal-to-root paths are allowed to turn only at Steiner points, then it is NP-hard to decide whether multiple rectilinear Steiner arborescences have a non-crossing drawing. The setting of angle-restricted Steiner arborescences is more subtle than the rectilinear case. Our NP-hardness result extends, but testing whether there exists a non-crossing drawing if the roots of both trees are free requires additional conditions to be fulfilled.
@article{non-crossing-geometric-steiner-arborescences:2017,
title = {Non-Crossing Geometric Steiner Arborescences},
author = {I. Kostitsyna and B. Speckmann and K.A.B. Verbeek},
bookTitle = {28th International Symposium on Algorithms and Computation, ISAAC 2017},
pp. 45—48, 2017.
Fréchet Isotopies to Monotone Curves
K.A. Buchin, E.W. Chambers, T.A.E. Ophelders, and B. Speckmann.
@article{fr-chet-isotopies-to-monotone-curves:2017,
title = {Fréchet Isotopies to Monotone Curves},
author = {K.A. Buchin and E.W. Chambers and T.A.E. Ophelders and B. Speckmann},
Non-Crossing Drawings of Multiple Geometric Steiner Arborescences
@article{non-crossing-drawings-of-multiple-geometric-steiner-arborescences:2017,
title = {Non-Crossing Drawings of Multiple Geometric Steiner Arborescences},
Area-Preserving Simplification and Schematization of Polygonal Subdivisions
Kevin Buchin, Wouter Meulemans, André Van Renssen, and Bettina Speckmann.
ACM Transactions on Spatial Algorithms and Systems , 2(1):1—36, 2016.
<p>In this article, we study automated simplification and schematization of territorial outlines. We present a quadratic-time simplification algorithm based on an operation called edge-move. We prove that the number of edges of any nonconvex simple polygon can be reduced with this operation. Moreover, edge-moves preserve area and topology and do not introduce new orientations. The latter property in particular makes the algorithm highly suitable for schematization in which all resulting lines are required to be parallel to one of a given set of lines (orientations). To obtain such a result, we need only to preprocess the input to use only lines that are parallel to one of the given set. We present an algorithm to enforce such orientation restrictions, again without changing area or topology. Experiments show that our algorithms obtain results of high visual quality.</p>
@article{area-preserving-simplification-and-schematization-of-polygonal-subdivisions:2016,
title = {Area-Preserving Simplification and Schematization of Polygonal Subdivisions},
author = {Kevin Buchin and Wouter Meulemans and André Van Renssen and Bettina Speckmann},
Convex-Arc Drawings of Pseudolines
D. Eppstein, M. van Garderen, B. Speckmann, and T. Ueckerdt.
A weak pseudoline arrangement is a topological generalization of a line arrangement, consisting of curves topologically equivalent to lines that cross each other at most once. We consider arrangements that are outerplanar---each crossing is incident to an unbounded face---and simple---each crossing point is the crossing of only two curves. We show that these arrangements can be represented by chords of a circle, by convex polygonal chains with only two bends, or by hyperbolic lines. Simple but non-outerplanar arrangements (non-weak) can be represented by convex polygonal chains or convex smooth curves of linear complexity.
@article{convex-arc-drawings-of-pseudolines:2016,
title = {Convex-Arc Drawings of Pseudolines},
author = {D. Eppstein and M. van Garderen and B. Speckmann and T. Ueckerdt},
Distance-Sensitive Planar Point Location
B. Aronov, M.T. de Berg, D. Eppstein, M.J.M. Roeloffzen, and B. Speckmann.
Let S be a connected planar polygonal subdivision with n edges that we want to preprocess for point-location queries, and where we are given the probability γ i that the query point lies in a polygon P i of S . We show how to preprocess S such that the query time for a point~p∈P i depends on~γ i and, in addition, on the distance from p to the boundary of~P i ---the further away from the boundary, the faster the query. More precisely, we show that a point-location query can be answered in time O(min(logn,1+logarea(P i )γ i Δ 2 p )) , where Δ p is the shortest Euclidean distance of the query point~p to the boundary of P i . Our structure uses O(n) space and O(nlogn) preprocessing time. It is based on a decomposition of the regions of S into convex quadrilaterals and triangles with the following property: for any point p∈P i , the quadrilateral or triangle containing~p has area Ω(Δ 2 p ) . For the special case where S is a subdivision of the unit square and γ i =area(P i ) , we present a simpler solution that achieves a query time of O(min(logn,log1Δ 2 p )) . The latter solution can be extended to convex subdivisions in three dimensions.
@article{distance-sensitive-planar-point-location:2016,
title = {Distance-Sensitive Planar Point Location},
author = {B. Aronov and M.T. de Berg and D. Eppstein and M.J.M. Roeloffzen and B. Speckmann},
Let SS be a connected planar polygonal subdivision with n edges that we want to preprocess for point-location queries, and where we are given the probability γiγi that the query point lies in a polygon PiPi of SS. We show how to preprocess SS such that the query time for a point p∈Pip∈Pi depends on γiγi and, in addition, on the distance from p to the boundary of PiPi—the further away from the boundary, the faster the query. More precisely, we show that a point-location query can be answered in time View the MathML sourceO(min(logn,1+logarea(Pi)γiΔp2)), where ΔpΔp is the shortest Euclidean distance of the query point p to the boundary of PiPi. Our structure uses O(n)O(n) space and O(nlogn)O(nlogn) preprocessing time. It is based on a decomposition of the regions of SS into convex quadrilaterals and triangles with the following property: for any point p∈Pip∈Pi, the quadrilateral or triangle containing p has area View the MathML sourceΩ(Δp2). For the special case where SS is a subdivision of the unit square and γi=area(Pi)γi=area(Pi), we present a simpler solution that achieves a query time of View the MathML sourceO(min(logn,log1Δp2)). The latter solution can be extended to convex subdivisions in three dimensions.
Strict Confluent Drawing
D. Eppstein, D. Holten, M. Löffler, M. Nöllenburg, B. Speckmann, and K.A.B. Verbeek.
Journal of Computational Geometry, 7(1):22—46, 2016.
We define strict confluent drawing, a form of confluent drawing in which the existence of an edge is indicated by the presence of a smooth path through a system of arcs and junctions (without crossings), and in which such a path, if it exists, must be unique. We prove that it is NP-complete to determine whether a given graph has a strict confluent drawing but polynomial to determine whether it has an outerplanar strict confluent drawing with a fixed vertex ordering (a drawing within a disk, with the vertices placed in a given order on the boundary).<br/>
@article{strict-confluent-drawing:2016,
title = {Strict Confluent Drawing},
author = {D. Eppstein and D. Holten and M. Löffler and M. Nöllenburg and B. Speckmann and K.A.B. Verbeek},
Visual Encoding of Dissimilarity Data Via Topology-Preserving Map Deformation
Q.W. Bouts, T. Dwyer, J. Dykes, B. Speckmann, S. Goodwin, N.H. Riche, S. Carpendale, and A. Liebman.
<p>We present an efficient technique for topology-preserving map deformation and apply it to the visualization of dissimilarity data in a geographic context. Map deformation techniques such as value-by-area cartograms are well studied. However, using deformation to highlight (dis)similarity between locations on a map in terms of their underlying data attributes is novel. We also identify an alternative way to represent dissimilarities on a map through the use of visual overlays. These overlays are complementary to deformation techniques and enable us to assess the quality of the deformation as well as to explore the design space of blending the two methods. Finally, we demonstrate how these techniques can be useful in several-quite different-applied contexts: travel-time visualization, social demographics research and understanding energy flowing in a wide-area power-grid.</p>
@article{visual-encoding-of-dissimilarity-data-via-topology-preserving-map-deformation:2016,
title = {Visual Encoding of Dissimilarity Data Via Topology-Preserving Map Deformation},
author = {Q.W. Bouts and T. Dwyer and J. Dykes and B. Speckmann and S. Goodwin and N.H. Riche and S. Carpendale and A. Liebman},
An Improved Lower Bound on the Minimum Number of Triangulations
O. Aichholzer, V. Alvarez, T. Hackl, A. Pilz, B. Speckmann, and B. Vogtenhuber.
32nd International Symposium on Computational Geometry, SoCG 2016, 14-17 June 2016, Boston, Massachusetts, pp. 7.1—7.16, 2016.
<p>Upper and lower bounds for the number of geometric graphs of specific types on a given set of points in the plane have been intensively studied in recent years. For most classes of geometric graphs it is now known that point sets in convex position minimize their number. However, it is still unclear which point sets minimize the number of geometric triangulations; the so-called double circles are conjectured to be the minimizing sets. In this paper we prove that any set of n points in general position in the plane has at least Ω(2.631<sup>n</sup>) geometric triangulations. Our result improves the previously best general lower bound of Ω(2.43<sup>n</sup>) and also covers the previously best lower bound of Ω(2.63<sup>n</sup>) for a fixed number of extreme points. We achieve our bound by showing and combining several new results, which are of independent interest: 1. Adding a point on the second convex layer of a given point set (of 7 or more points) at least doubles the number of triangulations. 2. Generalized configurations of points that minimize the number of triangulations have at most ⌊n/2⌋ points on their convex hull. 3. We provide tight lower bounds for the number of triangulations of point sets with up to 15 points. These bounds further support the double circle conjecture.</p>
@article{an-improved-lower-bound-on-the-minimum-number-of-triangulations:2016,
title = {An Improved Lower Bound on the Minimum Number of Triangulations},
author = {O. Aichholzer and V. Alvarez and T. Hackl and A. Pilz and B. Speckmann and B. Vogtenhuber},
bookTitle = {32nd International Symposium on Computational Geometry, SoCG 2016, 14-17 June 2016, Boston, Massachusetts},
Circles in the Water: Towards Island Group Labeling
A.I. van Goethem, M.J. van Kreveld, and B. Speckmann.
Proc. of the 9th International Conference on Geographic Information Science (GIScience), pp. 293—307, 2016.
Many algorithmic results are known for automated label placement on maps. However, algorithms to compute labels for groups of features, such as island groups, are largely missing. In this paper we address this issue by presenting new, efficient algorithms for island label placement in various settings. We consider straight-line and circular-arc labels that may or may not overlap a given set of islands. We concentrate on computing the line or circle that minimizes the maximum distance to the islands, measured by the closest distance. We experimentally test whether the generated labels are reasonable for various real-world island groups, and compare different options. The results are positive and validate our geometric formalizations.
@article{circles-in-the-water-towards-island-group-labeling:2016,
title = {Circles in the Water: Towards Island Group Labeling},
author = {A.I. van Goethem and M.J. van Kreveld and B. Speckmann},
bookTitle = {Proc. of the 9th International Conference on Geographic Information Science (GIScience)},
Geo Word Clouds
K.A. Buchin, D.J.A. Creemers, A. Lazzarotto , B. Speckmann, and J.J.H.M. Wulms.
2016 IEEE Pacific Visualization Symposium (PacificVis), 19-22 April 2016, Taipei, Taiwan , pp. 144—151, 2016.
Word clouds are a popular method to visualize the frequency of words in textual data. Nowadays many text-based data sets, such as Flickr tags, are geo-referenced, that is, they have an important spatial component. However, existing automated methods to generate word clouds are unable to incorporate such spatial information. We introduce geo word clouds: word clouds which capture not only the frequency but also the spatial relevance of words. Our input is a set of locations from one (or more) geographic regions with (possibly several) text labels per location. We aggregate word frequencies according to point clusters and employ a greedy strategy to place appropriately sized labels without overlap as close as possible to their corresponding locations. While doing so we "draw" the spatial shapes of the geographic regions with the corresponding labels. We experimentally explore trade-offs concerning the location of labels, their relative sizes and the number of spatial clusters. The resulting word clouds are visually pleasing and have a low error in terms of relative scaling and locational accuracy of words, while using a small number of clusters per label.
@article{geo-word-clouds:2016,
title = {Geo Word Clouds},
author = {K.A. Buchin and D.J.A. Creemers and A. Lazzarotto and B. Speckmann and J.J.H.M. Wulms},
bookTitle = {2016 IEEE Pacific Visualization Symposium (PacificVis), 19-22 April 2016, Taipei, Taiwan },
GlamMap: Geovisualization for e-Humanities
T.H.A. Castermans, B. Speckmann, K.A.B. Verbeek, M.A. Westenberg, A. Betti, and H. van den Berg.
Proc. 1st Workshop on Visualization for the Digital Humanities (VIS4DH), pp. 1—5, 2016.
This paper presents GlamMap, a visualization tool for large, multi-variate georeferenced humanities data sets. Our approach visualizes the data as glyphs on a zoomable geographic map, and performs clustering and data aggregation at each zoom level to avoid clutter and to prevent overlap of symbols. GlamMap was developed for the Galleries, Libraries, Archives, and Museums (GLAM) domain in cooperation with researchers in philosophy. We demonstrate the usefulness of our approach by a case study on history of logic, which involves navigation and exploration of 7100 bibliographic records, and scalability on a data set of sixty million book records.
@article{glammap-geovisualization-for-e-humanities:2016,
title = {GlamMap: Geovisualization for e-Humanities},
author = {T.H.A. Castermans and B. Speckmann and K.A.B. Verbeek and M.A. Westenberg and A. Betti and H. van den Berg},
bookTitle = {Proc. 1st Workshop on Visualization for the Digital Humanities (VIS4DH)},
GlamMapping Trove
A. Betti, T.H.A. Castermans, B. Speckmann, H. van den Berg, and K.A.B. Verbeek.
Proc. VALA, 2016.
This paper presents the current state of development of GlamMap, a visualisation tool that displays library metadata on an interactive, computer-generated geographic map. The focus in the paper is on the most crucial improvement achieved in the development of the tool: GlamMapping Trove. The visualisation of Trove's sixty-million book records is possible thanks to an improved database structure, more efficient data retrieval, and more scalable visualisation algorithms. The paper analyses problems encountered in visualising massive datasets, describes remaining challenges for the tool, and presents a use case demonstrating GlamMap's ability to serve researchers in the history of ideas.
@article{glammapping-trove:2016,
title = {GlamMapping Trove},
author = {A. Betti and T.H.A. Castermans and B. Speckmann and H. van den Berg and K.A.B. Verbeek},
bookTitle = {Proc. VALA},
Grouping Time-Varying Data for Interactive Exploration
A.I. van Goethem, M.J. van Kreveld, M. Löffler, B. Speckmann, and F. Staals.
Proc. of the 32nd annual Symposium on Computational Geometry (SoCG), pp. 1—16, 2016.
We present algorithms and data structures that support the interactive analysis of the grouping structure of one-, two-, or higher-dimensional time-varying data while varying all defining parameters. Grouping structures characterise important patterns in the temporal evaluation of sets of time-varying data. We follow Buchin et al. [JoCG 2015] who define groups using three parameters: group-size, group-duration, and inter-entity distance. We give upper and lower bounds on the number of maximal groups over all parameter values, and show how to compute them efficiently. Furthermore, we describe data structures that can report changes in the set of maximal groups in an output-sensitive manner. Our results hold in R^d for fixed d.
@article{grouping-time-varying-data-for-interactive-exploration:2016,
title = {Grouping Time-Varying Data for Interactive Exploration},
author = {A.I. van Goethem and M.J. van Kreveld and M. Löffler and B. Speckmann and F. Staals},
bookTitle = {Proc. of the 32nd annual Symposium on Computational Geometry (SoCG)},
Modeling Checkpoint-Based Movement with the Earth Mover's Distance
M. Duckham, M.J. van Kreveld, R. Purves, B. Speckmann, Y. Tao, K.A.B. Verbeek, and J. Wood.
Geographic Information Science , pp. 225—239, 2016.
<p>Movement data comes in various forms, including trajectory data and checkpoint data. While trajectories give detailed information about the movement of individual entities, checkpoint data in its simplest form does not give identities, just counts at checkpoints. However, checkpoint data is of increasing interest since it is readily available due to privacy reasons and as a by-product of other data collection. In this paper we propose to use the Earth Mover's Distance as a versatile tool to reconstruct individual movements or flow based on checkpoint counts at different times. We analyze the modeling possibilities and provide experiments that validate model predictions, based on coarse-grained aggregations of data about actual movements of couriers in London, UK. While we cannot expect to reconstruct precise individual movements from highly granular checkpoint data, the evaluation does show that the approach can generate meaningful estimates of object movements.</p>
@article{modeling-checkpoint-based-movement-with-the-earth-mover-s-distance:2016,
title = {Modeling Checkpoint-Based Movement with the Earth Mover's Distance},
author = {M. Duckham and M.J. van Kreveld and R. Purves and B. Speckmann and Y. Tao and K.A.B. Verbeek and J. Wood},
bookTitle = {Geographic Information Science },
K.A. Buchin, T.A.E. Ophelders, and B. Speckmann.
author = {K.A. Buchin and T.A.E. Ophelders and B. Speckmann},
pp. 7—10, 2016.
A Formal Approach to the Automated Labeling of Groups of Features
A. Reimer, A.I. Goethem, van, M. Rylov, M.J. Kreveld, van, and B. Speckmann.
Cartography and Geographic Information Science, 42(4):333—344, 2015.
A variety of recurring geographic entities form collections of point, line, or area features. Examples are groups of islands (Archipelago), relief features in deserts, periglacial lakes or geomorphological forms, such as drumlins and sinkholes. All these groups of features might be best identified with a single label. Surprisingly, the (automated) labeling of groups of features has received little attention so far. We propose a framework to determine formal measures that describe the geometric aspects of the cartographic design space for labeling feature groups. Our framework gives rise to a large variety of geometrically optimal labels. We list the optimal label positions and shapes for which labels can already be computed using existing algorithms. However, in many of the cases computing optimal label placements is still an open algorithmic question which can readily be investigated in future work. Once the necessary algorithms are developed, our framework provides an objective basis to investigate the geometric measures used by cartographers to label groups of features. We preliminarily explore the applicability of our framework using one of the geometric optimality choices. Keywords: automated cartography, labeling, groups of features
@article{a-formal-approach-to-the-automated-labeling-of-groups-of-features:2015,
title = {A Formal Approach to the Automated Labeling of Groups of Features},
author = {A. Reimer and A.I. Goethem, van and M. Rylov and M.J. Kreveld, van and B. Speckmann},
bookTitle = {Cartography and Geographic Information Science},
Algorithms for Necklace Maps
International Journal of Computational Geometry and Applications, 25(1):15—36, 2015.
Necklace maps visualize quantitative data associated with regions by placing scaled symbols, usually disks, without overlap on a closed curve (the necklace) surrounding the map regions. Each region is projected onto an interval on the necklace that contains its symbol. In this paper we address the algorithmic question how to maximize symbol sizes while keeping symbols disjoint and inside their intervals. For that we reduce the problem to a one-dimensional problem which we solve efficiently. Solutions to the one-dimensional problem provide a very good approximation for the original necklace map problem. We consider two variants: Fixed-Order, where an order for the symbols on the necklace is given, and Any-Order where any symbol order is possible. The Fixed-Order problem can be solved in O(n log n) time. We show that the Any-Order problem is NP-hard for certain types of intervals and give an exact algorithm for the decision version. This algorithm is fixed-parameter tractable in the thickness K of the input. Our algorithm runs in O(n log n + n2K4K) time which can be improved to O(n log n + nK2K) time using a heuristic. We implemented our algorithm and evaluated it experimentally. Keywords: Necklace maps; scheduling; automated cartography
@article{algorithms-for-necklace-maps:2015,
title = {Algorithms for Necklace Maps},
bookTitle = {International Journal of Computational Geometry and Applications},
Angle-Restricted Steiner Arborescences for Flow Map Layout
K. Buchin, B. Speckmann, and K.A.B. Verbeek.
We introduce a new variant of the geometric Steiner arborescence problem, motivated by the layout of flow maps. Flow maps show the movement of objects between places. They reduce visual clutter by bundling curves smoothly and avoiding self-intersections. To capture these properties, our angle-restricted Steiner arborescences, or flux trees, connect several targets to a source with a tree of minimal length whose arcs obey a certain restriction on the angle they form with the source. We study the properties of optimal flux trees and show that they are crossing-free and consist of logarithmic spirals and straight lines. Flux trees have the shallow-light property. We show that computing optimal flux trees is NP-hard. Hence we consider a variant of flux trees which uses only logarithmic spirals. Spiral trees approximate flux trees within a factor depending on the angle restriction. Computing optimal spiral trees remains NP-hard, but we present an efficient 2-approximation, which can be extended to avoid "positive monotone" obstacles. Keywords: Steiner arborescences; Flow maps; Computational geometry; Automated cartography
@article{angle-restricted-steiner-arborescences-for-flow-map-layout:2015,
title = {Angle-Restricted Steiner Arborescences for Flow Map Layout},
author = {K. Buchin and B. Speckmann and K.A.B. Verbeek},
Exploring Curved Schematization of Territorial Outlines
A.I. Goethem, van, W. Meulemans, B. Speckmann, and J.D. Wood.
Hand-drawn schematized maps traditionally make extensive use of curves. However, there are few automated approaches for curved schematization; most previous work focuses on straight lines. We present a new algorithm for areapreserving curved schematization of territorial outlines. Our algorithm converts a simple polygon into a schematic crossing-free representation using circular arcs.We use two basic operations to iteratively replace consecutive arcs until the desired complexity is reached. Our results are not restricted to arcs ending at input vertices. The method can be steered towards different degrees of "curviness": we can encourage or discourage the use of arcs with a large central angle via a single parameter. Our method creates visually pleasing results even for very low output complexities. To evaluate the effectiveness of our design choices, we present a geometric evaluation of the resulting schematizations. Besides the geometric qualities of our algorithm, we also investigate the potential of curved schematization as a concept. We conducted an online user study investigating the effectiveness of curved schematizations compared to straight-line schematizations. While the visual complexity of curved shapes was judged higher than that of straight-line shapes, users generally preferred curved schematizations. We observed that curves significantly improved the ability of users to match schematized shapes of moderate complexity to their unschematized equivalents. Keywords: Schematization; algorithm; circular arcs; user study
@article{exploring-curved-schematization-of-territorial-outlines:2015,
title = {Exploring Curved Schematization of Territorial Outlines},
author = {A.I. Goethem, van and W. Meulemans and B. Speckmann and J.D. Wood},
Improved Grid Map Layout by Point Set Matching
D. Eppstein, M.J. Kreveld, van, B. Speckmann, and F. Staals.
International Journal of Computational Geometry and Applications, 25(2):101—122, 2015.
Associating the regions of a geographic subdivision with the cells of a grid is a basic operation that is used in various types of maps, like spatially ordered treemaps and Origin-Destination maps (OD maps). In these cases the regular shapes of the grid cells allow easy representation of extra information about the regions. The main challenge is to find an association that allows a user to find a region in the grid quickly. We call the representation of a set of regions as a grid a grid map. We introduce a new approach to solve the association problem for grid maps by formulating it as a point set matching problem: Given two sets A (the centroids of the regions) and B (the grid centres) of n points in the plane, compute an optimal one-to-one matching between A and B. We identify three optimisation criteria that are important for grid map layout: maximise the number of adjacencies in the grid that are also adjacencies of the regions, minimise the sum of the distances between matched points, and maximise the number of pairs of points in A for which the matching preserves the directional relation (SW, NW, etc.). We consider matchings that minimise the L1-distance (Manhattan-distance), the ranked L1-distance, and the L22-distance, since one can expect that minimising distances implicitly helps to fulfill the other criteria. We present algorithms to compute such matchings and perform an experimental comparison that also includes a previous method to compute a grid map. The experiments show that our more global, matching-based algorithm outperforms previous, more local approaches with respect to all three optimisation criteria. Keywords: Grid map; point-set matching; visualization
@article{improved-grid-map-layout-by-point-set-matching:2015,
title = {Improved Grid Map Layout by Point Set Matching},
author = {D. Eppstein and M.J. Kreveld, van and B. Speckmann and F. Staals},
Mosaic Drawings and Cartograms
R.G. Cano, K. Buchin, T.H.A. Castermans, A. Pieterse, W.M. Sonke, and B. Speckmann.
Cartograms visualize quantitative data about a set of regions such as countries or states. There are several different types of cartograms and – for some – algorithms to automatically construct them exist. We focus on mosaic cartograms: cartograms that use multiples of simple tiles – usually squares or hexagons – to represent regions. Mosaic cartograms communicate well data that consist of, or can be cast into, small integer units (for example, electorial college votes). In addition, they allow users to accurately compare regions and can often maintain a (schematized) version of the input regions' shapes. We propose the first fully automated method to construct mosaic cartograms. To do so, we first introduce mosaic drawings of triangulated planar graphs. We then show how to modify mosaic drawings into mosaic cartograms with low cartographic error while maintaining correct adjacencies between regions. We validate our approach experimentally and compare to other cartogram methods.
@article{mosaic-drawings-and-cartograms:2015,
title = {Mosaic Drawings and Cartograms},
author = {R.G. Cano and K. Buchin and T.H.A. Castermans and A. Pieterse and W.M. Sonke and B. Speckmann},
Trajectory Grouping Structure
K.A. Buchin, M.E. Buchin, M.J. van Kreveld, B. Speckmann, and F. Staals.
The collective motion of a set of moving entities like people, birds, or other animals, is characterized by groups arising, merging, splitting, and ending. Given the trajectories of these entities, we define and model a structure that captures all of such changes using the Reeb graph, a concept from topology. The trajectory grouping structure has three natural parameters that allow more global views of the data in group size, group duration, and entity inter-distance. We prove complexity bounds on the maximum number of maximal groups that can be present, and give algorithms to compute the grouping structure efficiently. We also study how the trajectory grouping structure can be made robust, that is, how brief interruptions of groups can be disregarded in the global structure, adding a notion of persistence to the structure. Furthermore, we showcase the results of experiments using data generated by the NetLogo flocking model and from the Starkey project. The Starkey data describe the movement of elk, deer, and cattle. Although there is no ground truth for the grouping structure in this data, the experiments show that the trajectory grouping structure is plausible and has the desired effects when changing the essential parameters. Our research provides the first complete study of trajectory group evolvement, including combinatorial,<br/>algorithmic, and experimental results.<br/>
@article{trajectory-grouping-structure:2015,
title = {Trajectory Grouping Structure},
author = {K.A. Buchin and M.E. Buchin and M.J. van Kreveld and B. Speckmann and F. Staals},
Clustered Edge Routing
Q.W. Bouts and B. Speckmann.
2015 IEEE Pacific Visualization Symposium (PacificVis2015, Hangzhou, China, April 14-17, 2015), pp. 55—62, 2015.
The classic method to depict graphs is a node-link diagram where vertices (nodes) are associated with each object and edges (links) connect related objects. However, node-link diagrams quickly appear cluttered and unclear, even for moderately sized graphs. If the positions of the nodes are fixed then suitable link routing is the only option to reduce clutter. We present a novel link clustering and routing algorithm which respects (and if desired refines) user-defined clusters on links. If no clusters are defined a priori we cluster based on geometric criteria, that is, based on a well-separated pair decomposition (WSPD).We route link clusters individually on a sparse visibility spanner. To completely avoid ambiguity we draw each individual link and ensure that clustered links follow the same path in the routing graph. We prove that the clusters induced by the WSPD consist of compatible links according to common similarity measures as formalized by Holten and van Wijk [17]. The greedy sparsification of the visibility graph allows us to easily route around obstacles. Our experimental results are visually appealing and convey a sense of abstraction and order.
@article{clustered-edge-routing:2015,
title = {Clustered Edge Routing},
author = {Q.W. Bouts and B. Speckmann},
bookTitle = {2015 IEEE Pacific Visualization Symposium (PacificVis2015, Hangzhou, China, April 14-17, 2015)},
K. Buchin, T.A.E. Ophelders, and B. Speckmann.
Proc. 23rd Annual European Symposium on Algorithms (ESA), pp. 928—940, 2015.
In this paper we study similarity measures for moving curves which can, for example, model changing coastlines or glacier termini. Points on a moving curve have two parameters, namely the position along the curve as well as time. We therefore focus on similarity measures for surfaces, specifically the Fréchet distance between surfaces. While the Fréchet distance between surfaces is not even known to be computable, we show for variants arising in the context of moving curves that they are polynomial-time solvable or NP-complete depending on the restrictions imposed on how the moving curves are matched. We achieve the polynomial-time solutions by a novel approach for computing a surface in the so-called free-space diagram based on max-flow min-cut duality.
author = {K. Buchin and T.A.E. Ophelders and B. Speckmann},
bookTitle = {Proc. 23rd Annual European Symposium on Algorithms (ESA)},
Geometric K Shortest Paths
S. Eriksson-Bique, J. Hershberger, V. Polishchuk, B. Speckmann, S. Suri, T. Talvitie, K.A.B. Verbeek, and H. Yildiz.
Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'15, San Diego CA, USA, January 4-6, 2015), pp. 1616—1625, 2015.
We consider the problem of computing k shortest paths in a two-dimensional environment with polygonal obstacles, where the jth path, for 1 = j = k, is the shortest path in the free space that is also homotopically distinct from each of the first j – 1 paths. In fact, we consider a more general problem: given a source point s, construct a partition of the free space, called the kth shortest path map (k-SPM), in which the homotopy of the kth shortest path in a region has the same structure. Our main combinatorial result establishes a tight bound of T(k2h + kn) on the worst-case complexity of this map. We also describe an O((k3h + k2n) log (kn)) time algorithm for constructing the map. In fact, the algorithm constructs the jth map for every j = k. Finally, we present a simple visibility-based algorithm for computing the k shortest paths between two fixed points. This algorithm runs in O(m log n + k) time and uses O(m + k) space, where m is the size of the visibility graph. This latter algorithm can be extended to compute k shortest simple (non-self-intersecting) paths, taking O(k2 m(m + kn) log (kn)) time. We invite the reader to play with our applet demonstrating k-SPMs [10].
@article{geometric-k-shortest-paths:2015,
title = {Geometric K Shortest Paths},
author = {S. Eriksson-Bique and J. Hershberger and V. Polishchuk and B. Speckmann and S. Suri and T. Talvitie and K.A.B. Verbeek and H. Yildiz},
bookTitle = {Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'15, San Diego CA, USA, January 4-6, 2015)},
Towards Characterizing Graphs with a Sliceable Rectangular Dual
V. Kusters and B. Speckmann.
Graph Drawing and Network Visualization, pp. 460—471, 2015.
<p>Let G be a plane triangulated graph. A rectangular dual of G is a partition of a rectangle R into a set R of interior-disjoint rectangles, one for each vertex, such that two regions are adjacent if and only if the corresponding vertices are connected by an edge. A rectangular dual is sliceable if it can be recursively subdivided along horizontal or vertical lines. A graph is rectangular if it has a rectangular dual and sliceable if it has a sliceable rectangular dual. There is a clear characterization of rectangular graphs. However, a full characterization of sliceable graphs is still lacking. The currently best result (Yeap and Sarrafzadeh, 1995) proves that all rectangular graphs without a separating 4-cycle are slice- able. In this paper we introduce a recursively defined class of graphs and prove that these graphs are precisely the nonsliceable graphs with exactly one separating 4-cycle.</p>
@article{towards-characterizing-graphs-with-a-sliceable-rectangular-dual:2015,
title = {Towards Characterizing Graphs with a Sliceable Rectangular Dual},
author = {V. Kusters and B. Speckmann},
bookTitle = {Graph Drawing and Network Visualization},
Trajectory Grouping Structure Under Geodesic Distance
I. Kostitsyna, M.J. van Kreveld, M. Löffler, B. Speckmann, and F. Staals.
Proc. 31st International Symposium on Computational Geometry (SoCG), pp. 674—688, 2015.
<p>In recent years trajectory data has become one of the main types of geographic data, and hence algorithmic tools to handle large quantities of trajectories are essential. A single trajectory is typically represented as a sequence of time-stamped points in the plane. In a collection of trajectories one wants to detect maximal groups of moving entities and their behaviour (merges and splits) over time. This information can be summarized in the trajectory grouping structure. Significantly extending the work of Buchin et al. [WADS 2013] into a realistic setting, we show that the trajectory grouping structure can be computed efficiently also if obstacles are present and the distance between the entities is measured by geodesic distance. We bound the number of critical events: times at which the distance between two subsets of moving entities is exactly ", where " is the threshold distance that determines whether two entities are close enough to be in one group. In case the n entities move in a simple polygon along trajectories with τ vertices each we give an O(τn<sup>2</sup>) upper bound, which is tight in the worst case. In case of well-spaced obstacles we give an O(τ (n<sup>2</sup> + mλ<sub>4</sub>(n))) upper bound, where m is the total complexity of the obstacles, and λs(n) denotes the maximum length of a Davenport-Schinzel sequence of n symbols of order s. In case of general obstacles we give an O(τ min{n<sup>2</sup> + m<sup>3</sup>λ<sub>4</sub>(n), n<sup>2</sup>m<sup>2</sup>}) upper bound. Furthermore, for all cases we provide efficient algorithms to compute the critical events, which in turn leads to efficient algorithms to compute the trajectory grouping structure.</p>
@article{trajectory-grouping-structure-under-geodesic-distance:2015,
title = {Trajectory Grouping Structure Under Geodesic Distance},
author = {I. Kostitsyna and M.J. van Kreveld and M. Löffler and B. Speckmann and F. Staals},
bookTitle = {Proc. 31st International Symposium on Computational Geometry (SoCG)},
R.G. Cano, K.A. Buchin, T.H.A. Castermans, A. Pieterse, W.M. Sonke, and B. Speckmann.
author = {R.G. Cano and K.A. Buchin and T.H.A. Castermans and A. Pieterse and W.M. Sonke and B. Speckmann},
Optimal Straight-Line Labels for Island Groups
A.I. van Goethem, M.J. van Kreveld, Andreas W. Reimer, M. Rylov, and B. Speckmann.
Maps are used to solve a wide variety of tasks, ranging from navigation to analysis. Often, the quality of a map is directly related to the quality of its labelling. Consequently, a lot of research has focussed on the automatization of the labelling process. Surprisingly the (automated) labelling of island groups has received little attention so far. This is at least partially caused by the lack of cartographic principles. 31 Though extensive guidelines for map labelling exist, information on the labelling of groups of islands is surprisingly sparse. We define a formal framework for island labelling. The framework spawns a large series of unexplored computational geometry problems, which are interesting for the CG-community. In this paper we start by looking at a non-overlapping, straight label. We describe two algorithms for a straight-line label that is, or is not, allowed overlap with islands. Furthermore, we discus several extensions to these algorithms solving closely related problems.
@article{optimal-straight-line-labels-for-island-groups:2015,
title = {Optimal Straight-Line Labels for Island Groups},
author = {A.I. van Goethem and M.J. van Kreveld and Andreas W. Reimer and M. Rylov and B. Speckmann},
In this paper we study similarity measures for moving curves which can, for example, model changing coastlines or retreating glacier termini. Points on a moving curve have two parameters, namely the position along the curve as well as time. We therefore focus on similarity measures for surfaces, specifically the Fr\'echet distance between surfaces. While the Fr\'echet distance between surfaces is not even known to be computable, we show for variants arising in the context of moving curves that they are polynomial-time solvable or NP-complete depending on the restrictions imposed on how the moving curves are matched. We achieve the polynomial-time solutions by a novel approach for computing a surface in the so-called free-space diagram based on max-flow min-cut duality.
Analysis and Visualisation of Movement
U. Demšar, K.A. Buchin, F. Cagnacci, K. Safi, B. Speckmann, N. de Weghe, D. Weiskopf, and R. Weibel.
Movement Ecology, 3(5):, 2015.
<p>The processes that cause and influence movement are one of the main points of enquiry in movement ecology. However, ecology is not the only discipline interested in movement: a number of information sciences are specialising in analysis and visualisation of movement data. The recent explosion in availability and complexity of movement data has resulted in a call in ecology for new appropriate methods that would be able to take full advantage of the increasingly complex and growing data volume. One way in which this could be done is to form interdisciplinary collaborations between ecologists and experts from information sciences that analyse movement. In this paper we present an overview of new movement analysis and visualisation methodologies resulting from such an interdisciplinary research network: the European COST Action "MOVE - Knowledge Discovery from Moving Objects" (http://www.move-cost.info). This international network evolved over four years and brought together some 140 researchers from different disciplines: those that collect movement data (out of which the movement ecology was the largest represented group) and those that specialise in developing methods for analysis and visualisation of such data (represented in MOVE by computational geometry, geographic information science, visualisation and visual analytics). We present MOVE achievements and at the same time put them in ecological context by exploring relevant ecological themes to which MOVE studies do or potentially could contribute.</p>
@article{analysis-and-visualisation-of-movement:2015,
title = {Analysis and Visualisation of Movement},
author = {U. Demšar and K.A. Buchin and F. Cagnacci and K. Safi and B. Speckmann and N. de Weghe and D. Weiskopf and R. Weibel},
bookTitle = {Movement Ecology},
Automatische Schematisering Met Gebogen Lijnen
A.I. Goethem, van, H.J. Haverkort, W. Meulemans, A. Reimer, B. Speckmann, and J.D. Wood.
Geo-Info, 2014(2):10—13, 2014.
@article{automatische-schematisering-met-gebogen-lijnen:2014,
title = {Automatische Schematisering Met Gebogen Lijnen},
author = {A.I. Goethem, van and H.J. Haverkort and W. Meulemans and A. Reimer and B. Speckmann and J.D. Wood},
bookTitle = {Geo-Info},
Moving Beyond the Point: an Agenda for Research in Movement Analysis with Real Data
R.S. Purves, P. Laube, M. Buchin, and B. Speckmann.
Computers, Environment and Urban Systems, 47:1—4, 2014.
@article{moving-beyond-the-point-an-agenda-for-research-in-movement-analysis-with-real-data:2014,
title = {Moving Beyond the Point: an Agenda for Research in Movement Analysis with Real Data},
author = {R.S. Purves and P. Laube and M. Buchin and B. Speckmann},
bookTitle = {Computers, Environment and Urban Systems},
On the Number of Regular Edge Labelings
K. Buchin, B. Speckmann, and S. Verdonschot.
Discrete Mathematics and Theoretical Computer Science, 16(3):215—228, 2014.
We prove that any irreducible triangulation on n vertices has O (4:6807n ) regular edge labeling,s and that there are irreducible triangulations on n vertices with (3:0426n ) regular edge labelings. Our upper bound relies on a novel application of Shearer's entropy lemma. As an example of the wider applicability of this technique, we also improve the upper bound on the number of 2-orientations of a quadrangulation to O (1:87n ). Keywords: Counting; Regular edge labeling; Shearer's entropy lemma
@article{on-the-number-of-regular-edge-labelings:2014,
title = {On the Number of Regular Edge Labelings},
author = {K. Buchin and B. Speckmann and S. Verdonschot},
Plane Graphs with Parity Constraints
O. Aichholzer, T. Hackl, M. Hoffmann, A. Pilz, G. Rote, B. Speckmann, and B. Vogtenhuber.
Graphs and Combinatorics, 30(1):47—69, 2014.
Let S be a set of n points in general position in the plane. Together with S we are given a set of parity constraints, that is, every point of S is labeled either even or odd. A graph G on S satisfies the parity constraint of a point p¿S if the parity of the degree of p in G matches its label. In this paper, we study how well various classes of planar graphs can satisfy arbitrary parity constraints. Specifically, we show that we can always find a plane tree, a two-connected outerplanar graph, or a pointed pseudo-triangulation that satisfy all but at most three parity constraints. For triangulations we can satisfy about 2/3 of the parity constraints and we show that in the worst case there is a linear number of constraints that cannot be fulfilled. In addition, we prove that for a given simple polygon H with polygonal holes on S, it is NP-complete to decide whether there exists a triangulation of H that satisfies all parity constraints.
@article{plane-graphs-with-parity-constraints:2014,
title = {Plane Graphs with Parity Constraints},
author = {O. Aichholzer and T. Hackl and M. Hoffmann and A. Pilz and G. Rote and B. Speckmann and B. Vogtenhuber},
bookTitle = {Graphs and Combinatorics},
Similarity of Trajectories Taking into Account Geographic Context
M. Buchin, S. Dodge, and B. Speckmann.
Journal of Spatial Information Science, 9:101—124, 2014.
The movements of animals, people, and vehicles are embedded in a geographic context. This context influences the movement and may cause the formation of certain behavioral responses. Thus, it is essential to include context parameters in the study of movement and the development of movement pattern analytics. Advances in sensor technologies and positioning devices provide valuable data not only of moving agents but also of the circumstances embedding the movement in space and time. Developing knowledge discovery methods to investigate the relation between movement and its surrounding context is a major challenge in movement analysis today. In this paper we show how to integrate geographic context into the similarity analysis of movement data. For this, we discuss models for geographic context of movement data. Based on this we develop simple but efficient context-aware similarity measures for movement trajectories, which combine a spatial and a contextual distance. These are based on well-known similarity measures for trajectories, such as the Hausdorff, Fréchet, or equal time distance. We validate our approach by applying these measures to movement data of hurricanes and albatross. Keywords: movement data, geographic context, spatiotemporal analytics, similarity measures, trajectory analysis, modeling context, environmental factors, Fréchet distance, Hausdorff distance, equal time distance
@article{similarity-of-trajectories-taking-into-account-geographic-context:2014,
title = {Similarity of Trajectories Taking into Account Geographic Context},
author = {M. Buchin and S. Dodge and B. Speckmann},
bookTitle = {Journal of Spatial Information Science},
Stenomaps: Shorthand for Shapes
A.I. Goethem, van, A. Reimer, B. Speckmann, and J.D. Wood.
We address some of the challenges in representing spatial data with a novel form of geometric abstraction – the stenomap. The stenomap comprises a series of smoothly curving linear glyphs that each represent both the boundary and the area of a polygon. We present an efficient algorithm to automatically generate these open, C1-continuous splines from a set of input polygons. Feature points of the input polygons are detected using the medial axis to maintain important shape properties. We use dynamic programming to compute a planar non-intersecting spline representing each polygon's base shape. The results are stylised glyphs whose appearance may be parameterised and that offer new possibilities in the 'cartographic design space'. We compare our glyphs with existing forms of geometric schematisation and discuss their relative merits and shortcomings. We describe several use cases including the depiction of uncertain model data in the form of hurricane track forecasting; minimal ink thematic mapping; and the depiction of continuous statistical data.
@article{stenomaps-shorthand-for-shapes:2014,
title = {Stenomaps: Shorthand for Shapes},
author = {A.I. Goethem, van and A. Reimer and B. Speckmann and J.D. Wood},
Treemaps with Bounded Aspect Ratio
M.T. Berg, de, B. Speckmann, and V. Weele, van der.
Computational Geometry, 47(6):683—693, 2014.
Treemaps are a popular technique to visualize hierarchical data. The input is a weighted tree T where the weight of each node is the sum of the weights of its children. A treemap for T is a hierarchical partition of a rectangle into simply connected regions, usually rectangles. Each region represents a node of T and its area is proportional to the weight of the corresponding node. An important quality criterion for treemaps is the aspect ratio of its regions. One cannot bound the aspect ratio if the regions are restricted to be rectangles. In contrast, polygonal partitions, that use convex polygons, can have bounded aspect ratio. We are the first to obtain convex partitions with optimal aspect ratio O(depth(T)). However, depth(T) still depends on the input tree. Hence we introduce a new type of treemaps, namely orthoconvex treemaps, where regions representing leaves are rectangles, L-, and S-shapes, and regions representing internal nodes are orthoconvex polygons. We prove that any input tree, irrespective of the weights of the nodes and the depth of the tree, admits an orthoconvex treemap of constant aspect ratio. We also obtain several specialized results for single-level treemaps, that is, treemaps where the input tree has depth 1. Keywords: Convex treemap; Orthoconvex treemap; Aspect ratio
@article{treemaps-with-bounded-aspect-ratio:2014,
title = {Treemaps with Bounded Aspect Ratio},
author = {M.T. Berg, de and B. Speckmann and V. Weele, van der},
Triangulating and Guarding Realistic Polygons
G. Aloupis, P. Bose, V. Dujmovic, C.M. Gray, S. Langerman, and B. Speckmann.
Computational Geometry, 47(2, Part C):296—306, 2014.
We propose a new model of realistic input: k-guardable objects. An object is k-guardable if its boundary can be seen by k guards. We show that k-guardable polygons generalize two previously identified classes of realistic input. Following this, we give two simple algorithms for triangulating k-guardable polygons. One algorithm requires the guards as input while the other does not. Both take linear time assuming that k is constant and both are easily implementable.
@article{triangulating-and-guarding-realistic-polygons:2014,
title = {Triangulating and Guarding Realistic Polygons},
author = {G. Aloupis and P. Bose and V. Dujmovic and C.M. Gray and S. Langerman and B. Speckmann},
Column Planarity and Partial Simultaneous Geometric Embedding
W.S. Evans, V.J.J. Kusters, M. Saumell, and B. Speckmann.
Graph Drawing (22nd International Symposium, GD 2014, Würzburg, Germany, September 24-26, 2014, Revised Selected Papers), pp. 259—271, 2014.
We introduce the notion of column planarity of a subset R of the vertices of a graph G. Informally, we say that R is column planar in G if we can assign x-coordinates to the vertices in R such that any assignment of y-coordinates to them produces a partial embedding that can be completed to a plane straight-line drawing of G. Column planarity is both a relaxation and a strengthening of unlabeled level planarity. We prove near tight bounds for column planar subsets of trees: any tree on n vertices contains a column planar set of size at least 14n/17 and for any e¿>¿0 and any sufficiently large n, there exists an n-vertex tree in which every column planar subset has size at most (5/6¿+¿e)n. We also consider a relaxation of simultaneous geometric embedding (SGE), which we call partial SGE (PSGE). A PSGE of two graphs G 1 and G 2 allows some of their vertices to map to two different points in the plane. We show how to use column planar subsets to construct k-PSGEs in which k vertices are still mapped to the same point. In particular, we show that any two trees on n vertices admit an 11n/17-PSGE, two outerpaths admit an n/4-PSGE, and an outerpath and a tree admit a 11n/34-PSGE.
@article{column-planarity-and-partial-simultaneous-geometric-embedding:2014,
title = {Column Planarity and Partial Simultaneous Geometric Embedding},
author = {W.S. Evans and V.J.J. Kusters and M. Saumell and B. Speckmann},
bookTitle = {Graph Drawing (22nd International Symposium, GD 2014, Würzburg, Germany, September 24-26, 2014, Revised Selected Papers)},
Computing the Fréchet Distance with Shortcuts is NP-Hard
M. Buchin, A. Driemel, and B. Speckmann.
30th ACM Symposium on Computational Geometry (SoCG, Kyoto, Japan, June 8-11, 2014), pp. 367—376, 2014.
We study the shortcut Fréchet distance, a natural variant of the Fréchet distance, that allows us to take shortcuts from and to any point along one of the curves. The classic Fréchet distance is a bottle-neck distance measure and hence quite sensitive to outliers. The shortcut Fréchet distance allows us to cut across outliers and hence produces more meaningful results when dealing with real world data. Driemel and Har-Peled recently described approximation algorithms for the restricted case where shortcuts have to start and end at input vertices. We show that, in the general case, the problem of computing the shortcut Fréchet distance is NP-hard. This is the first hardness result for a variant of the Fréchet distance between two polygonal curves in the plane. We also present two algorithms for the decision problem: a 3-approximation algorithm for the general case and an exact algorithm for the vertex-restricted case. Both algorithms run in O(n3 log n) time.
@article{computing-the-fr-chet-distance-with-shortcuts-is-np-hard:2014,
title = {Computing the Fréchet Distance with Shortcuts is NP-Hard},
author = {M. Buchin and A. Driemel and B. Speckmann},
bookTitle = {30th ACM Symposium on Computational Geometry (SoCG, Kyoto, Japan, June 8-11, 2014)},
Exploring Curved Schematization
A.I. van Goethem, W. Meulemans, B. Speckmann, and J.D. Wood.
7th IEEE Pacific Visualization Symposium (PacificVis), pp. 1—8, 2014.
Hand-drawn schematized maps traditionally make extensive use of curves. However, there are few automated approaches for curved schematization most previous work focuses on straight lines. We present a new algorithm for area-preserving curved schematization of geographic outlines. Our algorithm converts a simple polygon into a schematic crossing-free representation using circular arcs. We use two basic operations to iteratively replace consecutive arcs until the desired complexity is reached. Our results are not restricted to arcs ending at input vertices. The method can be steered towards different degrees of 'curviness': we can encourage or discourage the use of arcs with a large central angle via a single parameter. Our method creates visually pleasing results even for very low output complexities. We conducted an online user study investigating the effectiveness of the curved schematizations compared to straight-line schematizations of equivalent complexity. While the visual complexity of the curved shapes was judged higher than those using straight lines, users generally preferred curved schematizations. We observed that curves significantly improved the ability of users to match schematized shapes of moderate complexity to their unschematized equivalents.
@article{exploring-curved-schematization:2014,
title = {Exploring Curved Schematization},
author = {A.I. van Goethem and W. Meulemans and B. Speckmann and J.D. Wood},
bookTitle = {7th IEEE Pacific Visualization Symposium (PacificVis)},
Geometric Kth Shortest Paths: the Applet
J. Hershberger, V. Polishchuk, B. Speckmann, and T. Talvitie.
30th ACM Symposium on Computational Geometry (SoCG, Kyoto, Japan, June 8-11, 2014), pp. 96—97, 2014.
@article{geometric-kth-shortest-paths-the-applet:2014,
title = {Geometric Kth Shortest Paths: the Applet},
author = {J. Hershberger and V. Polishchuk and B. Speckmann and T. Talvitie},
GlamMap: Visualising Library Metadata
A. Betti, D.H.P. Gerrits, B. Speckmann, and H. Berg, van den.
VALA 2014 (17th Biennual Conference and Exhibition, Melbourne, Australia, February 3-6, 2014), pp. 1—15, 2014.
Libraries provide access to large amounts of library metadata. Unfortunately, many libraries only offer textual interfaces for searching and browsing their holdings. Visualisations provide simpler, faster, and more efficient ways to navigate, search and study large quantities of metadata. This paper presents GlamMap, a visualisation tool that displays library metadata on an interactive, computer-generated geographic map. We provide detailed discussion of how GlamMap benefits the work of librarians and researchers. We show how geographic representations help librarians to perform tasks such as collection assessment and how geographic information helps researchers to identify important scientific resources.
@article{glammap-visualising-library-metadata:2014,
title = {GlamMap: Visualising Library Metadata},
author = {A. Betti and D.H.P. Gerrits and B. Speckmann and H. Berg, van den},
bookTitle = {VALA 2014 (17th Biennual Conference and Exhibition, Melbourne, Australia, February 3-6, 2014)},
Map Schematization with Circular Arcs
T. van Dijk, A.I. van Goethem, J.H. Haunert, W. Meulemans, and B. Speckmann.
8th International Conference on Geographic Information Science (GIScience), pp. 1—17, 2014.
We present an algorithm to compute schematic maps with circular arcs. Our algorithm iteratively replaces two consecutive arcs with a single arc to reduce the complexity of the output map and thus to increase its level of abstraction. Our main contribution is a method for replacing arcs that meet at high-degree vertices. This allows us to greatly reduce the output complexity, even for dense networks. We experimentally evaluate the effectiveness of our algorithm in three scenarios: territorial outlines, road networks, and metro maps. For the latter, we combine our approach with an algorithm to more evenly distribute stations. Our experiments show that our algorithm produces high-quality results for territorial outlines and metro maps. However, the lack of caricature (exaggeration of typical features) makes it less useful for road networks.
@article{map-schematization-with-circular-arcs:2014,
title = {Map Schematization with Circular Arcs},
author = {T. van Dijk and A.I. van Goethem and J.H. Haunert and W. Meulemans and B. Speckmann},
bookTitle = {8th International Conference on Geographic Information Science (GIScience)},
26th Canadian Conference on Computational Geometry (CCCG 2014), 11-13 August 2014, Halifax, Canada, pp. 233—238, 2014.
<p>We consider the following question: How many edgedisjoint plane spanning trees are contained in a complete geometric graph GK<sub>n</sub> on any set S of n points in general position in the plane?</p>
bookTitle = {26th Canadian Conference on Computational Geometry (CCCG 2014), 11-13 August 2014, Halifax, Canada},
Trajectory Grouping Structure: the Video
K. Buchin, M. Buchin, M.J. Kreveld, van, B. Speckmann, and F. Staals.
30th Annual Symposium on Computational Geometry (SOCG'14, Kyoto, Japan, June 8-11, 2014), pp. 88—89, 2014.
@article{trajectory-grouping-structure-the-video:2014,
title = {Trajectory Grouping Structure: the Video},
author = {K. Buchin and M. Buchin and M.J. Kreveld, van and B. Speckmann and F. Staals},
bookTitle = {30th Annual Symposium on Computational Geometry (SOCG'14, Kyoto, Japan, June 8-11, 2014)},
Travel-Time Maps: Linear Cartograms with Fixed Vertex Locations
K. Buchin, A.I. van Goethem, M. Hoffmann, M.J. van Kreveld, and B. Speckmann.
Proc. of the 8th International Conference on Geographic Information Science (GIScience), pp. 18—33, 2014.
Linear cartograms visualize travel times between locations, usually by deforming the underlying map such that Euclidean distance corresponds to travel time. We introduce an alternative model, where the map and the locations remain fixed, but edges are drawn as sinusoid curves. Now the travel time over a road corresponds to the length of the curve. Of course the curves might intersect if not placed carefully. We study the corresponding algorithmic problem and show that suitable placements can be computed efficiently. However, the problem of placing as many curves as possible in an ideal, centered position is NP-hard. We introduce three heuristics to optimize the number of centered curves and show how to create animated visualizations.
@article{travel-time-maps-linear-cartograms-with-fixed-vertex-locations:2014,
title = {Travel-Time Maps: Linear Cartograms with Fixed Vertex Locations},
author = {K. Buchin and A.I. van Goethem and M. Hoffmann and M.J. van Kreveld and B. Speckmann},
Distance-Sensitive Point Location Made Easy
B. Aronov, M.T. Berg, de, D. Eppstein, M.J.M. Roeloffzen, and B. Speckmann.
pp. 1—4, 2014.
Let S be a planar polygonal subdivision with n edges contained in the unit square. We present a data structure for point location in S where queries with points far away from any region boundary are answered faster. More precisely, we show that point location queries can be answered in time O(1 + min(log \frac{1}{\Delta_p} , log n)), where {\Delta_p} is the Euclidean distance of the query point p to the boundary of the region containing p. Our solution consists of a depth-bounded quadtree and a general point location structure, both of which can be constructed in O(n log n)time. We also show how to extend the result to convex polyhedral subdivisions in three dimensions.
@article{distance-sensitive-point-location-made-easy:2014,
title = {Distance-Sensitive Point Location Made Easy},
author = {B. Aronov and M.T. Berg, de and D. Eppstein and M.J.M. Roeloffzen and B. Speckmann},
Sea Regions for Rectangular Cartograms
K. Buchin, R.G. Cano, P.J. Rezende, de, C.C. Souza, de, and B. Speckmann.
In a rectangular cartogram, each region of a map is represented by a rectangle whose area is proportional to some statistical data of interest. Current techniques for constructing rectangular cartograms partition a large rectangle (the map) into a set of smaller rectangles which correspond to land or sea regions. The position and size of sea rectangles determine the outline of land masses. Therefore, sea regions have a direct impact on the recognizability and, thus, on the visual quality of cartograms. In this paper, we describe the first algorithm for the automated creation of sea regions for rectangular cartograms and present results obtained with our method.
@article{sea-regions-for-rectangular-cartograms:2014,
title = {Sea Regions for Rectangular Cartograms},
author = {K. Buchin and R.G. Cano and P.J. Rezende, de and C.C. Souza, de and B. Speckmann},
Flip Graphs of Bounded-Degree Triangulations
O. Aichholzer, T. Hackl, D. Orden, P. Ramos, G. Rote, A. Schulz, and B. Speckmann.
Graphs and Combinatorics, 29(6):1577—1593, 2013.
We study ¿ip graphs of triangulations whose maximum vertex degree is bounded by a constant k. In particular, we consider triangulations of sets of n points in convex position in the plane and prove that their ¿ip graph is connected if and only if k > 6; the diameter of the ¿ip graph is O(n2). We also show that, for general point sets, ¿ip graphs of pointed pseudo-triangulations can be disconnected for k = 9, and ¿ip graphs of triangulations can be disconnected for any k. Additionally, we consider a relaxed version of the original problem. We allow the violation of the degree bound k by a small constant. Any two triangulations with maximum degree at most k of a convex point set are connected in the ¿ip graph by a path of length O(n log n), where every intermediate triangulation has maximum degree at most k + 4. Keywords: Flip graphs - Triangulations - Rotation distance - Connectivity - Degree bounds
@article{flip-graphs-of-bounded-degree-triangulations:2013,
title = {Flip Graphs of Bounded-Degree Triangulations},
author = {O. Aichholzer and T. Hackl and D. Orden and P. Ramos and G. Rote and A. Schulz and B. Speckmann},
KelpFusion: a Hybrid Set Visualization Technique
W. Meulemans, N. Henry Riche, B. Speckmann, B. Alper, and T. Dwyer.
We present KelpFusion: a method for depicting set membership of items on a map or other visualization using continuous boundaries. KelpFusion is a hybrid representation that bridges hull techniques such as Bubble Sets and Euler Diagrams and line- and graph-based techniques such as LineSets and Kelp Diagrams. We describe an algorithm based on shortest-path graphs to compute KelpFusion visualizations. Based on a single parameter, the shortest-path graph varies from the minimal spanning tree to the convex hull of a point set. Shortest-path graphs aim to capture the shape of a point set and smoothly adapt to sets of varying densities. KelpFusion fills enclosed faces based on a set of simple legibility rules. We present the results of a controlled experiment comparing KelpFusion to Bubble Sets and LineSets. We conclude that KelpFusion outperforms Bubble Sets both in accuracy and completion time, and outperforms LineSets in completion time. Keywords: Information visualization, visualization techniques and methodologies
@article{kelpfusion-a-hybrid-set-visualization-technique:2013,
title = {KelpFusion: a Hybrid Set Visualization Technique},
author = {W. Meulemans and N. Henry Riche and B. Speckmann and B. Alper and T. Dwyer},
Maximizing Maximal Angles for Plane Straight-Line Graphs
O. Aichholzer, T. Hackl, M. Hoffmann, C. Huemer, F. Santos, B. Speckmann, and B. Vogtenhuber.
Computational Geometry, 46(1):17—28, 2013.
Let G=(S,E) be a plane straight-line graph on a finite point set S¿R2 in general position. The incident angles of a point p¿S in G are the angles between any two edges of G that appear consecutively in the circular order of the edges incident to p. A plane straight-line graph is called f-open if each vertex has an incident angle of size at least f. In this paper we study the following type of question: What is the maximum angle f such that for any finite set S¿R2 of points in general position we can find a graph from a certain class of graphs on S that is f-open? In particular, we consider the classes of triangulations, spanning trees, and spanning paths on S and give tight bounds in most cases.
@article{maximizing-maximal-angles-for-plane-straight-line-graphs:2013,
title = {Maximizing Maximal Angles for Plane Straight-Line Graphs},
author = {O. Aichholzer and T. Hackl and M. Hoffmann and C. Huemer and F. Santos and B. Speckmann and B. Vogtenhuber},
Topologically Safe Curved Schematisation
A.I. Goethem, van, W. Meulemans, A. Reimer, H.J. Haverkort, and B. Speckmann.
The Cartographic Journal, 50(3):276—285, 2013.
Traditionally schematised maps make extensive use of curves. However, automated methods for schematisation are mostly restricted to straight lines. We present a generic framework for topology-preserving curved schematisation that allows a choice of quality measures and curve types. The framework fits a curve to every part of the input. It uses Voronoi diagrams to ensure that curves fitted to disjoint parts do not intersect. The framework then employs a dynamic program to find an optimal schematisation using the fitted curves. Our fully-automated approach does not need critical points or salient features. We illustrate our framework with Bézier curves and circular arcs.
@article{topologically-safe-curved-schematisation:2013,
title = {Topologically Safe Curved Schematisation},
author = {A.I. Goethem, van and W. Meulemans and A. Reimer and H.J. Haverkort and B. Speckmann},
bookTitle = {The Cartographic Journal},
Accentuating Focus Maps Via Partial Schematization
21st ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems (ACM GIS), pp. 418—421, 2013.
We present an algorithm for schematized focus maps. Focus maps integrate a high detailed, enlarged focus region continuously in a given base map. Recent methods integrate both with such low distortion that the focus region becomes hard to identify. We combine focus maps with partial schematization to display distortion of the context and to emphasize the focus region. Schematization visually conveys geographical accuracy, while not increasing map complexity. We extend the focus-map algorithm to incorporate geometric proximity relationships and show how to combine focus maps with schematization in order to cater to different use cases.
@article{accentuating-focus-maps-via-partial-schematization:2013,
title = {Accentuating Focus Maps Via Partial Schematization},
bookTitle = {21st ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems (ACM GIS)},
F. Hurtado, M. Korman, M.J. Kreveld, van, M. Löffler, V. Sacristán, R.I. Silveira, and B. Speckmann.
Graph Drawing : 21st International Symposium, GD 2013, Bordeaux, France, September 23-25, 2013, Revised Selected Papers, pp. 280—291, 2013.
We study an algorithmic problem that is motivated by ink minimization for sparse set visualizations. Our input is a set of points in the plane which are either blue, red, or purple. Blue points belong exclusively to the blue set, red points belong exclusively to the red set, and purple points belong to both sets. A red-blue-purple spanning graph (RBP spanning graph) is a set of edges connecting the points such that the subgraph induced by the red and purple points is connected, and the subgraph induced by the blue and purple points is connected. We study the geometric properties of minimum RBP spanning graphs and the algorithmic problems associated with computing them. Specifically, we show that the general problem is NP-hard. Hence we give an (½ ¿ + 1)-approximation, where ¿ is the Steiner ratio. We also present efficient exact solutions if the points are located on a line or a circle. Finally we consider extensions to more than two sets.
author = {F. Hurtado and M. Korman and M.J. Kreveld, van and M. Löffler and V. Sacristán and R.I. Silveira and B. Speckmann},
bookTitle = {Graph Drawing : 21st International Symposium, GD 2013, Bordeaux, France, September 23-25, 2013, Revised Selected Papers},
B. Aronov, M.T. Berg, de, M.J.M. Roeloffzen, and B. Speckmann.
Algorithms and Data Structures (13th International Symposium, WADS 2013, London, ON, Canada, August 12-14, 2013. Proceedings), pp. 49—60, 2013.
Let be a connected planar polygonal subdivision with n edges and of total area 1. We present a data structure for point location in TeX where queries with points far away from any region boundary are answered faster. More precisely, we show that point location queries can be answered in time TeX, where ¿p is the distance of the query point p to the boundary of the region containing p. Our structure is based on the following result: any simple polygon P can be decomposed into a linear number of convex quadrilaterals with the following property: for any point p¿¿¿P, the quadrilateral containing p has area TeX.
author = {B. Aronov and M.T. Berg, de and M.J.M. Roeloffzen and B. Speckmann},
bookTitle = {Algorithms and Data Structures (13th International Symposium, WADS 2013, London, ON, Canada, August 12-14, 2013. Proceedings)},
6th IEEE Pacific Visualization Symposium (PacificVis, Sydney, Australia, February 26-March 1, 2013), pp. 25—32, 2013.
Associating the regions of a geographic subdivision with the cells of a grid is a basic operation that is used in various types of maps, like spatially ordered treemaps and OD maps. In these cases the regular shapes of the grid cells allows easy representation of extra information about the regions. The main challenge is to find an association that allows a user to find a region in the grid quickly. We call the representation of a set of regions as a grid a grid map.
bookTitle = {6th IEEE Pacific Visualization Symposium (PacificVis, Sydney, Australia, February 26-March 1, 2013)},
Kinetic 2-Centers in the Black-Box Model
M.T. Berg, de, M.J.M. Roeloffzen, and B. Speckmann.
Proc. 29th ACM Symposium on Computational Geometry (SoCG), pp. 145—154, 2013.
We study two versions of the 2-center problem for moving points in the plane. Given a set P of n points, the Euclidean 2-center problem asks for two congruent disks of minimum size that together cover P; the rectilinear 2-center problem correspondingly asks for two congruent axis-aligned squares of minimum size that together cover P. Our methods work in the black-box KDS model, where we receive the locations of the points at regular time steps and we know an upper bound d_{max} on the maximum displacement of any point within one time step. We show how to maintain the rectilinear 2-center in amortized sub-linear time per time step, under certain assumptions on the distribution of the point set P. For the Euclidean 2-center we give a similar result: we can maintain in amortized sub-linear time (again under certain assumptions on the distribution) a (1+e)-approximation of the optimal 2-center. In many cases---namely when the distance between the centers of the disks is relatively large or relatively small---the solution we maintain is actually optimal. We also present results for the case where the maximum speed of the centers is bounded. We describe a simple scheme to maintain a 2-approximation of the rectilinear 2-center, and we provide a scheme which gives a better approximation factor depending on several parameters of the point set and the maximum allowed displacement of the centers. The latter result can be used to obtain a 2.29-approximation for the Euclidean 2-center; this is an improvement over the previously best known bound of 8/p approx 2.55. These algorithms run in amortized sub-linear time per time step, as before under certain assumptions on the distribution.
@article{kinetic-2-centers-in-the-black-box-model:2013,
title = {Kinetic 2-Centers in the Black-Box Model},
author = {M.T. Berg, de and M.J.M. Roeloffzen and B. Speckmann},
bookTitle = {Proc. 29th ACM Symposium on Computational Geometry (SoCG)},
D. Eppstein, D.H.R. Holten, M. Löffler, M. Nöllenburg, B. Speckmann, and K.A.B. Verbeek.
Graph Drawing (21st International Symposium, GD 2013, Bordeaux, France, September 23-25, 2013. Revised Selected Papers), pp. 352—363, 2013.
We define strict confluent drawing, a form of confluent drawing in which the existence of an edge is indicated by the presence of a smooth path through a system of arcs and junctions (without crossings), and in which such a path, if it exists, must be unique. We prove that it is NP-complete to determine whether a given graph has a strict confluent drawing but polynomial to determine whether it has an outerplanar strict confluent drawing with a fixed vertex ordering (a drawing within a disk, with the vertices placed in a given order on the boundary).
author = {D. Eppstein and D.H.R. Holten and M. Löffler and M. Nöllenburg and B. Speckmann and K.A.B. Verbeek},
bookTitle = {Graph Drawing (21st International Symposium, GD 2013, Bordeaux, France, September 23-25, 2013. Revised Selected Papers)},
Algorithms and Data Structures (13th International Symposium, WADS 2013, London, ON, Canada, August 12-14, 2013. Proceedings), pp. 219—230, 2013.
The collective motion of a set of moving entities like people, birds, or other animals, is characterized by groups arising, merging, splitting, and ending. Given the trajectories of these entities, we define and model a structure that captures all of such changes using the Reeb graph, a concept from topology. The trajectory grouping structure has three natural parameters, namely group size, group duration, and entity inter-distance. These parameters allow us to obtain detailed or global views of the data. We prove complexity bounds on the maximum number of maximal groups that can be present, and give algorithms to compute the grouping structure efficiently. Furthermore, we showcase the results of experiments using data generated by the NetLogo flocking model and from the Starkey project. Although there is no ground truth for the groups in this data, the experiments show that the trajectory grouping structure is plausible and has the desired effects when changing the essential parameters. Our research provides the first complete study of trajectory group evolvement, including combinatorial, algorithmic, and experimental results.
We study the shortcut Fréchet distance, a natural variant of the Fréchet distance that allows us to take shortcuts from and to any point along one of the curves. We show that, surprisingly, the problem of computing the shortcut Fréchet distance exactly is NP-hard. Furthermore, we give a 3-approximation algorithm for the decision version of the problem.
Kinetic Euclidean 2-Centers in the Black-Box Model
We study the 2-center problem for moving points in the plane. Given a set P of n points, the Euclidean 2-center problem asks for two congruent disks of minimum size that together cover P. Our methods work in the black-box KDS model, where we receive the locations of the points at regular time steps and we know an upper bound d_max on the maximum displacement of any point within one time step. We show how to maintain a (1 + e)-approximation of the Euclidean 2-center in amortized sub-linear time per time step, under certain assumptions on the distribution of the point set P. In many cases --namely when the distance between the centers of the disks is relatively large or relatively small-- the solution we maintain is actually optimal.
@article{kinetic-euclidean-2-centers-in-the-black-box-model:2013,
title = {Kinetic Euclidean 2-Centers in the Black-Box Model},
Topologically Safe Curved Schematization
A.I. van Goethem, H.J. Haverkort, W. Meulemans, A. Reimer, and B. Speckmann.
Traditionally schematized maps make extensive use of curves. However, automated methods for schematization are mostly restricted to straight lines. We present a generic framework for topology-preserving curved schematization that allows a choice of quality measures and curve types. Our fully-automated approach does not need critical points or salient features. We illustrate our framework with Bézier curves and circular arcs.
@article{topologically-safe-curved-schematization:2013,
title = {Topologically Safe Curved Schematization},
author = {A.I. van Goethem and H.J. Haverkort and W. Meulemans and A. Reimer and B. Speckmann},
The collective motion of a set of moving entities like people, birds, or other animals, is characterized by groups arising, merging, splitting, and ending. Given the trajectories of these entities, we define and model a structure that captures all of such changes using the Reeb graph, a concept from topology. The trajectory grouping structure has three natural parameters that allow more global views of the data in group size, group duration, and entity inter-distance. We prove complexity bounds on the maximum number of maximal groups that can be present, and give algorithms to compute the grouping structure efficiently. We also study how the trajectory grouping structure can be made robust, that is, how brief interruptions of groups can be disregarded in the global structure, adding a notion of persistence to the structure. Furthermore, we showcase the results of experiments using data generated by the NetLogo flocking model and from the Starkey project. The Starkey data describe the movement of elk, deer, and cattle. Although there is no ground truth for the grouping structure in this data, the experiments show that the trajectory grouping structure is plausible and has the desired effects when changing the essential parameters. Our research provides the first complete study of trajectory group evolvement, including combinatorial, algorithmic, and experimental results.
D. Eppstein, M. Garderen, van, B. Speckmann, and T. Ueckerdt.
author = {D. Eppstein and M. Garderen, van and B. Speckmann and T. Ueckerdt},
Analysis and Visualization of Animal Movement
J. Shamoun-Baranes, E.E. Loon, van, R.S. Purves, B. Speckmann, D. Weiskopf, and C.J. Camphuysen.
Biology Letters, 8(1):6—9, 2012.
The interdisciplinary workshop 'Analysis and Visualization of Moving Objects' was held at the Lorentz Centre in Leiden, The Netherlands, from 27 June to 1 July 2011. It brought together international specialists from ecology, computer science and geographical information science actively involved in the exploration, visualization and analysis of moving objects, such as marine reptiles, mammals, birds, storms, ships, cars and pedestrians. The aim was to share expertise, methodologies, data and common questions between different fields, and to work towards making significant advances in movement research. A data challenge based on GPS tracking of lesser black-backed gulls (Larus fuscus) was used to stimulate initial discussions, cross-fertilization between research groups and to serve as an initial focus for activities during the workshop.
@article{analysis-and-visualization-of-animal-movement:2012,
title = {Analysis and Visualization of Animal Movement},
author = {J. Shamoun-Baranes and E.E. Loon, van and R.S. Purves and B. Speckmann and D. Weiskopf and C.J. Camphuysen},
bookTitle = {Biology Letters},
Area-Universal and Constrained Rectangular Layouts
D. Eppstein, E. Mumford, B. Speckmann, and K.A.B. Verbeek.
SIAM Journal on Computing, 41(3):537—564, 2012.
A rectangular layout is a partition of a rectangle into a finite set of interior-disjoint rectangles. These layouts are used as rectangular cartograms in cartography, as floorplans in building architecture and VLSI design, and as graph drawings. Often areas are associated with the rectangles of a rectangular layout and it is desirable for one rectangular layout to represent several area assignments. A layout is area-universal if any assignment of areas to rectangles can be realized by a combinatorially equivalent rectangular layout. We identify a simple necessary and sufficient condition for a rectangular layout to be area-universal: a rectangular layout is area-universal if and only if it is one-sided. We also investigate similar questions for perimeter assignments. The adjacency requirements for the rectangles of a rectangular layout can be specified in various ways, most commonly via the dual graph of the layout. We show how to find an area-universal layout for a given set of adjacency requirements whenever such a layout exists. Furthermore we show how to impose restrictions on the orientations of edges and junctions of the rectangular layout. Such an orientation-constrained layout, if it exists, may be constructed in polynomial time, and all orientation-constrained layouts may be listed in polynomial time per layout.
@article{area-universal-and-constrained-rectangular-layouts:2012,
title = {Area-Universal and Constrained Rectangular Layouts},
author = {D. Eppstein and E. Mumford and B. Speckmann and K.A.B. Verbeek},
bookTitle = {SIAM Journal on Computing},
Kelp Diagrams: Point Set Membership Visualization
K. Dinkla, M.J. Kreveld, van, B. Speckmann, and M.A. Westenberg.
We present Kelp Diagrams, a novel method to depict set relations over points, i.e., elements with predefined positions. Our method creates schematic drawings and has been designed to take aesthetic quality, efficiency, and effectiveness into account. This is achieved by a routing algorithm, which links elements that are part of the same set by constructing minimum cost paths over a tangent visibility graph. There are two styles of Kelp Diagrams to depict overlapping sets, a nested and a striped style, each with its own strengths and weaknesses. We compare Kelp Diagrams with two existing methods and show that our approach provides a more consistent and clear depiction of both element locations and their set relations.
@article{kelp-diagrams-point-set-membership-visualization:2012,
title = {Kelp Diagrams: Point Set Membership Visualization},
author = {K. Dinkla and M.J. Kreveld, van and B. Speckmann and M.A. Westenberg},
Kinetic Convex Hulls, Delaunay Triangulations and Connectivity Structures in the Black-Box Model
Journal of Computational Geometry, 3(1):222—249, 2012.
Over the past decade, the kinetic-data-structures framework has become thestandard in computational geometry for dealing with moving objects. A fundamental assumption underlying the framework is that the motions of the objects are known in advance. This assumption severely limits the applicability of KDSs. We study KDSs in the black-box model, which is a hybrid of the KDS model and the traditional time-slicing approach. In this more practical model we receive the position of each object at regular time steps and we have an upper bound on dmax, the maximum displacement of any point in one time step. We study the maintenance of the convex hull and the Delaunay triangulation of a planar point set P in the black-box model, under the following assumption on dmax: there is some constant k such that for any point p in P the disk of radius dmax contains at most k points. We analyze our algorithms in terms of ¿k, the so-called k-spread of P. We show how to update the convex hull at each time step in O(min(n, k¿klog n)log n) amortized time. For the Delaunay triangulation our main contribution is an analysis of the standard edge-flipping approach; we show that the number of flips is O(k2¿k2) at each time step.
@article{kinetic-convex-hulls-delaunay-triangulations-and-connectivity-structures-in-the-black-box-model:2012,
title = {Kinetic Convex Hulls, Delaunay Triangulations and Connectivity Structures in the Black-Box Model},
Shooting Permanent Rays Among Disjoint Polygons in the Plane
M. Ishaque, B. Speckmann, and Cs.D. Tóth.
SIAM Journal on Computing, 41(4):1005—1027, 2012.
We present a data structure for ray shooting and insertion in the free space between disjoint polygonal obstacles with a total of $n$ vertices in the plane, where each ray starts at the boundary of some obstacle. The portion of each query ray between the starting point and the first obstacle hit is inserted permanently as a new obstacle. Our data structure uses $O(n\log n)$ space and preprocessing time, and it supports $m$ successive ray shooting and insertion queries in $O((n+m)\log^2 n + m\log m)$ total time in the real RAM model of computation. We present two applications: (1) Our data structure supports efficient implementation of auto-partitions in the plane, that is, binary space partitions where each partition is done along the supporting line of an input segment. If $n$ input line segments are fragmented into $m$ pieces by an auto-partition, then it can now be implemented in $O(n\log^2n+m\log m)$ time. This improves the expected runtime of Patersen and Yao's classical randomized auto-partition algorithm for $n$ disjoint line segments in the plane to $O(n\log^2 n)$. (2) If we are given disjoint polygonal obstacles with a total of $n$ vertices in the plane, a permutation of the reflex vertices, and a half-line at each reflex vertex that partitions the reflex angle into two convex angles, then the convex partitioning algorithm draws a ray emanating from each reflex vertex in the prescribed order in the given direction until it hits another obstacle, a previous ray, or infinity. The previously best implementation (with a semidynamic ray shooting data structure) requires $O(n^{3/2-\varepsilon/2})$ time using $O(n^{1+\varepsilon})$ space for any $\varepsilon>0$. Our data structure improves the runtime to $O(n\log^2 n)$.
@article{shooting-permanent-rays-among-disjoint-polygons-in-the-plane:2012,
title = {Shooting Permanent Rays Among Disjoint Polygons in the Plane},
author = {M. Ishaque and B. Speckmann and Cs.D. Tóth},
Context-Aware Similarity of Trajectories
Geographic Information Science (7th International Conference, GIScience 2012, Columbus, OH, USA, September 18-21, 2012. Proceedings), pp. 43—56, 2012.
The movement of animals, people, and vehicles is embedded in a geographic context. This context influences the movement. Most analysis algorithms for trajectories have so far ignored context, which severely limits their applicability. In this paper we present a model for geographic context that allows us to integrate context into the analysis of movement data. Based on this model we develop simple but efficient context-aware similarity measures. We validate our approach by applying these measures to hurricane trajectories. Keywords: Movement data – geographic context – similarity measures
@article{context-aware-similarity-of-trajectories:2012,
title = {Context-Aware Similarity of Trajectories},
bookTitle = {Geographic Information Science (7th International Conference, GIScience 2012, Columbus, OH, USA, September 18-21, 2012. Proceedings)},
Evolution Strategies for Optimizing Rectangular Cartograms
A rectangular cartogram is a type of map where every region is a rectangle. The size of the rectangles is chosen such that their areas represent a geographic variable such as population or GDP. In recent years several algorithms for the automated construction of rectangular cartograms have been proposed, some of which are based on rectangular duals of the dual graph of the input map. In this paper we present a new approach to efficiently search within the exponentially large space of all possible rectangular duals. We employ evolution strategies that find rectangular duals which can be used for rectangular cartograms with correct adjacencies and (close to) zero cartographic error. This is a considerable improvement upon previous methods that have to either relax adjacency requirements or deal with larger errors. We present extensive experimental results for a large variety of data sets. Keywords: Rectangular cartogram – evolution strategy – regular edge labeling
@article{evolution-strategies-for-optimizing-rectangular-cartograms:2012,
title = {Evolution Strategies for Optimizing Rectangular Cartograms},
Kinetic Compressed Quadtrees in the Black-Box Model with Applications to Collision Detection for Low-Density Scenes
Algorithms - ESA 2012 (20th European Symposium on Algorithms, Ljubljana, Slovenia, September 10-12, 2012. Proceedings), pp. 383—394, 2012.
We present an efficient method for maintaining a compressed quadtree for a set of moving points in R d . Our method works in the black-box KDS model, where we receive the locations of the points at regular time steps and we know a bound d max on the maximum displacement of any point within one time step. When the number of points within any ball of radius d max is at most k at any time, then our update algorithm runs in O(nlogk) time. We generalize this result to constant-complexity moving objects in R d . The compressed quadtree we maintain has size O(n); under similar conditions as for the case of moving points it can be maintained in O(n log¿) time per time step, where ¿ is the density of the set of objects. The compressed quadtree can be used to perform broad-phase collision detection for moving objects; it will report in O((¿¿+¿k)n) time a superset of all intersecting pairs of objects.
@article{kinetic-compressed-quadtrees-in-the-black-box-model-with-applications-to-collision-detection-for-low-density-scenes:2012,
title = {Kinetic Compressed Quadtrees in the Black-Box Model with Applications to Collision Detection for Low-Density Scenes},
bookTitle = {Algorithms - ESA 2012 (20th European Symposium on Algorithms, Ljubljana, Slovenia, September 10-12, 2012. Proceedings)},
K. Buchin, M. Buchin, W. Meulemans, and B. Speckmann.
20th European Symposium on Algorithms (ESA), pp. 229—240, 2012.
The Fréchet distance is a metric to compare two curves, which is based on monotonous matchings between these curves. We call a matching that results in the Fréchet distance a Fréchet matching. There are often many different Fréchet matchings and not all of these capture the similarity between the curves well. We propose to restrict the set of Fréchet matchings to "natural" matchings and to this end introduce locally correct Fréchet matchings. We prove that at least one such matching exists for two polygonal curves and give an O(N^3 log N) algorithm to compute it, where N is the total number of edges in both curves. We also present an O(N^2) algorithm to compute a locally correct discrete Fréchet matching.
author = {K. Buchin and M. Buchin and W. Meulemans and B. Speckmann},
bookTitle = {20th European Symposium on Algorithms (ESA)},
Kinetic Collision Detection for Low-Density Scenes in the Black-Box Model
We present an efficient method for collision detection in the black-box KDS model for a set S of n objects in the plane. In this model we receive the object locations at regular time steps and we know a bound dmax on the maximum displacement of any object within one time step. Our method maintains, in O((¿+k)n) time per time step, a compressed quadtree on the bounding-box vertices of the objects; here ¿ denotes the density of S and k denotes the maximum number of objects that can intersect any disk of radius dmax. Collisions can then be detected by testing O((¿+k)2n) pairs of objects for intersection.
@article{kinetic-collision-detection-for-low-density-scenes-in-the-black-box-model:2012,
title = {Kinetic Collision Detection for Low-Density Scenes in the Black-Box Model},
The Fréchet distance is a metric to compare two curves, which is based on monotonous matchings between these curves. We call a matching that results in the Fréchet distance a Fréchet matching. There are often many different Fréchet matchings and not all of these capture the similarity between the curves well. We propose to restrict the set of Fréchet matchings to "natural" matchings and to this end introduce locally correct Fréchet matchings. We prove that at least one such a matching exists for two polygonal curves and give an algorithm to compute it.
One-To-One Point Set Matchings for Grid Map Layout
D. Eppstein, M.J. van Kreveld, B. Speckmann, and F. Staals.
We study several one-to-one point set matching problems which are motivated by layout problems for grid maps. We are given two sets A and B of n points in the plane, and we wish to compute an optimal one-to-one matching between A and B. We consider two optimisation criteria: minimising the sum of the L1-distances between matched points, and maximising the number of pairs of points in A for which the matching preserves the directional relation. We show how to minimise the total L1-distance under translation or scaling in O(n6 log3 n) time, and under both translation and scaling in O(n10 log3 n) time. We further give a 4-approximation for preserving directional relations by computing a minimum L1-distance matching in O(n2 log3 n) time.
@article{one-to-one-point-set-matchings-for-grid-map-layout:2012,
title = {One-To-One Point Set Matchings for Grid Map Layout},
author = {D. Eppstein and M.J. van Kreveld and B. Speckmann and F. Staals},
The Frechet distance is a metric to compare two curves, which is based on monotonous matchings between these curves. We call a matching that results in the Frechet distance a Frechet matching. There are often many different Frechet matchings and not all of these capture the similarity between the curves well. We propose to restrict the set of Frechet matchings to "natural" matchings and to this end introduce locally correct Frechet matchings. We prove that at least one such matching exists for two polygonal curves and give an O(N^3 log N) algorithm to compute it, where N is the total number of edges in both curves. We also present an O(N^2) algorithm to compute a locally correct discrete Frechet matching.
Guest Editor's Foreword (Special Issue with Selected Papers from the 19th International Symposium on Graph Drawing, GD 2011)
M.J. Kreveld, van and B. Speckmann.
Journal of Graph Algorithms and Applications, 16(3):631—633, 2012.
This issue of the Journal of Graph Algorithms and Applications is devoted to the nineteenth International Symposium on Graph Drawing, held September 19-21, 2011, in Eindhoven, the Netherlands.
@article{guest-editor-s-foreword-special-issue-with-selected-papers-from-the-19th-international-symposium-on-graph-drawing-gd-2011-:2012,
title = {Guest Editor's Foreword (Special Issue with Selected Papers from the 19th International Symposium on Graph Drawing, GD 2011)},
author = {M.J. Kreveld, van and B. Speckmann},
Connect the Dot: Computing Feed-Links for Network Extension
B. Aronov, K. Buchin, M. Buchin, B.M.P. Jansen, T. Jong, de, M.J. Kreveld, van, M. Löffler, J. Luo, R.I. Silveira, and B. Speckmann.
Journal of Spatial Information Science, 3:3—31, 2011.
Road network analysis can require distance from points that are not on the network themselves. We study the algorithmic problem of connecting a point inside a face (region) of the road network to its boundary while minimizing the detour factor of that point to any point on the boundary of the face. We show that the optimal single connection (feed-link) can be computed in O(¿7(n) log n) time, where n is the number of vertices that bounds the face and ¿7(n) is the slightly superlinear maximum length of a Davenport-Schinzel sequence of order 7. We also present approximation results for placing more feed-links, deal with the case that there are obstacles in the face of the road network that contains the point to be connected, and present various related results.
@article{connect-the-dot-computing-feed-links-for-network-extension:2011,
title = {Connect the Dot: Computing Feed-Links for Network Extension},
author = {B. Aronov and K. Buchin and M. Buchin and B.M.P. Jansen and T. Jong, de and M.J. Kreveld, van and M. Löffler and J. Luo and R.I. Silveira and B. Speckmann},
Empty Pseudo-Triangles in Point Sets
H.K. Ahn, S.W. Bae, M.J. Kreveld, van, I. Reinbacher, and B. Speckmann.
Discrete Applied Mathematics, 159(18):2205—2213, 2011.
We study empty pseudo-triangles in a set P of n points in the plane, where an empty pseudo-triangle has its vertices at the points of P, and no points of P lie inside. We give bounds on the minimum and maximum number of empty pseudo-triangles. If P lies inside a triangle whose corners must be the convex vertices of the pseudo-triangle, then there can be between T(n2) and T(n3) empty pseudo-triangles. If the convex vertices of the pseudo-triangle are also chosen from P, this number lies between T(n3) and T(n6). If we count only star-shaped pseudo-triangles, the bounds are T(n2) and T(n5). We also study optimization problems: minimizing or maximizing the perimeter or the area over all empty pseudo-triangles defined by P. If P lies inside a triangle whose corners must be used, we can solve these problems in O(n3) time. In the general case, the running times are O(n6) for the maximization problems and O(nlogn) for the minimization problems.
@article{empty-pseudo-triangles-in-point-sets:2011,
title = {Empty Pseudo-Triangles in Point Sets},
author = {H.K. Ahn and S.W. Bae and M.J. Kreveld, van and I. Reinbacher and B. Speckmann},
bookTitle = {Discrete Applied Mathematics},
Flow Map Layout Via Spiral Trees
K.A.B. Verbeek, K. Buchin, and B. Speckmann.
Flow maps are thematic maps that visualize the movement of objects, such as people or goods, between geographic regions. One or more sources are connected to several targets by lines whose thickness corresponds to the amount of flow between a source and a target. Good flow maps reduce visual clutter by merging (bundling) lines smoothly and by avoiding self-intersections. Most flow maps are still drawn by hand and only few automated methods exist. Some of the known algorithms do not support edgebundling and those that do, cannot guarantee crossing-free flows. We present a new algorithmic method that uses edge-bundling and computes crossing-free flows of high visual quality. Our method is based on so-called spiral trees, a novel type of Steiner tree which uses logarithmic spirals. Spiral trees naturally induce a clustering on the targets and smoothly bundle lines. Our flows can also avoid obstacles, such as map features, region outlines, or even the targets. We demonstrate our approach with extensive experiments.
@article{flow-map-layout-via-spiral-trees:2011,
title = {Flow Map Layout Via Spiral Trees},
author = {K.A.B. Verbeek and K. Buchin and B. Speckmann},
Geometric Simultaneous Embeddings of a Graph and a Matching
S. Cabello, M.J. Kreveld, van, G. Liotta, H. Meijer, B. Speckmann, and K.A.B. Verbeek.
Journal of Graph Algorithms and Applications, 15(1):79—96, 2011.
The geometric simultaneous embedding problem asks whether two planar graphs on the same set of vertices in the plane can be drawn using straight lines, such that each graph is plane. Geometric simultaneous embedding is a current topic in graph drawing and positive and negative results are known for various classes of graphs. So far only connected graphs have been considered. In this paper we present the first results for the setting where one of the graphs is a matching. In particular, we show that there exist a planar graph and a matching which do not admit a geometric simultaneous embedding. This strengthens an analogous negative result for a planar graph and a path. On the positive side, we describe algorithms that compute a geometric simultaneous embedding of a matching and a wheel, outerpath, or tree. Our drawing algorithms minimize the number of orientations used to draw the edges of the matching. Specifically, when embedding a matching and a tree, we can draw all matching edges horizontally. When embedding a matching and a wheel or an outerpath, we use only two orientations.
@article{geometric-simultaneous-embeddings-of-a-graph-and-a-matching:2011,
title = {Geometric Simultaneous Embeddings of a Graph and a Matching},
author = {S. Cabello and M.J. Kreveld, van and G. Liotta and H. Meijer and B. Speckmann and K.A.B. Verbeek},
On Planar Supports for Hypergraphs
K. Buchin, M.J. Kreveld, van, H. Meijer, B. Speckmann, and K.A.B. Verbeek.
A graph G is a support for a hypergraph H = (V, S) if the vertices of G correspond to the vertices of H such that for each hyperedge Si ¿ S the subgraph of G induced by Si is connected. G is a planar support if it is a support and planar. Johnson and Pollak [11] proved that it is NPcomplete to decide if a given hypergraph has a planar support. In contrast, there are lienar time algorithms to test whether a given hypergraph has a planar support that is a path, cycle, or tree. In this paper we present an e¿cient algorithm which tests in polynomial time if a given hypergraph has a planar support that is a tree where the maximal degree of each vertex is bounded. Our algorithm is constructive and computes a support if it exists. Furthermore, we prove that it is already NP-hard to decide if a hypergraph has a 2-outerplanar support.
@article{on-planar-supports-for-hypergraphs:2011,
title = {On Planar Supports for Hypergraphs},
author = {K. Buchin and M.J. Kreveld, van and H. Meijer and B. Speckmann and K.A.B. Verbeek},
A New Method for Subdivision Simplification with Applications to Urban-Area Generalization
K. Buchin, W. Meulemans, and B. Speckmann.
19th ACM SIGSPATIAL International Symposium on Advances in Geographic Information Systems (ACM GIS), pp. 261—270, 2011.
We introduce a local operation for polygons and subdivisions called an edge-move. Edge-moves do not change the edge orientations present in the input and are thus suitable for iterative simplification or even schematization. Based on edge-moves we present a new efficient method for area- and topology-preserving subdivision simplification. We show how to tailor this generic method towards the specific needs of building wall squaring and urban-area generalization. Our algorithm is guaranteed to make further progress on any subdivision that has two or more faces and/or reflex vertices. Furthermore, our method produces output of high visual quality and is able to generalize maps with approximately 1.8 million edges in a few hours.
@article{a-new-method-for-subdivision-simplification-with-applications-to-urban-area-generalization:2011,
title = {A New Method for Subdivision Simplification with Applications to Urban-Area Generalization},
author = {K. Buchin and W. Meulemans and B. Speckmann},
bookTitle = {19th ACM SIGSPATIAL International Symposium on Advances in Geographic Information Systems (ACM GIS)},
A Splitting Line Model for Directional Relations
K. Buchin, V.J.J. Kusters, B. Speckmann, F. Staals, and B.N. Vasilescu.
Proceedings 19th ACM SIGSPATIAL International Symposium on Advances in Geographic Information Systems (ACM-GIS 2011, Chicago IL, USA, November 1-4, 2011), pp. 142—151, 2011.
Directional relations are fundamental to spatial data queries, analysis and reasoning. Consequently there has been a significant amount of effort to determine directional relations between two regions. However, many existing methods do not perform well when the regions are neighboring or intertwined. In this paper we introduce a new model for directional relations which is based on a splitting line separating the two regions in question. We identify essential quality criteria for directional relation models and translate them into measurable properties of a given splitting line. We present an efficient algorithm that computes an optimal splitting line for two regions and perform extensive experiments. Our results show that the splitting line model captures directional relations very well and that it clearly outperforms existing approaches on pairs of neighboring or intertwined regions.
@article{a-splitting-line-model-for-directional-relations:2011,
title = {A Splitting Line Model for Directional Relations},
author = {K. Buchin and V.J.J. Kusters and B. Speckmann and F. Staals and B.N. Vasilescu},
bookTitle = {Proceedings 19th ACM SIGSPATIAL International Symposium on Advances in Geographic Information Systems (ACM-GIS 2011, Chicago IL, USA, November 1-4, 2011)},
Algorithms and Computation (22nd International Symposium, ISAAC 2011, Yokohama, Japan, December 5-8, 2011. Proceedings), pp. 250—259, 2011.
We introduce a new variant of the geometric Steiner arborescence problem, motivated by the layout of flow maps. Flow maps show the movement of objects between places. They reduce visual clutter by bundling lines smoothly and avoiding self-intersections. To capture these properties, our angle-restricted Steiner arborescences, or flux trees, connect several targets to a source with a tree of minimal length whose arcs obey a certain restriction on the angle they form with the source. We study the properties of optimal flux trees and show that they are planar and consist of logarithmic spirals and straight lines. Flux trees have the shallow-light property. Computing optimal flux trees is NP-hard. Hence we consider a variant of flux trees which uses only logarithmic spirals. Spiral trees approximate flux trees within a factor depending on the angle restriction. Computing optimal spiral trees remains NP-hard, but we present an efficient 2-approximation, which can be extended to avoid "positive monotone" obstacles.
bookTitle = {Algorithms and Computation (22nd International Symposium, ISAAC 2011, Yokohama, Japan, December 5-8, 2011. Proceedings)},
Delineating Imprecise Regions Via Shortest-Path Graphs
M.T. Berg, de, W. Meulemans, and B. Speckmann.
An imprecise region, also called a vernacular region, is a region without a precise or administrative boundary. We present a new method to delineate imprecise regions from a set of points that are likely to lie inside the region. We use shortest-path graphs based on the squared Euclidean distance which capture the shape of region boundaries well. Shortest-path graphs naturally adapt to point sets of varying density, and they are always connected. As opposed to neighborhood graphs, they use a non-local criterion to determine which points to connect. Furthermore, shortest- path graphs can easily be extended to take geographic context into account by modeling context as "soft" obstacles. We present efficient algorithms to compute shortest-path graphs with or without geographic context. We experimentally evaluate the quality of the imprecise regions computed with our method. To fairly compare our results to those obtained by the common KDE approach, we also show how to integrate context into KDE by again using soft obstacles.
@article{delineating-imprecise-regions-via-shortest-path-graphs:2011,
title = {Delineating Imprecise Regions Via Shortest-Path Graphs},
author = {M.T. Berg, de and W. Meulemans and B. Speckmann},
Kinetic Convex Hulls and Delaunay Triangulations in the Black-Box Model
Proceedings 27th Annual ACM Symposium on Computational Geometry (SoCG'11, Paris, France, June 13-15, 2011), pp. 244—253, 2011.
Over the past decade, the kinetic-data-structures framework has become the standard in computational geometry for dealing with moving objects. A fundamental assumption underlying the framework is that the motions of the objects are known in advance. This assumption severely limits the applicability of KDSs. We study KDSs in the black-box model, which is a hybrid of the KDS model and the traditional time-slicing approach. In this more practical model we receive the position of each object at regular time steps and we have an upper bound on dmax, the maximum displacement of any point in one time step. We study the maintenance of the convex hull and the Delaunay triangulation of a planar point set P in the black-box model, under the following assumption on dmax: there is some constant k such that for any point p ¿ P the disk of radius dmax contains at most k points. We analyze our algorithms in terms of ¿k , the so-called k-spread of P. We show how to update the convex hull at each time step in O(k¿k log2 n) amortized time. For the Delaunay triangulation our main contribution is an analysis of the standard edge-flipping approach; we show that the number of flips is O(k2 ¿k2) at each time step.
@article{kinetic-convex-hulls-and-delaunay-triangulations-in-the-black-box-model:2011,
title = {Kinetic Convex Hulls and Delaunay Triangulations in the Black-Box Model},
bookTitle = {Proceedings 27th Annual ACM Symposium on Computational Geometry (SoCG'11, Paris, France, June 13-15, 2011)},
Optimizing Regular Edge Labelings
Graph Drawing (18th International Symposium, GD'10, Konstanz, Germany, September 21-24, 2010. Revised selected papers), pp. 117—128, 2011.
A regular edge labeling (REL) of an irreducible triangulation G uniquely defines a rectangular dual of G. Rectangular duals find applications in various areas: as floor plans of electronic chips, in architectural designs, as rectangular cartograms, or as treemaps. An irreducible triangulation can have many RELs and hence many rectangular duals. Depending on the specific application different duals might be desirable. In this paper we consider optimization problems on RELs and show how to find optimal or near-optimal RELs for various quality criteria. Furthermore, we give upper and lower bounds on the number of RELs.
@article{optimizing-regular-edge-labelings:2011,
title = {Optimizing Regular Edge Labelings},
bookTitle = {Graph Drawing (18th International Symposium, GD'10, Konstanz, Germany, September 21-24, 2010. Revised selected papers)},
The 2x2 Simple Packing Problem
A.M. Renssen, van and B. Speckmann.
Proceedings of the 23rd Annual Canadian Conference on Computational Geometry (CCCG 2011), August 10-12, 2011, Toronto, Ontario, Canada, pp. 387—392, 2011.
We significantly extend the class of polygons for which the 22 simple packing problem can be solved in polynomial time.
@article{the-2x2-simple-packing-problem:2011,
title = {The 2x2 Simple Packing Problem},
author = {A.M. Renssen, van and B. Speckmann},
bookTitle = {Proceedings of the 23rd Annual Canadian Conference on Computational Geometry (CCCG 2011), August 10-12, 2011, Toronto, Ontario, Canada},
Treemaps are a popular technique to visualize hierarchical data. The input is a weighted tree where the weight of each node is the sum of the weights of its children. A treemap for is a hierarchical partition of a rectangle into simply connected regions, usually rectangles. Each region represents a node of and its area is proportional to the weight of the corresponding node. An important quality criterion for treemaps is the aspect ratio of its regions. One cannot bound the aspect ratio if the regions are restricted to be rectangles. In contrast, polygonal partitions, that use convex polygons, can have bounded aspect ratio. We are the first to obtain convex partitions with optimal aspect ratio O(depth()). However, depth() still depends on the input tree. Hence we introduce a new type of treemaps, namely orthoconvex treemaps, where regions representing leaves are rectangles, L-, and S-shapes, and regions representing internal nodes are orthoconvex polygons. We prove that any input tree, irrespective of the weights of the nodes and the depth of the tree, admits an orthoconvex treemap of constant aspect ratio.
Flow maps visualize the movement of objects between places. One or more sources are connected to several targets by arcs whose thickness corresponds to the amount of flow between a source and a target. Flow maps reduce visual clutter by merging (bundling) lines smoothly and by avoiding self-intersections. We present algorithms that compute crossing-free flows of high visual quality. To this end we introduce a new variant of the geometric Steiner arborescence problem. The goal is to connect the targets to a source with a tree of minimal length whose arcs obey a certain restriction on the angle they form with the source. Such an angle-restricted Steiner arborescence, or simply ow tree, naturally induces a clustering on the targets and smoothly bundles arcs. We study the properties of optimal flow trees and show that they consist of logarithmic spirals and straight lines. Computing optimal flow trees is NP-hard. Hence we consider a variant of flow trees which uses only logarithmic spirals, so called spiral trees. Spiral trees approximate flow trees within a factor depending on the angle restriction. Computing optimal spiral trees remains NP-hard. We present an efficient 2-approximation for spiral trees, which can be extended to avoid certain types of obstacles.
Area-Preserving c-Oriented Schematization
We define an edge-move operation for polygons and prove that every simple non-convex polygon P has a non-conflicting pair of complementary edge-moves that reduces the number of edges of P while preserving its area. We use this result to generate area-preserving C-oriented schematizations of polygons.
@article{area-preserving-c-oriented-schematization:2011,
title = {Area-Preserving c-Oriented Schematization},
Convex Treemaps with Bounded Aspect Ratio
M.T. de Berg, B. Speckmann, and V. van der Weele.
Treemaps are a popular technique to visualize hierarchical data. The input is a weighted tree T where the weight of each node is the sum of the weights of its children. A treemap for T is a hierarchical partition of a rectangle into simply connected regions, usually rectangles. Each region represents a node of T and the area of each region is proportional to the weight of the corresponding node. An important quality criterium for treemaps is the aspect ratio of its regions. Unfortunately, one cannot bound the aspect ratio if the regions are restricted to be rectangles. Hence Onak and Sidiropoulos introduced polygonal partitions, which use convex polygons. We are the first to obtain convex partitions with optimal aspect ratio O(depth(T )). We also consider the important special case that depth(T ) = 1, that is, single-level treemaps. We show how to construct convex single-level treemaps that use only four simple shapes for the regions and have aspect ratio at most 34/7.
@article{convex-treemaps-with-bounded-aspect-ratio:2011,
title = {Convex Treemaps with Bounded Aspect Ratio},
author = {M.T. de Berg and B. Speckmann and V. van der Weele},
Kinetic Convex Hulls in the Black-Box Model
M.T. de Berg, M.J.M. Roeloffzen, and B. Speckmann.
Over the past decade, the kinetic-data-structures framework has become the standard in computational geometry for dealing with moving objects. A fundamental assumption underlying the framework is that the motions of the objects are known in advance. This assumption severely limits the applicability of KDSs. We study KDSs in the black-box model, which is a hybrid of the KDS model and the traditional time-slicing approach. In this more practical model we receive the position of each object at regular time steps and we have an upper bound on d_max, the maximum displacement of any point in one time step. We study the maintenance of the convex hull of a planar point set P in the black-box model, under the following assumption on d_max: there is some constant k such that for any point p in P the disk of radius d_max contains at most k points. We analyze our algorithms in terms of \Delta_k, the so-called k-spread of P. We show how to update the convex hull at each time step in O(k \Delta_k log^2 n) amortized time.
@article{kinetic-convex-hulls-in-the-black-box-model:2011,
title = {Kinetic Convex Hulls in the Black-Box Model},
author = {M.T. de Berg and M.J.M. Roeloffzen and B. Speckmann},
Algorithmic Aspects of Proportional Symbol Maps
S. Cabello, H.J. Haverkort, M.J. Kreveld, van, and B. Speckmann.
Proportional symbol maps visualize numerical data associated with point locations by placing a scaled symbol—typically opaque disks or squares—at the corresponding point on a map. Overlapping symbols need to be drawn in such a way that the user can still judge their relative sizes accurately. We identify two types of suitable drawings: physically realizable drawings and stacking drawings. For these we study the following two problems: Max-Min—maximize the minimum visible boundary length of each symbol—and Max-Total—maximize the total visible boundary length over all symbols. We show that both problems are NP-hard for physically realizable drawings. Max-Min can be solved in O(n2 log n) time for stacking drawings, which can be improved to O(n log n) or O(n log2 n) time when the input has certain properties. We also experimented with four methods to compute stacking drawings: our solution to the Max-Min problem performs best on the data sets considered.
@article{algorithmic-aspects-of-proportional-symbol-maps:2010,
title = {Algorithmic Aspects of Proportional Symbol Maps},
author = {S. Cabello and H.J. Haverkort and M.J. Kreveld, van and B. Speckmann},
Finding the Most Relevant Fragments in Networks
K. Buchin, S. Cabello, J. Gudmundsson, M. Löffler, J. Luo, G. Rote, R.I. Silveira, B. Speckmann, and T. Wolle.
We study a point pattern detection problem on networks, motivated by applications in geographical analysis, such as crime hotspot detection. Given a network N (a connected graph with non-negative edge lengths) together with a set of sites, which lie on the edges or vertices of N, we look for a connected subnetwork F of N of small total length that contains many sites. The edges of F can form parts of the edges of N. We consider different variants of this problem where N is either a general graph or restricted to a tree, and the subnetwork F that we are looking for is either a simple path or a tree. We give polynomial-time algorithms, NP-hardness and NP-completeness proofs, approximation algorithms, and also fixed-parameter tractable algorithms.
@article{finding-the-most-relevant-fragments-in-networks:2010,
title = {Finding the Most Relevant Fragments in Networks},
author = {K. Buchin and S. Cabello and J. Gudmundsson and M. Löffler and J. Luo and G. Rote and R.I. Silveira and B. Speckmann and T. Wolle},
Necklace Maps
Statistical data associated with geographic regions is nowadays globally available in large amounts and hence automated methods to visually display these data are in high demand. There are several well-established thematic map types for quantitative data on the ratio-scale associated with regions: choropleth maps, cartograms, and proportional symbol maps. However, all these maps suffer from limitations, especially if large data values are associated with small regions. To overcome these limitations, we propose a novel type of quantitative thematic map, the necklace map. In a necklace map, the regions of the underlying two-dimensional map are projected onto intervals on a one-dimensional curve (the necklace) that surrounds the map regions. Symbols are scaled such that their area corresponds to the data of their region and placed without overlap inside the corresponding interval on the necklace. Necklace maps appear clear and uncluttered and allow for comparatively large symbol sizes. They visualize data sets well which are not proportional to region sizes. The linear ordering of the symbols along the necklace facilitates an easy comparison of symbol sizes. One map can contain several nested or disjoint necklaces to visualize clustered data. The advantages of necklace maps come at a price: the association between a symbol and its region is weaker than with other types of maps. Interactivity can help to strengthen this association if necessary. We present an automated approach to generate necklace maps which allows the user to interactively control the final symbol placement. We validate our approach with experiments using various data sets and maps.
@article{necklace-maps:2010,
title = {Necklace Maps},
Optimal BSPs and Rectilinear Cartograms
M. Berg, de, E. Mumford, and B. Speckmann.
A cartogram is a thematic map that visualizes statistical data about a set of regions like countries, states or provinces. The size of a region in a cartogram corresponds to a particular geographic variable, for example, population. We present an algorithm for constructing rectilinear cartograms (each region is represented by a rectilinear polygon) with zero cartographic error and correct region adjacencies, and we test our algorithm on various data sets. It produces regions of very small complexity—in fact, most regions are rectangles—while still ensuring both exact areas and correct adjacencies for all regions. Our algorithm uses a novel subroutine that is interesting in its own right, namely a polynomial-time algorithm for computing optimal binary space partitions (BSPs) for rectilinear maps. This algorithm works for a general class of optimality criteria, including size and depth. We use this generality in our application to computing cartograms, where we apply a dedicated cost function leading to BSPs amenable to the constructing of high-quality cartograms.
@article{optimal-bsps-and-rectilinear-cartograms:2010,
title = {Optimal BSPs and Rectilinear Cartograms},
author = {M. Berg, de and E. Mumford and B. Speckmann},
Pointed Binary Encompassing Trees: Simple and Optimal
M. Hoffmann, B. Speckmann, and Cs.D. Tóth.
For n disjoint line segments in the plane we construct in optimal O(nlogn) time and linear space an encompassing tree of maximum degree three such that at every vertex v all edges of the tree that are incident to v lie in a halfplane bounded by the line through the input segment which v is an endpoint of. In particular, this tree is pointed since every vertex has an incident angle greater than p. Such a pointed binary tree can be augmented to a minimum pseudo-triangulation. It follows that every set of disjoint line segments in the plane has a constrained minimum pseudo-triangulation whose maximum vertex degree is bounded by a constant.
@article{pointed-binary-encompassing-trees-simple-and-optimal:2010,
title = {Pointed Binary Encompassing Trees: Simple and Optimal},
author = {M. Hoffmann and B. Speckmann and Cs.D. Tóth},
Area-Preserving Subdivision Schematization
W. Meulemans, A.M. Renssen, van, and B. Speckmann.
6th International Conference on Geographic Information Science (GIScience), pp. 160—174, 2010.
We describe an area-preserving subdivision schematization algorithm: the area of each region in the input equals the area of the corresponding region in the output. Our schematization is axis-aligned, the final output is a rectilinear subdivision. We first describe how to convert a given subdivision into an area-equivalent rectilinear subdivision. Then we define two area-preserving contraction operations and prove that at least one of these operations can always be applied to any given simple rectilinear polygon. We extend this approach to subdivisions and showcase experimental results. Finally, we give examples for standard distance metrics (symmetric difference, Hausdorff- and Fréchet-distance) that show that better schematizations might result in worse shapes.
@article{area-preserving-subdivision-schematization:2010,
title = {Area-Preserving Subdivision Schematization},
author = {W. Meulemans and A.M. Renssen, van and B. Speckmann},
Graph Drawing (17th International Symposium, GD'09, Chicago, IL, USA, September 22-25, 2009. Revised Papers), pp. 183—194, 2010.
The geometric simultaneous embedding problem asks whether two planar graphs on the same set of vertices in the plane can be drawn using straight lines, such that each graph is plane. Geometric simultaneous embedding is a current topic in graph drawing and positive and negative results are known for various classes of graphs. So far only connected graphs have been considered. In this paper we present the first results for the setting where one of the graphs is a matching. In particular, we show that there exists a planar graph and a matching which do not admit a geometric simultaneous embedding. This generalizes the same result for a planar graph and a path. On the positive side, we describe algorithms that compute a geometric simultaneous embedding of a matching and a wheel, outerpath, or tree. Our proof for a matching and a tree sheds new light on a major open question: do a tree and a path always admit a geometric simultaneous embedding? Our drawing algorithms minimize the number of orientations used to draw the edges of the matching. Specifically, when embedding a matching and a tree, we can draw all matching edges horizontally. When embedding a matching and a wheel or an outerpath, we use only two orientations.
bookTitle = {Graph Drawing (17th International Symposium, GD'09, Chicago, IL, USA, September 22-25, 2009. Revised Papers)},
Homotopic Rectilinear Routing with Few Links and Thick Edges
LATIN 2010: Theoretical Informatics (Proceedings 9th Latin American Symposium, Oaxaca, Mexico, April 19-23, 2010), pp. 468—479, 2010.
We study the NP-hard problem of finding non-crossing thick minimum-link rectilinear paths which are homotopic to a set of input paths in an environment with rectangular obstacles. We present a 2-approximation that runs in O(n 3 + k_in log n+ k_out) time, where n is the total number of input paths and obstacles and k in and k out are the total complexities of the input and output paths. Our algorithm not only approximates the minimum number of links, but also simultaneously minimizes the total length of the paths. We also show that an approximation factor of 2 is optimal when using smallest paths as lower bound.
@article{homotopic-rectilinear-routing-with-few-links-and-thick-edges:2010,
title = {Homotopic Rectilinear Routing with Few Links and Thick Edges},
bookTitle = {LATIN 2010: Theoretical Informatics (Proceedings 9th Latin American Symposium, Oaxaca, Mexico, April 19-23, 2010)},
A graph G is a support for a hypergraph H=(V,S) if the vertices of G correspond to the vertices of H such that for each hyperedge Si e S the subgraph of G induced by Si is connected. G is a planar support if it is a support and planar. Johnson and Pollak [9] proved that it is NP-complete to decide if a given hypergraph has a planar support. In contrast, there are polynomial time algorithms to test whether a given hypergraph has a planar support that is a path, cycle, or tree. In this paper we present an algorithm which tests in polynomial time if a given hypergraph has a planar support that is a tree where the maximal degree of each vertex is bounded. Our algorithm is constructive and computes a support if it exists. Furthermore, we prove that it is already NP-hard to decide if a hypergraph has a 3-outerplanar support.
Treemaps are a popular technique to visualize hierarchical data. The input is a weighted tree $\tree$ where the weight of each node is the sum of the weights of its children. A treemap for $\tree$ is a hierarchical partition of a rectangle into simply connected regions, usually rectangles. Each region represents a node of $\tree$ and the area of each region is proportional to the weight of the corresponding node. An important quality criterium for treemaps is the aspect ratio of its regions. Unfortunately, one cannot bound the aspect ratio if the regions are restricted to be rectangles. Hence Onak and Sidiropoulos in SoCG 2008 introduced \emph{polygonal partitions}, which use convex polygons. We are the first to obtain convex partitions with optimal aspect ratio $O(\depth(\tree))$. Furthermore, we consider rectilinear partitions, which retain more of the schematized flavor of standard rectangular treemaps. Our rectilinear treemaps have constant aspect ratio, independent of $\depth(\tree)$ or the number and weight of the nodes. The leaves of $\tree$ are represented by rectangles, L-, and S-shapes and internal nodes by orthoconvex polygons. We also consider the important special case that $\depth(\tree)=1$, that is, single-level treemaps. We prove that it is strongly NP-hard to minimize the aspect ratio of a rectangular single-level treemap. On the positive side we show how to construct rectilinear and convex single-level treemaps with constant aspect ratio. Our rectilinear single-level treemaps use only rectangles and L-shapes and have aspect ratio at most $2 + 2 \sqrt{3}/3$. The convex version uses four different octilinear shapes and has aspect ratio at most 9/2.
Edges and Switches, Tunnels and Bridges
D. Eppstein, M.J. Kreveld, van, E. Mumford, and B. Speckmann.
Edge casing is a well-known method to improve the readability of drawings of non-planar graphs. A cased drawing orders the edges of each edge crossing and interrupts the lower edge in an appropriate neighborhood of the crossing. Certain orders will lead to a more readable drawing than others. We formulate several optimization criteria that try to capture the concept of a "good" cased drawing. Further, we address the algorithmic question of how to turn a given drawing into an optimal cased drawing. For many of the resulting optimization problems, we either find polynomial time algorithms or NP-hardness results.
@article{edges-and-switches-tunnels-and-bridges:2009,
title = {Edges and Switches, Tunnels and Bridges},
author = {D. Eppstein and M.J. Kreveld, van and E. Mumford and B. Speckmann},
Electronic Notes in Discrete Mathematics, 34:509—513, 2009.
We study flip graphs of triangulations whose maximum vertex degree is bounded by a constant k. Specifically, we consider triangulations of sets of n points in convex position in the plane and prove that their flip graph is connected if and only if k>6; the diameter of the flip graph is O(n2). We also show that for general point sets, flip graphs of triangulations with degree k can be disconnected for any k.
bookTitle = {Electronic Notes in Discrete Mathematics},
Kinetic Collision Detection for Convex Fat Objects
M.A. Abam, M. Berg, de, S.H. Poon, and B. Speckmann.
We design compact and responsive kinetic data structures for detecting collisions between n convex fat objects in 3-dimensional space that can have arbitrary sizes. Our main results are: (i) If the objects are 3-dimensional balls that roll on a plane, then we can detect collisions with a KDS of size O(nlog¿n) that can handle events in O(log¿2 n) time. This structure processes O(n 2) events in the worst case, assuming that the objects follow constant-degree algebraic trajectories. (ii) If the objects are convex fat 3-dimensional objects of constant complexity that are free-flying in R3, then we can detect collisions with a KDS of O(nlog¿6 n) size that can handle events in O(log¿7 n) time. This structure processes O(n 2) events in the worst case, assuming that the objects follow constant-degree algebraic trajectories. If the objects have similar sizes then the size of the KDS becomes O(n) and events can be handled in O(log¿n) time.
@article{kinetic-collision-detection-for-convex-fat-objects:2009,
title = {Kinetic Collision Detection for Convex Fat Objects},
author = {M.A. Abam and M. Berg, de and S.H. Poon and B. Speckmann},
Kinetic Kd-Trees and Longest-Side Kd-Trees
M.A. Abam, M. Berg, de, and B. Speckmann.
We propose a simple variant of kd-trees, called rank-based kd-trees, for sets of n points in Rd. We show that a rank-based kd-tree, like an ordinary kd-tree, supports orthogonal range queries in O(n1-1/d + k) time, where k is the output size. The main advantage of rank-based kd-trees is that they can be efficiently kinetized: the kinetic data structure (KDS) processes O(n2) events in the worst case, assuming that the points follow constant-degree algebraic trajectories; each event can be handled in O(log n) time, and each point is involved in O(1) certificates. We also propose a variant of longest-side kd-trees, called rank-based longest-side kd-trees, for sets of points in R2. Rank-based longest-side kd-trees can be kinetized efficiently as well, and like longest-side kd-trees, they support e-approximate nearest-neighbor, e-approximate farthest-neighbor, and e-approximate range queries with convex ranges in O((1/e) log2 n) time. The KDS processes O(n3 log n) events in the worst case, assuming that the points follow constant-degree algebraic trajectories; each event can be handled in O(log2 n) time, and each point is involved in O(log n) certificates.
@article{kinetic-kd-trees-and-longest-side-kd-trees:2009,
title = {Kinetic Kd-Trees and Longest-Side Kd-Trees},
author = {M.A. Abam and M. Berg, de and B. Speckmann},
Matched Drawings of Planar Graphs
E. Di Giacomo, W. Didimo, M.J. Kreveld, van, G. Liotta, and B. Speckmann.
A natural way to draw two planar graphs whose vertex sets are matched is to assign each matched pair a unique y-coordinate. In this paper we introduce the concept of such matched drawings, which is a relaxation of simultaneous geometric embeddings with mapping. We study which classes of graphs allow matched drawings and show that (i) two 3-connected planar graphs or a 3-connected planar graph and a tree may not be matched drawable, while (ii) two trees or a planar graph and a sufficiently restricted planar grap - such as an unlabeled level planar (ULP) graph or a graph of the family of "carousel graphs" - are always matched drawable.
@article{matched-drawings-of-planar-graphs:2009,
title = {Matched Drawings of Planar Graphs},
author = {E. Di Giacomo and W. Didimo and M.J. Kreveld, van and G. Liotta and B. Speckmann},
On Minimum Weight Pseudo-Triangulations
O. Aichholzer, F. Aurenhammer, T. Hackl, and B. Speckmann.
Computational Geometry, 42(6-7):627—631, 2009.
In this note we discuss some structural properties of minimum weight pseudo-triangulations of point sets.
@article{on-minimum-weight-pseudo-triangulations:2009,
title = {On Minimum Weight Pseudo-Triangulations},
author = {O. Aichholzer and F. Aurenhammer and T. Hackl and B. Speckmann},
On Rectilinear Duals for Vertex-Weighted Plane Graphs
Discrete Mathematics, 309(7):1794—1812, 2009.
Let G= (V, E) be a plane triangulated graph where each vertex is assigned a positive weight. A rectilinear dual of is a partition of a rectangle into |V| simple rectilinear regions, one for each vertex, such that two regions are adjacent if and only if the corresponding vertices are connected by an edge in E. A rectilinear dual is called a cartogram if the area of each region is equal to the weight of the corresponding vertex. We show that every vertex-weighted plane triangulated graph admits a cartogram of constant complexity, that is, a cartogram where the number of vertices of each region is constant. Furthermore, such a rectilinear cartogram can be constructed in O(nlogn) time where n=|V|.
@article{on-rectilinear-duals-for-vertex-weighted-plane-graphs:2009,
title = {On Rectilinear Duals for Vertex-Weighted Plane Graphs},
bookTitle = {Discrete Mathematics},
Polychromatic Colorings of Plane Graphs
N. Alon, R. Berke, K. Buchin, M. Buchin, P. Csorba, S. Shannigrahi, B. Speckmann, and P. Zumstein.
Discrete and Computational Geometry, 42(3):421—442, 2009.
We show that the vertices of any plane graph in which every face is incident to at least g vertices can be colored by ¿(3g-5)/4¿ colors so that every color appears in every face. This is nearly tight, as there are plane graphs where all faces are incident to at least g vertices and that admit no vertex coloring of this type with more than ¿(3g+1)/4¿ colors. We further show that the problem of determining whether a plane graph admits a vertex coloring by k colors in which all colors appear in every face is in P for k=2 and is -complete for k=3,4. We refine this result for polychromatic 3-colorings restricted to 2-connected graphs which have face sizes from a prescribed (possibly infinite) set of integers. Thereby we find an almost complete characterization of these sets of integers (face sizes) for which the corresponding decision problem is in P, and for the others it is -complete.
@article{polychromatic-colorings-of-plane-graphs:2009,
title = {Polychromatic Colorings of Plane Graphs},
author = {N. Alon and R. Berke and K. Buchin and M. Buchin and P. Csorba and S. Shannigrahi and B. Speckmann and P. Zumstein},
bookTitle = {Discrete and Computational Geometry},
Area-Universal Rectangular Layouts
Proceedings 25th Annual ACM Symposium on Computational Geometry (SoCG'09, Aarhus, Denmark, June 8-10, 2009), pp. 267—276, 2009.
A rectangular layout is a partition of a rectangle into a finite set of interior-disjoint rectangles. They are used as rectangular cartograms in cartography, as floorplans in building architecture and VLSI design, and as graph drawings. Often areas are associated with the rectangles of a rectangular layout and it is desirable for one rectangular layout to represent several area assignments. A layout is area-universal if any assignment of areas to rectangles can be realized by a combinatorially equivalent rectangular layout. We identify a simple necessary and sufficient condition for a rectangular layout to be area-universal: a rectangular layout is area-universal if and only if it is one-sided. We also investigate similar questions for perimeter assignments. The adjacency requirements for the rectangles of a rectangular layout can be specified in various ways, most commonly via the dual graph of the layout. We show how to find an area-universal layout for a given set of adjacency requirements whenever such a layout exists.
@article{area-universal-rectangular-layouts:2009,
title = {Area-Universal Rectangular Layouts},
bookTitle = {Proceedings 25th Annual ACM Symposium on Computational Geometry (SoCG'09, Aarhus, Denmark, June 8-10, 2009)},
Connect the Dot: Computing Feed-Links with Minimum Dilation
B. Aronov, K. Buchin, M. Buchin, M.J. Kreveld, van, M. Löffler, J. Luo, R.I. Silveira, and B. Speckmann.
Algorithms and Data Structures (Proceedings 11th International Workshop, WADS 2009, Banff, Alberta, Canada, August 21-23, 2009), pp. 49—60, 2009.
A feed-link is an artificial connection from a given location p to a real-world network. It is most commonly added to an incomplete network to improve the results of network analysis, by making p part of the network. The feed-link has to be "reasonable", hence we use the concept of dilation to determine the quality of a connection. We consider the following abstract problem: Given a simple polygon P with n vertices and a point p inside, determine a point q on P such that adding a feedlink minimizes the maximum dilation of any point on P. Here the dilation of a point r on P is the ratio of the shortest route from r over P and to p, to the Euclidean distance from r to p. We solve this problem in O(¿ 7(n)logn) time, where ¿ 7(n) is the slightly superlinear maximum length of a Davenport-Schinzel sequence of order 7. We also show that for convex polygons, two feed-links are always sufficient and sometimes necessary to realize constant dilation, and that k feed-links lead to a dilation of 1¿+¿O(1/k). For (a,ß)-covered polygons, a constant number of feed-links suffices to realize constant dilation.
@article{connect-the-dot-computing-feed-links-with-minimum-dilation:2009,
title = {Connect the Dot: Computing Feed-Links with Minimum Dilation},
author = {B. Aronov and K. Buchin and M. Buchin and M.J. Kreveld, van and M. Löffler and J. Luo and R.I. Silveira and B. Speckmann},
bookTitle = {Algorithms and Data Structures (Proceedings 11th International Workshop, WADS 2009, Banff, Alberta, Canada, August 21-23, 2009)},
Detecting Hotspots in Geographic Networks
Advances in GIScience (Proceedings of the 12th AGILE Conference, Hannover, Germany, June 2-5, 2009), pp. 217—231, 2009.
We study a point pattern detection problem on networks, motivated by geographical analysis tasks, such as crime hotspot detection. Given a network N (for example, a street, train, or highway network) together with a set of sites which are located on the network (for example, accident locations or crime scenes), we want to find a connected subnetwork F of N of small total length that contains many sites. That is, we are searching for a subnetwork F that spans a cluster of sites which are close with respect to the network distance. We consider different variants of this problem where N is either a general graph or restricted to a tree, and the subnetwork F that we are looking for is either a simple path, a path with self-intersections at vertices, or a tree. Many of these variants are NP-hard, that is, polynomial-time solutions are very unlikely to exist. Hence we focus on exact algorithms for special cases and efficient algorithms for the general case under realistic input assumptions.
@article{detecting-hotspots-in-geographic-networks:2009,
title = {Detecting Hotspots in Geographic Networks},
bookTitle = {Advances in GIScience (Proceedings of the 12th AGILE Conference, Hannover, Germany, June 2-5, 2009)},
Let S be a set of n points in general position in the plane. Together with S we are given a set of parity constraints, that is, every point of S is labeled either even or odd. A graph G on S satisfies the parity constraint of a point p¿¿¿S, if the parity of the degree of p in G matches its label. In this paper we study how well various classes of planar graphs can satisfy arbitrary parity constraints. Specifically, we show that we can always find a plane tree, a two-connected outerplanar graph, or a pointed pseudo-triangulation which satisfy all but at most three parity constraints. With triangulations we can satisfy about 2/3 of all parity constraints. In contrast, for a given simple polygon H with polygonal holes on S, we show that it is NP-complete to decide whether there exists a triangulation of H that satisfies all parity constraints.
Rectangular Cartograms: the Game
M.T. Berg, de, F.S.B. Nijnatten, van, B. Speckmann, and K.A.B. Verbeek.
Proceedings 25th Annual ACM Symposium on Computational Geometry (SoCG'09, Aarhus, Denmark, June 8-10, 2009), pp. 96—97, 2009.
Raisz [3] introduced rectangular cartograms in 1934 as a way of visualizing spatial information, such as population or economic strength, of a set of regions like countries or states. Rectangular cartograms represent geographic regions by rectangles; the positioning and adjacencies of the rectangles are chosen according to their geographic locations, while their areas are proportional to the numeric values being communicated by the cartogram. Rectangles have the advantage that the sizes (area) of the regions can be estimated easily. On the other hand the rectangular shape is less recognizable and imposes limitations on possible layouts of the cartogram. In recent years several algorithms [2, 4, 5] were developed to efficiently compute rectangular cartograms. This note describes a game that was inspired by these algorithmic results. The game was initially developed as part of the outreach effort of TU Eindhoven, and it has been used at TU Eindhoven's Open Day. The goal of the Open Day is to popularize the research activities of the various departments; the audience are often families with children in the age 6–15, or parents of (prospective) students. The goal of our game was thus to show in an entertaining way the difficulties one faces when developing algorithms for automatic cartogram construction. The game was quite popular, especially with children. It is an applet and can be found at http://www.win.tue.nl/~speckman/demos/game/. Following [4, 5] the cartograms used in the game are each based on a rectangular layout: a subdivision of a rectangle into finitely many interior-disjoint rectangles. Each region of the input map corresponds to a rectangle of the layout, in addition the layout may contain additional "sea rectangles" that help to preserve the original outline of the regions. See, for example, Figure 1: the lower left corner shows a layout for the currently loaded map, in this case, the provinces of the Netherlands. The layout precisely specifies the required rectangle adjacencies of the final cartogram.
@article{rectangular-cartograms-the-game:2009,
title = {Rectangular Cartograms: the Game},
author = {M.T. Berg, de and F.S.B. Nijnatten, van and B. Speckmann and K.A.B. Verbeek},
We present a data structure for ray shooting-and-insertion in the free space among disjoint polygonal obstacles with a total of $n$ vertices in the plane, where each ray starts at the boundary of some obstacle. The portion of each query ray between the starting point and the first obstacle hit is inserted permanently as a new obstacle. Our data structure uses O(n log n) space and preprocessing time, and it supports m successive ray shooting-and-insertion queries in O(n log2 n + m log m) total time. We present two applications for our data structure: (1) Our data structure supports efficient implementation of auto-partitions in the plane i.e. binary space partitions where each partition is done along the supporting line of an input segment. If n input line segments are fragmented into m pieces by an auto-partition, then it can now be implemented in O(n log2n+m log m) time. This improves the expected runtime of Patersen and Yao's classical randomized auto-partition algorithm for n disjoint line segments to O(n log2 n). (2) If we are given disjoint polygonal obstacles with a total of n vertices in the plane, a permutation of the reflex vertices, and a half-line at each reflex vertex that partitions the reflex angle into two convex angles, then the folklore convex partitioning algorithm draws a ray emanating from each reflex vertex in the prescribed order in the given direction until it hits another obstacle, a previous ray, or infinity. The previously best implementation (with a semi-dynamic ray shooting data structure) requires O(n3/2-e/2) time using O(n1+e) space. Our data structure improves the runtime to O(n log2 n).
Subdivision Drawings of Hypergraphs
M. Kaufmann, M.J. Kreveld, van, and B. Speckmann.
Graph Drawing (16th International Symposium, GD'08, Heraklion, Crete, Greece, September 21-24, 2008, Revised Papers), pp. 396—407, 2009.
We introduce the concept of subdivision drawings of hypergraphs. In a subdivision drawing each vertex corresponds uniquely to a face of a planar subdivision and, for each hyperedge, the union of the faces corresponding to the vertices incident to that hyperedge is connected. Vertex-based Venn diagrams and concrete Euler diagrams are both subdivision drawings. In this paper we study two new types of subdivision drawings which are more general than concrete Euler diagrams and more restricted than vertex-based Venn diagrams. They allow us to draw more hypergraphs than the former while having better aesthetic properties than the latter. This research was initiated during the Bertinoro Workshop on Graph Drawing, 2008. Bettina Speckmann is supported by the Netherlands Organisation for Scientific Research (NWO) under project no. 639.022.707.
@article{subdivision-drawings-of-hypergraphs:2009,
title = {Subdivision Drawings of Hypergraphs},
author = {M. Kaufmann and M.J. Kreveld, van and B. Speckmann},
bookTitle = {Graph Drawing (16th International Symposium, GD'08, Heraklion, Crete, Greece, September 21-24, 2008, Revised Papers)},
A rectangular layout is a partition of a rectangle into a finite set of interior-disjoint rectangles. A layout is area-universal if any assignment of areas to rectangles can be realized by a combinatorially equivalent rectangular layout. We identify a simple necessary and sufficient condition for a rectangular layout to be area-universal: a rectangular layout is area-universal if and only if it is one-sided. More generally, given any rectangular layout L and any assignment of areas to its regions, we show that there can be at most one layout (up to horizontal and vertical scaling) which is combinatorially equivalent to L and achieves a given area assignment. We also investigate similar questions for perimeter assignments. The adjacency requirements for the rectangles of a rectangular layout can be specified in various ways, most commonly via the dual graph of the layout. We show how to find an area-universal layout for a given set of adjacency requirements whenever such a layout exists.
We study the problem of finding non-crossing thick minimum-link rectilinear paths homotopic to a set of input paths in an environment with rectangular obstacles. This problem occurs in the context of map schematization under geometric embedding restrictions, for example, when schematizing a highway network for use as a thematic layer. We present a 2-approximation algorithm that runs in O(n3 +kin log n + kout) time, where n is the total number of input paths and obstacles and kin and kout are the total complexities of the input and output paths, respectively. Our algorithm not only approximates the minimum number of links, but also minimizes the total length of the paths. An approximation factor of 2 is optimal when using smallest paths as lower bound.
A rectangular layout is a partition of a rectangle into a finite set of interior-disjoint rectangles. Rectangular layouts appear in various applications: as rectangular cartograms in cartography, as floorplans in building architecture and VLSI design, and as graph drawings. Often areas are associated with the rectangles of a rectangular layout and it might hence be desirable if one rectangular layout can represent several area assignments. A layout is area-universal if any assignment of areas to rectangles can be realized by a combinatorially equivalent rectangular layout. We identify a simple necessary and sufficient condition for a rectangular layout to be area-universal: a rectangular layout is area-universal if and only if it is one-sided. More generally, given any rectangular layout L and any assignment of areas to its regions, we show that there can be at most one layout (up to horizontal and vertical scaling) which is combinatorially equivalent to L and achieves a given area assignment. We also investigate similar questions for perimeter assignments. The adjacency requirements for the rectangles of a rectangular layout can be specified in various ways, most commonly via the dual graph of the layout. We show how to find an area-universal layout for a given set of adjacency requirements whenever such a layout exists.
Kinetic Data Structures
B. Speckmann.
Encyclopedia of Algorithms, pp. 417—419, 2008.
@article{kinetic-data-structures:2008,
title = {Kinetic Data Structures},
author = {B. Speckmann},
bookTitle = {Encyclopedia of Algorithms},
Efficient Algorithms for Maximum Regression Depth
M.J. Kreveld, van, J.S.B. Mitchell, P.J. Rousseeuw, M. Sharir, J. Snoeyink, and B. Speckmann.
We investigate algorithmic questions that arise in the statistical problem of computing lines or hyperplanes of maximum regression depth among a set of n points. We work primarily with a dual representation and find points of maximum undirected depth in an arrangement of lines or hyperplanes. An O(n d ) time and O(n d-1) space algorithm computes undirected depth of all points in d dimensions. Properties of undirected depth lead to an O(nlog¿2 n) time and O(n) space algorithm for computing a point of maximum depth in two dimensions, which has been improved to an O(nlog¿n) time algorithm by Langerman and Steiger (Discrete Comput. Geom. 30(2):299–309, [2003]). Furthermore, we describe the structure of depth in the plane and higher dimensions, leading to various other geometric and algorithmic results. A preliminary version of this paper appeared in the proceedings of the 15th Annual ACM Symposium on Computational Geometry (1999)
@article{efficient-algorithms-for-maximum-regression-depth:2008,
title = {Efficient Algorithms for Maximum Regression Depth},
author = {M.J. Kreveld, van and J.S.B. Mitchell and P.J. Rousseeuw and M. Sharir and J. Snoeyink and B. Speckmann},
On the Number of Pseudo-Triangulations of Certain Point Sets
O. Aichholzer, D. Orden, F. Santos, and B. Speckmann.
Journal of Combinatorial Theory, Series A, 115(2):254—278, 2008.
We pose a monotonicity conjecture on the number of pseudo-triangulations of any planar point set, and check it on two prominent families of point sets, namely the so-called double circle and double chain. The latter has asymptotically 12n nT(1) pointed pseudo-triangulations, which lies significantly above the maximum number of triangulations in a planar point set known so far.
@article{on-the-number-of-pseudo-triangulations-of-certain-point-sets:2008,
title = {On the Number of Pseudo-Triangulations of Certain Point Sets},
author = {O. Aichholzer and D. Orden and F. Santos and B. Speckmann},
bookTitle = {Journal of Combinatorial Theory, Series A},
Placing Diagrams and Symbols on Maps
Nieuwsbrief van de Nederlandse Vereniging voor Theoretische Informatica, 12:14—24, 2008.
The most commonly used types of maps are road maps or topographic maps. However, to convey specialized information, such as statistical data associated with various regions, cartographers use special purpose maps. This note describes some algorithmic questions that arise from the automated generation of two types of these maps, namely maps that contain diagrams and proportional symbol maps. We also sketch some solution strategies and show the output generated by our algorithms.
@article{placing-diagrams-and-symbols-on-maps:2008,
title = {Placing Diagrams and Symbols on Maps},
bookTitle = {Nieuwsbrief van de Nederlandse Vereniging voor Theoretische Informatica},
Feed-Links for Network Extensions
Proceedings 16th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems (ACM-GIS'08, Irvine CA, USA, November 5-7, 2008), pp. 35—1/9, 2008.
Road network data is often incomplete, making it hard to perform network analysis. This paper discusses the problem of extending partial road networks with reasonable links, using the concept of dilation (also known as crow flight conversion coefficient). To this end, we study how to connect a point (relevant location) inside a polygon (face of the known part of the road network) to the boundary so that the dilation from that point to any point on the boundary is not too large. We provide algorithms and heuristics, and give a computational and experimental analysis.
@article{feed-links-for-network-extensions:2008,
title = {Feed-Links for Network Extensions},
bookTitle = {Proceedings 16th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems (ACM-GIS'08, Irvine CA, USA, November 5-7, 2008)},
Graph Drawing (15th International Symposium, GD'07, Sydney, Australia, September 23-26, 2007, Revised Papers), pp. 183—194, 2008.
A natural way to draw two planar graphs whose vertex sets are matched is to assign each matched pair a unique y-coordinate. In this paper we introduce the concept of such matched drawings, which are a relaxation of simultaneous geometric embeddings with mapping. We study which classes of graphs allow matched drawings and show that (i) two 3-connected planar graphs or a 3-connected planar graph and a tree may not be matched drawable, while (ii) two trees or a planar graph and a planar graph of some special families—such as unlabeled level planar (ULP) graphs or the family of "carousel graphs"—are always matched drawable. Research partially supported by the MIUR Project "MAINSTREAM: Algorithms for massive information structures and data streams".
bookTitle = {Graph Drawing (15th International Symposium, GD'07, Sydney, Australia, September 23-26, 2007, Revised Papers)},
Proceedings 24th Annual ACM Symposium on Computational Geometry (SoCG'08, College Park MD, USA, June 9-11, 2008), pp. 338—345, 2008.
We show that the vertices of any plane graph in which every face is of size at least g can be colored by (3g Àý 5)=4 colors so that every color appears in every face. This is nearly tight, as there are plane graphs that admit no vertex coloring of this type with more than (3g+1)=4 colors. We further show that the problem of determining whether a plane graph admits a vertex coloring by 3 colors in which all colors appear in every face is NP-complete even for graphs in which all faces are of size 3 or 4 only. If all faces are of size 3 this can be decided in polynomial time.
bookTitle = {Proceedings 24th Annual ACM Symposium on Computational Geometry (SoCG'08, College Park MD, USA, June 9-11, 2008)},
Proceedings 20th Canadian Conference on Computational Geometry (CCCG'08, Montréal, Québec, August 13-15, 2008), pp. 107—110, 2008.
We propose a new model of realistic input: k-guardable objects. An object is k-guardable if its boundary can be seen by k guards in the interior of the object. In this abstract, we describe a simple algorithm for triangulating k-guardable polygons. Our algorithm, which is easily implementable, takes linear time assuming that k is constant.
bookTitle = {Proceedings 20th Canadian Conference on Computational Geometry (CCCG'08, Montréal, Québec, August 13-15, 2008)},
Colour Patterns for Polychromatic Four-Colourings of Rectangular Subdivisions
H.J. Haverkort, M. Löffler, E. Mumford, M. O'Meara, J. Snoeyink, and B. Speckmann.
A non-degenerate rectangular subdivision is a subdivision of a rectangle into a set of non-overlapping rectangles S, such that no four rectangles meet in a point. We consider a problem that Katz and colleagues call strong polychromatic four-colouring: Colouring the vertices of the subdivision with four colours, such thateach rectangle of S has all colours among its four corners. By considering the possible colouring patterns, we can give short constructive proofs of colourabilityfor subdivisions that are sliceable or one-sided. We also present techniques and observations for nonsliceable, two-sided subdivisions.
@article{colour-patterns-for-polychromatic-four-colourings-of-rectangular-subdivisions:2008,
title = {Colour Patterns for Polychromatic Four-Colourings of Rectangular Subdivisions},
author = {H.J. Haverkort and M. Löffler and E. Mumford and M. O'Meara and J. Snoeyink and B. Speckmann},
Algorithmic Aspects of Cartogram Computation
Bulletin of the European Association for Theoretical Computer Science, EATCS, 92:33—43, 2007.
In this note we describe some recent algorithms for the computation of rectangular and rectilinear cartograms.
@article{algorithmic-aspects-of-cartogram-computation:2007,
title = {Algorithmic Aspects of Cartogram Computation},
bookTitle = {Bulletin of the European Association for Theoretical Computer Science, EATCS},
Decompositions, Partitions, and Coverings with Convex Polygons and Pseudo-Triangles
O. Aichholzer, C. Huemer, S. Kappes, B. Speckmann, and Cs.D. Tóth.
Graphs and Combinatorics, 23(5):481—507, 2007.
We propose a novel subdivision of the plane that consists of both convex polygons and pseudo-triangles. This pseudo-convex decomposition is significantly sparser than either convex decompositions or pseudo-triangulations for planar point sets and simple polygons. We also introduce pseudo-convex partitions and coverings. We establish some basic properties and give combinatorial bounds on their complexity. Our upper bounds depend on new Ramsey-type results concerning disjoint empty convex k-gons in point sets.
@article{decompositions-partitions-and-coverings-with-convex-polygons-and-pseudo-triangles:2007,
title = {Decompositions, Partitions, and Coverings with Convex Polygons and Pseudo-Triangles},
author = {O. Aichholzer and C. Huemer and S. Kappes and B. Speckmann and Cs.D. Tóth},
Editorial (Special Issue on 21st European Workshop on Computational Geometry (EWCG 2005), Eindhoven, the Netherlands, March 9-11, 2005)
M. Berg, de, J. Gudmundsson, R. Oostrum, van, and B. Speckmann.
Computational Geometry, 36(1):1—, 2007.
Without abstract.
@article{editorial-special-issue-on-21st-european-workshop-on-computational-geometry-ewcg-2005-eindhoven-the-netherlands-march-9-11-2005-:2007,
title = {Editorial (Special Issue on 21st European Workshop on Computational Geometry (EWCG 2005), Eindhoven, the Netherlands, March 9-11, 2005)},
author = {M. Berg, de and J. Gudmundsson and R. Oostrum, van and B. Speckmann},
Efficient Detection of Patterns in 2D Trajectories of Moving Points
J. Gudmundsson, M.J. Kreveld, van, and B. Speckmann.
GeoInformatica, 11(2):195—215, 2007.
Moving point object data can be analyzed through the discovery of patterns in trajectories. We consider the computational efficiency of detecting four such spatio-temporal patterns, namely flock, leadership, convergence, and encounter, as defined by Laube et al., Finding REMO—detecting relative motion patterns in geospatial lifelines, 201–214, (2004). These patterns are large enough subgroups of the moving point objects that exhibit similar movement in the sense of direction, heading for the same location, and/or proximity. By the use of techniques from computational geometry, including approximation algorithms, we improve the running time bounds of existing algorithms to detect these patterns.
@article{efficient-detection-of-patterns-in-2d-trajectories-of-moving-points:2007,
title = {Efficient Detection of Patterns in 2D Trajectories of Moving Points},
author = {J. Gudmundsson and M.J. Kreveld, van and B. Speckmann},
bookTitle = {GeoInformatica},
On Rectangular Cartograms
A rectangular cartogram is a type of map where every region is a rectangle. The size of the rectangles is chosen such that their areas represent a geographic variable (e.g., population). Good rectangular cartograms are hard to generate: The area specifications for each rectangle may make it impossible to realize correct adjacencies between the regions and so hamper the intuitive understanding of the map. We present the first algorithms for rectangular cartogram construction. Our algorithms depend on a precise formalization of region adjacencies and build upon existing VLSI layout algorithms. Furthermore, we characterize a non-trivial class of rectangular subdivisions for which exact cartograms can be computed efficiently. An implementation of our algorithms and various tests show that in practice, visually pleasing rectangular cartograms with small cartographic error can be generated effectively.
@article{on-rectangular-cartograms:2007,
title = {On Rectangular Cartograms},
Proceedings of the 10th International Workshop on Algorithms and Data Structures (WADS 2007) 15-17 August 2007, Halifax, Nova Scotia, Canada, pp. 77—88, 2007.
bookTitle = {Proceedings of the 10th International Workshop on Algorithms and Data Structures (WADS 2007) 15-17 August 2007, Halifax, Nova Scotia, Canada},
Proceedings of the 23rd Annual ACM Symposium on Computational Geometry (SoCG 2007) 6-8 June 2007, Geongju, South Korea, pp. 364—372, 2007.
We propose a simple variant of kd-trees, called rank-based kd-trees, for sets of points in Rd. We show that a rank-based kd-tree, like an ordinary kd-tree, supports range search queries in O(n1-1/d + k) time, where k is the output size. The main advantage of rank-based kd-trees is that they can be efficiently kinetized: the KDS processes O(n2) events in the worst case, assuming that the points follow constant-degree algebraic trajectories, each event can be handled in O(log n) time, and each point is involved in O(1) certificates. We also propose a variant of longest-side kd-trees, called rank-based longest-side kd-trees (RBLS kd-trees, for short), for sets of points in R2. RBLS kd-trees can be kinetized efficiently as well and like longest-side kd-trees, RBLS kdtrees support nearest-neighbor, farthest-neighbor, and approximate range search queries in O((1/e) log2 n) time. The KDS processes O(n3 log n) events in the worst case, assuming that the points follow constant-degree algebraic trajectories; each event can be handled in O(log2 n) time, and each point is involved in O(log n) certificates.
bookTitle = {Proceedings of the 23rd Annual ACM Symposium on Computational Geometry (SoCG 2007) 6-8 June 2007, Geongju, South Korea},
O. Aichholzer, T. Hackl, M. Hoffmann, C. Huemer, A. Pór, F. Santos, B. Speckmann, and B. Vogtenhuber.
Proceedings of the 10th International Workshop on Algorithms and Data Structures (WADS 2007) 15-17 August 2007, Halifax, Nova Scotia, Canada, pp. 458—469, 2007.
Let G¿=¿(S, E) be a plane straight-line graph on a finite point set in general position. The incident angles of a point p¿¿¿S in G are the angles between any two edges of G that appear consecutively in the circular order of the edges incident to p. A plane straight-line graph is called ¿-open if each vertex has an incident angle of size at least ¿. In this paper we study the following type of question: What is the maximum angle ¿ such that for any finite set of points in general position we can find a graph from a certain class of graphs on S that is ¿-open? In particular, we consider the classes of triangulations, spanning trees, and paths on S and give tight bounds in most cases.
author = {O. Aichholzer and T. Hackl and M. Hoffmann and C. Huemer and A. Pór and F. Santos and B. Speckmann and B. Vogtenhuber},
On (Pointed) Minimum Weight Pseudo-Triangulations
Proceedings of the 19th Canadian Conference on Computational Geometry (CCCG 2007) 20-22 August 2007, Ottawa, Canada, pp. 209—212, 2007.
In this note we discuss some structural properties of minimum weight (pointed) pseudo-triangulations.
@article{on-pointed-minimum-weight-pseudo-triangulations:2007,
title = {On (Pointed) Minimum Weight Pseudo-Triangulations},
bookTitle = {Proceedings of the 19th Canadian Conference on Computational Geometry (CCCG 2007) 20-22 August 2007, Ottawa, Canada},
On the Number of Empty Pseudo-Triangles in Point Sets
Proceedings of the 19th Canadian Conference on Computational Geometry (CCCG 2007) 20-22 August 2007, Ottawa, Canada, pp. 37—40, 2007.
We analyze the minimum and maximum number of empty pseudo-triangles defined by any planar point set. We consider the cases where the three convex vertices are fixed and where they are not fixed. Furthermore, the pseudo-triangles must either be star-shaped or can be arbitrary.
@article{on-the-number-of-empty-pseudo-triangles-in-point-sets:2007,
title = {On the Number of Empty Pseudo-Triangles in Point Sets},
Kinetic Kd-Trees
We propose a simple variant of kd-trees, called rankbased kd-trees, for sets of points in Rd. We show that a rank-based kd-tree, like an ordinary kd-tree, supports range search queries in O(n1-1/d + k) time, where k is the output size. The main advantage of rank-based kd-trees is that they can be efficiently kinetized: the KDS processes O(n2) events in the worst case, assuming that the points follow constantdegree algebraic trajectories, each event can be handled in O(log n) time, and each point is involved in O(1) certificates.
@article{kinetic-kd-trees:2007,
title = {Kinetic Kd-Trees},
Maximizing Maximal Angles for Plane Straight Line Graphs
pp. 98—101, 2007.
Let G = (S,E) be a plane straight line graph on a finite point set S R2 in general position. For a point p 2 S let the maximum incident angle of p in G be the maximum angle between any two edges of G that appear consecutively in the circular order of the edges incident to p. A plane straight line graph is called '-open if each vertex has an incident angle of size at least '. In this paper we study the following type of question: What is the maximum angle ' such that for any finite set S R2 of points in general position we can find a graph from a certain class of graphs on S that is '-open? In particular, we consider the classes of triangulations, spanning trees, and paths on S and give tight bounds in all but one cases.
title = {Maximizing Maximal Angles for Plane Straight Line Graphs},
Area-Preserving Approximations of Polygonal Paths
P. Bose, S. Cabello, O. Cheong, J. Gudmundsson, M.J. Kreveld, van, and B. Speckmann.
Journal of Discrete Algorithms, 4(4):554—566, 2006.
Let P be an x-monotone polygonal path in the plane. For a path Q that approximates P let WA(Q) be the area above P and below Q, and let WB(Q) be the area above Q and below P. Given P and an integer k, we show how to compute a path Q with at most k edges that minimizes WA(Q)+WB(Q). Given P and a cost C, we show how to find a path Q with the smallest possible number of edges such that WA(Q)+WB(Q)C. However, given P, an integer k, and a cost C, it is NP-hard to determine if a path Q with at most k edges exists such that max{WA(Q),WB(Q)}C. We describe an approximation algorithm for this setting. Finally, it is also NP-hard to decide whether a path Q exists such that |WA(Q)-WB(Q)|=0. Nevertheless, in this error measure we provide an algorithm for computing an optimal approximation up to an additive error.
@article{area-preserving-approximations-of-polygonal-paths:2006,
title = {Area-Preserving Approximations of Polygonal Paths},
author = {P. Bose and S. Cabello and O. Cheong and J. Gudmundsson and M.J. Kreveld, van and B. Speckmann},
bookTitle = {Journal of Discrete Algorithms},
A Linear Programming Approach to Rectangular Cartograms
B. Speckmann, M.J. Kreveld, van, and S. Florisson.
Progress in Spatial Data Handling (Proceedings 12th International Symposium, SDH'06, Vienna, Austria, July 12-14, 2006), pp. 529—546, 2006.
In [26], the first two authors of this paper presented the first algorithms to construct rectangular cartograms. The first step is to determine a representation of all regions by rectangles and the second—most important—step is to get the areas of all rectangles correct. This paper presents a new approach to the second step. It is based on alternatingly solving linear programs on the x-coordinates and the y-coordinates of the sides of the rectangles. Our algorithm gives cartograms with considerably lower error and better visual qualities than previous approaches. It also handles countries that cannot be present in any purely rectangular cartogram and it introduces a new way of controlling incorrect adjacencies of countries. Our implementation computes aesthetically pleasing rectangular and nearly rectangular cartograms, for instance depicting the 152 countries of the World that have population over one million.
@article{a-linear-programming-approach-to-rectangular-cartograms:2006,
title = {A Linear Programming Approach to Rectangular Cartograms},
author = {B. Speckmann and M.J. Kreveld, van and S. Florisson},
bookTitle = {Progress in Spatial Data Handling (Proceedings 12th International Symposium, SDH'06, Vienna, Austria, July 12-14, 2006)},
Algorithms - ESA 2006 (Proceedings 14th Annual European Symposium, Zürich, Switzerland, September 11-13, 2006), pp. 720—731, 2006.
bookTitle = {Algorithms - ESA 2006 (Proceedings 14th Annual European Symposium, Zürich, Switzerland, September 11-13, 2006)},
Mathematical Foundations of Computer Science (Proceedings 31st International Symposium, MFCS 2006, Stará Lesná, Slovakia, August 28-September 1, 2006), pp. 86—97, 2006.
We propose a novel subdivision of the plane that consists of both convex polygons and pseudo-triangles. This pseudo-convex decomposition is significantly sparser than either convex decompositions or pseudo-triangulations for planar point sets and simple polygons. We also introduce pseudo-convex partitions and coverings.We establish some basic properties and give combinatorial bounds on their complexity. Our upper bounds depend on new Ramsey-type results concerning disjoint empty convex k-gons in point sets.
bookTitle = {Mathematical Foundations of Computer Science (Proceedings 31st International Symposium, MFCS 2006, Stará Lesná, Slovakia, August 28-September 1, 2006)},
Algorithms - ESA 2006 (Proceedings 14th Annual European Symposium, Zürich, Switzerland, September 11-13, 2006), pp. 4—15, 2006.
We design compact and responsive kinetic data structures for detecting collisions between n convex fat objects in 3-dimensional space that can have arbitrary sizes. Our main results are: (i) If the objects are 3-dimensional balls that roll on a plane, then we can detect collisions with a KDS of size O(n log n) that can handle events in O(log n) time. This structure processes O(n2) events in the worst case, assuming that the objects follow constant-degree algebraic trajectories. (ii) If the objects are convex fat 3-dimensional objects of constant complexity that are free-flying in R3, then we can detect collisions with a KDS of O(n log6 n) size that can handle events in O(log6 n) time. This structure processes O(n2) events in the worst case, assuming that the objects follow constant-degree algebraic trajectories. If the objects have similar sizes then the size of the KDS becomes O(n) and events can be handled in O(1) time.
Graph Drawing (13th International Symposium, GD'05, Limerick, Ireland, September 12-14, 2005, Revised papers), pp. 61—72, 2006.
bookTitle = {Graph Drawing (13th International Symposium, GD'05, Limerick, Ireland, September 12-14, 2005, Revised papers)},
Proceedings 14th International Symposium on Advances in Geographic Information Systems (ACM-GIS'06, Arlington VA, USA, November 10-11, 2006), pp. 19—26, 2006.
A cartogram is a thematic map that visualizes statistical data about a set of regions like countries, states or provinces. The size of a region in a cartogram corresponds to a particular geographic variable, for example, population. We present an algorithm for constructing rectilinear cartograms (each region is represented by a rectilinear polygon) with zero cartographic error and correct region adjacencies, and we test our algorithm on various data sets. It produces regions of very small complexity---in fact, most regions are rectangles---while still ensuring both exact areas and correct adjacencies for all regions.Our algorithm uses a novel subroutine that is interesting in its own right, namely a polynomial-time algorithm for computing optimal binary space partitions (BSPs) for rectilinear maps. This algorithm works for a general class of optimality criteria, including size and depth. We use this generality in our application to computing cartograms, where we apply a dedicated cost function leading to BSP's amenable to the constructing of high-quality cartograms.
bookTitle = {Proceedings 14th International Symposium on Advances in Geographic Information Systems (ACM-GIS'06, Arlington VA, USA, November 10-11, 2006)},
Kinetic Collision Detection for Balls Rolling on a Plane
This abstract presents a first step towards kinetic col- lision detection in 3 dimensions. In particular, we design a compact and responsive kinetic data struc- ture (KDS) for detecting collisions between n balls of arbitrary sizes rolling on a plane. The KDS has size O(n log n) and can handle events in O(log n) time. The structure processes O(n2) events in the worst case, assuming that the objects follow low-degree al- gebraic trajectories. The full paper [1] presents ad- ditional results for convex fat 3-dimensional objects that are free-flying in R3.
@article{kinetic-collision-detection-for-balls-rolling-on-a-plane:2006,
title = {Kinetic Collision Detection for Balls Rolling on a Plane},
Computational Geometry: Fundamental Structures
M. Berg, de and B. Speckmann.
Handbook of Data Structures and Applications, pp. 62—1—62—20, 2005.
@article{computational-geometry-fundamental-structures:2005,
title = {Computational Geometry: Fundamental Structures},
author = {M. Berg, de and B. Speckmann},
bookTitle = {Handbook of Data Structures and Applications},
Allocating Vertex $\pi$-Guards in Simple Polygons Via Pseudo-Triangulations
B. Speckmann and Cs.D. Tóth.
@article{allocating-vertex-pi-guards-in-simple-polygons-via-pseudo-triangulations:2005,
title = {Allocating Vertex $\pi$-Guards in Simple Polygons Via Pseudo-Triangulations},
author = {B. Speckmann and Cs.D. Tóth},
Off-Line Admission Control for Advance Reservations in Star Networks
U. Adamy, T. Erlebach, D. Mitsche, I. Schurr, B. Speckmann, and E. Welzl.
Approximation and Online Algorithms (2nd International Workshop, WAOA 2004, Bergen, Norway, September 14-16, 2004, Revised Selected Papers), pp. 211—224, 2005.
Given a network together with a set of connection requests, call admission control is the problem of deciding which calls to accept and which ones to reject in order to maximize the total profit of the accepted requests. We consider call admission control problems with advance reservations in star networks. For the most general variant we present a constant-factor approximation algorithm resolving an open problem due to Erlebach. Our method is randomized and achieves an approximation ratio of 1/18. It can be generalized to accommodate call alternatives, in which case the approximation ratio is 1/24. We show how our method can be derandomized. In addition we prove that call admission control in star networks is -hard even for very restricted variants of the problem.
@article{off-line-admission-control-for-advance-reservations-in-star-networks:2005,
title = {Off-Line Admission Control for Advance Reservations in Star Networks},
author = {U. Adamy and T. Erlebach and D. Mitsche and I. Schurr and B. Speckmann and E. Welzl},
bookTitle = {Approximation and Online Algorithms (2nd International Workshop, WAOA 2004, Bergen, Norway, September 14-16, 2004, Revised Selected Papers)},
Rectangular Cartogram Computation with Sea Regions
Proceedings 22nd International Cartographic Conference (XXII ICC, A Coruña, Spain, July 9-16, 2005), CD-ROM, 2005.
@article{rectangular-cartogram-computation-with-sea-regions:2005,
title = {Rectangular Cartogram Computation with Sea Regions},
bookTitle = {Proceedings 22nd International Cartographic Conference (XXII ICC, A Coruña, Spain, July 9-16, 2005), CD-ROM},
Rectangular Cartograms: Construction and Animation
S. Florisson, M.J. Kreveld, van, and B. Speckmann.
Proceedings 14th Annual Multimedia Review of Computational Geometry (part of 21st Annual ACM Symposium on Computational Geometry, Pisa, Italy, June 6-8, 2005), pp. 372—373, 2005.
@article{rectangular-cartograms-construction-and-animation:2005,
title = {Rectangular Cartograms: Construction and Animation},
author = {S. Florisson and M.J. Kreveld, van and B. Speckmann},
bookTitle = {Proceedings 14th Annual Multimedia Review of Computational Geometry (part of 21st Annual ACM Symposium on Computational Geometry, Pisa, Italy, June 6-8, 2005)},
On Pseudo-Convex Decompositions, Partitions, and Coverings
O. Aichholzer, C. Huemer, S. Renkl, B. Speckmann, and Cs.D. Tóth.
We introduce pseudo-convex decompositions, partitions, and coverings for planar point sets. They are natural extensions of their convex counterparts and use both convex polygons and pseudo-triangles. We discuss some of their basic combinatorial properties and establish upper and lower bounds on their complexity.
@article{on-pseudo-convex-decompositions-partitions-and-coverings:2005,
title = {On Pseudo-Convex Decompositions, Partitions, and Coverings},
author = {O. Aichholzer and C. Huemer and S. Renkl and B. Speckmann and Cs.D. Tóth},
Convexity Minimizes Pseudo-Triangulations
O. Aichholzer, F. Aurenhammer, H. Krasser, and B. Speckmann.
Computational Geometry, 28(1):3—10, 2004.
@article{convexity-minimizes-pseudo-triangulations:2004,
title = {Convexity Minimizes Pseudo-Triangulations},
author = {O. Aichholzer and F. Aurenhammer and H. Krasser and B. Speckmann},
Bounds on the k-Neighborhood for Locally Uniform Sampled Surfaces
M. Andersson, J. Giesen, M. Pauly, and B. Speckmann.
Proceedings of Symposium on Point-Based Graphics (SPBG 2004, Zurich, Switzerland, June 2-4, 2004), pp. 167—171, 233, 2004.
Given a locally uniform sample set P of a smooth surface S. We derive upper and lower bounds on the number k of nearest neighbors of a sample point p that have to be chosen from P such that this neighborhood contains all restricted Delaunay neighbors of p. In contrast to the trivial lower bound, the upper bound indicates that a sampling condition that is used in many computational geometry proofs is quite reasonable from a practical point of view.
@article{bounds-on-the-k-neighborhood-for-locally-uniform-sampled-surfaces:2004,
title = {Bounds on the k-Neighborhood for Locally Uniform Sampled Surfaces},
author = {M. Andersson and J. Giesen and M. Pauly and B. Speckmann},
bookTitle = {Proceedings of Symposium on Point-Based Graphics (SPBG 2004, Zurich, Switzerland, June 2-4, 2004)},
Efficient Detection of Motion Patterns in Spatio-Temporal Data Sets
Proceedings 12th International Workshop on Geographic Information Systems (ACM-GIS 2004, Washington DC, November 12-13, 2004), pp. 250—257, 2004.
@article{efficient-detection-of-motion-patterns-in-spatio-temporal-data-sets:2004,
title = {Efficient Detection of Motion Patterns in Spatio-Temporal Data Sets},
bookTitle = {Proceedings 12th International Workshop on Geographic Information Systems (ACM-GIS 2004, Washington DC, November 12-13, 2004)},
M.J. van Kreveld and B. Speckmann.
Proceedings of the Tenth Annual Conference of the Advanced School for Computing and Imaging (ASCI 2004), Port Zélande, Ouddorp, The Netherlands, June 2-4, 2004, pp. 11—16, 2004.
author = {M.J. van Kreveld and B. Speckmann},
bookTitle = {Proceedings of the Tenth Annual Conference of the Advanced School for Computing and Imaging (ASCI 2004), Port Zélande, Ouddorp, The Netherlands, June 2-4, 2004},
Algorithms - ESA 2004 : proceedings 12th annual european symposium, Bergen, Norway, September 14-17, 2004, pp. 724—735, 2004.
A rectangular cartogram is a type of map where every region is a rectangle. The size of the rectangles is chosen such that their areas represent a geographic variable (e.g., population). Good rectangular cartograms are hard to generate: The area specifications for each rectangle may make it impossible to realize correct adjacencies between the regions and so hamper the intuitive understanding of the map. Here we present the first algorithms for rectangular cartogram construction. Our algorithms depend on a precise formalization of region adjacencies and are building upon existing VLSI layout algorithms. Furthermore, we characterize a non-trivial class of rectangular subdivisions for which exact cartograms can be efficiently computed. An implementation of our algorithms and various tests show that in practice, visually pleasing rectangular cartograms with small cartographic error can be effectively generated.
bookTitle = {Algorithms - ESA 2004 : proceedings 12th annual european symposium, Bergen, Norway, September 14-17, 2004},
Pointed Binary Encompassing Trees
Algorithm Theory - SWAT 2004 (Proceedings 9th Scandinavian Workshop on Algorithm Theory, Humlebaek, Denmark, July 8-10, 2004), pp. 442—454, 2004.
We show that for any set of disjoint line segments in the plane there exists a pointed binary encompassing tree T, that is, a spanning tree on the segment endpoints that contains all input segments, has maximum degree three, and every vertex v $\in$ T is pointed, that is, v has an incident angle greater than $\pi$. Such a tree can be completed to a minimum pseudo-triangulation. In particular, it follows that every set of disjoint line segments has a minimum pseudo-triangulation of bounded vertex degree.
@article{pointed-binary-encompassing-trees:2004,
title = {Pointed Binary Encompassing Trees},
bookTitle = {Algorithm Theory - SWAT 2004 (Proceedings 9th Scandinavian Workshop on Algorithm Theory, Humlebaek, Denmark, July 8-10, 2004)},
Tight Degree Bounds for Pseudo-Triangulations of Points
L. Kettner, D. Kirkpatrick, A. Mantler, J. Snoeyink, B. Speckmann, and F. Takeuchi.
Computational Geometry, 25(1-2):3—12, 2003.
@article{tight-degree-bounds-for-pseudo-triangulations-of-points:2003,
title = {Tight Degree Bounds for Pseudo-Triangulations of Points},
author = {L. Kettner and D. Kirkpatrick and A. Mantler and J. Snoeyink and B. Speckmann and F. Takeuchi},
Proceedings 14th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2003, Baltimore MD, USA, January 12-14, 2003), pp. 109—118, 2003.
bookTitle = {Proceedings 14th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2003, Baltimore MD, USA, January 12-14, 2003)},
Degree Bounds for Constrained Pseudo-Triangulations
O. Aichholzer, M. Hoffmann, B. Speckmann, and Cs.D. Tóth.
Proceedings of the 15th Canadian Conference on Computational Geometry (CCCG'03, Halifax, Nova Scotia, Canada, August 11-13, 2003), Electronic proceedings, pp. 155—158, 2003.
We introduce the concept of a constrained pointed pseudo-triangulation TG of a point set S with respect to a pointed planar straight line graph G = (S,E). For the case that G forms a simple polygon P with vertex set S we give tight bounds on the vertex degree of TG.
@article{degree-bounds-for-constrained-pseudo-triangulations:2003,
title = {Degree Bounds for Constrained Pseudo-Triangulations},
author = {O. Aichholzer and M. Hoffmann and B. Speckmann and Cs.D. Tóth},
bookTitle = {Proceedings of the 15th Canadian Conference on Computational Geometry (CCCG'03, Halifax, Nova Scotia, Canada, August 11-13, 2003), Electronic proceedings},
Proc. 15th Canadian Conference on Computational Geometry (CCCG), pp. 141—144, 2003.
We pose a monotonicity conjecture on the number of pseudo-triangulations of any planar point set, and check it in two prominent families of point sets, namely the so-called double circle and double chain. The latter has asymptotically 12nn£(1) pointed pseudo-triangulations, which lies signi¯cantly above the maximum number of triangulations in a planar point set known so far.
bookTitle = {Proc. 15th Canadian Conference on Computational Geometry (CCCG)},
The Zigzag Path of a Pseudo-Triangulation
O. Aichholzer, G. Rote, B. Speckmann, and I. Streinu.
Algorithms and Data Structures (Proceedings 8th International Workshop, WADS 2003, Ottawa, Ontario, Canada, July 30-August 1, 2003), pp. 377—388, 2003.
We define the zigzag path of a pseudo-triangulation, a concept generalizing the path of a triangulation of a point set. The pseudo-triangulation zigzag path allows us to use divide-and-conquer type of approaches for suitable (i.e., decomposable) problems on pseudo-triangulations. For this we provide an algorithm that enumerates all pseudo-triangulation zigzag paths (of all pseudo-triangulations of a given point set with respect to a given line) in O(n 2) time per path and O(n 2) space, where n is the number of points. We illustrate applications of our scheme which include a novel algorithm to count the number of pseudo-triangulations of a point set.
@article{the-zigzag-path-of-a-pseudo-triangulation:2003,
title = {The Zigzag Path of a Pseudo-Triangulation},
author = {O. Aichholzer and G. Rote and B. Speckmann and I. Streinu},
bookTitle = {Algorithms and Data Structures (Proceedings 8th International Workshop, WADS 2003, Ottawa, Ontario, Canada, July 30-August 1, 2003)},
Cutting a Country for Smallest Square Fit
Marc van Kreveld and Bettina Speckmann.
Algorithms and Computation - 13th International Symposium, ISAAC 2002, Proceedings, pp. 91—102, 2002.
<p>We study the problem of cutting a simple polygon with n vertices into two pieces such that - if we reposition one piece disjoint of the other, without rotation - they have the minimum possible bounding square. If we cut with a single horizontal or vertical segment, then we can compute an optimal solution for a convex polygon with n vertices in O(n) time. For simple polygons we give an O(n<sup>4</sup>α(n) logn) time algorithm.</p>
@article{cutting-a-country-for-smallest-square-fit:2002,
title = {Cutting a Country for Smallest Square Fit},
author = {Marc van Kreveld and Bettina Speckmann},
bookTitle = {Algorithms and Computation - 13th International Symposium, ISAAC 2002, Proceedings},
Kinetic Collision Detection for Simple Polygons
D. Kirkpatrick, J. Snoeyink, and B. Speckmann.
Proceedings 16th Annual ACM Symposium on Computational Geometry (Clear Water Bay, Hong Kong, June 12-14, 2000), pp. 322—330, 2000.
@article{kinetic-collision-detection-for-simple-polygons:2000,
title = {Kinetic Collision Detection for Simple Polygons},
author = {D. Kirkpatrick and J. Snoeyink and B. Speckmann},
bookTitle = {Proceedings 16th Annual ACM Symposium on Computational Geometry (Clear Water Bay, Hong Kong, June 12-14, 2000)},
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matrix rank calculator
Find the rank of matrix using using online calculator with gaussian elimination and montante's method step by step In linear algebra, Matrix rank is the maximum number of independent row or column vectors in the matrix. The rank of a matrix … Get the free "matrix rank" widget for your website, blog, Wordpress, Blogger, or iGoogle. The Matrix, Inverse. Use this online calculator to find the rank of NXN matrices. This corresponds to the maximal number of linearly independent columns of .This, in turn, is identical to the dimension of the vector space spanned by its rows. File name 33s-English-Manual-040130-Publication(Edition 2). There... Read More. This calculator uses basis minor method to find out matrix rank. The Rank of a Matrix. By using this website, you agree to our Cookie Policy. The Row Space Calculator will find a basis for the row space of a matrix for you, and show all steps in the process along the way. In linear algebra, the rank of a matrix is the dimension of the vector space generated (or spanned) by its columns. To calculate a rank of a matrix we need to do the following steps: Set the matrix as shown 3 x 3 matrix We choose the first element in the 1st column and eliminate all elements that are below the current one then choose the second element in the second column and carry out same operations till end. This website uses cookies to ensure you get the best experience. the row rank of A = the column rank of A. Return matrix rank of array using SVD method. Now if these 3 vectors are also linearly independent, can I say these vectors from image of A also span A's column space, so they are so also basis of range of A? If A and B are two equivalent matrices, we write A … To calculate a rank of a matrix you need to do the following steps. The most handy and user-friendly Matrix Calculator available on the store. Learn more Accept. Show Instructions In general, you can skip the multiplication sign, so `5x` is equivalent to `5*x`. he. Calculate determinant, rank and inverse of matrix Matrix size: Rows: x columns: Solution of a system of n linear equations with n variables Number of the linear equations . Matrix calculator supports matrices with up to 40 rows and columns. pt. The matrix is not considered to be full rank, since the default algorithm calculates the number of singular values larger than max (size (A))*eps (norm (A)). Supports fractions (1/3 instead of 0.33333) for more accurate results. Matrix rank is : About Calculator School Online calculators and converters have been developed to make calculations easy, these calculators are great tools for mathematical, algebraic, numbers, engineering, physics problems. Extended Keyboard; Upload; Examples; Random; Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. Free matrix calculator - solve matrix operations and functions step-by-step. Related Symbolab blog posts. Rank of a matrix refers to the number of linearly independent rows or columns of the matrix. Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. Enter coefficients of your system into the input fields. Rows of the matrix must end with a new line, while matrix elements in a … 3x3 Matrix Transpose, Inverse, Trace, Determinant and Rank. The rank of a matrix is the greatest order of any non-zero minor in the matrix (the order of a minor being the size of the square sub-matrix of which it is the determinant). Matrix Calculator computes a number of matrix properties: rank, determinant, trace, transpose matrix, inverse matrix and square matrix. You can use fractions for example 1/3. Solving systems of linear equations. The Rank of a Matrix. What is square matrix and rectangular matrix.???? Set the matrix. ... rank\:\begin{pmatrix}1 & 2 \\3 & 4\end{pmatrix} However, you can specify a different tolerance with the command rank(A,tol) . Matrix, the one with numbers, arranged with rows and columns, is extremely useful in most scientific fields. For this matrix, the small value on the diagonal is excluded since it is smaller than the tolerance. It is useful in letting us know if we have a chance of solving a system of linear equations : when the rank equals the number … Matrix Rank Calculator. Matrix rank calculator. Matrix Rank of Arrays in Coprimality Tensor. A minor of a matrix A is the determinant of some smaller square matrix, cut down from A by removing one or more of its rows or columns. For methods and operations that require complicated calculations a … Calcolo della somma, prodotto fra matrici, matrice inversa, calcolo del determinante e rango, matrice trasposta, riduzione delle matrici alla forma canonica di Jordan, calcolo delle matrici esponenziali When the rank equals the smallest dimension it is called "full rank", a smaller rank is called "rank deficient". Manipulability Ellipsoid of a … Null Space of Matrix Calculator Step 1: To Begin, select the number of rows and columns in your Matrix, and press the "Create Matrix" button. The Row Space Calculator will find a basis for the row space of a matrix for you, and show all steps in the process along the way. A matrix obtained from a given matrix by applying any of the elementary row operations is said to be equivalent to it. This calculator solves Systems of Linear Equations using Gaussian Elimination Method, Inverse Matrix Method, or Cramer's rule.Also you can compute a number of solutions in a system of linear equations (analyse the compatibility) using Rouché–Capelli theorem.. Enter a matrix, and this calculator will show you step-by-step how to calculate a Basis for the Row Space of that matrix. Much better than other graphic calculators and matrix solvers. The calculator will find the null space of the given matrix, with steps shown. Pick the 2nd element in the 2nd column and do the same operations up to the end (pivots may be shifted sometimes). Learn more Accept. The rank of a matrix or a linear transformation is the dimension of the image of the matrix or the linear transformation, corresponding to the number of linearly independent rows or columns of the matrix, or to the number of nonzero singular values of the map. Multiplying by the inverse... rank\:\begin{pmatrix}1 & 2 \\3 & 4\end{pmatrix}, rank\:\begin{pmatrix}1 & 2 & 3 \\4 & 5 & 6 \\7 & 8 & 9\end{pmatrix}, rank\:\begin{pmatrix}1 & 3 & 5 & 9 \\1 & 3 & 1 & 7 \\4 & 3 & 9 & 7 \\5 & 2 & 0 & 9\end{pmatrix}. With help of this calculator you can: find the matrix determinant, the rank, raise the matrix to a power, find the sum and the multiplication of matrices, calculate the inverse matrix. Pick the 1st element in the 1st column and eliminate all elements that are below the current one. In linear algebra, the rank of a matrix is defined as the maximum number of independent column vectors in the given matrix or the maximum number of independent row vectors in the given matrix. Calculator. The Matrix, Inverse. This corresponds to the maximal number of linearly independent columns of .This, in turn, is identical to the dimension of the vector space spanned by its rows. All the basic matrix operations as well as methods for solving systems of simultaneous linear equations are implemented on this site. Singular Value Decomposition (SVD) of a Matrix calculator - Online matrix calculator for Singular Value Decomposition (SVD) of a Matrix, step-by-step. Matrix, the one with numbers, arranged with rows and columns, is extremely useful in most scientific fields. Please try again using a different payment method. Because of this fact, there is no reason to distinguish between row rank and column rank; the common value is simply called the rank of the matrix. [Note: Since column rank = row rank, only two of the four columns in A — c … Addition and subtraction of matrices. {aij} It supports Matrices of maximum order of 20×20. you can apply this technique by checking maximum three minors for zero (you can stop as soon as you find non-zero minor) numpy.linalg.matrix_rank¶ linalg.matrix_rank (M, tol=None, hermitian=False) [source] ¶ Return matrix rank of array using SVD method. The rank is at least 1, except for a zero matrix (a matrix made of all zeros) whose rank is 0. Online calculator to perform matrix operations on one or two matrices, including addition, subtraction, multiplication, and taking the power, determinant, inverse, or transpose of a matrix. Basis minor method: In this method we try to find consequently the non-zero minors of the size 1, 2, …, adding on each Leave extra cells empty to enter non-square matrices. Related Symbolab blog posts. Matrix rank calculator to find the rank of a 3*3 matrix. Calculate the rank of the matrix again, but specify a tolerance of 1e-16. matrix.reshish.com is the most convenient free online Matrix Calculator. Online calculator to perform matrix operations on one or two matrices, including addition, subtraction, multiplication, and taking the power, determinant, inverse, or transpose of a matrix. This matrix calculator computes determinant, inverses, rank, characteristic polynomial, eigenvalues and eigenvectors.It decomposes matrix using LU and Cholesky decomposition. ?.....how to find the rank of the matrix??? matrix rank {{a, b}, {c, d}} 3x3 Matrix Explorer. This lesson introduces the concept of matrix rank and explains how the rank of a matrix is revealed by its echelon form.. You can input only integer numbers or fractions in this online calculator. Matrix Calculator: A beautiful, free matrix calculator from Desmos.com. If the determinant is zero, there are linearly dependent columns and the matrix is not full rank. Free matrix rank calculator - calculate matrix rank step-by-step This website uses cookies to ensure you get the best experience. The rank of a matrix is defined as the maximum number of linearly independent column vectors or row vectors. Rank of the array is the … Create a matrix just by swiping across the screen in order to add a row or column. Since this site already has the Matrix Rank calculator, it is used to determine rank of entered coordinates matrix, and if it equals to 1, points are collinear. More in-depth information read at these rules. The rank of a matrix A is computed as the number of singular values that are larger than a tolerance. For methods and operations that require complicated calculations a 'very detailed solution' feature has been made. ? ... rank\:\begin{pmatrix}1 & 2 \\3 & 4\end{pmatrix} Matrix Rank Calculator Choose the dimensions of the matrix, insert the values and click on "Compute Matrix Rank". Null Space Calculator. rank (A,1e-16) Using this online calculator, you will receive a detailed step-by-step solution to your problem, which will help you understand the algorithm how to find the rank of a matrix. Free matrix calculator - solve matrix operations and functions step-by-step. Welcome to the matrix rank calculator, where you'll have the opportunity to learn how to find the rank of a matrix and what that number means.In short, it is one of the basic values that we assign to any matrix, but, as opposed to the determinant, the array doesn't have to be square.The idea of matrix rank in linear algebra is connected with linear independence of vectors. This website uses cookies to ensure you get the best experience. Matrix calculator supports matrices with up to 40 rows and columns. matrix rank calculator. All the basic matrix operations as well as methods for solving systems of simultaneous linear equations are implemented on this site. By using this website, you agree to our Cookie Policy. This video explains " how to find RANK OF MATRIX " with an example of 4*4 matrix. Matrix Rank. You can think of an r x c matrix as a set of r row vectors, each having c elements; or you can think of it as a set of c column vectors, each having r … More than just an online determinant calculator. The Matrix… Symbolab Version. We use cookies to improve your experience on our site and to show you relevant advertising. It lets you perform matrix operations just like average calculators on numbers. The vector calculator is used according to the same principle for calculating the norm of a vector in a space of any dimension. Exercises. This lesson introduces the concept of matrix rank and explains how the rank of a matrix is revealed by its echelon form.. Example with proof of rank-nullity theorem: Consider the matrix A with attributes {X1, X2, X3} 1 2 0 A = 2 4 0 3 6 1 then, Number of columns in A = 3 R1 and R3 are linearly independent. Wolfram|Alpha is the perfect resource to use for computing determinants of matrices. The simplest way to find it is to reduce the matrix to its simplest form. Thanks for the feedback. Number of Rows: Number of Columns: Gauss Jordan Elimination. For the simplest case of three points in 2d space: with the matrix. It can also calculate matrix products, rank, nullity, row reduction, diagonalization, eigenvalues, eigenvectors and much more. (enter a data after click each cell in matrix) Matrix A. Multiply Two Matrices. Matrix Calculator ermöglicht es, eine Reihe von Eigenschaften der Matrix zu berechnen:Rang, Determinante, Spur Der relative Fehler ist die Norm dividiert durch die Norm der Inversen. Matrix Calculator matrix.reshish.com is the most convenient free online Matrix Calculator. This matrix rank calculator help you to find the rank of a matrix. Invert a Matrix. matrix-rank-calculator. In linear algebra, the rank of a matrix is the dimension of the vector space generated (or spanned) by its columns. Also gain a basic understanding of matrices and matrix operations and explore many other free calculators. a11 ×aij −ai,1 ×a1j P 0 → aij i ≠ 1,P 1 = a11 =(1 2 1 0 1 3 0 -1 -3)a22 × aij - ai,2 × a2j P1 → aij i ≠ 2, P2 = a22 = 1 (1 0 -5 0 1 3 0 0 0) rank(1 2 1 -2 -3 1 3 5 0) = rank(1 0 -5 0 1 3 0 0 0) = 2. Calculates the rank of a matrix. There... Read More. Following calculations can be carried out by this Matrix calculator: addition, subtraction, transpose, determinant, scalar product, and rank of Matrix. Welcome to OnlineMSchool. Find more Mathematics widgets in Wolfram|Alpha. Matrix Calculator computes a number of matrix properties: rank, determinant, trace, transpose matrix, inverse matrix and square matrix. More in-depth information read at these rules. system of equations solver by using Gaussian Elimination reduction calculator that will the reduced matrix from the augmented matrix step by step Assume I have a matrix A which has a full rank of 3, and from linear combo of its columns I got 3 vectors from A's image. Purpose of use To double-check my L2 norm calculations. Matrix Calculator: A beautiful, free matrix calculator from Desmos.com. This web site owner is mathematician Dovzhyk Mykhailo. The Matrix… Symbolab Version. You can think of an r x c matrix as a set of r row vectors, each having c elements; or you can think of it as a set of c column vectors, each having r … You can input only integer numbers, decimals or fractions in this online calculator (-2.4, 5/7, ...). Therefore, if A is m x n, it follows from the inequalities in (*) that. Message received. If you want to contact me, probably have some question write me email on [email protected], Matrix addition and subtraction calculator, Inverse matrix calculator (Gaussian elimination), Inverse matrix calculator (Matrix of cofactors). Matrix Solvers(Calculators) with Steps. The most handy and user-friendly Matrix Calculator available on the store. It lets you perform matrix operations just like average calculators on numbers. Rows of the matrix must end with a new line, while matrix elements in a … By using this website, you agree to our Cookie Policy. Thus, the rank of a matrix does not change by the application of any of the elementary row operations. This matrix rank calculator help you to find the rank of a matrix. To create your new password, just click the link in the email we sent you. Calculate Pivots. Rank is thus a measure of the "nondegenerateness" of the system of linear equations and linear transformation encoded by . A simple test for determining if a square matrix is full rank is to calculate its determinant. Rank of the array is the number of singular values of the array that are greater than tol. Bug report Incoorect L2 norm computed for the following matrix: 2 -1 0 0-1 2 -1 0 0 -1 2 -1 image/svg+xml. Use this free online algebra calculator to find the rank of a matrix of 3x3 dimension. Changed in version 1.14: Can now operate on … In general, then, to compute the rank of a matrix, perform elementary row operations until the matrix is left in echelon form; the number of nonzero rows remaining in the reduced matrix is the rank. By default, the tolerance is max(size(A))*eps(norm(A)) . scalar quantities. matrix-rank-calculator. By using this website, you agree to our Cookie Policy. $\endgroup$ – user3086871 Nov 30 '16 at 17:51 The calculator will perform symbolic calculations whenever it is possible. Matrix Calculator is a useful software to calculate basic Matrix operations. I designed this web site and wrote all the mathematical theory, online exercises, formulas and calculators. Matrix Rank Calculator. The rank tells us a lot about the matrix. 3x3 Matrix Rank Calculator. The rank is also the dimension of the image of the linear transformation. Using this online calculator, you will receive a detailed step-by-step solution to your problem, which will help you understand the algorithm how to find the rank of a matrix. image/svg+xml. Matrix Rank. Free matrix rank calculator - calculate matrix rank step-by-step This website uses cookies to ensure you get the best experience. Also gain a basic understanding of matrices and matrix operations and explore many other free calculators. matrix rank calculator. Just type matrix elements and click the button. How to Find Matrix Rank; For an M x N matrix; If M is less than N, then the maximum rank of the matrix is M. If M is greater than N, then the maximum rank of the matrix is N. Only a zero matrix has rank zero. What is not so obvious, however, is that for any matrix A, . Set the matrix. Matrix, the one with numbers, arranged with rows and columns, is extremely useful in most scientific fields. There... For matrices there is no such thing as division, you can multiply but can't divide. By browsing this website, you agree to our use of cookies. Rank is thus a measure of the "nondegenerateness" of the system of linear equations and linear transformation encoded by . This website uses cookies to ensure you get the best experience. Multivariate normal distribution - Maximum Likelihood Estimation. , you can specify a tolerance 2 -1 matrix rank calculator help you to find the rank tells a. All elements that are below the current one it can also calculate matrix products, rank, characteristic polynomial eigenvalues... The input fields t divide this online calculator ( -2.4, 5/7,... ) i designed this site... Of the array is the most handy and user-friendly matrix calculator available on the store Jordan Elimination...... Calculator - calculate matrix rank but specify a different tolerance with the command rank ( )! The multiplication sign, so ` 5x ` is equivalent to ` 5 * x.... However, is extremely useful in most scientific fields or row vectors a data after click each cell in )... 2 -1 0 0 -1 2 -1 0 0-1 2 -1 matrix is! About the matrix???????????????... Is max ( size ( a ) ) Inverse, Trace, determinant and rank a about! A zero matrix ( a matrix just by swiping across the screen order... Linear transformation encoded by calculator uses basis minor method to find the rank of using. Its columns transformation encoded by do the following steps it lets you matrix... Calculator: a beautiful, free matrix calculator available on the store it lets you perform matrix operations explore. Perform symbolic calculations whenever it is to reduce the matrix is revealed by its columns so ` 5x is! Create a matrix is defined as the number of independent row or column in. Is also the dimension of the array is the maximum number of independent row or column concept of rank... This website, you agree to our Cookie Policy detailed solution matrix rank calculator has. Rows and columns, is that for any matrix a is m x,... Matrix is the most handy and user-friendly matrix calculator - solve matrix operations as well as for. Graphic calculators and matrix operations as well as methods for solving systems of simultaneous linear equations linear... The dimensions of the matrix application of any dimension independent column vectors or row vectors free matrix calculator: beautiful... In matrix ) matrix a is computed as the maximum number of columns matrix rank calculator Gauss Jordan.! Implemented on this site, eigenvectors and much more as methods for solving of. Is m matrix rank calculator n, it follows from the inequalities in ( * ).! By using this website uses cookies to ensure you get the best experience and! Is computed as the maximum number of linearly independent rows or columns of the `` nondegenerateness of. * ) that create a matrix is defined as the maximum number of linearly independent rows or columns of system..., formulas and calculators maximum number of linearly independent column vectors or row vectors change the! Specify a tolerance of 1e-16 a beautiful, free matrix rank '' widget for your website you. * 3 matrix points in 2d space: with the matrix is not so obvious however... Computing determinants of matrices and matrix operations as well as methods for solving of! { c, d } } 3x3 matrix Explorer of your system into input! Order to add a row or column calculator help you to find rank... Of a vector in a … more than just an online determinant calculator this site order... Relevant advertising matrix of 3x3 dimension a different tolerance with the command (! Just by swiping across the screen in order to add a row or column vectors or vectors... ( m, tol=None, hermitian=False ) [ source ] ¶ Return matrix and! Best experience elements that are larger than a tolerance of 1e-16 3 3. ( 1/3 instead of 0.33333 ) for more accurate results free matrix calculator: a beautiful, free matrix available! Matrix solvers ( calculators ) with steps that require complicated calculations a … more than just an online determinant.! Must end with a new line, while matrix elements in a space of any dimension calculating. To show you relevant advertising ' feature has been made Wordpress, Blogger matrix rank calculator or iGoogle - solve operations! For matrices there is no such thing as division, you agree to use!, Blogger, or iGoogle t divide max ( size ( a ) ) eps! Of 3x3 dimension linearly dependent columns and the matrix such thing as division, you agree to our Cookie.. Experience on our site and to show you step-by-step how to calculate a rank of the `` nondegenerateness of... Matrix refers to the end ( pivots may be shifted sometimes ) -2.4, 5/7,... ) in! ( -2.4, 5/7,... ), the one with numbers, decimals or in. Calculations a 'very detailed solution ' feature has been made, the with. Same principle for calculating the norm of a calculators and matrix operations and step-by-step., free matrix calculator the email we sent you of simultaneous linear equations and linear transformation by. } } 3x3 matrix Transpose, Inverse, Trace, determinant and rank that matrix to 40 rows and,!, decimals or fractions in this online calculator to find rank of a matrix is not so obvious,,! Row or column the `` nondegenerateness '' of the matrix and Cholesky decomposition … than! Extremely useful in most scientific fields perform symbolic calculations whenever it is possible methods for solving of! To do the following matrix: 2 -1 0 0 -1 2 -1 matrix rank calculator help you find... Used according to the end ( pivots may be shifted sometimes ) operations as well as methods for solving of... 5X ` is equivalent to ` 5 * x ` ` 5 * x ` matrix Explorer L2... Are larger than a tolerance of 1e-16 the dimensions of the array is perfect! `` how to find the rank is thus a measure of the image of the array the... And explore many other free calculators lesson introduces the concept of matrix with. And to show you relevant advertising rank of a matrix of 3x3 dimension password, just click link... ' feature has been made SVD method in matrix ) matrix a is x... Instructions in general, you agree to our Cookie Policy fractions ( 1/3 instead of 0.33333 for. Detailed solution ' feature has been made smaller than the tolerance is max ( size (,. … more than just an online determinant calculator computed as the number of linearly independent rows or columns the! Is to reduce the matrix, the one with numbers, arranged rows! Smaller than the tolerance of simultaneous linear equations are implemented on this site values of the matrix, d }... Nullity, row reduction, diagonalization, eigenvalues and eigenvectors.It decomposes matrix using and... Characteristic polynomial, eigenvalues, eigenvectors and much more to find the rank is 0 and explore many free!, b }, { c, d } } 3x3 matrix Transpose, Inverse,,. There... for matrices there is no such thing as division, agree! Tolerance is max ( size ( a, b }, { c, d }. For the row rank of a matrix determinant is zero, there are linearly dependent and. With rows and columns, is extremely useful in most scientific fields number of singular values the! Echelon form with up to the number of linearly independent rows or columns of the system of equations. 2 -1 matrix rank calculator help you to find the rank of NXN.. Thus, the one with numbers, arranged with rows and columns, is extremely useful in most scientific.... Wolfram|Alpha is the maximum number of independent row or column ( calculators ) steps. We use cookies to improve your experience on our site and wrote all the matrix! To the end ( pivots may be shifted sometimes ) 2nd column eliminate! With an example of 4 * 4 matrix 0 -1 2 -1 0 0-1 2 -1 0 0 -1 -1! The most handy and user-friendly matrix calculator is thus a measure of the transformation. To show you step-by-step how to find the rank tells us a lot the!,... ) web site and to show you step-by-step how to find rank. Is also the dimension of the `` nondegenerateness '' of the matrix end!, nullity, row reduction, diagonalization, eigenvalues and eigenvectors.It decomposes matrix LU... Since it is to reduce the matrix is defined as the maximum of... Help you to find matrix rank calculator rank of a matrix if a is computed as the maximum of... It is possible mathematical theory, online exercises, formulas and calculators ) * eps norm! '' widget for your website, you can multiply but can ' t divide rank is 0 any..., eigenvalues, eigenvectors and much more for more accurate results to it system of linear equations are implemented this... Image of the vector calculator is used according to the number of linearly column... Current one is smaller than the tolerance for calculating the norm of a of... Trace, determinant and rank pivots may be shifted sometimes ) detailed solution ' has! Complicated calculations a … more than just an online determinant calculator to create new! On the diagonal is excluded since it is smaller than the tolerance is max ( size (,! Any dimension hermitian=False ) [ source ] ¶ Return matrix rank calculator you! Is also the dimension of the matrix again, but specify a tolerance of 1e-16 so ` 5x is...
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2020 matrix rank calculator
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Phytase production by Aspergillus niger NCIM 563 for a novel application to degrade organophosphorus pesticides
Parin C. Shah1,2,
V. Ravi Kumar1,3,
Syed G. Dastager1,2 &
Jayant M. Khire1,2
The production of phytase using Aspergillus niger NCIM 563 under submerged fermentation conditions was studied using protein rich chickpea flour as substrate. Employing a hybrid statistical media optimization strategy of Plackett–Burman and Box-Behnken experimental designs in shake-flasks gave an increased phytase activity from an initial 66 IU/mL in 216 h to 160 IU/mL in a reduced time of 132 h. Productivity, thus increased by 3.97 times from 7.3 to 29 IU/mL/day. Using the optimized media, the production was successfully scaled-up further and improved up to 164 IU/mL in 96 h by studies carried out employing 2 and 10-L fermenters. The enzyme supernatant was recovered using centrifugal separation of biomass and the stability of the produced phytase was tested for animal feed applications under gastric conditions. In vitro degradation studies of water soluble monocrotophos, methyl parathion and water insoluble chlorpyrifos, pesticides used extensively in agriculture was carried out. It was observed by HPLC analysis that phytase could degrade 72% of chlorpyrifos at pH 7.0, 35 °C. Comparable results were obtained with monocrotophos and methyl parathion. With chlorpyrifos at higher temperature 50 °C as much as 91% degradation could be obtained. The degradation of chlorpyrifos was further validated by spraying phytase on harvested green chilli (Capsicum annuum L) under normal conditions of pH 7.0, 35 °C and the degradation products obtained analyzed by LCMS. Thus, the present study brings out a potentially novel application of phytase for biodegradation of organophosphorus pesticides.
Over the past century, an increase in demand of food grains and vegetables has led to an extensive use of manmade pesticides in agriculture. In fact, crop protection in India is known to annually use nearly 40,000 metric tons of pesticides. Organophosphorus pesticides (OpP), are widely used in agriculture for controlling variety of sucking, chewing and boring insects, spider mites, aphides and pests. In particular, OpP cannot be easily removed by washing and rinsing with tap water (Vendan 2016) and this leads to bioaccumulation in the food chain.
Organophosphorus pesticides are esters of phosphoric acid, which include aliphatic, phenyl and heterocyclic derivatives (Baishya and Sarma 2015), which are known to be potent irreversible acetylcholinesterase (AChE) inhibitors by phosphorylation of the serine residue at the enzyme active site. This leads to adverse effects on the nervous system of exposed animals including humans (Mileson et al. 1998), and there exists a dire need to degrade OpP post-harvest, so as to prevent their entry into the food chain.
The currently used physicochemical processes for OpP remediation include incineration or disposal in landfills that are expensive, non-ecofriendly and the process is often incomplete leading to formation of toxic intermediates (Debarati et al. 2005). Alternatively, the use of whole-cell microorganisms is advantageous because it offers a safe, economic and eco-friendly green option (Rayu et al. 2012; Sutherland et al. 2004). The factors that impact bioremediation are mainly the availability of organic sources for microbial growth, optimal pH, bioavailability of inhibitory substrates and the satisfaction of regulatory norms for release of microbes into the environment (Boopathy 2000), which can be addressed by using cell-free microbial enzymes that can act on diverse pollutants (Scott et al. 2011).
Most widely studied cell-free enzymes for OpP degradation are from bacteria, viz., organophosphorus hydrolase (EC 8.1.3.1) (Gao et al. 2012), phosphotriesterase (EC 3.1.8.1) (Chino-Flores et al. 2012) and organophosphorus acid hydrolase (EC 3.1.8.1) (Theriot and Grunden 2011). There are few reports where, fungi have been studied for OpP degradation by cleaving the phosphate group (John and Shaike 2015; Wyss et al. 1999). Phytase (PYT) or myo-inositol hexakisphosphate phosphohydrolase (EC 3.1.3.8) from Aspergillus niger NCIM 563, is a good example of a hydrolytic enzyme that can release inorganic phosphorus by the degradation of phytic acid (Bhavsar et al. 2008). Current progress on PYT research is focused on phytic acid degradation for major animal feed supplementation, plant growth promotion and human nutrition (Dersjant-Li et al. 2015; Kumar et al. 2010). The GRAS cleared PYT enzyme has not been studied for its potential for biodegradation of toxic pesticides by phosphorous release and is therefore studied here. A positive result would bring out a novel and useful enzymatic application of PYT for pesticide detoxification.
Agricultural residues including rice bran, wheat bran, groundnut oil cake, corn starch, etc., are widely used as substrates for PYT production (Bhavsar et al. 2008; Alves et al. 2016; Buddhiwant et al. 2015). However, the cost of production increases with use of agricultural residue as it needs pre-treatment (Bhavsar et al. 2008; Rani et al. 2014). The need for cost effective sustainable production requires alternative substrate for PYT production. In this context, the use of protein-rich legume flour as substrate has advantages for sustainable low cost PYT production because it can avoid the pre-treatment step. India is the largest producers of chickpea (Cicer arietinum) (7.17 metric tons in 2014–2015) but around 20% of the cultivated chickpea seeds are rejected due to non-uniform growth, color and damage during harvesting and post-harvesting process (Torres-Fuentes et al. 2011). These rejected green seeds are however, rich in protein, carbohydrate, lipids and major dietary minerals such as calcium, phosphorus, magnesium, iron and potassium (Christodoulou et al. 2006) and has been used in poultry diet (Garsen et al. 2007). For the above reasons, in the present study, in view of its availability as agricultural waste, we used green chickpea flour (GrCf) as the substrate of choice for producing and maximizing PYT production. The increased production of PYT would facilitate studying for animal feed applications and as discussed earlier in conducting studies related to dephosphorylation potential of OpP by this extracellular enzyme.
In the present study for increasing the production of extracellular PYT from GrCf using A. niger, growth media optimization was envisaged using shake flasks under submerged fermentation condition (SmF) by employing an effective hybrid strategy that involves carrying out statistical sets of experiments (Shah et al. 2009). In this approach, we initially aim at identifying the significant factors influencing PYT production by implementing a Plackett–Burman Design (PBD) of experiments (Plackett and Burman 1956). Subsequently, a second set of experiments can then be conducted to further optimize the levels of the significant factors that maximize PYT production by application of a more rigorous experimental design such as the Box Behnken design (BBD) (Box et al. 1978; Khuri and Cornell 1987). Obtaining positive results would then suggest scale-up studies with 2 and 10-L fermenters so as to confirm maintenance or obtain improvements in the production levels for process feasibility. This would also require assessing the suitability and stability of the PYT produced by GrCf so that animal feed applications can become possible. As discussed above, from a novel application point of view, it would be interesting to study and test the enzyme effectiveness for in vitro pesticide degradation using for example a commercially available OpP such as water insoluble chlorpyrifos (CPyF) (Dursban 2E©) and water soluble monocrotophos (MCP) and methyl parathion (MP). Again, a positive result would suggest carrying out studies with a test system such as post-harvest fresh green chillies (Capsicum annum L) treated with CPyF. The results obtained by carrying out suitable studies for the above objectives and plan are discussed in this work.
Phytic acid sodium salt, 3, 5, 6-trichloro-2-pyridinol (TCP) and diethyl thiophosphate (DETP) was purchased from Sigma Chemical Company (St. Louis, MO, USA). Acetonitrile (ACN) of HPLC grade was purchased from Merck. All other chemicals used were of analytical grade. CPyF (Dursban 2E©, 20%), MCP (36%) and MP (50%), harvested green chilli, soybean meal and seeds of green chickpea were purchased from a local market. The seeds were minced in a grinder to obtain green chickpea flour (GrCf).
Microorganism and production in basal media
Aspergillus niger NCIM 563, used in the present study was obtained from National Collection of Industrial Microorganisms (NCIM), CSIR-National Chemical Laboratory (CSIR-NCL), Pune, India, which was maintained on potato dextrose agar (PDA) slants and stored at 4 °C. A time course of the PYT production was studied using the basal media (pH 5.5) in triplicates. 100 mL media comprising of (g%): 1.0 GrCf; 5.0 glucose; 0.86 NaNO3; 0.05 KCl; 0.05 MgSO4·7H2O; 0.01 FeSO4·7H2O was dispensed in 250 mL Erlenmeyer flask and sterilized by autoclaving at 121 °C for 20 min. Spores from 7 days old PDA slant were gently scraped using sterile wire loop with 25 mL sterile saline solution containing 0.01% Tween 80. The spore suspension was collected in sterile tube and homogenized by vortexing for 1 min. 1 × 107 spores (using Neubaur chamber) was used as spore inoculum (Sp-I) for inoculating the basal media and incubated at 28 °C at 170 rpm under aerated culture conditions. Vegetative inoculum (Ve-I) was prepared by inoculating 5 mL basal media with 1 × 107 spores under same aerated culture conditions for 10 h which was used for inoculating 95 mL basal media. The samples were withdrawn every 24 h and centrifuged. The supernatant was checked for total residual glucose and PYT activity.
PYT assay
The PYT analysis solution consisting of 3 mM sodium phytate with 100 mM glycine–HCl buffer (pH 2.5) and 100 μL of liquid enzyme extract solution was incubated for 30 min at 50 °C. The liberated inorganic phosphate was measured by the ammonium molybdate method (Heinohen and Lathi 1981). A freshly prepared solution of acetone, 5 N H2SO4, 10 mM ammonium molybdate (2:1:1, v/v) and 400 μL citric acid (1 M) was added to the enzyme-substrate solution and absorbance was measured at 370 nm against blank consisting of buffer and substrate. One unit of PYT activity (IU) was expressed as the amount of enzyme that liberates 1 μmol phosphorus/min/mL under standard assay conditions while enzyme production was expressed as PYT activity IU/mL.
The influence of pH and temperature on PYT activity was determined by assaying in the pH range of 1.5 and 9.0 using 100 mM buffers: glycine–HCl (pH 2.0–3.0), sodium acetate (pH 4.0–6.0), Tris–HCl (pH 7.0–8.0), and glycine–NaOH (pH 9.0) at 50 °C. The pH stability using the same buffer solutions was also determined by pre-incubating enzyme samples at 35 °C for the period of 12 h considering PYT activity at zero time as 100%. The studies on optimum temperature were carried out in the temperature range of 30–70 °C, while the temperature stability was determined by incubating the enzyme samples over the above temperature range for a period of 1 h on comparing with the control without incubation. The stability of PYT was also checked at gastric conditions of poultry. One gram soybean meal was dissolved in 9 mL of simulated gastric fluid (250 mM glycine–HCl containing 2.0 mg/mL NaCl and 3.2 mg/mL of pepsin) and the pH was adjusted over a range from 1.5 to 6.5 using HCl and NaOH as required. The solutions were incubated at 37 °C for 30 min, as the poultry gut temperature varies from 37–39 °C (Lei and Stahl 2001). 40 IU of PYT was added to the solution and incubated at 37 °C for 60 min. The released phosphorus was determined as described in PYT assay.
Biodegradation of OpP using PYT
The potential of extra-cellular PYT, produced in basal media, was studied for biodegradation of CPyF, a water insoluble organophosphate. 1 mL stock solution of CPyF (10,000 ppm) was incubated with 100 µL mycelial free PYT (100 IU; specific activity 53 IU/mg) for 2 h at 35 °C, pH 7 as well as its optimum conditions (50 °C, pH 2.5). The selection of a high concentration of CPyF was employed for sensing and quantifying the degradation metabolites. The treated sample was diluted 10 times with mobile phase and the amount of residual CPyF was monitored using HPLC as mentioned in analytical methods. For other OpPs (MCP and MP), studies were carried out in similar way.
Concentration of the protein (mg) was determined using Lowry method with bovine serum albumin as standard (Lowry et al. 1951) and specific activity (IU/mg) using enzyme units and protein concentration were calculated. Di-nitro salicylic acid (DNSA) method (Miller 1959) was used to calculate total residual reducing sugar concentration.
The analysis of OpPs were carried out using HPLC (Dionex-ASI 100, auto sampler series) with reverse-phase column (C18 −4.6 × 250 mm, Waters) maintained at 40 °C. The mobile phase used was mixture of ACN: water (70:30 v/v), with flow rate of 0.5 mL/min. The detection was done at 230 nm and injection volume was 50 µL. Standard solutions of OpP was transferred in vials to reach final concentrations in range of 100–1000 ppm using mobile phase. The analysis of CPyF and its degraded products on green chilly were carried out using LC–MS (Waters-Xevo TQD-USA) with reverse phase C18 column (Acquity—UTLC BEH—2.1 × 100 mm) maintained at 45 °C. The mobile phase used was mixture of ACN: water (70:30 v/v), with flow rate of 0.3 mL/min. Injection volume of sample was 5 µL. Standard solutions of each was transferred in vials to reach concentration in range of 0.2–1 ppm using mobile phase.
Application of PYT on harvested green chilli (Capsicum annuum L)
The potential of PYT to degrade CPyF on harvested green chilly was studied at 35 °C, pH 7.0. Green chillies (250 g), obtained from local market was sprayed with CPyF (20 ppm) and dried. One part of the chillies (test) was treated with crude mycelial—free PYT (80 IU) at 35 °C and pH 7.0 for 2 h keeping the second part untreated (control). Both the parts were separately cut into small pieces and homogenized with a household mill (equipped with stainless steel knives). 10 mL of ACN was added to 10 ± 0.1 g sample and vortexed for 1 min to which 10 g sodium sulfate was added and homogenized at 15,000 rpm for 1 min. The samples were centrifuged at 5000 rpm for 5 min and 5 mL of the supernatant were transferred to a 15 mL PTFE tube. 10 mg graphitized carbon black and 25 mg primary secondary amine were added to remove colored impurities. The extract was shaken using a vortex mixer for 30 s and centrifuged at 10,000 rpm for 5 min. 2 mL of the supernatant was used to analyze the presence of CPyF and its degraded metabolites using LC–MS as mentioned in analytical methods.
Media optimization in shake flasks
Carrying out a PBD of experiments offers a rapid multifactor way to screen and identify the most significant factors (Plackett and Burman 1956). The potential effect of 10 variables (Additional file 1: Table S1) on PYT production, were evaluated in 12 PBD runs at two levels, low level (−) and high level (+). The choice of the above variables was made based on reports available for enhanced PYT production by solid state fermentation (Bhavsar et al. 2013). The complete PBD matrix for screening was designed using a standard Plackett–Burman orthogonal array constructed using Design Expert Software (DES) Version 7.1.2, Stat-Ease, Minneapolis, MN, USA. The response values of PYT produced in IU/mL were analyzed to obtain a best-fit linear mathematical model that could be further analyzed by ANOVA for acceptability. Subsequently, a BBD of experiments was generated by DES and studies at three different levels −1, 0, +1 (Additional file 1: Table S2) were carried out to further optimize enzyme production levels with respect to the major factors identified by PBD. The less significant factors were maintained at the average of the high and low levels used in the PBD study. A best-fit model for enzyme production was further studied by ANOVA to test for its statistical significance.
Fermenter scale production
Scale-up studies in a batch fermenter (New Brunswick BioFlo 110) with 2-L media was carried out on the basis of optimized media formulation. Successive fermentation batches at different rpm (400, 500 and 600 rpm) were performed with constant aeration of 0.5 vvm and temperature of 28 °C. The fermenter containing 1.8-L medium that was earlier sterilized in situ was inoculated with 36 h old Ve-I (200 mL). The dissolved oxygen (DO) was measured using Mettler Toledo oxygen probe. Samples from the fermenter were withdrawn at regular intervals and analyzed for PYT activity and residual glucose on biomass separation. The process was further scaled up using a 10-L capacity fermenter with 1-L, 36 h old Ve-I.
PYT production in basal media
The PYT production was tested using two types of inoculum; spore (Sp-I) and vegetative (Ve-I). The results showed, slow and gradual increase in PYT production using Sp-I, with the maximum activity of 66 ± 3.3 IU/mL on 9th day, (i.e., 216 h) (Fig. 1). The glucose concentration also showed a gradual depletion, with complete utilization by 10th day (i.e., 240 h). In fact, using Ve-I, higher PYT activity of 86 ± 4.3 IU/mL was observed in a lower production time of 6 days (i.e., 144 h) with glucose depletion in 7 days (i.e., 168 h). Thus, the productivity [units of enzyme produced per day (IU/mL/day)] increased from 7.3 IU/mL/day obtained using Sp-I to 14.3 IU/mL/day using Ve-I. Thus, the type of inoculum has marked effect on the production of PYT and all further studies were therefore carried out using Ve-I.
Comparison of phytase production using spore inocula (Sp-I) and vegetative inocula (Ve-I). Experiments were carried out in triplicate (mean ± SD)
Stability studies of PYT
The conditions for optimum activity and stability of the PYT was assessed by carrying out studies with respect to temperature and pH. Optimum temperature studies ranged from 30 to 70 °C. It was observed that, the optimum temperature for maximum PYT activity is 50 °C (Additional file 1: Figure S1). On considering the activity at 50 °C to be 100%, we observe that at 35 °C, the activity reduces to 30% of maximum activity and at 60 °C to 60% of maximum activity. Temperature stability studies were then carried out, which showed 100% stability in 1 h from 30 to 50 °C (Additional file 1: Figure S1) while 20% activity was reduced at 60 °C, 50% at 65 °C and 100% at 70 °C. Studies with varying pH interestingly showed that the enzyme showed high activity at a low pH value of 2.5 (Additional file 1: Figure S2). Considering the activity at pH 2.5 to be 100%, only 50 and 10% activity was observed at pH 4.5 and 7.0, respectively. It may be noted that the PYT showed overall broad pH stability from pH 2.5–9 (Additional file 1: Figure S2). Experiments showed that the enzyme activity was retained for 12 h in the pH range studied. The pH and temperature stability profile was also determined under poultry gut conditions as described in methods. High efficacy of phosphate release was shown by the PYT in simulated gastric fluid in pH ranging from 2.0 to 4.5 (Additional file 1: Figure S3).
The ability of crude PYT, (100 IU) produced using basal media, to degrade CPyF was studied both under normal conditions (35 °C, pH 7.0) as well as under optimum enzyme conditions (50 °C, pH 2.5). HPLC analyses, showed a single major peak for the sample containing only CPyF (control sample) at 35 °C, pH 7.0 (Fig. 2a) as well as at 50 °C, pH 2.5 (Fig. 2c) and having a retention time of 3.62 min with a relative area of 97%. On the other hand, for a PYT treated sample at 35 °C, pH 7.0, multiple peaks were observed. Notably, it was observed that there was decrease in relative area by 72% at the retention time of CPyF (Fig. 2b). A similar study with PYT at 50 °C, pH 2.5, the relative peak area seen at retention time of CPyF showed an even higher decrease in relative area by 91% (Fig. 2d). The positive result for CPyF suggested degradation studies with other OpPs, namely, MCP and MP by PYT would be useful. HPLC analysis with MCP having a retention time of 5.5 min for the control sample (Additional file 1: Figure S4a) at 35 °C, pH 7.0 showed that a higher unit of PYT (250 IU) to obtain 53% degradation in 4 h (Additional file 1: Figure S4b) when compared to CPyF (PYT 100 IU, 72% degradation) (Fig. 2) in 2 h. Better degradation results with MP (HPLC retention time of 9.5 min for the control sample, Additional file 1: Figure S4c) in comparison with MCP were obtained by a higher decrease in peak area (77%) on treatment with PYT (250 IU) (Additional file 1: Figure S4d) in 4 h. Thus, all the above results with CPyF, MCP and MP corroborate the finding that PYT has the ability to effectively act on OpP and degrade them.
Reduction of CPyF using phytase at different conditions a, c. CPyF before phytase action, b, d. CPyF after phytase action
Phytase shows dephosphorylation action by breaking the phospho-ester bond to release phosphate from substrate (Joshi 2014). As per food safety and standards authority of India (FSSAI), limit of CPyF on vegetables is 0.2 ppm (FSSAI notification 2011). Detoxification of CPyF is achieved by cleaving the phospho-ester bond, generating TCP as the major product along with DETP (Chen et al. 2012; Hanley et al. 2000; Bicker et al. 2005), which are both water soluble. Human studies show that, both the metabolites are considered as urinary markers of CPyF exposure and are easily excreted through urine within 12 h. Studies on rat show that, TCP and DETP are the predominant urinary metabolites of CPyF catabolism (Bicker et al. 2005). In the present study, the applicability of PYT applied on post-harvest chillies to degrade water insoluble CPyF was therefore studied with respect to the formation of TCP and DETP water soluble degradation products.
LC–MS analysis of standard CPyF showed that it eluted with a retention time (RT) of 5.56 min and m/z of 349.90 while DETP and TCP eluted at 0.72 and 3.48 min with m/z values of 169.17 and 198, respectively. In PYT untreated sample (control), a single peak was detected after LC–MS analysis with a RT of 5.56 min and m/z of 349.90 indicating it to be CPyF (Fig. 3a). In PYT treated sample, 3 peaks (Fig. 3b–d) were observed at RT of 0.72, 3.48 and 5.56 min. MS analysis of these peaks showed m/z values of 169.17, 198 and 349.90, respectively. On comparing with the standards, the 3 peaks were ascertained to be DETP, TCP and CPyF, respectively. The percent degradation of CPyF using PYT from A. niger NCIM 563 can vary depending on pH and temperature prevailing at the field. Peak area analysis shows 8% degradation of CPyF at 35 °C and pH 7.0 in 2 h using PYT (80 IU). To increase the degradation, higher units of phytase or reaction time may thus be required. On using higher units of PYT (250 IU) to degrade CPyF on green chilli our results in fact showed that 90% degradation was possible in 12 h (Additional file 1: Figure S5a, b). It is reported that TCP is not fetotoxic and teratoxic in either rat or rabbits at dosage levels of 100 ppm. TCP is shown to have moderate toxicity to salmonoids at LD50 value of 1.8 ppm (Marino et al. 1999). Studies of TCP showed that a minimum concentration of 0.6 ppm when exposed for 24 h is toxic during the multiple developmental stages of zebra fish (Suvarchala and Philip 2016). Our observation is that PYT can degrade CPyF present on raw agricultural products. Thus, development of a potential new way that prevents toxic OpP from entering the food chain by forming easily removable metabolites could become possible.
Analysis of CPyF and its degraded metabolites using LC–MS. a CPyF on green chilli before phytase action. b CPyF on green chilli after phytase action. c Release of TCP after phytase action. d Release of DETP after phytase action
Media optimization for enhanced PYT production in shake flask
For the 12 PBD runs the experimentally obtained maximum response values of PYT activity (IU/mL) obtained on the 6th day (i.e., 144 h) are reported in Additional file 1: Table S3. Run number 10 showed a maximum PYT production value of 132 ± 6.6 IU/mL. Regression analysis of the response values obtained for the PBD runs yielded a best fit linear model, viz.,
$$\begin{aligned} {\text{Phytase activity }} = { 111}.0 9 - 30. 6 7 { } \times \left[ {{\text{NaNO}}_{ 3} } \right] + 4 5. 3 3\times \left[ {\text{GrCf}} \right]{-} 1 6 4 4. 4 4 { } \times \left[ {{\text{MnSO}}_{ 4} \cdot{\text{H}}_{ 2} {\text{O}}} \right] \, \hfill \\ \, {-} 1 90.0 \times \left[ {{\text{CaCl}}_{ 2} \cdot 2 {\text{H}}_{ 2} {\text{O}}} \right] \hfill \\ \end{aligned}$$
The suitability of the model was further corroborated by ANOVA tests. Thus, the obtained model F-value of 11.35 implies the model is significant and that there is only a 0.35% chance that the model F-value could occur due to noise. The values of Prob > F was less than 0.05, for the four variables showing their significance. The coefficient of determination R2 = 0.87 provided a satisfactory measure for the variability in the observed response that could be explained by the model. The Pred R2 of 0.61 is in reasonable agreement with Adj-R2 of 0.79. The adeq precision, a measure of the signal-to-noise ratio, is found to have a high value of 9.72 and this indicated the signal strength to be strong. All the above tests confirm that the model Eq. (1) can be used to navigate the design space. In fact, using the above model, it was found that, the above four factors accounted for 86.68% of the total contribution to the estimates of the response values. The remaining variables then accounted for only 13.32% and thus PBD identified them to be less significant. The ANOVA results were complemented by the fact that four out of the 10 factors studied in the PBD, namely, NaNO3, GrCf, MnSO4·H2O and CaCl2·2H2O were significant on comparing their t-values using a Pareto chart (Additional file 1: Figure S6).
The maximum contributory factors identified by PBD for PYT production were further optimized by a BBD of experiments (29 runs) generated by DES employing three chosen levels for each variable. Additional file 1: Table S4 gives the BBD for the four significant variables along with the experimentally obtained PYT activity. We observe that the optimization studies showed that run number 25 remarkably improved the PYT activity (160 ± 8.0 IU/mL). The wide variation in activity reported in all the runs bring out the process sensitivity to the experimentally chosen conditions and shows the usefulness of having carried out this systematic optimization study. The response data of BBD runs was regressed successfully using actual factors and interestingly showed linear dependency without interacting terms, namely,
$${\text{Phytase activity}} = 2 8 6. 7 4 - 4 9. 1 7\times \left[ {{\text{NaNO}}_{ 3} } \right]{-} 1 20. 8 3\times \left[ {\text{GrCf}} \right]$$
The ANOVA analysis of the above model satisfied the statistical tests with an obtained model F-value of 27.95 implying that, the model is significant with only a 0.01% chance that the model F-value could arise due to noise. The value of correlation coefficient (Pred R2 = 0.62) for PYT production suggested a good agreement between the observed and model predicted response values. The coefficient of determination (R2 = 0.68), suggests that 68% of the variability in the data was explained by Eq. 2. The obtained signal-to-noise ratio value of 15.04 brought out the presence of an adequate signal.
The final formulation of ten variables in the range studied in PBD and BBD showed that the highest activity of 160 ± 8.0 IU/mL was obtained in 132 h with 100 mL media comprising of (g%): 4.0 glucose; 0.4 NaNO3; 0.075 MgSO4·7H2O; 0.075 KCl; 0.015 FeSO4·7H2O; 0.015 Tween 80; 1.0 GrCf; 0.35 dextrin; 0.02 MnSO4·H2O and 0.1 CaCl2·2H2O. To validate the formulation, a time course experiment for PYT production was carried out using this optimized condition. The results obtained after showed that a 1.86 fold enhancement in PYT activity from 86 ± 4.3 IU/mL to 160 ± 8.0 IU/mL with glucose completely utilized in 5.5 days (i.e., 132 h) was achieved by adopting the outlined hybrid media optimization approach (Fig. 4). The effect of varying the phosphate concentration by addition of sodium phytate and KH2PO4 using the optimized media formulation with 1% GrCf was thus studied. Results showed that the PYT activity decreased to 79 IU/mL on addition of sodium phytate (0.004 g%) and to 151 IU/mL on addition of KH2PO4 (0.002 g%). This may be compared to PYT activity of 160 IU/mL obtained using the optimized media with 1% GrCf and suggests that it optimally provides the requirements of phosphate.
Validation of phytase production by using unoptimized and optimized media. Experiments were carried out in triplicate (mean ± SD)
Scale-up studies of PYT production in batch fermenters
The optimized media formulation was studied in scaled-up larger volume batches using fermenters to ensure maintenance of the PYT production throughput. The results of fermentation studies carried out with 2 and 10-L media scaled-up volume batches are discussed below. It is expected that when operating with larger volumes, the dissolved oxygen (DO) is an important factor to consider and this would depend on both the aeration rate and the agitation speed. For the present studies, we chose to keep the aeration rate constant at 0.5 vvm while varying the initial agitation speeds (400, 500, 600 rpm). The behavior in time of DO and pH were simultaneously monitored. For the 400 rpm run, a decrease in DO was observed till 36 h. 50% DO was maintained by gradual increase in rpm and it was thus maintained at the higher rpm. The pH of the media gradually decreased from 5.0 to 2.3. For the 2-L batch study with an initial rpm of 400, the maximum activity of 97 ± 4.8 IU/mL was obtained in 132 h. PYT production of 122 ± 6.1 IU/mL and 158 ± 7.9 IU/mL was achieved at initial rpm of 500 and 600 respectively, in 132 h. Thus, PYT production was successfully scaled up to 2-L production from the shake flask experiment (158 ± 7.9 IU/mL at 600 rpm) (Fig. 5a).
Fermenter scale production a 2-L (at 600 rpm) and b 10-L (at 500 rpm) depicting phytase activity, pH and glucose concentration. Experiments were carried out in triplicate (mean ± SD)
Successful production at 2-L scale, paved way to studying the feasibility of enzyme production in a 10-L volume fermenter to bring out the process biotech potential. Similar to studies in the 2-L scale and varying the initial agitation speed (400, 500 and 600 rpm), the DO and pH were monitored in 10-L scale. DO and pH pattern were again observed, but with higher decreasing rates than in the 2-L scale. Remarkably, the maximum PYT activity of 164 ± 8.2 IU/mL was maintained but it is important to note that the maximum activity was obtained in a much shorter time of 96 h at 500 rpm (Fig. 5b).
Phytase is widely used to act on phytic acid for the aim of increasing the bioavailability of phosphorus, proteins and essential minerals in animal diets. It is commercially produced by employing a submerged fermentation process using spore inoculum (Coban and Demirci 2014; Krishna and Nokes 2001). We considered it desirable to carry out studies that use vegetative inoculum for PYT production over spore inoculum that is commonly employed for PYT production. Results obtained suggest that PYT production may in fact be enhanced using vegetative inoculum. This result is of considerable significance. Similar results of enhanced PYT production have been observed using vegetative inoculum in solid state fermentation (Krishna and Nokes 2001). Our studies also show that the use of GrCf as substrate for PYT production with A. niger NCIM 563 gives activity and productivity higher than earlier reports under submerged fermentation conditions using different strains and substrates (Table 1). Our observation is that the PYT produced using GrCf is active and highly stable over wide range of temperature and pH that simulates gastric conditions and brings out its superiority for use in animal feed applications. A novel application of PYT in degrading CPyF, MCP and MP is also discussed here to bring out its potential as an agent for pesticide degradation. It needs to be noted that FDA has approved A. niger as a generally-recognized-as-safe (GRAS) organism (Schuster et al. 2002). Together with the laboratory study and analysis report obtained from Central Avian Research Institute, Bareilly (UP), India confirms that A. niger NCIM 563 is a non-mycotoxin producer and the PYT obtained can thus be advantageously exploited.
Table 1 Comparing phytase productivity of A. niger NCIM 563 with other Aspergillus species
Statistical experimentation for media optimization provides a time saving approach for enhancing PYT production (Bhavsar et al. 2013) and provides the base experimental conditions maintaining/improving the productivity for scale-up. Our results with this approach showed that the PYT productivity obtained in a 10-L fermenter working volume (41.0 IU/mL/day) was improved by a factor of 2.87 and 1.41 times from shake flask (14.3 IU/mL/day) and 2-L scale (29 IU/mL/day) experiments, respectively. The increase in productivity with reduction in production time at 10-L fermenter scale may be due to the better maintenance of fermentation parameters viz., agitation, temperature, aeration, etc. This observation further supports the biotech potential of the present PYT production process. Studies of submerged PYT production using S. thermophile in cane molasses medium showed that the productivity obtained in the 10-L working volume fermenter (5.2 IU/mL/day) was improved by a factor of 2.2 times from shake flask (2.5 IU/mL/day) experiments and having optimum activity at pH 5.5 and 45 °C (Singh and Satyanarayana 2008). Similar trend of increased productivity by 1.5 times for glucoamylase production was observed by Kumar et al. (2007).
In conclusion, the present study shows an efficient process of producing PYT, which has wide applications prospects in animal feed and agriculture. Our studies obtained high yields of PYT from A. niger NCIM 563 using GrCf. The PYT from A. niger was found to be stable over wide range of temperature and pH and has thus shown the necessary potential for use as animal feed supplement as well as on crops for field applications. The PYT from A. niger NCIM 563 beneficially shows biodegradation of CPyF when tested on green chillies. Studies show biodegradation of CPyF is maximum at conditions chosen close to the enzyme optimum conditions for high PYT activity (50 °C and pH 2.5). These findings thus bring out an interesting and new applications for PYT produced from GrCf and A. niger NCIM 563. Although, the above optimal conditions do not correspond to field, they do justify the need to carry out studies that would improve the efficacy of PYT for biodegradation of OpP compounds.
BBD:
Box-Behnken design
CPyF:
GrCf:
green chickpea flour
MCP:
monocrotophos
methyl parathion
organophosphorus pesticide
PYT:
phytase
PBD:
Plackett–Burman design
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PS carried out the experiments. JM participated in the design of the study, supervised the research work and reviewed the manuscript. VR designed the statistical optimization studies, drafted and edited the original manuscript. SD supervised the research work, administered the project and funding. All authors have read and approved the final manuscript.
PS would like to thank Council of Scientific and Industrial Research (CSIR) for Senior Research Fellowship. The authors thank Dr. Ameeta Ravikumar, Institute of Bioinformatics and Biotechnology (IBB), Savitribai Phule Pune University (SPPU) for scientific discussions on the pesticide degradation studies. We thank Maarc Labs Pvt. Ltd., Pune, recognized by the National Accreditation Board for Testing and Calibration, for carrying out the reported LCMS studies.
The datasets supporting the conclusions of this article are included within the article and its additional file.
The funding for the pesticide degradation analysis carried out at Maarc labs Pvt. Ltd., was provided by Department of Biotechnology (DBT) funded project (GAP 313226).
Academy of Scientific and Innovative Research (AcSIR), Council of Scientific and Industrial Research-National Chemical Laboratory, CSIR-NCL, Pune, 411008, India
Parin C. Shah, V. Ravi Kumar, Syed G. Dastager & Jayant M. Khire
National Collection of Industrial Micro-organisms (NCIM) Resource Center, Biochemical Sciences Division, CSIR-NCL, Pune, 411008, India
Parin C. Shah, Syed G. Dastager & Jayant M. Khire
Chemical Engineering and Process Development Division (CEPD), CSIR-NCL, Pune, 411008, India
V. Ravi Kumar
Parin C. Shah
Syed G. Dastager
Jayant M. Khire
Correspondence to Jayant M. Khire.
Additional file 1.
Additional figures and tables.
Additional data.
Shah, P.C., Kumar, V.R., Dastager, S.G. et al. Phytase production by Aspergillus niger NCIM 563 for a novel application to degrade organophosphorus pesticides. AMB Expr 7, 66 (2017). https://doi.org/10.1186/s13568-017-0370-9
Received: 16 December 2016
Fermenter scale
Submerged fermentation
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CommonCrawl
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KDiamend: a package for detecting key drivers in a molecular ecological network of disease
Volume 12 Supplement 1
Selected articles from the 16th Asia Pacific Bioinformatics Conference (APBC 2018): systems biology
Mengxuan Lyu1,
Jiaxing Chen1,
Yiqi Jiang1,
Wei Dong1,
Zhou Fang1 &
Shuaicheng Li1
BMC Systems Biology volume 12, Article number: 5 (2018) Cite this article
Microbial abundance profiles are applied widely to understand diseases from the aspect of microbial communities. By investigating the abundance associations of species or genes, we can construct molecular ecological networks (MENs). The MENs are often constructed by calculating the Pearson correlation coefficient (PCC) between genes. In this work, we also applied multimodal mutual information (MMI) to construct MENs. The members which drive the concerned MENs are referred to as key drivers.
We proposed a novel method to detect the key drivers. First, we partitioned the MEN into subnetworks. Then we identified the most pertinent subnetworks to the disease by measuring the correlation between the abundance pattern and the delegated phenotype—the variable representing the disease phenotypes. Last, for each identified subnetwork, we detected the key driver by PageRank. We developed a package named KDiamend and applied it to the gut and oral microbial data to detect key drivers for Type 2 diabetes (T2D) and Rheumatoid Arthritis (RA). We detected six T2D-relevant subnetworks and three key drivers of them are related to the carbohydrate metabolic process. In addition, we detected nine subnetworks related to RA, a disease caused by compromised immune systems. The extracted subnetworks include InterPro matches (IPRs) concerned with immunoglobulin, Sporulation, biofilm, Flaviviruses, bacteriophage, etc., while the development of biofilms is regarded as one of the drivers of persistent infections.
KDiamend is feasible to detect key drivers and offers insights to uncover the development of diseases. The package is freely available at http://www.deepomics.org/pipelines/3DCD6955FEF2E64A/.
Assessment and characterization of microbiota are prevalent in human disease studies [1–3]. When the species within the microbial community interact with each other in equilibrium, serving as co-adapted colonists and providing beneficial goods and services, disruption of such alliances may induce health issue [4]. For example, the imbalance in the community could lead to bacterial overgrowth and the development of respiratory infections [5]. In this case, network analysis, for instance, differential network analysis, which identifies biomarker candidates by detecting changes in the correlation relationships between different experimental conditions [6], provides a better understanding towards disease. Thereafter, in microbiome area, molecular ecological networks (MENs) [7] can be constructed to perform network analysis for different types of actors within the microbial community, for examples, species, taxons, or phylogenetic gene markers, and they are referred to as phylogenetic molecular ecological networks (pMENs) where phylogenetic gene markers serve as the actors [8]. Similar to co-expression networks, Deng et al. proved that the MENs are scale-free and small world [7].
In a MEN, the removal of some species could be disproportionately deleterious. These species are referred to as keystone species. Keystone species are topologically important molecules in the MEN. Berry et al. has studied the detection of keystone species in MENs thoroughly [9]. They applied a brute-force leave-one-out strategy to evaluate the keystoneness of a species in a given MEN, and demonstrated the impact of the keystone species on species richness. They also classified keystone species according to their topological properties using linear discriminant analysis. Deng et al. proposed a method to detect keystone species from the MENs by integrating phenotype information [10]. They identified keystone species by calculating the correlation between a phenotype variable and the abundance pattern of species clusters. Researchers also considered species connected to many others in MENs as keystone species (also referred to as hub nodes) [11].
Key drivers, which are major components that drive the disease concerned MENs, provide hints to understand the mechanisms of disease and are intensively studied with RNA data. There are various of methods to identify the key drivers in a co-expression network. One method is to incorporate the annotation of genes and pathways of diseases in order to locate the key drivers by considering enrichment of statistic of genes neighborhood [12, 13]. Another category of method distinguishes important MENs by calculating associations between gene modules with meta information like phenotype and GWAS analysis [14, 15], and then detects the key drivers by measuring the genes topology effect. For example, MEGENA [16] did multiscale hub analysis and Zhang et al. examined the number of N-hob downstream nodes [17]. Those methods on detecting key drivers in RNA data analysis can be adopted to detect key drivers in MENs. Even though Portune et al. locates important microbial species and genes with the assistance of gene annotation to study the MENs [18], the annotation for microbial genes and species yet demands intensive efforts and the pathways of diseases are incomplete.
The distinction between keystone species and key drivers is that the keystone species are only topologically important, while key drivers motivate disease associated networks. MENs of diseases can be different compared to those from healthy individuals. By analyzing the factors driving the differences, we can uncover the development of the disease.
Inspired by key drivers analysis with RNA data and keystone species studies in MENs, we proposed a method to perform key drivers analysis without the availability of annotation information. Given a microbial abundance profile, we first construct the MEN, in which the nodes represent the microbial species or phylogenetic gene markers and the edges capture the associations between their respective nodes. Then we divide the MEN into multiple subnetworks and extract the subnetworks that are most relevant to the disease by calculating the associations between subnetworks and phenotype variables. A single phenotype variable could be insufficient to capture the changes in disease networks from healthy networks and it can be biased. To address this issue, we applied principal component analysis to extract delegated phenotype, which is more robust. Last, our method detects the key driver based on PageRank, which utilizes node topological properties within each extracted subnetwork. It captures the global link structure of subnetworks thus outperforms statistical algorithms that only use local information.
There are multiple ways to calculate inference of MENs, of which two of the most popular ways are Pearson correlation coefficient (PCC) and mutual information (MI) [19]. A review of correlation detection strategies in MENs [20] suggests that although some methods outperform others, the inference calculating method still needs further improvement. To reduce the effect of the high proportion of zero counts, Paulson et al. applied a mixture model that implemented a zero-inflated Gaussian (ZIG) distribution of mean group abundance for each taxonomic feature to do differential abundance analysis. Experiments show the improvement of mixture model compared to other models, for instance, DESeq, edgeR and Kruskal-Wallis test [21, 22]. Inspired by the above trials of solving rare microbes issues with mixture models, we proposed to construct the network by multimodal mutual information (MMI) [23] under the assumption of the Gaussian mixture model. In KDiamend, we implemented both PCC and MMI to infer the associations between nodes in the MENs. However, correlation-based methods, like PCC, have their limitations. To be more specific, it is hard to distinguish correlation with causation [24]. There are many other arbitrary methods to construct networks, like Bayesian network [24] and WGCNA [25], which apply topology overlaps to measure the similarities between nodes. These various methods can be implemented to construct the network as potential options in our framework. Nevertheless, it is out of the range of this work.
Our main contribution is that we refined the framework of key driver detection, and proposed delegated phenotype to capture the changes in disease networks from healthy networks. To validate our method, we performed experiment based on simulated data. Then, we tested KDiamend with two real microbiome datasets. We conducted key drivers analysis on Type 2 diabetes (T2D) and Rheumatoid Arthritis (RA), whose data are from gut microbiome and oral microbiome respectively. For each disease, we also compared experiment using PCC and MMI as two different inference methods, and acquired both consensus and divergence. Experiments of the two inference methods identified multiple identical phylogenetic gene markers and identified consensus pattern of disease-associated networks, indicating the robustness of our framework. On the other hand, the two different inference methods also led to specific findings, providing us with various aspects to study the mechanisms of diseases. We detected six T2D-relevant subnetworks and identified key drivers for each of them correspondingly. The identified key drivers include IPR006047, IPR018485 and IPR003385 related to the carbohydrate metabolic process, while the carbohydrate metabolic process is an important issue during the development of T2D [26]. In addition, we also detected key drivers for RA. Both PCC and MMI experiments located multiple InterPro matches (IPRs) which are related membrane and infection. Six subnetworks were extracted by PCC, containing IPRs concerned with immunoglobulin, Sporulation. Three subnetworks were detected by MMI, with IPRs about biofilm, Flaviviruses, bacteriophage, etc. The result is inspiring since the development of biofilms is regarded as one of the drivers of persistent infections [27] and some biofilms-growing bacterias contribute to RA [28].
Our method is to detect the key drivers which drive the diseases related networks in the microbial community. The key drivers can be microbial species or phylogenetic gene markers. For simplicity, we present our method with nodes as genes in the subsequent descriptions.
The detection of key drivers consists of following steps (see Fig. 1). First, we construct a MEN to represent the relationship between genes based on microbial abundance profiles and infer the weight of each edge. Second, we cluster the genes and partition the MEN into multiple subnetworks. Third, we analyze the phenotype variables and extract the delegated phenotype. By computing the associations between subnetworks and delegated phenotype, we obtain subnetworks that are most related to the disease. Last, based on PageRank, we identify actors with top influence over others in each subnetwork as key drivers.
Flowchart. First, we build a MEN and cluster genes into multiple subnetworks. After that, we summarize the phenotype variables and connect it to subnetworks. Then, we locate key drivers through PageRank
Inference method
In KDiamend, we provide two ways to compute distances between genes. The first one is PCC, which is the most popular way to capture similarities between genes. In addition, inspired from the inference of gene regulatory network in RNA analysis and mixture models in the microbiome analysis, we adopted the MMI and normalization processes in Context Likelihood of Relatedness(CLR) [29]. MI, which uses the mutual dependency and common uncertainty as for the measurement of connection between genes, does not assume linear, or continuous dependence like correlation [19, 30], so it can detect interactions which might be missed by PCC. MMI, under the assumption of the Gaussian mixture model, is for dealing with the high proportion of zero counts issue. The adopted CLR, which considers the context of the whole network and eliminates noises from the background, makes MMI more tolerant for noises when measuring the interactions.
At the beginning, we have an abundance matrix E, which contains abundance value of n genes in m samples. For each gene i, we have a vector of X i =(Ei1,Ei2,…,E im ). The strength of the relationship between gene i and gene j can be measured by PCC:
$$ PCC(i,j)=\frac{cov(X_{i},X_{j})}{\sigma_{i},\sigma_{j} }, $$
where cov(X i ,X j )is the covariance of X i and X j , and σ i is the standard deviation of X i . The adjacency matrix A of network can be generated from A ij =PCC(i,j). Then the distance between gene i and gene j can be interpreted as D ij =1−|PCC(i,j)|.
Apart from PCC, we also implemented MMI [23]. First, we decomposed X i into c i bins as Xi,1, …, \(X_{i, c_{i}}\). In this case, X i was distributed according to the following function:
$$ f_{X_{i}}(x)=\sum_{k=1}^{c_{i}}\pi_{i, k}g_{X_{i, k}}(x), $$
where \(g_{X_{i, k}}(x)\) (1≤k≤c i ) denotes the density function for Ci,k, and πi,k is the proportion for each sample in Ci,k. As proved in former work [23], assuming that Xi,k fits in a Gaussian distribution, we can estimate the mutual information between X i and X j as:
$$ MMI(X_{i},X_{j})=MMI^{O}(X_{i},X_{j})+MMI^{I}(X_{i},X_{j}). $$
The "outer" MI, MMIO(X i ,X j ), captures discretized dependency, while the "inner" MI, MMII(X i ,X j ), refers to the weighted aggregation of MI for each bin.
After computing MMI between all the genes and get a matrix M, we normalized the distance between gene i and gene j by: \(CLR(Z_{i},Z_{j}) = \sqrt {\left (Z_{i}^{2},Z_{j}^{2}\right)}\) where Z i and Z j are z-scores of M ij taking M i and M j as background respectively [29]. Then we applied hierarchical clustering and partitioned the MEN into multiple subnetworks according to the distance between genes.
Delegated phenotype
To best capture phenotype change in disease networks from healthy networks, we generated delegated phenotype by rotating Coordinates in the PC space of phenotype variable matrix S. Each row in S represents a sample while each column refers to a phenotype variable, such as gender, age, disease state, etc. If the properties are nonnumerical, we converted data into numbers before further analysis. Suppose one column v in S indicates whether each sample is collected from a person with or without this disease. That is v=(v1,v2,…,v k ,…,v m ),v k ∈(Y,N),1<k<m where m is the number of samples, Y indicates that this sample is collected from a person with the disease, and N means not.
We applied principal component analysis (PCA) to conclude S. We consider the first two principal components (PCs) to be enough for explaining disease variability, since the number of phenotype variables is relatively small in our test data. When phenotype data are more complicated, we may need extra analysis to decide the number of PCs we use to conclude delegated phenotype. We investigated the first two PCs and plotted samples in the coordinate of PC1 and PC2, regarding every sample as a point. Consequently, we got m points and each point refers to one sample. The coordinates for point k, is expressed as (x k ,y k ). We rotated PC to PC′ and make sure it has the largest correlation with v upon rotation. PC′ can best explain variability related to the disease. The angle of rotation is the θ that maximize f(θ).
$$\begin{array}{*{20}l} {}f(\theta)\! =\!\! \sum_{k, v_{k} \in N}(x_{k} cos \theta_{k} + y_{i} sin \theta_{k}) \,-\, \sum_{k, v_{k} \in Y}(x_{k} cos \theta_{k} + y_{k} sin \theta_{k}) \end{array} $$
That is to say, the angle between PC1 and PC1′ is:
$$\begin{array}{*{20}l} \theta=arctan \frac{\sum_{k, v_{k} \in N}x_{k}-\sum_{k, v_{k} \in Y} x_{k}}{\sum_{k, v_{k} \in N}y_{k}-\sum_{k, v_{k} \in Y} y_{k}} \end{array} $$
For example (see delegated phenotype in Fig. 1), blue points represent v k =Y and red points represent v k =N. By calculating θ, we got the line which implies the direction most correlated to the disease state.
We acquired delegated phenotype PC′, which has the largest correlation with the disease state and outperforms other single variables on explaining the variability of phenotype information at the meantime. For every subnetwork, we concluded the abundance pattern of genes as eigengenes [31]. Then we bridged the subnetworks to phenotype information by calculating the correlation between eigengenes and PC′. Subnetworks which have strong relationships to PC′ are extracted as disease-relevant subnetworks.
Identifying Key driver
Last, in every extracted disease relevant subnetwork, we applied network topology analysis and assigned every gene a PageRank score. PageRank (PR) ranks the nodes in a graph according to the structures of links with others and is used by Google's search engine to compute rankings of websites. In this algorithm, the score for one node can be affected by its neighbors [32] and if one's neighbors have high scores, its score increases iteratively [33].
As stated in [34], letting F u be the nodes linking to the node, B u be the nodes linked from it, and N u =|F u | be the magnitude of F u . Besides, considering there might be other factors towards the ranking, let E(u) be the vector concerned with some of the rank. Then, the PageRank of the node is defined as.
$$ R(u)=c\sum_{\upsilon \in B_{u}} \frac{R(\upsilon)}{N_{\upsilon} }+cE(u) $$
We tested our method with real microbiome datasets and compared PCC with MMI in this framework. First, In order to detect key drivers for T2D, we downloaded processed InterPro matches (IPR) abundance data from EBI (SRP008047), which is gut metagenome (microbiome) data from Chinese samples. InterPro [35] provides a functional analysis of protein sequences by classifying them into families and predicting the presence of domains and important sites. The phenotype information of the dataset is provided in related paper [1]. We used the 145 samples from stage one.
We first trimmed IPRs with low abundance in relative abundance matrix and then applied quantile normalization [36, 37]. By computing PCC and MMI between pairs of genes, we reconstructed a MEN and conducted clustering to partition the MEN into multiple subnetworks. For all subnetworks, eigengenes were calculated by selecting the first PC of abundance. The eigengene was used for summarizing the abundance pattern in each subnetwork and to bridge it with phenotype information. On the other hand, we digitalized the phenotype matrix and applied PCA to it. We generated delegated phenotype by rotating coordinates in the PC space to best capture the phenotype change in disease networks from healthy networks. We extracted three subnetworks for PCC experiment and three subnetworks for MMI according to the correlation between eigengenes and delegated phenotype and the corresponding p-value (see Fig. 2). The p-value for correlation of each subnetwork was calculated by permuting the same number of genes from the dataset and calculating the correlation between the permuted eigengene and the delegated phenotype. After repeating 1000 times, the rank of the real correlation for the subnetwork is regarded as p-value.
Experiment results for type 2 diabetic. a The delegated phenotype. b, c Distribution for correlation between subnetworks and delegated phenotype. Every point refers to a subnetwork. The x-axis of the point refers to the correlation between this subnetwork and the delegated phenotype. The y-axis of the point refers to the 1 −p-value for the correlation. The y-axis on the right is for the histogram of these correlation. b is for PCC experiment and c is for MMI experiment. d, e Plots for eigengene and delegated phenotype. Blue line refers to the z-score of eigengene for that subnetwork. X-axis refers to samples. d is for PCC experiment and E is for MMI experiment
PCC experiment and MMI experiment detected 11 consensus IPRs which scattered in three subnetworks for MMI and two subnetworks for PCC. Consequently, the interaction generated from two types of inference connects these five extracted subnetworks together and merges them into one large community (see Fig. 3). Most of the consensus IPRs in the merged community are associated with the metabolic process and the catalytic activity which implies that the process is relevant to the disease. More specifically, two experiments both identified IPR018485 which participates in the carbohydrate metabolic process with the phosphotransferase activity and is active in carrying out ATP-dependent phosphorylation [38]. The extracted disease-relevant subnetworks in T2D are about the carbohydrate metabolic process and phosphorelay.
Extracted networks for T2D. Detected T2D-relevant subnetworks for PCC and MMI experiment. Blue and yellow nodes refer to IPRs identified by PCC and MMI, while green ones refer to IPRs identified by both methods. The weight calculated by PCC and MMI was normalized to the same scale. The corresponding subnetworks are labeled. Nodes with top PageRank in each subnetwork are enlarged
In addition, for PCC experiment, key drivers in subnetwork 89 and subnetwork 208 are related to the carbohydrate metabolic process, including IPR006047, IPR018485, and IPR003385. The key driver in subnetwork 166 is IPR001789, which plays a role in phosphorelay signal transduction system. MMI also detected IPR018211, IPR005538, IPR003501, and IPR001790 which are related to phosphorelay. PCC and MMI both identified IPRs related to the carbohydrate metabolic process and phosphorelay.
Oral microbiome
We applied our method to oral microbiome to detect the key drivers in microbial community related to dysbiosis in Rheumatoid Arthritis (RA). The abundance data was downloaded from EBI (ERP006678). Information of phenotype variables for different individuals was acquired from published paper [2]. We mapped the samples downloaded from EBI with the individual ID and got 49 oral microbial samples in total. Among them, 27 samples were collected from patients with RA in different disease states, 22 samples, used as the control, were collected from people without RA. 21 of them are saliva samples and 28 are dental samples.
We first conducted filtering and then applied normalization to avoid noises. After that, we constructed the MEN by computing similarities between all pairs of IPRs. Then we partition the MEN into multiple subnetworks by clustering.
Similar to the analysis for T2D, we processed the phenotype matrix and detected subnetworks most related to RA. First, we removed phenotype variables with more than 1/3 missing values. Then, for remaining phenotype variables, we conducted imputation using R package MICE [39]. By computing correlation between delegated phenotype and eigengenes of subnetworks, we extracted six subnetworks most related to RA using PCC and three subnetworks using MMI. Finally, we identified key drivers for detected disease associated subnetworks correspondingly.
We applied key drivers analysis for RA using PCC and MMI as two different inference methods respectively. Both experiments show IPRs, in extracted associated subnetworks, have higher abundance in disease state than in normal state (see Fig. 4). For PCC experiment, annotation shows that most IPRs in subnetwork 335 and 63 are about cell membrane while most IPRs in subnetwork 676, 128, 679 and 680 are about replication and cell growth. Functions for IPRs were inferred according to keywords and Gene Ontology (GO) mentioned in InterPro [35]. Moreover, subnetwork 335 also contains IPR014879(Sporulation initiation factor Spo0A, C-terminal) and IPR013783 (Immunoglobulin-like fold). IPR013783 is about immunoglobulin molecules and T-cell receptor antigen [40, 41], while RA is a disease caused by compromised immune systems [42].
Experiment results for Oral experiment. a The delegated phenotype. b, c Distribution for correlation between subnetworks and delegated phenotype. Every point refers to a subnetwork. The x-axis of the point refers to the correlation between this subnetwork and the delegated phenotype. The y-axis of the point refers to the 1 −p-value for the correlation. The y-axis on the right is for the histogram of these correlation. b is for PCC experiment and c is for MMI experiment. d, e Bar plot for the average abundance of extracted subnetworks. The relative abundance is different in different disease states
For MMI experiment, subnetwork 1642, which has the largest correlation with delegated phenotype, contains multiple IPRs about biofilm: IPR024487, IPR019669, and IPR010344. There are totally 24 IPRs in this subnetwork and top 5 of them are IPR003496, IPR024205, IPR008542, IPR010344, and IPR019669. Specifically, IPR010344 plays a role in biofilm formation and IPR019669 participates in single-species biofilm formation on the inanimate substrate. The development of biofilms is one of the drivers of persistent infections [27]. Some bacteria, when growing in the biofilm, e.g., Porphyromonas gingivalis in dental plaque, can become destructive and may contribute to RA [28]. Besides, subnetwork 1642 also contains IPR013756, associated with Flaviviruses, and IPR009774, related to hypothetical Streptococcus thermophilus bacteriophage, which hints the infection process in this subnetwork.
Noise tolerance of delegated phenotype
To test whether our delegated phenotype is robust when phenotypes are deficient, we tried every combination of phenotypes with removing 1,2,3,6,10 of them from the phenotype variables matrix of RA, and generated delegated phenotype for each of them. Then we calculated the correlations between those generated delegated phenotypes and extracted subnetworks. The result is promising and these extracted subnetworks have high correlation values in most cases (see Fig. 5).
Noise tolerance of delegated phenotype. a The first box plot refers to combination with removal of one, two, or three phenotype variables. b The second refers to the removal of six phenotype variables. c The third refers to the removal of ten phenotype variables. d The last one refers to the overall performance with all these combinations
Performance of PageRank in searching the key driver
We tested the performance of PageRank on a simulated dataset. At the beginning, we named the driven relationship as sub-gene relationship. We simplified the network by assuming that one gene could only be driven by one gene. Linear function is used to represent the driven relationship. i.e. y=Ax+n, where x and y denotes expression levels for gene and its sub-gene and n is the noise following the normal distribution with 0 mean. There are three parameters for the simulation algorithm: the number of sub-genes for each gene, the depth of the network and the noise level. Here, we used the variance to define the noise level and the variance of noise is βx. The structure of the simulated network and three parameters are shown in Fig. 6.
The structure of simulated network. The simplified network can be transferred into a tree
We generated the simulated data with various parameters. For each parameter group, 100 samples were produced. We compared the performance of PageRank with the degree algorithm that locates the key driver with the highest degree. As shown in Fig. 7, the noise level has little effect on prediction precision. The result of the degree algorithm also follows this pattern. To compare these two algorithms, we collected the cases where only one algorithm correctly found the key driver and the result is shown in C. Since when the number of sub-genes is large, both algorithms have high prediction precision, we focus more on cases where the sub-gene number is relatively small. In this situation, the PageRank has better performance.
Testing results for PageRank algorithm to find the key driver. The first two figures show the prediction precision of PageRank under different simulation parameters. a is related to the network depth and the number of sub-genes. b is for the noise level and the number of sub-genes. The third figure compares the performances of PageRank and degree algorithm. We counted cases where only one algorithm found the key driver correctly
Application to Alzheimer's disease
To further validate our method is capable of detecting key drivers of disease, we applied KDiamend to Alzheimer's Disease (AD) with analyzing RNA expression profiles, which were downloaded from GEO(GSE44770) [17]. Both of PCC and MMI experiments identified FBXL16 and OLFM1. FBXL16 related pathways are Innate Immune System and Class I MHC mediated antigen processing and presentation, while researches have shown that the activation of the Innate Immune System plays a crucial role in promoting AD [43]. OLFM1 is related to nervous system development and Neuroblastoma [44]. Besides, PCC experiment also identified RPS4Y1 and PITPNB as key drivers for extracted disease associated subnetworks. MMI experiment also identified KAZALD1, OR4A47, RNASE11, TXNDC2, 7-Mar, RTN4, TSPAN9, PCNP and PPP2R2C. More specifically, RTN4 is related to Demyelinating Disease [45] and KAZALD1 is related to Lobar Holoprosencephaly [46]. These experiments show a possible application of our method. It is capable of detecting key drivers in the network inferred from not only the microbial abundance profile but also other kinds of abundance data, like RNA expression or proteomics.
We proposed a novel method to detect key actors who drive the disease concerned MENs, which helps to understand microbial factors relevant to the certain disease. We divided the MENs into multiple subnetworks and then, instead of detecting important genes based on pathways or gene annotations, we extracted subnetworks which are most relevant to disease by utilizing the correlation between the patterns of abundance profiles and the delegated phenotype. Lastly, we identified key drivers based on PageRank.
We tested our method with two real microbial datasets. We detected that the disease-relevant subnetworks in T2D are related to the carbohydrate metabolic process and phosphorelay, while RA-relevant subnetworks are related to membrane, cell growth, and infection. The extracted subnetworks for RA include IPRs concerned with immunoglobulin, Sporulation, biofilm, Flaviviruses, bacteriophage, etc. Then we located corresponding key drivers for extracted disease-relevant subnetworks. Besides microbial data, we also tested our method with gene expression profiles to identify key drivers for AD and the outcome was inspiring. Experiments show our method is capable of detecting key drivers and providing hints to understand the mechanisms of diseases.
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We would like to express sincere gratitude to our colleagues who greatly assisted the research. We thank Tan Bowen for offering assistance during designing the algorithm and Zhao Zicheng, Xu Chang for comments that greatly improved the manuscript.
This work and the publication of this article were sponsored by the GRF Project from RGC General Research Fund at City University of Hong Kong, Kowloon, HKSAR under Grant 9042181 (CityU 11203115).
Package is available at http://www.deepomics.org/pipelines/3DCD6955FEF2E64A/.
About this supplement
This article has been published as part of BMC Systems Biology Volume 12 Supplement 1, 2018: Selected articles from the 16th Asia Pacific Bioinformatics Conference (APBC 2018): systems biology. The full contents of the supplement are available online at https://bmcsystbiol.biomedcentral.com/articles/supplements/volume-12-supplement-1.
Department of Computer Science, City University of Hong Kong, Tat Chee Avenue, Hong Kong, China
Mengxuan Lyu, Jiaxing Chen, Yiqi Jiang, Wei Dong, Zhou Fang & Shuaicheng Li
Mengxuan Lyu
Jiaxing Chen
Yiqi Jiang
Wei Dong
Zhou Fang
Shuaicheng Li
LYU was responsible for the implementation of the package. Chen and LYU together designed the algorithm, conducted experiments and completed the manuscript. JIANG provided expertise on metagenomics and provided the data. DONG helped to improve the algorithm, conduct experiments and complete the manuscript. Fang provided assistance for testing and deploying the pipeline on the platform. LI provided insights and also polished the manuscript. All of the authors have read and approved of the final manuscript.
Correspondence to Shuaicheng Li.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver(http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Lyu, M., Chen, J., Jiang, Y. et al. KDiamend: a package for detecting key drivers in a molecular ecological network of disease. BMC Syst Biol 12 (Suppl 1), 5 (2018). https://doi.org/10.1186/s12918-018-0531-8
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Optimal control problems for some ordinary differential equations with behavior of blowup or quenching
Feedback stabilization with one simultaneous control for systems of parabolic equations
September 2018, 8(3&4): 789-808. doi: 10.3934/mcrf.2018035
Weak stability of a laminated beam
Yanfang Li 1, , Zhuangyi Liu 2, and Yang Wang 3,,
College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China
Department of Mathematics and Statistics, University of Minnesota, Duluth, MN 55812, USA
College of Information Science and Technology, Donghua University, School of Mathematics, Physics and Statistics, Shanghai University of Engineering Science, Shanghai 201620, China
* Corresponding authorr: Yang Wan
Received October 2017 Revised May 2018 Published September 2018
In this paper, we consider the stability of a laminated beam equation, derived by Liu, Trogdon, and Yong [6], subject to viscous or Kelvin-Voigt damping. The model is a coupled system of two wave equations and one Euler-Bernoulli beam equation, which describes the longitudinal motion of the top and bottom layers of the beam and the transverse motion of the beam. We first show that the system is unstable if one damping is only imposed on the beam equation. On the other hand, it is easy to see that the system is exponentially stable if direct damping are imposed on all three equations. Hence, we investigate the system stability when two of the three equations are directly damped. There are a total of seven cases from the combination of damping locations and types. Polynomial stability of different orders and their optimality are proved. Several interesting properties are revealed.
Keywords: Laminated beam, polynomial stability, exponential stability, optimal decay rate, semigroup.
Mathematics Subject Classification: Primary: 35B35, 47D03; Secondary: 93D05.
Citation: Yanfang Li, Zhuangyi Liu, Yang Wang. Weak stability of a laminated beam. Mathematical Control & Related Fields, 2018, 8 (3&4) : 789-808. doi: 10.3934/mcrf.2018035
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Yanfang Li Zhuangyi Liu Yang Wang
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CommonCrawl
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SEQdata-BEACON: a comprehensive database of sequencing performance and statistical tools for performance evaluation and yield simulation in BGISEQ-500
Yanqiu Zhou1 na1,
Chen Liu1 na1,
Rongfang Zhou1 na1,
Anzhi Lu1,
Biao Huang1,
Liling Liu1,
Ling Chen1,
Bei Luo1,
Jin Huang1 &
Zhijian Tian1
BioData Mining volume 12, Article number: 21 (2019) Cite this article
The sequencing platform BGISEQ-500 is based on DNBSEQ technology and provides high throughput with low costs. This sequencer has been widely used in various areas of scientific and clinical research. A better understanding of the sequencing process and performance of this system is essential for stabilizing the sequencing process, accurately interpreting sequencing results and efficiently solving sequencing problems. To address these concerns, a comprehensive database, SEQdata-BEACON, was constructed to accumulate the run performance data in BGISEQ-500.
A total of 60 BGISEQ-500 instruments in the BGI-Wuhan lab were used to collect sequencing performance data. Lanes in paired-end 100 (PE100) sequencing using 10 bp barcode were chosen, and each lane was assigned a unique entry number as its identification number (ID). From November 2018 to April 2019, 2236 entries were recorded in the database containing 65 metrics about sample, yield, quality, machine state and supplies information. Using a correlation matrix, 52 numerical metrics were clustered into three groups signifying yield-quality, machine state and sequencing calibration. The distributions of the metrics also delivered information about patterns and rendered clues for further explanation or analysis of the sequencing process. Using the data of a total of 200 cycles, a linear regression model well simulated the final outputs. Moreover, the predicted final yield could be provided in the 15th cycle of the early stage of sequencing, and the corresponding R2 of the 200th and 15th cycle models were 0.97 and 0.81, respectively. The model was run with the test sets obtained from May 2019 to predict the yield, which resulted in an R2 of 0.96. These results indicate that our simulation model was reliable and effective.
Data sources, statistical findings and application tools provide a constantly updated reference for BGISEQ-500 users to comprehensively understand DNBSEQ technology, solve sequencing problems and optimize run performance. These resources are available on our website http://seqBEACON.genomics.cn:443/home.html.
Next generation sequencing (NGS, also known as high-throughput sequencing) has led us into the genomic era. In the past 15 years, the development of sequencing technology has been mainly committed to a reduction in cost and improvements in throughput, accuracy and read lengths. Currently, the sequencer manufacturers Illumina and Beijing Genomics Institute (BGI) provide high throughput and accuracy, while Pacific Bioscience and Oxford Nanopore offer long read lengths [1, 2]. The first BGI sequencer BGISEQ-500 was launched in 2015 (http://en.mgitech.cn/), which was based on two key technologies: DNA nanoball (DNB) and Combinatorial Probe-Anchor Synthesis (cPAS). In library preparation, DNA is fragmented, end-repaired and ligated with adapters. The ligation product is amplified by PCR for several cycles to become DNA libraries. These libraries are then circularized by DNA ligase with a splint oligo that was reverse complemented to one strand. Next, the DNA library circles are replicated with polymerase phi29 to form a single-stranded DNA molecule called a DNB by rolling circle amplification (RCA). The DNBs are dispersed and immobilized on a photolithographically etched, patterned flow cell by a loader machine. When sequencing, a probe is first annealed to a DNA molecular anchor on the DNB. In each cycle, DNA polymerase incorporates one base labeled with a fluorescence group, and the four colored light signals emitted by the bases are collected via a high-resolution imaging system and converted into bases after basecalling [3].
Recently, BGISEQ-500 platform has been widely used in a variety of sequencing applications, such as whole genome sequencing (WGS) [4], whole exome sequencing (WES) [5], RNA-seq [6], small RNA and metagenomics [7, 8]. BGISEQ-500 sequencing platform has not only participated in the transcriptome analysis of plant nitrogen metabolism [9] but also in human clinical applications for cancer genome sequencing and TP53 mutation detection in high-grade serous ovarian cancer [10, 11]. In addition, it has been reported that BGISEQ-500 has good performance at the single-cell resolution, such as in scRNA-seq and scCAT-seq [12,13,14]. Compared with the Illumina platforms, the BGISEQ-500 is cost-effective and has low error and duplication rates [1, 13]. The throughput per lane of BGISEQ-500 is approximately twice as HiSeq4000 in the PE100 sequencing type. The cost per gigabase (Gb) is 40–60% of that of the Illumina HiSeq4000 platform. The generation of DNBs is based on rolling circle amplification, which effectively prevents errors from PCR amplification. The sequencing data derived by BGISEQ-500 are compatible with widely used bioinformatics tools and pipelines such as GATK, bwa, HISAT, DEseq2 and SnpEff [15]. The only difference is the setting of some software parameters. To date, the results obtained from data generated from Illumina and BGISEQ-500 platforms have been comparable. For example, BGISEQ-500 demonstrates comparable SNP detection accuracy in WGS [10], similar consistency in variation detection in WES [5], and high concordance in transcriptome and metagenomic studies [16, 17]. Therefore, DNBSEQ technology provides a new choice for resolving issues in scientific research and agriculture, environment and clinical applications.
The performance of sequencers is very important for the throughput, quality and reliability of data generated in each run. Among the sequencing performance metrics in BGISEQ-500, yield (e.g., Reads, the number of DNBs recognized by the Basecall software) and quality (e.g., Q30, the percentage of bases with an error rate below 0.001) are of the highest concern. Other metrics regarding chemical reaction and instrument state are also recorded in the sequencing summary. However, we still lack a profound understanding of these metrics, especially the connections between them. As one of the world's largest sequencing service providers, BGI performs thousands of sequencing runs each year. Accompanied with enormous amounts of nucleotide data, massive sequencing performance data are highly valuable for illustrating this complicated process and for troubleshooting. Unfortunately, this type of datasets or databases are presently rare, and a comprehensive database is required to integrate the abundant run performance data.
In this study, we designed a database, SEQdata-BEACON, to comprehensively collect sequencing performance data from BGISEQ-500, including sample, yield, quality, machine state and supplies. We calculated Pearson's correlation coefficients for 52 numerical metrics for hierarchical clustering and analyzed their distribution patterns. We also used linear regression to establish yield simulation models to investigate the connections among yield correlated metrics and attempted to predict the final yield at the early stage of sequencing. All the data and statistical analysis results are available on our open access website. These resources can be used as an updating reference dataset for BGISEQ-500 users in different enterprises or schools to gain a deeper understanding of DNBSEQ technology.
Data collection and database construction
DNA libraries were loaded onto a patterned array. After successive chemical reactions, signal acquisition and basecalling, the sequencers normally generate a series of folders and files in each cycle or at the end of the sequencing process. In BGISEQ-500, more than 10 files recorded the run configuration data of the entire sequencing process, such as InputInfo_*.txt, RunInfo.txt, summaryReport.html, fovReport.QC.txt, BarcodeStat.txt and fq.fqStat.txt. We chose 60 BGISEQ-500 sequencers in the BGI-Wuhan lab and collected all available files generated following chemical reactions and basecalling. The criteria for selecting metrics from these files are as follows: 1) metrics of great concern and closely related to the sequencing process [18]; 2) metrics covering information on the sequencing type, the optical path state of the machine etc.; 3) metrics related to traceable information for troubleshooting. Based on these, we used flow cell identifier (FC) as an index to extract 64 metrics from data resources to create a database 'SEQdata-BEACON'. Each entry was assigned a unique identification number (ID). We accumulated lanes in paired-end 100 (PE100) sequencing since it is the major sequencing type in BGISEQ-500. No detailed sample information was entered into the database in order to protect the privacy of customers. The database was constructed on a MySQL server (version 8.0).
Web visual interface and statistical analysis
A user-friendly interface for SEQdata-BEACON was built on Apache (version 2.4.33 win64 VC15 server). The Google Chrome web browser (version 68.0.3440.106) is suggested to access the website. Statistical analysis and figures in this study were generated based on data obtained from our database with R software (version 3.5.0 x 64) with the ability to install additional packages as needed.
Yield simulation model
We used yield-related independent variables to construct the yield simulation model. First, considering the sequencing principle, DNB was the definitive source for yield. The number of successfully fixed DNBs on the patterned array determined the ability to produce reads [3, 18]. TotalEsr (Total Effective Spot Rate) multiplied by Dnbnumber excluded those spots without DNBs or with dark DNBs that showed no light signal and represented the maximum signal that could be successfully collected. Second, metrics that changed with the sequencing process were also included, such as BIC (Basecall information content), accGRR (Accumulated Good Reads Rate), SNR (Signal to Noise Ratio), FIT (indicates the distribution of differences between signal and noise for each base), Intensity (Light intensity of each base (ATCG) for each cycle), Runon and Lag (the percentage of read strands being out of phase with the current cycle). These metrics reflect the loss in final yield resulting from chemical reactions, light intensity, camera acquisition and basecalling in the sequencing process. The loss in yield is primarily composed of filtered reads. Third, the values of these metrics were collected from our database as the datasets. The correlations of each metric with yield were studied, and the ones that were linearly correlated with yield were chosen (Additional file 1: Figure S1). We used Tukey's boxplot to pinpoint the possible outliers, which were defined as observations that fell below Q1–1.5 IQR (interquartile range) or above Q3 + 1.5 IQR and excluded from the model construction. Finally, we constructed a linear regression (LR) model with TotalEsr*Dnbnumber, BIC, accGRR, SNR and FIT as input variables and yield (Reads) as the output variable. The LR model formula was as follows (Eq. (1)):
$$ Y={\beta}_0+{\beta}_1 TD+{\beta}_2B+{\beta}_3G+{\beta}_4S+{\beta}_5F+\varepsilon, \upvarepsilon \sim \mathrm{Normal}\left(0,\upsigma \right) $$
Y is the value of the dependent variable yield. T is the value corresponding to the current cycle in TotalEsr. D is the value of Dnbnumber at the beginning of the sequencing. B, G, S and F are the average values of BIC, accGRR, SNR and FIT, respectively, for the first 200 cycles. ε is the observed error and obeys a normal distribution. The model parameters βn are the coefficient values estimated using a regression model.
Furthermore, we simulated the final yield based on the metric values obtained every 5 cycles. The formula was as follows (Eq. (2)):
$$ {Y}_i={\beta}_{0,i}+{\beta}_{1,i}{T}_i{D}_i+{\beta}_{2,i}{B}_i+{\beta}_{3,i}{G}_i+{\beta}_{4,i}{S}_i+{\beta}_{5,i}{F}_i+{\varepsilon}_i,{\varepsilon}_i\sim \mathrm{Normal}\left(0,\upsigma \right) $$
Yi is the value of the dependent variable yield. Ti is the value corresponding to the current cycle i in TotalEsr. Di is the value of Dnbnumber at the beginning of the sequencing. Bi, Gi, Si and Fi are the average values of BIC, accGRR, SNR and FIT, respectively, for the first i cycles. εi is the observed error in the i cycle and obeys a normal distribution. The model parameters βn, i are the coefficient values estimated using a regression model. The coefficient of determination (R2), which ranges from 0 to 1, was used to measure the accuracy of our model; a value closer to 1 means better performance of the model.
Evaluation of the linear regression model was performed with test sets to test the reliability of the prediction model. The test sets were used in the model formula to obtain prediction results. An R2 for the test sets was used to evaluate the constructed model. Both linear regression and the backward elimination method of stepwise regression and evaluation were conducted in R.
The DNA sequencing process generated a series of files used to record sequencing performance information. In this study, we extracted 65 metrics to construct a database 'SEQdata-BEACON' to store the massive data source. The architecture of the database and applications is shown in Fig. 1. From November 2018 to April 2019, 'SEQdata-BEACON' accumulated a total of 2236 entries in PE100 sequencing using a 10 bp barcode from 60 sequencers in the BGI-Wuhan lab. The database primarily collected information on sample, yield, quality, machine status and supplies. Sequencing type and other sample information were stored as Sample. Yield and quality are the gold standards of the reliability of our sequencing data. Regular quality control metrics such as Reads, Bases—related to yield and Q30—related to quality were stored. Machine performance reflects the stable environment of the sequencing process, so machine states such as Signal, Intensity and Theta were stored in the database. Supplies provides traceable information to solve problems in troubleshooting, so sequencing reagents and sequencing time were also stored in the database. Furthermore, we explored our database to describe the statistical results of metric features and to construct a yield simulation model based on yield-related metrics.
Schematic architecture of SEQdata-BEACON. The architecture of the database and applications contains four parts. "Sequencing Files" shows the files generated from BGISEQ-500 sequencers. "SEQdata-BEACON" shows the information of metrics. "Yield simulation model" shows the special function of our database to predict the final yield from input metrics. "Web services" shows three applications in our website to provide interactive functions for users--Predicting, Browsing and Data querying
Web visual interface
To provide open access to our data, we designed a comprehensive website 'SEQdata-BEACON' with Home, Browse, Tools, Download and Guide pages to display the database and data-mining applications (Fig. 2a). The 'Home' page gives users an introduction to our website and a schematic architecture of our database. The 'Browse' page allows users to look through the numerical metric features, including a heatmap of Pearson's correlation coefficients and the metric distributions. For example, the distribution of FIT and its changes per cycle are both illustrated in charts for observing the distribution patterns and fluctuations. Users can choose the name of their metric of interest in the drop-down menu, and the corresponding distribution chart will be shown at the bottom of the webpage (Fig. 2b; see the section 'Statistical findings: metric features' for details). The 'Download' page allows users to obtain the data in EXCEL format according to our update time; all the data and analysis results will be updated every 2 months (Fig. 2c). The 'Tools' page allows users to test our simulation model. Users can enter specific metric values on our website in the example format and click 'Start', then the expected yield confidential intervals will be shown (Fig. 2d; see the section 'Statistical findings: yield simulation model' for details). The 'Guide' page supplies a guideline for regular operation.
Web visual interface of SEQdata-BEACON. The screen shots show views of functional modules. a The navigator bar on the "Home" page. b The drop-down menu and distribution charts of metrics are shown on the "Browse" page. c The download sources are listed on our website's "Download" page. d The input windows and the results of the yield simulation model are shown on the "Tools" page. Please note that not all fields are shown
Statistical findings: metric features
Based on the data from November 2018 to April 2019, Pearson's correlation coefficients of the numerical metrics were calculated, and the resulting 52*52 correlation matrix is shown as a heatmap (Fig. 3). The metrics were mainly clustered into three groups with 20, 15 and 17 metrics; the first group could be further divided into two branches that represent yield and quality. The other two groups represent machine state (e.g., Theta indicates the angle between the moving direction of the array stage and the track line on the array) and sequencing calibration (e.g., Lag, Runon). Red blocks indicate positive relationship and blue blocks indicate negative relationship. Yield was positively correlated with quality, and the sequencing calibration was negatively correlated with both. It can also be seen that five metrics with a Pearson correlation of 1 with Reads were redundant; these were deleted from further analysis. In addition, totalCG% (the percent of CG based in sequencing lane) is a conceptual summary of R1CG% and R2CG%, and only one metric was reserved according to analytical needs.
Numerical metrics correlation. Hierarchical clustering of the Pearson's correlation matrix between 52 metrics. The three branches Yield and Quality, Machine State, and Sequencing Calibration are marked with purple, orange and yellow, respectively, on the right side of the Y-axis in the figure. Red blocks in the heatmap indicate positive relationship, blue blocks indicate negative relationship, and white blocks indicate no relationship
Next, we investigated the distribution patterns of the above metrics. It can be seen in the scatterplot of Q30 versus Reads that yield ranged from 350 to 850 M, and Q30 was above 70%. The histograms above and to the right of the chart also show the distributions of Reads and Q30, respectively (Fig. 4a). During normal performance of BGISEQ-500, Reads is greater than 650 M and Q30 is over 85% in each lane (http://en.mgitech.cn/). In the scatter plot, a total of 2026 lanes were located in the normal range, and the proportion of outliers was less than 10%, which suggests that the instrument performance was relatively stable. The distribution pattern of FIT was also plotted in a histogram, and it was mostly around 0.80 (Fig. 4b). The FIT value was calculated cycle by cycle, and it was found to slowly decrease from 0.811 to 0.757 in read1 and from 0.835 to 0.763 in read2, with more deviation at the beginning of both reads and less deviation toward the end (Fig. 4c). The metric FIT suggested the distribution of differences between signal and noise for each base and the performance of the optical path during the sequencing process [18]. The deviation in each cycle may reflect the changing pattern of FIT in the sequencing process and indirectly hint at the status of the signal or optical path.
Distribution of Metrics in SEQdata-BEACON. a Q30 versus Reads, histogram of Q30 and Reads shown in gray, density profiles shown in blue. b Histogram of FIT in all entries. c Scatterplot of FIT through 200 cycles
Statistical findings: yield simulation model
In this study, considering the BGISEQ-500 sequencing principle, the meanings of the metrics and the linear correlations, the linear regression model included the variables TotalEsr, Dnbnumber, BIC, accGRR, SNR and FIT as predictors. Statistical program R was applied to construct the model using training sets obtained from November 2018 to April 2019 and to test the model with test sets obtained from May 2019. The LR model is shown in Eq. (3) and Table 1 displays the standard error, t-value and p-value of this function in the regression results. The results indicate that the contribution of accGRR was not significant. Subsequently, the backward elimination method of stepwise regression was used to build the model to find an optimum solution, and the final LR model is shown in Eq. (4). The regression results of this function indicate that the contributions of TotalEsr*Dnbnumber, BIC, SNR, FIT were significant at the 1% probability level (Table 1). Here, we compared the yield with the predicted results in our model, obtaining an R2 of 0.97. Then, the model was run with the test sets to predict the yield, which resulted in an R2 of 0.96. The performance of the real test meets the expectations of constructed model. These results indicate that our simulation model was reliable and effective (Fig. 5). It also suggests that five metrics, TotalEsr*Dnbnumber, BIC, SNR, FIT, successfully simulated yield in the LR model. In addition, we expected that this simulation model could be used to predict the final yield at the early stage of sequencing. During the sequencing process, the yield was assumed to be a function of the values of TotalEsr, Dnbnumber, BIC, accGRR, SNR and FIT at each running cycle. The model was run every 5 cycles using backward elimination regression with R, and a total of 40 model runs were performed from cycle 1 to 200. The residuals of the prediction models fluctuated at the beginning of read1 and read2, which was mainly due to the establishment of the algorithm matrix by the sequencer (Additional file 2: Figure S2). However, the R2 of the model in the 15th cycle was 0.81, indicating that the final yield could be effectively simulated at this stage. The corresponding LR model is shown in Eq. (5).
$$ Y=-129.692+0.983 TD+1.400B+0.143G-3.334S+26.997F $$
$$ Y=-129.787+0.983 TD+1.402B-3.334S+27.027F $$
$$ {Y}_{15}=-141.021+1.270 TD+4.743B-604.450G-3.465S+28.882F $$
Table 1 Model summaries of linear regressions for predicting yield outputs
A comparison of predicted vs. actual yield in training sets and test sets. a Training sets. b Test sets. The linear regression line is shown in blue
Continuous improvement in DNBSEQ technology has introduced more efficient sequencing platforms, such as MGISEQ-2000 and DNBSEQ-T7. Compared to the Illumina platform, BGISEQ-500 is cheap and has a low sequencing error rate. Sequencing costs are often affected by geographic, institutional, personnel, and reagent costs, and continue to decline as technology updates. According to recent research statistics [13], taking the PE100 sequencing type as an example and not concerning the physical loss of the sequencer, the cost per Gb for the BGISEQ-500 is half that of the HiSeq4000 platform. And BGISEQ-500 sequencing data showed a lower error rate than Illumina (< 0.1 and 0.1%, respectively) [1]. Moreover, the sequencing data produced from BGISEQ-500 are compatible with widely used bioinformatics tools and pipelines. The data analysis results showed comparable accuracy and reproducibility. Recent investigations have reported that MGISEQ-2000 has comparable single nucleotide polymorphism (SNP) detection accuracy in WGS and high gene detection in scRNA-seq to Illumina platforms [19, 20].
In fact, the stable performance of sequencers in massively parallel sequencing guarantees the utility of data and the cost efficiency of each run. Some tools have been reported to evaluate sequencing run performance by analyzing sequencing data quality. For example, FASTQC is a commonly used tool for quality control of sequencing data and the generation of a comprehensive QC report [21]. It can also be incorporated into an analysis pipeline to represent the quality of raw data in easy-to-browse HTML reports [22]. The whole NGS workflow included library and template preparation, enrichment, sequencing and data analysis, but quality control (QC) checkpoints for sequencing performance were often performed in the data quality check rather than in the sequencing process [23]. Different from data quality evaluation, the run performance metrics brought us a wealth of information that could be used to effectively assess the sequencing process and its results. To gain insight into the sequencing performance in BGISEQ-500, we established the first-reported BGISEQ-500 sequencing performance database and website to comprehensively collect performance data.
There were 2236 entries with 65 metrics containing information on sample, yield, quality, machine state and supplies in 'SEQdata-BEACON'. The method of automatically collecting metric values from sequencing configuration files could effectively lighten human labor, shorten time costs and improve data accuracy. The run data we collected covered libraries from most species and major types of sequencing applications. At present, in our 60 BGISEQ-500 sequencers in the BGI-Wuhan lab, PE100 sequencing using a 10 bp barcode is suitable for WGS, WES and RNA-seq, and the libraries are derived from DNA or RNA samples of plants, animals, microbes and humans. In the Q30 versus Reads scatterplot, 90.6% of the lanes had reads greater than 650 M and values of Q30 above 85%, which shows that BGISEQ-500 was stable and reliable in massively parallel sequencing. Therefore, without the risk of index hopping, DNBSEQ can generate excellent sequencing data with fewer duplications and errors [24] and has extensive application in population-scale sequencing projects, such as the 10KP (10,000 Plants) Genome Sequencing Project [25]. To study the correlation of yield-associated metrics, we used the backward elimination method of stepwise regression and established a yield simulation model with an R2 of 0.97. The model produced a good simulation, which suggests that TotalEsr, Dnbnumber, BIC, SNR and FIT contributed to the yield. While predicting the final production, we used all six parameters to construct 40 prediction models using stepwise regression. From the residual deviation of the models, it was shown that the final yield could be predicted in the 15th cycle at the early stage of sequencing, and the small changes in the residuals within read1 and read2 implied little fluctuation of the metrics during the sequencing process. The linear regression model is a common statistical technique for simulating the associations between variables, but whether other methods may produce better simulation results cannot be ruled out. Furthermore, we wanted to investigate quality-associated metrics and establish a quality simulation model. Combined with the yield simulation model, these two models may effectively simulate the sequencing results and bring us more ideas for increasing the sequencing performance.
Recently, the sequencer manufacturer Illumina revealed a new service named "Proactive Instrument Monitoring", which is a proactive support service that involves remote instrument monitoring in real time [26]. By sending instrument performance data to Illumina, the support team can monitor the instrument and resolve issues more quickly. Apart from monitoring the instrument performance, our study paid more attention to data accumulation and was expected to explore data patterns by statistical analysis and interpret sequencing results. In the future, we plan to gather more sequencing platforms built on DNBSEQ technology, which will provide an integrated performance reference for BGI sequencers and will be beneficial to fully understand this series of instruments. Moreover, we also want to add the PacBio Sequel II and Oxford Nanopore PromethION sequencers to obtain a deeper understanding of single-molecule sequencing technology. We expect SEQdata-BEACON to be a comprehensive platform: with data accumulation, it can demonstrate the actual performance of the sequencing platforms; by developing more data-mining applications, it can enrich functional tools such as QC metrics models and metrics standards; by presenting data and statistical results on the website, it can also give users useful optimization and troubleshooting suggestions to solve their problems.
Widespread application of NGS has resulted in a large amount of data, including nucleotide sequences and sequencing process performance. We designed a database, SEQdata-BEACON, to accumulate run performance data from BGISEQ-500 containing 65 metrics with information on sample, yield, quality, machine state and supplies. A correlation matrix of 52 numerical metrics was clustered into three groups: yield-quality, machine state and sequencing calibration. The distribution of numerical metrics presented some features and provided clues for further interpreting the meanings of these metrics and their analysis. We also constructed linear regression models to accurately simulate the final yield using metric values in the 200th and 15th cycles of the runs. The data sources, statistical findings and application tools are all available on our website (http://seqBEACON.genomics.cn:443/home.html), which can facilitate BGISEQ-500 users from enterprises or schools to understand DNBSEQ technology and interpret their sequencing results.
The datasets generated and/or analysed during the current study are available in the SEQdata-BEACON site, [http://seqBEACON.genomics.cn:443/home.html].
accGRR:
Accumulated Good Reads Rate
BGI:
Beijing Genomics Institute
Basecall information content
cPAS:
Combinatorial Probe-Anchor Synthesis
DNB:
DNA nanoball
Eq.:
FC:
Flow cell identifier
Gb:
Gigabase
IQR:
Interquartile range
LR:
NGS:
PE100:
Paired-end 100
RCA:
Rolling circle amplification
scCAT-seq:
Single-cell chromatin accessibility and transcriptome sequencing
scRNA-seq:
Single cell RNA sequencing
SNP:
Single nucleotide polymorphism
TotalEsr:
Total Effective Spot Rate
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Zhu FY, Chen MX, Ye NH, Qiao WM, Gao B, Law WK, et al. Comparative performance of the BGISEQ-500 and Illumina HiSeq4000 sequencing platforms for transcriptome analysis in plants. Plant Methods. 2018;14:69. https://doi.org/10.1186/s13007-018-0337-0.
Fang C, Zhong H, Lin Y, Chen B, Han M, Ren H, et al. Assessment of the cPAS-based BGISEQ-500 platform for metagenomic sequencing. Gigascience. 2018;7(3):1–8. https://doi.org/10.1093/gigascience/gix133.
Wang O, Chin R, Cheng X, Wu KYM, Mao Q, Tang J, et al. Efficient and unique co-barcoding of second-generation sequencing reads from long DNA molecules enabling cost effective and accurate sequencing, haplotyping, and de novo assembly. Genome Res. 2019. https://doi.org/10.1101/gr.245126.118.
Gorbachev A, Kulemin N, Naumov V, Belova V, Kwon D, Rebrikov D, et al. Comparative analysis of novel MGISEQ-2000 sequencing platform vs Illumina HiSeq 2500 for whole-genome sequencing. BioRxiv. 2019. https://doi.org/10.1101/577080.
Senabouth A, Anderson S, Shi Q, Shi L, Jiang F, Zhang W, et al. Comparative performance of the BGI and Illumina sequencing technology for single-cell RNAsequencing. BioRxiv. 2019. https://doi.org/10.1101/552588.
Andrews S. FastQC: a quality control tool for high throughput sequence data. http://www.bioinformatics.babraham.ac.uk/projects/fastqc/. Accessed 18 Nov 2018.
Icay K, Chen P, Cervera A, Rantanen V, Lehtonen R, Hautaniemi S. SePIA: RNA and small RNA sequence processing, integration, and analysis. BioData Min. 2016;9:20. https://doi.org/10.1186/s13040-016-0099-z.
Endrullat C, Glokler J, Franke P, Frohme M. Standardization and quality management in next-generation sequencing. Appl Transl Genom. 2016;10:2–9. https://doi.org/10.1016/j.atg.2016.06.001.
Li Q, Zhao X, Zhang W, Wang L, Wang J, Xu D, et al. Reliable multiplex sequencing with rare index mis-assignment on DNB-based NGS platform. BMC Genomics. 2019;20(1):215. https://doi.org/10.1186/s12864-019-5569-5.
Cheng S, Melkonian M, Smith SA, Brockington S, Archibald JM, Delaux PM, et al. 10KP: a phylodiverse genome sequencing plan. Gigascience. 2018;7(3):1–9. https://doi.org/10.1093/gigascience/giy013.
Illumina Proactive Instrument Monitoring. https://www.illumina.com/services/instrument-services-training/product-support-services/instrument-monitoring.html. Accessed 20 May 2019.
We would like to acknowledge the ongoing contributions and support of all our BGI employees.
We thank Zetao Bai (Oil Crops Research Institute, Chinese Academy of Agriculture Sciences, Wuhan, China) for her assistance editing this manuscript.
No funding agency has funded this work.
Yanqiu Zhou, Chen Liu and Rongfang Zhou contributed equally to this work.
BGI-Wuhan Clinical Laboratories, Building B2, No.666 Gaoxin Road, Wuhan East lake Hi-tech Development zone, Wuhan, 430074, China
Yanqiu Zhou
, Chen Liu
, Rongfang Zhou
, Anzhi Lu
, Biao Huang
, Liling Liu
, Ling Chen
, Bei Luo
, Jin Huang
& Zhijian Tian
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CL designed and constructed the database, and developed web application; YZ collected data, performed data examination and analysis, and prepared figures; RZ prepared tables, performed data and application examination, and wrote the manuscript; AL, BH, LL, LC, BL contributed materials and analysis tools. JH, ZT conceived and designed the experiments, approved the final draft of the manuscript submitted for review and publication. All the authors have read and approve the manuscript.
Corresponding authors
Correspondence to Jin Huang or Zhijian Tian.
Additional file 1: Figure S1. The correlation of each metric with yield (ReadsM). The linear regression line is shown in blue.
Additional file 2: Figure S2. Residual deviation of the LR model every 5 cycles. The box plot displays the residuals of all 40 LR models; each box shows the median and first and third quartiles, and a star indicates the standard deviation.
Zhou, Y., Liu, C., Zhou, R. et al. SEQdata-BEACON: a comprehensive database of sequencing performance and statistical tools for performance evaluation and yield simulation in BGISEQ-500. BioData Mining 12, 21 (2019) doi:10.1186/s13040-019-0209-9
BGISEQ-500
Sequencing run performance
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CommonCrawl
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Patrick Fasano
Graduate student, nuclear theorist, breaker of supercomputers.
Emergent Sp(3,ℝ) Dynamical Symmetry in the Nuclear Many-Body System from an Ab Initio Description
Published in Physical Review Letters, 2020
DOI: 10.1103/PhysRevLett.125.102505 | arXiv: 2008.05522
Recommended citation: A. E. McCoy, M. A. Caprio, T. Dytrych, and P. J. Fasano, Phys. Rev. Lett. 125, 102505 (2020). (download)
Ab initio nuclear theory provides not only a microscopic framework for quantitative description of the nuclear many-body system, but also a foundation for deeper understanding of emergent collective correlations. A symplectic $\mathrm{Sp}(3,\mathbb{R}) \supset \mathrm{U}(3)$ dynamical symmetry is identified in ab initio predictions, from a no-core configuration interaction approach, and found to provide a qualitative understanding of the spectrum of 7Be. Low-lying states form an Elliott $\mathrm{SU}(3)$ spectrum, while an $\mathrm{Sp}(3,\mathbb{R})$ excitation gives rise to an excited rotational band with strong quadrupole connections to the ground state band.
© 2022 Patrick Fasano. Powered by Jekyll & AcademicPages, a fork of Minimal Mistakes.
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CommonCrawl
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death of a red dwarf star / minimum mass needed for a white dwarf?
OK, first, I know there's a variety of sizes and types of red dwarf stars and the universe is too young for any of them to have reached the end of their main sequence phase yet, so it's all theoretical and/or modeling.
http://en.wikipedia.org/wiki/Red_dwarf
But what is the theoretical size needed for a star to undergo the electron degeneracy process which turns a small star from at least Jupiter sized, usually bigger, into to earth sized super dense object, where, as I understand it, the electrons are squeezed off the nuclei - electron degeneracy.
It would seem to me that a 7.5% solar mass star, which gradually burns hydrogen into helium but doesn't burn helium, might not have the mass to compact into a true white dwarf but might end it's life looking more like a brown dwarf / super Jupiter - well, talking appearance, not really, cause super Jupiter & Brown dwarfs are mostly hydrogen while and an end of life red dwarf should be mostly helium - which, in and of itself, might make the difference.
It's just my curiosity whether all red dwarfs turn into white dwarfs at the end of their burning phase or is there a theoretical mass that's needed for that level of shrinkage to occur?
stellar-evolution stellar-astrophysics
userLTKuserLTK
Stars that have a mass lower than about $0.5 M_{\odot}$ will not ignite helium in their cores, in an analogous fashion to the way that stars with $M<8M_{\odot}$ have insufficiently massive cores that never reach high enough temperatures to ignite carbon.
The cause in both cases is the onset of electron degeneracy pressure, which is independent of temperature and allows the core to cool at constant pressure and radius. [A normal gas would contract and become hotter as it loses energy!]
The end result for a $0.5M_{\odot}$ star will be a helium white dwarf with a mass (depending on uncertain details of the mass-loss process) of around $0.2M_{\odot}$. Such things do exist in nature now, but only because they have undergone some kind of mass transfer event in a binary system that has accelerated their evolution. The collapse to a degenerate state would be inevitable even for the lowest mass stars (which would of course then be very low-mass white dwarfs). As an inert core contracts it loses heat and cools - a higher density and lower temperate eventually lead to degenerate conditions that allow the core to cool without losing pressure.
The lowest mass stars ($<0.3 M_{\odot}$) do get there via a slightly different route - they are fully convective, so the "core" doesn't exist really, it is always mixed with the envelope. They do not develop into red giants and thus I guess will suffer much less mass loss.
The remnant would be a white dwarf in either case and is fundamentally different from a brown dwarf both in terms of size and structure, because it would be made of helium rather than (mostly) hydrogen. This should have an effect in two ways. For the same mass, the brown dwarf should end up bigger because the number of mass units per electron is smaller (1 vs 2) and also because the effects of a finite temperature are larger in material with fewer mass units per particle - i.e. its outer, non-degenerate layer would be more "puffed up". NB: The brown dwarfs we see today are Jupiter-sized, but are still cooling. They will get a bit smaller and more degenerate.
A simple size calculation could use the approximation of an ideal, cold, degenerate gas. A bit of simple physics using the virial theorem gives you $$ \left(\frac{R}{R_{\odot}}\right) \simeq 0.013\left(\frac{\mu_e}{2}\right)^{-5/3} \left(\frac{M}{M_{\odot}}\right)^{-1/3},$$ where $\mu_e$ is the number of atomic mass units per electron. Putting in appropriate numbers I get $0.32\ R_{Jup}$ for a $0.07M_{\odot}$ Helium white dwarf versus $1.01\ R_{Jup}$ for a $0.07M_{\odot}$ completely degenerate Hydrogen brown dwarf (in practice it would be a bit smaller because it isn't all hydrogen).
However, it would be interesting to see some realistic calculations of what happens to a $0.07M_{\odot}$ brown dwarf versus a $0.08M_{\odot}$ star in a trillion years or so. I will update the answer if I come across such a study.
EDIT: I knew I'd seen something on this. Check out Laughlin et al. (1997), which studies the long-term evolution of very low-mass stars. Low-mass stars do not pass through a red giant phase, remain fully convective and can thus convert almost all their hydrogen into helium over the course of $10^{13}$ years and end up cooling as degenerate He white dwarfs.
Not the answer you're looking for? Browse other questions tagged stellar-evolution stellar-astrophysics or ask your own question.
Black dwarf binary systems
Is a white dwarf hotter than a Red Giant?
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How massive does a main sequence star need to be to go type 1 supernova?
How could a neutron star collapse into a black hole?
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Why does shell fusion produce more energy than core fusion?
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CommonCrawl
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What are the uses of Dimensional Equations? and what is Conversion of one system of units into another?
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Uses of Dimensional Equations
Following are the uses of dimensional equations.
1. Conversion of one system of units into another
2. Checking the accuracy of various formulas or equation
3. Derivation of Formula.
Conversion of one system of units into another
Let the numerical values are $n_{1}$ and $n_{2}$ of a given quantity $Q$ in two-unit system and the units are-
$U_{1}=M_{1}^{a} L_{1}^{b} T_{1}^{c}$ and $U_{2}=M_{2}^{a} L_{2}^{b} T_{2}^{c}$
Therefore, By the principle $\mathrm{nu}=$ constant
$\Rightarrow \mathrm{n}_{2} \mathrm{U}_{2}=\mathrm{n}_{1} \mathrm{u}_{1}$
$\mathrm{n}_{2}\left[\mathrm{M}_{2}^{a} \mathrm{~L}_{2}^{b} \mathrm{~T}_{2}^{c}\right]=\mathrm{n}_{1}\left[\mathrm{M}_{1}^{a} L_{1}^{b} \mathrm{~T}_{1}^{c}\right]$
$\Rightarrow \quad \mathrm{n}_{2}=\frac{\mathrm{n}_{1}\left[\mathrm{M}_{1}^{\mathrm{a}} \mathrm{L}_{1}^{\mathrm{b}} \mathrm{T}_{1}^{\mathrm{c}}\right]}{\left[\mathrm{M}_{2}^{\mathrm{a}} \mathrm{L}_{2}^{\mathrm{b}} \mathrm{T}_{2}^{\mathrm{c}}\right]}$
$\Rightarrow \quad n_{2}=\left[\frac{M_{1}}{M_{2}}\right]^{a}\left[\frac{L_{1}}{L_{2}}\right]^{b}\left[\frac{T_{1}}{T_{2}}\right]^{c} n_{1}$
Principle of Homogeneity
The dimensions of both sides in an equation are the same.
Ex. $\mathrm{s}=\mathrm{ut}+\frac{1}{2} \mathrm{gt}^{2}$
$[L]=\left[L T^{-1} \cdot T\right]+\left[L T^{-2} \cdot T^{2}\right]$
$[\mathrm{L}]=[\mathrm{L}]+[\mathrm{L}]$
JEE Physics Notes
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conversion of one system of unit into another examples
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uses of dimensional equation with examples
uses of dimensional formula class 11
what are the uses of dimensional equations
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CommonCrawl
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What would be the temperature of earth if there was no atmosphere?
I do know that the atmosphere protects life on Earth by absorbing ultraviolet solar radiation, warming the surface through heat retention (greenhouse effect),and reducing temperature extremes between day and night (the diurnal temperature variation). I wonder what temperature would earth reach if there was no atmosphere?.
atmosphere temperature
plannapus
Praveen KadambariPraveen Kadambari
$\begingroup$ My recollection is an average sufrace temp of 255 K based on radiative equilibrium. I'll make an answer if I can back that number and how it is determined. $\endgroup$ – casey♦ May 7 '14 at 3:59
$\begingroup$ The question is a little ambiguous. Are you asking about the temperature difference between the earth with surface as it is now, but without an atmosphere? Or about the earth with no atmosphere and with a bare surface? It makes a significant difference to the answer. If you're just interested in how much difference the greenhouse effect gives, then it's best just to ask that explicitly - and the answer then, as casey says, is a surface temp of 255 K, and a temp diff of about +33 K $\endgroup$ – EnergyNumbers May 7 '14 at 7:28
$\begingroup$ @EnergyNumbers What I meant to ask was usual temperature one can read on thermometer. $\endgroup$ – Praveen Kadambari May 7 '14 at 8:01
$\begingroup$ OK. Of the earth exactly as the surface is now, just in the absence of the atmosphere (that is, what it would look like the instant the atmosphere vanished)? Or of the earth without the atmosphere, and if the Earth's surface was like the moon's: no water, no snow, no forests, etc (that is, what the surface would come to look like after centuries of no atmosphere)? $\endgroup$ – EnergyNumbers May 7 '14 at 10:06
$\begingroup$ Assuming no change to the Sun, and after the atmosphere vanishes we've allowed enough time for all water to disappear, all ice to sublimate, all dead flora and fauna to crumble to dust, etc.? Then I would expect temperatures to be very similar to the Moon's, modified by a slightly higher albedo (the Moon is quite dark, and there could be a bright coating of salt where the oceans used to be) (-, but might radiate less at night) and a little bit more heat flow from the Earth's interior (+). A lot depends on the history of the planet -- did it ever have oceans, etc. $\endgroup$ – Phil Perry May 7 '14 at 16:28
According to Wikipedia an approximate average surface temperature for a bare earth is 274.5 K. This scenario is quite reasonable in my opinion as stripping the atmosphere without changing much else would (on a geological timescale) rather quickly result in a bare earth without ice caps or vegetation, causing circumstances quite close to those on the moon. (I assume the Earth's magnetic field protecting both the atmosphere and life below it has disappeared as well)
This is estimated by comparing the black body radiation of the Earth and Moon, which is then corrected for albedo (fraction of incoming radiation that is reflected) and emissivity (ability of a material to emit radiation), which are properties of a material. Since the Earth and moon are both at the same distance to the sun and made up from the same material on average, measurements of the albedo and emissivity of the moon can than be used as estimations of these properties for the Earth.
The black body radiation of the sun is calculated with the Stefan-Boltzmann law:
$$P_{\text{S,emit}} = 4\pi R_S^2 \sigma T_S^4$$
$P_{\text{S,emit}}$ is the emitted energy by the sun, $R_S$ is the radius of the sun, and $T_S$ is the temperature of the sun. The fraction of this energy recieved by the Earth is then proportional to the circular surface area facing the sun and the energy density at the distance $D$ between the Earth and the sun.
$$P_{\text{SE}} = P_{\text{S,emit}}\left(\frac{\pi R_E^2}{4\pi D^2}\right)$$
$R_E$ is the Earth's radius. Using albedo $\alpha$ the absorbed energy can be calculated:
$$P_{\text{E,abs}} = (1-\alpha)P_{\text{SE}}$$
Applying the Stefan-Boltzman law to the Earth, corrected for the emissivity $\overline{\epsilon}$, the emitted energy is then:
$$P_{\text{E,emit}} = \overline{\epsilon} 4\pi R_E^2 \sigma T_E^4$$
Assuming energy equilibrium $P_{\text{E,abs}} = P_{\text{E,emit}}$ we can now calculate $T_E$:
$$\begin{aligned} \frac{(1-\alpha)4\pi R_S^2 \sigma T_S^4\pi R_E^2}{4\pi D^2} & = \overline{\epsilon}4\pi R_E^2 \sigma T_E^4 \\ T_E^4 & = \frac{(1-\alpha)4\pi R_S^2 \sigma T_S^4\pi R_E^2}{\overline{\epsilon}4\pi D^2 4\pi R_E^2 \sigma} \\ T_E^4 & = \frac{(1-\alpha) R_S^2 T_S^4}{ 4\overline{\epsilon}D^2 } \\ T_E & = \left( \frac{(1-\alpha) R_S^2 T_S^4}{4 \overline{\epsilon}D^2 }\right)^{\frac{1}{4}} \\ T_E & = T_S \left( \frac{(1-\alpha) R_S^2}{4 \overline{\epsilon} D^2 }\right)^{\frac{1}{4}} \\ T_E & = T_S \sqrt{ \frac{ R_S \sqrt{\frac{1-\alpha}{\overline{\epsilon}}} }{2 D } } \end{aligned}$$
Finally we only need to insert the correct values:
$R_S = 6.96\times 10^8$ m
$T_S = 5778$ K
$D = 1.496\times 10^{11}$ m
$\alpha = 0.1054$ (assuming value of the moon)
$\overline{\epsilon} = 0.95$ (assuming value of the moon)
This gives us a temperature of 274.5 K.
Note that there are many factors that can cause local and temporal variations. For example, incoming radiation varies with latitude and season, and if the removal of the atmosphere would be caused by a dying sun that grows to engulf the earth temperatures would be much higher than this. All in all, to account for all those factors a very large model must be made that can analyse the influence of each factor, including the decrease in temperature of a dying sun etc., but that would be nearly impossible to build if only for the resources it would take to do so.
Since one of the most contested factors is the albedo after the atmosphere is removed the following graph shows how the average surface temperature changes with albedo. At an albedo of zero all incoming solar radiation is absorbed, while at 1 all radiation is reflected. Note that the temperature of 0K is an effect of the assumed equilibrium between incoming and emitted radiation, which will not hold at that point. As said above, the albedo for a bare earth will be approximately 0.1, while current values on average range from 0.3-0.4, largely contributed to by clouds. An average for the albedo of the Earth in its current vegetated state, but without clouds I haven't been able to find.
As stated by @ardie-j in his answer, another possible fate of the Earth could be that it gets covered in ice, as another Snowball Earth Event. In that case the albedo would rise to levels ranging from 0.4-0.9, resulting in a drastically cooler Earth.
hugovdberghugovdberg
$\begingroup$ According to your wikipedia link, the temp is 254-255 K. But you've reported it as saying 274.5 K. Any idea where that discrepancy comes from? $\endgroup$ – EnergyNumbers May 7 '14 at 7:04
$\begingroup$ @EnergyNumbers Please read the wikipedia article more carefully. The 254K is the BLACK BODY temperature, the value of 274K I provide here is corrected for albedo and emissivity of a bare earth without atmosphere and thus a more realistic value than the plain black body radiation. $\endgroup$ – hugovdberg May 7 '14 at 7:08
$\begingroup$ @EnergyNumbers fixed a couple of copy paste errors, and it doesn't say 274.5K but 1.36C instead (a couple of lines below the 254K), but I prefered to use Kelvin throughout my answer. $\endgroup$ – hugovdberg May 7 '14 at 7:13
$\begingroup$ But isn't the question about earth in the absence of the atmosphere, rather than a bare earth? Now, I agree there's an ambiguity there in the question because of that. Do you think it might be worth explicitly dealing with that ambiguity? $\endgroup$ – EnergyNumbers May 7 '14 at 7:26
$\begingroup$ @EnergyNumbers I think an earth without atmosphere would quickly turn into a bare earth, or never had plants over it. Besides, large parts of the albedo are caused by clouds and ice coverage, removing the atmosphere would at the very least remove the clouds, so still decrease the albedo, and cause a higher temperature, although that is probably hard to qualify. $\endgroup$ – hugovdberg May 7 '14 at 7:30
Actual climatologist here.
The earth would be covered in ice. That is the only answer you need.
Others attempting to calculate a change in global temperature seem to lack a basic understanding of how the global system functions (greenhouse gases are only one aspect that controls global temperature).
Those who seem to think it would be fried or barren neglect to understand the difference between the atmosphere and earth's protective magnetic field. If we had no magnetic field, in addition to many, many other problems, our planet would fry regardless of the presence and chemical composition of an atmosphere. Without an atmosphere and greenhouse gases, however, the planet would freeze. With too many greenhouse gases under other optimal climate conditions, it would look like Venus.
Please don't trust someone who gets their answers from Wikipedia. If you prefer more mass-media resources, you can cite the following articles: "Snowball Earth" Confirmed: Ice Covered Equator (National Geographic, 5 Mar 2010)
Alternatively, you can use more science-based research: Did the Snowball Earth Have a Slushball Ocean? (NASA GIS, Oct 2002)
No atmosphere = Snowball earth.
Jan Doggen
Ardie JArdie J
$\begingroup$ Thanks for your answer, given the hypothetical nature of the question a large range of answers is possible. I still stand by my answer that it would turn to a bare earth, even though it does indeed base on the hidden assumption the magnetic field has disappeared as well. I will expand my answer to include that assumption, since I think it is more reasonable that the atmosphere was blown away by lack of magnetic protection than because it just disappeared. Also, were the ice indeed covered in earth that still fits my answer, albeit with a higher albedo, which is accomodated for in the plot. $\endgroup$ – hugovdberg Sep 13 '16 at 20:06
$\begingroup$ The links you provide don't back up your hypothesis. Can you provide other sources or elaborate in your answer why a lack of greenhouse gases result in a global snow/ice cover? $\endgroup$ – ye-ti-800 Sep 14 '16 at 14:04
$\begingroup$ If we had no atmosphere but still had an ocean (which you imply by saying the earth will be locked in ice), then why wouldn't solar heating create a water vapor atmosphere? Plus your second link involves reducing solar output by 4%; that is not part of the OP's problem statement. Your argument seems to have plenty of holes and little evidence. $\endgroup$ – kingledion Nov 21 '16 at 13:14
$\begingroup$ No atmosphere implies no water since the vapor pressure of ice and liquid water is greater than zero. No water implies no ice. ;-) $\endgroup$ – aventurin Sep 16 '19 at 19:47
Sunlight is 393 Kelvin. The earth is irradiated at half the surface area that emits the radiation. And the radiation is absorbed in a spherical volume. So: $((σ393^4/(4/3))/2=510W/m^2$ or 308 Kelvin(32 degrees celsius).
Emil JunvikEmil Junvik
$\begingroup$ Do you have a source for "Sunlight is 393 Kelvin". That seems an odd statement. $\endgroup$ – userLTK May 5 '17 at 22:20
$\begingroup$ Sunlight at the top of the atmosphere is 1361W/m^2. Use the Stefan-Boltzmann equation to find the temperature: σ=0.0000000567>$σ393^4=1361$ > take the fourth root of 1361/0.0000000567 $\endgroup$ – Emil Junvik May 6 '17 at 16:24
$\begingroup$ How can people downvote a calculation of the simplest radiative transfer and geometry. It is not a single error in that calculation. Mind boggling $\endgroup$ – Emil Junvik May 6 '17 at 16:29
$\begingroup$ @Emil "How can people downvote.." .This is an online forum, there is no entry qualification, you can use mathematical and scientific nonsense here and only some of it will be pointed out, you can claim to be an actual climate scientist here or a guy from NASA and nobody will know that you are just a paid shill, you can hold religious views and pretend they are science, you can be here for political reasons, to further an agenda ,you can be stupid and pretend otherwise but most importantly my friend here on Stack Exchange YOU CAN LIE and quite often find lying friends too ! $\endgroup$ – user7733 May 14 '17 at 23:11
$\begingroup$ Emil Junvik, you are badly confused. Sunlight does not have a "temperature." Sunlight does have a spectrum which is characteristic of the emitting source, which is the photosphere of the Sun. That average temperature is 5,777 K (5,504 °C), but it is in no way to be considered "the temperature of" sunlight. TSI (about 1361 W/m²) is entirely different. It is a function of Earth's distance from the Sun and the intensity of the light emitted from the Sun. $\endgroup$ – Dave Burton May 29 '17 at 12:30
Not the answer you're looking for? Browse other questions tagged atmosphere temperature or ask your own question.
How does the Sun heat the Earth?
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CommonCrawl
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Conditional variational principle for the irregular set in some nonuniformly hyperbolic systems
Integrability of vector fields versus inverse Jacobian multipliers and normalizers
Longtime behavior of the semilinear wave equation with gentle dissipation
Zhijian Yang 1, , Zhiming Liu 2, and Na Feng 2,
Department of Mathematics, Zhengzhou University, No.100, Science Road, Zhengzhou 450001
School of Mathematics and Statistics, Zhengzhou University, No.100, Science Road, Zhengzhou 450001, China, China
Received December 2015 Revised June 2016 Published August 2016
The paper investigates the well-posedness and longtime dynamics of the semilinear wave equation with gentle dissipation: $u_{tt}-\triangle u+\gamma(-\triangle)^{\alpha} u_{t}+f(u)=g(x)$, with $\alpha\in(0,1/2)$. The main results are concerned with the relationships among the growth exponent $p$ of nonlinearity $f(u)$ and the well-posedness and longtime behavior of solutions of the equation. We show that (i) the well-posedness and longtime dynamics of the equation are of characters of parabolic equations as $1 \leq p < p^* \equiv \frac{N + 4\alpha}{(N-2)^+}$; (ii) the subclass $\mathbb{G}$ of limit solutions has a weak global attractor as $p^* \leq p < p^{**}\equiv \frac{N+2}{N-2}\ (N \geq 3)$.
Keywords: gentle dissipation, exponential attractor., global attractor, well-posedness, Semilinear wave equation.
Mathematics Subject Classification: Primary: 35B41, 35B33; Secondary: 35B40, 35B65, 37L3.
Citation: Zhijian Yang, Zhiming Liu, Na Feng. Longtime behavior of the semilinear wave equation with gentle dissipation. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6557-6580. doi: 10.3934/dcds.2016084
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Special Publication 330
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SP 330 - Foreword
SP 330 - Preface to the 9th edition
SP 330 - Section 1
SP 330 - Appendix 1
SP 330 - Appendix 2 & 3
SP 330 - List of acronyms used in the present volume
SP 330 - Version History
Historical notes on the development of the International System of Units and its base units
Part 1. The historical development of the realization of SI units
Experimental methods used for the realization of units and which use equations of physics are known as primary methods. The essential characteristic of a primary method is that it allows a quantity to be measured in a particular unit directly from its definition by using only quantities and constants that themselves do not contain that unit.
Traditionally, a unit for a given quantity was taken to be a particular example of that quantity, which was chosen to provide numerical values of common measurements of a convenient size. Before the rise of modern science, units were necessarily defined in terms of material artifacts, notably the meter and kilogram for length and mass, or the property of a particular object, namely the rotation of the earth for the second. Even at the origin of the metric system at the end of the 18th century it was recognized that a more desirable definition of a unit of length for example would be one based on a universal property of nature such as the length of a pendulum beating seconds. Such a definition would be independent of time and place and would in principle be accessible all over the world. At the time, practical considerations resulted in the simpler, artifact definitions for the meter and the kilogram and the second remained linked to the rotation of the Earth. It was only in 1960 that the first non-material definition was adopted, namely the wavelength of a specified optical radiation for the meter.
Since then, definitions of the ampere, kelvin, mole and candela have been adopted that do not refer to material artifacts. In the case of the ampere it refers to a specified electric current required to produce a given electromagnetic force and, in the case of the kelvin, to a particular thermodynamic state, namely the triple point of water. Even the atomic definition of the second was in terms of a specified transition of the cesium atom. The kilogram has always stood out as the one unit that had resisted the transformation from an artifact. The definition that opened the way to real universality was that of the meter in 1983. This definition implied, although it did not state, a fixed numerical value for the speed of light. The definition was worded, however, in the traditional form and stated essentially that the meter was the distance travelled by light in a specified time. In this way it reflected the other definitions of the base units of the SI each of which has the same form, for example "the ampere is the current which…" and "the kelvin is a fraction of a specified temperature." Such definitions can be called explicit unit definitions.
Although these definitions meet many of the requirements for universality and accessibility, and a variety of realizations are often possible, they nevertheless constrain practical realizations to experiments that are directly or indirectly linked to the particular conditions or states specified in each definition. In consequence, the accuracy of realization of such definitions can never be better than the accuracy of realization of the particular conditions or states specified in the definitions.
This is a particular problem with the present definition of the second, which is based on a microwave transition of an atom of cesium. Frequencies of optical transitions of different atoms or ions are now demonstrably more reproducible, by some orders of magnitude, than the defined frequency of cesium.
In the present definition of the SI based on the set of defining constants, instead of each definition specifying a particular condition or state, which sets a fundamental limit to the accuracy of realization, any convenient equation of physics that links the particular constant or constants to the quantity we want to measure may be used. This is a much more general way of defining the basic units of measurement. It is one that is not limited by today's science or technology as future developments may lead to as yet unknown equations that could result in different ways of realizing units with a much higher accuracy. When defined in this way, there is, in principle, no limit to the accuracy with which a unit can be realized. The exception remains the definition of the second in which the original microwave transition of cesium remains, for the time being, the basis of the definition.
The difference between an explicit unit and an explicit constant definition can be clearly illustrated using the two previous definitions of the meter that depended upon a fixed numerical value of the speed of light and secondly the two definitions of the kelvin. The original 1983 definition of the meter states, in effect, that "the meter is the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second". The new definition simply states that the meter is defined by taking the constant that defines the second, the specified cesium frequency and the fixed numerical value of the speed of light expressed in units m·s−1. We can thus use any equation of physics including, of course, that indicated by the former definition, the time taken to travel the given distance which is used for astronomical distances, but also the simple equation relating frequency and wavelength to the speed of light. The former definition of the kelvin based on a fixed numerical value for the temperature of the triple point of water requires ultimately a measurement at the triple point of water. The new definition, based on the fixed numerical value for the Boltzmann constant, is much more general in that any thermodynamic equation in which k appears can in principle be used to determine a thermodynamic temperature at any point on the temperature scale. For example, by determining the total radiant exitance of a black body at temperature T, equal to (2π5k4/15c2h3) T4, in Wm−2 we can determine T directly.
For the kilogram, the unit whose definition has undergone the most fundamental change, realization can be through any equation of physics that links mass, the Planck constant, the velocity of light and the cesium frequency. One such equation is that which describes the operation of an electro-mechanical balance, previously known as a watt balance, more recently known as a Kibble11 balance. With this apparatus, a mechanical power, measured in terms of a mass, m, the local acceleration due to gravity, g, and a velocity, v, can be measured in terms of an electrical power measured in terms of an electric current and voltage measured in terms of the quantum Hall and Josephson effects respectively. The resulting equation is mgv = Ch where C is a calibration constant that includes measured frequencies and h is Planck's constant.
Another method that can be used for a primary realization of the kilogram is through the determination of the number of atoms in a silicon sphere and using the equation:
$$m\,=\,\frac{8V}{a^3_0}\,\frac{2 R_\infty h}{c \alpha^2}\,\frac{m_{Si}}{m_e}$$
with the mass m and volume V of the sphere, lattice parameter a0, Rydberg constant R∞, fine structure constant α, and the masses of a silicon atom (averaged over the three isotopes used for the sphere) mSi, and the electron me, respectively. The first fraction corresponds to the number of atoms in the sphere, the second to the electron mass and the third fraction is the ratio of the mass of the (isotopically averaged) silicon atom to the electron mass.
Another possibility for measuring mass through the new definition, but this time at the microscopic level, is through measurements of atomic recoil using the relation that includes h/m.
All these provide a striking illustration of the generality of the new way of defining units. Detailed information on the current realization of the base and other units is given on the BIPM website.
Part 2. The historical development of the International System
The 9th CGPM (1948, Resolution 6; CR 64) instructed the CIPM:
to study the establishment of a complete set of rules for units of measurement;
to find out for this purpose, by official enquiry, the opinion prevailing in scientific, technical and educational circles in all countries;
to make recommendations on the establishment of a practical system of units of measurement suitable for adoption by all signatories to the Meter Convention.
The same CGPM also laid down, in Resolution 7 (CR 70), 'general principles for the writing of unit symbols' and listed some coherent derived units that were assigned special names.
The 10th CGPM (1954, Resolution 6; CR 80) adopted as base quantities and units for this practical system the following six quantities: length, mass, time, electric current, thermodynamic temperature and luminous intensity, as well as the six corresponding base units: meter, kilogram, second, ampere, kelvin and candela. After a lengthy discussion between physicists and chemists, the 14th CGPM (1971, Resolution 3, CR 78 and Metrologia 1972, 8, 36) added amount of substance, unit mole, as the seventh base quantity and unit.
The 11th CGPM (1960, Resolution 12; CR 87) adopted the name Système international d'unités, with the international abbreviation SI, for this practical system of units and laid down rules for prefixes, derived units and the former supplementary units, as well as other matters; it thus established a comprehensive specification for units of measurement. Subsequent meetings of the CGPM and the CIPM have added to and modified the original structure of the SI to take account of advances in science and of the changing needs of users.
The historical sequence that led to these important decisions may be summarized as follows.
The creation of the decimal metric system at the time of the French Revolution and the subsequent deposition of two platinum standards representing the meter and the kilogram, on 22 June 1799, in the Archives de la République in Paris, which can be seen as the first step that led to the present International System of Units.
In 1832, Gauss strongly promoted the application of this metric system, together with the second defined in astronomy, as a coherent system of units for the physical sciences. Gauss was the first to make absolute measurements of the earth's magnetic field in terms of a decimal system based on the three mechanical units millimeter, gram and second for, respectively, the quantities length, mass and time. In later years Gauss and Weber extended these measurements to include other electrical phenomena.
These applications in the field of electricity and magnetism were further extended in the 1860s under the active leadership of Maxwell and Thomson through the British Association for the Advancement of Science (BAAS). They formulated the requirement for a coherent system of units with base units and derived units. In 1874 the BAAS introduced the CGS system, a three-dimensional coherent unit system based on the three mechanical units centimeter, gram and second, using prefixes ranging from micro to mega to express decimal sub-multiples and multiples. The subsequent development of physics as an experimental science was largely based on this system.
The sizes of the coherent CGS units in the fields of electricity and magnetism proved to be inconvenient, so in the 1880s the BAAS and the International Electrical Congress, predecessor of the International Electrotechnical Commission (IEC), approved a mutually coherent set of practical units. Among them were the ohm for electrical resistance, the volt for electromotive force, and the ampere for electric current.
After the signing of the Meter Convention on 20 May 1875, which created the BIPM and established the CGPM and the CIPM, work began on establishing new international prototypes for the meter and the kilogram. In 1889 the 1st CGPM sanctioned the international prototypes for the meter and the kilogram. Together with the astronomical second as the unit of time, these units constituted a three‑dimensional mechanical unit system similar to the CGS system, but with the base units meter, kilogram and second, known as the MKS system.
In 1901 Giorgi showed that it is possible to combine the mechanical units of this MKS system with the practical electrical units to form a coherent four-dimensional system by adding to the three base units a fourth unit, of an electrical nature such as the ampere or the ohm, and also rewriting the equations occurring in electromagnetism in the so-called rationalized form. Giorgi's proposal opened the path to a number of new developments.
After the revision of the Meter Convention by the 6th CGPM (1921), which extended the scope and responsibilities of the BIPM to other fields in physics and the subsequent creation of the Consultative Committee for Electricity (CCE) by the 7th CGPM (1927), the Giorgi proposal was thoroughly discussed by the IEC, the International Union of Pure and Applied Physics (IUPAP) and other international organizations. This led the CCE to propose in 1939 the adoption of a four‑dimensional system based on the meter, kilogram, second and ampere, the MKSA system, a proposal approved by the CIPM in 1946.
Following an international enquiry by the BIPM, which began in 1948, the 10th CGPM (1954), approved the further introduction of the kelvin and the candela, as base units for thermodynamic temperature and luminous intensity, respectively. The name International System of Units, with the abbreviation SI, was given to the system by the 11th CGPM (1960). Rules for prefixes, derived units, the former supplementary units as well as other matters, were established, thus providing a comprehensive specification for all units of measurement.
At the 14th CGPM (1971) a new base unit, the mole, symbol mol, was adopted for the quantity amount of substance. This followed a proposal from the International Organization for Standardization originating in a proposal from the Commission on Symbols, Units and Nomenclature (SUN Commission) of IUPAP, which was supported by the International Union for Pure and Applied Chemistry (IUPAC). This brought the number of base units of the SI to seven.
Since then, extraordinary advances have been made in relating SI units to truly invariant quantities such as the fundamental constants of physics and the properties of atoms. Recognizing the importance of linking SI units to such invariant quantities, the 24th CGPM (2011), adopted the principles of a new definition of the SI based on using a set of seven such constants as references for the definitions. At the time of the 24th CGPM, experiments to determine their values in terms of the then base units were not completely consistent but by the time of the 26th CGPM (2018) this had been achieved and the new definition of the SI was adopted in Resolution 1. This is the basis of the definition presented in this brochure and it is the simplest and most fundamental way of defining the SI.
The SI was previously defined in terms of seven base units and derived units defined as products of powers of the base units. The seven base units were chosen for historical reasons, as the metric system, later the SI, evolved and developed over the last 130 years. Their choice was not unique, but it has become established and familiar over the years, not only by providing a framework for describing the SI, but also for defining the derived units. This role for the base units continues in the present SI even though the SI itself is now defined in terms of the seven defining constants. In this brochure therefore, definitions of the seven base units can still be found but are henceforth based on the seven defining constants: the cesium hyperfine frequency ΔνCs; the speed of light in vacuum c; the Planck constant h; elementary charge e; Boltzmann constant k; Avogadro constant NA; and the luminous efficacy of a defined visible radiation Kcd.
The definitions of the seven base units can be related unambiguously to the numerical values of the seven defining constants. However, there is not a one-to-one relationship between the seven defining constants and the seven base units as many of the base units call upon more than one of the defining constants.
Part 3. Historical perspective on the base units
Unit of time, second
Before 1960, the unit of time the second, was defined as the fraction 1/86 400 of the mean solar day. The exact definition of "mean solar day" was left to astronomers. However measurements showed that irregularities in the rotation of the Earth made this an unsatisfactory definition. In order to define the unit of time more precisely, the 11th CGPM (1960, Resolution 9, CR, 86) adopted a definition given by the International Astronomical Union based on the tropical year 1900. Experimental work, however, had already shown that an atomic standard of time, based on a transition between two energy levels of an atom or a molecule, could be realized and reproduced much more accurately. Considering that a very precise definition of the unit of time is indispensable for science and technology, the 13th CGPM (1967-1968, Resolution 1, CR, 103 and Metrologia, 1968, 4, 43) chose a new definition of the second referenced to the frequency of the ground state hyperfine transition in the cesium-133 atom. A revised more precise wording of this same definition now in terms of a fixed numerical value of the unperturbed ground-state hyperfine transition frequency of the cesium-133 atom, ΔνCs, was adopted in Resolution 1 of the 26th CGPM (2018).
Unit of length, meter
The 1889 definition of the meter, namely, the length of the international prototype of platinum-iridium, was replaced by the 11th CGPM (1960) using a definition based on the wavelength of the radiation corresponding to a particular transition in krypton 86. This change was adopted in order to improve the accuracy with which the definition of the meter could be realized, this being achieved using an interferometer with a travelling microscope to measure the optical path difference as the fringes were counted. In turn, this was replaced in 1983 by the 17th CGPM (Resolution 1, CR, 97, and Metrologia, 1984, 20, 25) with a definition referenced to the distance that light travels in vacuum in a specified interval of time, as presented in 2.3.1. The original international prototype of the meter, which was sanctioned by the 1st CGPM in 1889 (CR, 34-38), is still kept at the BIPM under conditions specified in 1889. In order to make clear its dependence on the fixed numerical value of the speed of light, c, the wording of the definition was changed in Resolution 1 of the 26th CGPM (2018).
Unit of mass, kilogram
The 1889 definition of the kilogram was simply the mass of the international prototype of the kilogram, an artifact made of platinum-iridium. This was, and still is, kept at the BIPM under the conditions specified by the 1st CGPM (1889, CR, 34-38) when it sanctioned the prototype and declared that "this prototype shall henceforth be considered to be the unit of mass". Forty similar prototypes were made at about the same time and these were all machined and polished to have closely the same mass as the international prototype. At the 1st CGPM (1889), after calibration against the international prototype, most of these "national prototypes" were individually assigned to Member States, and some also to the BIPM. The 3rd CGPM (1901, CR, 70), in a declaration intended to end the ambiguity in common usage concerning the use of the word "weight", confirmed that "the kilogram is the unit of mass; it is equal to the mass of the international prototype of the kilogram". The complete version of these declarations appears on p. 70 of the above-mentioned CGPM proceedings.
By the time of the second verification of national prototypes in 1946 it was found that on average the masses of these prototypes were diverging from that of the international prototype. This was confirmed by the third verification carried out from 1989 to 1991, the median difference being about 25 micrograms for the set of original prototypes sanctioned by the 1st CGPM (1889). In order to assure the long-term stability of the unit of mass, to take full advantage of quantum electrical standards and to be of more utility to modern science, a new definition for the kilogram based on the value of a fundamental constant, for which purpose the Planck constant h was chosen, was adopted by Resolution 1 of the 26th CGPM (2018).
Unit of electric current, ampere
Electric units, called "international units," for current and resistance were introduced by the International Electrical Congress held in Chicago in 1893 and definitions of the "international ampere" and "international ohm" were confirmed by the International Conference in London in 1908.
By the time of the 8th CGPM (1933) there was a unanimous desire to replace the "international units" by so-called "absolute units". However because some laboratories had not yet completed experiments needed to determine the ratios between the international and absolute units, the CGPM gave authority to the CIPM to decide at an appropriate time both these ratios and the date at which the new absolute units would come into effect. The CIPM did so in 1946 (1946, Resolution 2, PV, 20, 129-137), when it decided that the new units would come into force on 1 January 1948. In October 1948 the 9th CGPM approved the decisions taken by the CIPM. The definition of the ampere, chosen by the CIPM, was referenced to the force between parallel wires carrying an electric current and it had the effect of fixing the numerical value of the vacuum magnetic permeability μ0 (also called the magnetic constant). The numerical value of the vacuum electric permittivity ε0 (also called the electric constant) then became fixed as a consequence of the new definition of the meter adopted in 1983.
However the 1948 definition of the ampere proved difficult to realize and practical quantum standards (based on Josephson and quantum-Hall effects), which link both the volt and the ohm to particular combinations of the Planck constant h and elementary charge e, became almost universally used as a practical realization of the ampere through Ohm's law (18th CGPM (1987), Resolution 6, CR 100). As a consequence, it became natural not only to fix the numerical value of h to redefine the kilogram, but also to fix the numerical value of e to redefine the ampere in order to bring the practical quantum electrical standards into exact agreement with the SI. The present definition based on a fixed numerical value for the elementary charge, e, was adopted in Resolution 1 of the 26th CGPM (2018).
Unit of thermodynamic temperature, kelvin
The definition of the unit of thermodynamic temperature was given by the 10th CGPM (1954, Resolution 3; CR 79) which selected the triple point of water, TTPW, as a fundamental fixed point and assigned to it the temperature 273.16 K, thereby defining the kelvin. The 13th CGPM (1967-1968, Resolution 3; CR, 104 and Metrologia, 1968, 4, 43) adopted the name kelvin, symbol K, instead of "degree kelvin," symbol ºK, for the unit defined in this way. However, the practical difficulties in realizing this definition, requiring a sample of pure water of well-defined isotopic composition and the development of new primary methods of thermometry, led to the adoption of a new definition of the kelvin based on a fixed numerical value of the Boltzmann constant k. The present definition, which removed both of these constraints, was adopted in Resolution 1 of the 26th CGPM (2018).
Unit of amount of substance, mole
Following the discovery of the fundamental laws of chemistry, units called, for example, "gram-atom" and "gram molecule", were used to specify amounts of chemical elements or compounds. These units had a direct connection with "atomic weights" and "molecular weights", which are in fact relative atomic and molecular masses. The first compilations of "Atomic weights" were originally linked to the atomic weight of oxygen, which was, by general agreement, taken as being 16. Whereas physicists separated the isotopes in a mass spectrometer and attributed the value 16 to one of the isotopes of oxygen, chemists attributed the same value to the (slightly variable) mixture of isotopes 16, 17 and 18, which for them constituted the naturally occurring element oxygen. An agreement between the International Union of Pure and Applied Physics (IUPAP) and the International Union of Pure and Applied Chemistry (IUPAC) brought this duality to an end in 1959-1960. Physicists and chemists had agreed to assign the value 12, exactly, to the so-called atomic weight, correctly referred to as the relative atomic mass Ar, of the isotope of carbon with mass number 12 (carbon-12, 12C). The unified scale thus obtained gives the relative atomic and molecular masses, also known as the atomic and molecular weights, respectively. This agreement is unaffected by the redefinition of the mole.
The quantity used by chemists to specify the amount of chemical elements or compounds is called "amount of substance". Amount of substance, symbol n, is defined to be proportional to the number of specified elementary entities N in a sample, the proportionality constant being a universal constant which is the same for all entities. The proportionality constant is the reciprocal of the Avogadro constant NA, so that n = N/NA. The unit of amount of substance is called the mole, symbol mol. Following proposals by the IUPAP, IUPAC and ISO, the CIPM developed a definition of the mole in 1967 and confirmed it in 1969, by specifying that the molar mass of carbon 12 should be exactly 0.012 kg/mol. This allowed the amount of substance nS(X) of any pure sample S of entity X to be determined directly from the mass of the sample mS and the molar mass M(X) of entity X, the molar mass being determined from its relative atomic mass Ar (atomic or molecular weight) without the need for a precise knowledge of the Avogadro constant, by using the relations
$$n_s(X) = m_s/M(X),\,and\,M(X) = A_r(X)\, g/mol$$
Thus, this definition of the mole was dependent on the artifact definition of the kilogram.
The numerical value of the Avogadro constant defined in this way was equal to the number of atoms in 12 grams of
carbon-12. However, because of recent technological advances, this number is now known with such precision that a simpler and more universal definition of the mole has become possible, namely, by specifying exactly the number of entities in one mole of any substance, thus fixing the numerical value of the Avogadro constant. This has the effect that the new definition of the mole and the value of the Avogadro constant are no longer dependent on the definition of the kilogram. The distinction between the fundamentally different quantities 'amount of substance' and 'mass' is thereby emphasized. The present definition of the mole based on a fixed numerical value for the Avogadro constant, NA, was adopted in Resolution 1 of the 26th CGPM (2018).
Unit of luminous intensity, candela
The units of luminous intensity, which were based on flame or incandescent filament standards in use in various countries before 1948, were replaced initially by the "new candle" based on the luminance of a Planckian radiator (a black body) at the temperature of freezing platinum. This modification had been prepared by the International Commission on Illumination (CIE) and by the CIPM before 1937 and the decision was promulgated by the CIPM in 1946. It was then ratified in 1948 by the 9th CGPM, which adopted a new international name for this unit, the candela, symbol cd; in 1954 the 10th CGPM established the candela as a base unit; In 1967 the 13th CGPM (Resolution 5, CR, 104 and Metrologia, 1968, 4, 43-44) amended this definition.
In 1979, because of the difficulties in realizing a Planck radiator at high temperatures, and the new possibilities offered by radiometry, i.e. the measurement of optical radiation power, the 16th CGPM (1979, Resolution 3, CR, 100 and Metrologia,1980, 16, 56) adopted a new definition of the candela.
The present definition of the candela uses a fixed numerical value for the luminous efficacy of monochromatic radiation of frequency 540 ×1012 Hz, Kcd, adopted in Resolution 1 of the 26th CGPM (2018).
11 To recognize Bryan Kibble's invention of the watt balance.
Metrology and Weights and measures
David B. Newell
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Eite Tiesinga
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Created September 3, 2019, Updated June 2, 2021
‹ SP 330 - Appendix 2 & 3
SP 330 - List of acronyms used in the present volume ›
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The fundamental building blocks of linear algebra are vectors. Vectors are defined as quantities having both direction and magnitude, compared to scalar quantities that only have magnitude. In order to have direction and magnitude, vector quantities consist of two or more elements of data. The dimensionality of a vector is determined by the number of numerical elements in that vector. For example, a vector with four elements would have a dimensionality of four.
Let's take a look at examples of a scalar versus a vector. A car driving at a speed of 40mph is a scalar quantity. Describing the car driving 40mph to the east would represent a two-dimensional vector quantity since it has a magnitude in both the x and y directions.
Vectors can be represented as a series of numbers enclosed in parentheses, angle brackets, or square brackets. In this lesson, we will use square brackets for consistency. For example, a three-dimensional vector is written as:
v=[xyz]v = \begin{bmatrix}
x \\
y \\
\end{bmatrix}v=⎣⎡xyz⎦⎤
The magnitude (or length) of a vector, ||v||, can be calculated with the following formula:
∣∣v∣∣=∑i=1nvi2||v|| = \sqrt{\sum\limits_{i=1}^{n} v_{i}^2}∣∣v∣∣=i=1∑nvi2
This formulates translates to the sum of each vector component squared, which can be also written out as:
∣∣v∣∣=v12+v22+⋯+vn2||v|| = \sqrt{v_{1}^2 + v_{2}^2 + \dots + v_{n}^2}∣∣v∣∣=v12+v22+⋯+vn2
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Convert the velocities into a vector, and find the total speed of the ball. (Hint: the speed of the ball is the magnitude of the velocity vector!)
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Use this applet and/or mathematical calculation to find the magnitudes of the following vectors:
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b=[026]b = \begin{bmatrix}
26 \\
\end{bmatrix}b=[026]
c=[260]c = \begin{bmatrix}
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d=[−1213]d = \begin{bmatrix}
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Home › Azimuth Project › - Strategy
This week's progress
December 2016 edited February 2019 in - Strategy
I've been writing weekly progress reports to my grad students. It may make sense to copy them here. Let me give it a try.
17 November 2016:
Daniel Cicala points out the math conference December 3rd and 4th at Union College in upstate NY. There will be talks on category theory organized by Susan Niefeld - she does this every year, and I've spoken there once. If you can go, do it!
(If you read the category theory mailing list you can stay up to date on such conferences.)
Here is this week's progress:
1) Blake and I went to San Diego and learned about Metron's "ExAMS" software for designing complex systems. Then John Foley came up and, with help from Joseph Moeller, we figured out a bunch of stuff.
I blogged about ExAMS here:
Complex Adaptive System Design (Part 2)
This software raises lots of interesting questions. I believe it's based on "timed hierarchical colored Petri nets with guards". I would like to make sure this is true, and understand this kind of network category-theoretically. In case anyone wants to help me, here's an intro:
Wil M. P. van der Aalst, Christian Stahl, and Michael Westergaard, Strategies for modeling complex processes using colored Petri nets.
When we met, Tom Mifflin at Metron seemed pretty eager for our work to go in this direction.
2) I went to the Mathematical Association of America conference and gave a talk on The answer to the ultimate question of life, the universe and everything. Brandon and Daniel also went there.
3) I finally blogged about Brendan's thesis:
Open and interconnected systems.
As you can see, it's a lazy blog article - yet it still manages to give a detailed introduction to his work! If you haven't yet learned everything that Brendan is doing, this is a good place to start.
Comment Source:17 November 2016: Hi - Daniel Cicala points out the [math conference December 3rd and 4th at Union College](http://www.math.union.edu/%7Etoddg/ucc/) in upstate NY. There will be talks on category theory organized by Susan Niefeld - she does this every year, and I've spoken there once. If you can go, do it! (If you read the category theory mailing list you can stay up to date on such conferences.) Here is this week's progress: 1) Blake and I went to San Diego and learned about Metron's "ExAMS" software for designing complex systems. Then John Foley came up and, with help from Joseph Moeller, we figured out a bunch of stuff. I blogged about ExAMS here: * [Complex Adaptive System Design (Part 2)](https://johncarlosbaez.wordpress.com/2016/10/18/complex-adaptive-system-design-part-2/) This software raises lots of interesting questions. I believe it's based on "timed hierarchical colored Petri nets with guards". I would like to make sure this is true, and understand this kind of network category-theoretically. In case anyone wants to help me, here's an intro: * Wil M. P. van der Aalst, Christian Stahl, and Michael Westergaard, [Strategies for modeling complex processes using colored Petri nets](http://wwwis.win.tue.nl/%7Ewvdaalst/publications/p710.pdf). When we met, Tom Mifflin at Metron seemed pretty eager for our work to go in this direction. 2) I went to the Mathematical Association of America conference and gave a talk on [The answer to the ultimate question of life, the universe and everything](http://math.ucr.edu/home/baez/42/). Brandon and Daniel also went there. 3) I finally blogged about Brendan's thesis: * [Open and interconnected systems](https://johncarlosbaez.wordpress.com/2016/10/23/open-and-interconnected-systems/). As you can see, it's a lazy blog article - yet it still manages to give a detailed introduction to his work! If you haven't yet learned everything that Brendan is doing, this is a good place to start.
Lots of progress this week:
1) Jason uploaded his thesis to the arXiv! That's great, because it establishes priority - now he can prove he did this stuff before anyone else, even if it takes a while for him to publish a paper.
2) Blake gave a talk about open chemical reaction networks at the Santa Fe Institute workshop on Statistical Mechanics, Information Processing and Biology:
Compositional frameworks for open systems.
It was a real hit. Susanne Still said it was "great" - she works on Markov processes and the information bottleneck method for predictive inference. Jim Crutchfield liked it so much he invited Blake to give a talk up at U. C. Davis! He's a real bigshot: the Wikipedia article on him says
Over the last three decades Prof. Crutchfield has worked in the areas of nonlinear dynamics, solid-state physics, astrophysics, fluid mechanics, critical phenomena and phase transitions, chaos, and pattern formation. His current research interests center on computational mechanics, the physics of complexity, statistical inference for nonlinear processes, genetic algorithms, evolutionary theory, machine learning, quantum dynamics, and distributed intelligence. He has published over 100 papers in these areas.
If Blake can strike up a relationship with Crutchfield and maybe work on a project, that'll be excellent.
3) Joshua Tan, a grad student at Oxford (and friend of Brendan), invited me to join a bunch of people in writing a grant proposal.
It's for an NSF grant called "Smart & Connected Communities", and part of the plan would be to model cities as composable, open systems using category theory. Here are the other people involved in writing the proposal:
Dennis Frenchman is a professor at MIT and an expert in building digital tools for cities. He is the likely PI.
Sokwoo Rhee is a director at NIST managing 100+ smart cities projects and will be collaborating directly with us, but he is a silent partner due to federal rules.
Stephen Walter is a program director at the City of Boston, Mayor's Office of New Urban Mechanics.
Matthew Claudel is a student of Dennis' who has been working with me to write the proposal. His research is in urban innovation.
Possibly: Eric Gordon, a professor at the Engagement Lab at Emerson College, who works on civic participation.
Other personnel attached to the project include Elizabeth Christoforetti (Harvard, MIT urban planner) and Nissia Sabri (startup, hardware specialist).
This would be a great step toward my ultimate goal: using network theory for studying complex systems like biological systems and designing systems to deal with climate change.
Comment Source:23 November 2016: Lots of progress this week: 1) Jason [uploaded his thesis to the arXiv](https://arxiv.org/abs/1611.07591)! That's great, because it establishes priority - now he can prove he did this stuff before anyone else, even if it takes a while for him to publish a paper. 2) Blake gave a talk about open chemical reaction networks at the Santa Fe Institute workshop on [Statistical Mechanics, Information Processing and Biology](http://www.santafe.edu/gevent/detail/science/2452/): * [Compositional frameworks for open systems](https://johncarlosbaez.wordpress.com/2016/11/27/compositional-frameworks-for-open-systems/). It was a real hit. [Susanne Still](http://www2.hawaii.edu/%7Esstill/) said it was "great" - she works on Markov processes and the information bottleneck method for predictive inference. [Jim Crutchfield](http://csc.ucdavis.edu/%7Echaos/) liked it so much he invited Blake to give a talk up at U. C. Davis! He's a real bigshot: the Wikipedia article on him says > Over the last three decades Prof. Crutchfield has worked in the areas of nonlinear dynamics, solid-state physics, astrophysics, fluid mechanics, critical phenomena and phase transitions, chaos, and pattern formation. His current research interests center on computational mechanics, the physics of complexity, statistical inference for nonlinear processes, genetic algorithms, evolutionary theory, machine learning, quantum dynamics, and distributed intelligence. He has published over 100 papers in these areas. If Blake can strike up a relationship with Crutchfield and maybe work on a project, that'll be excellent. 3) [Joshua Tan](http://www.joshuatan.com/research/), a grad student at Oxford (and friend of Brendan), invited me to join a bunch of people in writing a grant proposal. It's for an NSF grant called "[Smart & Connected Communities](https://www.nsf.gov/pubs/2016/nsf16610/nsf16610.htm#pgm_desc_txt)", and part of the plan would be to model cities as composable, open systems using category theory. Here are the other people involved in writing the proposal: * Dennis Frenchman is a professor at MIT and an expert in building digital tools for cities. He is the likely PI. * Sokwoo Rhee is a director at NIST managing 100+ smart cities projects and will be collaborating directly with us, but he is a silent partner due to federal rules. * Stephen Walter is a program director at the City of Boston, Mayor's Office of New Urban Mechanics. * Matthew Claudel is a student of Dennis' who has been working with me to write the proposal. His research is in urban innovation. * Possibly: Eric Gordon, a professor at the Engagement Lab at Emerson College, who works on civic participation. * Other personnel attached to the project include Elizabeth Christoforetti (Harvard, MIT urban planner) and Nissia Sabri (startup, hardware specialist). This would be a great step toward my ultimate goal: using network theory for studying complex systems like biological systems and designing systems to deal with climate change.
1) Daniel Cicala passed his oral exam today! He spoke about this paper that he put on the arXiv last week:
Spans of cospans.
Abstract. We introduce the notion of a span of cospans and define, for them, horizontal and vertical composition. These compositions satisfy the interchange law if working in a topos C and if the span legs are monic. A bicategory is then constructed from C-objects, C-cospans, and doubly monic spans of C-cospans. The primary motivation for this construction is an application to graph rewriting.
2) Tobias Fritz is visiting us! He'll be speaking in the network theory seminar tomorrow and also joining our group meeting on Wednesday at 11 am. Here's his talk:
Inferring hidden network structure: the case of causal inference.
Abstract. The problem of causal inference is to determine if a given probability distribution on observed variables is compatible with some hypothetical Bayesian network structure. In the presence of hidden nodes (unobserved variables), this is a challenging problem for which no exact methods are known. The inflation technique of http://arxiv.org/abs/1609.00672 provides a new practical tool for approaching this problem. It has the potential to be generalized to other kinds of networks, in particular those that live in semicartesian monoidal categories.
3) My former grad student Chris Rogers will be giving a special seminar on symplectic stuff on Thursday 3:40-5:00, either in the Undergraduate Study Room or in some better room like room 284 or 268 - it's not exactly clear, but I'll try to inform you when I find out.
It will be very good for Brandon and Adam to attend this, since they're doing symplectic stuff. However, Chris will blow them out of the water with his erudition.
From Hamiltonian mechanics to homotopy Lie theory
Abstract. In Hamiltonian mechanics, physicists model the phase space of a physical system using symplectic geometry, and they use Lie algebras to describe the space's infinitesimal symmetries. Given such a Lie algebra of symmetries, the geometry naturally produces a new Lie algebra called a "central extension''. This central extension plays a crucial role, especially in quantum mechanics. The famous Heisenberg algebra, for example, arises precisely in this way.
In this talk, I will explain how the above recipe can be enhanced to geometrically produce examples of "homotopy Lie algebras''. A homotopy Lie algebra is a topologist's version of a Lie algebra: a chain complex equipped with structures which satisfy the axioms of a Lie algebra only up to chain homotopy. They provide important tools for rational homotopy theory and deformation theory. The homotopy Lie algebras produced from our construction turn out to have interesting relationships with the theory of loop groups and what are called "string structures'' in algebraic topology
Comment Source:28 November 2016: 1) Daniel Cicala passed his oral exam today! He spoke about this paper that he put on the arXiv last week: * [Spans of cospans](https://arxiv.org/abs/1611.07886). > **Abstract.** We introduce the notion of a span of cospans and define, for them, horizontal and vertical composition. These compositions satisfy the interchange law if working in a topos C and if the span legs are monic. A bicategory is then constructed from C-objects, C-cospans, and doubly monic spans of C-cospans. The primary motivation for this construction is an application to graph rewriting. 2) Tobias Fritz is visiting us! He'll be speaking in the network theory seminar tomorrow and also joining our group meeting on Wednesday at 11 am. Here's his talk: * [Inferring hidden network structure: the case of causal inference](https://simons.berkeley.edu/talks/tobias-fritz-12-06-2016). > **Abstract.** The problem of causal inference is to determine if a given probability distribution on observed variables is compatible with some hypothetical Bayesian network structure. In the presence of hidden nodes (unobserved variables), this is a challenging problem for which no exact methods are known. The inflation technique of [http://arxiv.org/abs/1609.00672](http://arxiv.org/abs/1609.00672) provides a new practical tool for approaching this problem. It has the potential to be generalized to other kinds of networks, in particular those that live in semicartesian monoidal categories. 3) My former grad student Chris Rogers will be giving a special seminar on symplectic stuff on Thursday 3:40-5:00, either in the Undergraduate Study Room or in some better room like room 284 or 268 - it's not exactly clear, but I'll try to inform you when I find out. It will be very good for Brandon and Adam to attend this, since they're doing symplectic stuff. However, Chris will blow them out of the water with his erudition. * From Hamiltonian mechanics to homotopy Lie theory > **Abstract.** In Hamiltonian mechanics, physicists model the phase space of a physical system using symplectic geometry, and they use Lie algebras to describe the space's infinitesimal symmetries. Given such a Lie algebra of symmetries, the geometry naturally produces a new Lie algebra called a "central extension''. This central extension plays a crucial role, especially in quantum mechanics. The famous Heisenberg algebra, for example, arises precisely in this way. > In this talk, I will explain how the above recipe can be enhanced to geometrically produce examples of "homotopy Lie algebras''. A homotopy Lie algebra is a topologist's version of a Lie algebra: a chain complex equipped with structures which satisfy the axioms of a Lie algebra only up to chain homotopy. They provide important tools for rational homotopy theory and deformation theory. The homotopy Lie algebras produced from our construction turn out to have interesting relationships with the theory of loop groups and what are called "string structures'' in algebraic topology
10 December 2016:
Some very good news this time:
1) Brendan Fong has accepted a postdoc at MIT working with David Spivak. Having seen them discuss math together, I think we can expect great things!
2) I'm 99% sure that Daniel Cicala has been accepted to the Kan Extension Seminar, a high-powered online course on category theory. This time it'll be about functorial semantics - you can see the papers they'll discuss by clicking the link.
3) I gave a talk on Compositionality in network theory at this week's workshop on Compositionality at the Simons Institute for the Theory of Computing. You can see a video by clicking the link. I explained Brendan's theory of decorated cospans, illustrating it with a paper that Blake and I are writing about Petri nets.
4) Brendan gave a talk on Modelling interconnected systems with decorated corelations at the same workshop. This goes further, introducing decorated corelations, which are a generalization of decorated cospans. Again you can see a video by clicking the link.
These talks seem to have gone over well, along with other closely connected talks by David Spivak, Ross Duncan, Pawel Sobocinski and others. I was invited by Michael Mislove, who edits a column on semantics at the journal Logic, Semantics and Theory of Programming, to contribute a column. The whole lot of us were invited to participate more in various conferences on logic and computer science, since what we're doing seems to fit into that heading.
5) Blake's work on Markov processes was cited in at least two talks, and Prakash Panagaden gave me a draft of his paper on a bicategory of Markov processes, which I append here - Blake, Kenny and Daniel should read it!
I think we can and should do better, but we'll have to avoid stepping on Prakash's toes. For one thing, we can build a symmetric monoidal bicategory. For another thing, they are doing discrete-time Markov processes, with 2-morphisms being maps called 'simulations'. We can do something else. Daniel's work on perfect measure spaces should come into this, as well as what Kenny has been doing on bicategories with coarse-grainings as 2-morphisms.
6) Some negative news: the grant proposal I mentioned recently, engineered by Joshua Tan, has fallen through. I'm not too upset.
Comment Source:10 December 2016: Some very good news this time: 1) Brendan Fong has accepted a postdoc at MIT working with David Spivak. Having seen them discuss math together, I think we can expect great things! 2) I'm 99% sure that Daniel Cicala has been accepted to the [Kan Extension Seminar](https://golem.ph.utexas.edu/category/2016/10/the_kan_extension_seminar_retu.html), a high-powered online course on category theory. This time it'll be about functorial semantics - you can see the papers they'll discuss by clicking the link. 3) I gave a talk on [Compositionality in network theory](https://johncarlosbaez.wordpress.com/2016/11/29/compositionality-in-network-theory/) at this week's workshop on Compositionality at the Simons Institute for the Theory of Computing. You can see a video by clicking the link. I explained Brendan's theory of decorated cospans, illustrating it with a paper that Blake and I are writing about Petri nets. 4) Brendan gave a talk on [Modelling interconnected systems with decorated corelations](https://johncarlosbaez.wordpress.com/2016/12/09/modelling-interconnected-systems-with-decorated-corelations/) at the same workshop. This goes further, introducing decorated corelations, which are a generalization of decorated cospans. Again you can see a video by clicking the link. These talks seem to have gone over well, along with other closely connected talks by David Spivak, Ross Duncan, Pawel Sobocinski and others. I was invited by Michael Mislove, who edits a column on semantics at the journal Logic, Semantics and Theory of Programming, to contribute a column. The whole lot of us were invited to participate more in various conferences on logic and computer science, since what we're doing seems to fit into that heading. 5) Blake's work on Markov processes was cited in at least two talks, and Prakash Panagaden gave me a draft of his paper on a bicategory of Markov processes, which I append here - Blake, Kenny and Daniel should read it! I think we can and should do better, but we'll have to avoid stepping on Prakash's toes. For one thing, we can build a symmetric monoidal bicategory. For another thing, they are doing discrete-time Markov processes, with 2-morphisms being maps called 'simulations'. We can do something else. Daniel's work on perfect measure spaces should come into this, as well as what Kenny has been doing on bicategories with coarse-grainings as 2-morphisms. 6) Some negative news: the grant proposal I mentioned recently, engineered by Joshua Tan, has fallen through. I'm not too upset.
Finally catching up to today, 13 December 2016:
1) Kenny Courser's paper A bicategory of decorated cospans was accepted by Theory and Applications of Categories!
The referee wants him to say more about examples. That makes a lot of sense. I'd also like to deal with this issue: in our favorite examples, the 2-morphisms in Kenny's bicategory are a bit too restrictive.
For example, in the bicategory of cospans of finite sets where the apex is decorated by a graph, the 2-morphism do not allow arbitrary graph morphisms, only those that are "cocartesian lifts" of maps between finite sets.
2) On December 14th, Brendan is giving a talk called "All hypergraph categories are decorated corelation categories" at Macquarie University in Australia.
Brendan: make sure to say hi to Ross Street and my old friend James Dolan!
3) At Berkeley, it became clear that the stuff we do fits into "theoretical computer science", which is a very broad subject by now.
All of us were invited to submit papers to CALCO 2017, a conference on algebra and coalgebra in computer science. Daniel reminded me of this, saying:
Actually, a few of the gang could probably submit, since their interests include:
String Diagrams and Network Theory
- Combinatorial approaches
- Theory of PROPs and operads
- Rewriting problems and higher-dimensional approaches
- Automated reasoning with string diagrams
- Applications of string diagrams
- Connections with Control Theory, Engineering and Concurrency
So, think of submitting papers here! Daniel has a plan to do this.
Comment Source:Finally catching up to today, 13 December 2016: 1) Kenny Courser's paper [A bicategory of decorated cospans](https://arxiv.org/abs/1605.08100) was accepted by _Theory and Applications of Categories_! The referee wants him to say more about examples. That makes a lot of sense. I'd also like to deal with this issue: in our favorite examples, the 2-morphisms in Kenny's bicategory are a bit too restrictive. For example, in the bicategory of cospans of finite sets where the apex is decorated by a graph, the 2-morphism do not allow arbitrary graph morphisms, only those that are "cocartesian lifts" of maps between finite sets. 2) On December 14th, Brendan is giving a talk called "All hypergraph categories are decorated corelation categories" at Macquarie University in Australia. Brendan: make sure to say hi to Ross Street and my old friend James Dolan! 3) At Berkeley, it became clear that the stuff we do fits into "theoretical computer science", which is a very broad subject by now. All of us were invited to submit papers to [CALCO 2017](http://coalg.org/calco17/index.html), a conference on algebra and coalgebra in computer science. Daniel reminded me of this, saying: > Actually, a few of the gang could probably submit, since their interests include: > * String Diagrams and Network Theory > - Combinatorial approaches > - Theory of PROPs and operads > - Rewriting problems and higher-dimensional approaches > - Automated reasoning with string diagrams > - Applications of string diagrams > - Connections with Control Theory, Engineering and Concurrency So, think of submitting papers here! Daniel has a plan to do this.
For some reason dates aren't showing up in Azimuth Forum entries, so it's good I included the dates in the entries here! I have some new entries...
Comment Source:For some reason dates aren't showing up in Azimuth Forum entries, so it's good I included the dates in the entries here! I have some new entries...
Here are two things that happened this week:
1) Brandon and Brendan's paper Corelations are the prop for extraspecial commutative Frobenius monoids has been accepted for publication by Theory and Applications of Categories subject to making some small corrections.
2) It doesn't really count as mathematics, but I've started the Azimuth Backup Project to help back up climate data before Trump becomes president - because almost all his big hires are people who deny the importance of global warming.
Other teams are doing this too, and you can get the basic idea in this article of mine:
Saving Climate Data (Part 1).
You can see our team's progress here:
Azimuth Backup Project (Part 1).
We've got a great team, including a guy who used to drive a Mars rover for NASA, and so far we've backed up about a terabyte of data! In a couple of days I'll start a Kickstarter campaign to raise funds to store the data. We'll try to store it at least until larger institutions accept this responsibility.
Comment Source:22 December 2016: Here are two things that happened this week: 1) Brandon and Brendan's paper [Corelations are the prop for extraspecial commutative Frobenius monoids](https://arxiv.org/abs/1601.02307) has been accepted for publication by _Theory and Applications of Categories_ subject to making some small corrections. 2) It doesn't really count as mathematics, but I've started the Azimuth Backup Project to help back up climate data before Trump becomes president - because almost all his big hires are people who deny the importance of global warming. Other teams are doing this too, and you can get the basic idea in this article of mine: * [Saving Climate Data (Part 1)](https://johncarlosbaez.wordpress.com/2016/12/13/saving-climate-data/). You can see our team's progress here: * [Azimuth Backup Project (Part 1)](https://johncarlosbaez.wordpress.com/2016/12/16/azimuth-backup-project/). * [Azimuth Backup Project (Part 2)](https://johncarlosbaez.wordpress.com/2016/12/20/azimuth-backup-project-part-2/). We've got a great team, including a guy who used to drive a Mars rover for NASA, and so far we've backed up about a terabyte of data! In a couple of days I'll start a Kickstarter campaign to raise funds to store the data. We'll try to store it at least until larger institutions accept this responsibility.
We had a hugely productive meeting on Wednesday and I'm really excited about the new ideas:
The connection between Brandon Coya's work on bond graphs and Ross Street's work on weak bimonoids, noticed by Brendan, is really fascinating - it implies that there's a "quantum groupoid" associated to electrical circuits, and it implies that our conjectured list of axioms characterizing the category of bond graphs was missing some highly nonobvious relations.
Daniel Cicala is revisiting Franciscus Rebro's work on the bicategory of cobordisms and will prove it's a symmetric monoidal bicategory.
Kenny Courser had the smart idea of revisiting Jeffrey Morton and Jamie Vicary's work on Khovanov's categorified Heisenberg algebra and making it rigorous using our new ability to get ahold of symmetric monoidal bicategories, and I realized we can actually do this.
Adam Yassine seems to have proved that there's a bicategory of symplectic manifolds and cospans whose legs are Poisson fibrations — good for the study of open systems in classical mechanics.
And that's not all! In our Metron project,
Blake Pollard and John Foley are developing a new framework for search and rescue operations (and many other distributed optimization problems).
Joseph Moeller created a new algebraic structure generalizing the "operad for communication networks", and I think we can prove this new structure has an elegant category-theoretic description.
It's all great stuff. But these weekly reports are supposed to be about things that have been completed, just to focus your attention on getting things finished. So here are two things like that:
The math department at U.C. Riverside is hosting the Fall Meeting of the AMS Western Section on Saturday and Sunday, Nov. 4 and 5, 2017. My proposal for a special session on Applied Category Theory has been accepted! I hope you submit proposals for talks — if you're able to come despite the fact that, as usual for such meetings, we have no money. I'll say in a while how you can propose a talk: there will be a webpage where you can do this.
U. C. Riverside has agreed to be a repository of climate data collected by the Azimuth Backup Project. This means we don't have to figure out how to hold this data permanently. We've raised over $10,000 by now, so we're fine in the short term.
Comment Source:13 January 2017: We had a hugely productive meeting on Wednesday and I'm really excited about the new ideas: * The connection between Brandon Coya's work on bond graphs and Ross Street's work on weak bimonoids, noticed by Brendan, is really fascinating - it implies that there's a "quantum groupoid" associated to electrical circuits, and it implies that our conjectured list of axioms characterizing the category of bond graphs was missing some highly nonobvious relations. * Daniel Cicala is revisiting Franciscus Rebro's work on the bicategory of cobordisms and will prove it's a symmetric monoidal bicategory. * Kenny Courser had the smart idea of revisiting Jeffrey Morton and Jamie Vicary's work on Khovanov's categorified Heisenberg algebra and making it rigorous using our new ability to get ahold of symmetric monoidal bicategories, and I realized we can actually do this. * Adam Yassine seems to have proved that there's a bicategory of symplectic manifolds and cospans whose legs are Poisson fibrations — good for the study of open systems in classical mechanics. And that's not all! In our Metron project, * Blake Pollard and John Foley are developing a new framework for search and rescue operations (and many other distributed optimization problems). * Joseph Moeller created a new algebraic structure generalizing the "operad for communication networks", and I think we can prove this new structure has an elegant category-theoretic description. It's all great stuff. But these weekly reports are supposed to be about things that have been completed, just to focus your attention on getting things finished. So here are two things like that: * The math department at U.C. Riverside is hosting the Fall Meeting of the AMS Western Section on Saturday and Sunday, Nov. 4 and 5, 2017. My proposal for a special session on **Applied Category Theory** has been accepted! I hope you submit proposals for talks — if you're able to come despite the fact that, as usual for such meetings, we have no money. I'll say in a while how you can propose a talk: there will be a webpage where you can do this. * U. C. Riverside has agreed to be a repository of climate data collected by the Azimuth Backup Project. This means we don't have to figure out how to hold this data permanently. We've raised over $10,000 by now, so we're fine in the short term.
1) Kenny Courser dealt with the referee's comments on his paper A bicategory of decorated cospans, resubmitted it to Theory and Applications of Categories, and put the new improved version on the arXiv.
2) Daniel Cicala has submitted his paper Spans of cospans to Theory and Applications of Categories. Is that true, Daniel? If not, make it so. Also, it probably makes sense to update the version on the arXiv.
(Kenny and Daniel are now teaming up to write a wonderful paper on bicategories of spans and cospans, including the "maps-of-spans" case already dealt with more generally by Mike, but more importantly the "spans-of-spans" case and "spans-of-cospans" case.)
3) Joseph Moeller has written up the statement of a theorem on generalizations of the operad of communication networks, suitable for inclusion in the paper that Blake is working on for DARPA. This paper is due the day after tomorrow!
4) I've been invited to join another project run by the US Defense Department!
I got an email from Michael A. Smith, a microbiologist who's the director of the Defense Biological Product Assurance Office. These are the people who check whether mysterious white powders showing up in people's mail are actually anthrax... and they also keep track of diseases that suddenly spring up, like Ebola or MIRS. They used to be called the Critical Reagents Program.
Smith is interested in whether modifications of the network of hospitals and laboratories in Africa could help them better keep track of new epidemics that show up on this continent. He read my book with Jacob Biamonte on Petri nets, and he thinks I might able to help. I'll start by listening in to a phone conversation he's having with Gary Kobinger, who helped develop a vaccine for Ebola, and who
developed and pioneered use of small mobile laboratories — a lab in a suitcase, essentially — that have changed the way testing is done during Ebola outbreaks.
I'm already way too busy, but this sounds interesting! Michael Smith said that his higher-ups mainly evaluate him on whether he spends money fast enough. I could certainly help with that... but it would also be cool if network ideas could help combat disease.
5) I arranged to visit the Institute for Scientific Interchange in Turin, Italy during the last week of June. This center is very interested in the mathematical foundations of complex systems, including networks.
I ran a workshop there in May 2015 and the director, Mario Rasetti, said I was welcome to visit any time. I decided to see if this offer was real: I'm visiting the category theorist Marco Grandis in nearby Genoa in mid-June, so I emailed Rasetti and asked if I could visit after that.
It turns out he's been reading our papers with interest! He'd actually like me to visit for a sabbatical or maybe even get some long-term affiliation with this center. I'll find out more in June.
Comment Source:21 January 2017: 1) Kenny Courser dealt with the referee's comments on his paper [A bicategory of decorated cospans](https://arxiv.org/abs/1605.08100), resubmitted it to _Theory and Applications of Categories_, and put the new improved version on the arXiv. 2) Daniel Cicala has submitted his paper [Spans of cospans](https://arxiv.org/abs/1611.07886) to _Theory and Applications of Categories_. Is that true, Daniel? If not, make it so. Also, it probably makes sense to update the version on the arXiv. (Kenny and Daniel are now teaming up to write a wonderful paper on bicategories of spans and cospans, including the "maps-of-spans" case already dealt with more generally by Mike, but more importantly the "spans-of-spans" case and "spans-of-cospans" case.) 3) Joseph Moeller has written up the statement of a theorem on generalizations of the operad of communication networks, suitable for inclusion in the paper that Blake is working on for DARPA. This paper is due the day after tomorrow! 4) I've been invited to join another project run by the US Defense Department! I got an email from [Michael A. Smith](https://www.linkedin.com/in/michael-smith-b6468844), a microbiologist who's the director of the [Defense Biological Product Assurance Office](https://globalbiodefense.com/2017/01/02/dod-seeks/). These are the people who check whether mysterious white powders showing up in people's mail are actually anthrax... and they also keep track of diseases that suddenly spring up, like Ebola or MIRS. They used to be called the Critical Reagents Program. Smith is interested in whether modifications of the network of hospitals and laboratories in Africa could help them better keep track of new epidemics that show up on this continent. He read [my book with Jacob Biamonte on Petri nets](http://math.ucr.edu/home/baez/stoch_stable.pdf), and he thinks I might able to help. I'll start by listening in to a phone conversation he's having with Gary Kobinger, who helped develop a vaccine for Ebola, and who > developed and pioneered use of small mobile laboratories — a lab in a suitcase, essentially — that have changed the way testing is done during Ebola outbreaks. I'm already way too busy, but this sounds interesting! Michael Smith said that his higher-ups mainly evaluate him on whether he spends money fast enough. I could certainly help with that... but it would also be cool if network ideas could help combat disease. 5) I arranged to visit the [Institute for Scientific Interchange](https://en.wikipedia.org/wiki/Institute_for_Scientific_Interchange) in Turin, Italy during the last week of June. This center is very interested in the [mathematical foundations of complex systems](http://www.isi.it/en/research/mathematics-foundation-of-complex-systems), including networks. I ran a workshop there in May 2015 and the director, Mario Rasetti, said I was welcome to visit any time. I decided to see if this offer was real: I'm visiting the category theorist Marco Grandis in nearby Genoa in mid-June, so I emailed Rasetti and asked if I could visit after that. It turns out he's been reading our papers with interest! He'd actually like me to visit for a sabbatical or maybe even get some long-term affiliation with this center. I'll find out more in June.
1) I gave a talk called Biology as Information Dynamics at a workshop called Biological Complexity: Can it be Quantified? at the Beyond Center at Arizona State University. The last result in my talk is new: it's an improved version of Fisher's fundamental theorem of natural selection, which is more general than the usual version, and phrased in terms of information geometry.
I talked about gene regulatory networks with Sara Walker, who works there. A gene regulatory network is a very simple thing: a finite set of 'genes' with two kinds of arrows from some genes to some other genes: 'express' and 'repress'. It would be nice to study 'open' gene regulatory networks using the decorated cospan formalism. There seems to be a fairly simple semantics for these networks where time goes in integer steps and at each time each gene is either on (expressed) or off (repressed).
I also had some conversations with Jim Crutchfield, who wants Blake Pollard to visit him up at U.C. Davis.
2) Tom Mifflin of Metron said that DARPA likes our work so much that they're starting two new programs related to these ideas. One will be at the Strategic Technology Office, and it will be big. Tom suggested I could get more money for grad students. The other is a small "seedling" program, just $500,000, for Metron to help develop autonomous vehicles for the Navy.
3) Blake noticed that Jason Erbele's paper Categories in control was cited by Dominique Luzeaux in a paper on the category-theoretic foundations of systems engineering. The main interesting thing about this paper is that Lzueaux is the Deputy Director of the Joint Directorate for Networks, Infrastructure and Information Systems, part of the French defense department.
It may seem weird that the military is interested in category theory and operads, but it makes sense. They have a lot of money, they're willing to experiment to stay ahead of other countries, and they have huge organizational/strategic problems that involve complex networked systems. Thus, they've traditionally been at the forefront of "systems of systems engineering".
Comment Source:14 February 2017: 1) I gave a talk called [Biology as Information Dynamics](http://math.ucr.edu/home/baez/bio_asu/) at a workshop called [Biological Complexity: Can it be Quantified?](https://beyond.asu.edu/workshop/biological-complexity-can-it-be-quantified) at the [Beyond Center](https://beyond.asu.edu/) at Arizona State University. The last result in my talk is new: it's an improved version of Fisher's fundamental theorem of natural selection, which is more general than the usual version, and phrased in terms of information geometry. I talked about gene regulatory networks with Sara Walker, who works there. A gene regulatory network is a very simple thing: a finite set of 'genes' with two kinds of arrows from some genes to some other genes: 'express' and 'repress'. It would be nice to study 'open' gene regulatory networks using the decorated cospan formalism. There seems to be a fairly simple semantics for these networks where time goes in integer steps and at each time each gene is either on (expressed) or off (repressed). I also had some conversations with Jim Crutchfield, who wants Blake Pollard to visit him up at U.C. Davis. 2) Tom Mifflin of Metron said that DARPA likes our work so much that they're starting two new programs related to these ideas. One will be at the Strategic Technology Office, and it will be big. Tom suggested I could get more money for grad students. The other is a small "seedling" program, just $500,000, for Metron to help develop autonomous vehicles for the Navy. 3) Blake noticed that Jason Erbele's paper [Categories in control](http://www.tac.mta.ca/tac/volumes/30/24/30-24abs.html) was cited by [Dominique Luzeaux](https://scholar.google.com/citations?user=8G2BKfEAAAAJ&hl=en) in a paper on the category-theoretic foundations of systems engineering. The main interesting thing about this paper is that Lzueaux is the Deputy Director of the Joint Directorate for Networks, Infrastructure and Information Systems, part of the French defense department. It may seem weird that the military is interested in category theory and operads, but it makes sense. They have a lot of money, they're willing to experiment to stay ahead of other countries, and they have huge organizational/strategic problems that involve complex networked systems. Thus, they've traditionally been at the forefront of "[systems of systems engineering](https://en.wikipedia.org/wiki/System_of_systems_engineering)".
Four pieces of news:
1) Brendan and Brandon made all the required changes to their paper "Corelations are the prop for extraspecial commutative Frobenius monoids", so this paper has now been published by Theory and Applications of Categories. You can see it here:
http://www.tac.mta.ca/tac/index.html#vol32
It's in the same issue as two papers by Voevodsky, inventor of homotopy type theory, along with papers by Clemens Berger, Michael Batanin and Jiri Adamek — all famous category theorists! So, congratulations!
2) Blake applied to give a talk on his work at the American Physical Society March Meeting in New Orleans on March 13-17. They asked him to present a poster. He applied to the Grad Student Association for funding to go on this trip, but it looks like he'll have to foot some of the bill himself. (Sadly, while the Metron project he and I are working on pays him enough money so he doesn't have to be a teaching assistant, it doesn't give us any travel money, and the Grad Student Association gives rather limited funds.)
3) Blake has also gotten an invitation from the famous information theorist Jim Crutchfield to speak about our work at U. C. Davis. I mentioned this earlier, but now it's really come true! I forget when Blake is going up there.
4) Not exactly academic, but: the Azimuth Climate Data Backup Project's Kickstarter campaign has succeeded! Our original goal was $5000. We got $20,427 of donations, and after Kickstarter took its cut we received $18,590.96. I wrote thank-you notes to all 627 contributors, and compiled a list thanking everyone who was okay with having their name made public.
We are close to reaching our goal of backing up 40 terabytes of data. The next step will be to put copies in several secure locations. 3 places have volunteered to hold copies: the Princeton math department, a UNESCO center in France, and the U. C. Riverside department of computing and communications. It will take a while to accomplish these transfers.
It looks like I'll be interviewed about this tomorrow by Amy Harmon of the New York Times. But reporters are often busy and distracted, so I'll believe it when it happens.
Comment Source:27 February 2017: Four pieces of news: 1) Brendan and Brandon made all the required changes to their paper "Corelations are the prop for extraspecial commutative Frobenius monoids", so this paper has now been published by Theory and Applications of Categories. You can see it here: [http://www.tac.mta.ca/tac/index.html#vol32](http://www.tac.mta.ca/tac/index.html#vol32) It's in the same issue as two papers by Voevodsky, inventor of homotopy type theory, along with papers by Clemens Berger, Michael Batanin and Jiri Adamek — all famous category theorists! So, congratulations! 2) Blake applied to give a talk on his work at the American Physical Society March Meeting in New Orleans on March 13-17. They asked him to present a poster. He applied to the Grad Student Association for funding to go on this trip, but it looks like he'll have to foot some of the bill himself. (Sadly, while the Metron project he and I are working on pays him enough money so he doesn't have to be a teaching assistant, it doesn't give us any travel money, and the Grad Student Association gives rather limited funds.) 3) Blake has also gotten an invitation from the famous information theorist [Jim Crutchfield](http://csc.ucdavis.edu/~chaos/) to speak about our work at U. C. Davis. I mentioned this earlier, but now it's really come true! I forget when Blake is going up there. 4) Not exactly academic, but: the Azimuth Climate Data Backup Project's [Kickstarter campaign](https://www.kickstarter.com/projects/592742410/azimuth-climate-data-backup-project) has succeeded! Our original goal was $5000. We got $20,427 of donations, and after Kickstarter took its cut we received $18,590.96. I wrote thank-you notes to all 627 contributors, and compiled [a list thanking everyone who was okay with having their name made public](https://johncarlosbaez.wordpress.com/2017/02/18/azimuth-backup-project-part-4/). We are close to reaching our goal of backing up 40 terabytes of data. The next step will be to put copies in several secure locations. 3 places have volunteered to hold copies: the Princeton math department, a UNESCO center in France, and the U. C. Riverside department of computing and communications. It will take a while to accomplish these transfers. It looks like I'll be interviewed about this tomorrow by [Amy Harmon](https://www.nytimes.com/by/amy-harmon) of the _New York Times_. But reporters are often busy and distracted, so I'll believe it when it happens.
AndriusKulikauskas
John, Congratulations on such a great response at Kickstarter and all the other good news!
Comment Source:John, Congratulations on such a great response at Kickstarter and all the other good news!
Comment Source:Thanks!
1) Jason Erbele and Daniel Cicala have applied to attend the AMS workshop on homotopy type theory in the "breathtaking mountain setting" of Snowbird Resort in Utah, June 4th to 10th. It's being run by my friends Dan Christensen, Mike Shulman and Emily Riehl along with Chris Kapulkin and Dan Licata (who I don't know so well).
2) Next week Blake Pollard is going to the American Physical Society meeting in New Orleans and presenting a poster on his work on open systems.
3) My book with Jacob Biamonte, Quantum Techniques for Stochastic Processes, seems to have been accepted for publication by World Scientific Press.
4) Less significant, but fun: the Australian science fiction writer Greg Egan and I proved an interesting equation expressing pi in terms of the golden ratio:
Pi and the golden ratio, Azimuth, 7 March 2017.
Comment Source:8 March 2017: 1) Jason Erbele and Daniel Cicala have applied to attend the AMS workshop on homotopy type theory in the "breathtaking mountain setting" of Snowbird Resort in Utah, June 4th to 10th. It's being run by my friends Dan Christensen, Mike Shulman and Emily Riehl along with Chris Kapulkin and Dan Licata (who I don't know so well). 2) Next week Blake Pollard is going to the American Physical Society meeting in New Orleans and presenting a poster on his work on open systems. 3) My book with Jacob Biamonte, _[Quantum Techniques for Stochastic Processes](http://math.ucr.edu/home/baez/stoch_stable.pdf)_, seems to have been accepted for publication by World Scientific Press. 4) Less significant, but fun: the Australian science fiction writer Greg Egan and I proved an interesting equation expressing pi in terms of the golden ratio: * [Pi and the golden ratio](https://johncarlosbaez.wordpress.com/2017/03/07/pi-and-the-golden-ratio/), _Azimuth_, 7 March 2017.
1) On Friday, Blake and John Foley and I finished off two essays for our Complex Adaptive Composition and Design Environment project.
The first is called "Compositional Tasking: an operad-based approach to adaptive behaviors for distributed systems-of-systems and planning under uncertainty." This is a general overview of our plan to use operads to design and "task" (boss around) networks while moving up and down levels of abstraction as desired. I find this really exciting! With luck the folks at Metron will create some software to illustrate these ideas
The second, called "Compositional Tasking", has more of the mathematical details. Joseph played a key role in this by checking that the Grothendieck construction can create the operads we need. I hope we continue to improve this essay and publish it in one or more papers, which I hope to write this summer.
2) Prakash Panangaden, an expert in categories and computer science at McGill University in Montreal, has come out with a paper which takes the category-theoretic characterization of relative entropy that Tobias Fritz and I found and extends it from finite sets to more general measurable spaces (sets with a sigma-algebra of subsets):
Abstract. The inspiration for the present work comes from two recent developments. The first is the beginning of a categorical understanding of Bayesian inversion and learning, the second is a categorical reconstruction of relative entropy. The present paper provides a categorical treatment of entropy in the spirit of Baez and Fritz in the setting of Polish spaces, thus setting the stage to explore the role of entropy in learning.
"Polish spaces" are a nice class of measurable spaces, loved by the analysts who lived in Poland before the Nazis invaded.
3) Prakash has also written a paper about a bicategory where the morphisms are open Markov processes. This is based on a paper that Brendan, Blake and I wrote:
Abstract. We construct bicategories of Markov processes where the objects are input and output sets, the morphisms (one-cells) are Markov processes and the two-cells are simulations. This builds on the work of Baez, Fong and Pollard, who showed that a certain kinds of finite-space continuous-time Markov chain (CTMC) satisfying a detailed-balance condition can be viewed as morphisms in a category. This view allows a compositional description of their CTMCs. Our contribution is to develop a notion of simulation between processes and construct a bicategory where the two-cells are simulation morphisms. Our version is for processes that are essentially probabilistic transition systems with discrete time steps and which do not satisfy a detailed balance condition. We have also extended the theory to continuous space processes.
In short, Prakash is moving in and starting to offer competition in our field of work. A lot of people in computer science respect his work, so they will start to read our stuff and explore similar ideas. This means we can't laze around when it comes to publishing ideas we have... but it's basically good, because it means more people will be inclined to hire my grad students!
Comment Source:27 March 2017: 1) On Friday, Blake and John Foley and I finished off two essays for our [Complex Adaptive Composition and Design Environment](https://johncarlosbaez.wordpress.com/2016/10/02/complex-adaptive-system-design-part-1/) project. The first is called "Compositional Tasking: an operad-based approach to adaptive behaviors for distributed systems-of-systems and planning under uncertainty." This is a general overview of our plan to use operads to design and "task" (boss around) networks while moving up and down levels of abstraction as desired. I find this really exciting! With luck the folks at Metron will create some software to illustrate these ideas The second, called "Compositional Tasking", has more of the mathematical details. Joseph played a key role in this by checking that the Grothendieck construction can create the operads we need. I hope we continue to improve this essay and publish it in one or more papers, which I hope to write this summer. 2) Prakash Panangaden, an expert in categories and computer science at McGill University in Montreal, has come out with a paper which takes the category-theoretic characterization of relative entropy that Tobias Fritz and I found and extends it from finite sets to more general measurable spaces (sets with a sigma-algebra of subsets): > **Abstract**. The inspiration for the present work comes from two recent developments. The first is the beginning of a categorical understanding of Bayesian inversion and learning, the second is a categorical reconstruction of relative entropy. The present paper provides a categorical treatment of entropy in the spirit of Baez and Fritz in the setting of Polish spaces, thus setting the stage to explore the role of entropy in learning. "Polish spaces" are a nice class of measurable spaces, loved by the analysts who lived in Poland before the Nazis invaded. 3) Prakash has also written a paper about a bicategory where the morphisms are open Markov processes. This is based on a paper that Brendan, Blake and I wrote: > **Abstract**. We construct bicategories of Markov processes where the objects are input and output sets, the morphisms (one-cells) are Markov processes and the two-cells are simulations. This builds on the work of Baez, Fong and Pollard, who showed that a certain kinds of finite-space continuous-time Markov chain (CTMC) satisfying a detailed-balance condition can be viewed as morphisms in a category. This view allows a compositional description of their CTMCs. Our contribution is to develop a notion of simulation between processes and construct a bicategory where the two-cells are simulation morphisms. Our version is for processes that are essentially probabilistic transition systems with discrete time steps and which do not satisfy a detailed balance condition. We have also extended the theory to continuous space processes. In short, Prakash is moving in and starting to offer competition in our field of work. A lot of people in computer science respect his work, so they will start to read our stuff and explore similar ideas. This means we can't laze around when it comes to publishing ideas we have... but it's basically good, because it means more people will be inclined to hire my grad students!
John, congratulations to you and your colleagues/students for inspiring others with your work!
Comment Source:John, congratulations to you and your colleagues/students for inspiring others with your work!
April 2017 edited May 2017
5 April 2017:
1) Gheorghe Craciun visited UCR and gave a talk today on his proof of the Global Attractor Conjecture, which until recently was one of the biggest open questions in mathematical chemistry.
2) Blake Pollard and I put our paper A compositional framework for reaction networks on the arXiv today. The conclusions summarize a lot of the work our group has done so far, and fits it into a big commutative diagram. Craciun and I have already used these ideas to construct a large new class of reaction networks with nice stability properties.
Abstract. Reaction networks, or equivalently Petri nets, are a general framework for describing processes in which entities of various kinds interact and turn into other entities. In chemistry, where the reactions are assigned "rate constants", any reaction network gives rise to a nonlinear dynamical system called its "rate equation". Here we generalize these ideas to "open" reaction networks, which allow entities to flow in and out at certain designated inputs and outputs. We treat open reaction networks are morphisms in a category. Composing two such morphisms connects the outputs of the first to the inputs of the second. We construct a functor sending any open reaction network to its corresponding "open dynamical system". This provides a compositional framework for studying the dynamics of reaction networks. We then turn to statics: that is, steady state solutions of open dynamical systems. We construct a "black-boxing" functor that sends any open dynamical system to the relation that it imposes between input and output variables in steady states. This extends our earlier work on black-boxing for Markov processes.
3) My former student Brendan Fong, who developed the "decorated cospan" and "decorated corelation" approach to network theory in this thesis, put related two papers onto the arXiv: Decorated corelations and A universal construction for (co)relations. I need to blog about these!
4) Brendan also gave an expository talk about "The mathematics of system composition" at BAE Systems, a British defense company.
5) My former student Mike Stay wrote two papers with Greg Meredith on the use of categories in computer science: Name-free combinators for concurrency and Representing operational semantics with enriched Lawvere theories.
6) My student Daniel Cicala got invited to the American Mathematical Society conference on Homotopy Type Theory that will take place in Snowbird, Utah on June 4-10.
Comment Source:5 April 2017: 1) Gheorghe Craciun visited UCR and gave a talk today on his proof of the [Global Attractor Conjecture](https://sinews.siam.org/Details-Page/discussing-the-proof-of-the-global-attractor-conjecture-1), which until recently was one of the biggest open questions in mathematical chemistry. 2) Blake Pollard and I put our paper [A compositional framework for reaction networks](https://arxiv.org/abs/1704.02051) on the arXiv today. The conclusions summarize a lot of the work our group has done so far, and fits it into a big commutative diagram. Craciun and I have already used these ideas to construct a large new class of reaction networks with nice stability properties. > **Abstract.** Reaction networks, or equivalently Petri nets, are a general framework for describing processes in which entities of various kinds interact and turn into other entities. In chemistry, where the reactions are assigned "rate constants", any reaction network gives rise to a nonlinear dynamical system called its "rate equation". Here we generalize these ideas to "open" reaction networks, which allow entities to flow in and out at certain designated inputs and outputs. We treat open reaction networks are morphisms in a category. Composing two such morphisms connects the outputs of the first to the inputs of the second. We construct a functor sending any open reaction network to its corresponding "open dynamical system". This provides a compositional framework for studying the dynamics of reaction networks. We then turn to statics: that is, steady state solutions of open dynamical systems. We construct a "black-boxing" functor that sends any open dynamical system to the relation that it imposes between input and output variables in steady states. This extends our earlier work on black-boxing for Markov processes. 3) My former student Brendan Fong, who developed the "decorated cospan" and "decorated corelation" approach to network theory in this thesis, put related two papers onto the arXiv: [Decorated corelations](https://arxiv.org/abs/1703.09888) and [A universal construction for (co)relations](https://arxiv.org/abs/1703.08247). I need to blog about these! 4) Brendan also gave an expository talk about "The mathematics of system composition" at [BAE Systems](https://en.wikipedia.org/wiki/BAE_Systems), a British defense company. 5) My former student Mike Stay wrote two papers with Greg Meredith on the use of categories in computer science: [Name-free combinators for concurrency](https://arxiv.org/abs/1703.07054) and [Representing operational semantics with enriched Lawvere theories](https://arxiv.org/abs/1704.03080). 6) My student Daniel Cicala got invited to the American Mathematical Society conference on [Homotopy Type Theory](https://homotopytypetheory.org/2016/10/04/hott-mrc/) that will take place in Snowbird, Utah on June 4-10.
1) This week Blake and I visited Metron Scientific Solutions. I only visited for a few hours, but Blake stayed for several days, including a hackathon where the Metron team tried to solve a search and rescue problem posed by DARPA!
One thing I did during my short stay was try to sell people on the vision of using operads to design systems 'a bit at a time'. Here 'a bit at a time' is meant in two ways. The basic idea is that a 'system' is an element of some algebra $A$ of some operad $O$. Then:
a) We can design small systems and stick them together using the operations of $O$ to get bigger systems, and
b) We can design systems at a high level of abstraction (meaning: not much detail) and then move to a lower level of abstraction (meaning: add extra details) with the help of a map of operads $O' \to O$. We do this by starting with an element of some algebra $A$ of $O$, and then choosing a way to 'lift' it to an element of some algebra $A'$ of $O'$. Here the primed guys are the ones with more detail.
The second point here is subtler than the first and I'm not really trying to explain it here, so don't feel bad if it doesn't make enough sense!
There were a lot of interesting discussions after I left. Blake summarized them for me... but very very briefly, it seems that:
a) we need to think harder about how to model "levels of abstraction", because this phrase means several different things,
b) we need to think more about how our "operads of communication networks" are connected to Spivak's "operad of wiring diagrams",
c) we should continue thinking about Petri nets. The operad of wiring diagrams has an algebra whose elements are open Petri nets. These have a bit of computational power, at least if we equip the Petri nets with suitable bells and whistles. Thus, we can use them to model many real-world gadgets, which may be handy as we move forwards in this project.
Comment Source:24 April 2017: 1) This week Blake and I visited Metron Scientific Solutions. I only visited for a few hours, but Blake stayed for several days, including a hackathon where the Metron team tried to solve a search and rescue problem posed by DARPA! One thing I did during my short stay was try to sell people on the vision of using operads to design systems 'a bit at a time'. Here 'a bit at a time' is meant in two ways. The basic idea is that a 'system' is an element of some algebra $A$ of some operad $O$. Then: a) We can design small systems and stick them together using the operations of $O$ to get bigger systems, and b) We can design systems at a high level of abstraction (meaning: not much detail) and then move to a lower level of abstraction (meaning: add extra details) with the help of a map of operads $O' \to O$. We do this by starting with an element of some algebra $A$ of $O$, and then choosing a way to 'lift' it to an element of some algebra $A'$ of $O'$. Here the primed guys are the ones with more detail. The second point here is subtler than the first and I'm not really trying to explain it here, so don't feel bad if it doesn't make enough sense! There were a lot of interesting discussions after I left. Blake summarized them for me... but very very briefly, it seems that: a) we need to think harder about how to model "levels of abstraction", because this phrase means several different things, b) we need to think more about how our "operads of communication networks" are connected to Spivak's "operad of wiring diagrams", c) we should continue thinking about Petri nets. The operad of wiring diagrams has an algebra whose elements are open Petri nets. These have a bit of computational power, at least if we equip the Petri nets with suitable bells and whistles. Thus, we can use them to model many real-world gadgets, which may be handy as we move forwards in this project.
1) Kenny Courser passed his oral exam! He gave a talk called A bicategory of decorated cospans, based on his paper with the same title. He survived my questions and also the fact that two members of committee didn't remember to come until he went and grabbed them. (One them was me.)
Congratulations, Kenny!
2) I gave a talk at the Stanford Complexity Group, called Biology as information dynamics. You can see the slides by clicking on the link, and you can also see a video here.
It went a lot better than my similar talk Arizona State University, probably because this audience was more interested in the subject, I covered more ground, and I was more confident.
Marc Harper attended - he came up with some of the math I discussed, and now he's working at Google. So did Vaughn Pratt - the computer scientist who helped come up with Pratt certificates and the Knuth-Morris-Pratt algorithm. He used to be very active on the category theory mailing list, but not much lately.
3) Daniel Cicala put a paper on the arXiv:
Categorifying the zx-calculus.
Abstract. This paper presents a symmetric monoidal and compact closed bicategory that categorifies the zx-calculus developed by Coecke and Duncan. The 1-cells in this bicategory are certain graph morphisms that correspond to the string diagrams of the zx-calculus, while the 2-cells are rewrite rules.
He is submitting this to CALCO, the Conference on Algebra and Coalgebra in Computer Science. After he put it on the arXiv, Duncan asked him to submit it to QPL, the conference on Quantum Physics and Logic run by Duncan, Coecke and others. (Even I am involved in it, slightly, and Jason Erbele spoke there in 2015.) Luckily he can submit an "extended abstract" to QPL and have the best of both worlds.
Comment Source:28 April 2017: 1) Kenny Courser passed his oral exam! He gave a talk called A bicategory of decorated cospans, based on his paper with the same title. He survived my questions and also the fact that two members of committee didn't remember to come until he went and grabbed them. (One them was me.) Congratulations, Kenny! 2) I gave a talk at the Stanford Complexity Group, called [Biology as information dynamics](http://math.ucr.edu/home/baez/bio_asu/). You can see the slides by clicking on the link, and you can also [see a video here](https://www.youtube.com/watch?v=IKetDJof8pk). It went a lot better than my similar talk Arizona State University, probably because this audience was more interested in the subject, I covered more ground, and I was more confident. Marc Harper attended - he came up with some of the math I discussed, and now he's working at Google. So did Vaughn Pratt - the computer scientist who helped come up with Pratt certificates and the Knuth-Morris-Pratt algorithm. He used to be very active on the category theory mailing list, but not much lately. 3) Daniel Cicala put a paper on the arXiv: * [Categorifying the zx-calculus](https://arxiv.org/abs/1704.07034). > **Abstract.** This paper presents a symmetric monoidal and compact closed bicategory that categorifies the zx-calculus developed by Coecke and Duncan. The 1-cells in this bicategory are certain graph morphisms that correspond to the string diagrams of the zx-calculus, while the 2-cells are rewrite rules. He is submitting this to [CALCO](http://coalg.org/mfps-calco2017/), the Conference on Algebra and Coalgebra in Computer Science. After he put it on the arXiv, Duncan asked him to submit it to [QPL](http://qpl.science.ru.nl/), the conference on Quantum Physics and Logic run by Duncan, Coecke and others. (Even I am involved in it, slightly, and Jason Erbele spoke there in 2015.) Luckily he can submit an "extended abstract" to QPL and have the best of both worlds.
3 May 2017:
This week's progress is all about the virtues of initiative: finding opportunities and seizing them.
1) Blake found an interesting ad on MathJobs for a research position at the Bay Area Complexity Institute. The institute is so new that it doesn't even exist yet, but the job description looks like it was written specially for Blake:
You must have a working knowledge of graduate-level category theory, measure theory, and probability theory. Bonus points if you are experienced with any of: statistical mechanics, dynamical systems, Markov chains, various theories and models of computation, organismal evolution, or complexity in any formal setting. You must also enjoy research and writing, and have the personality traits of a theory builder rather than a problem solver, but have the skills of both. Perhaps most importantly, you should be excited and passionate about working on a small team to pursue the connection between information-theoretic complexity, thermodynamics and entropy, and the evolution of complexity in cosmology and living systems.
He applied, and they sound interested! He's going to meet the people running this institute on May 11th, right after his talk to Crutchfield's group at U.C. Davis.
Moral: look around for jobs. Apply for jobs that sound cool.
2) Jason had been invited by Tobias Fritz to a conference at the Perimeter Institute, in Canada. That's where I used to hang out when I worked on quantum gravity - and indeed, my former student Derek Wise is giving a talk at this conference about his wonderful work on Hopf algebras and quantum gauge theories!
One problem: Tobias didn't offer him travel money. An invitation without money is a sad thing. But Jason hustled and overcame this problem:
I initially asked Tobias Fritz and his group if I could get any travel funds through the Perimeter Institute. The answer was a fairly definite "probably not." I tried various avenues at Victor Valley College, where I am currently teaching, all of which came back No. So I bought my plane tickets, reserved a hotel and rental car, etc. and informed Lucy Zhang, the person in Tobias' group with whom I have been primarily in contact. Two days later I had a sponsor: Daniel Gottesman. So I cancelled my hotel and rental car reservations, the cost of the plane tickets will be reimbursed, the entry fee is waived, and more. I think the only thing that isn't covered is parking at LAX. There is still some time, so I am looking into alternatives to that.
Daniel Gottesman is a bigshot in quantum computation at the Perimeter Institute.
Moral: it pays to ask for help - you just might get it!
3) As for me, I'm not doing anything useful - just hanging out in Hong Kong, checking out temples. But I got here because I wrote about Guowu Meng's amazing work connecting special relativity to Newton's inverse square force law... and he's here in Hong Kong, so he invited me here!
Moral: blog about cool stuff. If you explain it well, people will pay attention.
Comment Source:3 May 2017: This week's progress is all about the virtues of initiative: finding opportunities and seizing them. 1) Blake found an interesting ad on MathJobs for a [research position at the Bay Area Complexity Institute](https://www.mathjobs.org/jobs/jobs/10174). The institute is so new that it doesn't even exist yet, but the job description looks like it was written specially for Blake: > You must have a working knowledge of graduate-level category theory, measure theory, and probability theory. Bonus points if you are experienced with any of: statistical mechanics, dynamical systems, Markov chains, various theories and models of computation, organismal evolution, or complexity in any formal setting. You must also enjoy research and writing, and have the personality traits of a theory builder rather than a problem solver, but have the skills of both. Perhaps most importantly, you should be excited and passionate about working on a small team to pursue the connection between information-theoretic complexity, thermodynamics and entropy, and the evolution of complexity in cosmology and living systems. He applied, and they sound interested! He's going to meet the people running this institute on May 11th, right after his talk to Crutchfield's group at U.C. Davis. **Moral: look around for jobs. Apply for jobs that sound cool.** 2) Jason had been invited by Tobias Fritz to a conference at the Perimeter Institute, in Canada. That's where I used to hang out when I worked on quantum gravity - and indeed, my former student Derek Wise is giving a talk at this conference about his [wonderful work on Hopf algebras and quantum gauge theories](https://arxiv.org/abs/1512.03966)! One problem: Tobias didn't offer him travel money. An invitation without money is a sad thing. But Jason hustled and overcame this problem: > I initially asked Tobias Fritz and his group if I could get any travel funds through the Perimeter Institute. The answer was a fairly definite "probably not." I tried various avenues at Victor Valley College, where I am currently teaching, all of which came back No. So I bought my plane tickets, reserved a hotel and rental car, etc. and informed Lucy Zhang, the person in Tobias' group with whom I have been primarily in contact. Two days later I had a sponsor: Daniel Gottesman. So I cancelled my hotel and rental car reservations, the cost of the plane tickets will be reimbursed, the entry fee is waived, and more. I think the only thing that isn't covered is parking at LAX. There is still some time, so I am looking into alternatives to that. Daniel Gottesman is a bigshot in quantum computation at the Perimeter Institute. **Moral: it pays to ask for help - you just might get it!** 3) As for me, I'm not doing anything useful - just hanging out in Hong Kong, checking out temples. But I got here because I wrote about [Guowu Meng's amazing work connecting special relativity to Newton's inverse square force law](http://math.ucr.edu/home/baez/gravitational.html)... and he's here in Hong Kong, so he invited me here! **Moral: blog about cool stuff. If you explain it well, people will pay attention.**
1) You may recall that Daniel Cicala got into the Kan Extension Seminar, which is an advanced online course on category theory run by Emily Riehl, Alexander Campbell and our very own Brendan Fong. Participants read famous papers on category theory, discuss them and blog about them. Daniel recently blogged about the paper he read:
A discussion on notions of Lawvere theory.
2) Blake got offered a 3-month internship at the Princeton branch of Siemens, a big engineering firm! The advertisement was pretty interesting:
PhD Intern – Next-Generation Engineering with Category Theory and Sheaves
Develop concepts for unifying different engineering formalisms that describe different domains and aspects of complex systems using Category Theory.
Implement the developed concepts in software prototypes to demonstrate novel languages and algorithms to represent and extract knowledge across engineering disciplines.
Investigate the use of Sheaves for the creation of hybrid engineering models that combine data-driven and physics-based representations.
Participate in the preparation of scientific publications.
Required Knowledge/Skills, Education, and Experience
PhD student in Mathematics or Physics.
Programming skills in Python and/or JavaScript and ability to quickly prototype in these languages.
Good written and verbal communication skills in English are required, as well as excellent interpersonal skills in multi-cultural environments.
Team player who can also be independent, prioritize work and thrive in a fast-paced dynamic environment.
The successful candidate must be able to work with controlled technology in accordance with US Export Control Law.
The guy in charge is named Arquimedes Canedo, and he seems potentially interested in hiring Blake or other applied category theory people (hint hint!) for a longer-term project.
Comment Source:6 May 2017: 1) You may recall that Daniel Cicala got into the [Kan Extension Seminar](https://golem.ph.utexas.edu/category/2016/10/the_kan_extension_seminar_retu.html), which is an advanced online course on category theory run by Emily Riehl, Alexander Campbell and our very own Brendan Fong. Participants read famous papers on category theory, discuss them and blog about them. Daniel recently blogged about the paper he read: * [A discussion on notions of Lawvere theory](https://golem.ph.utexas.edu/category/2017/05/a_discussion_on_notions_of_law.html). 2) Blake got offered a 3-month internship at the Princeton branch of Siemens, a big engineering firm! The [advertisement](https://jobs.siemens-info.com/jobs/207833/PhD+Intern+%E2%80%93+Next-Generation+Engineering+with+Category+Theory+and+Sheaves?lang=en-US) was pretty interesting: > **PhD Intern – Next-Generation Engineering with Category Theory and Sheaves** > **Responsibilities** > * Develop concepts for unifying different engineering formalisms that describe different domains and aspects of complex systems using Category Theory. > * Implement the developed concepts in software prototypes to demonstrate novel languages and algorithms to represent and extract knowledge across engineering disciplines. > * Investigate the use of Sheaves for the creation of hybrid engineering models that combine data-driven and physics-based representations. > * Participate in the preparation of scientific publications. > **Required Knowledge/Skills, Education, and Experience** > * PhD student in Mathematics or Physics. > * Programming skills in Python and/or JavaScript and ability to quickly prototype in these languages. > * Good written and verbal communication skills in English are required, as well as excellent interpersonal skills in multi-cultural environments. > * Team player who can also be independent, prioritize work and thrive in a fast-paced dynamic environment. > * The successful candidate must be able to work with controlled technology in accordance with US Export Control Law. The guy in charge is named [Arquimedes Canedo](https://scholar.google.com/citations?user=r2zo9HQAAAAJ&hl=en), and he seems potentially interested in hiring Blake or other applied category theory people (hint hint!) for a longer-term project.
WebHubTel
That first paper by Canedo is context-modeling, which is something we worked on via a DARPA project a few years ago
https://www.researchgate.net/publication/283579370_C2M2L_Final_Report
Basically covering the same territory and I recognize all the players
Comment Source:That first paper by Canedo is context-modeling, which is something we worked on via a DARPA project a few years ago https://www.researchgate.net/publication/283579370_C2M2L_Final_Report Basically covering the same territory and I recognize all the players
Interesting, WebHubTel! I will tell my student Blake Pollard about this, since he is going to do that 3-month internship with Canedo this summer.
Comment Source:Interesting, WebHubTel! I will tell my student Blake Pollard about this, since he is going to do that 3-month internship with Canedo this summer.
Lots of good news:
1) Brandon Coya got a "dissertation year fellowship" which offers him $7200 of funding for one quarter next year. All of you UCR grad students should apply for this at the appropriate time.
2) Blake also got one! He'll use it this summer.
3) I gave a plenary talk at the Hong Kong Mathematical Society, on the dodecahedron, the icosahedron and E8. Check it out - it's fun! I'm coming back to the US now.
Comment Source:28 May 2017: Lots of good news: 1) Brandon Coya got a "dissertation year fellowship" which offers him $7200 of funding for one quarter next year. All of you UCR grad students should apply for this at the appropriate time. 2) Blake also got one! He'll use it this summer. 3) I gave a plenary talk at the Hong Kong Mathematical Society, on [the dodecahedron, the icosahedron and E8](https://johncarlosbaez.wordpress.com/2017/05/16/the-dodecahedron-the-icosahedron-and-e8/). Check it out - it's fun! I'm coming back to the US now.
2 June 2017:
More good news:
1) Adam Yassine passed his oral exam, speaking on "Open systems in classical mechanics".
2) Blake Pollard passed his thesis defense, speaking on "Open Markov processes and reaction networks".
Comment Source:2 June 2017: More good news: 1) Adam Yassine passed his oral exam, speaking on "Open systems in classical mechanics". 2) Blake Pollard passed his thesis defense, speaking on "[Open Markov processes and reaction networks](http://math.ucr.edu/home/baez/thesis_defense_pollard.pdf)".
1) Kenny Courser's paper A bicategory of decorated cospans got accepted for publication in Theory and Applications of Categories!
He had to go through a couple rounds of revisions demanded by the referee. In the last round, the referee demanded that Kenny explain the difference between his use of double categories and Lerman and Spivak's use of double categories to study dynamical systems. Kenny wrote and explanation and sent a new version of his paper to the editor. He didn't hear back for many weeks. Finally, it turned out that the editor had never sent the new version to the referee! When Kenny discovered this, the matter was quickly resolved.
Moral: if you sent an email to a journal editor and don't hear back, it's possible they haven't read your email, or forgot to do anything about it. Don't be too shy to politely ask them what's up.
2) Daniel Cicala's paper on Categorifying the zx-calculus, rejected by CALCO, was accepted by QPL.
In computer science conference papers are considered more important than papers in journals - the opposite from math. It really helps your career to get papers accepted by important conferences. CALCO is the Category on Algebra and Coalgebra in Computer Science. QPL is Quantum Physics and Logic. Daniel's paper is a good fit for QPL since that conference is organized by people like Bob Coecke and Ross Duncan, who helped develop the zx-calculus (a diagrammatic method for dealing with certain categories that come up in quantum computation). So, if you write any sort of paper on diagrammatic methods for dealing with categories that show up in physics or engineering - signal flow diagrams, bond graphs, electrical circuits, etc. - you should consider submitting it to QPL. It comes around once every summer, but papers can be submitted earlier.
(I helped referee a bunch of papers for QPL, but I'm not allowed to referee papers by my own students.)
3) Daniel Cicala is now at Snowbird, Utah, learning about homotopy type theory.
It turns out two of the organizers of this workshop are friends of mine - Mike Shulman and Dan Christensen. Furthermore, they helped me answer a question that came up in a paper Daniel and Kenny are writing: is any functor from a groupoid to itself equivalent to an isofibration? It turns out the answer is yes and that this was not previously known (at least not by them).
Comment Source:9 June 2017: More good news: 1) Kenny Courser's paper [A bicategory of decorated cospans](https://arxiv.org/abs/1605.08100) got accepted for publication in _Theory and Applications of Categories_! He had to go through a couple rounds of revisions demanded by the referee. In the last round, the referee demanded that Kenny explain the difference between his use of double categories and Lerman and Spivak's use of double categories to study dynamical systems. Kenny wrote and explanation and sent a new version of his paper to the editor. He didn't hear back for many weeks. Finally, it turned out that the editor had never sent the new version to the referee! When Kenny discovered this, the matter was quickly resolved. Moral: if you sent an email to a journal editor and don't hear back, it's possible they haven't read your email, or forgot to do anything about it. Don't be too shy to politely ask them what's up. 2) Daniel Cicala's paper on [Categorifying the zx-calculus](https://arxiv.org/abs/1704.07034), rejected by CALCO, was accepted by QPL. In computer science conference papers are considered more important than papers in journals - the opposite from math. It really helps your career to get papers accepted by important conferences. [CALCO](http://coalg.org/mfps-calco2017/cfp-calco.html) is the Category on Algebra and Coalgebra in Computer Science. [QPL](http://qpl.science.ru.nl/) is Quantum Physics and Logic. Daniel's paper is a good fit for QPL since that conference is organized by people like Bob Coecke and Ross Duncan, who helped develop the zx-calculus (a diagrammatic method for dealing with certain categories that come up in quantum computation). So, if you write any sort of paper on diagrammatic methods for dealing with categories that show up in physics or engineering - signal flow diagrams, bond graphs, electrical circuits, etc. - you should consider submitting it to QPL. It comes around once every summer, but papers can be submitted earlier. (I helped referee a bunch of papers for QPL, but I'm not allowed to referee papers by my own students.) 3) Daniel Cicala is now at Snowbird, Utah, learning about [homotopy type theory](https://homotopytypetheory.org/2016/10/04/hott-mrc/). It turns out two of the organizers of this workshop are friends of mine - Mike Shulman and Dan Christensen. Furthermore, they helped me answer a question that came up in a paper Daniel and Kenny are writing: is any functor from a groupoid to itself equivalent to an isofibration? It turns out the answer is yes and that this was not previously known (at least not by them).
14 June 2017:
1) It looks like we'll be having a new member on our team this fall! Christina Vasilakopoulou is being offered a visiting assistant professorship at UCR, and seems likely to accept. She was a student of Martin Hyland, the category theorist at Cambridge University who also advised Tom Leinster, Eugenia Cheng and Aaron Lauda (who worked with me when he was an undergrad at UCR). Since then she's worked with various people including David Spivak. She's done a mix of "pure" category theory (applied to algebra, actually) and "applied" category theory such as this:
David I. Spivak, Christina Vasilakopoulou, Patrick Schultz, Dynamical systems and sheaves.
Patrick Schultz, David I. Spivak, Christina Vasilakopoulou, Ryan Wisnesky, Algebraic databases.
I'm sure she'll bring a lot of energy and new ideas to our team.
2) Blake and I gave talks here:
Dynamics, Thermodynamics and Information Processing in Chemical Networks, 13-16 June 2017, Complex Systems and Statistical Mechanics Group, University of Luxembourg. Organized by Massimiliano Esposito and Matteo Polettini.
You can see my talk slides here:
The mathematics of open reaction networks.
There are a lot of interesting people here, including Luca Peliti, who talked about an analogy I'm really interested in:
Luca Peliti, On the value of information in gambling, evolution and thermodynamics.
Abstract. The connection between the information value of a message and capital gain was made by Kelly in 1953. In 1965 Kimura tried to evaluate the rate of information intake by a population undergoing Darwinian evolution by equating it with the substitutional load. Recently, the analogy between Kelly's scheme and work extraction was pointed out in the context of stochastic thermodynamics. I shall try to connect these threads, highlighting analogies and differences between the meaning of information and its value in the different contexts.
and Hong Qian - Blake kept running into his work while working on his thesis:
Hong Qian, The mathematical foundation of a landscape theory for living matter and life.
Abstract. The physicists' notion of energy is derived from Newtonian mechanics. The theory of thermodynamics is developed based on that notion, and the realization of mechanical energy dissipation in terms of heat. Since the work of L. Boltzmann, who trusted that atoms were real as early as in 1884, the heat became intimately related to the stochastic motion of the invisible atoms and molecules. In this talk, starting from a stochastic description of a class of rather general dynamics that is not limited to mechanics, we show a notion of energy can be derived mathematically, in the limit of vanishing stochasticity, based on the Kullback-Leibler divergence, or relative entropy associated with the stochastic, Markov processes. With the emergent notion of an energy function, e.g., "landscape", a mathematical structure inherent to the stochastic dynamics, which is akin to thermodynamics, is revealed. This analysis implies that an abstract "mathematicothermodynamics" structure exists, and can be formulated, for dynamics of complex systems independent of classical thermal physics, for example, in ecology.
I've gotten a bunch of ideas for new projects, which I'm listing in a notebook.
Comment Source:14 June 2017: 1) It looks like we'll be having a new member on our team this fall! [Christina Vasilakopoulou](https://arxiv.org/find/math/1/au:+Vasilakopoulou_C/0/1/0/all/0/1) is being offered a visiting assistant professorship at UCR, and seems likely to accept. She was a student of Martin Hyland, the category theorist at Cambridge University who also advised Tom Leinster, Eugenia Cheng and Aaron Lauda (who worked with me when he was an undergrad at UCR). Since then she's worked with various people including David Spivak. She's done a mix of "pure" category theory (applied to algebra, actually) and "applied" category theory such as this: * David I. Spivak, Christina Vasilakopoulou, Patrick Schultz, [Dynamical systems and sheaves](https://arxiv.org/abs/1609.08086). * Patrick Schultz, David I. Spivak, Christina Vasilakopoulou, Ryan Wisnesky, [Algebraic databases](https://arxiv.org/abs/1602.03501). I'm sure she'll bring a lot of energy and new ideas to our team. 2) Blake and I gave talks here: * [Dynamics, Thermodynamics and Information Processing in Chemical Networks](https://luxcnworkshop.wordpress.com/), 13-16 June 2017, Complex Systems and Statistical Mechanics Group, University of Luxembourg. Organized by Massimiliano Esposito and Matteo Polettini. You can see my talk slides here: * [The mathematics of open reaction networks](http://math.ucr.edu/home/baez/networks_luxembourg/open_reaction_networks_web.pdf). There are a lot of interesting people here, including Luca Peliti, who talked about an analogy I'm really interested in: * [Luca Peliti](http://www.peliti.org/), On the value of information in gambling, evolution and thermodynamics. > **Abstract.** The connection between the information value of a message and capital gain was made by Kelly in 1953. In 1965 Kimura tried to evaluate the rate of information intake by a population undergoing Darwinian evolution by equating it with the substitutional load. Recently, the analogy between Kelly's scheme and work extraction was pointed out in the context of stochastic thermodynamics. I shall try to connect these threads, highlighting analogies and differences between the meaning of information and its value in the different contexts. and Hong Qian - Blake kept running into his work while working on his thesis: * [Hong Qian](https://arxiv.org/find/math-ph/1/au:+Qian_H/0/1/0/all/0/1), The mathematical foundation of a landscape theory for living matter and life. > **Abstract.** The physicists' notion of energy is derived from Newtonian mechanics. The theory of thermodynamics is developed based on that notion, and the realization of mechanical energy dissipation in terms of heat. Since the work of L. Boltzmann, who trusted that atoms were real as early as in 1884, the heat became intimately related to the stochastic motion of the invisible atoms and molecules. In this talk, starting from a stochastic description of a class of rather general dynamics that is not limited to mechanics, we show a notion of energy can be derived mathematically, in the limit of vanishing stochasticity, based on the Kullback-Leibler divergence, or relative entropy associated with the stochastic, Markov processes. With the emergent notion of an energy function, e.g., "landscape", a mathematical structure inherent to the stochastic dynamics, which is akin to thermodynamics, is revealed. This analysis implies that an abstract "mathematicothermodynamics" structure exists, and can be formulated, for dynamics of complex systems independent of classical thermal physics, for example, in ecology. I've gotten a bunch of ideas for new projects, which I'm listing in a notebook.
Here is this week's progress, as far as I know:
1) I'm visiting the University of Genoa. It's home to 3 well-known category theorists:
Marco Grandis (who works on double and n-tuple categories),
Giuseppe Rosolini (who does functorial semantics for programming languages) and
Eugenio Moggi (the guy who introduced monads in computer science - a big deal in Haskell and some other languages).
Marco Grandis told me something very interesting. His advisor, Gabriele Darbo, introduced a "theory of devices" in 1970, based on the category of corelations! He applied it to linear electrical circuits using the "add currents, duplicate voltages" rule. All this is VERY similar to Brendan's thesis work and also some of my work with Blake. But it's also different!
Yesterday I summarized the ideas here:
The theory of devices.
I haven't had time yet to think hard about how his formalism connects to ours. Darbo's work was mostly ignored, perhaps because it's all in Italian. I think we can still learn something from it, even though we've gone further.
2) Today I gave a general talk for the math and science faculty here at Genoa:
Tales of the dodecahedron: from Pythagoras to Plato to Poincaré.
3) I also gave a talk to the math department:
Applied category theory.
Comment Source:22 June 2017: Here is this week's progress, as far as I know: 1) I'm visiting the University of Genoa. It's home to 3 well-known category theorists: * Marco Grandis (who works on double and n-tuple categories), * Giuseppe Rosolini (who does functorial semantics for programming languages) and * Eugenio Moggi (the guy who introduced monads in computer science - a big deal in Haskell and some other languages). Marco Grandis told me something very interesting. His advisor, Gabriele Darbo, introduced a "theory of devices" in 1970, based on the category of corelations! He applied it to linear electrical circuits using the "add currents, duplicate voltages" rule. All this is VERY similar to Brendan's thesis work and also some of my work with Blake. But it's also different! Yesterday I summarized the ideas here: * [The theory of devices](https://johncarlosbaez.wordpress.com/2017/06/20/the-theory-of-devices/). I haven't had time yet to think hard about how his formalism connects to ours. Darbo's work was mostly ignored, perhaps because it's all in Italian. I think we can still learn something from it, even though we've gone further. 2) Today I gave a general talk for the math and science faculty here at Genoa: * [Tales of the dodecahedron: from Pythagoras to Plato to Poincaré](http://math.ucr.edu/home/baez/dodecahedron/genoa_talk/1.html). 3) I also gave a talk to the math department: * [Applied category theory](http://math.ucr.edu/home/baez/control/applied_category_theory.pdf).
1) Daniel Cicala submitted an abstract for my AMS special session on Applied Category Theory:
A bicategorical syntax for pure state qubit quantum mechanics
Abstract. We begin by constructing a framework used to study open networks modeled by graphs and their rewritings. This consists of a symmetric monoidal compact closed bicategory built by combining spans and cospans inside a topos. Into this bicategorical framework, we fit Coecke and Duncan's zx-calculus, a graphical language used to reason about pure state qubit quantum mechanics. After viewing the zx-calculus through this lens, we highlight several benefits over the 1-categorical approach: the presence of a symmetric monoidal compact closed structure and a better representation of rewriting information. (Received June 22, 2017)
2) So did Brendan Fong:
Black boxes and decorated corelations
Abstract. Consider an electric circuit. Suppose this circuit has chosen terminals, which we may connect with the terminals of another circuit. That is to say, consider that we may compose two circuits to obtain another circuit. This suggests we might model circuits as morphisms in a category. Next, suppose I want to compose a circuit with a resistor of resistance 2 ohms. If I have no such resistors, I could substitute with a pair of 1 ohm resistors in series. This suggests a coarser representation of circuits, one that keeps track of only how the circuit behaves, and not their [its] constituent components. In this talk I shall introduce decorated corelations as a tool for constructing categories that model circuits, and constructing 'black box' functors that shift between these models. This framework is applicable not only to circuits, but to open systems in general. (Received June 23, 2017)
Comment Source:24 June 2017: 1) Daniel Cicala submitted an abstract for my AMS special session on Applied Category Theory: * A bicategorical syntax for pure state qubit quantum mechanics > **Abstract.** We begin by constructing a framework used to study open networks modeled by graphs and their rewritings. This consists of a symmetric monoidal compact closed bicategory built by combining spans and cospans inside a topos. Into this bicategorical framework, we fit Coecke and Duncan's zx-calculus, a graphical language used to reason about pure state qubit quantum mechanics. After viewing the zx-calculus through this lens, we highlight several benefits over the 1-categorical approach: the presence of a symmetric monoidal compact closed structure and a better representation of rewriting information. (Received June 22, 2017) 2) So did Brendan Fong: * Black boxes and decorated corelations > **Abstract.** Consider an electric circuit. Suppose this circuit has chosen terminals, which we may connect with the terminals of another circuit. That is to say, consider that we may compose two circuits to obtain another circuit. This suggests we might model circuits as morphisms in a category. Next, suppose I want to compose a circuit with a resistor of resistance 2 ohms. If I have no such resistors, I could substitute with a pair of 1 ohm resistors in series. This suggests a coarser representation of circuits, one that keeps track of only how the circuit behaves, and not their [its] constituent components. In this talk I shall introduce decorated corelations as a tool for constructing categories that model circuits, and constructing 'black box' functors that shift between these models. This framework is applicable not only to circuits, but to open systems in general. (Received June 23, 2017)
More abstracts for our AMS special session on Applied Category Theory on November 4-5.
1) Kenny Courser:
A bicategory of coarse-grained Markov processes.
Abstract. If C is a category with finite colimits, D is a symmetric monoidal category and F is a lax symmetric monoidal functor from C to D, Fong has developed a theory of F-decorated cospans which are suitable for representing open dynamical systems. Indeed, Fong has shown the existence of a symmetric monoidal category consisting of objects of C and isomorphism classes of F-decorated cospans in C as morphisms. One application of this result is given by Baez, Fong and Pollard in which they construct a symmetric monoidal category whose morphisms are given by isomorphism classes of open Markov processes. Using a result of Shulman, we present a symmetric monoidal bicategory consisting of finite sets as objects,open Markov processes as morphisms and coarse-grainings of open Markov processes as 2-morphisms. (Received June 25, 2017)
2) Adam Yassine:
Open systems in classical mechanics
Abstract. Using the framework of category theory, we formalize the heuristic principles that physicists employ in constructing the Hamiltonians for open classical systems as sums of Hamiltonians of subsystems. First we construct a category where the objects are symplectic manifolds and the morphisms are spans whose legs are surjective Poisson maps. Using a slight variant of Fong's theory of "decorated" cospans, we then decorate the apices of our spans with Hamiltonians. This gives a category where morphisms are open classical systems, and composition allows us to build these systems from smaller pieces. (Received June 26, 2017)
Keep 'em coming! I'm still hoping that Brandon, Joseph and Christina will submit abstracts. John Foley may submit one too: it would be great to have him give a talk about how our Metron project is using operads for "compositional tasking". This would be a nice sequel to a talk by Joseph on the underlying math - namely, "network models" and the operads they give rise to.
Comment Source:27 June 2017: More abstracts for our AMS special session on Applied Category Theory on November 4-5. 1) Kenny Courser: * A bicategory of coarse-grained Markov processes. > **Abstract.** If C is a category with finite colimits, D is a symmetric monoidal category and F is a lax symmetric monoidal functor from C to D, Fong has developed a theory of F-decorated cospans which are suitable for representing open dynamical systems. Indeed, Fong has shown the existence of a symmetric monoidal category consisting of objects of C and isomorphism classes of F-decorated cospans in C as morphisms. One application of this result is given by Baez, Fong and Pollard in which they construct a symmetric monoidal category whose morphisms are given by isomorphism classes of open Markov processes. Using a result of Shulman, we present a symmetric monoidal bicategory consisting of finite sets as objects,open Markov processes as morphisms and coarse-grainings of open Markov processes as 2-morphisms. (Received June 25, 2017) 2) Adam Yassine: * Open systems in classical mechanics > **Abstract.** Using the framework of category theory, we formalize the heuristic principles that physicists employ in constructing the Hamiltonians for open classical systems as sums of Hamiltonians of subsystems. First we construct a category where the objects are symplectic manifolds and the morphisms are spans whose legs are surjective Poisson maps. Using a slight variant of Fong's theory of "decorated" cospans, we then decorate the apices of our spans with Hamiltonians. This gives a category where morphisms are open classical systems, and composition allows us to build these systems from smaller pieces. (Received June 26, 2017) Keep 'em coming! I'm still hoping that Brandon, Joseph and Christina will submit abstracts. John Foley may submit one too: it would be great to have him give a talk about how our Metron project is using operads for "compositional tasking". This would be a nice sequel to a talk by Joseph on the underlying math - namely, "network models" and the operads they give rise to.
I'm way behind on these, but I'll continue where I left off:
We're getting more abstracts for our AMS special session on Applied Category Theory on November 4-5! I'm happy to announce that David Spivak has submitted one, because he generally doesn't like to travel.
1) Joseph Moeller submitted one on his work with the Metron project:
Operads for modeling networks
Abstract. A network is a complex of interacting systems which can often be represented as a graph equipped with extra structure. Networks can be combined in many ways, including by overlaying one on top of the other or sitting one next to another. We introduce network models - which are formally a simple kind of lax symmetric monoidal functor - to encode these ways of combining networks. By applying a general construction to network models, we obtain operads for the design of complex networked systems. (Received June 29, 2017)
2) Christina Vasilakopoulou submitted on on her work with David Spivak and Patrick Schultz:
Abstract dynamical systems
Abstract. We describe a categorical framework of modeling and analyzing systems in a broad sense. The latter can be thought of as 'machines' with inputs and outputs, carrying some sort of signal that occurs through some notion of time; special cases include discrete and continuous dynamical systems. Modeling them as algebras for the wiring diagram operad, a central goal is to understand the behavior of composite systems, formed as arbitrary interconnections of component subsystems. This shall be accomplished using lax monoidal functors, which provide a coherent formalization of systems, as well as sheaf theory, which captures the crucial notion of time. (Received June 29, 2017)
3) David Spivak submitted one:
A higher-order temporal logic for dynamical systems
Abstract. We consider a very general class of dynamical systems—including discrete, continuous, hybrid, deterministic, non- deterministic, etc.—based on sheaves. We call these sheaves behavior types: they tell us the set of possible behaviors over any interval of time. A machine can be construed as a wide span of such sheaves, and these machines can be composed as morphisms in a hypergraph category. The topos of sheaves has an internal language, which we use as a new sort of higher-order internal logic for talking about behaviors. We can use this logic to prove properties about a composite system of systems from properties of the parts and how they are wired together. (Received June 28, 2017)
4) A third-year statistics grad student at Stanford named Evan Patterson submitted an interesting one:
Knowledge representation in bicategories of relations
Abstract. We introduce the relational ontology log, or relational olog, a categorical framework for knowledge representation based on the category of sets and relations. It is inspired by Spivak and Kent's olog, a knowledge representation system based on the category of sets and functions. Relational ologs interpolate between ologs and description logic, the dominant formalism for knowledge representation today. On a practical level, we demonstrate that relational ologs have an intuitive yet fully precise graphical syntax, derived from the string diagrams of monoidal categories. We explain several other useful features of relational ologs not possessed by most description logics, such as a type system and a rich, flexible notion of instance data. In a more theoretical vein, we draw on categorical logic to show how relational ologs can be translated to and from logical theories in a fragment of first-order logic. (Received June 29, 2017)
On other news, I got back to Riverside yesterday! I have some big news to report from my visit to Turin, but I'll report that separately. If anyone at UCR wants to talk to me in person, I'll be here until around June 14th - set up an appointment! We've got a lot of projects going that are worth talking about. I'm especially eager to accelerate Kenny's work on coarse-graining Markov processes, and help Brandon finish his increasingly deep and interesting paper on bond graphs, and help Joseph finish a paper on network models, and get him started on a paper on compositional tasking.
Comment Source:I'm way behind on these, but I'll continue where I left off: 30 June 2017 We're getting more abstracts for our [AMS special session on Applied Category Theory](http://math.ucr.edu/home/baez/ACT2017) on November 4-5! I'm happy to announce that David Spivak has submitted one, because he generally doesn't like to travel. 1) Joseph Moeller submitted one on his work with the Metron project: * Operads for modeling networks > **Abstract.** A network is a complex of interacting systems which can often be represented as a graph equipped with extra structure. Networks can be combined in many ways, including by overlaying one on top of the other or sitting one next to another. We introduce network models - which are formally a simple kind of lax symmetric monoidal functor - to encode these ways of combining networks. By applying a general construction to network models, we obtain operads for the design of complex networked systems. (Received June 29, 2017) > 2) Christina Vasilakopoulou submitted on on her work with David Spivak and Patrick Schultz: * Abstract dynamical systems > **Abstract.** We describe a categorical framework of modeling and analyzing systems in a broad sense. The latter can be thought of as 'machines' with inputs and outputs, carrying some sort of signal that occurs through some notion of time; special cases include discrete and continuous dynamical systems. Modeling them as algebras for the wiring diagram operad, a central goal is to understand the behavior of composite systems, formed as arbitrary interconnections of component subsystems. This shall be accomplished using lax monoidal functors, which provide a coherent formalization of systems, as well as sheaf theory, which captures the crucial notion of time. (Received June 29, 2017) 3) David Spivak submitted one: * A higher-order temporal logic for dynamical systems > **Abstract.** We consider a very general class of dynamical systems—including discrete, continuous, hybrid, deterministic, non- deterministic, etc.—based on sheaves. We call these sheaves behavior types: they tell us the set of possible behaviors over any interval of time. A machine can be construed as a wide span of such sheaves, and these machines can be composed as morphisms in a hypergraph category. The topos of sheaves has an internal language, which we use as a new sort of higher-order internal logic for talking about behaviors. We can use this logic to prove properties about a composite system of systems from properties of the parts and how they are wired together. (Received June 28, 2017) 4) A third-year statistics grad student at Stanford named Evan Patterson submitted an interesting one: * Knowledge representation in bicategories of relations > **Abstract.** We introduce the relational ontology log, or relational olog, a categorical framework for knowledge representation based on the category of sets and relations. It is inspired by Spivak and Kent's olog, a knowledge representation system based on the category of sets and functions. Relational ologs interpolate between ologs and description logic, the dominant formalism for knowledge representation today. On a practical level, we demonstrate that relational ologs have an intuitive yet fully precise graphical syntax, derived from the string diagrams of monoidal categories. We explain several other useful features of relational ologs not possessed by most description logics, such as a type system and a rich, flexible notion of instance data. In a more theoretical vein, we draw on categorical logic to show how relational ologs can be translated to and from logical theories in a fragment of first-order logic. (Received June 29, 2017) On other news, I got back to Riverside yesterday! I have some big news to report from my visit to Turin, but I'll report that separately. If anyone at UCR wants to talk to me in person, I'll be here until around June 14th - set up an appointment! We've got a lot of projects going that are worth talking about. I'm especially eager to accelerate Kenny's work on coarse-graining Markov processes, and help Brandon finish his increasingly deep and interesting paper on bond graphs, and help Joseph finish a paper on network models, and get him started on a paper on compositional tasking.
Three pieces of news. The third one is a big deal for me.
1) Brandon has submitted an abstract for our AMS special session on the weekend of November 4-5. ) * Frobenius monoids, weak bimonoids, and corelations
Abstract. In this talk we consider object 2 in the category FinCorel, whose objects are finite sets and whose morphisms are "corelations." The object 2 can be equipped with two different Frobenius monoid structures. We show that the two Frobenius monoids interact to form a "weak bimonoid" as defined by Pastro and Street. Baez and Fong have shown that FinCorel is useful for modeling circuits made of wire as morphisms in a category. In this analogy the object 1 is viewed as a single wire. We show how the two Frobenius monoids associated to the object 2 relate to placing pairs of wires into series and parallel connections. (Received July 06, 2017)
2) Kenny and Daniel have finished writing a paper and put it on the arXiv.
Bicategories of spans and cospans
Abstract. If C is a category with chosen pullbacks and a terminal object then, using a result of Shulman, we obtain a fully dualizable and symmetric monoidal bicategory Sp(Sp(C))} whose objects are those of C whose morphisms are spans in C, and whose 2-morphisms are isomorphism classes of spans of spans in C. If C is a topos, the first author has previously constructed a bicategory MonicSp(Csp(C)) whose objects are those of C, whose morphisms are cospans in C, and whose 2-morphisms are isomorphism classes of spans of cospans in C with monic legs. We prove this bicategory is also symmetric monoidal and even compact closed. We discuss applications of such bicategories to graph rewriting as well as to Morton and Vicary's combinatorial approach to Khovanov's categorified Heisenberg algebra.
3) At the end of my trip to Europe I spent a week in Turin at the Institute for Scientific Interchange. This is a research institute with about 56 scientists run by Mario Rasetti, a physicist who used to do statistical mechanics at the Institute for Advanced Studies. 30 years ago he went back to Italy to set up a roughly similar institute in his home town. They study complex networks and data science.
He's been reading our work and liking it. He asked me to set up a mathematics group there!
If I do this, I'll be hiring a number of postdocs to work on networks, applied category theory and the like. That could be some of you - though of course I don't want to hire just my own students; I want to hire the best people I can find. There should also be opportunities for shorter visits, like conferences and workshops, which some of you might attend.
If I do this I won't be giving up my position at UCR, at least not soon. Instead, I'll visit the place repeatedly, for example during summers. Turin is a wonderful city, so I like this idea. But mainly I'm excited at the idea of being able to put together a team of people working on the kinds of math I like.
It's not 100% certain this will happen: I need to decide I really want to do it, and write a proposal, which the board of the ISI will then read, etc.
Comment Source:7 July 2017: Three pieces of news. The third one is a big deal for me. 1) Brandon has submitted an abstract for our [AMS special session](http://math.ucr.edu/home/baez/ACT2017) on the weekend of November 4-5. ) * Frobenius monoids, weak bimonoids, and corelations > **Abstract.** In this talk we consider object 2 in the category FinCorel, whose objects are finite sets and whose morphisms are "corelations." The object 2 can be equipped with two different Frobenius monoid structures. We show that the two Frobenius monoids interact to form a "weak bimonoid" as defined by Pastro and Street. Baez and Fong have shown that FinCorel is useful for modeling circuits made of wire as morphisms in a category. In this analogy the object 1 is viewed as a single wire. We show how the two Frobenius monoids associated to the object 2 relate to placing pairs of wires into series and parallel connections. (Received July 06, 2017) 2) Kenny and Daniel have finished writing a paper and put it on the arXiv. * [Bicategories of spans and cospans](https://arxiv.org/abs/1707.02098v1) > **Abstract.** If C is a category with chosen pullbacks and a terminal object then, using a result of Shulman, we obtain a fully dualizable and symmetric monoidal bicategory Sp(Sp(C))} whose objects are those of C whose morphisms are spans in C, and whose 2-morphisms are isomorphism classes of spans of spans in C. If C is a topos, the first author has previously constructed a bicategory MonicSp(Csp(C)) whose objects are those of C, whose morphisms are cospans in C, and whose 2-morphisms are isomorphism classes of spans of cospans in C with monic legs. We prove this bicategory is also symmetric monoidal and even compact closed. We discuss applications of such bicategories to graph rewriting as well as to Morton and Vicary's combinatorial approach to Khovanov's categorified Heisenberg algebra. 3) At the end of my trip to Europe I spent a week in Turin at the Institute for Scientific Interchange. This is a research institute with about 56 scientists run by Mario Rasetti, a physicist who used to do statistical mechanics at the Institute for Advanced Studies. 30 years ago he went back to Italy to set up a roughly similar institute in his home town. They study complex networks and data science. He's been reading our work and liking it. He asked me to set up a mathematics group there! If I do this, I'll be hiring a number of postdocs to work on networks, applied category theory and the like. That could be some of you - though of course I don't want to hire just my own students; I want to hire the best people I can find. There should also be opportunities for shorter visits, like conferences and workshops, which some of you might attend. If I do this I won't be giving up my position at UCR, at least not soon. Instead, I'll visit the place repeatedly, for example during summers. Turin is a wonderful city, so I like this idea. But mainly I'm excited at the idea of being able to put together a team of people working on the kinds of math I like. It's not 100% certain this will happen: I need to decide I really want to do it, and write a proposal, which the board of the ISI will then read, etc.
This week's progress:
1) I just finished a cool paper with Brandon Coya and Franciscus Rebro:
Props in network theory
Abstract. Long before the invention of Feynman diagrams, engineers were using similar diagrams to reason about electrical circuits and more general networks containing mechanical, hydraulic, thermodynamic and chemical components. We can formalize this reasoning using props: that is, strict symmetric monoidal categories where the objects are natural numbers, with the tensor product of objects given by addition. In this approach, each kind of network corresponds to a prop, and each network of this kind is a morphism in that prop. A network with m inputs and n outputs is a morphism from m to n, putting networks together in series is composition, and setting them side by side is tensoring. Here we work out the details of this approach for various kinds of electrical circuits, starting with circuits made solely of ideal perfectly conductive wires, then circuits with passive linear components, and then circuits that also have voltage and current sources. Each kind of circuit corresponds to a mathematically natural prop. We describe the 'behavior' of these circuits using morphisms between props. In particular, we give a new proof of the black-boxing theorem proved by Fong and the first author; unlike the original proof, this new one easily generalizes to circuits with nonlinear components. We also give use a morphism of props to clarify the relation between circuit diagrams and the signal-flow diagrams in control theory. Mathematically, the key tools are the Rosebrugh-Sabadini-Walters result relating circuits to special commutative Frobenius monoids, the monadic adjunction between props and signatures, and a result saying which symmetric monoidal categories are equivalent to props.
I started this project with Franciscus over a year ago, so it's great to be done.
Except we're not done! There are probably lots of mistakes and suboptimalities, so I'd be really grateful if all of you could look over it and send me comments. Also, at one point I promised Jason that the appendix would contain some lemmas on props that he needed. It doesn't have those yet - just the monadic adjunction between props and signatures, which is the key to all those further lemmas.
2) Daniel Cicala is doing lots of good stuff:
Also, just to keep you updated, I got back last week from the QPL conference, which was lots of fun. My talk seemed to go well. Next week, I'm traveling to Vancouver for CT 2017 where I'm giving two talks. The first is a ten minute expository talk attached to the Kan seminar. It's on Lack & Rosicky's Notions of Lawvere theories paper. The second is an accepted talk about my span of cospans stuff.
3) Brendan is doing lots of stuff - but I can't keep track of it all, so I invite him to tell us.
4) I'm going to Singapore tonight, and staying at the Centre of Quantum Technologies until September 15th.
5) I got a referee's report on a paper I'd almost given up on. It's so old most of you guys probably don't know it:
Quantum techniques for reaction networks
Abstract. Reaction networks are a general formalism for describing collections of classical entities interacting in a random way. While reaction networks are mainly studied by chemists, they are equivalent to Petri nets, which are used for similar purposes in computer science and biology. As noted by Doi and others, techniques from quantum field theory can be adapted to apply to such systems. Here we use these techniques to study how the 'master equation' describing stochastic time evolution for a reaction network is related to the 'rate equation' describing the deterministic evolution of the expected number of particles of each species in the large-number limit. We show that the relation is especially strong when a solution of master equation is a 'coherent state', meaning that the numbers of entities of each kind are described by independent Poisson distributions.
I finished writing this in 2013, as a kind of 'prequel' to a more interesting paper that Brendan and I wrote. Since then it's been in journal hell - or perhaps purgatory or limbo. It's a long and boring story, but in 2014 it was solicited for a special issue in Natural Computing, and I've been trying to get a referee's report from them ever since. I finally got it - and luckily, they say they'll accept it after I make some small changes, which are really small.
Moral: when you're trying to publish something, never give up.
Comment Source:13 July 2017: This week's progress: 1) I just finished a cool paper with Brandon Coya and Franciscus Rebro: * [Props in network theory](https://arxiv.org/abs/1707.08321) > **Abstract.** Long before the invention of Feynman diagrams, engineers were using similar diagrams to reason about electrical circuits and more general networks containing mechanical, hydraulic, thermodynamic and chemical components. We can formalize this reasoning using props: that is, strict symmetric monoidal categories where the objects are natural numbers, with the tensor product of objects given by addition. In this approach, each kind of network corresponds to a prop, and each network of this kind is a morphism in that prop. A network with m inputs and n outputs is a morphism from m to n, putting networks together in series is composition, and setting them side by side is tensoring. Here we work out the details of this approach for various kinds of electrical circuits, starting with circuits made solely of ideal perfectly conductive wires, then circuits with passive linear components, and then circuits that also have voltage and current sources. Each kind of circuit corresponds to a mathematically natural prop. We describe the 'behavior' of these circuits using morphisms between props. In particular, we give a new proof of the black-boxing theorem proved by Fong and the first author; unlike the original proof, this new one easily generalizes to circuits with nonlinear components. We also give use a morphism of props to clarify the relation between circuit diagrams and the signal-flow diagrams in control theory. Mathematically, the key tools are the Rosebrugh-Sabadini-Walters result relating circuits to special commutative Frobenius monoids, the monadic adjunction between props and signatures, and a result saying which symmetric monoidal categories are equivalent to props. I started this project with Franciscus over a year ago, so it's great to be done. Except we're not done! There are probably lots of mistakes and suboptimalities, so I'd be really grateful if all of you could look over it and send me comments. Also, at one point I promised Jason that the appendix would contain some lemmas on props that he needed. It doesn't have those yet - just the monadic adjunction between props and signatures, which is the key to all those further lemmas. 2) Daniel Cicala is doing lots of good stuff: > Also, just to keep you updated, I got back last week from the QPL conference, which was lots of fun. My talk seemed to go well. Next week, I'm traveling to Vancouver for CT 2017 where I'm giving two talks. The first is a ten minute expository talk attached to the Kan seminar. It's on Lack & Rosicky's Notions of Lawvere theories paper. The second is an accepted talk about my span of cospans stuff. 3) Brendan is doing lots of stuff - but I can't keep track of it all, so I invite him to tell us. 4) I'm going to Singapore tonight, and staying at the Centre of Quantum Technologies until September 15th. 5) I got a referee's report on a paper I'd almost given up on. It's so old most of you guys probably don't know it: * [Quantum techniques for reaction networks](http://arxiv.org/abs/1306.3451) > **Abstract.** Reaction networks are a general formalism for describing collections of classical entities interacting in a random way. While reaction networks are mainly studied by chemists, they are equivalent to Petri nets, which are used for similar purposes in computer science and biology. As noted by Doi and others, techniques from quantum field theory can be adapted to apply to such systems. Here we use these techniques to study how the 'master equation' describing stochastic time evolution for a reaction network is related to the 'rate equation' describing the deterministic evolution of the expected number of particles of each species in the large-number limit. We show that the relation is especially strong when a solution of master equation is a 'coherent state', meaning that the numbers of entities of each kind are described by independent Poisson distributions. I finished writing this in 2013, as a kind of 'prequel' to a more interesting paper that Brendan and I wrote. Since then it's been in journal hell - or perhaps purgatory or limbo. It's a long and boring story, but in 2014 it was solicited for a special issue in Natural Computing, and I've been trying to get a referee's report from them ever since. I finally got it - and luckily, they say they'll accept it after I make some small changes, which are really small. Moral: when you're trying to publish something, never give up.
1) The paper Blake and I wrote on A compositional framework for reaction networks has been accepted by Reviews in Mathematical Physics. The best part: it seems no corrections were demanded!
2) More news from Daniel, who is at the big annual category theory conference:
Some updates from Vancouver:
I ran into Rick Blute who was my masters adviser and the editor of TAC that I sent my Spans of Cospans paper to. He told me he recently sent an email to the referee reminding them to get me a report. We'll see how that goes.
I gave my talk on Lack and Rosicky's Notions of Lawvere Theories paper today. I got great feedback from Emily Riehl, so that was nice. I presented believing the entire time that this white haired and white bearded man sitting front and center was Bill Lawvere (he was supposed to be here) but it turned out to be Michael Barr. Oddly, I was not the only one of the Kan seminar group to make this mistake.
Us Kan folk, along with some alumni from the first instance of the online seminar went out to dinner and it turns out Christina Vasilakopoulou was one of these alumni, so we got to chat a bit.
3) James Haydon is applying decorated cospans to computer science:
What I've done is set up a framework for composing coroutines (= asynchronous cooperating processes) using a category of decorated cospans. Furthermore, I've implemented the whole thing in code: actual composition of concurrent processes via pushouts!
As an underlying category I take typed channel contexts; this represents a support for a pi-calculus process: channel names and types which it may read and write from. The morphisms map names while respecting the typing structure.
For such a context \(X\), I define a restricted pi-calculus \(\Pi(X)\), which is the set of well typed pi-calculi processes that may only read and write to the channel names specified in \(X\).
This defines a monoidal functor
$$ \Pi : (\mathrm{TyCh}, +) \to (\mathrm{Set}, \times) $$ with the required properties to form a category of decorated cospans. I have implemented all this in the Idris programming language, the source code is here:
https://github.com/jameshaydon/cospanProc
I've experimented with several examples and I think this provides a nice framework for organising code, and composing processes in a safe way. While you compose the processes, you simultaneously compute, via the pushout, the communication interface the resulting process will expose.
Comment Source:20 July 2017: 1) The paper Blake and I wrote on [A compositional framework for reaction networks](https://arxiv.org/abs/1704.02051) has been accepted by _Reviews in Mathematical Physics_. The best part: it seems no corrections were demanded! 2) More news from Daniel, who is at the big annual category theory conference: > Some updates from Vancouver: > I ran into Rick Blute who was my masters adviser and the editor of TAC that I sent my Spans of Cospans paper to. He told me he recently sent an email to the referee reminding them to get me a report. We'll see how that goes. > I gave my talk on Lack and Rosicky's Notions of Lawvere Theories paper today. I got great feedback from Emily Riehl, so that was nice. I presented believing the entire time that this white haired and white bearded man sitting front and center was Bill Lawvere (he was supposed to be here) but it turned out to be Michael Barr. Oddly, I was not the only one of the Kan seminar group to make this mistake. > Us Kan folk, along with some alumni from the first instance of the online seminar went out to dinner and it turns out Christina Vasilakopoulou was one of these alumni, so we got to chat a bit. 3) James Haydon is applying decorated cospans to computer science: > What I've done is set up a framework for composing coroutines (= asynchronous cooperating processes) using a category of decorated cospans. Furthermore, I've implemented the whole thing in code: actual composition of concurrent processes via pushouts! > As an underlying category I take typed channel contexts; this represents a support for a pi-calculus process: channel names and types which it may read and write from. The morphisms map names while respecting the typing structure. > For such a context \\(X\\), I define a restricted pi-calculus \\(\Pi(X)\\), which is the set of well typed pi-calculi processes that may only read and write to the channel names specified in \\(X\\). > This defines a monoidal functor > \[ \Pi : (\mathrm{TyCh}, +) \to (\mathrm{Set}, \times) \] > with the required properties to form a category of decorated cospans. I have implemented all this in the Idris programming language, the source code is here: > https://github.com/jameshaydon/cospanProc > I've experimented with several examples and I think this provides a nice framework for organising code, and composing processes in a safe way. While you compose the processes, you simultaneously compute, via the pushout, the communication interface the resulting process will expose.
This week's progress!
1) First, some good news from Jason Erbele:
A. Just a quick reminder that I will be attending the Hopf Algebras etc. conference at the Perimeter Institute this coming week. Based on the list on the conference website, just under 50 people are participating. The relatively small crowd should make it a bit easier for me to meet people and network, despite not being one of the speakers.
B. I haven't gotten word one way or the other yet regarding the Air Force Research Laboratory postdoc in Dayton, OH, but I have been invited to spend a couple of days at the beginning of September at U. Penn. to get to know some of the people that I would be working with, assuming I do get the job. It'll be Tuesday and Wednesday, September 5-6. They also want me to give a talk, and Dan Koditschek had this to say about it:
"[M]y suggestion would be that you aim for a 45 min. talk with your major effort being to explain the nature and impact (as distinct from the details of the technical accomplishments) of your ideas and results to an audience interested in but very unfamiliar with your area. My group is typically quite active and you will likely be interrupted throughout with questions."
You will likely recall that Dan Koditschek is the head of the group that Brendan was with before he took David Spivak's offer at MIT.
The second item is especially promising. He's telling you to focus on the big picture and not sink into too many technical details, and he's telling you to discuss the "impact" of your work, which means you have to make it sound like a big deal. I urge you to get some advice from Brendan about how to do this, because he knows Koditschek. He also knows Sobocinski and company, who seem to be doing a reasonable job of convincing people of the importance of their work.
In trying to sell your ideas and accomplishments, it will help to interpret "your" rather broadly, to include the work of Bonchi, Sobocinski and Zanasi on similar themes. I don't mean to pretend you did their work - you should of course credit them. But you should learn their work, and if they've done interesting things, especially "practical" things, you should feel free to explain those things in your talk.
In short, think of yourself not as a mere individual, but as a representative of a movement. If the movement is doing interesting things, and you represent it ably, Koditschek may want to hire you - to have "someone who knows that stuff" around, especially now that Brendan is no longer there.
2) Second, Nina Otter is at a workshop on Macaulay2, which is a software package for algebraic geometry. I get the feeling that this workshop is focused on using Macaulay2 in statistics, phylogenetics and other areas - "applied algebraic geometry".
3) Third, Blake and I put our paper on A compositional framework for Markov processes into the suitable format for Reviews in Mathematical Physics and gave it to that journal. They should give us the proofs back in a week or two.
4) Fourth, Brandon and Franciscus and I put our paper on Props in network theory onto the arXiv. Shortly afterward, I got a comment from the category theorist Thomas Holder. He said:
Please excuse me for making a couple of comments on your arXiv.1707.08321. For one your first reference in the bibliography is empty.
This proves that the first thing people do is look at your bibliography. I fixed this.
The other thing is, since you pay considerable attention to the history of the subject, I'd like to point out that the use of monoidal categories in network theory including the use of string diagrams was pioneered by the German computer scientist Günter Hotz in 1965/66 and that by the end of the 1960s there was considerable activity by his group in Saarbrücken with some nice results e.g. Hans Langmaarck discovered the first ordering of the braid group in the process of giving normal forms for the morphisms in 1969. At the same time this was taken up by a research group in East Germany around the algebraicist Lothar Budach, a former student of Krull's, in collaboration with Hans-Joachim Hoehnke leading to a monumental monograph on 'Automaten und Funktoren' in 1975. David Benson at that time also worked in this framework.
Most of this work is unfortunately in German (e.g. a monograph of Hotz on 'Schaltkreistheorie' or Boolean circuit theory in 1974) - you find some of the relevant references at the nLab entry on Hotz.
Permit me to unload here an observation which for a lack of proper understanding of QFT I've never known much what to make of: One of the results in the Budach-Hoehnke book is a generalization of the Chomsky normal form for context-free grammars which says that the morphisms (=syntactical derivations) in the monoidal category corresponding to your (not necessarily context-free) grammar have a normal form provided the rewriting rules that generate your category don't include at the same time a (creation) rule that rewrites the empty string to a non empty string e->X1...Xn and the reverse annihilation rule X1...Xn->e. This suggests that (at least to a non cognoscente like me) that renormalization of a QFT is tied to the normalization of the corresponding theory of Feynman diagrams.
thanks for your patience&best wishes Thomas Holder
Personally I think the relation between "normalization of strings" and "renormalization in QFT" is mainly just a pun, but the remarks on Hotz are interesting and I may decide to add something about his work to our paper.
Comment Source:28 July 2017: This week's progress! 1) First, some good news from Jason Erbele: > A. Just a quick reminder that I will be attending the [Hopf Algebras etc.](https://www.perimeterinstitute.ca/conferences/hopf-algebras-kitaevs-quantum-double-models-mathematical-connections-gauge-theory) conference at the Perimeter Institute this coming week. Based on the list on the conference website, just under 50 people are participating. The relatively small crowd should make it a bit easier for me to meet people and network, despite not being one of the speakers. > B. I haven't gotten word one way or the other yet regarding the Air Force Research Laboratory postdoc in Dayton, OH, but I have been invited to spend a couple of days at the beginning of September at U. Penn. to get to know some of the people that I would be working with, assuming I do get the job. It'll be Tuesday and Wednesday, September 5-6. They also want me to give a talk, and Dan Koditschek had this to say about it: > > "[M]y suggestion would be that you aim for a 45 min. talk with your major effort being to explain the nature and impact (as distinct from the details of the technical accomplishments) of your ideas and results to an audience interested in but very unfamiliar with your area. My group is typically quite active and you will likely be interrupted throughout with questions." > You will likely recall that Dan Koditschek is the head of the group that Brendan was with before he took David Spivak's offer at MIT. The second item is especially promising. He's telling you to focus on the big picture and not sink into too many technical details, and he's telling you to discuss the "impact" of your work, which means you have to make it sound like a big deal. I urge you to get some advice from Brendan about how to do this, because he knows Koditschek. He also knows Sobocinski and company, who seem to be doing a reasonable job of convincing people of the importance of their work. In trying to sell your ideas and accomplishments, it will help to interpret "your" rather broadly, to include the work of Bonchi, Sobocinski and Zanasi on similar themes. I don't mean to pretend you did their work - you should of course credit them. But you should learn their work, and if they've done interesting things, especially "practical" things, you should feel free to explain those things in your talk. In short, **think of yourself not as a mere individual, but as a representative of a movement**. If the movement is doing interesting things, and you represent it ably, Koditschek may want to hire you - to have "someone who knows that stuff" around, especially now that Brendan is no longer there. 2) Second, Nina Otter is at a workshop on [Macaulay2](https://sites.google.com/view/macaulay2-gatech-2017/home), which is a software package for algebraic geometry. I get the feeling that this workshop is focused on using Macaulay2 in statistics, phylogenetics and other areas - "applied algebraic geometry". 3) Third, Blake and I put our paper on [A compositional framework for Markov processes](https://arxiv.org/abs/1704.02051) into the suitable format for _Reviews in Mathematical Physics_ and gave it to that journal. They should give us the proofs back in a week or two. 4) Fourth, Brandon and Franciscus and I put our paper on [Props in network theory](https://arxiv.org/abs/1707.08321) onto the arXiv. Shortly afterward, I got a comment from the category theorist Thomas Holder. He said: > Please excuse me for making a couple of comments on your arXiv.1707.08321. For one your first reference in the bibliography is empty. This proves that the first thing people do is look at your bibliography. I fixed this. > The other thing is, since you pay considerable attention to the history of the subject, I'd like to point out that the use of monoidal categories in network theory including the use of string diagrams was pioneered by the German computer scientist Günter Hotz in 1965/66 and that by the end of the 1960s there was considerable activity by his group in Saarbrücken with some nice results e.g. Hans Langmaarck discovered the first ordering of the braid group in the process of giving normal forms for the morphisms in 1969. At the same time this was taken up by a research group in East Germany around the algebraicist Lothar Budach, a former student of Krull's, in collaboration with Hans-Joachim Hoehnke leading to a monumental monograph on 'Automaten und Funktoren' in 1975. David Benson at that time also worked in this framework. > Most of this work is unfortunately in German (e.g. a monograph of Hotz on 'Schaltkreistheorie' or Boolean circuit theory in 1974) - you find some of the relevant references at the nLab entry on Hotz. > Permit me to unload here an observation which for a lack of proper understanding of QFT I've never known much what to make of: One of the results in the Budach-Hoehnke book is a generalization of the Chomsky normal form for context-free grammars which says that the morphisms (=syntactical derivations) in the monoidal category corresponding to your (not necessarily context-free) grammar have a normal form provided the rewriting rules that generate your category don't include at the same time a (creation) rule that rewrites the empty string to a non empty string e->X1...Xn and the reverse annihilation rule X1...Xn->e. This suggests that (at least to a non cognoscente like me) that renormalization of a QFT is tied to the normalization of the corresponding theory of Feynman diagrams. > thanks for your patience&best wishes Thomas Holder Personally I think the relation between "normalization of strings" and "renormalization in QFT" is mainly just a pun, but the remarks on Hotz are interesting and I may decide to add something about his work to our paper.
1) Nina Otter is at Banff International Research Station, the place in Canada where they have tons of math workshops, attending one on Topological Data Analysis: Developing Abstract Foundations. Topological data analysis uses things like persistent homology to calculate the topology of clouds of data points. It's very fashionable these days.
2) Blake and I gave Reviews in Mathematical Physics a final version of our paper "A compositional framework for reaction networks". I celebrated by blogging about it all over:
Category theory in chemistry, on G+, a "pop" explanation that got lots of likes but also lots of interesting comments.
A compositional framework for reaction networks, on Azimuth, a more technical explanation that also got some good comments.
A compositional framework for reaction networks, on the n-Category Cafe, an almost identical technical explanation that got completely different comments, including a really fascinating one from Mike Shulman: he's been looking at open Petri nets in logic, without knowing that's what they're called! Instead of chemicals reacting to give other chemicals, he's looking at assumptions "reacting" to give conclusions! In other words, he's looking at proofs. /This is the kind of analogy Mike Stay and I really like, and I bet it will produce interesting new ways of looking at both chemistry and logic.
3) Nina and I just sent a revised version of our paper "Operads and phylogenetic trees" to Steve Lack, who is an editor at TAC. The referee had wanted tons of big changes, so this was a lot of work. Our reply letter listing what we've done is 11 pages long! I won't be surprised if the referee points out mistakes in this.
But the paper is better now - if you were too scared to read it before, read it now! It's about Markov processes and operads.
The referee had earlier said "Overall, I believe that an improved version of this paper would make an excellent, unusual and interesting contribution to TAC", so I hope they accept it now. We've been working on this paper since December 2013.
Moral: it sometimes takes insane amounts of persistence to publish things.
Comment Source:2 August 2017: This week's progress: 1) Nina Otter is at Banff International Research Station, the place in Canada where they have tons of math workshops, attending one on [Topological Data Analysis: Developing Abstract Foundations](https://www.birs.ca/events/2017/5-day-workshops/17w5108). Topological data analysis uses things like [persistent homology](https://en.wikipedia.org/wiki/Persistent_homology) to calculate the topology of clouds of data points. It's very fashionable these days. 2) Blake and I gave _Reviews in Mathematical Physics_ a final version of our paper "A compositional framework for reaction networks". I celebrated by blogging about it all over: * [Category theory in chemistry](https://plus.google.com/u/0/+johncbaez999/posts/hhyzhu1mB7F), on G+, a "pop" explanation that got lots of likes but also lots of interesting comments. * [A compositional framework for reaction networks](https://johncarlosbaez.wordpress.com/2017/07/30/a-compositional-framework-for-reaction-networks/), on Azimuth, a more technical explanation that also got some good comments. * [A compositional framework for reaction networks](https://golem.ph.utexas.edu/category/2017/07/a_compositional_framework_for_2.html), on the n-Category Cafe, an almost identical technical explanation that got completely different comments, including a really fascinating one from Mike Shulman: he's been looking at open Petri nets in logic, without knowing that's what they're called! Instead of chemicals reacting to give other chemicals, he's looking at assumptions "reacting" to give conclusions! In other words, he's looking at proofs. /This is the kind of analogy Mike Stay and I really like, and I bet it will produce interesting new ways of looking at both chemistry and logic. 3) Nina and I just sent a revised version of our paper "[Operads and phylogenetic trees](http://math.ucr.edu/home/baez/phylo.pdf)" to Steve Lack, who is an editor at TAC. The referee had wanted tons of big changes, so this was a lot of work. Our reply letter listing what we've done is 11 pages long! I won't be surprised if the referee points out mistakes in this. But the paper is better now - if you were too scared to read it before, read it now! It's about Markov processes and operads. The referee had earlier said "Overall, I believe that an improved version of this paper would make an excellent, unusual and interesting contribution to TAC", so I hope they accept it now. We've been working on this paper since December 2013. **Moral: it sometimes takes insane amounts of persistence to publish things.**
1) Jason Erbele wrote an entertaining blog article about the conference he attended at the Perimeter Institute:
Hopf Algebras in Kitaev's Quantum Double Models: Mathematical Connections from Gauge Theory to Topological Quantum Computing and Categorical Quantum Mechanics.
2) Nina Otter gave a talk at a week-long conference in Sapporo, Japan, called Applied Algebraic Topology 2017:
New invariants for multi-parameter persistent homology.
Abstract. Topological data analysis (TDA) is a field that lies at the intersection of data analysis, algebraic topology, computational geometry, computer science, and statistics. The main goal of TDA is to use ideas and results from geometry and topology to develop tools for studying qualitative features of data. One of the most successful methods in TDA is persistent homology (PH), a method that stems from algebraic topology, and has been used in a variety of applications from different fields, including robotics, material science, biology, and finance.
PH allows to study qualitative features of data across different values of a parameter, which one can think of as scales of resolution, and provides a summary of how long individual features persist across the different scales of resolution. In many applications, data depend not only on one, but several parameters, and to apply PH to such data one therefore needs to study the evolution of qualitative features across several parameters. While the theory of 1-parameter persistent homology is well understood, the theory of multi-parameter PH is hard, and it presents one of the biggest challenges of TDA.
In this talk I will briefly introduce persistent homology, give an overview of the complexity of the theory in the multi-parameter case, and then discuss how tools from commutative algebra give invariants able to capture homology classes with large persistence.
No prior knowledge on the subject is assumed. This talk is based on joint work with Heather Harrington, Henry Schenck, and Ulrike Tillmann.
3) I also gave a talk at this conference:
The rise and spread of algebraic topology
Abstract. As algebraic topology becomes more important in applied mathematics, it is worth looking back to see how this subject has changed our outlook on mathematics in general. When Noether moved from working with Betti numbers to homology groups, she forced a new outlook on topological invariants: namely, they are often functors, with two invariants counting as "the same" if they are naturally isomorphic. To formalize this it was necessary to invent categories, and to formalize the analogy between natural isomorphisms between functors and homotopies between maps it was necessary to invent 2-categories. These are just the first steps in the "homotopification" of mathematics, a trend in which algebra more and more comes to resemble topology, and ultimately abstract "spaces" (for example, homotopy types) are considered as fundamental as sets. It is natural to wonder whether topological data analysis is a step in the spread of these ideas into applied mathematics, and how the importance of "robustness" in applications will influence algebraic topology.
Quite a few other talks used category theory! But as this slide from Tom Leinster's talk, they did so with a bit of hesitancy:
Comment Source:10 August 2017: This week's progress: 1) Jason Erbele wrote an entertaining blog article about the conference he attended at the Perimeter Institute: * [Hopf Algebras in Kitaev's Quantum Double Models: Mathematical Connections from Gauge Theory to Topological Quantum Computing and Categorical Quantum Mechanics](https://jasonmaths.wordpress.com/2017/08/09/haikqdmmcfgtttqcacqm-postmortem/). 2) Nina Otter gave a talk at a week-long conference in Sapporo, Japan, called Applied Algebraic Topology 2017: * New invariants for multi-parameter persistent homology. > **Abstract.** Topological data analysis (TDA) is a field that lies at the intersection of data analysis, algebraic topology, computational geometry, computer science, and statistics. The main goal of TDA is to use ideas and results from geometry and topology to develop tools for studying qualitative features of data. One of the most successful methods in TDA is persistent homology (PH), a method that stems from algebraic topology, and has been used in a variety of applications from different fields, including robotics, material science, biology, and finance. > PH allows to study qualitative features of data across different values of a parameter, which one can think of as scales of resolution, and provides a summary of how long individual features persist across the different scales of resolution. In many applications, data depend not only on one, but several parameters, and to apply PH to such data one therefore needs to study the evolution of qualitative features across several parameters. While the theory of 1-parameter persistent homology is well understood, the theory of multi-parameter PH is hard, and it presents one of the biggest challenges of TDA. > In this talk I will briefly introduce persistent homology, give an overview of the complexity of the theory in the multi-parameter case, and then discuss how tools from commutative algebra give invariants able to capture homology classes with large persistence. > No prior knowledge on the subject is assumed. This talk is based on joint work with Heather Harrington, Henry Schenck, and Ulrike Tillmann. 3) I also gave a talk at this conference: * [The rise and spread of algebraic topology](http://math.ucr.edu/home/baez/alg_top/) > **Abstract.** As algebraic topology becomes more important in applied mathematics, it is worth looking back to see how this subject has changed our outlook on mathematics in general. When Noether moved from working with Betti numbers to homology groups, she forced a new outlook on topological invariants: namely, they are often functors, with two invariants counting as "the same" if they are naturally isomorphic. To formalize this it was necessary to invent categories, and to formalize the analogy between natural isomorphisms between functors and homotopies between maps it was necessary to invent 2-categories. These are just the first steps in the "homotopification" of mathematics, a trend in which algebra more and more comes to resemble topology, and ultimately abstract "spaces" (for example, homotopy types) are considered as fundamental as sets. It is natural to wonder whether topological data analysis is a step in the spread of these ideas into applied mathematics, and how the importance of "robustness" in applications will influence algebraic topology. Quite a few other talks used category theory! But as this slide from Tom Leinster's talk, they did so with a bit of hesitancy: <center><img width = "600" src = "http://math.ucr.edu/home/baez/alg_top/leinster_talk.jpg"></center>
1) Kenny's first paper, A bicategory of decorated cospans, has been published by Theory and Applications of Categories. He put a lot of work into this! It looks great in its final form.
2) Nina Otter is visiting me at the Centre for Quantum Technologies - we got some money from the center for her to do this. We're trying to write a quick, short paper on the relation between persistent homology and magnitude homology.
3) I wrote a blog article that starts explaining how our project with Metron is using operads to design networks:
Complex adaptive system design (part 3).
I'll need to write more to really explain the idea! DARPA is having a seminar on operads on September 14th and they may use some of these blog articles as reading material.
4) And now for the really big news: there will be a workshop on Applied Category Theory at the Lorentz Center in Leiden, the Netherlands, April 23 to May 4, 2018. This is being organized by Bob Coecke, Aleks Kissinger, Martha Lewis and Josh Tan... and our very own Brendan Fong!
They've put together an ambitious program where grad students will join online seminars in January and start doing projects. Nina Otter is helping organize this. This phase will culminate in a few days of sessions at the Lorentz Center, which come right before the actual conference.
At the conference there will be speakers roughly like this:
Samson Abramsky
Mikhail Gromov
Jean Krivine
Tom Leinster
Nicoletta Sabadini
Mehrnoosh Sadrzadeh
This list is likely to change a bit, so please don't spread it around. (Gromov is such a bigshot that I'll be amazed if he comes, but it would be cool.)
I hope all of you apply when the application opens up, though I can't promise all of you will be accepted. I'll be running one of the seminars, but for my grad students this will be a great chance to learn from someone else and - if you're well-organized - even write a paper. It should be a wonderful experience: I don't think there's ever been a conference that covered such wide-ranging applications of category theory.
Now I'm supposed to apply for an NSF grant to help pay for US grad students to attend this conference. Brendan should keep pestering me to actually do this.
There's a bit more information on the conference below. I imagine some of this is tentative.
Comment Source:17 August 2017: 1) Kenny's first paper, <a href = "http://www.tac.mta.ca/tac/volumes/32/29/32-29.pdf">A bicategory of decorated cospans</a>, has been published by <i>Theory and Applications of Categories</i>. He put a lot of work into this! It looks great in its final form. 2) Nina Otter is visiting me at the Centre for Quantum Technologies - we got some money from the center for her to do this. We're trying to write a quick, short paper on the relation between persistent homology and magnitude homology. 3) I wrote a blog article that starts explaining how our project with Metron is using operads to design networks: * <a href = "https://johncarlosbaez.wordpress.com/2017/08/17/complex-adaptive-system-design-part-3/">Complex adaptive system design (part 3)</a>. I'll need to write more to really explain the idea! DARPA is having a seminar on operads on September 14th and they may use some of these blog articles as reading material. 4) And now for the really big news: there will be a workshop on Applied Category Theory at the <a href = "http://www.lorentzcenter.nl/">Lorentz Center</a> in Leiden, the Netherlands, April 23 to May 4, 2018. This is being organized by Bob Coecke, Aleks Kissinger, Martha Lewis and Josh Tan... and our very own Brendan Fong! They've put together an ambitious program where grad students will join online seminars in January and start doing projects. Nina Otter is helping organize this. This phase will culminate in a few days of sessions at the Lorentz Center, which come right before the actual conference. At the conference there will be speakers roughly like this: * Samson Abramsky * John Baez * Mikhail Gromov * Kathryn Hess * Jean Krivine * Tom Leinster * Nicoletta Sabadini * Mehrnoosh Sadrzadeh This list is likely to change a bit, so please don't spread it around. (Gromov is such a bigshot that I'll be amazed if he comes, but it would be cool.) I hope all of you apply when the application opens up, though I can't promise all of you will be accepted. I'll be running one of the seminars, but for my grad students this will be a great chance to learn from someone else and - if you're well-organized - even write a paper. It should be a wonderful experience: I don't think there's ever been a conference that covered such wide-ranging applications of category theory. Now I'm supposed to apply for an NSF grant to help pay for US grad students to attend this conference. Brendan should keep pestering me to actually do this. There's a bit more information on the conference below. I imagine some of this is tentative.
1) Jason Erbele submitted an abstract for the Nov. 4-5 special session on applied category theory at UCR:
Controllability and observability: diagrams and duality
Abstract. Diagrams of systems appear in many different fields of study, and for good reason: they can dramatically simplify communication of and calculations with those systems. In many cases, large diagrams can be viewed as coming from piecing together smaller diagrams in ways that preserve important data, and complicated diagrams can be rewritten to produce simpler diagrams that represent the same behavior. Category theory provides a framework to reason with diagrams as mathematical objects that can be composed and transformed by rewrite rules. In particular, for linear, time independent control systems, the dual notions of controllability and observability can be expressed in terms of a the dual notions of epimorphism and monomorphism, as applied to certain composite diagrams. (Received August 25, 2017)
2) Nina Otter put a new paper on the arXiv with Hal Schenk and her two advisors, the mathematical biologist Heather Harrington and the homotopy theorist Ulrike Tillmann:
Stratifying multiparameter persistent homology
Abstract. A fundamental tool in topological data analysis is persistent homology, which allows extraction of information from complex datasets in a robust way. Persistent homology assigns a module over a principal ideal domain to a one-parameter family of spaces obtained from the data. In applications data often depend on several parameters, and in this case one is interested in studying the persistent homology of a multiparameter family of spaces associated to the data. While the theory of persistent homology for one-parameter families is well-understood, the situation for multiparameter families is more delicate. Following Carlsson and Zomorodian we recast the problem in the setting of multigraded algebra, and we propose multigraded Hilbert series, multigraded associated primes and local cohomology as invariants for studying multiparameter persistent homology. Multigraded associated primes provide a stratification of the region where a multigraded module does not vanish, while multigraded Hilbert series and local cohomology give a measure of the size of components of the module supported on different strata. These invariants generalize in a suitable sense the invariant for the one-parameter case.
3) Joseph Moeller has been visiting Metron Scientific Solutions in Reston since last Friday. He's talking to John Foley about our project, for example the use of "graphic monoids" for describing networks of commitments among a collection of agents.
Graphic monoids were introduced by Lawvere. These are monoids obeying the identity xyx = xy, which means that if you try to commit to x, then to y, then to x it's the same as trying to commit to x and then trying to commit to y. You may or may not be able to make these commitments, which is why I say "try". But if you succeed in committing to x the first time, committing to it a second time doesn't change anything, even if you've made some other commitments in the meantime!
I'm hoping Joseph will get pulled into the more applied aspects of our project when he's there, since the cutting edge right now is using our math to get things done. But the applied aspects, done right, will probably involve a lot of brand new pure math, like what Joseph and John are doing right now with graphic monoids.
4) I wrote another blog article about our Metron project:
This is about the simplest example of a "network operad" and its simplest algebra.
Comment Source:26 August 2017: Here is this week's progress: 1) Jason Erbele submitted an abstract for the Nov. 4-5 special session on applied category theory at UCR: * Controllability and observability: diagrams and duality > <b>Abstract.</b> Diagrams of systems appear in many different fields of study, and for good reason: they can dramatically simplify communication of and calculations with those systems. In many cases, large diagrams can be viewed as coming from piecing together smaller diagrams in ways that preserve important data, and complicated diagrams can be rewritten to produce simpler diagrams that represent the same behavior. Category theory provides a framework to reason with diagrams as mathematical objects that can be composed and transformed by rewrite rules. In particular, for linear, time independent control systems, the dual notions of controllability and observability can be expressed in terms of a the dual notions of epimorphism and monomorphism, as applied to certain composite diagrams. (Received August 25, 2017) 2) Nina Otter put a new paper on the arXiv with Hal Schenk and her two advisors, the mathematical biologist Heather Harrington and the homotopy theorist Ulrike Tillmann: * <a href = "https://arxiv.org/abs/1708.07390">Stratifying multiparameter persistent homology</a> > **Abstract.** A fundamental tool in topological data analysis is persistent homology, which allows extraction of information from complex datasets in a robust way. Persistent homology assigns a module over a principal ideal domain to a one-parameter family of spaces obtained from the data. In applications data often depend on several parameters, and in this case one is interested in studying the persistent homology of a multiparameter family of spaces associated to the data. While the theory of persistent homology for one-parameter families is well-understood, the situation for multiparameter families is more delicate. Following Carlsson and Zomorodian we recast the problem in the setting of multigraded algebra, and we propose multigraded Hilbert series, multigraded associated primes and local cohomology as invariants for studying multiparameter persistent homology. Multigraded associated primes provide a stratification of the region where a multigraded module does not vanish, while multigraded Hilbert series and local cohomology give a measure of the size of components of the module supported on different strata. These invariants generalize in a suitable sense the invariant for the one-parameter case. 3) Joseph Moeller has been visiting Metron Scientific Solutions in Reston since last Friday. He's talking to John Foley about our project, for example the use of "graphic monoids" for describing networks of commitments among a collection of agents. <a href = "https://ncatlab.org/nlab/show/graphic+category">Graphic monoids</a> were introduced by Lawvere. These are monoids obeying the identity xyx = xy, which means that if you try to commit to x, then to y, then to x it's the same as trying to commit to x and then trying to commit to y. You may or may not be able to make these commitments, which is why I say "try". But if you succeed in committing to x the first time, committing to it a second time doesn't change anything, even if you've made some other commitments in the meantime! I'm hoping Joseph will get pulled into the more applied aspects of our project when he's there, since the cutting edge right now is using our math to get things done. But the applied aspects, done right, will probably involve a lot of brand new pure math, like what Joseph and John are doing right now with graphic monoids. 4) I wrote another blog article about our Metron project: * <a href = "https://johncarlosbaez.wordpress.com/2017/08/22/complex-adaptive-system-design-part-4/">Complex adaptive system design (part 4)</a> This is about the simplest example of a "network operad" and its simplest algebra.
1) I made some final changes to a paper with Brandon Coya and Franciscus Rebro, Props in network theory, and we submitted it to Theory and Application of Categories.
The hardest part was proving a result Jason Erbele wanted for his thesis: if we have a prop P presented by some generators and relations, and we add some extra generators, we get a new prop P' having P as a sub-prop.
There's clearly a morphism P → P'. The challenge is to show it's monic. This seems obvious, but I needed Todd Trimble to help me prove it, and eventually Charles Rezk explained why the proof is subtle: while this result is true for props, it's not true for all algebraic gadgets described by multi-sorted Lawvere theories.
Yes, sometimes adding an extra generator can make an algebraic gadget "collapse" and become smaller in some ways! This doesn't happen for the familiar gadgets that have just one sort of element, like groups or rings, so it seems bizarre, but Rezk came up with a cool example of what can go wrong when we have more sorts.
The story is tersely told in Appendix A.2 of our paper, but you can see the whole saga in this discussion on the n-Café:
A puzzle on multi-sorted Lawvere theories
Moral: it's good to solve problems in public, on blogs. More brains make things go faster, it's more fun, and people get to know you, which is essential for getting jobs. If you have questions you want help with, write a blog article and I'll post it.
2) Brendan Fong visited Blake Pollard at Princeton, where Blake is interning with Arquimedes Canedo at the engineering firm Siemens. Brendan gave a talk. Blake wrote a nice description of what happened, but the most important part for you grad students is:
Canedo wants to hire more interns in the future: ideally people who know some applied category theory and can do some scientific programming! This could be YOU.
The word "Princeton" looks really good on your CV - believe me. Blake writes:
Brendan's visit and accompanying talk went really well. Afterwards (over beers), Arqui proposed that the Siemens Princeton office become the first corporate 'Applied Category Theory Hub.' He is also interested in coming to the Netherlands workshop next summer, or at least to contribute some new problems from the Siemens domain.
It was great to get Brendan, Dimitris, and myself together in a room with Arqui to discuss the potential role that applied category theory could play in helping Siemens realize this vision of a kind of `universal knowledge base' for their engineering and manufacturing processes and systems.
It is clear that their vision is still a ways out, but in addition to trying to formalize such a knowledge base, riding the current wave of interest in applied category theory might help the Siemens folk continue to secure both internal and government funding to realize such a vision. Apparently there is a long-time Siemens guy in Germany who is also interested in exploring potential applications of category theory.
I think we've done a good job helping shift the perceived benefits of category theory from 'provides magic wand' to 'provides new, potentially unifying perspective.'
Some good analogies were thrown around with respect to the question: What is category theory buying us?
This is like asking: What has set theory done for us?
Or if you headed back and showed calculus to the folks designing and building the Roman aqueducts, they'd most likely respond, 'So what? Check out those aqueducts.'
Brendan's talk went really well. I convinced him to include his result translating the condition of controllability for linear time invariant systems to a diagrammatic condition and I think people liked that a lot. Feeling that you understand something more intuitively by looking at pictures is a big initial draw for category theory.
Afterwards we all got to chat a bit about categories and where exactly they might be useful.
I like the perspective that we're trying to find the analogue of the functional programming style in engineering.
Arqui hasn't had any 'ah ha' moments yet, but he's a dreamer, he sees the potential, and now he understands better the work required to get to a more realistic vision of how categories can help corporate technology. I really hope we can get beyond 'category theory as a way to secure DARPA funds.'
Arqui wants more interns in the future exploring this stuff. So tell the gang! Ideally they should know basic scientific programming for prototyping ideas. I'm trying to provide evidence that folks with my kind of background are the ones you want around, not just to dream about categories, but also to push forward more practical research projects which are only tangentially related (hence my current interlude with machine learning). I'm giving a talk tomorrow about my work so far on my internship as kind of an interview for a full-time position.
I invited Jason to give a talk next week since he'll be in Philly. The Siemens gang is up for a weekly category theory seminar!
Also Brendan and I revived the idea of a Banff Oaxaca workshop 'Sticking things together in Mexico.' I think the applications are due in late September.
I forgot about that workshop. I'd be happy to help a bit with applying, though I'm getting insanely busy.
Finally, some non-progress. Last week Lisa and I took a vacation in Ubud, the "cultural capital" of Bali. We listened to music:
hiked through rice paddies:
and much more. We'd like to rent a house and spend a month there sometime.
Comment Source:1 September 2017: Here is this week's progress, as far as I know: 1) I made some final changes to a paper with Brandon Coya and Franciscus Rebro, <a href = "https://arxiv.org/abs/1707.08321">Props in network theory</a>, and we submitted it to <i>Theory and Application of Categories</i>. The hardest part was proving a result Jason Erbele wanted for his thesis: if we have a prop P presented by some generators and relations, and we add some extra generators, we get a new prop P' having P as a sub-prop. There's clearly a morphism P → P'. The challenge is to show it's monic. This seems obvious, but I needed Todd Trimble to help me prove it, and eventually Charles Rezk explained why the proof is subtle: while this result is true for props, it's not true for all algebraic gadgets described by multi-sorted Lawvere theories. Yes, sometimes adding an extra generator can make an algebraic gadget "collapse" and become smaller in some ways! This doesn't happen for the familiar gadgets that have just one sort of element, like groups or rings, so it seems bizarre, but Rezk came up with a cool example of what can go wrong when we have more sorts. The story is tersely told in Appendix A.2 of our paper, but you can see the whole saga in this discussion on the n-Café: * <a href = "https://golem.ph.utexas.edu/category/2017/08/a_puzzle_on_multisorted_lawver.html">A puzzle on multi-sorted Lawvere theories</a> <b>Moral: it's good to solve problems in public, on blogs.</b> More brains make things go faster, it's more fun, and people get to know you, which is essential for getting jobs. If you have questions you want help with, write a blog article and I'll post it. 2) Brendan Fong visited Blake Pollard at Princeton, where Blake is interning with Arquimedes Canedo at the engineering firm Siemens. Brendan gave a talk. Blake wrote a nice description of what happened, but the most important part for you grad students is: <b>Canedo wants to hire more interns in the future: ideally people who know some applied category theory and can do some scientific programming! This could be YOU.</b> The word "Princeton" looks really good on your CV - believe me. Blake writes: > Brendan's visit and accompanying talk went really well. Afterwards (over beers), Arqui proposed that the Siemens Princeton office become the first corporate 'Applied Category Theory Hub.' He is also interested in coming to the Netherlands workshop next summer, or at least to contribute some new problems from the Siemens domain. > It was great to get Brendan, Dimitris, and myself together in a room with Arqui to discuss the potential role that applied category theory could play in helping Siemens realize this vision of a kind of `universal knowledge base' for their engineering and manufacturing processes and systems. > It is clear that their vision is still a ways out, but in addition to trying to formalize such a knowledge base, riding the current wave of interest in applied category theory might help the Siemens folk continue to secure both internal and government funding to realize such a vision. Apparently there is a long-time Siemens guy in Germany who is also interested in exploring potential applications of category theory. > I think we've done a good job helping shift the perceived benefits of category theory from 'provides magic wand' to 'provides new, potentially unifying perspective.' > Some good analogies were thrown around with respect to the question: What is category theory buying us? > This is like asking: What has set theory done for us? > Or if you headed back and showed calculus to the folks designing and building the Roman aqueducts, they'd most likely respond, 'So what? Check out those aqueducts.' > Brendan's talk went really well. I convinced him to include his result translating the condition of controllability for linear time invariant systems to a diagrammatic condition and I think people liked that a lot. Feeling that you understand something more intuitively by looking at pictures is a big initial draw for category theory. > Afterwards we all got to chat a bit about categories and where exactly they might be useful. > I like the perspective that we're trying to find the analogue of the functional programming style in engineering. > Arqui hasn't had any 'ah ha' moments yet, but he's a dreamer, he sees the potential, and now he understands better the work required to get to a more realistic vision of how categories can help corporate technology. I really hope we can get beyond 'category theory as a way to secure DARPA funds.' > Arqui wants more interns in the future exploring this stuff. So tell the gang! Ideally they should know basic scientific programming for prototyping ideas. I'm trying to provide evidence that folks with my kind of background are the ones you want around, not just to dream about categories, but also to push forward more practical research projects which are only tangentially related (hence my current interlude with machine learning). I'm giving a talk tomorrow about my work so far on my internship as kind of an interview for a full-time position. > I invited Jason to give a talk next week since he'll be in Philly. The Siemens gang is up for a weekly category theory seminar! > Also Brendan and I revived the idea of a Banff Oaxaca workshop 'Sticking things together in Mexico.' I think the applications are due in late September. I forgot about that workshop. I'd be happy to help a bit with applying, though I'm getting insanely busy. Finally, some non-progress. Last week Lisa and I took a vacation in Ubud, the "cultural capital" of Bali. We listened to music: <center><img width = "600" src = "http://math.ucr.edu/home/baez/diary/bali/drummer_barong_dance_in_batu_balan_small.jpg"></center> hiked through rice paddies: <center><img width = "600" src = "http://math.ucr.edu/home/baez/diary/bali/ubud_rice_fields_small.jpg"></center> and <a href = "http://math.ucr.edu/home/baez/diary/september_2017.html">much more</a>. We'd like to rent a house and spend a month there sometime.
Jim Stuttard
Reading that Siemens has an outpost at Princeton and might be into ACT I thought I'd mention Schneider Electric which is a European leader in energy and environmental control. People might like to look at some of their circuits for practical ideas.
Comment Source:Reading that Siemens has an outpost at Princeton and might be into ACT I thought I'd mention [Schneider Electric](https://www.schneider-electric.co.uk/en/) which is a European leader in energy and environmental control. People might like to look at some of their circuits for practical ideas.
It would be great if someone at Schneider Electric were willing to talk. Usually it's just one or two people at a firm who are interested in applied category theory... but that's all it takes for something to happen! Arquimedes Canedo not only gave my student Blake a summer internship at Siemens, he's also looking for more math students to work on applied category theory as interns there. I haven't managed to get any of my students to do it this summer, mainly because I found this out so late.
Comment Source:It would be great if someone at Schneider Electric were willing to talk. Usually it's just one or two people at a firm who are interested in applied category theory... but that's all it takes for something to happen! Arquimedes Canedo not only gave my student Blake a summer internship at Siemens, he's also looking for more math students to work on applied category theory as interns there. I haven't managed to get any of my students to do it this summer, mainly because I found this out so late.
"It would be great if someone at Schneider Electric were willing to talk." That's what I thought so I'll try and find if there's somebody there who might be interested.
Comment Source:"It would be great if someone at Schneider Electric were willing to talk." That's what I thought so I'll try and find if there's somebody there who might be interested.
Jim what due you mean with ACT? The https://en.wikipedia.org/wiki/ACT_(NASDAQ) ?
yes schneider electric seems to have automated reporting tools that is on:
https://www.schneider-electric.com/en/about-us/investor-relations/share-information/share-price.jsp
they write:
The Schneider Electric share price and the CAC 40 index are 15 minutes delayed and are automatically updated during the trading day from 9:00 am to 5:30 pm, (Paris time). Note that we are not responsible for any typing or transmission errors. Source : SYMEX ECONOMICS
or do you mean with ACT something completely different? Maybe: https://en.wikipedia.org/wiki/Advanced_Concepts_Team ?
Schneider Electric has a lot of products, as it seems so to say from A (-ct(Nasdaq)) to Z-ealandian dairy that is on the front page they write:
To go from local milk co-op to global industry giant is a bold idea. Fonterra redefined New Zealand's milk industry and became one of the most efficient dairy plants in the world, optimizing EFFICIENCY with Schneider EcoStruxure™ Plant.
in Dahlem Dorf they have an Agroculture Museum they may be interested in new dairy tech, but John?
Comment Source:Jim what due you mean with ACT? The https://en.wikipedia.org/wiki/ACT_(NASDAQ) ? yes schneider electric seems to have automated reporting tools that is on: https://www.schneider-electric.com/en/about-us/investor-relations/share-information/share-price.jsp they write: >The Schneider Electric share price and the CAC 40 index are 15 minutes delayed and are automatically updated during the trading day from 9:00 am to 5:30 pm, (Paris time). Note that we are not responsible for any typing or transmission errors. Source : SYMEX ECONOMICS or do you mean with ACT something completely different? Maybe: https://en.wikipedia.org/wiki/Advanced_Concepts_Team ? Schneider Electric has a lot of products, as it seems so to say from A (-ct(Nasdaq)) to Z-ealandian dairy that is on the front page they write: >To go from local milk co-op to global industry giant is a bold idea. Fonterra redefined New Zealand's milk industry and became one of the most efficient dairy plants in the world, optimizing EFFICIENCY with Schneider EcoStruxure™ Plant. in <a href="https://www.google.com/maps/@52.4519219,13.2860502,17z">Dahlem Dorf </a> they have an <a href="https://en.wikipedia.org/wiki/Dahlem_Manor">Agroculture Museum</a> they may be interested in new dairy tech, but John?
@nad Sorry to send you up the garden of the forking paths*
ACT = Applied Category theory.
Schneider's work is interesting, isn't it?
(Borgesian p-value fishing)
Comment Source:@nad Sorry to send you up the garden of the forking paths* ACT = Applied Category theory. Schneider's work is interesting, isn't it? * (Borgesian p-value fishing)
Two bits of progress:
1) Brendan Fong has arrived at MIT and is starting to enjoy the 3:30 math department teas, and Jacob Lurie's weekly seminar on higher categories and homotopy theory.
2) There's a postdoc for someone working in applied category theory! I strongly urge Jason to apply for this, because it looks exactly like the stuff he does. So, Jason, even if you're hoping to get the U. Penn job, you should apply for this too. I told the guy who emailed me, Spencer Breiner, that you would be good for this position.
Here's the email I got about this:
Dr. Baez,
Hello. My name is Spencer Breiner, and I am a researcher at the US National Institute for Standards and Technology.
I am writing because my colleague, Eswaran Subrahmanian, has recently obtained funding for a one-year postdoc in applied category theory. We are currently searching for candidates, and I am hoping that you would be willing to post an advertisement on the n-Category Café and/or your Azimuth blog. The project itself will draw from your own work with Brendan Fong on passive linear networks.
An announcement for the position is listed below. If you have any question or requests please let me know and I will be happy to provide additional information.
Spencer Breiner
One Year Postdoc Position at Carnegie Mellon/NIST
We are seeking an early-career researcher with a background in category theory, functional programming and/or electrical engineering for a one-year post-doctoral position supported by an Early-concept Grant (EAGER) from the NSF's Systems Science program. The position will be managed through Carnegie Mellon University [PI: Eswaran Subrahmanian], but the position itself will be located at the US National Institute for Standards and Technology (NIST), located in Gaithersburg, MD outside of Washington, DC.
The project aims to develop a compositional semantics for electrical networks which is suitable for system prediction, analysis and control. This work will extend existing methods for linear circuits (featured on this blog!) to include (i) probabilistic estimates of future consumption and (ii) top-down incentives for load management. We will model a multi-layered system of such ``distributed energy resources'' including loads and generators (e.g., solar array vs. power plant), different types of resource aggregation (e.g., apartment to apartment building), and across several time scales. We hope to demonstrate that such a system can balance local load and generation in order to minimize expected instability at higher levels of the electrical grid.
This post is available full-time (40 hours/5 days per week) for 12 months, and can begin as early as October 1st.
For more information on this position, please contact Dr. Eswaran Subrahmanian ([email protected]) or Dr. Spencer Breiner ([email protected]).
Comment Source:7 September 2017: Two bits of progress: 1) Brendan Fong has arrived at MIT and is starting to enjoy the 3:30 math department teas, and Jacob Lurie's weekly seminar on higher categories and homotopy theory. 2) There's a postdoc for someone working in applied category theory! I strongly urge Jason to apply for this, because it looks exactly like the stuff he does. So, Jason, even if you're hoping to get the U. Penn job, you should apply for this too. I told the guy who emailed me, Spencer Breiner, that you would be good for this position. Here's the email I got about this: > Dr. Baez, > Hello. My name is Spencer Breiner, and I am a researcher at the US National Institute for Standards and Technology. > I am writing because my colleague, Eswaran Subrahmanian, has recently obtained funding for a one-year postdoc in applied category theory. We are currently searching for candidates, and I am hoping that you would be willing to post an advertisement on the n-Category Café and/or your Azimuth blog. The project itself will draw from your own work with Brendan Fong on passive linear networks. > An announcement for the position is listed below. If you have any question or requests please let me know and I will be happy to provide additional information. > Thanks for your help, > Spencer Breiner > NIST > One Year Postdoc Position at Carnegie Mellon/NIST > We are seeking an early-career researcher with a background in category theory, functional programming and/or electrical engineering for a one-year post-doctoral position supported by an Early-concept Grant (EAGER) from the NSF's Systems Science program. The position will be managed through Carnegie Mellon University [PI: Eswaran Subrahmanian], but the position itself will be located at the US National Institute for Standards and Technology (NIST), located in Gaithersburg, MD outside of Washington, DC. > The project aims to develop a compositional semantics for electrical networks which is suitable for system prediction, analysis and control. This work will extend existing methods for linear circuits (featured on this blog!) to include (i) probabilistic estimates of future consumption and (ii) top-down incentives for load management. We will model a multi-layered system of such ``distributed energy resources'' including loads and generators (e.g., solar array vs. power plant), different types of resource aggregation (e.g., apartment to apartment building), and across several time scales. We hope to demonstrate that such a system can balance local load and generation in order to minimize expected instability at higher levels of the electrical grid. > This post is available full-time (40 hours/5 days per week) for 12 months, and can begin as early as October 1st. > For more information on this position, please contact Dr. Eswaran Subrahmanian ([email protected]) or Dr. Spencer Breiner ([email protected]).
June 2018 edited August 2018
11 September 2017:
1) You can now see the webpage for the workshop and "summer school" that Brendan and others are running on Applied Category Theory next April in the Netherlands.
Click the link to read more! You grad students should apply for the summer school now. If the math department were serving pizza from 12 to 1 you would not show up at 12:50. Apply the same level of intelligence to this situation.
Accompanying the workshop will be a 4-day summer school for a limited number of early-career researchers, also at the Lorentz Center from April 23 to April 27, 2018, as well as a 16-week series of online seminars for up to 16 PhD students and postdocs called the Kan Extension Lab. Applications are due November 1, and admissions will be notified on November 15.
While attendance at the summer school is not required to attend the online seminar, or vice versa, our intention is for participants to attend both. Participants will have the opportunity to work with established mentors in the field, and will have the opportunity to present their research at the full workshop. Applications are due on November 1.
As I mentioned before, I'll be one of those "established mentors" - but if you get in you should work with someone else, to expand your scope.
2) It hasn't happened yet, but I'll be flying back from Singapore when it does, so:
On Friday, John Foley and Tom Mifflin will give a presentation to some DARPA bigshots about our work on operads. With luck, this will get the more interested in using modern mathematics to design complex networked systems. Here is John's outline, just so you get the idea:
A) Tom Mifflin: Why operads? (5 min)
B) John Foley: An introduction to operads and their algebras (10 min)
a. Operads
b. Algebras of an operad
C) Examples: point-to-point communications (15 min)
a. An operad based on simple graphs
b. An algebra to compose simple graphs
c. An algebra to compose entities with location
d. An algebra for range-limited communications
D) Working at multiple levels with operads (10 min)
a. Layers of abstraction: granularity and detail
b. Formalizing a tool chain to construct operads and compose networks
c. Example of the potential of operads: the recognition principle (time permitting)
E) Conclusion: An R&D path provided by operads (3 min)
a. Match a composition formalism to your domain
b. Leverage the formalism to automate 'recipe authorship'
c. Exploit composition and attempt to integrate with optimization
Comment Source:11 September 2017: 1) You can now see the webpage for the workshop and "summer school" that Brendan and others are running on <a href = "http://www.appliedcategorytheory.org/">Applied Category Theory</a> next April in the Netherlands. Click the link to read more! You grad students should apply for the summer school now. If the math department were serving pizza from 12 to 1 you would not show up at 12:50. Apply the same level of intelligence to this situation. > Accompanying the workshop will be a 4-day summer school for a limited number of early-career researchers, also at the Lorentz Center from April 23 to April 27, 2018, as well as a 16-week series of online seminars for up to 16 PhD students and postdocs called the Kan Extension Lab. Applications are due November 1, and admissions will be notified on November 15. > While attendance at the summer school is not required to attend the online seminar, or vice versa, our intention is for participants to attend both. Participants will have the opportunity to work with established mentors in the field, and will have the opportunity to present their research at the full workshop. Applications are due on November 1. As I mentioned before, I'll be one of those "established mentors" - but if you get in you should work with someone else, to expand your scope. 2) It hasn't happened yet, but I'll be flying back from Singapore when it does, so: On Friday, John Foley and Tom Mifflin will give a presentation to some DARPA bigshots about our work on operads. With luck, this will get the more interested in using modern mathematics to design complex networked systems. Here is John's outline, just so you get the idea: A) Tom Mifflin: Why operads? (5 min) B) John Foley: An introduction to operads and their algebras (10 min) a. Operads b. Algebras of an operad C) Examples: point-to-point communications (15 min) a. An operad based on simple graphs b. An algebra to compose simple graphs c. An algebra to compose entities with location d. An algebra for range-limited communications D) Working at multiple levels with operads (10 min) a. Layers of abstraction: granularity and detail b. Formalizing a tool chain to construct operads and compose networks c. Example of the potential of operads: the recognition principle (time permitting) E) Conclusion: An R&D path provided by operads (3 min) a. Match a composition formalism to your domain b. Leverage the formalism to automate 'recipe authorship' c. Exploit composition and attempt to integrate with optimization
I've finalized the schedule for our Applied Category Theory special session on the weekend of November 4th and 5th. See below for when your talk will take place! Unless your name is Brendan Fong or David Spivak, your talk is 20 minutes long.
We got a couple more talks, including one from Peter Gates on databases built using algebraic theories. Peter works for a company with the delightful name Categorical Informatics, which David Spivak is also involved with.
Brendan, David, Dmitry and perhaps others will be staying on until Monday, Tuesday or Wednesday and talking with us. So, be prepared for lots of interesting conversations!
Special Session SS 4A, Applied Category Theory
Saturday November 4, AM:
9:00-9:40am: David Spivak, 1134-18-32
10:00-10:20am: Dmitry Vagner, 1134-18-21
10:30-10:50am: Christina Vasilakopoulou, 1134-18-34
Saturday, PM:
3:00-3:40pm: Brendan Fong, 1134-18-27
4:00-4:20pm: Blake Pollard, 1134-18-37
4:30-4:50pm: Kenny Courser, 1134-18-30
5:00-5:20pm: Daniel Michael Cicala, 1134-18-26
5:30-5:50pm: Adam Yassine, 1134-18-31
Sunday November 5, AM:
9:00-9:20am: Jason Erbele, 1134-93-103
10:30-10:50 am: Brandon Coya, 1134-18-38
10:00-10:20 am: John Foley, 1134-18-182
10:30-10:50 am: Joseph Moeller, 1134-94-36
Sunday, PM:
2:00-2:40 pm: Vin de Silva, 1134-18-122
3:00-3:20 pm: Evan Patterson, 1134-18-35
3:30-3:50 pm: David P. Ellerman, 1134-94-9
4:00-4:20pm: Ralph L. Wojtowicz, 1134-60-11
4:30-4:40pm: Peter Y. Gates, 1134-68-336
Comment Source:13 September 2017: I've finalized the schedule for our Applied Category Theory special session on the weekend of November 4th and 5th. See below for when your talk will take place! Unless your name is Brendan Fong or David Spivak, your talk is 20 minutes long. We got a couple more talks, including one from Peter Gates on databases built using algebraic theories. Peter works for a company with the delightful name Categorical Informatics, which David Spivak is also involved with. Brendan, David, Dmitry and perhaps others will be staying on until Monday, Tuesday or Wednesday and talking with us. So, be prepared for lots of interesting conversations! Special Session SS 4A, Applied Category Theory Saturday November 4, AM: 9:00-9:40am: David Spivak, 1134-18-32 10:00-10:20am: Dmitry Vagner, 1134-18-21 10:30-10:50am: Christina Vasilakopoulou, 1134-18-34 Saturday, PM: 3:00-3:40pm: Brendan Fong, 1134-18-27 4:00-4:20pm: Blake Pollard, 1134-18-37 4:30-4:50pm: Kenny Courser, 1134-18-30 5:00-5:20pm: Daniel Michael Cicala, 1134-18-26 5:30-5:50pm: Adam Yassine, 1134-18-31 Sunday November 5, AM: 9:00-9:20am: Jason Erbele, 1134-93-103 10:30-10:50 am: Brandon Coya, 1134-18-38 10:00-10:20 am: John Foley, 1134-18-182 10:30-10:50 am: Joseph Moeller, 1134-94-36 Sunday, PM: 2:00-2:40 pm: Vin de Silva, 1134-18-122 3:00-3:20 pm: Evan Patterson, 1134-18-35 3:30-3:50 pm: David P. Ellerman, 1134-94-9 4:00-4:20pm: Ralph L. Wojtowicz, 1134-60-11 4:30-4:40pm: Peter Y. Gates, 1134-68-336
1) Blake finished his thesis and turned it in last Friday!
2) Blake got two offers of postdoc positions! One is from Spencer Breiner at the National Institute of Standards in Gaithersburg Maryland, and one at Siemens - I guess at Princeton? But he may instead go surfing in Peru.
3) John Foley's talk to the DARPA bigshots went well! DARPA may start up more projects connected to operads and categories.
4) Daniel and Kenny finished a paper, put it on the arXiv, and submitted it to Susan Niefield for publication in Theory and Applications of Categories!
Spans of cospans in a topos
Abstract. For a topos T, there is a bicategory MonicSp(Csp(T)) whose objects are those of T, morphisms are cospans in T, and 2-morphisms are isomorphism classes of monic spans of cospans in T. Using a result of Shulman, we prove that MonicSp(Csp(T)) is symmetric monoidal, and moreover, that it is compact closed in the sense of Stay. We provide an application which illustrates how to encode double pushout rewrite rules as 2-morphisms inside a compact closed sub-bicategory of MonicSp(Csp(Graph).
5) Here's what I wrote on Azimuth about our special session on category theory - you can see people's abstracts by clicking on their talk titles. I hope you're all getting ready for this big pow-wow!
Applied Category Theory at UCR (Part 2)
Comment Source:21 September 2017: 1) Blake finished his thesis and turned it in last Friday! 2) Blake got two offers of postdoc positions! One is from Spencer Breiner at the National Institute of Standards in Gaithersburg Maryland, and one at Siemens - I guess at Princeton? But he may instead go surfing in Peru. 3) John Foley's talk to the DARPA bigshots went well! DARPA may start up more projects connected to operads and categories. 4) Daniel and Kenny finished a paper, put it on the arXiv, and submitted it to Susan Niefield for publication in _Theory and Applications of Categories_! * [Spans of cospans in a topos](https://arxiv.org/abs/1707.02098) > **Abstract.** For a topos T, there is a bicategory MonicSp(Csp(T)) whose objects are those of T, morphisms are cospans in T, and 2-morphisms are isomorphism classes of monic spans of cospans in T. Using a result of Shulman, we prove that MonicSp(Csp(T)) is symmetric monoidal, and moreover, that it is compact closed in the sense of Stay. We provide an application which illustrates how to encode double pushout rewrite rules as 2-morphisms inside a compact closed sub-bicategory of MonicSp(Csp(Graph). 5) Here's what I wrote on Azimuth about our special session on category theory - you can see people's abstracts by clicking on their talk titles. I hope you're all getting ready for this big pow-wow! * [Applied Category Theory at UCR (Part 2)](https://johncarlosbaez.wordpress.com/2017/09/21/applied-category-theory-at-ucr-part-2/)
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CommonCrawl
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Yebei, Jepchumba Beatrice; Nyakundi, James Nyambane; Pham, Ba Toan (2019). Academic library as social network site for climate change: Poster presented at IFLA WLIC 2019, Athens, Greece.
Academic Library as a Social network site is made up of an interconnection among students, faculty, researchers, and educators with public areas to exploit Information and Repositories of Climate Change. USIU-A library and information center librarians designed a programme to highlight the effects of climate change to the current society. The library conducted climate literacy classes within the library and in classes. Provided mobile library service that promoted demand driven acquisition of books on climate change. This poster shares the experience of academic libraries being champions of climate change in the user communities they serve and also offers further recommendations to strengthen the campaign through inclusion of NGO's, Volunteer groups, donors among others.
International Federation of Library Associations_and_Institutions (IFLA) (2016). Acces and opportunity for all: How libraries contribute to the United Nations 2030 Agenda.
Jain, Priti; Jibril, Lynn (2018). Achieving sustainable development through libraries: Some preliminary observations from Botswana public libraries: Paper presented at IFLA WLIC 2018, Kuala Lumpur, Malaysia.
Sustainable development in the 21st century is based on access to information and knowledge. Like their counterparts, African libraries have recognized the key role of libraries in achieving sustainable Development Goals (SDGs).The National Library of Uganda provides ICT training to female farmers to access weather forecasts, crop prices, and to set up online markets in their local languages. Nigerian Librarians developed a SDGs Action Group on Facebook to achieve SDGs. Similarly, public libraries in Botswana are actively engaged in achieving sustainable development goals. For example, supporting Goal 8, some public libraries provide basic ICT training to empower community so as to increase access to information for career and job opportunities leading to social and economic development. To support Goal 4: Quality education, some Public Libraries provide homework assistance to children to build strong educational foundation. To address Goal 5, Gender Equality, some public libraries in Botswana organize gender related activities to support girls and women. The aim of this paper is to share Botswana public libraries' initiatives toward sustainable development. The data for this paper was collected through interviews with the librarians, who are actively involved in SDGs and observations.
Kavuri-Mutuku, Purity (2018). Action to combat climate change and its impact: Green library initiatives at the Kenya National Library Service. In Hauke, Petra; Charney, Madeleine; Sahavirta, Harri (Hrsg.), Going green: implementing sustainable strategies in libraries around the world (S. 86-93). De Gruyter Saur. DOI: 10.1515/9783110608878-009
Meyers, Jane Kinney (2016). Addressing SDGs and library relevance by serving the majority: introducing innovative child/youth public library services in Zambia: Poster presented at: IFLA WLIC 2016, Columbus, OH.
Library services to children and teens have been barely existent in Zambia, even though young people account for nearly 70% of the country's population. Undergraduate and master's level LIS education is offered at the University of Zambia, but the programs include no courses on children or youth services. Public libraries in the nation have been poorly supported and not seen to be playing an important role in the lives of its citizens. Libraries do not have acquisitions budgets because of the misperception that book donation programs can provide needed materials, and thus the librarians do not perform the core library activities and functions of defining user needs, collection development, selection and acquisition of materials. Technology and innovation have also evaded the public library sector in Zambia. The poster will show how Lubuto Library Partners (LLP) has collaborated with Zambia's library community and stakeholders for over a decade to address these issues and, in partnership with Government and the Zambia Library Service, create dynamic, impactful, heavily used and highly visible libraries serving children and youth. Photos and data will illustrate LLP innovations, their enthusiastic acceptance by society and the library community, and the role the libraries play in helping Zambia achieve the U.N. Sustainable Development Goals.
Garcia-Febo, Loida; Kim, Ju; Sailo, Lallaisangzuali; Jain, Vinita; Tugwell, Yolanda (2017). Advancing the United Nations Sustainable Development Goals: Experiences of international academic and research libraries. College & Research Libraries News, 78(9), 516-520. DOI: 10.5860/crln.78.9.516
In this article, the author focuses on the experiences of international academic and research libraries. It mentions the launch of the Sustainable Development Goals (SDGs) by the United Nations (UN) to guide development efforts worldwide. It also mentions the working of International Federation of Library Associations and Institutions (IFLA) information and communication technologies (ICTs).
Senyolo, Mokgadi Rebecca; Matolong, Harry (2017). Advocacy for libraries: A South African perspective: Poster presented at IFLA WLIC 2017, Wrocław, Poland.
Libraries have been presented with an unprecedented opportunity to make a contribution to the development agendas, principally the United Nations (UN) 2030 Agenda and the Sustainable Development Goals (SDGs). Libraries need to talk about the valuable work they are doing in their communities by making use of activities, data/ statistics, and stories, to relate to the SDGs and to align themselves with national development plans. Most South Africans still do not have adequate access to libraries or information services. Communities in rural parts of the country have been largely excluded from any meaningful development and suffer from a lack of infrastructure. Illiteracy, unemployment, health care, economic development and education remain major national challenges. Having access to libraries and information could make all the difference to the quality of lives and circumstances for all people. Libraries have a critical role to play in the lives of information deprived communities. Not only do libraries nurture and grow literacy, but also they are better positioned to serve community development and enhance social cohesion.
Bats, Raphaëlle; Delaune, Camille; Insaurralde, Gustavo (2019). Agenda 2030 et les bibliothèques françaises: Poster présenté à IFLA WLIC 2019, Athènes, Grèce.
Faire de sa bibliothèque un pilier pour le développement durable local : un exemple d'advocacy en France. Point d'accès à l'information (comme le rappelle la Déclaration de Lyon (2014), les 65000 bibliothèques et centres de documentation français (source : public libraries 2020) sont des acteurs essentiels du développement durable. Leurs missions leur permettent de répondre à tous les objectifs de développement durable de l'Agenda 2030 de l'ONU. Cependant, en être convaincu ne suffit pas, en avoir la preuve chaque jour au sein de nos établissements ne suffit pas non plus. Il convient de savoir le dire, le montrer et le prouver aussi bien auprès des élus et des publics que des bibliothécaires eux-mêmes. C'est dans cet objectif que l'IFLA a mis en place le programme IAP (international advocacy programme), grâce auquel deux bibliothécaires français ont été formés pour développer des projets d'advocacy basés sur l'Agenda 2030 dans le contexte français. Ce poster vise à présenter les actions menées suite à cette formation pour s'organiser, collecter des données, sensibiliser les bibliothécaires, produire des contenus et mener des premières actions de lobbying et d'advocacy. Le poster montrera l'importance de s'organiser en un groupe de travail efficace, capable d'avoir une audience nationale et de mettre en place un programme d'action pour les années à venir ; en d'autres termes une bonne illustration de l'objectif 17 de l'Agenda 2030. Une bonne organisation ne peut avancer sans matériel sur quoi baser son travail. Aussi, l'étape suivante aura été la collecte d'exemples et d'histoires, pour montrer la participation active des bibliothèques envers tous les objectifs de l'Agenda 2030. Dotés de témoignages, nous avons accompagné nos collègues à la maîtrise d'un discours de plaidoyer sur les bibliothèques actrices du développement durable. Pour cela, nous avons fait des formations et créé un serious game collaboratif. La facilité à défendre les bibliothèques à l'oral n'épargne pas un travail sur les documents qui peuvent être fournis aux élus et décideurs. Aussi, le groupe de travail a créé une base de données, un site web et un fil d'actualités, ainsi qu'une brochure de témoignages d'actions françaises menées pour chacun des objectifs de l'Agenda 2030. Ainsi prêts, nous avons mis en pratique nos acquis dans trois situations de lobbying à Genève (ONU), à New York (Ambassades auprès de l'ONU) et à Bruxelles (Parlement Européen). Ces premières expériences nous ont permis de faire un petit guide pour mieux se préparer à ces rencontres parfois courtes avec les décideurs. Le travail n'est pas fini. Il nous reste encore des outils de sensibilisation (vidéo) à développer, des contacts à consolider (avec une carte de voeux), des formations à assurer (au niveau francophone), etc. Ce poster présente donc un point d'étape de l'action menée pendant deux années pour sensibiliser les professionnels de l'information français à l'Agenda 2030 et à l'expression de leur rôle dans la réalisation de ses 17 objectifs.
Maddaluno, Paola (2019). Agenda 2030: Pratiche bibliotecarie e obiettivi di sviluppo sostenibile: AIB Notizie
AIB Associazione italiana biblioteche (2020). Agenda 2030: la rubrica su ''AIB Notizie'' promossa da AIB, Gruppo per l'attuazione degli obiettivi di sviluppo sostenibile dell'Agenda ONU (SDGs)
Gaspar Pinto, Leonor; Ochôa, Paula (2018). Alinhamento e evidências do contributo para o desenvolvimento sustentável: Guia para bibliotecas.
Este Guia para bibliotecas insere-se no âmbito das ações do Projeto Bibliotecas para o Desenvolvimento e a Agenda 2030 da ONU (abril-julho 2018), uma iniciativa da B.A.D. - Associação Portuguesa de Bibliotecários, Arquivistas e Documentalistas em articulação com a IFLA - International Federation of Federation of Library Associations and Institutions (International Advocacy Programme) para valorização do contributo das bibliotecas portuguesas para o desenvolvimento sustentável a nível local, regional e nacional. Foi desenvolvido por uma equipa de investigadoras do CHAM-Centro de Humanidades (NOVA FCSH e Universidade dos Açores). Alignment and Evidences of the Contribution for Sustainable Development: Guide for Libraries is one of the activities undertaken to complete the Project Libraries for Development and the UN Agenda 2030 /April- Jully 2018) - an initiative of the B.A.D. -- the Portuguese Association of Librarians, Archivists and Documentalists in conjunction with IFLA - International Federation of Library Associations and Institutions (International Advocacy Program) to enhance the contribution of Portuguese libraries to sustainable development at local, regional and national levels. This Guide for Libraries was developed by a team of researchers from CHAM - Centre for the Humanities (NOVA FCSH and University of the Azores).
Smith Aldrich, Rebekkah; Bollerman, Mathew (2019). All together...how? Building capacity & commitment for change: Paper presented at IFLA WLIC 2019, Athens, Greece.
Changing the mindset of a profession needs a strategy: deliberate choices to do things differently, building on our unique strengths for a specific outcome. The New York Library Association's Sustainability Initiative has set out to do just that and has employed techniques to accelerate its success to help professionals, institutions and communities adopt the concept of "sustainable thinking." This paper explores the beliefs, philosophies and guiding principles that accelerated the work of this initiative that are replicable for wider adoption of sustainability work in our profession.
Jenkin, Tracy A.; Webster, Jane; McShane, Lindsay (2011). An agenda for `Green' information technology and systems research. Information and Organization, 21(1), 17-40. DOI: 10.1016/j.infoandorg.2010.09.003
Green information technologies and systems refer to initiatives and programs that directly or indirectly address environmental sustainability in organizations. Although practitioners have begun to focus on `Green IT', there is little research in this area. To set the stage for this research, we develop a multilevel research framework to guide future research. To do so, we review the existing green information technology and systems literature, and also draw more broadly from research that addresses environmental sustainability in the management, environmental psychology, and social marketing domains. From this review, we identify important research gaps and present a set of propositions to guide future research.
Chowdhury, Gobinda (2012). An agenda for green information retrieval research. Information Processing & Management, 48(6), 1067-1077. DOI: 10.1016/j.ipm.2012.02.003
Nowadays we use information retrieval systems and services as part of our many day-to-day activities ranging from a web and database search to searching for various digital libraries, audio and video collections/services, and so on. However, IR systems and services make extensive use of ICT (information and communication technologies) and increasing use of ICT can significantly increase greenhouse gas (GHG, a term used to denote emission of harmful gases in the atmosphere) emissions. Sustainable development, and more importantly environmental sustainability, has become a major area of concern of various national and international bodies and as a result various initiatives and measures are being proposed for reducing the environmental impact of industries, businesses, governments and institutions. Research also shows that appropriate use of ICT can reduce the overall GHG emissions of a business, product or service. Green IT and cloud computing can play a key role in reducing the environmental impact of ICT. This paper proposes the concept of Green IR systems and services that can play a key role in reducing the overall environmental impact of various ICT-based services in education and research, business, government, etc., that are increasingly being reliant on access and use of digital information. However, to date there has not been any systematic research towards building Green IR systems and services. This paper points out the major challenges in building Green IR systems and services, and two different methods are proposed for estimating the energy consumption, and the corresponding GHG emissions, of an IR system or service. This paper also proposes the four key enablers of a Green IR viz. Standardize, Share, Reuse and Green behavior. Further research required to achieve these for building Green IR systems and services are also mentioned.
Ahn, In-Ja; Kwak, Chul-Wan; Noh, Younghee (2013). An analysis of literature trends in green library. Journal of the Korean BIBLIA Society for library and Information Science, 24(1), 189-205. DOI: 10.14699/kbiblia.2013.24.1.189
The purpose of this study is to describe the trends and contents of literature in green library and to suggest the research direction on green library. Earlier studies focus on the green library architecture and necessity, after then green library management studies have proliferated. Green library architecture studies have emphasized on 'energy and atmosphere' and 'indoor environmental quality' based upon 5 categories of LEED evaluation system. Green library management studies have focused on library resources recycling and utilizing energy through effective management of library collection. Based on the analysis of literature, five research directions are suggested, such as the number of libraries in an area, library shelf position and space, library space program, collection ratio for library storage, and user studies.
Morehart, Phil (2020). Answering the sustainability question: How libraries can embed this core value into their work. American Libraries, 51(January 25)
In 2019, the American Library Association (ALA) adopted sustainability as a core value of the library profession. But what does that mean, and how can it be implemented? Rebekkah Smith Aldrich and Matthew Bollerman tackled those questions and more at ``Sustainability Is Now a Core Value. So \($\ldots$\) Now What?'', a Symposium on the Future of Libraries session on Saturday, January 25, at the ALA 2020 Midwinter Meeting & Exhibits in Philadelphia.
Sánchez, R. A.; Geada, M. R.; Rodríguez, M. G.; Marraud, G. (2014). Aportaciones al estudio de la sostenibilidad bibliotecaria: las emisiones de CO₂ del servicio de apertura de fin de semana de la biblioteca de la Universidad de Vigo. Anales de Documentación, 17(1), 1-11. DOI: 10.6018/analesdoc.17.1.183881
Resumen: La Biblioteca de la Universidad de Vigo y la Ofici na de Medio Ambiente llevaron a cabo un estudio sobre los impactos ambientales del servicio de aper turas en fin de semana de la biblioteca central dur ante el período de exámenes extraordinarios de julio de 2013. El es tudio pone de manifiesto que las emisiones de gases de efecto invernadero (GEI) asociadas a las aperturas provien en sobre todo del consumo de energía eléctrica del edificio y de los desplazamientos de los usuarios hasta la biblio teca. Asimismo propone una metodología de cálculo d e la huella ecológica de este tipo de servicio con objeto de fa cilitar la toma en consideración de los criterios d e sostenibilidad en la programación y organización de este tipo de servicio.
Haridasan, Sudharma; Firdaus, Shamama (2021). Application of web-based sources and services for sustainable development in university libraries. Library Philosophy and Practice, 22(February)
This study aims to explore the provision of web-based sources and services through the websites of university libraries in India. The present study adopted a content analysis method for examining various web-based library services and web-based sources provided through the university library websites with a checklist. A total of 50 top-ranked Indian universities were listed from the National Institutional Ranking Framework (NIRF) ranking list for the year 2020 in the category university. From these universities to be included as a sample for the present study; each university must have an independent library website. Secondly, the university library should be offering web-based library services and resources through the library website. Based on these criteria, 42 library websites were shortlisted as a sample for this study. The study found that majority of the libraries were offering traditional library services such as referral service, and library catalogue in a web-based environment more extensively compared to innovative and modern present-day services like library application (App), Instant Messaging (IM), and virtual library tours. Moreover, libraries are yet to explore the full potential of social media services and web 2.0 tools. A majority of the library websites were lagging in the provision of these services. It is suggested that libraries should make efforts to provide social media services extensively as the current generation are the predominant users of social media; hence it is evident on the libraries' part to make efforts to reach out to a wide population of their library users. This research is the first study of its kind highlighting the significance of university library websites in the provision of web-based services and sources for sustainability, and exploring the role of academic libraries in the promotion of sustainable development by processes like collection development on sustainable development themes, developing information literacy skills, spreading awareness, and organizing events on sustainable development and related areas.
Anglada, Lluís (2014). Are libraries sustainable in a world of free, networked, digital information?. El Profesional de la Informacion, 23(6), 603-611. DOI: 10.3145/epi.2014.nov.07
The evolution of libraries through three stages –modernization, automation and digitization- is analysed. A formula is pre-sented to evaluate the importance of libraries to our society, and it is applied both retrospectively and futuristically, extra-polating a 2030 scenario. The conclusion is that if the current generation of librarians does not introduce radical changes in the role of libraries, their future is seriously threatened.
Metibogun, Lesley; Potangaroa, Regan (2019). Assessment of occupants' thermal condition and the risk of overheating in sustainable public libraries in Auckland Region: Paper presented at IFLA WLIC 2019, Athens, Greece.
This paper examines the thermal condition of occupants and the risk of overheating in two award winning sustainable public libraries in Auckland region of New Zealand through long term environmental monitoring of the building performance and occupants' survey. The survey was performed on eight staff of the library who have occupied the buildings for more than two years. The data were further contrasted with the risk of overheating assessment which were based on thermal comfort design parameters for free-running buildings. The standards that define the risk of overheating used in this study are CIBSE A 2006 standard the British Standard EN 1525; the TM52 CIBSE; and Building Bulleting 101 (BB101). The results demonstrate that none of the libraries are at the risk of overheating during the summer season. However, the building occupants claimed that in certain periods, the buildings can get overheated, even though, adaptive comfort criteria was used to design these libraries. The discrepancies between the overheating risk methods and the surveys can be further explained by the fact that none of the overheating methods considered the effects of the water content in the air. Such parameter is proven to have an effect on the capacity of the body to reduce its own temperature though the evaporative effect in the skin. Therefore, a standard that takes into account the humidity ratio such as the ASHRAE 55 might shed light to understand the thermal comfort responses of the occupants. The study provides insight into the socio-technical issues in overheating risk assessment in the context of public libraries in temperate climates.
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Method | Open | Published: 22 January 2019
Fast Adipogenesis Tracking System (FATS)—a robust, high-throughput, automation-ready adipogenesis quantification technique
Chengxiang Yuan1,
Smarajit Chakraborty1,
Krishna Kanth Chitta2,
Subha Subramanian1,
Tau En Lim1,
Weiping Han1,
K. N. Bhanu Prakash2 &
Shigeki Sugii ORCID: orcid.org/0000-0002-2537-95341,3
Stem Cell Research & Therapyvolume 10, Article number: 38 (2019) | Download Citation
Adipogenesis is essential in in vitro experimentation to assess differentiation capability of stem cells, and therefore, its accurate measurement is important. Quantitative analysis of adipogenic levels, however, is challenging and often susceptible to errors due to non-specific reading or manual estimation by observers. To this end, we developed a novel adipocyte quantification algorithm, named Fast Adipogenesis Tracking System (FATS), based on computer vision libraries. The FATS algorithm is versatile and capable of accurately detecting and quantifying percentage of cells undergoing adipogenic and browning differentiation even under difficult conditions such as the presence of large cell clumps or high cell densities. The algorithm was tested on various cell lines including 3T3-L1 cells, adipose-derived mesenchymal stem cells (ASCs), and induced pluripotent stem cell (iPSC)-derived cells. The FATS algorithm is particularly useful for adipogenic measurement of embryoid bodies derived from pluripotent stem cells and was capable of accurately distinguishing adipogenic cells from false-positive stains. We then demonstrate the effectiveness of the FATS algorithm for screening of nuclear receptor ligands that affect adipogenesis in the high-throughput manner. Together, the FATS offer a universal and automated image-based method to quantify adipocyte differentiation of different cell lines in both standard and high-throughput workflows.
Fat, or adipose tissue, is well known as an important endocrine and energy storage organ. Abnormally high or low levels of the tissue cause predisposition to diseases such as diabetes, inflammation, and even cancer [1]. In vitro adipogenesis is an essential technique for the study of fat tissue and is frequently used in the study of human disease modelling and drug screening. The adipogenesis protocol is also an important tool for examining trilineage differentiation of mesenchymal stem cells (MSCs), which are actively investigated for clinical applications [2]. Adipose-derived stem cells (ASCs) are also popular clinical sources of MSCs, due to their ease of isolation and abundance [3, 4]. One of the standard criteria to determine authenticity of MSCs/ASCs is their good differentiation capacity into adipocytes, along with chondrocytes and osteoblasts. However, while the adipogenesis protocol is relatively well established, there is no systematic method established for robust quantification of cells undergoing differentiation.
Induced pluripotent stem cells (iPSCs) make up a recent powerful addition to the arsenal of the cell and lipid biologist. Due to their ability to de-differentiate from and also differentiate into cells from all three germ layers, they can be used to study adipogenesis in ways that were not previously possible. For example, iPSCs have been successfully used to study human lipodystrophy caused by the BSCL2 gene mutation, which manifests as the near-complete absence of fat deposits in patients, resulting in the inability to extract adipose tissues from patients for analysis [5]. iPSCs can be used to overcome this issue by converting other cells, such as skin fibroblasts, into iPSCs, which can then be differentiated into adipocytes. This allows for the adipogenic effect of the mutations to be studied in vitro.
However, iPSCs also introduce new challenges in accurately quantifying adipogenesis. Despite the relatively high specificity of the available dyes in staining lipid droplets, they are susceptible to several kinds of off-target staining. This is especially visible when the dyes are applied to cell types that naturally form clumps such as embryoid bodies (EBs) derived from embryonic stem cells (ESCs) and iPSCs, due to nonspecific uptake of the dyes into the cell clumps. Additionally, ESCs/iPSCs also exhibit highly heterogeneous patterns of differentiation due to the random differentiation process of EB formation introducing additional uncertainty into the adipogenesis process [6]. Therefore, it is often difficult to quantify the fraction of ESC/iPSC-derived adipogenic cells in a precise and comprehensive manner.
Nile Red and Oil Red O are widely used neutral lipid stains which are useful in detecting the adipogenesis of various types of cultured cells [7]. While both dyes have been important in the analysis of adipogenesis in either cell culture or fixed tissue samples, there has been relatively little literature on accurate and reliable methods to quantify their signals in cell culture samples. Oil Red O is a member of the Sudan red family of dyes and has been used since the 1950s as an efficient stain of neutral lipids [8]. Its mechanism of action is based on the high solubility of Oil Red O in lipids versus the isopropanol solution in which the dye is dissolved [9]. Nile Red, also marketed as AdipoRed™, is a fluorescent dye identified in the 1980s. The dye fluoresces strongly while in lipophilic environments, but is quenched by aqueous environments, allowing it to act as a lipid-specific stain [7].
Existing protocols of quantification generally involve either dye extraction for Oil Red O [10] and fluorescence measurements for Nile Red stains by taking the ratio of nuclear stain strength to lipid stain strength and area [11, 12]. More often, however, the quantification is done by manual counting or visual estimation, which can be very time-consuming, prone to observer bias, and only capable of covering small samples sizes. This often leads to inaccurate measurements of adipogenesis due to the heterogeneous nature of differentiation requiring large samples to obtain statistically relevant results. The problem of these methods gets worse because standard adipogenesis protocols usually let cells become overconfluent, which makes identification of individual cells more difficult.
Currently available software techniques are capable of detecting and quantifying the round lipid droplets found in adipogenic cells [13, 14], but their application in ESCs/iPSCs are limited due to the high density of overlapping cells in EB outgrowths preventing the detection of lipid droplets. Therefore, these techniques are limited to detecting adipogenesis in monolayer cultures with relatively low confluence. Furthermore, these techniques utilise the absolute quantity of fluorescence or absorbance from a particular field and, as a result, are vulnerable to differences in cell density and changes in background fluorescence levels due to nonspecific staining.
To the best of our knowledge, this is the first image analysis technique which is specifically designed to tackle the difficult issue of measuring adipogenesis of ESCs/iPSCs by taking adipogenic degrees of individual cells into account, while maintaining its detection accuracy in monolayer cell lines such as ASCs and 3T3-L1 preadipocytes. Additionally, this technique can also recognise image data from different types of stains and be programmable to fully automated functions, which is useful for analysis of high-throughput drug screening.
Cell culture and differentiation
All cells were cultured in various culture containers in humidified 5% CO2 incubators at 37 °C. Use of human patients-derived cells was conducted with informed consent obtained for each subject, approved by the National Healthcare Group Domain Specific Review Board, Singapore, and performed in accordance with its relevant regulations.
Differentiation of 3T3-L1 cells
Mouse 3T3-L1 preadipocytes were cultured following the conditions described by Yang et al. [15]. In brief, cells were proliferated to 70–80% confluence in T75 or T175 flasks using Dulbecco's modified Eagle's media (DMEM) supplemented with 10% heat inactivated newborn calf serum, and were passaged upon reaching 70% confluence using a 0.25% solution of trypsin/EDTA (Gibco). Differentiation was carried out on gelatin or poly-l-lysine-coated 96-well plates, by first seeding the cells (20,000 per well in 96-well plates) and letting them grow to confluence. On day 0, (2 days after the cells have reached confluence), adipogenesis was induced with 3T3 Induction Medium 1, consisting of DMEM supplemented with 10% heat-inactivated fetal bovine serum (FBS), 1 μM dexamethasone, 0.5 mM 3-isobutyl-1-methylxanthine (IBMX), and 1 μM insulin. On day 2, the cells had their media changed to 3T3 Induction Medium 2, consisting of DMEM, 10% FBS, and 1 μM insulin. On days 4 and 7, the cells had their media changed to 3T3 maintenance media (DMEM with 10% FBS), and the cells were ready for imaging on day 10.
Differentiation of human iPSCs and ESCs
Human iPSCs and ESCs were cultured following the conditions described by Sugii et al. [16]. Briefly, iPSCs and ESCs were maintained under feeder-free conditions in mTESR1 media (Stem Cell Technologies) on Matrigel (Corning) in 6-well plates. The cells had their media changed daily, except on weekends, when double the quantity of media was added and the cells were fed after 48 h. Passaging of the cells was performed every 5–7 days, usually when the iPSC clumps reached a size of 2 mm, and was carried out enzymatically by treatment of the cells with a 0.2% solution of dispase (Thermo Fisher) in DMEM-F12 medium for 6–7 min.
Differentiation of the iPSCs and ESCs was carried out following a modified version of the conditions described by Mohsen-Kanson et al. [17]. The cells were first developed into embryoid bodies (EBs) by detaching the cells using collagenase and transferring them onto ultra-low attachment 6-well plates. The EBs were then subjected to formation medium (EB medium) consisting of DMEM-F12 base media with 20% Knockout Serum Replacement (KOSR). On days 3, 4, and 5 following the detachment, the cells were fed with EB medium supplemented with 1 μM of retinoic acid. After the treatment, the cells had their media changed with EB medium every 2 days. On day 11, the EBs were plated onto gelatin-coated 24-well plates and maintained in EB media, changing the medium every 2 days. On day 20, the cells had their media changed to EB adipogenic differentiation media (DMEM/F12, 10% KOSR, 1 μg/ml insulin, 0.5 mM IBMX, 0.25 μM dexamethasone, 0.2 nM triiodothyronine (T3), and 1 μM rosiglitazone). The media were then changed every 3 days. On day 30, the cells were ready for imaging.
Derivation of mesenchymal stem cells from iPSCs and ESCs
The iPSC-derived MSCs were first treated in an identical manner to the EB protocol above, except that on day 20, the cells were dissociated completely with trypsin and transferred to T75 plates. The media was then changed to MSC maintenance medium (MSCmm), consisting of DMEM, 15% FBS, 0.5 ng/ml basic fibroblast growth factor (bFGF), and 1X Non-essential amino acids (Gibco). The cells were then passaged twice, with residual EB clumps removed by aspiration during media changes. The resulting iPSC/ESC-derived MSCs were then treated identically with ASCs below for culture purposes.
Immortalisation of human adipose-derived stem cells
Subcutaneous and visceral human ASCs (previously described in Ong et al. and Takeda et al. [18, 19]) were immortalised using the protocol detailed in Chen et al. [20]. Briefly, lentiviruses were produced using the pLVX-puro-Myc plasmid and the Lenti-X HT Packaging System. The culture media was collected after 48 h, filtered with a 0.2-μm filter, and then used to infect the ASCs with the presence of 4 μg/ml polybrene. The ASCs were then selected with 2 μg/ml puromycin for 3 days, following which they were continuously cultured in MSCmm. The immortalised ASCs retained normal differentiation levels up to passage 30 (data not shown).
Differentiation of human MSCs and ASCs
Human ASCs and iPSC-derived MSCs were proliferated to 80–90% confluence in T75 or T175 flasks using MSCmm and were passaged every 3–6 days upon reaching 90% conference by a 0.25% trypsin/EDTA solution.
Differentiation was carried out on gelatin or poly-l-lysine-coated 96-well plates, by first seeding the cells (20,000 per well in 96-well plates) and letting them grow to confluence. Two days after the cells have reached confluence (day 0), adipogenesis was induced with ASC induction media 1 (DMEM, 10% FBS, 1 μM dexamethasone, 0.5 mM IBMX, 1 μM insulin). On day 3, the cells were further induced with ASC induction media 2 (DMEM, 10% FBS, 1 μM insulin), and the cells then had their media changed on days 6 and 9 to maintenance media (DMEM, 10% FBS). To induce browning, differentiated ASCs on day 12 were treated with either 10 μM Forskolin or DMSO control for 6 h. The cells were ready for imaging on day 12.
Drug screening using nuclear receptor ligand library
Differentiation was carried out as per the 3T3-L1 and iPSC-derived MSCs protocols above, except that after the cells have become confluent 2 days before adipogenic induction (day 2), the cells were treated with 10 μM each of nuclear receptor ligands (Enzo Life Sciences). The treatment was continued for a second time during the addition of the first differentiation medium till day 2, and the adipogenesis protocol was carried out as per normal to completion. Two independent experiments were carried out in 3T3-L1 cells, each in triplicate plates, and average adipogenic scores were calculated from total of six plates. Adipogenic scores from human iPSC-derived cells were estimated from a single plate.
Fluorescent dye staining of cells
Cells were first washed with 1X phosphate-buffered saline (PBS) with Ca2+ and Mg2+, and then a mix containing 10 μg/ml of Hoechst 33342 and a 40× dilution of Nile Red (AdipoRed, Lonza) was used to stain the cells. The cells were covered in aluminium foil and incubated at room temperature on a rocking incubator for 15 min before fluorescence imaging. The phase contrast images were taken from the same fields and adjusted for the dynamic range (contrast adjusted in Photoshop to take up the entire dynamic range of the image).
Oil Red O/haematoxylin staining of cells
Oil Red O stock solution was prepared by dissolving 30 mg of Oil Red O powder in 100 ml of 100% isopropanol and readied by dilution to 60% with water and passing through a 0.2-μm syringe filter.
Cells were washed with 1X PBS with Ca2+ and Mg2+ and then incubated with 10% neutral buffered formalin for 60 min. The cells were then washed with distilled water three times, incubated in 60% isopropanol for 5 min at room temperature, and then incubated in Oil Red O working solution for another 5 min. The cells were finally destained with distilled water until no more Oil Red O was dissolved into the water. Haematoxylin counterstain was applied by incubation with modified Harris haematoxylin for 1 min at room temperature, and then the cells were destained with distilled water until no more haematoxylin was dissolved into the water.
Fluorescence measurements
Fluorescence intensity of adipocyte staining was measured with a Molecular Devices SpectraMax M2 plate reader for both 96 well and 24 well plates. For the Nile Red lipid stain, an excitation/emission wavelength of 485/572 nm was used, and for the Hoechst 33342 nucleus stain, an excitation/emission wavelength of 353/483 nm was used.
High-throughput fluorescence imaging
Images were acquired using a Molecular Devices ImageXpress Micro system at × 10 or × 20 magnification. The DAPI fluorescence channel was used to image the nuclei stained by Hoechst 33342, while the FITC fluorescence channel was used to image the lipid droplets stained by Nile Red. The Nile Red fluorescence peak was also broad enough to be visualised using either the TRITC or the Texas Red fluorescence channels, with no significant losses in sensitivity.
Colour imaging of Oil Red O stains
Colour images were acquired using a Nikon N-STORM microscope system at × 20 magnification. In order to convert the Oil Red O colour images into fluorescent images, the three colour channels of the image (red, green, and blue) were analysed in order to find regions of the image that were red (indicating presence of lipid droplets stained by Oil Red O) or blue (indicating presence of nuclei stained by haematoxylin). The difference images were then fed into the FATS algorithm to be analysed.
FATS algorithm and image analysis
The overall flow chart for the FATS algorithm and analysis is depicted in Fig. 1. We made use of lipid droplets staining by Nile Red and cell nuclei staining by Hoechst 33342 for our image-based algorithm, FATS. These fluorescent dyes would generally be much brighter than the background and have a sharp rise in brightness near the edges. However, when the cell densities become very high, which is often the case with overconfluency procedure of the adipogenesis protocol, the background staining also increases, in some cases getting brighter than the signal in other areas of the same image. To counteract this effect, a local intensity filter was used to detect local high-intensity regions, which greatly reduces the negative impact of background signals. The Python-OpenCV (https://github.com/itseez/opencv) and Mahotas [21] libraries were used to perform the image analysis. For ease of use, the iPython/Jupyter Notebook integrated development environment (IDE) [22] was used to simplify operation. The results were outputted and formatted using the Matplotlib libraries [23].
Flow chart of the FATS algorithm. The nuclei channel (blue) and the lipid channel (green) are generated from either the fluorescence or the colour images by calculation of the blue and red bias in the colour channels. The resulting two-channel image is then binarised, the nuclei identified, and the lipid scores calculated
Binarisation algorithm
Background subtraction and correction is an essential step in any automatic image analysis. A 2D Gaussian kernel G with dimensions z × z (where z is an odd integer) is generated from the image (Eq. 1). The value of z depends on the average sizes of the objects being detected, in this case the nuclei and groups of lipid droplets.
$$ {\mathrm{G}}_{x,y}=\alpha {e}^{-\frac{x^2+{y}^2}{2{\sigma}^2}},\sigma =0.3\left(\frac{z-1}{2}-1\right)+0.8 $$
The sizes of the kernels depend on the average sizes of the objects, and we found empirically that kernels with sizes two to three times the size of the average nucleus group produced the best results for 3T3-L1, MSC, and iPSC cells at high cell densities. In this case, we used a kernel size of 33 μm for nuclei and 66 μm for lipid droplet detection.
The factor α works as the correction factor that normalises sum of the entire Gaussian kernel to 1, where e is the base of the natural logarithm. The Gaussian kernel is convolved with overlapping sub-regions of the image In (Eq. 2), and the threshold value Tx,y for each sub-region is computed as the mean of the pixels in the convolution product.
$$ {\mathrm{T}}_{x,y}=\frac{1}{z^2}\sum \limits_{a=-\frac{z-1}{2}}^{\frac{z-1}{2}}\sum \limits_{b=-\frac{z-1}{2}}^{\frac{z-1}{2}}{\mathrm{G}}_{a,b}{\mathrm{In}}_{x-a,y-b} $$
This process is then iterated over the entire input image, producing the threshold image T.
$$ {\mathrm{Out}}_{x,y}=\left\{\begin{array}{c}255,{\mathrm{if}\ \mathrm{In}}_{x,y}>{\mathrm{T}}_{x,y}\\ {}0,\mathrm{otherwise}\end{array}\right. $$
Finally, the threshold image T is compared to the input image In (Eq. 3). If the threshold value for a given pixel is above that of the original input image, the pixel is considered to have a value of 255 (the maximum); otherwise, it has a value of 0. Since the thresholding takes into account the relative prominence of the pixel above its neighbours, it will be able to detect positive stains even in the presence of heavy background staining. While this step improves specificity of detection, it also loses some resolution due to the convolution process. As a result, the algorithm requires a pixel pitch of < 2 μm/pixel for optimal processing of the nuclei and lipid droplets.
After thresholding and removing the background, the nuclei were detected and segmented into individual regions. User-definable filters were applied on the aspect ratio (length to width) as well as sizes of the regions identified after binarisation. Regions that are too large, too small, or incorrectly shaped were excluded from the analysis. For example, dead cells often have highly misshapen nuclei distinguishable from live cells and could be removed through the filter. Then, a watershed algorithm was applied to divide nuclei that were joined together due to their proximity.
$$ \mathrm{S}={\mathrm{P}}_T\times {\mathrm{I}}_s $$
The algorithm then searched for any lipid droplets in the vicinity of the nuclei using a user-definable distance outside of the bounding box of the nucleus. During our testing, we found that a radius of 3 μm was the optimal distance for balancing false positives and negatives when the cells were present in high densities. A lipid score of each nucleus S was calculated by a proportionate amount based on the percentage of area above threshold PT and intensity of the stain Is (Eq. 4). The lipid score of all the nuclei in the well was then tabulated, producing a histogram of adipogenic populations within the well. If the lipid score of a nucleus exceeds a pre-set gate threshold, the cell would be marked as differentiated, and otherwise the cell would be marked as non-differentiated. An example of analysis of differentiated cells is shown in Fig. 2, along with intermediate steps in the algorithm.
Representative analysis of 3T3-L1 cells after adipogenic stimulation. The representative phase-contrast image (a), binarised green lipid channel (b), and blue nuclei (c) channels are shown. The final calculated score for each of the detected nuclei is shown in d, where the number within each nucleus indicates the calculated lipid score for each of the nuclei. Note that blue-stained regions that were too small were excluded from the calculations. Finally, a histogram is created for all of the nuclei counted within the well, and the percentage of nuclei that have lipid scores above the threshold (in green dots) is indicated as adipogenic score in the upper right corner (e). For all the histograms, X-axis is the lipid score and Y-axis is the number of cells with the particular score. The vertical grey line indicates mean lipid fluorescence levels per nuclei for that cell population. Scale bars of images, 100 μm
FATS analysis of 3T3-L1 preadipocytes in the time-dependent manner
We first tested the FATS algorithm on a time course assay using the 3T3-L1 preadipocyte line, which is commonly used to study adipogenesis. The cells were treated with the differentiation protocol for a varying number of days, then imaged and analysed (Fig. 3). The lipid score was computed for each nucleus by measuring the intensity and area of lipid droplets fluorescence around it. Histograms were created that indicate the lipid fluorescence level in X-axis and the count of cell nuclei in Y-axis. The adipogenic score was estimated by counting the percentage of nuclei having the lipid score above a gate threshold. The vertical grey line indicates mean lipid fluorescence levels per nuclei for that cell population.
Time course assay of 3T3-L1 adipogenesis. The representative fields of phase-contrast (top row), nuclei and lipid droplets detection (middle row), and histograms from a whole well (bottom row) are shown for the five samples: undifferentiated (a), day 2 (b), day 4 (c), day 7 (d), and day 10 (e). Scale bars 100 μm. f The left graph shows average fluorescence measurements of Nile Red staining by spectroscopy (blue bars; left y-axis; n = 8) and average adipogenic scores (orange bars; right y-axis; n = 8–16) for each time points. The right graph indicates correlation between the spectroscopic fluorescence measurements (standard) and adipogenic scores (by FATS). Results are presented as mean ± standard error of mean (SEM)
There was a clear increase in the adipogenesis levels determined by the FATS algorithm during the time course. Consistent with known adipogenic development of 3T3-L1 cells, undifferentiated and day 2 cells showed minimal levels of lipid droplets (0.29% and 0.08%), a small number of day 4 cells started to develop lipid droplets (1.04%), substantial percentage of day 7 cells contained significant levels of lipid droplets (49.96%), and most of the day 10 cells have fully differentiated into mature adipocytes (66.01%). Additionally, the average adipogenic score of day 10 cells also increased versus day 7. In order to validate our method, fluorescence intensities of lipid droplet staining were measured by spectroscopy, which has been a commonly used standard method. The results indicate good correlation (R2 = 0.96267) of spectroscopic lipid droplet measurements with adipogenic scores by FATS analysis for all the time points (Fig. 3f). Besides the spectroscopic analysis, we have performed quantitative PCR to measure expression of five common adipogenic genes from days 0 to 12 of adipogenesis (PPARG, FABP4, CD36, ADIPONECTIN and LEPTIN; Additional file 1: Figure S1). Our correlation analysis indicates that there is also good association between each gene expression and adipogenic score (Additional file 1: Figure S1). It is also possible to modify the analysis method and measure mean or ranges of radius per lipid droplet. The distribution of lipid droplet radii indicates that adipogenesis is associated with increases in mean droplet sizes and emergence of larger droplets, especially between days 4 and 7, which is consistent with changes in the adipogenic scores (Additional file 1: Figure S2).
This result thus validates applications of the FATS in estimating percentage of cells having mature lipid droplets in the real time-dependent manner. Unlike the commonly practised means to manually and roughly estimate percentage of mature adipocytes by observers, the FATS analysis is capable of calculating the adipogenic percentage more in the subjective and precise manner.
FATS can determine adipogenesis and browning levels of cell lines besides 3T3-L1
In order to confirm applicability of the FATS algorithm in other cell lines than 3T3-L1, we subsequently ran the analysis on two human immortalised ASC cell lines and three MSC cell lines derived from iPSCs and ESCs (Fig. 4). It is known that under standard differentiation protocols in vitro, human subcutaneous fat-derived ASCs differentiate into adipocytes much better than visceral fat-derived ASCs [18, 19]. The FATS analysis proved and quantified these differences in subcutaneous ASCs (63.46%; Fig. 4a) and visceral ASCs (10.0%; Fig. 4b). We also tested and validated the FATS on adipogenesis of three MSC-like cell lines derived from pluripotent stem cell lines including iPSCs and ESCs (26.67–40.14%; Fig. 4c–e). The Nile Red staining used for the FATS is clearly more sensitive in detecting lipid droplets, compared to recognising lipid droplet structures in phase contrast images. Similar to 3T3-L1 cells, time course of adipocyte differentiation was studied in human iPSC-derived MSCs. The result indicates time-dependent increases in adipogenic scores during adipogenesis of these cells, consistent with those from 3T3-L1 cells (Additional file 1: Figure S3). In order to test if the FATS can be used to measure browning differentiation as well, subcutaneous-derived ASCs were differentiated into white adipocytes (Normal), followed by browning treatment (Browning). Additional file 1: Figure S4A shows the FATS analysis of representative staining images from normal and browning adipocytes. The results indicate that the FATS reliably quantifies significant drop in percentage of cells containing positive lipid droplets in browning adipocytes compared with white adipocytes, which is in line with standard quantification by spectroscopy (Additional file 1: Figure S4B and S4C). Together, these data provide applicability of the FATS algorithm beyond 3T3-L1 cell lines or standard adipogenesis.
Adipogenesis analysis of human ASCs and iPSC-derived MSCs. The nuclei/lipid droplet detection of representative fields (top row), phase contrast fields (middle row), and whole well histograms (bottom row) are shown for the five samples: human immortalised subcutaneous ASCs (a), human immortalised visceral ASCs (b), human MSCs derived from iPSC line 1 (c), and human MSCs derived from iPSC line 2 (d), and human MSCs derived from HUES6 ESC line (e). Scale bars, 100 μm
FATS is particularly useful for characterising adipogenesis of pluripotent stem cell lines
Next, we carried out the FATS analysis in direct adipogenesis of ESC/iPSC-derived EBs, which has been especially challenging to quantify. The result with Hoechst and Nile Red staining indicated successful measurement of ESC/iPSC-derived cells undergoing adipogenesis (15.73–28.07%; Fig. 5). As an additional benefit in the use of the FATS algorithm, dead cells are excluded for quantification by the analysis (see Fig. 5b as an example). Dead or dying cells often have highly misshapen nuclei and are therefore distinguishable from live cells when a nuclear stain is applied. The "small nucleus" filter of the FATS algorithm can then remove dead cells from consideration.
Analysis of direct adipogenesis of EBs derived from pluripotent stem cells. The fluorescence image analysis (top half) and Oil Red O image analysis (bottom half) of the entire well (top row) and of the representative microscopic field (middle rows) are shown along with their histograms (bottom row) for the 3 samples: human iPSC line 1 (a), human iPSC line 2 (b), human HUES6 ESC line (c). Phase contrast images are also shown as references for fluorescent analysis in the top half. The clump of cells to the left in the top row image of (b) are dead or dying, and therefore their nuclei fail to register and are excluded from consideration by the FATS analysis. Red scale bars: 100 μm, yellow scale bars: 1 mm
It is also possible to analyse with the FATS algorithm cells stained with more commonly used Oil Red O (for lipid droplets) and haematoxylin (for nuclei), albeit with significantly decreased sensitivity and inability to detect adipogenesis in high-density clumps of cells (Fig. 5). Comparing the two modes of imaging, it was observed that the fluorescence imaging was significantly superior to Oil Red O imaging as the fluorescence channels were independent of each other, whereas the red/blue bias were reduced if there was overlap between the red and blue channels in the colour image. This resulted in lower measured values for adipogenesis (5.55–19.36%) under Oil Red O images for all of the three samples.
In order to determine accuracy of the FATS analysis over the conventional quantification method, we investigated two sets of cell samples derived from human iPSCs. One sample set contains mostly non-differentiated fibroblastic cells that were spread out of a large EB clump in the centre (Fig. 6a). The second set contains well-differentiated adipocytes that were derived from and surrounding a smaller EB in the centre (Fig. 6b). The spectroscopic measurements of these two cell samples indicate high reading values of the EB clump sample relatively comparable to the EB adipocyte sample, due to non-specific adsorbing of Nile Red dyes by the EB clump structure (Fig. 6c). In contrast, the FATS analysis ignored non-specific staining caused by EB clump and more correctly calculated percentage of adipocytes for the EB clump sample (1.49%) and that for the EB adipocyte sample (23.15%). Together, these data support improved accuracy of the FATS system for estimation of adipogenesis derived from pluripotent stem cells.
Improved accuracy of FATS analysis to estimate adipogenesis of EBs derived from pluripotent stem cells. An example of EB clump sample that is surrounded by non-differentiated fibroblastic cells spread out of EB and has little amount of differentiated adipocytes, yet adsorbs lipid droplets dye non-specifically (a). The FATS algorithm excludes such non-specific staining caused by cell clump, calculating an adipogenic score as 1.49%. The case of an EB sample that contains significant amount of differentiated adipocytes in the surrounding area gives rise to estimation of a good adipogenic score as 23.15% by FATS (b). This is in contrast to artificially high values of the EB clump sample (826.3) given by fluorescent spectroscopic reading (c). Red scale bars, 100 μm; yellow scale bars, 1 mm
FATS is suitable for high-throughput image analysis of drug screening
Finally, we did a proof-of-concept study to examine usefulness of the FATS in the high-throughput drug screening assay. As a test example, nuclear receptor ligand library composed of 76 compounds was chosen because a number of nuclear receptor superfamily members have been implicated in regulating the process of adipocyte differentiation [24,25,26]. The high-throughput assay was performed on 3T3-L1 cells, in two independent experiments, each consisting of triplicate plates (total of six plates). Although there was an inter-plate variation and a smaller amount of intra-plate variation, overall, the FATS algorithm was able to detect significant changes in the adipogenic percentage caused by drug treatments (Fig. 7, Additional file 1: Table S1). The top four hits included troglitazone (+ 34.99% on average compared to DMSO control), GW7647 (+ 27.80%), pioglitazone (+ 26.73%), and mifepristone (+ 24.16%) whereas the bottom four hits were 3-methylcholanthrene (3MCA; − 35.52%), 6-formylindolo [3,2-B] carbazole (FICZ; − 25.37%), 25-hydroxyvitamin D3 (25HVD3; − 22.21%), and adapalene (− 16.04%). In addition, library of nuclear receptor ligands were also investigated for adipogenesis of human iPSC-derived cells (Additional file 1: Figure S5 and Additional file 1: Table S2). The results indicate that all of the top four hits from the 3T3-L1 screening (i.e. troglitazone, GW7647, pioglitazone, and mifepristone) also increased adipogenesis whereas three out of four bottom hits (i.e. 3MCA, FICZ, and Adapalene, but not 25HVD3) decreased adipogenesis of iPSC-derived cells. This suggests that a good number of nuclear receptor ligands exhibit common adipogenic effects in different cell types. Collectively, these results demonstrate applicability of the FATS analysis to high-throughput image-based screening of adipogenesis.
High-throughput screening assay of nuclear receptor ligands in 3T3-L1 cells analysed by FATS. Adipogenic scores for the top four samples (a) and the bottom four samples (b) are shown. Error bars are standard error of the mean (SEM) for n = 6 plates. Histograms indicate representative samples from a single plate
While in vitro adipocyte differentiation has been widely used as an assay to study adipogenic processes and screen for potential therapeutic compounds, the methodology to quantify its differentiation degrees remains premature. There have been several methods previously proposed to analyse adipocyte biology. The most commonly used ones include gene expression, biochemical assays, microscopy, flow cytometry, and newer techniques such as laser-scanning cytometry [27] and coherent anti-Stokes Raman Scattering (CARS) [28]. Flow cytometry is still a widely used technique to analyse cells due to its highly quantitative nature, but the necessity of detaching cells from their growth surfaces prior to analysis makes flow cytometry difficult to apply to adipocytes [29]. In addition, adipocytes are very fragile and buoyant due to the presence of large low-density lipid droplets, and thus careful handling procedure and special setup are necessary, although an attempt has indeed been made to observe adipogenesis using flow cytometry [11].
The necessity of dissociating cells during one-dimensional flow cytometry techniques led to the development of laser-scanning cytometry (LSC), which has many similarities with fluorescence microscopy. However, fluorescence microscopy is still superior to LSC in that it can achieve much higher rates of measurement. For example, a single image at the recommended setting of × 20 magnification can cover 500–1000 cells per second, and the rate can be further increased if lower magnifications are used. In comparison, LSC achieves up to 5000 cells per minute under optimum conditions [30]. While the dynamic range of measurements is lower in fluorescence microscopy, the observed dynamic range is more than sufficient to detect well-stained adipocytes.
Dye-free detection from phase contrast/bright field images has also been proposed as a method which can detect lipid droplets in adipocytes by Varinli et al. [14] and by Sims et al. [31]. This method uses contour edge detection to identify the circular lipid droplets in monolayer culture cells such as 3T3-L1 cells and can quantitate the size and quantity distributions of lipid droplets within the image analysed. However, as shown in the corresponding phase contrast/bright field images, authentic lipid droplet structures are sometimes invisible and may thus not be recognised by the phase contrast-based analysis. Another main drawback of this method is that high cell densities and the presence of partially overlapping cells, due to 'overconfluent cells' nature of standard adipogenesis protocols, can confound the contour edge detection. This is especially problematic for analysis of ESC/iPSC-derived adipogenesis because in the vicinity of EB outgrowths, lipid droplet counting becomes very difficult, and fluorescence microscopy is much more effective at detecting adipogenesis in such cases. Another paper by Surdo et al. [32] used fluorescent dyes Hoechst and Nile Red for specifically analysing bone marrow-derived mesenchymal stem cells. It made use of the macro function of existing NIS-Elements software, where the Nile Red signals are enhanced to the level enough to overlay the nuclei signals. While the method is convenient to run and likely suitable for comparable analysis of similar samples, it can oversimplify the estimation, not reflecting actual changes in lipid droplet accumulation during adipogenesis and particularly problematic for estimating ES/iPS-derived adipogenesis. There was no evidence shown for correlation with conventional analytic methods. In contrast, our technology does not depend on particular commercial software (capable of taking existing images), not require lipid droplets overlaid with nuclei (instead taking their proximity to nuclei into account), and can deal with other staining methods such as Oil Red O and haematoxylin. The FATS analysis overcomes difficulties of measuring ESC/iPSC adipogenesis by disregarding the areas of cell clumps that can non-specifically adsorb lipid droplet dyes, as exemplified by Fig. 6. More technically, the system excludes the multilayers from consideration due to the inability to correctly resolve nuclei. This is because the multilayered clumps in the middle of EB differentiation take up dyes non-specifically, and therefore, they are usually stained a uniform cyan colour (maxed out blue and green fluorescence). This means that the post-segmentation nuclear regions and lipid regions will exceed the maximum pixel size limit and therefore become excluded from the calculations. As the outgrowth layers become less highly confluent (generally ≤ 3 layers) and the nuclei become resolvable, classification will then proceed as per normal.
The global obesity epidemic has resulted in the increasing need for drugs which can be used to treat obesity and associated complications. The identification of substances that increase or decrease the process of adipogenesis can offer further insight into the adipogenesis mechanism and therapeutic responses of adipose tissue. The effects of these drugs could manifest through either increase in cell size (hypertrophy) or numbers (hyperplasia). Both of these effects can be measured in a high-throughput manner in vitro using the FATS algorithm. As a proof of concept, we demonstrated usefulness of the FATS in assessing the effect of ligands for nuclear receptor superfamily using both 3T3-L1 preadipocytes and human iPSC-derived MSCs. Among the top hits, troglitazone and pioglitazone are well-known agonists for PPARγ, which is a master regulator of adipogenesis [33, 34]. Similarly, PPARα agonist GW7467 was previously shown to promote adipogenesis [35]. The effect of progesterone receptor antagonist mifepristone/RU486 is less clear, but this compound was reported to induce adiponectin production and glucose uptake in differentiated adipocytes [36]. In contrast to top four hits, the effects of three out of four bottom hits—aryl hydrocarbon receptor agonists, 3MCA and FICZ, and retinoid adapalene—in adipogenesis are unclear and warrant further study. In contrast, 25HVD3 was shown to inhibit adipogenesis and its level in circulation is inversely correlated with obesity in human [37, 38]. However, the inhibitory effect of 25HVD3 was not observed in human iPSC-derived cells. Together, these data validate the FATS analysis for identifying known and novel compounds that affect adipogenesis in the high-throughput manner.
Our analytical method can be modified to measure other parameters. In one modification, we demonstrated changes of size distribution and mean per lipid droplet during adipogenesis. Despite potential versatility, our method can be further improved to fully realise the potential. For example, measurement of mean fluorescence intensity per LD was relatively challenging due to heterogeneities in staining degrees, brightness, and contrast across different image fields even within a well. This is why, we made binarisation and background subtraction/correction of images, which resulted in normalisation of fluorescence intensities. Authentic estimation of other potential parameters such as LD counts per cell and total LD areas per cell would be possible if we find and incorporate a good cell surface staining dye that works well in mature adipocytes. Meanwhile, X-axis of our FATS histograms indicates arbitrary LD fluorescence levels surrounding each nucleus, which may be approximation to measures of LD area per cell. We believe that future improvement depends on better image acquisition system from imaging equipment and reliable cell surface demarcation. This is particularly important for estimating ESC/iPSC-derived adipogenesis where non-specific staining areas due to large cell clumps and high cell densities must be unequivocally excluded.
In conclusion, we have demonstrated that the FATS algorithm is a robust approach for measuring adipogenesis in a wide range of cell types and that it can be universally used as a high-throughput image-based screening method for detecting drugs that affect the process of adipogenesis. The system is versatile, accurate, and capable of distinguishing dead cells, non-specific cell clump, and immature adipocytes with scant lipid droplets. Although we demonstrate in this paper that the FATS algorithm can analyse images of cells stained by Nile Red or Oil Red O, it can be applicable to other lipid droplet staining methods, such as those by lipophilic dyes of BODIPY or LipidTOX species. While the FATS can exclude dead cells based on the misshapen nuclei, it is also possible to modify the algorithm with a live/dead staining, such as propidium iodide on a third fluorescence channel, to completely rule out the likelihood of including dead cells.
25HVD3:
25-Hydroxyvitamin D3
3MCA:
3-Methylcholanthrene
ASC:
Adipose-derived stem cell
CARS:
Coherent anti-Stokes Raman Scattering
EB:
Embryoid body
ESC:
Embryonic stem cell
Fast Adipogenesis Tracking System
FICZ:
6-Formylindolo [3,2-B] carbazole
iPSC:
LSC:
Laser-scanning cytometry
MSC:
Mesenchymal stem cell
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The authors would like to acknowledge Shi Yi, Edmund Tan, Derryn Chan and members of Laboratory of Metabolic Medicine at SBIC for help in laboratory work, Drs. Asim Shabbir, Sue-Anne Toh and colleagues at National University Health System for sourcing human samples, and Nikon Imaging Centre at Biopolis for the usage of microscopy equipment.
The work was supported by intramural funding from Biomedical Research Council (BMRC) of A*STAR Singapore.
The datasets used and/or analysed during the current study are available from the corresponding author upon request.
Laboratory of Metabolic Medicine, Singapore Bioimaging Consortium (SBIC), Agency for Science, Technology and Research (A*STAR), 11 Biopolis Way #02-02, Singapore, 138667, Singapore
Chengxiang Yuan
, Smarajit Chakraborty
, Subha Subramanian
, Tau En Lim
, Weiping Han
& Shigeki Sugii
Signal and Image Processing Group, Singapore Bioimaging Consortium (SBIC), Agency for Science, Technology and Research (A*STAR), 11 Biopolis Way #02-02, Singapore, 138667, Singapore
Krishna Kanth Chitta
& K. N. Bhanu Prakash
Duke-NUS Medical School, Singapore, Singapore
Shigeki Sugii
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CY, WH, and SS designed the experiments. CY wrote programmes for the FATS algorithm with advice from BPK and SS. CY and SS drafted the main manuscript text. SC, SS, KC, TEL, WH, and BPK critically revised the article. CY, SC, S, KC, and TEL performed the cell culture experiments, data collection, and analysis. All authors approved the final version for publication.
Correspondence to Shigeki Sugii.
Use of human patients-derived cells was conducted with informed consent obtained for each subject, approved by the National Healthcare Group Domain Specific Review Board, Singapore, and performed in accordance with its relevant regulations.
S. Sugii is a co-founder and shareholder of Celligenics Pte. Ltd., of which this study is not under any financial or scientific influence. Other authors declare that they have no competing interests.
Additional file 1:
Figure S1. Correlation of adipogenic scores and adipogenic genes expression in 3T3-L1 cells. Figure S2. Sizes per lipid droplet from Nile Red staining were estimated with the modified method using background subtraction and circular Hough transform during adipogenesis of 3T3-L1 cells. Figure S3. Adipogenesis time course of human iPSC-derived MSCs. Figure S4. FATS analysis of browning in subcutaneous fat-derived ASCs. Figure S5. High-throughput adipogenic screening assay of nuclear receptor ligands using human iPSC-derived MSCs was analyzed by FATS. Table S1. List of all the nuclear receptor ligands that were tested for high-throughput screening assay for adipogenesis in 3T3-L1 cells. Table S2. Results of high-throughput screening assay for adipogenesis of human iPS-derived MSCs in a single plate using nuclear receptor ligands library. Table S3. List of RT-qPCR primers-Oligos 5' to 3'.(DOCX 5748 kb)
Adipocytes/obesity
Microscopy/fluorescence
Adipose-derived mesenchymal stromal cells (ASC/ADSC/A-MSC)
Adipogenesis and browning assay
iPS and ES cell
Oil Red O/Nile Red
High-throughput imaging-based quantitation
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December 2010, 5(4): 765-782. doi: 10.3934/nhm.2010.5.765
Groundwater flow in a fissurised porous stratum
Michiel Bertsch 1, and Carlo Nitsch 2,
Istituto per le Applicazioni del Calcolo "M. Picone", Consiglio Nazionale delle Ricerche, Via dei Taurini 19, 00185 Roma
Dipartimento di Matematica e Applicazioni "R. Caccioppoli", Università degli Studi di Napoli "Federico II", Via Cintia, Monte S. Angelo, I-80126 Napoli, Italy
Received February 2010 Published November 2010
In [2] Barenblatt e.a. introduced a fluid model for groundwater flow in fissurised porous media. The system consists of two diffusion equations for the groundwater levels in, respectively, the porous bulk and the system of cracks. The equations are coupled by a fluid exchange term. Numerical evidence in [2, 8] suggests that the penetration depth of the fluid increases dramatically due to the presence of cracks and that the smallness of certain parameter values play a key role in this phenomenon. In the present paper we give precise estimates for the penetration depth in terms of the smallness of some of the parameters.
Keywords: Groundwater flow, system of partial differential equations, free boundary., fissurized porous medium.
Mathematics Subject Classification: Primary: 35K55; Secondary: 35B30, 35Q35, 35R35, 76S0.
Citation: Michiel Bertsch, Carlo Nitsch. Groundwater flow in a fissurised porous stratum. Networks & Heterogeneous Media, 2010, 5 (4) : 765-782. doi: 10.3934/nhm.2010.5.765
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María Anguiano, Francisco Javier Suárez-Grau. Newtonian fluid flow in a thin porous medium with non-homogeneous slip boundary conditions. Networks & Heterogeneous Media, 2019, 14 (2) : 289-316. doi: 10.3934/nhm.2019012
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Michiel Bertsch Carlo Nitsch
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The cognitive impairment and risk factors of the older people living in high fluorosis areas: DKK1 need attention
Chao Ren1,2 na1,
Peng Zhang1,3 na1,
Xiao-Yan Yao2 na1,
Hui-Hua Li4,
Rui Chen5,
Cai-Yi Zhang3 &
De-Qin Geng1
BMC Public Health volume 21, Article number: 2237 (2021) Cite this article
To evaluate cognitive impairment and risk factors of elders in high fluoride drinking water areas and investigate whether DKK1 is involved in this disorder.
MoCA-B and AD-8 were used to measure the cognitive functions of 272 and 172 subjects over the age of 60 came from the high and normal fluoride drinking water areas respectively, general information and peripheral blood were collected, the level of SOD, GSH and MDA were measured, mRNA level of DKK1, the concentration of blood fluoride and the polymorphism of APOE were tested.
The blood fluoride concentration, mRNA level of DKK1 and ratio of abnormal cognitive function of subjects in high fluorine drinking water areas were higher than those in normal areas. The level of SOD of subjects in high fluorine drinking water was low compared with those in normal areas. The level of MDA and GSH had no difference between the two crowds in different fluorine drinking water areas. There were differences in cigarette smoking, education, dental status, hypertension, hyperlipidaemia and APOE results between the two crowds in different fluorine drinking water areas. The mRNA level of DKK1 and the level of cognitive function showed a positive correlation and DKK1 was one of five risk factors involved in cognitive impairment of older people living in high fluorosis areas.
The cognitive functions could be impaired in the older people living in high fluoride drinking water areas, and DKK1 may as a potential intervention point of this brain damage process need attention.
Fluoride can easily be found in our daily life, and humans may suffer from damage induced by fluorosis [1,2,3]. Some reports have suggested that excessive fluoride can cause impairments in many systems of the human body. Dental fluorosis and bone damage can easily occur. However, it takes a longer time for side effects related to fluoride to manifest in the central nervous system. Fluoride exposure can affect emotions and cognition in animal models and in children during development [4,5,6]. One study reported that the intelligence quotient (IQ) and executive functions were lower in individuals living in high fluoride drinking water areas [7]. In our previous study, we found that cognitive disorders resulted from drinking water with high fluoride levels in rat model [8]. The specific mechanism of fluorosis in cognitive disorders was unclear, but some studies showed that oxidative stress damages, neuron apoptosis, changes in neurotransmitters, and synaptic dysfunctions are involved [9,10,11]. It suggested that oxidative stress damages deserve attention, for the fragmentation and redistribution of mitochondria and the imbalance between the mitochondrial fusion and fission were found in the neurons of the rat exposure to chronic fluoride. Under this circumstance, the respiratory electron transport chain was disturbed and more superoxide radicals were produced. These changes will result in the high level of oxidative stress and contribute to the brain injury [12]. For instance, we found that the antioxidant substance such as superoxide dismutase (SOD) and glutathione (GSH) decreased significantly in rat model of fluorosis in our previous study [8]. It indicated the linkage between cognitive impairment induced by fluoride and oxidative stress.
Recent studies have indicated that cognitive dysfunction is related to many different cell signaling pathways. One study reported that the canonical Wnt signalling pathway was involved in the development of Alzheimer's disease (AD) [13]. The Wnt signalling pathway plays an important role in neurogenesis, neuroplasticity, memory and learning, which may be potential mechanisms for cognitive impairments [14,15,16]. Dickkopf-1 (DKK1), a canonical Wnt signalling pathway inhibitor, may bind to and sequester LRP5/6 and then disrupt Wnt- Frizzled-LRP6 complex formation, which would cause changes in down-stream pathway activities [17]. Our research team have recently demonstrated that the canonical Wnt pathway was involved in the fluoride-induced impairment of PC-12 cells [18] and BV2 cells [19], and we found the expression level of DKK1 was significantly higher in the fluoride group than that in control group. Excitingly, more and more studies proved DKK1 expression level increased significantly in the cerebrospinal fluid, plasma and brain tissue of AD patients and AD transgenic mice [20, 21]. In our recent review, we supposed DKK1 may be a key mediator and potential risk factor for AD development, and it may also be as a novel intervention point of brain damage prevention that need attention [22].
Some reports suggested the cognition was impaired in high fluoride drinking water areas [23, 24]. However, the difference of risk factors of cognitive impairment between induced by fluorosis and AD in older people were not clear, not to speak of whether DKK1 is involved in this disorder (cognitive impairment induced by fluoride). Based on these, we investigated the cognitive level of older people and the related risk factors in the high fluoride drinking water areas in China. In brief, our aim was to investigate the risk factors for cognitive disorders induced by high fluoride drinking water and the relationship between cognitive function and the expression of DKK1.
Methods and materials
Study design and sample size
The study was carried out with an observational cross-sectional survey design and performed from 1-Jan-2016 to 28-Feb-2017. The sample size was calculated by the PASS software. The calculation formula is below:
$$\mathrm{n}=\frac{2\overline{p}\ \overline{q}{\left({Z}_{\alpha }+{Z}_{\beta}\right)}^2}{{\left(P1-P2\right)}^{2.}}$$
P1 = 0.2, P2 = 0.4, α = 0.05, Z0.05 = 1.96, β = 0.9, Zβ = 1.28, nmin = 110.
Study of subjects
The fluoride concentration of drinking water was high in the Feng County, Xuzhou City, Jiangsu Province, China, where dental fluorosis was very common. However, there was no data available regarding the cognitive function of the individuals who lived in this area. In this study, a total of 272 subjects from the high fluoride drinking water community (water fluoride concentration > 2 mg/L) of Feng County, Xuzhou City, Jiangsu Province, China, were randomly enrolled in this study. And a total of 172 subjects, from the normal fluoride drinking water community (water fluoride concentration < 0.8 mg/L) of Suining County, Xuzhou City, Jiangsu Province, China, were randomly chosen as the control group. These two counties have a similar culture, lifestyle and economic development level. Individuals with cerebral ischaemia, brain tumours and psychiatric disorders were excluded. This study was approved by the Ethics Committee of Affiliated Xuzhou Oriental Hospital of Xuzhou Medical University, China.
Socio-demographic and personal information was collected from each subject. These data included gender, age, education, alcohol drinking, smoking, dental status, hypertension, hyperlipidaemia, diabetes and family history (psychiatric diseases or dementia). Age was categorized into 3 groups as follows: 60–69 years of age, 70–80 years of age and older than 80 years of age. Education was categorized into the following groups: illiterate, primary school, middle school, and high school or higher. Dental status was categorized as follows: dental fluorosis, normal and dentures.
Cognitive function tests
The Montreal Cognitive Assessment-Basic (MoCA-B) and AD-8 were used to investigate the cognitive functions of the subjects. The MoCA-B had excellent validity in screening for mild cognitive impairment in poorly educated older adults regardless of literacy [25]. The AD-8 is an 8-item informant-based questionnaire, which was designed to detect changes in the fields of memory, orientation, judgement and executive function [26].
Blood sampling and pretreatment
A total of 4 tubes of venous blood (5 ml each) were collected and centrifuged at 5000 rpm/min for 5 min. Sera were stored at − 80 °C. These samples were used for testing the mRNA level of DKK1, fluoride concentration, SOD, GSH, malondialdehyde (MDA) concentration and apolipoprotein E (APOE) gene polymorphism.
Biochemical tests
Blood fluoride concentration was measured using the method of fluoride ion selective electrode method. Briefly, different concentrations of Fluoride Standard Liquid reagent were used to make a standard line, and then the concentration of the blood samples was adjusted according to the standard line.
The different oxidative stress status was evaluated by measuring levels of SOD, GSH and MDA according to the manufacturer's instructions in the reagent kit (Nanjing, Jiancheng, China).
The mRNA level of DKK1 was measured using qRT-PCR. Briefly, total human blood RNA was isolated with Trizol reagent (ProbeGene, China), and the concentration was measured using ultraviolet spectrophotometry. Reverse transcription was achieved using the cDNA Synthesis Kit (ProbeGene, China), and qRT-PCR amplification was performed using the SYBR-Green Master mix (Probegene MQ051, China) with the following amplification conditions: 95 °C for 10 min, 40 cycles of 95 °C for 15 s, 60 °C for 30 s, and 72 °C for 2 min. The amplification primer sequences were F: 5′- TCA TAG CAC CTT GGA TGG GTA TTC - 3′, and R: 5′- TTG GAC CAG AAG TGT CTA GCA CAA - 3′. The results were analysed using the ABI2720 PCR System (Applied Bio-systems, USA).
For analysis of APOE gene polymorphisms (also known as genotype), genomic DNA was extracted from collected venous blood samples using a commercial kit (QIAamp DNA Blood Mini Kit, Qiagen, Shanghai, China) beforehand. Then, the potentially mutated positions in 112 (rs429358) and 158 (rs7412) of the APOE gene were conducted by a detection kit (GeneChip Assay, Sinochip, Zhuhai, China) based on genomic DNA. First, all samples were amplified with the Verti™ DX Thermal Cycler (Life Technologies, Singapore) (45 cycles, 94 °C for 30 s and 65 °C for 45 s); then the amplified products were assayed by the fully automated GeneChip detection system (Sinochip, Zhuhai, China). All genotyping results were obtained from the GeneChip automated analysis system.
Socio-demographic factors and numeration data were analysed using the chi-square (χ2) test. The nonparametric rank sum test was used to compare the quantitative data between two groups. The degree of association between fluoride-induced cognitive impairment and risk factors was analysed using binary logistic regression analysis. Bivariate correlation analysis was performed using Spearman correlation analysis. SSPS16.0 software was used to analyse all data. A p-level of 0.05 was considered statistically significant.
Demographic characteristics of subjects in the normal fluoride group and high fluoride group
The demographic characteristics of subjects are displayed in Table 1. The majority of subjects are female (N = 119, 69.2% VS N = 166, 61% in the two areas, respectively). Over 70% (N = 122) of the subjects did not receive education in normal drinking area. The rate of fluorosis teeth (N = 266, 97%), hypertension (N = 103, 37.7%) and hyperlipemia (N = 34,12.5%) are higher in the participants who lived in the high fluoride drinking area than these who lived in the normal fluoride drinking area.
Table 1 The socio-demographic characteristics of subjects
The blood fluoride concentration in subjects from normal and high fluoride drinking water areas
The blood fluoride concentration of subjects was higher in the high fluoride drinking water areas [0.04 (0.027–0.049) mg/L] compared with the fluoride concentration in subjects from the normal fluoride drinking water areas [0.02 (0.016–0.029) mg/L] (Fig. 1).
The blood fluoride concentration in subjects from high and normal fluoride drinking water areas is shown. The blood fluoride concentration was higher in subjects form the high fluoride drinking water areas than those from normal fluoride drinking water areas. ***P < 0.001
Associations between fluoride and oxidative stress
The level of SOD, an anti-oxidative factor, increased significantly in subjects from the high fluoride water drinking areas [60.66(50.73–70.62)U/mL] (Fig. 2A). The level of MDA, a pro-oxidative factor, and GSH, another anti-oxidative factor had no significant difference between the high and normal fluoride water drinking areas (Fig. 2B, C).
The level of SOD, MDA and GSH in subjects from high and normal fluoride drinking water areas: A The level of SOD, B The level of MDA, C The level of GSH. The level of SOD decreased significantly in the high fluoride drinking water areas. The level of MDA and GSH had no significant difference between the high and normal fluoride water drinking areas. ***P < 0.001. GSH: glutathione, MDA: malondialdehyde, SOD: superoxide dismutase
Fluoride increased the mRNA level of DKK1
DKK1 is an inhibitor of the canonical Wnt signalling pathway and is often associated with different diseases. Our previous studies (cellular and animal models) indicated that there was some relationship between DKK1 and fluoride. In this study, we measured the mRNA level of DKK1 by qRT-PCR. The mRNA level of DKK1 was significantly higher in subjects form the high fluoride drinking water areas [24.47(23.19–25.47)] than those from normal fluoride drinking water areas [22.05(20.99–23.24)] (Fig. 3).
The mRNA level of DKK1 in the high and low fluoride drinking water groups is shown. The mRNA level of DKK1 was higher in subjects form the high fluoride drinking water areas than those from normal fluoride drinking water areas. ***P < 0.001. DKK1: dickkopf-1
The incidence of cognitive impairment in older people increased in high fluoride drinking water areas
The MoCA-B and AD-8 were used to measure the cognitive function of subjects. The results suggested that the ratio of abnormal cognitive function of study population in high fluorine drinking water areas were higher (N = 124, 45.6%)than those (N = 26, 15.1%) in normal areas (Table 2).
Table 2 Statistical analysis of the incidence of cognitive impairment measured by the AD-8 and MoCA-B
The effect of education on cognitive function in subjects from different fluoride drinking water areas
We investigated the effects of different levels of education on cognitive function in both groups. The incidence of cognitive impairment was higher in study population with illiterate (N = 52, 38%) and primary school education level (N = 47, 53.4%) who lived in the high fluoride drinking water area compared with those (illiterate N = 12, 9.8%, primary school N = 7, 22.6%)from normal fluoride drinking water areas. However, the incidence of cognitive impairment in the study population with middle school (N = 4, 30.8%; N = 24, 54.5%) or high and over high middle school (N = 3, 50%; N = 1, 33.%) education level was not significantly different between the two groups (Table 3).
Table 3 The effect of level of education on cognitive impairment in subjects from different fluoride drinking water areas
In this study, spearman correlation analysis was used to investigate the relationship between some of observed variables (fluoride, DKK1and MoCA-B). The result was showed in the Table 4. There was a positive correlation (r = 0.313) between the level of DKK1 and the concentration of fluoride(r = 0.313, p < 0.05), and MoCA-B (r = 0.320, p < 0.05).
Table 4 A correlation of matrix of DKK1, blood concentration of Fluoride and MoCA-B
Risk factors of cognitive impairment were analysed using binary logistic regression analysis. As showed in Table 5, age, education, fluoride, DKK1 and dental fluorosis may be the risk factors of cognitive impairment in older people who lived in high fluorosis areas.
Table 5 The risk factors of cognitive impairment were analyzed by regression logistics analysis
Fluoride is a necessary element for the development and growth of organs in the human body, and it is often found in our environment [27, 28]. Frequently fluoride is added to toothpaste for the prevention of tooth decay. Since the 1980s, the use of fluoride in dental products has increased significantly. Furthermore, drinking water is a main source of fluoride, and the concentration of fluoride ranges from 0.1–0.8 mg/L [29]. However, the safety margin of fluoride is very narrow. Therefore, it is easy to suffer from fluorosis in high fluoride drinking water areas. The problems of high fluoride drinking water associated with dental and bone fluorosis had frequently been reported. Recently, some reports suggested that the excessive intake of fluoride could induce cognitive impairment in a rat model and in individuals who live in high fluoride drinking water areas [24]. Children who were exposed to high fluoride drinking water in China and India showed decreases in IQ [5, 30].
General information is shown in Table 1. As shown in Table 1, differences in the level of education, hypertension, and APOE polymorphism were observed between the fluoride group and the control group. The blood fluoride concentration was tested and found to be higher in subjects from high drinking water areas than in subjects from normal drinking water areas, as shown in Fig. 1. However, toxic blood fluoride concentrations were not detected, which means that even in high fluoride drinking water areas, the blood fluoride concentrations in older people were considered safe. That may be the reason that the cognitive impairment induced by fluoride was not serious. Since a previous study showed that fluoride (700 μmol/L NaF) could stimulate the ability of cellular anti- oxidative effects [31]. Therefore, it could be inferred that a certain low dose of fluoride exposure may play a protective rather than adverse role on cognitive impairment through the mechanisms of stimulating cells viability and anti-oxidative ability [24]. Moreover, fluorosis was associated with some other factors including age, kidney function, and sex [32]. The results showed that the incidence of dentures was higher in subjects in the normal fluoride drinking water areas. Fluoride is often added to toothpaste for the prevention of cavities, although the absence of fluoride does not necessarily contribute to the development of cavities. Nevertheless, fluoride had a protective effect against damage to the teeth caused by acid.
Oxidative stress is very common in patients with cognitive impairment, as usual, anti-oxidative factors decreased and pro-oxidative factors increased significantly. In our previous study, damage associated with oxidative stress from fluorosis was observed in a rat model [8]. In the present study, we measured two factors related to the oxidative stress. The results indicated that the level of SOD increased significantly in subjects from high fluoride drinking water areas. There was no significant difference in the level of GSH and MDA between the two groups. This result was not completely consistent with the results of our previous animal experiment [8]. The reason may be that the higher concentration of fluoride (not a toxic concentration) stimulated the activation of the oxidative system, which led to an increase in the level of the anti-oxidative stress factor SOD. These findings could indicate that a protective and reactive mechanism was engaged in response to the increased blood fluoride concentration [31]. What's more, it further supported the viewpoint presented by Li et al. that the certain low doses of fluoride intake may be a potential protective rather than a harmful factor for cognitive function; however, high fluoride exposure is a potential risk factor for cognitive impairment in older population [24].
As shown in Tables 1 and 3, although the prevalence of subjects who were illiterate was lower in the high fluoride drinking water areas, the incidence of abnormal cognitive function was higher. This result suggested that fluoride could increase the incidence of cognitive impairment. Additionally, we found that less education was associated with a higher incidence of cognitive impairment. For subjects who received more than 8 years of education (above middle school in China), no significant difference in cognitive function was observed between subjects from high fluoride drinking water areas and those from normal fluoride drinking water areas. This finding possibly suggested that the severity of cognitive impairment induced by fluoride was less in person with the higher level of education. More and more researches suggest that people with more education have lower prevalence of dementia [33]. Some scholars believe that this is benefit from the high cognitive reserve in individuals with high education [34]. But this protective effect of a high-education level may only be stronger in the early stage of disease [35]. The older persons with cognitive impairment induced by fluorosis in our study may be in the early stages of the disease. Meanwhile, other studies have found that high education may offer protection against tauopathy in patients with mild cognitive impairment [36].
DKK1, an inhibitor of the canonical Wnt signalling pathway, has been reported to be involved in cognitive impairment [37]. Increased DKK1 is associated with AD [38]. A positive correlation exists between cognitive function and DKK1 expression in diabetic rats, and there is also a positive correlation between DKK1-reducing therapy and improved cognitive function in rats [39]. Our recent research found that the level of DKK1 increased significantly in PC-12 cells [18] and BV2 cells [19] exposed to fluoride. However, the expression level of DKK1 in older people from high fluoride drinking water areas is unknown. The mRNA level of DKK1 was measured in both groups. The level of DKK1 increased significantly in subjects from high fluoride drinking water areas compared with that in subjects from normal fluoride drinking water areas. And we also investigated the correlation between observed variability. The result implied that DKK1was correlated with cognition and Fluoride concentration, although the correlation was weak. This finding suggested that DKK1 could be an indicator of cognitive impairment induced by fluoride. The function of DKK1 in the process of this disorder is unknown. Scientifically, we also need to calmly consider few limitations in this study. Firstly, the samples were chosen from one province in China and more high fluoride drinking water areas should be involved in the future study. Secondly, we could not clear which specific aspect of the cognition, such as memory, language, calculations, orientation, attention and concentration, executive functions, visuoconstructional skills, and conceptual thinking, was affected by fluoride via the disfunction of DKK1 and more researches should be performed in the future.
In conclusion, cognitive impairment could be induced by high fluoride drinking water, and DKK1 may be involved in this process. However, the specific mechanisms are unclear. More studies are needed to determine the function of DKK1, which may as a potential intervention point of this brain damage process that need attention.
The datasets used or analysed during the current study are available from the corresponding author on reasonable request.
APOE:
DKK1:
Dickkopf-1
MoCA-B:
Montreal Cognitive Assessment-Basic
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Thanks to all the staff and subjects who participated in the questionnaire.
This study was supported by Natural Science Foundation of Jiangsu Province (BK20151159), Shandong Provincial Key Research & Development Project (2017GSF218043), National Natural Science Foundation of China (81501185), Jiangsu Provincial Medical Youth Talent (QNRC2016369) and Xuzhou Medical Talents Project and Xuzhou technological and scientific project (KC14SH050).
Chao Ren, Peng Zhang and Xiao-Yan Yao contributed equally to this work.
Department of Neurology, The Affiliated Hospital of Xuzhou Medical University, Xuzhou, 221006, China
Chao Ren, Peng Zhang & De-Qin Geng
Department of Neurology , Department of Neurology Yantai Yuhuangding Hospitalof Qingdao University, Yantai, 264000, China
Chao Ren & Xiao-Yan Yao
Department of Psychiatry and Psychology, The Affiliated Xuzhou Oriental Hospital of Xuzhou Medical University, Xuzhou, 221000, China
Peng Zhang & Cai-Yi Zhang
Zhenjiang Mental Health Center, The Fifth People's Hospital of Zhenjiang City, Zhenjiang, 212000, China
Hui-Hua Li
Department of Neurology, The Affiliated Huai'an Hospital of Xuzhou Medical University, Huai'an, 223002, China
Rui Chen
Chao Ren
Peng Zhang
Xiao-Yan Yao
Cai-Yi Zhang
De-Qin Geng
CR implemented epidemiological investigation, found the references and drafted the manuscript. PZ and XY Y read the literature, summarized the information and helped to draft the manuscript. HH L and RC made data collation and analysis. RC and PZ drew the tables and Figs. CY Z and DQ G designed the study, modified the manuscript and guided the above work. All authors read and approved the final manuscript.
Correspondence to Cai-Yi Zhang or De-Qin Geng.
The study was granted ethical approval by the Affiliated Xuzhou Oriental Hospital of Xuzhou Medical University, and was performed in accordance with the guidelines of the Helsinki Declaration and the CIOMS's international ethical guidelines. The written informed consent was obtained from all the participants.
Ren, C., Zhang, P., Yao, XY. et al. The cognitive impairment and risk factors of the older people living in high fluorosis areas: DKK1 need attention. BMC Public Health 21, 2237 (2021). https://doi.org/10.1186/s12889-021-12310-6
DKK1
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CommonCrawl
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Erratum to: "On a functional satisfying a weak Palais-Smale condition"
Liouville type theorem for nonlinear elliptic equation with general nonlinearity
November 2014, 34(11): 4967-4986. doi: 10.3934/dcds.2014.34.4967
The Cauchy problem for a generalized $b$-equation with higher-order nonlinearities in critical Besov spaces and weighted $L^p$ spaces
Shouming Zhou 1,
College of Mathematics Science, Chongqing Normal University, Chongqing 401331, China
Received August 2013 Revised March 2014 Published May 2014
This paper deals with the Cauchy problem for a generalized $b$-equation with higher-order nonlinearities $y_{t}+u^{m+1}y_{x}+bu^{m}u_{x}y=0$, where $b$ is a constant and $m\in\mathbb{N}$, the notation $y:= (1-\partial_x^2) u$, which includes the famous $b$-equation and Novikov equation as special cases. The local well-posedness in critical Besov space $B^{3/2}_{2,1}$ is established. Moreover, a lower bound for the maximal existence time is derived. Finally, the persistence properties in weighted $L^p$ spaces for the solution of this equation are considered, which extend the work of Brandolese [L. Brandolese, Breakdown for the Camassa-Holm equation using decay criteria and persistence in weighted spaces, Int. Math. Res. Not. 22 (2012), 5161-5181] on persistence properties to more general equation with higher-order nonlinearities.
Keywords: local well-posedness, blow-up, Persistence properties, weighted $L^p$ spaces..
Mathematics Subject Classification: Primary: 35G25, 35L05; Secondary: 35Q5.
Citation: Shouming Zhou. The Cauchy problem for a generalized $b$-equation with higher-order nonlinearities in critical Besov spaces and weighted $L^p$ spaces. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4967-4986. doi: 10.3934/dcds.2014.34.4967
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Shouming Zhou
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CommonCrawl
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The COVID-19 pandemic, well-being, and transitions to post-secondary education
Malte Sandner1,
Alexander Patzina ORCID: orcid.org/0000-0001-5039-70161,2,
Silke Anger1,2,3,
Sarah Bernhard1 &
Hans Dietrich1
Review of Economics of the Household (2022)Cite this article
This study examines the immediate and intermediate effects of the COVID-19 pandemic on the well-being of two high school graduation cohorts (2020 and 2021) and how changes in well-being affect students' educational plans and outcomes. Our unique panel data on 3697 students from 214 schools in 8 German federal states contain prospective survey information on three dimensions of well-being: mental health problems, self-rated health, and life satisfaction. Data is collected several months before (fall 2019), shortly before and soon after (spring 2020) as well as several months after (fall/winter 2020/21) the beginning of the COVID-19 pandemic. Applying difference-in-differences designs, random effect growth curve models, and linear regression models, we find that school closures had a positive immediate effect on students' well-being. Over the course of the pandemic, however, well-being strongly declined, mainly among the 2021 graduation cohort. We show that a strong decline in mental health is associated with changes in educational and career plans and transition outcomes. As adverse life experiences in adolescence are likely to accumulate over the life course, this study is the first to exhibit potential long-lasting negative effects of the COVID-19 pandemic on education and careers of young individuals.
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The COVID-19 pandemic and the related policies to stop the spread of the coronavirus, particularly school closures, present a severe shock to mental and physical well-being for millions of young individuals worldwide. Distancing measures may affect the mental and physical health and life satisfaction of young individuals, as these measures massively change the schooling, learning processes and leisure activities of students, such as physical activity, time spent in front of screens, social contacts, substance use, and sleep time (Belot et al., 2021; Champeaux et al., 2020; Davis, 2021; Hisler & Twenge, 2021; Orgilés et al., 2020; Shanahan et al., 2020). Families with younger kids suffered from decreased well-being (Pisano et al., 2020; Stassart et al., 2021) but pandemic policies may especially impact the well-being of young individuals who are close to graduation because the measures not only affected schooling and leisure but also strongly reduced students' perceived career security and job and educational opportunities. As students' well-being presents a crucial resource in the process of educational decision-making and socioeconomic attainment (Haas, 2006), shocks to well-being may disrupt the transition from upper secondary to post-secondary education. Such transition disruptions at this stage may have negative consequences on future educational and labour market success, lifetime earnings and later life health (e.g., Leopold, 2018; Tamborini et al., 2015; Oreopoulos, 2007).
However, thus far, no empirical evidence exists on how school closures and the COVID-19 pandemic affect the well-being of students in their final high school years, and how effects on their mental and physical conditions relate to their educational and career plans and transition outcomes. We fill this research gap by using large-scale panel data on well-being, educational and career plans and transition outcomes of 3697 German high school students we sampled from the 2020 and 2021 graduation cohorts. These data have two key features. First, they entail three detailed indicators of well-being, i.e., mental health (10-item Hopkins Symptom Checklist; Derogatis et al., 1974), self-rated physical health (5-point scale; e.g., Mossey & Shapiro, 1982), and life satisfaction (11-point scale; e.g., Lucas, 2007). Second, these data contain both pre-pandemic information and information during the pandemic, as they stem from three survey waves in fall 2019, spring 2020, and fall/winter 2020/21.
Drawing on these data enables us to investigate (i) the immediate effects of nationwide school closures on students' well-being in spring 2020; (ii) the intermediate effects of the COVID-19 pandemic in general on students' well-being in fall/winter 2020/21; (iii) the heterogeneous effects of the COVID-19 pandemic on high school graduates who transition to post-secondary education and students still enrolled in high school; and (iv) the impact of decline in mental health on career and educational plans and educational decisions of graduates.
In the first step of our analysis, we separately investigate immediate and intermediate effects of the COVID-19 pandemic on well-being. This separation is important, particularly with respect to school policies, since students may perceive school closures as holidays or health protection in the short run (Helliwell & Wang, 2014), while in the long run, stressors due to adverse health, learning achievement, distancing measures or uncertainty about the future may prevail. To evaluate the immediate effects of nationwide school closures, the data allow the application of a difference-and-differences design exploiting the fact that some students within the second survey wave responded just before and other students shortly after the school closures. To elaborate on the intermediate effects of the pandemic (i.e., the developments prior to and during the crisis), we employ linear growth curve modelling.
In a second step, we investigate whether the COVID-19 pandemic has different effects on students who spend most of their two final high school years in times of the pandemic (2021 graduation cohort) and students who graduated from high school shortly after the outbreak of the pandemic (2020 graduation cohort). Differences in well-being between graduation cohorts might occur because students still enrolled in high school may face greater uncertainty about future decisions than graduates who already realized their transition to post-secondary education. However, school graduates face a completely unknown situation at their new educational institutions, since universities and vocational schools have similarly introduced distance learning (Crawford et al., 2020), which hardly enabled any interactions with new fellow students and apprentices. Additionally, the pandemic has reduced the available vocational training positions as alternative educational paths after high school as well as the number of student jobs, which may affect the financial situations of university students (Yükselen et al., 2020). Thus, it remains open whether the pandemic and related distancing measures affect students still at school or school graduates differently.
In a final analysis step, this study investigates to what extent a severe decrease in mental health leads to changes in educational and career plans and transition outcomes. Investigating such associations is important because earlier work showed that pre-transition health positively influences university enrolment decisions (Zheng, 2017). Furthermore, poor mental health increases the probability of educational dropout (Cornaglia, Crivellaro, & McNally, 2015). One potential mechanism explaining the importance of mental health for educational decision making might be that, for instance, depressive symptoms alter perceptions of the future (e.g., Leykin et al., 2011; Roepke & Seligman, 2016). Thus, students with decreasing mental health may lose confidence in their educational and career plans or opt for transitions they would not have made with a better mental health status. As other major societal crisis, e.g., the Great Depression, have cumulative negative effects for individuals over the life course (Hale, 2017), investigating changes in mental health and educational plans and outcomes may provide important insights on the potential long-term consequences of the COVID-19 pandemic.
In analysing the immediate and intermediate effects of the COVID-19 pandemic on students' well-being in two graduation cohorts and how changes in well-being relate to educational and career plans and transition outcomes, we extend the existing and rapidly emerging research on the effects of the COVID-19 pandemic on the well-being and mental health of young adults and teenagers (e.g., Elmer et al., 2020; Giuntella et al., 2021; Shanahan et al., 2020). Furthermore, we contribute to the literature on how lockdowns affect educational and career plans, which until now has concentrated on university students or employed individuals (Aucejo et al., 2020; Fiaschi & Tealdi, 2021). We combine these strands of literature and demonstrate that students who are close to the transition to post-secondary education are most vulnerable to shocks to their well-being due to the COVID-19 pandemic and that such shocks are related to educational and career plans and transition outcomes.
The participants of this study attended the highest track of secondary school in Germany, "Gymnasium", in the final 2 years. The educational system comprises three tracks of secondary school: the lower and intermediate tracks prepare students for vocational training, whereas the highest (academic) track results—after successful completion—in the high school diploma "Abitur", which qualifies students both to enrol at university or at a post-secondary vocational education or training. This academic track usually ends with final examinations at grade 12/13. These exams largely take place in March, and students receive their graduation diploma in the summer before they enrol at university or start vocational training in the fall.
After the outbreak of the COVID-19 pandemic at the beginning of 2020 and as one of the first nationwide pandemic prevention measures, all schools were closed after March 13th, 2020 in Germany. On April 23rd, 2020, the German federal states partly started reopening schools, albeit with very large regional and institutional variations: Since educational policy is the responsibility of the 16 federal states, there was no uniform school opening policy in Germany. Furthermore, local developments of the pandemic affected the closing of whole schools, class levels, or single classes. After the summer break, schools started on a regular basis and then went gradually back to limited schooling in November and December 2020, first by allowing only alternating groups of students from each classroom and then from January 2021 switching back to complete distance schooling.
The data used in this study were collected for the BerO study, which evaluates the effectiveness of intensive job counselling for students in the highest secondary school track. The baseline survey (wave 1) was conducted as a paper-and-pencil interview (PAPI) in 214 schools in 8 of 16 German federal states. Students in these schools could voluntary complete the questionnaire in school between September and November 2019 and were instructed by a professional data collection team.Footnote 1 In addition to these data, our analyses draw on data from two follow-ups, which took place outside the school context as a computer-assisted web or telephone interview (CAWI/CATI). Students were interviewed from February to June 2020 (wave 2) during the first wave of infection with some students answering before and others after school closures. Survey wave 3 took place from November to January 2021 during the second COVID-19 wave.Footnote 2 Fig. 1 gives an overview of the timeline of the data collection.
Timeline of data collection and COVID-19 infections in Germany
To answer the first and second research questions on the immediate and intermediate effects of the COVID-19 pandemic on the mental and physical well-being of young individuals, this study investigates changes over time in three outcomes: (i) mental health problems, (ii) life satisfaction, and (iii) self-rated health. The first three rows of Table 1 give an overview of the descriptive statistics of these outcome variables.
First, as a widespread measure for mental health, in waves 2 and 3, this study employs data from a subscale of the well-established Hopkins Symptom Checklist (HSCL-58; Derogatis et al., 1974) to approximate individuals' risk for anxiety disorders and depression. The employed 10-item version (HSCL-10) has been shown to be a very good proxy for the longer HSCL-25 (Haavet et al., 2011; Schmalbach et al., 2021). This study uses a scale of four categories for each question ("No," "Yes, a little," "Yes, quite slightly," and "Yes, extremely," rated 1 to 4, respectively) and employs a binary measure, which indicates 1 if an individual's average score on the 10-item scale exceeds the widely used cut-off point of 1.85. We used this binary predictor because trials have indicated clinically relevant symptoms of anxiety and depression above this cut-off point (Strand et al., 2003).Footnote 3
Second, life satisfaction refers to "the degree to which an individual judges the overall quality of his/her own life as a whole favorably." (Veenhoven, 2012: 66). For waves 1, 2 and 3, we employ the established 11-point scale (e.g., Lucas, 2007) and rely on answers to the following question: "How satisfied are you currently with your life in general?" Respondents could answer on a scale ranging from 0 ("totally dissatisfied") to 10 ("totally satisfied"). Prior research on life satisfaction and mortality (e.g., Diener & Chan, 2011) indicates that such cognitive evaluations of individuals' lives predict mortality.
Third, for waves 1, 2 and 3, this study examines self-rated health. Empirically, we rely on the question "How would you describe your current state of health?" Respondents could answer on a five-point likert-type scale ranging from 1 ("poor") to 5 ("very well"). This question is a widely used item in many health studies in the social sciences and has been shown to be a strong predictor of mortality because it proxies general physical well-being (e.g., Mossey & Shapiro, 1982).
To elaborate on reasons for potential heterogeneity in well-being between graduation cohorts (i.e., between school students and school graduates), we additionally investigate whether students from the two cohorts perceive the COVID-19 situation differently. To this end, we explore whether differences in the current situation or whether worries about the future explain potential cohort variation. To approximate young individuals' current situation, we use questions addressing enjoyment with learning and the extent to which young individuals are burdened by distancing measures. We approximate worries about the future in using questions asking about students' worries about career plans (the descriptive statistics of these outcomes are shown in Table A1).
To answer the last research question, we identified severe drops in mental health between spring 2020 and fall/winter 2020/21 independent of the baseline value of mental health and the chosen cut-off point. A strong decrease is coded as 1 if values in the individuals' HSCL-10 scores changed by at least 0.4 scale points. This applies to a quarter of the sample.
To investigate the influence of these severe mental health drops on educational and career plans, we rely on five different measures. For graduation cohort 2020 analysed educational and career plans comprise probabilistic beliefs about finishing the current post-secondary education. For graduation cohort 2021 analysed plans comprise probabilistic beliefs about successfully finishing potential university education, the certainty about the future educational pathway, expected final grade point average (GPA), and probabilistic beliefs of studying a science, technology, engineering and mathematics (STEM) major. While the probability of studying STEM directly measures one important educational aspiration, probabilistic beliefs and GPA expectations address—according to rational action theory (e.g., Breen & Goldthorpe, 1997; Cameron & Heckman, 1998)—important determinants of educational decisions.
To investigate the influence of decreasing mental health on educational transitions, we rely on three different outcomes after graduation, which were measured in the 2020 graduation cohort. Analysed measures comprise satisfaction with the overall educational decision, satisfaction with the new learning institution, and satisfaction with the location of this institution. Analysing satisfaction measures appears important because research shows that satisfaction outcomes are associated with educational dropout (e.g., Sarra, Fontanella, & Di Zio, 2019).
Analytical strategy
Estimating the immediate effects of the first school closures
When investigating the effects of school closures, a pure outcome comparison between the students who answered before and after the school closures in March 2020 may be biased because the two groups of students may have different characteristics that could be related to our well-being outcomes under study. To solve this problem, we use the panel dimension of our data and apply a difference-in-differences estimation using data from waves 1 and 2 shown in Eq. 1:
$$Y_{it} = \beta _1SC_i + \beta _2W_{it} + \beta _3\left( {SC_iW_{it}} \right) + \varepsilon _{it}$$
where Yit is the outcome of interest of individual i at wave t (life satisfaction and self-rated health, which are both available for waves 1 and 2). SCi (School Closure) is a binary variable that takes a value of 1 for students who answered in March 2020 post school closure and 0 for students who answered in March 2020 pre school closure, and β1 captures the difference between those individuals. Wit (Wave) contains a wave dummy for Wave 2 interviews, where β2 captures the corresponding coefficient. εit is a standard error term. Finally, SCiWit is the interaction term of SCi and Wit that takes a value of 1 for students in wave 2 who answered the questionnaire post school closure. The coefficient β3 then measures the divergence in the outcome between those who answered post school closure, i.e., the treatment group, and those who answered pre school closure, i.e., the control group, which indicates the average treatment effect on the treated (ATT). This is the effect of the school closures.Footnote 4
Only for the Hopkins scale we do not have information at wave 1. Therefore, for this outcome, we compare students who answered before and after the school closures, including a rich set of individual characteristics, as controls (shown in Table 1). For the analysis of all three outcomes, we restrict the time window to individuals who responded in the second survey wave prior to school closures and during closures. In doing so, the difference-in-differences analysis excludes students who participate in the interviews after school reopenings. Before the closures, all students answered within a time window of two weeks. Therefore, we argue that it is rather unlikely that pandemic factors, such as the infection rate, explain differences in well-being before and after school closures.
Estimating the development of well-being during the COVID-19 pandemic
To investigate the development of the examined well-being outcomes during the COVID-19 pandemic, we use the following specifications of linear random effects growth curve modelsFootnote 5:
$$Y_{it} = \alpha + \beta _n\mathop {\sum}\limits_{n = 2}^3 {W_{nit}} + \lambda {^{\prime}}X_i + \mu {^{\prime}}X_{it} + \theta _i + \varepsilon _{it}$$
$$Y_{it} = \alpha + \beta _n\mathop {\sum }\limits_{n = 2}^3 W_{nit} + \gamma C_i + \delta _n\left( {C_i \times \mathop {\sum }\limits_{n = 2}^3 W_{nit}} \right) + \lambda {^{\prime}}X_i + \mu {^{\prime}}X_{it} + \theta _i + \varepsilon _{it}$$
In both equations, Yit represents either life satisfaction (0 to 10), self-rated health (1 to 5) or mental health problems, which are approximated by the risk for anxiety disorders and depression (0 vs. 1). θi represents a person-specific error term, which is modelled as a random variable. εit constitutes an idiosyncratic error term. Wnit indicates dummy variables for each survey wave. Xit indicates a vector with time-varying confounding variables, whereas Xi captures time-constant confounders. In Eq. 3, we introduce Ci indicating whether a respondent stems from the 2021 or 2020 graduation cohort. To allow for variation across graduation cohorts, we interact Ci with each wave dummy. While the multiplicative effect of γ captures heterogeneity between cohorts at wave 1, δn captures heterogeneity in well-being between cohorts over the course of the COVID-19 pandemic.
Finally, we apply two sets of ordinary least square regressions. In the first set, we specify a model to elaborate on differences between graduation cohorts 2020 and 2021 at wave 3 (fall/winter 2020/21). In the second set, we identify individuals with strong decreases in mental health between survey waves 2 and 3 to generate a binary variable (reference group: slight or no decrease in mental health) and regress educational and career plans and transition outcomes at wave 3 on this binary indicator.Footnote 6 In correlating these measures, we elaborate on the potential long-term impact of the COVID-19 pandemic. Although this procedure constitutes a correlative workaround due to potential reversed causality, we can rule out parts of endogenous selection bias by using our rich data. To this end, our model specifications condition on a vast set of individual characteristics as control variables, described in "Sample characteristics and control variables", and they also include the baseline level (i.e., survey responses given in fall 2019 that are independent of any COVID-19-related factors) of each dependent variable and the baseline level of mental health (measured at wave 2).
Sample characteristics and control variables
From the BerO baseline sample (N = 7192), we construct a balanced sample and restrict our analyses to students who participated in all three waves (fall 2019, spring 2020 and fall/winter 2020/21) with nonmissing information on our outcomes or metric control variables. If information is missing in a categorical control variable, we create a missing information category. As one main question of this study is as to what extent mental well-being declines correlate with perceived post-secondary transition outcomes, we further exclude individuals from the analytical sample that transitioned from high school to a "gap year" or standby state. In this year young individuals typically bridge the time between high school graduation and enrolling at university or starting vocational training. Since such a sample truncation has the potential to distort our main findings (e.g., Elwert and Winship, 2014), we re-run all analyses on a full sample. This robustness check reveals that our main results also hold if we include graduates in a "gap year" (Figures A2 to A4 in the Online Appendix). Overall, our balanced sample consists of 3697 students. 2451 students stem from graduation cohort 2021 and 1246 students from graduation cohort 2020.
The set of individual characteristics that we use as controls includes socio-demographics, i.e., cohort (graduation cohort 2021 dummy), gender (male dummy), migration background (1st/2nd generation migrant), parental education (at least one parent with university education), and educational achievement (GPA better than 2.5 on average on a scale from 1 – best grade to 6 – failed). For these characteristics, the strong gender imbalance is notable. This imbalance is partly because women are in general overrepresented in German high schools and because women tend to participate in questionnaires more often (e.g., Becker, 2021). Moreover, we use a rich set of preferences, i.e., risk aversion and myopia, and personality traits, for which we use constructs based on multiple items to measure self-efficacy, grit, and the Big Five personality dimensions openness, conscientiousness, extraversion, agreeableness, and neuroticism. We also include a dummy variable for interview mode to account for possible mode effects and, finally, rely on school fixed effects to take into account institutional and geographical variation in the data.
Table 1 presents the sample characteristics in the three waves, with the spring 2020 wave split into individuals who answered before and after the school closures. The first rows show the means of our overall well-being measures. They show strong variation among the waves and before and after the school closures. We will investigate these changes in detail in the next sections. The next rows depict the sociodemographic characteristics, educational achievement and educational choice, as well as preferences and personality traits of the sample. As we use a balanced sample, there are no differences in these characteristics between wave 1 (fall 2019) and wave 3 (fall/winter 2020/21). However, the figures reveal that the characteristics between those students who answered before and after the school closures differ; for example, more males and slightly worse performing students answered the questionnaire after the school closures compared to students who participated in the survey before the school closures. This finding supports our strategy to apply a difference-in-differences approach to rule out biases by this selection.
Appendix Table A1 shows the descriptives of the measures, which are only available for wave 3 (fall/winter 2020/21). The variables contain information on attitudes and worries, i.e., how students deal with the COVID-19 pandemic and distancing measures, as well as on educational and career plans and further sociodemographic characteristics. Appendix Table A2 shows the aggregated values over the three waves for the variables shown in Table 1.
Table 1 Sample characteristics by wave
Immediate effects of first school closures on well-being
Table 2 presents the immediate effects of school closures on the three well-being outcomes. Using the difference-in-differences approach explained in Eq. 1 demonstrates that while school closures did not affect life satisfaction, self-rated health weakly increased after school closures by 0.21 standard deviations (see online Appendix Figure A1 for a graphical illustration of the effects). Investigating the immediate effect of school closure on mental health based on a within-wave 2-comparison reveals that the risk of mental health problems—while controlling for the variables introduced in "Sample characteristics and control variables"—is 5 percentage points lower for the students who participated in the survey after the closures. Overall, these results indicate that school closures had a positive effect on overall health in the first weeks after their implementation, as indicated by improvements in two of the three measures.
Table 2 Immediate effects of school closures on well-being: Results from difference-in-differences and OLS regressions
To test whether our treatment indicator actually picks up the effect of school closures, we ran a placebo test. In doing so, we assigned all individuals who answered in survey wave 2 in calendar week eleven (one week before the school closures) and onwards to the post school closure group. As a consequence the post school closure comprises a large amount of respondents (around 1700) who were not affected by school closures. The results from this robustness check suggest that the increase in self-rated health and the decrease in mental health problems are substantially smaller and statistically non-significant. Thus, our main specification appears to pick up the immediate effects of nationwide school closures.
Development of well-being before and during the first and second waves of the COVID-19 pandemic
In this section, we investigate the effects of school closures and distancing regulations eight months after the pandemic started. Figure 2 shows the development of the three outcomes at wave 1, at wave 2 before and after the school closures and at wave 3 calculated by applying Eq. 2 (see Appendix Table A3 for point estimates and significance levels). In line with the previous section, we see an immediate increase in self-rated health and a decrease in mental health problems in wave 2 after the school closures. However, from spring to fall, we observe a strong decrease in life satisfaction and self-rated health and a particularly strong increase in mental health problems. Overall, the data suggest that after students' overall health improved in the short term, it strongly declined in the longer term.
Development of well-being since fall 2019. Results from random effect growth curve models. Notes: Outcomes: Life satisfaction (0 to 10); self-rated health (1 to 5); Dummy for being above the clinical threshold for a high anxiety and depression risk (HSCL-10). N Life Satisfaction = 11,091; N SRH = 11,091; N HSCL-10 = 7394. Controls: school fixed effects, gender, migration status, parental education, school performance at wave 1, self-efficacy, Grit, Big Five personality traits, graduation cohort, risk aversion, time preferences, and interview mode. Data: BerO study wave 1 to 3
COVID-19 effects on mental and physical well-being by graduation cohorts 2020 and 2021
We now investigate how the effects on the three well-being measures differ over time between the 2020 and 2021 graduation cohorts. Using Eq. 3, Fig. 3 shows that none of the three measures differed significantly at wave 1 or 2 (before and after the school closures) between the two cohorts (see Appendix Table A4 for the point estimates and significance levels). However, at wave 3 (fall/winter 2020/21), the graduation cohort 2021 (graduating in spring 2021) showed significantly worse outcomes for all three well-being measures. The difference was most pronounced for mental health problems, for which the increase for graduation cohort 2021 from wave 2 to wave 3 amounts to almost 20 pp, while the increase was 5 pp for the 2020 graduation cohort.
Development of mental and physical well-being by graduation cohort. Results from random effect growth curve models. Notes: Outcomes: Life satisfaction (0 to 10); self-rated health (1 to 5); Dummy for being above the clinical threshold for a high anxiety and depression risk (HSCL-10). N Life Satisfaction = 11,091; N SRH = 11,091; N HSCL = 7394. Controls: school fixed effects, gender, migration status, parental education, school performance at wave 1, self-efficacy, Grit, Big Five personality traits, risk aversion, time preferences, and interview mode. Data: BerO study waves 1 to 3
After showing substantial variation between graduation cohorts in fall/winter 2020/21, we now investigate two potential mechanisms that might explain cohort differences. In particular, we test whether differences in the current enjoyment of education and burdens induced by distancing measures or whether worries about the future can explain cohort variation. We assume that the mechanisms under study are important because young individuals spend a great amount of their daily time with being at school and with school work (Anger et al., 2020). Moreover, research indicates that perceived career insecurity is detrimental to well-being (e.g., Kopasker, Montagna, & Bender, 2019).
For this purpose, Columns 1 and 2 of Table 3 present the results of two sets of questions that the students in both cohorts answered at wave 3, and Column 3 shows the differences between the two graduation cohorts. Students who had just started their final school year (graduation cohort 2021) reported higher levels of burden due to distancing measures and less enjoyment of learning in fall 2020 than those who already had left school (graduation cohort 2020) and had already attended university study (84%) or vocational training (16%). Analysing respondents' worries about their future reveals that students from graduation cohort 2021 were more worried than those in graduation cohort 2020 about their occupational futures and expected negative effects of distancing policies on their future careers. They also claim deficits with respect to receiving relevant career information.
Table 3 Attitudes of the 2020 and 2021 graduation cohorts regarding distancing measures and worries (Fall 2020)
Next, we investigate to what extent these higher concerns of the graduation cohort 2021 can explain the observed differences in well-being between the two cohorts. Table 4 shows the results of the estimations in which we regress the three well-being measures on the control variables (column 1) and on the two sets of questions that may explain the difference between the cohorts in well-being (Columns 2 and 3). In line with the results in Fig. 3, Column 1 demonstrates that graduation cohort 2021 reported significantly worse outcomes in the health measures than graduation cohort 2020. However, the results in column 2 show that including the present attitudes completely absorbs the difference between the cohorts for all outcomes and explains the largest share of the difference in life satisfaction, self-rated health, and mental health between the cohorts. The present attitudes explain even more of the gap than future worries, which also reduces the effect of the graduation cohort but does not fully absorb it (column 3).
Table 4 Mechanisms explaining differences in dimensions of well-being between graduation cohorts 2020 and 2021
Associations between mental health decreases and educational and career plans and transition outcomes
In this section, we analyse whether and to what extent the decrease in mental health that we observe from spring 2020 to fall/winter 2020/21 is related to students' educational and career plans and transition outcomes. We focus on mental health, for which we find the strongest decrease between spring 2020 and fall/winter 2020/21. As described in "Analytical Strategy", we regress educational and career plans and transition outcomes, measured in fall/winter 2020/21, on a dummy that takes a value of 1 if a student showed a strong decline in mental health (i.e., an increase on the HSCL-10 scale above a value of 0.4, which represents the upper quartile) from spring to fall/winter. The first model controls for student characteristics, while the second model additionally uses the panel dimension of our data and includes mental health values at wave 2 (i.e., the first measure of mental health available) and the dependent variable of the model, i.e., the educational and career plans at wave 1 (i.e., the first measure that is independent of the COVID-19 situation) as control variables. Models investigating transition outcomes include life satisfaction at wave 1 as a control. These two additional controls exclude the possibility that our estimates merely capture the effect of a student's generally low mental health and that those students with a decrease in mental health would have already stated less ambitious educational and career plans and dissatisfaction before the decrease.
Table 5 shows the results of the estimations for the 2020 graduation cohort (Columns 1 and 2) and for the 2021 graduation cohort (Columns 3 and 4). In both cohorts, a strong decline in mental health was related to a lower success probability of the current educational path or potential future university education. The next rows present the results for educational and career plans among graduation cohort 2021 and transition outcomes among graduation cohort 2020. For the 2020 graduation cohort, the results reveal that students with a strong decline in mental health were less satisfied with their overall current educational decision, location, and institution than students with a lower or no decline. In line with these results, students with a strong decline in mental health in the 2021 graduation cohort stated that they felt less secure about their future career paths, they expected a worse GPA, and they had a lower probability of wanting to study STEM subjects. Columns 2 and 4 demonstrate that these results hold in the very tight specification, which includes lagged mental health and lagged educational and career plans and satisfaction. The sizes of the effects (between 10 and 15% of a standard deviation) are higher for the 2020 graduation cohort than for the 2021 cohort. However, for the 2021 cohort, the effects were in a relevant range, with approximately five percent of a standard deviation.
Table 5 Associations between strong decreases in mental health and educational and career plans and transition outcomes in fall/winter 2020/21
We start the discussion of the results with the question of why school closures create positive effects on well-being in the short run. Our first explanation for the positive short-run effects refers to the idea that students perceived school closures as a relief, hence resembling additional holidays. This explanation is based on the finding that individuals show higher well-being on weekends and during holidays (e.g., Ryan et al., 2010), which in the case of students may be caused by the fact that studying provides less well-being than other leisure activities (Helliwell & Wang, 2014). Additional studies have shown that if high school students are not at school, they feel less stressed because of reduced pressure and bullying, which in extreme cases even leads to less suicide during the holiday months (Hansen & Lang, 2011, Kim & Leventhal, 2008). The finding that students spend much fewer hours studying during school closures than in normal times supports the holiday explanation (Anger et al., 2020 for the present sample; Grewenig et al., 2020, Grätz & Lipps, 2021).
An alternative explanation for the increase in well-being might be that students perceive school closures as a measure to protect their health and relieve their fear of becoming infected with COVID-19. The finding that a positive effect exists for self-rated health and mental health problems while the effect is absent for life satisfaction may support this explanation. In addition, the decline in self-rated health and life satisfaction from Wave 1 to Wave 2 for the students who answered the questionnaire before the school closures supports the health protection explanation, as the decline may have resulted from COVID-19 fear. Finally, self-rated health for students who answered the questionnaire after the school closures remained at the same level as in wave 1 and did not increase. In the case of a holiday effect, we may have expected an increase in this outcome. However, interview mode or honesty-in-reporting effects may also explain the decline in life satisfaction and self-rated health from wave 1 to wave 2, which supports the holiday explanation (Chadi, 2013; Warren & Halpern-Manners, 2012). Irrespective of the final explanation for the increase in well-being shortly after school closures, the findings are policy relevant, as they demonstrate that short-term school closures are not harmful to students' well-being.
Next, we discuss why well-being declines in the long run and why this decline is stronger for students who are still enrolled at high schools. In the longer run, the burdens of school closures and other distancing measures may accumulate over time because students suffer more from social distancing and home schooling and may be afraid about a loss of human capital. Additionally, students increasingly perceive the pandemic not only as a short-term event but also as a long-term condition. This is an important finding, because our results implicate that physical and mental well-being developed very dynamically during the pandemic and measuring well-being at one point in time during a crisis may be misleading. Svaleryd & Vlachos, (2022) describe positive and negative effects on mental health and well-being in their literature review. For example, Sachser et al., (2021) also found positive immediate effects of lockdown measures on mental health in a representative sample of the German population without assessing longer-term outcomes. In contrast, there are studies which also found short-term decreases in well-being, particularly for mothers and working parents in general (e.g., Cheng et al., 2021; Huebener et al., 2021). Taken together, the results suggest the importance of differentiating by mechanisms, groups, and time points when researching mental health and well-being.
As an explanation for the observed long-run decline in well-being of the 2021 graduation cohort, we find that students who were still in school were more worried about their future careers and were more burdened by the current COVID measures, in particular school closures, social distancing measures during phases of reopened schools and perceived future career uncertainty. These differences in perceptions explain the difference in decline in overall well-being almost completely.
The existence and explanation of the difference between the cohorts are surprising, as the high school graduates from the 2020 cohort were also strongly affected by distancing measures, such as having online lectures or prohibitions of freshman events, and uncertainties about the future. However, in the first post-high school year, individuals from graduation cohort 2020 appeared to cope better with the situation. One explanation for the finding may be that freshmen were not familiar with universities, vocational schools or training firms without distancing measures and that therefore, they did not miss anything, for example, on-site lectures. Additionally, after having made a successful transition, worries about the future may have decreased. In contrast, high school students still have to make the transition out of school and are therefore exposed to greater uncertainty. Overall, the results indicate that students who were locked down while still in school are most vulnerable to an overall health decline, which should be considered in prevention or support services.
Finally, it is important that the decline in well-being is related to educational and career plans and satisfaction with the chosen educational path. As students from the 2020 graduation cohort with a decline in mental health reported less satisfaction with their choices, they were more likely to drop out, causing high costs for the individual and society. The same is true for the 2021 graduation cohort: our results, i.e., that these students feel less secure about the future, suggest that they will make decisions that they would have not made without the pandemic, which may also lead to high individual and societal costs.
Although our study has many strengths, it also shows limitations. First, we cannot state whether the COVID-19 distancing measures causally generated the decline in physical and mental well-being, as we do not have a control group who was not affected by the pandemic policies. However, it is very unlikely that only time, seasonal effects or any other event caused the reduction, as the decline was too strong for these explanations. Similar declines only occur in regions where a war started or in individuals who have experienced a stroke of fate, such as becoming widowed or disabled (Coupe & Obrizan, 2016; Infurna et al., 2017; Oswald & Powdthavee, 2008). Furthermore, other studies showed that well-being remained stable during the final years of high school education for cohorts who graduated before the pandemic (Herke et al., 2019). A second limitation might be that the size of the relationship between the decline in mental health and the transition outcomes in our estimates is not very large. Nevertheless, the relationships are meaningful because a rich body of research based on the notion of cumulative (dis)advantage shows that even small changes in this critical stage of the life course can have long-lasting accumulating effects over the life course (e.g., DiPrete & Eirich, 2006).
This paper analyses how the COVID-19 pandemic and the related measures to reduce the spread of the coronavirus have affected the well-being, educational and career plans and transition outcomes of students from the 2020 and 2021 high school graduation cohorts. The results show that after an immediate increase in physical and mental health around the time of the first school closures, well-being strongly declined in the longer run, particularly for students in the 2021 graduation cohort, who were still in school at the time of the survey in fall/winter 2020/21. Additionally, our results demonstrate that this decline in well-being was related to several educational and career plans and transition outcomes. The results clearly indicate that the COVID-19 pandemic, including school closures and distancing measures, has had negative effects on current graduation cohorts, which may cause life-long harm. Next, disentangling the effects of different pandemic policies, e.g., school closures or shutdown of leisure activities, would be crucial to evaluate the true costs and benefits of such policies. This is particularly important because the effectiveness of school closures as one of the main policies to prevent COVID-19 infections is disputed (Courtemanche et al., 2021; Isphording et al., 2021; van Bismarck-Osten et al., 2021). Finally, we address the question of intergenerational equality, as our findings demonstrate that the younger generation is likely to bear the long-term costs of pandemic policies, while the benefits of distancing measures in terms of lower infections are likely to be higher for older individuals.
The voluntary participation in the first wave within schools and in the follow-up waves outside the school context means that the survey data is not necessarily representative for the whole population of German high school students in their final years. However, we compare some of the outcome measures (self-rated health, life satisfaction) of our sample with data from the representative SOEP Youth Survey and find similar pre-pandemic values. Furthermore, to account for different response rates and any other characteristics which may differ between schools, we include school fixed effects.
To investigate whether attrition between the panel waves may be selective, we analyzed the determinants of sample inclusion in the balanced panel. In particular, individuals from the younger graduation cohort (cohort 2021) were more likely to stay in the sample, which can be explained by the higher mobility of the older graduation cohort (cohort 2020) after their high school degree. While there seems a slightly positive selection of individuals in the balanced panel (e.g., high-performing students), there is no indication that our outcome variables determine the participation in all three waves.
As a robustness check, we also conducted the analysis using the original variable for mental health to allow for more variation in the outcome variable, which confirmed the results of our main analysis.
As those individuals who were exposed to the school closures also have experienced a significantly longer exposure to the pandemic in general, our school closure effects may also capture effects of the pandemic in general. To address this threat, in Table A6 in the Appendix we show that intra-individual differences in interview timing between wave 2 and 3 after the outbreak of the COVID-19 pandemic (see Figure A5 for an overview of differences in response timing), which we use as a proxy for pandemic exposure, has no substantial effect on well-being outcomes under study. In respect to our Diff. in Diff. approach, the results presented in Table A6 suggest that the positive short-term school closure effects on wellbeing outcomes are a lower bound as the time effect points in another direction and leads to a decline in well-being (although mainly not statistically significant and very small in size). As the employed fixed-effects approach uses a subsample of respondents who answered in survey wave 2 after school closures and in survey wave 3 (N=1,405) and relies on strong exogeneity assumptions, we want to stress that these results constitute rather stylized findings on pandemic exposure on our well-being outcomes. Nevertheless, they strengthen our results retrieved from Eq. 1.
We use linear random effects growth curve models mainly because these models allow for individual-specific changes in the outcomes under study over time (i.e., they incorporate an individual- and time-specific random effect). To this end, the random effect model uses individual-specific and between-person variation thereby allowing to investigate how group differences emerge over time, a special feature of the model we heavily rely on in our study. The use of within-person variation constitutes a major advantage compared to pooled OLS. The advantage over FE models constitutes the fact that RE growth curve models allow investigating group differences (of course at the cost of potential bias due to individual-specific time constant unobserved heterogeneity). In Table A7 in the appendix we show, however, that results from RE, OLS and FE models do hardly differ.
We show results of linear probability models, as these are more intuitive and easy to interpret. However, to take into account that using a non-linear binary choice estimator may be more accurate for binary outcome variables, we also conduct analyses using an ordered logit model for life satisfaction and self-rated health, as well as a logit model for mental health. The coefficients of these non-linear models are very similar to those revealed by our main analyses.
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Open Access funding enabled and organized by Projekt DEAL.
Institute for Employment Research (IAB), Nuremberg, Germany
Malte Sandner, Alexander Patzina, Silke Anger, Sarah Bernhard & Hans Dietrich
University of Bamberg, Bamberg, Germany
Alexander Patzina & Silke Anger
Institute of Labor Economics (IZA), Bonn, Germany
Silke Anger
Malte Sandner
Alexander Patzina
Sarah Bernhard
Hans Dietrich
Correspondence to Alexander Patzina.
Sandner_etal_COVID_Wellbeing_HighSchoolStudents_Appendix_
Sandner, M., Patzina, A., Anger, S. et al. The COVID-19 pandemic, well-being, and transitions to post-secondary education. Rev Econ Household (2022). https://doi.org/10.1007/s11150-022-09623-9
High school graduates
Mental and physical well-being
School-to-work transition
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CommonCrawl
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Aqueous garlic extract improves renal clearance via vasodilatory/antioxidant mechanisms and mitigated proteinuria via stabilization of glomerular filtration barrier
Christian Eseigbe Imafidon ORCID: orcid.org/0000-0002-3806-964X1,2,
Rufus Ojo Akomolafe2,
Omotayo Alaba Eluwole3,
Isiaka Ayofe Adekunle2 &
Ruby Adebusola Agbaje4
Lead (Pb) remains an apparently indispensable material in several industrial processes. It is a potent environmental toxin with associated deleterious biological effects. The study investigated the effects of aqueous garlic extract (AGE) on renal clearance and proteinuria in Wistar rats with Pb-induced kidney injury.
Thirty male Wistar rats were divided into six groups of five rats each such that exposure to Pb (35 mg/kg i.p) for 10 consecutive days was either followed by 30 days recovery period (without treatment) or 30 days post-treatment with oral graded doses of AGE at 100, 200 and 400 mg/kg while comparisons where made against a control (2 ml/kg NORMAL SALINE) at p < 0.05. The phytochemical constituents of the extract were determined using conventional standard protocols before administration to the rats.
Pb toxicity induced deleterious alterations of renal function biomarkers (creatinine, urea and total protein) in the plasma and urine, indicators of oxidative stress and lipid peroxidation (GSH, SOD, CAT and TBARS) in the kidney tissues as well as significantly lowered plasma and kidney NO level (p < 0.05) of the rats. It also significantly lowered creatinine clearance and fractional excretion of urea while urine total protein-creatinine ratio was significantly increased in the rats. Kidney histology showed evidence of Pb-induced glomerular atrophy with tubular and interstitial vacuolation. However, AGE administration was associated with significant normalization of the aforementioned biochemical parameters (p < 0.05) as well as kidney histoarchitectural improvement. The pharmacological effects of AGE were attributed to its determined phytochemical constituents.
AGE normalized renal clearance through vasodilatory and antioxidant mechanisms with associated mitigation of proteinuria through stabilization of glomerular filtration barrier.
Kidney, the principal organ of homeostasis, performs both excretory and regulatory functions [1, 2]. Renal dysfunction results in bioaccumulation of metabolic wastes which could be terminal (resulting in death), if left unchecked [3]. While some forms of kidney disorder are idiopathic, others have been associated with known causes namely genetic, lifestyle or chemically-induced (toxicological) [4,5,6]. Heavy metal toxicity, of which lead (Pb) toxicity is important, plays a significant role in the pathogenesis of kidney disorder in many population [5, 7, 8].
Lead (Pb), occurring in both organic and inorganic forms in the environment, is a toxic heavy metal of environmental and occupational concern [9, 10]. Due to its unique inherent qualities namely low melting point, resistant to corrosion, malleability, ductility and softness, Pb has found its relevance in several industrial processes such as in the production of paints, automobiles, water pipes, ceramics as well as electric storage batteries [9, 11]. Nevertheless, it is a non-biodegradable toxicant which when absorbed into the body (either by inhalation or ingestion) has an estimated biological half-life of about 10 years, thus enhancing bioaccumulation [10, 12]. Acting through disturbance of the antioxidant system, Pb-induced oxidative stress is known to occur via any of the following two pathways in order to cause toxic effects; generation of reactive oxygen species (ROS) and depletion of antioxidant reserves through the generation of ROS [13, 14]. According to literature, Pb toxicity is associated with deleterious biological effects including hemopoietic [15], nervous [12], reproductive [9], cardiovascular [16], gastrointestinal [10], hepatic [17] and kidney [18, 19] dysfunctions. Since the basic mechanism of Pb-induced toxicity is disruption of the antioxidant system, therapeutic interventions are usually geared towards application of chelating agents namely dimercaprol (BAL), Calcium Disodium EDTA (CaNa2EDTA) and succimer (2, 3–meso-dimercaptosuccinic acid or DMSA). These chelating agents are expensive, not readily available and sometimes burdened with undesirable side effects [9, 20]. This study aimed at exploring the therapeutic potential of a local (easily accessible) medicinal plant in an experimental model of Pb-induced kidney injury.
It was reported by WHO that, in health care aids, medicinal plants are used among 80% of the world's population either as the plant extract or in the form of their active components due to their health beneficial effects [21]. Besides forming the basis for almost all medicinal therapy, herbs are contained in about 40% of prescriptions and interest for herbal remedies instead of synthetic drugs is on the increase due to their associated relatively lesser side effects [22, 23].
Garlic (Allium sativum) is a common and readily accessible plant that is cultivated both sexually and asexually [24]. It is a nutritional plant that is used as spice in food and has vast medicinal properties [8, 9]. Some of its pharmacological activities namely anti-oxidant, anti-inflammatory, anti-biotic and anti-thrombotic effects have been experimentally demonstrated in conditions such as prostate cancer, cardiovascular disease, stroke, reproductive toxicity and kidney injury [8, 9, 25]. However, there is dearth of literature on the effects of garlic extract on the mechanisms of renal clearance and proteinuria in models of Pb-induced kidney injury. This study aims to bridge this gap in knowledge.
Metabolic cages
Metabolic cages were fabricated by Central Technological Laboratory and Workshops (CTLW), Obafemi Awolowo University (OAU), Ile-Ife, Osun state, Nigeria [3, 26].
Plant material, lead salt, biochemical kits and chemicals
Fresh garlic bulbs where purchased from a commercial supplier at Lagere market, Ile-Ife, Osun State, Nigeria and certified by a Botanist at the Department of Botany, Obafemi Awolowo University (OAU) where a voucher specimen was deposited.
Lead, in the form of lead acetate salt, was procured from Guangzhou Fischer Chemical Co., Ltd. (Guangdong, China).
Assay kits for renal function tests were purchased from Randox Laboratories Ltd. (United Kindgdom) while 1, 1-Diphenyl-2-picryl-hydrazy (DPPH) was procured from Sigma Chemicals Co. (USA). All other chemicals used were of analytical grade, available commercially.
Extraction process of aqueous garlic extract (AGE)
The garlic bulbs were peeled, rinsed in tap water and thereafter weighed (W1). The weighed bulbs were pulverized in distilled water with the aid of a Waring blender (Waring Commercial, Torrington, CT). With the aid of an electric shaker, the resulting mixture was subjected to constant shaking for 12 h and thereafter filtered using < 2 μm pore sieve. The filtrate, without been concentrated with a rotary evaporator, was directly fixed-dried in a lyophilizer at − 40 °C (Ilshin Lab. Co. Ltd., Seoul, Republic of Korea). The yield obtained (aqueous garlic extract) was weighed (W2) and thereafter kept in a desiccator until when needed. The percentage yield of aqueous garlic extract (AGE) was calculated using the formula below; [3, 27, 28].
$$ \mathrm{Percentage}\ \mathrm{yield}\ \mathrm{of}\ \mathrm{AGE}\ \left(\%\right)=\frac{\mathrm{Yield}\ \mathrm{of}\ \mathrm{extract}\ \mathrm{in}\ \mathrm{gram}\ \left(\mathrm{W}2\right)}{\mathrm{Weight}\ \mathrm{of}\ \mathrm{freshly}\ \mathrm{peeled}\ \mathrm{bulbs}\ \mathrm{in}\ \mathrm{gram}\ \left(\mathrm{W}1\right)}\times 100\% $$
Note: The extraction process of aqueous garlic extract (AGE) was carried out without the application of heat.
Phytochemical screening of AGE
Phytochemical screening of the extract was carried out in accordance with established standard laboratory protocols, described as follows;
The presence of alkaloids were qualitatively determined by the method of Halilu and co-workers [29] and quantified according to the method of Harbone [30]. Flavonoids were qualitatively determined by the method of Halilu and co-workers [29] and quantified by the method of Obadoni and Ochuko [31]. The presence of tannins was determined by the method of Halilu and co-workers [29] and thereafter quantified by the method of Allen and co-workers [32]. The presence of saponins was determined using Froth test as described by Benmehdi and co-workers [33] and thereafter quantified by the method of Obadoni and Ochuko [31]. The presence of phenolics was determined and quantified by the method of Edeoga and co-workers [34]. The presence of terpenoids was qualitatively determined by the method of Benmehdi and co-workers [33]. Qualitative analysis of cardiac glycoside was by Keller-Kiliani test as described by Anjali and Sheetal [35] while the quantitative analysis was by the method of Harbone [30].
Determination of total flavonoid and total phenolic contents of AGE
Total phenolic content of AGE was determined by the method of Singleton and Rossi [36] and as described by Gulcin and co-workers [37] using Folin–Ciocalteu's phenol reagent which is an oxidizing reagent. 0.2 ml of Folin–Ciocalteu's phenol reagent was added to a mixture of 0.1 ml of the sample and 0.9 ml of distilled water. The resulting mixture was voltexed. After 5 min of standing, 1.00 ml of 7% (w/w) Na2CO3 solution was added and thereafter made up to 2.5 ml with distilled water before the resulting mixture was incubated for 90 min at room temperature. Using an ultraviolet (UV)-Vis spectrophotometer (Labtronics, India; Model LT-290), the absorbance was read at a wavelength of 750 nm against a negative control which contained 1 ml of distilled water. The gallic acid equivalent (GAE) of AGE was determined using gallic acid at 0.1 mg/ml as a standard, after preparing a calibration curve.
Total flavonoid content of AGE was determined using aluminum chloride colorimetric assay method according to Zhilen and co-workers [38] and as described by Miliauskas and co-workers [39]. Standard quercetin with varying concentrations 0.1, 0.2, 0.3, 0.4 and 0.5 mg/ml was used as standard in comparison to the AGE sample. 0.4 ml of DW was added to 0.1 ml of AGE/the standard, followed by 0.1 ml of 5% sodium nitrate solution. After 5 min, 0.1 ml of 10% aluminum chloride, and 0.2 ml of sodium hydroxide solutions were added to the resulting mixture after which the volume was made up to 2.5 ml with distilled water. The absorbance, at a wavelength of 510 nm, was read against blank using a UV–vis spectrophotometer (Labtronics, India; Model LT-290).
The aforementioned tests were performed in triplicate, and the final results were expressed as mean ± Standard Error of Mean in mg quercetin/GAE per gram of AGE using the formula below; [37, 39].
$$ \mathrm{X}=\mathrm{q}\ \left(\mathrm{V}/\mathrm{w}\right) $$
X = total content of flavonoid or phenolic compound in quercetin or GAE respectively;
q = concentration of quercetin or gallic acid established from the standard curve;
V = volume of AGE (ml); and.
w = weight of AGE sample.
Radical scavenging activity of AGE
The extract was evaluated for radical scavenging activity using Table 1, 1-diphenyl-2-picryl-hydrazy (DPPH) according to the method of Blois [40]. Butylated hydroxyl anisole (BHA) solutions as well as varied concentrations of the extract (2.5, 5, 10, and 20 g/ml) were taken into different test tubes. Four millilitres (4 ml) of 0.1 mM of DPPH was added to the test tubes, voltexed and allowed to stand at 27 °C for 20 min. Absorbance of the samples was measured at 517 nm against the absorbance of the control and their radical scavenging activities were calculated as follows; [40].
Table 1 Experimental protocol and dose regimen
$$ \mathrm{Radical}\ \mathrm{scavenging}\ \mathrm{activity}\ \left(\%\right)=\frac{1\hbox{-} \mathrm{Asample}}{\mathrm{Acontrol}}\times 100 $$
Determination of acute oral lethal dose (LD50) of AGE
Acute oral lethal dose (LD50) of AGE was determined by the method of Lorke [41] as modified by Imafidon and co-workers [27]. The modification of Lorke's procedure was the use of 2 rats per group in the second phase of the study; Lorke proposed the use of 1 rat per group at the second phase of LD50 determination.
A total of 17 adult Wistar rats were used for the determination of LD50. In the first phase of the experiment, 9 rats were divided into 3 groups of 3 rats each and thereafter received graded doses of AGE at 10 mg/kg, 100 mg/kg and 1000 mg/kg orally. The rats were observed for 24 h after which the first phase of the study was terminated.
In the second phase, 8 rats were divided into 4 groups of 2 rats each after which they received graded doses of AGE at 750 mg/kg, 1500 mg/kg, 3000 mg/kg and 6000 mg/kg orally. They were also observed for 24 h after which the second phase of the study was terminated. Thereafter, the oral LD50 of AGE was determined using the formula below;
$$ {\mathrm{LD}}_{50}=\surd \mathrm{a}\ x\ \mathrm{b} $$
Where a = least dose that killed a rat;
b = highest dose that did not kill a rat.
Preparation of stock solution of Pb and AGE
Lead (Pb), in the form of lead acetate [Pb (CH3COO)2], was used for the study. It was ensured that each 100 g rat received 0.2 ml of any of the solutions in order to avoid fluid overload.
The stock solution of Pb was prepared by dissolving 35 mg of lead acetate in 20 ml of normal saline so that each 100 g rat received intraperitoneal injection of 0.2 ml of Pb from the stock solution (35 mg/kg of Pb) for 10 consecutive days.
The stock solution for 100 mg/kg of AGE was prepared by dissolving 1 g of AGE in 20 ml of normal saline. The stock solutions for 200 mg/kg and 400 mg/kg of AGE were prepared by dissolving 2 g and 4 g of AGE each in 20 ml of normal saline respectively. The stock solutions for AGE were refrigerated after use while fresh samples were prepared every 48 h throughout the period of study.
Animal management and experimental protocol
The experimental protocols were in strict compliance with the guidelines for animal research, as detailed in NIH Guidelines for the Care and use of Laboratory Animals [42] and approved by local Institutional Research Committee.
Thirty male Wistar rats (aged 8–9 weeks old) weighing 120 – 150 g were used for this study. They were purchased from the Animal Holdings Unit of the College of Health Sciences, OAU, Ile-Ife where the study was carried out. Each rat was housed in a separate metabolic cage and was allowed access to standard rat chow (ACE Feeds Plc, Osogbo, Nigeria) and water ad libitum. The rats were allowed to acclimatize to life in the metabolic cages for two weeks before the study commenced. The rats were thereafter divided into six groups of five rats each as follows; Group 1 received normal saline (2 ml/kg) throughout the study period, after which they were sacrificed under ketamine anesthesia (65 mg/kg i.m). Group 2 received 35 mg/kg of Pb (i.p) for 10 consecutive days after which they were euthanized under ketamine anesthesia. Group 3 were pretreated as group 2 and thereafter left for a recovery period of 30 days (without treatment) before they were also euthanized. Groups 4, 5 and 6 were pretreated as group 2 and thereafter received graded doses of AGE at 100, 200 and 400 mg/kg respectively for 30 days before they were also euthanized (Table 1). Before the rats were euthanized, their 24 h urine samples were collected inside the metabolic cages. Blood sample from each rat was collected by cardiac puncture (under ketamine anesthesia) into separate lithium heparinized bottles and centrifuged at 4000 rpm for 15 min using a cold centrifuge (Centrium Scientific, model 8881) at − 4 °C. The plasma obtained was decanted into separate plain bottles using separate sterile syringes. The left kidney of each rat was excised and kept in a cooler for the preparation of homogenates while their right kidneys were fixed in 10% formal-saline solution for histological examination.
Assessment of kidney function biomarkers, total protein concentration and plasma nitric oxide level
The biomarkers of kidney function namely creatinine and urea were assayed using Randox standard laboratory kits (Randox, UK) according to the manufacturer's instructions. However, creatinine clearance was calculated using the formula below;
$$ \mathrm{Creatinine}\ \mathrm{clearance}\ \left(\mathrm{ml}/\min \right)=\frac{{\mathrm{U}}_{\mathbf{c}}\mathrm{V}}{{\mathrm{P}}_{\mathbf{c}}} $$
Where Uc = urine creatinine concentration;
V = urine flow rate = volume of urine/time; and.
Pc = plasma creatinine concentration.
The concentration of total protein was determined by the method of Lowry and co-workers [43] while nitric oxide level was determined by the method of Grisham and co-workers [44].
Note: For the conversion of S.I. units from dl/ml to mg/ml in the determination of urine total protein-creatinine ratio, the following system of conversion was used; [26].
$$ 1\ \mathrm{dl}=100\ \mathrm{g} $$
Determination of fractional excretion of urea (FEurea)
Fractional excretion of urea (FEurea) was determined using the formula below; [45].
$$ \mathrm{FEurea}\ \left(\%\right)=\frac{{\mathrm{U}}_{\mathrm{urea}}\times {\mathrm{P}}_{\mathrm{Cr}}}{{\mathrm{P}}_{\mathrm{urea}}\times {\mathrm{U}}_{\mathrm{Cr}}}\times 100 $$
Where Uurea = urea concentration in the urine; Purea = urea concentration in the plasma; UCr = creatinine concentration in the urine; and PCr = creatinine concentration in the plasma.
Assessment of oxidative stress status and lipid peroxidation
With the aid of an electric homogenizer (SI601001), 10% homogenate in phosphate buffer (100 mM) was prepared with the left kidney tissue at a pH of 7.4. Thereafter, the homogenate was centrifuged at 3000 rpm for 20 min and the supernatant was collected for the assessment of the following indices of oxidative stress;
Reduced glutathione (GSH) level was determined by the method of Beutler and co-workers [46], the activity of superoxide dismutase was determined by the method of McCord and Fridovich [47], catalase (CAT) activity was by the method of Sinha [48] while the level of thiobartiburic acid reactive substances (TBARS) was determined according to the method of Ohkawa and co-workers [49].
The right kidney of each rat was fixed in 10% formal-saline solution. Each kidney was thereafter dehydrated in graded alcohol and embedded in paraffin wax. Sections of about 7–8 μm thick were subjected to Haematoxylin and Eosin (H & E) staining technique for photomicrographic assessment using Leica DM 750 camera microscope at × 400 magnification.
Data were analyzed by one-way analysis of variance and expressed as mean ± Standard Error of Mean. Thereafter, they were subjected to Newman Keul's post-hoc test and the level of significance was set at p < 0.05. Differences between two variables was assessed using student's t-test. The statistical analysis was carried out using graph pad prism 5.03 (Graph Pad Software Inc., CA, USA).
Percentage yield and phytochemical screening of AGE
The percentage yield of AGE was determined to be 69.47 ± 0.51% (n = 3) (Table 2) while phytochemical screening detected the presence of flavonoids, phenolics, cardiac glycosides, alkaloids, tannins and saponin in the extract (Table 3).
Table 2 Percentage yield of AGE
Table 3 Qualitative and quantitative determination of phytochemicals in AGE
Total flavonoid content, total phenolic content, radical scavenging activity and oral lethal dose of AGE
The extract was observed to have high amount of total flavonoid and phenolic contents in gram per milligram of their respective standards (Table 4) while the radical scavenging capacity was observed to increase with increasing doses (Fig. 1). Oral lethal dose (LD50) of AGE was determined to be greater than 6000 mg/kg (Table 5).
Table 4 Total Phenolic and total flavonoid content of AGE
Radical scavenging activity of aqueous garlic extract (AGE) at different concentrations. Graph showing concentration-dependent radical scavenging activity of AGE (p < 0.0001; F = 650.85). Each value represents mean ± Standard Error of Mean (n = 3); RSA = radical scavenging activity; AGE = aqueous garlic extract; BHA = butylated hydroxyl anisole
Table 5 Acute oral toxicity test (LD50) of AGE
Effects of AGE on plasma and urine concentrations of creatinine, urea and total protein of Wistar rats with Pb-induced kidney injury
Pb administration was associated with a significant increase in the plasma creatinine concentration (mg/dl) of group 2 when compared with group 1 (p < 0.05). That of the AGE-treated groups 4, 5 and 6 were significantly lower when compared with groups 2 and 3, with no significant difference shown when compared with group 1. The urine creatinine levels of groups 2 and 3 were significantly lower than that of group 1. Groups 4, 5 and 6 showed a significantly higher level of urine creatinine when compared with group 1 (p < 0.05) (Table 6).
Table 6 Effects of AGE on plasma and urine concentrations of creatinine, urea and total protein in Wistar rats with Pb-induced kidney injury
There was a significant increase in plasma urea level (mg/dl) of groups 2 and 3 when compared with group 1 (p < 0.05). However, that of groups 4, 5 and 6 were not significantly different from group 1 (p > 0.05). On the other hand, the urine urea concentration of groups 2 and 3 were significantly lower than that of group 1 while no significant difference was recorded in that of AGE-treated groups 5 and 6 when compared with group 1 (Table 6).
The plasma total protein concentration [× 10− 2 (mg/ml)] was significantly lowered in groups 2 and 3 when compared with group 1 (p < 0.05). That of groups 4, 5 and 6 were significantly higher than groups 2 and 3 but showed no significant difference when compared with group 1. On the other hand, the urine total protein concentration of groups 2 and 3 was significantly higher than that of group 1 (p < 0.05). However, the AGE-treated groups 5 and 6 showed no significant difference in urine total protein level when compared with group 1 (Table 6).
Effects of AGE plasma and kidney nitric oxide (NO) concentration, creatinine clearance and fractional excretion of urea (FEurea) of Wistar rats with Pb-induced kidney injury
The plasma NO level (μM) was significantly lowered in groups 2 and 3 (0.39 ± 0.03 and 0.42 ± 0.03 respectively) when compared with group 1 (1.14 ± 0.04) (p < 0.05). The AGE-treated groups 4, 5 and 6 (0.94 ± 0.03; 1.12 ± 0.03 and 1.25 ± 0.02 respectively) showed significantly higher plasma NO level when compared with group 2 (0.39 ± 0.03) (p < 0.05) (Fig. 2).
Effects of AGE on plasma and kidney nitric oxide concentrations, creatinine clearance and fractional excretion of urea of Wistar rats with Pb-induced kidney injury. Each bar represents mean ± Standard Error of Mean (S.E.M.) at p < 0.05. a = significantly different from control group [1], b = significantly different from Pb group [2], c = significantly different from Pb + recovery group [3], d = significantly different from Pb + 100 mg/kg group [4], and e = significantly different from Pb + 200 mg/kg group [5]
The kidney NO level (μM) was significantly lowered in groups 2 and 3 (0.20 ± 0.01 and 0.24 ± 0.01 respectively) when compared with group 1 (0.80 ± 0.03) (p < 0.05). The AGE-treated groups 5 and 6 (0.76 ± 0.02 and 0.75 ± 0.02 respectively) were significantly higher than groups 2 and 3 (0.20 ± 0.01 and 0.24 ± 0.01 respectively) but showed no significant difference when compared with group 1 (0.80 ± 0.03) (p < 0.05) (Fig. 2).
Groups 2 and 3 (1.09 ± 0.05 and 1.13 ± 0.06 respectively) showed a significantly lowered creatinine clearance [× 10− 2 (ml/min)] when compared with group 1 (3.38 ± 0.22) (p < 0.05). However, the AGE-treated groups 5 and 6 (3.37 ± 0.12 and 3.55 ± 0.13 respectively) showed no significant difference in creatinine clearance when compared with group 1 (3.38 ± 0.22) (p > 0.05) but were significantly higher than groups 2 and 3 (1.09 ± 0.05 and 1.13 ± 0.06 respectively) (Fig. 2).
The fractional excretion of urea (%) was significantly lowered in groups 2 and 3 (19.62 ± 2.13 and 27.20 ± 1.80 respectively) when compared with group 1 (54.29 ± 1.00) (p < 0.05). Groups 5 and 6 (58.18 ± 1.91 and 59.57 ± 1.39 respectively), however, showed a significantly higher fractional excretion of urea when compared with group 1 (54.29 ± 1.00) (p < 0.05) (Fig. 2).
Effects of AGE on indicators of oxidative stress (GSH, SOD, CAT) and lipid peroxidation (TBARS) of Wistar rats with Pb-induced kidney injury
Kidney GSH level (μg/mg protein) was significantly lowered in groups 2 and 3 when compared with group 1 as well as the AGE-treated groups 4, 5 and 6 (p < 0.05). However, the AGE-treated groups 5 and 6 showed no significant difference in the GSH level when compared with group 1 (Table 7).
Table 7 Effects of AGE on kidney indicators of oxidative stress and lipid peroxidation in Wistar rats with Pb-induced kidney injury
Although the AGE-treated groups 4 and 5 showed a significantly lowered kidney SOD level (mM/mg protein) when compared with group 1 (p < 0.05), all AGE-treated groups 4, 5 and 6 showed a significantly higher SOD level when compared with groups 2 and 3 (p < 0.05). SOD level of groups 2 and 3 was significantly lower than group 1 (p < 0.05) (Table 7).
Groups 2 and 3 had a significantly lower kidney CAT level (μmol/min/mg protein) when compared with group 1 as well as the AGE-treated groups 4, 5 and 6 (p < 0.05). However, the AGE-treated groups 5 and 6 showed no significant difference in CAT level when compared with group 1 (p > 0.05) (Table 7).
Kidney TBARS level (nmol/mg protein) was significantly higher in groups 2 and 3 when compared with group 1 as well as the AGE-treated groups 4, 5 and 6 (p < 0.05). However, the AGE-treated groups 5 and 6 showed no significant difference in TBARS level when compared with group 1 (p > 0.05) (Table 7).
Effects of AGE on urine total protein-creatinine ratio of Wistar rats with Pb-induced kidney injury
Groups 2 and 3 (3.16 ± 0.07 and 3.18 ± 0.11 respectively) showed a significantly higher urine total protein-creatinine ratio (mg/g) when compared with group 1 (2.82 ± 0.04) and the AGE-treated groups 4, 5 and 6 (2.75 ± 0.16; 2.55 ± 0.04 and 2.33 ± 0.06 respectively) (p < 0.05). However, that of groups 4 and 5 (2.75 ± 0.16 and 2.55 ± 0.04 respectively) showed no significant difference when compared with group 1 (2.82 ± 0.04) (p > 0.05) while the AGE-treated group 6 (2.33 ± 0.06) had a significantly lower urine total protein-creatinine ratio when compared with group 1 (2.82 ± 0.04) (p < 0.05) (Fig. 3).
Effects of AGE on urine total protein – creatinine ratio of Wistar rats with Pb-induced kidney injury. Each bar represents mean ± Standard Error of Mean (S.E.M.) at p < 0.05. a = significantly different from control group [1], b = significantly different from Pb group [2], c = significantly different from Pb + recovery group [3], and d = significantly different from Pb + 100 mg/kg group [4]
Histological effects of AGE on the kidney of Wistar rats with Pb-induced kidney injury
Pb administration was associated with histoarchitectural distortion that was characterized by atrophied and shrunken glomerulus as well as tubular and interstitial vacuolation in groups 2. Also, the recovery group 3 showed evidence of sustained kidney histoarchitectural distortion with apparent vacuolation of renal tubules and medullary interstitium. Although representative micrograph of the AGE-treated group 4 showed evidence of atrophied glomerulus with mild tubular vacuolation, those of groups 5 and 6 had similar features with the micrographic evidence of group 1 which was characterized by an apparently normal glomerulus, renal tubules (proximal and distal) and apparently intact interstitium (Fig. 4).
Histological effects of AGE on the kidney of Wistar rats with Pb-induced kidney injury. Magnification = × 400; [1] to [6] = groups 1 to 6; AGE = aqueous garlic extract; Pb = Lead; G = glomerulus; P = proximal tubule; D = distal tubule; Black arrow = atrophied glomerulus; Red arrow = tubular and interstitial vacuolation
The study demonstrated the therapeutic effects of aqueous garlic extract (AGE) on the renal function of Wistar rats with Pb-induced kidney injury. Worthy of note is the fact that preparation of the crude extract was without application of heat (direct lyophilizing of filtrate) in order to preserve any possible heat-labile constituent in the extract.
Abnormal levels of plasma and urine biomarkers of renal function that were associated with Pb administration, as shown in this study, are clear indications that this heavy metal (Pb) induces both glomerular and tubular defects. This fact was well corroborated by the representative micrographic evidence which showed glomerular defects with tubular and interstitial vacuolation. Deleterious effects on the kidney's filtering ability are usually associated with a significant increase in plasma levels of renal function biomarkers, hence the determination of renal clearance is essential for the assessment of glomerular filtration rate [50, 51]. This study showed that renal clearance was significantly reduced following Pb administration, with a consequent increase in the plasma level of creatinine and urea as well as significantly lowered urinary excretion of these renal function biomarkers. This study, therefore, suggests that AGE has the potential to ameliorate Pb-induced kidney injury via modulation of glomerular and renal tubular activities in order to bring about normalization of renal clearance. These effects were, however, enhanced by AGE-induced increase in renal perfusion via increased secretion of vasodilatory chemokine(s) (NO); facts that were demonstrated in this study by the increased levels of NO and fractional excretion of urea (FEurea). While increased vasodilatory effect of AGE was characterized by increased plasma and kidney NO level, the AGE-induced increase in renal perfusion was corroborated by a significant increase in FEurea following Pb administration. According to literature, NO is a potent vasodilator that has been reported to be essential for normal kidney function due to its vital role in renal mechanisms including renin release, extracellular fluid regulation, tubulo-glomerular feedback as well as regulation of glomerular and medullary hemodynamics [52, 53] while reduced FEurea level is a reflection of pre-renal effects as a result of reduced renal perfusion [54, 55].
Furthermore, the Pb-induced reduction in renal perfusion was associated with derangements of the antioxidant system as demonstrated by kidney activities of GSH, SOD, CAT as well as deleterious TBARS level (an index of lipid peroxidation). This suggests a possible renal ischemia reperfusion injury. Basically, ischemia reperfusion injury refers to cell injury or damage that results from the return of blood supply after a period of inadequate blood supply to any part of the body (ischemia) [56]. Chen and co-workers [57] reported that reperfusion injury induces reactive oxygen species (ROS) generation as a result of injury or damage to mitochondrial complexes. Since the kidney has capacity to auto regulate itself; maintaining a fairly constant blood supply despite fluctuations in arterial supply [1], it can be inferred that the observed significantly lowered circulating NO level must have coincided with compromised auto regulation capacity by the kidney of the Pb-treated group for renal ischemia reperfusion injury to have occurred. The findings of this study suggests that Pb administration elicits deleterious effects on the renal antioxidant system through renal ischemia reperfusion injury. This mechanism apparently produced a secondary deleterious effect on renal clearance, as representative micrographic evidence showed Pb-induced kidney histoarchitectural distortions that were characterized by atrophied glomerulus as well as tubular and interstitial vacuolation. It has been reported that the basic mechanism of Pb-induced deleterious biological effects is disruption of the antioxidant system via ROS generation [9]. The significantly lowered endogenous antioxidants following Pb administration can be attributed to the increased usage of these antioxidants by the kidney to scavenge free radicals (ROS) and or reduced capacity of the renal system to replenish the used-up antioxidants in a corresponding rate at which they are being utilized. As predetermined in this study, AGE demonstrated a concentration-dependent free radical scavenging capacity (Fig. 1); making it an extract with potent antioxidant capacity with increasing doses. The administration of AGE was found to be associated with normalization of renal clearance via improved antioxidant system and significantly lowered kidney lipid peroxidation. Apparently, this presents the extract as a potential therapeutic choice in the adjuvant treatment or management of patients with renal oxidative stress that is associated with renal ischemia reperfusion injury. The pharmacological activities of the extract, as demonstrated by vasodilatory and antioxidant-boosting potentials were conferred by its important phytochemicals such as flavonoids, phenolics alkaloids and tannins (Table 3). According to literature, these phytochemicals are reputed to demonstrate anti-oxidant, anti-inflammatory and membrane-stabilizing properties both invivo and invitro [58,59,60].
Pb administration was associated with reduced plasma total protein concentration and a significantly higher urine total protein excretion when compared with the control. It is not unlikely that the decreased plasma total protein concentration may be a direct consequence of increased urine total protein excretion; since representative micrographic evidence showed apparently compromised glomerular filtration barrier. Nevertheless, this study demonstrated that Pb-induced kidney injury elicits decreased protein synthesis (by the liver) into the circulation and or increased urinary excretion of total protein. Although, under apparently normal condition, Wistar rats excrete protein in their urine [26, 61, 62], proteinuria was showed to be significantly higher in the toxic group following exposure to Pb toxicity when compared with the control. According to literature, minimal change nephropathy has been found to be associated with loss of negative charges that are normally present in the glomerular capillary basement membrane [63]. Basically, the physiologic relevance of the negatively charged basement membrane is to repel the negatively charged plasma proteins since like-charge repels [63]. Subject to further verification, it can be inferred that Pb toxicity enhances the rapid or increased loss of normal negative charges in the basement membrane since this loss is usually associated with the unhindered passage of plasma protein through the glomerular membrane into the urine. Besides the loss of negative charge, Imafidon and co-workers [26] reported that changes in renal histomorphometry (Bowman's capsular space, thickness and size of the glomerulus and Bowman's capsule as well as the endothelia fenestrae of the glomerular capillaries) reflects the capacity to express proteinuria in a relationship that is directly proportional; that is, increased size of these features is directly proportional to increased capacity to express proteinuria. Apparently, this translates to increased urine excretion of total protein due to defect(s) of the glomerular filtration barrier; apparently lager Bowman's capsular space. The representative micrographic evidence of the AGE-treated groups showed apparently improved or normal kidney histoarchitecture when compared with both the control and toxic groups, suggesting that the extract potentially stabilizes the glomerular filtration barrier via normalization of electric charges of the basement membrane as well as histological strengthening of the filtration barrier integrity to repel plasma protein. These pharmacological effects demonstrate that the extract possesses potential stabilizing effects on the glomerular filtration barrier as its administration was associated with mitigation of the Pb-induced proteinuria; a finding that is worthy of further scientific exploration. In clinical models of kidney injury, a significantly increased urine total protein – creatinine ratio is an indication of a compromised renal function [64, 65]. Furthermore, the urine total protein – creatinine ratio trends with the level of proteinuria and allows for an effective monitoring of the progression of renal condition [65,66,67]. AGE administration significantly mitigated the Pb-induced increase in urine total protein–creatinine ratio. This portrays the extract as a potential choice in the adjuvant treatment or management of nephropathies that are associated with proteinuria. Since Pb is a toxicant with a potential to bio-accumulate [10, 12], it may be inferred that a vital mechanism for the ameliorating effect of the extract is the inhibition of Pb bioaccumulation via increased (urinary) excretion from the body. This is, however, subject to further scientific verification as this study did not include urinary Pb quantification to its scope.
Based on the findings of this study, a mechanistic view of Pb-induced kidney injury and the ameliorating effects of AGE have been summarized in Fig. 5.
Apparent mechanism of Pb-induced kidney injury and the effects of AGE on the mechanism of action. + = positive feedback mechanism; Pb = lead acetate; AGE = aqueous garlic extract; NO = nitric oxide; FEurea = fractional excretion of urea
Although this study does not indicate a clear-cut dose-dependent renal effects of aqueous garlic extract, it however demonstrates that the therapeutic effects of the extract (on Pb-induced kidney injury) increases with increasing doses. Additionally, at 400 mg/kg (the adopted highest dose for this study based on a pre-determined oral LD50 > 6000 mg/kg) the extract produced therapeutic effects on the renal function of Wistar rats. This implies that in an average human of 75 kg body weight, sub-chronic ingestion of the extract up to 30 g daily may be sufficient to produce therapeutic effects in conditions of Pb-induced kidney injury. This is, however, subject to further scientific verification and human trials.
In order to reduce the risk of sustained/irreversible kidney injury or the progression of renal dysfunction to end stage renal damage, it is recommended that Pb-exposed subjects should resort to prompt and efficacious treatment or management therapy as recovery period without any form of treatment was shown, by this study, to sustain renal dysfunction. Based on the findings of this study, the consumption of garlic (to be used as spice in food or consumed raw) should be encouraged as it promotes healthy status through its beneficial biological effects. Further verification on the effects of the extract in surgically-induced renal ischemia reperfusion injury is also recommended. Although this study provides a template for further scientific exploration of effects of AGE on the glomerular filtration barrier, novel studies that takes into account the assessment or quantification of the negative charges on the glomerular basement membrane, special staining techniques for the assessment of the filtration membranes as well as novel appraisal of the glomerular filtration barrier using electron microscopy are highly recommended in this scope of research on environmental toxicology and pharmacology.
It was concluded that aqueous garlic extract normalized renal clearance through vasodilatory and antioxidant mechanisms as well as caused the mitigation of proteinuria through stabilizing effects on the glomerular filtration barrier in Wistar rats with Pb-induced kidney injury.
Aqueous garlic extract
CAT:
Catalase
DPPH:
1,1-Diphenyl-2-picryl-hydrazy
FEurea:
Fractional excretion of urea
I.P.:
Intraperitoneal
LD50 :
NIH:
OAU:
Pb:
ROS:
RPM:
TBARS:
Thiobarbituric acid reactive substances
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The authors acknowledge members of staff of Professor Obuotor's Laboratory, Department of Biochemistry and Molecular Biology, Obafemi Awolowo University (OAU), Ile-Ife and the Central Technological Laboratory and Workshops (CTLW), OAU, Ile-Ife, Osun State, Nigeria for their kind support and technical assistance.
The authors received no form of funding support to carry out this research. The total cost of completing the research work was saddled by the authors.
Renal Research Laboratory, Department of Physiology, Faculty of Basic Medical and Health Sciences, Bowen University Iwo, Iwo, Osun State, Nigeria
Christian Eseigbe Imafidon
Department of Physiological Sciences, Faculty of Basic Medical Sciences, Obafemi Awolowo University, Ile-Ife, Osun State, Nigeria
Christian Eseigbe Imafidon, Rufus Ojo Akomolafe & Isiaka Ayofe Adekunle
Department of Medical Pharmacology and Therapeutics, Faculty of Basic Medical Sciences, Obafemi Awolowo University, Ile-Ife, Osun State, Nigeria
Omotayo Alaba Eluwole
Department of Biochemistry and Molecular Biology, Faculty of Sciences, Obafemi Awolowo University, Ile-Ife, Osun State, Nigeria
Ruby Adebusola Agbaje
Rufus Ojo Akomolafe
Isiaka Ayofe Adekunle
ICE conceptualized the study. ICE, EOA and AIA were responsible for funding acquisition. ARO supervised and proof-read the manuscript for intellectual content. ARA and ICE carried out the biochemical analyses. All authors conducted the research, are responsible for data analyses, improvement of intellectual content and approval of final manuscript.
Correspondence to Christian Eseigbe Imafidon.
This study was approved by Health Research Ethic Committee (HREC) of the Institute of Public Health, Obafemi Awolowo, University, Ile-Ife, Osun state, Nigeria.
The authors declare that there are no competing interests regarding the publication of this paper.
Imafidon, C.E., Akomolafe, R.O., Eluwole, O.A. et al. Aqueous garlic extract improves renal clearance via vasodilatory/antioxidant mechanisms and mitigated proteinuria via stabilization of glomerular filtration barrier. Clin Phytosci 5, 27 (2019). https://doi.org/10.1186/s40816-019-0118-y
Received: 29 April 2019
Renal clearance
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CommonCrawl
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Graphene saturable absorbers applications in fiber lasers
Xi Peng1 &
Yixin Yan1
Two-dimensional materials are widely used in a variety of fields, such as medical treatment, industrial preparation, machining, etc. In this review, we have made a detailed description of the development of fiber lasers as well as the evolution of two-dimensional materials, especially graphene. In addition, we describe the optical properties of graphene and its preparations, for instance, chemical exfoliatio, liquid phase exfoliation, electrochemical technique, chemical vapor deposition, supercritical fluid exfoliation, and thermal exfoliation. Meanwhile, we also summarized several types of graphene saturable absorbers like all fiber, D-shaped, and optical deposition. Furthermore, we summarize the optical applications of fiber lasers based on graphene. Finally, we also take a look at the future perspectives of graphene and discuss the future applications of graphene in the field of optics. It is note worth that future fiber lasers will use more heterostructures or gas-solid mixtures to prepare saturable absorbers.
Fiber laser is a mature technology that has become an indispensable tool facilitating a wide range of science, medical diagnosis, biological sensor and industrial applications [1,2,3,4,5,6,7]. In the past decade, with the in-depth research of fiber lasers, the fiber lasers mixed with two-dimensional materials have been developed to further meet people's production and living needs, such as Er-doped fiber lasers, Yb-doped fiber lasers, Ho-doped fiber lasers, Tm-doped fiber lasers, Pr-doped fiber laser, and Nd:YOV4-doped fiber lasers. Moreover, it should be pointed out that the recently proposed D-shaped fiber lasers [7,8,9,10], which has been widely accepted because its outstanding performance. As for fiber laser, there are two kinds of fiber lasers setup structure, including ring cavity and line cavity, while ring cavity is used more often. Advances in ultrafast fiber lasers have been applied as essential tools in a wide range of fields, such as laser physics, nanotechnology and medicine [1, 2, 11].
More recently, an increasing number of attentions have been paid to the pulse light source based on optical fiber, due to its many practical advantages. On the other hand, the passive approach using a saturable absorber (SA) device attracts tremendous interest since it has the advantages of simple structure, low-cost, shorter achievable pulse durations and high repetition rate.
In the field of lasers, there are two types of SA devices, for instance, real SAs and artificial SAs. As for real SAs, including semiconductor saturable absorber mirrors (SESAM) and low-dimensional nanomaterial, the materials themselves possess light intensity dependent nonlinear absorption, in terms of artificial SAs, the devices use the nonlinear refractive index or birefringent properties to induce an intensity dependent nonlinear absorption, mimicking the response of real SAs. Take into account the growing interest and achievements in the field of materials science. We mainly focus on the real SAs with a particular emphasis on 2D nanomaterial in this article.
The early research of using SA devices as an ultrafast optical switch were demonstrated 4 years after the first laser emission reported by Maiman, where a colored glass filter and a reversibly bleachable dye were applied in Q-switched bulk lasers for short-pulse generation. Since then, SA technology has attracted a lot of attention from researchers, so it can be widely used in optical fiber lasers.
With the development of low-loss optical fiber, mode-locked lasers based on actively-doped fiber amplifiers emerged, including an early 1983 report of unstable mode-locking of a Nd:YOV4-doped fiber fiber laser using a dye SA. However, until the semiconductor saturation absorption mirror (SESAM) was proposed in the early 1990s, generating stable mode-locked pulses using passive mode-locked SA in optical fiber systems remained a challenge, SESAMs quickly became a highly successful technology for generating ultrafast mode-locked pulses and high-energy Q-switched emission from fiber lasers. Because of the optical properties of SESAMs, for example, narrow operating bandwidth, require costly fabrication and packing, and the relaxation speed is limited to picosecond timescales, so as to promote the development of new materials, including metals nanostructures, carbon nanotubes (CNTs) [12], topological insulators (TIs), transition metal oxides (TMO) [13], transition metal dichalcogenides (TMDs) [14,15,16,17,18,19], grapheme [20] and black phosphorous (BP). Whereas experimental research into the photonics properties of such material remains at an early stage, it has offered new opportunities for photonic and optoelectronic applications in compact ultrafast light sources.
More recently, graphene has triggered rapid growing interests in both academic research and potential applications since its remarkable properties. Few layer graphene presents layer-count-dependent properties, which is a typical feature for 2D nanomaterials. For instance, it has been shown that we make different saturable absorbers by changing the number of graphene layers, such as from 9 to 28 layers, then applying them to different fiber laser will produce different light source outputs. Nowadays, there are several types of graphene saturable absorbers, such as D-shaped, all-fiber lasers, optical deposition grapheme SA, but the graphene-based D-shaped saturable absorber technology has been demonstrated to produce shorter pulse-high repetitive frequency light source output, with practical advantages of simple structure, low cost and superior performance, which prompts the extensive investigation of this material, enabling its fabrication into versatile photonic and optoelectronic devices for desirable applications.
In this review, we summarized the current development situation of graphene based photonics devices, including their fabrication and integration process. Besides, we also summarized the structural types of graphene saturated absorbers and applications as SA devices for graphene-based fiber lasers. Based on these advances, a conclusion and outlook of new potential opportunities of graphene-based devices in future optoelectronic and photonic technologies are highlighted.
Material optical properties and preparations
Graphene optical properties
Graphene is the name given to a fiber at monolayer of carbon atoms tightly packed into a two-dimensional (2D) honeycomb lattice [21], and is a basic building block for graphitic materials of all other dimensionalities, as shown in the inset of Fig. 1. Due to its flexible structure can be formed into many shapes, so it is widely used in different fields of material science for developing nanocomposites, sensors, supercapacitors, hydrogen storage and optoelectronic devices [23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43]. In this regard, it is important to highlight that graphene has a variety of properties, such as electronic properties, mechanical properties, optical properties and thermal properties. The function of graphene in burgeoning applications is based on one or more fundamental properties [23, 44,45,46,47,48,49,50].
The structure of grapheme. Taken from [22]
Compared with other two-dimensional materials, saturated absorption appliances made of graphene have the advantages of low cost, good stability and environmental protection, especially the excellent performance of optical properties. As shown in Fig. 2, this constant transparency has been experimentally observed for graphene in the visible range and the transmittance linearly decreases with the number of layers for n-layer graphene.
Graphene optical properties. Taken from [51]
Since its unique optical properties, for example, ultra-broad tuning range, graphene may be widely used as a saturable absorber for all types of fiber lasers [21]. Other outstanding optical properties, such as very low saturable absorption threshold level and high modulation depth, provide new possibilities in terms of ultrafast mode-locked fiber lasers development [52]. It is the basic unit in order to build other dimensional carbon materials (such as zero-dimensional fullerenes, one-dimensional carbon nanotubes, and three-dimensional graphite). In 2009, Bao et al. first confirmed the use of atomic-level graphene as a saturable absorber for mode-locking of Er-doped fiber lasers [53]. Most of the research has focused on the optimization of graphene mode-locking and Q-switching performance in wavebands as saturable absorbers [15, 16, 32, 54, 55].
The Raman spectra of as-produced graphene samples are shown in Fig. 3. The monolayer and 3-layer graphene samples are plotted in black solid line and red short dash line, respectively. From the two curves, we can see three typical peaks of graphene around 1340 cm, 1590 cm and 2660 cm, which represent D band, G band and 2D band [34]. The much sharper and higher 2D band peak shows the characterization of monolayer graphene and that of 3-layer graphene is lower. The intensity ratios of D and G bands are very low, which indicate the high quality of the two graphene samples. In addition, this free-carrier response of graphene supports the 2D plasmon mode, which exhibits unusually strong confinement and a distinctive dependence on carrier concentration. The ability to control broadband graphene absorption and plasmon excitation through electrostatic gating has enabled many electro-optic modulator designs based on graphene which function at the terahertz to visible wavelengths [57].
The Raman spectra of monolayer, bilayer, three layers, and four layers grapheme on quartz (a) and SiO2 (300 nm)/Si substrate (b). The enlarged 2D-band regions with curve fit are also shown in panels c and d. Taken from [56]
The mechanical properties of monolayer graphene including Young's modulus and fracture strength have been investigated by numerical simulations such as molecular dynamics. The Young's modulus of few-layer graphene was experimentally investigated with force-displacement measurements by atomic force microscopy (AFM) on a strip of graphene suspended over trenches. Recently, the elastic properties and intrinsic breaking strength of free-standing monolayer graphene were measured by nanoindentation using AFM shown in Fig. 4.
Mechanical properties of grapheme. a Scanning electron microscopy (SEM) image of a graphene flake spanning an array of circular holes (scale bar, 3 μm) and b Schematic illustration of nanoindentation on membranes; c and d show graphene oxide paper and its cross-section in SEM. Reproduced with permission from (a and b) and (c and d). Copyright: 2008 American Association for the Advancement of Science (a and b) and 2007 Nature Publishing Group (c and d). Taken from [58]
Graphene has high electron carrier mobility, but it is a zero band gap semimetal. When graphene is used in the SiO2 sub-strate to manufacture a transistor, performance of grapheme decreases due to roughness of the surface, charged surface and SiO2 surface optical phonons. Therefore, graphene-based heterostructures are preferred instead of using pure graphene especially in transistor applications.
Material preparations
2D materials used in the preparation of saturated absorber, such as disulfide, black phosphorus, graphene, transition metal disulfide. In the case of graphene, various methods have been used to prepare graphene including chemical exfoliation [59], liquid phase exfoliation (LPE) [13, 60], electrochemical technique [61,62,63], chemical vapor deposition (CVD) [64, 65], supercritical fluid exfoliation [66,67,68,69], thermal exfoliation [70], etc. However, liquid phase exfoliation (LPE) is one of the most important methods among top down graphene production techniques. This procedure includes direct exfoliation of graphite to three graphene layers by shear forces through ultra sonication, which is a simple, cheap, and environmental friendly approach without using hazardous materials such as strong acids and hydrazine derivatives. The schematic representation of the liquid-phase exfoliation is depicted in Fig. 5.
Mechanism of graphite exfoliation through two methods and a combination of both in order to increase the yield and reuse of all the products of milling. Taken from [60]
In this section, we will focus on the CVD method [19, 55, 71,72,73,74,75,76,77], because this method is widely used in all kinds of experiments. Graphene saturable absorbers may be formed due to chemical methods, such as dispersing graphite sheets in different solvents (polyvinyl alcohol— PVA, dimethyl formamide—DMF), CVD on Ni/Si substrates [38], or epitaxial growth on SiC substrates. The another efficient method is based on mechanical exfoliation of graphene from pure graphite [78], most likely highly ordered pyrolytic graphite (HOPG). By employing CVD method, samples of graphene were prepared as follows: firstly, monolayer and three-layer graphene thin films were grown on Cu foils separately, then we use ammonium persulfate to etch the copper. After thorough rinsing in the distilled water, the graphene samples were floating on the water. Finally, the samples were fished by quartz substrate and dried in nitrogen gas [38]. Nowadays, an increasing number of research using Cryogenic liquid stripping in the laboratory, which low cost, easy to make and stable performance.
Graphene-based saturable absorbers
D-shaped
These D-shaped fiber laser with this structure is mainly using ring cavity. We present graphene saturable absorbers on D-shaped optical fiber for fast and ultrashort pulse generation [79]. Furthermore, the materials are obtained by CVD, liquid–phase exfoliation or mechanical exfoliation of graphite. Figure 6 illustrates the schematic structure of the graphene-based SA. The monolayer graphene film is directly synthesized by the CVD method on polycrystalline Cu substrate [38]. The polymer clad resin is uniformly adhered to the graphene film on a Cu substrate without an air bubble in it, and is then cured by ultraviolet (UV) light. After 24 h, the polymer/graphene/Cu layers are soaked with 0.05 mg∕mlFeCl3 solution to remove the Cu layer. Then the ferric icon is washed away from polymer or graphene layers using distilled water. The length of the graphene is 10 mm. Finally, after cleaning the polished surface of the D-shaped fiber with 99.5% propyl alcohol, the polymer-supported monolayer graphene film is transferred onto the flat surface of the D-shaped fiber for interaction with the evanescent field. Such a structure is used as the graphene-based SA in fiber laser system. In addition, it should be noted that the thickness of D-shaped fiber should be between 67 μm (on top of fiber core) and 77 μm, in order to obtain a low loss and strong evanescent field simultaneously. Considering the interaction length of 10 mm of graphene, the 72 μm thickness of D-shaped fiber is appropriate to structure.
Schematic structure of the graphene-based SA; the right schematic diagram is the cross section. Taken from [80]
The experimental setup of the Er-doped fiber laser with total length of 15.4 m [80], as shown in the Fig. 7. It consists of a 2 m length Er-doped fiber with absorption coefficient − 33.8 dB/m and dispersion coefficient of − 57.0 ps/nm/km at 1550 nm, a 980 nm semiconductor pump laser coupled in co-propagating configuration through a 980/1550 nm WDM, an isolator with 50 dB isolation and 0.07 dB loss at 1550 nm, a polarization controller and an output coupler of 15.3%. The cavity average dispersion was 6 ps/ km/nm and the accumulated dispersion was 100 fs/nm. The experimental results obtained from the D-shaped fiber lasers on passively mode-locking is shown in Fig. 8, where the central wavelength, pulse duration and repetition rate are 1565 nm, 256 fs and 12.29 MHz, respectively [81] (Table 1).
Passively mode locked by graphene saturable absorber on D-shaped optical fiber. Taken from [81]
The experimental results obtained from the D-shaped fiber lasers on passively mode-locking. a Output linear spectrum (inset – log scale spectrum) and b autocorrelation trace (inset – cavity fundamental repetition rate). Taken from [82]
Table 1 Typical properties of D-shaped fiber lasers
In a word, the fiber laser with D-shaped saturation absorber has the advantages of high output pulse frequency, high large output power and short pulse duration. Besides, the laser structure is simple, low cost and easy to manufacture. In the future, due to better optical properties, it is will be more likely to attract the attention of researchers and be widely used in a variety of fields [79,80,81, 83,84,85,86, 96].
At present, graphene saturated absorbers in all-fiber form are widely used in fiber lasers. In this review, we summarize various of the most common all-fiber lasers in recent years, for instance, Er-doped [97], Yb-doped [98],Ho-doped [94], Tm-doped [11], Pr-doped [49, 75, 77] and Nd:YVO4-doped [39]. Meanwhile, the Er-doped fiber lasers are used in a growing number of academic research and optical field, due to its many great optical properties, such as light absorption, high repetition frequency as well as the pulse duration is short. However, with the rapid development of fiber lasers, an increasing amount of 2D materials are used in the research of ultrafast lasers.
By analyzing these common fiber lasers, the preparation methods of graphene saturated absorbers are various, including CVD, ME and LPE. However, from the perspective of production difficulty, performance stability and production cost, it can be concluded that the graphene saturated absorber made by CVD method has good stability, low cost and simply make.
Overall, the central frequency of all-fiber laser is concentrated at about 1.5um, and it can also generate a light source with a central frequency of 2um. Most all-fiber lasers produce light sources of MHz repetition frequency and pulse duration from femtosecond to microsecond. In the future laser research field, there will be a new type of optical fiber laser composed of all-fiber and various noble gases with good optical properties.
Optical deposition
To place graphene onto fiber end to fabricate graphene saturable absorber (GSA), various methods have been proposed, for instance, chemical vapor deposition, polymer nanocomposite and optical deposition [47, 48, 74, 99,100,101]. Among the methods mentioned above, what's more, optically driven deposition is the simplest and effective way to deposit graphene onto the fiber end. The optical deposition method was first proposed and performed to deposit carbon nanotubes in 2007. Subsequently, it was also shown that graphene could be optically deposited on the fiber end as a SA. However, more detailed information about the factors that influence the process of optical deposition of graphene was not provided in early investigations.
Optical deposition is an effective way to attract graphene sheets to be deposited on a fiber end. The injected laser intensity, deposition time and solution concentration are the key parameters that influence the final result of the optical deposition. According to these parameters, there are two main mechanisms that explain the different results: the optical trapping effect (acting on the micro-particles) and the thermal effect (acting on the solution), as illustrated in Fig. 9. As the incident laser intensity increases, the transmission of the graphene film also increases, which is shown in Fig. 10.
Mechanism of the optical deposition of graphene. Taken from [74]
Transmission curve for a saturable absorber. Taken from [74]
The experimental setup for optical deposition is shown in Fig. 11. A fiber laser diode (LD) at 980 nm generated the laser beam used for optical deposition. Then, the continuous wave traveled through the optical fiber into the 3 dB coupler (50:50) and split into two laser beams coupled into a single mode fiber (SMF; Corning SMF −28e the mode field diameter 10.9 μm at 1550 nm). These two SMF are cleaved by fiber cutter to ensure the smoothness of fiber ends and then equipped with fiber ferrule. During the deposition process, one SMF was used to monitor the reflected power through an optical power meter. The other SMF was used for optical deposition, which was vertically immersed into the graphene dispersion solution. Finally, the laser beam was injected from the cleaved fiber end into the solution, driving the graphene sheets to be deposited onto the SMF end to fabricate GSA films. The shapes of the GSA films attached on the fiber end were measured by a microscope. The images of the fiber ends which deposited with graphene sheets and the transmission curves are presented in the next section. The images of the fiber ends are observed by microscope. The transmission curves are measured using a homemade mode-locked laser that operates at central wavelength of 1553 nm with a repetition rate of 20 MHz and a pulse duration of 500 ps.
Experimental setup for graphene optical deposition. Taken from [74]
However, it is worthwhile mentioning that optically deposited graphene saturated absorbers are affected by a number of factors, firstly, Influence of light intensity, with the increasingly injected laser intensity, under a certain range, the modulation depth increases and the transmission rate slightly increases, but the saturable intensity is less changed. In addition, influence of deposition time, with the increasing deposition time, an increasing amount of energy was absorbed by the solution, causing a larger temperature gradient. Lastly, influence of the concentration of the graphene solution, the effect of the decreasing solution concentration might be due to the combination of the decreasing of laser intensity and deposition time. Additionally, the graphene particles in the low concentration solution are smaller, which may be the reason for the increase in the saturable intensity [101].
This review studies and summarizes the properties of graphene, the preparation method of graphene saturated absorbers, and the various shapes of graphene saturated absorbers. Compared with other two-dimensional materials, graphene has the advantages of low absorption rate, extremely high thermal conductivity, good ductility and flexibility [102]. Due to these physical properties of graphene, it can be made into saturated absorbers with different shapes., And then widely used in lasers.
Optical applications
Hitherto, research on fiber lasers with 2D materials has attracted huge interest, and has translated into tremendous progress over the past few years. The inclusion of a SA into a fiber laser can initiate pulsation by Q-switching or mode-locking, where the output properties depend on the cavity design and saturable absorber properties. So far, graphene and graphene oxide have been widely investigated for applications in optoelectronic and photonic devices, since its unique electric and optical properties [22]. Moreover, the fast recovery time enables graphene to be used as an efficient saturable absorption element for fiber lasers. In this part, we will then focus on the optical property of graphene-based devices as well as their applications as SA devices for many kinds of fiber lasers, including Er-doped, Yb-doped, Ho-doped, Tm-doped, Pr-doped, and Nd:YVO4-doped optical fiber lasers.
In the pulse modulation process of fiber lasers, Q-switching and mode-locking technology can excite short-pulse lasers, and Q-switching and mode-locking are realized by a two-dimensional material saturable absorber. The use of two-dimensional materials as the modulation element for Q-switching and mode-locking in the laser not only enables the process of Q-switching and mode-locking without external control signals, but also has a simple and reliable structure. For graphene saturable absorbers, both mode-locking and Q-switching technologies have been applied and a large number of research results have been published. This review summarizes the use of graphene as a saturable absorber in six types of fiber lasers.
Er-doped fiber lasers
Saturated absorbers made from graphene and its oxides are widely used in a variety of types of fiber lasers. It is important to highlight that Er-doped fiber lasers are widely used because of its low cost, superior performance as well as simple structure. It is found that Er-doped fiber laser is the most experimentally verified fiber laser [91, 97, 103, 104]. Table 2 summarizes the properties of different output pulses of typical Er-doped fiber lasers. A lot of experiments have validated the mode-locking capability of graphene in the 3 μm wavelength region, mode-locked pulses at 2.8 μm with an average output power of 18 mW at a repetition rate of 25.4 MHz, corresponding to a pulse energy of 0.7nJ [107]. Cao et al. demonstrated a wide-band tunable passively Q-switched fiber laser by using a graphene-based SA, the stable Q-switched pulse with a tunable range from 1519.3 nm to 1569.9 nm was achieved, covering a wavelength range of over 50.6 nm [108]. In addition, it can be seen from the Tables 2 and 3 that the central wavelength of Er-doped fiber laser is concentrated at 1.5um. Besides, Er-doped lasers generate laser pulses with a duration from femtosecond to microsecond and high repetition rate.
Table 2 Typical properties of mode-locked Er-doped fiber lasers
Table 3 Typical properties of Q-switched Er-doped fiber laser
The typical circuit structure of Er-doped fiber laser is shown in Fig. 12. It consists of long highly doped-erbium fiber, a fiber isolator, single-mode WDM coupler, in-line fiber polarization controller, 10% output coupler and the graphene-based saturable absorber placed between two FC/APC connectors [106]. The center wavelength of the output pulse of the Er-doped fiber laser is concentrated at 1.56um, as shown in Fig. 13 [106].
Typical circuit structure of Er-doped fiber laser. Taken from [106]
Optical spectrum of Er-doped fiber laser. Taken from [106]
For fiber lasers, its pulse output characteristics mainly depend on the fiber material, cavity structure, working mode, pump power and type and other factors used. Then the modulation depth and saturation also have a certain influence on the pulse output of the laser, so we need to measure the size of its parameters and understand their influence on the pulse output of the fiber laser. J. SOTOR et al. proposed a graphene saturable absorber mode-locked erbium-doped fiber laser [78]. Graphene is obtained by mechanical exfoliation, with a pulse center wavelength of 1562 nm, a half-width band of 9 nm, a duration of 630 fs, and a repetition frequency of 41.9 MHz. In the experiment, a femtosecond laser was used as the light source for them, and the signal first passed through a variable optical attenuator (VOA), and then passed through a fiber coupler. The measurement result of power dependent transmission is shown in Fig. 14.
Non-liner saturated absorption characteristic curve of graphene absorber. Taken from [78]
In order to calculate the parameters of SA, the measured data is substituted into the following formula for calculation to obtain the result.
$$ \alpha (I)=\frac{\alpha_0}{1+\left(I/{I}_{sat}\right)}+{\alpha}_{ns} $$
Where α(I) is the absorption coefficient, I is the light intensity, Isat is the saturation intensity, α0 andαns are saturable and non-saturable absorption, respectively.
Yb-doped fiber lasers
Recently, the graphene used in many experiments was prepared by the chemical reduction process of graphene oxide (GO) [129]. Furthermore, graphene oxide is widely used in Yb-doped fiber lasers. As a graphene derivative, graphene oxide not only has all the characteristics of ultrafast recovery time and broadband saturable absorption, but also is much easier and cheaper to be obtained [130]. The graphene oxide for the experiment was prepared through the vertical evaporation method from chemical oxidized graphite, the same as in the previous works [131]. The flake of the oxidized graphite is about 1 to 3 atomic layers and 0.1 μm to 5.0 μm of the diameter.
Up to date, saturated absorbers made of graphene oxide have been used in Yb-doped fiber lasers, The results of Yb-doped fiber laser research show that Yb-doped fiber laser based on graphene oxide is dominated by ring cavity. In this part, the fiber lasers as well as output pulse performance of the ring cavity are introduced. The fiber laser was schematically shown in Fig. 14.
Tables 4 and 5 summarizes the nonlinear optical saturable absorption properties and their applications in laser cavities using graphene oxide in the literature to date, including the nonlinear properties of few-layer graphene oxide device to mode-lock laser cavities and great progresses for demonstrated BP-based SA devices for Q-switched lasers. Zhao et al. experimentally verified an Yb-doped fiber laser based on a graphene saturating absorber could generate double rectangular pulses. The operating wavelength of graphene oxide-based ultrafast lasers has expended since then, covering from 1 μm to 3 μm. The performances of the output pulses are shown in Fig. 15. Compared with Er-doped fiber laser, this can be clearly seen in Fig. 15, and Tables 4 and 5, first of all, Yb-doped fiber laser is mainly composed of ring cavity. In the second place, the output pulse repetition frequency generated by Yb-doped fiber laser is lower than Er-doped fiber laser. Finally, the central wavelength of ytterbium doped fiber laser is at 1um (Fig. 16).
Table 4 Typical properties of mode-locked Yb-doped fiber lasers
Table 5 Typical properties of Q-switched Yb-doped fiber lasers
The schematic configuration of the laser cavity integrated with the GOSA. PC, polarization controller; WDM, wavelength division multiplexer; YDF, Yb-doped fiber; ISO, isolator; GOSA, graphene oxide saturable absorber; SMF, single mode fiber. Taken from [133]
Output pulse sequence diagram and output pulse characteristic curve. Taken from [133]
Nonlinear absorbance of the utilized graphene oxide saturable absorber piece. Taken from [146]
In this experiment, the researchers also tested the saturation characteristics of its graphene, and the results are shown in Fig. 17. The results show that the modulation depth is 25.2%, the saturation intensity is 8.5 MW/cm2, and the unsaturated loss is about 42.1%. In short, the greater the modulation depth, the shorter the output laser pulse.
Tm-doped fiber lasers
With the discovery of an increasing number of optical materials through research, Tm-doped fiber has been gradually applied to ultrafast fiber lasers. Tm-doped fiber laser operating in the 1.8 μm to 2.0 μm wavelength range is currently one of the most important branches of laser technology and experienced tremendous progress over the last decade [146]. Moreover, it is worthwhile mentioning that all lasers have similar circuit structure. The laser cavity is schematically shown in Fig. 18. It comprises all-fiber integrated components for an environmentally robust and compact regime. Tm-doped fiber amplifier is used to amplify laser pulse signal. The optical isolator is a passive optical device which only allows one-way light to pass through. The use of optical isolator is to prevent the backward transmission light in the optical path caused by various reasons from adversely affecting the light source as well as the optical communication system. Between semiconductor laser source and optical transmission system installed an isolator, it can largely reduce the reflected light of the negative effects of spectral power output stability of light source in a high-speed direct modulation direct detection of optical fiber communication system, the transfer will generate additional noise and light, which will degrade the performance of the system, so it also need isolator to eliminate. The function of band pass filter is to let the light signal pass and suppress the interference light source. Graphene saturable absorbers are used to absorb light signals in preparation for the next output. Finally, the direction of light transmission is corrected by adjusting the polarization controller to output similar laser signals.
TDFA-Tm-doped fiber amplifier; ISO-isolator; BPF-bandpass filter; OC-output coupler; GSA-graphene-SA; PC-polarization controller. Taken from [147]
Tables 6 and 7 summarizes the properties of different output pulses of typical Tm-doped fiber laser. It is found that mode-locking of a thulium-doped fiber laser operating at 1.94 μm, using a graphene-polymer based saturable absorber [11]. This is a simple, low-cost, stable and convenient laser oscillator for applications where eye-safe and low-photon-energy light sources are required, such as sensing and biomedical diagnostics [162]. The autocorrelation of the output pulse, and the corresponding optical spectrum are plotted in Fig. 19.
Table 6 Typical properties of mode-locked Tm-doped fiber lasers
Table 7 Typical properties of Q-locked Tm-doped fiber lasers
a Autocorrelation, b optical spectrum. Taken from [147]
In general, Tm-doped fiber lasers, with five representative spectra at the wavelength of 1876, 1908, 1943, 1953, and 2023 nm presented [11, 148, 149, 157, 162]. In previous research experiments, the graphene saturation characteristic measurement device and the result with the center wavelength of the output pulse of ~ 2000 nm are shown in Fig. 20. The results show that the modulation depth is 1.5% and the saturation intensity is 0.0035 MW/cm2 [161].
a Setup for the nonlinear absorption measurement. b Curve-fitted non-linear saturable absorption property of the graphene SA. Taken from [161]
Ho-doped fiber lasers
Ultrafast fiber laser operating below 2 μm based on different mode-locking mechanisms: nonlinear amplifying loop mirror, nonlinear polarization evolution (NPE), semiconductor saturable mirrors (SESAM), single wall carbon nanotubes (SWCNT), graphene, topological insulators, and black phosphorus were demonstrated [94, 163]. However, the number of laser sources operating at wavelengths longer than 2 μm is very limited. Recently, the first all-fiber Ho-doped oscillator based on a real SA has been used in fiber lasers at the first time.
The setup of the all-fiber Ho-doped mode-locked oscillator is depicted in Fig. 21. The 1 m long piece of Ho-doped gain fiber (HDF) was pumped by 1950 nm and 2080 nm wavelength division multiplexer (WDM) (both components are custom) by a continuous wave (CW) fiber laser delivering 0.5 W at 1950 nm. The resonator was arranged in a ring configuration, where the codirectional propagation of the signal and pump was forced by a fiber isolator. The signal was coupled out from the cavity using fiber output couplers with coupling ratios (CR) varying from 10% to 70%. Because the designed resonator was based on standard single-mode fibers, a polarization controller (PC) was spliced to the cavity to initiate the laser operation in mode-locking regime. The GSA was made by a graphene/poly (methyl methacrylate) (PMMA) composite inserted between two fiber connectors [163].
Setup of the all-fiber Ho-doped mode-locked laser. Taken from [163]
Figure 22 depicts the use of a 1550 nm all-fiber device to measure the power-dependent light transmittance of a saturated absorber containing 3 to 6 layers of graphene. The results show that the modulation depth increases as the number of graphene layers increases, while the saturation intensity decreases.
Measured energy dependent transmittance of GSAs containing of 3, 4, 5, and 6 layers of graphene. Taken from [163]
Table 8 summarizes the properties of different output pulses of typical Ho-doped fiber lasers. We can learn from the data in the table that all fiber Ho-doped mode-locked laser utilizedfew-layer graphene as saturable absorber. The laser was capable of generating optical solitons in the 2067 nm to 2081 nm spectral range with the shortest duration of 810 fs. The average output power and the pulse energy at the level of 44 mW and 1.3nJ were obtained, respectively. The pulse energy is the highest reported for the Ho-doped laser using GSA, while the 811 fs pulses are the shortest generated from a Ho-doped soliton laser.
Table 8 Typical properties of Ho-doped fiber lasers
Pr-doped fiber lasers
Since its discovery, graphene has been widely used as a saturation absorber in several common fiber lasers. In addition, it is also involved in Pr-doped fiber lasers. For this part, we will introduce the Pr-doped fiber laser. The technique used in Pr-doped fiber laser is Q–switching, it is experimental setup of a Q-switched Pr-doped fiber laser with graphene is shown in Fig. 23. On the one hand, the fiber laser adopts linear cavity structure, which has the advantages of simple structure and low cost. The graphene has a transmittance of 95.2% at 636 nm, indicating that it has two or more layers. The Pr-doped fiber laser has a central wavelength of 637 nm as shown in Fig. 24. However, because the amplitude of its output light source is not equal, the development of this kind of fiber laser is limited to a large extent. Although this kind of laser has some defects, perhaps after people's thorough research and discussion, this kind of fiber laser can be widely used in many fields [49, 75, 77] (Fig. 25).
Experimental Setup of Q-switching Pr-doped fiber laser. Taken from [77]
Spectra of 637-nm-band Q-switched pulse laser oscillation. Taken from [77]
Pulse waveform of 637 nm band Q-switched pulse laser oscillation. Taken from [77]
Nd:YOV4-doped fiber lasers
To date, an increasing number of materials can be used in a variety of fiber lasers. Furthermore, Tengfei Dai et al. demonstrated the Nd:YOV4-doped fiber laser application in optical filed [39]. In this section, we discuss the optical source properties of this kind of fiber laser. Frist of all, the experimental setup of the Nd:YOV4-doped fiber laser is shown in Fig. 26. Compared with several other types of fiber lasers, its structure is very simple and uses the linear cavity.
Experimental layout of the passively Q-switched laser based on Graphene-SA. Taken from [165]
In order to further study the characteristics of the output laser, a digital oscilloscope is used to observe and record the time trajectory of the pulse. The output waveform of the single pulse and the corresponding pulse sequence are shown in Fig. 27. When the pump power is 13.2 W, the pulse time path is observed and recorded by the digital oscilloscope. A stable Q-switched pulse with a pulse width of 280 ns, and a repetition rate of 1.29 MHz is obtained. Figure 28 is the output laser spectrum of continuous wave and Q-Switched operation when the pump power is 2.2w. The median wave length of the output spectrum of the laser is 1063.78 nm.
Output characteristics of pulsed laser at the absorbed pump power of 13.2 W. a Single pulse waveform of 200 ns/div; b Pulse sequence of 5 μs/div. Taken from [165]
Laser output spectrogram. Taken from [165]
Analysis and summary of multi-wavelength
In this review, it is found through summary that using different types of pump diodes and adjusting their power can achieve dual-wavelength pulse output. This article summarizes part of the research results as shown in Table 9, most of them use single-walled carbon nanotube saturable absorbers, erbium-doped fiber lasers and other optical devices to form ring cavity lasers. It can be seen that the center wavelength of the two pulses is distributed ~ 1500 nm, the repetition frequency ranges from a few MHz to tens of MHz, and the pulse duration ranges from femtoseconds to microseconds. Among the research results that have been summarized, Xin Zhang et al. published a passively mode-locked multi-wavelength output laser based on a single-walled carbon nanotube saturable absorber (SWNT-SA) [183]. The experimental device is shown in Fig. 29. The experimental device uses a ring-shaped cavity, including several commonly used and more important optical components. The purpose of connecting the isolater in the fiber ring cavity is to ensure that the light can travel in one direction in the ring cavity and eliminate the backscatter of the laser. The polarization controller realizes different polarization orientation states. The pump diode provides a light source of specific wavelength and power, and realizes the output of multiple wavelength pulses by adjusting its power. Erbium-doped fiber and single-mode fiber are used as gain media. The experimental instruments used to measure the output pulse mainly include power meters, spectrum analyzers, digital oscilloscopes, photodiode detectors and radio frequency analyzers.
Table 9 Summary of dual-wavelength fiber laser pulse output
Multi-wavelength pulse output experimental device setup. Taken from [183]
In this review, we summarize six different doped fiber lasers and three types of graphene-based saturable absorbers, which are the most widely used graphene saturated absorbers (GSA). Firstly more importantly, it is more essential that the six kinds of fiber lasers could generate well optical properties of optical source, such as high repetition frequency, essay preparation and low cost. Besides, most of fiber lasers use mode-locking and ring cavity, and the fiber lasers structure all most includes PC, GSA (graphene saturable absorber), coupler, LD, SMF, and doped fiber. Meanwhile, the fiber lasers are used in various filed.
Many researches on two-dimensional materials have demonstrated the properties of various materials, providing a large platform to future optical applications and promoting the development of fiber lasers. In ultrafast fiber laser applications, the properties of a variety of two-dimensional nanomaterials, due to their excellent photonic and optoelectronic properties are particularly significant for the fabrication of multifunctional SA devices in an simple and efficient manner.
Graphene and its oxides have attracted much attention due to their optical properties, for instance, super fast recovery time, simple structure and low cost. Therefore, the reported graphene SA photonics can be used in mode-locked and Q-switched fiber lasers to generate short pulses at the KHz-MHz repetition rate, or even at the GHz frequency, with the pulse duration ranging from fs to us, applications in various doped fiber lasers. In terms of device integration strategies, an increasing number of processing technologies including embedded transparent polymer films as well as photoconductive optical deposition on optical devices have been demonstrated on various platforms.
The growing interest of exploring the broad and increasing catalogue of available 2D nanomaterials encourages the researchers to search for new physics and technology breakthroughs. An increasing number of 2D nanomaterials have been demonstrated as ultra-fast wideband optical switches for mode-locked and Q-switched fiber lasers, since the reliability of these 2D nanomaterial based ultrafast optical switches are required to be proven to the same degree as incumbent saturable absorber technology like SESAM. Furthermore, it is worth mentioning that the D-shaped graphene saturable absorber is embedded in a ring cavity to generate a stable and well performance light source pulse, central wavelength focus on 1.5um, repetition rate is above MHz, with the pulse duration ranging from femtosecond to microsecond. A number of problems are pressing research problems to be addressed, including the relatively high unsaturated loss and laser cavity length of two-dimensional saturated nanoabsorbent materials, which limits the efficiency of fiber based light sources.
In the future, the use of graphene saturable absorbers in fiber lasers will transform the laser pulse generation by changing the number of layers. Moreover, heterogeneous structures made of different single-layer stacks may be the frontier and direction of future fiber laser source research, as they combine multiple different two-dimensional nanomaterials layers to create a new material system with significant optical properties that can be used in ideal applications. The development of quantum mechanics and materials science has provided theoretical tools to analyze size-dependent behavior on the nanoscale of such materials. On the other hand, graphene can be combined with a variety of gases or liquids with good optical properties to make saturated absorbers for fiber lasers, which will become the key direction and development trend of future research. So far, researchers have considered how to improve manufacturing techniques to scale up to accommodate integration with various optical devices. Two-dimensional nanomaterials will have revolutionary technological implications in photonics, such as graphene and its oxides, in line with current research interests.
SESAM:
Semiconductor saturation absorption mirror
Saturable absorber
CNTs:
TMDs:
Transition metal dichalcogenides
Tis:
Topological insulators
TMO:
Transition metal oxides
BP:
Black phosphorous
Two-dimensional
AFM:
Atomic force microscopy
LPE:
Liquid phase exfoliation
CVD:
Chemical vapor deposition
PVA:
DMF:
Dimethyl formamide
HOPG:
Highly ordered pyrolytic graphite
LD:
Laser diode
WDM:
Wavelength Division Multiplexer
Optically driven deposition
CE:
Chemically exfoliated
Mechanically exfoliate
EE:
Electrochemical exfoliation
Vertical evaporation
FH:
Facile hydrothermal
CW:
Continuous wave
CR:
Coupling ratios
PC:
Polarization controller
PMMA:
Graphene/poly (methyl methacrylate)
SWNT:
Single-wall carbon nanotube
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Liu, L., Zhao, X., Zheng, Z., Wang, Q.: Fast, long-scan-range pump-probe measurement using a dual-wavelength mode-locked fiber laser. FiO/LS Tech. Dig. (2012). https://doi.org/10.1364/FIO.2012.FW2A.1
Zhao, X., Gong, Z., Liu, Y., Yang, Y., Hu, G., Zheng, Z.: Coherent dual-comb mode-locked fiber laser based on a birefringent ring cavity. Front. Optics/Laser Sci. (2015). https://doi.org/10.1364/FIO.2015.FW3C.3
Chen, J., et al.: Low-power consumption dual-comb spectroscopy based on a battery-powered, free-running dual-comb laser system. Front. Opt. (2017). https://doi.org/10.1364/FIO.2017.JTu3A.17
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Zheng, Z., Zhao, X.: High-Resolution, Dual-Comb Asynchronous Sampling Enabled by Dual-Wavelength Ultrafast Fiber Lasers and its Applications. In: Conference on Lasers and Electro-Optics Pacific Rim (CLEO-PR) (2013)
Chen, J., et al.: Dual-wavelength, dual-comb fiber laser based on a nearly-adiabatic fiber-taper filter. Opt. Soc. Am. (2016). https://doi.org/10.1364/FIO.2016.JTh2A.112
Hu, G.: Multiwavelength, subpicosecond pulse generation from a SWNT-SA mode-locked ring birefringent fiber laser. Nonlin. Opt. Fibers. (2015)
Liu, Y., et al.: Multi-wavelength dissipative soliton, single-wall carbon nanotube mode-locked fiber laser. FIO/ LS Tech. Dig. (2011)
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The author would like to thank all colleagues for their hard work.
This work is supported in part by Heilongjiang Province Key Laboratory of Laser Spectroscopy Technology and Application, Harbin University of Science and Technology.
Heilongjiang Province Key Laboratory of Laser Spectroscopy Technology and Application, Harbin University of Science and Technology, Harbin, 150080, China
Xi Peng & Yixin Yan
Xi Peng
Yixin Yan
XP is the main author of the current paper. XP contributed to the development of the ideas, design of the study, result analysis, and article writing. YY finished the analysis and interpretation of data and drafted the manuscript. All authors read and approved the final manuscript.
Authors' information
Xi Peng is currently a master student at Harbin University of Science and Technology. His current research interests include optical fiber and communication and information systems.
Yixin Yan received a PhD in measurement and control technology and instrumentation from Harbin University of Science and Technology in 2013. Her current research interests include embedded systems and applications, data mining and applications, software engineering, optics and its applications in medicine.
Correspondence to Xi Peng or Yixin Yan.
Peng, X., Yan, Y. Graphene saturable absorbers applications in fiber lasers. J. Eur. Opt. Soc.-Rapid Publ. 17, 16 (2021). https://doi.org/10.1186/s41476-021-00163-w
Accepted: 07 July 2021
Two-dimensional materials
Fiber lasers
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Why is there so much debate over the US Supreme Court?
The death of liberal US Supreme Court Justice Ruth Bader Ginsburg has huge implications for the future of law and life in the United States, using Republican President Donald Trump the opportunity to cement a 6-3 conservative bulk on the court.The United
States was barrelling along a rollercoaster of an election project currently, with its dissentious partisan politics, a failing economy, race riots and an aggravating COVID death toll soaring previous 200,000.
Now the departure of Ginsburg, a cultural and feminist icon in the US (nicknamed RBG), will see both sides mobilising in a gradually fractious environment to maximise possible political benefits.Why is the Supreme Court such a big offer in the US? How are judges chosen? And what difference would a brand-new face on the bench make to life in America?What's the United States Supreme Court and why is the choice of judges such a big deal?They are informally known as the Supremes– not a brand-new super-hero franchise nor a reworked variation of the '60s all-female singing group, but a great deal of superstars in the United States judicial firmament.Advertisement Post III of the Constitution established the federal judiciary, with the very first conference of the court in 1790. 9 judges, described as justices, rest on the greatest court in the US and often supply the last word on extremely
controversial laws, arguments in between states and the federal government, and last draw in remain executions.They make landmark choices, says Jared Mondschein, a United States Investigates Centre senior expert at the University of Sydney, including that these decisions can" basically transform the nation, such as its 1954 judgment that the segregation of publicschools was unconstitutional". That option suggested that trainees might no longer be excluded on the basis of race. A Supreme Court justice is a life time assessment. Australian justices need to retire at the age of 70. It's a function similar to the one carried out by the Australian High Court, however it's a lot more partisan body.There are 3 essential aspects driving this partisanship.One, the United States president provides a nomination for any job that emerges in the court, which is
examined and approved by the Senate. In Australia, it's the Governor-General who selects
a new justice on the guidance of the Commonwealth Attorney-General and the federal cabinet.Advertisement Second, becoming a Supreme Court justice is a life time consultation. Australian justices require to retire at the age of 70. Filling Third, the court, as in Australia, can consider laws unconstitutional. In the United States, that can form up as overruling progressive legislation enacted by Congress in case Democrats have the votes to pass significant costs on issues such as environment modification. A conservative court would be even less most likely to accept
causes, such as ending the death sentence, although its existing 6-3 ruling in favour of LGBT employee rights recommends that concern might be an exception in specific circumstances.As a result, the elevation of a brand-new United States justice is a hotly challenged political area, subject to the combative pulling and pressing of the political representatives of the time. Ruth Bader Ginsburg talks with John Roberts, the Chief Justice of the Supreme Court, before a ceremony at the White House in 2018. Credit: Getty Images; image transformed Why are the Republicans in such a rush to change RBG?With Trump routing in the polls, and less than seven weeks up until election day, Politico's Tim Alberta says the Trump group has a view that they can" lock down" the judicial branch of the federal government, and energise their advocate base in the taking place protest.Advertisement In 2016, exit tally revealed that Supreme Court judge elections were the most important concern for 26 percent of the people who chose Trump, particularly for conservatives who want to enforce a spiritual program. Just 18 percent of Clinton fans said the same.At the really same time, the Republican Celebration understands they don't have much time. Supreme Court elections have in fact taken about 70 days to move through the Senate and the most current, for Brett Kavanaugh, took 89 days. The election is on November 3.
Yet there are no
set guidelines for how long the treatment need to take; some have moved quicker. Ginsburg's was 42 days, but not in an election year.It will come down to election project politics and the arcane workings of the SENATE. Which's where it gets intriguing. Trump's nomination for the court
can still be authorized even if he loses the election. Congress has really fulfilled in lame-duck sessions, which could include selecting a brand-new Supreme Court justice after the 2020 election. In a peculiarity of the US system, the Senate sits after the November election and prior to the start of the brand-new Congress for a minimum of 2 months, in what is known as lame-duck sessions.Prior to the adoption
of the 20th change to the Constitution in 1933, these lame-duck sessions may last for more than a year.Since that time, Congress has actually satisfied in lame-duck sessions to conclude immediate or insufficient organization, which may consist of selecting a new Supreme Court justice after the 2020 election.For some Republicans who currently manage the SENATE, this opportunity is more essential than a Trump success because they can impose their world viewpoint for generations through an effective court nomination.Advertisement What does it require to confirm a nominee?Only a bulk vote of the Senate. Republicans control the Senate by a 53-47 margin, indicating they might lose roughly 3 votes and still validate a justice, if Vice-President Mike Pence were to break a 50-50 tie.Supreme Court elections used to need 60 pick confirmation if any senator objected, but Republican political leader Senate leader Mitch McConnell modified Senate standards in 2017 to enable the confirmation of justices with 51 votes. Then Supreme Court justice prospect Merrick Garland in 2016. Credit: Getty Images; image customized Are elections constantly so controversial?The death of Ginsburg, at the age of 87, left an opening on the Supreme Court 46 days prior to the governmental election. There have in fact been 16 Supreme Court vacancies in US governmental election years that appeared prior to election day– just one other happened closer to the election. Chief Justice Roger B. Taney passed away 27 days prior to the 1864 presidential election and president Abraham Lincoln delayed his nomination till after Lincoln had actually won re-election. Advertisement Throughout history,
presidents have actually nominated Supreme Court prospects in spite of whether it was an election year. However, the death of Justice Antonin Scalia in 2016 saw the Republican-controlled Senate led by McConnell decline to consider then president Barack Obama's election of Merrick Garland to the court, specifying that not given that 1932 had actually the Senate verified in a governmental election year a Supreme Court candidate to a job occurring in that year.Today, despite claims of hypocrisy, McConnell is leading a push for holding a Supreme Court vote on a Republican candidate. The difference from 2016: the Republicans still manage the Senate and can verify President Trump's option if they stay united.How may a change of judge change the United States?Two words: abortion rights.With the Supreme Court now including 3 judges appointed by Democratic presidents and five check outs by Republicans, the possible replacement of the progressive Ginsburg by a conservative judge may have generational implications for issues such as access to abortion." There's a basic concern in this, which is Roe versus Wade, the American legal option which has actually been the necessary structure of the issue of abortion in the United States," says Rick Wilson, Republican strategist. "And there's a belief that Roe versus Wade will fall if there's a 6-3 conservative bulk in the Supreme Court." If Trump changes Ginsburg with a staunch conservative, the chances of the court considerably reducing abortion rights becomes ever most likely. Given That the Supreme Court legalised abortion in its 1973 Roe versus
Wade judgment, conservative activists have actually searched for to overturn the decision and have actually constantly fallen simply short. If Trump replaces Ginsburg with a strong conservative, the opportunities of the court drastically suppressing abortion rights winds up being ever more likely.Similarly, the conservative wing of the
court might be emboldened to make sweeping proceed other social problems, which could consist of broadening gun rights, strengthening specific spiritual rights and suppressing ballot rights.In the short-term, Ginsburg's absence may be felt most keenly when the court hears oral arguments on November 10 on the current difficulty by conservatives to the Obamacare health law, enacted in 2010 in a quote to offer budget-friendly health care to all Americans, and formerly supported by the Supreme Court on a 5-4 vote in 2012. Ginsburg was among the five justices in the bulk then, which means that her replacement may tilt the balance. A March for Life saw pro-life and pro-choice advocates voicing their views on the 31st anniversary of Roe v Wade outside the Supreme Court in 2004. Credit: Getty Images; image modified How have' the Supremes' found out an US election before?In 2000, the Supreme Court successfully selected the outcome of the governmental election when it overthrew the Supreme Court of Florida, which had actually permitted states sought by the Democratic governmental prospect Al Gore's task. In an associated, 5-4 vote, the judges ruled that no option method of recount might be established in a timely manner, which suggested Republican contender George W. Bush won in Florida, supplying him the winning votes in the Electoral College over his( partly )more popular opponent.Why is the primary justice's role so crucial too?For the past 2 years, following the retirement of conservative Justice Anthony Kennedy in 2018, Chief Justice John Roberts has in fact been the pivotal figure on the court, however his position of impact will be weakened if a Trump prospect changes Ginsburg.At the ideological centre of the nine-member court, Roberts has had the alternative of siding with the four liberal justices to his left or 4 conservatives to his right to protect a bulk opinion.Roberts, described as a protector of the court as a company and champ of the judiciary as an independent branch of government, sided with Ginsburg and the court's other 3 liberals in essential cases. Roberts has, on occasion, tried to find compromises in huge cases, in many cases to the dismay of his more conservative colleagues. In June, he assisted overrule a restrictive Louisiana abortion law and fended off Trump's quote to rescind defenses for numerous countless restricted immigrants called" Dreamers "who went into the United States as children.Roberts has, on celebration, looked for compromises in huge cases, in some cases to the discouragement of his more conservative
partners. In July, for example, he made up both choices as the court ruled that a New York district attorney could try to get Trump's monetary records but prevented Democratic-led Legislature committees from immediately getting comparable documents.Without Ginsburg, Roberts loses the ability to move the balance on his own.What did RBG state on the matter?Shortly before Ruth Bader Ginsburg passed away, she made a demand about what requires to occur to her seat on the Supreme Court." My most impassioned dream is that I will not be changed till a new president is established," she mentioned, with National Public Radio reporting that she dictated the note to her granddaughter, Clara Spera
, in the last days of her life.It's gradually most likely that Ginsburg's want her replacement will never ever be satisfied, but it's totally possible she may get the new president she so desired.Strap yourself in, this is going to be a rough flight.– with Chris Zappone, Reuters, AP, Bloomberg, The New York Times Homages to Ruth Bader Ginsburg outside the United States Supreme Court on September 21. Credit: Getty Images; image transformed Let us discuss If you 'd like some professional background on an issue or a news occasion, drop us a line at [email protected]!.?.! or [email protected]!.?.!. Discover more explainers here. Heath Gilmore is the United States Votes 2020 editor.< img
alt =" Tributes to Ruth Bader Ginsburg outside the United States Supreme Court on September 21." src=" https://static.ffx.io/images/$zoom_0.4069%2C$multiply_0.3541%2C$ratio_1%2C$width_1059%2C$x_412%2C$y_0/t_crop_custom/q_86%2Cf_auto/3f88ef449cd2f655ce78473b03543791a2fceb7d,%20https://static.ffx.io/images/$zoom_0.4069%2C$multiply_0.7082%2C$ratio_1%2C$width_1059%2C$x_412%2C$y_0/t_crop_custom/q_62%2Cf_auto/3f88ef449cd2f655ce78473b03543791a2fceb7d%202x"/ > Filling.
US-Qatar Partnership Objectives to Discover Buried Water in Earth's Deserts– Environment Change: Important Signs of the World
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Living Reviews in Solar Physics
December 2011 , 8:4 | Cite as
Magnetic Structure of Sunspots
Juan M. Borrero
Kiyoshi Ichimoto
First Online: 09 September 2011
In this review we give an overview about the current state-of-knowledge of the magnetic field in sunspots from an observational point of view. We start by offering a brief description of tools that are most commonly employed to infer the magnetic field in the solar atmosphere with emphasis in the photosphere of sunspots. We then address separately the global and local magnetic structure of sunspots, focusing on the implications of the current observations for the different sunspots models, energy transport mechanisms, extrapolations of the magnetic field towards the corona, and other issues.
Optical Depth Flux Tube Magnetic Field Vector Horizontal Magnetic Field Spectropolarimetric Observation
Supplementary material is available for this article at 10.12942/lrsp-2011-4.
1.1 Role of magnetic field in cosmic bodies
The role of the magnetic field has become firmly recognized in astrophysics as humans discover the rich variety of phenomena present in the universe. The largest constituent of cosmic bodies is in the state of plasma, i.e., ionized gas, which interacts with magnetic fields. Through this interaction the magnetic field is responsible for many of the structures and dynamics that we observe with modern instrumentation. The fundamental role of magnetic fields can be summarized as follows:
By trapping charged particles, the magnetic field guides the plasma motions (for example, it confines plasma or suppresses the convective fluid motions) and generates a variety of density structures in universe.
The magnetic pressure causes the plasma to expand and makes it buoyant, thereby driving the emergence of magnetic loops into the tenuous 'outer' atmosphere against the action of gravity.
The magnetic field guides (MHD) waves and transports energy and disturbances from a site of energy injection to other locations.
Magnetic fields inhibit the thermal conduction across them, and make the presence of multitemperature structures possible in tenuous, high conductivity gas like stellar corona.
The magnetic field stores and releases energy that produces transient dynamic phenomena like flare explosions and plasma ejections.
The magnetic field also plays a crucial role in accelerating non-thermal particles to the relativistic regime in tenuous plasmas.
The magnetic field is, therefore, one of the fundamental ingredients of the universe. In the case of the Sun, our nearest star, we observe spectacular active and dynamic phenomena driven by magnetic fields with their spatial, temporal, and spectral structures in detail. Because of this, the Sun serves as an excellent plasma laboratory and provides us a unique opportunity to study the fundamental processes of the cosmical magnetohydrodynamics. Let us make an addition on the list of the role of the magnetic field, i.e.:
The magnetic field produces or modifies the polarization property of the light emitted from, or absorbed by, cosmical bodies, making themselves measurable.
1.2 Discovery of sunspot's magnetic field
The key physical process that makes human be aware of the existence of magnetic fields in sunspots is the interaction between atoms and the magnetic field, i.e., the so called Zeeman effect, discovered in laboratory by the young physicist Pieter Zeeman in The Netherlands by the end of the 19th century (Zeeman, 1897). This effect describes how the electronic energy levels of an atom split in the presence of a magnetic field, giving raise to several absorption/emission spectral lines where there was only one spectral line in the absence of the magnetic field. In addition, the magnetic field modifies the polarization properties of the emitted/absorbed photons in a manner that depends on the viewing angle between the observer and the magnetic field vector.
An initial hint of the presence of the Zeeman effect in the spectra of sunspots is actually found in a historic record prior to the discovery of the Zeeman effect by Lockyer in 1866, describing "thick spectral lines in sunspots". Cortie in 1896 mentioned a reversal (bright core) of an absorption line in sunspot spectra, which would obviously be a manifestation of the Zeeman effect under a strong magnetic field.
A more concrete evidence of the presence of magnetic field in sunspots was established by George Hale in a paper entitled "On the Probable Existence of a Magnetic Field in Sun-Spots" (Hale, 1908). He observed line splitting and polarization in sunspot spectra observed by the newly constructed 35-feet solar tower at the Mount Wilson Observatory. By comparing the separation between the spectral components in the observed lines in sunspots and in sparks in laboratory experiments, he deduced that the magnetic field strength in sunspots was about 2600–2900 Gauss. This was the first detection of the extraterrestrial magnetic field, which opened the way for measuring the magnetic field on the Sun and on other astronomical objects. A more detailed description of the discovery of magnetic fields in sunspots can also be found in Toro Iniesta (1996).
1.3 Current tools to infer sunspot's magnetic field
Not much has changed since Hale's discovery of magnetic fields in sunspots (Hale, 1908). The broadening of the intensity profiles of spectral lines he saw on his photographic plates was produced by the Zeeman splitting of the atomic energy levels in the presence of the sunspot's magnetic field. Hale estimated a magnetic field strength of about 2600–2900 Gauss. This basic technique is still widely used nowadays. The addition of the polarization profiles: Stokes Q, U, and V, besides the intensity or Stokes I, allows us to determine not only the strength of the magnetic field but the full magnetic field vector B. This is done thanks to the radiative transfer equation (RTE):
$$\frac{{d{{I}_{\lambda }}\left( {X\left[ {{{\tau }_{c}}} \right]} \right)}}{{d{{\tau }_{c}}}}=\left. {{{{\hat{\mathcal{K}}}}_{\lambda }}\left( {X\left[ {{{\tau }_{c}}} \right]} \right)\left[ {{{I}_{\lambda }}\left( {X\left[ {{{\tau }_{c}}} \right]} \right)} \right]-{{S}_{\lambda }}\left( {X\left[ {{{\tau }_{c}}} \right]} \right)} \right]$$
, where \(I_\lambda (X[\tau _c ]) = (I,Q,U,V)\)†1 is the Stokes vector at a given wavelength λ. The variation of the Stokes vector with optical depth τ c appears on the right-hand side of Equation (1). The dependence of Iλ with τ c arises from the fact that X is a function of the optical depth itself: X = X[τ c ]. Here X represents the physical parameters that describe the solar atmosphere:
$$X(\tau _c ) = [B(\tau _c ),T(\tau _c ),P_g (\tau _c ),P_e (\tau _c ),\rho (\tau _c ),V_{los} (\tau _c ),V_{mic} (\tau _c ),V_{mac} (\tau _c )],$$
where B(τ c ) is the magnetic field vector, T(τ c ) is the temperature stratification, P g (τ c ) and P e (τ c ) are the gas and electron pressure stratification, ρ(τ c ) is the density stratification, and Vlos(τ c ) is the stratification with optical depth of the line-of-sight velocity. In addition, macro-turbulent Vmac(τ c ) and micro-turbulent Vmic(τ c ) velocities are often employed to model velocity fields occurring at spatial scales much smaller than the resolution element. Finally, on the right-hand side of Equation (1) we have the propagation matrix \({{\hat{\mathcal{K}}}_{\lambda }}\left( {X\left[ {{{\tau }_{c}}} \right]} \right)\) and the source function \(S_\lambda (X[\tau _c ])\) at a wavelength λ. The latter is always non-polarized and, therefore, only contributes to Stokes I:
$$S_\lambda (X[\tau _c ]) = (S_\lambda (X[\tau _c ]),0,0,0)\dag .$$
The radiative transfer equation has a formal solution in the form:
$${{I}_{\lambda }}\left( {X\left[ {{{\tau }_{c}}} \right]} \right)=\int\nolimits_{0}^{\infty } {{{\mathcal{O}}_{\lambda }}\left( {X\left[ {{{\tau }_{c}}} \right]} \right){{{\hat{\mathcal{K}}}}_{{c\lambda }}}\left( {X\left[ {{{\tau }_{c}}} \right]} \right){{S}_{\lambda }}\left( {X\left[ {{{\tau }_{c}}} \right]} \right)d{{\tau }_{c}}} ,$$
where \({{\hat{\mathcal{O}}}_{\lambda }}\left( {0,{{\tau }_{c}}} \right)\) is the evolution operator, which needs to be evaluated at every layer in order to perform the integration. During the 1960s and early 1970s, the first numerical solutions to the radiative transfer equation for polarized light became available (Beckers, 1969a,b; Stenflo, 1971; Landi Degl'Innocenti and Landi Degl'Innocenti, 1972; Wittmann, 1974a; Auer et al., 1977). Techniques to solve Equation (1) have continued to be developed even during the past two decades (Rees et al., 1989; Bellot Rubio et al., 1998; López Ariste and Semel, 1999a,b).
Figure 1 shows an example of how the Stokes vector varies when the magnetic field vector changes. In that movie we use spherical coordinates to represent the three components of the magnetic field vector: B = (B, γ, ϕ), where B is the strength of the magnetic field, γ is the inclination of the magnetic field with respect to the observer, and ϕ is the azimuth of the magnetic field in the plane perpendicular to the observer's line-of-sight. In Figure 1 we assume that the observer looks down along the z -axis, but this does not need to be always the case.
mpg-Movie (908.0 KB) Still from a movie showing The change of the synthetic emergent Stokes profiles (I, Q, U, V) when the magnetic field present in the solar plasma varies. The magnetic field vector is expressed in spherical coordinates: B moduli of the magnetic field vector, γ inclination of the magnetic field vector with respect to the observer's line-of-sight (z-axis in this case), and ϕ azimuth of the magnetic field vector in the plane perpendicular to the observer's line-of-sight. Results have been obtained under the Milne-Eddington approximation. (For video see appendix)
A major milestone was reached when these methods to solve the RTEs (1) and (4) were implemented into efficient minimization algorithms that allow for the retrieval of magnetic field vector in an automatic way (Ruiz Cobo and del Toro Iniesta, 1992; Ruiz Cobo, 2007; Socas-Navarro, 2002; del Toro Iniesta, 2003a; Bellot Rubio, 2006). This retrieval is usually done by means of non-linear minimization algorithms that iterate the free parameters of the model X(τ c ) (Equation (2)) while minimizing the difference between the observed and theoretical Stokes profiles (measured by the merit function X 2). The X(τ c ) that minimizes this difference is assumed to correspond to the physical parameters present in the solar atmosphere:
$$\chi ^2 = \frac{1} {{4M - L}}\sum\limits_{i = 1}^4 {\sum\limits_{k = 1}^M {\left[ {\frac{{I_i^{obs} (\lambda _k ) - I_i^{syn} (\lambda _k ,X[\tau _c ])}} {{\sigma _{ik} }}} \right]^2 } } . $$
Here \(I_i^{obs} (\lambda _k )\) and \(I_i^{syn} (\lambda _k ,X[\tau _c ])\) represent the observed and theoretical (i.e., synthetic) Stokes vector, respectively. The latter is obtained from the solution of the RTE (4) given a particular set of free parameters X (Equation (2)). The letter L represents the total number of free parameters in X and, thus, the term 4M-L represents the total number of degrees of freedom of the problem (number of data points minus the number of free parameters). In Equation (5), indexes i and k run for the four components of the Stokes vector (I, Q, U, V) and for all wavelengths, respectively. Finally, σ ik represents the error (e.g., noise) in the observations \(I_i^{obs} (\lambda _k )\).
Traditionally, the X 2-minimization has been carried out by minimization algorithms such as the Levenberg-Marquardt method (Press et al., 1986). However, more elaborated methods have also been employed in recent years: genetic algorithms (Charbonneau, 1995; Lagg et al., 2004), Principal-Component Analysis (Rees et al., 2000; Socas-Navarro et al., 2001), and Artifical Neural Networks (Carroll and Staude, 2001; Socas-Navarro, 2003).
Due to our limited knowledge of the line-formation theory, these investigations have usually been limited to the study of the photospheric magnetic field, where Local Thermodynamic Equilibrium and Zeeman effect apply for the most part. However, recent advancements in the line-formationtheory under NLTE conditions (Mihalas, 1978), scattering polarization (Trujillo Bueno et al., 2002; Manso Sainz and Trujillo Bueno, 2003; Landi Degl'Innocenti and Landolfi, 2004), etc., allow us to extend these techniques to the study of the chromosphere (Socas-Navarro et al., 2000a; Asensio Ramos et al., 2008; Trujillo Bueno, 2010; Casini et al., 2009). Indeed, some recent works have appeared where the magnetic structure of sunspots in the chromosphere is being investigated (Socas-Navarro et al., 2000b; Socas-Navarro, 2005a,b; Orozco Suarez et al., 2005).
Techniques to study the coronal magnetic field from polarimetric measurements of spectral lines are also becoming available nowadays (Tomczyk et al., 2007, 2008). These observations, carried out mostly with near-infrared spectral lines, are recorded using coronographs (to block the large photospheric contribution coming from the solar disk) and, therefore, limited to the solar limb. Other possibilities to observe polarization on the solar disk involve EUV (Extreme Ultra Violet) lines, which are only accessible from space, and radio observations of Gyroresonance and Gyrosynchrotron emissions, which can show large polarization signals: White (2001, 2005), Brosius et al. (2002), and Brosius and White (2006). Unfortunately, so far radio measurements have allowed only to infer the magnetic field strength in the solar corona. Interestingly, opacity effects in the gyroresonance emission (see Equations (1) and (2) in White, 2001) might also permit to infer the inclination of the magnetic field vector with respect to the observer's line-of-sight, i.e., γ However, this possibility has not been yet successfully exploited.
1.3.1 Formation heights
According to Equations (1) and (2) the solution to the radiative transfer equation depends on the stratification with optical depth τ c of the physical parameters. The range of optical depths in which the solution X(τ c ) will be valid depends on the region of the photosphere in which the analyzed spectral lines are formed. In the future we will refer to this range as \(\bar \tau = [\tau _{c,\min } ,\tau _{c,\max } ]\). \(\bar \tau \) can be determined by means of the so-called contribution functions (Grossmann-Doerth et al., 1988; Solanki and Bruls, 1994) and the response functions (Landi Degl'Innocenti and Landi Degl'Innocenti, 1977; Ruiz Cobo and del Toro Iniesta, 1994). In the literature, it is usually considered that the range of optical depths, that a given spectral lines is sensitive to, is so narrow that the physical parameters do not change significantly over [τc,min, τc,max]. This can be mathematically expressed as:
$$\frac{{X_f (\tau _{c,\max } ) - X_f (\tau _{c,\min } )}} {{X_f (\tau _{c,\max } ) + X_f (\tau _{c,\min } )}} < < 1, $$
where X f refers to the f-component of X (Equation (2)). When the conditions in Equation (6) are met for all f's, a Milne-Eddington-like (ME) inversion can be applied. The advantage of MEcodes is that an analytical solution for the RTE (1) exists in this case. ME-codes assume that the physical parameters are constant in the range \(\bar \tau \). One way to determine the magnetic field at different heights in the solar atmosphere is to perform ME-inversions of spectropolarimetric data in several spectral lines that are formed at different average optical depths \(\bar \tau \)'s, with each spectral line yielding information in a plane at a different height above the solar surface.
As an example of the results retrieved by a Milne-Eddington-like inversion code we show, in Figures 2 and 3, the three components of the magnetic field vector, for two different sunspots, in the observer's reference frame. B or magnetic field strength is shown in the upper-right panels, γ or the inclination of the magnetic field vector with respect to the observer's line-of-sight in the lower-left panels, and finally, ϕ or the azimuthal angle of the magnetic field vector in the plane perpendicular to the observer's light-of-sight in the lower-right panels. The first sunspot, AR 10923 (Figure 2), was observed very close to disk center (Θ ≃ 9°) on November 14, 2006. The second sunspot, AR 10933 (Figure 3), was observed on January 9, 2007 very close to the solar limb (Θ ≃ 50°). In both cases, the magnetic field vector was obtained from the VFISV Milne-Eddington-type inversion (Borrero et al., 2010) of the Stokes vector recorded with the spectropolarimeter on-board the Japanese spacecraft Hinode (Suematsu et al., 2008; Tsuneta et al., 2008; Ichimoto et al., 2008a). The observed Stokes vector corresponds to the Fe i line pair at 630 nm, which are formed in the photosphere. As explained above, Milne-Eddington inversion codes assume that, among others, the magnetic field vector does not change with optical depth: B ≠ f(τ c ) (see Equation (2)). Therefore, Figures 2 and 3 should be interpreted as the averaged magnetic field vector over the region in which the employed spectral lines are formed: \(\bar \tau \simeq [1,10^{ - 3} ]\).
These plots show the magnetic field vector in the sunspot AR 10923, observed on November 14, 2006 close to disk center (Θ = 8.7° at the umbral center). The upper-left panel displays the normalized (to the quiet Sun value) continuum intensity at 630 nm. The upper-right panel displays the total magnetic field strength, whereas the lower-left and lower-right panels show the inclination of the magnetic field vector γ with respect to the observer's line-of-sight, and the azimuth of the magnetic field vector in the plane perpendicular to the line-of-sight ϕ, respectively. The white contours on the colored panels indicate the umbral boundary, defined as the region in the top-left panel where I/Iqs < 0.3. These maps should be interpreted as the average over the optical depth range in which the employed spectral lines are formed: \(\bar \tau \simeq [1,10^{ - 3} ]\).
Same as Figure 2 but for the sunspot AR 10933, observed on January 9, 2007 close to the limb (Θ = 49.0° at the umbral center).
When the conditions in Equation (6) are not met, it is not possible to perform a ME-line inversion. If we do, the results should be interpreted accordingly, that is, the inferred values for X correspond to an average over the region \(\bar \tau = [\tau _{c,\min } ,\tau _{c,\max } ]\) where the spectral line is formed. A different approach consists in the application of inversion codes for the radiative transfer equation that consider the full τ c dependence of the physical parameters X. In this case, the solution of the radiative transfer equation can only be found numerically (cf. López Ariste and Semel, 1999b). Examples of these codes are: SIR (Ruiz Cobo and del Toro Iniesta, 1992), SPINOR (Frutiger et al., 1999), and LILIA (Socas-Navarro, 2002). This allows to obtain the optical depth dependence (τ c -dependence) of the physical parameters with one single spectral line. Ideally, in order to increase the range of validity of the inferred models, one still wants to employ different spectral lines.
1.3.2 Azimuth ambiguity
The elements of the propagation matrix \({{\hat{\mathcal{K}}}_{\lambda }}\left( {X\left[ {{{\tau }_{c}}} \right]} \right)\) (Equation (1)) for the linear polarization (see, e.g., del Toro Iniesta, 2003b, Chapter 7.5) can be written as:
$$ {\eta _Q \propto \cos 2\phi } $$
$$\min \left( {\frac{{B \cdot r}} {{\left| {B \cdot r} \right|}} \pm 1} \right), $$
where ϕ corresponds to the azimuthal angle of the magnetic field vector in the plane perpendicular to the observer's line-of-sight. Equations (7) and (8) also hold for the dispersion profiles (magnetooptical effects) ρ Q and ρ U present in the propagation matrix \({{\hat{\mathcal{K}}}_{\lambda }}\). Note that these matrix elements remain unchanged if we take ϕ + π instead of ϕ. Because of this the radiative transfer equation cannot distinguish between these two possible solutions for the azimuth: [ϕ, ϕ + π]. This is the so-called 180°-ambiguity problem in the azimuth of the magnetic field. Because of this ambiguity, the azimuthal angle of the magnetic field ϕ (as retrieved from the inversion of spectrolarmetric data) in Figures 2 and 3 (lower-right panels) is displayed only between 0° and 180°. A number of techniques have been developed to solve this problem. These techniques can be classified in terms of the auxiliary physical quantity that is employed:
Acute-angle methods: these techniques minimize the angle between the magnetic field vector inferred from the observations (see Section 1.3) and the magnetic field vector obtained from a given model. The question is, therefore, how is the model magnetic field obtained. Traditionally, it is obtained from potential or force-free extrapolations of the observed longitudinal component of the magnetic field: Blos = B cos γ, which is ϕ independent. The extrapolation yields the horizontal component of the magnetic field, which is then compared with the two possible ambiguous solutions: ϕ and ϕ+π Whichever is closer to the extrapolated horizontal component is then considered to be the correct, ambiguity-free, solution. Potential field and force-free extrapolations can be obtained employing Fourier transforms (Alissandrakis, 1981; Gary, 1989). Some methods that solve the 180°-ambiguity employing this technique have been presented by Wang (1997) and Wang et al. (2001). In addition, Green's function can also be used for the extrapolations and to solve the ambiguity (Sakurai, 1982; Abramenko, 1986; Cuperman et al., 1990, 1992).
Current free and null divergence methods: these methods select the solution, ϕ or ϕ + π, that minimizes the current vector j and/or the divergence of the magnetic field: ∇ · B. The calculation of these quantities makes use of the derivatives of the three components of the magnetic field vector. Because the vertical (z-axis) derivatives are usually not available through a Milne-Eddington inversion (see Sections 1.3.1, 2.1, and 2.3) only the vertical component of the current j z is employed. In addition, the term \(\partial B_z /\partial z\) is neglected in the calculation of the divergence of the magnetic field. The minimization of the aforementioned quantities can be done locally or globally. Finally, note that current free and null divergence methods usually rely on initial solutions given by acute-angle methods and potential field extrapolations.
In recent reviews by Metcalf et al. (2006) and Leka et al. (2009) several of these techniques are compared against each other, employing previously known magnetic field configurations and measuring their degree of success employing different metrics when recovering the original one. It is important to mention that in these reviews, some other very successful methods (which do not necessarily fall into the aforementioned categories) are also employed2: the non-potential magnetic field calculation method by Georgoulis (2005) and the manual utility AZAM by Lites et al. (private communication), which is part of the ASP routines (Elmore et al., 1992). In those reviews it is found that acute-angle methods perform well only if the configuration of the magnetic field is simple, whereas interactive methods (AZAM) tend to fail in the presence of unresolved structures below the resolution element of the observations. Current free and null divergence methods tend to work better when both conditions (Canfield et al., 1993; Metcalf, 1994) are applied instead of only one (Gary and Demoulin, 1995; Crouch and Barnes, 2008), with local minimization being more prone to propagate errors than global minimization techniques.
Several of these techniques are very suitable to study complex regions, in particular outside sunspots. However, in regular sunspots (excluding those with prominent light bridges or δ-sunspots3) the magnetic field is highly organized, with filaments that are radially aligned in the penumbra. We can use this fact to resolve the 180°-ambiguity in the determination of the azimuthal angle ϕ. This is done by finding the coordinates of the magnetic field vector B in the local reference frame: {e α , e β , e ρ }4 and taking whichever solution, B(ϕ) or B(ϕ + π), minimizes the following quantity:
$$\begin{gathered} \zeta = \cos ^{ - 1} \left[ {\frac{{B_\rho }} {{\sqrt {B_\alpha ^2 + B_\beta ^2 } }}} \right], \hfill \\ \Psi = \tan ^{ - 1} \left[ {\frac{{B_\beta }} {{B_\alpha }}} \right]. \hfill \\ \end{gathered} $$
where the vector r corresponds to the radial direction in the sunspot or, in other words, r is the vector that connects the center of the umbra with the point of observation. Because the condition of radial magnetic fields (Equation (9)) can only be safely applied in the local reference frame, it is important to describe how B and r are obtained. A detailed account is provided in Appendix 5 of this paper.
Because we aim at minimizing the above value (Equation (9)) this method can be considered as an acute-angle method where the reference magnetic field is not obtained from a potential extrapolation but rather assumed to be radial. Note that if the sunspot has positive polarity, the magnetic field vector and the radial vector tend to be parallel: Br > 0 and, therefore, the - (minus) sign should be used in Equation (9). If the sunspot has negative polarity, then the magnetic field vector and the radial vector are anti-parallel and, therefore, the sign + (plus) should be employed. However, this is only a convention: we can choose to represent the magnetic field vector as if a sunspot had a different polarity as the one indicated by Stokes V.
As an example of the method depicted here we show, in Figures 4, 5, and 6, the vertical B ρ and horizontal B β and B α components of the magnetic field vector (Equation (45)), once the 180°-ambiguity has been resolved for two sunspots: AR 10923 and AR 10933 (same as in Figures 2 and 3). B ρ , B β , and B α are the components of the magnetic field vector in the local reference frame. Note that strictly speaking, the unit vectors e β and e α shown in these figures correspond to the unit vectors at the umbral center. Although differences are small, at other points in the image the unit vectors have different directions since those points have different (X c ,Y c ) and (α, β) coordinates (Equations (28)–(32)). Once the 180°-ambiguity has been solved we can obtain, in the local reference frame, the inclination and the azimuth of the magnetic field, ζ (Figure 7) and Ψ (Figure 8) as:
$$d_{\tau _c } = - \rho (x_\rho )\chi _c [T(x_\rho ),P_g (x_\rho ),P_e (x_\rho )]dx_\rho ,$$
$$\frac{{dP_g (x_\rho )}} {{dx_\rho }} = - g\rho (x_\rho ), $$
Vertical component of the magnetic field B ρ in the local reference frame in two different sunspots: AR 10923 (top; Θ = 8.7°) and AR 10933 (bottom: Θ = 49.0°). The black contours highlight the regions where the magnetic field points downwards towards the solar center: B ρ < 0. The white contours surround the umbral region, defined as the region where the continuum intensity (normalized to the quiet Sun intensity) I/Iqs < 0.3. The horizontal and vertical directions in these plots correspond to the e β and e α directions, respectively.
Same as Figure 4 but for the B β component of the magnetic field vector in the local reference frame. The arrow field indicates the direction of the magnetic field vector in the plane tangential to the solar surface.
Same as Figure 5 but for the B α component of the magnetic field vector.
Same as Figure 4 but for the inclination of the magnetic field with respect to the normal vector to the solar surface e ρ : ζ (see Equation (10)). The black contours indicate the regions where ζ > 90° and coincide with the regions, in Figure 4, where B ρ < 0.
Same as Figure 7 but for the azimuthal angle of the magnetic field in the plane of the solar surface: Ψ (see Equation (11)).
It is important to notice that because the ambiguity has now been solved, the angle Ψ varies between 0° and 360° (see Figure 8), whereas before, lower-right panels in Figures 2 and 3, ϕ ranged only between 0° and 180°.
As already mentioned, the method we have described here works very well for regular (e.g., round) sunspots. There is, however, one important caveat: when the retrieved inclination γ (in the observer's reference frame) is close to 0, the azimuth ϕ is not well defined. In this case, applying Equation (9) does not make much sense. Here we must resort to other techniques (Metcalf et al., 2006) to solve the ambiguity. The region where γ = 0° occurs usually at the center of the umbra for sunspots close to disk center, and it shifts towards the center-side penumbra as the sunspot is closer to the limb. A similar coordinate transformation as the one depicted here have been described in Hagyard (1987) and Venkatakrishnan et al. (1988), with the difference that no attempt to solve the 180°-ambiguity was made. Bellot Rubio et al. (2004) and Sánchez Almeida (2005a) employ a smoothness condition to solve the 180°-ambiguity, however their coordinate transform is done in two dimensions, whereas here we consider the Sun's spherical shape. In addition, only one heliocentric angle Θ was considered in their transformation, whereas here Θ changes for each point on the solar surface (Equation (38)). One might think that the variation of the angle Θ across the field-of-view (FOV) are negligible. However, for a FOV with 100 × 100 arcsec2 this variation can be as large as 4–5°. These differences can be important, for instance, when searching for regions in the sunspot penumbra where the magnetic field points down into the solar surface: B ρ < 0.
1.3.3 Geometrical height and optical depth scales
Traditionally, inversion codes for the RTE (1) such as: SIR (Ruiz Cobo and del Toro Iniesta, 1992) and SPINOR (Frutiger et al., 1999), provide the physical parameters as a function of the optical depth, X(τ c ) (Equation (2)). The optical depth is evaluated at some wavelength where there are no spectral lines (continuum), hence the sub-index c. When this is done for each pixel in an observed two-dimensional map, the inversion code yields X(\(x_\beta \), \(x_\alpha \), τ c ). However, it is oftentimes convenient to express them as a function of the geometrical height \(x_\rho \). To that end, the following relationship is employed:
$$\frac{{dP_g (\tau _c )}} {{d\tau _c }} = \frac{g} {{\chi _c (\tau _c )}}. $$
where χc is the opacity evaluated at a continuum wavelength and depends on the temperature, gas pressure, and electron pressure. Now, these thermodynamic parameters barely affect the emergent Stokes profiles Iλ and, therefore, are usually not obtained from the inversion of the polarization profiles themselves. Instead, other kind of constraints are usually employed to determine them, being the most common one, the application of the vertical hydrostatic equilibrium equation:
$$P_{g,1} (\tau _{\min - 1} ) = P_g (\tau _{\min } ) - \frac{g} {{\chi _c (\tau _{\min } )}}[\tau _{\min } - \tau _{\min - 1} ] $$
which after applying Equation (12) becomes:
$$\frac{{dP_g (\tau _c )}} {{d\tau _c }} = \frac{g} {{\chi _c (\tau _c )}} $$
Note that, since Equations (13) and (14) do not depend on (\(x_\beta \), \(x_\alpha \)), they can be applied independently for each pixel in the map. Hence, the geometrical height scale (at each pixel) can be obtained by following the next steps:
Given a boundary condition for the gas pressure in the uppermost layer of the atmosphere, P g (τmin), we can employ the fixed-point iteration described in Wittmann (1974a) and Mihalas (1978) to obtain the electron pressure in this layer: P e (τmin).
From the inversion, the full temperature stratification T(τ c ) and, thus, T(τmin) are known. Since the continuum opacity χc depends on the electron pressure, gas pressure, and temperature, it is therefore possible to obtain χc (τmin).
A predictor-corrector method is employed to integrate downwards Equation (14) and obtain P g (τmin-1). This is done by first assuming that χc is constant between τmin and τmin-1:
$$\rho (v\nabla )v = - \nabla P_g + \frac{1} {c}j \times B + \rho g. $$
and with Pg,1(τmin-1), we apply step #1 to calculate Pe,1(τmin-1).
Since we also know T(τmin-1), we repeat step #2 to recalculate χc (τmin7#x002D;1), which is then employed to re-integrate Equation (14) as:
$$hydrostatic:\left\{ {\begin{array}{*{20}c} {dP_g /dx_\rho = - \rho g} \\ {dP_g /dx_\beta = 0} \\ {dP_g /dx_\alpha = 0.} \\ \end{array} } \right. $$
Step #4 is repeated k-times until convergence: \(\left| {P_{g,k} (\tau _{\min - 1} ) - P_{g,k - 1} (\tau _{\min - 1} )} \right| < \varepsilon \).
We now have P g (τmin-1). In addition, T(τ c ) and, thus, T(τmin) are known. Consequently, we can repeat steps #1 to #3 in order to infer P g (τmin-2).
Thus, repeating steps #1 through #5 yields: P g (τ c ), ....(τ c ), and χc P e (τ c ).
The equation of ideal gases can be now employed to determine ρ(τ c ). And, finally, the integration of Equation (12) yields the geometrical depth scale as: τ c (\(x_\rho \)). To integrate this equation, a boundary condition is needed. This is usually taken as \(x_\rho (\tau _c = 1) = 0\), which sets an offset to the geometrical height such that the continuum level τ = 1 coincides with \(x_\rho = 0\).
Applying the condition of hydrostatic equilibrium to obtain the density, gas pressure, and the geometrical height scale z is strictly valid only when the Lorentz force are small and the velocities are much smaller than the speed of sound. In the chromosphere and corona this is certainly not the case. In the solar photosphere the assumption of hydrostatic equilibrium is, in general, well justified. One exception are sunspots, where the large velocities and magnetic fields might break down this assumption. In these case, a more general momentum (force balance) equation must be employed5:
$$\frac{{dB_\rho }} {{dx_\rho }} \approx \frac{{B_{\rho ,2} - B_{\rho ,1} }} {{x_{\rho ,2} - x_{\rho ,1} }}[Gkm^{ - 1} ], $$
Trying to solve this equation to obtain the gas pressure, density, and geometrical height scale is not an easy task. In the hydrostatic case, the horizontal derivatives did not play any role, thus simplifying Equation (17) into:
However, if the Lorentz force j × B and the advection term (vΔ)v cannot be neglected, the horizontal components of the momentum equation must be considered. In addition, the horizontal derivatives of the gas pressure mix the results of the magnetic field and velocity from nearby pixels. Thus, the determination of the gas pressure, density, and geometrical height scale cannot be achieved individually for each pixel of the map. Instead, a global technique must be employed. This can be done by shifting the z-scale at each pixel in the map (effectively changing the boundary condition mentioned in step #7 above) in order to globally minimize the imbalances in the three components of the momentum equation and the term ∇ · B. The shift at each pixel, \(Z_W (x_\beta ,x_\alpha )\), represents the Wilson depression. This kind of approach has been followed by Maltby (1977), Solanki et al. (1993), Martínez Pillet and Vazquez (1993), and Mathew et al. (2004). However, changing the boundary condition in step #7 does not change the fact that the vertical stratification of the gas pressure still complies with hydrostatic equilibrium (Equation (13)). A way out of this problem has not been figured out until very recently with the work of Puschmann et al. (2010a,b), who have devised a technique that takes into account the general momentum equation (17) when determining the gas pressure and establishing a common z-scale. Figure 9 shows a map for the Wilson depression in a small region of the inner penumbra of a sunspot (adapted from Puschmann et al., 2010b). Another interesting technique has been proposed recently by Carroll and Kopf (2008), where the vertical height scale can be obtained, instead of a posteriori as in Puschmann et al. (2010b), directly during the inversion of the Stokes profiles. This is achieved by performing the inversion employing Artificial Neural Networks (ANNs; Carroll and Staude, 2001, see Section 1.3) that have been previously trained with snapshots of MHD simulations, which are given in the z-scale.
Map of the Wilson depression Z w (x, y) in a small region of the inner penumbra in AR 10953 observed on May 1, 2007 with Hinode/SP. The white contours enclose regions where upflows are present: Vlos > 0.3 km s−1. Negative values of Zw correspond to elevated structures. In this figure x and y correspond to our coordinates x β and x α , respectively (from Puschmann et al., 2010b, reproduced by permission of the AAS).
2 Global Magnetic Structure
In this section, we will discuss the global structure of the magnetic field vector in sunspots. Even though sunspot's magnetic fields are organized at very small scales (see, for example, Figures 4–8), there are many questions that can be addressed considering mainly its global structure: wave propagation (Khomenko and Collados, 2008; Moradi and Cally, 2008), helioseismology (Moradi et al., 2010; Cameron et al., 2011), extrapolations to obtain the coronal magnetic field (Schrijver et al., 2008; Metcalf et al., 2008; DeRosa et al., 2009). In the former cases, small-scale magnetic structures do not interact with typical helioseimology waves (p and f-modes) because their wavelengths are much larger than the typical sizes of the magnetic structures. In the latter case, small-scale horizontal magnetic structures do not affect the coronal magnetic structure because they produce loops that close at photospheric and chromospheric levels (Wiegelmann et al., 2010).
Other branch where observational inferences of sunspot's global magnetic structure are needed is in theoretical modeling of sunspots (i.e., magneto-hydrostatic; Low, 1975, 1980; Osherovich and Lawrence, 1983; Pizzo, 1986, 1990; Jahn and Schmidt, 1994b). These models employ the magnetic field configuration inferred from observations as boundary conditions in their equations, as well as employing the observations to validate their final results.
The first section of this chapter will be devoted to study the magnetic field configuration as seenat a constant optical depth or σ-level, whereas the second section will study the vertical variationsof the magnetic field. These two can be employed as Dirichlet or Neumann boundary conditions,respectively, in theoretical models and extrapolations. The rest of the sections in this chapter will focus on other issues such as the plasma-β, potentiality of the magnetic field, thermal-magnetic relation, and so forth.
2.1 As seen at constant τ-level
In Figures 2 and 3 in Section 1.3, and Figures 4–8 in Section 1.3.2, we have presented the 3 components of the magnetic field both in the observer's reference frame and in the local reference frame. Those maps were obtained from the inversion of spectropolarimetric observations employing a Milne-Eddington (ME) atmospheric model (see Section 1.3.1). This means that the results from a ME inversion should be interpreted as an average of the magnetic field vector over the region where the lines are formed \(\bar \tau :B_\beta (x_\beta ,x_\alpha ,\bar \tau )\), \(B_\alpha (x_\beta ,x_\alpha ,\bar \tau )\), and \(B_\rho (x_\beta ,x_\alpha ,\bar \tau )\). This makes the results from the ME inversion ideal to study the magnetic field at a constant τ-level. The coordinates x α and x β refer to the local reference frame: {e β , e α , e ρ } as described in Section 1.3.2. Note that the optical depth σ is employed instead of x ρ , which is the coordinate representing the geometrical height. For convenience let us now consider polar coordinates in the αβ-plane: (r, θ) with r being the radial distance between any point in the sunspot to the center of the umbra. θ is defined as the angle between the radial vector that connects this point with the umbra center and the e β axis (see, for example, Figure 4). With this transformation we now have: \(B_h (r,\theta ,\bar \tau ) = \sqrt {B_\beta ^2 (r,\theta ,\bar \tau ) + B_\alpha ^2 (r,\theta ,\bar \tau )} \) (horizontal component of the magnetic field) and \(B_\rho (r,\theta ,\bar \tau )\) (vertical component of the magnetic field).
We will now focus on the radial variations of the Ψ-azimuthally averaged (see Equation (11)) components of the magnetic field vector. Since sunspots are not usually axisymmetric we will employ ellipses, as illustrated in Figure 10, to calculate those averages. The ellipses are determined by first obtaining the coordinates of the center of the umbra: {x β,u ; x α,u }, and then fitting ellipses with different major and minor semi-axes, such that the outermost blue ellipses in Figure 10 provides a good match to the boundary between the penumbra and the quiet Sun. The upper panel in Figure 10 shows the ellipses for AR 10923 observed on November 14, 2006 at Θ = 8.7°, whereas the lower panel shows AR 10933 observed on January 9, 2007 at Θ = 49.0°.
Map of the continuum intensity for two sunspots. The top panel shows AR 10923 observed at Θ = 8.7°, whereas the bottom panel shows AR 10933, observed at Θ = 49.0°. These are the same sunspots as discussed in Sections 1.3 and 1.3.2. The blue ellipses are employed to determine the azimuthal averages (Ψ-averages) of the magnetic field vector. Note that the outermost ellipse tries to match the boundary between the penumbra and the quiet Sun. The orange arrow points towards the center of the solar disk.
The radial variation of the azimuthal averages is presented in Figure 11. The vertical bars in this figure represent the standard deviation for all considered points along each ellipse's perimeter. Note that, although the scatter is significant, the radial variation of the different components of the magnetic field vector are very well defined. Furthermore, both sunspots (AR 10923 in the upper panels; AR 10933 in the lower panels) show very similar behaviors of the magnetic field vector with r/R s (R s refers to the total sunspot radius). This happens for all relevant physical quantities: the total magnetic field strength \(B_{tot} (r,\bar \tau )\) (green curve), the vertical component of the magnetic field vector \(B_\rho (r,\bar \tau )\) (red curve), as well as for the horizontal component of the magnetic field vector \(B_h (r) = \sqrt {B_\beta ^2 (r,\bar \tau ) + B_\alpha ^2 (r,\bar \tau )} \) (blue curve). Consequently, the radial variation of the inclination of the magnetic field vector with respect to the vertical on the solar surface ζ, which can be obtained from B ρ and B h (Equation (10)) is also very similar for both sunspots (right panels in Figure 11).
Left panels: Azimuthally averaged components of the magnetic field vector as a function of the normalized radial distance r/R s from the sunspot's center. The magnetic field corresponds to a constant τ-level. In green the total magnetic field strength \(B_{tot} (r,\bar \tau )\) is presented while red and blue refer to the vertical B ρ and horizontal B h components of the magnetic field. Top panel shows the radial variations for AR 10923 and the bottom panel refers to AR 10933 (see Figure 10 for details). Right panels: inclination at a constant τ-level of the magnetic field vector with respect to the vertical direction on the solar surface, as a function of the normalized radial distance from the sunspot's center: \(\zeta (r,\bar \tau )\) (see Equation (10)). The horizontal dashed line is placed at ζ = 90°, indicating when the magnetic field points downwards on the solar surface. The vertical dashed line at r/R s ≃ 0.4 is placed at the boundary between the umbra and the penumbra.
The vertical component of the magnetic field vector B ρ monotonously decreases with the radial distance from the center of the umbra, while the transverse component of the magnetic field, B h , first increases until r/R s ∼ 0.5, and decreases afterward. In both sunspots the vertical and the transverse component become equally strong close to r/R s ∽ 0.5, which results in an inclination for the magnetic field vector of ζ ≃ 45° exactly in the middle of the sunspot radius (right panels in Figure 11). This location is very close to the umbra-penumbra boundary, which occurs at approximately r/R s ≃ 0.4 (vertical dashed lines in Figure 11). The inclination of the magnetic field ζ monotonously increases from the center of the sunspot, where it is considerably vertical (ζ ≃ 10–20°), to the outer penumbra, where it becomes almost horizontal (ζ ≃ 80°). Furthermore, the inclination at individual regions at large radial distances from the sunspot's center can be truly horizontal (ζ = 90°) or, as indicated by the vertical bars in Figure 11, the magnetic field vector can even point downwards in the solar surface, with B ρ < 0 at certain locations. This is also clearly noticeable in the black contours in Figures 4 and 7. Before Hinode/SP data became available, detecting these patches where the magnetic field returns into the solar surface (Bellot Rubio et al., 2007b) was not possible unless more complex inversions (not ME-like) were carried out (see Section 2.2). Nowadays with Hinode7#x2019;s 0.32" resolution, these patches which sometimes can be as long as 3–4 Mm, are detected routinely (see Figure 4; also Figure 4 in Bellot Rubio et al., 2007b). Note that theoretical models for the sunspot magnetic field allow for the possibility of returning-flux at the edge of the sunspot (Osherovich, 1982; Osherovich and Lawrence, 1983; Osherovich, 1984).
All these results are consistent with previous results obtained from Milne-Eddington inversions such as: Lites et al. (1993); Stanchfield II et al. (1997); Bellot Rubio et al. (2002, 2007b). Although most of these inversions were also obtained from the analysis of spectropolarimetric data in the Fe I line pair at 630 nm, a few of them also present maps of the magnetic field vector obtained from other spectral lines such as C I 538.0 nm and Fe I 537.9 nm (Stanchfield II et al., 1997), or Fe I 1548 nm in Bellot Rubio et al. (2002). Analysis of spectropolarimetric data employing other techniques such as the magnetogram equation, which yields the vertical component of the magnetic field at a constant τ-level, have also been carried out by other authors (Bello González et al., 2005). The consistency between all the aforementioned results is remarkable, specially if we consider that each work studied different sunspots and employed different spectral lines.
The picture of a sunspot that one draws from these radial variations is that of a vertical flux tube, with a diameter of 30–40 Mm (judging from Figure 10), where the magnetic field is very strong and vertical at the flux tube's axis (umbral center), while it becomes weaker and more horizontal as we move towards the edges of the flux tube. Even though these results were obtained only for a fixed τ-level on the solar photosphere, they clearly indicate that the flux tube is expanding with height as the magnetic field encounters a lower density plasma. The overall radial variations of the components of the magnetic field seem to be independent of the sunspot size, although the maximum field strength (which occurs at the sunspot's center) clearly does, as illustrated by Figure 11, where the magnetic field strength for AR 10923 peaks at about 3300 Gauss (large sunspot), whereas for AR 10933 (small sunspot) it peaks at around 2900 Gauss. This has been further demonstrated by several works that employed data from many different sunspots (Ringnes and Jensen, 1960; Brants and Zwaan, 1982; Kopp and Rabin, 1992; Collados et al., 1994; Livingston, 2002; Jin et al., 2006).
As explained in Section 1.3.1, Figures 4, 5, 6, and 11 refer to the average magnetic field vector in the photosphere: \(\bar \tau \in [1,10^{ - 3} ]\). This is because they were obtained from the Milne-Eddington inversion of spectropolarimetric data for the line Fe i pair at 630 nm. The investigations of the magnetic field vector in the chromosphere is far more complicated, since Non-Local Thermodynamic Equilibrium (NLTE) conditions make the interpretation of the Stokes parameters more difficult. However, in the last years a number of works have addressed some of these issues. For example, Orozco Suarez et al. (2005) analyzes data from the Si I and He I spectral lines at 1083 nm, which are formed in the mid-photosphere and upper-chromosphere, respectively. They find very similar radial variations of the magnetic field vector in the chromosphere and the photosphere, with the main difference being a reduction in the total magnetic field strength. Furthermore, Socas-Navarro (2005a) has presented an actual NLTE inversion of the Ca II lines at 849.8 and 854.2 nm. These two spectral lines are formed in the photosphere and chromosphere: \(\bar \tau \in [1,10^{ - 6} ]\).
2.2 Vertical-τ variations
The determination of the vertical variations of the magnetic field in sunspots has been a recurrent topic in Solar Physics for decades. Traditionally, this determination had been done through a combination of spectropolarimetric observations, where the magnetic field is measured at different heights in the solar atmosphere (Kneer, 1972; Wittmann, 1974b), and theoretical consideration such as employing a given sunspot model, applying the ∇ · B = 0 condition, etcetera (Hagyard et al., 1983; Osherovich, 1984, and references therein). Those first attempts were usually limited to the vertical component of the magnetic field B ρ :
$$\frac{{dB_\rho }} {{d\tau }} \approx \frac{{B_{\rho ,2} - B_{\rho ,1} }} {{\bar \tau _2 - \bar \tau _1 }}[G]. $$
where x ρ is the coordinate along the direction that is perpendicular to the solar surface (see Figure 42) and has been referred to as z in Section 1.3.3. In those early works, inferences of the vertical gradient of the vertical component of the magnetic field could differ by as much as an order of magnitude: 1–10 G km−1 (Kotov, 1970), 0.5–2 G km−1 (Makita and Nemoto, 1976). Here we will refer, however, to the gradients of the magnetic field in terms of the optical depth scale (Equation (12)):
$$\left. {X_k (\tau _c ) = X_k (\log \tau _c = 0) + \log \tau _c \frac{{dX_k }} {{d\tau _c }}} \right|_{\log \tau _c = 0} , $$
If xρ,2 and xρ,1 (or alternatively \(\bar \tau _2 \) and \(\bar \tau _1 \)) are sufficiently far apart (> 1000 km), the gradient refers to the average gradient between the chromosphere and the photosphere. This can be done, for example, employing pairs of lines where one of them is photospheric and another one is chromospheric: Fe I 525.0 nm and C IV 154.8 nm (Hagyard et al., 1983), Fe I 1082.8 nm and He I 1083.0 nm (Kozlova and Somov, 2009), Fe I 630.2 nm and Na I 589.6 nm (Leka and Metcalf, 2003). Through a Milne-Eddington-like inversion (or applying a magnetrogram calibration) the vertical component of the magnetic field can be inferred separately for each line and, thus, separately for xρ1 and xρ,2. Another way is to employ a single spectral line whose formation range is very wide. Examples of such lines are: Ca II 393.3 nm or Ca II 854.2 nm. These lines are sensitive to \(\bar \tau \in [1,10^{ - 6} ]\), with τ c = 1 being the photosphere and τ c = 10−6 the chromosphere (Socas-Navarro, 2005a,b).
Since the theory of spectral line formation in the chromosphere is not currently fully understood (see Section 1.3), in this review we will focus mostly in the photospheric gradient of the magnetic field. To that end, we will employ the Fe I line pair at 630 nm observed with Hinode/SP. These two spectral lines are both formed within a range of optical depths of \(\bar \tau \in [1,10^{ - 3} ]\). We perform an inversion of the Stokes vector in these two spectral lines, assuming that each of the physical parameters in X (Equation (2)) change linearly with the logarithm of the optical depth:
$$j_\rho = (\nabla \times B)_z = \frac{1} {{\mu _0 }}\left[ {\frac{{dB_\alpha }} {{dx_\beta }} - \frac{{dB_\beta }} {{dx_\alpha }}} \right]. $$
where X k refers to the k-component of X. Note that the inversion cannot be carried out with a Milne-Eddington-like inversion code, since those assume that the physical parameters do not change with optical depth: X k ≠ f(τ c ) (see Sections 1.3.1 and 1.3.2). Instead, we employ an inversion code that allows for the inclusion of gradients in the physical parameters. In this case we have used the SIR inversion code (Ruiz Cobo and del Toro Iniesta, 1992), but we could have also employed SPINOR (Frutiger et al., 1999) or LILIA (Socas-Navarro, 2002). Applying this inversion code allows us to determine two-dimensional maps of the three components of the magnetic field vector B(x β , x α ), γ(x β , x α ), and ϕ(x β , x α ) at different optical depths τ c (cf. Figures 2 and 3).
Once those maps are obtained, the 180°-ambiguity in the azimuth of the magnetic field ϕ can be resolved at each optical depth following the prescriptions given in Section 1.3.2. This allows to obtain \(B_\rho (x_\beta ,x_\alpha ,\tau _c )\) (vertical component of the magnetic field on the solar surface) and \(B_h (x_\beta ,x_\alpha ,\tau _c ) = \sqrt {B_\alpha ^2 (x_\beta ,x_\alpha ,\tau _c ) + B_\beta ^2 (x_\beta ,y_\alpha ,\tau _c )} \) (horizontal component of the magnetic field). By the same method as in Section 2.1 we then employ ellipses to determine the angular averages of these physical parameters as a function of the normalized radial distance in the sunspot: r/R s . However, as opposed to the previous section, it is now possible to determine this radial variations at different optical depths. The results are presented in Figure 12, in red color for the deep photosphere (log τ c = 0), blue for the mid-photosphere (log τ c = -1.5), and green for the highphotosphere (log τ c = -3)6.
Azimuthally averaged components of the magnetic field vector as a function of the normalized radial distance in the sunspot r/R s : total magnetic field strength B (upper-left), vertical component of the magnetic field B ρ (upper-right), horizontal component of the magnetic field B h (lower-left), inclination of the magnetic field vector with respect to the vertical direction on the solar surface ζ (lower-right). Each panel contains three curves, representing different optical depths: red is for the deep photosphere or continuum level (log τ c = 0), blue is the mid-photosphere (log τ c = -1.5), and green is the upperphotosphere (log τ c = -3). The vertical dashed line at r/R s ≈ 0.4 indicates the separation between the umbra and the penumbra. These results correspond to the sunspot AR 10923 observed on November 14, 2006 at Θ = 8.7° (see also Figure 2; upper panels in Figures 4, 5, 6, and 11).
Figure 12 shows two distinct regions. The first one corresponds to the inner part of the sunspot: r/R s < 0.5, where the total magnetic field strength Btot (upper-left panel) decreases from the deep photosphere (red color) upwards. This is caused by an upwards decrease of the vertical B ρ (upperright), and horizontal B h (lower-left) components of the magnetic field. Also, in this region the inclination of the magnetic field vector ζ (lower-right) remains constant with height. From the middle-half of the sunspot and outwards, r/R s > 0.5, the situation, however, reverses. The total magnetic field strength, as well as the vertical and horizontal components of the magnetic field, increase from the deep photosphere (log τ c = 0) to the higher photosphere (log τ c = -3). In this region, the inclination of the magnetic field vector ζ no longer remains constant with τ c but it decreases towards the higher photospheric layers. The actual values of the gradients are given in Figure 13. These values are close to the lower limits (< 1 G km−1) obtained in early works7 (Kotov, 1970; Makita and Nemoto, 1976; Osherovich, 1984, and references therein). However, Figures 12 and 13 extend those results for the three components of the magnetic field vector and not only for its vertical component B ρ . In addition, these figures show a clear distinction between the inner and the outer sunspot. Although Figures 12 and 13 show only the results for AR 10923, the other analyzed sunspot (AR 10933) presents very similar features.
Top panel: vertical derivatives of the different components of the magnetic field vector as a function of the normalized radial distance in a sunspot: r/R s . Total field strength dBtot/dx ρ (green), horizontal component of the magnetic field dB h /dx ρ (r) (blue), vertical component of the magnetic field dB ρ /dx ρ (red). Bottom panel: same as above but for the inclination of the magnetic field vector with respect to the vector perpendicular to the solar surface: dζ/dx ρ . The vertical dashed line at r/R s ≃ 0.4 represents the umbra-penumbra boundary. The vertical solid lines gives an idea about the standard deviation (from all pixels across a given ellipse in Figure 10). These results correspond to AR 10923, observed on November 14, 2006 at Θ = 8.7°.
Similar studies have been carried out in a number of recent works. For instance, our results are in very good agreement with those from Westendorp Plaza et al. (2001b) (see their Figure 9) in the value and sign of the gradients in the different components of the magnetic field. In our case, as well as theirs, the total magnetic field strength decreases towards the deep photosphere for r/R s > 0.6. At the same time the inclination (with respect to the vertical) ζ increases towards deeper photospheric layers. This can be interpreted in terms of the existence of a canopy (see also Leka and Metcalf, 2003), and is perfectly consistent with a picture in which sunspots are vertical flux tubes where the magnetic field lines fan out with increasing height as they meet a plasma with lower densities. Another interesting result concerns the fact that, once the physical parameters are allowed to vary with optical depth τ (Equation (20)), the evidence for return-flux (ζ > 90°) in the deep photosphere becomes more clear: compare lower-right panels in Figures 11 and 12. It is important to note that Westendorp Plaza et al. (2001b) also employed in their inversions spectropolarimetric data from the Fe I line pair at 630 nm.
Other spectral lines, such as the Fe I line pair at 1564.8 nm were employed by Mathew et al. (2003), who instead found that the magnetic field strength increases towards deeper layers in the photosphere at all radial distances in the sunspot: dBtot/dτ > 0 (see their Figure 15). In addition, they found that the inclination of the magnetic field ζ decreases towards deep layers: dζ/dτ < 0 at all radial distances. These results are, therefore, consistent with ours as far as the inner part of the sunspot is concerned, but they are indeed opposite to ours (and to Westendorp Plaza et al., 2001b) for the sunspot's outer half. Furthermore, Sánchez Cuberes et al. (2005), as well as Balthasar and Gömöry (2008), analyzed two Fe I lines and one Si I line at 1078.3 nm to study the magnetic structure of a sunspot. From their spectropolarimetric analysis (see their Figure 11) they inferred a total magnetic field strength that was stronger in the deep photospheric layers: dBtot/dτ > 0 at all radial distances from the sunspot's center (in agreement with Mathew et al., 2003). As far as the inclination ζ of the magnetic field is concerned, Sánchez Cuberes et al. (2005) obtained different behaviors depending on the scheme employed to treat the stray light in the instrument. However, they lend more credibility to the results obtained with a constant amount of stray light. In this case, they concluded that dζ/dτ ≈ 0 for r/R s < 0.5 and dζ/dτ > 0 for r/R s > 0.5, which supports our results and those from Westendorp Plaza et al. (2001b), but not Mathew et al. (2003). Results from all the aforementioned investigations are summarized in Table 1. It is important to mention that, although Balthasar and Gömöry (2008) did not find dBtot/dx ρ > 0 in the outer half of the sunspot (considered as evidence for a canopy), they did indeed find this trend outside the visible boundary of the sunspot.
Same as Figure 9 but for the vertical component of the current density vector: j z (or j ρ ). The arrows indicate the horizontal component of the current density vector: j β and j α . The white contours enclose the area where the vertical component of the magnetic field, B ρ , is equal to 650 G (solid) and 450 G (dashed) (from Puschmann et al., 2010c, reproduced by permission of the AAS).
Sign of the gradients of the different components of the magnetic field vectors: total magnetic field strength Btot, vertical component of the magnetic field vector Bρ, horizontal component of the magnetic field vector B h , and inclination of the magnetic field vector with respect to the vertical direction on the solar surface ζ The sign of the gradients are split in two distinct regions: inner sunspot r/R s < 0.5, and outer sunspot r/R s > 0.5. Figure 13 gives the actual values.
This work and Westendorp Plaza et al. (2001b)
r/R s
dBtot/dxρ
dBρ/dxρ
dB h /dxρ
dζ/dxρ
≈ 0
> 0.5
Mathew et al. (2003)
dBtot/dτ
dζ/dτ
dζ/dxτ
Sánchez Cuberes et al. (2005); Balthasar and Gömöry (2008)
In the light of these opposing results it is critical to ask ourselves where do these differences come from. One possible source is the spatial resolution of the observations. Westendorp Plaza et al. (2001b), Mathew et al. (2003), and Sánchez Cuberes et al. (2005) employed spectropolarimetric observations at low spatial resolution (about 1"). The data employed here (Hinode/SP) possess much better resolution: 0.32". However, this should not be very influential to the study of the global properties of the sunspot, since we are discussing azimuthally or Ψ-averaged quantities. Another possible explanation lies in the different formation heights of the employed spectral lines. Since each set of spectral lines samples a slightly different \(\bar \tau \) region in the solar photosphere, they might be sensing slightly different magnetic fields, thereby yielding gradients (see Table 1). This is a plausible explanation because the Fe I line pair at 1564.8 nm sample a deep and narrow photospheric layer: \(\bar \tau \in [3,3 \times 10^{ - 2} ]\), as compared to \(\bar \tau \in [1,10^{ - 3} ]\) for the Fe I lines at 630 nm (see, for example, Figures 3 and 4 in Mathew et al., 2003, and Figure 3 in Bellot Rubio et al., 2000). Indeed, the different formation heights have been exploited by numerous authors (Bellot Rubio et al., 2002; Mathew et al., 2003; Borrero et al., 2004; Borrero and Solanki, 2008) in order to explain the opposite gradients obtained from different sets of spectral lines in terms of penumbral flux tubes and the fine structure of the sunspot (see, also, Sections 3.2.1 and 3.2.5). The role of the sunspot's fine structure is emphasized by the fact that the scatter bars (produced by the inversion of individual pixels; see Figures 12 and 13) are of the order of, or even larger than, the differences between the magnetic field at the different atmospheric layers chosen for plotting. To solve this problem one would like to analyze, ideally, many different spectral lines formed at different heights (Section 1.3.1). This approach has been already followed by the recent works of Cabrera Solana et al. (2008) and Beck (2011), where simultaneous and co-spatial spectrolarimetric observations in Fe I 630 nm and Fe I 1564.8 nm where analyzed. Their results further emphasize the role of the fine structure of the sunspot in the determination of the vertical gradients of the magnetic field vector.
A final possibility to explain the difference in the gradients obtained by different authors could be the different treatments employed to model the scattered light in the instrument. Arguments in favor of this possibility are given by Sánchez Cuberes et al. (2005) and Solanki (2003). Arguments against the results being affected by the treatment of the scattered light have been presented in Borrero and Solanki (2008). Moreover, in Cabrera Solana et al. (2006), Cabrera Solana (2007), and Cabrera Solana et al. (2008) a careful correction for scattered light was performed, and still the fine structure of the sunspot had to be invoked to explain the observed gradients in the magnetic field vector.
2.3 Is the sunspot magnetic field potential?
The potentiality of the magnetic field vector in sunspots is often studied by means of the current density vector \(j = \frac{1} {{\mu _0 }}\nabla \times B \) (in SI units). Theoretical models for sunspots usually come in two distinct flavors attending to the vector j: those where the currents are localized at the boundaries of the sunspot (current sheets) and the magnetic field vector is potential elsewhere (Simon and Weiss, 1970; Meyer et al., 1977; Pizzo, 1990), and those where there are volumetric currents distributed everywhere inside the sunspot (Pizzo, 1986). From an observational point of view, in order to evaluate j it is necessary to calculate the vertical derivatives of the three components of the magnetic field vector: dB α /dx ρ , dB β /dx ρ , and dB ρ /dx ρ . This is not possible through a Milne-Eddington inversion, because it assumes that the magnetic field vector is constant with height: τ c or x ρ (Equation (12)). In this case it is only possible to determine the vertical component of the current density vector, j ρ (aka j z ), because it involves only the horizontal derivatives:
$$\beta = \frac{{P_g }} {{P_m }} = \frac{{8\pi P_g }} {{B^2 }}(in cgs units). $$
An example of the vertical component of the current density vector, j ρ , obtained from a ME inversion is presented in Figure 14. This corresponds to the sunspot observed in November 14, 2006 with Hinode/SP at Θ = 8.7°. The derivatives in Equation 21 have been obtained from Figures 5 and 6. Because the magnetic field vector was obtained from a ME inversion, these derivatives of the magnetic field vector refer to a constant optical depth \(\bar \tau \) in the atmosphere. As long as the \(\bar \tau (x_\rho )\)-surface (Wilson depression) is not very corrugated (small pixel-to-pixel variations) and that the vertical-τ variations of the magnetic field vector are not very strong (Equation (6)), it is justified to assume that the maps in Figures 5 and 6 also correspond to a constant geometrical height x ρ If these assumptions are in fact not valid, artificial currents in j ρ might appear as a consequence of measuring the magnetic field at different heights from one pixel to the next one.
Same as Figures 4–8 but for the vertical component of the current density vector j z (or j ρ ) in sunspot AR 10923.
Note that prior to the calculation of currents, the 1807#x00B0;-ambiguity in the azimuth of the magnetic field vector must be solved (see Section 1.3.2). The final results for j ρ are displayed in Figure 14, where it can be seen that the vertical component of the current density vector is highly structured in radial patterns resembling penumbral filaments. The values of the current density are of the order of |j ρ | < 75 mA m−2. This value is consistent with previous results obtained with different instruments and, therefore, different spectral lines and spatial resolutions: |j ρ | < 50 mA m−2 (Figure 10 in Li et al., 2009; 2" and Fe I 630 nm), |j ρ | < 40 mA m−2 (Figure 8 in Balthasar and Gömöry, 2008; ≈ 0.9" and Si/Fe I 1078 nm), |j ρ | < 100 mA m−2 (Figure 3 in Shimizu et al., 2009; 0.32" and Fe I 630 nm). These various results show a weak tendency for the current to increase with increasing spatial resolution. However, this result is to be taken cautiously, since at low spatial resolutions two competing effects can play a role. On the one hand, a better spatial resolution can detect larger pixel-to-pixel variations in the magnetic field and, thus, yields larger values for j z . On the other hand, a worse spatial resolution can leave certain magnetic structures unresolved and, in this case, the finite-differences involved in the Equation (21) can produce artificial currents where originally there were none.
A curious effect worth noticing is the large and negative values of j ρ in Figure 14 around the sunspot center that describe an oval shape (next to the light bridges). This is an artificial result produced by an incorrect solution to the 180°-ambiguity in the azimuth of the magnetic field close to the umbral center (see Section 1.3.2). An incorrect choice between ϕ and ϕ + π (see for instance Equation (26)) can lead to very large and unrealistic pixel-to-pixel variations in dB α /dx β or dB β /dx α . Thus, regions where large values of j ρ are consistently obtained can sometimes be used to identify places where the solution to the 180°-ambiguity was not correct. Indeed, many methods to solve the 180°-ambiguity minimize j ρ in order to choose between the two possible solutions in the azimuth of the magnetic field vector (Metcalf et al., 2006, see also Section 1.3.2).
In order to compute the horizontal component of the current density vector \(j_h = \sqrt {j_\alpha ^2 + j_\beta ^2 } \), it is necessary to analyze the spectropolarimetric data employing an inversion code that allows to retrieve the stratification with optical depth in the solar atmosphere (see Section 2.2). Even in this case, the derivatives must be evaluated in terms of the geometrical height x ρ instead of the optical depth τ c . Because the conversion from these two variables assuming hydrostatic equilibrium is not reliable in sunspots (see Section 1.3.3), j h is not something commonly found in the literature. As a matter of fact, most inferences of j h were performed through indirect means (Ji et al., 2003; Georgoulis and LaBonte, 2004). Very recently, however, Puschmann et al. (2010c) have been able to determine the full current density vector j from purely observational means (inversion of Stokes profiles including τ c -dependence) plus a proper conversion between τ c and x ρ (Puschmann et al., 2010b, see also Section 1.3.3). In the latter two works, the authors found that the horizontal component of the current density vector is about 3–4 times larger than the vertical one: j h ≈ 4j ρ Figure 15 reproduces Figure 1 from Puschmann et al. (2010c), which shows the j vector in a region of the penumbra. j ρ (they refer to it as j z ) also shows radial patterns as in our Figure 14. More importantly, j h is strongest in the vicinity of the regions where j ρ is large.
Currents in the chromosphere have also been studied, although to a smaller extent, by Solanki et al. (2003) and Socas-Navarro (2005b). The latter author finds values for the vertical component of the current density vector in the chromosphere which are compatible with those in the photosphere: |j z | < 250 mA m−2. In addition, the detected currents are distributed in structures that resemble vertical current sheets, spanning up to 1.5 Mm in height. The mere presence of large currents within sunspots clearly implies that the magnetic field vector is not potential: ∇ × B ≠ 0.
2.4 What is the plasma-β in sunspots?
In the previous Section 2.3 we have argued that the magnetic field vector in sunspots is nonpotential. However, in order to establish its degree of non-potentiality it is important to develop this statement further. The way this has been traditionally done is through the study of the plasma-β parameter. The plasma-β is defined as the ratio between the gas pressure and the magnetic pressure:
In the solar atmosphere, if β >> 1 the dynamics of the system are dominated by the plasma motions, which twist and drag the magnetic field lines while forcing them into highly non-potential configurations. If β << 1 the opposite situation occurs, that is, the magnetic field is not influenced by the plasma motions. In this case, the magnetic field will evolve into a state of minimum energy which happens to coincide with a potential configuration (see Chapter 3.4 in Priest, 1982). Therefore, many works throughout the literature focus on the plasma-β in order to study the potentiality of the magnetic field. Here, we will employ our results from the inversion of spectropolarimetric data in Section 2.2 to investigate the value of the plasma-β parameter in a sunspot. Figure 16 shows the variation of the azimuthally averaged plasma-β (along ellipses in Figure 10) as a function of the normalized radial distance in the sunspot r/R s . This figure displays β at four different optical depths, from the deep photosphere τ c = 1 (yellow) to the high-photosphere τ c = 10−3 (blue). This figure shows that β << 1 above τ c ≤ 10−2 and, thus, the magnetic field can be considered to be nearly potential (or at least force-free) in these high layers. At τ c = 1 (referred to as continuum) β ≥ 1 and, therefore, the magnetic field is non-potential. At the intermediate layer of τ c = 0.1 (around 100 kilometers above the continuum) the magnetic field is nearly potential in the umbra, but it cannot be considered this way in the penumbra: r/R s > 0.4.
Similar to Figure 11 but for the plasma-β as a function of the normalized radial distance in the sunspot: r/R s . The different curves refer to different optical depths in the sunspot: τ c = 1 (yellow), τ c = 0.1 (red), τ c = 10−2 (green), and τ c = 10−3 (blue). The vertical dashed line at r/R s ≃ 0.4 indicates the umbra-penumbra boundary.
In Figure 16 the gas pressure was obtained under the assumption of hydrostatic equilibrium (Section 1.3.3), which we know not to be very reliable in sunpots. A more realistic approach was followed by Mathew et al. (2004, and references therein), where an attempt to consider the effect of the magnetic field in the force balance of the sunspot was made. Their results for the deep photosphere (τ c = 1) obtained from the inversion of the Fe I line pair at 1564.8 nm are consistent with our Figure 16 (obtained from the inversion of the Fe I line pair at 630 nm), with β ≈ 1 close to the continuum everywhere in the sunspot. Similar results were also obtained by Puschmann et al. (2010c, see their Figure 4), who performed an even more realistic estimation of the geometrical height scale, considering the three components of the Lorenz force term (j 7#x00D7; B; Equation (17)). In Figure 17 we reproduce their results, which further confirm that the β ≈ 1 in the deep photospheric layers of the penumbra.
Same as Figures 9 and 15 but for the plasma-β at z = 0 in the inner penumbra of a sunspot. The white contours are the same as in Figure 15: B ρ = 650 (solid white) and B ρ = 450 (dashed white). This sunspot is AR 10953 observed on May 1st, 2007 with Hinode/SP (from Puschmann et al., 2010c, reproduced by permission of the AAS).
These results have important consequences for magnetic field extrapolations from the photosphere towards the corona, because they imply that those extrapolations cannot be potential. In addition, as pointed out by Puschmann et al. (2010c) the magnetic field is not force-free because in many regions the current density vector j and the magnetic field vector B are not parallel. Unfortunately, extrapolations cannot deal thus far with non-force-free magnetic field configurations. Considering that it has now become possible to infer the full current density vector j, developing tools to perform non-force free magnetic field extrapolations will be a necessary and important step for future investigations. These results also have important consequences for sunspot's helioseismology, because of the deep photospheric location of the β = 1 region, which is the region where most of the conversion from sound waves into magneto-acoustic waves takes place.
In the chromosphere of sunspots, the magnetic field strength is about half of the photospheric value (see Figure 4 in Orozco Suarez et al., 2005). Therefore, the magnetic pressure in the chromosphere is only about 25% of the photospheric value. However, the density and gas pressure are at least 2–3 orders of magnitude smaller. Thus, the chromosphere of sunspots is clearly a low-β (β << 1) environment, which in turn means that the magnetic field configuration is nearly potential.
2.5 Sunspots' thermal brightness and thermal-magnetic relation
The Eddington-Barbier approximation can be employed to relate the observed intensity from any solar structure with a temperature close to the continuum layer: τ = 2/3. This is done by assuming that the observed intensity is equal to the Planck's function, and solving for the temperature:
$$ I_{obs} \sim \frac{{2hc^2 }} {{\lambda ^5 }}\exp \left\{ { - \frac{{hc}} {{\lambda KT}}} \right\} $$
Variations in the observed intensity can be related to a change in the temperature through:
$$ \frac{{\Delta I_{obs} }} {{\Delta T}} \sim \frac{{dI_{obs} }} {{dT}} = \frac{{hc}} {{\lambda KT^2 }} $$
The observed brightness of a sunspot umbra at visible wavelengths is about 5–25% of the observed brightness of the granulation at the same wavelength: Iumb ≈ 0.05–0.25Iqs. In the penumbra this number is about 65–85% of the granulation brightness: Ipen ≈ 0.65–0.85Iqs. Assuming that the temperature at τ = 2/3 for the quiet Sun is about 6050 K, the numbers we obtain from Equation (24) are: Tumb(τ = 2/3) ≈ 4800 K and Tpen(τ = 2/3) ≈ 5650 K.
At infrared wavelengths the difference in the brightness between quiet Sun and umbra or penumbra is greatly reduced: Iumb ≈ 0.4–0.6Ic,qs and Ipen ≈ 0.7–0.9Ic,qs (see Figure 1 in Mathew et al., 2003). This happens as a consequence of the behavior of the Planck's function B(λ, T), whose ratio for two different temperatures decreases towards larger wavelengths. All numbers mentioned thus far are strongly dependent on the spatial resolution and optical quality of the instruments. For example, large amounts of scattered light tend to reduce the intensity contrast and, therefore, temperature differences between different solar structures.
In Figure 18, we present scatter plots showing the relationship between the sunspot's thermal brightness and the components of the magnetic field vector. These plots have been adapted from Figure 4 in Mathew et al. (2004). They show T(τ = 1): vs. B (total magnetic field strength; upperleft), vs. ζ (zenith angle - Equation (10) - upper-right), vs. B z (or our B ρ vertical component of the magnetic field; lower-left), and vs. B r (or our B h horizontal component of the magnetic field; lower-right). As expected, the thermal brightness anti-correlates with the total field strength B since the latter is larger (see Figures 4 and 11) in the darkest part of the sunspot: the umbra. However, the inclination of the magnetic field ζ correlates well with the thermal brightness. Again, this was to be expected (see Figures 7 and 11) since the inclination of the magnetic field increases towards the penumbra, which is brighter (see Figures 2 and 3). As we will discuss intensively throughout Section 3, these trivial results have important consequences for the energy transport in sunspots.
Upper-left: scatter plot of the total field strength vs. temperature at τ c = 1. Upper-right: inclination of the magnetic field with respect to the vertical direction on the solar surface (ζ; see Equation (10)) versus temperature at τ c = 1. Bottom-left: vertical component of the magnetic field B z (called B ρ in our Section 2.1) vs. temperature at τ c = 1. Bottom-right: horizontal component of the magnetic field B r (called B h in Section 2.1) versus the temperature at τ c = 1. In all these panels circles represent umbral points, whereas crosses and triangles correspond to points in the umbra-penumbra boundary and penumbral points, respectively (from Mathew et al., 2004, reproduced by permission of the ESO).
2.6 Twist and helicity in sunspots' magnetic field
Let us define the angle of twist of a sunspot's magnetic field, Δ, as the angle between the magnetic field vector B at a given point of the sunspot and the radial vector that connects that particular point with the sunspot's center, r (Equation (46)):
$$\Delta = \cos ^{ - 1} \left[ {\frac{{Br}} {{|B||r|}}} \right]. $$
Note that in Section 1.3.2 this angle Δ is precisely the quantity that was being minimized when solving the 180°-ambiguity (Equation (9)). However, minimizing it does not guarantee that Δ will be zero. This is, therefore, the origin of the twist: a deviation from a purely radial (i.e., parallel to r) magnetic field in the sunspot. Figure 19 shows maps of the twist angle Δ for two different sunspots at two different heliocentric angles. These two examples illustrate that the magnetic field vector is radial throughout most of the sunspot, but there are regions where significant deviations are observed. These deviations could be already seen in the arrows in Figures 5 and 6 representing the magnetic field vector in the plane of the solar surface. In addition, in these two examples the sign of the twist (wherever it exists) remains constant for the entire sunspot.
Same as Figure 7 but for twist angle of the magnetic field in the plane of the solar surface: Δ (Equation (25)).
Twisted magnetic fields in sunspots have been observed for a very long time, going back to the early works of Hale (1925, 1927) and Richardson (1941), who observed them in H α filaments. They established what is known as Hale's rule, which states that sunspots in the Northern hemisphere have a predominantly counter-clockwise rotation, whereas it is clockwise in the Southern hemisphere. However, sunspots violating Hale's rule are common if we attend only at H α filaments (Nakagawa et al., 1971). A better estimation of the twist in the magnetic field lines can be obtained from spectropolarimetric observations. To our knowledge, the first attempts in this direction were performed by Stepanov (1965).
Twist in sunspots can also be studied by means of the α-parameter in non-potential force-free magnetic configurations: ∇ × B = αB. Another commonly used parameter is the helicity: H = ∫ V A · BdV, where B is the magnetic field vector and A represents the magnetic vector potential. As demonstrated by Tiwari et al. (2009a) the value of α corresponds to twice the degree of twist per unit axial length. In addition, α and H posses the same sign. Thus, any of these two parameters can be also employed to study the sign of the twist in the magnetic field vector. Using these parameters Pevtsov et al. (1994) and Abramenko et al. (1996) found a good correlation (up to 90%) between the sign of the twist and the hemisphere where the sunspot appear (Hale's rule).
Recent works, however, find large deviations from Hale's rule (Pevtsov et al., 2005; Tiwari et al., 2009b). It has been hypothesized that these deviations from Hale's rule might indicate a dependence of the twist with the solar cycle (Choudhuri et al., 2004). Other possible explanations for the twist of the magnetic field in sunspots, in terms of the solar rotation and Coriolis force, have been offered by Peter (1996) and Fan and Gong (2000). In particular, the former work is also able to explain the deviations from Hale's rule observed in Hα filaments. However, a definite explanation is yet to be identified. This might be more complicated than it seems at first glance because different twisting mechanisms might operate in different regimes and atmospheric layers. This is supported by recent spectropolarimetric observations that infer a twist in the magnetic field that can change sign from the photosphere to the chromosphere (Socas-Navarro, 2005a).
3 The Era of 0.1–0.5" Resolution: Small-Scale Magnetic Structures in Sunspots
In Section 2, we focused on sunspot's global magnetic structure. To that end we studied the radial variation of the azimuthally averaged magnetic properties: three components of the magnetic field vector, plasma-β, potentiality, currents, etcetera. In this section, we will investigate the small-scale structure of the magnetic field. This will help us understand and identify some of the basic building blocks of the sunspot's magnetic field, as well as the fundamental physical processes that occur in sunspots. Another difference with Section 2, where only the magnetic field structure was discussed, is that in this section we will also address the velocity field since they are both intimately linked at small scales (e.g., Evershed flow; Evershed, 1909).
In this section spectropolarimetric observations at the highest spatial resolution will be employed and, instead of discussing averaged quantities, we will focus mostly in their pixel-to-pixel variations. In the first part of this section we will address the fine structure of the umbral magnetic field, whereas the second will be devoted to the penumbral magnetic field. This division is somewhat artificial because the current paradigm points towards a clear relationship between the small-scale structure in these two different regions (Rimmele, 2008). However, there is one important difference between these two regions (umbra and penumbra), and it has to do with the mean inclination of the ambient magnetic field ζ: in the umbra the magnetic field is mostly vertical, whereas in the penumbra is highly inclined (see Figure 11). This difference leads to a somewhat different interaction between the convective motions and the magnetic field.
As we saw in Section 2.5, typical temperatures at τ = 2/3 in the umbra and penumbra are approximately 4500 K and 6000 K, respectively. Plasma heated up to this temperature loses energy in the form of radiation. If the brightness of the umbra and penumbra is to remain constant, the energy loses due to radiation must be compensated by some other transport mechanism that will bring energy from the convection zone into the photosphere. The mechanism usually invoked is convection. However, the strong magnetic field present in the sunspots (see Figure 11) inhibits convective flows (Cowling, 1953). This inhibited convection is, therefore, the reason why umbra and penumbra posses a reduced brightness compared to the granulation. How do these convective movements take place? In the umbra the answer to this question is to be found in the so-called umbral dots, whereas in the penumbra convection occurs within the penumbral filaments.
3.1 Sunspot umbra and umbral dots
3.1.1 Central and peripheral umbral dots
Umbral dots appear as small-scale regions of enhanced brightness within the umbral core (see Figure 20). Sizes and lifetimes of umbral dots have been extensively discussed in the literature. The current consensus points towards a large selection bias. Although it is clear that umbral dots are detected at spatial scales smaller than 1" and temporal scales larger than 2 minutes, it is not well established whether they posses a typical size or lifetime, since more and more are detected as the spatial resolution of the observations increases (Sobotka and Hanslmeier, 2005; Riethmüller et al., 2008b).
Map of a sunspot (AR 11072) umbra and inner penumbra obtained with the Swedish 1-m Solar Telescope (SST). This sunspot was observed on May 23, 2010 at 7Θ = 15°. The image was taken with a 10 Å filter located between the Ca H and Ca K spectral lines. It was subsequently restored using Multi-Object Multi-Frame Blind Deconvolution (MOMFBD) technique. The red circles surround a local intensity enhancement in the umbral core: umbral dot (UD), and a portion of a light bridge (LB) (adapted from Henriques et al., 2011; in preparation).
Traditionally, umbral dots have been sub-categorized in central (CUDs) and peripheral umbral dots (PUDs) (Loughhead et al., 1979; Grossmann-Doerth et al., 1986). This distinction is based upon the location of the umbral dots: CUDs appear mostly close to the darkest region of the umbra, whereas PUDs appear commonly at the umbral and penumbral boundary. Although sometimes disputed (see, e.g., Sobotka et al., 1997), there are many works that claim that these two families of umbral dots posses very different proper motions (Molowny-Horas, 1994; Sobotka et al., 1995; Riethmüller et al., 2008b; Watanabe et al., 2009a), with the peripheral ones exhibiting the largest velocities and apparently being related to inner bright penumbral grains. The physical similarities between peripheral umbral dots and penumbral grains have been studied by Sobotka and Jurčák (2009).
3.1.2 Thermal and magnetic structure of umbral dots
The large continuum intensities, as compared to the umbral dark surroundings, immediately implies (see, for example, Section 2.5) that the temperature in umbral dots at τ c = 2/3 is larger than the temperature at the same layer in the umbral background. Old and current estimates all coincide in a temperature difference that ranges from 500 K (Grossmann-Doerth et al., 1986; Riethmüller et al., 2008a) up to 1500 K (Tritschler and Schmidt, 1997; Socas-Navarro et al., 2004). This temperature difference almost vanishes about 200–250 km above τ c = 1: see Figure 8 in Socas-Navarro et al. (2004), and Figure 4 in Riethmüller et al. (2008a), which is reproduced here (in Figure 21).
Vertical stratification (in optical depth τ c -scale) of the temperature (left panel) and the total magnetic field strength (right panel). The blue curves shows the stratification for the diffuse umbral background, where the red curves correspond to the vertical stratification along an umbral dot. Close to the continuum, τ c = 1, the umbral dot is much hotter and possesses a weaker magnetic field than the umbra. These differences disappear about 100–200 km higher in the photosphere: log τ c ∼ −2 (from Riethmüller et al., 2008a, reproduced by permission of the AAS).
The strength of the magnetic field inside umbral dots has been a somewhat controversial subject, with some works finding no large differences between umbral dots and the umbral background (Lites et al., 1989; Tritschler and Schmidt, 1997), and other works finding a clear reduction of the field strength both in central and peripheral umbral dots (Wiehr and Degenhardt, 1993; Socas-Navarro et al., 2004; Riethmüller et al., 2008a). However, as pointed out by the the latter works this could again be τ-dependent, with the differences in the magnetic field being small a few hundred kilometers above τ c = 1, but fairly large close to this level. Here the difference can be such that the magnetic field inside the umbral dot is only a few hundred Gauss (see Figure 21). The inclination of the magnetic field ζ has been found to be only slightly larger than in the mean umbral background (see Figure 8 in Socas-Navarro et al., 2004, and Figure 4 in Bharti et al., 2009), which is itself very much vertical (see Figures 7 and 11). This will be a recurrent topic in future sections (Sections 3.2.5, 3.2.7, and 3.2.6) when discussing the differences/similarities between umbral dots, penumbral filaments, and light bridges.
The smaller field strengths inside umbral dots leads to an enhanced gas pressure as compared to the surrounding umbra. This is consistent with the larger temperatures found inside UDs. These numbers can be employed to derive a Wilson depression of about 100–200 km, that is, the τ c = 1 level is formed about 100–200 km higher in umbral dots than in the surrounding umbra (Socas-Navarro et al., 2004). This value is similar to the height difference for the continuum level between penumbral spines and intraspines (see Figure 9). This has important consequences because the measured differences in the thermal and magnetic structure correspond to τ c = 1. If the continuum level is actually formed higher (in the geometrical height scale) inside UDs than in the umbra, this means that the differences, if measured at the same geometrical height, would be much larger than the numbers previously cited. This effect applies indeed, not only to umbral dots, but also in any other structure in the solar photosphere that is elevated with respect to its surroundings. Note also that the Wilson depression between the umbra and umbral dots can be inferred from purely geometrical considerations of sunspot observations close to the limb (Lites et al., 2004; Watson et al., 2009).
Note that, the presence of regions inside the sunspot umbra where the magnetic field is strongly reduced and the temperature and gas pressure enhanced around τ c = 1, goes along the same lines as Section 2.4, where we concluded that close to the continuum level the plasma-β is larger than unity. As explained in Sect 2.3 this leads to non-potential configurations for the sunspot magnetic field because the convective motions are strong enough to drag and twist the magnetic field lines.
3.1.3 Signatures of convection in umbral dots
As mentioned in Section 3.1, there must exist some form of convection operating in the umbra of sunspots. The main candidate for this are the umbral dots. This was motivated by the fact that umbral dots show enhanced brightness with respect to the umbral background and, therefore, must be heated more efficiently. In addition, numerical simulations of umbral magneto-convection (Schüssler and Vögler, 2006) predict the existence of upflows at the center of umbral dots and downflows at its edges. As it occurs in the case of penumbral filaments (see Section 3.2.4), the search for convective-like velocity patterns in umbral dots has been hindered by the limited spatial resolution of the observations. For instance, while upflows ranging from 0.4–1.0 km s−1 at the center of umbral dots have been known for quite some time (Rimmele, 2004; Socas-Navarro et al., 2004; Watanabe et al., 2009b), downflows have been much more difficult to detect. However, in the past few years there have been a few positive detections of downflows at the edges of umbral dots (Bharti et al., 2007; Ortiz et al., 2010). The latter work presents evidence that supports the numerical simulations of umbral convection in great detail, with umbral dots that show upflows along their central dark lane and strong downflows at the footpoints of the dark lanes (see Figure 3 in Ortiz et al., 2010). This agreement is evident if we compare the observations from Ortiz et al. (2010) in Figure 22 with the simulations from Schüssler and Vögler (2006) in Figure 23.
Results from spectropolarimetric observations. Left panel: continuum intensity map inside the umbra of a sunspot. The circles denote the location of several umbral dots (see as intensity enhancements; see also 20). The largest circle encircles two large umbral dots that show prominent central dark lanes. Right panel: map of the line-of-sight velocity in deep layers. This map shows an upflow (blueshift) along the central dark lane and downflows (redshift) at the footpoints of the dark lane (from Ortiz et al., 2010, reproduced by permission of the AAS).
Results from 3D MHD simulations. Left panel: continuum intensity in the umbra of a sunspot. Right panel: map of the line-of-sight velocity. This panel shows upflows (blueshift) along the central dark lane of umbral dots. Downflows (redshift) are also visible all around the central dark lane, although they are stronger at the footpoints of the dark lane (from Schüssler and Vögler, 2006, reproduced by permission of the AAS).
The lower magnetic field inside umbral dots mentioned in Section 3.1.2 is a direct consequence of the convective motions described here. In the sunspot umbra, convective motions push the magnetic field lines towards the boundary of the convective cell, thereby creating a region where the vertical component of the magnetic field vector is strongly reduced. Since the ambient magnetic field is vertical, this automatically yields a very small field inside the umbral dot. At the top of the convective cell the magnetic field forms a cusp or canopy, preventing the material from continuing to flow upwards. The pile-up of material at this point creates a region of locally enhanced density, which is responsible for the appearance of the central dark lane inside umbral dots (Schüssler and Vögler, 2006).
3.1.4 Light bridges
Besides umbral dots, the most striking manifestation of convection in the umbra appears in the form of light bridges. These are elongated bright features that often split the umbra in two sections connecting two different sides of the penumbra (see Figure 20). Light bridges and umbral dots share many similarities. For instance, both feature a central dark lane and bright edges. Indeed, light bridges can be considered as an extreme form of elongated umbral dots. Their larger sizes have actually allowed for the detection of both blue and redshifted velocities with only a moderate spatial resolution of 1" (Sobotka et al., 1995; Leka, 1997; Rimmele, 1997).
Recent observations at much better spatial resolution have also been able to establish a clear connection between upflows and the central dark lane in light bridges, as well as downflows and the bright edges of the light bridge (Hirzberger et al., 2002; Berger and Berdyugina, 2003; Rouppe van der Voort et al., 2010). In addition, as it also occurs with umbral dots (see Section 3.1.2), the magnetic field is weaker and slightly more inclined in light bridges as compared to the surrounding umbra (Beckers and Schröter, 1969b; Rueedi et al., 1995; Jurčák et al., 2006). The nature of the central dark lane in light bridges is the same as in umbral dots (Section 3.1.3).
3.1.5 Subsurface structure of sunspots: cluster vs. monolithic models
The presence of several convective features in the umbra of sunspots immediately posses the question of whether the convective upflows and downflows in umbral dots and light bridges extend deep into the solar interior or, on the contrary, are only a surface effect. Two distinct theoretical models are usually cited to showcase these two possibilities: the cluster model (Parker, 1979) and the monolithic model (Gokhale and Zwaan, 1972; Meyer et al., 1974, 1977). In both cases, convective upflows at the plume's center reach the photosphere, where they lose their energy via radiative cooling and sink back into the Sun at the edges of the umbral dots or light bridges. In the monolithic model the vertical extension of the plumes is small, leading to a situation in which the plume is completely surrounded by the sunspot's magnetic field. However, in the cluster model convective plumes reach very deep into the solar interior, connecting with field-free convection zone below the sunspot. In the latter model, what appears as a single flux tube in the photosphere splits into many smaller flux tubes deeper down, leaving intrusions of field-free plasma in between the smaller tubes. Inside this intrusions is where the convection takes place.
It is not possible to distinguish between these two models employing spectropolarimetric observations because, below τ c = 1, the plasma is so opaque that no photon can travel from that depth without being absorbed. Currently, the only observational tool at our disposal that can allow us to infer the subsurface structure of sunspots is local helioseismology (Gizon and Birch, 2005; Moradi et al., 2010). Although this technique is still under development, it will hopefully shed some light on this subject in the near future.
An alternative way of studying the subsurface structure of sunspots is by means of numerical simulations of solar magneto-convection. Some recent studies (Schüssler and Vögler, 2006) show that convection can occur in the umbra in the form of plumes that do not reach more than 1 Mm beneath the solar surface. These convective plumes are completely surrounded by the sunspot's magnetic field and manifest themselves in the photosphere in the form of umbral dots. Furthermore, they also transport sufficient amounts of energy as to account for the observed umbral brightness (see Sections 2.5 and 3.1; see also Figure 23): 10–30% of the quiet Sun. At first glance these simulations seem to lend support to the monolithic sunspot model. However, the depth of the simulation box in Schüssler and Vögler (2006) is only 1.6 Mm. New simulations with deeper domains have been presented by Rempel (2011) and Cheung et al. (2010), with boxes of 6.1 and 8.2 Mm depth, respectively. In these new simulations, umbral dots present a very similar topology as with shallower boxes. However, light bridges appear to be rooted very deep, with convective plumes that reach more than 2 Mm into the Sun (see, for example, Figure 12 in Cheung et al., 2010). Further work is, therefore, needed since the current simulations are not sufficient to completely rule out the cluster model.
3.2 Sunspot penumbra and penumbral filaments
3.2.1 Spines and intraspines
The filamentary structure of sunspot penumbra was recognised early in the 19th century in visual observations (see review by Thomas and Weiss, 2008). The progress of observational techniques to attain higher spatial resolution revealed that the sunspot penumbra consists of radially elongated filaments with a width of 0.2–0.3" as seen in continuum images (e.g., Danielson, 1961a; Muller, 1976). Resolving the structure of the magnetic field with such high resolution is much more difficult because polarimetric measurements require multiple images taken in different polarization states, and a longer exposure time in a narrow wavelength band to isolate the Zeeman signal in a spectral line. For this reason, until recently many of the investigations of the magnetic field in the penumbra have reported contradictory results.
A hint of fluctuation in the magnetic field, in association with the penumbral filamentary structure, was first reported by Beckers and Schröter (1969a), who reported that the magnetic field was stronger and more horizontal in dark regions of the penumbra. Wiehr and Stellmacher (1989), however, found no general relationship between brightness and the strength of the magnetic field.
Advancement of large solar telescopes at locations with a good seeing conditions made it possible to better resolve the penumbral filamentary structure in spectroscopic and polarimetric observations, and a number of papers on the small-scale magnetic field structures in sunspot penumbra were published in early 1990s. Lites et al. (1990), using the Swedish Vacuum Solar Telescope (SVST) in La Palma, found a rapid change in the inclination of the magnetic field between some dark and light filaments near the edge of the penumbra, while the field strength showed only a gradual variation across the filaments. Degenhardt and Wiehr (1991), using the Gregory Coudé Telescope in Tenerife, found fluctuations in the inclination of the magnetic field vector in the penumbra by 7–14°, with steeper (more vertical) regions having a stronger magnetic field.
Schmidt et al. (1992), using the German Vacuum Tower Telescope (VTT) in Tenerife, found more horizontal field lines in dark filaments, while the strength of the magnetic field did not differ between bright and dark penumbral filaments. Title et al. (1993), using a series of Dopplergrams and line-of-sight magnetograms taken by a tunable narrowband filter equipped on SVST, found variations in the inclination of the magnetic field of about ± 18° across penumbral filaments. Lites et al. (1993), using the Advanced Stokes Polarimeter (ASP) on the Dunn Solar Telescope at Sacramento Peak, identified radial narrow lanes in the penumbra where the magnetic field is more vertical and stronger, thereby naming such regions as spines. Their results indicated that spines feature an azimuthal expansion of the magnetic field towards the sunspot's border. In addition, they found no clear evidence for a spatial correlation between spines and brightness. The correlation between the magnetic field strength and field inclination (i.e., stronger field in spines) was confirmed by Stanchfield II et al. (1997) using ASP data.
With a highly resolved spectrum in the Fe I 684.3 nm spectral line, which is formed in the deep photosphere, Wiehr (2000) found that darker penumbral lanes correlate with a stronger and more horizontal magnetic field, though the slit of the spectrograph sampled only a portion of the penumbrae. Better defined polarization maps of spines were taken with the Swedish 1-m Solar Telescope employing adaptive optics (Langhans et al., 2005), demonstrating that spines are regions with stronger and more vertical magnetic field, and that they are associated with bright penumbral filaments.
High quality vector magnetograms with a high spatial resolution are now routinely obtained by the spectropolarimeter (SP) on-board Hinode. Figure 24 (panels a and b) show the continuum intensity and the inclination of the magnetic field for a sunspot observed on January 5, 2007 (AR 10933), located very close to the center of the solar disk (Θ ≈ 2.9°). The field inclination was derived by a Milne-Eddington inversion (Section 1.3) as the angle between the magnetic field vector and the line-of-sight, γ (see Equation 26), but because of the proximity of the sunspot to the disk center, the inclination can be regarded as the inclination of the magnetic field, ζ (Equation 10), with respect to the local normal to the solar surface: e ρ (see Figure 42). It is obvious in the inclination map that the penumbrae consists of radial channels that have alternative larger and smaller field inclination. A close comparison with the continuum image shows that more horizontal field channels in panel b (also called intraspines) tend to be bright filaments in inner penumbra but to be dark filaments in outer penumbra. Panels c and d in Figure 24 show the divergence of transverse component of the magnetic field vector (∇ · B h ) and the total field strength B, respectively, obtained from the Milne-Eddington inversion. It is confirmed that spines have stronger field than intraspines, as well as a positive field divergence. Also noticeable is the presence of a number of patches that have opposite polarity to the sunspot around the outer border of the penumbra (see also Figures 4 and 7).
Sunspot AR 10933 observed at Θ = 2.9° on January 5, 2007 with the spectropolarimeter SOT/SP on-board Hinode. Displayed are: a) continuum intensity I c , b) magnetic field inclination γ, c) divergence of the horizontal component of the magnetic field vector ∇ · B h ., and d) total field strength B. All parameters were obtained from a Milne-Eddington inversion of the recorded Stokes spectra. The green arrow in panel a indicates the direction of the center of the solar disk. The yellow box surrounds the sunspot region displayed in Figure 34.
Thus, the penumbral magnetic field consists of two major components: spines where the magnetic field is stronger and more vertical with respect to the direction perpendicular to the solar surface, and intraspines where the magnetic field is weaker and more horizontal. Whereas the magnetic field of the spines possibly connect with regions far from the sunspot to form coronal loops over the active region, the magnetic field in the intraspines turns back into the photosphere at the outer border of the sunspot or extend over the photosphere to form a canopy (Solanki et al., 1992; Rueedi et al., 1998). The filamentary structure of the penumbra persists even after averaging a time series of continuum images over 2–4.5 hours (Balthasar et al., 1996; Sobotka et al., 1999). This suggests that the two magnetic field components are more or less exclusive to each other (Thomas and Weiss, 2004; Weiss, 2006) except for a possible interaction through reconnection at the interface between them in the photosphere (Katsukawa et al., 2007). Such structure of the penumbral magnetic field, i.e., magnetic fields with two distinct inclinations interlaced with each other in the azimuthal direction, is referred to as uncombed penumbra (Solanki and Montavon, 1993) or interlocking comb structure (Thomas and Weiss, 1992). The fact that the magnetic field is weakened in the intraspines, as compared with the spines, can also be employed to deduce through total pressure balance considerations (as we already did in the case of umbral dots and light bridges; see Section 3.1.2) that the intraspines are elevated with respect to the spines.
To account for the filamentary structure of penumbra with the uncombed magnetic fields, some distinguished models, that are under a hot discussion nowadays, were proposed. One of these models, the embedded flux tube model is an empirical model proposed by Solanki and Montavon (1993), in which nearly horizontal magnetic flux tubes forming the intraspines are embedded in more vertical background magnetic fields (spines) in the penumbra (Figure 25, left panel). The downward pumping mechanism (Thomas et al., 2002) was proposed to explain the origin of field lines that return back into the solar surface at the outer penumbra (Figures 4, 7, and 24). In this scenario, submergence of the outer part of flux tubes occurs as a result of the downward pumping by the granular convection outside the sunspots, and such magnetic fields form the low-laying horizontal flux tubes. Another idea to account for the penumbral filaments is the field-free gap model initially proposed by Choudhuri (1986) and later refined by Spruit and Scharmer (2006) and Scharmer and Spruit (2006). Here, the penumbral bright filaments are regarded as manifestations of the protrusion of non-magnetized, convecting hot gas into the background oblique magnetic fields of the penumbra. Due to the continuity condition of the normal component of the magnetic field across the boundary between the background field and the protruding non-magnetized gas, the vertical component of the magnetic field vector, B ρ , in the background magnetic field must vanish right on top of the non-magnetic gas. This immediately yields a region, above the penumbral filaments, where the magnetic field is almost horizontal (i.e., intraspines).
Models for explaining the uncombed penumbral structure. Upper-left: embedded flux tube model (from Solanki and Montavon, 1993, reproduced by permission of the ESO); lower-left: rising flux tube model (from Schlichenmaier et al., 1998a, reproduced by permission of the ESO); right: field-free gap model (from Spruit and Scharmer, 2006, reproduced by permission of the ESO).
All the aforementioned models attempt to explain, with different degrees of success, the configuration of the magnetic field in the penumbra. However, the appearance of a penumbra is always associated with a distinctive gas flow, i.e., the Evershed flow and, therefore, this must also be taken into account by these models. In the next section we will address this issue.
3.2.2 Relation between the sunspot magnetic structure and the Evershed flow
The Evershed flow was discovered in 1909 by John Evershed at the Kodaikanal Observatory in India as red and blue wavelength shifts in the spectra of absorption lines in the limb-side and disk-centerside of the penumbra, respectively. This feature can be explained by a nearly horizontal outflow in the photosphere of the penumbra (Evershed, 1909). Under an insufficient spatial resolution, it appears as a stationary flow with typical speeds of 1–2 km s−1, where the magnitude of the flow velocity increases with optical depth τ c (towards the deep photosphere; Bray and Loughhead, 1979).
An outstanding puzzle about the Evershed flow lies in the relation between the velocity vector and the magnetic field vector in the penumbra. Since the averaged magnetic field in the penumbra has a significant vertical component, with an angle with respect to the normal vector on the solar surface between ζ ≈ 40–80° (see Figure 11 in Section 2.1), and the Evershed flow is apparently horizontal, this would mean that the flow would move across the magnetic field. Under these circumstances, the sunspot's magnetic field (which is frozen-in to the photospheric gas) would be removed away within a few hours.
It is highly plausible that there is a close relationship between the Evershed flow and the filamentary structure of the penumbra. Indeed, it was recognized in the 1960s that the flow is not spatially uniform but concentrated in narrow channels in penumbra; e.g., Beckers (1968) reported that the flow originates primarily in dark regions between bright penumbral filaments. Two models were proposed to account for the nature of penumbral filaments and the Evershed flow before 1990. One is the elevated dark filament model in which the penumbral dark regions are regarded as elevated fibrils with nearly horizontal magnetic field overlaying the normal photosphere and carry the Evershed flow in them (Moore, 1981; Cram and Thomas, 1981; Thomas, 1988; Ichimoto, 1988). The other is the rolling convection model in which penumbral filaments are regarded as convective elements radially elongated by a nearly horizontal magnetic field in penumbra and where the Evershed flow is confined in dark lanes that are analogous to the intergranular dark lanes (Danielson, 1961b; Galloway, 1975). Both models assume a nearly horizontal magnetic field in the penumbra and, therefore, contradict the observational fact that a significant fraction of sunspot's vertical magnetic flux comes out through the penumbra (Solanki and Schmidt, 1993).
The long-lasting enigma on the Evershed flow was finally solved by the discovery of the interlocking comb structure of the penumbral magnetic field (Section 3.2.1). Under this scenario, the Evershed flow is confined in nearly horizontal magnetic field channels in penumbra (i.e., intraspines), while out of the flow channels (i.e., in the spines) the magnetic field is more vertical. Both components, when averaged together, make the spatially averaged magnetic field far from completely horizontal (Figures 11 and 26; see also Title et al., 1993). The relationship between the Evershed flow and the horizontal magnetic field in the penumbra has been highlighted in many works in the past: Stanchfield II et al. (1997, Figure 7) or Mathew et al. (2003, Figure 12). The latter two works were obtained with spectropolarimetric data at 1" resolution. A more updated result, employing Hinode/SP data with 0.3" resolution, has been presented by Borrero and Solanki (2008, see Figure 27). This figure demonstrates that the Evershed flow (seen as large positive or redshifted line-of-sight velocities; middle panel) is concentrated along the intraspines: regions where the magnetic field is horizontal (γ ≈ 90°; bottom panel) and weaker (upper panel).
Geometry of magnetic field and Evershed flow in penumbra. Magnetic field lines are shown by inclined and colored cylinders, while the Evershed flow is indicated by white arrows in dark penumbral channels. Note that the Evershed flow concentrates along the more horizontal magnetic field lines (white cylinders) (from Title et al., 1993, reproduced by permission of the AAS).
Variation of the physical parameters at τ c = 1 (continuum) along an azimuthal cut around the limb-side penumbra (i.e., along one of the blue ellipses in Figure 10). From top to bottom: magnetic field strength B, line-of-sight velocity Vlos, and inclination of the magnetic field γ. Dotted curves in each panel show the scattered light fraction obtained from the inversion algorithm. Note that the velocity (Evershed flow) is strongest in the regions where the magnetic field is weak and horizontal (intraspines), while it avoids the regions with more less inclined and stronger magnetic field (spines) (from Borrero and Solanki, 2008, reproduced by permission of the AAS).
In the embedded flux-tube model (Solanki and Montavon, 1993), the Evershed flow is supposed to be confined in the horizontal magnetic flux tubes embedded in more vertical background magnetic field of the penumbra. In such picture, the siphon flow mechanism was proposed as the driver of the Evershed flow (Meyer and Schmidt, 1968; Thomas, 1988; Degenhardt, 1991; Montesinos and Thomas, 1993): a difference in the magnetic field strength between two footpoints of a flux tube causes a difference of gas pressure, and drives the flow in a direction towards the footpoint with a higher field strength (i.e., the footpoint outside the sunspot to account for the Evershed outflow). Schlichenmaier et al. (1998b) and Schlichenmaier et al. (1998a) investigated the dynamical evolution of a thin magnetic flux tube8 embedded in a penumbral stratification (Jahn and Schmidt, 1994a) and proposed the hot rising flux tube model, in which a radial and thin flux tube containing hot plasma raises towards the solar surface due to buoyancy. As the flux tube reaches the τ c = 1-level it cools down due to radiation, thereby producing a gradient on the gas pressure along the flux tube and, thus, driving the Evershed flow along the tube's axis (i.e., radial direction in the penumbra).
By performing an inversion of Stokes profiles of three infrared spectral lines at 1565 nm and using a two component penumbral model in which two different magnetic atmospheres are interlaced horizontally, Bellot Rubio et al. (2004) found a perfect alignment of the magnetic field vector and the velocity vector in the component that contains the Evershed flow. This picture was supported by Borrero et al. (2004) who also performed Stokes inversions of the same infrared lines. With a further elaborated analysis, Borrero et al. (2005) found that the penumbral flux tubes are hotter and not completely horizontal in the inner part of the penumbra, while they become gradually more horizontal and cooler with increasing radial distance. This is accompanied by an increase in the flow velocity and a decrease of the gas pressure difference between flux tube and the background component, with the flow speed eventually exceeding the critical value to form a shock front at large radial distances (V > 6–7 km s−1). They argued that these results strongly support the siphon flow as the physical mechanism responsible for the Evershed flow.
Until recently the relationship between the Evershed flow and the brightness of the penumbral filaments has been somewhat controversial. Many authors (Beckers, 1968; Title et al., 1993; Shine et al., 1994; Rimmele, 1995a; Balthasar et al., 1996; Stanchfield II et al., 1997; Rouppe van der Voort, 2002) have presented evidence that the Evershed flow is concentrated in dark filaments, while some studies claimed that there is no correlation (Wiehr and Stellmacher, 1989; Lites et al., 1990; Hirzberger and Kneer, 2001). Rimmele (1995a) showed that the correlation becomes better when one compares the intensity and velocity originating from the same height, and also gave a hint that the correlation is different between inner and outer penumbra. Schlichenmaier et al. (2005), Bellot Rubio et al. (2006), and Ichimoto et al. (2007a) presented evidence that the Evershed flow takes place preferentially in bright filaments in the inner penumbra, but in dark filaments in the outer penumbra.
Figure 28 presents the spatial correlation between penumbral filaments and the Evershed flow. The correlation coefficient between the Doppler shift (Vlos) and the elevation angle of magnetic field vector from the solar surface9 as a function of the radial position in the penumbra, is displayed in the upper-right panel, whereas the correlation between the Doppler shift (Vlos) and continuum intensity is shown in the lower-right panel. In these plots, the results for limb-side penumbra are shown in red color and for disk-center-side are shown in blue color. The abscissa in this figure spans from the umbra-penumbra boundary (left) to the outer border of the penumbra (right). The curves along which the correlation coefficients are obtained are shown for both limb-side and disk-center-side penumbra in the left panels. The data employed for this figure was obtained by SOT/SP when the sunspot was located at the heliocentric angle of Θ = 31°, thus, the Doppler shift is mainly produced by the horizontal Evershed flow. In this plot, line-of-sight velocities are taken in absolute value such that there is no difference between the redshifts in the limb-side and the blueshifts in the center side that are characteristic of the Evershed flow. In Figure 28, we notice that the Evershed flow correlates with more horizontal magnetic fields throughout the entire penumbra, while it correlates with bright filaments in the inner penumbra but with dark filaments in the outer penumbra. These results are consistent with the idea that penumbral filaments, which harbor a nearly horizontal magnetic field, are brighter in inner penumbra but darker in outer penumbra. The correlation between Doppler shift and intensity shows an asymmetric distribution between the disk center-side and limb-side penumbra (lower right). This suggests that overposed to the Evershed flow, which is mainly horizontal, there exists a vertical component in the velocity vector in the penumbra. This vertical component will be discussed in detail in Sections 3.2.3 and 3.2.4.
Spatial correlation between penumbral filaments and the Evershed flow. Correlation coefficients between the Evershed flow and brightness (lower-right), and between the Evershed flow and the elevation angle of magnetic field from the solar surface (upper-right) are shown as a function of radial distance from the sunspot center. Arc-segments along which the correlation is calculated are shown in the left panel. The sunspot was located at the heliocentric angle of Θ = 31°. The direction to the center of the solar disk is shown by an arrow in the left panel. The data employed here was recorded with the spectropolarimeter on-board Hinode (SOT/SP). Red lines show the results for the limb-side penumbra, whereas blue corresponds to the center-side penumbra.
So far we have discussed only investigations that were carried out with ME-inversion codes and, therefore, referred only to the physical parameters of the sunspot penumbra at a constant τ-level (see Sections 1.3.1 and 2.1). In order to investigate the depth dependence of the line-of-sight velocity and magnetic field vector in the penumbra, τ-dependent inversion codes (see Sections 1.3.1 and 2.2) must be applied to observations of the polarization signals in spectral lines. This has been addressed by a number of authors, such as Jurčák et al. (2007), who applied the SIR inversion code (Ruiz Cobo and del Toro Iniesta, 1992) to the spectropolarimetric data obtained by SOT/SP on Hinode and found that a weaker and more horizontal magnetic field is associated with an increased line-of-sight velocity in the deep layers of the bright filaments in the inner penumbra. In the outer penumbra, however, stronger flows and more horizontal magnetic fields tend to be located in dark filaments (Jurčáak and Bellot Rubio, 2008). With a further application of the SIR inversion on SOT/SP data, Borrero et al. (2008) found that the magnetic field in the spines wraps around the horizontal filaments (i.e., intraspines). Some results from the latter two works are presented in Figures 29 and 30.
Depth structure of penumbra derived from Stokes inversions of spectro-polarimetric data. Showns are vertical cuts across the penumbral filaments. On the left, from top to bottom, are temperature T, field strength B, field inclination γ, and line-of-sight velocity Vlos (from JurČáak et al., 2007, reproduced by permission of the PASP).
Vertical stratification (optical depth τ c ) of the physical parameters in the penumbra. The horizontal axis is the azimuthal direction around the penumbra and, therefore, it is perpendicular to the radial penumbral filaments. Upper-left panel: line-of-sight velocity Vlos. Upper-right: total magnetic field strength B. Lower-left: inclination of the magnetic field vector with respect to the normal vector to the solar surface ζ. (see Equation 10). Lower-right: azimuth of the magnetic field vector Ψ (Equation 11). This plot demonstrates that the strong and vertical magnetic field of the spines extends above the intraspines (indicated by the index i), where the Evershed flow is located where the magnetic field is rather horizontal and weak. It also shows that the azimuth of the magnetic field changes sign above the intraspines, indicating that the magnetic field of the spines wraps around the intraspines. The arrows in this figure show the direction of the magnetic field in the plane perpendicular to the axis of the penumbral filaments (from Borrero et al., 2008, reproduced by permission of the ESO).
3.2.3 The problem of penumbral heating
One important issue that needs to be addressed to understand the origin of the penumbra is how the energy transport takes place. Whatever mechanism exists, it must supply enough energy to maintain the penumbral surface brightness to a level of 70–80% of the quiet Sun granulation (see Sections 2.5 and 3). Since the most efficient form of energy transport in the solar photosphere is convection, the key question is therefore to identify how convective motions in the presence of a rather strong, B ≈ 1500 G, and horizontal, ζ ≈ 40–80°, magnetic field (see Figure 11) occur.
In search for these convective motions we have to examine the predictions that the different models (see Section 3.2.2) make about vertical flows in the penumbra. The hot rising flux-tube model (see Section 3.2.2) predicts the presence of upflows at the inner footpoints of the flux tubes and downflows at their outer footpoints (Schlichenmaier, 2002). Provided that the flux tubes are evenly distributed, this would yield a preference for upflows in the inner penumbra, whereas downflows would dominate at large radial distances. Because of these features, we will refer to this form of convection as radial convection, with convective flows occurring along the penumbral filament (see upper panel in Figure 31). Schlichenmaier and Solanki (2003) examined the possible heat transport in the context of this model and found that the heat supplied by this model is sufficient only if the upflowing hot plasma at the inner flux tube's footpoint travels only a small radial distance \(\mathcal{L}\) before tuning into a downflow (see upper panel in Figure 31), with a new flux tube appearing immediately after. This implies that there should be a significant magnetic flux and mass flux returning to and emerging from the photosphere in penumbra. This is a natural consequence of the rapid cooling that the hot rising plasma suffers once it reaches the τ c = 1-level (Schlichenmaier et al., 1999).
Possible patterns of convection present in the sunspot penumbra. The upper panel corresponds to a pattern of radial convection, where upflows are presented at the inner footpoints of the penumbral filaments and downflows at the outer footpoints. This pattern is predicted by the embedded flux-tube model and the hot rising flux-tube model. The lower panel shows a pattern of azimuthal or overturning convection, where the upflows/downflows alternate in the direction perpendicular to the filaments' axis. This is the flow pattern predicted by the field-free gap model.
Contrary to the aforementioned models, the field-free gap model provides a very efficient heat transport since here convective motions are present over the entire length along the bright penumbral filaments, with upflows at the center of the filaments and downflows at the filaments' edges. This strongly resembles to the convective motions discussed in Sections 3.1.3 and 3.1.4 in the context of umbral dots and light bridges and, therefore, provides a connection between the different small-scale features in sunspots. The field-free gap model does not predict any particular radial preference for upflows and downflows in the penumbra. It however predicts that alternating upflows/downflows should be detected in the direction perpendicular to the filaments (i.e., azimuthally around the penumbra). Because of this feature we will refer to this type of convection as azimuthal convection or overturning convection (see lower panel in Figure 31). Note that the field-free gap model does not readily offer an explanation for the Evershed flow. This is an important point that will be addressed in Section 3.2.7.
3.2.4 Vertical motions in penumbra and signature of convection
In order to distinguish between the different proposed models that attempt to explain the heating mechanism in the penumbra (Section 3.2.3) we need to address the origin of the Evershed flow, as well as identifying the sources and sinks associated with this flow and its mass balance. Since its discovery, the Evershed flow has been recognized as a horizontal motion of the photospheric gas. However, as already mentioned in Section 3.2.2, the Evershed flow is not purely horizontal as it possesses a vertical component. A clear evidence for the vertical motions appeared only after the 1990s when high spatial resolution became available in spectroscopic observations.
Since then, a number of observations have been reported regarding the vertical component of the flow in the penumbra. On the one hand, upflows in the penumbra have been reported by Johannesson (1993), Schlichenmaier and Schmidt (1999, 2000), and Bellot Rubio et al. (2005), and with much higher spatial resolution by Rimmele and Marino (2006). On the other hand, downflows have been observed in and around the outer edge of penumbra by, among others, Rimmele (1995b), Westendorp Plaza et al. (1997), del Toro Iniesta et al. (2001), Schlichenmaier et al. (2004), Bellot Rubio et al. (2004), and Sánchez Cuberes et al. (2005). Both down- and upflows have been simultaneously observed in the penumbra by Schmidt and Schlichenmaier (2000), Schlichenmaier and Schmidt (2000), Westendorp Plaza et al. (2001a), Tritschler et al. (2004), Sánchez Almeida et al. (2007), Ichimoto et al. (2007a), and Franz and Schlichenmaier (2009). Figure 32 highlights some observations that clearly show the downflow patches around a sunspot with an opposite polarity (Westendorp Plaza et al., 1997, left panel) and upflow patches at the leading edge of penumbral bright filaments (Rimmele and Marino, 2006, right panel). These last features correspond to the solar called bright penumbral grains and they are related to the peripheral umbral dots (Sobotka et al., 1999).
Selected observations of vertical motions in sunspots. Left panels: Discovery of downflows around the outer border of a sunspot. The sunspot is located near the center of the solar disk. Top is the continuum image and bottom is the magnetic field inclination overlaid with velocity contours. Blue regions have a magnetic field polarity opposite to the sunspot, while white contours associated with these regions show downflows with + 3 km s−1 (from Westendorp Plaza et al., 1997, reproduced by permission of Macmillan Publishers Ltd: Nature). Right panels: Close-up of the inner part of a limb-side penumbra. Top and bottom are filtergram (intensity) and Dopplergram (Vlos) in the FeI 5576 Å line. Each Evershed flow channel (white filaments in the Dopplergram) is associated with a bright grain and upflow (dark point in the Dopplergram) (from Rimmele and Marino, 2006, reproduced by permission of the AAS).
According to the picture drawn by the present observations, most material carried by the Evershed flow is, thus, supposed to flow back into the photosphere at the downflow patches (Westendorp Plaza et al., 2001a), while some fraction (∼ 10%) of the material may continue to flow across the penumbral outer edge along the elevated magnetic field to form a canopy (Solanki and Bruls, 1994; Solanki et al., 1999). The mass flux balance between up- and downflows in a sunspot observed near the disk center was also inferred under the MISMA hypothesis, though individual flow regions were not spatially resolved (Sáanchez Almeida, 2005b). This model postulates that the magnetic field varies rapidly (in all three directions) at scales much smaller than the mean free path of the photon (Sáanchez Almeida and Landi Degl'Innocenti, 1996; Sáanchez Almeida et al., 1996).
The configuration of the magnetic field is affected by the aforementioned vertical flows. In fact, some of the magnetic field lines plunge back into the deep photosphere at the outer edge of the sunspot and its surroundings (see Figures 4 and 7). The relationship between vertical motions and the magnetic field vector in the penumbra is clearly demonstrated by spectropolarimetric data of a sunspot near disk center. Figure 33 shows maps of Stokes V (circular polarization) at Δλ = ±(100, 300) mÅ away from the line center of the Fe I 6302.5 Å spectral line. The sunspot in this figure is the same one as in Figure 24 (Θ = 2.9°). The sign of Stokes V is reversed for Δλ = (100, 300) mÅ. If there are no mass motions in the sunspot, Stokes V maps in the blue and red wings are expected to be identical since the Zeeman effect produces anti-symmetric Stokes V profiles around the line center. This is the case of the main lobes of the Stokes V profiles at ± 100 mÅ. However, the maps in ± 300 mÅ are remarkably different from each other: a number of small and elongated structures with the same polarity of the sunspot are visible in the −300 mÅ V map (middle-left panel in Figure 33) over the penumbra, but with a slight preference to appear in inner penumbra, whereas a number of patches with the opposite polarity of the sunspot are seen in the +300 mÅ V map (middle-right panel in Figure 33), preferentially in the mid and outer parts of the penumbra. As is confirmed by the Dopplergram in the line-wing of Stokes I (bottom-left panel in Figure 33), the former features are associated with upward motions while the later correspond to strong downflows. The typical line-of-sight velocities of the blueshifted regions and the redshifted regions are approximately 1 km s−1 and ∼ 47#x2013;7 km s−1, respectively. The presence of very fast downflows in the mid and outer regions of the penumbra has been reported previously by del Toro Iniesta et al. (2001) and Bellot Rubio et al. (2004), who suggest that many of those downflow patches (where the magnetic field also turns back into the solar photosphere) harbor supersonic velocities.
Upper and middle panels: Stokes V maps of a sunspot near the solar disk center (Θ = 2.9° same sunspot as in Figures 24 and 34) at two different wavelengths (shown in each panel) from the center of the Fe I 6302.5 Å spectral line. The sign of Stokes V is reversed for +(100,300) mÅ. Bottom panels: line-of-sight velocity (Doppler velocity) measured in the wings (left) and on the core (right) of the spectral line.
Figure 34 shows enlargement of a penumbral region indicated by a box in Figure 24a, where contours for upflow and downflow regions are overlaid on continuum intensity (panel a) and field inclination γ (panel b) maps. It is obvious in panel b that the upflow and downflow patches are aligned with nearly horizontal field channels (filaments with light appearance in the inclination map) that carry the Evershed flow, and that small-scale upflows are preferentially located near the inner penumbra, while downflows dominate at the outer ends of the horizontal field channels (see also Figure 1 in Ichimoto, 2010 and Figure 5 in Franz and Schlichenmaier, 2009). Thus, the upflow and downflow patches seen here can be regarded as the sources and sinks of the elementary Evershed flow embedded in deep penumbral photosphere. When all the aforementioned results and observations for the velocity and the magnetic field vector are put together, the picture of the penumbra that emerges is that of Figure 35.
Continuum intensity I c (panel a) and field inclination γ (panel b) in the penumbral region shown as a yellow box in Figure 24a. Overlaid are contours for upflow regions with 0.8 km s−1 (blue) and downflow regions with V/I c = 0.01 in the far red wing of Fe I 6302.5 7#x00C5; line (red). The sunspot shown here was located almost at disk center: Θ = 2.9°.
Cartoon of penumbral magnetic field and the Evershed flow structure.
These results seem to support the idea that convective motions occur in a radial pattern along the penumbral filaments (radial convection; see upper panel in Figure 31). In principle, this lends a strong support to the hot rising flux-tube model (see Section 3.2.1). However, a closer look at Figure 34 reveals that the radial distance \(\mathcal{L}\) between upflows in the inner penumbra and downflows in the outer penumbra is typically several megameters. This would imply that the energy carried by the upflows is not sufficient to heat the penumbra according to the argument of Schlichenmaier and Solanki (2003). Interestingly, Ruiz Cobo and Bellot Rubio (2008) revisited this problem with the embedded flux-tube model, and argued that the significant portion of brightness of the penumbra can be explained with the hot Evershed flow taking place at the inner footpoints of rather thick flux tubes (> 200 km; see Footnote 8) even for large values of \(\mathcal{L}\).
Notice, however, that the fact that one type of convection is detected, does not immediately rule out the existence of the other type, azimuthal/overturning convection which is proposed by the field-free gap model (lower panel in Figure 31). Indeed, the search for an azimuthal convective pattern, i.e., upflows at the center of penumbral filaments and downflows at their edges, has intensified in the past few years. Some works, employing continuum images, have provided compelling evidence that the azimuthal/overturning convection does indeed also exist (Márquez et al., 2006; Ichimoto et al., 2007b; Bharti et al., 2010, see also Section 3.2.5), at least in the inner penumbra. Unfortunately, results based on spectroscopic measurements have been contradictory so far. Whereas some works (Rimmele, 2008; Zakharov et al., 2008) report on positive detections of such downflows and upflows (of up to 1 km s−1), others claim that at the present resolution that convective pattern does not exist (Franz and Schlichenmaier, 2009; Bellot Rubio et al., 2010). It is, therefore, of the uttermost importance to provide a conclusive detection (or ruling out) of an azimuthal/overturning convective flow in the penumbra that might help us settle once and for all the problem of the penumbral heating. A number of reasons have been put forward in order to explain the lack of evidence supporting an azimuthal/overturning convection. One of the reasons is the lack of sufficient spatial resolution to resolve the velocity fields inside the penumbral filaments. However, this does not explain why azimuthal/overturning convective motions are already detected in umbral dots at the present resolution (see Section 3.1.3) but not in penumbral filaments. Another reason that has been advocated, within the context of the field-free gap model, has been that the τ c = 1 level is formed above the convective flow rendering it invisible to spectropolarimetric observations. This argument, however, fails to explain why is then the azimuthal/overturning convective pattern seen in umbral dots since there the τ c = 1 level should be formed even higher above the convective flow than in penumbral filaments (Borrero, 2009). The most adequate explanation, therefore, for the lack of evidence supporting an azimuthal/overturning convective flow pattern (if it exists) lies in the large magnitude of the Evershed flow, which overshadows the contribution from the convective up/downflows on the measured line-of-sight velocity as soon as the observed sunspot is slightly away from disk-center (Θ = 0°).
Regardless of which form of convection takes place, it is very clear that this is indeed the mechanism that is responsible for the energy transport in the penumbra. This is emphasized by the very close relationship existing between upflows and bright grains in the penumbra as seen in Figure 34. In this figure we display the continuum intensity I c (panel a) and the inclination of the magnetic field vector γ (panel b) for the southern part of the sunspot shown in Figure 24 overlaid with contours showing the vertical motions. The blue contours show blueshifts equal or larger than 0.8 km s−1 in the wing of Stokes I of Fe I 6301.5 Å, while the red contours show V/I c > 0.01 at Fe I 6302.5 Å + 0.365 Å representing strong downflow regions with opposite magnetic polarity to the spot. The fact that upflows (blue contours) correlate so well with bright penumbral regions, strongly suggest that the vertical component of the Evershed flow supplies the heat to maintain the penumbral brightness even though a quantitative evaluation of the heat flux is not available (Ichimoto et al., 2007a). Puschmann et al. (2010b) supports this scenario in a more quantitative manner based on their 3D empirical penumbral model derived from the Stokes inversion of the Hinode/SP data, i.e., the penumbral brightness can be explained by the energy transfer of the ascending mass carried by the Evershed flow if the obtained physical quantities are extrapolated to slightly deeper layer below the observable depth (τ c = 1).
Figure 36 shows typical Stokes profiles in upflow (panel a) and downflow (panel b) regions in the penumbra, respectively. These profiles correspond to the sunspot AR 10944 on February 28, 2007 very close to the center of the solar disk: Θ = 1.1°. It is noticeable that the upflow region shows a blue hump in Stokes V with the same polarity of the main lobe in the blue wing, while the downflow region shows a strong third lobe with opposite polarity in the far red wing of Stokes V. These asymmetric Stokes V profiles imply the presence of a strong velocity (and magnetic field) gradient along the line-of-sight: Vlos(τ c ) and B(τ c ). The solid curves show the best-fit profiles produced by a Milne-Eddington (ME) inversion algorithm. A ME-inversion assumes that the physical parameters are constant with optical depth (see Section 1.3) and, therefore, it always produces anti-symmetric Stokes V profiles, thereby failing to properly fit the highly asymmetric observed circular polarization signals. Sánchez Almeida and Ichimoto (2009) reproduced the redlobe profiles using the MISMA model (see Section 3.2.4), and suggested that reverse polarity patches result from aligned magnetic field lines and mass flows that bend over and return to the solar interior at very small scales all throughout the penumbra. While this scenario does not help to distinguish between a radial or azimuthal/overturning form of convection (Figure 31) it certainly emphasizes the presence of small-scale convection, which in turn is needed to sustain the penumbral brightness. Other works have also pointed out the relationship between the polarity of the vertical component of the magnetic field and the upflow/downflow regions in the penumbra. For instance, Sainz Dalda and Bellot Rubio (2008) found small-scale, radially elongated, bipolar magnetic structures in the mid-penumbra aligned with intraspines. They move radially outward and were interpreted by these authors as manifestations of the sea-serpent field lines that harbor the Evershed flow (Schlichenmaier, 2002) and, eventually, leave the spot to form moving magnetic features. Martínez Pillet et al. (2009) found a continuation of such magnetized Evershed flow outside sunspots at supersonic speeds.
Stokes profiles (observed with Hinode/SP) of Fe I 6301.5 Å and 6302.5 Å spectral lines in an upflow (panel a) and downflow (panel b) regions in the penumbra. Solid curves show results of a Milne-Eddington fitting algorithm (see Section 1.3.). These profiles correspond to the sunspot observed very close to disk center (Θ = 1.1°) on February 28, 2007 (AR 10944).
3.2.5 Inner structure of penumbral filaments
Improvements in the spatial resolution in ground-based optical observations revealed further details about the rich variety of fine-scale structures in the penumbra. Scharmer et al. (2002) discovered a notable feature in penumbral filaments at 0.1" resolution with the Swedish 1-m Solar Telescope at La Palma, i.e., bright penumbral filaments in the inner penumbra often show internal substructure in the form of two bright edges separated by a central dark core (Figure 37). The temporal evolution of these structures shows that the dark core and lateral bright edges move together as a single entity. Some of the dark features are not in parallel to penumbral filaments, but they form oblique dark streaks crossing penumbral filaments. These streaks make the filaments look as if they are twisting with several turns along their length.
Bright penumbral filaments showing a dark central core. Image was taken in G-band at 430.5 nm with the Swedish 1-m Solar Telescope. Tickmarks have a scaling of 1000 km on the Sun (from Scharmer et al., 2002, reproduced by permission of Macmillan Publishers Ltd: Nature).
The visibility of the dark cores in the filaments is not uniform over the penumbra when the sunspot is located outside the disk center: dark cores are more clearly identified in disk centerside penumbra while they are hardly seen in limb side penumbra (Sütterlin et al., 2004; Langhans et al., 2007). In addition, dark cores are better defined in G-band images than in continuum, which suggests that dark cores are structures that are elevated above the continuum formation height τ c = 1 (Rimmele, 2008).
The typical lifetime of dark-cored penumbral filaments was estimated as < 45 minutes (Sütterlin et al., 2004) while some dark cores last longer than 90 minutes (Langhans et al., 2007). The first spectroscopic observation of dark cores was reported by Bellot Rubio et al. (2005), who found a significant enhancement of the Doppler shift which they interpreted as an upflow in the dark cores. It is also found by spectroscopic (Bellot Rubio et al., 2007a) and filtergram (Langhans et al., 2007) observations, that dark-cored filaments are more prominent in polarized light than in continuum intensity, and that dark cores are associated with a weaker and a more horizontal magnetic field than their lateral brightenings and harbor an enhanced radial Evershed outflow. These features are to be considered on top of the already weak and horizontal magnetic field that characterizes the penumbral intraspines (see Section 3.2.1).
Based on a stratified atmosphere consisting of nearly horizontal magnetic flux tubes embedded in a stronger and more vertical field Borrero (2007), as well as Ruiz Cobo and Bellot Rubio (2008), performed radiative transfer calculations to show that these models reproduce the appearance of the dark-cored penumbral filaments. In these models, the origin of the dark cores is attributed to the presence of the higher density region inside the tubes, which shifts the surface of optical depth unity towards higher (cooler) layers.
From time series of continuum images taken by Hinode/SOT, Ichimoto et al. (2007b) found a number of penumbral bright filaments revealing twisting motions around their axes. As it also happens with the dark core at the center of the penumbral filaments, the twisting motions are well observed only in particular portions in the penumbra but, in this case, the locations in which penumbral filaments are oriented parallel to the limb. The direction of the twist (lateral motions of dark streaks that run across filaments diagonally) is always from the limb-side to disk-center side (see Figure 38). Therefore, the twisting feature is not likely a real twist or turn of filaments but, rather, are a manifestation of their dynamical nature, so that their appearance depends on the viewing angle. Overturning/azimuthal convection (see Figure 31) at the source region of the Evershed flow (Ichimoto et al., 2007b; Zakharov et al., 2008) has been proposed as the origin of such features. Such picture with overturning/azimuthal convection causing the observed twisting motions is supported by a positive correlation between the speed of twisting motion and the brightness of penumbral filament in space and time (Bharti et al., 2010). Spruit et al. (2010) interpret the oblique striations that propagate outward to produce the twisting appearance of the filaments as a corrugation of the boundary between the convective flow inside the bright filament and the magnetic field wrapping around it. On the other hand, there are some arguments that some of filaments have intrinsic twist originated from the screw pinch instability (Ryutova et al., 2008; Su et al., 2010).
A sunspot located at Θ = 30°, east-ward from the center of the solar disk. Space-time plots along the slits across inner penumbral filaments are shown on both sides. The position of the slits are indicated at the top of each space-time plot with partial images whose locations are shown by dashed lines on the sunspot image. Twist (or turning motion) of penumbral bright filament is seen as helical structures of bright filaments in the space-time plot.
3.2.6 The Net Circular Polarization in sunspots
The net circular polarization (NCP) is defined as \(\mathcal{N} = \int {V(\lambda )d\lambda } \) with the Stokes V signal integrated over a spectral line. \(\mathcal{N} = 0\) in a perfectly anti-symmetric Stokes V profile, as the area of the blue lobe compensates the area of the red lobe (see solid lines in Figure 36). However, the Stokes V profiles deviate from purely anti-symmetric (see dots in the same figure) and, therefore, \(\mathcal{N} \ne 0\) if there exists a gradient of plasma motion along the line-of-sight: Vlos(τ c ). In general, Vlos(τ c ) can produce only small amounts of the NCP. For the large values observed in sunspots, a coupling of a velocity gradient and a gradient in the magnetic field vector B(τ c ) within the line-forming region \(\bar \tau \) (see Section 1.3.1) is required (Auer and Heasley, 1978; Landolfi and Landi Degl'Innocenti, 1996). Consequently, the NCP provides a valuable tool to diagnose the magnetic field and velocity structures along the optical depth τ c in the sunspot's atmosphere. Observation of the NCP in sunspots were first reported by Illing et al. (1974a) and Illing et al. (1974b), and were followed by Henson and Kemp (1984) and Makita and Ohki (1986). From these early observations a number of basic features and properties of the NCP at low spatial resolution were inferred (see, e.g., Martínez Pillet, 2000):
The largest NCP occurs in the limb-side penumbra around the apparent magnetic neutral line with the same sign as the umbra's blue lobe of the Stokes V profile.
The disk center-side penumbra also shows NCP but in the opposite sign to that of the limbside penumbra and with less magnitude.
The penumbra of sunspots at disk center show a NCP with the same sign to that of the limb-side penumbra.
Besides a gradient in Vlos(τ c ), which is a necessary condition to produce a non-vanishing NCP, the works from Sánchez Almeida and Lites (1992) and Landolfi and Landi Degl'Innocenti (1996) show that a gradient in any of the three components of the magnetic field vector will also enhance the amount of NCP. These gradients are often referred to as the ΔB, Δγ and Δϕ mechanisms, with Δ indicating a variation of the physical quantity with optical depth τ c . The NCP in sunspots was first interpreted in terms of the ΔB-effect by Illing et al. (1975), who employed a magnetic field strength and line-of-sight velocity that increased with optical depth in the penumbra. Makita (1986) interpreted the NCP in sunspots by means of the Δϕ-effect, i.e., the sunspot's magnetic field is twisted and unwound along its axis, and has azimuthal rotation along the line-of-sight in the penumbra.
Nowadays, the most successful scenario to reproduce the NCP of sunspots is based on the Δγ-effect. Sánchez Almeida and Lites (1992) explained the NCP employing a penumbral model in which the Evershed flow increases with depth and where magnetic field lines become progressively more horizontal with depth in the penumbra. They argued, however, that the large gradient in the inclination of the magnetic field needed to explain the observed NCP was not consistent with a sunspot's magnetic field in magnetohydrostatic equilibrium. Solanki and Montavon (1993) addressed this problem, and proposed that the needed gradients to reproduce the NCP could be achieved without affecting the sunspot's equilibrium if they assumed the presence of a horizontal flux tube carrying the Evershed flow embedded in a more vertical background that wraps around it: embedded flux-tube model. In this model, the gradients in Vlos and γ are naturally produced as the line-of-sight crosses the boundary between the background and the horizontal flux tube. Those works were followed by more elaborated models by Martínez Pillet (2000) and Borrero et al. (2006). In order to explain the NCP observed in penumbra at disk-center, Solanki and Montavon (1993) and Martínez Pillet (2000) assumed an upflow in the background magnetic field to make the Δγ-effect to operate with the deeply embedded flux tubes that carries the horizontal Evershed flow. Schlichenmaier et al. (2002), Müller et al. (2002), and Müller et al. (2006) further developed this idea and provided three-dimensional penumbral models in which horizontal flux tubes are embedded in a more vertical penumbral background magnetic field, to successfully account for the observed azimuthal (i.e., variation around the sunspot) distribution of the NCP at low spatial resolution (≃ 1") over the penumbra located outside the disk center (see Figure 39). In these models, the Δϕ-effect also plays an important role to reproduce the asymmetric distribution of the NCP around the line connecting the disk center and the sunspot's center, in particular for the signals observed in the Fe I 15648 Å spectral line (Schlichenmaier and Collados, 2002). Additional improvements were implemented by Borrero et al. (2007), who incorporated a more realistic configuration of the flux tubes and surrounding magnetic fields. The azimuthal distribution and center-to-limb variation of NCP in Fe I 6302.5 Å and Fe I 15648 Å lines were again reproduced successfully. Borrero and Solanki (2010) further studied the effect of azimuthal/overturning convective motions in penumbral filaments on the NCP, and found that these convective motions are less significant than Evershed flow for the generation of net circular polarization.
Spatial distribution of the net circular polarization in sunspots reproduced by the embedded flux tube model and observed in Fe I 6302.5 Å and Fe I 15648 Å spectral lines (from Müller et al., 2002, 2006, reproduced by permission of the ESO).
High resolution observations (< 0.5") of the NCP in sunspots were first reported by Tritschler et al. (2007). They demonstrated the filamentary distribution of NCP in the penumbra, although the spatial correlation with the Evershed flow channels was not conclusive. Using Hinode/SP data Ichimoto et al. (2008b) found that, as expected, the NCP with the same sign as the umbral blue-lobe is associated with the Evershed flow channels in limb-side penumbra. Remarkably these authors also found that the Evershed flow channels in the disk-center-side penumbra show again the same sign of the NCP, whereas the opposite sign was observed in disk-center-side penumbra in the inter-Evershed flow channels (spines; see Section 3.2.1). This is indicated in Figure 40.
Upper panels: spatial distribution of NCP observed by SOT/SP in the Fe I 6302.5 Å spectral line for a sunspot close to disk center (Θ = 2.9°; same as sunspot in Figure 24). Lower panels: same as above but for a sunspot at Θ = 27.4°. In panels b) and d), the velocity contours are plotted over the orginal NCP distributions shown in panels a) and c), respectively. Color contours indicate velocities of −1.8 km s−1 (blue), −0.6 km s−1 (green), −0.6 km s−1 (pink), and 1.8 km s−1 (red), with negative and positive values indicating up- and downflows, respectively.
When the sunspot is close to disk center, the NCP in both upflow and downflow regions is associated with the same sign of the umbral blue-lobe (see panels a and b in Figure 40). These results appear to be inconsistent with the current explanation of the NCP by means of the Δγ-effect associated with the presence of the Evershed flow in the deep layers of the penumbra. It rather suggests a positive correlation between the magnetic field strength and the flow velocity as the cause of the NCP, and also serves as a strong evidence for the presence of gas flows in inter-Evershed flow channels (spines). The presence of plasma motions in spines inferred from the net circular polarization informs us that the current simple two component penumbral models consisting of Evershed flow channels and the spines (with no mass motion) are not compatible with the observations, and strongly suggests that there are dynamic features in penumbra that remain unresolved with the current observations.
It is also noticeable that the NCP associated with the upflow regions in disk-center-side penumbra is created by a hump in blue wing of Stokes V profiles (see panel a in Figure 36). Such Stokes V profiles obviously infer that the upflows in the inner penumbra posses a strong magnetic field compared with that in the surrounding penumbral atmosphere, and is apparently inconsistent with numerical simulations that infer upflowing gas with a weaker magnetic field (see Section 3.2.7). Similar conclusions have been reached by Tritschler et al. (2007) and Borrero and Solanki (2008) for the outer penumbra.
3.2.7 Unified picture and numerical simulations of the penumbra
As described in previous sections, two opposing ideas have been proposed to account for the penumbral uncombed structures (see Section 3.2.1): the embedded flux-tube model and the field-free gap model. The embedded flux-tube model, or the rising hot flux tube with the dynamic evolution of the flux tube, explains a number of observational aspects about the fine scale features of the penumbra such as the origin of Evershed flow, inward migration of penumbral grains, and asymmetric Stokes profiles observed in penumbra (see Section 3.2.6), but it faces difficulties when attempting to explain the heat transport to the penumbral surface (see Section 3.2.3). Some observational results support the finite vertical extension of the flux tube; i.e., Doppler shifts in multiple spectral lines formed at different height infer elevated Evershed flow channels (Ichimoto, 1987; Rimmele, 1995a; Stanchfield II et al., 1997), and some SIR-like inversions10 of Stokes profiles in spectral lines claim to have detected the lower boundary of the flux tube (Borrero et al., 2006). However, these results do not necessary provide a concrete evidence of the presence of thin and elevated flux tubes in the inner or middle penumbra and, actually, most observations suggest a monotonic increase in the magnitude of the Evershed flow towards the deeper photospheric layers, while finding no evidence for a lower discontinuity of the magnetic field in the observable layers (see also Figures 29 and 30). The flux-tube models by Borrero et al. (2007) and Ruiz Cobo and Bellot Rubio (2008) also suggest that penumbral flux tubes are not necessarily thin since the τ c = 1 level is located inside the tubes and, therefore, the lower boundary will not be visible. Thus, there is no definite observational evidence for the presence of a lower boundary in the flux tubes, at least in inner and middle penumbra, and the concept of narrow, elevated flux tubes embedded in the penumbra is not a scheme with strong observational bases.
In the field-free gap penumbral model, the gap is formed by a convecting hot and field-free gas protruding upward into the background oblique magnetic fields of the penumbra, and is supposed to be the region that harbors the Evershed flow. Contrary to the previous flux-tube model, the field-free gap penumbral model (Spruit and Scharmer, 2006; Scharmer and Spruit, 2006) has an advantage in explaining the heat transport to penumbral surface and, possibly, in explaining the twisting appearance of penumbral bright filaments. It does so thanks to the azimuthal/overturning convective pattern described in Section 3.2.3 (see also lower panel in Figure 31). However, it does not address the origin of the Evershed flow nor physical nature of the inner and outer ends of penumbral filaments. Furthermore, it is obvious from the highly Doppler-shifted polarization signals in spectral lines in penumbra that the flowing gas is not field-free. From the SIR inversion of a spectro-polarimetric data, Borrero and Solanki (2008) argued that the magnetic field strength in the Evershed flow channels (intraspines) increases with the depth below τ c = 0.1 level, and there exist strong magnetic fields near the continuum formation level that is not compatible with the field-free gap model.
Thus, both the embedded flux-tube model and the field-free gap model have their own advantages but also have considerable shortcomings. It would be natural, therefore, to modify these two penumbral models as follows; in the flux-tube model, we may consider vertically elongated flux tubes (or slabs) rather than the round cross section, and add an azimuthal/overturning convective flow pattern inside the penumbral filaments in addition to the horizontal Evershed flow. This would allow this model to transport sufficient energy through convection as to explain the penumbral brightness. In the field-free gap model we propose to add a rather strong, ≃ 1000 G (yet still weaker than in the spines), and nearly horizontal magnetic field inside the field-free gap. This has the consequence that the presence of rather strong horizontal magnetic field inside the gap allows this model to explain the observed net circular polarization in sunspots (see Section 3.2.6; see also Borrero and Solanki, 2010) as well as featuring a magnetized Evershed flow.
After the proposed modifications, we find that there are no fundamental differences between the two pictures as far as geometry of the inner penumbra is concerned. Note that in both the rising flux-tube model and the field-free gap model, the rising motions of hot gas in the Evershed flow channels are driven by the buoyancy force in the superadiabatic stratification of the penumbral atmosphere. Such unified picture has been already discussed by Scharmer et al. (2008), Borrero (2009), and Ichimoto (2010). In this concept, the Evershed flow could be understood as a consequence of the thermal convection with the gas flow deflected horizontally outward under a strong and inclined magnetic field.
Although the proposed modifications are observationally driven, recent 3D MHD simulations of sunspots (Heinemann et al., 2007; Rempel et al., 2009a,b; Kitiashvili et al., 2009; Rempel, 2011) present a magnetic field configuration that closely follows the inner structure for penumbral filaments that we have proposed above. The results from these simulations are able to reproduce the radial filamentary structure of the penumbra as seen in continuum images, the uncombed structure of the magnetic field, Evershed outflows along the filaments with a nearly horizontal magnetic field, and overturning convective motions in upwelling plumes. In addition, a detailed inspection of the numerical simulations provides great insights on the physical processes taking place in the penumbra. According to Rempel (2011), the Evershed flow is driven by vertical pressure forces in upflows that are deflected into the horizontal direction through the Lorentz-force generated by the horizontally stretched magnetic fields in flow channels, and the radial flow velocity reaches up to 8 km s−1 at the depth of τ c = 1 with a rapid decline toward the higher atmospheric layers.
Figure 41 shows a vertical cross section of the filaments in the inner penumbra from the MHD simulations by Rempel (2011). Remarkable features are the sharp enhancement of the radial component of the magnetic field around τ c = 1 level, where the upflow of convective gas protrudes and creates a narrow boundary layer with a concentration of a strong horizontal Lorenz force that acts as the engine that drives the horizontal Evershed flow. The connectivity, or the presence of the outer footpoints, of the magnetic field in the flowing channel are rather a consequence of the fast outflow than its cause as is assumed in the siphon-flow picture.
Vertical variation, according to the MHD simulations by Rempel (2011), of the physical parameters across a cut perpendicular to the penumbral filaments in the inner penumbra. Displayed are: a) radial and b) vertical components of the magnetic field vector, c) inclination of the magnetic field vector with respect to the vertical direction z. The bottom panels show: d) radial and e) vertical components of the velocity vector, and f) the energy conversion by the component, along the direction of the filaments, of the Lorentz force. The two solid lines indicate the τ c = 1 and τ c = 0.01 levels (from Rempel, 2011, reproduced by permission of the AAS).
Thus, the recent MHD simulations have begun to reproduce many details of fine scale dynamics and structure of the magnetic field observed in the penumbra. The most essential physical processes that form the penumbra take place near or beneath the τ c = 1. Therefore, the detection of the vertical gradients of the magnetic field and velocity vectors in the deep layers of the penumbra is an important target for future observations. Another important target for observations is to find the downflows or returning (inward) flows that could be associated with the azimuthal/overturning convection but still not have been detected yet at the spatial resolutions of Hinode (Franz and Schlichenmaier, 2009) and of the Swedish 1-m Solar Telescope (Bellot Rubio et al., 2010). This had been already discussed in Section 3.2.4.
The dynamic interaction between magnetic fields and granular convection around outer edge of penumbra is discussed as the formation mechanism of the interlocking comb structure of penumbra in the simulations by Thomas et al. (2002), Weiss et al. (2004), and Brummell et al. (2008). In this picture, the magnetic fields in the intraspines, which plunge below the solar surface near the edge of the spot (see Section 3.2.1), are created as a consequence of the submergence of the magnetic field lines due to the downward pumping-mechanism by small-scale granular convection outside the sunspot. Stochastic flows could be driven along such magnetic fields by the convective collapse caused by the the localized submergence of the magnetic fields. This scenario may capture an essential point of the dynamical convective process, but it is questionable if the entire penumbral structure is controlled by such processes taking place outside the sunspots.
1The symbol † indicates the transpose.
2Many of the codes that have been compared in these papers are publicly available to the community. Minimum energy method: http://www.cora.nwra.com/AMBIG; Non-Potential field calculation: http://sd-www.jhuapl.edu/FlareGenesis/Team/Manolis/codes/ambiguity_resolution/; and AZAM: http://www.csac.hao.ucar.edu/csac/visualize.jsp
3δ-sunspots are commonly defined as those where the umbra possesses two difference polarities.
4The unit vectors of the local reference frame are defined as follows: e ρ is the unit vector that is perpendicular to the tangential plane on the solar surface at the point of observation, while e α and e β are inside this plane (see Figure 42).
Sketch showing the geometry of the problem and the different reference frames employed in Section 1.3.2. The reference frame {e x , e y , e z } is centered at the Sun's center 'O'. The observed point at the Sun's surface is denoted by 'P'. The observer is located at the point 'E', denoting the Earth. The vector OP is parallel to e ρ , while EP is parallel to e l * .
5Note that the most general momentum equation would also include, in the right-hand term of Equation (17), the terms corresponding to the viscous forces: \(\mu _1 \nabla ^2 v\) and \([\mu _2 + (1/3)\mu _1 ]\nabla (\nabla v)\), where the coefficients μ1 and μ2 are often referred to as shear viscosity and bulk viscosity, respectively.
6According to Equation (12), z and τ c have opposite signs. This indicates that τ c decreases when z increases and, therefore, τ c decreases from the photosphere to the corona.
7We shall mention here that, in order to provide the values of the derivatives in terms of the geometrical height instead of the optical depth, we have assumed that hydrostatic equilibrium holds (see Section 1.3.3).
8Thin in this context means that the thin flux-tube approximation (Spruit, 1981) can be applied. This reduces the problem to a 1D problem: it does so by assuming that the flux tube's radius is much smaller than the pressure scale height.
9The elevation angle is defined as (1-ζ) (see Equation 10). Thus, small elevation angles correspond to horizontal magnetic fields (contained in the solar surface) and large elevation angles indicate magnetic fields that are rather vertical (perpendicular to the solar surface).
10SIR-like inversions refer to inversions of spectropolarimetric data where the physical parameters are allowed to vary with optical depth X(τ c ) (Equation 2). This is discussed in some detail in Section 1.3.
The authors are very grateful to K.D. Leka and R.H. Cameron, as well as to two anonymous referees, for carefully reading the manuscript and their many suggestions that helped improve this paper. This work has made extensive use of Hinode/SOT data. Hinode is a Japanese mission developed and launched by ISAS/JAXA, collaborating with NAOJ as a domestic partner, NASA and STFC (UK) as international partners. Scientific operation of the Hinode mission is conducted by the Hinode science team organized at ISAS/JAXA. This team mainly consists of scientists from institutes in the partner countries. Support for the post-launch operation is provided by JAXA and NAOJ (Japan), STFC (U.K.), NASA, ESA, and NSC (Norway). This research has made use of NASA's Astrophysics Data System. Finally, we would like to thank our colleagues and Astrophysical Journal, Astronomy and Astrophysics, Annual Reviews in Astronomy and Astrophysics, and Nature for granting us the rights to reproduce many previously published figures.
41116_2015_4_MOESM1_ESM.mpg (908 kb)
mpg-Movie (908.0 KB) Still from a movie showing The change of the synthetic emergent Stokes profiles (I, Q, U, V) when the magnetic field present in the solar plasma varies. The magnetic field vector is expressed in spherical coordinates: B moduli of the magnetic field vector, γ inclination of the magnetic field vector with respect to the observer's line-of-sight (z-axis in this case), and ϕ azimuth of the magnetic field vector in the plane perpendicular to the observer's line-of-sight. Results have been obtained under the Milne-Eddington approximation.
In this Appendix we describe the coordinate transformation that allows us to solve the 180°-ambiguity in the azimuth ϕ of the magnetic field vector (see Section 1.3.2). To that end let us define three different reference frames: {e x , e y , e z }, {e α , e β , e ρ }, and \(\left\{ {e_l^* ,e_x^* ,e_y^* } \right\}\). The first frame is a Cartesian frame centered at the Sun's center. The second frame is a curvilinear frame located at the point of observation P, where e ρ is the unit vector that is perpendicular to the plane tangential to the solar surface at this point. This plane contains the unit vectors e α and e β . This is the so-called local reference frame. The third reference frame is the observer's reference frame. It is centered at the point of observation P, with the unit vector \(e_l^* \) referring to the line-of-sight. As mentioned in Section 1.3 the inversion of the radiative transfer equation (1) yields the magnetic field vector in the observer's reference frame:
$$\begin{gathered} B=B\cos \gamma e_{1}^{*}+B\sin \gamma \cos \varphi e_{x}^{*}+B\sin \gamma \sin \gamma e_{y}^{*}= \hfill \\ \left( {\begin{array}{*{20}{c}} {B\cos \gamma } \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & {B\sin \gamma \cos \varphi } \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & {B\sin \gamma \sin \varphi } \hfill \\ \end{array}} \right)\left( {\begin{array}{*{20}{c}} {e_{1}^{*}} \hfill \\ {e_{x}^{*}} \hfill \\ {e_{y}^{*}} \hfill \\ \end{array}} \right)=\hat{\mathcal{B}}\left( {\begin{array}{*{20}{c}} {e_{1}^{*}} \hfill \\ {e_{x}^{*}} \hfill \\ {e_{y}^{*}} \hfill \\ \end{array}} \right) \hfill \\ \end{gathered}$$
. The key point to solve the 180°-ambiguity is to find the coordinates of the magnetic field vector into the local reference frame, where we will apply the condition that the magnetic field vector in a sunspot must spread radially outwards. In order to do so, we will obtain the coordinates of {e α , e β , e ρ } and \(\left\{ {e_l^* ,e_x^* ,e_y^* } \right\}\) into the reference frame at the Sun's center: {e x , e y , e z }.
Let us focus first on {e x , e y , e z }: the unit vectors defining a coordinate system at the Sun's center. The unit vector e z connects the Earth with the Sun's center, while e y corresponds to the proyection of the Sun's rotation axis on the plane perpendicular to e z (see Figure 42). In this coordinate system, the vector connecting the observer's and the Sun's center is OE (E means Earth):
$$OE = Ae_z ,$$
where A = 1.496 × 1011 m is the Astronomical Unit. Let us now suppose that we observe a point P on the solar surface. The coordinates of this point are usually given in the reference frame of the observer with the values X c and Y c (usually in arcsec). These two values refer to the angular distances of the point P measured from the center of the solar disk as seen from the Earth (see Figure 42). From this figure we can extract two triangles: \(\widehat{{\text{OEP}}_1}\) and \(\widehat{{\text{OEP}}_2}\) (see Figure 42) that can be employed to obtain the values of ± and β given X c and Y c . From the latter triangle we obtain:
$$\sin \alpha = \frac{A} {{R_ \odot }}\tan Y_c , $$
where R⊙ = 6.955 × 108 m is the Sun's radius. Once α has been obtained, we can employ the cosine theorem in the triangle \(\widehat{{\text{OEP}}_1}\) to determine as:
$$\cos \beta = \frac{{ - b \pm \sqrt {b^2 - 4ac} }} {{2a}}, $$
$$a = \frac{{R_ \odot ^2 \cos ^2 \alpha }} {{\sin ^2 X_c }}, $$
$$b = - 2R_ \odot A\cos \alpha ,$$
$$c = A^2 - \frac{{R_ \odot ^2 \cos ^2 \alpha }} {{\sin ^2 X_c }}. $$
Note that the obtained values for α and β will have to be modified depending upon the signs of X c and Y c (which determine the quadrant on the solar disk). Equation (29) shows that there are two possible values for β, however, one of them always corresponds to an angle |β| > π/2 and, therefore, can be neglected. It is also important to bear in mind that in Figure 42 the points labeled as P1 and P2 are the projections of the observed point on the solar surface, P, onto the planes y = 0 and z = 0 respectively. In fact, once that β and α are known, the coordinates of P in the reference frame of {e x , e y , e z } (the vector OP) can be written as:
$$OP = R_ \odot \cos \alpha \sin \beta e_x + R_ \odot \sin \alpha e_y + R_ \odot \cos \alpha \cos \beta e_z .$$
Another vector that will be useful later is the unit vector from the Earth to the observation point P. This can be written as follows:
$$PE = - R_ \odot \cos \alpha \sin \beta e_x - R_ \odot \sin \alpha e_y + [A - R_ \odot \cos \alpha \cos \beta ]e_z .$$
The next step is to realize that OP (Equation (33)) is parallel to the radial vector in the solar surface and, therefore, it is perpendicular to the tangential plane on the solar surface at the point of observation. This means that the vector e ρ (see Figure 42) belonging to the local frame can be obtained as:
$$\begin{gathered} e_\rho = \frac{{OP}} {{|OP|}} = \cos \alpha \sin \beta e_x + \sin \alpha e_y + \cos \alpha \cos \beta e_z \hfill \\ = e_{\rho x} e_x + e_{\rho y} e_y + e_{\rho z} e_z . \hfill \\ \end{gathered} $$
With this data we can now define the local reference frame as {e α , e β , e ρ }. This coordinate system is defined on the plane that is tangential to the solar surface at the point P, with e ρ being perpendicular to this plane, and e α and e β being contained in this plane. e α and e β are the tangential vectors to the β =cons and α =cons curves, respectively. The relation between the local reference frame and the one located at the Sun's center can be easily derived from Figure 42:
$$\left( {\begin{array}{*{20}{c}} {{{e}_{\alpha }}} \hfill \\ {{{e}_{\beta }}} \hfill \\ {{{e}_{\rho }}} \hfill \\ \end{array}} \right)=\left( {\begin{array}{*{20}{c}} {-\sin \alpha \sin \beta } \hfill & {\cos \alpha } \hfill & {-\sin \alpha \cos \beta } \hfill \\ {\cos \beta } \hfill & 0 \hfill & {-\sin \beta } \hfill \\ {\cos \alpha \sin \beta } \hfill & {\sin \alpha } \hfill & {\cos \alpha \cos \beta } \hfill \\ \end{array}} \right)\left( {\begin{array}{*{20}{c}} {{{e}_{x}}} \hfill \\ {{{e}_{y}}} \hfill \\ {{{e}_{z}}} \hfill \\ \end{array}} \right)=\hat{\mathcal{M}}\left( {\begin{array}{*{20}{c}} {{{e}_{x}}} \hfill \\ {{{e}_{y}}} \hfill \\ {{{e}_{z}}} \hfill \\ \end{array}} \right)$$
The third reference frame we have mentioned is the observer's reference frame: \(\left\{ {e_l^* ,e_x^* ,e_y^* } \right\}\). Note that the first of the unit vectors can be directly obtained (see Figure 42 and Equation (34)) as:
$$e_l^* = \frac{{PE}} {{|PE|}} = e_{lx}^* e_x + e_{ly}^* e_y + e_{lz}^* e_z . $$
The heliocentric angle Θ is defined as the angle between the normal vector to the solar surface at the point of observation P (aka e ρ ) and the line-of-sight (aka \(e_l^* \)). Therefore, the scalar product between these two already known vectors (Equations (34) and (35)) yields the heliocentric angle cos \(\Theta = e_\rho \cdot e_l^* \):
$$\cos \Theta = \frac{{A\cos \alpha \cos \beta - R_ \odot }} {{|PE|}}. $$
The other two vectors of this coordinate system, i.e., \(e_x^* \) and \(e_y^* \) must be perpendicular to the line-of-sight. The first one, \(e_x^* \), can be obtained from Equations (34), (35), and (37) as:
$$e_x^* = \frac{{e_\rho \times e_l^* }} {{|e_\rho \times e_l^* |}} = \frac{1} {{|e_\rho \times e_l^* |}}\left\{ {[e_{\rho y} e_z^* - e_{\rho z} e_{ly}^* ]e_x + [e_{\rho x} e_{ly}^* - e_{\rho y} e_{lx}^* ]e_x } \right\} = e_{xx}^* e_x + e_{xy}^* e_y + e_{xz}^* e_z . $$
Finally, the vector \(e_y^* \) can be obtained employing the following conditions:
$$\begin{gathered} \left\{ {\begin{array}{*{20}{c}} {e_{x}^{*}\cdot e_{y}^{*}=0} \hfill \\ {e_{1}^{*}\cdot e_{y}^{*}=0} \hfill \\ {\left| {e_{y}^{*}} \right|=1} \hfill \\ \end{array}} \right. \hfill \\ e_{y}^{*}=e_{{yx}}^{*}{{e}_{x}}+e_{{yy}}^{*}{{e}_{y}}+e_{{yz}}^{*}{{e}_{z}} \hfill \\ \end{gathered}$$
. This represents a system of three equations with three unknowns: \(e_{yx}^* ,e_{yy}^* ,e_{yz}^* \). These represent the proyections of the \(e_y^* \) unit vector on the base located at the Sun's center: {e x , e y , e z }. Solving the previous system of equations yields the following solutions:
$$e_{yz}^* = \left[ {\left( {\frac{{e_{yz}^* e_{ly}^* - e_{xy}^* e_{lz}^* }} {{e_{xy}^* e_{lx}^* - e_{xx}^* e_{ly}^* }}} \right)^2 + \left( {\frac{{e_{xz}^* e_{lx}^* - e_{xx}^* e_{lz}^* }} {{e_{xy}^* e_{lx}^* - e_{xx}^* e_{ly}^* }}} \right)^2 + 1} \right]^{ - \frac{1} {2}} , $$
$$e_{yx}^* = e_{yz}^* \left( {\frac{{e_{xz}^* e_{ly}^* - e_{xy}^* e_{lz}^* }} {{e_{xy}^* e_{lx}^* - e_{xx}^* e_{ly}^* }}} \right), $$
$$e_{yy}^* = - e_{yz}^* \left( {\frac{{e_{xz}^* e_{lx}^* - e_{xx}^* e_{lz}^* }} {{e_{xy}^* e_{lx}^* - e_{xx}^* e_{ly}^* }}} \right). $$
All the three components of \(e_y^* \) — Equations (41), (42), and (43) — can be obtained through Equations (37) and (39). We are, therefore, able to construct the matrix \(\hat{\mathcal{N}}\), which transforms the observer's reference frame into the reference frame at the Sun's center:
$$\left( {\begin{array}{*{20}{c}} {e_{1}^{*}} \hfill \\ {e_{x}^{*}} \hfill \\ {e_{y}^{*}} \hfill \\ \end{array}} \right)=\left( {\begin{array}{*{20}{c}} {e_{{lx}}^{*}} \hfill & {e_{{ly}}^{*}} \hfill & {e_{{lz}}^{*}} \hfill \\ {e_{{xx}}^{*}} \hfill & {e_{{xy}}^{*}} \hfill & {e_{{xz}}^{*}} \hfill \\ {e_{{yx}}^{*}} \hfill & {e_{{yy}}^{*}} \hfill & {e_{{yz}}^{*}} \hfill \\ \end{array}} \right)\left( {\begin{array}{*{20}{c}} {{{e}_{x}}} \hfill \\ {{{e}_{y}}} \hfill \\ {{{e}_{z}}} \hfill \\ \end{array}} \right)=\hat{\mathcal{N}}\left( {\begin{array}{*{20}{c}} {{{e}_{x}}} \hfill \\ {{{e}_{y}}} \hfill \\ {{{e}_{z}}} \hfill \\ \end{array}} \right)$$
. An important point to consider is that the local reference frame, {e α , e β , e ρ }, and the observer's reference frame, \(\left\{ {e_l^* ,e_x^* ,e_y^* } \right\}\), vary with the observed point P on the solar surface. This means that each of these coordinates systems must be recalculated for each point of an observed 2-dimensional map.
We can now express the magnetic field vector in Equation (26), in the local reference frame:
$$B=\underbrace{{\hat{\mathcal{B}}\left( {\begin{array}{*{20}{c}} {e_{1}^{*}} \hfill \\ {e_{x}^{*}} \hfill \\ {e_{y}^{*}} \hfill \\ \end{array}} \right)}}_{{Eq.\;\left( {26} \right)}}=\underbrace{{\hat{\mathcal{B}}\hat{\mathcal{N}}\left( {\begin{array}{*{20}{c}} {{{e}_{x}}} \hfill \\ {{{e}_{y}}} \hfill \\ {{{e}_{z}}} \hfill \\ \end{array}} \right)}}_{{Eq.\;\left( {44} \right)}}=\underbrace{{\hat{\mathcal{B}}\hat{\mathcal{N}}\,{{{\hat{\mathcal{M}}}}^{{-1}}}\left( {\begin{array}{*{20}{c}} {{{e}_{\alpha }}} \hfill \\ {{{e}_{\beta }}} \hfill \\ {{{e}_{\rho }}} \hfill \\ \end{array}} \right)}}_{{Eq.\;\left( {36} \right)}}={{B}_{\alpha }}{{e}_{\alpha }}+{{B}_{\beta }}{{e}_{\beta }}+{{B}_{\rho }}{{e}_{\rho }}$$
. Note that because of the 180°-ambiguity in the determination of ϕ, the two possible solutions in the magnetic field vector also exist (in an intricate manner) in the local reference frame {e α , e β , eρ}: B(ϕ) and B(ϕ + φ) (Equation (45)). In order to distinguish which one of the two is the correct one, we will consider that the magnetic field in a sunspot is mostly radial from the center of the sunspot. This means that we will take the solution that minimizes Equation (9). To evaluate that equation we already know the coordinates of the magnetic field vector B in the local reference frame. However, we must still find the coordinates, in this same reference frame, of the vector r which, as already mentioned in Section 1.3.2, is the vector that connects the center of the umbra (denoted by U) with the point P of observation:
$$r = OP - OU = R_ \odot [(\cos \alpha _p \sin \beta _p - \cos \alpha _u \sin \beta _u )e_x + (\sin \alpha _p - \sin \alpha _u )e_y + (\cos \alpha _p \cos \beta _p - \cos \alpha _u \cos \beta _u )e_z ]r_x e_y + r_z e_z ,$$
where we have distinguished between (α p ,β p ) and (α u ,β u ) to differentiate the coordinates of the point of observation P and the umbral center U, respectively. For convenience, we re-write Equation (46) as follows:
$$r=\underbrace{{\left( {\begin{array}{*{20}{c}} {{{r}_{z}}} \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & {{{r}_{y}}} \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & {{{r}_{z}}} \hfill \\ \end{array}} \right)}}_{{\hat{\mathcal{R}}}}\left( {\begin{array}{*{20}{c}} {{{e}_{x}}} \hfill \\ {{{e}_{y}}} \hfill \\ {{{e}_{z}}} \hfill \\ \end{array}} \right)=\hat{\mathcal{R}}\left( {\begin{array}{*{20}{c}} {{{e}_{x}}} \hfill \\ {{{e}_{y}}} \hfill \\ {{{e}_{z}}} \hfill \\ \end{array}} \right)$$
. Note that Equations (46) and (47) refer to the reference frame at the Sun's center. However, in order to calculate its scalar product with the magnetic field vector (Equation (9)) we need to express it in the local reference frame {e α , e β , e ρ } at the point of observation P:
$$r=\hat{\mathcal{R}}\left( {\begin{array}{*{20}{c}} {{{e}_{x}}} \hfill \\ {{{e}_{y}}} \hfill \\ {{{e}_{z}}} \hfill \\ \end{array}} \right)=\hat{\mathcal{R}}\,\underbrace{{{{{\hat{\mathcal{M}}}}^{{-1}}}\left( {\begin{array}{*{20}{c}} {{{e}_{\alpha }}} \hfill \\ {{{e}_{\beta }}} \hfill \\ {{{e}_{\rho }}} \hfill \\ \end{array}} \right)}}_{{Eq.\;\left( {36} \right)}}$$
, where the inverse matrix \({{\hat{\mathcal{M}}}^{{-1}}}\) (Equation (36)) must be obtained employing α p and β p (coordinates on the solar surface for the observed point P). Once B and r are known in the local reference frame (Equations (45) and (48), respectively), it is now possible to evaluate Equation (9). This allows us to determine which solution for the azimuth of the magnetic field, υ or υ + π, yields a magnetic field vector which is closer to be radially aligned.
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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
1.Kiepenheuer-Institut für SonnenphysikFreiburgGermany
2.Kwasan and Hida ObservatoriesKyoto UniversityYamashina, KyotoJapan
Borrero, J.M. & Ichimoto, K. Living Rev. Sol. Phys. (2011) 8: 4. https://doi.org/10.12942/lrsp-2011-4
First Online 09 September 2011
DOI https://doi.org/10.12942/lrsp-2011-4
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CommonCrawl
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Integral Test
If the individual terms of a series (in other words, the terms of the series' underlying sequence) do not converge to zero, then the series must diverge. This Special Purpose Rifle mount comes in QD and Bolt-On Models. The p-test gives a rough borderline of what is 'fast enough. See differential Differential, and Integration. 1 and Import Test Diag Located at $8000- in memory and use harness 251169-02. Viewed 4k times 3. sh integral-table the configuration file here, and the shell scripts ht5mjlatex and makejax. Integral Ad Science (IAS) is the global market leader in digital ad verification, offering technologies that drive high-quality advertising media. Serioes of this type are called p-series. You can use the links below to visit resources for AS/A level Further Mathematics and for Mechanics or Statistics for AS/A level Mathematics, and also STEP resources. (a) The general Riemann Sum is found using the formula:. An early form of this was discovered in India by Madhava of Sangamagramma in the 14th century. Each integral is worth 1 point. If the integral gives a finite value, then the series is convergent. There's rarely a right "one". This page was last edited on 5 October 2019, at 16:34. Learn exactly what happened in this chapter, scene, or section of Calculus BC: Series and what it means. Then generalize to define an entire class of series called p-series, and prove a theorem showing when they converge. The Integral Test Let \(f\left( x \right)\) be a function which is continuous, positive, and decreasing for all \(x\) in the range \(\left[ {1, + \infty } \right). The Integral Test (14 minutes, SV3 » 41 MB, H. Currently taught in over a dozen countries in North America, Europe, and Asia, ISP is a comprehensive approach to embodiment based on Western and Eastern psychology. If you searching to test Cadet 60 X 32 Air Whirlpool Bathtub With Hydro Massage System L Integral Apron And Right Hand Outlet By American Standard CmhNNrqIS 00975nlXPuKcp Cadet 60 X 32 Air Whirlpool Bathtub With Hydro Massage System L Integral Apron And Right Hand Outlet By American Standard CmhNNrqIS 00975nlXPuKcp price. Hence, if the improper integral from 1 to infinity of the series converges, then the series converges. The series diverges by the Integral Test. Weisstein 1999-05-26. For example, enter 3x+2=14 into the text box to get a step-by-step explanation of how to solve 3x+2=14. In CUDA C, why does the following. Definition of the definite integral as the limit of a Properties of the definite integral. If we did not pull out the negative sign, we would not be able to apply the integral test, because this test can only be applied when all terms in the series are positive. The Root Test Lecture Notes So far, we have learned how to use the limit comparison test to determine whether a series converges or diverges. This page will give you the numerical answer to an integral. (a)Verify that the integral test can be used to decide if this series converges and then use the integral test to show it converges. Dimensional Data (inches and [ mm ]) are Subject to Manufacturing Tolerances and Change Without Notice ® CF2982 Torque Set Closet Flange w/ Integral Test Cap TAG _____ Rev. On Screen Display. It doesn't matter whether we compute the two integrals on the left and then subtract or compute the single integral on the right. INTEGRAL_TEST can handle both unweighted and weighted integration schemes. The first thing that confused me is that I've yet to hear the term tail sum, so I still am unsure of what that means. Solution: The function is continuous, positive, decreasing function on [1,∞) so we use the Integral Test: Since is a convergent integral and so, by the Integral test, the series is convergent. Improper integral definition is - a definite integral whose region of integration is unbounded or includes a point at which the integrand is undefined or tends to infinity. (a) This is a case where you want to think about the. Water Analysis Laboratory Services Water, the source of life, the human body is made up of approximately 60% water – which is the major reason that we can survive weeks without food but only days without water. The harmonic series, which is a special case of p-series, is defined. (since C++11) Checks whether T is an integral type. Input LAG tester is a Diva exclusive add-on. You have 20 minutes to solve these 25 integrals. Indeed, since when , then we have Because is divergent (by the p-test), then the limit test implies that the integral is divergent. We'll end up using this corollary more than the integral test itself. Calculus II , Final (practice test) 9:00-12:00 noon, Friday, Dec. Comparing f(x) = [ ln(x) / x ] with g(x) = (1/x) also shows divergence. If we did not pull out the negative sign, we would not be able to apply the integral test, because this test can only be applied when all terms in the series are positive. It is also known as Maclaurin-Cauchy Test. _____ Problem 2: a) Determine the proper benchmark for using the Comparison Test to find whether. Find the region E for which the triple integral (1—212 —7y2 —2=2) dV is a maximum. Then use the Integral Test to determine the convergence or divergence of the series. On Screen Display. (Integral Test) Suppose is a series in which the terms are positive. Wolfram Problem Generator » Unlimited random practice problems and answers with built-in Step-by-step solutions. org dictionary, synonyms and antonyms. We can apply the integral test, as the function satisfies the conditions for the integral test. Of course, the key point is that the first few terms will not affect divergence or convergence - it is the ultimate behavior which counts and this is measured by the integral. Proofs are given in the appendix. Integrated Health Plan was acquired by MultiPlan in 2011, and the IHP Network is now part of MultiPlan's comprehensive suite of healthcare cost management solutions. Factor out a -1. If so, use the integral test to determine whether the series converges or diverges. The integral test tells us that if the improper integral is convergent (that is, it is equal to a finite number), then the infinite series is convergent. C-64 Diagnostic Rev x ROMs. It is the base class for the C++ type traits. • Available in a range of working pressures up to 15,000psi. sareen New member. It will diverge when x>1/2 or when x<-1/2. The integral test for convergence is a method used to test infinite series of non-negative terms for convergence. Then converges if and only if the improper integral converges. The winners of this round will be finalists in the main tournament. Proof - The Integral Test Contact Us If you are in need of technical support, have a question about advertising opportunities, or have a general question, please contact us by phone or submit a message through the form below. integral test. This site contains high school calculus video lessons from four experienced high school math teachers. In terms of area the Comparison Test makes a lot of sense. Vle Data Thermodynamic Consistency Integral Area Test. Perfect for acing essays, tests, and quizzes, as well as for writing lesson plans. Veitch 1 1 2 Z 1 1 1 x dx therefore, we have the following relation X1 n=1 1 n2 1 12 Z 1 1 1 x2 dx The natural question at this point is, does. Integral Test Remainder For a series that converges by the integral test , this is a quantity that measures how accurately the nth partial sum estimates the overall sum. As we discussed in lecture, even though the limit of the sequence of partial sums of a sequence by de nition determines whether a series converges or diverges, in practice we almost never actually nd the limit of a sequence. That is, if the improper integral converges then the series converges and if the improper integral diverges, the series diverges. It can be invaluable, whether you're familiar with the new content, or if you're teaching some aspects, such as mechanics or statistics, for the first time. The S90V blade sports a tall grind to deliver high-performance cutting action, with a smooth ball bearing pivot. Integral Test: Let Sum from n = 1 to INF {An} be a series. This is the nth term test for divergence. Uniform Continuity; Sequences and Series of Functions 6 8. For many positive series, the question of convergence for the series can be replaced by a question of convergence for a closely related integral. View more articles from The American Mathematical Monthly. The Integral Test: formally stated The integral test stated formally f(x) is a continuous, positive decreasing function in the interval [a,∞) and f(n)=u n then The P-series: sum of 1/n p. Attending to all four quadrants and recognizing various lines will give us a very broad view of what is involved in being human. Calculus: Integral Test and Estimates of Sums (English) UCI Math 2B: Single-Variable Calculus (Fall 2013) Lec 21. We will first do some questions that require you to use the integral test instead of p-series test. (since C++11) Checks whether T is an integral type. The two vent valves allow for 100% capture of vented or drained process, and simplified in-process calibration capability. You can also get a better visual and understanding of the function and area under the curve using our graphing tool. Uniform Continuity; Sequences and Series of Functions 6 8. Free online storage and sharing with Screencast. INTEGRAL TEST AND ESTIMATES OF SUMS 3 4. If this integral had diverged to infinity, we would have said the series diverged to infinity. Free math lessons and math homework help from basic math to algebra, geometry and beyond. Clearly fis positive, continuous, and decreasing on [1;1). Be able to evaluate a given line integral over a curve Cby rst parameterizing C. ) f is continuous. The idea is to take the general term as a function in terms of x, and then integrate it. The above conclusion is still valid. If you searching to test Cadet 60 X 32 Air Whirlpool Bathtub With Hydro Massage System L Integral Apron And Right Hand Outlet By American Standard CmhNNrqIS 00975nlXPuKcp Cadet 60 X 32 Air Whirlpool Bathtub With Hydro Massage System L Integral Apron And Right Hand Outlet By American Standard CmhNNrqIS 00975nlXPuKcp price. C-64 Diagnostic Rev 3. This labor-saving compression gasket is compatible with plastic, steel, no-hub, extra heavy and service weight cast iron soil pipe. In fact, by extending our work on X1 n=1 1 n and X1 n=1 1 n5 we use the integral test on any xed power of n: X1 n=1 1 np converges if p > 1 and diverges if p 1: 9. If the terms look like a function you know how to integrate, try it. Integral test. Improper integral diverges so the. Infinite Series: Integral Test For Convergence The integral test for convergence is a method used to test infinite series of non-negative terms for convergence. Why doesn't the integral test for convergence work on negative, increasing, and continuous functions? Mathematics I know the answer is probably somewhat obvious but if the test for convergence works for positive, decreasing, and continuous functions, why doesn't it also work for neg. 1), as long as this integral is well-defined for a. Problem 1 Evaluate the following integrals R dx x2+2x+5 Solution: 1 2 tan −1(x+1. (Hint: Use integration by parts to evaluate the integral. Designed to develop deep mathematical understanding and all the skills students need for their AS/A level studies and beyond. We focus on the decision-making process rather then on the mechanics of integration. 5 feet) in height attached to the tank may be filled with water to accomplish the 35 kPa (5 psig) test. Since the main contribution to the integral comes from a region of a point c at which the phase g(t) is stationary, (2) is called the stationary phase approximation. Solve the differential equation. The Integral Test The observations from the previous examples give us a new convergence test called the integral test : Integral Test Suppose that is a sequence, and suppose that is an eventually continuous, positive, and decreasing function with for all , where is an integer. Learn how it works in this video. Integral Building & Design is an active member of the USGBC and provides Green Rating verification services for the LEED for Homes Program. , ATHLET, RELAP5-3D, KORSAR, and TECH-M) and institutions performed on the. 190 Chapter 9 Applications of Integration It is clear from the figure that the area we want is the area under f minus the area under g, which is to say Z2 1 f(x)dx− Z2 1 g(x)dx = Z2 1 f(x)−g(x)dx. This is a team made up of mental health professionals who help adults and children having a mental health crisis. It is the base class for the C++ type traits. 9) Explain why the comparison test cannot be used to decide if the following series converge or diverge. 3, Positive Series: The Integral Test Homework: 9. Pharm, BCA, M. Hence we conclude that the given series is convergent precisely when p > 1. Introduction to the Integral Test by – Sean Turkington Activity overview This activity introduces the Integral Test for series convergence by utilizing the built-in CAS capabilities of the TI-Nspire CAS. This integral can be actually evaluated using partial fractions, but it is easier to answer this question using the Comparison test. Solution of Example 2. In terms of area the Comparison Test makes a lot of sense. For more about how to use the Integral Calculator, go to "Help" or take a look at the examples. However, to really master curl and the meaning of its components, you need to understand the basis of curl from the circulation that is captured by line integrals. If so, use the integral test to determine whether the series converges or diverges. 5 meters (11. However, to really master curl and the meaning of its components, you need to understand the basis of curl from the circulation that is captured by line integrals. Factor out a -1. What does integral mean? Proper usage and audio pronunciation (plus IPA phonetic transcription) of the word integral. Free Integration Online Practice Tests. So I'm doing a report on convergence tests in LaTeX, and would like to produce a diagram demonstrating the integral test for convergence. Integral Test The integral test provides a means to testing whether a series converges or diverges. Integral Test Lecture 22 Section 11. The Integral Test can be used on a infinite series provided the terms of the series are positive and decreasing. The qualifying round will consist of a written test of 25 integrals, for which participants will have 20 minutes. The Integral Map - Part 9 : Levels, States and Types of Development. The test is also called the Cauchy integral test or Maclaurin integral test. Includes full solutions and score reporting. If you're seeing this message, it means we're having trouble loading external resources on our website. Testing our code is (or should be) an integral part of our daily work. integral test - WordReference English dictionary, questions, discussion and forums. 358 8 Integral Operators That is, if f is a measurable function on R, then Lkf is the function defined by equation (8. The Integral Test The observations from the previous examples give us a new convergence test called the integral test : Integral Test Suppose that is a sequence, and suppose that is an eventually continuous, positive, and decreasing function with for all , where is an integer. Find the number of terms that you would need to ensure an estimate that is accurate to the rst 3 decimal. The integral test for convergence is a method used to test the infinite series of non-negative terms for convergence. To understand why, it is useful to understand the general idea of the proof of the integral test. Okay, the integral from the last step is a divergent integral and so by the Integral Test the series must also be a divergent series. Improper integral diverges so the. The Integral Test. 963938092e91). Definition of the definite integral as the limit of a Properties of the definite integral. NOTE: The Integral Test says that the Series and the Improper Integral either both converge or both diverge. In other words, the behavior of the improper. As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students. Let N be a natural number (non-negative number), and it is a monotonically decreasing function, then the function is defined as. We can get a pretty good intuition behind the formula for the components of the curl by just visualizing spinning spheres immersed in fluid. Paul's Online Notes. 5}\,\text{d}x$$ To solve this using Importance Sampling MC integration, one needs to select an importance pdf that is approximately the same as the function plot; My R code to solve the same is this :. _____ Problem 2: a) Determine the proper benchmark for using the Comparison Test to find whether. integral, the crack-tip opening displacement (CTOD) and the crack-tip opening angle (CTOA) from the basic concept, definition, to experimental estimation, test methods and ASTM standardizing practices. The Definite Integral Quiz Web resources available Questions This quiz tests the work covered in Lecture 21 and corresponds to Section 5. Learn vocabulary, terms, and more with flashcards, games, and other study tools. Answer to: Use the Integral Test to determine whether the series converges or diverges. The qualifying round will consist of a written test of 25 integrals, for which participants will have 20 minutes. Patheos Explore the world's faith through different perspectives on religion and spirituality! Patheos has the views of. The Integral Test The observations from the previous examples give us a new convergence test called the integral test : Integral Test Suppose that is a sequence, and suppose that is an eventually continuous, positive, and decreasing function with for all , where is an integer. Currently taught in over a dozen countries in North America, Europe, and Asia, ISP is a comprehensive approach to embodiment based on Western and Eastern psychology. Analogously, to calculate the area between two curves using horizontal elements, subtract the left function from the right function. Integral test. Ask Question Asked 4 years, 8 months ago. Best Answer: The integral test says that if An, 1/(2n+3) is positve on x>=1, decreasing, and continuous on x>=1, then the integral of the function and the series will both be the same: convergent or divergent. Remember that the integral of a constant is the constant times the integral. Attention is paid to guidelines on how to choose an appro-priate fracture parameter to characterize fracture toughness for the material of interest,. MATH 1D, WEEK 3 { THE RATIO TEST, INTEGRAL TEST, AND ABSOLUTE CONVERGENCE INSTRUCTOR: PADRAIC BARTLETT Abstract. The test is designed specifically to assess key skills needed for success in law school, including reading comprehension, analytical reasoning, and logical reasoning. QUADPACK is a FORTRAN90 library which estimates integrals using numerical quadrature, by Piessens, deDoncker-Kapenga, Ueberhuber, and Kahaner. ∑ n=0 ∞ ne−n the integral converges, so the series converges 2. Integral is the world's most advanced cloud-based eFX platform combining features for liquidity management, pricing, distribution and risk management. Indeed, since when , then we have Because is divergent (by the p-test), then the limit test implies that the integral is divergent. If f(x) is larger than g(x) then the area under f(x) must also be larger than the area under g(x). Chapter 3 • Integral Relations for a Control Volume 177 P3. Read "Commissioning of the ATLAS thermal-hydraulic integral test facility, Annals of Nuclear Energy" on DeepDyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. integral test - WordReference English dictionary, questions, discussion and forums. The integral calculator gives chance to count integrals of functions online free. Integral calculator This is a calculator which computes the definite and indefinite integrals (antiderivative) of a function with respect to a variable x. converges or diverges. So far, we have had to compute the limit of the sequence of the partial sums to determine if a series converges or diverges: X1 i=1 a i = lim n!1 s n: We now develop a powerful tool for detecting convergence or divergence of series whose terms are positive and decreasing, i. Convergence Tests for Infinite Series In this tutorial, we review some of the most common tests for the convergence of an infinite series $$ \sum_{k=0}^{\infty} a_k = a_0 + a_1 + a_2 + \cdots $$ The proofs or these tests are interesting, so we urge you to look them up in your calculus text. If ƒ is a function that is positive and decreasing for positive x , then the infinite series with n th term ƒ and the integral of ƒ from 1 to ∞ are either. 4 A fire hose has a 5-inch inside diameter and is flowing at 600 gal/min. The side entry sub is manufactured with box by pin connections to match the drill pipe by an integral 2" 1502 union side outlet. We use the integral test. Recognize a p-series and use the value of pto make a conclusion about the convergence of the series. This lightweight, premier manifold comes in a 2-, 3- and 5-valve configuration and allows instrument isolation, controlled venting and added protection from leaking to ensure measurement integrity. When the improper integral in convergent then we say that the function f(t) possesses a Laplace transform. Given a conservative vector eld, F, be able to nd a potential function fsuch that F = rf. Thread starter sareen; Start date Dec 2, 2009; S. Integral Somatic Psychology was developed by Raja Selvam, PhD. This ¿rst step focuses on developing a basic understandi. You will then be told whether the answer is correct or not. The integral test is another way to test to prove if a series converges or diverges. All qualified applicants will receive consideration for employment without regard to race, color, sex, sexual orientation, gender identity, religion, national origin, disability, veteran status, age, marital status, pregnancy, genetic information, or other legally protected status. assumptions: The terms are decreasing and positive and this is a known integral, so the series behaves the same as the integral Z 1 1 e 2x dx= lim b!1 Z b 1 e 2x dx= lim b!1 e 2x 2 b 1 = lim b!1 e 2 2 e 2 1 2 = 0 + e 2 2 So the series converges since the integral does. INTEGRAL TEST AND ESTIMATES OF SUMS 3 4. The series diverges by the Integral Test. Cauchy's test for convergence Explanation of Cauchy integral test Cauchy integral test | Article about Cauchy integral test by The Free Dictionary. integral test Consider a sequence ( a n ) = { a 0 , a 1 , a 2 , a 3 , … } and given M ∈ ℝ consider any monotonically nonincreasing function f : [ M , + ∞ ) → ℝ which extends the sequence, i. Integral includes features such as teacher forums and student tracking. Improper Integrals 5 7. 31 Word ID. The integral test is a test that allows us to relate the convergence of a series to that of an improper integral. 5 feet) in height attached to the tank may be filled with water to accomplish the 35 kPa (5 psig) test. √ Find the indefinite integral and check the result by differentiation. I went to the tutor lab on campus and. Variable substitution allows you to integrate when the Sum Rule, Constant Multiple Rule, and Power Rule don't work. There are routines for nonadaptive or adaptive integration, finite, semi-infinite or fully infinite integration regions, integrands with singularities, and integrands that include a factor of SIN(X) or COS(X). Recognizing these types will help you decide which tests or strategies will be most useful in finding. " (Matt Damon) teams up with iconoclastic test driver Ken. Perfect for acing essays, tests, and quizzes, as well as for writing lesson plans. The graph of a decreasing, positive function f is drawn, where f has domain [1,¥). This is easier than actually nding the sum of an in - nite series, which is possible only in special cases. The side entry sub is manufactured with box by pin connections to match the drill pipe by an integral 2" 1502 union side outlet. It provides two blocking valves, two test/vent valves, and one equalizing valve. assumptions: The terms are decreasing and positive and this is a known integral, so the series behaves the same as the integral Z 1 1 e 2x dx= lim b!1 Z b 1 e 2x dx= lim b!1 e 2x 2 b 1 = lim b!1 e 2 2 e 2 1 2 = 0 + e 2 2 So the series converges since the integral does. All qualified applicants will receive consideration for employment without regard to race, color, sex, sexual orientation, gender identity, religion, national origin, disability, veteran status, age, marital status, pregnancy, genetic information, or other legally protected status. Remember, you can always split the integral up into two integrals, one finite integral that contains the conditions where the Integral Test cannot be applied, and another infinite integral where you can apply the Integral Test on. Thus the series and the integral converge or diverge together. The Integral Test is a version of Riemann Sums for series, assuming that the partitions are of unit length (length = 1). For the integral test, when we say that f must be positive and decreasing, it is actually enough that f is EVENTUALLY always positive and decreasing. 3, Positive Series: The Integral Test Homework: 9. Desktop & mobile. Definite integral. You can also check your answers! Interactive graphs/plots help visualize and better understand the functions. The integral test helps us determine a series convergence by comparing it to an improper integral, which is something we already know how to find. Barriers to integration have included inadequate hardware and software, difficulties in securing sufficient funding, inadequate staff development, and deficiencies in planning. The integral test is a test that allows us to relate the convergence of a series to that of an improper integral. Okay, the integral from the last step is a divergent integral and so by the Integral Test the series must also be a divergent series. And therefore, we can evaluate the improper integral as a limit of the partial sums. 1), as long as this integral is well-defined for a. • Thestudent is asked to Continuum identify two tar-get words in context per line of text, with. WITH INTEGRAL TEST CAP The Zurn Neo-Loc Gasket with Integral Test Cap is a unique pipe connection designed to secure a drain fixture to the drain line. Use the integral test to determine whether or not the following series converge or diverge. Read "Commissioning of the ATLAS thermal-hydraulic integral test facility, Annals of Nuclear Energy" on DeepDyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. 2018 was an exceptional year at Integral Powertrain, two of our high-power density Integral e-Drive motors powered the record-breaking run at Pikes Peak in the Rocky Mountains where the times for electric vehicles and the overall speed record were smashed. Another way to say that is that you can pass a constant through the integral sign. For continuous function f that is positive and decreasing for x ≥ 1 with f (n) = a n, n = 1, 2, 3,. The Integral Test: Suppose that f is a continuous, positive, decreasing function on 1, and let a n f n. For many positive series, the question of convergence for the series can be replaced by a question of convergence for a closely related integral. The Integral Test takes an infinite series and transforms it into an Improper Integral. ∫xndx=n+1xn+1+K (This is true as long as n≠−1) For the integral of a power of x: add 1 to the power and divide by the new number. 963938092e91). Date: 08/24/09 C. Integral test. We also offer a five-valve integral manifold with a metering pattern for Natural Gas installations. We focus on the decision-making process rather then on the mechanics of integration. We note that for positive x,. Remainder Estimate for the Integral Test. JPMCC Frankfurt 2019 The Fintegral team from the Frankfurt office took part at the JPMCC in Frankfurt. Derivatives and the Mean Value Theorem 3 4. In other words, as long as f is always positive and decreasing after a certain point, we can use the Integral Test. Best Answer: If the function is continuous, and is positive and decreasing for sufficiently large x, you can apply the Integral Test. The Integral Test: formally stated The integral test stated formally f(x) is a continuous, positive decreasing function in the interval [a,∞) and f(n)=u n then The P-series: sum of 1/n p. c) Find an upper bound for the value of the improper integral. Declare a variable u , set it equal to an algebraic expression that appears in the integral, and then substitute u for this expression in the integral. If the Integral Test can be applied to the series, enter CONV if it converges or DIV if it diverges. I have never learned about the Laurent series expansion I'm supposed to be using the Remainder Estimate for Integral Test I am assuming that I am not using it correctly since I am not getting the same answer. If the integral test cannot be applied to the series, enter NA. Where f(x) = Ax Here An = 1 / n^p. SAMPLE QUESTIONS FOR PRELIMINARY REAL ANALYSIS EXAM VERSION 2. the hair test uses the DNA of the hair follicle to find the cause and the best holistic health treatment of your problems. THE INTEGRAL TEST Chapter 11. Next, we see that Z 1 2 1 x(lnx)2 dx = lim b!1 Z b 2 1 x(lnx)2 dx = lim b!1 1 lnx b 2! = lim b!1 1 ln2 1 lnb = 1 ln2: Since the improper integral converges, so does the corresponding series. Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube. Best Answer: The integral test says that if An, 1/(2n+3) is positve on x>=1, decreasing, and continuous on x>=1, then the integral of the function and the series will both be the same: convergent or divergent. It is a test which is useful for +1 and +2 exams. The integral calculator gives chance to count integrals of functions online free. In terms of area the Comparison Test makes a lot of sense. Step (2) There are only three criteria we need to check before applying the integral test. Test (Look at dominating erms). On Screen Display. Partial hint. All qualified applicants will receive consideration for employment without regard to race, color, sex, sexual orientation, gender identity, religion, national origin, disability, veteran status, age, marital status, pregnancy, genetic information, or other legally protected status. These are the lecture notes from week 3 of Ma1d, the Caltech. The study is based on an experimental test executed in the BETHSY integral test facility, identified as 9. Interactivity in the cloud. MCAS Results. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture. In other words, as long as f is always positive and decreasing after a certain point, we can use the Integral Test. The Integral Test. 3: The Integral Test and Estimates of Sums The Integral Test. Our mission is to provide a free, world-class education to anyone, anywhere. ; Detailed Solution:Here For problems 10 { 20, determine if the series converges or diverges by applying the Divergence Test, Integral Test, or noting that the series is a p-series. a) dt t+t3 0 ⌠3 ⌡ ⎮ Improper at x = 0, where the t is much larger than the t3, so this "looks like" the p-type dt 0t ⌠1 ⌡ ⎮ which converges since p < 1. The EU-wide stress test will be launched in January 2020 while results of the test will be published by the end of July. The integral test helps us determine a series convergence by comparing it to an improper integral, which is something we already know how to find. Concept of antiderivatives Basic integration formulas Integration by substitution (use of identities, change Distance and velocity from acceleration with initial nos i f out Sl o. Chapter 3 • Integral Relations for a Control Volume 177 P3. 3 Group Work For sequences EXPLAIN or SHOW WORK documenting why your answer is correct: (a) does it. Use the algebraic properties of series. Perfect for acing essays, tests, and quizzes, as well as for writing lesson plans. = + infinity. The series can be compared to an integral to establish convergence or divergence. "Note on Cauchy's Integral Test" is an article from The American Mathematical Monthly, Volume 18. There's rarely a right "one". UCL Architecture students choose Integral LED for Summer Showcase. All three books are great, my personal favorite is the flrst one. Maclaurin-Cauchy integral test. There are of course certain conditions needed to apply the integral test. Each integral is worth. Hence, if the improper integral from 1 to infinity of the series converges, then the series converges. Perfect for acing essays, tests, and quizzes, as well as for writing lesson plans. Solution: The function is continuous, positive, decreasing function on [1,∞) so we use the Integral Test: Since is a convergent integral and so, by the Integral test, the series is convergent. Moreover, we have is convergent if and only if p <1 is convergent if and only if p >1 In the next pages, we will see how some easy tests will help in deciding whether an improper integral is convergent or divergent. At this site is a good illustration of the Riemann sums being taken and how they actually relate to the infinite series (this is one of the nice examples when the proof is essentially the picture). Use the integral test to compare the series to an appropriate improper integral, then use a comparison test to show the integral converges or diverges and conclude whether the initial series converges or diverges. "Use the integral test to find upper and lower bounds for the following tail sum: ___< Σ(1/n(n 2 +7)) < ___ (With the bounds of the series being "n=3" to infinity. GeoGebra Math Apps Get our free online math tools for graphing, geometry, 3D, and more!. Find more Mathematics widgets in Wolfram|Alpha. Integrally definition, of, relating to, or belonging as a part of the whole; constituent or component: integral parts. 963938092e91). Then converges if and only if the improper integral converges. To understand why, it is useful to understand the general idea of the proof of the integral test. Uniform Continuity; Sequences and Series of Functions 6 8. Integral Software redefines the way light is measured and revolutionizes how lighting measurements and control are synthesized into useful information. 10 hours ago · Mixing sound and music is integral to several period Oscar contenders, including "The Irishman," "Rocketman," and "Ford v Ferrari. ∫xndx=n+1xn+1+K (This is true as long as n≠−1) For the integral of a power of x: add 1 to the power and divide by the new number. We will first do some questions that require you to use the integral test instead of p-series test. Since the integral converges (this we remember, see Properties and Examples), by the Comparison test, also the integral in question converges. , ATHLET, RELAP5-3D, KORSAR, and TECH-M) and institutions performed on the. RE: Integral test for the Casio Fx-CG500 (09-02-2017 01:34 AM) Gamo Wrote: How about this integral I try on HP Prime App, HP 15C App for Android and HP 15C Virtual Calculator for Windows PC. A summary of The Integral Test in 's Calculus BC: Series. Later the Furnishing Division was inaugurated on 2nd October, 1962 and the production of fully furnished coaches steadily increased over the years. Integrals Tutorial General Integrals Test on iLrn Advanced Integrals Test. 3 The Integral Test and p-Series 619 p-Series and Harmonic Series In the remainder of this section, you will investigate a second type of series that has a simple arithmetic test for convergence or divergence.
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Computational study of the effects of arterial bifurcation on the temperature distribution during cryosurgery
Yong-Chang Zheng1 na1,
Jun-Hong Wu2 na1,
Zhi-Zhu He3 &
Shao-Jiong Huang3
BioMedical Engineering OnLine volume 17, Article number: 4 (2018) Cite this article
Thermally significant blood flows into locally cooled diseased tissues and warm them during cryosurgery so that the iceball is often hard to cover the whole diseased volume. This paper is aimed at investigating the effects of large arterial bifurcation on the temperature distribution during cryosurgery through simulation method.
A parametric geometry model is introduced to construct a close-to-real arterial bifurcation. The three-dimensional transient conjugate heat transfer between bifurcated artery and solid tissues with phase change during cryosurgery is performed by finite volume method.
The discussion was then made on the effects of the relative position between cryoprobe and artery bifurcation, the inlet velocity of root artery and the layout of multiple cryoprobes on the temperature distribution and iceball evolution. The results show that the thermal interaction between blood flow and iceball growth near bifurcation is considerable complex. The thermal effects of bifurcation could modulate the iceball morphology, severely weaken its freezing volume and prevent the blood vessel from being frozen.
The present work is expected to be valuable in optimizing cryosurgery scheme of the situation that the bifurcated artery is embedded into the disease tissue.
Cryosurgery has been widely demonstrated as an excellent therapeutic approach to destroy the diseased tissues (such as tumor) due to its minimal invasiveness [1,2,3]. In order to freeze and kill the target cells, largely decreasing the temperature of target tissues to below 0 °C is necessary. The temperature distribution in target tissue is the major parameter to evaluate the cryosurgical output. Accordingly, it is very important to accurately control the temperature distribution in order to enhance the destruction of diseased tissues and avoid the injury of healthy tissues.
Blood flow could remarkably affect the temperature distributions during freezing, especially with the presence of large blood vessels (larger than 0.5 mm in diameter) [4]. Thermally significant blood flows into locally cooled diseased tissues and warm them during cryosurgery so that the iceball is often hard to cover the whole diseased volume. Recently the study on thermal behavior of large blood vessels has attracted much attention in the cryosurgery area. Deng et al. [5] adopted infrared thermography system to investigate the thermal effects of large vessels during cryosurgery based on simulation and animal experiments. The results showed that the heating nature of the flowing blood in the large vessels could produce steep temperature gradients and inadequate cooling to the frozen tissues. Numerical simulation method was also used to study the thermal effects of the large blood vessel. In the early research [6], the blood velocity in the line-like vessel was simply considered as constant. The cylindrical cryoprobes and blood vessel were both approximated as cubes. Such simplification would induce numerical errors to the temperature distribution [7]. Latterly, a finite element method based on FEM commercial software was introduced to obtain a more accurate numerical solution of temperature field near blood vessel [8]. However, compared with the investigation for the thermal effects of large vessels on the hyperthermia ablation [9,10,11,12,13,14], study on the similar issues in cryosurgery is still rare.
Compared to single large line-like artery, the bifurcated artery (such as inlet artery in liver) has complicated structure so that the complex blood flow distribution would induce heterogeneous heat transfer surrounding artery bifurcation. In recent years, some investigations focus on the cooling effects of artery bifurcation on the hyperthermia ablation, such as microwave ablation [15, 16] and radio-frequency ablation [17]. However, few investigation contributes to the warm effects of artery bifurcation on the cryosurgery. The details of three-dimensional transient temperature distribution during cryosurgery surrounding artery bifurcation, which is more real and complex, are still unknown. The phase-change heat transfer combined with the convective mechanism of blood flow in arterial bifurcation during cryosurgery would be more complex than that during hyperthermia ablation. Tremendous contributions are needed to probe into such important issues, which are very useful for optimizing cryosurgery scheme of the situation that the bifurcated artery is embedded into the disease tissues. The aim of this paper is to disclose the detailed temperature characteristics of cryosurgery in the vicinity of an arterial bifurcation.
In present study, a parametric geometry model [18] is introduced to construct a close-to-real bifurcated artery. The steady state blood flow is considered here. The three-dimensional transient conjugate heat transfer between bifurcated artery and solid tissues during cryosurgery is performed by finite volume method. Then the relative position between cryoprobe and artery bifurcation, the changes of root vessel inlet velocity and the layout of multiple cryoprobes would be considered to investigate the detailed temperature distribution between bifurcation and cryoprobe. The mechanism of heat transfer between artery bifurcation and solid tissues, the iceball edge evolution would be then revealed to evaluate the thermal effects of large artery bifurcation on the tissue temperature distribution and iceball growth during cryosurgery.
Geometric and mathematical model
Geometric model
The arterial bifurcation is constructed by a parametric model [18], which is represented by two curved tubes with the same size attached to a straight root tube. It could reproduce effectively realistic configuration of arterial bifurcation. The thickness of artery wall is omitted here. The detailed geometry of symmetric arterial bifurcation is illustrated in Fig. 1a, where L1 = 20 mm is the distance from root vessel inlet to bifurcation, L2 = 80 mm is the height of daughter vessel along vertical direction, D0 = 10 mm is the diameter of root vessel [17], \({\text{D}}_{1} = {\text{D}}_{2} = \sqrt[3]{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}}{\text{D}}_{0}\) 7.94 mm is the diameter of daughter vessels, Ф = 60° is the angle of bifurcation. The arterial bifurcation is embedded in cylindrical solid tissues with diameter D t = 120 mm and length L t = L1 + L2 = 100 mm (see Fig. 1b). Single cylindrical cryoprobe with diameter Dp = 4 mm inserts the solid tissue near the bifurcation site along horizontal direction. The distance from cryoprobe center to bifurcation point denotes as Ld (see Fig. 1a). The probe shape is illustrated in Fig. 1c. The cryoprobe has two parts, adiabatic shaft and active tip with length Lp = 22 mm, which could lead to an extremely low temperature (such as − 196 °C of liquid nitrogen).
Schematic diagram of the bifurcated artery (a) and computational field (b). The shape of probe is illustrated in (c)
Governing equation
During cryosurgery, the solid tissues consist of unfrozen region (T > T u ), transition field (T l ≤ T ≤ T u ) and frozen area (T < T l ) according to temperature distribution, where T l and T u denote respectively the lower and upper phase transition temperature of solid tissues. Then the Pennes bioheat transfer equation for whole solid tissues based on the effective heat capacity method [6] could be written as
$$\hat{C}\frac{{\partial T_{t} }}{\partial t}\, = \,\nabla \cdot \left( {\hat{\kappa }\nabla T_{t} } \right)\, + \,\hat{\omega }_{cb} C_{b} \left( {T_{cb} \, - \,T_{t} } \right)\, + \,\hat{Q}_{m} ,$$
T t is the temperature of solid tissues and T cb denotes the temperature of blood perfusion from capillary vessel. C b is the heat capacity of blood. In addition, the detailed expression of \(\hat{C}\) the effective tissue heat capacity, \(\hat{\kappa }\) the effective tissue thermal conductivity, \(\hat{\omega }_{cb}\) the effective blood perfusion and \(\hat{Q}_{m}\) the effective tissue metabolic heat generation are determined by solid tissue state, which is given by
$$\hat{C}\, = \,\left\{ \begin{aligned} C_{f} \quad \quad T\, < \,T_{l} \hfill \\ \frac{{Q_{f} }}{{T_{u} \, - \,T_{l} }}\, + \,\left[ {C_{f} + \left( {C_{u} \, - \,C_{f} } \right)\frac{{T\, - \,T_{l} }}{{T_{u} \, - \,T_{l} }}} \right]\quad T_{l} \, \le \,T\, \le \,T_{u} \hfill \\ C_{t} \quad \quad T\, > \,T_{u} \hfill \\ \end{aligned} \right.,$$
$$\hat{\kappa }\, = \,\left\{ \begin{aligned} & \kappa_{f} \quad \quad T\, < \,T_{l} \\ & \kappa_{f} \, + \,\left( {\kappa_{u} \, - \,\kappa_{f} } \right)\frac{{T\, - \,T_{l} }}{{T_{u} \, - \,T_{l} }}\quad T_{l} \, \le \,T\, \le \,T_{u} \\ & \kappa_{t} \quad \quad T\, > \,T_{u} \\ \end{aligned} \right.,$$
$$\hat{\omega }_{b} \, = \,\left\{ \begin{aligned} & 0\quad \quad T\, < \,T_{l} \hfill \\ & 0\quad \quad T_{l} \, \le \,T\, \le \,T_{u} \hfill \\ & \omega_{b} \quad \quad T\, > \,T_{u} \hfill \\ \end{aligned} \right.,\quad \hat{Q}_{m} \, = \,\left\{ \begin{aligned} & 0\quad \quad T\, < \,T_{l} \hfill \\ & 0\quad \quad T_{l} \, \le \,T\, \le \,T_{u} \hfill \\ & Q_{m} \quad \quad T\, > \,T_{u} \hfill \\ \end{aligned} \right.,$$
The thermo-physical parameters involved in the above equation could be found in [6]:C t = C b = 3.6 MJ/m3K, C f = 1.8 MJ/m3K, ω cb = 5 × 10−4/s, Q m = 420 J/m3, κ f = 2 W/mK, κ t = κ b = 0.5 W/mK, L f = 250 MJ/m3, Tcb = 37 °C, T u = − 1 °C and T l = − 8 °C. Blood viscosity is μ = 2.5 × 10−3 Ns/m2.
The blood is modeled as incompressible and Newtonian flow, which is governed by Navier–Stokes equation. In the present study, the blood in bifurcated artery domain has large undercooling and could not be frozen due to flow. Then the blood velocity here is considered as steady and independent on the temperature. The energy equation for artery domain could be given as
$$C_{b} \left( {\frac{{\partial T_{b} }}{\partial t}\, + \,V \cdot \nabla T_{b} } \right)\, = \,\nabla \cdot \left( {\kappa_{b} \nabla T_{b} } \right)$$
where T b denotes the temperature in bifurcated artery. The heat transfer in solid tissues described by Eq. (1) and bifurcated artery presented in Eq. (5) could be combined by the interface between them, which is a typical conjugate heat transfer problem.
Boundary condition
For solid tissues boundary, the thermal condition is considered as adiabatic wall boundary. The inlet velocity of the root artery adopts parabolic velocity profile u(r) = 2 V[1 − (2r/D0)2], where V is average velocity and r the radial position. The inlet temperature of the root artery assumes as constant T0 = 37 °C. The outlet boundary of the daughter bifurcation artery adopts the pressure boundary, where the reference pressure is set as zero. The interface between solid tissues and cryoprobes tip is set as T p = − 196 °C.
Finite volume analysis tool
Firstly, the geometrical model of bifurcated artery constructed from software Solidworks imports into mesh generation software Gambit. Mesh size in artery domains is 1 mm so that about 10,000 tetrahedral elements and 20,000 nodes are obtained. The nonuniform mesh is used to map the solid tissues. The mesh size 0.6 mm is used for the field close to active cryoprobe, where the large temperature gradient happens during cryosurgery. The mesh size 2 mm is used for the field far away from the active cryoprobe. Thus there are about 1,500,000 tetrahedral elements and 270,000 nodes for the whole solid tissues. After creating the mesh, Gambit generates an input file to the finite volume package Fluent 6.3, which has been extensively utilized to address a variety of practical engineering problems nowadays. The blood flow and temperature analysis are solved according to SIMPLE and second order up wind algorithms.
For every numerical simulation case, the steady blood velocity profile is firstly performed and considered as input parameters to solve the transient temperature field evolution. The second order upwind scheme is used to discretize the convectional term in Eq. (5). The first order implicit scheme is used for time evolution. The nonuniform time step is applied to improve the stability of numerical simulation. At the beginning of freezing, the temperature distribution near cryoprobe has large gradient and changes violently from 37 to − 196 °C in a short time. Thus the small time step is applied to improve the stability of numerical simulation.
$$\Delta t\, = \,\left\{ \begin{aligned} & 0.005\,s\quad \,\,\,t\, < \,1\,s \\ & 0.01\,s\quad \quad 1\,s\, \le \,t\, \le \,5\,s \\ & 0.05\,s\quad \quad 5\,s\, < \,t\, \le \,100\,s \\ & 0.1\,s\quad \quad \,\,\,100\,s\, < \,t \\ \end{aligned} \right.$$
Steady blood velocity distribution
Before solving the temperature evolution during cryosurgery, we have to obtain the velocity distribution in artery. Figure 2 shows the contours of blood velocity magnitude in x = 0, y = 0 and different height cross-section planes of bifurcated artery. A fully developed laminar blood with parabola velocity profile (V = 0.20 m/s) flows into the root vessel. With the increase of cross-section area, the amplitude of velocity parabola velocity distribution becomes weaker. When the blood flow approaches to bifurcation, the velocity is close to zero, which lead to a low velocity field near bifurcation. A boundary layer formed near the inside wall downstream from the flow bifurcation, with the maximum axial velocity just outside this domain. Another low velocity field happens at the outer wall of two daughter vessels inlet segment, where the blood flow presents the recirculation pattern. The blood flow entering into daughter vessels presents skewed structure toward the inner wall and develops finally a similar parabola profile. One can clearly find from Fig. 2 that compared to the simple blood flow with parabola pattern in single large line-like artery, the flow pattern in the bifurcation presents the complicated structure. This complexity would induce the inhomogeneous convective heat transfer between solid tissues and arterial bifurcation. The heat transfer would be enhanced in high velocity and weaken in low velocity fields.
Contours of blood velocity magnitude in the planar plane: x = 0, y = 0 and different height cross-section planes of bifurcated artery, where V0 = 0.20 m/s
Thermal interaction between iceball evolution and arterial bifurcation
Firstly, a case with small distance Ld = 20 mm between cryoprobe and bifurcation point is considered here. Figure 3 shows the temperature distribution in the plane x = 0, y = 0 and z = − 30 mm of bifurcated artery (Fig. 3a) and the iceball pattern view from x axis direction and temperature contours in the plane x = 0 (Fig. 3b) at freezing time t = 20 min. The iceball boundary in present study is defined as iso-surface of T = 0 °C. In order to focus on the temperature variation in the whole bifurcated artery, the contour interval is here set as (32.5, 37.5 °C) (Fig. 3a). In fact, the lowest temperature for artery domains, which happens on the inner wall of branching artery closest to cryoprobe, could reach 5 °C. This temperature could not decrease with freezing time evolution. It indicates that the warm blood flow prevents the artery from being frozen. However, one should note that the blood in the artery may be frozen for a shorter distance Ld or a smaller inlet velocity of the root artery. The temperature in the vicinity of bifurcation point keeps at a relatively high level about 35 °C. From Fig. 3a, one can see that the bifurcated artery domains close to cryoprobe has a lower temperature distribution so that the following catchment area would be cooling. From Fig. 3b, we can see that the iceball stops evolution when encountering the arterial bifurcation. The large temperature gradient between iceball and artery wall is helpful to convective heat transfer which would enhance the thermal effects of blood flow. Different from line-like artery, the arterial bifurcation would lead to a more irregular iceball, which may make the target tissue frozen incompletely.
The temperature distribution in the plane x = 0, y = 0 and z = − 30 mm of bifurcated artery (a) and the iceball pattern view from x axis direction and temperature contours in the plane x = 0 (b) at freezing time t = 20 min (Dp = 4 mm) position for Ld = 20 mm
Figure 4 shows the iceball pattern view from z axis direction and temperature contours in the center plane of cryoprobe (Dp = 4 mm) for its different positions, Ld = 20 mm (Fig. 4a), Ld = 30 mm (Fig. 4b) and Ld = 40 mm (Fig. 4c) at freezing time t = 20 min. It is easy to find from Fig. 4 that the intensity of thermal interaction between iceball and artery bifurcation becomes weaker with increase of the relative distance between cryoprobe and bifurcation. The shorter distance between cryoprobe and artery would advance their thermal interaction during freezing. The iceball formation induced by cryoprobe has a limited volume. Thus the thermal effects from warm blood flow on the iceball would disappear when their distance is beyond available range. Clinical practice for cryosurgery has demonstrated that the most serious cryoinjury is achieved when the target tissue undergo below so-called lethal temperature, which is considered currently as in the range of − 50 to − 40 °C [3]. Table 1 presents the volume of iceball and lethal domains (where the temperature is below − 40 °C) for different cryoprobe positions. Both the volume of iceball and lethal domain increase when the cryoprobe is far away from arterial bifurcation.
The iceball pattern view from z axis direction and temperature contours in the cryoprobe center plane for different cryoprobe (Dp = 4 mm) positions, a Ld = 20 mm, b Ld = 30 mm, c Ld = 40 mm, at freezing time t = 20 min
Table 1 The iceball volume (T = 0 °C) and the lethal volume (T = − 40 °C) for different cryoprobe positions at freezing time t = 20 min
Figure 5 records the temperature evolution of the point (located on the center-line of cylinder) with a distance 12 mm from cryoprobe centerline for different cryoprobe (Dp = 4 mm) positions Ld = 20 mm, Ld = 30 mm and Ld = 40 mm. The temperature remains at a high level for Ld = 20 mm due to strong thermal effects of warm blood flow. At the beginning of the freezing process, the temperature of the recorded point has the consistent cooling rate for each position. Such consistence would keep longer time with increase of Ld. The reason is that at freezing initial stage a large distance between the iceball and artery leads to a weak thermal interaction. Thus the temperature of the recorded point close to cryoprobe is mainly determined by iceball at initial freezing stage. Figure 6 shows the total heat flux evolution of bifurcated artery surface for different cryoprobe (Dp = 4 mm) positions. With increase of Ld, the total heat flux of bifurcated artery surface significantly decrease. This is mainly because the longer distance weaken the thermal effects of the blood flow.
The temperature evolution of the point (located on the line Ld) with distance 12 mm from cryoprobe centerline for different cryoprobe (Dp = 4 mm) positions Ld = 20 mm, Ld = 30 mm and Ld = 40 mm
The total heat flux evolution of bifurcated artery surface for different cryoprobe (Dp = 4 mm) positions Ld = 20 mm, Ld = 30 mm and Ld = 40 mm
In order to evaluate the effects of arterial bifurcation on the iceball growth, different inlet velocities of artery root are considered. Figure 7 represents the temperature distribution of the line: ([− 60 60], 0, − 50) mm for different inlet average velocities V = 0.01, 0.05, 0.10, 0.20 and 0.30 m/s with the same cryoprobe position Ld = 30 mm and freezing time t = 20 min. Figure 8 illustrates the total heat flux evolution of bifurcated artery surface for different inlet average velocities. Both Figs. 7 and 8 indicate that the smaller velocity would induces more weak thermal effects of arterial bifurcation. In addition, with inlet velocity increase, the artery thermal effects become flats, such as cases V = 0.20 and 0.30 m/s. The results are also demonstrated by evaluating both the iceball and lethal volumes for different inlet velocity cases (shown in Table 2).
The temperature distribution of the line: ([− 60 60], 0, − 50) mm for different inlet velocity V = 0.01, 0.05, 0.10, 0.20 and 0.30 m/s with cryoprobe position Ld = 30 mm and freezing time t = 20 min
The total heat flux evolution of bifurcated artery surface for different inlet velocity V = 0.01, 0.05, 0.10, 0.20 and 0.30 m/s with cryoprobe position Ld = 30 mm
Table 2 The iceball volume (T = 0 °C) and the lethal volume (T = − 40 °C) for different inlet velocity with the same cryoprobe position Ld = 30 mm and freezing time t = 20 min
For cryosurgery practice, except for killing tumor cells, another important task is to prevent organs and large vessels from being damaged. Table 3 lists the treatment time of cryosurgery when the minimum temperature of the bifurcated artery surface approaches to the freezing temperature (T = 0 °C) with different cryoprobe positions and inlet velocities of artery root. As can be seen from Table 3, different cryoprobe positions and inlet velocities of artery root have strong impacts on the treatment time of cryosurgery. When Ld equals to 10 mm and 15 mm, the treatment time of cryosurgery should not exceed 65 s. And when the large cryoprobe distance is adopted, the bifurcated artery could avoid the injury owing to the thermal effects of blood flow.
Table 3 The treatment time of cryosurgery when the minimum temperature of the bifurcated artery surface approaches to the freezing temperature (T = 0 °C) according to different cryoprobe positions and inlet velocities of artery root
Single cryoprobe often fails to overcome the large and irregular diseased tissues. Here, three cryoprobes array is used to freeze large fields. Three cryoprobes with diameter Dp = 2 mm and active length Lp = 20 mm are distributed uniformly on the circumference with radius 20 mm. Its center locates on the center-line of cylinder with Ld = 30 mm. Figure 9 shows that the iceball view from x direction and y direction for three cryoprobes at different freezing time. Three single iceballs are formed at initial cooling stage and combined fast to a large ingle iceball under a high-strength freezing. It can be seen from Fig. 9 that the three iceball induced by different cryoprobes have different growth rates due to varied cryoprobe positions and the inhomogeneous convective heat transfer between solid tissue and arterial bifurcation.
Fig 9
The iceball view form x direction and y direction for three cryoprobes with freezing evolution: a t = 5 min, b t = 15 min, c t = 25 min, other parameters Dp = 2 mm, where V = 0.20 m/s
Figure 10 shows the iceball volume and lethal volume for three cryoprobes cryosurgery with freezing time evolution. Both iceball and lethal volumes' growth rate is high at initial freezing stage and becomes lower at later freezing stage. In fact, this growth rate would approach zero after a longer freezing time, while it could not be observed in present short freezing time.
Fig 10
The iceball volume a and lethal volume b for three-cryoprobe cryosurgery for freezing time evolution
In summary, the present paper has adopted three dimensional numerical simulation method to investigate the thermal effects of arterial bifurcation on temperature responses during cryosurgery based on single and multiple cryoprobe system. We have investigated in detail that the blood velocity distribution in arterial bifurcation and its effects on the iceball growth. The results indicate that complex blood velocity distribution could induce the inhomogeneous convective heat transfer between solid tissue and arterial bifurcation. Thus the iceball near arterial bifurcation presents strong irregular geometry. The blood flow of bifurcated artery has significant heating effects on the target freezing domains. It is also noteworthy that the artery wall is easily suffering from cold injury, which should be paid by special attention. In order to protect the artery wall, nanoparticle and external fields [19, 20] could be applied to enhance heat transfer near artery wall with low temperature.
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YCZ and JHW performed the computations, analyzed the data, and wrote the draft paper. SJH analyzed the data and wrote the draft paper. ZZH conceived and designed this study. All authors read and approved the final manuscript.
This work is supported by the National Natural Science Foundation of China under Grant Nos. 51476181 and 61675236.
The authors declare that this study does not involve human subjects, human material and human data.
Yong-Chang Zheng and Jun-Hong Wu contributed equally to this work
Department of Liver Surgery, Peking Union Medical College Hospital, Chinese Academy of Medical Sciences and Peking Union Medical College, Beijing, 100730, China
Yong-Chang Zheng
College of Environmental and Energy Engineering, Beijing University of Civil Engineering and Architecture, Beijing, 100044, China
Jun-Hong Wu
Vehicle Engineering, College of Engineering, China Agricultural University, Beijing, 100083, China
Zhi-Zhu He & Shao-Jiong Huang
Zhi-Zhu He
Shao-Jiong Huang
Correspondence to Zhi-Zhu He or Shao-Jiong Huang.
Zheng, YC., Wu, JH., He, ZZ. et al. Computational study of the effects of arterial bifurcation on the temperature distribution during cryosurgery. BioMed Eng OnLine 17, 4 (2018). https://doi.org/10.1186/s12938-018-0438-z
Bioheat transfer
Numerical simulation
Arterial bifurcation
Blood thermal effect
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CommonCrawl
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On the generic construction of identity-based signatures with additional properties
On cycle-free lattices with high rate label codes
Amin Sakzad 1, and Mohammad-Reza Sadeghi 1,
Department of Mathematics and Computer Science, Amirkabir University of Technology, No. 424, Hafez Avenue, Tehran 15914, Iran, Iran
Received October 2009 Revised June 2010 Published November 2010
Etzion et al. have shown that high rate codes based on cycle-free Tanner graphs have minimum distance at most $2$. This result was extended by Sadeghi et al. to a small class of lattices based on Construction $D'$ only. In this paper, we prove a key theorem which relates the minimum distance of every lattice to the minimum distance of its label code. Then, using this powerful tool along with some new bounds on minimum distance of cycle-free group codes, we generalize those results to a large class of lattices here called RPS and PFP lattices. More importantly, we show that this class of cycle-free lattices are not so good in the view of coding gain.
Keywords: group code, Tanner graph, label code., Lattice.
Mathematics Subject Classification: Primary: 11H31, 94B75; Secondary: 11H7.
Citation: Amin Sakzad, Mohammad-Reza Sadeghi. On cycle-free lattices with high rate label codes. Advances in Mathematics of Communications, 2010, 4 (4) : 441-452. doi: 10.3934/amc.2010.4.441
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Amin Sakzad Mohammad-Reza Sadeghi
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CommonCrawl
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Results for 'R. T. Rathod'
Critical Examination on the Problem of Our Knowledge of Other Minds.R. T. Rathod - forthcoming - Indian Philosophical Quarterly.details
The philosophical problem of knowledge of other minds is rational justifiction. this paper covers n malcolm, h h price, j mill, strawson, hamshire, l wittgenstein and a j ayer's controversial thought. philosophical scepticism holds that it is logically impossible to know mental experiences. "i know, i have a pain." how do i know that other people also can have similar pain? it provides as ideal knowledge of mental events. when i say, "i have a pain," i can have certain knowledge (...) of this because i have conclusive individual evidence for it. (shrink)
The Problem of Other Minds in Philosophy of Mind
The T-Schema is Not a Logical Truth.R. T. Cook - 2012 - Analysis 72 (2):231-239.details
It is shown that the logical truth of instances of the T-schema is incompatible with the formal nature of logical truth. In particular, since the formality of logical truth entails that the set of logical truths is closed under substitution, the logical truth of T-schema instances entails that all sentences are logical truths.
Deflationism about Truth in Philosophy of Language
The End of the Timeless God.R. T. Mullins - 2016 - Oxford University Press UK.details
The End of the Timeless God considers two approaches to the philosophy of time, presentism and eternalism. It is often held that God cannot be timeless if presentism is true, but can be if eternalism is true. R. T. Mullins draws on recent work in the philosophy of time as well as the work of classical Christian thinkers such as Augustine, Anselm, and Aquinas to contend that the Christian God cannot be timeless in either case.
Divine Eternity in Philosophy of Religion
Divine Immutability in Philosophy of Religion
Divine Simplicity in Philosophy of Religion
Incarnation in Philosophy of Religion
Philosophy of Time, Misc in Metaphysics
Theories of Omniscience in Philosophy of Religion
What is Wrong with Cantor's Diagonal Argument?R. T. Brady & P. A. Rush - 2008 - Logique Et Analyse 51 (1):185-219..details
We first consider the entailment logic MC, based on meaning containment, which contains neither the Law of Excluded Middle (LEM) nor the Disjunctive Syllogism (DS). We then argue that the DS may be assumed at least on a similar basis as the assumption of the LEM, which is then justified over a finite domain or for a recursive property over an infinite domain. In the latter case, use is made of Mathematical Induction. We then show that an instance of the (...) LEM is intrumental in the proof of Cantor's Theorem, and we then argue that this is based on a more general form than can be reasonably justified. We briefly consider the impact of our approach on arithmetic and naive set theory, and compare it with intuitionist mathematics and briefly with recursive mathematics. Our "Four Basic Logical Issues" paper would provide useful background, the current paper being an application of the some of the ideas in it. (shrink)
Logical Consequence and Entailment in Logic and Philosophy of Logic
Divine Temporality, the Trinity, and the Charge of Arianism.R. T. Mullins - 2016 - Journal of Analytic Theology 4:267-290.details
Divine temporality is all the rage in certain theological circles today. Some even suggesting that the doctrine of the Trinity entails divine temporality. While I find this claim a bit strong, I do think that divine temporality can be quite useful for developing a robust model of the Trinity. However, not everyone agrees with this. Paul Helm has offered an objection to the so-called Oxford school of divine temporality based on the Christian doctrine of the Trinity. He has argued that (...) this form of divine temporality entails Arianism. In other words, divine temporality suffers from an inadequate doctrine of the Trinity. In this paper I shall first articulate the so-called Oxford school of divine temporality. From there I shall develop some of the Oxford school's theological benefits that help flesh out the doctrine of the Trinity, and assuage the charge of Arianism. Then I shall offer an examination and refutation of the Arian charge to divine temporality in order to show that the divine temporalist can maintain a robust Trinitarian theology. (shrink)
The Trinity in Philosophy of Religion
Hasker on the Divine Processions of the Trinitarian Persons.R. T. Mullins - 2017 - European Journal for Philosophy of Religion 9 (4):181-216.details
Within contemporary evangelical theology, a peculiar controversy has been brewing over the past few decades with regard to the doctrine of the Trinity. A good number of prominent evangelical theologians and philosophers are rejecting the doctrine of divine processions within the eternal life of the Trinity. In William Hasker's recent Metaphysics and the Tri-Personal God, Hasker laments this rejection and seeks to offer a defense of this doctrine. This paper shall seek to accomplish a few things. In section I, I (...) shall first set the stage for a proper understanding of the discussion. Section II will articulate the basic Trinitarian desiderata that must be satisfied by any model of the doctrine of the Trinity. This will help one understand the debate between Hasker and the procession deniers. Section III will offer an articulation of what the doctrine of divine processions teaches. Section IV will examine Hasker's defense of the doctrine point by point. I shall argue that his defense of the doctrine of the divine processions fails. (shrink)
The Nature and Limits of Authority.R. T. DeGEORGE - 1985details
Divine Perfection and Creation.R. T. Mullins - 2016 - Heythrop Journal 57 (1):122-134.details
Proclus (c.412-485) once offered an argument that Christians took to stand against the Christian doctrine of creation ex nihilo based on the eternity of the world and God's perfection. John Philoponus (c.490-570) objected to this on various grounds. Part of this discussion can shed light on contemporary issues in philosophical theology on divine perfection and creation. First I will examine Proclus' dilemma and John Philoponus' response. I will argue that Philoponus' fails to rebut Proclus' dilemma. The problem is that presentism (...) is incompatible with divine simplicity, timelessness, and a strong doctrine of immutability. From there I will look at how this discussion bears on contemporary understandings of divine perfection and creation, and argue that there are at least two possible ways contemporary philosophical theologians can try to get around the dilemma. One option is to adopt four-dimensional eternalism and maintain the traditional account of the divine perfections. I argue that this option suffers from difficulties that are not compatible with Christian belief. The other option is to keep presentism and modify the divine perfections. I argue that this option is possible and preferable since our understanding of the divine perfections must be modified in light of divine revelation and the incarnation. (shrink)
Presentism in Metaphysics
God and Emotion.R. T. Mullins - 2020 - Cambridge University Press.details
An introductory exploration on the nature of emotions, and examination of some of the critical issues surrounding the emotional life of God as they relate to happiness, empathy, love, and moral judgments. Covering the different criteria used in the debate between impassibility and passibility, readers can begin to think about which emotions can be predicated of God and which cannot.
The Aloneness Argument Against Classical Theism.Joseph C. Schmid & R. T. Mullins - forthcoming - Religious Studies:1-19.details
We argue that there is a conflict among classical theism's commitments to divine simplicity, divine creative freedom, and omniscience. We start by defining key terms for the debate related to classical theism. Then we articulate a new argument, the Aloneness Argument, aiming to establish a conflict among these attributes. In broad outline, the argument proceeds as follows. Under classical theism, it's possible that God exists without anything apart from Him. Any knowledge God has in such a world would be wholly (...) intrinsic. But there are contingent truths in every world, including the world in which God exists alone. So, it's possible that God (given His omniscience) contingently has wholly intrinsic knowledge. But whatever is contingent and wholly intrinsic is an accident. So, God possibly has an accident. This is incompatible with classical theism. Finally, we consider and rebut several objections. (shrink)
Beyond Liberalism.R. T. Allen - 1999 - Tradition and Discovery 26 (1):16-18.details
This is a brief response to S. Jacob's review of Beyond Liberalism.
Liberalism in Social and Political Philosophy
Kit Fine, The Limits of Abstraction Oxford, Clarendon Press, 2002, Cloth 18.99/US $25.00 ISBN: 0-19-924618-1. [REVIEW]R. T. Cook - 2004 - British Journal for the Philosophy of Science 55 (4):791-800.details
Critical Notice of The Limits of abstraction by Kit Fine, Oxford: Clarendon Press, 2002, pp.216. ISBN 9780191567261.
Mathematical Neo-Fregeanism in Philosophy of Mathematics
Flint's 'Molinism and the Incarnation' is Still Too Radical — A Rejoinder to Flint.R. T. Mullins - 2017 - Journal of Analytic Theology 5:515-532.details
I greatly appreciate Thomas Flint's reply to my paper, "Flint's 'Molinism and the Incarnation' is too Radical." In my original paper I argue that the Christology and eschatology of Flint's paper "Molinism and the Incarnation" is too radical to be considered orthodox. I consider it an honor that a senior scholar, such as Flint, would concern himself with my work in the first place. In this response to Flint's reply I will explain why I still find Flint's Christology and eschatology (...) to be too radical. Below I shall attempt to address various issues raised by Flint in his reply. (shrink)
Heaven and Hell in Philosophy of Religion
The Cognitive Functions of Emotion.R. T. Allen - 2000 - Appraisal 3:38.details
Emotions in Philosophy of Mind
Theories of Emotion in Philosophy of Mind
Completeness Proofs for RM3 and BN4.R. T. Brady - 1982 - Logique Et Analyse 25:9-32.details
'Because I Say So!' Some Limitations Upon the Rationalisation of Authority.R. T. Allen - 1987 - Journal of Philosophy of Education 21 (1):15–24.details
Philosophy of Education in Philosophy of Social Science
The Reality of Responses to Fiction.R. T. Allen - 1986 - British Journal of Aesthetics 26 (1):64-68.details
Fiction, Misc in Aesthetics
The Mechanism of Cavitation in Magnesium During Creep.R. T. Ratcliffe & G. W. Greenwood - 1965 - Philosophical Magazine 12 (115):59-69.details
The Meaning of Life and Education.R. T. Allen - 1991 - Journal of Philosophy of Education 25 (1):47–58.details
The Meaning of Life in Value Theory, Miscellaneous
How 'Decent' Is a Decent Minimum of Health Care?R. T. Meulen - 2011 - Journal of Medicine and Philosophy 36 (6):612-623.details
This article tries to analyze the meaning of a decent minimum of health care, by confronting the idea of decent care with the concept of justice. Following the ideas of Margalith about a decent society, the article argues that a just minimum of care is not necessarily a decent minimum. The way this minimum is provided can still humiliate individuals, even if the end result is the best possible distribution of the goods as seen from the viewpoint of justice. This (...) analysis is combined with an analysis from the perspective of solidarity, particularly of reflective solidarity, as a way to develop decent care, which is care that does not humiliate individuals and maintains their dignity. (shrink)
Biomedical Ethics in Applied Ethics
Public Health in Applied Ethics
Genetics and the Origin of the Species.R. T. Eddison - 1954 - Philosophy of Science 21 (3):272-272.details
Genetics in Philosophy of Biology
In Pursuit of the Functions of theWnt Family of Developmental Regulators: Insights fromXenopus Laevis.R. T. Moon - 1993 - Bioessays 15 (2):91-97.details
Biological Sciences in Natural Sciences
Jeanine Diller and Asa Kasher, Eds., Models of God and Alternative Ultimate Realities.R. T. Mullins - 2014 - Journal of Analytic Theology 2:288-293.details
Rational Autonomy: The Destruction of Freedom.R. T. ALlen - 1982 - Journal of Philosophy of Education 16 (2):199–207.details
Rational Autonomy: The Destruction of Freedom.R. T. ALlen - 1982 - Philosophy of Education 16 (2):199-207.details
Surprise as a Factor in the von Restorff Effect.R. T. Green - 1956 - Journal of Experimental Psychology 52 (5):340.details
The Moral Dimension of Children's and Adolescents' Conceptualisation of Tolerance to Human Diversity.R. T. Witenberg - 2007 - Journal of Moral Education 36 (4):433-451.details
This study examined the kinds of justifications children and adolescents used to support tolerant and intolerant judgements about human diversity. For the tolerant responses, three main belief categories emerged, based on the beliefs that others should be treated fairly , empathetically and that reason/logic ought to govern judgements . Fairness emerged as the most used belief to support tolerant judgements and the most commonly used combination of beliefs was found to be fairness/empathy, linking tolerance to moral reasoning, rules and values. (...) Specifically noticeable was that 6-7-year-olds appealed to fairness more often in comparison to the 11-12 and 15-16-year-olds. Older students used a larger repertoire of beliefs to support tolerance, indicating developing cognitive maturity. There was also a tendency for females to appeal to fairness/empathy more often than males. The major constraint to positive tolerance was not prejudice toward the target groups but the adolescents' beliefs in freedom of speech as a democratic right, pointing to a conflict in values between tolerance and other human rights. (shrink)
Toleration in Applied Ethics in Social and Political Philosophy
Is Coming to Believe in God Reasonable or Unreasonable?R. T. Herbert - 1991 - Faith and Philosophy 8 (1):36-50.details
Arguments Against Theism in Philosophy of Religion
The Number of Gods in Philosophy of Religion
On Not Understanding Prayer.R. T. Allen - 1971 - Sophia 10 (3):1-7.details
Prayer in Philosophy of Religion
Learning Without Awareness of What is Being Learned or Intent to Learn It.E. L. Thorndike & R. T. Rock - 1934 - Journal of Experimental Psychology 17 (1):1.details
Conscious and Unconscious Learning in Philosophy of Cognitive Science
Unconscious and Conscious Processes in Philosophy of Cognitive Science
The Metaphysics of Natural Right in Spinoza.John R. T. Grey - forthcoming - Oxford Studies in Early Modern Philosophy 10.details
In the Tractatus Theologico-Politicus (TTP), Spinoza argues that an individual's natural right extends as far as their power. Subsequently, in the Tractatus Politicus (TP), he offers a revised argument for the same conclusion. Here I offer an account of the reasons for the revision. In both arguments, an individual's natural right derives from God's natural right. However, the TTP argument hinges on the claim that each individual is part of the whole of nature (totius naturae), and for this reason inherits (...) part of the natural right of that whole. Using several analogous cases from the Ethics, I show that this form of argument from division is not compatible with Spinoza's considered metaphysical views. The revised argument, by contrast, avoids the pitfalls of his earlier efforts. It also better reveals the deep roots by which the monistic metaphysics of the Ethics feeds into Spinoza's conception of natural right. (shrink)
Spinoza: Metaphysics, Misc in 17th/18th Century Philosophy
Spinoza: Natural Law in 17th/18th Century Philosophy
Spinoza: Rights in 17th/18th Century Philosophy
New Consecution Calculi for R→T.Katalin Bimbó & J. Michael Dunn - 2012 - Notre Dame Journal of Formal Logic 53 (4):491-509.details
The implicational fragment of the logic of relevant implication, $R_{\to}$ is one of the oldest relevance logics and in 1959 was shown by Kripke to be decidable. The proof is based on $LR_{\to}$ , a Gentzen-style calculus. In this paper, we add the truth constant $\mathbf{t}$ to $LR_{\to}$ , but more importantly we show how to reshape the sequent calculus as a consecution calculus containing a binary structural connective, in which permutation is replaced by two structural rules that involve $\mathbf{t}$ (...) . This calculus, $LT_\to^{\text{\textcircled{$\mathbf{t}$}}}$ , extends the consecution calculus $LT_{\to}^{\mathbf{t}}$ formalizing the implicational fragment of ticket entailment . We introduce two other new calculi as alternative formulations of $R_{\to}^{\mathbf{t}}$ . For each new calculus, we prove the cut theorem as well as the equivalence to the original Hilbert-style axiomatization of $R_{\to}^{\mathbf{t}}$ . These results serve as a basis for our positive solution to the long open problem of the decidability of $T_{\to}$ , which we present in another paper. (shrink)
Nonclassical Logics in Logic and Philosophy of Logic
Proof Theory in Logic and Philosophy of Logic
Scepticism and Neoplatonism.R. T. Wallis - 1987 - In Wolfgang Haase (ed.), Philosophie, Wissenschaften, Technik. Philosophie. De Gruyter. pp. 911-954.details
$35.00 used $428.92 new $559.00 from Amazon (collection) Amazon page
Particle Size Distribution.R. T. DeHoff & F. N. Rhines - 1968 - In Robert T. DeHoff & Frederick N. Rhines (eds.), Quantitative Microscopy. New York: Mcgraw-Hill.details
Quantum Mechanics in Philosophy of Physical Science
$29.99 used (collection) Amazon page
When Loyalty No Harm Meant.R. T. Allen - 1989 - Review of Metaphysics 43 (2):281 - 294.details
LOYALTY HAS NOT HAD A BAD PRESS, but, as far as Anglo-Saxon philosophy is concerned, very little press. It has merited entries in the Encyclopaedia of Religion and Ethics and the Encyclopedia of Philosophy, and a short one in Macquarrie's A Dictionary of Christian Ethics. Of course, there is also Josiah Royce's The Philosophy of Loyalty. I propose to argue that these discussions of loyalty tend to assimilate it to faithfulness to a promise, and so omit what is distinctive of (...) it. I shall also argue that the likely reason for this distortion is the modern view of man as a self-defining subject who has autonomously to invent his own law or way, for the universe presents none to him. (shrink)
Virtues and Vices in Normative Ethics
The Meaning of Life and Education.R. T. Allen - 1991 - Philosophy of Education 25 (1):47-58.details
Flint's 'Molinism and the Incarnation' is Too Radical.R. T. Mullins - 2015 - Journal of Analytic Theology 3:109-123.details
In a series of papers, Thomas P. Flint has posited that God the Son could become incarnate in any human person as long as certain conditions are met (Flint 2001a, 2001b). In a recent paper, he has argued that all saved human persons will one day become incarnated by the Son (Flint 2011). Flint claims that this is motivated by a combination of Molinism and orthodox Christology. I shall argue that this is unmotivated because it is condemned by orthodox Christology. (...) Flint has unknowingly articulated a version of the heresy called Origenism that is condemned by the Fifth Ecumenical Council. After arguing that Flint's account is unmotivated because it is condemned, I shall offer some reflections and prolegomena on the relationship between contemporary analytic theology and the ecumenical creeds. (shrink)
The Arousal and Expression of Emotion by Music.R. T. Allen - 1990 - British Journal of Aesthetics 30 (1):57-61.details
Aesthetics and Emotions in Aesthetics
Music and Emotion in Aesthetics
Doing Hard Time: Is God the Prisoner of the Oldest Dimension?R. T. Mullins - 2014 - Journal of Analytic Theology 2:160-185.details
In this paper I shall consider an objection to divine temporality called "The Prisoner of Time" objection. I shall begin by distinguishing divine timelessness from divine temporality in order to clear up common misunderstandings and caricatures of divine temporality. From there I shall examine the prisoner of time objection and explain why the prisoner of time objection fails to be a problem for the Christian divine temporalist.
The Pursuit of Magnetic Shadows: The Formal-Empirical Dipole Field of Early-Modern Geomagnetism.Art R. T. Jonkers - 2008 - Centaurus 50 (3):254-289.details
The State and Civil Society as Objects of Aesthetic Appreciation.R. T. Allen - 1976 - British Journal of Aesthetics 16 (3):237-242.details
Civil Society in Social and Political Philosophy
Topics in Aesthetics in Aesthetics
Supererogation Revised.R. T. Allen - 1981 - Sophia 20 (2):5-11.details
Religious Topics in Philosophy of Religion
Supererogation in Normative Ethics
Studies in Arabic Philosophy.R. T. Blackwood - 1970 - Philosophy East and West 20 (2):199-201.details
Governance by Emotion.R. T. Allen - 1991 - Journal of the British Society for Phenomenology 22 (2):15-29.details
Notes on Dares and Dictys.R. T. Clark - 1914 - Classical Quarterly 8 (01):17-.details
C. i., p. 2, 12 dicit Peliae regi se eo uelle ire si uires sociique non deessent. Pelias … Argum … iussit … nauim aedificaret.Considering the next sentence read perhaps n a u e s for uires.C. ii., p. 3, 25. Graeci aduentare nauibus. mittit ad portam.M reads nauibus uti. May this conceal e t i t a ? cf. p. II , 2. For change of tense cf. opening lines of C. iii.
Classics in Arts and Humanities
Patient Self-Determination Act: An African American Perspective.R. T. Tucker - 1994 - Cambridge Quarterly of Healthcare Ethics 3 (3):417.details
African and African-American Philosophy in African/Africana Philosophy
African-American Philosophy in African/Africana Philosophy
Topics in African-American Philosophy in African/Africana Philosophy
A Metaphysics for the Future, by R.E. Allinson.R. T. Allen - 2002 - Journal of the British Society for Phenomenology 33 (1):110-111.details
The Relativity of Simultaneity.R. T. Herbert - 1987 - Philosophy 62 (242):455 - 471.details
In connection with the special theory of relativity, Einstein made use of a now familiar thought experiment1 involving two lightning flashes, a railway train, and an embankment. Whether he used it merely to help explain the theory to others or whether it played a role in the theory's very generation as well is perhaps a matter of conjecture. However, physicist Richard Feynman, for one, believes that Einstein first conceived his theories in the visualizations of thought experiments and developed their mathematical (...) formulations afterwards. According to a recent magazine essay, 'Einstein came to an understanding about relativity by imagining people going up in elevators and beaming light back and forth between rocket ships. (shrink)
History of Physics in Philosophy of Physical Science
Special Relativity in Philosophy of Physical Science
Thought Experiments in Metaphilosophy
Conway's Ontological Objection to Cartesian Dualism.John R. T. Grey - 2017 - Philosophers' Imprint 17:1-19.details
Anne Conway disagrees with substance dualism, the thesis that minds and bodies differ in nature or essence. Instead, she holds that "the distinction between spirit and body is only modal and incremental, not essential and substantial". Yet several of her arguments against dualism have little force against the Cartesian, since they rely on premises no Cartesian would accept. In this paper, I show that Conway does have at least one powerful objection to substance dualism, drawn from premises that Descartes seems (...) bound to accept. She argues that two substances differ in nature only if they differ in their "original and peculiar" cause ; yet all created substances have the same original and peculiar cause; so, all created substances have the same nature. As I argue, the Cartesian is under a surprising amount of pressure to accept Conway's argument, since its key premise is motivated by a conception of substance similar to one endorsed by Descartes in his Principles of Philosophy. (shrink)
Anne Conway in 17th/18th Century Philosophy
Cambridge Platonism in 17th/18th Century Philosophy
Structural Depths in Indian Thought.R. T. Raju - 1987 - Philosophy East and West 37 (2):211-214.details
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Body mass index–measured adiposity and population attributability of associated factors: a population-based study from Buea, Cameroon
Leopold Ndemnge Aminde1,2,
Jeannine A. Atem3,
Andre Pascal Kengne2,4,
Anastase Dzudie2,3,5 &
J. Lennert Veerman1
BMC Obesity volume 4, Article number: 1 (2017) Cite this article
Obesity is currently a global health challenge driven by a mix of behavioural, environmental and genetic factors. Up to date population-based disease burden estimates are needed to guide successful prevention and control efforts in African countries. We investigated the prevalence and population attributable fractions of overweight and obesity in Buea, the Southwest region of Cameroon.
Data are from a community-based cross-sectional study involving randomly selected adults conducted in 2016. Body mass index (BMI) was categorized according to the WHO classification. Multivariable logistic regressions were used to investigate factors independently associated with obesity. Corresponding population attributable fractions were estimated.
Among the 1,139 participants, age-standardized prevalence (95% CI) of overweight and obesity were; 36.5 (33.7–39.3) and 11.1 (9.3–12.9) percent respectively. Mean BMI was 25.3 ± 4.3 kg/m2; women were heavier than men (25.8 vs. 24.4 kg/m2; p <0.0001). Factors associated with obesity were; female gender [odds ratio 3.20 (95% CI: 1.93–5.59)], age > 31 years [3.21 (1.86–5.28)] and being married [2.10 (1.60–3.51)]. At the population level; older age, being married, low level of education, high monthly income and physical inactivity accounted respectively for 11.9%, 21.8%, 11.6%, 6.4% and 8.7% of overweight and obesity among the women, while older age and being married explained 9.2% and 28.3% of overweight and obesity in men.
The prevalence of overweight and obesity in this semi-urban Cameroonian population is high, affecting over a third of individuals. Community-based interventions to control weight would need to take into account gender specificities and socio-economic status.
Overweight (body mass index [BMI] > 25 kg/m2) and obesity (BMI > 30 kg/m2) are conditions of excessive fat accumulation that may impair health. According to the World Health Organization (WHO) 2014 estimates, about 2 billion of the world's adult population are overweight with over 600 million being obese [1]. Overweight and obesity are recognized public health problems especially in industrialized countries [2–4] with prevalence up to almost 40% [5]. The prevalence is now rapidly increasing in low-income settings and Africa in particular, due to rapid urbanization and the adoption of western lifestyles [6, 7]. Current global trends from pooled analysis of population based studies around the world suggest an almost three-fold increase in age-standardized prevalence of obesity (from 3.2–10.8% and 6.4–14.9% for men and women respectively) over the last four decades [8].
In sub-Saharan Africa, South Africa seems to carry a significant burden of obesity [9] with adult population prevalence of over 30%, largely due to its comparatively higher level of urbanization. In Nigeria, Okpechi et al. using the WHO STEPwise approach in a population-based cross-sectional study in Abia State found prevalence of overweight or obesity of 33.7% [10]. Similarly, a population survey among 1,521 Nigerian adults found overweight and obesity rates of 32.2% and 19.6% for men and 29.8% and 36% for women respectively [11]. Data from Ghana among financial institution workers showed over half of participants were either overweight (37.8%) or obese (17.8%), with physical inactivity, being married, female gender and increasing age being independently associated with increased body mass [12]. In Cameroon, Fezeu et al. showed an almost doubling of the age-standardized prevalence of overweight and obesity in rural settings between 1994 and 2003 [13]. More recently, Fouda and colleagues found a prevalence of obesity of 23.4% among workers in Douala [14]. While studies in Cameroon have explored obesity, the majority are from specific sub-populations [14–17] and a few from the general population conducted over a decade ago [13, 18]. The most recent population level estimates for adult overweight and obesity in Cameroon are from a study conducted in 2010 in an urban setting [19].
In SSA, nutrition transition, rapid urbanization and increasing sedentary behaviours have been earmarked as drivers for the obesity epidemic [20, 21]. Cameroon, a lower middle income country like other SSA countries is experiencing similar trends in urbanization (urban growth rate of 3.5%) [22] and westernization of diets. Data from a 2007 nationally representative household consumption and expenditures survey (ECAM3) among women and children in Cameroon suggested consumption of sugar of up to 68.7% (for household) and 78.3% (individually) during the week prior to survey [23]. In addition, data from Food and Agricultural Organization (FAO) suggest that per capita dietary intake increased from 2,001 kilocalories per day (Kcal/Day) in 1962 to 2,269 (Kcal/Day) in 2007, during which an almost double fat consumption was noticed (26–43 g per day). In 2011, dietary intake had further increased to 2586 (Kcal/Day) with sugar and oil intakes of 104 and 132 (Kcal/Day) respectively [24]. Socioeconomic status and physical inactivity are other important risk factors of the obesity epidemic and poor metabolic health previously described in Cameroon [13, 18, 25]. These observations highlight that obesity is rapidly taking a significant position as a cause of disease burden in Cameroon and may worsen if nothing is done. The population attributable risk estimates for these and other potential drivers in Cameroon have not been investigated. Population attributable fraction (PAF) or population attributable risk percent refers to the proportion of cases (disease or outcome) that would not occur in a population if the factor (risk factor or exposure) were eliminated [26]. Its calculation requires the prevalence of exposure to a given risk factor in the population, and the relative risk (RR) of the outcome associated with that exposure. BMI is the most widely used parameter to define overweight and obesity, especially for clinical practice and for epidemiological studies. In this study, we assessed the current adult prevalence of BMI-defined overweight and obesity in Buea, Cameroon and the population attributable impact of its determinants.
During April 2016, we conducted a community based cross-sectional study in Buea, a semi-urban town and Capital of the Southwest region of Cameroon. The study population consisted of adults (age ≥ 18 years) residing in Buea. The Buea Health District (BHD) has seven health areas and an overall population of about 200,000 inhabitants [27]. The estimated sample size was generated using an online sample size calculator [28] and the following criteria: confidence level of 95%, precision of 3%. This gave an estimated sample size of 1,061 subjects.
Via a multi-stage sampling technique, eligible participants were recruited. Systematic random sampling was used to identify households (every fourth house was selected) in each health area to be included in the survey. Thereafter, for each selected household, a simple random sampling via balloting of eligible members was used to select one participant from each house. Data collectors alternated between daytime and evening visits to homes during recruitment to optimize inclusion of individuals who were likely to be at work place during the day.
Assessment of demographic and health parameters
Data were collected by trained final year medical students to minimise error. This included data on socio-economic and demographic characteristics (age, sex, marital status, level of education, monthly income), and lifestyle and personal health (alcohol use, smoking history, family history of cardiovascular disease, consumption of healthy food (rich in vegetables and fruits, low in salt and saturated fats). Participants who had difficulties with respect to details of food content were further clarified by the data collectors. Physical measurements were done using standard methods. Weight was measured using Seca® scales while height was measured using adult Leicester® stadiometers with participants in light clothing and barefoot. With the stadiometers against the wall, participants stood upright without shoes and their heels and occiput on the stadiometer. Measurement was to one decimal place (for weight in kg) and to the nearest 0.5 cm (for height). Body mass index of participants was calculated via the following formula: weight (kg) / height (m) 2 . Participants' BMI was further categorised according to the WHO International Classification of adult normal, overweight and obesity: 18.5 ≤ BMI (kg/m2) ≤ 24.99 as normal weight, 25.0 ≤ BMI (kg/m2) ≤ 29.99 as pre-obese, 30.0 ≤ BMI (kg/m2) ≤ 34.99 as Obese class I, 35.0 ≤ BMI (kg/m2) ≤ 39.99 as Obese class II and BMI ≥ 40.0 kg/m2 as obese class III. Overall, participants with BMI ≥ 25.0 kg/m2 and BMI ≥ 30.0 kg/m2 were considered overweight and obese respectively [29].
Ethical issues
This study was approved by the Ethical Committee of the Southwest Regional Delegation of the Ministry of Public Health (MOH) of Cameroon. The components and purpose of the study were explained to participants and only those who freely provided consent were included. Confidentiality was maintained and the study adhered to the Helsinki declarations.
We used IBM-Statistical Package for Social Sciences statistical software v.23 for Windows (SPSS Inc., Chicago, IL) and STATA software version 12 SE for analysis of the data. Via direct standardization, we used the Cameroon National population in 2016 [30] as standard population to calculate the age-standardized prevalence of overweight and obesity. Overall, categorical variables were summarised with counts and percentages while continuous variables were summarized using means and standard deviations. The chi square test and independent samples t-test were used for group comparisons where appropriate. To determine factors associated with overweight and obesity, we used binary logistic regressions. In the first step, bivariate analysis was done using the following predictor variables; age (dichotomized as <32 years and ≥ 32 years, to explore the effect of obesity on either side of the mean population age), gender (male vs. female), monthly income (<50,000FCFA ~ USD100, 50,000–100,000FCFA ~ USD 100–200, and >100,000FCFA ~ USD 200), level of education [low-moderate (up to secondary/high school) vs. high (diploma/university)], marriage (yes or no), smoking (never, former or current smoker), alcohol consumption (yes or no) and physical activity status (≥3 times per week vs. <2 times per week). All variables with 25% significance (i.e. p ≤ 0.25) were included in the multivariable model to determine independent associations. We have presented adjusted odds ratios (aOR) and 95% confidence intervals of significant associations in the multivariable analysis. Statistical significance was set at p < 0.05.
Population attributable fraction calculation
To obtain the RR of each of the significant independent factors identified, the above model was similarly constructed as a Poisson regression with robust variance estimator. These adjusted RRs were used in the estimation of the PAF. This was done because the odds ratio obtained from a logistic regression of a relatively common disease or outcome potentially overestimates the RR [31]. Further to this, we used a formula (eq. 1) which provides valid estimates of PAF in the presence of confounding [26, 32]. We estimated the uncertainty (95% confidence intervals) around our PAF estimates based on the Bonferroni inequality with formula (eq. 2) [33].
$$ \mathbf{P}\mathbf{A}\mathbf{F} = \mathbf{P}\ \left(\mathbf{R}\mathbf{R}-\mathbf{1}/\mathbf{R}\mathbf{R}\right) $$
where P = prevalence of outcome exposed to given risk factor (exposure) and RR = adjusted relative risk.
$$ \left[{\mathbf{P}}_{\mathbf{L}}\left(\mathbf{R}{\mathbf{R}}_{\mathbf{L}}-\mathbf{1}/\mathbf{R}{\mathbf{R}}_{\mathbf{L}}\right),\ {\mathbf{P}}_{\mathbf{U}}\left(\mathbf{R}{\mathbf{R}}_{\mathbf{U}}-\mathbf{1}/\mathbf{R}{\mathbf{R}}_{\mathbf{U}}\right)\right] $$
where PL is the lower limit of the 97.5% CI of P, RRL is the lower limit of the 97.5% CI of RR, PU and RRU the respective upper limits.
General characteristics of the study population
Socio-demographic characteristics of the study population are presented in Table 1. Overall, 1,139 of the 1,250 adults invited were analysed in this study (response rate = 91.1%). Overall, the mean age was 32 years with the majority (68%) being young adults (aged 18–34 years). Women accounted for 61% of the study population. A quarter of the participants were married and 46% had less than secondary or high school. Over a tenth of the population were unemployed, affecting mostly women (11.6% vs. 12.3%; p < 0.0001). With respect to monthly income, over two-thirds (64%) of the population had a monthly income less than XAF 50,000 (~ USD 100), women being disadvantaged (66.1% vs. 60.7%; p < 0.0001).
Table 1 Socio demographic characteristics of study population according to gender
Prevalence and age-trend in overweight and obesity
The overall age-standardized prevalence of overweight and obesity was 36.5% (33.7–39.3) and 11.1% (9.3–12.9) while mean BMI was 25.3 ± 4.3 kg/m2. Women were heavier than the men with overweight and obesity rates of 37.4% and 13.2% vs. 30.6% and 5.1%; mean BMI of 25.8 kg/m2 vs. 24.4 kg/m2; both p < 0.0001. Overweight and obesity were more frequent with increasing monthly income (p < 0.0001). Similarly, there was higher prevalence in those with a family history of CVD (47.7% vs. 43.2%; p = 0.005). An increasing trend in prevalence of overweight and obesity existed across age groups with highest prevalence for men at age 45–54 years (53.6%) and women at age 55–64 years (65.7%), and declines thereafter (p-trend < 0.0001), see Table 2 and Fig. 1.
Table 2 Overall prevalence of overweight and obesity and according to demographic and clinical characteristics
Trend in prevalence of overweight and obesity by sex and age group
Factors associated with overweight and obesity
In the overall population, factors associated with overweight and obesity in multivariable analysis were; female gender, age >31 years, low level of education, being married and high monthly income (Table 3). In gender stratified analysis for women; low education [aOR: 1.91 (1.34–2.71), p <0.0001], age > 31 years [1.85 (1.23–2.55), p = 0.001], being married [2.30 (1.61–3.41), p <0.0001], high monthly income [2.25 (1.19–4.24), 0.012], physical inactivity [1.62 (1.20–2.21)), p = 0.005] were independently associated with overweight and obesity while older age [2.43 (1.50–3.73), p <0.0001] and being married [1.85 (1.13–2.89), p = 0.011] were associated with obesity alone. In men; older age [1.83 (1.11–3.02), p = 0.018] and being married [3.42 (1.90–6.13), p < 0.0001] for overweight and obesity, and being married [3.07 (1.20–8.71), p = 0.032] for obesity alone, were the independent associations in multivariable analysis (Tables 4 and 5).
Table 3 Factors associated with overweight and obesity in the overall study population
Table 4 Factors associated with overweight and obesity in adult men and women in Buea, Southwest region, Cameroon s2016
Table 5 Variables associated with obesity alone in adult men and women from Buea, Southwest region of Cameroon, 2016
Population attributable fraction estimates
Our estimates for PAF showed that overweight and obesity among women in the population could be attributed to marriage (21.8%), low level of education (11.8%), physical inactivity (8.7%) and high monthly income (6.4%), while 7.8%, 6% and 10.8% of the obesity among women could be attributed to marriage, having a family history of CVD and older age respectively. In men, 28.3% of overweight and obesity could be attributed to marriage, which accounted for about 7% of obesity alone (Table 6).
Table 6 Population attributable fraction estimates for obesity by gender in Buea, Southwest region, Cameroon
Determinants for overweight and obesity have previously been explored in the Cameroonian population; however, how much overweight and obesity in the general population can be attributed to these determinants has not been studied in Cameroon. In this population-based study, we found that over a third of participants were overweight or obese. The overweight and obesity burden in this population was partly attributable to low levels of education, older age, high monthly income, physical inactivity and being married.
These findings suggest an increasing trend in overweight and obesity in Cameroon, as previous reports in Cameroon showed just over a quarter of the population was overweight [34, 35]. Our findings are similar to reports of a recent systematic review from Nigeria which found rates of 20–35% and 8.1–22% for overweight and obesity respectively [36]. Higher rates of obesity have been reported in South Africa, with rates twice as high as our findings [9]. As expected, women were found to be significantly heavier than men, which confirm previous local reports [18, 30, 32] and findings elsewhere [10, 31, 33]. With respect to age, we noticed an overall increasing trend in prevalence of overweight and obesity that peaked at age group 45–54 years and 55–64 years for men and women respectively. Similar trends were also observed elsewhere in Cameroon [35, 37], and in Uganda [38]. Most of these studies were from urban settings, with a few involving rural participants, while our study was composed mainly of semi-urban dwellers, who tend to have a higher prevalence of obesity. Furthermore, an overall tendency to overweight was observed in our study population (mean BMI of 25.3 kg/m2). This was marginally higher than previously reported in Cameroon [34, 35]. While the variance may not be striking, it should be noted that the majority of our study population was made up of young adults, almost a decade younger than previous reports in Cameroon [34]. This suggests an expansion (including younger people) in the burden of overweight and obesity and consequently an increased risk of NCDs in the younger population. This could translate into a high disease burden in the future, if nothing is done to curtail the rise in overweight and obesity rates.
We found that older age, being married, a low level of education, high monthly income and physical inactivity were potential determinants of overweight and obesity among the women, while being married was the main determinant in men. Physical inactivity explained about a tenth of the overweight and obesity among women in this population. This could be due to the adoption of western lifestyles and sedentary behaviours seen to accompany the epidemiologic transition most SSA countries are currently facing. Previous subnational studies in Cameroon [13, 18] have shown an association between physical inactivity and overall obesity prevalence, though less clearly in women, which contrasts our findings. A recent study from South Africa found that lack of exercise accounted for 15% of the obesity [9]. Larger nationally representative studies in Cameroon exploring sex specificities in relation to physical activity levels, obesity and overall metabolic health have a potential to better inform understanding.
With respect to education, low levels of education accounted for 12% of the overweight and obesity among women. Studies from other African countries have found similar associations between low level of education and obesity [9, 39]. This could in part be explained by the fact that more literate individuals are likely to be more informed and potentially tend to adopt healthier lifestyles [40]. Interestingly, we found in our study that though women with low education were more affected, a high monthly income explained more of the overweight and obesity. This is in line with prior studies in Cameroon [13] and Tanzania [41] which found a positive relationship between adiposity and socio-economic status. Our findings are analogous to those of Roskam et al. who found that individuals with low levels of education were more likely to have increased waist lines in Europe [42]. Thus, our results suggest a particular situation with patterning of overweight and obesity as regards educational levels of Cameroonians, with individuals having low levels of education being disadvantaged. Larger nationwide studies are however welcome to explore from a broader perspective, the relationship of obesity and education in the Cameroonian population.
Studies elsewhere have suggested that increasing socio-economic development is associated with increased affordability of cheap energy-dense foods with the major consequence of these being among those in the lower socio-economic groups [43]. Cameroon has steadily experienced increasing per capita dietary intake from 2,001 Kcal/Day in 1962–2,269 Kcal/Day in 2007 and 2,586 Kcal/Day in 2011. Similarly, per capita consumption of sugar and starchy foods was 104 and 419Kcal/day respectively in 2011 [24]. This recent surge and increasing trend in dietary energy intake with relatively low activity levels paralleling increase in other related risk factors like high blood pressure [44], support the unhealthy influence (adoption of 'western' diets and physical activity patterns) of urbanization in Cameroon and other low income countries. However, contrary to western countries where women in the higher socio-economic groups have the tendency to maintain small sizes [45], women with highest monthly income in our study had over twice the odds of being overweight or obese compared to those with low incomes.
While there is evidence about the potential health benefits associated with marriage [46], our findings suggest that marriage is also associated with overweight and obesity. Being married explained about a quarter of overweight and obesity and 15% of obesity alone, in both men and women. This was comparable to findings from South Africa, where being married was a significant determinant for obesity, especially among males [9], and to findings from a study among Greek adults [47]. Whilst there have been suggestions explaining this association in western countries [47], there is need for more in depth exploration of this relationship in the Cameroonian or African context.
Factors such as ageing seem particular in terms of preventive interventions as it is a non-modifiable risk factor, essentially reflecting the potential cumulative exposure to other behavioural and environmental factors. However, adequate counselling of young couples (and even prenuptial) with genetic risk (family histories) to adopt healthy lifestyles may assist in reducing the obesity burden in subsequent generations.
Our study has some limitations. The cross sectional design used precluded the actual exploration of causality for obesity in the population but rather obtained associations using odds ratios, which inherently seem inappropriate for PAF estimations. We therefore used rigorous statistical methods to obtain relative risks that were adjusted for potential confounders, and used these to estimate PAFs and the uncertainty around the estimates. In addition, we used a formula for PAF calculations which accounts for confounding. It is possible that with some household visits for recruitment occurring during the day, we may have missed some adults who were at their places of work. However, we attempted to avert this by alternating between day and evening visits to maximise participation in the study. Our study provides recent updates on the burden of obesity in semi-urban Cameroon and is among the few to have explored population attributable fractions for determinants of obesity in sub-Saharan Africa, and the first of its kind in Cameroon.
Our study revealed a high prevalence of overweight and obesity affecting a third of the population and has provided pioneer evidence on its determinants. Population-based interventions promoting physical activity and health education targeting people with relatively low education, high income earners and couples are potential avenues to control the rapidly growing overweight and obesity epidemic in low-income countries.
CVD:
PAF:
population attributable fraction
RR:
relative risk
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We thank all the participants who took part in this study as well as the medical students who assisted in the data collection. Sincere thanks to Dr. Mark Jones and Solomon M Woldeyohannes at School of Public Health, The University of Queensland, Australia for proof reading and their advice.
The authors can be contacted regarding inquiries to the data.
LNA conceived and designed the study, contributed to data acquisition, did the statistical analysis, interpretation of results and wrote the draft manuscript. JAA contributed to data acquisition and reviewed the manuscript. APK & AD contributed to review of the draft manuscript and interpretation of results. JLV contributed to the analysis and interpretation of results and critically revised the manuscript for intellectual content. All authors read and approved the final manuscript.
LNA: Physician and PhD Candidate at the School of Public Health, University of Queensland, Brisbane, Australia.
JAA: House Officer, Faculty of Health Sciences, University of Buea, Buea, Cameroon.
APK: Professor of Cardiovascular Epidemiology and Prevention, Director, Non-Communicable Diseases Research Unit, South African Medical Research Council and University of Cape Town, South Africa.
AD: Senior Consultant Internist-Cardiologist, Department of Medicine, Douala General Hospital and Faculty of Health Sciences, University of Buea, Cameroon.
LV: Program Director, Master of Public Health Program & Head, Burden of Disease & Cost-effectiveness Unit, School of Public Health, The University of Queensland, Brisbane, Australia.
Not applicable for this study.
This study has been performed in accordance with the declaration of Helsinki. The Ethics committee of the South West Regional Delegation of the Ministry of Public Health, Cameroon approved the study and written informed consent was obtained from all participants.
The University of Queensland, School of Public Health, Herston, QLD, 4006, Australia
Leopold Ndemnge Aminde & J. Lennert Veerman
Non-communicable Diseases Unit, Clinical Research Education, Networking & Consultancy (CRENC), Douala, Cameroon
Leopold Ndemnge Aminde, Andre Pascal Kengne & Anastase Dzudie
Faculty of Health Sciences, University of Buea, Buea, Cameroon
Jeannine A. Atem & Anastase Dzudie
South African Medical Research Council and University of Cape Town, Cape Town, South Africa
Andre Pascal Kengne
Department of Medicine, Douala General Hospital and Faculty of Medicine & Biomedical Sciences, University of Yaoundé, Yaoundé, Cameroon
Anastase Dzudie
Leopold Ndemnge Aminde
Jeannine A. Atem
J. Lennert Veerman
Correspondence to Leopold Ndemnge Aminde.
Aminde, L.N., Atem, J.A., Kengne, A.P. et al. Body mass index–measured adiposity and population attributability of associated factors: a population-based study from Buea, Cameroon. BMC Obes 4, 1 (2017). https://doi.org/10.1186/s40608-016-0139-8
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Journal of the American Mathematical Society
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The Nitsche conjecture
by Tadeusz Iwaniec, Leonid V. Kovalev and Jani Onninen PDF
J. Amer. Math. Soc. 24 (2011), 345-373 Request permission
The Nitsche conjecture is deeply rooted in the theory of doubly-connected minimal surfaces. However, it is commonly formulated in slightly greater generality as a question of existence of a harmonic homeomorphism between circular annuli \[ h \colon \mathbb A = A(r,R) \overset {\text {onto}}{\longrightarrow } A(r_\ast , R_\ast ) =\mathbb A^*. \] In the early 1960s, while attempting to describe all doubly-connected minimal graphs over a given annulus $\mathbb A^*$, J. C. C. Nitsche observed that their conformal modulus cannot be too large. Then he conjectured, in terms of isothermal coordinates, even more:
A harmonic homeomorphism $h\colon \mathbb {A} \overset {\text {onto}}{\longrightarrow } \mathbb {A}^\ast$ exists if and only if \[ \frac {R_\ast }{r_\ast } \geqslant \frac {1}{2} \left (\frac {R}{r}+ \frac {r}{R}\right ). \]
In the present paper we provide, among further generalizations, an affirmative answer to his conjecture.
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Retrieve articles in Journal of the American Mathematical Society with MSC (2010): 31A05, 58E20, 30C20
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Tadeusz Iwaniec
Affiliation: Department of Mathematics, Syracuse University, Syracuse, New York 13244 and Department of Mathematics and Statistics, University of Helsinki, Finland
Email: [email protected]
Leonid V. Kovalev
Affiliation: Department of Mathematics, Syracuse University, Syracuse, New York 13244
MR Author ID: 641917
Email: [email protected]
Jani Onninen
Email: [email protected]
Received by editor(s): November 3, 2009
Published electronically: November 10, 2010
Additional Notes: The first author was supported by the NSF grant DMS-0800416 and the Academy of Finland grant 1128331.
The second author was supported by the NSF grant DMS-0913474.
The third author was supported by the NSF grant DMS-0701059.
The copyright for this article reverts to public domain 28 years after publication.
Journal: J. Amer. Math. Soc. 24 (2011), 345-373
MSC (2010): Primary 31A05; Secondary 58E20, 30C20
DOI: https://doi.org/10.1090/S0894-0347-2010-00685-6
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OSA Publishing > JOSA A > Volume 36 > Issue 12 > Page D31
Iterative phase retrieval for digital holography: tutorial
Tatiana Latychevskaia
Tatiana Latychevskaia1,2
1Department of Physics, University of Zurich, 8057 Zurich, Switzerland
2Paul Scherrer Institute, 5232 Villigen, Switzerland ([email protected])
Tatiana Latychevskaia https://orcid.org/0000-0001-6693-8681
T Latychevskaia
pp. D31-D40
•https://doi.org/10.1364/JOSAA.36.000D31
Tatiana Latychevskaia, "Iterative phase retrieval for digital holography: tutorial," J. Opt. Soc. Am. A 36, D31-D40 (2019)
Digital holographic imaging
Phase measurement
Phase retrieval
Original Manuscript: July 16, 2019
Applied Optics Joint Feature Issue in Applied Optics and Journal of the Optical Society of America A: Digital Holography and 3D Imaging 2019 (2019) JOSA A Joint Feature Issue in Applied Optics and Journal of the Optical Society of America A: Digital Holography and 3D Imaging 2019 (2019)
This paper provides a tutorial of iterative phase retrieval algorithms based on the Gerchberg–Saxton (GS) algorithm applied in digital holography. In addition, a novel GS-based algorithm that allows reconstruction of 3D samples is demonstrated. The GS-based algorithms recover a complex-valued wavefront using wavefront back-and-forth propagation between two planes with constraints superimposed in these two planes. Iterative phase retrieval allows quantitatively correct and twin-image-free reconstructions of object amplitude and phase distributions from its in-line hologram. The present work derives the quantitative criteria on how many holograms are required to reconstruct a complex-valued object distribution, be it a 2D or 3D sample. It is shown that for a sample that can be approximated as a 2D sample, a single-shot in-line hologram is sufficient to reconstruct the absorption and phase distributions of the sample. Previously, the GS-based algorithms have been successfully employed to reconstruct samples that are limited to a 2D plane. However, realistic physical objects always have some finite thickness and therefore are 3D rather than 2D objects. This study demonstrates that 3D samples, including 3D phase objects, can be reconstructed from two or more holograms. It is shown that in principle, two holograms are sufficient to recover the entire wavefront diffracted by a 3D sample distribution. In this method, the reconstruction is performed by applying iterative phase retrieval between the planes where intensity was measured. The recovered complex-valued wavefront is then propagated back to the sample planes, thus reconstructing the 3D distribution of the sample. This method can be applied for 3D samples such as 3D distribution of particles, thick biological samples, and other 3D phase objects. Examples of reconstructions of 3D objects, including phase objects, are provided. Resolution enhancement obtained by iterative extrapolation of holograms is also discussed.
Tatiana Latychevskaia, "Iterative phase retrieval for digital holography: tutorial: publisher's note," J. Opt. Soc. Am. A 37, 45-45 (2020)
http://proxy.osapublishing.org/josaa/abstract.cfm?uri=josaa-37-1-45
22 November 2019: A correction was made to the title. Typographical corrections were made in the abstract and in Section 4.A.
Iterative phase-retrieval-assisted off-axis terahertz digital holography
Yuchen Zhao, Jean-François Vandenrijt, Murielle Kirkove, and Marc Georges
From Fienup's phase retrieval techniques to regularized inversion for in-line holography: tutorial
Fabien Momey, Loïc Denis, Thomas Olivier, and Corinne Fournier
J. Opt. Soc. Am. A 36(12) D62-D80 (2019)
Iterative algorithm of phase determination in digital holography for real-time recording of real objects
Tatsunori Nakamura, Kouichi Nitta, and Osamu Matoba
Three-dimensional display system using near on-axis, phase-only digital holography
Samsheerali P. T., Kedar Khare, and Joby Joseph
Appl. Opt. 54(3) 451-457 (2015)
Research on object-plane constraints and hologram expansion in phase retrieval algorithms for continuous-wave terahertz inline digital holography reconstruction
Jiaqi Hu, Qi Li, and Shanshan Cui
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Allen, L. J.
Almoro, P.
Arfire, C.
Balbekin, N. S.
Belashov, A. V.
Bergoend, I.
Chapman, H. N.
Charalambous, P.
Chen, C.
Cohen, O.
Depeursinge, C.
Duker, H.
Eldar, Y. C.
Fienup, J. R.
Fink, H.-W.
Formanek, P.
Gabor, D.
Gehri, F.
Gerchberg, R. W.
Goodman, J. W.
Gorodetsky, A.
Guo, C.
Haine, M. E.
Huang, H.
Joyeux, D.
Jueptner, W.
Kan, X. C.
Kirkland, E. J.
Kirz, J.
Koch, C. T.
Koren, G.
Kulya, M. S.
Latychevskaia, T.
Lawrence, R. W.
Leith, E. N.
Li, G. B.
Li, J. C.
Li, L.
Li, Q.
Li, Z.
Li, Z. Y.
Liu, G.
Liu, S. T.
Liu, Z. J.
Lubk, A.
McBride, W.
Miao, J.
Miao, J. W.
Mollenstedt, G.
Mulvey, T.
O'Leary, N. L.
Osten, W.
Oxley, M. P.
Pavillon, N.
Pedrini, G.
Petrov, N. V.
Picart, P.
Polack, F.
Qin, Y.
Ratcliffe, J. A.
Rong, L.
Saxton, W. O.
Sayre, D.
Scherzer, O.
Schiske, P.
Schnars, U.
Scott, P. D.
Segev, M.
Shechtman, Y.
Shen, C.
Tan, J. B.
Tiziani, H. J.
Upatnieks, J.
Wang, D.
Yu, Z.
Zhang, Y.
Zhou, X.
Zhou, Z.
Acta Crystallogr. Sec. A (1)
IEEE Signal Process. Mag. (1)
J. Opt. Soc. Am. (2)
J. Opt. Soc. Am. A (3)
Opt. Commun. (1)
Optik (1)
Proc. R. Soc. London, Ser. A (1)
Reports on progress in physics (1)
Ultramicroscopy (2)
Z. Phys. (3)
Fig. 1. Position of the object and its twin image during recording and reconstruction with (a), (b) spherical waves and (c), (d) plane waves.
Fig. 2. Schematic drawing of Gerchberg–Saxton iterative phase retrieval algorithm from two intensity measurements, adapted from [10]. Measured intensities: $ |{u_0}|^2 $ is the intensity in the object plane, and $ |{U_0}|^2 $ is the intensity in the diffraction plane. From these two measured intensities, the algorithm recovers the complex-valued distributions in the object plane ($ {u_0}\exp (i{\varphi _0}) $) and in the diffraction plane ($ {U_0}\exp (i{\Phi _0}) $). (a) The algorithm starts in the object plane, where the initial complex-valued distribution is created by combining the measured amplitude distribution with the random phase distribution. (b) The Fourier transform of the object distribution gives the complex-valued distribution in the diffraction plane (c). The amplitude distribution in the diffraction plane is replaced with the measured amplitude distribution, creating an updated distribution of the complex-valued wavefront in the diffraction plane (d). (e) An inverse Fourier transform gives the complex-valued distribution in the object plane (f). The amplitude distribution in the object plane is replaced with the measured amplitude distribution, creating an updated object distribution for the next iteration starting at (a).
Fig. 3. General scheme of iterative phase retrieval from a single-shot intensity measurement (hologram), adapted from [11]. (a) The algorithm starts in the hologram plane, where the initial complex-valued distribution is created by combining the measured amplitude distribution with the phase of the reference wave. (b) The wavefront propagates from the hologram plane to the sample plane, where it gives the distribution complex-valued transmission function $ t(x,y) $. (c) Constraints in the sample plane are applied, and the updated transmission function $ t^\prime (x,y) $ is obtained (d). (e) The wavefront is propagated from the sample plane to the detector plane (f). The amplitude of the wavefront distribution in the hologram plane is replaced with the measured amplitude. The complex-valued wavefront distribution in the detector plane is updated for the next iteration starting at (a).
Fig. 4. 200 keV in-line hologram of latex sphere and its reconstruction. (a) In-line hologram of the sphere recorded at the defocus 180 µm. The blue lines mark the area outside which the transmission was set to 1 during the iterative reconstruction. (b) Retrieved amplitude distribution of the object wave. (c) Retrieved phase distribution of the object wave. Adapted from [27].
Fig. 5. Object with absorbing and phase-shifting properties. (a) Distributions of transmittance (top) and phase (bottom) of the object. (b) Simulated hologram (top) and phase distributions at the detector plane (bottom). (c) Reconstructed amplitude (top) and phase (bottom) distributions of the transmission function. (d) Iteratively reconstructed amplitude (top) and phase (bottom) distributions in the object plane. The blue curves are the line scans through the centers of the corresponding images. Adapted from [12].
Fig. 6. Reconstruction of 3D objects from two or more intensity measurements. (a) Experimental arrangement. The 3D sample is represented by a set of planes at different $ z $ positions. Here, the sample is sampled with four planes. Two holograms are acquired at different distances from the sample, $ {H_1} $ and $ {H_2} $. (b) Reconstructed amplitude distributions at the four planes within the sample distribution. Parameters of the simulations: wavelength is 532 nm, sample size is $ 1000 \;{\unicode{x00B5}{\rm m}} \times 1000\;{\unicode{x00B5}{\rm m}} $, sampled with $ 1000 \times 1000 $ pixels, distances between the planes within the sample are $ 50\;\unicode{x00B5}{\rm m} $, and $ {H_1} $ and $ {H_2} $ are acquired at distances 200 µm and 300 µm from the sample, respectively. In (b), only the central parts of the reconstructed distributions, $ 150\;{\unicode{x00B5}{\rm m}} \times 150\;{\unicode{x00B5}{\rm m}} $, sampled with $ 150 \times 150 $ pixels, are shown.
Fig. 7. Reconstruction of 3D phase objects from two or more intensity measurements. (a) Experimental arrangement. The 3D sample is represented by a set of planes at different $z$ positions. Here, the sample is sampled with four planes. Two holograms are acquired at different distances from the sample, H1 and H2. (b) Reconstructed amplitude distributions at the four planes within the sample distribution. Parameters of the simulations are the same as in Fig. 6; the diameter of the spherical objects is 10 µm.
Fig. 8. Resolution enhancement in digital holography by self-extrapolation of hologram. (a) Scanning electron microscope image of the sample. (b) $ 1000 \times 1000 $ pixels experimental optical hologram of the sample and its reconstruction (bottom). (c) Self-extrapolation of a piece of the hologram. The selected $ 500 \times 500 $ pixels part of the hologram is padded with zeros up to $ 1000 \times 1000 $ pixels and the corresponding reconstruction (bottom). (d) $ 1000 \times 1000 $ pixels self-extrapolated hologram from (c) after 300 iterations and its reconstruction (bottom). The details of the experiment and reconstruction procedure are available in Ref. [38].
(1) H ( X , Y ) = | R ( X , Y ) + O ( X , Y ) | 2 = | R ( X , Y ) | 2 + | O ( X , Y ) | 2 + R ∗ ( X , Y ) O ( X , Y ) | + R ( X , Y ) O ( X , Y ) ∗ .
(2) R ( X , Y ) H ( X , Y ) ∝ | R ( X , Y ) | 2 O ( X , Y ) | + R 2 ( X , Y ) O ( X , Y ) ∗ .
(3) t ( x , y ) = exp [ − a ( x , y ) ] exp [ i φ ( x , y ) ] ,
(4) u ( x , y ) = u 0 ( x , y ) t ( x , y ) .
(5) t ( x , y ) = 1 + o ( x , y ) ,
(6) A u 0 ( x , y ) t ( x , y ) = A u 0 ( x , y ) [ 1 + o ( x , y ) ] → A [ R ( X , Y ) + O ( X , Y ) ] .
(7) H ( X , Y ) = | A | 2 | R ( X , Y ) + O ( X , Y ) | 2 .
(8) u z 2 = F T − 1 { F T ( u z 1 ) exp ( 2 π i Δ z λ 1 − α 2 − β 2 ) } ,
(9) F T ( u ) = ∬ u ( x , y ) exp [ − 2 π i z ( α λ x + β λ y ) ] d x d y .
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CommonCrawl
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Curves for the Mathematically Curious
Julian Havil has already produced several popular math books. Some of them have been reviewed here: Impossible? (2008), The Irrationals (2012) and John Napier (2014). In this book, containing an anthology of ten iconic curves, he takes another angle of approach to tell more stories about mathematics. Havil's popular math books are more of the recreational kind. I mean that while telling his story, he is not hiding away the mathematics. There can be many formulas and derivations that are however easy to follow with some background in basic calculus.
The curves selected have names. That is because they are in some sense important. If it is the name of a mathematician, it is, like often in mathematics, not always the name of the one who first studied the object. This is again illustrated by Havil in this book when he explores the history underlying the origin of the curve. There is one chapter for every curve. Sometimes it is just one particular curve described by a unique formula like the catenary, but often these curves have parameters or it is just a whole collection of curves with a special property like space filling curves. "Why these ten?" is an obvious question to ask, and Havil has anticipated this because he opens every chapter with a section that explains why he has chosen this curve. Whatever reason he gives, what is important for the reader is that there is always a story or stories worth telling that can be connected to that curve and in some cases these also have a very long history. Let me illustrate the yeast of the book by a telegraphic survey of the ten chapters.
1. The Euler spiral. Its parametrizations are analyzed and the connection with elastic curves and Fresnel integrals. It is also known under other names (e.g. Coru spiral and clothoid), and Havil also explains the history of how and why this has happened.
2. The Weierstrass curve. This is defined as an infinite sum and it is probably the first fractal ever described: a continuous function that is nowhere differentiable. The proof of Weierstrass for these properties is included.
3. Bézier curves. This is an introduction to the characterization of these curves and how they are constructed by the Casteljau algorithm. There are two fun stories connected to these curves. One is about a Bézier curve called Lump which is the name of a dachshund as it was sketched by Picasso caught in one smooth Bézier curve. Havil provides its control points. Another story on the side is about how these curves are used to design letter fonts.
4. The rectangular hyperbola. This is an excellent occasion to tell the history of how logarithms were invented. This is of course described in much more detail in Havil's book about John Napier.
5. The quadratrix of Hippia. The history of this curve is connected to the classic Greek problem of trisecting an angle using only compass and straight edge, but the story would not be complete if one did not recall also the other "impossible" problems of squaring the circle and doubling the cube. The quadratrix is formed by the intersection of two moving lines one translating and another rotating at constant speed. If one could construct that curve, then trisecting an angle and squaring the circle became possible as well as constructing segments whose length is a unit fraction or a square root. The latter are examples of how Havil manages to add some extra mathematics of his own to a well known story.
6. Two space-filling curves. Cantor, Hilbert, and Peano, are three names connected with these curves. The construction of these curves is of course related to the study of cardinality. The Peano curve is a continuous map from a unit interval to a unit square but it is not surjective.
7. Curves of constant width. These are curves like the Reuleaux triangle that looks like a triangle that is slightly inflated, and yet shares many properties with a circle. If it is used as a drill, it will produce square holes (with slightly rounded corners). But there are several generalizations to study. Again, the latter are typical examples of Havil's mathematical extras.
8. The normal curve. This bell shaped curve is probably best known since it represents the normal probability distribution and it is related to the accumulation of rounding errors in long computations. No introduction to probability or statistics is possible without it. There are a few less known names of mathematicians that show up in the birth history of this curve.
9. The catenary. This is the curve formed by a chain loosely hanging from its fixed extremes. It looks deceptively like a parabola, but it isn't and that has fooled some mathematicians of the past. It is of course a place to discuss also the other hyperbolic functions. This is one of the curves that has been used to shape bridges and arches. It is also the shape of the road on which one can smoothly drive with square wheels.
10. Elliptic curves. These are the most complex curves of the book. They are related to Diophantine equations and they are most famous for their use in cryptography.
It is clear that the variety of topics is very broad: form constructions with compass and straight edge to cryptography and from the foundations of mathematics to the design on fonts with Bézier curves and the Casteljau algorithm. There are also seven short appendices explaining some preliminaries or expanding on some topics. However the first appendix is a surprise. On one of the very first pages of the book (page ii, before the title page) there are two 13 × 41 blocks of decimal digits or a number N of over 500 digits spread over 13 lines. No reference, no explanation. The explanation comes in the first appendix. It shows a complicated formula whose main ingredient is a modulo 2 formula for an expression depending on x and y. It thus gives a 0 or 1 depending on x and y which are assumed within certain bounds. The bounds for y depend on a number N, It turns out that it describes the pixel values within a rectangle of a page that will reproduce a pixelated image of the formula on a 106 × 17 pixel grid. Thus the N is the decimal representation of the binary number with 106 × 17 = 1802 bits giving the bit pattern of the pixmap one wants to generate. The two blocks at the beginning of the book give the two N values needed to reproduce the title of this book in pixel-form. The idea is from a 2001 paper by computer scientist Jeff Tupper.
A few pages further at the beginning of the book on page vi shortly after the title page, there is a mathematical doodle with nine wild curves symmetrically arranged in a 3 x 3 matrix, and a trigonometric formula. No further explanation, hence leaving it as a puzzle and a challenge to tease the reader.
There is more serious mathematics to be found in some other relatively long excursions in the chapters. Many of them are following some historical evolution of the problem. For example in the chapter on the normal distribution there is a lot of formula manipulation to move from a binomial distribution, via summing binomial coefficients and Bernoulli numbers, to finally arrive at the exponential expression. The discussion that a bijective map from the unit interval to the unit square cannot be continuous is illustrated by following the steps of the proof of continuity and non-differentiability as given for the Peano curve. The move from an parametrization of the Euler spiral to a simple one, parametrized by arc length, is fully explained and variations in the parametrization can produce very frivolous curves. And there are more not-so-trivial derivations in other chapters that can set the reader on a DIY path for further exploration. The fun items on page ii and vi will certainly trigger the interest of the mathematical puzzlers to find explanations or variations. The conclusion of the book is that $x^2+(\frac{5}{4}y-\sqrt{|x|})^2=1$ is the most important curve of all and it is indeed a lovely one.
Adhemar Bultheel
In this book Julian Havil selects ten iconic curves to tell entertaining stories about mathematics. The stories are written for a broad audience, but still there is also a lot of juggling with formulas. Some basic background in mathematical calculus should however suffice.
julian havil
9780691180052 (hbk), 9780691197784 (ebk)
£ 24.00 (hbk)
https://press.princeton.edu/books/hardcover/9780691180052/curves-for-the-mathematically-curious
Algebraic and Complex Geometry
Mathematics Education and Popularization of Mathematics
Submitted by Adhemar Bultheel | 2 / Dec / 2019
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CommonCrawl
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Lmendes (talk | contribs)
Note that for HFI the resulting pointing model cannot easily be directly compared to a physical/optical model: in particular, it includes a phase shift in the scan direction from the convolution and deconvolution of the detector transfer function, which is complex in the Fourier domain (see {{BibCite|planck2013-p03c}} {{P2013|7}} ). This phase shift was not measured during normal operations, but a short campaign during which the spacecraft was spun at a higher rate will be used to determine these offsets in future date releases. Comparison with the initial optical model indicates that the in-scan change due to this phase shift is of the order of 1 arcminute. Note also that aberration is corrected in all observations.
The final pointing model is measured to be better than 2 arcsecond rms in the co-scan and cross-scan directions averaged over ten-day periods, as shown below. Note that there are larger hourly drifts of up to 10 arcseconds due to interference from the radiometer electronics box assembly (REBA) as discussed more fully in {{BibCite|planck2013-p03}} {{P2013|6}}).
Prior to modeling of systematic changes in the Planck pointing, we find secular drifts of order one arcminute over the course of the nominal mission. We monitor this by making point-source catalogs based on single Planck surveys, only counting those objects which are observed over the course of less than seven days (this limits us to observations away from the ecliptic poles where Planck's observing strategy is highly cross-linked). We cross-match these catalog positions to the known IRAS point-source catalog {{BibCite|Lingyu2009}} and average the deviations in ten-day blocks (the individual measurements are shown as lighter points; error bars assume equal weighting for all points, but the results are not sensitive to any imposed S/N cutoff):
The profiles and locations of the beams are determined from the four observations of Jupiter listed in Table 1. Details are given in {{BibCite|planck2013-p02d}}.
The origin of the focal plane is the optical line of sight defined in {{BibCite|tauber2010b}}. In the LFI RIMO, the beam centres are given by four numbers, $\theta_{\mathrm uv}$, $\phi_{\mathrm uv}$, $\psi_{\mathrm uv}$, and $\psi_{\mathrm pol}$. Only $\theta_{\mathrm uv}$ and $\phi_{\mathrm uv}$, which are the beam pointing in spherical coordinates referred to the line of sight, can be determined with Jupiter observations. The polarization orientation of the beams, defined by $\psi_{\mathrm uv} + \psi_{\mathrm pol}$, is not estimated from flight data but is derived from main beam simulations based on ground measurements.
single scan. Results are reported, with errors, in {{BibCite|planck2013-p02d}}. In addition, the beams are stacked in pairs (J1J2 and J3J4) and all together (J1J2J3J4) in order to improve the signal-to-noise ratio of the measurements. Before the stacking, each beam is artificially repointed along the direction given by the arithmetic average of the centre of each beam to be stacked. Then a fit is performed again on the stacked beams and the resulting parameters recorded. For single scans it has been found that there is an agreement within 2 arcsec in the pointing direction between J1 and J2. The same agreement occurs between J3 and J4. In contrast, a 15 arcsec systematic deviation of the beam centre was detected when comparing J1J2 to J3J4. Figure below shows the reconstructed beam positions and
errors in the line-of-sight frame magnified by a factor of 100. The shift is evident for the 70 GHz beams, as well as in all
the J1J2 and J3J4 stacked beam centres. The change in the location has been found mainly in the scan direction (i.e., $v$-coordinate). To account for this pointing shift, we apply two pointing solutions for LFI. The first focal plane calibration is valid from OD91 to OD540 and is based on the J1J2 beam pointing determination. The second calibration is valid from OD541 to OD563 and is based on the J3J4 beam
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CommonCrawl
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Sequential Outage Checkers for Analyzing Cascading Outages and Preventing Large Blackouts
Hur, Jin;Joung, Man-Ho;Baldick, Ross 585
https://doi.org/10.5370/JEET.2011.6.5.585 PDF KSCI
This paper presents the use of sequential outage checkers to identify the potential cascading processes that might lead to large blackouts. In order to analyze cascading outages caused by a combination of thermal overloads, low voltages, and under-frequencies following an initial disturbance, sequential outage checkers are proposed. The proposed sequential outage checkers are verified using the AEP 9-bus system, New England 39-bus system, and IEEE 118-bus system.
Harmonic Current Compensation based on Three-phase Three-level Shunt Active Filter using Fuzzy Logic Current Controller
Salim, Chennai;Benchouia, M.T.;Golea, A. 595
A three-phase three-level shunt active filter controlled by fuzzy logic current controller which can compensate current harmonics generated by nonlinear loads is presented. Three-level inverters and fuzzy controllers have been successfully employed in several power electronic applications these past years. To improve the conventional pwm controller performance, a new control scheme based on fuzzy current controller is adopted for three-level (NPC) shunt active filter. The scheme is designed to improve compensation capability of APF by adjusting the current error using a fuzzy rule. The inverter current reference signals required to compensate harmonic currents use the synchronous reference detection method. This technique is easy to implement and achieves good results. To maintain the dc voltage across capacitor constant and reduce inverter losses, a proportional integral voltage controller is used. The simulation of global system control and power circuits is performed using Matlab-Simulink and SimPowerSystem toolbox. The results obtained in transient and steady states under various operating conditions show the effectiveness of the proposed shunt active filter based on fuzzy current controller compared to the conventional scheme.
Communication Architecture of the IEC 61850-based Micro Grid System
Yoo, Byong-Kwan;Yang, Seung-Ho;Yang, Hyo-Sik;Kim, Won-Yong;Jeong, Yu-Seok;Han, Byung-Moon;Jang, Kwang-Soo 605
As the power grids are integrated into one big umbrella called a "smart grid," communication protocol plays a key role in successful operations. The successful deployment of smart grid interoperability is a major hurdle that must be overcome. The micro grid, a small power system that distributes energy resource, is operated in diverse regions. Different vendors use different communication protocols in the operation of the micro grid. Recently, the IEC 61850 has been legislated to solve the interoperability problems in power utility automation. The present paper presents a micro grid system based on the IEC 61850 protocol. It consists of a micro grid monitoring system, a protocol converter that transforms serial data to IEC 61850 data, and distributed energy resource controllers for diverse DER nodes. A developed communication gateway can be deployed for DER controllers with serial links to exchange data with IEC 61850-based devices. The gateway can be extended to IEC 61850-based distribution automation systems, substation automation systems, or SCADA.
Dynamic Characteristic Analysis of Electric Actuator for 1 kV/3.2 kA Air Circuit Breaker Based on the Three-link Structure
Lee, Seung-Min;Kang, Jong-Ho;Kwak, Sang-Yup;Kim, Rae-Eun;Jung, Hyun-Kyo 613
In the present paper, a new type of electrical actuator, an electromagnetic force driving actuator (EMFA), applicable to air circuit breaker is developed and analyzed. Transient analysis is performed to obtain the dynamic characteristics of EMFA. The distribution of static magnetic flux is obtained using the finite element method. The coupled problems of electrics and mechanics governing equations are solved using the time-difference method. According to the interception rate of each contactor, investigation of the contactor spring load condition is conducted and applied to the threelink system. Comparisons of the dynamic characteristics of the three-link simulation and experimental data are performed.
Design and Analysis of Double Excited 3-Degree-of-Freedom Motor for Robots
Kwon, Byung-Il;Kim, Young-Boong 618
This paper presents a double excited three degree-of-freedom (3DOF) motor. The proposed 3DOF motor is designed with a laminated structure, making it easy to manufacture. In addition, it has windings on the stator and rotor, and does not require an expensive permanent magnet. We explain the structure, principle of motion, and design of the proposed motor, and perform an analysis of the static characteristics using the two- and three-dimensional finite element methods (3D FEM). The feasibility of 3D FEM analysis is confirmed by comparing the 3D FEM analysis and experimental results for the rolling and pitching motion. We also confirm the occurrence of holding torque in every motion.
A Study on Fast Maximum Efficiency Control of Stator-Flux-oriented Induction Motor Drives
Shin, Myoung-Ho 626
This paper presents a novel maximum efficiency control scheme for convergence improvement in stator-flux-oriented induction motor drives. Three input powers are calculated at three different flux levels, respectively. A quadratic curve is obtained using the quadratic interpolation method using the three points. The flux level at the lowest point of the interpolated curve is calculated, which is not the real minimum input power of the motor, but an estimated one. Hence, the quadratic interpolations are repeated with three new points chosen using the selection method for new points for refitting until the convergence criteria are satisfied. The proposed method is verified by simulation results.
Magnetic Saturation Effect on the Rotor Core of Synchronous Reluctance Motor
Kim, Ki-Chan 634
This paper presents a study on the design parameters that consider the magnetic saturation effect in a rotor core of a synchronous reluctance motor. Two important design parameters in a rotor are selected to analyze the saturation effect of a synchronous reluctance motor, particularly in a rotor core. The thickness of the main segment, which is the main path of the d-axis flux, and the end rip, which affects the q-axis flux, are analyzed using the d-axis and q-axis inductances. Moreover, the characteristics of torque and torque ripple when magnetic saturation takes place are analyzed. The saturation effect is verified by comparing the reluctance torque between the experiment and FEM simulation.
Performance of High-Speed 4/2 Switched Reluctance Motor
Lee, Dong-Hee;Ahn, Jin-Woo 640
The current study presents the design and performance of a novel 4/2 switched reluctance motor (SRM) for a high-speed air blower. With a comparative study of some rotor structures for a high-speed drive, a stepper-type rotor is optimized to produce a continuous torque and a low torque ripple. Rotor pole arc is modified to have a wide continuous output torque region, and air gap is determined to develop less torque ripple. The rotor radius is determined to reduce torque ripple with a reiterative FEM analysis. The designed rotor has three regions: short uniform, long uniform, and nonuniform air-gap region. The positive torque region is wider than a conventional 4/2 SRM without any torque dead zone. A prototype is tested and the efficiency is up to 72[%] at 30,000[rpm], 600[w] output.
Sliding Mode Control with Fixed Switching Frequency for Four-wire Shunt Active Filter
Hamoudi, Farid;Chaghi, A. Aziz;Amimeur, Hocine;Merabet, El Kheir 647
The present paper proposes a sliding mode control with fixed switching frequency for three-phase three-leg voltage source inverter based four-wire shunt active power filter. The aim is to improve phase current waveform, neutral current mitigation, and reactive power compensation in electric power distribution system. The performed sliding mode for active filter current control is formulated using elementary differential geometry. The discrete control vector is deduced from the sliding surface accessibility using the Lyapunov stability. The problem of the switching frequency is addressed by considering hysteresis comparators for the switched signals generation. Through this method, a variable hysteresis band has been established as a function of the sliding mode equivalent control and a predefined switching frequency in order to keep this band constant. The proposed control has been verified with computer simulation which showed satisfactory results.
Novel Fast Peak Detector for Single- or Three-phase Unsymmetrical Voltage Sags
Lee, Sang-Hoey;Cha, Han-Ju 658
In the present paper, a novel fast peak detector for single- or three-phase unsymmetrical voltage sags is proposed. The proposed detector is modified from a single-phase digital phase-locked loop based on a d-q transformation using an all-pass filter (APF). APF generates a virtual phase with $90^{\circ}$ phase delay. However, this virtual phase cannot reflect a sudden change of the grid voltage in the moment of voltage sag, which causes a peak value to be significantly distorted and to settle down slowly. Specifically, the settling time of the peak value is too long when voltage sag occurs around a zero crossing, such as phase $0^{\circ}$ and $180^{\circ}$. This paper describes the operating principle of the APF problem and proposes a modified all-pass filter (MAPF) to mitigate the inherent APF problem. In addition, a new fast peak detector using MAPF is proposed. The proposed detector is able to calculate a peak value within 0.5 ms, even when voltage sag occurs around zero crossing. The proposed fast peak detector is compared with the conventional detector using APF. Results show that the proposed detector has faster detection time in the whole phase range. Furthermore, the proposed fast peak detector can be effectively applied to unsymmetrical three-phase voltage sags. Simulation and experimental results verify the advantages of the proposed detector and MAPF.
A Method for Indentifying Broken Rotor Bar and Stator Winding Fault in a Low-voltage Squirrel-cage Induction Motor Using Radial Flux Sensor
Youn, Young-Woo;Hwang, Don-Ha;Sun, Jong-Ho;Kang, Dong-Sik 666
In this paper, a method for detecting broken rotor bar and stator winding fault in a low voltage squirrel-case induction motor using an air-gap flux variation analysis is proposed to develop a simple and low cost diagnosis technique. To measure the leakage flux in radial direction, a radial flux sensor is designed as a search coil and installed between stator slots. The proposed method is able to identify two kinds of motor faults by calculating load condition of motors and monitoring abnormal signals those are related with motor faults. Experimental results obtained on 7.5kW three-phase squirrel-cage induction motors are discussed to verify the performance of the proposed method.
Modeling and Investigation of Multilayer Piezoelectric Transformer with a Central Hole for Heat Dissipation
Thang, Vo Viet;Kim, In-Sung;Jeong, Soon-Jong;Kim, Min-Soo;Song, Jae-Sung 671
A multilayer square-type piezoelectric transformer with a hole at the center was investigated in this paper. Temperature distribution at the center was improved by using this construction, therefore increasing input voltage and output power. This model was simulated and investigated successfully by applying a finite element method (FEM) in ATILA software. An optimized structure was then fabricated, examined, and compared to the simulation results. Electrical characteristics, including output voltage and output power, were measured at different load resistances. The temperature distribution was also monitored using an infrared camera. The piezoelectric transformer operated at first radial vibration mode and a frequency area of 70 kHz. The 16 W output power was achieved in a three-layer transformer with 96% efficiency and $20^{\circ}C$ temperature rise from room temperature under 115 V driving voltage, 100 ${\Omega}$ matching load, $28{\times}28{\times}1.8mm$ size, and 2 mm hole diameter. With these square-type multilayer piezoelectric transformers, the temperature was concentrated around the hole and lower than in piezoelectric transformers without a hole.
Setup of standard PD calibrator and its uncertainties
Kim, Kwang-Hwa;Yi, Sang-Hwa;Lee, Heun-Jin;Kang, Dong-Sik 677
The present paper describes the setup of standard partial discharge calibrator for measuring partial discharge and estimating uncertainties. The standard PD calibrator was designed and set up, consisting of a digital pulse generator, capacitor modules, and a digital oscilloscope controlled by software developed in the laboratory. Using this software, averages of charges and rising times and their standard deviations in the measured pulses can also be calculated. The standard PD calibrator generates five types of pulses: single, double, random, oscillating, and long-rising. The coefficient sensitivities to estimate the uncertainties of pulses were extracted in the model circuit of the standard PD calibrator. The uncertainties of charges and rising times in pulses of the standard PD calibrator were estimated with single pulses. These values were 0.3%-1.4% in charges and 1.9%-7.0% in rising time; however, these values are lower than the limit values in IEC 60270.
Comparison of transparent conductive indium tin oxide, titanium-doped indium oxide, and fluorine-doped tin oxide films for dye-sensitized solar cell application
Kwak, Dong-Joo;Moon, Byung-Ho;Lee, Don-Kyu;Park, Cha-Soo;Sung, Youl-Moon 684
In this study, we investigate the photovoltaic performance of transparent conductive indium tin oxide (ITO), titanium-doped indium oxide (ITiO), and fluorine-doped tin oxide (FTO) films. ITO and ITiO films are prepared by radio frequency magnetron sputtering on soda-lime glass substrate at $300^{\circ}C$, and the FTO film used is a commercial product. We measure the X-ray diffraction patterns, AFM micrographs, transmittance, sheet resistances after heat treatment, and transparent conductive characteristics of each film. The value of electrical resistivity and optical transmittance of the ITiO films was $4.15{\times}10^{-4}\;{\Omega}-cm$. The near-infrared ray transmittance of ITiO is the highest for wavelengths over 1,000 nm, which can increase dye sensitization compared to ITO and FTO. The photoconversion efficiency (${\eta}$) of the dye-sensitized solar cell (DSC) sample using ITiO was 5.64%, whereas it was 2.73% and 6.47% for DSC samples with ITO and FTO, respectively, both at 100 mW/$cm^2$ light intensity.
Atomic Layer Deposition-incorporated Catalyst Deposition for the Vertical Integration of Carbon Nanotubes
Jung, Sung-Hwan 688
Carbon nanotubes (CNTs) are vertically grown inside high-aspect-ratio vertical pores of anodized aluminum oxide. A CNT catalyst layer is introduced by atomic layer deposition to the bottom of the pores, after which the CNTs are successfully grown from the layer using chemical vapor deposition. The CNTs formed a complete vertical conductive path. The conductivity of the CNT-vertical path is also measured and discussed. The present atomic layer deposition-incorporated catalyst deposition is predicted to enable the integration of CNTs with various challenging configurations, including high-aspect-ratio vertical channels or vertical interconnects.
Resistive and Inductive Loading Techniques on Microstrip Antenna for Wideband Application
Jeon, Sang-Bong;Ahn, Chang-Hoi 693
In this work, an exponentially tapered microstrip antenna was implemented using a resistive loading technique in order to suppress the internal reflections. The inductive loading was realized by introducing slits on the antenna to improve radiation efficiency. Compared with a resistive-loaded antenna, the proposed antenna had an average improvement of about 6.2% in radiation efficiency within the range of 2-10.5 GHz. In addition, the highest peak of the radiated short pulse from the proposed antenna became 45% greater than that of an antenna with resistive loading only.
State-Space Analysis on The Stability of Limit Cycle Predicted by Harmonic Balance
Lee, Byung-Jin;Yun, Suk-Chang;Kim, Chang-Joo;Park, Jung-Keun;Sung, Sang-Kyung 697
In this paper, a closed-loop system constructed with a linear plant and nonlinearity in the feedback connection is considered to argue against its planar orbital stability. Through a state space approach, a main result that presents a sufficient stability criterion of the limit cycle predicted by solving the harmonic balance equation is given. Preliminarily, the harmonic balance of the nonlinear feedback loop is assumed to have a solution that determines the characteristics of the limit cycle. Using a state-space approach, the nonlinear loop equation is reformulated into a linear perturbed model through the introduction of a residual operator. By considering a series of transformations, such as a modified eigenstructure decomposition, periodic averaging, change of variables, and coordinate transformation, the stability of the limit cycle can be simply tested via a scalar function and matrix. Finally, the stability criterion is addressed by constructing a composite Lyapunov function of the transformed system.
Necessary and Sufficient Conditions for the Existence of Decoupling Controllers in the Generalized Plant Model
Park, Ki-Heon;Choi, Goon-Ho 706
Necessary and sufficient conditions for the existence of diagonal, block-diagonal, and triangular decoupling controllers in linear multivariable systems for the most general setting are presented. The plant model in this study is sufficiently general to accommodate non-square plant and non-unity feedback cases with one-degree-of-freedom (1DOF) or two-degree-of-freedom (2DOF) controller configuration. The existence condition is described in terms of rank conditions on the coefficient matrices in partial fraction expansions.
New Stability Criteria for Linear Systems with Interval Time-varying State Delays
Kwon, Oh-Min;Cha, Eun-Jong 713
In the present paper, the problem of stability analysis for linear systems with interval time-varying delays is considered. By introducing a new Lyapunov-Krasovskii functional, new stability criteria are derived in terms of linear matrix inequalities (LMIs). Two numerical examples are given to show the superiority of the proposed method.
Rapid Acquisition of CM and CL Code for GPS L2C Software Receivers
Kwon, Keum-Cheol;Shim, Duk-Sun 723
The GPS modernization program offered a new civil signal on the L2 band, and the first modernized GPS Block IIR satellite was launched in September 2005. Currently, eight GPS Block IIRM satellites and two Block IIF satellites transmit L2C signal. The L2C signal contains two codes of CM and CL that are much longer than the L1 C/A code. Thus, the acquisition of the CM and CL codes takes more time compared with that of L1 C/A code. Under the assumption that the L2C signal is strong enough for detection, this paper suggests rapid acquisition methods for the GPS L2C signals for software receivers and compares its performance with that of other methods.
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CommonCrawl
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On the inverse Laplace transform of the product of a general class of polynomials and the multivariable H-function
K C Gupta R C Soni
Volume 100 Issue 1 April 1990 pp 21-24
https://www.ias.ac.in/article/fulltext/pmsc/100/01/0021-0024
The Laplace transform; the multivariable H-function; a general class of polynomials
In this paper we evaluate the inverse Laplace transform of$$\begin{gathered} s^{ - \eta } (s^{l_1 } + \lambda _1 )^{ - \sigma } (s^{l_2 } + \lambda _2 )^{ - \rho } \hfill \\ \times S_n^m [xs^{ - W} (S^{l_1 } + \lambda _1 )^{ - \upsilon } (S^{l_2 } + \lambda _2 )^{ - w} ]S_{n'}^{m'} [ys^{ - w'} (S^{l_1 } + \lambda _1 )^{ - \upsilon '} (S^{l_2 } + \lambda _2 )^{ - w_r } ] \hfill \\ \times H[z_1 s^{ - W_1 } (S^{l_1 } + \lambda _1 )^{ - \upsilon _1 } (S^{l_2 } + \lambda _2 )^{ - w_1 } ,...,z_r s^{ - w_r } (S^{l_1 } + \lambda _1 )^{ - \upsilon _r } (S^{l_2 } + \lambda _2 )^{ - w'} ] \hfill \\ \end{gathered} $$
Due to the general nature of the multivariable H-function involved herein, the inverse Laplace transform of the product of a large number of special functions involving one or more variables, occurring frequently in the problems of theoretical physics and engineering sciences can be obtained as simple special cases of our main findings. For the sake of illustration, we obtain here the inverse Laplace transform of a product of the Hermite polynomials, the Jacobi polynomials andr different modified Bessel functions of the second kind. A theorem obtained by Srivastava and Singh[7] follows as a special case of our main result.
K C Gupta1 R C Soni1
Department of Mathematics, M. R. Engineering College, Jaipur - 302 017, India
Proceedings – Mathematical Sciences | News
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CommonCrawl
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communications physics
Gain-controlled broadband tuneability in self-mode-locked Thulium-doped fibre laser
Towards visible-wavelength passively mode-locked lasers in all-fibre format
Jinhai Zou, Chuchu Dong, … Zhengqian Luo
Intense few-cycle visible pulses directly generated via nonlinear fibre mode mixing
R. Piccoli, J. M. Brown, … L. Razzari
Photonic Generation of High Power, Ultrastable Microwave Signals by Vernier Effect in a Femtosecond Laser Frequency Comb
Khaldoun Saleh, Jacques Millo, … Yann Kersalé
Highly tunable repetition-rate multiplication of mode-locked lasers using all-fibre harmonic injection locking
Chan-Gi Jeon, Shuangyou Zhang, … Jungwon Kim
Towards high-power mid-IR light source tunable from 3.8 to 4.5 µm by HBr-filled hollow-core silica fibres
Zhiyue Zhou, Zefeng Wang, … Yingying Wang
Femtosecond Mode-locked Fiber Laser at 1 μm Via Optical Microfiber Dispersion Management
Lizhen Wang, Peizhen Xu, … Limin Tong
Power controllable gain switched fiber laser at ~ 3 μm and ~ 2.1 μm
Yiwen Shi, Jianfeng Li, … Yong Liu
Raman-assisted broadband mode-locked laser
Shota Kimura, Shuntaro Tani & Yohei Kobayashi
Ultrafast soliton and stretched-pulse switchable mode-locked fiber laser with hybrid structure of multimode fiber based saturable absorber
Fengyan Zhao, Yishan Wang, … Kaiming Zhou
Dennis C. Kirsch ORCID: orcid.org/0000-0002-6930-65131,
Anastasia Bednyakova2,
Petr Varak3,
Pavel Honzatko3,
Benoit Cadier4,
Thierry Robin4,
Andrei Fotiadi5,6,
Pavel Peterka3 &
Maria Chernysheva ORCID: orcid.org/0000-0003-4905-431X1
Communications Physics volume 5, Article number: 219 (2022) Cite this article
Mode-locked lasers
Ensuring self-driven mode-locking and broadband wavelength tuneability in all-fibre-integrated femtosecond laser sources enables a new level of their versatility and extends areas of their applications. Principle limitations for this are traditionally available ultrafast modulators and tuneability techniques. Here, we exploit Thulium-doped fibre to perform three roles in the cavity: laser gain, saturable absorber, and tuneability element via controlling its excitation level. We confirmed that Tm-doped fibre saturable absorption is defined by a reinforced quenching of Tm3+ pairs. As a result, we present both numerically and experimentally a highly stable sub-picosecond pulse generation with a ~90 nm tuneability range spanning from 1873 to 1962 nm via adjusting the cavity feedback. The maximum laser efficiency corresponds to 25% cavity feedback, enabling the highest output energy of 1 nJ in 600-fs solitons at 1877 nm. Overall, the presented laser system establishes a compact and straightforward approach for ultrafast generation, which can be translated to other fibre laser operation wavelengths.
Strengthening of laser technology's position in scientific, industrial and daily life applications relies on the system performance, specifically on how flexible it can adapt to the broad range of requirements. In this perspective, Thulium-doped fibres enable the exploration of a highly desirable broad wavelength range, spanning over ~350 nm around 2 μm. Such unique operation bandwidth has placed these fibre laser systems at the forefront of diverse applications. Environmental monitoring of greenhouse gases1, polymer or semiconductor machining2, optical coherence tomography3, nonlinear microscopy4, and optical communication5—these are just a handful of the ultrafast laser tasks which have been enabled by or enhanced with the Tm-doped fibre systems development. Versatile, tuneable, and highly integrated turnkey ultrafast fibre laser solutions are of high demand to empower emerging technological platforms and reinforce their rapid progress towards hand-held instruments. Yet, the development of such instruments is mainly obstructed by the imperfections and limitations of specific essential components that exhibit saturable absorption and wavelength tuneability behaviour.
The most notable saturable absorbers, which ensure phase-locking of longitudinal cavity modes, are SESAMs (e.g. GaSb)6, carbon nanomaterials7, transition metal dichalcogenides8, or MXene9. However, besides poor long-term reliability and power-handling, their performance in the short-wave infra-red (SWIR) wavelength range lags behind their operation at Near-IR. At the same time, widely used all-fibre artificial modulators based on the optical Kerr effect, such as nonlinear polarisation evolution (NPE), nonlinear loop mirrors or nonlinear multimode interference, suffer from the diminished nonlinearity of optical fibres at the SWIR. This limitation leads to a high self-starting threshold and the necessity to design complex cavity configurations10.
In such context, the visionary idea of mode-locked fibre lasers involving no apparent saturable absorber in the cavity has been validated through nonlinear coupling between cores in multi-core fibres11, inter-mode beating12, or self-absorption in a gain fibre. Conventionally, the latter has been considered undesirable instability in continuous-wave fibre lasers13 and was only later studied as an ultrashort pulse formation mechanism. The saturable absorption behaviour of unexcited rare-earth-doped fibres has been mainly exploited at bandwidths where their absorption cross-section covers the emission cross-section of separate gain fibre14,15. Due to the slow recovery time of such a saturable absorption mechanism, the reciprocity of self-phase modulation (SPM) and anomalous group-delay dispersion became imperative for soliton shaping16. So far, only a few works have reported the gain fibre saturable absorption dynamics to enable the formation of few-picosecond mode-locked pulses as long as was assisted by SPM accumulated over tens of metres long laser cavities17,18,19.
At the same time, a broad range of laser generation wavelength tuneability up to 300 nm (1733–2033 nm), demonstrated in Tm-doped fibre lasers20, is generally implemented using bulk acousto-optic tunable filters21, diffraction gratings22, volume Bragg gratings23, or planar semiconductor chips24. For this reason, the benefits of laser operation tuneability come at the cost of generation stability and substantial insertion losses. Thus, the possibilities of shaping and manipulating the laser generation through the interplay of intrinsic intracavity phenomena have recently become a topic of intensive experimental and theoretical investigation25,26,27. It is worth mentioning that bending of active fibre presents another technique for the operation wavelength variation, demonstrating 152 nm wavelength tuneability from 1740 to 1892 nm28. However, the fundamental principle of such tuneability relies on wavelength-selective losses, suppressing spontaneous emission at the longer edge of the gain spectrum. Such tuneability mechanism has been ensured by the proper design of a bend-sensitive W-type active fibre, in which cladding comprises a depressed ring with a lower refractive index. Importantly, like any other technique relying on loss introduction, these results came at the price of reduced laser efficiency. With up to 1.3 W pump power, the maximum average output power reached only 4.5 mW. Additionally, the realisation of continuous sweeping of the central laser wavelength by changing the fibre coiling radius introduces another level of complexity to the system.
Specifically, in Tm-doped fibre as quasi-three-level systems, the equilibrium between emission and absorption and, thus, the profile of the gain spectrum g(λ) is governed by the lower N1 and excited laser level N2 population fractions (Supplementary Note 1). Supplementary Fig. S1b predicts spectral gain profiles of Tm-doped fibre (used in further investigations) to red-shift at higher excitation level29,30. The control of the laser cavity Q-factor allows efficient governing of an excitation level of the gain, leveraging output power and laser generation wavelength. The control of the Q-factor can be implemented, for example, by changing the active fibre length or its doping concentration31, or by altering cavity loss through implemented attenuation32,33 or controlling laser feedback34. The cavity feedback control has been investigated mainly for the relatively narrow band of the Erbium gain34,35,36. Active fibres with broader gain spectra hold the true potential since the actual tuning range is built by the shape and overlap of the emission and absorption cross-sections. Nevertheless, the presented attempts to control wavelength-independent losses have not demonstrated remarkable tuneability ranges in Tm-doped fibre lasers. While theoretical work predicted the possibility of achieving a 105 nm tuneability in a continuous-wave generation, only 3632 and 48 nm33 tuneability has been achieved in the experiment. Still, continuous wavelength tuneability in ultrafast Tm-doped fibre lasers remains largely unexplored.
In this work, we explore the phenomenon of self-mode-locking and enable tuneable ultrashort pulse generation by manipulating the gain excitation level, omitting the application of intracavity filters or saturable absorbers. The proposed concept employs a single Tm-doped fibre as the gain medium, saturable absorber and element responsible for filter-less wavelength tuneability simultaneously. Experimental studies show that a ring fibre oscillator comprising such fibre and only a few conventional non-polarisation-maintaining components can support the generation of near transform-limited 350-fs soliton pulses with ~1.2 nJ energy at ~45 MHz repetition rate. Importantly, we explore the impact of Tm3+ ion-pair clusters on the saturable absorption properties of doped silica fibres. Moreover, the ultrafast fibre laser system demonstrates effective continuous wavelength tuneability within the range spanning from 1873 to 1962 nm by controlling the cavity feedback and maintaining high quantum efficiency. For this, we suggest employing a variable output coupler, which determines the intracavity energy and, therefore, the excitation level of fibre gain. The rigorous theoretical investigation confirms experimental observations of the pulse formation and clarifies the origin of the tuneability effects.
Overall, the elegance and low complexity of the laser system, together with one order of magnitude shorter pulse duration and more than a twofold increase of the tuneability range are the advancements where we focus the key claims of the current work. Another advantage of the demonstrated technique is that it can be translated to other laser configurations and wavelength ranges, which currently lack an extensive selection of components to develop robust fibre-based systems. The demonstrated technique holds high potential in exploring the Mid-IR range, where rare-earth fibres demonstrate broad gain spectra (e.g. Dy3+ ions allow nearly 600 nm tuneability around 3.1 μm37).
Investigations of saturable absorption origins in Tm-doped gain fibre
The origin of the saturable absorption in rare-earth-doped fibres is associated with ion pairs excited-state absorption38,39,40 and upconversion interaction41,42. In the context of Tm-doped fibres, the work of Jackson et al. has concluded that 3F4, 3F4 → 3H6, 3H4 (Supplementary Fig. S1a) energy transfer upconversion in unpumped ions plays a key role in establishing the saturable absorption behaviour43. The energy difference of this transition is relatively high, −1500 cm−1, and, therefore, a quite high degree of clustering is required for the Tm-doped silica fibres to achieve ion separations that are short enough to provide a sufficient level of saturable absorption.
To reveal the nature of saturable absorption, we examined non-polarisation-maintaining highly Tm-doped fibre with a 2.9 μm core radius, 0.18 numerical aperture and a cut-off wavelength of 1389 nm. The glass matrix of the fibre core comprises an aluminosilicate host with 2.85 mol.% Al2O3 and 0.33 mol.% Tm2O3, thus providing an 8.6:1 Al:Tm doping ratio. It is worth mentioning that our measurements provided only indicative values of the ion concentration due to the limited resolution of the electron microscope and the small size of the fibre core. Nevertheless, these estimations suggest that the Al concentration is not sufficient to prevent the clustering of Tm ions44. A transverse refractive index measurement can be seen together with scanning electron microscope images of the fibre facet in Fig. 1a. Judging from these measurements, the fibre exhibits a pronounced core circularity and concentricity with a characteristic dip in the core centre due to the decrease of Al3+ concentration due to the formation of volatile AlF345. Furthermore, the refractive index profile features are depressed by the F-doping cladding area.
Fig. 1: Tm-doped fibre parameters.
a Refractive index profile (RIP). Inset: Scanning electron microscopy image of the fibre core; b pump saturable absorption of 18-cm fibre section at 1550 nm. c Fluorescent lifetime of the 3H4 and 3F4 manifold against pump power for 1.5-mm fibre length; d normalised nonlinear absorption for four length of the Tm-doped fibre.
It is worth noting that we also investigated another fibre sample with a germanaoalumosilicate matrix of the core with ~13 mol.% GeO2, 1.58 mol.% Al2O3 and 0.54 mol.% Tm2O3, resulting in Ge:Al:Tm ion concentration ratio of 12:3:1 (Supplementary Note 2). However, despite high doping concentration and potential ion clustering, this fibre did not enable self-mode-locking generation in the above-discussed laser configuration. Therefore, further measurements of its properties are presented in Supplementary Materials (Supplementary Fig. S2).
Fluorescence lifetime
The time-resolved luminescence properties, especially the decay time-power dependency, can provide important information about the Tm3+ ion incorporation in the optical fibre matrix and the formation of clusters and ion pairs. The fluorescence lifetimes of the 3F4 and 3H4 levels were measured by direct in-band pumping using 1618 and 792 nm diodes. The emission was collected at around 2 μm and 800 nm using InGaAs and Si photodiode detectors, as discussed by Kamrádek et al.46. In order to suppress the influence of light travelling along the waveguide, namely the amplified spontaneous emission and reabsorption, the measured fibre was only 1.5 mm long, and the emission was detected transversely. The decay time, τ, was obtained from the e−1 value on the normalised decay curve. The fluorescence lifetime, τ0, was retrieved by extrapolation of the decay time to zero excitation power, assuming the behaviour according to ref. 47:
$$\tau =\frac{{\tau }_{{{{{{{{\rm{0}}}}}}}}}}{1+\left(\frac{{\tau }_{0}}{{\tau }_{{{{{{{{\rm{sat}}}}}}}}}}-1\right){\left(\frac{P}{P\,+\,{P}_{{{{{{{{\rm{crit}}}}}}}}}}\right)}^{2}},$$
wherein τsat is the saturated lifetime, extrapolated to infinite excitation power, P is the excitation power, and Pcrit is the critical power of the energy-transfer processes.
Figure 1c shows the lifetime values of the Tm-doped fibre under the study. The value 425 μs of the 3F4 level lifetime is characteristic for highly doped Tm fibres48. Importantly, the steep decrease in decay time with increasing excitation power suggests a high rate of energy-transfer upconversion. The experimentally obtained value of 3H4 lifetime is around 21.4 μs, which is also typical for highly doped Tm fibres. Conventionally, the 3H4 decay time of highly doped fibres with homogeneously distributed Tm3+ ions should increase with excitation power due triggering the cross-relaxation effect 3H4,3H6 → 3F4,3F4. As observed in our fibre sample, the decrease of decay time with excitation power suggests a suppressed cross-relaxation and, therefore, further supports the hypothesis of clustering and pair-induced quenching of Tm ions.
Pump saturable absorption
For estimating the concentration of Tm3+ pairs in our active fibres, we used the approach proposed by Myslinski et al.49 for measuring the saturable loss of the pump absorption. Departing from the measurement of the pump absorption coefficient as a function of the irradiated pump power (Fig. 1b), we model the non-saturable loss growing with the fraction of Tm-pairs using RP Fiber Power software.
A 18-cm long section of studied Tm-doped fibre demonstrated a small-signal absorption, α0, of ~15.3 dB. Towards high irradiation, the experimentally recorded pump absorption coefficients saturate at 2.5 dB, following the trend described as:
$${\alpha }_{{{{{{\rm{sat}}}}}}}=mk{\alpha }_{0}\left\{1-\frac{{\sigma }_{{{{{{{{\rm{a}}}}}}}}}+{\sigma }_{{{{{{{{\rm{e}}}}}}}}}}{m{\sigma }_{{{{{{{{\rm{a}}}}}}}}}+{\sigma }_{{{{{{{{\rm{e}}}}}}}}}}\right\},$$
here k is the proportion of Tm3+ clusters in the total Tm3+ concentration, m is the number of ions per cluster50. To define the model, we used parameters of Tm-doped fibre, recorded during the fluorescence lifetime measurement and the upconversion coefficients from Kamradek et al.46. The fibre indicated the significant contribution of upconversion processes to the 1G4 level, corresponding to 480 nm. Since the pump at 1550 nm wavelength could not provide single-photon excitation to the 1G4, our simulation accounted for the multiphoton or cooperative process as a third-order term quenching, which is limited by used RP Fibre Power software. We assumed cubic quenching coefficient of 2.8 ⋅ 10−73 m3 s−1 in the simulations. This third-order quenching coefficient ensured considerably better agreement with experimentally measured data, preventing numerical absorption curves from saturating far too fast at lower launched pump powers (Fig. 1b). The detailed numerical simulation allowed estimation of ~20% of Tm3+ pairs in the studied gain fibre.
Nonlinear saturable absorption
To measure the saturable absorption of the active fibre via the twin-detector approach, we used a self-build Tm-doped fibre laser generating 500-fs pulses at 1890 nm. The laser output power, controlled via an external variable optical attenuator, was split into two arms and launched into an unpumped fibre under test and a reference detector. Since the saturable absorption behaviour is considered to occur in unexcited rare rare-earth-doped optical fibre, we investigated nonlinear dynamics at 1, 3, 5 and 10-cm long fibre sections. Figure 1d shows the normalised intensity-dependent absorption measurement data with the approximation according to the two-level energy model51:
$$\alpha (I)=\frac{{\alpha }_{0}}{1+\frac{I}{{I}_{{{{{{{{\rm{sat}}}}}}}}}}}+{\alpha }_{{{{{{{{\rm{ns}}}}}}}}},$$
where α0 and αns are the modulation depth and non-saturable losses, correspondingly. I is the launched pulse intensity, and Isat is the saturation intensity, which corresponds to the twice reduced absorption of the test sample. As seen in Fig. 1d, the saturation intensity of Tm-doped fibre drops from 250 to 72 MW cm−2 with the reduction of the active fibre length. The active fibre demonstrates high modulation depth spanning from 34 to 9.5%, when trimmed from 10 to 3 cm. These values are comparable with the parameters of conventional material saturable absorbers, operating at SWIR wavelength range52. However, the saturation intensity of Tm-doped fibre is significantly higher, which can be explained by a high energy difference between the initial and final upconversion energy states. Moreover, the peak power of the available ultrafast fibre laser was not sufficient to fully saturate the Tm-doped fibre section with the 10-cm length. Therefore, we anticipate that the longer section would not act as an efficient saturable absorber or require considerable laser gain to compensate for the high saturation threshold. Similarly, the decrease of the length to ~1 cm cannot provide benefits to self-mode-locking initiation, as the modulation depth drops to nearly 1% with the saturation intensity on only ~40 MW cm−2.
Further, to assess the influence of laser operation wavelength on saturable absorption of the Tm-doped gain fibre, we have investigated the power-dependent absorption at various wavelengths within the tuneability range from 1880 to 1947 nm. The summary of the saturable absorption parameters of 5-cm long Tm-doped fibre section at 1880, 1900, and 1920 nm wavelengths is presented in Supplementary Fig. S3. It is worth noting that the trend of the measured values of the saturable loss coincides well with the relative absorption slope of Tm3+ ions. The saturation intensity depends on the upper state lifetime τ and the absorption cross-section σabs expressed as \({I}_{{{{{{{{\rm{sat}}}}}}}}}=hc{(2\lambda \tau {\sigma }_{{{{{{{{\rm{abs}}}}}}}}})}^{-1}\), such that the modulation depth is proportional to the absorption cross-section, taking Supplementary Eq. (S1) into account.
Experimental configuration
To examine the self-mode-locking and tuneability dynamics of the ultrashort pulse generation in the laser cavity with a variable Q-factor, we assembled the Tm-doped fibre laser setup as presented schematically in Fig. 2. The ring fibre laser cavity employed a 1550-nm pump laser with 1-W power (HPFL-300, BKtel), a 1550/2000 wavelength division multiplexer, an isotropic isolator, and a squeezing polarisation controller to restore the polarisation state after each round trip. A 0.5 m section of a non-polarisation maintaining Tm-doped silica fibre, discussed in the previous chapter, provided both active gain and saturable absorption to enable self-mode-locking. The fibre small-signal gain was estimated as ~30 dB (for 0.95 W pumping and 65 μW seeding a 44 cm piece). Next, the active fibre group velocity dispersion, third-order dispersion and nonlinearity were estimated as β2 = −20 ps2/km, β3 = 0.27 ps3/km, γ = 2 W−1 km−1, correspondingly, at around 1.9 μm wavelength range. The rest of the cavity comprises 4.2 metres-long standard single-mode fibre ports of the fiberised laser components with the following parameters: β2 = −59 ps2/km, β3 = 0.28 ps3/km, γ = 1.3 W−1 km−1 at Tm-doped fibre laser operation wavelength. It is important to stress that the oscillator consists fully of polarisation-insensitive elements and concentric, cylindrical silica fibres. Therefore, any other mode-locking mechanism, particularly NPE, could be not considered due to short cavity length and negligible polarisation dependent loss53.
Fig. 2: Schematic of the ultrafast Thulium-doped fibre laser.
The oscillator comprises a single-mode pump laser (Er-doped fibre laser at 1.55 μm), a wavelength division multiplexer (WDM), Tm-doped fibre, a polarisation-insensitive isolator, an in-line variable output coupler (VOC) and a squeezing type polarisation controller. All passive fibres are polarisation-insensitive SMF28e. Note, the VOC is replaced by a fixed, fused coupler with 20% feedback for the section "Pulse duration and laser optimisation". The inset depicts the nonlinear course of the coupling coefficient against the VOC's gauge.
The tuneability of the laser generation was established by introducing no filtering elements into the cavity, but only through including an in-line variable fibre-optic coupler (Evanescent Optics) based on D-shaped polished fibres, interacting via the evanescence field. The separation of these fibres changes through rotating a knob controller, thus, modifying the coupling efficiency from one waveguide into another, and overall laser cavity feedback from 8 to 93% with nearly 0.3–0.1 dB excess loss. The performance of the variable coupler was characterised by reference laser transmission measurements at 1.95 μm and is shown in the inset in Fig. 2.
Numerical simulations
To validate the possibility of the self-mode-locking and broadband tuneability in the presented Tm-doped fibre laser, we developed a numerical model which describes consequent pulse propagation through different cavity elements. The pulse propagation along the passive fibre is governed by the Nonlinear Schrödinger equation. The following system of coupled equations for continuous-wave pump and pulsed signal generation, taking into account the effects of dispersion and nonlinearity, was considered to describe the signal amplification54,55:
$$\frac{\partial {A}_{{{{{{{{\rm{s}}}}}}}}}(z,t)}{\partial z} = -i\frac{{\beta }_{2}}{2}\frac{{\partial }^{2}{A}_{{{{{{{{\rm{s}}}}}}}}}(z,t)}{\partial {t}^{2}}+i\gamma | {A}_{{{{{{{{\rm{s}}}}}}}}}(z,t){| }^{2}{A}_{{{{{{{{\rm{s}}}}}}}}}(z,t) \\ \quad +\int\nolimits_{-\infty }^{\infty }\frac{{g}_{s}(\omega ,z)}{2}{\tilde{A}}_{{{{{{{{\rm{s}}}}}}}}}(z,\omega ){{{{{\rm{exp}}}}}}(-i\omega t)d\omega ,$$
$$\frac{\partial {P}_{{{{{{{{\rm{p}}}}}}}}}(z)}{\partial z}={g}_{{{{{{{{\rm{p}}}}}}}}}(z){P}_{{{{{{{{\rm{p}}}}}}}}}(z),$$
where As(z, t) is the slowly varying envelope associated with the signal, Pp(z) is the average power of continuous-wave pump, β2 is the group velocity dispersion, γ is the Kerr nonlinearity, gs and gp are signal and pump gain/loss coefficients, correspondingly. Equation (4) was numerically solved by the split-step Fourier method. The spectral window considered in the model extended from 1300 to 2800 nm. The temporal window was equal to 140 ps.
The wavelength dependence of the gain is considered in the frequency domain, where optical field \(\tilde{A}(z,\omega )\) is multiplied by the gain profile gs(ω, z). Each spectral component of the gain gs(λi, z) (i = 1, .., Nω, where Nω is the number of the discrete frequencies in simulations) and pump gain/loss coefficient at each step along the fibre were found from the rate equations in the stationary case dN2,3/dt = 0:
$${g}_{s}({\lambda }_{i},z)= \; \left({\sigma }_{21}^{s}({\lambda }_{i})-{\sigma }_{23}^{s}({\lambda }_{i})\right){\rho }_{s}({\lambda }_{i}){N}_{2}(z)-{\sigma }_{12}^{s}({\lambda }_{i}){\rho }_{s}({\lambda }_{i}){N}_{1}(z) \\ +{\sigma }_{32}^{s}({\lambda }_{i}){\rho }_{s}({\lambda }_{i}){N}_{3}(z),\,i=1,...,{N}_{\omega }$$
$${g}_{p}(z)={\sigma }_{21}^{p}{\rho }_{p}{N}_{2}(z)-{\sigma }_{12}^{p}{\rho }_{p}{N}_{1}(z),$$
$$\frac{d{N}_{2}(z)}{dt}= \; \left({\sigma }_{12}^{p}{\rho }_{p}\frac{{P}_{p}(z)}{h{\nu }_{p}}+\mathop{\sum }\limits_{k=1}^{k}\left({\sigma }_{12}^{s}({\lambda }_{k}){\rho }_{s}({\lambda }_{k})\frac{{P}_{s}({\lambda }_{k},z)}{h{\nu }_{k}}\right)\right){N}_{1}(z) \\ +\left(\mathop{\sum }\limits_{k=1}^{k}\left({\sigma }_{32}^{s}({\lambda }_{k}){\rho }_{s}({\lambda }_{k})\frac{{P}_{s}({\lambda }_{k},z)}{h{\nu }_{k}}\right)\right){N}_{3}(z)\\ -\left({\sigma }_{21}^{p}{\rho }_{p}\frac{{P}_{p}(z)}{h{\nu }_{p}}+\mathop{\sum }\limits_{k=1}^{k}\left({\sigma }_{21}^{s}({\lambda }_{k}){\rho }_{s}({\lambda }_{k})\frac{{P}_{s}({\lambda }_{k},z)}{h{\nu }_{k}}\right)\right. \\ \left.+\mathop{\sum }\limits_{k=1}^{k}\left({\sigma }_{23}^{s}({\lambda }_{k}){\rho }_{s}({\lambda }_{k})\frac{{P}_{s}({\lambda }_{k},z)}{h{\nu }_{k}}\right)+\frac{1}{{T}_{2}}\right){N}_{2}(z)-2{k}_{2231}{N}_{2}^{2}+2{k}_{3122}{N}_{1}{N}_{3},$$
$$\frac{d{N}_{3}(z)}{dt}= \; \left(\mathop{\sum }\limits_{k=1}^{k}\left({\sigma }_{23}^{s}({\lambda }_{k}){\rho }_{s}({\lambda }_{k})\frac{{P}_{s}({\lambda }_{k},z)}{h{\nu }_{k}}\right)\right){N}_{2}(z)\\ -\left(\mathop{\sum }\limits_{k=1}^{k}\left({\sigma }_{32}^{s}({\lambda }_{k}){\rho }_{s}({\lambda }_{k})\frac{{P}_{s}({\lambda }_{k},z)}{h{\nu }_{k}}\right)+\frac{1}{{T}_{3}}\right){N}_{3}(z) \\ +{k}_{2231}{N}_{2}^{2}-{k}_{3122}{N}_{1}{N}_{3},$$
$${N}_{1}(z)=N-{N}_{2}(z)-{N}_{3}(z),$$
here N1,2,3 are population densities in the energy levels 3H6, 3F4 and 3H4 correspondingly, N = 4.05938 ⋅ 1015 m−1 is the total number of Tm-ions integrated over the fibre mode cross-section, \({P}_{s}({\omega }_{k},z)=| \tilde{A}(z,{\omega }_{k}){| }^{2}\) is the signal power at the frequency ωk and position z along the fibre, T2 = 425 μs and T3 = 21.4 μs are fluorescence lifetimes. The effective pump absorption and emission cross-sections at pump wavelengths are \({\sigma }_{12}^{p}=1.5630\cdot 1{0}^{-25}\) m2 and \({\sigma }_{21}^{p}=5.1005\cdot 1{0}^{-27}\) m2. The absorption and emission cross-section spectra in the considered spectral window (shown in Supplementary Fig. S4) are described by \({\sigma }_{12}^{s}({\lambda }_{i})\), \({\sigma }_{23}^{s}({\lambda }_{i})\), \({\sigma }_{21}^{s}({\lambda }_{i})\) and \({\sigma }_{32}^{s}({\lambda }_{i})\). The normalised pump and signal power distributions through the fibre cross-section are marked ρp,s = Γp,s/πa2, where a = 2.65 μm is the core radius of a single-mode fibre, Γp(Γs) corresponds to the modal overlap factor between the pump (signal) mode and the ion distribution. Γp = 1 for core pumping, Γs = 1 − exp(−2a2/w2), w is 1/e electric field radius of the equivalent Gaussian spot. The energy transfer coefficients k3122 = 2.52 ⋅ 10−22 m3/s and k2231 = 3.44 ⋅ 10−24 m3/s describe the cross-relaxation process 3H4, 3H6 → 3F4, 3F4 and energy transfer upconversion 3F4, 3F4 → 3H4, 3H6 correspondingly48.
The used approach dNi/dt = 0 means that this evolution occurs with the roundtrips quite slowly, i.e., during one round trip the optical field does not change the populations N1, N2, N3 significantly. At the end of the active fibre, the pump is depleted, and the last fibre segment plays the role of a saturable absorber. Equations (4)–(9) are also applicable to describe the optical losses (α = −gs) experienced by the signal beam in the absence of the pump power. In this case, the condition dNi/dt = 0 sets an instantaneous response of the saturable absorber on the pulse power within each round-trip. It results in the non-saturated losses αns determined by the ion quenching and up convention processes and the saturated absorption fraction (Eq. (3)), expressed through the modulation depth α0, saturation intensity Isat and with I(t) = ∣A(t)∣2. Specifically, for our numerical simulations, we have simplified the model considering the active fibre as a combination of the amplifier described by the wavelength-dependent gain profile gs(λi, z) distributed over the whole fibre length and an absorber described as the time-dependent optical losses α(t, z) = −gs(t, z) localised in a rare end of fibre section. Such consideration has allowed us to separate contributions of the amplification and saturable absorption to the laser dynamics and utilise for simulations the characteristics of Tm-doped fibre directly measured in the experiment (Fig. 1d). In fact, an analogous approach based on a combination of the wavelength-dependent and time-dependent gain has been recently applied for accurate modelling of optical pulse propagation in Yb-doped fibres56. However, the used gain presentation as a product of wavelength-dependent and time-dependent factors was rather artificial to approximate a more complicated mathematical function.
The initial field at the first round trip was modelled with "white" Gaussian noise. As Fig. 3a illustrates, a lasing self-start could not be obtained using saturable absorber with saturation intensity Isat = 46 MW/cm2, corresponding to the shortest 1-cm long fibre samples. The optical field tends to zero in this case. The optimal value of saturation intensity is Isat = 150 MW/cm2, which match the parameters of 5-cm section of Tm-fibre used in the experiments. Here, we observed the formation of stable pulse generation (Fig. 3b). Further increase of saturation intensity leads to the distortion of the pulse, significant growth of the Kelly sidebands (Fig. 3c) and pulse break-up (Fig. 3d). Therefore, in further simulations, we fixed parameters of saturable absorber at values corresponding to 5-cm long Tm-doped fibre saturable absorber, i.e. α0 = 0.15 and αns = 0.85, Psat = 100 W.
Fig. 3: Theoretical simulation of spectral evolution during pulse formation with variable saturable absorber properties.
The first 200 round trip of the pulse formation are evaluated in a spectral domain. For a–d increasing values of the Tm-doped fibre saturation intensity (Isat) is assumed from 46 to 52 MW/cm2. Vertical axis corresponds to normalised intensity (in arb. units).
Figure 4a shows simulated output spectra at different gain excitation levels. The increase in cavity feedback leads to a red-shift of the pulse central wavelength from 1872 to 1961 nm, resulting in wavelength tuneability of about 90 nm. The output spectra have pronounced Kelly sidebands characteristic to periodically amplified average solitons.
Fig. 4: Numerical simulations results of pulse tuneability.
a Calculated output spectra corresponding to varying cavity feedback R = 0.05–0.9; b convergence to steady-state regime of pulse generation from different seed parameters in simulations; c evolution of the gain profile along amplifier for R = 0.05 (i) and R = 0.9 (ii). Insets: white lines depict the pulse spectrum; d gain spectra at different points ("1", "2" and "3") along the fibre at 5 and 90% cavity feedback.
We also investigated field evolution towards the steady-state mode-locking regime using initial seed pulse with different parameters to confirm the reproducibility and uniqueness of solution. First, a hyperbolic secant pulse with 1.13 pJ energy, 1 W peak power, 1 ps duration and varying wavelength was used as a seed. The simulation results show that the output pulse wavelength and energy in the stable regime do not depend on the wavelength and character of energy evolution of the seed pulse. Figure 4b shows the example of simulation with the feedback ratio of 30%, where four initial pulses with different wavelengths converge to the same attractor with the increase of the cavity round trips number. Different colours of the lines in Fig. 4b correspond to different wavelengths of the seed pulses at the fixed ~1 pJ initial energy. Such dynamics verifies the uniqueness of the steady-state solution. Overall, this study justifies that the tuneability of the central operation wavelength in the filter-less laser cavity under investigation is governed solely by the wavelength-dependent gain distributed over the fibre length.
To better understand the nature of the tuneability dynamics, we considered the evolution of the gain gs(λ) along the active fibre corresponding to the cavity feedback values R = 5% and R = 90% (Fig. 4c). The pulse spectrum in steady-state is depicted by white lines in Fig. 4c. A feedback increase from 5 to 90% leads to higher pulse energy inside the cavity and faster gain saturation. The gain maximum shifts towards a higher wavelength and causes the shift of the pulse spectrum to find energy balance. Thus, absorption and amplification actively reshape the gain spectrum along the fibre, driving the pulse wavelength. Gain spectra measured at the points "1", "2", and "3" along the fibre (Fig. 4d) qualitatively agree with the spectra shown in Supplementary Fig. S1.
The system of the coupled Eqs. (4)–(9) allows to describe amplification of a narrow sub-picosecond pulse accompanied by dynamically evolving gain spectrum. Note, that in the conventional approach for modelling of Tm-doped amplifiers the system of population inversion rate equations is written for the continuous-wave signal and pump and describes evolution of the average powers43,46,48. The spectral dependence of the gain and nonlinear pulse propagation along the fibre are not taken into account. Here we solve the population inversion rate equations simultaneously with the pulse evolution, where spectral dependence of the gain is calculated at each step along the fibre. Therefore, wavelength tuneability of the output pulse can be demonstrated.
Experimental laser characterisation
We next examined the self-mode-locked Tm-doped fibre laser experimentally. The application of 50 cm of Tm-doped fibre in the ring laser cavity allowed achieving stable self-starting ultrashort pulse generation. Notably, the self-mode-locked operation was observed within the entire range of the cavity feedback tuneability from 8 to 93%, yet featured a different threshold. In general, the mode-locking threshold was relatively higher than one for the schemes with conventional material saturable absorbers. This can be explained by the high saturation intensity (Fig. 1d) of Tm- doped fibre when it acts as a saturable absorber. In contrast to conventional ultrafast lasers, the efficiency of the mode-locked regime is lower than that of continuous-wave generation. We attribute this to the nature of the saturable absorption mechanism in Tm-doped fibre associated with the reabsorption of the laser signal. The intracavity polarisation controller required proper initial adjustment, while later, after several switching on-off cycles, the self-mode-locking regime could self-start without polarisation tuning.
The continuous central wavelength tuneability of the self-mode-locked generation was observed in the range spanning from 1873 and 1962 nm (Fig. 5a) without further alignment. As the numerical simulation predicted, the tuneability was highly reproducible with the feedback variation even after several switch-on and off cycles. Here, we would like to stress that to investigate tuneability dynamics of the single-pulse generation under the same conditions, the pump power and polarisation state in the cavity were fixed, while only the feedback was altered. Naturally, at higher feedback, hence, higher intracavity energy, the fundamental soliton generation regime tended to break into multi-pulsing with the pump power increase. Therefore, we limited the pump power at 0.7 W. Supplementary Movie 1 demonstrates smooth wavelength tuneability of the self-mode-locked generation with no pulse break up or instabilities. The variation of output average power at 0.7 W pump power is presented in Fig. 5b (blue scatters). Its trend is in good agreement with the laser efficiency dynamics predicted numerically using the Rigrod model57 (Supplementary Fig. S5), which showed the highest efficiency at 20% cavity feedback. While the efficiency is decreasing for longer laser operation wavelengths, this deterioration can be efficiently mitigated by appropriate optimisation of background losses according to Rigrod analysis.
Fig. 5: Experimental study of feedback-controlled laser tuneability.
Variation of output pulse characteristics with the variable coupler providing 8 to 93% feedback at fixed polarisation state and 0.7 W pump power. a Wavelength tuneability of the optical spectrum; b variance of the pump threshold (blue, left axis) and output power (orange, right axis); c alteration of pulse duration (blue, left axis) and time-bandwidth product (green, right axis).
Figure 5b, c demonstrates self-mode-locked pulse parameters at the laser output with the different outcoupling ratio. Thus, the pulse duration broadened from 550 to 860 fs with the feedback increase. At the same time, the spectral full width at half-maximum ranges only from 7.4 to 7.7 nm with the increase of the cavity feedback. Consequently, the pulses evolve from nearly transform-limited below 12% cavity feedback to slightly chirped with decreased outcoupling. At high feedback values (over 85%) and, therefore, high intracavity intensities, the time-bandwidth product rises to 1.172, indicating that nonlinear effects do not balance the cavity dispersion. With the increase in feedback, the pump power threshold for achieving self-mode-locking decreased from 538 to 289 mW (Fig. 5b). At the same time, the upper threshold for stable single-pulse operation reduces with higher feedback due to gain saturation.
With careful adjustment of the polarisation controller, the maximum average output power of 68 mW could be obtained in a single-pulse generation regime with 25% feedback and maximum available pump power of 1240 mW, resulting in 5.5% optical conversion efficiency. It is important to notice that neither saturation of the output power nor degradation of any components were observed, while the only limitation for further power increase was the availability of a pump source. Figure 6 demonstrates the output pulse parameters at the highest average power. With its central wavelength at 1889 nm, the optical spectrum displays a bandwidth of 6.8 nm with pronounced Kelly-sidebands, as depicted in Fig. 6a. Figure 6b demonstrates an autocorrelation trace with sech2 approximation and full width at half-maximum of ~870 fs, resulting in a soliton pulse duration of ~600 fs. The fundamental repetition rate complies with the fibre laser cavity length at around 44 MHz (Fig. 6c). Note that the RF spectrum was retrieved numerically via the fast Fourier transform of the pulse train, recorded by the oscilloscope with the dynamic range of 5.5 bits. This limited the signal-to-noise ratio to ~34 dB. Overall, it yields a peak power of 1.6 kW, corresponding to 1.0 nJ pulse energy, bearing in mind that nearly 26% of the total power belongs to the sidebands.
Fig. 6: Measured output pulse characteristics with highest average power.
a Optical spectrum centred at 1889 nm; b autocorrelation trace with sech2 fit; c RF spectrum over 40 GHz (i) and a close-up of the fundamental frequency (ii).
Furthermore, the self-mode-locked generation regime can be adapted by using the feedback variation. Supplementary Fig. S6 shows the map of possible generation regimes with fixed polarisation controller and with its adjustment, ensuring single-pulse generation when possible. The laser regime can be switched from the generation of fundamental solitons to soliton complexes, stable high-harmonics generation (up to fourth order), and chaotic behaviour.
Pulse duration and laser optimisation
As we mentioned before, the self-mode-locking dynamics rely on the fibre cavity nonlinearity, being assisted by SPM. To validate this fact, we have replaced the tuneable coupler in the laser cavity with a fused one, enabling 20% cavity feedback as predicted by Rigrod analysis (Supplementary Fig. S5). In this case, the maximum output power reached 83 mW at 903 mW pump power, resulting in the laser slope efficiency of 14.5%. The output autocorrelation trace, spectrum and pulse train are shown in Supplementary Fig. S7. In brief, the output optical spectrum is centred at 1888 nm. The intensity autocorrelation features a symmetric sech2 peak without secondary signals or a substantial pedestal, which estimates the pulse duration of 350 fs. We tested the self-mode-locked Tm-doped fibre laser generation at maximum output performance over 49 h of continuous operation to confirm its stability. Supplementary Fig. S7d justifies that no pulse breakup occurred, merely a negligible variance in the intensity of sidebands and ~1 nm the central wavelength blue-shift due to temperature fluctuations in the laboratory.
To get a deeper insight into the saturable absorption performance of the Tm-doped fibre and separate this role from the one of the gain medium, we studied the self-mode-locking regime while gradually cutting back the length of the active fibre. Our findings demonstrate that the mode-locking dynamics are sensitive to the length of the active fibre. In the experiments, the ultrashort pulse generation could still be achieved with the length of Tm-doped fibre reduced from 50 down to 47.8 cm with the fine-tuning of the polarisation controller. However, Q-switch intensity modulation became more pronounced and affected the quality of the generation. The cut-back studies demonstrated a reduction of the spectral bandwidth of the generated solitons down to 0.95 nm, as shown in Supplementary Fig. S8. It also reveals the rise of the mode-locking threshold from 610 mW pump power at the original length, over 690–810 mW. In addition, laser efficiency increased with the active fibre length reduction, which can be explained by reduced losses due to signal reabsorption in the unexcited rear part of the active fibre. Overall, the experimental studies supported by numerical simulations presented in Fig. 3 allow us to conclude that only a rear unexcited section of Tm-doped fibre (with a length of around 5–7 cm) can efficiently operate as a saturable absorber.
Our comprehensive investigation of highly Tm-doped fibre properties has confirmed that highly Tm-doped fibre can efficiently operate as a laser gain and saturable absorber and provide generation wavelength tuneability. While comparing two fibre samples with different core glass matrixes and comparable Tm concentrations, we concluded that the refractive index profile is not a decisive criterion for fibre performance as a saturable absorber. In contrast, the important aspects for saturable absorption are the strongly quenched lifetime for the 3F4 level, a rather low cross-relaxation action for the 3H4 level and certain concentration of Tm3+ pairs. These features justify reinforced interionic energy transfer mechanisms between 3F4 ⇀ 3H6: 3H6 ⇀ 3F4 levels. Our study confirms that the estimated concentration of Tm3+ ion pairs of 20% equips aluminosilicate gain fibre with 23% modulation depth and high saturation intensity (95 MW cm−2). These parameters appeared to be sufficient for establishing effective self-mode-locking generation. On the other hand, the germanoaluminosilicate Tm-doped fibre demonstrated a more complicated quenching of Tm ions, and our measurements did not confirm the presence of ion pairs. This fibre featured excessively high saturation intensity and could not initiate mode-locked generation in our laser configuration. Nevertheless, detailed spectroscopic and pump-probe investigation of various Thulium-doped fibre compositions would be beneficial for in-depth evaluation of the impact of clustering and quenching processes on the relaxation and saturable absorption behaviour.
Furthermore, the presented experimental and numerical studies of the self-mode-locked Tm-doped fibre laser with variable feedback demonstrated broadband central wavelength tuneability. The results of numerical simulations agree well with the experiment, demonstrating the same tuneability range. Both investigations have confirmed the variation of the gain excitation level through alteration of cavity feedback to be the primary mechanism of the tuneability of the ultrashort pulse spectrum. Our numerical simulations have also confirmed that the range of observed gain-controlled tuneability through laser feedback variation is determined by the emission cross-section and bandwidth of used Tm-doped fibre (see Supplementary Fig. S4). It is important to note that the amplification and absorption, responsible for different aspects of the laser dynamics, are localised in different parts of the fibre. Wavelength tuneability is provided by wavelength-dependent gain distributed over the fibre length, whereas the saturable absorption localised at the fibre rear end is responsible for the pulse formation.
In conclusion, the experimental and theoretical investigations reported in the current work have extended the understanding of gain dynamics in ultrafast fibre lasers. The self-mode-locked laser setup enabled the generation of stable self-starting ultrashort pulses with the duration of 350 fs at 45 MHz repetition rate delivering ~1.2 nJ of the energy in the main pulse peak. To the best of our knowledge, this is the first demonstration of femtosecond pulse generation directly from the self-mode-locked Thulium-doped fibre laser. Moreover, through the alteration of cavity feedback in the ultrafast Tm-doped fibre laser, we have recorded a central wavelength shift within the range spanning from 1873 to 1962 nm. The control of the excitation level and gain through variation of out-coupled power proved to be more advantageous than loss management. Together with a broad emission spectrum and high cross-section, this became enabling aspects for more than doubled tuneability range. From an instrument development viewpoint, our results have provided a further example of the incredible flexibility of ultrafast fibre lasers. The next step to provide a higher stability and avoid polarisation instabilities could be a realisation of all-polarisation maintaining laser cavity employing the concept of feedback-controlled self-mode-locking. Yet, this would require first and foremost the development of a Tm-doped fibre with properties similar and, particularly, glass matrix to the one studied here. Although the current studies were focused on Tm-doped fibres as the gain medium, the key underlying phenomenon of the suggested tuneability method is versatile. It could be translated to other wavelengths, where the majority of fibreised laser components, including filters, are currently unavailable. In particular, this refers to exploring the Mid-IR wavelength range, where Dy and Er-doped fluoride-based fibres also offer exceptionally broad gain spectra.
The measuring equipment is made up of a 10 GHz extended InGaAs photodetector (ET-5000, EOT) next to a 25 GS/s oscilloscope (DPO 70604C, Tektronix) for electric characterisation. For optical characterisation an optical spectrum analyser with down to 50 pm resolution and a 1.1–2.5 μm spectral coverage is used (AQ6375, Yokogawa). With a frequency doubling autocorrelator (PulseCheck, APE), the time dependence has been examined. For quantifying the mean optical power, an integrating sphere type sensor with InGaAs photodiode and 1 nW resolution is employed (S148C, Thorlabs).
Experimental data underlying the results presented in this paper are available in ref. 58.
Computer code for the numerical simulations is available upon reasonable request from the authors.
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D.C.K. and M.C. acknowledge the support of the Deutsche Forschungsgemeinschaft (DFG—German Research Foundation, Project No. CH 2600_1-1). A.B. acknowledges the support of the Ministry of Science and Higher Education of the Russian Federation (Project No. FSUS-2021-0015). A.F. acknowledges the support of the Ministry of Science and Higher Education of the Russian Federation (Project No. 075-15-2021-581) and Russian Science Foundation (Project No. 18-12-00457P).
Open Access funding enabled and organized by Projekt DEAL.
Ultrafast Fibre Laser, Leibniz Institute of Photonic Technology, Albert-Einstein-Str. 9, 07745, Jena, Germany
Dennis C. Kirsch & Maria Chernysheva
Novosibirsk State University, 1 Pirogova str., Novosibirsk, 630090, Russia
Anastasia Bednyakova
Institute of Photonics and Electronics, The Czech Academy of Sciences, Chaberska 57, 18251, Prague, Czechia
Petr Varak, Pavel Honzatko & Pavel Peterka
iXblue Photonics, Rue Paul Sabatier, 22300, Lannion, France
Benoit Cadier & Thierry Robin
Ulyanovsk State University, 42, Leo Tolstoy Street, Ulyanovsk, 432017, Russia
Andrei Fotiadi
Electromagnetism and Telecommunication Department, University of Mons, B-7000, Mons, Belgium
Dennis C. Kirsch
Petr Varak
Pavel Honzatko
Benoit Cadier
Thierry Robin
Pavel Peterka
Maria Chernysheva
D.C.K. and M.C. conceived the experiment(s), D.C.K. conducted laser development experiments, A.B. and A.F. built the numerical model, A.B. performed numerical simulations. B.C. and T.R. developed highly Tm-doped fibres. D.C.K., P.V., P.H. and P.P. have investigated gain fibre properties. All authors contributed to results discussions and writing the manuscript.
Correspondence to Dennis C. Kirsch.
Communications Physics thanks Xiaomin Liu and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.
Kirsch_PR File
Description of Additional Supplementary Files
Supplementary Movie 1
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
Kirsch, D.C., Bednyakova, A., Varak, P. et al. Gain-controlled broadband tuneability in self-mode-locked Thulium-doped fibre laser. Commun Phys 5, 219 (2022). https://doi.org/10.1038/s42005-022-00989-x
DOI: https://doi.org/10.1038/s42005-022-00989-x
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Developing Walvis Bay Port into a logistics gateway for southern Africa: Issues, challenges and the potential implications for Namibia's future
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Many developing countries wish to become the 'gateway' to a region or part of a continent.One strategy involves encouraging logistics cluster development. These hubs support [...]
Modeling road accident injury under-reporting in Europe
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Quantifying the Benefits from a Major Infrastructure Improvement: The Case of Thessaloniki Western Ring Road Upgrade to Eliminate at Grade Signalised Intersections
E. Bouhouras, M. Miltiadou
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Collision risk-capacity tradeoff analysis of an en-route corridor model
M. Hu, J. Shortle, B. Ye
Flow corridors are a new class of trajectory-based airspace which derives from the next generation air transportation system concept of operations. Reducing the airspace complexity and increasing the capacity are the main purposes of the en-route corridor. This paper analyzes the collision risk-capacity tradeoff using a combined discrete–continuous simulation method. A basic two-dimensional en-route flow corridor with performance rules is designed as the operational environment. A second-order system is established by combining the point mass model and the proportional derivative controller together to simulate the self-separation operations of the aircrafts in the corridor and the operation performance parameters from the User Manual for the Base of Aircraft Data are used in this research in order to improve the reliability. Simulation results indicate that the aircrafts can self-separate from each other efficiently by adjusting their velocities, and rationally setting the values of some variables can improve the rate and stability of the corridor with low risks of loss of separation. Document type: Article
Flow corridors are a new class of trajectory-based airspace which derives from the next generation air transportation system concept of operations. Reducing the airspace complexity [...]
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The current literature in the field of cycle lanes has often shown contradictory evidence as to the benefits and risks of cycle lanes and previous work has specifically shown [...]
The Geopolitical Energy Security Evaluation Method and a China Case Application Based on Politics of Scale
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monitoring, specifically infrastructure monitoring such as water distribution pipelines, is becoming increasingly critical for utility owners who face new challenges due to [...]
A new methodology for approaching motorcycle riders' behavior at curved road sections
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The present paper focuses on the investigation of motorcycle riders' behavior at curved road sections by introducing a new methodology based on global positioning system (GPS) technology. In the frame of the research, the determination of the optimum regression curve between the curve radius' and the corresponding velocities, was investigated. Within the context of the paper field measurements were conducted, with the use of appropriate velocity recording equipment in order to confirm the efficiency of the proposed methodology. The measurements were conducted taking into account various factors that potentially influence riders' behavior such as the different light conditions, the difference on riding experience level, the familiarity of the riders with the routes, the presence of pillion and the different road environments, such as width/condition of the road pavement, roadside land use, right/left hand curves etc. The experimental environment that served the needs of the experiment was mountain Pelion in Magnesia region in Greece and was based upon four primary conditions: the location, the type of the road, the weather conditions and finally, the time and date that the experiment would be conducted. The validation of the proposed methodology was performed by recruiting two motorcyclists. Their selection was based on demographic, psychometric and experience criteria. The research showed among others, that the regression curves could be used as a curve classification mean. Moreover, a significant variation was detected on the riders' behavior when carrying a pillion related to their experience levels. Document type: Article
The present paper focuses on the investigation of motorcycle riders' behavior at curved road sections by introducing a new methodology based on global positioning system [...]
An overview of heavy oil properties and its recovery and transportation methods
R. Santos, W. Loh, A. Bannwart, O. Trevisan, 0. 0000-0002-8049-3321, 0. 0000-0003-1618-3449, S. informação
Submitted by Susilene Barbosa da Silva ([email protected]) on 2020-07-09T13:35:58Z No. of bitstreams: 0 Made available in DSpace on 2020-07-09T13:35:58Z (GMT). No. of bitstreams: 0 Previous issue date: 2014 Unconventional oils - mainly heavy oils, extra heavy oils and bitumens - represent a significant share of the total oil world reserves. Oil companies have expressed interest in unconventional oil as alternative resources for the energy supply. These resources are composed usually of viscous oils and, for this reason, their use requires additional efforts to guarantee the viability of the oil recovery from the reservoir and its subsequent transportation to production wells and to ports and refineries. This review describes the main properties of high-viscosity crude oils, as well as compares traditional and emergent methods for their recovery and transportation. The main characteristics of viscous oils are discussed to highlight the oil properties that affect their flowability in the processes of recovery and pipeline transportation. Chemical composition is the starting point for the oil characterization and it has major impact on other properties, including key properties for their dynamics, such as density and viscosity. Next, enhanced oil recovery (EOR) methods are presented, followed by a discussion about pipeline and transportation methods. In addition, the main challenges to achieve viable recovery and transportation of unconventional oils are compared for the different alternatives proposed. The work is especially focused on the heavy oils, while other hydrocarbon solid sources, such as oil sands and shale oil, are outside of the scope of this review 31 3 571 590 Document type: Article
Submitted by Susilene Barbosa da Silva ([email protected]) on 2020-07-09T13:35:58Z No. of bitstreams: 0 Made available in DSpace on 2020-07-09T13:35:58Z (GMT). No. of bitstreams: [...]
Research on Congestion Pricing in Multimode Traffic considering Delay and Emission
R. Zhao, H. Dai, E. Yao
Rapid development of urbanization and automation has resulted in serious urban traffic congestion and air pollution problems in many Chinese cities recently. As a traffic demand management strategy, congestion pricing is acknowledged to be effective in alleviating the traffic congestion and improving the efficiency of traffic system. This paper proposes an urban traffic congestion pricing model based on the consideration of transportation network efficiency and environment effects. First, the congestion pricing problem under multimode (i.e., car mode and bus mode) urban traffic network condition is investigated. Second, a traffic congestion pricing model based on bilevel programming is formulated for a dual-mode urban transportation network, in which the delay and emission of vehicles are considered. Third, an improved mathematical algorithm combining successive average method with the genetic algorithm is proposed to solve the bilevel programming problem. Finally, a numerical experiment based on a hypothetical network is performed to validate the proposed congestion pricing model and algorithm. Document type: Article
Rapid development of urbanization and automation has resulted in serious urban traffic congestion and air pollution problems in many Chinese cities recently. As a traffic [...]
Environmental Factors and Intermodal Freight Transportation: Analysis of the Decision Bases in the Case of Spanish Motorways of the Sea
M. López-Navarro
Today, there is widespread consensus about the notable, yet simultaneously growing, negative environmental impacts generated by the transportation sector. Experts working in a number of different fields consider the current situation to be unsustainable and possible measures to reduce emissions and foster sustainability are being encouraged. The European Commission has highlighted the need to shift away from unimodal road transport toward a greater use of intermodal transport through, for example, motorways of the sea, in light of the evidence that the former makes a significant contribution to increased CO 2 emissions. However, although there is a general perception that sea transport is environmentally preferable to road transport, recent studies are beginning to question this assumption. Moreover, little research has been conducted to quantify environmental aspects and incorporate them into the decision-making processes involved in the modal shift. This study first reviews the existing literature to examine the extent to which environmental aspects are relevant in the modal choice in the case of short sea shipping and motorways of the sea. Related to this, the study also evaluates the role that different agents may play in making decisions about choice of mode, taking into consideration environmental aspects. Secondly, we use the values the European Commission provides to calculate external costs for the Marco Polo freight transport project proposals (call 2013) to estimate the environmental costs for several routes (a total of 72), comparing the use of road haulage with the intermodal option that incorporates the Spanish motorways of the sea. The results of this comparative analysis show that the intermodal option is not always the best choice in environmental terms. Consequently, the traditional environmental argument to justify this alternative must be used carefully. Document type: Article
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Consensus between Pipelines in Structural Brain Networks
S. Ourselin, M. Dayan, M. Dayan, J. Clayden, C. Clark, P. Daga, C. Parker, C. Parker, M. Cardoso, M. Modat, F. Deligianni
Structural brain networks may be reconstructed from diffusion MRI tractography data and have great potential to further our understanding of the topological organisation of brain structure in health and disease. Network reconstruction is complex and involves a series of processesing methods including anatomical parcellation, registration, fiber orientation estimation and whole-brain fiber tractography. Methodological choices at each stage can affect the anatomical accuracy and graph theoretical properties of the reconstructed networks, meaning applying different combinations in a network reconstruction pipeline may produce substantially different networks. Furthermore, the choice of which connections are considered important is unclear. In this study, we assessed the similarity between structural networks obtained using two independent state-of-the-art reconstruction pipelines. We aimed to quantify network similarity and identify the core connections emerging most robustly in both pipelines. Similarity of network connections was compared between pipelines employing different atlases by merging parcels to a common and equivalent node scale. We found a high agreement between the networks across a range of fiber density thresholds. In addition, we identified a robust core of highly connected regions coinciding with a peak in similarity across network density thresholds, and replicated these results with atlases at different node scales. The binary network properties of these core connections were similar between pipelines but showed some differences in atlases across node scales. This study demonstrates the utility of applying multiple structural network reconstrution pipelines to diffusion data in order to identify the most important connections for further study. Document type: Article
Structural brain networks may be reconstructed from diffusion MRI tractography data and have great potential to further our understanding of the topological organisation of [...]
A Proposed Scalable Design and Simulation of Wireless Sensor Network-Based Long-Distance Water Pipeline Leakage Monitoring System
A. Alotaibi, A. Almazyad, Y. Seddiq, A. Obeid, M. BenSaleh, S. Qasim, A. Al-nasheri
nomalies such as leakage and bursts in water pipelines have severe consequences for the environment and the economy. To ensure the reliability of water pipelines, they must be monitored effectively. Wireless Sensor Networks (WSNs) have emerged as an effective technology for monitoring critical infrastructure such as water, oil and gas pipelines. In this paper, we present a scalable design and simulation of a water pipeline leakage monitoring system using Radio Frequency IDentification (RFID) and WSN technology. The proposed design targets long-distance aboveground water pipelines that have special considerations for maintenance, energy consumption and cost. The design is based on deploying a group of mobile wireless sensor nodes inside the pipeline and allowing them to work cooperatively according to a prescheduled order. Under this mechanism, only one node is active at a time, while the other nodes are sleeping. The node whose turn is next wakes up according to one of three wakeup techniques: location-based, time-based and interrupt-driven. In this paper, mathematical models are derived for each technique to estimate the corresponding energy consumption and memory size requirements. The proposed equations are analyzed and the results are validated using simulation. Document type: Article
nomalies such as leakage and bursts in water pipelines have severe consequences for the environment and the economy. To ensure the reliability of water pipelines, they must [...]
Research on Gas Hydrate Plug Formation under Pipeline-Like Conditions
T. Scholz, F. Merkel, C. Schmuck, H. Schultz, S. Wolinski
Hydrates of natural gases like methane have become subject of great interest over the last few decades, mainly because of their potential as energy resource. The exploitation of these natural gases from gas hydrates is seen as a promising mean to solve future energetic problems. Furthermore, gas hydrates play an important role in gas transportation and gas storage: in pipelines, particularly in tubes and valves, gas hydrates are formed and obstruct the gas flow. This phenomenon is called "plugging" and causes high operational expenditure as well as precarious safety conditions. In this work, research on the formation of gas hydrates under pipeline-like conditions, with the aim to predict induction times as a mean to evaluate the plugging potential, is described. Document type: Article
Hydrates of natural gases like methane have become subject of great interest over the last few decades, mainly because of their potential as energy resource. The exploitation [...]
Innovative Tools of Sustainable Mobility in European Urban Areas: Experience with Evaluation and Role of Political Barriers
H. Brůhová-Foltýnová, R. Jordová
The presented article describes a European initiative for supporting innovative measures in sustainable transport in urban areas Civitas, particularly its latest completed edition Civitas Plus (2008-2012). The article presents the evaluation process of the performed measures and shows an overview of the main obstacles the Civitas Plus cities struggled with (with the emphasis on political barriers). In addition, it mentions drivers which helped to implement the sustainable mobility measures. It concludes with examples from European cities on how to avoid these barriers and prevent their occurrence. Document type: Article
The presented article describes a European initiative for supporting innovative measures in sustainable transport in urban areas Civitas, particularly its latest completed [...]
A Study of Feature Combination for Vehicle Detection Based on Image Processing
J. Arróspide, L. Salgado
Video analytics play a critical role in most recent traffic monitoring and driver assistance systems. In this context, the correct detection and classification of surrounding vehicles through image analysis has been the focus of extensive research in the last years. Most of the pieces of work reported for image-based vehicle verification make use of supervised classification approaches and resort to techniques, such as histograms of oriented gradients (HOG), principal component analysis (PCA), and Gabor filters, among others. Unfortunately, existing approaches are lacking in two respects: first, comparison between methods using a common body of work has not been addressed; second, no study of the combination potentiality of popular features for vehicle classification has been reported. In this study the performance of the different techniques is first reviewed and compared using a common public database. Then, the combination capabilities of these techniques are explored and a methodology is presented for the fusion of classifiers built upon them, taking into account also the vehicle pose. The study unveils the limitations of single-feature based classification and makes clear that fusion of classifiers is highly beneficial for vehicle verification. This work was partially supported by the Ministerio de Economía y Competitividad of the Spanish Government under Project TEC2010-20412 (Enhanced 3D TV). Document type: Article
Video analytics play a critical role in most recent traffic monitoring and driver assistance systems. In this context, the correct detection and classification of surrounding [...]
Experimental Correlations with Calculus Parameters for a Dynamic System Equipped with Antiseismic Elastomeric Devices
P. Bratu
Rezumat"/jats:title" "jats:p" Lucrarea prezintă rezultatul încercărilor efectuate la cicluri cinematice cu deplasări instantanee armonice pentru elemente elastomerice cu rol de izolare dinamică în conformitate cu SR EN 1337-3 şi SR EN 15129. Valorile amortizării interne determinate pe standuri de încercare în condiţiile din documentele de referinţă trebuie corelate cu valorile sistemului de dispozitive antiseismice elastomerice ce echipează un sistem dinamic supus acţiunilor seismice exterioare."/jats:p" "jats:p"În acest caz, pentru fiecare situaţie reală, proiectantul trebuie să realizeze corelaţia dintre cele două valori ale amortizării obţinute în laborator cu amortizarea specifică sistemului structural care este supus acţiunilor dinamice exterioare. Din lucrare rezultă echivalenţa necesară astfel încât, valorile experimentale să poată fi corectate în mod riguros şi să poată reprezenta parametrii reali ce trebuie luaţi în calcul. Document type: Article
Rezumat"/jats:title" "jats:p" Lucrarea prezintă rezultatul încercărilor efectuate la cicluri cinematice cu deplasări instantanee armonice pentru elemente elastomerice [...]
The impact of high-speed rail and low-cost carriers on European air passenger traffic
R. Clewlow, J. Sussman, H. Balakrishnan
The expansion of high-speed passenger rail service is often argued as a potentially effective, lower-carbon substitute for intercity air travel. Previous studies on the impact of high-speed rail on air travel in Europe and Asia have primarily examined the impact of travel time and price on market share for a specific city pair (or a handful of city pairs). There has been little focus on the extent to which high-speed rail (HSR) has reduced total short-haul air travel demand (versus market share), or on the potential impacts of high-speed rail on system-wide air travel demand. This paper presents an empirical, econometric analysis of air travel demand in Europe, utilizing an expanded data set to explore: (1) the impact of rail travel times, population density, and market characteristics on air traffic; and (2) the impact of high-speed rail and low-cost-carriers on system-wide air traffic. Although improvements in rail travel times have resulted in reductions in short-haul air travel, variations in city and airport characteristics significantly influence the substitution between air and rail. This paper also finds that HSR substitution has resulted in a modest reduction in system-wide air travel demand, whereas the expansion of low-cost carriers has led to a significant increase in total European air traffic. As concerns about the climate impacts of transportation grow, these results have significant implications for future transport and energy policy. National Science Foundation (U.S.) (National Academy of Sciences (U.S.). Grant OISE-738129) Document type: Article
The expansion of high-speed passenger rail service is often argued as a potentially effective, lower-carbon substitute for intercity air travel. Previous studies on the impact [...]
Oil Pollution in the Southeastern Baltic Sea in 2009-2011
A. Kostianoy, M. Mityagina, O. Lavrova, A. Semenov
"jats:p"From January 2009 to April 2012 a satellite survey of the central and southeastern parts of the Baltic Sea was carried out by the Space Radar Laboratory at the Space Research Institute of Russian Academy of Sciences (RAS). The main attention was focused on the detection of oil pollution as well as biogenic and anthropogenic surfactant films. The basic data are high resolution radar images obtained by advanced synthetic aperture radar (ASAR) on board of the Envisat satellite of the European Space Agency. Remotely sensed data in visual and infrared (IR) bands acquired by sensors MERIS Envisat, MODIS-Terra and -Aqua, and AVHRR NOAA nearly simultaneously with the ASAR images, were processed and analysed in order to facilitate the discrimination between different types of surface pollutants, to understand a comprehensive features of meteorological and hydrodynamic processes in the sea area of investigation, and to reveal factors determining pollutants spread and drift. The regions of the most intense oil pollution are outlined. Document type: Article
"jats:p"From January 2009 to April 2012 a satellite survey of the central and southeastern parts of the Baltic Sea was carried out by the Space Radar Laboratory at the Space [...]
Assessing driver's ability to estimate compliance rates to in-car, advisory driver support
M. Martens, M. Risto
Purpose In-car support systems focus increasingly on improving traffic flow and throughput. Advisory systems allow for fast market penetration, advising drivers how to drive in order to improve general flow. By following the advice, drivers cannot create a beneficial effect by themselves but rely on other road users to comply as well. Drivers who sense a low compliance among other road users may be discouraged to use the system themselves. The present experiment investigated whether drivers are able to distinguish between various compliance rates to Connected Cruise Control (CCC), an advisory driver support system that gives headway, speed and lane advice to improve throughput on motorways. Method Forty-two participants estimated the compliance of other road users to CCC in a driving simulator. Actual system compliance was varied between 10, 50 and 90 %. Half of the participants received detailed information about the advice and the manifestation of compliant behaviour in traffic. Results Compliance estimates showed no effect of actual compliance rates. Overall compliance ratings were higher for participants who had not received additional information about the system. Difference scores between compliance estimate and actual compliance indicate that additional information did not improve estimation accuracy, neither did it increase participants' confidence with their estimate. Conclusions When actual compliance is low, drivers still show high compliance estimates which can have beneficial effect on system acceptance. Additional information does not improve compliance estimates. Document type: Article
Purpose In-car support systems focus increasingly on improving traffic flow and throughput. Advisory systems allow for fast market penetration, advising drivers how to drive [...]
Towards Life Cycle Sustainability Assessment of Alternative Passenger Vehicles
O. Tatari, M. Kucukvar, N. Onat
Sustainable transportation and mobility are key components and central to sustainable development. This research aims to reveal the macro-level social, economic, and environmental impacts of alternative vehicle technologies in the U.S. The studied vehicle technologies are conventional gasoline, hybrid, plug-in hybrid with four different all-electric ranges, and full battery electric vehicles (BEV). In total, 19 macro level sustainability indicators are quantified for a scenario in which electric vehicles are charged through the existing U.S. power grid with no additional infrastructure, and an extreme scenario in which electric vehicles are fully charged with solar charging stations. The analysis covers all life cycle phases from the material extraction, processing, manufacturing, and operation phases to the end-of-life phases of vehicles and batteries. Results of this analysis revealed that the manufacturing phase is the most influential phase in terms of socio-economic impacts compared to other life cycle phases, whereas operation phase is the most dominant phase in the terms of environmental impacts and some of the socio-economic impacts such as human health and economic cost of emissions. Electric vehicles have less air pollution cost and human health impacts compared to conventional gasoline vehicles. The economic cost of emissions and human health impact reduction potential can be up to 45% and 35%, respectively, if electric vehicles are charged through solar charging stations. Electric vehicles have potential to generate income for low and medium skilled workers in the U.S. In addition to quantified sustainability indicators, some sustainability metrics were developed to compare relative sustainability performance alternative passenger vehicles. BEV has the lowest greenhouse gas emissions and ecological land footprint per $ of its contribution to the U.S. GDP, and has the lowest ecological footprint per unit of its energy consumption. The only sustainability metrics that does not favor the BEV is the water-energy ratio, where the conventional gasoline vehicle performed best. Document type: Article
Sustainable transportation and mobility are key components and central to sustainable development. This research aims to reveal the macro-level social, economic, and environmental [...]
Urban Policies On Diversity In Antwerp, Belgium
A. Saeys, Y. Albeda, N. Van Puymbroeck, S. Oosterlynck, G. Verschraegen, D. Dierckx
Collection of open reports in transport research (2014). 63
Critical analysis of existing urban policy programmes and discourses in Antwerp, Belgium. Includes overview of political systems and governance structures, key shifts in national discourses, and approaches to policy over migration, citizenship, and diversity.
Critical analysis of existing urban policy programmes and discourses in Antwerp, Belgium. Includes overview of political systems and governance structures, key shifts in national [...]
SIMPATO-The safety impact assessment tool of interactive
M. Noort, T. Bakri, F. Fahrenkrog, J. Dobberstein
One step in the development of safety oriented Advanced Driver Assistance Systems (ADAS) is an ex ante assessment of the expected safety impacts. This requires a careful analysis combining models and data from various sources. This paper describes the Safety IMPact Assessment Tool, called SIMPATO, that was developed in the interactIVe project. This tool performs «what if» analysis for accident scenarios to determine the effect of an ADAS on the outcome. The unique quality of the tool is that it requires very little data on the ADAS itself, and uses in-depth accident data to obtain a representative result.
One step in the development of safety oriented Advanced Driver Assistance Systems (ADAS) is an ex ante assessment of the expected safety impacts. This requires a careful analysis [...]
Quantitative assessment of environmental risk due to accidental spills from onshore pipelines
G. Antonioni, P. Morra, V. Cozzani, S. Bonvicini
The transport of hazardous materials by pipeline is widely used for the transfer of significant quantities of oil and chemicals. Due to the extremely low frequency of spills, pipelines are considered the safest mode for the land transportation of hazardous substances. Accident records, while confirming that Loss of Containment (LOC) events are rare, also point out the major-accident hazard of pipelines, due to the extremely severe potential consequences of spills. Quantitative Risk Analysis (QRA) techniques have been applied to pipelines since many years with the aim of evaluating risk for workers or exposed population. However, releases of liquids, as oil and oil products, also create an hazard to the environment, due to the potential of extensive soil and groundwater contamination. An integrated model was developed for the environmental Risk Analysis of spills from pipelines. Specific environmental risk indexes were defined, expressing the risk of soil and groundwater contamination, both in physical and economic terms. A case-study is presented and discussed to illustrate the features of the methodology. The results confirmed that the proposed model may be considered an important tool within a comprehensive approach to the management of risk related to onshore pipelines.
The transport of hazardous materials by pipeline is widely used for the transfer of significant quantities of oil and chemicals. Due to the extremely low frequency of spills, [...]
Airport congestion pricing when airlines price discriminate
A. Czerny, A. Zhang
This paper extends the literature on airport congestion pricing by allowing carriers to price-discriminate between the business and leisure passengers when operating costs are the same for all passengers. The main results are: First, the second-best discriminating business fare exceeds the first-best uniform fare (which equals the external part of the marginal congestion costs), while the second-best discriminating leisure fare is lower than the first-best uniform fare. Second, the optimal airport charge implements the first-best uniform or second-best discriminating fares. Importantly, this charge can always be higher than what would be expected when all passengers were treated as having the same time valuation. This result provides some support to the finding that the welfare losses associated with an atomistic airport congestion charge may be low. © 2014 Elsevier Ltd.
This paper extends the literature on airport congestion pricing by allowing carriers to price-discriminate between the business and leisure passengers when operating costs [...]
Determination of the dip angles of Karayaka and Gok double dagger e fault planes in Turkey by seismic reflection method
F. Saroglu, G. Aldas, B. Ecevitoğlu
WOS: 000360068200081 Along its track between Samsun and Adana, an oil pipeline was planned to cross two active faults among others: Gok double dagger e Fault located at 7 km south of Tokat and Karayaka Fault located at 45 km northeast of Tokat. Geological investigations suggest that both faults reveal reverse fault properties. Determination of the dip angles of the fault planes is crucial in pipeline construction. Fault plane dip angles can be best determined by seismic reflection methods. Although first few metres from the Earth's surface are important in pipeline construction, few hundred metres of penetration depth are required for reliable fault identifications and fault-parameter computations. Two seismic profiles were accomplished on Gok double dagger e and Karayaka faults. Computations based on seismic depth sections obtained in this study reveal that the apparent dip angle of Gok double dagger e Fault is 84.6A degrees, and the apparent dip angle of Karayaka Fault is 72.4A degrees. ENVY Inc. This work has been performed within the cooperative work of the Ankara University Geophysics and Energy and Environmental Investments Inc. (ENVY). We acknowledge with gratitude the technical and financial support of ENVY Inc. The authors also give thanks to the stuff of General Directorate of Disaster Affairs Earthquake Research Department for providing seismic recording instruments and their technical support in the field.
WOS: 000360068200081 Along its track between Samsun and Adana, an oil pipeline was planned to cross two active faults among others: Gok double dagger e Fault located at 7 [...]
Theoretical modeling and experimental investigations for the improvement of the mechanical efficiency in sliding vane rotary compressors
G. Bianchi, R. Cipollone
Positive displacement compressors lead the market of compressed air production for industrial applications. Among them, sliding vane rotary compressors represent an energetically virtuous alternative to the current compression technologies. In the present work, the effects of compressor design parameters were investigated through a comprehensive approach that aimed at addressing more efficient machines to promote sliding vane compressors as the key enabling technology in compressed air systems. A comprehensive mathematical model was developed to study the main phenomena occurring in this kind of compressors. The model provides the cell volume evolution over a whole rotation during which filling, compression and discharge processes occur. The first and latter phases are described by the quasi-propagatory approach that represents the inertial, capacitive and resistive features of one-dimensional unsteady flows. The dynamics of the compressor blades led to four different arrangements inside the rotor slots while an analysis of the hydrodynamic lubrication established between blade tip and stator wall focused on the oil film thickness evolution to prevent dry contacts. An extensive experimental campaign on a mid-size industrial compressor allowed the model validation at different outlet pressure levels and revolution speeds using a direct measurement of mechanical power and the reconstruction of the indicator diagram from piezoelectric pressure transducers. The friction coefficient at the contact points between blades with stator and rotor was estimated in 0.065 and further improvements of the mechanical efficiency were eventually addressed considering the roles of compressor aspect ratio, revolution speed, and blade tilt. The first two theoretical optimizations might lead to an increase of the compressor efficiency of 2 and 9 percentage points respectively. On the other hand, acting on the blade tilt would not produce relevant improvements. The Authors acknowledge Ing. E. Mattei S.p.A. and particularly its CEO, Dr. Giulio Contaldi, for continuous research funding and support. The work has been done also under the FP7 Project ''Complete Vehicle Energy-Saving CONVENIENT'' founded by the European Commission.
Positive displacement compressors lead the market of compressed air production for industrial applications. Among them, sliding vane rotary compressors represent an energetically [...]
A study of challenges to the success of the safety management system in aircraft maintenance organizations in Turkey
E. Gerede
WOS: 000348088600013 The ICAO is an international regulatory authority put in place, in addition to current prescriptive regulatory approaches based on regulatory compliance that use reactive tools, performance based approaches that focus on, processes, proactivity, productivity and safety performance, to reduce the number of accidents and fatalities irrespective of the volume of air traffic. The safety management system (SMS), a product of this new approach, requires transformations which are likely to create certain challenges to its performance. This study sets out to investigate the challenges to the successful implementation of SMS in aircraft maintenance organizations, the degree of priority of these challenges, the major problems affecting the performance of SMS, the factors causing the problems, and the ensuing results. During a two-day workshop, a problem analysis was conducted with experts using the nominal group technique. At the end of the study, it was identified that 'just culture' problems would be the main challenge to the success of the SMS. It was predicted that, impairing the reporting process, these problems are likely to have an adverse impact on information acquisition within an organization, organizational learning, efficiency of predictive tools and proactivity
WOS: 000348088600013 The ICAO is an international regulatory authority put in place, in addition to current prescriptive regulatory approaches based on regulatory compliance [...]
Book 'Moving towards low carbon mobility' review
E. Bruun
"Book 'Moving towards low carbon mobility' review" Transport, 29(3), pp. 342-344 First Published Online: 22 Sep 2014
A real-time multi-sensor fusion platform for automated driving application development
T. Bijlsma, M. Mnatsakanyan, M. Kwakkernaat
dvanced Driver Assistance Systems (ADAS) become standard and sometimes mandatory for vehicles, e.g. autonomous emergency braking. Future vehicles will include multiple ADAS that assist with safety-critical operations. For efficiency and effectiveness, these ADAS should share resources, information and functionalities. Additionally, ADAS performing safety-critical functionality require predictability for the execution of their processes. This paper presents the integration of a layer-based multi-sensor fusion and processing platform into a real-time system that also supports non-critical processes, i.e. a mixed-criticality system. A suitable system is selected, the behavior of the platform and its interfaces is described and tests are performed to validate the predictable behavior, by examining the difference in jitter and execution latency. The real-time layer-based platform is suitable for the development and testing of multiple integrated safety-critical ADAS. cop. 2015 IEEE.
dvanced Driver Assistance Systems (ADAS) become standard and sometimes mandatory for vehicles, e.g. autonomous emergency braking. Future vehicles will include multiple ADAS [...]
Changes in Oil Transportation in the Years 2020 and 2030 – The Case of the Gulf of Finland
O. Brunila, J. Storgård
This paper covers the current state of maritime oil transportation in the Baltic Sea and the development of oil transportation in the 2000s, as well as estimations of transported oil volumes in 2020 and 2030 in the Gulf of Finland. The scenarios were formulated on the basis of a current state analysis, energy and transportation strategies and scenarios and expert assessments. The study showed that the volumes of oil transportation in the Gulf of Finland will increase only moderately compared to the current status: 9.5-33.8 %, depending on the scenario. Green energy policy favours renewable energy sources, which can be seen in the smaller volumes of transported oil in the 2030 scenarios compared to the 2020 scenarios. In the Slow development 2020 scenario, oil transport volumes for 2020 are expected to be 170.6 Mt (million tonnes), in the Average development 2020 187.1 Mt and in the Strong development 2020 201.5 Mt. The corresponding oil volumes for the 2030 scenarios were 165 Mt for the Stagnating development 2030 scenario, 177.5 Mt for the Towards a greener society 2030 scenario and 169.5 Mt in the Decarbonising society 2030 scenario.
This paper covers the current state of maritime oil transportation in the Baltic Sea and the development of oil transportation in the 2000s, as well as estimations of transported [...]
Reinterpreting EU Air Transport Deregulation: A Disaggregated Analysis of the Spatial Distribution of Traffic in Europe, 1990-2009
G. Burghouwt, X. Fageda, P. Suau-Sanchez
This paper analyses the spatial distribution of seat capacity in the EU from 1990 to 2009 and sheds light on the contrasting results in the literature. It contributes to the debate on the deregulation and whether the rise of hub-and-spoke networks and the success of low-cost carriers lead to concentration or deconcentration. We use the Gini index and its decomposition to evaluate the contribution of airport subgroups and airline networks to the overall concentration of seat capacity. We conclude that, overall, seat capacity follows a spatial deconcentration pattern. While intra-EU seat capacity became more spatially deconcentrated, extra-EU seat capacity concentrated. However, our results do not support the general view that network carriers tend to increase concentration levels and low-cost carriers to decrease them, leading us to a reinterpretation of the impacts of air transport deregulation. The results show the increasing importance of foreign carriers and new strategies such as hub-bypassing.
This paper analyses the spatial distribution of seat capacity in the EU from 1990 to 2009 and sheds light on the contrasting results in the literature. It contributes to the [...]
Understanding mode choice in the Chinese context: the case of Nanjing Metropolitan Area
J. Feng, J. Feng, M. Dijst, B. Wissink, J. Prillwitz
In post-reform China, rapid motorisation causes various problems like traffic congestion, diminishing road safety and air pollution. Adequate policies necessitate an understanding of the forces behind changing mode choices, but the rapidly developing literature is not complete yet. This paper aims to help fill that gap with an analysis of mode choice for commuting and shopping-leisure trips in Nanjing. Using the Nanjing Residents Travel Survey, we find that models with the same independent variables explain mode choice in Nanjing better than in other cities in the world. Comparatively, members of 'adult families' use public transport and walking more often than the private car and bicycle. And inhabitants of danwei neighbourhoods walk more often than residents in commodity housing estates. These conclusions suggest that ongoing socio-spatial transformations will push mode choice in China further towards private car use.
In post-reform China, rapid motorisation causes various problems like traffic congestion, diminishing road safety and air pollution. Adequate policies necessitate an understanding [...]
Leveraging MPLS Backup Paths for Distributed Energy-Aware Traffic Engineering
N. Wang, K. Moessner, R. Schmidt, F. Francois, S. Georgoulas
Backup paths are usually pre-installed by network operators to protect against single link failures in backbone networks that use multi-protocol label switching. This paper introduces a new scheme called Green Backup Paths (GBP) that intelligently exploits these existing backup paths to perform energy-aware traffic engineering without adversely impacting the primary role of these backup paths of preventing traffic loss upon single link failures. This is in sharp contrast to most existing schemes that tackle energy efficiency and link failure protection separately, resulting in substantially high operational costs. GBP works in an online and distributed fashion, where each router periodically monitors its local traffic conditions and cooperatively determines how to reroute traffic so that the highest number of physical links can go to sleep for energy saving. Furthermore, our approach maintains quality-of-service by restricting the use of long backup paths for failure protection only, and therefore, GBP avoids substantially increased packet delays. GBP was evaluated on the point-of-presence representation of two publicly available network topologies, namely, GÉANT and Abilene, and their real traffic matrices. GBP was able to achieve significant energy saving gains, which are always within 15% of the theoretical upper bound. © 2004-2012 IEEE.
Backup paths are usually pre-installed by network operators to protect against single link failures in backbone networks that use multi-protocol label switching. This paper [...]
State of charge estimation of high power lithium iron phosphate cells
G. Ludovici, G. Lutzemberger, T. Huria
This paper describes a state of charge (SOC) evaluation algorithm for high power lithium iron phosphate cells characterized by voltage hysteresis. The algorithm is based on evaluating the parameters of an equivalent electric circuit model of the cell and then using a hybrid technique with adequate treatment of errors, through an additional extended Kalman filter (EKF). The model algorithm has been validated in terms of effectiveness and robustness by several experimental tests.
This paper describes a state of charge (SOC) evaluation algorithm for high power lithium iron phosphate cells characterized by voltage hysteresis. The algorithm is based on [...]
A modified reinforcement learning algorithm for solving coordinated
S. Haldenbilen, H. Ceylan, O. Baskan, C. Ozan
This study proposes Reinforcement Learning (RL) based algorithm for finding optimum signal timings in Coordinated Signalized Networks (CSN) for fixed set of link flows. For this purpose, MOdified REinforcement Learning algorithm with TRANSYT-7F (MORELTRANS) model is proposed by way of combining RL algorithm and TRANSYT-7F. The modified RL differs from other RL algorithms since it takes advantage of the best solution obtained from the previous learning episode by generating a sub-environment at each learning episode as the same size of original environment. On the other hand, TRANSYT-7F traffic model is used in order to determine network performance index, namely disutility index. Numerical application is conducted on medium sized coordinated signalized road network. Results indicated that the MORELTRANS produced slightly better results than the GA in signal timing optimization in terms of objective function value while it outperformed than the HC. In order to show the capability of the proposed model for heavy demand condition, two cases in which link flows are increased by 20% and 50% with respect to the base case are considered. It is found that the MORELTRANS is able to reach good solutions for signal timing optimization even if demand became increased. (C) 2015 Elsevier Ltd. All rights reserved.
This study proposes Reinforcement Learning (RL) based algorithm for finding optimum signal timings in Coordinated Signalized Networks (CSN) for fixed set of link flows. For [...]
Climate Change Adaptation. Challenges and Opportunities for a Smart Urban Growth
A. Galderisi
Climate change is one of the main environmental issues challenging cities in the 21th century. At present, more than half of the world population lives in cities and the latter are responsible for 60% to 80% of global energy consumption and greenhouse gas (GHG) emissions, which are the main causes of the change in climate conditions. In the meantime, they are seriously threatened by the heterogeneous climate-related phenomena, very often exacerbated by the features of the cities themselves. In the last decade, international and European efforts have been mainly focused on mitigation rather than on adaptation strategies. Europe is one of the world leaders in global mitigation policies, while the issue of adaptation has gained growing importance in the last years. As underlined by the EU Strategy on adaptation to climate change, even though climate change mitigation still remains a priority for the global community, large room has to be devoted to adaptation measures, in order to effectively face the unavoidable impacts and related economic, environmental and social costs of climate change (EC, 2013). Thus, measures for adaptation to climate change are receiving an increasing financial support and a growing number of European countries are implementing national and urban adaptation strategies to deal with the actual and potential climate change impacts. According to the above considerations, this paper explores strengths and weaknesses of current adaptation strategies in European cities. First the main suggestions of the European Community to improve urban adaptation to climate change are examined; then, some recent Adaptation Plans are analyzed, in order to highlight challenges and opportunities arising from the adaptation processes at urban level and to explore the potential of Adaptation Plans to promote a smart growth in the European cities.
Climate change is one of the main environmental issues challenging cities in the 21th century. At present, more than half of the world population lives in cities and the latter [...]
Pruning and ranking the Pareto optimal set, application for the dynamic multi-objective network design problem
M. Bliemer, L. Wismans, T. Brands, E. Berkum
Solving the multi-objective network design problem (MONDP) resorts to a Pareto optimal set. This set can provide additional information like trade-offs between objectives for the decision making process, which is not available if the compensation principle would be chosen in advance. However, the Pareto optimal set of solutions can become large, especially if the objectives are mainly opposed. As a consequence, the Pareto optimal set may become difficult to analyze and to comprehend. In this case, pruning and ranking becomes attractive to reduce the Pareto optimal set and to rank the solutions to assist the decision maker. Because the method used, may influence the eventual decisions taken, it is important to choose a method that corresponds best with the underlying decision process and is in accordance with the qualities of the data used. We provided a review of some methods to prune and rank the Pareto optimal set to illustrate the advantages and disadvantages of these methods. The methods are applied using the outcome of solving the dynamic MONDP in which minimizing externalities of traffic are the objectives, and dynamic traffic management measures are the decision variables. For this, we solved the dynamic MONDP for a realistic network of the city Almelo in the Netherlands using the non-dominated sorting genetic algorithm II. For ranking, we propose to use a fuzzy outranking method that can take uncertainties regarding the data quality and the perception of decision makers into account; and for pruning, a method that explicitly reckons with significant trade-offs has been identified as the more suitable method to assist the decision making process.
Solving the multi-objective network design problem (MONDP) resorts to a Pareto optimal set. This set can provide additional information like trade-offs between objectives [...]
Determination of the economical optimum insulation thickness for VRF (variable refrigerant flow) systems
M. Ersöz, A. Yildiz
This study deals with the investigation into optimum insulation thickness of installed inside building pipe network of VRF (variable refrigerant flow) systems. Optimum insulation thickness, energy savings over a lifetime of 10 years and payback periods are determined for high pressure gas pipelines, low pressure gas pipelines and low pressure liquid pipelines under the heating-only and cooling-only modes of the three-pipe VRF system using R-410A as refrigerant. By using the P1-P2 method, the value of the amount of the net energy savings is calculated. Under heating mode of VRF system, while the optimum insulation thickness varies between 16 and 20 mm depending on the pipe sections of high pressure gas pipeline, it varies from 11 to 13 mm for the pipe sections of low pressure liquid pipeline. Under cooling mode of VRF system, the optimum insulation thickness varies between 7 and 8 mm for pipe sections of low pressure gas pipeline and low pressure liquid pipeline. © 2015 Elsevier Ltd.
This study deals with the investigation into optimum insulation thickness of installed inside building pipe network of VRF (variable refrigerant flow) systems. Optimum insulation [...]
Ranking Links in a Road Transport Network: A Practical Method for the Calculation of Link Importance
A. Danesi, S. Bernardi, G. Rossi, S. Angelini, F. Rupi
This paper deals with the issue of road network vulnerability and describes the implementation of a methodology which ranks the links of a network according to their importance for maintaining a proper connectivity between all origin-destination pairs. Such a ranking can be easily used by practitioners and decision-makers for prioritising maintenance investments along the links of a road transport network. In this regard, following a conceptual approach well consolidated in transport literature, vulnerability is assumed to be related to the concept of importance, i.e. a measure of the consequences of the collapse of a network's element. In the present study, the definition of importance – with respect to a given link –simultaneously includes two aspects: the level of usage, i.e. how many people typically use the link for their trips, and the impact that the closure of the link itself can have on the general functionality of the network as a whole. These two aspects are considered in the link importance index formulation, as two different functions that can be properly weighted by means of different coefficients. The methodology proposed has been implemented in the framework of Paramount EU project, to obtain a ranking of importance for the links of a real-scale network, i.e. the road network of Bolzano, a highly mountainous province located in the Italian Alps. The application of the methodology led to satisfactory results represented by a ranking of links, in decreasing order importance scores. Furthermore, Spearman's rank correlation coefficient has been used to quantify the variation of the importance ranking caused by the variation of its coefficients.
This paper deals with the issue of road network vulnerability and describes the implementation of a methodology which ranks the links of a network according to their importance [...]
Applying complexity science to air traffic management
S. Miccichè, R. Mantegna, D. Rivas, A. Cook, R. Vazquez, F. Lillo, M. Zanin, H. Blom
Complexity science is the multidisciplinary study of complex systems. Its marked network orientation lends itself well to transport contexts. Key features of complexity science are introduced and defined, with a specific focus on the application to air traffic management. An overview of complex network theory is presented, with examples of its corresponding metrics and multiple scales. Complexity science is starting to make important contributions to performance assessment and system design: selected, applied air traffic management case studies are explored. The important contexts of uncertainty, resilience and emergent behaviour are discussed, with future research priorities summarised.
Complexity science is the multidisciplinary study of complex systems. Its marked network orientation lends itself well to transport contexts. Key features of complexity science [...]
Security Economics: an adversarial risk analysis approach to airport security
A. Tedeschi, U. Turhan, J. Cano, D. Insua
WOS: 000385425300018 We analyze the case of protecting an airport, in which there is concern with terrorist threats against the Air Traffic Control Tower. To deter terrorist actions, airport authorities rely on various protective measures, which entail multiple consequences. By deploying them, airport authorities expect to reduce the probabilities and potential impacts of terrorist actions. We aim at giving advice to the airport authorities by devising a security resource allocation plan. We use the framework of adversarial risk analysis to deal with the problem. European Union [285223]; Spanish Ministry of Economy and Innovation program [MTM2011-28983-C03-01]; Government of Madrid RIESGOS-CM program [S2009/ESP-1685]; AXA-ICMAT Chair on Adversarial Risk Analysis; URJC's postdoctoral program This project has received funding from the European Union's Seventh Framework Programme for Research, Technological Development and Demonstration under grant agreement no 285223. Work has been also supported by the Spanish Ministry of Economy and Innovation program MTM2011-28983-C03-01, the Government of Madrid RIESGOS-CM program S2009/ESP-1685 and the AXA-ICMAT Chair on Adversarial Risk Analysis. We are grateful to airport experts and stakeholders for fruitful discussions about modeling issues. This work was completed while the first author was visiting Uppsala University, supported by a grant from URJC's postdoctoral program.
WOS: 000385425300018 We analyze the case of protecting an airport, in which there is concern with terrorist threats against the Air Traffic Control Tower. To deter terrorist [...]
Estimation of the damage in a natural gas flow line caused by the motion of methane hydrates
A. Soprano, F. Cascetta, G. Lamanna, G. Rotondo, F. Caputo
Gas storage and petroleum Companies invest every year large sums of money to prevent the formation of gas hydrates plugs in flow lines as the safety requirements and the costs related to the blocking of pipelines suggest to invest in research studies to avoid the hydrate formation or to detect its presence. Gas hydrates are aggregates where hydrocarbons gas such as methane are embedded in a crystalline cage of water molecules: they can develop abruptly and block pipelines. The nucleation of hydrates and their growth occurs when methane comes in contact with water under suitable thermodynamic conditions, such as high pressures and low temperatures. Methane hydrates induce the most relevant troubles as they form with no warning and can produce partial or even full obstructions in a pipeline thus reducing flow, increasing both back- and differential pressures. If a line section is obstructed the differential pressure across the hydrate can set the plug in motion and accelerate it quickly up to a speed approaching that of sound. In that case, the moving mass can cause serious mechanical damages at locations where the plug meets such constrictions or obstacles as valves, elbows or tees. In this paper a gas pipeline failure case due to the presence of a moving hydrate plug has been investigated by using appropriate analytical and numerical models to analyse the motion of a hydrate plug as well as its impact against the pipe wall.
Gas storage and petroleum Companies invest every year large sums of money to prevent the formation of gas hydrates plugs in flow lines as the safety requirements and the costs [...]
Car2x with software defined networks, network functions virtualization and supercomputers technical and scientific preparations for the Amsterdam Arena telecoms fieldlab
R. Koning, M. Makkes, P. Jackson, R. Cushing, R. Meijer, S. Klous, A. Meerwijk, C. Laat
In the invited talk 'Car2x with SDN, NFV and supercomputers' we report about how our past work with SDN [1, 2] allows the design of a smart mobility fieldlab in the huge parking lot the Amsterdam Arena. We explain how we can engineer and test software that handle the complex conditions of the Car2X case. The talk starts by describing the engineering challenges that developers of smart car telecommunications and computing infrastructures face. We concentrate on the development of software defined networks (SDN) that support smart cars optimally and securely over a heterogeneous, dynamic and developing ICT infrastructure. The goal here is to enable smart cars to profit maximally from any bit of information available from fixed and moving objects as well as persons. For example, in a low tech situation, the lack of other options makes that one has to fall back to GPRS to download only traffic jam locations. As a contrast, in an advanced telecom environment video streams from multiple cars are transmitted via 5G pico cells to computers a few millisecond nearby. These computers fuse the video information to generate a local traffic model. In an ultimate situation, cars use all the communications infrastructures that are available including the numerous WiFi hotspots, all generations of mobile telecommunications, the developing car-to-car communications technologies and even the smart phone of a passing person. SDN technologies deal with the complexities of such communication environment. © 2015 IEEE.
In the invited talk 'Car2x with SDN, NFV and supercomputers' we report about how our past work with SDN [1, 2] allows the design of a smart mobility fieldlab in the huge parking [...]
Optimising data placement and traffic routing for energy saving in Backbone Networks
Z. Yang, J. Lioret, Z. Wang, D. Zhang, W. Fang
The energy consumption of network elements has become a big concern due to the exponential traffic growth and the rapid expansion of communication infrastructures. To deal with this problem, we propose a new approach called Backbone network Energy Saving based on Traffic engineering (BEST), which reduces the power consumption of network elements at the backbone level through jointly optimising data placement and traffic routing. Based on analysis on traffic characteristics, BEST firstly optimises the placement of data services such that the pairwise traffic demands can be better coordinated with the pairwise network costs, in order to minimise the traffic burden imposed on the network elements. Then, BEST optimises the routing of traffic flows and tries to find the minimum-power network subset that must remain active to fulfill the traffic requirements. Efficient heuristics are given by BEST to find an admissible solution when the problem size is very large. The simulation results illustrate the efficacy and efficiency of BEST in energy conservation on backbone networks This work was supported in part by the National Science Foundation of China (grant nos. 61202430, 61303245 and 61103185), the Science and Technology Foundation of Beijing Jiaotong University (grant no. 2012RC040). Fang, W.; Wang, Z.; Lloret, J.; Zhang, D.; Yang, Z. (2014). Optimising data placement and traffic routing for energy saving in Backbone Networks. Transactions on Emerging Telecommunications Technologies. 25(9):914-925. doi:10.1002/ett.2774 S 914 925 25 9
The energy consumption of network elements has become a big concern due to the exponential traffic growth and the rapid expansion of communication infrastructures. To deal [...]
State of science: mental workload in ergonomics
K. Brookhuis, P. Hancock, M. Young, C. Wickens
Mental workload (MWL) is one of the most widely used concepts in ergonomics and human factors and represents a topic of increasing importance. Since modern technology in many working environments imposes ever more cognitive demands upon operators while physical demands diminish, understanding how MWL impinges on performance is increasingly critical. Yet, MWL is also one of the most nebulous concepts, with numerous definitions and dimensions associated with it. Moreover, MWL research has had a tendency to focus on complex, often safety-critical systems (e.g. transport, process control). Here we provide a general overview of the current state of affairs regarding the understanding, measurement and application of MWL in the design of complex systems over the last three decades. We conclude by discussing contemporary challenges for applied research, such as the interaction between cognitive workload and physical workload, and the quantification of workload 'redlines' which specify when operators are approaching or exceeding their performance tolerances.
Mental workload (MWL) is one of the most widely used concepts in ergonomics and human factors and represents a topic of increasing importance. Since modern technology in many [...]
Key research themes on travel behavior, lifestyle, and sustainable urban mobility
V. Acker, F. Witlox, P. Goodwin
The concept of lifestyle adds a behavioral component to travel models that used to be dominated by engineering and econometric traditions. This article presents an overview of how lifestyle is defined and measured in transport studies, and how travel behavior is influenced by lifestyles. Lifestyles are often used pragmatically rather than theoretically in the behavior studies. Nevertheless, some important theoretical contributions have been made, especially in sociology by scholars such as Weber, Bourdieu, Ganzeboom, and Schulze who agree on the communicative character of lifestyles: individuals express their social position through specific patterns of behavior, consumption, and leisure. These behavioral patterns are shaped by underlying opinions and orientations, including beliefs, interests, and attitudes. Thus, travel behavior is not simply determined by price, speed, and comfort but is also related to attitudes, status, and preferences. Because lifestyle has many different dimensions, a variety of measurement approaches exists. Nevertheless, most studies suggest that travel behavior is conditioned by specific lifestyles. How lifestyles themselves can be modified to promote more sustainable patterns of transport has not received much attention to date. This article argues that lifestyles need to be considered as dynamic rather than as static and given, and that future research could delve more deeply into this area.
The concept of lifestyle adds a behavioral component to travel models that used to be dominated by engineering and econometric traditions. This article presents an overview [...]
Aircraft routing with minimal climate impact: the REACT4C climate cost function modelling approach (V1.0)
P. Hullah, K. Shine, P. Jöckel, T. Champougny, K. Dahlmann, T. Berntsen, S. Brinkop, E. Tsati, S. Dietmüller, C. Frömming, J. Fuglestvedt, M. Ponater, S. Matthes, O. Søvde, H. Garny, V. Grewe, E. Irvine
Abstract. In addition to CO2, the climate impact of aviation is strongly influenced by non-CO2 emissions, such as nitrogen oxides, influencing ozone and methane, and water vapour, which can lead to the formation of persistent contrails in ice-supersaturated regions. Because these non-CO2 emission effects are characterised by a short lifetime, their climate impact largely depends on emission location and time; that is to say, emissions in certain locations (or times) can lead to a greater climate impact (even on the global average) than the same emission in other locations (or times). Avoiding these climate-sensitive regions might thus be beneficial to climate. Here, we describe a modelling chain for investigating this climate impact mitigation option. This modelling chain forms a multi-step modelling approach, starting with the simulation of the fate of emissions released at a certain location and time (time-region grid points). This is performed with the chemistry–climate model EMAC, extended via the two submodels AIRTRAC (V1.0) and CONTRAIL (V1.0), which describe the contribution of emissions to the composition of the atmosphere and to contrail formation, respectively. The impact of emissions from the large number of time-region grid points is efficiently calculated by applying a Lagrangian scheme. EMAC also includes the calculation of radiative impacts, which are, in a second step, the input to climate metric formulas describing the global climate impact of the emission at each time-region grid point. The result of the modelling chain comprises a four-dimensional data set in space and time, which we call climate cost functions and which describes the global climate impact of an emission at each grid point and each point in time. In a third step, these climate cost functions are used in an air traffic simulator (SAAM) coupled to an emission tool (AEM) to optimise aircraft trajectories for the North Atlantic region. Here, we describe the details of this new modelling approach and show some example results. A number of sensitivity analyses are performed to motivate the settings of individual parameters. A stepwise sanity check of the results of the modelling chain is undertaken to demonstrate the plausibility of the climate cost functions. Document type: Article
Abstract. In addition to CO2, the climate impact of aviation is strongly influenced by non-CO2 emissions, such as nitrogen oxides, influencing ozone and methane, and water [...]
Differences between HCM Procedures and Fundamental Diagram Calibration for Operational LOS Assessment on Italian Freeways
F. Rupi, A. Pompigna
A not clear national framework and the uncertainty as to the transferability of the U.S. HCM require an assessment of standard methodologies more calibrated on Italian freeways. The study is aimed at testing the Levels of Service assessment for a sample freeway segment on the basis of a calibrated Fundamental Diagram, and at evaluating its consistency with respect to the most recent methodologies from HCM2010. The research shows a test calibration of the Fundamental Diagram according to the Longitudinal Control Model and Van Aerde Model. The comparative analysis shows how standard procedures and ranges could underestimate operational congestion levels on the test section. Therefore, the results suggest that operators should to use carefully HCM standard procedures and that transferability issues should be further analyzed.
A not clear national framework and the uncertainty as to the transferability of the U.S. HCM require an assessment of standard methodologies more calibrated on Italian freeways. [...]
Color enhanced pipelines for reality-based 3D modeling of on site medium sized archeological artifacts
F. Apollonio, M. Gaiani, M. Ballabeni
The paper describes a color enhanced processing system - applied as case study on an artifact of the Pompeii archaeological area - developed in order to enhance different techniques for reality-based 3D models construction and visualization of archaeological artifacts. This processing allows rendering reflectance properties with perceptual fidelity on a consumer display and presents two main improvements over existing techniques: a. the color definition of the archaeological artifacts; b. the comparison between the range-based and photogrammetry-based pipelines to understand the limits of use and suitability to specific objects. Document type: Article
The paper describes a color enhanced processing system - applied as case study on an artifact of the Pompeii archaeological area - developed in order to enhance different [...]
Business models for sustainable technologies: Exploring business model evolution in the case of electric vehicles
A. Kolk, J. Pinkse, R. Bohnsack
International audience; Sustainable technologies challenge prevailing business practices, especially in industries that depend heavily on the use of fossil fuels. Firms are therefore in need of business models that transform the specific characteristics of sustainable technologies into new ways to create economic value and overcome the barriers that stand in the way of their market penetration. A key issue is the respective impact of incumbent and entrepreneurial firms' path-dependent behaviour on the development of such new business models. Embedded in the literature on business models, this paper explores how incumbent and entrepreneurial firms' path dependencies have affected the evolution of business models for electric vehicles. Based on a qualitative analysis of electric vehicle projects of key industry players over a five-year period (2006-2010), the paper identifies four business model archetypes and traces their evolution over time. Findings suggest that incumbent and entrepreneurial firms approach business model innovation in distinctive ways. Business model evolution shows a series of incremental changes that introduce service-based components, which were initially developed by entrepreneurial firms, to the product. Over time there seems to be some convergence in the business models of incumbents and entrepreneurs in the direction of delivering economy multi-purpose vehicles.
International audience; Sustainable technologies challenge prevailing business practices, especially in industries that depend heavily on the use of fossil fuels. Firms are [...]
Queue Profile Estimation in Congested Urban Networks with Probe Data
N. Geroliminis, M. Ramezani
Queues at signalized intersections are the main cause of traffic delays and travel time variability in urban networks. In this article, we propose a method to estimate queue profiles that are traffic shockwave polygons in the time-space plane describing the spatiotemporal formation and dissipation of queues. The method integrates the collective effect of dispersed probe vehicle data with traffic flow shockwave analysis and data mining techniques. The proposed queue profile estimation method requires position and velocity data of probe vehicles; however, any explicit information of signal settings and arrival distribution is indispensable. Moreover, the method captures interdependencies in queue evolutions of successive intersections. The significance of the proposed method is that it is applicable in oversaturated conditions and includes queue spillover identification. Numerical results of simulation experiments and tests on NGSIM field data, with various penetration rates and sampling intervals, reveal the promising and robust performance of the proposed method compared with a uniform arrival queue estimation procedure. The method provides a thorough understanding of urban traffic flow dynamics and has direct applications for delay analysis, queue length estimation, signal settings estimation, and vehicle trajectory reconstruction.
Queues at signalized intersections are the main cause of traffic delays and travel time variability in urban networks. In this article, we propose a method to estimate queue [...]
Planning Pharmacies: An Operational Method to Find the Best Location
S. Tondelli, S. Fatone
The theme of the spatial distribution of the pharmacies on the territory is closely connected to urban planning and to services supply. In Italy, the regulatory change that took place in 2012, triggered partly by the need to adhere more to a constantly changing economic system, has led to a revision of the existing situation, consisting both on the method to quantify the pharmacies distribution and on the efficiency of the service. If Law 27/2012 has effectively allowed municipalities to increase the number of pharmacies that they can settle on the municipal territory, it has also started a process of rethinking the logic of pharmacies location and of their catchment areas. In this framework, the present paper aims to discuss the merits of a regulatory evolutions that sparked the process of liberalization of locations, integrating the law guidelines and goals with an operating logic process, usable and useful to translate goals into planning actions in a continuous dialogue between law and territory, constraints and opportunities, equity and accessibility of the care services. Following this logic operations, we have investigated the urban context of Castelfranco Emilia, assuming the location of new offices on the basis of pharmaceutical analyzes.
The theme of the spatial distribution of the pharmacies on the territory is closely connected to urban planning and to services supply. In Italy, the regulatory change that [...]
Airport traffic complexity and environment efficiency metrics for evaluation of ATM measures
T. Simic, O. Babic
For a given (current or planned) traffic demand, different air traffic management measures could result in different airport traffic complexity and efficiency. This paper presents the research on the relationship between airport traffic complexity and time and environmental efficiency for different air traffic control (ATC) tactics applied to the given or planned airport layout. Emphasis is placed on the evaluation of airport traffic complexity, aircraft fuel consumption, gas emissions and time efficiency for different ATC tactics and/or airport airfield layouts. For busy airports during peak hours, arrival queuing delays, taxi-in, taxi-out times and departure queuing delays increase, which induces additional unnecessary fuel consumption, gas emission and time inefficiency. In order to find a tool which could indicate potential delay generators, a measure of airport traffic complexity – called Dynamic Complexity is proposed. Experiments were performed for airports with different airfield layouts, for different traffic demands and ATC applied tactics using SIMMOD simulation model. Traffic situations were analyzed and delays were measured. The values of airport traffic complexity, fuel consumption and gas emissions were also determined. A comparative analysis of the results show: first, the proposed airport traffic complexity metric quite satisfactorily reflects the influence of traffic characteristics upon the environmental state of the system, and second, different ATM strategic and tactical measures (airport airfield infrastructure development and applied ATC tactics) could significantly reduce traffic complexity and increase time and environmental efficiency at the airport.
For a given (current or planned) traffic demand, different air traffic management measures could result in different airport traffic complexity and efficiency. This paper [...]
Location of logistics companies : a stated preference study to disentangle the impact of accessibility
P. Goos, T. Zijlstra, A. Verhetsel, J. Cant, R. Kessels, N. Blomme
© 2014 Elsevier Ltd. Due to the globalization and the fragmentation of industrial production processes, the logistics sector, organizing the linkages between different production plants and the market, is growing fast. This results in an increasing demand for suitable new business locations. Previous research has indicated that accessibility is a key factor in the location decision making process. Though the literature on this subject is extensive, little research has been done to quantify the impact of the different dimensions of accessibility on the location decision process of logistics companies. This paper aims to fill this void in the literature by means of a revealed preference study (using a Geographic Information System (GIS) analysis) and a stated preference study (using a designed discrete choice experiment) in Flanders (Belgium). The results of the revealed preference study served as input to the design of the choice situations in the stated preference study. In the stated preference study, the respondents were confronted with a series of choice situations described by means of accessibility variables as well as land rent information. An analysis of the resulting data by means of discrete choice modeling revealed that land rent is the most important factor in the location choice of logistics companies in Flanders. Access to a port is the second most important factor, followed by access to a motorway, the location in a business park and an inland navigation terminal, which are all about equally important. Access to a rail terminal plays no significant role in the location choice of logistics companies in Flanders. ispartof: Journal of Transport Geography vol:42 pages:110-121 status: published
© 2014 Elsevier Ltd. Due to the globalization and the fragmentation of industrial production processes, the logistics sector, organizing the linkages between different production [...]
Development of an internal air cooling sprayed oil injection technique for the energy saving in sliding vane rotary compressors through theoretical and experimental methodologies
G. Bianchi, G. Contaldi, R. Cipollone, S. Murgia
The present work highlights the energy saving potential of the lubricant fluid supplied in Sliding Vane Rotary air Compressors. A Lagrangian theoretical model of a sprayed oil injection technology assessed the cooling effect of the lubricant due to the high surface to volume ratio of the oil droplets and predicted a reduction of the indicated power. The model validation was carried out through a test campaign on a mid-size sliding vane compressor equipped with a series of pressure swirl atomizers. The oil injections took place along the axial length of the compressor. The reconstruction of the indicator diagram and the direct measurement of the mechanical power revealed a reduction of the energy consumption close to 7% using an injection pressure of 20 bar. A parametric analysis on the injection pressure and temperature and on the cone spray angle was eventually carried out to identify an optimal set of operative injection parameters. The work has been done under the FP7 Project "Complete Vehicle Energy-Saving CONVENIENT" founded by the European Commission.
The present work highlights the energy saving potential of the lubricant fluid supplied in Sliding Vane Rotary air Compressors. A Lagrangian theoretical model of a sprayed [...]
Investigation of a method for real time quantification of gas bubbles in pipelines
K. Baik, T. Leighton, J. Jiang
The need to measure the dynamic void fraction (the proportion of flowing bubbly liquid that is gas) is common across many power, processing and manufacturing industries. Many such pipelines and liquids are optically opaque, and work on margins that require a low cost solution that is not commensurate with the size of the challenge. Such a solution will therefore be a compromise, and in this paper costs are reduced by using a narrowband acoustic solution that cannot, on its own, contain enough information to characterise the void fraction in real time unambiguously. The ambiguity is reduced using likely estimates of the general shape of the bubble size distribution so that, with a single source-receiver pair attached to the outside of the pipe, the absolute gas content can be estimated. Whilst the data that are required a priori (the general shape of the bubble size distribution) are not identical to the output of the inversion (the absolute void fraction of gas entrained as bubbles in the flow), the requirement for such a priori information could limit the usefulness of the technique in industry.
The need to measure the dynamic void fraction (the proportion of flowing bubbly liquid that is gas) is common across many power, processing and manufacturing industries. Many [...]
Urban sustainable mobility. Part 1: Rationality in transport planning
A. Carteni'
The impact of the transport sector is in the range of 20%-40% in terms of consumption of fossil fuels and emissions of greenhouse gases and particulate matter. In this context, policies aimed at reducing these effects are very important. Many urban areas are trying to adopt planning strategies aimed to a sustainable use of resources often referred to as sustainable mobility. These policies are very different in terms of costs and expected benefits, and the effects of these policies and their combinations are difficult to anticipate on a purely intuitive basis and sometimes the end effect could be contrary to intuitive expectations (e.g. policies aimed to reduce pollution, ending up in increasing it). In this context, the concept of eco-rational planning assumes a central role. This means identifying the right mixture of interventions to be implemented on the transport system that is: rational for the transport system (e.g. reduction in terms of congestion, traffic accidents, travel time) and sustainable for people's health and for the environmental (e.g. emissions reduction) and requires minimal economic resources (e.g. lower monetary cost per unit of CO2 saved). The paper discusses the importance of rational decisions in transport planning.
The impact of the transport sector is in the range of 20%-40% in terms of consumption of fossil fuels and emissions of greenhouse gases and particulate matter. In this context, [...]
Improved cost models for optimizing CO2 pipeline configuration for point-to-point pipelines and simple networks
A. Ramírez, A. Faaij, M. Knoope, W. Guijt
In this study, a new cost model is developed for CO2 pipeline transport, which starts with the physical properties of CO2 transport and includes different kinds of steel grades and up-to-date material and construction costs. This pipeline cost model is used for a new developed tool to determine the configuration leading to the lowest levelized costs for CO2 transport, for point-to-point pipelines as well as for simple networks on different types of terrain and for different time frames. The model optimizes inlet pressure, diameter, steel grade and number of pumping stations. Results show that gaseous CO2 transport can give lower levelized costs than liquid CO2 transport for point-to-point pipelines and for simple networks, if the CO2 is stored in a reservoir with a low required injection pressure, like depleted natural gas fields. However, for storage fields with a required injection pressure of 8 MPa or higher (like aquifers), CO2 can be better transported in a liquid form. For onshore pipelines transporting liquid CO2, the optimal inlet pressure is 9-13 MPa and pumping stations are installed roughly every 50-100 km. For offshore pipelines, pumping stations are not an option and the inlet pressure is determined by the length of the pipeline. The maximum inlet pressure is about 25 MPa and for even longer pipelines, a larger diameter is selected. The levelized costs (excluding initial compression) for transporting 100 kg/s (about 3 Mt/y) over 100 km are between 1.8 and 33 (sic)/t for liquid and 4.0-6.4 (sic)/t for gaseous CO2 transport. For larger mass flows the costs are decreasing, for instance transporting 200 kg/s (about 6 Mt/y) over 100 km are 1.2-1.8 (sic)/t for liquid and 3.0-3.8 (sic)/t for gaseous CO2 transport. Furthermore, results show that higher steel grades lead to lower investment costs for onshore pipelines transporting liquid CO2. Using X120 in the long term reduces the pipeline costs up to 17%. For gaseous CO2 transport, lower steel grades (like X42 and X52) are the best option. Also offshore pipelines do not benefit from the development of higher steel grades over time because the thickness should be at least 2.5% of the outer diameter. The results indicate that oversizing the pipeline, to transport CO2 from an additional source that is coming available later, is not always cost-attractive. This strongly depends on the time span of which further CO2 sources are available and on the mass flows. Oversizing appears earlier cost-attractive compared to two point-to-point pipelines if the source with the largest mass flow becomes available first. (C) 2014 Elsevier Ltd. All rights reserved.
In this study, a new cost model is developed for CO2 pipeline transport, which starts with the physical properties of CO2 transport and includes different kinds of steel grades [...]
In-Order Delivery Approach for 2D and 3D NoCs
S. Dytckov, J. Plosila, M. Ebrahimi, M. Daneshtalab
In many applications, it is critical to guarantee the in-order delivery of requests from the master cores to the slave cores, so that the requests can be executed in the correct order without requiring buffers. Since in NoCs packets may use different paths and on the other hand traffic congestion varies on different routes, the in-order delivery constraint cannot be met without support. To guarantee the in-order delivery, traditional approaches either use dimension-order routing or employ reordering buffers at network interfaces. Dimension-order routing degrades the performance considerably while the usage of reordering buffers imposes large area overhead. In this paper, we present a mechanism allowing packets to be routed through multiple paths in the network, helping to balance the traffic load while guaranteeing the in-order delivery. The proposed method combines the advantages of both deterministic and adaptive routing algorithms. The simple idea is to use different deterministic algorithms for independent flows. This approach neither requires reordering buffers nor limits packets to use a single path. The algorithm is simple and practical with negligible area overhead over dimension-order routing. The concept is investigated in both 2D and 3D mesh networks.
In many applications, it is critical to guarantee the in-order delivery of requests from the master cores to the slave cores, so that the requests can be executed in the correct [...]
Application of new human factors tool in an air traffic management organization
A. Teperi, A. Leppänen, N. Norros, L. Leena
m: An easy-to-use human factors (HF) tool for air traffic control (ATC) operators was applied in an air traffic management (ATM) organization to ensure ATC operators' learning and commitment to HF, which is seen as a critical factor in improving safety. Methods: ATC operators analyzed the positive and negative causal factors of 3163 incident reports in 2008-2010 using the HF tool in 27 ATC units. We studied the active usage of the HF tool and causal factors of incidents. Users' experiences of the HF tool were assessed using questionnaires with open questions. Results: Although the usage of the tool varied from year to year and across units, it helped ATC units analyze both positive and negative HF-related causal factors of incidents. It also enabled ATC operators to learn about the new field of HF. Discussion: The target ATM organization accepted the new HF tool. The benefits of the tool were its visuality, user-friendliness and the congruence of its contents with existing HF tools. The lessons learnt revealed the need for more extensive training, clearer instructions for the users of the HF tool, and publicizing the actions based on the findings. Conclusions: The application of a new HF tool in the target ATM organization was supported by the fact that it could be included as one of the core processes of ATC work (reporting system). Improving HF competence in the organization is recommended, to further improve ATC work and the safety of ATC operations. The HF tool would support this.
m: An easy-to-use human factors (HF) tool for air traffic control (ATC) operators was applied in an air traffic management (ATM) organization to ensure ATC operators' learning [...]
Sustainability as a Change Agent? Lessons From the European Airspace Regulation
M. Mölders
The contribution at hand follows the sociologist Niklas Luhmann in framing ecological problems as communicative ones. This approach offers valuable insights for the difficulties in finding society-wide accepted solutions. Reflexive steering (RS) as well as reflexive governance (RG) took up basic assumptions but also added proposals for working towards solutions. Because actors from different contexts cannot escape from a "vicious circle of first-order reflexivity", they have to be forced to take into account the big picture (claim 1). In the RG literature it is argued that notions such as "sustainable development" may ease triggering communication across different societal domains by working as a "change agent" (claim 2). Both of these claims are examined by introducing a case study on European airspace regulation ("Single European Sky"). It shows that economy – usually the system that is intended to be changed – makes use of sustainability as a change agent in order to redirect the pressure to reflect and adapt towards others. It is concluded that, on the one hand, public credibility is a powerful means to induce changes within idiosyncratic societal entities. On the other hand, terms such as "sustainable development" are so widely diffused that they allow for being used by almost any societal actor able to communicate publicly. This multi-directionality has been neglected in the literature so far which calls for thinking of new communicative solutions – especially beyond procedural proposals which are favored by many approaches from RS to RG.
The contribution at hand follows the sociologist Niklas Luhmann in framing ecological problems as communicative ones. This approach offers valuable insights for the difficulties [...]
A dendrochronological reassessment of three Roman boats from Utrecht (the Netherlands) : evidence of inland navigation between the lower-Scheldt region in Gallia Belgica and the limes of Germania inferior
M. Kosian, K. Haneca, E. Jansma
This study addresses the provenance of two Roman river barges and a Roman punt excavated along the limes of Germania inferior near the Dutch city of Utrecht ( De Meern 1 , 4 and 6 ). To establish the geographical origin of these vessels, their tree-ring series are compared to 1452 dated oak ( Quercus robur/petraea ) growth patterns from Roman-period sites in the current Netherlands and Belgium. The strong resemblance to patterns of oak used in the civitas Menapiorum and the bordering region of civitas Nerviorum in Gallia Belgica indicates that the ships were built with oak from the lower-Scheldt region in present-day Flanders (north-western Belgium). Given the absence of Flemish oak in Roman land-based constructions along the Dutch limes , this provenance implies that the vessels were constructed in the lower-Scheldt region. The geographical location of the final wreck sites of De Meern 1 , 4 and 6 points at inland-navigation between this region and the Rhine-based limes of Germania inferior .
This study addresses the provenance of two Roman river barges and a Roman punt excavated along the limes of Germania inferior near the Dutch city of Utrecht ( De Meern [...]
Threat-based sensor management for target tracking
H. Driessen, A. Yarovoy, F. Katsilieris
sensor management scheme that focuses on managing the uncertainty in the threat level of targets is proposed. The scheme selects the best sensing mode such that the uncertainty in the threat level of targets is minimized. The main advantage of the proposed scheme is that it opens the possibility for incorporation of the operational context when performing Bayes-optimal sensor management. Different aspects of threat can be meaningfully aggregated making this flexible approach a favorite choice for multifunctional systems. The proposed scheme is demonstrated in simulated scenarios, both simple and advanced, where the data association problem is taken into account. In the multitarget example, the proposed scheme outperforms the other schemes considered in this manuscript, both naive and adaptive. The proposed scheme can be used in target tracking applications, such as air traffic management or area surveillance.
sensor management scheme that focuses on managing the uncertainty in the threat level of targets is proposed. The scheme selects the best sensing mode such that the uncertainty [...]
The Modal Split of Cities: A Workplace-Based Mixed Modelling Perspective
T. Vanoutrive
Many transport planners and policy-makers compare cities on the basis of travel behaviour or transport system characteristics. These comparisons are then used to promote cities as cycling capitals or public transport paradises. However, data-related and methodological issues make comparisons across cities difficult. This paper presents an alternative to the dominant atomistic-residential perspective which takes a random sample of residents in a city as input. We conceptualise cities as concentrations of workplaces and compare the commuting behaviour of employees active in a selection of sectors. This paper presents a new indicator to measure the modal split of cities and explains the level of car use in Belgian cities using mixed models. The presence of a railway station is associated with lower levels of car use, while highway access increases the share of the car. The car is also less popular in city centres, subcentres and in more dense urban areas.
Many transport planners and policy-makers compare cities on the basis of travel behaviour or transport system characteristics. These comparisons are then used to promote cities [...]
Decentralized Signal Control for Urban Road Networks
N. Walton, P. Kovacs, L. Andrew, H. Vu, S. Hoogendoorn, T. Le
We propose in this paper a decentralized traffic signal control policy for urban road networks. Our policy is an adaptation of a so-called BackPressure scheme which has been widely recognized in data network as an optimal throughput control policy. We have formally proved that our proposed BackPressure scheme, with fixed cycle time and cyclic phases, stabilizes the network for any feasible traffic demands. Simulation has been conducted to compare our BackPressure policy against other existing distributed control policies in various traffic and network scenarios. Numerical results suggest that the proposed policy can surpass other policies both in terms of network throughput and congestion. Comment: Public technical repor
We propose in this paper a decentralized traffic signal control policy for urban road networks. Our policy is an adaptation of a so-called BackPressure scheme which has been [...]
A GIS-based tool for the management of industrial accidents triggered by volcanic ash fallouts
G. Ancione, M. Milazzo, G. Maschio, E. Salzano
In this last decade, worldwide attention has been focused on the hazards derived from the interaction between extreme natural phenomena and critical infrastructures and/or chemical and process industry (natural–technological hazards or Na-Tech). Due to the recent occurrence of significant events, great attention has also been given to Na-Tech hazards triggered by volcanic eruptions; in particular, the eruption of the Icelandic volcano alarmed the European community due to the ash fallout over the continent, which caused significant problems for the population, road, rail and air traffic and production activities. This study aims at defining a procedure for the representation of the vulnerability of industrial facilities to potential volcanic ash fallouts. Its implementation on a Geographical Information System has also been executed and a semi-automatic procedure for the vulnerability mapping has been constructed. © 2014 Taylor & Franci
In this last decade, worldwide attention has been focused on the hazards derived from the interaction between extreme natural phenomena and critical infrastructures and/or [...]
In the wake of liberalisation: long-term developments in the EU air transport market
G. Burghouwt, J. Wit
Using a 24-year analysis period (1990-2013), a new perspective is offered on long-term first- and second-order developments following liberalisation of the intra-EU air transport market. The focus of the analysis is on supply-side issues, such as airline output, structure of supply, yields, business models, and the position of (former) flag carriers. We find that air transport liberalisation in the EU internal market has facilitated significant growth in both the number and frequency of routes. Greater competition and lower fares at the route level went hand in hand with ongoing consolidation. The first and second liberalisation packages have enabled former flag carriers to develop into full-service hub-and-spoke carriers. It was only after the third package (1993-1997) that low-cost carriers were able to fully develop Europe-wide, point-to-point networks. They rapidly achieved a substantial market share after 2000. Following the adoption of the third package of liberalisation measures, charter operators launched scheduled operations as leisure airlines and adopted in part the new low-cost business model. Until 2013, direct competition between low-cost carriers and former flag carriers was limited due to different route development strategies. This picture has changed in recent years since more low-cost carriers have begun to focus on primary hubs. The competitive impact of the rapidly expanding new generation of hub carriers from Turkey and the Gulf has further increased competitive pressure on the former European flag carriers. Consequently, consolidation of the European airline industry is likely to continue, as well as the ongoing effort to reduce costs. Finally, rising competitive pressure may incentivise EU member states to develop a more protective national aviation policy.
Using a 24-year analysis period (1990-2013), a new perspective is offered on long-term first- and second-order developments following liberalisation of the intra-EU air transport [...]
New environmental demands and the future of the Helsinki−Tallinn freight route
O. Hilmola, H. Lorentz, D. Rhoades
The environmental friendliness of short sea shipping has been justified in Europe by the ensuing lower congestion at hinterlands and unneeded large-scale infrastructure investments on roads and railways. However, the attractiveness of short sea shipping is about to change. This is because of increasing environmental regulations (International Maritime Organization (IMO) sulfur regulation in the Baltic Sea and planned CO2 emissions trading) and increased world market oil prices. In this research, we analyze this potential change using data envelopment analysis on the existing transportation chain alternatives in the Helsinki (Finland)−Tallinn (Estonia) short sea route (chains using either roro, ropax or container ships). The analysis also includes the planned railway tunnel between the two cities. On the basis of our findings, the current truck and semi-trailer-based transportation is challenged by containers, irrespective of how they are carried (ship type). In the long term, for reasons of emissions and oil independency, the possibility of tunnel construction would make it vital to have container ship operations available along this route. The forthcoming change is not radical, but rather evolutionary and long term oriented.
The environmental friendliness of short sea shipping has been justified in Europe by the ensuing lower congestion at hinterlands and unneeded large-scale infrastructure investments [...]
Classification for Safety-Critical Car-Cyclist Scenarios Using Machine Learning
I. Cara, E. Gelder
The number of fatal car-cyclist accidents is increasing. Advanced Driver Assistance Systems (ADAS) can improve the safety of cyclists, but they need to be tested with realistic safety-critical car-cyclist scenarios. In order to store only relevant scenarios, an online classification algorithm is needed. We demonstrate that machine learning techniques can be used to detect and classify those scenarios based on their trajectory data. A dataset consisting of 99 realistic car-cyclist scenarios is gathered using an instrumented vehicle. We achieved a classification accuracy of the gathered data of 87.9%. The execution time of only 45.8 us shows that the algorithm is suitable for online purposes. cop. 2015 IEEE.
The number of fatal car-cyclist accidents is increasing. Advanced Driver Assistance Systems (ADAS) can improve the safety of cyclists, but they need to be tested with realistic [...]
On certain anisotropic elliptic equations arising in congested optimal transport: local gradient bounds
G. Carlier, L. Brasco
Motivated by applications to congested optimal transport problems, we prove higher integrability results for the gradient of solutions to some anisotropic elliptic equations, exhibiting a wide range of degeneracy. The model case we have in mind is the following: \\[ \\partial_x \\left[(|u_{x}|-\\delta_1)_+^{q-1}\\, \\frac{u_{x}}{|u_{x}|}\\right]+\\partial_y \\left[(|u_{y}|-\\delta_2)_+^{q-1}\\, \\frac{u_{y}}{|u_{y}|}\\right]=f, \\] for $2\\le q"\\infty$ and some non negative parameters $\\delta_1,\\delta_2$. Here $(\\,\\cdot\\,)_+$ stands for the positive part. We prove that if $f\\in L^\\infty_{loc}$, then $\ abla u\\in L^r_{loc}$ for every $r\\ge 1$.
Motivated by applications to congested optimal transport problems, we prove higher integrability results for the gradient of solutions to some anisotropic elliptic equations, [...]
The Potential Impact of Vehicle-to-Vehicle and Sensor-to-Vehicle Communication in Urban Parking
K. Martens, R. Heijden, G. Tasseron
Studies have shown that up to thirty percent of all traffic in crowded urban areas can be cruising for parking. Information provision to drivers can potentially decrease cruising time for individual drivers and subsequently improve the performance of the overall system. While most cities provide drivers with information on the occupancy rates of off-street parking facilities, information on single on-street parking places was non-existing until recently. Recent technological advances have made it possible to provide such information.
Studies have shown that up to thirty percent of all traffic in crowded urban areas can be cruising for parking. Information provision to drivers can potentially decrease cruising [...]
Marine spatial planning and Good Environmental Status: : a perspective on spatial and temporal dimensions
A. Gilbert, K. Alexander, R. Sardá, R. Brazinskaite, C. Fischer, K. Gee, M. Jessopp, P. Kershaw, H. Los, D. Morla, C. O'Mahony, M. Pihlajamäki, S. Rees, V. Varjopuro, R. Riku
Accident risk and factors regarding non-motorised road users - a central road safety challenge with deficient data
N. Agerholm, C. Andersen
most half of all traffic fatalities worldwide are non-motorised road users (NMRUs). In Denmark, the number has increased with about 30%. NMRUs consist of about 63% of the injured in the Danish traffic. Much has been done to reduce the number of injured NMRUs with counterparts, while little effort is put into the reduction of the vast majority of the accidents, NMRU single accidents, which are about 90% of all injured NMRUs. There are no efficient tools available to reduce this number. A significantly better designed, maintained, and illuminated road network would most likely help. However, that is expensive and not possible for most road authorities. Despite this, the challenges with NMRUs in single accidents need more attention, if road safety is to be improved. The situation in Denmark is more than likely the case in many other countries as well; although the documentation is scarce. Almost half of all traffic fatalities worldwide are non-motorised road users (NMRUs). In Denmark, the number has increased with about 30%. NMRUs consist of about 63% of the injured in the Danish traffic. Much has been done to reduce the number of injured NMRUs with counterparts, while little effort is put into the reduction of the vast majority of the accidents, NMRU single accidents, which are about 90% of all injured NMRUs. There are no efficient tools available to reduce this number. A significantly better designed, maintained, and illuminated road network would most likely help. However, that is expensive and not possible for most road authorities. Despite this, the challenges with NMRUs in single accidents need more attention, if road safety is to be improved. The situation in Denmark is more than likely the case in many other countries as well; although the documentation is scarce.
most half of all traffic fatalities worldwide are non-motorised road users (NMRUs). In Denmark, the number has increased with about 30%. NMRUs consist of about 63% of the [...]
Flow–structure–seabed interactions in coastal and marine environments
B. Sumer
Flow–structure–seabed interaction in coastal and marine environments is a rapidly growing area of research and applications. In this vision paper, this area is discussed with a view of identifying its state of the art and current research challenges. The discussion draws attention to key issues related to structures such as marine pipelines, offshore windfarms, and multiuse offshore platforms. Tsunamis, which received considerable attention after two recent extreme events (2004 Indonesia tsunami and 2011 Japan tsunami) are also included in the discussion. Marine hydro-geomechanics is highlighted, among other areas, as an emerging branch of Marine Civil Engineering. Predictions of the field development for the forthcoming years are also briefly outlined."br
Flow–structure–seabed interaction in coastal and marine environments is a rapidly growing area of research and applications. In this vision paper, this area is discussed [...]
Roads to Nowhere:The Accuracy of Travel Demand Forecasts for Do-nothing Alternatives
P. Næss, M. Nicolaisen
Impact appraisals of major transport infrastructure projects rely extensively on the accuracy of forecasts for the expected construction costs and aggregate travel time savings. The latter of these further depend on the accuracy of forecasts for the expected travel demand in both the do-something and do-nothing alternatives, in order to assess the impact of implementing new projects compared to doing nothing or postponing the decision. Previous research on the accuracy of travel demand forecasts has focused exclusively on the do-something alternatives, where inaccuracies have been revealed in the form of large imprecision as well as systematic biases. However, little or no attention has been given to the accuracy of demand forecasts for the do-nothing alternatives, which are equally important for impact appraisals. This paper presents the first ex-post evaluation of demand forecast accuracy for do-nothing alternatives, based on an empirical study of 35 road projects in Denmark and England. The results show a tendency for systematic overestimation of travel demand in the do-nothing alternatives, which is in contrast to the systematic underestimation of travel demand observed in previous studies of do-something alternatives. The main implication for planning practice is that the severity of future congestion problems is systematically overestimated. As a consequence, impact appraisals of road construction as a means of congestion relief appear overly beneficial.
Impact appraisals of major transport infrastructure projects rely extensively on the accuracy of forecasts for the expected construction costs and aggregate travel time savings. [...]
Towards smart grids: Identifying the risks that arise from the integration of energy and transport supply chains
T. Onna, J. Eising, F. Alkemade
This paper identifies the risks for the functionality and reliability of the grid that arise from the integration of the transport and supply chain. The electrification of transport is a promising option for the transition to a low carbon energy and transport system. But on the short term, the electrification of transport also creates risks. More specifically, when promising technological such as vehicle-to-grid and smart-grids are not yet available on a large scale, the rapid diffusion of electric vehicles and the recharging behaviour of consumers can create risks for grid functioning. In order to assess these risks, this paper present a GIS-based simulation method that assesses electricity demand and supply on the neighbourhood level. The paper combines local level electric vehicle diffusion forecasts, with neighbourhood level data about the grid additional capacity. Application of the model to the Netherlands shows that risks for grid functioning already appear as early as 2015. More specifically, the diffusion of electric vehicles is found to compromise the functioning of the grid on the short term in densely populated areas such as Amsterdam. In these neighbourhoods early and fast adoption of electric vehicles coincides with the presence of an older grid with less additional capacity. The model provides insights for grid operators as well as for policy makers that seek to stimulate the transition to sustainable energy and transport systems, and can be used as a strategic tool to plan (smart) grid investments.
This paper identifies the risks for the functionality and reliability of the grid that arise from the integration of the transport and supply chain. The electrification of [...]
A three-dimensional macroscopic fundamental diagram for mixed bi-modal urban networks
N. Zheng, N. Geroliminis, K. Ampountolas
Recent research has studied the existence and the properties of a macroscopic fundamental diagram (MFD) for large urban networks. The MFD should not be universally expected as high scatter or hysteresis might appear for some type of networks, like heterogeneous networks or freeways. In this paper, we investigate if aggregated relationships can describe the performance of urban bi-modal networks with buses and cars sharing the same road infrastructure and identify how this performance is influenced by the interactions between modes and the effect of bus stops. Based on simulation data, we develop a three-dimensional vehicle MFD (3D-nu MFD) relating the accumulation of cars and buses, and the total circulating vehicle flow in the network. This relation experiences low scatter and can be approximated by an exponential-family function. We also propose a parsimonious model to estimate a three-dimensional passenger MFD (3D-pMFD), which provides a different perspective of the flow characteristics in hi-modal networks, by considering that buses carry more passengers. We also show that a constant Bus-Car Unit (BCU) equivalent value cannot describe the influence of buses in the system as congestion develops. We then integrate a partitioning algorithm to cluster the network into a small number of regions with similar mode composition and level of congestion. Our results show that partitioning unveils important traffic properties of flow heterogeneity in the studied network. Interactions between buses and cars are different in the partitioned regions due to higher density of buses. Building on these results, various traffic management strategies in hi-modal multi-region urban networks can then be integrated, such as redistribution of urban space among different modes, perimeter signal control with preferential treatment of buses and bus priority. (C) 2014 Elsevier Ltd. All rights reserved.
Recent research has studied the existence and the properties of a macroscopic fundamental diagram (MFD) for large urban networks. The MFD should not be universally expected [...]
Comparison of Electric Vehicles Charging Strategies and their Impact on Network Capacity
F. Malik, L. Lehtonen, M. Matti
Identification of effective visual problem solving strategies in a complex visual domain
J. Merriënboer, J. Merriënboer, S. Brand-Gruwel, L. Meeuwen, H. Jarodzka, J. Bock, P. Kirschner
Students in complex visual domains must acquire visual problem solving strategies that allow them to make fast decisions and come up with good solutions to real-time problems. In this study, 31 air traffic controllers at different levels of expertise (novice, intermediate, expert) were confronted with 9 problem situations depicted on a radar screen. Participants were asked to provide the optimal order of arrival of all depicted aircrafts. Eye-movements, time-on-task, perceived mental effort, and task performance were recorded. Eye-tracking data revealed that novices use inefficient means-end visual problem solving strategies in which they primarily focus on the destination of aircraft. Higher levels of expertise yield visual problem solving strategies characterized by more efficient retrieval of relevant information and more efficient scan paths. Furthermore, experts' solutions were more similar than intermediates' solutions and intermediates' solutions were more similar than novices' solutions. Performance measures showed that experts and intermediates reached better solutions than novices, and that experts were faster and invested less mental effort than intermediates and novices. These findings may help creating eye-movement modeling examples for the teaching of visual problem solving strategies in complex visual domains.
Students in complex visual domains must acquire visual problem solving strategies that allow them to make fast decisions and come up with good solutions to real-time problems. [...]
Reference architecture for interoperability testing of Electric Vehicle charging
F. Lehfuss, E. Werkmany, J. Lopezz, M. Nöhrer, E. Zabalaz
This paper presents a reference architecture for interoperability testing of electric vehicles as well as their support equipment with the smart grid and the e-Mobility environment. Pan-European Electric Vehicle (EV)-charging is currently problematic as there are compliance and interoperability issues on different communication levels and among the different domains comprising the eMobility system. The discussed reference architecture is composed out of three layers that enable addressing a direct mapping of interfaces, functions and services, as well as real world actors and/or laboratory equipment. Utilizing this architecture enables cross-domain co-simulation for interoperability within the electric mobility and the smart grid environment. Given the future challenges that rise to interoperability testing and the fact that certain aspects are yet unknown or rarely defined, the discussed architecture presents a very modular and extendable approach. © 2015 IEEE.
This paper presents a reference architecture for interoperability testing of electric vehicles as well as their support equipment with the smart grid and the e-Mobility environment. [...]
Seismic vulnerability of gas and liquid buried pipelines
F. Magistris, G. Fabbrocino, E. Salzano, G. Lanzano
Lifelines play a crucial and essential role in human life and in economic development. The resilience of those systems under extreme events as earthquakes is a primary requirement, especially when large amount of toxic and flammable material are transported. In this work, the seismic vulnerability of buried gas and liquid pipelines has been analyzed, starting from a large number of damage data to pipelines collected from post-earthquake reconnaissance reports.Seismic fragility formulations and threshold values for the earthquake intensity with respect to the release of content from different types of pipelines have been derived. The main outcome of the work is therefore a novel seismic assessment tool which is able to cover the needs of industrial risk assessment procedures and land use planning requirements. © 2013 Elsevier Ltd.
Lifelines play a crucial and essential role in human life and in economic development. The resilience of those systems under extreme events as earthquakes is a primary requirement, [...]
Prices, prices and prices: A study in the airline sector
J. García, S. Forgas-Coll, Y. Narangajavana, F. Garrigos-Simon
[EN] This article creates and empirically analyzes a dynamic pricing framework. The model includes the main theories on prices in the literature, and dynamically analyzes how companies change their pricing policies according to different circumstances. Through the use of regressions, the article shows the relative long and short-term impact on pricing equations (strategic versus operational matters), and on production conditions versus demand perspectives in the setting of operational prices. In addition, it highlights how the proportions of these pricing perspectives change in the airline sector according to the effect of the environment on the configuration of the strategy (determined by monthly and hourly seasonality), and also according to the nature of this competitive strategy (determined by the characteristics of the company and the destination airport). The results confirm our framework, and show how firms combine the different perspectives in their pricing strategies. We acknowledge financial support from the Valencian Economic Research Institute (IVIE, Valencian Government), and the Universitat Politècnica de València. Narangajavana, Y.; Garrigós Simón, FJ.; Sánchez García, J.; Forgas-Coll, S. (2014). Prices, prices and prices: A study in the airline sector. Tourism Management. 41:28-42. doi:10.1016/j.tourman.2013.08.008 S 28 42 41
[EN] This article creates and empirically analyzes a dynamic pricing framework. The model includes the main theories on prices in the literature, and dynamically analyzes [...]
Friction power modeling and measurements in sliding vane rotary compressors
In compressed air systems, mechanical and organic losses account for 15% of compressor energy consumption. In the current research, the energy saving potential achievable through friction power reduction in sliding vane rotary compressors was investigated using experimental and modeling approaches. Tests on a new mid-size industrial compressor operating at different steady conditions (outlet pressure 9, 12.5, 14.5 bar at 1000 and 1500 RPM) assessed the machine performance through measurement of mechanical power and the reconstruction of the pressure-volume diagram. An experimental methodology was also developed to quantify the power lost by friction and its measurement uncertainty using the concept of indicated mean effective pressure. Modeling the compressor blade dynamics allowed a friction power decomposition while an analysis of the hydrodynamic lubrication at the most severe friction location, namely between blade tip and stator wall, additionally provided the oil film thickness evolution along the contact surface. The agreement between modeling and experimental data identified a value for the friction coefficient of 0.065. Design suggestions on existing machines and new design solutions were eventually outlined varying blade mass, revolution speed and compressor aspect ratio. These improved configurations predicted an efficiency increase up to 6%. The authors acknowledge Ing. Enea Mattei S.p.A. and particularly its CEO, Dr. Giulio Contaldi, for continuous research funding and support. The work has been done also under the FP7 Project "Complete Vehicle Energy-Saving CONVENIENT" funded by the European Commission.
In compressed air systems, mechanical and organic losses account for 15% of compressor energy consumption. In the current research, the energy saving potential achievable [...]
A comparative analysis of the effects of economic policy instruments in promoting environmentally sustainable transport
F. Ramjerdi, R. Elvik
This paper presents a comparative analysis of the effects of economic policy instruments in promoting environmentally sustainable transport. Promoting environmentally sustainable transport is defined as follows: (1) Reducing the volume of motorised travel; (2) Transferring travel to modes generating less external effects, and (3) Modifying road user behaviour in a way that will reduce external effects of transport. External effects include accidents, congestion, traffic noise and emissions to air. Four economic policy instruments are compared: (1) Prices of motor fuel; (2) Congestion charges; (3) Toll schemes; (4) Reward systems giving incentives to reduce driving or change driver behaviour. The effects of these policy instruments are stated in terms of elasticities. All four economic policy instruments have negative elasticities, which means that they do promote environmentally sustainable transport. Long-term elasticities tend to be larger than short term elasticities. The long-term elasticities of reward systems are unknown. (C) 2014 Elsevier Ltd. All rights reserved.
This paper presents a comparative analysis of the effects of economic policy instruments in promoting environmentally sustainable transport. Promoting environmentally sustainable [...]
Trade-offs between mobility and equity maximization under environmental capacity constraints: A case study of an integrated multi-objective model
T. Feng, H. Timmermans
This paper investigates the performance of a policy decision tool proposed for multi-objective decision under different policy interventions. This tool deals with the trade-off between mobility and equity maximization under environmental capacity constraints. Two system objectives, maximization of mobility and equity, are formulated in terms of the sum of total car ownership and number of trips, and the differences in accessibility between zones. Environmental capacities are based on production efficiency theory in which the frontier emission under maximum system efficiency is taken as environmental capacity. To examine the performance of the proposed model, three types of hypothetical policies (network improvement, population increase and urban sprawl) are formulated. Effects are simulated using data pertaining to Dalian City, China. Results show that the proposed model is capable of representing the trade-offs between mobility and equity based on different policy interventions. Compared with two extreme cases with the single objective of mobility maximization or equity maximization, the Pareto-optimal solutions provide more interesting practical options for decision makers. Taking the solution based on the maximum equity as an example, the policy of urban sprawl yields the most significant improvement in both emission and accessibility of the three scenarios.
This paper investigates the performance of a policy decision tool proposed for multi-objective decision under different policy interventions. This tool deals with the trade-off [...]
To Fuel or Not to Fuel? Is that the Question?
E. Ayra, J. Cano, D. Insua
ording to the International Air Transport Association, the industry fuel bill accounts for more than 25% of the annual airline operating costs. In times of severe economic constraints and increasing fuel costs, air carriers are looking for ways to reduce costs and improve fuel efficiency without putting flight safety into jeopardy. In particular, this is inducing discussions on how much additional fuel to put in a planned route to avoid diverting to an alternate airport due to Air Traffic Flow Management delays. We provide here a general model to support such decisions. We illustrate it with a case study and provide comparison with the current practice, showing the relevance of our approach. Peer Reviewed
ording to the International Air Transport Association, the industry fuel bill accounts for more than 25% of the annual airline operating costs. In times of severe economic [...]
Seismic damage to pipelines in the framework of Na-Tech risk assessment
F. Magistris, G. Fabbrocino, G. Lanzano, E. Salzano
The structural integrity of pipelines undergone seismic waves is crucial for industrial installation and for the distributed transportation networks of gaseous and liquid fluids. However, it is nowadays proved that the definition of seismic vulnerability based on purely, structural-derived limit states or on return-to-service or even on the purely economic repair rate indications, is not sufficient for the holistic analysis of risks. On the other hand, detailed numerical studies based on full analyses (including fluid/soil/structure interaction) are too expensive for the aims of risk assessment and simplified methodologies are still needed.In this paper, a large database of earthquake-induced damage for steel and non-steel pipelines is presented. Each case was analyzed and collected from post-earthquake reconnaissance, seismic engineering reports and technical papers. The database may be adopted for the definition of specific vulnerability function (fragility curves), which are commonly implemented in multi-hazard analyses, and more in general for the assessment of Na-Tech risks (Natural events triggering Technological disasters). © 2014 Elsevier Ltd.
The structural integrity of pipelines undergone seismic waves is crucial for industrial installation and for the distributed transportation networks of gaseous and liquid [...]
The adoption of sustainable innovations: Driven by symbolic and environmental motives
Personalised feedback and eco-driving: An explorative study
B. Holleman, T. Hof, A. Stuiver, R. Brouwer, J. Pauwelussen, L. Kroon
Conventional road transport has negative impact on the environment. Stimulating eco-driving through feedback to the driver about his/her energy conservation performance has the potential to reduce CO"inf"2"/inf" emissions and promote fuel cost savings. Not all drivers respond well to the same type of feedback. Research has shown that different drivers are attracted to different types of information and feedback. The goal of this paper is to explore which different driver segments with specific psychographic characteristics can be distinguished, how these characteristics can be used in the development of an ecodriving support system and whether tailoring eco-driving feedback technology to these different driver segments will lead to increased acceptance and thus effectiveness of the eco feedback technology. The driver segments are based on the value orientation theory and learning orientation theory. Different possibilities for feedback were tested in an exploratory study in a driving simulator. An explorative study was selected since the choice of the display (how and when the information is presented) may have a strong impact on the results. This makes testing of the selected driver segments very difficult. The results of the study nevertheless suggest that adapting the display to a driver segment showed an increase in acceptance in certain cases. The results showed small differences for ratings on acceptation, ease of use, favouritism and a lower general rating between matched (e.g., learning display with learning oriented drivers) and mismatched displays (e.g., learning display with performance oriented drivers). Using a display that gives historical feedback and incorporates learning elements suggested a non-verifiable increase in acceptance for learning oriented drivers. However historical feedback and learning elements may be less effective for performance oriented drivers, who may need comparative feedback and game elements to improve energy conserving driving behaviour. © 2015.
Conventional road transport has negative impact on the environment. Stimulating eco-driving through feedback to the driver about his/her energy conservation performance has [...]
Frequency modelling and solution of fluid–structure interaction in complex pipelines
Z. Jiao, D. Johnston, A. Plummer, Y. Xu, Y. Xu
Complex pipelines may have various structural supports and boundary conditions, as well as branches. To analyse the vibrational characteristics of piping systems, frequency modelling and solution methods considering complex constraints are developed here. A fourteen-equation model and Transfer Matrix Method (TMM) are employed to describe Fluid–Structure Interaction (FSI) in liquid-filled pipes. A general solution for the multi-branch pipe is proposed in this paper, offering a methodology to predict frequency responses of the complex piping system. Some branched pipe systems are built for the purpose of validation, indicating good agreement with calculated results.
Complex pipelines may have various structural supports and boundary conditions, as well as branches. To analyse the vibrational characteristics of piping systems, frequency [...]
European Cities Dealing with Climate Issues: Ideas and Tools for a Better Framing of Current Practices.
Contemporary cities have to deal with numerous challenges, from the growth and aging of urban populations to the scarcity of resources; from environmental degradation to climate change. The latter, also due to the increasing severity of climate-related impacts on urban areas, is widely considered one of the most urgent challenges for urban development in the near future: cities are the main contributors to energy consumption and GHG emissions, paying, at the same time, the highest price for the climate impacts. Thus, climate issues have gained increasing importance in the last decades, both in terms of the metaphors coined by scholars relative to urban future (low-carbon cities, transition cities, smart cities, resilient cities, etc.) and in terms of the initiatives undertaken on different institutional levels. Unfortunately, mitigation and adaptation are generally regarded as two different approaches, neglecting the potential synergies and trade-offs between the related strategies. Hence, based on the growing awareness of the need for mainstreaming mitigation and adaptation policies at city level, this study will provide an overview of the state of the art of the mitigation and adaptation initiatives in Italian metropolitan cities. Then, focusing on the concepts of the "smart" and the "resilient" city – recognized as key concepts for reducing CO2 emissions and improving the ability of cities to respond to climate impacts – and with reference to a conceptual framework for building up a smart and resilient urban system carried out in previous research works (Papa et al., 2015), the study will examine case studies of the cities of Rotterdam and Barcelona, highlighting how this framework may improve our understanding and, above all, contribute to better integration of the fragmented on-going strategies and initiatives.
Contemporary cities have to deal with numerous challenges, from the growth and aging of urban populations to the scarcity of resources; from environmental degradation to climate [...]
Temporal Performance of Advanced Driver Assistance Systems vis-á-vis Human Driving Behavior in Dense Traffic
R. Hamberg, T. Hendriks, T. Bijlsma
dvanced Driver Assistance Systems (ADAS) are becoming ubiquitous, and gradually take over the role of human drivers in the vision of the automotive sector. Humans are different from most systems: while in general humans exhibit a much higher error rate when performing specific functions, they are also unmatched in their adaptability, and their ability to recognize patterns and anticipate on these. In this paper we derive temporal requirements on future ADAS operation, needed to at least match human driving behavior in dense traffic. We examine Adaptive Cruise Control and Automated Emergency Braking systems at highway speeds, derive temporal requirements, and show that in dense traffic situations intent communication has a significant benefit to improve systems operation. The resulting requirements will challenge ADAS developments in the coming years. cop. 2015 IEEE.
dvanced Driver Assistance Systems (ADAS) are becoming ubiquitous, and gradually take over the role of human drivers in the vision of the automotive sector. Humans are different [...]
Impacts of a green-driving application in city buses on fuel consumption, speeding and passenger comfort
S. Innamaa, M. Penttinen
The impacts of a green-driving application on fuel consumption, speeding and passenger comfort were assessed on a frequently operated bus route in the Helsinki metropolitan area. The main results of the study show that use of a green-driving application in buses significantly reduces fuel consumption and speeding and increases passenger comfort. Novel users of the system drove more economically than those who had used the system longer, but the impact was smaller than found in previous studies. In addition to savings on fuel, the application encouraged driving within the speed limit. A transfer effect on journeys without the system was found for long-term users but not for novel users. Furthermore, passengers of drivers who used the green-driving application gave better grades for decelerations and the driver's service attitude in peak traffic. The authors main conclusion is that use of a green-driving application is beneficial even after years of use for reducing both fuel consumption and speeding. The authors recommended that drivers be regularly encouraged to use the system and that green driving be included in driver training.
The impacts of a green-driving application on fuel consumption, speeding and passenger comfort were assessed on a frequently operated bus route in the Helsinki metropolitan [...]
Design of lightweight magnesium car body structure under crash and vibration constraints
M. Rais-Rohani, M. Kiani, K. Motoyama, I. Gandikota
Car body design in view of structural performance and lightweighting is a challenging task due to all the performance targets that must be satisfied such as vehicle safety and ride quality. In this paper, material replacement along with multidisciplinary design optimization strategy is proposed to develop a lightweight car body structure that satisfies the crash and vibration criteria while minimizing weight. Through finite element simulations, full frontal, offset frontal, and side crashes of a full car model are evaluated for peak acceleration, intrusion distance, and the internal energy absorbed by the structural parts. In addition, the first three fundamental natural frequencies are combined with the crash metrics to form the design constraints. The wall thicknesses of twenty-two parts are considered as the design variables. Latin Hypercube Sampling is used to sample the design space, while Radial Basis Function methodology is used to develop surrogate models for the selected crash responses at multiple sites as well as the first three fundamental natural frequencies. A nonlinear surrogate-based optimization problem is formulated for mass minimization under crash and vibration constraints. Using Sequential Quadratic Programming, the design optimization problem is solved with the results verified by finite element simulations. The performance of the optimum design with magnesium parts shows significant weight reduction and better performance compared to the baseline design. Document type: Article
Car body design in view of structural performance and lightweighting is a challenging task due to all the performance targets that must be satisfied such as vehicle safety [...]
Evaluation of Strain Distribution for the P91 Steel under Static Load Using Espi System
D. Kukla, P. Grzywna
The goal of the research was to evaluate the change of displacement/strain phases in the P91 steel under static load conditions. Undertaken tests were aimed at estimation and analysis of the impact of the material state, which was subjected to loading conditions, on the distribution of stress pattern using ESPI system. Specimen made of high temperature creep resistant steel X10CrMoVNb9-1 (P91) used as a construction material for boiler steam feed heaters, vapor tanks, pressure vessels and vapor pipelines, is used in the service conditions of temperature range up of 650°C. Test samples were taken from two P91 steel pipes. One sample came from a segment of a pipeline transporting fresh vapor in time 80 000 h, under the pressure of 8.4 MPa and temperature 540 °C. The second sample was the same material but in the delivery state. Document type: Article
The goal of the research was to evaluate the change of displacement/strain phases in the P91 steel under static load conditions. Undertaken tests were aimed at estimation [...]
Traffic Information Interface Development in Route Choice Decision
S. Chowdhury, B. Haque, G. Sarwar
"jats:p"In this paper, a method has been developed based on historic traffic data (vehicle speed), which helps the commuters to choose routes by their intelligence knowing the traffic conditions in Google maps. Data has been collected on basis of video analysis from several segments between Tuker Bazar and Bandar Bazar route. For each of the video footage, a reference length has been recorded with measurement tape for use in video analysis. Software has been also developed based on Java language to get the traffic information from historic data, which shows the output as images consisting of traffic speed details on the available routes by giving day and time limit as inputs. The developed models provide useful insights and helpful for the policy makers that can lead to the reduction of traffic congestion and increase the scope of intelligence of the road users, at least for the underdeveloped or developing country where navigation is still unavailable. Document type: Article
"jats:p"In this paper, a method has been developed based on historic traffic data (vehicle speed), which helps the commuters to choose routes by their intelligence knowing [...]
Reduced carbon and energy footprint in highway operations: the Highway Energy Assessment (HERA) methodology
S. Hernandez, A. Monzón, N. Sobrino
Global demand for mobility is increasing and the environmental impact of transport has become an important issue in transportation network planning and decision-making, as well as in the operational management phase. Suitable methods are required to assess emissions and fuel consumption reduction strategies that seek to improve energy efficiency and furthering decarbonization. This study describes the development and application of an improved modeling framework – the HERA (Highway EneRgy Assessment) methodology – that enables to assess the energy and carbon footprint of different highways and traffic flow scenarios and their comparison. HERA incorporates an average speed consumption model adjusted with a correction factor which takes into account the road gradient. It provides a more comprehensive method for estimating the footprint of particular highway segments under specific traffic conditions. It includes the application of the methodology to the Spanish highway network to validate it. Finally, a case study shows the benefits from using this methodology and how to integrate the objective of carbon footprint reductions into highway design, operation and scenario comparison. Document type: Article
Global demand for mobility is increasing and the environmental impact of transport has become an important issue in transportation network planning and decision-making, as [...]
Event Detection and Localization in Urban Water Distribution Network
T. Zan, B. Lee, K. Wong, A. Whittle, H. Lim
Urban water supply and distribution infrastructure is aging rapidly and the frequency of pipe burst increases. These events can be very expensive due to water supply disruptions, and damage to surrounding properties and infrastructures. Therefore, methods of detecting and localizing underground burst events in real time can be very helpful in mitigating these impacts. This paper is a part of the WaterWiSe@SG project in Singapore. In this paper, a cost-effective wireless sensor network was developed for a real-time monitoring, analyzing, and modeling of urban water distribution systems. This paper presents an application of joint time-frequency analysis (JTFA) for detecting events in water distribution pipelines. The idea behind this method is based on the detection of pressure fluctuations induced by the burst. This proposed approach for event detection employs a spectrogram, one of the JTFA approaches. The feasibility of the proposed method is tested through emulated leak-off experiments and validated with monitoring data in an operational system. The results demonstrate that the proposed method has the potential to assist in the management of water infrastructure by monitoring existing conditions and providing real-time feedback in case of the failure. Document type: Article
Urban water supply and distribution infrastructure is aging rapidly and the frequency of pipe burst increases. These events can be very expensive due to water supply disruptions, [...]
Numerical approach to a low pressure gas-injection scroll compressor
L. Liu, H. Li, Z. Dengke, T. Chen, P. Wei, G. Zhou
In order to solve problems arising in ordinary heat pump system for pure electric vehicles at ultra-low temperature, a low pressure gas-injection scroll compressor is designed, and a mathematical model is established. Comparison with the experimental results shows good accuracy of the theoretical prediction. Document type: Article
In order to solve problems arising in ordinary heat pump system for pure electric vehicles at ultra-low temperature, a low pressure gas-injection scroll compressor is [...]
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CommonCrawl
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Which Professor? Suppose Professor Alpha and Professor Omega each teach Introductory Biology. You need to Decide which professor to take the class from and have just completed your introductory statistics course. Records obtained from the past students indicated that students in professor alphas class have a mean score of 80% with a standard deviation of 5%, while past students in professor Omega's class have a mean score of 80% with a standard deviation of 10%. Decide which professor to take for introductory biology using a statistical argument.
Keonn'a
It indicates is that the variability is greater in Omega's class. If you are an "A" student, you need 90% to score at the 97.5 percentile in Alpha's class, while you would need 100% to score at the 97.5 percentile for Omega.
PsyDAG
You are on the roof of the physics building, 46 m above the ground. Your physics professor, who is 1.80 m tall, is walking alongside the building at a constant speed of 1.20 m/s. If you wish to drop an egg on your professor's
A professor orders 250 mL of house coffee at precisely 95°C. She then adds enough milk at 10°C to drop the temperature of the coffee to exactly 90° C. Calculate the amount of milk (in mL) the professor must add to reach this
Pysics
Realizing that she often doesn't have her students' full attention during class, a professor devises an elaborate device on which to stand while she lectures. The pulley is placed so that the string makes a 45.0-degree angle with
precalculus, complex numbers
Let $\omega$ be a complex number such that $\omega^7 = 1$ and $\omega \neq 1$. Let $\alpha = \omega + \omega^2 + \omega^4$ and $\beta = \omega^3 + \omega^5 + \omega^6$. Then $\alpha$ and $\beta$ are roots of the quadratic \[x^2 +
Read this sentence from Virginia Woolf's A Room of One's Own. "They had no money evidently; according to Professor Trevelyan they were married whether they liked it or not before they were out of the nurssery, at fifteen or
A college professor told her students, the purpose of a management course is to teach students about management ,not to teach them to be managers. Do you agree or disagree with this statement? Discuss.
7. Professors' Salaries The average salary for a Queens College full professor is $85,900. If the average salaries are normally distributed with a standard deviation of $11,000, find these probabilities. a. 0.3557 (TI: 0.3547)
(2) Suppose A is the set of students currently registered at the University of Calgary, B is the set of professors at the University of Calgary, and C is the set of courses currently being offered at the University of Calgary.
In a classroom demonstration, a 79.2-kg physics professor lies on a "bed of nails." The bed consists of a large number of evenly spaced, relatively sharp nails mounted in a board so that the points extend vertically outward
Professor Baird wants to mix a solution containing 10% acid with one containing 15% acid to obtain a 20 ounce solution containing 12% acid. How many ounces of the 10% solution should the professor use? Translate the word problem
The untimely death of Professor Hathaway halted his groundbreaking research into the uses of solar power. It therefore appears that Hathaway's research will not result in practical applications. Which one of the following, if
Professor Martin is considering leaving the university and opening a consultant business. His services as a consultant he would be paid $75,000 a year. To open the business professor Martin must convert a house from which he
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CommonCrawl
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Proceedings of the 29th International Conference on Genome Informatics (GIW 2018): systems biology
Analysis of significant protein abundance from multiple reaction-monitoring data
Jongsu Jun1,
Jungsoo Gim2,
Yongkang Kim1,
Hyunsoo Kim3,4,
Su Jong Yu5,
Injun Yeo3,
Jiyoung Park3,
Jeong-Ju Yoo5,
Young Youn Cho5,
Dong Hyeon Lee5,
Eun Ju Cho5,
Jeong-Hoon Lee5,
Yoon Jun Kim5,
Seungyeoun Lee6,
Jung-Hwan Yoon5,
Youngsoo Kim3,4 &
Taesung Park1,7
Discovering reliable protein biomarkers is one of the most important issues in biomedical research. The ELISA is a traditional technique for accurate quantitation of well-known proteins. Recently, the multiple reaction-monitoring (MRM) mass spectrometry has been proposed for quantifying newly discovered protein and has become a popular alternative to ELISA. For the MRM data analysis, linear mixed modeling (LMM) has been used to analyze MRM data. MSstats is one of the most widely used tools for MRM data analysis that is based on the LMMs. However, LMMs often provide various significance results, depending on model specification. Sometimes it would be difficult to specify a correct LMM method for the analysis of MRM data. Here, we propose a new logistic regression-based method for Significance Analysis of Multiple Reaction Monitoring (LR-SAM).
Through simulation studies, we demonstrate that LMM methods may not preserve type I error, thus yielding high false- positive errors, depending on how random effects are specified. Our simulation study also shows that the LR-SAM approach performs similarly well as LMM approaches, in most cases. However, LR-SAM performs better than the LMMs, particularly when the effects sizes of peptides from the same protein are heterogeneous. Our proposed method was applied to MRM data for identification of proteins associated with clinical responses of treatment of 115 hepatocellular carcinoma (HCC) patients with the tyrosine kinase inhibitor sorafenib. Of 124 candidate proteins, LMM approaches provided 6 results varying in significance, while LR-SAM, by contrast, yielded 18 significant results that were quite reproducibly consistent.
As exemplified by an application to HCC data set, LR-SAM more effectively identified proteins associated with clinical responses of treatment than LMM did.
Discovering protein disease biomarkers is an urgently pressing issue in biomedical research [1]. Historically, the enzyme-linked immunosorbent assay (ELISA) is a highly accurate protein quantitation technique [2], representing the "gold standard" for measuring levels of specific proteins [3]. However, recent discoveries of many novel proteins, having no available antibodies, now limits the use of the ELISA method [4]. Moreover, for newly discovered proteins, development of high quality ELISA assays requires considerable time and resources [5]. Together, these shortcomings have created a need for a different technique of targeted protein quantitation [6]. One such technique, quantitative mass spectrometry of proteins has advanced significantly over the last decade, and several methods have been developed for relative quantification of targeted proteins, using this technique [7].
In recent years, multiple reaction monitoring (MRM) mass spectrometry has been developed as an attractive tool for targeted proteins and now represents a promising alternative to ELISA for quantification of proteins. To that end, MRM uses sequence-specific tandem mass spectrometer fragmentations of peptides that can provide highly selective measurements for peptide-containing proteins [8]. Thus, without enrichment or fractionation approaches, MRM assays allow for the quantitation of protein ions within ranges of low (μg/mL) to high (ng/mL) levels [5, 9]. Additionally, development time for MRM assays is relatively shorter and less expensive than that for ELISA, with no requirements (and thus no costs) for antibody development.
Due to these many advantages, MRM assays are being increasingly used in systems biology and clinical investigations [10,11,12,13]. There are some analysis tools to analyze MRM data. Skyline is representative tool to create and analyze MRM dataset [14]. ProteoSign gives some simple statistical analysis results to find differentially expressed proteins [15]. roHits-viz is a web-based tool for visualizing interaction of protein data [16]. However, the development of statistical methods to determine significantly abundant proteins by the MRM assay has not received enough attention, compared to improvement of the MRM assay technology itself [17]. For MRM data analysis, two-sample t-tests or paired t-tests have been applied to identify proteins that change in abundance between two groups [18,19,20]. To test for multiple groups, one-way analysis of variance (ANOVA) has been employed [21, 22].
Recently, a linear mixed model (LMM) approach was proposed for MRM data analysis, and was implemented in MSstats (v3.7.1) [17], resulting in its widespread adoption [23]. LMM approaches include a single parameter for a group effect, peptide effects, and group and peptide interaction effects, along with subject and run effects. LMMs include both subject and run effects that can be designated as either random or fixed, to identify significantly evident proteins by testing the group effect. Recently, we observed that the LMM approach often provides different p-values for the group effect, from the same data, depending on which effects are treated as random or fixed. In particularly, the LMM test results vary considerably when there is an interaction effect between group and peptide.
In this report, we propose a new logistic regression-based method for Significance Analysis of Multiple Reaction Monitoring (LR-SAM). Unlike LMMs, our LR-SAM approach uses a much smaller number of parameters. Moreover, our LR-SAM does not require inclusion of all the effects related to the run. Accordingly, our model does not need to specify run effects as random or fixed.
Since LR-SAM uses log2-transformed relative intensity values, it does not need to include run effects. We consider two separate cases when the effects of peptides in a protein are homogeneous or heterogeneous. For the significance test for proteins, we consider Wald type tests, likelihood ratio tests (LRTs), and score tests [24].
Through various simulation studies, we compared the performance of LMMs to our approach. We first observed cases in which an LMM approach did not preserve type I error, and we also examined cases in which LR-SAM methods performed better. Some tests, based on the LR-SAM method, were more powerful when the expression pattern of peptide between groups was heterogeneous.
To further establish the translational relevance of our method, we compared LMMs to our approaches in a clinical study to identify candidate serum biomarkers of sorafenib response and prognosis in patients with hepatocellular carcinoma (HCC). Sorafenib is a multikinase inhibitor mainly used for the treatment of various solid tumors including HCC [25]. However, the response rate of sorafenib is about 8% for HCC patients. Due to its high cost, it would be more economical and more beneficial to patients if Sorafenib is applied only to the patients with high chance of response. Moreover, there are no good markers to predict the patients' response to sorafenib [26]. From May 2013 to August 2014, 115 HCC patient serum samples were collected from Seoul National University Hospital as part of an ongoing HCC study. One hundred twenty-four candidate protein biomarkers, and hepatic disease-associated proteins, were chosen from 50,265 proteins, based on the LiverAtlas database [27]. Sorafenib responses were measured according to the modified response evaluation criteria in solid tumors (mRECIST) guidelines [28]. Patients with complete responses, partial responses, or stable disease were categorized as responders, while those with progressive disease were categorized non-responders. Both LMM approaches and LR-SAM methods were employed to analyze the MRM data.
MRM dataset structure
Table 1 shows the structure of MRM dataset. Sample ID indicates individual, Run the shared run membership of the endogenous, and Area Ratio the expression of peptide detected. The MRM data has a hierarchical structure such that one protein contains several peptides. Since our goal is to identify proteins that are significantly different between the group, we need to combine the summarized expression of peptides from the same protein efficiently. Also, the batch effect would occur when MRM experiments were performed in different batches separately. These characteristics of MRM make it difficult to use the standard analyses such as two sample t-test.
Table 1 MRM dataset structure
Review of LMM approach
For a given protein, suppose it is comprised of K peptides. Let yi, j(i), k, l denote the log2(intensity) value of the j-th subject, nested in the i-th group of the k-th peptide and the l-th run. Then, the LMM used in MSstats is given as follows:
$$ {\mathrm{y}}_{\mathrm{i},\mathrm{j}\left(\mathrm{i}\right),\mathrm{k},\mathrm{l}}=\mu +{G}_i+S{(G)}_{j(i)}+{P}_k+{R}_l+{\left(\mathrm{G}\times \mathrm{P}\right)}_{\mathrm{i},\mathrm{k}}+{\left(P\times R\right)}_{k,l}+{\epsilon}_{i,j(i),k,l}, $$
where μ is the global mean; Gi is the i-th group effect; S(G)j(i) representing the j-th subject effect nested in the i-th group; Pk stands for the k-th peptide effect; Rl stands for the l-th run effect, (G × P)ik stands for the interaction effect between the i-th group and the k-the peptide; and (P × R)kl signifies the interaction effect between the k-th peptide and the l-th run. When all the effects are treated as fixed, these parameters have the following restrictions: \( \sum \limits_{i=0}^2{G}_i=0 \), \( \sum \limits_{j(i)=1}^{J(i)}S{(G)}_{j(i)}=0 \), \( \sum \limits_{k=1}^K{P}_k=0 \), \( \sum \limits_{l=1}^L{R}_l=0 \), \( \sum \limits_{i=0}^2{\left(G\times P\right)}_{i,k}=0 \), \( \sum \limits_{k=1}^K{\left(G\times P\right)}_{i,k}=0 \), \( \sum \limits_{k=1}^K{\left(P\times R\right)}_{k,l}=0,\sum \limits_{l=1}^L{\left(P\times R\right)}_{k,l}=0 \), and \( {\epsilon}_{i,j(i),k,l}\sim N\left(0,{\sigma}_{\epsilon}^2\right) \). Here, G0 stands for the effect of a reference group of MRM data. When the subject and run effects are treated as random, the restrictions on S(G)j(i), Rl, and (P × R)kl are replaced by \( S{(G)}_{j(i)}\sim N\left(0,{\sigma}_S^2\right) \), \( {R}_l\sim N\left(0,{\sigma}_R^2\right) \) and \( {\left(P\times R\right)}_{kl}\sim N\left(0,{\sigma}_{P\times R}^2\right) \), respectively.
For most MRM data analyses, the investigator's interest lies in determining which proteins differ in abundance between two groups. Thus, the hypothesis of interest is given below for comparing two groups:
$$ {H}_0:{G}_1={G}_2,{\mathrm{H}}_1:{\mathrm{G}}_1\ne {\mathrm{G}}_2 $$
The MSstats uses the t-test for this hypothesis.
LR-SAM approach
Our proposed LR-SAM approach uses a log2-transformed relative intensity value instead of the original log2-transformed intensity value itself. A log2-transformed relative intensity value was derived as yi, j(i), k, l − y0, 0(0), k, l. In that scenario, Pk, Rl and (P × R)kl effects that share the same k and l values are removed. If we denote yj, k as a log2-transformed relative intensity value of the j-th subject and the k-th peptide, where j = 1,J(1),J(1) + 1,…,J(1) + J(2). Then, the log2-transformed relative intensity value yj, k stands for the y1, j(1), k, l − y0, 0(0), k, l for j = 1,…,J(1), and it stands for the y2, j(2), k, l − y0, 0(0), k, l for j = J(1) + 1,…,J(1) + J(2).
LR-SAM with fixed effect
Consider a logistic regression model using log2-transformed relative intensity value, as follows:
$$ logit\left(P\left({Z}_j=1\right)\right)=\alpha +{\beta}_1{y}_{j,1}+\dots +{\beta}_K{y}_{j,K}, $$
where Zj is a group indicator of the j-th subject that is assumed to follow a Bernoulli distribution; α is an intercept; and βk is the coefficient of the k-th peptide. Note that the βk values are related to G1 − G2 + (G × P)1, k − (G × P)2, k of model (1). Therefore, if we treat all βk's as fixed, the hypothesis for comparing the two groups is:
$$ {H}_0:{\beta}_1=\cdots ={\beta}_K=0 $$
To test (4), we consider the likelihood ratio test (L) with K degrees of freedom, as follows:
$$ \mathrm{L}=-2\left({l}_0\left({\widehat{\alpha}}_0\right)-l\left(\widehat{\alpha},{\widehat{\beta}}_1,\dots, {\widehat{\beta}}_K\right)\right) $$
Here, \( {\widehat{\alpha}}_0 \) is the maximum likelihood estimate (MLE) of α under the null hypothesis (see above). \( {l}_0\left({\widehat{\alpha}}_0\right) \) is the maximum likelihood value under the null hypothesis, and \( \widehat{\alpha},{\widehat{\beta}}_1,\dots, {\widehat{\beta}}_K \) are the MLEs of α, β1, …, βK, respectively. It is also known that L asymptotically follows a chi-square distribution, with K degrees of freedom.
A Wald type test (W) statistic for analysis, is given below:
$$ W={\left(\begin{array}{c}{\widehat{\beta}}_1\\ {}\vdots \\ {}{\widehat{\beta}}_K\end{array}\right)}^{\prime } Var{\left({\widehat{\beta}}_1,\dots, {\widehat{\beta}}_K\right)}^{-1}\left(\begin{array}{c}{\widehat{\beta}}_1\\ {}\vdots \\ {}{\widehat{\beta}}_K\end{array}\right) $$
Here, W also asymptotically follows a chi-square distribution, with K degrees of freedom under the null hypothesis. If we assume that the βk values are homogeneous, a Wald test with 1 degree of freedom (W1) can be considered as follows:
$$ {W}_1={\widehat{\beta}}_p{A}^{-1}{\widehat{\beta}}_p, $$
in which \( {\widehat{\beta}}_p \) is the weighted average of \( {\widehat{\beta}}_k \) as \( \sum \limits_{k=1}^K{t}_k{\widehat{\beta}}_k \), where tk is a weight given as \( \frac{1/ Var\left({\widehat{\beta}}_k\right)}{\sum_{k=1}^K1/ Var\left({\widehat{\beta}}_k\right)} \), and A is the variance of \( {\widehat{\beta}}_p \), given as \( {\left(\begin{array}{c}{t}_1\\ {}\vdots \\ {}{t}_K\end{array}\right)}^{\prime } Var\left({\widehat{\beta}}_1,\dots, {\widehat{\beta}}_K\right)\left(\begin{array}{c}{t}_1\\ {}\vdots \\ {}{t}_K\end{array}\right) \). Thus, W1 asymptotically follows a chi-square distribution, with 1 degree of freedom, under the null hypothesis. Moreover, if the βkvalues are homogeneous, Eq. (3) (shown above) can be reduced to the following model:
$$ logit\left(P\left({Z}_j=1\right)\right)=\alpha +{\beta}^{\ast}\sum \limits_{k=1}^K{y}_{j,k} $$
The hypothesis of interest from this model (8) is H0 : β∗ = 0, and a Wald test (WS) is given by:
$$ {W}_S={\widehat{\beta}}^{\ast }\ Var{\left({\widehat{\beta}}^{\ast}\right)}^{-1}\ {\widehat{\beta}}^{\ast } $$
Here, \( {\widehat{\beta}}^{\ast } \) is the maximum likelihood estimate of β∗ from the model (8), and WS asymptotically follows a chi-square distribution, with 1 degree of freedom.
LR-SAM with random effects
When effects of peptides were heterogeneous, we could detect significant heterogenous effect by assuming random effects on coefficients βk. This assumption is commonly used in meta-analysis or rare variant analysis of genetic study [24, 29]. If we assume that βk follows a normal distribution, with mean 0 and variance wkτ (where wk is a known prior weight), then the logistic regression model (3) is expanded to a mixed effect model, and the hypothesis (4) is equivalent to:
$$ {H}_0:\tau =0,{H}_1:\uptau \ne 0 $$
Since the hypothesis H0 : τ = 0 is on the boundary of the parameter space, the variance-component score test can be considered. The score test statistic of the variance-component for (10) is:
$$ {S}_{VC}={\left(\boldsymbol{Z}-{\widehat{\boldsymbol{\mu}}}_0\right)}^{\prime}\boldsymbol{K}\left(\boldsymbol{Z}-{\widehat{\boldsymbol{\mu}}}_0\right) $$
Here, \( {\widehat{\boldsymbol{\mu}}}_0 \) is the estimated probability under H0; K = YWY′, where Y = [Y1, …, YK] and Yk = (y1k, …, ynk)′; Z means the group indicator vector, and W is a diagonal matrix with the k-th element as wk.
It is known that SVC follows a mixture of chi-square distributions \( {\sum}_{k=1}^K{\uplambda}_k{\chi_{1,k}}^2 \), where χ1, k2's are independent chi-square distributions with 1 degree of freedom, and λk is the k-th eigenvalue of P1/2KP1/2 [24]. Here, \( \boldsymbol{P}={\widehat{\boldsymbol{V}}}^{-\mathbf{1}}-{\widehat{\boldsymbol{V}}}^{-\mathbf{1}}\mathbf{1}\left({\mathbf{1}}^{\prime }{\widehat{\boldsymbol{V}}}^{-\mathbf{1}}\mathbf{1}\right){\mathbf{1}}^{\prime }{\widehat{\boldsymbol{V}}}^{-\mathbf{1}} \), where \( \widehat{\boldsymbol{V}} \) is a diagonal matrix with the k-th element as \( {\widehat{\mu}}_{0k}\left(1-{\widehat{\mu}}_{0k}\right) \). For simplicity, we assume a flat prior weight given by wk = 1 for k = 1, …, K.
Simulation design
We next performed simulation studies to investigate the power of the LMM and LR-SAM approaches for comparing two groups, and whether they preserve type I error. For analysis, there were four LMMs available, depending on how the random or fixed effects were specified: (i) LMM(FF), with fixed subject and run effects, (ii) LMM(FR), with fixed subject effect and random run effects, (iii) LMM(RF), with random subject and fixed run effects, and (iv) LMM(RR), with random subject and run effects. For each simulated dataset, the best LMM, LMM(best), was selected among four LMMs with the smallest Akaike information criterion (AIC) value [30]. Thus, there were five LR-SAM test statistics, L, W, W1, WS, and SVC, for comparison.
Simulation data was generated from the model, using 1000 repetitions. The global mean, μ, was arbitrarily set to 15, while the reference effects, G0, S(G)0, (0), and (G × P)0, k, for k = 1, …, K, were set to 0. The normal distribution, with mean 0 and variance 0.5, was set as the error distribution, and the number of peptides was set to 4. For the random subject effect, we generated S(G)j(i) from the identical normal distribution independently, with mean 0 and variance 0.25 for i = 1, 2. For the fixed subject effect, we set S(G)j(i) as follows:
$$ S{(G)}_{j(i)}=-{e}_S+\frac{2\left(j(i)-1\right)}{J(i)-1}{e}_S\ \mathrm{for}\kern0.5em j(i)=1,\dots, J(i)\kern0.75em \mathrm{and}\kern0.5em i=1,2,\mathrm{where}\kern0.5em {e}_S>0\ \kern0.5em \mathrm{and}\kern0.5em {e}_S=\sqrt{3\frac{J(i)-1}{J(i)+1}}{\sigma}_S,\kern0.5em \mathrm{with}\kern0.5em {\sigma}_S^2=0.25. $$
The peptide effect, Pk, was considered as a fixed effect, and was set as follows:
$$ {P}_k=-{e}_P+\frac{2\left(k-1\right)}{K-1}{e}_P\kern0.5em \mathrm{for}\kern0.5em k=1,\dots, K,\kern0.5em \mathrm{where}\kern0.5em {e}_P>0\kern0.5em \mathrm{and}\kern0.5em {e}_P=\sqrt{3\frac{K-1}{K+1}}{\sigma}_P\kern0.5em \mathrm{with}\kern0.5em {\sigma}_P^2=0.1 $$
For the random run effect, we generated Rl from the identical normal distribution, independently, with mean 0 and variance 0.25. For the fixed run effect, we set Rl as follows:
\( {R}_l=-{e}_R+\frac{2\left(m-1\right)}{M-1}{e}_R\kern0.5em \mathrm{for}\kern0.5em m=1,\dots, M,\kern0.5em \mathrm{where}\kern0.5em {e}_R>0\kern0.5em \mathrm{and}\kern0.5em {e}_R=\sqrt{3\frac{M-1}{M+1}}{\sigma}_R,\kern0.5em \mathrm{with}\kern0.5em {\sigma}_R^2=0.25 \)
We then considered the peptide×run interaction effect as either random or fixed, followed by the type of run effect. For the random peptide by run interaction effect, we generated (P × R)k, l from the identical normal distribution independently, with mean 0 and variance 0.1. For the fixed peptide×run interaction effect, we set (P × R)k, l as follows:
$$ {\left(P\times R\right)}_{k,l}=\left({e}_{PR}-\frac{2\left(k-1\right)}{K-1}{e}_{PR}\right){\left(-1\right)}^l\kern0.5em \mathrm{for}\kern0.5em l=1,\dots, L\kern0.5em \mathrm{and}\kern0.5em k=1,\dots, K,\kern0.5em \mathrm{where}\kern0.75em {e}_{PR}>0\kern0.5em \mathrm{and}\kern0.5em {e}_{PR}=\sqrt{3\frac{K-1}{K+1}}{\sigma}_{PR},\kern0.5em \mathrm{with}\kern0.5em {\sigma}_{PR}^2=0.1 $$
Provided those parameters, we further considered four group effect scenarios (GSs), as follows:
$$ \mathrm{GS}\ 0:\kern0.5em {G}_i=0\kern0.5em \mathrm{for}\ \mathrm{all}\ \mathrm{i}\ \mathrm{and}\ \mathrm{GS}\ 1:\kern0.5em {G}_2-{G}_1=1/3 $$
For detecting the group×peptide interaction effect, we set (G × P)i, k, as follows:
$$ {\left(G\times P\right)}_{i,k}=\left({e}_{GP}-\frac{2\left(k-1\right)}{K-1}{e}_{GP}\right){\left(-1\right)}^i\kern0.5em \mathrm{for}\kern0.5em i=1,2\kern0.5em \mathrm{and}\kern0.5em k=1,\dots, K,\kern0.5em \mathrm{where}\kern0.5em {e}_{GP}>0\kern0.5em \mathrm{and}\kern0.5em {e}_{GP}=\sqrt{3\frac{K-1}{K+1}}{\sigma}_{GP}. $$
Here, using σGP values determined from the squared average of the interaction effect, we considered three interaction effect scenarios (ISs) for σGP, as follows:
$$ \mathrm{IS}\ 1:\kern0.5em {\sigma}_{GP}^2=0.05\kern0.5em \mathrm{and}\kern0.5em \mathrm{IS}\ 2:\kern0.5em {\sigma}_{GP}^2=0.1 $$
To identify candidate serum hepatocellular carcinoma (HCC) biomarkers for prognosis and response to the tyrosine kinase inhibitor sorafenib, data from 115 HCC patients who had undergone continuous administration of sorafenib for more than 6 weeks, were collected from Seoul National University Hospital, as part of an ongoing study between May 2013 and August 2014. HCC was diagnosed by histological or radiological evaluation, with reference to the American Association for the Study of Liver Diseases (AASLD) [31] or the European Association for the Study of the Liver (EASL) [32] guidelines. All procedures/analyses were approved by the Seoul National University Hospital Institutional Review Board (IRB protocol No. 0506–150-005). Sorafenib response was evaluated using the modified response evaluation criteria in solid tumors (mRECIST) [28], using independent radiologic assessments. Patients with complete response, partial response, and stable disease were categorized as responders, while those with progressive disease were categorized as non-responders.
Toward these objectives, a total of 115 serum samples were randomly separated into two batches: the 1st batch consisted of 65 samples and the 2nd batch consisted of 50 samples. Of those, the total number of responders was 40, and the number of non-responders was 75. One hundred twenty-four candidate protein biomarkers, known hepatic disease-associated proteins, were chosen from 50,265 proteins, based on the LiverAtlas database [27]. One to seven peptides, comprised within each protein, were measured, and counted (Table 2).
Table 2 The number of proteins of each number of peptides
Since there were some cases in which LMM(FF), LMM(FR), and L did not preserve type I error in our simulation studies, we applied LMM(RF), LMM(RR), W, W1, WS and SVC to analyze the MRM data. Quantile normalization, provided within the MSstats package [20], was employed for preprocessing. To adjust the mean differences between the two batches, batch indicators were included in models (1), (3), and (8) for MRM data analysis.
Simulation results
The type I error and the empirical power were estimated as the proportion of p-values under 0.05, out of 1000 repetitions. The type I error rates of LMMs and LR-SAM are summarized in Table 3. The true type of subject effect more strongly affected the type I error rate of LMMs, as compared to the run effect. Analogously, when the true subject effect was fixed, the four LMMs controlled the type I errors. Among the five LR-SAM tests, L could not control the type I error rate when the sample size was 20, while the other four LR-SAM tests could. When the sample sizes were 50 and 100, all LR-SAM tests controlled type I error. When the true subject effect was random, LMM(FF) and LMM(FR) could not control type I error, whereas LMM(RF) and LMM(RR) could. L did not control the type I error rate when the sample sizes were 20 and 50, while the other four LR-SAM tests did. The AIC value of LMM(FF) tended to be the smallest among the four LMMs, under any conditions. Thus, LMM(FF) was most frequently selected as the best LMM.
Table 3 Estimated type I error of LMMs and LR-SAM methods
Some LR-SAM tests showed increased power as the interaction effects became large. When there was only an interaction effect, without group effects, the power of the LMMs W1, and WS did not increase, as shown in Table 4. As the sample size and the interaction effect became large, W and SVC showed increased power. Moreover, SVC provided higher power than W, when the sample size was 50, while W provided higher power than SVC, when the sample size was 100.
Table 4 Estimated power of LMMs and LR-SAM methods for GS 0. Bolded number indicates power of methods were more than 0.8
The size of the group effect showed large effects on all LMMs and LR-SAM methods. The estimated powers of the five LMMs, and the LR-SAM methods for GS 1, are shown in Table 5. The powers of L, W, W1, and SVC were all affected by the size of the interaction effect, when the group effect scenario was GS 1, while those of the five LMMs did not. As the interaction effect increased, the powers of L, W, and SVC increased, while the powers of four of the LMMs, and WS, did not, and the power of the W1 method even decreased. When the interaction scenario was IS 1, the powers of the L and SVC methods were higher than those of the LMMs, for a sample size of 100. For the IS 2 scenario, once the sample size exceeded 50, the L and SVC methods provided higher powers than those of the LMMs.
Collectively, as the size of group effects became large, all methods provided increased power. As the interaction effect became large, only the L, W, and SVC methods provided increased power.
HCC data analysis
Using the approaches described above, the maximum –log10-transformed p-values of the linear mixed models (LMMs) were higher than those of the LR-SAM methods. However, the correlation of the transformed p-values of LMM(RF) and LMM(RR) was lower than that of LR-SAM methods, as shown in Fig. 1. The correlation between LMM(RF) and LMM(RR) was 0.6558, while the minimum correlation among the LR-SAM methods was 0.784. The correlation between WS and SVC was the highest (0.9627), among the LR-SAM methods.
Pairwise scatter plot of –log10-transformed p-values from the LMM(RF), LMM(RR), W, W1, WS and SVC models based on multiple reaction monitoring (MRM) data. Vertical and horizontal dashed red lines represent Bonferroni-corrected significance levels, −log10(0.05/124). Diagonally dashed gray line represents one to one slope
The lowest p-values of LMM(RF) and LMM(RR) were lower than those of the LR-SAM methods, valued at 1.87 × 10−18 and 1.09 × 10−30, respectively. The lowest p-value of W, W1, and WS was 2 × 10−7, and the lowest p-value of SVC was 3.49 × 10−14 (Table 6). Although the lowest p-values of the LR-SAM methods were less significant than those of the LMMs, and for many proteins, LR-SAM methods provided higher significance results (Fig. 1).
Table 6 List of proteins and their p-values simultaneously identified by LMM or LR-SAM methods
The LMM(RF) and LMM(RR) methods identified 11 and 37 proteins, respectively. There were six (14.29%) proteins simultaneously identified by LMM(RF) and LMM(RR). For LR-SAM methods, W, W1, WS, and SVC identified 28, 19, 22, and 29 proteins, respectively. Among these proteins 18 (52.94%) were simultaneously identified by four methods. Finally, there were four (7.84%) proteins identified by all six methods of LMM and LR-SAM (Fig. 2).
Venn diagram of proteins identified from LMM (RF, RR) and LR-SAM (W, W1, WS and SVC), under a Bonferroni correction significance level of −log10(0.05/124)
All LMMs and LR-SAM methods simultaneously provided significance results for the presence of the proteins GPX3, IGHG1, IGHG3 and IGJ. Among these, IGJ (immunoglobulin J chain, linker protein for immunoglobulin alpha and mu) was previously reported as having a significant difference of expression in HCC tumors [33], while GPX3 (glutathione peroxidase 3) was reported as a tumor suppressor in HCC [34].
There were 14 proteins that the LR-SAM methods, but not the LMMs, simultaneously provided significance results of expression (Table 6). Among those 14, seven were previously reported, including FBLN1 (Fibulin-1), a tumor suppressor gene in HCC [35], SHBG (sex hormone-binding globulin), a prediagnostic risk marker for HCC [36], and LG3BP (galectin-3-binding protein), a potential marker in six cancer types, including HCC, lymphoma, NPC (nasopharyngeal carcinoma), CRC (colorectal carcinoma), and oral cancers [37]. Additionally, CATB (Cathepsin B) was previously reported as a potential candidate cancer biomarker in HCC [38], HPT (haptoglobin) was reported to associate with tumor progression in HCC [39], while POSTN (periostin) was reported as a marker for malignant transformation of hepatocytes [40]. Similarly, 14–3-3S (14–3-3 protein sigma) was reported as downregulated in HCC [41].
There were two proteins that LMMs simultaneously provided a significance result, while LR-SAM methods did not. Resultant p-values for the simultaneously identified 20 proteins are shown in Table 6. Two proteins that demonstrated significance by LMM(RF) and LMM(RR) were also previously reported: CD5L (CD5 antigen-like), reported as differentially expressed in hepatitis C patients [42]: and APOA4, (apolipoprotein A-IV), reported as misexpressed in liver metabolic disorders [43].
We examined the performance of LMMs and LR-SAM methods, through extensive simulation studies. When the true subject effect was random, LMM(FF) and LMM(FR) did not preserve type I error (Table 3). We also applied Akaike information criterion (AIC), for model selection, to check whether or not the best-performing LMM preserved type I error. However, our empirical study showed that LMM(FF) had the smallest AIC value for most simulation settings, which made it difficult to use AIC as a model selection criterion. On the other hand, LR-SAM methods, except L, well preserved type I error under any type of subject and run effects.
Since the hypothesis (2) did not consider the interaction effect, the power of the LMM approach was not affected under any size of the interaction effect. However, when the interaction effect (without the group effect) was considered, hypothesis (2) could not perform well, as we observed through our simulation studies. On the other hand, testing hypotheses (8) with W and (10) with SVC, changes under any size group effect and interaction effect were detectable. Additionally, LR-SAM methods did not provide higher power than LMMs, when there were no or only weak interaction effects.
Although LMM provided more protein significance results than LR-SAM methods, the proteins identified by the LMM(RF) and LMM(RR) approaches were very different from each other. On the other hand, LR-SAM methods provided more consistent significance results than those of the LMMs. Moreover, LR-SAM methods provided greater significance results than those identified by LMM(RF), for most proteins, and several proteins were identified only by LR-SAM methods. However, there was a still performance difference between the choices of fixed and random effect models in LR-SAM methods, as was in the LMM. This difference was caused by different testing hypotheses between W and SVC in LR-SAM. In addition, note that SVC allows the heterogenous effects of peptides, while W does not.
LMMs have been widely used to identify proteins with significantly altered abundance, in distinct disease states, based on MRM assays [39]. However, we found that LMM approaches provide inconsistent significance results for the same MRM data, depending on which effects are treated as random or fixed by simulation results. It is a well-known property of LMMs that the variance of model parameters are underestimated, when the fixed effect model is fitted but the true effects are random, and vice versa [44]. As a result, the protein significance results of LMMs may vary, depending on whether the true effect is random or fixed, and we also observed this phenomenon, as shown in Fig. 1 and Table 6. Thus, it is highly important to correctly specify the effect as random or fixed.
In this paper, we propose a new logistic regression-based method for Significance Analysis of Multiple Reaction Monitoring (LR-SAM). Unlike LMMs, our LR-SAM approach uses a much smaller number of parameters. Moreover, our LR-SAM does not require inclusion of all the effects related to the run. Accordingly, our model does not need to specify run effects as random or fixed.
In simulation study, our LR-SAM preserved type I error when the true subject effect was random, while LMM(FF) and LMM(FR) did not. In real data analysis, although LMM provided more protein significance results than LR-SAM methods, LR-SAM methods provided more consistent significance results than those of the LMMs. Thus, our proposed method, LR-SAM could give more reliable results than previous protein studies.
AASLD:
American Association for the Study of Liver Diseases
Akaike information criterion
EASL:
European Association for the Study of the Liver
ELISA:
GS:
Group effect scenarios
Interaction effect scenarios
LMM:
Linear mixed model
mRECIST:
Modified response evaluation criteria in solid tumors
MRM:
Multiple reaction monitoring
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This research was supported by a grant of the Korea Health Technology R&D Project through the Korea Health Industry Development Institute (KHIDI), funded by the Ministry of Health & Welfare, Republic of Korea (grant number: HI16C2037, HI15C2165). Publication of this article was sponsored by HI16C2037 grant.
This article has been published as part of BMC Systems Biology Volume 12 Supplement 9, 2018: Proceedings of the 29th International Conference on Genome Informatics (GIW 2018): systems biology. The full contents of the supplement are available online at https://bmcsystbiol.biomedcentral.com/articles/supplements/volume-12-supplement-9.
Department of Statistics, Seoul National University, Seoul, South Korea
Jongsu Jun
, Yongkang Kim
& Taesung Park
Graduate School of Public Health, Seoul National University, Seoul, South Korea
Jungsoo Gim
Department of Biomedical Engineering, Seoul National University College of Medicine, Seoul, South Korea
Hyunsoo Kim
, Injun Yeo
, Jiyoung Park
& Youngsoo Kim
Institute of Medical and Biological Engineering, Medical Research Center, Seoul National University College of Medicine, Seoul, South Korea
Department of Internal Medicine and Liver Research Institute, Seoul National University, Seoul, South Korea
Su Jong Yu
, Jeong-Ju Yoo
, Young Youn Cho
, Dong Hyeon Lee
, Eun Ju Cho
, Jeong-Hoon Lee
, Yoon Jun Kim
& Jung-Hwan Yoon
Department of Mathematics and Statistics, Sejong University, Seoul, South Korea
Seungyeoun Lee
Interdisciplinary program in Bioinformatics, Seoul National University, Seoul, South Korea
Taesung Park
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JJ and TP developed statistical method. JJ performed statistical analysis. HK, SJY, EJC, J-H Y, YSK and TP conceived and planned the experiments. HK, JP, and J-JY carried out the experiments. JJ, JG. YK and TP planned and carried out the simulations. YYC, DHL, YC, EJC, J-HL, and YJK contributed to sample preparation. JJ, JG, YK, YSK, TP contributed to the interpretation of the results. JJ and TP took the lead in writing the manuscript. All authors have read and approved the final manuscript.
Correspondence to Taesung Park.
To identify candidate serum hepatocellular carcinoma (HCC) biomarkers for prognosis and response to the tyrosine kinase inhibitor sorafenib, data from 115 HCC patients who had undergone continuous administration of sorafenib for more than 6 weeks, were collected from Seoul National University Hospital, as part of an ongoing study between May 2013 and August 2014. HCC was diagnosed by histological or radiological evaluation, with reference to the American Association for the Study of Liver Diseases (AASLD) [31] or the European Association for the Study of the Liver (EASL) [32] guidelines. All procedures/analyses were approved by the Seoul National University Hospital Institutional Review Board (IRB protocol No. 0506–150-005).
Jun, J., Gim, J., Kim, Y. et al. Analysis of significant protein abundance from multiple reaction-monitoring data. BMC Syst Biol 12, 123 (2018) doi:10.1186/s12918-018-0656-9
Multiple reaction-monitoring (MRM)
Logistic regression-based method for significance analysis of multiple reaction monitoring (LR-SAM)
Sorafenib response
|
CommonCrawl
|
Some properties for integro-differential operator defined by a fractional formal
Zainab E. Abdulnaby1,
Rabha W. Ibrahim2 &
Adem Kılıçman3
Recently, the study of the fractional formal (operators, polynomials and classes of special functions) has been increased. This study not only in mathematics but extended to another topics. In this effort, we investigate a generalized integro-differential operator \(\mathfrak {J}_{m}(z)\) defined by a fractional formal (fractional differential operator) and study some its geometric properties by employing it in new subclasses of analytic univalent functions.
The subject of fractional calculus (integral and derivative of any arbitrary real or complex order) has acquired significant popularity and major attention from several authors in various science due mainly to its direct involvement in the problems of differential equations in mathematics, physics, engineering and others for example Baskonus and Bulut (2015), Yin et al. (2015) and Bulut (2016). The fractional calculus has gained an interesting area in mathematical research and generalization of the (derivative and integral) operators and its useful utility to express the mathematical problems which often leads to problems to be solved see Yao et al. (2015), Baskonus (2016) and Kumar et al. (2016). Specifically, it utilized to define new classes and generalized many geometric properties and inequalities in complex domain. In another words, these operators are play an important role in geometric function theory to define new generalized subclasses of analytic univalent and then study their properties. By using the technique of convolution or Hadamard product, Sălăgean (1981) defined the differential operator \(\mathcal {D}_{n}\) of the class of analytic functions and it is well known as S\(\check{a}\)l\(\check{a}\)gean operator. Followed by Al-Oboudi differential operator see Al-Oboudi (2004). Several authors have used the S\(\check{a}\)l\(\check{a}\)gean operator to define and consider the properties of certain known and new classes of analytic univalent functions. We refer here some of them in recent years. Najafzadeh (2010) investigated a new subclass of analytic univalent functions with negative and fixed finitely coefficient based on S\(\check{a}\)l\(\check{a}\)gean and Ruscheweyh differential operators. Aouf et al. (2012) gave some results for certain subclasses of analytic functions based on the definition for S\(\check{a}\)l\(\check{a}\)gean operator with varying arguments. El-Ashwah (2014) used S\(\check{a}\)l\(\check{a}\)gean operator to define a new subclass of analytic functions and derived some subordination results for this subclass in open unit disk. Breaz et al. (2008) investigated a new general integral operator for certain holomorphic functions based on the S\(\check{a}\)l\(\check{a}\)gean differential operator and studied some properties for this integral operator on some subclasses of univalent function. Also, Deniz et al. (2012) defined a new general integral operator by considering the Hadamard product and gave new sufficient conditions for this operator to be univalent in \(\mathbb {U}\). Breaz et al. (2014) defined two general integral operators \(F_{\lambda }(z)\) and \(G_{\lambda }(z)\) and investigated some geometric properties for these operators on subclasses of analytic function in open unit disk.
In this paper, we define a generalized mixed integro-differential operator \(\mathfrak {J}_{m}(z)\) based on the concept of Breaz integral operator as well as the fractional differential operator and study some their geometric properties on some new subclasses in open unit disk.
Let \(\mathcal {A}\) denote the class of all functions of the form
$$f(z)= z+ \sum _{n=2}^{\infty } a_{n} z^{n}$$
which are analytic function in the open unit disk \(\mathbb {U}= \lbrace z: |z|<1\rbrace\) and usually normalized by \(f(0)= f^{\prime }(0)-1=0\). Also, let \(\mathcal {S}\) be the subclass of \(\mathcal {A}\) consisting of functions f of form (1) which are univalent in \(\mathbb {U}\). We denote by \(\mathcal {S}^{*}(\beta )\) and \(\mathcal {K}(\beta )\), \(0 \le \beta < 1\), the classes of starlike function and convex function in \(\mathbb {U}\), respectively. For \(f \in \mathcal {A}\), Esa et al. (2016a) introduced the following differential operator \(\mathcal {T}^{\alpha ,\delta }: \mathcal {A}\rightarrow \mathcal {A},\)
$$\begin{aligned} \mathcal {T}^{\alpha ,\delta }f(z) := z + \sum ^{\infty }_{n=2} \frac{\Gamma (\delta +1)}{\Gamma (\alpha +1)} \frac{\Gamma (n+\alpha )}{\Gamma (n+\delta )} a_{n} z^{n} \quad (z\in \mathbb {U}), \end{aligned}$$
for some \((0< \alpha \le 1)\), \((0 < \delta \le 1)\) and \(n \in \mathbb {N}{\setminus }\lbrace 0, 1 \rbrace\). If \(\alpha =\delta\) in (2), then we get
$$\mathcal {T}_{z}^{\alpha ,\delta } f(z)= f(z) \quad (z \in \mathbb {U})$$
for more details see Esa et al. (2016b). Now, let define a new fractional differential operator \(\mathcal {D}_{\lambda }^{k}:\mathcal {A}\rightarrow \mathcal {A}\) as follows
$$\begin{aligned} \mathcal {D}_{\lambda }^{0}f(z)&= f(z) \\ \mathcal {D}^{1}_{\lambda }f(z)&= (1-\lambda )\mathcal {T}^{\alpha ,\delta }f(z)+\lambda z\left( \mathcal {T}^{\alpha ,\delta }f(z)\right) ^{\prime }, \\ \mathcal {D}_{\lambda }^{k} f(z)&= \mathcal {D}_{\lambda }\left( \mathcal {D}_{\lambda }^{k-1} f(z)\right) \quad (k \in \mathbb {N}, \, z\in \mathbb {U}). \end{aligned}$$
In general, we write
$$\begin{aligned} D_{\lambda }^{k}f(z)= z+ \sum _{n=2}^{\infty }\Phi _{n,k}(\alpha ,\delta ,\lambda )a_{n} z^{n}\quad k \in =\lbrace 0,1,2,\ldots \rbrace \end{aligned}$$
$$\begin{aligned} \Phi _{n,k}(\alpha ,\delta ,\lambda )=\left[ \frac{\Gamma (\delta +1)}{\Gamma (\alpha +1)}\frac{\Gamma (n+\alpha )}{\Gamma (n+\delta )}(1+(n-1)\lambda )\right] ^{k}. \end{aligned}$$
When \(\alpha = \delta\), \(\lambda =1\) and \(k=1\), we get Sǎlǎgean operator see Sălăgean (1981) and when \(\alpha =\delta\), we have Al-Oboudi differential operator see Al-Oboudi (2004). Afterwards, we introduce some new subclasses of \(\mathcal {A}\) as follows.
Let \(\mathcal {S}^{k}( \lambda ,\phi )\) denote the class of functions \(f \in \mathcal {A}\) which satisfies the following condition:
$$\begin{aligned} \mathfrak {R}\left\{ \frac{z \mathcal {D}_{\lambda }^{k+1} f(z)}{\mathcal {D}_{\lambda }^{k}f(z)} \right\} > \phi \quad (z \in \mathbb {U}) \end{aligned}$$
for some \(0 \le \phi < 1,\, \lambda \ge 0\) and \(k \in \lbrace 0,1,\ldots \rbrace\). Let \(\mathcal {K}^{k}( \lambda ,\phi )\) denote the class of functions \(f \in \mathcal {A}\) which satisfies the following condition
$$\begin{aligned} \mathfrak {R}\left\{ 1+ \frac{z \mathcal {D}_{\lambda }^{k+2} f(z)}{\mathcal {D}_{\lambda }^{k+1}f(z)} \right\} > \phi \quad (z \in \mathbb {U}) \end{aligned}$$
for some \(0 \le \phi < 1,\, \lambda \ge 0\) and \(k \in \lbrace 0,1,\ldots \rbrace\). It is clear that, when \(k=0\) in (4) and (5), then we have the well known function classes
$$\begin{aligned} \mathcal {S}^{0}(\lambda ,\phi )=\mathcal {S}^{*}(\phi )\quad \text {and} \quad \mathcal {K}^{0}(\lambda ,\phi )=\mathcal {K}(\phi ). \end{aligned}$$
Further, let \(\mathcal {N}^{k}(\lambda ,\psi )\) the subclass of \(\mathcal {A}\), consisting of the functions f, which satisfies the following
$$\begin{aligned} \mathfrak {R}\left\{ 1+ \frac{z \mathcal {D}_{\lambda }^{k+2} f(z)}{\mathcal {D}_{\lambda }^{k+1} f(z)} \right\} < \psi \quad (z \in \mathbb {U}) \end{aligned}$$
and let \(\mathcal {M}^{k}(\lambda ,\psi )\) be subclass of \(\mathcal {A}\) consisting of the functions f which satisfies the following
$$\begin{aligned} \mathfrak {R}\left\{ \frac{z \mathcal {D}_{\lambda }^{k+1} f(z)}{\mathcal {D}_{\lambda }^{k}f(z)} \right\} < \psi \quad (z \in \mathbb {U}) \end{aligned}$$
for some \(\psi >1\), \(\lambda \ge 0\) and \(k \in \lbrace 0,1,\ldots \rbrace\). It is obvious that, when \(k=0\) in (6) and (7), then we obtain the following classes
$$\begin{aligned} \mathcal {M}^{0}(\lambda ,\psi )= \mathcal {M}(\psi )\quad \text {and} \quad \mathcal {N}^{0}(\lambda ,\psi )=\mathcal {N}(\psi ) \end{aligned}$$
were interested by Owa and Srivastava (2002), Dixit and Chandra (2008) and recently studied by Porwal (2011). Let a function f is said to be in the class \(\mathcal {K}^{k}\mathcal {L}(\rho ,\varphi )\), if
$$\begin{aligned} \mathfrak {R}\left\{ 1+ \frac{z\mathcal {D}_{\lambda }^{k+2}f(z)}{\mathcal {D}_{\lambda }^{k+1}f(z)} \right\} \ge \rho \left| \frac{z\mathcal {D}_{\lambda }^{k+2}f(z)}{\mathcal {D}_{\lambda }^{k+1}f(z)}\right| +\varphi \quad (0 \le \varphi < 1) \end{aligned}$$
for some \(\rho ,\, \lambda \ge 0\) and for all \(z \in \mathbb {U}\). When \(k=0\) in (8), we have the function class studied in Shams and Kulkarni (2004). For \(f_{j},\, g_{j} \in \mathcal {A}\) and \(\nu _{j},\, \beta _{j}\) be positive real numbers, \(j=\lbrace 1,2,\ldots ,m \rbrace\), we define the integral operator \(\mathfrak {J}_{m}(z):\mathcal {A}^{m} \rightarrow \mathcal {A}\) by
$$\begin{aligned} \mathfrak {J}_{m}(z)= \int _{0}^{z}\prod _{j=1}^{m} \left( \frac{\mathcal {D}_{\lambda }^{k}f_{j}(t)}{t}\right) ^{\beta _{j}}\big (\mathcal {D}_{\lambda }^{k+1}g_{j}(t)\big )^{\nu _{j}} dt. \end{aligned}$$
Note that, this integral operator is generalization of the integral operator recently defined by Stanciu and Breaz (2014). Also, the integral operator \(\mathfrak {J}_{m}(z)\) is generalizes the following operators defined and investigated by several researchers:
For \(k=0\) and \(f^{\prime }_{j}= g_{j}^{\prime }, \, j=\lbrace 1,2,\ldots ,m\rbrace\), we have integral operator defined as follows:
$$\begin{aligned} \mathcal {F}_{\nu ,\beta }f(z)= \int _{0}^{z} \prod _{j=1}^{m}\left( \frac{f_{j}(t)}{t}\right) ^{\beta _{j}}\left( {f_{j}^{\prime }(t)}\right) ^{\nu _{j}} dt. \end{aligned}$$
studied and considered by Frasin (2011).
For \(k=0\) and \(\nu _{j}=0,\, j=\lbrace 1,2,\ldots ,m\rbrace\), we have the following integral operator
$$\begin{aligned} \mathcal {I}_{{m}}(z)= \int _{0}^{z} \prod _{j=1}^{m}\left( \frac{f_{j}(t)}{t}\right) ^{\beta _{j}}dt. \end{aligned}$$
was considered by Breaz and Breaz (2002).
For \(k=0\), and \(\beta _{j}=0,\, j=\lbrace 1, 2,\ldots , m\rbrace\), we have the integral operator
$$\begin{aligned} \mathcal {I}_{\nu _{m}}(z)= \int _{0}^{z} \prod _{j=1}^{m}\left( {g_{j}^{\prime }(t)}\right) ^{\nu _{j}}dt. \end{aligned}$$
which studied by Breaz et al. (2009). In particular, for \(m=1, \nu _{1}=\nu , \beta _{1}=0\) and \(g_{1} = g\), we have the integral operator
$$\begin{aligned} \mathcal {I}_{\nu }f(z)= \int _{0}^{z}\left( {g^{\prime }(t)}\right) ^{\nu }dt. \end{aligned}$$
which was considered by Pascu and Pescar (1990).
For \(k=0,\, m=1\), \(\nu _{1}=0, \beta _{1}=\beta\) and \(f_{1}=f\), we have the following operator
$$\begin{aligned} \mathcal {I}_{\beta }f(z)= \int _{0}^{z}\left( \frac{f(t)}{t}\right) ^{\beta }dt. \end{aligned}$$
investigated by Miller et al. (1978). In particular, for \(\beta =1\), we have the Alexander's integral operator
$$\begin{aligned} \mathcal {I}f(z)= \int _{0}^{z}\frac{f(t)}{t}dt. \end{aligned}$$
which was studied by Alexander (1915).
We start our first result.
Let \(\nu _{j},\, \beta _{j}\) be positive real numbers, \(j=\lbrace 1,2,\ldots , m\rbrace\). If \(f_{j} \in \mathcal {M}^{k}(\lambda , \psi _{j} ),\, \psi _{j} > 1\) and \(g_{j} \in \mathcal {N}^{k}(\lambda , \eta _{j}), \,\eta _{j} > 1, j= \lbrace 1,2, \ldots , m \rbrace\), then the integral operator \(\mathfrak {J}_{m} (z)\) given by (9) is in the class \(\mathcal {N}^{k} (\lambda ,\rho )\), where
$$\begin{aligned} \rho = 1+\sum _{j=1}^{m}[ \beta _{j}(\psi _{j}-1)+\nu _{j}(\eta _{j}-1)]. \end{aligned}$$
On successive differentiation of \(\mathfrak {J}_{m}(z)\) defined in (9), we obtain
$$\begin{aligned} \mathfrak {J}^{\prime }_{m}(z)= \prod _{j=1}^{m} \left( \left( \frac{\mathcal {D}_{\lambda }^{k}f_{j}(z)}{z}\right) ^{\beta _{j}}\left (\mathcal {D}_{\lambda }^{k+1}g_{j}(z)\right )^{\nu _{j}}\right) \end{aligned}$$
$$\begin{aligned} \mathfrak {J}^{\prime \prime }_{m}(z)&= \sum _{j=1}^{m}\bigg [ \beta _{j}\left( \frac{\mathcal {D}_{\lambda }^{k}f_{j}(z)}{z}\right) ^{\beta _{j}-1} \bigg ( \frac{z\mathcal {D}_{\lambda }^{k+1}f_{j}(z)-\mathcal {D}_{\lambda }^{k}f_{j}(z)}{z^2} \bigg ) \big (\mathcal {D}_{\lambda }^{k+1}g_{j}(z)\big )^{\nu _{j}}\bigg ]\nonumber \\&\quad \times \prod _{\begin{array}{c} \ell =1 \\ \ell \ne j \end{array} }^{m} \left( \left( \frac{\mathcal {D}_{\lambda }^{k} f_{\ell }(z)}{z}\right) ^{\beta _{\ell }} (\mathcal {D}_{\lambda }^{k+1}g_{j}(z))^{\nu _{\ell }} \right) + \sum _{j=1}^{m}\bigg [ \bigg ( \frac{\mathcal {D}_{\lambda }^{k}f_{j}(z)}{z}\bigg )^{\beta _{j}} \nu _j \big (\mathcal {D}_{\lambda }^{k+1}g_{j}(z)\big )^{\nu _{j}-1} \nonumber \\&\quad \times \mathcal {D}_{\lambda }^{k+2}g_{j}(z)\bigg ] \prod _{\begin{array}{c} \ell =1 \\ \ell \ne j \end{array} }^{m} \left( \bigg ( \frac{\mathcal {D}_{\lambda }^{k} f_{\ell }(z)}{z}\bigg )^{\beta _{\ell }} (\mathcal {D}_{\lambda }^{k+1}g_{j}(z))^{\nu _{\ell }} \right) . \end{aligned}$$
By a calculation, we have
$$\begin{aligned} \frac{z\mathfrak {J}^{\prime \prime }_{m}(z)}{\mathfrak {J}^{\prime }_{m}(z)}= \sum _{j=1}^{m}\left[ \beta _{j} \left ( \frac{z\mathcal {D}_{\lambda }^{k+1}f_{j}(z)}{\mathcal {D}_{\lambda }^{k}f_{j}(z)}-1 \right ) + \nu _{j} \frac{z\mathcal {D}_{\lambda }^{k+2}g_{j}(z)}{\mathcal {D}_{\lambda }^{k+1}g_{j}(z)} \right] . \end{aligned}$$
The Eq. (14) is equivalent to
$$\begin{aligned} \frac{z\mathfrak {J}^{\prime \prime }_{m}(z)}{\mathfrak {J}^{\prime }_{m}(z)} + 1= \sum _{j=1}^{m}\left[ \beta _{j} \left ( \frac{z\mathcal {D}_{\lambda }^{k+1}f_{j}(z)}{\mathcal {D}_{\lambda }^{k}f_{j}(z)}-1 \right ) + \nu _{j} \frac{z\mathcal {D}_{\lambda }^{k+2}g_{j}(z)}{\mathcal {D}_{\lambda }^{k+1}g_{j}(z)} \right] +1. \end{aligned}$$
By calculating the real part of both expressions in (15), we have
$$\begin{aligned} \mathfrak {R}\left\{ \frac{z\mathfrak {J}^{\prime \prime }_{m}(z)}{\mathfrak {J}^{\prime }_{m}(z)} + 1\right\} = \sum _{j=1}^{m}\left[ \beta _{j}\mathfrak {R}\frac{z\mathcal {D}_{\lambda }^{k+1}f_{j}(z)}{\mathcal {D}_{\lambda }^{k}f_{j}(z)}-\beta _{j} + \nu _{j}\mathfrak {R}\frac{z\mathcal {D}_{\lambda }^{k+2}g_{j}(z)}{\mathcal {D}_{\lambda }^{k+1}g_{j}(z)} \right] +1, \\ = \sum _{j=1}^{m}\left [ \beta _{j}\mathfrak {R}\frac{z\mathcal {D}_{\lambda }^{k+1}f_{j}(z)}{\mathcal {D}_{\lambda }^{k}f_{j}(z)}-\beta _{j} + \nu _{j}\mathfrak {R}\left (\frac{z\mathcal {D}_{\lambda }^{k+2}g_{j}(z)}{\mathcal {D}_{\lambda }^{k+1}g_{j}(z)}+1\right )-\nu _{j} \right ]+1. \end{aligned}$$
Since \(f_{j} \in \mathcal {M}^{k}(\lambda ,\, \psi _{j}), \psi _{j} > 1\) and \(g_{j} \in \mathcal {N}^{k}( \lambda ,\,\eta _{j} ),\,\eta _{j} > 1,\, j=\lbrace 1,2,\ldots ,m\rbrace\), we have
$$\begin{aligned} \mathfrak {R}\left\{ \frac{z\mathfrak {J}^{\prime \prime }_{m}(z)}{\mathfrak {J}^{\prime }_{m}(z)} + 1\right\}&<\sum _{j=1}^{m}[ \beta _{j}\psi _{j}-\beta _{j}+\nu _{j}\eta _{j}-\nu _{j}]+1 \\&< \sum _{j=1}^{m}[ \beta _{j}(\psi _{j}-1)+\nu _{j}(\eta _{j}-1)]+1. \end{aligned}$$
Therefore, \(\mathfrak {J}_{m}(z)\in \mathcal {N}^{k}(\lambda ,\rho )\), where \(\rho = 1+\sum _{j=1}^{m}[ \beta _{j}(\psi _{j}-1)+\nu _{j}(\eta _{j}-1)]\).\(\square\)
Let \(k=0, m=1\) in Theorem 1, we have
Let \(\beta ,\, \nu\) be positive real numbers. If \(f \in \mathcal {M}(\psi ), \psi > 1\) and \(g \in \mathcal {N}(\eta ), \eta > 1\), then
$$\begin{aligned} \mathfrak {J}(z)=\int _{0}^{z} \left( \frac{f(t)}{t}\right) ^{\beta } (g^{\prime }(t))^{\nu } dt \end{aligned}$$
is in the class \(\mathcal {N}(\rho )\), where \(\rho = 1+ \beta (\psi -1)+\nu (\eta -1)\).
Let \(\beta _{j},\,\nu _{j}\) be positive real numbers, \(j= \lbrace 1,2,\ldots ,m\rbrace\). We assume that \(f_{j},\,j=\lbrace 1,2,\ldots ,m\rbrace\) are starlike functions by order \(\frac{1}{\beta _{j}}\) and that is \(f_{j} \in \mathcal {S}^{k}(\lambda ,\,\frac{1}{\beta _{j}})\) and \(g_{j}\in \mathcal {K}^{k}\mathcal {L}(\rho _{j},\, \eta _{j})\), \(\rho _{j} \ge 1,\,0 \le \eta _{j} < 1, j=\lbrace 1,2,\ldots ,m \rbrace\). If
$$\begin{aligned} \sum _{j=1}^{m}[\nu _{j}(1-\eta _{j})+ \beta _{j}]-m <1 \end{aligned}$$
then \(\mathfrak {J}_{m}(z)\) given by (9) is in the class \(\mathcal {K}^{k}(\lambda ,\omega )\) where
$$\begin{aligned} \omega = 1+ m + \sum _{j=1}^{m}[\nu _{j}(\eta _{j}-1)- \beta _{j}]. \end{aligned}$$
By following same methods as in Theorem 1, we have
$$\begin{aligned} \frac{z\mathfrak {J}^{\prime \prime }_{m}(z)}{\mathfrak {J}^{\prime }_{m}(z)}&= \sum _{j=1}^{m}\left[ \beta _{j} \bigg ( \frac{z\mathcal {D}_{\lambda }^{k+1}f_{j}(z)}{\mathcal {D}_{\lambda }^{k}f_{j}(z)}-1 \bigg ) + \nu _{j} \frac{z\mathcal {D}_{\lambda }^{k+2}g_{j}(z)}{\mathcal {D}_{\lambda }^{k+1}g_{j}(z)} \right] \nonumber \\&= \sum _{j=1}^{m}\left[ \beta _{j} \frac{z\mathcal {D}_{\lambda }^{k+1}f_{j}(z)}{\mathcal {D}_{\lambda }^{k}f_{j}(z)}- \beta _{j} + \nu _{j} \frac{z\mathcal {D}_{\lambda }^{k+2}g_{j}(z)}{\mathcal {D}_{\lambda }^{k+1}g_{j}(z)} \right] , \end{aligned}$$
we can see that, (18) is equivalent to
$$\begin{aligned} \frac{z\mathfrak {J}^{\prime \prime }_{m}(z)}{\mathfrak {J}^{\prime }_{m}(z)} +1 = \sum _{j=1}^{m}\left[ \beta _{j} \frac{z\mathcal {D}_{\lambda }^{k+1}f_{j}(z)}{\mathcal {D}_{\lambda }^{k}f_{j}(z)}- \beta _{j} + \nu _{j} \frac{z\mathcal {D}_{\lambda }^{k+2}g_{j}(z)}{\mathcal {D}_{\lambda }^{k+1}g_{j}(z)} \right] +1 \end{aligned}$$
then by taking the real part of (19), we have
$$\begin{aligned}&\mathfrak {R}\left\{ \frac{z\mathfrak {J}^{\prime \prime }_{m}(z)}{\mathfrak {J}^{\prime }_{m}(z)} +1\right\} \nonumber \\&\quad =\, \sum _{j=1}^{m}\left[ \beta _{j}\mathfrak {R}\frac{z\mathcal {D}_{\lambda }^{k+1}f_{j}(z)}{\mathcal {D}_{\lambda }^{k}f_{j}(z)}- \beta _{j}+ \nu _{j}\mathfrak {R}\bigg ( \frac{z\mathcal {D}_{\lambda }^{k+2}g_{j}(z)}{\mathcal {D}_{\lambda }^{k+1}g_{j}(z)}+ 1 \bigg ) - \nu _{i} \right] +1, \end{aligned}$$
but \(f_{j} \in \mathcal {S}^{k}(\lambda ,\,\frac{1}{\beta _{j}})\), that means \(\mathfrak {R}\left\{ \tfrac{\mathcal {D}_{\lambda }^{k+1}f_{j}(z)}{\mathcal {D}_{\lambda }^{k}f_{j}(z)}\right\} > \frac{1}{\beta _{j}}\) and \(g_{j} \in \mathcal {K}^{k}\mathcal {L}(\rho _{j},\eta _{j})\), \(\rho _{j} \ge 0\) and \(0 \le \eta _{j}< 1\), \(j=\lbrace 1,2,\ldots ,m\rbrace\), from (20), we have
$$\begin{aligned} \mathfrak {R}\left\{ \frac{z\mathfrak {J}^{\prime \prime }_{m}(z)}{\mathfrak {J}^{\prime }_{m}(z)} +1\right\}&>\sum _{j=1}^{m}\left[ 1 - \beta _{j}+ \nu _{j}\left (\rho _{i}\left | \frac{z\mathcal {D}_{\lambda }^{k+2}g_{j}(z)}{\mathcal {D}_{\lambda }^{k+1}g_{j}(z)}\right | + \eta _{j} \right )-\nu _{j} \right] +1 \\&> 1+ m -\sum _{j=1}^{m}\beta _{j} +\sum _{j=1}^{m} \nu _{j}\rho _{i}\left | \frac{z\mathcal {D}_{\lambda }^{k+2}g_{j}(z)}{\mathcal {D}_{\lambda }^{k+1}g_{j}(z)}\right | + \sum _{j=1}^{m}\nu _{j}(\eta _{j}-1). \end{aligned}$$
Since, \(\nu _{j}\,\rho _{j}\,\big | \tfrac{z\mathcal {D}_{\lambda }^{k+2}g_{j}(z)}{\mathcal {D}_{\lambda }^{k+1}g_{j}(z)}\big |>0\), we have
$$\begin{aligned} \mathfrak {R}\left\{ \frac{z\mathfrak {J}^{\prime \prime }_{m}(z)}{\mathfrak {J}^{\prime }_{m}(z)} +1\right\}&>1+ m - \sum _{j=1}^{m}\beta _{j} + \sum _{j=1}^{m}\nu _{j}(\eta _{j}-1),\\&> 1+ m + \sum _{j=1}^{m}[\nu _{j}(\eta _{j}-1)- \beta _{j}]. \end{aligned}$$
By using the condition in (17), we have that \(\mathfrak {J}_{m}(z) \in \mathcal {K}^{k}(\lambda ,\omega )\), where
\(\square\)
By setting \(k=0\) and \(m=1\) in Theorem 2, we have the following result.
Let \(\beta , \nu\) be positive real numbers, We assume that \(f \in \mathcal {S}^{*}(\frac{1}{\beta })\), \(g \in \mathcal {K}\mathcal {L}(\rho ,\, \eta ), \rho \ge 0\) and \(0 \le \eta < 1\). If
$$\begin{aligned} \beta + \nu (1-\eta ) < 2 \end{aligned}$$
then the integral operator
$$\begin{aligned} \mathfrak {J}(z)=\int _{0}^{z}\left( \frac{f(t)}{t}\right) ^{\beta }( g^{\prime }(z))^{\nu } dt \end{aligned}$$
is in the class \(\mathcal {K}(\omega )\) where \(\omega =2+\nu (\eta -1)-\beta .\)
In geometric function theory, we defined and studied a new integro -differential operator \(\mathfrak {J}_{m}(z)\), with a new classes of analytic and univalent functions. This operator is generalized and modified recent various fractional differential operators.
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The authors, ZA, RI and AK contributed equally to the writing of this paper. All authors read and approved the final manuscript.
The authors would like to thank the referees for valuable suggestions and comments, which helped the authors to improve this article substantially.
Department of Mathematics, Faculty of Science, Universiti Putra Malaysia (UPM), 43400, Serdang, Selangor, Malaysia
Zainab E. Abdulnaby
Faculty of Computer Science and Information Technology, University Malaya, 50603, Kuala Lumpur, Malaysia
Rabha W. Ibrahim
Department of Mathematics, Institute for Mathematical Research, Universiti Putra Malaysia (UPM), 43400, Serdang, Selangor, Malaysia
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Correspondence to Adem Kılıçman.
Abdulnaby, Z.E., Ibrahim, R.W. & Kılıçman, A. Some properties for integro-differential operator defined by a fractional formal. SpringerPlus 5, 893 (2016) doi:10.1186/s40064-016-2563-0
Fractional calculus
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Based on the idea and the provided source code of Andrej Karpathy (arxiv-sanity)
Search for Gravitational Waves Associated with Gamma-Ray Bursts During the First Advanced LIGO Observing Run and Implications for the Origin of GRB 150906B (1611.07947)
LIGO Scientific Collaboration, Virgo Collaboration, IPN Collaboration: B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Abernathy, F. Acernese, K. Ackley, C. Adams, T. Adams, P. Addesso, R. X. Adhikari, V. B. Adya, C. Affeldt, M. Agathos, K. Agatsuma, N. Aggarwal, O. D. Aguiar, L. Aiello, A. Ain, P. Ajith, B. Allen, A. Allocca, P. A. Altin, A. Ananyeva, S. B. Anderson, W. G. Anderson, S. Appert, K. Arai, M. C. Araya, J. S. Areeda, N. Arnaud, K. G. Arun, S. Ascenzi, G. Ashton, M. Ast, S. M. Aston, P. Astone, P. Aufmuth, C. Aulbert, A. Avila-Alvarez, S. Babak, P. Bacon, M. K. M. Bader, P. T. Baker, F. Baldaccini, G. Ballardin, S. W. Ballmer, J. C. Barayoga, S. E. Barclay, B. C. Barish, D. Barker, F. Barone, B. Barr, L. Barsotti, M. Barsuglia, D. Barta, J. Bartlett, I. Bartos, R. Bassiri, A. Basti, J. C. Batch, C. Baune, V. Bavigadda, M. Bazzan, C. Beer, M. Bejger, I. Belahcene, M. Belgin, A. S. Bell, B. K. Berger, G. Bergmann, C. P. L. Berry, D. Bersanetti, A. Bertolini, J. Betzwieser, S. Bhagwat, R. Bhandare, I. A. Bilenko, G. Billingsley, C. R. Billman, J. Birch, R. Birney, O. Birnholtz, S. Biscans, A. Bisht, M. Bitossi, C. Biwer, M. A. Bizouard, J. K. Blackburn, J. Blackman, C. D. Blair, D. G. Blair, R. M. Blair, S. Bloemen, O. Bock, M. Boer, G. Bogaert, A. Bohe, F. Bondu, R. Bonnand, B. A. Boom, R. Bork, V. Boschi, S. Bose, Y. Bouffanais, A. Bozzi, C. Bradaschia, P. R. Brady, V. B. Braginsky, M. Branchesi, J. E. Brau, T. Briant, A. Brillet, M. Brinkmann, V. Brisson, P. Brockill, J. E. Broida, A. F. Brooks, D. A. Brown, D. D. Brown, N. M. Brown, S. Brunett, C. C. Buchanan, A. Buikema, T. Bulik, H. J. Bulten, A. Buonanno, D. Buskulic, C. Buy, R. L. Byer, M. Cabero, L. Cadonati, G. Cagnoli, C. Cahillane, J. Calder'on Bustillo, T. A. Callister, E. Calloni, J. B. Camp, M. Canepa, K. C. Cannon, H. Cao, J. Cao, C. D. Capano, E. Capocasa, F. Carbognani, S. Caride, J. Casanueva Diaz, C. Casentini, S. Caudill, M. Cavagli`a, F. Cavalier, R. Cavalieri, G. Cella, C. B. Cepeda, L. Cerboni Baiardi, G. Cerretani, E. Cesarini, S. J. Chamberlin, M. Chan, S. Chao, P. Charlton, E. Chassande-Mottin, B. D. Cheeseboro, H. Y. Chen, Y. Chen, H.-P. Cheng, A. Chincarini, A. Chiummo, T. Chmiel, H. S. Cho, M. Cho, J. H. Chow, N. Christensen, Q. Chu, A. J. K. Chua, S. Chua, S. Chung, G. Ciani, F. Clara, J. A. Clark, F. Cleva, C. Cocchieri, E. Coccia, P.-F. Cohadon, A. Colla, C. G. Collette, L. Cominsky, M. Constancio Jr., L. Conti, S. J. Cooper, T. R. Corbitt, N. Cornish, A. Corsi, S. Cortese, C. A. Costa, M. W. Coughlin, S. B. Coughlin, J.-P. Coulon, S. T. Countryman, P. Couvares, P. B. Covas, E. E. Cowan, D. M. Coward, M. J. Cowart, D. C. Coyne, R. Coyne, J. D. E. Creighton, T. D. Creighton, J. Cripe, S. G. Crowder, T. J. Cullen, A. Cumming, L. Cunningham, E. Cuoco, T. Dal Canton, G.D'alya, S. L. Danilishin, S. D'Antonio, K. Danzmann, A. Dasgupta, C. F. Da Silva Costa, V. Dattilo, I. Dave, M. Davier, G. S. Davies, D. Davis, E. J. Daw, B. Day, R. Day, S. De, D. DeBra, G. Debreczeni, J. Degallaix, M. De Laurentis, S. Del'eglise, W. Del Pozzo, T. Denker, T. Dent, V. Dergachev, R. De Rosa, R. T. DeRosa, R. DeSalvo, J. Devenson, R. C. Devine, S. Dhurandhar, M. C. D'iaz, L. Di Fiore, M. Di Giovanni, T. Di Girolamo, A. Di Lieto, S. Di Pace, I. Di Palma, A. Di Virgilio, Z. Doctor, V. Dolique, F. Donovan, K. L. Dooley, S. Doravari, I. Dorrington, R. Douglas, M. Dovale 'Alvarez, T. P. Downes, M. Drago, R. W. P. Drever, J. C. Driggers, Z. Du, M. Ducrot, S. E. Dwyer, T. B. Edo, M. C. Edwards, A. Effler, H.-B. Eggenstein, P. Ehrens, J. Eichholz, S. S. Eikenberry, R. A. Eisenstein, R. C. Essick, Z. Etienne, T. Etzel, M. Evans, T. M. Evans, R. Everett, M. Factourovich, V. Fafone, H. Fair, S. Fairhurst, X. Fan, S. Farinon, B. Farr, W. M. Farr, E. J. Fauchon-Jones, M. Favata, M. Fays, H. Fehrmann, M. M. Fejer, A. Fern'andez Galiana, I. Ferrante, E. C. Ferreira, F. Ferrini, F. Fidecaro, I. Fiori, D. Fiorucci, R. P. Fisher, R. Flaminio, M. Fletcher, H. Fong, S. S. Forsyth, J.-D. Fournier, S. Frasca, F. Frasconi, Z. Frei, A. Freise, R. Frey, V. Frey, E. M. Fries, P. Fritschel, V. V. Frolov, P. Fulda, M. Fyffe, H. Gabbard, B. U. Gadre, S. M. Gaebel, J. R. Gair, L. Gammaitoni, S. G. Gaonkar, F. Garufi, G. Gaur, V. Gayathri, N. Gehrels, G. Gemme, E. Genin, A. Gennai, J. George, L. Gergely, V. Germain, S. Ghonge, Abhirup Ghosh, Archisman Ghosh, S. Ghosh, J. A. Giaime, K. D. Giardina, A. Giazotto, K. Gill, A. Glaefke, E. Goetz, R. Goetz, L. Gondan, G. Gonz'alez, J. M. Gonzalez Castro, A. Gopakumar, M. L. Gorodetsky, S. E. Gossan, M. Gosselin, R. Gouaty, A. Grado, C. Graef, M. Granata, A. Grant, S. Gras, C. Gray, G. Greco, A. C. Green, P. Groot, H. Grote, S. Grunewald, G. M. Guidi, X. Guo, A. Gupta, M. K. Gupta, K. E. Gushwa, E. K. Gustafson, R. Gustafson, J. J. Hacker, B. R. Hall, E. D. Hall, G. Hammond, M. Haney, M. M. Hanke, J. Hanks, C. Hanna, J. Hanson, T. Hardwick, J. Harms, G. M. Harry, I. W. Harry, M. J. Hart, M. T. Hartman, C.-J. Haster, K. Haughian, J. Healy, A. Heidmann, M. C. Heintze, H. Heitmann, P. Hello, G. Hemming, M. Hendry, I. S. Heng, J. Hennig, J. Henry, A. W. Heptonstall, M. Heurs, S. Hild, D. Hoak, D. Hofman, K. Holt, D. E. Holz, P. Hopkins, J. Hough, E. A. Houston, E. J. Howell, Y. M. Hu, E. A. Huerta, D. Huet, B. Hughey, S. Husa, S. H. Huttner, T. Huynh-Dinh, N. Indik, D. R. Ingram, R. Inta, H. N. Isa, J.-M. Isac, M. Isi, T. Isogai, B. R. Iyer, K. Izumi, T. Jacqmin, K. Jani, P. Jaranowski, S. Jawahar, F. Jim'enez-Forteza, W. W. Johnson, D. I. Jones, R. Jones, R. J. G. Jonker, L. Ju, J. Junker, C. V. Kalaghatgi, V. Kalogera, S. Kandhasamy, G. Kang, J. B. Kanner, S. Karki, K. S. Karvinen, M. Kasprzack, E. Katsavounidis, W. Katzman, S. Kaufer, T. Kaur, K. Kawabe, F. K'ef'elian, D. Keitel, D. B. Kelley, R. Kennedy, J. S. Key, F. Y. Khalili, I. Khan, S. Khan, Z. Khan, E. A. Khazanov, N. Kijbunchoo, Chunglee Kim, J. C. Kim, Whansun Kim, W. Kim, Y.-M. Kim, S. J. Kimbrell, E. J. King, P. J. King, R. Kirchhoff, J. S. Kissel, B. Klein, L. Kleybolte, S. Klimenko, P. Koch, S. M. Koehlenbeck, S. Koley, V. Kondrashov, A. Kontos, M. Korobko, W. Z. Korth, I. Kowalska, D. B. Kozak, C. Kr"amer, V. Kringel, B. Krishnan, A. Kr'olak, G. Kuehn, P. Kumar, R. Kumar, L. Kuo, A. Kutynia, B. D. Lackey, M. Landry, R. N. Lang, J. Lange, B. Lantz, R. K. Lanza, A. Lartaux-Vollard, P. D. Lasky, M. Laxen, A. Lazzarini, C. Lazzaro, P. Leaci, S. Leavey, E. O. Lebigot, C. H. Lee, H. K. Lee, H. M. Lee, K. Lee, J. Lehmann, A. Lenon, M. Leonardi, J. R. Leong, N. Leroy, N. Letendre, Y. Levin, T. G. F. Li, A. Libson, T. B. Littenberg, J. Liu, N. A. Lockerbie, A. L. Lombardi, L. T. London, J. E. Lord, M. Lorenzini, V. Loriette, M. Lormand, G. Losurdo, J. D. Lough, G. Lovelace, H. L"uck, A. P. Lundgren, R. Lynch, Y. Ma, S. Macfoy, B. Machenschalk, M. MacInnis, D. M. Macleod, F. Magana-Sandoval, E. Majorana, I. Maksimovic, V. Malvezzi, N. Man, V. Mandic, V. Mangano, G. L. Mansell, M. Manske, M. Mantovani, F. Marchesoni, F. Marion, S. M'arka, Z. M'arka, A. S. Markosyan, E. Maros, F. Martelli, L. Martellini, I. W. Martin, D. V. Martynov, K. Mason, A. Masserot, T. J. Massinger, M. Masso-Reid, S. Mastrogiovanni, F. Matichard, L. Matone, N. Mavalvala, N. Mazumder, R. McCarthy, D. E. McClelland, S. McCormick, C. McGrath, S. C. McGuire, G. McIntyre, J. McIver, D. J. McManus, T. McRae, S. T. McWilliams, D. Meacher, G. D. Meadors, J. Meidam, A. Melatos, G. Mendell, D. Mendoza-Gandara, R. A. Mercer, E. L. Merilh, M. Merzougui, S. Meshkov, C. Messenger, C. Messick, R. Metzdorff, P. M. Meyers, F. Mezzani, H. Miao, C. Michel, H. Middleton, E. E. Mikhailov, L. Milano, A. L. Miller, A. Miller, B. B. Miller, J. Miller, M. Millhouse, Y. Minenkov, J. Ming, S. Mirshekari, C. Mishra, S. Mitra, V. P. Mitrofanov, G. Mitselmakher, R. Mittleman, A. Moggi, M. Mohan, S. R. P. Mohapatra, M. Montani, B. C. Moore, C. J. Moore, D. Moraru, G. Moreno, S. R. Morriss, B. Mours, C. M. Mow-Lowry, G. Mueller, A. W. Muir, Arunava Mukherjee, D. Mukherjee, S. Mukherjee, N. Mukund, A. Mullavey, J. Munch, E. A. M. Muniz, P. G. Murray, A. Mytidis, K. Napier, I. Nardecchia, L. Naticchioni, G. Nelemans, T. J. N. Nelson, M. Neri, M. Nery, A. Neunzert, J. M. Newport, G. Newton, T. T. Nguyen, A. B. Nielsen, S. Nissanke, A. Nitz, A. Noack, F. Nocera, D. Nolting, M. E. N. Normandin, L. K. Nuttall, J. Oberling, E. Ochsner, E. Oelker, G. H. Ogin, J. J. Oh, S. H. Oh, F. Ohme, M. Oliver, P. Oppermann, Richard J. Oram, B. O'Reilly, R. O'Shaughnessy, D. J. Ottaway, H. Overmier, B. J. Owen, A. E. Pace, J. Page, A. Pai, S. A. Pai, J. R. Palamos, O. Palashov, C. Palomba, A. Pal-Singh, H. Pan, C. Pankow, F. Pannarale, B. C. Pant, F. Paoletti, A. Paoli, M. A. Papa, H. R. Paris, W. Parker, D. Pascucci, A. Pasqualetti, R. Passaquieti, D. Passuello, B. Patricelli, B. L. Pearlstone, M. Pedraza, R. Pedurand, L. Pekowsky, A. Pele, S. Penn, C. J. Perez, A. Perreca, L. M. Perri, H. P. Pfeiffer, M. Phelps, O. J. Piccinni, M. Pichot, F. Piergiovanni, V. Pierro, G. Pillant, L. Pinard, I. M. Pinto, M. Pitkin, M. Poe, R. Poggiani, P. Popolizio, A. Post, J. Powell, J. Prasad, J. W. W. Pratt, V. Predoi, T. Prestegard, M. Prijatelj, M. Principe, S. Privitera, G. A. Prodi, L. G. Prokhorov, O. Puncken, M. Punturo, P. Puppo, M. P"urrer, H. Qi, J. Qin, S. Qiu, V. Quetschke, E. A. Quintero, R. Quitzow-James, F. J. Raab, D. S. Rabeling, H. Radkins, P. Raffai, S. Raja, C. Rajan, M. Rakhmanov, P. Rapagnani, V. Raymond, M. Razzano, V. Re, J. Read, T. Regimbau, L. Rei, S. Reid, D. H. Reitze, H. Rew, S. D. Reyes, E. Rhoades, F. Ricci, K. Riles, M. Rizzo, N. A. Robertson, R. Robie, F. Robinet, A. Rocchi, L. Rolland, J. G. Rollins, V. J. Roma, R. Romano, J. H. Romie, D. Rosi'nska, S. Rowan, A. R"udiger, P. Ruggi, K. Ryan, S. Sachdev, T. Sadecki, L. Sadeghian, M. Sakellariadou, L. Salconi, M. Saleem, F. Salemi, A. Samajdar, L. Sammut, L. M. Sampson, E. J. Sanchez, V. Sandberg, J. R. Sanders, B. Sassolas, B. S. Sathyaprakash, P. R. Saulson, O. Sauter, R. L. Savage, A. Sawadsky, P. Schale, J. Scheuer, E. Schmidt, J. Schmidt, P. Schmidt, R. Schnabel, R. M. S. Schofield, A. Sch"onbeck, E. Schreiber, D. Schuette, B. F. Schutz, S. G. Schwalbe, J. Scott, S. M. Scott, D. Sellers, A. S. Sengupta, D. Sentenac, V. Sequino, A. Sergeev, Y. Setyawati, D. A. Shaddock, T. J. Shaffer, M. S. Shahriar, B. Shapiro, P. Shawhan, A. Sheperd, D. H. Shoemaker, D. M. Shoemaker, K. Siellez, X. Siemens, M. Sieniawska, D. Sigg, A. D. Silva, A. Singer, L. P. Singer, A. Singh, R. Singh, A. Singhal, A. M. Sintes, B. J. J. Slagmolen, B. Smith, J. R. Smith, R. J. E. Smith, E. J. Son, B. Sorazu, F. Sorrentino, T. Souradeep, A. P. Spencer, A. K. Srivastava, A. Staley, M. Steinke, J. Steinlechner, S. Steinlechner, D. Steinmeyer, B. C. Stephens, S. P. Stevenson, R. Stone, K. A. Strain, N. Straniero, G. Stratta, S. E. Strigin, R. Sturani, A. L. Stuver, T. Z. Summerscales, L. Sun, S. Sunil, P. J. Sutton, B. L. Swinkels, M. J. Szczepa'nczyk, M. Tacca, D. Talukder, D. B. Tanner, M. T'apai, A. Taracchini, R. Taylor, T. Theeg, E. G. Thomas, M. Thomas, P. Thomas, K. A. Thorne, E. Thrane, T. Tippens, S. Tiwari, V. Tiwari, K. V. Tokmakov, K. Toland, C. Tomlinson, M. Tonelli, Z. Tornasi, C. I. Torrie, D. T"oyr"a, F. Travasso, G. Traylor, D. Trifir`o, J. Trinastic, M. C. Tringali, L. Trozzo, M. Tse, R. Tso, M. Turconi, D. Tuyenbayev, D. Ugolini, C. S. Unnikrishnan, A. L. Urban, S. A. Usman, H. Vahlbruch, G. Vajente, G. Valdes, N. van Bakel, M. van Beuzekom, J. F. J. van den Brand, C. Van Den Broeck, D. C. Vander-Hyde, L. van der Schaaf, J. V. van Heijningen, A. A. van Veggel, M. Vardaro, V. Varma, S. Vass, M. Vas'uth, A. Vecchio, G. Vedovato, J. Veitch, P. J. Veitch, K. Venkateswara, G. Venugopalan, D. Verkindt, F. Vetrano, A. Vicer'e, A. D. Viets, S. Vinciguerra, D. J. Vine, J.-Y. Vinet, S. Vitale, T. Vo, H. Vocca, C. Vorvick, D. V. Voss, W. D. Vousden, S. P. Vyatchanin, A. R. Wade, L. E. Wade, M. Wade, M. Walker, L. Wallace, S. Walsh, G. Wang, H. Wang, M. Wang, Y. Wang, R. L. Ward, J. Warner, M. Was, J. Watchi, B. Weaver, L.-W. Wei, M. Weinert, A. J. Weinstein, R. Weiss, L. Wen, P. Wessels, T. Westphal, K. Wette, J. T. Whelan, B. F. Whiting, C. Whittle, D. Williams, R. D. Williams, A. R. Williamson, J. L. Willis, B. Willke, M. H. Wimmer, W. Winkler, C. C. Wipf, H. Wittel, G. Woan, J. Woehler, J. Worden, J. L. Wright, D. S. Wu, G. Wu, W. Yam, H. Yamamoto, C. C. Yancey, M. J. Yap, Hang Yu, Haocun Yu, M. Yvert, A. Zadro.zny, L. Zangrando, M. Zanolin, J.-P. Zendri, M. Zevin, L. Zhang, M. Zhang, T. Zhang, Y. Zhang, C. Zhao, M. Zhou, Z. Zhou, S. J. Zhu, X. J. Zhu, M. E. Zucker, J. Zweizig, R. L. Aptekar, D. D. Frederiks, S. V. Golenetskii, D. V. Golovin, K. Hurley, M. L. Litvak, I. G. Mitrofanov, A. Rau, A. B. Sanin, D. S. Svinkin, A. von Kienlin, X. Zhang
June 21, 2017 gr-qc, astro-ph.HE
We present the results of the search for gravitational waves (GWs) associated with $\gamma$-ray bursts detected during the first observing run of the Advanced Laser Interferometer Gravitational-Wave Observatory (LIGO). We find no evidence of a GW signal for any of the 41 $\gamma$-ray bursts for which LIGO data are available with sufficient duration. For all $\gamma$-ray bursts, we place lower bounds on the distance to the source using the optimistic assumption that GWs with an energy of $10^{-2}M_\odot c^2$ were emitted within the $16$-$500\,$Hz band, and we find a median 90% confidence limit of 71$\,$Mpc at 150$\,$Hz. For the subset of 19 short/hard $\gamma$-ray bursts, we place lower bounds on distance with a median 90% confidence limit of 90$\,$Mpc for binary neutron star (BNS) coalescences, and 150 and 139$\,$Mpc for neutron star-black hole coalescences with spins aligned to the orbital angular momentum and in a generic configuration, respectively. These are the highest distance limits ever achieved by GW searches. We also discuss in detail the results of the search for GWs associated with GRB 150906B, an event that was localized by the InterPlanetary Network near the local galaxy NGC 3313, which is at a luminosity distance of 54$\,$Mpc ($z=0.0124$). Assuming the $\gamma$-ray emission is beamed with a jet half-opening angle $\leq 30^{\circ}$, we exclude a BNS and a neutron star-black hole in NGC 3313 as the progenitor of this event with confidence $>99$%. Further, we exclude such progenitors up to a distance of 102$\,$Mpc and 170$\,$Mpc, respectively.
GitHub 0
A Missing-Link in the Supernova-GRB Connection: The Case of SN 2012ap (1402.6336)
Sayan Chakraborti, Alicia Soderberg, Laura Chomiuk, Atish Kamble, Naveen Yadav, Alak Ray, Kevin Hurley, Raffaella Margutti, Dan Milisavljevic, Michael Bietenholz, Andreas Brunthaler, Giuliano Pignata, Elena Pian, Paolo Mazzali, Claes Fransson, Norbert Bartel, Mario Hamuy, Emily Levesque, Andrew MacFadyen, Jason Dittmann, Miriam Krauss, M. S. Briggs, V. Connaughton, K. Yamaoka, T. Takahashi, M. Ohno, Y. Fukazawa, M. Tashiro, Y. Terada, T. Murakami, J. Goldsten, S. Barthelmy, N. Gehrels, J. Cummings, H. Krimm, D. Palmer, S. Golenetskii, R. Aptekar, D. Frederiks, D. Svinkin, T. Cline, I. G. Mitrofanov, D. Golovin, M. L. Litvak, A. B. Sanin, W. Boynton, C. Fellows, K. Harshman, H. Enos, A. von Kienlin, A. Rau, X. Zhang, V. Savchenko
April 23, 2015 astro-ph.CO, astro-ph.HE
Gamma Ray Bursts (GRBs) are characterized by ultra-relativistic outflows, while supernovae are generally characterized by non-relativistic ejecta. GRB afterglows decelerate rapidly usually within days, because their low-mass ejecta rapidly sweep up a comparatively larger mass of circumstellar material. However supernovae, with heavy ejecta, can be in nearly free expansion for centuries. Supernovae were thought to have non-relativistic outflows except for few relativistic ones accompanied by GRBs. This clear division was blurred by SN 2009bb, the first supernova with a relativistic outflow without an observed GRB. Yet the ejecta from SN 2009bb was baryon loaded, and in nearly-free expansion for a year, unlike GRBs. We report the first supernova discovered without a GRB, but with rapidly decelerating mildly relativistic ejecta, SN 2012ap. We discovered a bright and rapidly evolving radio counterpart driven by the circumstellar interaction of the relativistic ejecta. However, we did not find any coincident GRB with an isotropic fluence of more than a sixth of the fluence from GRB 980425. This shows for the first time that central engines in type Ic supernovae, even without an observed GRB, can produce both relativistic and rapidly decelerating outflows like GRBs.
IPN localizations of Konus short gamma-ray bursts (1301.3740)
V. D. Pal'shin, K. Hurley, D. S. Svinkin, R. L. Aptekar, S. V. Golenetskii, D. D. Frederiks, E. P. Mazets, P. P. Oleynik, M. V. Ulanov, T. Cline, I. G. Mitrofanov, D. V. Golovin, A. S. Kozyrev, M. L. Litvak, A. B. Sanin, W. Boynton, C. Fellows, K. Harshman, J. Trombka, T. McClanahan, R. Starr, J. Goldsten, R. Gold, A. Rau, A. von Kienlin, V. Savchenko, D. M. Smith, W. Hajdas, S. D. Barthelmy, J. Cummings, N. Gehrels, H. Krimm, D. Palmer, K. Yamaoka, M. Ohno, Y. Fukazawa, Y. Hanabata, T. Takahashi, M. Tashiro, Y. Terada, T. Murakami, K. Makishima, M. S. Briggs, R. M. Kippen, C. Kouveliotou, C. Meegan, G. Fishman, V. Connaughton, M. Boer, C. Guidorzi, F. Frontera, E. Montanari, F. Rossi, M. Feroci, L. Amati, L. Nicastro, M. Orlandini, V. Connaughton, E. Del Monte, E. Costa, I. Donnarumma, Y. Evangelista, I. Lapshov, F. Lazzarotto, L. Pacciani, M. Rapisarda, P. Soffitta, G. Di Cocco, F. Fuschino, M. Galli, C. Labanti, M. Marisaldi, J.-L. Atteia, R. Vanderspek, G. Ricker
Aug. 5, 2013 astro-ph.HE
Between the launch of the \textit{GGS Wind} spacecraft in 1994 November and the end of 2010, the Konus-\textit{Wind} experiment detected 296 short-duration gamma-ray bursts (including 23 bursts which can be classified as short bursts with extended emission). During this period, the IPN consisted of up to eleven spacecraft, and using triangulation, the localizations of 271 bursts were obtained. We present the most comprehensive IPN localization data on these events. The short burst detection rate, $\sim$18 per year, exceeds that of many individual experiments.
The Interplanetary Network Supplement to the Fermi GBM Catalog of Cosmic Gamma-Ray Bursts (1301.3522)
K. Hurley, V. D. Pal'shin, R. L. Aptekar, S. V. Golenetskii, D. D. Frederiks, E. P. Mazets, D. S. Svinkin, M. S. Briggs, V. Connaughton, C. Meegan, J. Goldsten, W. Boynton, C. Fellows, K. Harshman, I. G. Mitrofanov, D. V. Golovin, A. S. Kozyrev, M. L. Litvak, A. B. Sanin, A. Rau, A. von Kienlin, X. Zhang, K. Yamaoka, Y. Fukazawa, Y. Hanabata, M. Ohno, T. Takahashi, M. Tashiro, Y. Terada, T. Murakami, K. Makishima, S. Barthelmy, T. Cline, N. Gehrels, J. Cummings, H. A. Krimm, D.M. Smith, E. Del Monte, M. Feroci, M. Marisaldi
June 21, 2013 astro-ph.HE
We present Interplanetary Network (IPN) data for the gamma-ray bursts in the first Fermi Gamma-Ray Burst Monitor (GBM) catalog. Of the 491 bursts in that catalog, covering 2008 July 12 to 2010 July 11, 427 were observed by at least one other instrument in the 9-spacecraft IPN. Of the 427, the localizations of 149 could be improved by arrival time analysis (or triangulation). For any given burst observed by the GBM and one other distant spacecraft, triangulation gives an annulus of possible arrival directions whose half-width varies between about 0.4' and 32 degrees, depending on the intensity, time history, and arrival direction of the burst, as well as the distance between the spacecraft. We find that the IPN localizations intersect the 1 sigma GBM error circles in only 52% of the cases, if no systematic uncertainty is assumed for the latter. If a 6 degree systematic uncertainty is assumed and added in quadrature, the two localization samples agree about 87% of the time, as would be expected. If we then multiply the resulting error radii by a factor of 3, the two samples agree in slightly over 98% of the cases, providing a good estimate of the GBM 3 sigma error radius. The IPN 3 sigma error boxes have areas between about 1 square arcminute and 110 square degrees, and are, on the average, a factor of 180 smaller than the corresponding GBM localizations. We identify two bursts in the IPN/GBM sample that did not appear in the GBM catalog. In one case, the GBM triggered on a terrestrial gamma flash, and in the other, its origin was given as uncertain. We also discuss the sensitivity and calibration of the IPN.
GRB 080407: an ultra-long burst discovered by the IPN (1301.5203)
V. Pal'shin, K. Hurley, J. Goldsten, I. G. Mitrofanov, W. Boynton, A. von Kienlin, J. Cummings, M. Feroci, R. Aptekar, D. Frederiks, S. Golenetskii, E. Mazets, D. Svinkin, D. Golovin, M. L. Litvak, A. B. Sanin, C. Fellows, K. Harshman, R. Starr, A. Rau, V. Savchenko, X. Zhang, S. Barthelmy, N. Gehrels, H. Krimm, D. Palmer, E. Del Monte, M. Marisaldi
Jan. 22, 2013 astro-ph.HE
We present observations of the extremely long GRB 080704 obtained with the instruments of the Interplanetary Network (IPN). The observations reveal two distinct emission episodes, separated by a ~1500 s long period of quiescence. The total burst duration is about 2100 s. We compare the temporal and spectral characteristics of this burst with those obtained for other ultra-long GRBs and discuss these characteristics in the context of different models.
Extremely long hard bursts observed by Konus-Wind (1301.4829)
V. Pal'shin, R. Aptekar, D. Frederiks, S. Golenetskii, V. Il'Inskii, E. Mazets, K. Yamaoka, M. Ohno, K. Hurley, T. Sakamoto, P. Oleynik, M. Ulanov, I. G. Mitrofanov, D. Golovin, M. L. Litvak, A. B. Sanin, W. Boynton, C. Fellows, K. Harshman, C. Shinohara, R. Starr
We report the observations of the prompt emission of the extremely long hard burst, GRB 060814B, discovered by Konus-Wind and localized by the IPN. The observations reveal a smooth, hard, ~40-min long pulse followed by weaker emission seen several hours after the burst onset. We also present the Konus-Wind data on similar burst, GRB 971208, localized by BATSE/IPN. And finally we discuss the different possible origins of these unusual events.
Inverse Compton X-ray Emission from Supernovae with Compact Progenitors: Application to SN2011fe (1202.0741)
R. Margutti, A.M. Soderberg, L. Chomiuk, R. Chevalier, K. Hurley, D. Milisavljevic, R.J. Foley, J.P. Hughes, P. Slane, C. Fransson, M. Moe, S. Barthelmy, W. Boynton, M. Briggs, V. Connaughton, E. Costa, J. Cummings, E. Del Monte, H. Enos, C. Fellows, M. Feroci, Y. Fukazawa, N. Gehrels, J. Goldsten, D. Golovin, Y. Hanabata, K. Harshman, H. Krimm, M. L. Litvak, K. Makishima, M. Marisaldi, I. G. Mitrofanov, T. Murakami, M. Ohno, D. M. Palmer, A. B. Sanin, R. Starr, D. Svinkin
Feb. 3, 2012 astro-ph.HE
We present a generalized analytic formalism for the inverse Compton X-ray emission from hydrogen-poor supernovae and apply this framework to SN2011fe using Swift-XRT, UVOT and Chandra observations. We characterize the optical properties of SN2011fe in the Swift bands and find them to be broadly consistent with a "normal" SN Ia, however, no X-ray source is detected by either XRT or Chandra. We constrain the progenitor system mass loss rate to be lower than 2x10^-9 M_sun/yr (3sigma c.l.) for wind velocity v_w=100 km/s. Our result rules out symbiotic binary progenitors for SN2011fe and argues against Roche-lobe overflowing subgiants and main sequence secondary stars if >1% of the transferred mass is lost at the Lagrangian points. Regardless of the density profile, the X-ray non-detections are suggestive of a clean environment (particle density < 150 cm-3) for (2x10^15<R<5x10^16) cm around the progenitor site. This is either consistent with the bulk of material being confined within the binary system or with a significant delay between mass loss and supernova explosion. We furthermore combine X-ray and radio limits from Chomiuk et al. 2012 to constrain the post shock energy density in magnetic fields. Finally, we searched for the shock breakout pulse using gamma-ray observations from the Interplanetary Network and find no compelling evidence for a supernova-associated burst. Based on the compact radius of the progenitor star we estimate that the shock break out pulse was likely not detectable by current satellites.
The Interplanetary Network Supplement to the Fermi GBM Catalog - An AO-2 and AO-3 Guest Investigator Project (1110.6470)
K. Hurley, M. Briggs, V. Connaughton, C. Meegan, A. von Kienlin, A. Rau, X. Zhang, S. Golenetskii, R. Aptekar, E. Mazets, V. Pal'shin, D. Frederiks, S. Barthelmy, T. Cline, J. Cummings, N. Gehrels, H. A. Krimm, I. G. Mitrofanov, D. Golovin, M. L. Litvak, A. B. Sanin, W. Boynton, C. Fellows, K. Harshman, R. Starr, D. M. Smith, W. Hajdas, K. Yamaoka, M. Ohno, Y. Fukazawa, T. Takahashi, M. Tashiro, Y. Terada, T. Murakami, K. Makishima, D. M. Palmer, J. Goldsten, E. Del Monte, M. Feroci, M. Marisaldi
Oct. 28, 2011 astro-ph.HE
In the first two years of operation of the Fermi GBM, the 9-spacecraft Interplanetary Network (IPN) detected 158 GBM bursts with one or two distant spacecraft, and triangulated them to annuli or error boxes. Combining the IPN and GBM localizations leads to error boxes which are up to 4 orders of magnitude smaller than those of the GBM alone. These localizations comprise the IPN supplement to the GBM catalog, and they support a wide range of scientific investigations.
The Interplanetary Network Supplement to the HETE-2 Gamma-Ray Burst Catalog (0907.2709)
K. Hurley, J.-L. Atteia, C. Barraud, A. Pelangeon, M. Boer, R. Vanderspek, G. Ricker, E. Mazets, S. Golenetskii, D. D. Frederiks, V. D. Pal'shin, R. L. Aptekar, D.M. Smith, C. Wigger, W. Hajdas, A. Rau, A. von Kienlin, I. G. Mitrofanov, D. V. Golovin, A. S. Kozyrev, M. L. Litvak, A. B. Sanin, W. Boynton, C. Fellows, K. Harshman S. Barthelmy, T. Cline, J. Cummings, N. Gehrels, H. Krimm, K. Yamaoka, M. Ohno, Y. Fukazawa, Y. Hanabata, T. Takahashi, M. Tashiro, Y. Terada, T. Murakami, K. Makishima, C. Guidorzi, F. Frontera, C. E. Montanari, F. Rossi, J. Trombka, T. McClanahan, R. Starr J. Goldsten, R. Gold
Sept. 22, 2010 astro-ph.HE
Between 2000 November and 2006 May, one or more spacecraft of the interplanetary network (IPN) detected 226 cosmic gamma-ray bursts that were also detected by the FREGATE experiment aboard the HETE-II spacecraft. During this period, the IPN consisted of up to nine spacecraft, and using triangulation, the localizations of 157 bursts were obtained. We present the IPN localization data on these events.
The Interplanetary Network Supplement to the BeppoSAX Gamma-Ray Burst Catalogs (1004.1650)
K. Hurley, C. Guidorzi, F. Frontera, E. Montanari, F. Rossi, M. Feroci, E. Mazets, S. Golenetskii, D. D. Frederiks, V. D. Pal'shin, R. L. Aptekar, T. Cline, J. Trombka, T. McClanahan, R. Starr, J.-L. Atteia, C. Barraud, A. Pelangeon, M. Boer, R. Vanderspek, G. Ricker, I. G. Mitrofanov, D. V. Golovin, A. S. Kozyrev, M. L. Litvak, A. B. Sanin, W. Boynton, C. Fellows, K. Harshman, J. Goldsten, R. Gold, D.M. Smith, C. Wigger, W. Hajdas
April 9, 2010 astro-ph.HE
Between 1996 July and 2002 April, one or more spacecraft of the interplanetary network detected 787 cosmic gamma-ray bursts that were also detected by the Gamma-Ray Burst Monitor and/or Wide-Field X-Ray Camera experiments aboard the BeppoSAX spacecraft. During this period, the network consisted of up to six spacecraft, and using triangulation, the localizations of 475 bursts were obtained. We present the localization data for these events.
Integrating the Fermi Gamma-Ray Burst Monitor into the 3rd Interplanetary Network (0912.4294)
K. Hurley, M. Briggs, V. Connaughton, C. Meegan, T. Cline, I. Mitrofanov, D. Golovin, M. L. Litvak, A. B. Sanin, W. Boynton, C. Fellows, K. Harshman, R. Starr, S. Golenetskii, R. Aptekar, E. Mazets, V. Pal'shin, D. Frederiks, D. M. Smith, C. Wigger, A. Rau, A. von Kienlin, K. Yamaoka, M. Ohno, Y. Fukazawa, T. Takahashi, M. Tashiro, Y. Terada, T. Murakami, K. Makishima, S. Barthelmy, J. Cummings, N. Gehrels, H. Krimm, J. Goldsten, E. Del Monte, M. Feroci, M. Marisaldi
Dec. 21, 2009 astro-ph.HE
We are integrating the Fermi Gamma-Ray Burst Monitor (GBM) into the Interplanetary Network (IPN) of Gamma-Ray Burst (GRB) detectors. With the GBM, the IPN will comprise 9 experiments. This will 1) assist the Fermi team in understanding and reducing their systematic localization uncertainties, 2) reduce the sizes of the GBM and Large Area Telescope (LAT) error circles by 1 to 4 orders of magnitude, 3) facilitate the identification of GRB sources with objects found by ground- and space-based observatories at other wavelengths, from the radio to very high energy gamma-rays, 4) reduce the uncertainties in associating some LAT detections of high energy photons with GBM bursts, and 5) facilitate searches for non-electromagnetic GRB counterparts, particularly neutrinos and gravitational radiation. We present examples and demonstrate the synergy between Fermi and the IPN. This is a Fermi Cycle 2 Guest Investigator project.
A new analysis of the short-duration, hard-spectrum GRB 051103, a possible extragalactic SGR giant flare (0907.2462)
K. Hurley, A. Rowlinson, E. Bellm, D. Perley, I. G. Mitrofanov, D. V. Golovin, A. S. Kozyrev, M. L. Litvak, A. B. Sanin, W. Boynton, C. Fellows, K. Harshmann, M. Ohno, K. Yamaoka, Y. E. Nakagawa, D. M. Smith, T. Cline, N.R. Tanvir, P.T. O'Brien, K. Wiersema, E. Rol, A. Levan, J. Rhoads, A. Fruchter, D. Bersier, J.J. Kavelaars, N. Gehrels, H. Krimm, D. M. Palmer, R. C. Duncan, C. Wigger, W. Hajdas, J.-L. Atteia, G. Ricker, R. Vanderspek, A. Rau, A. von Kienlin
Nov. 23, 2009 astro-ph.HE
GRB 051103 is considered to be a candidate soft gamma repeater (SGR) extragalactic giant magnetar flare by virtue of its proximity on the sky to M81/M82, as well as its time history, localization, and energy spectrum. We have derived a refined interplanetary network localization for this burst which reduces the size of the error box by over a factor of two. We examine its time history for evidence of a periodic component, which would be one signature of an SGR giant flare, and conclude that this component is neither detected nor detectable under reasonable assumptions. We analyze the time-resolved energy spectra of this event with improved time- and energy resolution, and conclude that although the spectrum is very hard, its temporal evolution at late times cannot be determined, which further complicates the giant flare association. We also present new optical observations reaching limiting magnitudes of R > 24.5, about 4 magnitudes deeper than previously reported. In tandem with serendipitous observations of M81 taken immediately before and one month after the burst, these place strong constraints on any rapidly variable sources in the region of the refined error ellipse proximate to M81. We do not find any convincing afterglow candidates from either background galaxies or sources in M81, although within the refined error region we do locate two UV bright star forming regions which may host SGRs. A supernova remnant (SNR) within the error ellipse could provide further support for an SGR giant flare association, but we were unable to identify any SNR within the error ellipse. These data still do not allow strong constraints on the nature of the GRB 051103 progenitor, and suggest that candidate extragalactic SGR giant flares will be difficult, although not impossible, to confirm.
Observations of the Prompt Gamma-Ray Emission of GRB 070125 (0710.4590)
Eric C. Bellm, Kevin Hurley, Valentin Pal'shin, Kazutaka Yamaoka, Mark S. Bandstra, Steven E. Boggs, Soojing Hong, Natsuki Kodaka, A. S. Kozyrev, M. L. Litvak, I. G. Mitrofanov, Yujin E. Nakagawa, Masanori Ohno, Kaori Onda, A. B. Sanin, Satoshi Sugita, Makoto Tashiro, V. I. Tretyakov, Yuji Urata, Claudia Wigger
July 28, 2008 astro-ph
The long, bright gamma-ray burst GRB 070125 was localized by the Interplanetary Network. We present light curves of the prompt gamma-ray emission as observed by Konus-WIND, RHESSI, Suzaku-WAM, and \textit{Swift}-BAT. We detail the results of joint spectral fits with Konus and RHESSI data. The burst shows moderate hard-to-soft evolution in its multi-peaked emission over a period of about one minute. The total burst fluence as observed by Konus is $1.79 \times 10^{-4}$ erg/cm$^2$ (20 keV--10 MeV). Using the spectroscopic redshift $z=1.548$, we find that the burst is consistent with the ``Amati'' $E_{peak,i}-E_{iso}$ correlation. Assuming a jet opening angle derived from broadband modeling of the burst afterglow, GRB 070125 is a significant outlier to the ``Ghirlanda'' $E_{peak,i}-E_\gamma$ correlation. Its collimation-corrected energy release $E_\gamma = 2.5 \times 10^{52}$ ergs is the largest yet observed.
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CommonCrawl
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A semi-analytical performance prediction of a digital communication system using Fourier transform inversion
Fatima Ezzahra Naamane1,
Mohamed Et-tolba2 &
Mostafa Belkasmi1
EURASIP Journal on Wireless Communications and Networking volume 2015, Article number: 236 (2015) Cite this article
Analytical performance evaluation of a digital communication system remains a serious problem especially when a sophisticated digital signal processing is considered. Moreover, it is difficult to obtain the expected performance of such system using the Monte Carlo simulation method. In this paper, we propose a new semi-analytical approach for predicting error probability in a digital communication system. This approach is based on Fourier transform inversion formula to estimate the probability density function (pdf) of the observed soft sample at the receiver. Furthermore, we applied a bootstrap method for selecting the optimal smoothing parameter to make the proposed semi-analytical method more accurate. Simulation results show that the obtained semi-analytical error probability is close to the one measured using Monte Carlo simulation and provides a significant gain in terms of computing time. Besides, the use of the bootstrap method decreases the squared error between the true pdf and the estimated one.
Advanced wireless communication systems use sophisticated digital modulation schemes as well as space–time diversity in order to provide high data rates. The transmission quality of these systems is determined by the performance evaluation, which can be made using metrics such as the bit error probability (BEP), the block error probability (BLEP), or throughput. However, unified analytical expressions of these metrics are not available for several digital communication systems. The common method used to fix this problem is Monte Carlo simulation in which one has to simulate the transmitter, the transmission channel, and the receiver. Unfortunately, in complex systems, this method becomes very prohibitive in terms of computation time, and it requires a very large number of transmitted samples to estimate very low error probabilities. As a solution, semi-analytical performance prediction (SPP) has been proposed in recent years and it has been the subject of numerous studies. In [1], the authors have proposed the importance sampling (IS) method for BER prediction. It has been found that for simple memoryless systems (e.g., a BPSK modem [2]), the efficiency of the IS technique is high and its implementation is relatively easier. However, its accuracy can be severely degraded, especially when a complex system receiver is used. For this reason, Abdi et al. have proposed in [3] a low complexity prediction technique for turbo-like codes. It is based on estimating the probability density function (pdf) of the log-likelihood ratio (LLR) at the output of the decoder using a normal density as a reference. Nevertheless, it does not allow reducing the complexity of the iterative decoding algorithm. In [4], the authors have derived a semi-analytical expression of the bit error probability using a non-parametric estimation of the probability density of the observed samples. It has been shown that the accuracy of the pdf estimator is sensitive to the choice of the smoothing parameter. The method we have proposed in [5] considers the estimation of the pdf using kernel estimator [6] which uses an efficient technique for selecting the smoothing parameter. In [7], we have compared some methods to make up for the optimum smoothing parameter choice. The first is the minimum integrated squared error (MISE) [8], which exhibited a significant squared error between the true pdf and the estimated one. In the second method, the smoothing parameter is estimated using a cross-validation (CV) method [9, 10]. Simulation studies have concluded that the method called cross validation outperforms the other method in terms of squared error. Nevertheless, this technique can lead to inconsistent estimator and requires too much computing time.
In this paper, we propose a new semi-analytical approach based on Fourier transform inversion to derive a semi-analytical expression of error probability. In this method, the probability density of the decision variable at the matched filter output is estimated from the characteristic function via Fourier transform inversion. This is due to the fact that the characteristic function is defined as the Fourier transform of the probability density function. In addition, Fourier integrals can be numerically evaluated by the fast Fourier transform (FFT) algorithm. Furthermore, in order to control the behavior of the probability density estimator, we applied a bootstrap method for selecting the optimum smoothing parameter. This leads to an accurate semi-analytical error probability due to the bootstrap approach efficiency.
The remainder of the paper is organized as follows. In Section 2, we describe the system model considered in this work. In Section 3, a new semi-analytical expression of the error probability is derived, using Fourier inversion approach. Some methods for selecting the smoothing parameter are given in Section 4. Simulations and numerical results are given in Section 5. Then, concluding remarks are made in Section 6.
The digital communication system considered in this work is shown in Fig. 1. It consists of a transmitter, a transmission channel, and a receiver. At the transmitter end, a digital source delivers a bit-stream represented by the binary sequences denoted by b=[b 1,b 2,…,b L ] and each has length L. The sequences of bits are then passed to a digital modulation scheme which converts them into sequences of symbols, each has length M and whose elements take values in constellation set Ω. The digital modulation can perform binary phase-shift keying (BPSK), quadrature phase-shift keying (QPSK), or high order modulation such as 16-quadrature amplitude modulation (QAM) and 64-QAM. Other techniques, such as single carrier frequency division multiple access (SC-FDMA) or orthogonal frequency division multiplexing (OFDMA), can be included in the transmitter to improve the system reliability. After bit-to-symbol mapping, the modulation transforms the symbol stream into an analog signal suitable to be sent through the transmission channel which can degrade the signal quality.
General system model
At the receiver, the channel output is passed to a matched filter to reduce the noise effect. After that, the demodulation is performed for symbol-to-bit conversion. Finally, the receiver makes a decision to detect the information bits.
Semi-analytical error probability derivation
Bit error probability definition
The receiver observes a set of N samples C={x 1,x 2,…,x N } at the output of the matched filter and makes a decision to estimate the information bits. Due to the channel effect, this decision can be erroneous. So, it is important to measure the communication system efficiency in terms of bit error probability (BEP). According to the system model presented in Fig. 1, this bit error probability is defined to be the conditional probability that the receiver makes a wrong decision on a transmitted information bit. Assuming that the ith bit is transmitted, the error probability is expressed as follows:
$$ \begin{aligned} P_{b} &= Pr\left[\text{Error} ~|~ b_{i}~\text{sent}\right] \\ &= Pr\left[\widetilde{b}_{i} \neq b_{i} ~|~ b_{i}~\text{sent}\right], \end{aligned} $$
Let X be the random variable whose realizations are the observed samples at the matched filter output and define the decision region associated to the information bit b i as
$$ \begin{aligned} Z_{i}=\left\{ X \in \mathbb{R};~Pr\left[\widetilde{b}_{i} = b_{i}~|~ X\right] > Pr\left[\widetilde{b}_{i} \neq b_{i} ~|~ X\right] \right\}. \end{aligned} $$
where \(\widetilde {b}_{i}\) is the estimation of the ith information bit at the receiver end. The probability of error on the bit b i defined in (1) is then re-expressed as
$$ \begin{aligned} P_{b} &= Pr\left[X \notin Z_{i} ~|~ b_{i}~\text{sent}\right], \end{aligned} $$
We can express this error probability in terms of the probability density of X to get
$$ \begin{aligned} P_{b} &= \int_{X \notin Z_{i}}^{} f_{X}(x ~|~ b_{i}~\text{sent})\,dx, \end{aligned} $$
To obtain the expression of the average bit error probability P e , we divide the set of the observed samples C into two subsets C 0 and C 1. The first subset contains N 0 observed samples which corresponds to the transmission of b i =0. The second subset consist of N 1 observed samples when the bit b i =1 is transmitted. In this manner, the probability density function of X can be viewed as a mixture of two probability densities \(f_{X}^{(1)}(x)\) and \(f_{X}^{(0)}(x)\) of the observed samples corresponding to the transmitted information bits b i =1 and b i =0, respectively. Then, the average bit error probability is written as
$$ {\fontsize{9.1}{6}\begin{aligned} P_{e} &= P_{1}.Pr\left[X\notin Z_{1} | b_{i}=1\right] +P_{0}. Pr\left[X\notin Z_{0} | b_{i}=0\right] \\ &= P_{1}. \int_{-\infty}^{0} f_{X}^{(1)}\left(x | b_{i}=1\right)dx + P_{0}. \int_{0}^{+\infty} f_{X}^{(0)}\left(x | b_{i}=0\right)\,dx. \end{aligned}} $$
where \(P_{k} = \frac {N_{k}}{N}, k=0,1\), is the probability that b i =k is transmitted.
For equally likely transmitted information bits, the average BEP is finally given by
$$\begin{array}{@{}rcl@{}} P_{e} &=& \int_{0}^{+\infty} f_{X}^{(0)}\left(x | b_{i}=0\right)\,dx \\ &=& \int_{-\infty}^{0} f_{X}^{(1)}\left(x | b_{i}=1\right)\,dx. \end{array} $$
Accordingly, for predicting the error probability P e , one has to estimate the probability densities \({f_{X}^{1}}(x)\) and \({f_{X}^{0}}(x)\). In this paper, we will focus on the use of Fourier inversion approach and its use for estimating error probability.
Probability density function estimation
Various techniques for estimating the probability density function have been developed in literature. The most known of these methods is that based on kernel estimator [11]. The approach we propose in this paper is based on the fact that the pdf can be found from the characteristic function of a random variable X via Fourier transform inversion. This is expressed as follows:
$$\begin{array}{@{}rcl@{}} \widetilde{f}(x) &=& \frac{1}{2\pi}\int_{-\infty}^{+\infty} e^{-jtx} {\varphi}_{X}(t)\,dt, \end{array} $$
where φ X is the characteristic function of a random variable X, defined as
$$\begin{array}{@{}rcl@{}} {\varphi}_{X}(t) &=& \mathbb{E}\left[e^{jtX}\right] \\ &=& \int_{-\infty}^{+\infty} e^{jtx} f_{X}(x)\,dx, \end{array} $$
Given N observed samples {x 1,x 2,…,x N }, the expectation in (8) can be approximated by a finite sum. Hence, the characteristic function φ X can be written as
$$\begin{array}{@{}rcl@{}} \widetilde{\varphi}_{X}(t) &=& \frac{1}{N} \sum\limits_{i=1}^{N} e^{{jtx}_{i}}, \end{array} $$
Consequently, the probability density function can be estimated according to (7) by using the approximation of φ X (t) given in (9). However, the Fourier integral in (7) can exhibit divergence for large values of the time variable t. To solve this limitation, the characteristic function estimator \(\widetilde {\varphi }_{X}(t)\) is multiplied by a damping function ψ h (t)=ψ(h t) to control the smoothness of the estimated probability density function.
Therefore, the characteristic function expression becomes
$$\begin{array}{@{}rcl@{}} \widetilde{\varphi}_{X}(t) &=& \frac{1}{N} \sum\limits_{i=1}^{N} e^{{jtx}_{i}} \psi_{h}(t). \end{array} $$
where h is a smoothing parameter.
It follows that the estimated probability density function is given by (see proof in Appendix A)
$$\begin{array}{@{}rcl@{}} \widetilde{f}(x;h) &=& \frac{1}{Nh} \sum\limits_{i=1}^{N} v\left(\frac{x-x_{i}}{h}\right), \end{array} $$
$$\begin{array}{@{}rcl@{}} v(x) &=& \frac{1}{2\pi} \int_{-\infty}^{+\infty} e^{-jtx}\psi(t)\,dt. \end{array} $$
The most common choice for the damping function ψ(t) is the Gaussian function \(\psi (t)= e^{-\pi t^{2}}\). Then, the semi-analytical probability density function is done as
$$\begin{array}{@{}rcl@{}} \widetilde{f}(x;h) &=& \frac{1}{Nh} \sum\limits_{i=1}^{N} \frac{1}{2\pi} \int_{-\infty}^{+\infty} e^{-jt\frac{(x-x_{i})}{h} }e^{-\pi t^{2}}\,dt \\ &=& \frac{1}{Nh} \sum\limits_{i=1}^{N} \frac{1}{2\pi} e^{-\left(\frac{x-x_{i}}{2\sqrt{\pi}h}\right)^{2}}. \end{array} $$
After replacing the probability densities using the estimation above, and evaluating the integral in (5), the semi-analytical bit error probability can be finally re-expressed as (see proof in Appendix B)
$$\begin{array}{@{}rcl@{}} {}P_{e} = \frac{P_{1}}{N_{1}} \sum\limits_{i=1}^{N_{1}} Q\left(\frac{(x_{i})_{1}}{\sqrt{2\pi}h_{1}}\right) + \frac{P_{0}}{N_{0}} \sum\limits_{i=1}^{N_{0}} Q\left(\frac{-(x_{i})_{0}}{\sqrt{2\pi}h_{0}}\right). \end{array} $$
where (x i )0 and (x i )1 are the observed samples corresponding to the transmitted bits b i =0 and b i =1, respectively. h 1 (respectively, h 0) is the smoothing parameter which depends on the number of observed samples, i.e., N 1 (respectively, N 0). Q(:) denotes the complementary unit cumulative Gaussian distribution, that is
$$\begin{array}{@{}rcl@{}} Q(x)=\frac{1}{\sqrt{2\pi}} \int_{x}^{+\infty} e^{-t^{2}/2}\,dt. \end{array} $$
From (14), it is clear that the accuracy of bit error probability estimation depends on the choice of the optimal smoothing parameter.
Smoothing parameter selection
As already mentioned, the important task in semi-analytical BEP derivation is the selection of the smoothing parameter which impacts the precision of the estimator given in (14). The optimal smoothing parameter is defined to be the value of h that minimizes the error between the estimated pdf and the true pdf. The most common metric to represent this error is the mean integrated squared error (MISE) which is expressed as [12]
$$\begin{array}{@{}rcl@{}} \text{MISE}(h)=\mathbb{E}\left[\int_{-\infty}^{+ \infty} \left[\,\widetilde{f}(x;h) - f(x) \right]^{2} \,dx\right]. \end{array} $$
The optimal smoothing parameter is selected so that it minimizes MISE with respect to h:
$$\begin{array}{@{}rcl@{}} h_{\text{opt}} = \operatorname*{arg\,min}_{h}(\text{MISE}(h)), \end{array} $$
Based on kernel estimator [11], the smoothing parameter is calculated as (see Appendix C)
$$\begin{array}{@{}rcl@{}} h_{\text{opt}}&=&\left(\frac{R(K)}{{\mu_{2}}^{2}(K)R\left(f^{\prime\prime}\right)}\right)^{1/5}.N^{-1/5}, \end{array} $$
where \(R(g)= \int _{}^{} g^{2}(u) \, du\), \(\mu _{k}(g)= \int u^{k} g(u) \, du\), and K(.) represents the kernel function. Until now, it is difficult to measure h opt since it depends on the unknown quantity R(f ′′). To solve this problem, several types of MISE-based methods have been suggested in literature. Hereafter, we detail the most popular ones.
Rule-of-thumb method
The idea of rule-of-thumb [13] is to replace the unknown probability density, f, in (18) by a standard normal distribution that has mean μ and variance σ 2, i.e., \(\mathcal {N}(\mu,\sigma ^{2})\). In this manner, we get
$$ \begin{aligned} R\left(\,f^{\prime\prime}\right)= (8 \sqrt{\pi}/3)^{1/5} \sigma. \end{aligned} $$
Consequently, a Gaussian kernel function \(K(x)=\frac {1}{\sqrt {2\pi }} e^{-x^{2}/2}\) leads to
$$ \begin{aligned} R(K) = \left(2 \sqrt{\pi}\right)^{-1/5}~~; ~~{\mu_{2}}^{2}(K)=1, \end{aligned} $$
It follows that the smoothing parameter is done as
$$ \begin{aligned} h_{\mathrm{opt,ROT}} =(4/3)^{1/5} \sigma N^{-1/5} = 1.06 \sigma N^{-1/5}. \end{aligned} $$
Cross-validation method
In cross-validation (CV) method, instead of using a reference probability density, the idea is to estimate the unknown quantity R(f ′′) in h opt formula. Furthermore, CV approach considers the integrated squared error (ISE) to select the optimal smoothing parameter. This error metric is expressed as [14]
$$ \begin{aligned} \text{ISE} &=\int_{-\infty}^{+ \infty} \left[\widetilde{f}_{X}(x;h) - f(x) \right]^{2} \,dx \\ &=\int_{-\infty}^{+ \infty} \widetilde{f}_{X}^{2}(x;h)\,dx- 2 \int_{-\infty}^{+ \infty} \widetilde{f}_{X}(x;h)f(x)\,dx \\ &\quad + \int_{-\infty}^{+ \infty} f^{2}(x)\,dx. \end{aligned} $$
The third term \(\int _{-\infty }^{+ \infty } f^{2}(x)\,dx\) does not depend on the sample or on the smoothing parameter. Moreover, the new function used to estimate h is called least squares cross-validation (LSCV)-based method [15] expressed as
$$ \begin{aligned} {}\text{LSCV}(h)=\int_{-\infty}^{+ \infty} \widetilde{f}_{X}^{2}(x;h)\,dx - 2 \int_{-\infty}^{+ \infty} \widetilde{f}_{X}(x;h)f(x)\,dx, \end{aligned} $$
An approximately unbiased estimator of (23) is given by [16]
$$ \begin{aligned} \text{LSCV}(h)= \int_{-\infty}^{+ \infty} {\widetilde{f}}_{X}^{2}(x,h)\,dx -\frac{2}{N} \sum_{i=1}^{N}~\widetilde{f}_{X,-i}\left(x_{i},h\right). \end{aligned} $$
where, \(\widetilde {f}_{-i}(x_{i})\), i=1,…,N, is the estimated density using all the original observations except for x i .
It is well known that LSCV(h) is an unbiased estimator of \(\text {MISE}(h)-\int _{-\infty }^{+ \infty } f^{2}(y)\,dy\). This is expressed as :
$$ {\fontsize{9.1}{6}\begin{aligned} \mathbb{E}\left(\text{LSCV}(h)\right) &= \!\mathbb{E}\left[\int_{-\infty}^{+ \infty} \! \left[\,\widetilde{f}(x;h) - f(x) \right]^{2} dx\right] -\! \int_{-\infty}^{+ \infty} \!\!f^{2}\!(x)\,dx \\ &= \text{MISE}(h)-\int_{-\infty}^{+ \infty} f^{2}(y)\,dy, \end{aligned}} $$
As developed in Appendix C, the \(\mathbb {E}(\text {LSCV}(h))\) estimator is re-written as [17]
$$ {\fontsize{9.1}{6}\begin{aligned} \mathbb{E}(\text{LSCV}(h))= \frac{1}{N h} R(K)+\frac{h^{4}}{4} \mathbf{\mu}_{2}^{2}(K) R\left(\,f^{\prime\prime}\right)-R(f) + O\!\left(N^{-1}\right)\!. \end{aligned}} $$
A new method called biased cross-validation (BCV) [18, 19] considers only the asymptotic MISE to estimate h:
$$ \begin{aligned} \text{AMISE}= \frac{R(K)}{N h} +\frac{h^{4}}{4} \mathbf{\mu}_{2}^{2}(K) R\left(\,f^{\prime\prime}\right). \end{aligned} $$
Its idea is to replace the unknown quantity R(f ′′) by the estimator:
$$\begin{array}{@{}rcl@{}} \widetilde{R\left(\,f^{\prime\prime}\right)}&=& R\left(\,\widetilde{f}_{X}^{\prime\prime}\right)- \frac{1}{N.h^{5}}.R\left(K^{\prime\prime}\right) \\ &=& \frac{1}{N^{2}} \sum_{i \neq j}^{} \sum_{}^{} {K_{h}}^{\prime\prime} \ast {K_{h}}^{\prime\prime} \left(x_{i} - x_{j}\right). \end{array} $$
where \(\widetilde {f}_{X}^{\prime \prime }\) is the second derivative of the kernel density estimate and \(K_{h}(x)=\frac {1}{h} K\left (\frac {x}{h}\right)\). The operator ∗ indicates the convolution product.
By substituting (28) in (27), the BCV-based method is presented as
$$ \begin{aligned} {}\text{BCV}(h) &= \frac{1}{Nh} R(K)+ \frac{h^{4} \mathbf{\mu}_{2}^{2}(K)}{2N^{2}} \sum_{i \neq j}^{} \sum_{}^{} {K_{h}}^{\prime\prime} \!\ast {K_{h}}^{\prime\prime} (x_{i} \,-\, x_{j}). \end{aligned} $$
Finally, the smoothing parameter based on cross-validation method is done by
$$ \begin{aligned} h_{\mathrm{opt,CV}}= \operatorname*{arg\,min}_{h}(\text{BCV}(h)). \end{aligned} $$
Bootstrap method
Bootstrap procedures for selecting the smoothing parameter have been studied in previous work [20–22]. The idea is to estimate the MISE using the bootstrap and then minimize it with respect to h. Let \(\widetilde {f}_{X}(x;g)\) be the estimate of f(x) obtained from {x 1,…,x N }, with a pilot smoothing parameter g.
The straight forward approach to use the bootstrap method would be to resample \(\left \{x_{1}^{*},\ldots, x_{N}^{*}\right \}\) from \(\widetilde {f}_{X}(x;g)\) and then construct bootstrap estimates \(\widetilde {f}_{X}^{*}(x;h)\) [23]. The bootstrap estimator of the MISE is defined as
$$\begin{array}{@{}rcl@{}} {}\text{MISE}^{*}(h)&=& \mathbb{E}\left[\int_{-\infty}^{+ \infty} \left[\,\widetilde{f}_{X}^{*}(x;h) - \widetilde{f}_{X}(x;g) \right]^{2} \,dx\right], \end{array} $$
According to (13), \(\widetilde {f}_{X}^{*}(x;h)\) can be replaced by \(\frac {1}{Nh} \sum _{i=1}^{N} \frac {1}{2\pi } e^{-\left (\frac {x-x_{i}^{*}}{2\sqrt {\pi }h}\right)^{2}}\). Then, Taylor expansion of \(\widetilde {f}_{X}(x;g)\), under the assumption that h→0 as N→∞, leads to an asymptotic approximation to MISE∗ [24] as:
$$ {\fontsize{9.1}{6}\begin{aligned} \text{MISE}^{*}(h) = \frac{1}{2Nh\sqrt{2 \pi}} \left[2^{1/2}+1-\frac{4}{3^{1/2}}+ (N-1)h(2\pi)^{1/2} \right.\\ \left\{ 4\int_{}^{} h^{4}~\widetilde{f}_{X}^{(4)}(x;g)\,\widetilde{f}_{X}(x;g)\,dx- \frac{9}{2}\int h^{4}~\widetilde{f}_{X}^{(4)}(x;g)\,\widetilde{f}_{Y}(x;g)\,dy \right.\\ \left.\left.+\int_{}^{} h^{4}~\widetilde{f}_{X}^{(4)}(x;g)\,\widetilde{f}_{Y}(x;g)\,dx\right\}\right] +O\left(h^{6}\right), \end{aligned}} $$
After calculus simplification, this approximation can be written as [24]:
$$ {\fontsize{9.1}{6}\begin{aligned} \text{MISE}^{*}(h)=\frac{1.074}{2Nh\sqrt{\pi}} + \frac{h^{4}}{4}\int_{}^{} \widetilde{f}_{X}^{(4)}(x;g)\,\widetilde{f}_{X}(x;g)\,dx +O\left(h^{6}\right), \end{aligned}} $$
By using some standard properties of a density function, we re-express MISE∗(h) as
$$ \begin{aligned} {}\text{MISE}^{*}(h)=\frac{1.074}{2Nh\sqrt{\pi}} + \frac{h^{4}}{4}\int_{}^{} \left(\,\widetilde{f}_{X}^{\prime\prime}(x;g)\right)^{2}\,dx +O(h^{6}). \end{aligned} $$
The optimal smoothing parameter h opt,boot is obtained by minimizing MISE∗(h) with respect to h:
$$ \begin{aligned} h_{\mathrm{opt,boot}}=\operatorname*{arg\,min}_{h}\left(\text{MISE}^{*}(h)\right), \end{aligned} $$
The smoothing parameter based on bootstrap method h opt,boot obtained from (35) is given as (see proof in Appendix D):
$$ \begin{aligned} h_{\mathrm{opt,boot}}=\left(\frac{1.074}{2\sqrt{\pi}\int_{}^{}\left(\,\widetilde{f}_{X}^{\prime\prime}(x;g)\right)^{2}\,dx}\right)^{1/5}.N^{-1/5}. \end{aligned} $$
As it can be seen from this equation, the optimal h opt,boot value depends on the second derivative of the estimate pdf \(\int _{}^{}(\,\widetilde {f}_{X}^{\prime \prime }(x;g))^{2}\,dx\) where the pilot smoothing parameter g is selected using least squares the cross-validation method [25]. This parameter is chosen so as to minimize
$$ \begin{aligned} \text{LSCV}(g)=\int_{}^{}\left(\,\widetilde{f}_{X}(x;g)\right)^{2}\,dx -\frac{2}{N} \sum_{i=1}^{N}~f_{X,-i}(x_{i},g). \end{aligned} $$
where f N,−i (x i ,g) is the density estimate based on all of data expect x i . To justify the choice of the bootstrap method for selecting the optimal smoothing parameter, we have presented the integrated squared error as a function of the smoothing parameter h. Figure 2 shows the obtained results with bootstrap, cross-validation, and rule-of-thumb methods. It is seen that the bootstrap method outperforms the other methods in terms of the integrated squared error between the true probability density and the estimated density.
Standard error criterion comparison
Simulations and results
In order to verify the obtained semi-analytical expression of error probability, computer simulations were done using the system model presented in Fig. 1. We first validated the probability density estimation using Fourier inversion. We, then, used it to predict the semi-analytical bit error probability of several transmission scenarios. This probability is compared with the BER evaluated using Monte Carlo simulation which considers a 95 % confidence interval for all scenarios.
To measure the semi-analytical probability density of the received sample, we have considered a digital modulation scheme which uses bit-phase-shift keying (BPSK) for bit-to-symbol conversion. The symbol stream is then sent through an AWGN channel. At the matched filter output, the receiver observes N=10,000 samples and estimates the probability density using Fourier inversion method. The obtained probability density is compared to the theoretical density as shown in Fig. 3. It is seen that the density curve corresponding to Fourier inversion method is close to theoretical density curve. Moreover, we have evaluated the semi-analytical bit error probability (BEP) using the expression given in (14) in terms of signal to noise ratio (SNR). The simulation results obtained from the semi-analytical method are compared with those from Monte Carlo simulation, as well as from the analytical method. Besides, the analytical BEP is expressed as
$$ \begin{aligned} {P}_{\text{th-bpsk}}=0.5 erfc\left(\sqrt{\text{SNR}}\right). \end{aligned} $$
pdf result from Fourier inversion
where \(erfc(x)=\frac {2}{\sqrt {\pi }} \int _{x}^{+\infty } {e}^{{-x}^{2}}\,dx\).
The simulation results are presented in Fig. 4. It is shown that the proposed Fourier inversion-based semi-analytical method offers the same performance as the other methods. It is also observed that a significant gain in terms of computing time is obtained (see Table 1). In addition, to reach the bit error probability of 10−4, Monte Carlo simulation requires a number of 1,000,000 samples while Fourier inversion uses only 10,000 observed samples.
BEP performance prediction for BPSK modulation over AWGN channel
Table 1 Computing time comparison. This table summarizes an experiment comparing the time (in seconds) to obtain bit error probability
Furthermore, we have applied the proposed semi-analytical approach to a transmission scenario that employs SC-FDMA technique [26] to transmit the symbol stream at the output of the BPSK modulation scheme. The number of subcarrier is taken to be equal to 512. Figure 5 shows the results of the semi-analytical bit error probability in terms of SNR. From the result, it is observed that the proposed semi-analytical approach is accurate compared to the Monte Carlo method with a significant gain in terms of computing time (see Table 1).
BEP performance comparison for SC-FDMA system over AWGN channel
After that, the semi-analytical performance prediction (SPP) has been extended to a digital communication system which performs the digital modulation using four-state pulse amplitude modulation (4-PAM). The simulations have been carried assuming a transmission through an AWGN channel and with a number of the observed samples equal to 10,000. The measured semi-analytical bit error probability is depicted in Fig. 6. It has been compared to that estimated by Monte Carlo simulation and given analytically:
$$ \begin{aligned} P_{\mathrm{th-pam}}=0.75\,erfc\left(\sqrt{0.2 \,\text{SNR}}\right). \end{aligned} $$
BEP performance prediction for 4-PAM modulation over AWGN channel
We notice that the Fourier inversion approach provides the same performance as the Monte Carlo simulation and the analytical method. Besides, it has been proven that the computing time is significantly reduced with the Fourier inversion approach (see Table 1). Indeed, to reach a bit error probability of 10−3, Monte Carlo simulation requires a number of samples equal to 100,000 while the proposed method uses only N=10,000 samples. In addition, the same performance in terms of bit error probability has been obtained when quadrature phase-shift keying (QPSK) modulation is considered. The simulation results are presented in Fig. 7.
BEP performance prediction for QPSK modulation over AWGN channel
In another transmission scenario, we have considered that the BPSK symbol stream is sent through a Rayleigh channel generated using two independent Gaussian random variables each with mean zero and variance 0.5. Also, we have assumed that communication is done with the receiver diversity. The number of receiver antennas equals 2. To recover the transmitted information symbols, the outputs of the receiver antennas are combined using maximum ratio combining (MRC). We have evaluated the semi-analytical bit error probability at the output of the MRC combiner. Figure 8 presents the BEP results for N=20,000 observed samples. As for all scenarios, Fourier inversion curves are very close to Monte Carlo simulation curves and analytical method curves where its analytical expression is done by
$$ \begin{aligned} P_{\mathrm{bpsk-mrc}}=p^{2} \,\left(1+2.(1-p)\right). \end{aligned} $$
BEP performance for BPSK with MRC over Rayleigh channel
where \(p=\frac {1}{2} - \frac {1}{2}.\left (1+\frac {1}{\text {SNR}}\right)^{-\frac {1}{2}} \). Also, it is observed that a reduced computing time is obtained. This presents a major strength of the proposed approach and very promising for many practical systems.
In this paper, we have considered a new semi-analytical method for estimating the error probability for any digital communication system. We have shown that the problem of error probability estimation is equivalent to estimate the conditional probability density function (pdf) of the observed soft samples at the receiver output. The proposed method is based on Fourier inversion approach for predicting the pdf. It has been shown that the accuracy of this approach is very sensitive to the optimum smoothing parameter selection. Furthermore, we have applied the bootstrap method for selecting the optimal smoothing parameter which makes the proposed semi-analytical method more accurate. The simulation results have concluded that with either the Monte Carlo (MC) simulation technique or the new proposed semi-analytical approach, we have the same performance. Moreover, the use of the bootstrap method can decrease the squared error between the true pdf and the estimated one.
Appendix A Proof of (11)
Using the definition of Fourier transform inversion (7), the probability density function is done as:
$$\begin{array}{@{}rcl@{}} \widetilde{f}(x;h) &=& \frac{1}{2\pi}\int_{-\infty}^{+\infty} e^{-jtx} \widetilde{\varphi}_{X}(t)\,dt, \end{array} $$
((A.1))
where \(\widetilde {\varphi }_{X}\) is the characteristic function defined in (10), so we get :
$$\begin{array}{@{}rcl@{}} \widetilde{f}(x;h) &=& \frac{1}{2\pi}\int_{-\infty}^{+\infty} e^{-jtx} \frac{1}{N} \sum_{i=1}^{N} e^{{jtx}_{i}} \psi_{h}(t)\,dt \\ &=& \frac{1}{N} \sum\limits_{i=1}^{N} \frac{1}{2\pi} \int_{-\infty}^{+\infty} e^{-jt(x-x_{i})}\psi_{h}(t)\,dt \\ &=& \frac{1}{N} \sum\limits_{i=1}^{N} \frac{1}{2\pi} \int_{-\infty}^{+\infty} e^{-jt(x-x_{i})}\psi(ht)\,dt \\ &=& \frac{1}{Nh} \sum\limits_{i=1}^{N} \frac{1}{2\pi} \int_{-\infty}^{+\infty} e^{-jt\left(\frac{x-x_{i}}{h}\right)}\psi(t)\,dt, \end{array} $$
Let us define
$$\begin{array}{@{}rcl@{}} v(x) &=& \frac{1}{2\pi} \int_{\infty}^{+\infty} e^{-jtx}\psi(t)\,dt. \end{array} $$
It follows that the expression of the semi-analytical probability density function is expressed as
$$\begin{array}{@{}rcl@{}} \widetilde{f}(x;h) &=& \frac{1}{Nh} \sum\limits_{i=1}^{N} v\left(\frac{x-x_{i}}{h}\right). \end{array} $$
Appendix B Proof of (14)
Let us recall that the semi-analytical bit error probability is given by
$$ {\fontsize{9}{6}\begin{aligned} P_{e} &=& P_{1}. \int_{-\infty}^{0} f_{X}^{(1)}(x | b_{i}=1)\,dx + P_{0}. \int_{0}^{+\infty} f_{X}^{(0)}(x | b_{i}=0)\,dx, \end{aligned}} $$
((B.1))
where \(\widetilde {f}_{1}(x)\) and \(\widetilde {f}_{0}(x)\) are the estimated probability density function of the observed samples (x i )1 and (x i )0, respectively, which corresponds to transmitted information bits b i =1 and b i =0, respectively. By using the obtained semi-analytical probability density function in (13), we can define
$$\begin{array}{@{}rcl@{}} \widetilde{f}_{X}^{(1)}(x;h) = \frac{1}{N_{1}h_{1}} \sum\limits_{i=1}^{N_{1}} \frac{1}{2\pi} e^{-j\left(\frac{x-(x_{i})_{1}}{2 \sqrt{\pi} h_{1}}\right)}, \end{array} $$
where h 1 (respectively, h 0) is the smoothing parameter which depends on the number of observed samples, i.e., N 1 (respectively, N 0). By substituting the estimated pdf \(\widetilde {f}_{X}^{(1)}\) and \(\widetilde {f}_{X}^{(0)}\) in (B.1), we get
$$ \begin{aligned} P_{e} &= P_{1}. \int_{-\infty}^{0} \frac{1}{N_{1} h_{1}} \sum\limits_{i=1}^{N_{1}} \frac{1}{2\pi} e^{-\left(\frac{x-({x_{i}})_{1}}{2\sqrt{\pi}h_{1}}\right)^{2}} \,dx \\ &\qquad + P_{0}. \int_{0}^{+\infty} \frac{1}{N_{0} h_{0}} \sum\limits_{i=1}^{N_{0}} \frac{1}{2\pi} e^{-\left(\frac{x-({x_{i}})_{0}}{2\sqrt{\pi}h_{0}}\right)^{2}} \,dx \\ &= \frac{P_{1}}{N_{1} h_{1}} \sum\limits_{i=1}^{N_{1}} \frac{1}{2\pi} \int_{-\infty}^{0} e^{-\left(\frac{x-({x_{i}})_{1}}{\sqrt{2\pi}h_{1}}\right)^{2}/2} \,dx \\ &\qquad + \frac{P_{0}}{N_{0} h_{0}} \sum\limits_{i=1}^{N_{0}} \frac{1}{2\pi} \int_{0}^{+\infty} e^{-\left(\frac{x-({x_{i}})_{0}}{\sqrt{2\pi}h_{0}}\right)^{2}/2} \,dx, \end{aligned} $$
Using the following change of variable \(t_{1}= \frac {x-({x_{i}})_{1}}{\sqrt {2\pi }h_{1}} \) and \(t_{0}= \frac {x-({x_{i}})_{0}}{\sqrt {2\pi }h_{0}} \), we have
$$ \begin{aligned} P_{e} &= \frac{P_{1}}{N_{1}} \sum\limits_{i=1}^{N_{1}} \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\frac{-({x_{i}})_{1}}{\sqrt{2\pi}h_{1}}} e^{-{t_{1}}^{2}/2} \,{dt}_{1} \\ &\qquad+ \frac{P_{0}}{N_{0}} \sum\limits_{i=1}^{N_{0}} \frac{1}{\sqrt{2\pi}} \int_{\frac{-({x_{i}})_{0}}{\sqrt{2\pi}h_{0}}}^{+\infty} e^{-{t_{0}}^{2}/2} \,{dt}_{0} \\ &= \frac{P_{1}}{N_{1}} \sum\limits_{i=1}^{N_{1}} Q\left(\frac{(x_{i})_{1}}{\sqrt{2\pi}h_{1}}\right)\\ &\qquad+ \frac{P_{0}}{N_{0}} \sum\limits_{i=1}^{N_{0}} Q\left(\frac{-(x_{i})_{0}}{\sqrt{2\pi}h_{1}}\right). \end{aligned} $$
Appendix C Proof of (18)
We can prove the expression of the smoothing parameter using MISE method.
$$\begin{array}{@{}rcl@{}} \text{MISE} &=& \mathbb{E} \left[ \int_{}^{} \left\{\,\widetilde{f}(x;h) - f(x) \right\}^{2}\,dx\right], \end{array} $$
((C.1))
By using the theory of "Konig Huyghens", we have
$$ {\fontsize{9}{6}\begin{aligned} \mathbb{E} \left\{\,\widetilde{f}(x;h) - f(x) \right\}^{2} = \text{var}\left(\,\widetilde{f}(x;h)\right) +\left(\mathbb{E}\left(\,\widetilde{f}(x;h)\right) - f(x) \right)^{2}, \end{aligned}} $$
Let us use kernel estimator to estimate the probability density function \(\widetilde {f}\). We define the kernel function K(.) as any function satisfies \(\int _{}^{} K(x) \, dx =1 \) and:
$$\begin{array}{@{}rcl@{}} \widetilde{f}(x;h) &=& \frac{1}{N h} \sum\limits_{i=1}^{N} K\left(\frac{x-x_{i}}{h}\right). \end{array} $$
Estimation bias: Let us consider that the expectation of kernel function can be written as integrals of the convolution of the kernel density and the true density function:
$$\begin{array}{@{}rcl@{}} \mathbb{E}\left(\,\widetilde{f}(x;h)\right) &=& \frac{1}{N} \sum\limits_{i=1}^{N} \mathbb{E} \left(\frac{1}{h} K\left(\frac{x-x_{i}}{h}\right)\right) \\ &=& \int_{}^{} K\left(\frac{z-x}{h}\right) f(z) \, dz, \end{array} $$
By using \(u=\frac {z-x}{h}\), we have
$$\begin{array}{@{}rcl@{}} \mathbb{E}\left(\,\widetilde{f}(x;h)\right) &=& \int_{}^{} K(u) f(x+hu)\,du, \end{array} $$
So, we use a Taylor expansion of f(x+h u) in the argument hu and with h→0. For a ν ′ th-order kernel, we take the expansion out to the ν ′ th-term to solve this integral:
$$\begin{array}{@{}rcl@{}} f(x+hu) &=& f(x) + f^{(1)}(x)hu+\frac{1}{2}f^{(2)}(x)h^{2}u^{2} \\ & &+\frac{1}{3!}\,f^{(3)}(x)h^{3}u^{3} +\ldots \\ & & + \frac{1}{\nu!}\,f^{(\nu)}(x)h^{\nu} u^{\nu}+O(h^{\nu}), \end{array} $$
where \(\mu _{\nu }(K)=\int _{}^{} u^{\nu } K(u)\,du\).
So, integrating term by term and using that \(\int _{}^{} K(x)dx =1 \), to get
$$ {\fontsize{8.6}{6}\begin{aligned} \int_{}^{} \!K(u)\,f(x+hu)\,du &= f(x) + f^{(1)}(x)h \mu_{1}(K)\,+\,\frac{1}{2!}\,f^{(2)}(x)h^{2} \mu_{2}(K) \\ & \quad+\frac{1}{3!}\,f^{(3)}(x)h^{3} \mu_{3}(K) \\ & \quad+\ldots+\frac{1}{\nu!}\,f^{(\nu)}(x)h^{\nu} \mu_{\nu}(K)+O(h^{\nu}) \\ &= f(x)+ \frac{1}{\nu!}\,f^{(\nu)}(x)h^{\nu} \mu_{\nu}(K)+O(h^{\nu}), \end{aligned}} $$
$$\begin{array}{@{}rcl@{}} {}\mathbb{E}\left(\,\widetilde{f}(x;h)\right) &=& \sum\limits_{i=1}^{n} \mathbb{E}\left(\frac{1}{h} K\left(\frac{x_{i}-x}{h}\right)\right) \\ &=& f(x)+ \frac{1}{\nu!}\,f^{(\nu)}(x)h^{\nu} \mu_{\nu}(K)+O(h^{\nu}). \end{array} $$
The bias of \(\widetilde {f}_{h}(x)\) is then
$$\begin{array}{@{}rcl@{}} {}\text{Bias}\left(\,\widetilde{f}(x;h)\right) &=& \mathbb{E}\left(\,\widetilde{f}(x;h)\right) -f(x) \\ &=& \frac{1}{\nu!}\,f^{(\nu)}(x)h^{\nu} \mu_{\nu}(K)+O(h^{\nu}), \end{array} $$
To simplify the calculus, we take
$$\begin{array}{@{}rcl@{}} {}\text{Bias}\left(\,\widetilde{f}(x;h)\right) &=& \frac{1}{2}\,f^{(2)}(x)h^{2} \mu_{2}(v)+O(h^{2}). \end{array} $$
((C.10))
Estimation variance: Let us compute the variance of \(\widetilde {f}(x;h)\) for a density estimator:
$$ {\fontsize{7.9}{6}\begin{aligned} var\left(\,\widetilde{f}(x;h)\right) &= \mathbb{E}\left[\,\widetilde{f}(x;h) - \mathbb{E}\left(\,\widetilde{f}(x;h)\right) \right]^{2} \\ &= \mathbb{E}\left[\left(\,\widetilde{f}(x;h)\right)^{2} -2\,\widetilde{f}(x;h)\mathbb{E}\left(\,\widetilde{f}(x;h)\right)+ \left(\mathbb{E}\,\widetilde{f}(x;h)\right)^{2} \right] \\ &= \mathbb{E}\left(\!\left(\,\widetilde{f}(x;h)\right)^{2}\right) \,-\, 2\mathbb{E}\left(\,\widetilde{f}(x;h)\right)\!\mathbb{E}\left(\,\widetilde{f}(x;h)\right)\,+\, \left(\mathbb{E}\,\widetilde{f}(x;h)\!\right)^{2} \\ &= \mathbb{E}\left(\left(\,\widetilde{f}(x;h)\right)^{2}\right) -2\left(\mathbb{E}\left(\,\widetilde{f}(x;h)\right)\right)^{2} + \left(\mathbb{E}\,\widetilde{f}(x;h)\right)^{2} \\ &= \mathbb{E}\left(\left(\,\widetilde{f}(x;h)\right)^{2}\right) -\left(\mathbb{E}\left(\,\widetilde{f}(x;h)\right)\right)^{2}, \end{aligned}} $$
The kernel estimator is a linear estimate, so
$$\begin{array}{@{}rcl@{}} {}\text{var}\left(\,\widetilde{f}(x;h)\right) &=& \frac{1}{Nh^{2}} \mathbb{E}\left(K\Big(\frac{x_{i}-x}{h}\Big)\right)^{2} \\ &&-\frac{1}{N}\left(\frac{1}{h}\mathbb{E}\left(K\left(\frac{x_{i}-x}{h}\right)\right)\right)^{2}. \end{array} $$
As developed in the bias, we have \(\frac {1}{h} \mathbb {E}\left (K\left (\frac {x_{i}-x}{h}\right)\right) = f(x) + O(1)\) So, \(\frac {1}{N} \left (\frac {1}{h} \mathbb {E}\left (K\left (\frac {x_{i}-x}{h}\right)\right)\right)^{2}\) is \(O\left (\frac {1}{N}\right) \) For the first term of the variance, we can write the expectation of kernel function as integrals of the convolution of the kernel density and the true density and then use a first-order Taylor expansion, to get
$$\begin{array}{@{}rcl@{}} \frac{1}{h}\mathbb{E}\left(K\left(\frac{x_{i}-x}{h}\right)\right)^{2} &=& \frac{1}{h} \int_{}^{} \left\{K\left(\frac{z-x}{h}\right)\right\}^{2} f(z)\,dz \\ &=& \int_{}^{} K(u)^{2} f(x+hu)\,du \\ &=& f(x)\int_{}^{} K(u)^{2} \,du + O(h) \\ &=& f(x)R(K) + O(h), \end{array} $$
where \(R(K) =\int _{}^{} K(u)^{2} \,du \). Together, the estimation variance is written as
$$\begin{array}{@{}rcl@{}} \text{var}\left(\,\widetilde{f}(x;h)\right)&=& \frac{f(x)R(K)}{Nh} + O\left(\frac{1}{N}\right). \end{array} $$
Mean-squared error As defined, the mean squared error (MSE) is done as
$$\begin{array}{*{20}l} \text{MSE} &= \mathbb{E} \left\{\,\widetilde{f}(x;h) - f(x) \right\}^{2} \\ &= \text{var}\left(\,\widetilde{f}(x;h)\right) +\left(\text{Bias}\left(\,\widetilde{f}(x;h)\right)\right)^{2} \\ &= \frac{f(x)R(K)}{Nh} + \frac{1}{4}\left(\,f^{(2)}(x)\right)^{2}h^{4}{\mu_{2}}^{2}(K). \end{array} $$
By integrating the MSE, the mean integrated squared error (MISE) is done as
$$\begin{array}{*{20}l} \text{MISE} &= \mathbb{E} \int_{}^{} \left\{\,\widetilde{f}(x;h) - f(x) \right\}^{2}\,dx \\ &= \int_{}^{} {\text{Bias}\left(\,\widetilde{f}(x;h)\right)}^{2}\,dx + \int_{}^{} \text{var}\left(\,\widetilde{f}(x;h)\right)\,dx, \end{array} $$
Under an integrability assumption on f, we have
$$\begin{array}{@{}rcl@{}} \text{MISE} &=& \frac{R(K)}{Nh} + \frac{1}{4}h^{4}{\mu_{2}^{2}}(K)R\left(\,f^{\prime\prime}\right). \end{array} $$
where \(R(\,f^{\prime \prime })=\int _{}^{} {f^{\prime \prime }(u)}^{2}\,du\)
The expression (C.17) is the measure that we use to quantify the performance of the estimator. We can find the optimal smoothing parameter by minimizing the expression of (C.17) with respect to h. The first derivative is given by
$$ \begin{aligned} \frac{d(\text{MISE}(h))}{dh}=-\frac{R(K)}{2Nh^{2}} + h^{3}{\mu_{2}^{2}}(K)R\left(\,f^{\prime\prime}\right), \end{aligned} $$
Putting this equal to zero, we will have the optimal smoothing parameter:
$$\begin{array}{@{}rcl@{}} h_{\text{opt}}&=&\left(\frac{R(K)}{{\mu_{2}}^{2}(K)R(\,f^{\prime\prime})}\right)^{1/5}.N^{-1/5}. \end{array} $$
Appendix D Proof of (36)
In this Appendix, we provide further details related to the asymptotic expressions for the smoothing parameter using the bootstrap method. Here the normal kernel is used.
$$\begin{array}{@{}rcl@{}} \text{MISE}^{*}(h)&=& \mathbb{E} \int_{}^{} \left\{\, \widetilde{f^{*}}_{X}(x;h) - \widetilde{f}_{X}(x;g) \right\}^{2}\,dx \\ &=& \int_{}^{} \text{Bias}^{*}\left\{\,\widetilde{f^{*}}_{X}(x;h)\right\}^{2} \,dx \\ &&- \int_{}^{} Var^{*}\left\{\,\widetilde{f^{*}}_{X}(x;h)\right\} \,dx, \end{array} $$
((D.1))
where \(\mathbb {E}^{*}\), Bias∗, and Var∗ all involve expectations conditionally upon \(x^{*}_{1}, x^{*}_{2},\ldots, x^{*}_{N}\) and all x ∗ are sampled from the smoothed distribution \(\widetilde {f}_{X}(x;h)\). Making a substitution followed by a Taylor series expansion, this assumes that h→0 as N→∞, gives an asymptotic approximation:
$$ {\fontsize{9.2}{6}\begin{aligned} \text{MISE}^{*}(h)&=\frac{1}{2Nh\sqrt{2 \pi}} \left[2^{1/2}+1-\frac{4}{3^{1/2}}+ (N-1)h(2\pi)^{1/2}\right. \\ &\qquad\qquad\quad\left\{ 4\int_{}^{} h^{4} \,\widetilde{f}_{X}^{(4)}(x;g)\,\widetilde{f}_{X}(x;g)\,dx\right.\\ &\qquad\qquad- \frac{9}{2}\int_{}^{} h^{4} \,\widetilde{f}_{X}^{(4)}(x;g)\,\widetilde{f}_{X}(x;g)\,dx\\ &\qquad\quad~~~\left. \left. + \int_{}^{} h^{4} \,\widetilde{f}_{X}^{(4)}(x;g)\,\widetilde{f}_{X}(x;g)\,dx\right\}\right] +O(h^{6}), \end{aligned}} $$
To simplify this, the approximation can be written as
Using the condition that any probability density function satisfies
$$ \begin{aligned} \left[\,\widetilde{f}_{X}^{\prime\prime\prime}(x;g)\,\widetilde{f}_{X}(x;g)\right]_{-\infty}^{+\infty}=\left[\,\widetilde{f}_{X}^{\prime\prime}(x;g)\,\widetilde{f}_{X}^{'}(x;g)\right]_{-\infty}^{+\infty}=0. \end{aligned} $$
The asymptotic expression for bootstrap estimator of MISE is
$$ \begin{aligned} {}\text{MISE}^{*}(h)=\frac{1.074}{2Nh\sqrt{\pi}} + \frac{h^{4}}{4}\int_{}^{} \left(\,\widetilde{f}_{X}^{\prime\prime}(x;g)\right)^{2}\,dx +O\left(h^{6}\right), \end{aligned} $$
The optimal smoothing parameter is selected so that minimizing the expression of (D.5) with respect to h. The first derivative is given by
$$ {\fontsize{9.2}{6}\begin{aligned} \frac{d(\text{MISE}^{*}(h))}{dh}=-\frac{1.074}{2Nh^{2}\sqrt{\pi}} + h^{3} \int_{}^{} \left(\,\widetilde{f}_{X}^{\prime\prime}(x;g)\right)^{2}dx +O(h^{6}). \end{aligned}} $$
Putting this equal to zero, we will have the optimal smoothing parameter
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Department of Communication Networks, ENSIAS, Mohammed V University in Rabat, BP 713, Rabat, Morocco
Fatima Ezzahra Naamane
& Mostafa Belkasmi
Department of Communications Systems, INPT, Rabat, Morocco
Mohamed Et-tolba
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Correspondence to Fatima Ezzahra Naamane.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Naamane, F.E., Et-tolba, M. & Belkasmi, M. A semi-analytical performance prediction of a digital communication system using Fourier transform inversion. J Wireless Com Network 2015, 236 (2015) doi:10.1186/s13638-015-0467-2
Semi-analytical approach
Fourier transform inversion
Bit error probability
Smoothing parameter
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A content-adaptive unstructured grid based integral equation method with the TV regularization for SPECT reconstruction
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February 2020, 14(1): 1-26. doi: 10.3934/ipi.2019061
Artifacts in the inversion of the broken ray transform in the plane
Yang Zhang ,
Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA
* Corresponding author: Yang Zhang
Received July 2018 Revised June 2019 Published November 2019
Fund Project: The first author is supported by NSF grant DMS-1600327.
Figure(14)
We study the integral transform over a general family of broken rays in $ \mathbb{R}^2 $. One example of the broken rays is the family of rays reflected from a curved boundary once. There is a natural notion of conjugate points for broken rays. If there are conjugate points, we show that the singularities conormal to the broken rays cannot be recovered from local data and therefore artifacts arise in the reconstruction. As for global data, more singularities might be recoverable. We apply these conclusions to two examples, the V-line transform and the parallel ray transform. In each example, a detailed discussion of the local and global recovery of singularities is given and we perform numerical experiments to illustrate the results.
Keywords: Broken ray transform, artifacts, recovery of singularities.
Mathematics Subject Classification: Primary: 35R30, 44A12; Secondary: 65R32.
Citation: Yang Zhang. Artifacts in the inversion of the broken ray transform in the plane. Inverse Problems & Imaging, 2020, 14 (1) : 1-26. doi: 10.3934/ipi.2019061
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Figure 1. Left: a general broken ray, where $ l_1 $ and $ l_2 $ are related by a diffeomorphism. Right: a broken ray in the reflection case
Figure 2. The small neighborhood $ U_k $ and $ (x_k, \xi^k) $, for $ k = 1, 2 $
Figure 3. A sketch of a broken ray reflected on a smooth boundary and the notation
Figure 4. Two broken rays intersect when $ \alpha_2 $ increases as $ \alpha_1 $ increases
Figure 5. In (a) and (b), the bold part is the intersection region where the incoming rays hit there and reflect with conjugate points
Figure 6. Artifacts and caustics. Form left to right: $ f $, $ {{B}}^*\Lambda {{B}} f $, and caustics caused by reflected light
Figure 7. Local reconstruction by Landweber iteration
Figure 8. Inside a circular mirror, a sequence of broken rays and conjugate points on them
Figure 9. Reconstruction of $ f_1 $ and $ f_2 $ from global data, where $ e = \frac{\|f - f^{(100)}\|_2}{\|f\|_2} $ is the relative error
Figure 10. Reconstruction from global data for Modified Shepp-Logan phantom $ f_3 $, where $ e = \frac{\|f - f^{(100)}\|_2}{\|f\|_2} $ is the relative error
Figure 11. The error plot for the reconstruction of $ f_1, f_2, f_3 $ in order. The first two has the same range of color bar
Figure 12. Reconstruction of two coherent states. Left to right: true $ f $, the envelopes (caused by trajectories that carry singularities and are reflected only once), $ f^{(100)} $ (where $ e = \frac{\|f - f^{(100)}\|_2}{\|f\|_2} $), the error
Figure 13. Another case of radial singularities. Left to right: true $ f $, reconstruction $ f^{(100)} $, error for $ f $ with radial singularities after 100 iterations. The relative error $ e $ is defined as before
Figure 14. Left to right: true $ f $, backprojection $ f^{(1)} $, $ f^{(100)} $
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Nicholas Hoell, Guillaume Bal. Ray transforms on a conformal class of curves. Inverse Problems & Imaging, 2014, 8 (1) : 103-125. doi: 10.3934/ipi.2014.8.103
Melody Alsaker, Benjamin Bladow, Scott E. Campbell, Emma M. Kar. Automated filtering in the nonlinear Fourier domain of systematic artifacts in 2D electrical impedance tomography. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021066
Daniel Fusca. The Madelung transform as a momentum map. Journal of Geometric Mechanics, 2017, 9 (2) : 157-165. doi: 10.3934/jgm.2017006
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Understanding Hodgkin-Huxley's model and activation variables
This question is about the Hodgkin-Huxley model as introduced in Eugene M. Izhikevich, Dynamical Systems in Neuroscience, p.33 ff.
I'm having trouble to understand and interpret the differential equation for the activation variable $m$:
$$\dot{m} = (m_\infty(V) - m)/\tau(V)$$
which enters via
$$p = m^a h^b$$
into the equation
$$I = \bar{g} p (V − E)$$
for the net current $I$ generated by a large population of identical channels where $p$ is the average proportion of channels in the open state, $\bar{g}$ is the maximal conductance of the population, and $E$ is the reverse potential of the current.
$m$ is the probability of one of $a$ activation gates to be open. Interchangeably: $m^a$ is the proportion of open activation channels (assuming all of its activation gates must be open simultaneously).
The differential equation might give us $m$ as an explicit function of time but it – explicitly – involves $V$ which is – implicitly – another function of time, which in turn depends on the number of open gates. Things are horribly complicated!
On the other hand — since it's about voltage-gated channels and there are "immediate" voltage-sensitive regions in the channel protein which presumably don't have memory ("the single channel has no memory about the duration of its own state"), but possibly a time lag — I expected the being-open-probability of an activation gate to be a "pure" (possibly time-lagged) function of $V$.
My question comes in two disguises
(1) Given two explicit functions $m_\infty(V)$ and $\tau(V)$ like these:
together with $m(0)$ and $V(0)$, how could we ever arrive at an explicit solution for $m(t)$, assuming that $V(t)$ depends somehow on $m(t')$ for $t'\leq t$, but possibly also on some injected currents.
(2) How can an intuitive and sensible interpretation of the terms in
be given? What does the time constant $\tau(V)$ and its dependence on $V$ mean? In which respect and by which hypothesised mechanism is the gate "faster", when $\tau$ is smaller? How can the voltage-sensitive steady-state activation function $m_\infty(V)$ be intuitively explained other than by "giving the asymptotic value of $m$ when the potential $V$ is fixed (voltage-clamp)" or by some complicated description of experiments to determine it? What does $m_\infty(V) - m$ mean, i.e. "the deviation of the current activation from the steady-state activation"?
neuroscience neurophysiology action-potential
Hans-Peter Stricker
Hans-Peter StrickerHans-Peter Stricker
1) In practice, no one attempts to obtain explicit solutions for m(t), especially considering the presence of many other ion channel species, nor is there any need to. The standard of art is to use numerical simulations with a sampling interval dT that is sufficiently smaller than any relevant time constants.
2) I'm not sure what intuitive and sensible interpretation you are looking for; it may not make sense to come up with an intuitive explanation for each and every subset of terms. The use of time constants implies that the system is modeled as a linear time invariant system at a given voltage.
What does the time constant τ(V) and its dependence on V mean? In which respect and by which hypothesised mechanism is the gate "faster", when τ is smaller?
The time constant is a scalar that describes the rate at which the system approaches some equilibrium. You could think of the changes in channel state as chemical reactions of the form:
Open <=> Closed
which has some equilibrium and some rate constant - both the rate constant and equilibrium conditions can be a function of voltage, as depicted in the figure you provide. The physical meaning of a rate constant being voltage dependent would be that the activation energy for the transition between states is a function of voltage; when the activation energy is lower, the reaction can progress more quickly (in either direction).
How can the voltage-sensitive steady-state activation function m∞(V) be intuitively explained other than by "giving the asymptotic value of mm when the potential V is fixed (voltage-clamp)" or by some complicated description of experiments to determine it?
That's exactly what it means, and the descriptions are not complicated: the steady state value for a reversible reaction is the point at which the number of units moving from open to closed would be the same as the number moving from closed to open, so on average you see no more change: that's what steady-state means.
What does m∞(V)−m mean, i.e. "the deviation of the current activation from the steady-state activation"?
Yes, "the deviation of the current activation from the steady-state activation" is accurate. The equation is an equation for the change of m, and the change of m is proportional to the "distance from equilibrium" - this is a characteristic of any process that follows linear time invariant dynamics. Note that there are other ways to write this equation that involve a "forward rate", a "backward rate", and the concentration of units in each state (the Wikipedia Hodgkin-Huxley page you linked has these equations), but you can simplify these to the equation you show here because some proportion of the "forward rate" process is cancelled by the "backward rate": your equation is sufficient to describe the net change.
Bryan Krause♦Bryan Krause
$\begingroup$ Thanks for your answer (which I edited slightly)! Please let me announce a reply to your answer which will show that it is definitely worth the effort to obtain explicit solutions for $m(t)$ (and to play around with them). Among other things: to see how action potentials happen to be created (beyond a not so well-defined threshold), or how being open (of a channel) cannot follow fast changing voltages.Please stay tuned! $\endgroup$ – Hans-Peter Stricker Dec 13 '17 at 0:31
$\begingroup$ With regard to $\tau(V)$: It's all about the inertia of gates, let it be activation, inactivation, or deactivation gates. I never found time constants being called "inertia coefficients" or the like, but in the case of ion channels, this would hit the nail on the head. (Note, how inertia has to do with delay.) $\endgroup$ – Hans-Peter Stricker Dec 13 '17 at 0:59
$\begingroup$ If you are interested in a preview: send an email to [email protected]. $\endgroup$ – Hans-Peter Stricker Dec 13 '17 at 1:02
$\begingroup$ Thanks explicitly and especially for your link to Linear time-invariant theory, which was new to me. $\endgroup$ – Hans-Peter Stricker Dec 13 '17 at 1:09
$\begingroup$ "to see how action potentials happen to be created (beyond a not so well-defined threshold), or how being open (of a channel) cannot follow fast changing voltages" - all of these things can be answered with simulations. $\endgroup$ – Bryan Krause♦ Dec 13 '17 at 1:27
Not the answer you're looking for? Browse other questions tagged neuroscience neurophysiology action-potential or ask your own question.
How do firing patterns arise from the activity of many ion channels?
How does a change in the potential across a neuron's membrane get turned into a signal that is sent down the axon?
What exactly is the time constant of an ion channel gating?
How do you derive steady-state conditions from the open-state differential equation of ion channels?
Causes of the stochastic behaviour of voltage-gated ion channels
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Artificial Intelligence Meta
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How are these two versions of the Bellman optimality equation related?
I saw two versions of the optimality equation for $V_{*}(s)$ and $Q_{*}(s,a)$.
The first one is:
$$ V_{*}(s)=\max _{a} \sum_{s^{\prime}} P_{s s^{\prime}}^{a}\left(r(s, a)+\gamma V_{*}\left(s^{\prime}\right)\right) $$
$$ Q_{*}(s, a)=\sum_{s^{\prime}} P_{s s^{\prime}}^{a}\left(r(s, a)+\gamma \max _{a^{\prime}} Q_{*}\left(s^{\prime}, a^{\prime}\right)\right) $$
The second one is:
$$ V_{*}(s)=\max _{a \in \mathcal{A}}\left(R(s, a)+\gamma \sum_{s^{\prime} \in \mathcal{S}} P_{s s^{\prime}}^{a} V_{*}\left(s^{\prime}\right)\right) $$
and for $Q_*$
$$ Q_{*}(s, a)=R(s, a)+\gamma \sum_{s^{\prime} \in \mathcal{S}} P_{s s^{\prime}}^{a} \max _{a^{\prime} \in \mathcal{A}} Q_{*}\left(s^{\prime}, a^{\prime}\right) $$
If following distributive property to get from the first to the second expression. Why there is no summation term for the reward, for example, $$V_{*}(s) = \max_{a}(\sum_{s'}P^{a}_{ss'}r(s,a)+\gamma\sum_{s'}P^{a}_{ss'}V_{*}(s'))$$?
My guess is that $r(s,a)$ is the constant so it can be moved out of the summation, leaving $$r(s,a)\sum_{s'}P^{a}_{ss'} = r(s,a).$$
But is it always the case that $r(s,a)$ is independent of $s'$? I think the reward of moving from state $s$ to $s'$ may vary.
reinforcement-learning reward-functions bellman-equations dynamic-programming
nbro♦
KronicKronic
My guess is that $r(s,a)$ is the constant so it can be moved out of the summation, leaving $r(s,a)\sum_{s'}P^{a}_{ss'} = r(s,a)$
Yes, this is the case. More specifically:
$r(s,a)$ is the expected reward after taking action $a$ in state $s$.
Reward may depend on the state arrived in, $s'$, but that is ignored in the equations.
Reward may vary randomly, but by using the expected reward, this can be ignored.
The first equations you quote, which sum over $s'$ but use $r(s,a)$ inside that sum, are very misleading IMO, since the individual terms may not represent anything meaningful within the MDP. That is the term $r(s,a) + \gamma V^*(s')$ does not correspond to any part of the trajectory of the agent.
Although the sum is still mathematically sound, it is more normal to see a different term $r(s,a,s')$ (the expected reward similar to $r(s,a)$ but also conditional on $s'$) where the expected reward is used inside the sum of next states. The term $r(s,a,s') + \gamma V^*(s')$ does correspond to nodes on the trajectory of the agent. It is the expected future return from $s,a$ conditional on the state transitioning to $s'$.
but is it always the case that $r(s,a)$ is independent of $s'$. I think the reward of moving from state $s$ to $s'$ may vary.
Yes $r(s,a)$ is independent of $s'$. Although individual rewards may vary stochastically, and may depend on $s'$ too, the term is already the expected reward when taking the action $a$ in state $s$. So it already includes any effects of random state transition and random reward. For the Bellman equations to work as written, the expectation needs to be independent of the policy $\pi$ thus a property of the environment, and this is the case.
I think both sets of equations are a little bit awkward from using a combination of expected reward, yet summing up expectations over the state transition matrix. I prefer the notation used in second edition of Sutton & Barto's Reinforcement Learning: An Introduction:
$$v^*(s) = \text{max}_a \sum_{r,s'} p(r,s'|s,a)(r + \gamma v^*(s'))$$
Where $p(r, s'|s,a)$ is the conditional probability of observing reward $r$ and next state $s'$ given initial state $s$ and action $a$. The $p(r, s'|s,a)$ function replaces the combination of state transition matrices $P_{ss'}^a$ and the expected reward (either $r(s,a)$ or $r(s,a,s')$). Those objects can be derived from $p(r,s'|a,s)$ if you want, but personally I find the newer notation easier to follow.
Neil SlaterNeil Slater
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Apart from the state and state-action value functions, what are other examples of value functions used in RL?
Equivalence between expected parameter increments in "Off-Policy Temporal-Difference Learning with Function Approximation"
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#002 RNN – Architecture, Mapping, and Propagation
Strahinja Zivkovic Deep Learning 24.09.2020 | 0
Highlights: Recurrent Neural Networks (RNN) are sequence models that are a modern, more advanced alternative to traditional Neural Networks. Right from Speech Recognition to Natural Language Processing to Music Generation, RNNs have continued to play a transformative role in handling sequential datasets.
In this blog post, we will learn how to build and map a Recurrent Neural Network with some interesting examples. In addition, we will represent basic RNN models using the mathematical notations of Forward Propagation and Back Propagation techniques. So, shall we begin?
Tutorial Overview:
Brief Overview of RNN Fundamentals
Mapping Input and Output
Drawbacks of Using Standard Neural Networks
The architecture of a Recurrent Neural Network
Unidirectional Recurrent Neural Networks
Forward Propagation in RNN
Back Propagation in RNN
1. Brief Overview of RNN Fundamentals
Before we begin learning about how to build a Recurrent Neural Network, let us quickly recap the basic definition of RNN, as we learned in an earlier blog post.
Recurrent Neural Networks (RNN) are an alternative to traditional neural networks, especially used in cases where either the input data or the output data or both are sequential. By building sequence models such as RNNs, we can gain deeper insights into the problems of Speech Recognition, Machine Translation, Natural Language Processing, Music Generation, and many more.
Let us, now, move forward and understand how these sequence models or Recurrent Neural Networks are built, by training them how to map the input \(x \) and output \(y \).
2. Mapping Input and Output
The first task at hand for us is to try and see how much of the solution can standard neural networks provide us in our sequential dataset problems.
First, let us recall the example we used in the previous blog post.
A standard neural network for this example will look something like this:
In our example, there are a total of nine input words. We could use nine One-Hot Vectors for each of these nine words and, along with a few hidden layers, feed it into a standard neural network. The output of this standard neural network will be in the form of nine values, either 0 or 1, where 1 denotes the word which is part of a person's name.
However, there are a couple of problems that will arise eventually with such standard neural networks.
Not all datasets we come across will have the same lengths of Input $latex T_{x} $ and Output $latex T_{y} $. We can try capping the input and output lengths to a set maximum value, but it will still not be a good representative solution.
For a naïve neural network architecture, sharing features learned across different text positions is important. For example, if the neural network has learnt that the word "Luke" appearing in a certain position in a sequence is a part of a person's name, it should also learn automatically that the word "Luke" appearing elsewhere in the sequence, say at \(x^{\left \langle t \right \rangle} \), is also part of a person's name. This is quite similar to Convolutional Neural Networks, where we wanted the network to generalize results for one part of the image to other parts of the image as well. However, working with even a small 10,000-dimensional one-hot vector will make the input layer very large and we would end up having an enormous number of parameters.
Recurrent Neural Networks solve the above two key problems that arise if we start to think about using Standard Neural Networks to solve our sequential data problems.
We are now ready to see the framework of building a Recurrent Neural Network.
Understanding why standard neural networks fail to address the problems of interpreting sequential data is important to dig deeper into the blueprint of RNNs. The advantages of using Recurrent Neural Networks are significant but first, let us understand how these RNNs are built.
Let us look at the network below, where the length of the input and output sequence is the same. We will slowly move towards more scenarios and gain deeper knowledge about the benefits that RNN provides us.
The first input to the first layer of the above Neural Network is \(x^{\left \langle 1 \right \rangle} \). This input goes through a hidden layer and predicts an output whether the first word is part of a person's name or not. The role of the Recurrent Neural Network comes into play when the second word of the input sentence, \(x^{\left \langle 2 \right \rangle} \), is being read. Instead of just predicting \(\hat{y}^{\left \langle 2 \right \rangle} using only \)latex x^{\left \langle 2 \right \rangle} $, it also receives a second input in the form of an activation value from Time Step 1. The same process is repeated for the next time steps as well, until the last time step, where the input is \(x^{\left \langle T_{x} \right \rangle} \) and the output is \(\hat{y}^{\left \langle T_{y} \right \rangle} \). For the initial time step 1, the activation value is kept at 0, which is nothing but a vector of zeros with a dummy time step 0. There are some other ways adopted by few researchers who initialize the activation value, \(a^{\left \langle 0 \right \rangle } \), randomly but having a vector of zeroes is the most common practice.
Like we mentioned earlier and as you can see too that in this example, the length of the input sequence \(T_{x} \) is equal to the length of the output sequence \(T_{y}\). The architecture of the Recurrent Neural Network will change once the input and output sequence lengths start to differ.
Apart from the above graphical representation of a Recurrent Neural Network, there is another way of representing RNN as seen in some research papers. In such representations, a loop is drawn on the time step indicating that the layer feeds back to itself.
The above RNN reads the input data from left to right while sharing parameters at each time step. We will describe the parameters of the hidden layers in detail ahead but for now, let's look at the parameters that govern the connection between \(x^{\left \langle 1 \right \rangle} \) and the hidden layer. This parameter, which is the same for every time step, can be denoted as \(w_{aa} \). And similarly, the common parameter that governs the output predictions is denoted as \(w_{ya} \).
Consider the output parameter \(\hat{y}^{\left \langle 3 \right \rangle} \). This output or this prediction is made not just from this time step's input \(x^{\left \langle 3 \right \rangle} \), but also from the information received from the previous inputs \(x^{\left \langle 1 \right \rangle} \) and \(x^{\left \langle 2 \right \rangle} \) which is being passed along with activation values and the parameters governing them.
In all the graphical representations above, we can see the flow of information is in one direction only, i.e., from left to right. Such sequence models or RNNs pose some limitations which we will learn ahead.
In our discussions above, we mentioned how information from previous inputs in the sequence is used when predicting, say, \(\hat{y}^{\left \langle 3 \right \rangle} \). Having said that, we can also notice how it only used the previous information and doesn't take into consideration information from the words ahead in the sequence, such as \(x^{\left \langle 4 \right \rangle} \), \(x^{\left \langle 5 \right \rangle} \), \(x^{\left \langle x \right \rangle} \) and so on.
Consider a new sentence: "Bear Grylls loves adventure." For the network to judge that "Bear" is a part of a person's name, not only will it be useful for it to have information from the first two words but the rest of the sentence as well. Why we say this is because the sentence could also have been: "Bear with me for some time please." How is a neural network supposed to judge whether "Bear" is a part of a person's name or just an ordinary word? In the first example, it is a name while in the second it is not.
This limitation arises with the use of Unidirectional Neural Network architecture and can only be solved using Bidirectional Recurrent Neural Networks (BRNN), which modify the architecture slightly to incorporate information from all input words of the sentence at each time step. You will learn more about BRNNs in our future posts. For the sake of simplicity and understanding key concepts, we will be working with Unidirectional Recurrent Neural Networks only, for now.
Now we know how a basic RNN architecture is created. Let us continue building RNNs, this time using mathematical notations, and understanding how forward propagation works in Recurrent Neural Networks.
3. Forward Propagation in RNN
After a thorough understanding of the basic architectural framework of RNNs, we will learn how the whole process works and can be represented mathematically. We will start with the computational formulas used in the process of Forward Propagation in Recurrent Neural Networks (RNN).
Let us look at the RNN model again.
So, as we learnt earlier, the process starts off with the first 'fake' initial activation input, \(a^{\left \langle 0 \right \rangle}= \vec{0} \). Followed by this initial value, the subsequent forward propagation of input activation values can be written in the form of weights and biases as:
$$ a^{\left \langle 1 \right \rangle}= g\left ( w_{aa}\times a^{\left \langle 0 \right \rangle}+w_{ax}\times x^{\left \langle 1 \right \rangle}+b_{a} \right ) $$
Again, in order to compute \(\hat{y}^{\left \langle 1 \right \rangle} \), there will be yet another activation function internally, which will have the following mathematical formula:
$$ \hat{y}^{\left \langle 1 \right \rangle}= g\left ( w_{ya}\times a^{\left \langle 1 \right \rangle}+b_{y} \right ) $$
If we look at the notation \(w_{ax} \), the \(x \) means that the weight is to be multiplied by $latex x^{\left \langle 1 \right \rangle} $, and the \(a \) means that the notation \(w_{ax} \) is being used to calculate a value like \(a \). In the same way, when we write $latex w_{ay} $, it means that it is going to be multiplied by the value $latex a $ and will be used to calculate the value \(y \).
The activation function, as mentioned above, will most commonly be a Tanh. While sometimes for RNNs, ReLu can also be used but Tanh is the preferred option to solve the vanishing gradient problem. In the case of our output \(y \), choosing an activation function is an important step as well. If the output is a binary classification, we will use a Sigmoid activation function. Else, if the output is a K-classification problem, we shall go with Softmax. Therefore, for the Name Entity Recognition problem where w was either 0 or 1, \(g \) can be taken as a Sigmoid activation function. Writing in more general terms, we could denote \(a^{\left \langle t \right \rangle}\) at time \(t \) as:
$$ a^{\left \langle t \right \rangle}= g\left ( w_{aa}\times a^{\left \langle t-1 \right \rangle}+w_{ax}\times x^{\left \langle t \right \rangle}+b_{a} \right ) $$
And \(\hat{y}^{\left \langle t \right \rangle} \) is:
$$ \hat{y}^{\left \langle 1 \right \rangle}= g\left ( w_{ya}\times a^{\left \langle t \right \rangle}+b_{y} \right ) $$
So, these were the essential equations that define Forward Propagation in Recurrent Neural Networks. We start with an initial activation value \(a^{\left \langle 0 \right \rangle} \) which is a vector of all zero values. Then, we use this \(a^{\left \langle 0 \right \rangle}\) and the first input \(x^{\left \langle 1 \right \rangle} \) to predict our first output \(\hat{y}^{\left \langle 1 \right \rangle} \) and the next activation value \(a^{\left \langle 1 \right \rangle} \). Similarly, in the second time step, we use the input \(x^{\left \langle 2 \right \rangle}\) and the previous activation value \(a^{\left \langle 1 \right \rangle}\) to calculate the next activation value \(a^{\left \langle 2 \right \rangle} \) and a new output \(\hat {y}^{\left \langle 2 \right \rangle} \). And so we continue with our forward propagation from left to right in our Recurrent Neural Network.
We agree the equations are a little lengthy and pose a challenge when we think about using them in complex neural network systems. Let us, therefore, simplify these equations for a much more friendly RNN Notation.
$$ \hat{y}^{<t>}=g\left(W_{y a} a^{<t>}+b_{y}\right) $$
We can even write this Forward Propagation notation in a matrix form to arrive at a neatly stacked formula such as this:
Notice how we are defining \(w_{a}\ \) by stacking two matrices \(w_{aa}\ \) and \(w_{ax}\ \) horizontally, side by side. So, for example, if the vector \(a \) is 100-dimensional and our input vector \(x \) is 10,000-dimensional (because of dictionary size), then, the resultant matrices \(w_{aa}\ \) and \(w_{ax}\ \) would be of sizes \(100\times 100 \) and \(100\times 10,000 \), respectively.
Upon stacking \(w_{aa}\ \) and \(w_{ax}\ \) as above, the total number of elements in the matrix will thus, become 10,100 and hence, the resultant matrix \(w_{a}\ \) will be a \(100\times 10,100 \) dimensional matrix.
We could also write the above scenario in general form, by stacking the vectors \(a^{\left \langle t-1 \right \rangle} \) and \(x^{\left \langle t \right \rangle} \) to form a 10,100-dimensional vector, something like this:
You can even validate the above equation by working backward to see if you can reach the original Forward Propagation equation using this. Let us see how that works out.
$$ \begin{bmatrix}w_{aa} & w_{ax}\end{bmatrix}\times \begin{bmatrix}a^{\left \langle t-1 \right \rangle}\\x^{\left \langle t \right \rangle}\end{bmatrix}= w_{aa}\times a^{\left \langle t-1 \right \rangle}+w_{ax}\times x^{\left \langle t \right \rangle} $$
There you go! By working our way backward from the simplified formula, we have reached our original notation for Forward Propagation. This proves our point that rather than using two-parameter matrices \(w_{aa} \) and \(w_{ax} \), we can simply compress them into a single parameter matrix \(w_{a} \) and make our lives easy while working on more complex models.
Now we can rewrite our original equations using the simplified notation in the following way:
$$ \hat{y}^{\left \langle 1 \right \rangle}= g\left ( w_{a}\times a^{\left \langle t \right \rangle}+b_{y} \right ) $$
Look how neatly we have denoted Forward Propagation by using simple subscripts \(w_{y} \) and $latex b_{y} $. Here, \(w_{y} \) represents the weight of the matrix being computed and $latex b_{y} $ represents the type of output that we are computing. And, \(w_{a} \) and \(b_{a} \) represent the parameters for computing the input and output activation values.
Great! We have successfully learned how to notate Forward Propagation, i.e., moving from left to right through time, for basic Recurrent Neural Networks. However, as we have seen with other neural network models, moving forward in time is just not enough to see if our model is working accurately. This is the reason why Back Propagation is equally, if not more, important to understand even though the programming framework automatically takes care of it at the back-drop.
4. Back Propagation in RNN
We have reached an important stage in our understanding of Recurrent Neural Networks and how the basic models of RNN work. Now that we have seen how we move with time using various Input and Output parameters, we shall take the recursive path of going from right to left through time, in our basic RNN model.
In the section above, we saw how Forward Propagation works. Let us now see how the whole system of RNN model works, including Forward as well as Back Propagation.
In the graphical representation above, the blue arrows represent Forward Propagation and the red arrows represent Back Propagation. Going forward, we start with the input sequence \(x^{\left \langle 1 \right \rangle} \), \(x^{\left \langle 2 \right \rangle} \), \(x^{\left \langle 1 \right \rangle} \) up to \(x^{\left \langle T_{x} \right \rangle} \). We, then, use \(x^{\left \langle 1 \right \rangle} \) and \(a^{\left \langle 0 \right \rangle} \) to calculate the activation at time step 1, i.e., \(a^{\left \langle 1 \right \rangle} \). Next, we use this activation \(a^{\left \langle 1 \right \rangle} \) along with the next input \(x^{\left \langle 2 \right \rangle} \) to calculate the activation for the next time step, which is \(a^{\left \langle 2 \right \rangle} \). And, so on, up to \(a^{\left \langle T_{x} \right \rangle} \).
Now, earlier we mentioned how the initial activation value is just assumed to be a vector of zeros. However, if we were to actually compute this initial activation \(a^{\left \langle 0 \right \rangle} \), we would need the parameters \(w_{a} \) and \(b_{a} \). Similarly, we need these two parameters to calculate activation values \(a^{\left \langle 1 \right \rangle} \), \(a^{\left \langle 2 \right \rangle} \), \(a^{\left \langle 3 \right \rangle} \) and so on, for all subsequent time steps.
The outputs \(\hat{y} \) are calculated in a similar fashion as we computed the activation values. The first activation value \(a^{\left \langle 1 \right \rangle} \) is used to predict the first output \(\hat{y}^{\left \langle 1 \right \rangle} \), and the pattern follows for prediction of other outputs \(\hat{y}^{\left \langle 2 \right \rangle} \), \(\hat{y}^{\left \langle 3 \right \rangle} \) and so on up to $latex \hat{y}^{\left \langle T_{y} \right \rangle} $. For the RNN to compute these outputs \(\hat{y} \), it needs a set of weight and bias parameters too, which as represented as \(w_{y} \) and \(b_{y} \).
Now, for us to compute Back Propagation, first we need to determine a Loss Function. Let us start by defining an element-wise Loss Function. We know that if for a certain word in our sequence, to be a part of a person's name, the output \(\hat{y}^{\left \langle t \right \rangle} \) will be 1. Our Neural Network will output some probability value, either 0 or 1, indicating if it thinks that the word is part of a person's name. This can be perceived as a Standard Logistic Regression Loss, also called the Cross-Entropy Loss. This element-wise loss for a word is associated with a single prediction at a certain position or time step \(t \). We can write this mathematically like this:
$$ L^{\left \langle t \right \rangle}\left ( \hat{y}^{\left \langle t \right \rangle},y^{\left \langle t \right \rangle} \right )= -y^{\left \langle t \right \rangle}log\hat{y}^{\left \langle t \right \rangle}-\left ( 1-y^{\left \langle t \right \rangle} \right )log\left ( 1-\hat{y}^{\left \langle t \right \rangle} \right ) $$
The above equation represents Loss for a single word of the sequence. Let us see how we can extrapolate this to compute the overall loss of the entire sequence, represented as \(L \).
$$ L\left ( \hat{y},y \right )= \sum_{t= 1}^{T_{x}}L^{\left \langle t \right \rangle}\left ( \hat{y}^{\left \langle t \right \rangle},y^{\left \langle t \right \rangle} \right ) $$
The Total Loss of the entire input sequence is nothing but the sum of element-wise losses at each time step. Also, note that in our example, \(T_{x} \) and \(T_{y} \) are equal. This is how the overall computational graph of the Sequence's Loss Function looks like:
The computation problem, as we can see, is to pass messages in the opposite direction as compared to Forward Propagation. These reverse-sent messages will allow you to compute appropriate quantities, which will, in turn, help you derive and update all parameters using gradient descent.
This algorithm is interestingly named "Back Propagation through time" due to the nature of its significant messages arriving from right to left, working backwards through the time steps of Recurrent Neural Networks. If only RNN could go back in actual time and attend Stephen Hawking's Time Traveller Tea Party, but, oh well!
This brings us to the end of this tutorial post. We hope you have learned a great deal about how Recurrent Neural Networks are structured and mapped, and how forward and backpropagation works in basic RNN models. An important observation that we have mentioned earlier as well is that in the entire tutorial, we have considered the lengths of both input and output sequences to be equal. However, you can expect a new tutorial moving ahead, where we will understand a much wider range of RNN architectures tackling a variety of applications.
Forward and Back Propagation in Recurrent Neural Networks
RNNs work better than Standard Neural Networks
RNN architecture contains hidden layers that have memory
Simplified Forward Propagation methods
Element-wise loss calculated for Back Propagation
Wide range of real-life applications
RNNs are really an exciting topic to venture into, aren't they? We wish to keep continuing with our tutorials to help you build your own models soon. In no time, you will be capable enough to train your own Neural Network to identify speech, names, music, and whatever else you wish. So keep reading, keep learning, and keep propagating! See you 😊
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How far would I have to be from a nuclear detonation in space in order to survive it?
Assume someone detonates a Hiroshima-sized nuclear bomb in space. Since in space there's no air, the bomb will behave differently than on Earth. In particular, there will not be an air pressure wave, and certainly no "mushroom" cloud — the energy will be sent in all directions equally, as kinetic energy of the bomb fragments, and as radiation (of all sorts). Therefore, the damage pattern on Earth probably is very different from the damage pattern in space.
Now the question is: How far do I have to be away from the bomb in order to survive it without injury? Let's assume that at the time of the explosion I'm in space with the thinnest space suit possible.
science-based space weapon-mass-destruction
$\begingroup$ I can certainly say without much consideration that it would depend on the type of bomb (fission versus fusion) and its size. The rest is a wild guess for all to make. Nuclear fallout in space would be very very different from that on earth due to lack of winds to carry the death wave. $\endgroup$ – Youstay Igo Oct 8 '15 at 21:40
$\begingroup$ You could be a bazillion miles away, but if a chunk of shrapnel hits you and it hasn't slowed down, adios muchacho. $\endgroup$ – BrettFromLA Oct 8 '15 at 23:19
$\begingroup$ As I wrote to Jim2B, shock waves can very well be a problem. Supernova shock waves can compress gas clouds enough to trigger star formation. This would obviously be on a much smaller scale, but it could still be quite important. Space isn't a vacuum; it's filled with gas and dust. $\endgroup$ – HDE 226868♦ Oct 8 '15 at 23:35
$\begingroup$ @YoustayIgo - the question states "Hiroshima-sized" which pretty firmly establishes both type and yield. $\endgroup$ – WhatRoughBeast Oct 9 '15 at 0:26
$\begingroup$ it is like uncontrolled particle accelerator pointing in every direction and accordingly to Newton's first law all the products in this case the photons and other particles will move away from the point of explosion in a straight line unless otherwise subjected to an external force such as dust or gravity etc. Using inverse square law we can calculate the density and the distance from the point of explosion and confirm if the radiation dose is fatal or not btw this is a purely physic question. $\endgroup$ – user6760 Oct 9 '15 at 2:01
Nuclear bombs produce 3 effects on Earth
Thermal flash
Blast (caused by conversion of X-rays into heat)
In space, you need only concern yourself with the neutrons and X-rays.
Radiation Enhanced Bombs
According to Atomic Rockets: Radiation Flux:
A one megaton Enhanced-Radiation warhead (AKA "neutron bomb") will deliver a threshold fatal neutron dose to an unshielded human at 300 kilometers.
Normal Nuclear Weapons
A non-radiation enhanced bomb produces much less neutron radiation but more X-ray radiation. A 1 kton nuclear bomb is borderline survivable at a range of 30 km due to the X-ray flux.
The survivability range of nuclear bombs scales (roughly) linearly with bomb yield and as the inverse square of distance between bomb and victim. Meaning a 1 mton bomb would be borderline survivable at a range of ~900 km due to X-ray flux.
All numbers are for unshielded humans. Shielding can significantly alter these numbers.
The effectiveness of X-Ray shielding is primarily determined by the amount of mass it contains (high atomic mass materials work slightly better than low atomic mass ones).
The effectiveness of neutron shielding is dependent upon the number of low atomic mass nuclei between the bomb and the victim. High atomic mass nuclei in your radiation shielding can make neutron radiation more difficult to manage.
Edit 10/09/2015:
I concur with Thucydides, anyone interested in this topic should go to Atomic Rockets and read all relevant sections. It includes a description of what a nuclear detonation would look like, what effects it'd have on spacecraft, etc.
As for survivability, my answer only considers a person wearing a minimal spacesuit for protection. The actual physical damage a 1 kton weapon would inflict on a body (human or otherwise) at a range of 30 km would be minimal. A person at that range would be hit with a lethal dose of radiation. Without medical care it might take them days or longer to die in an extremely unpleasant manner.
You should also realize that being outside the "deadly" zone listed above does not necessarily mean you will live. Radiation sickness is nasty and you'll require intense medical treatment in order to survive a large dose. Atomic Rockets has a Acute Radiation Syndrome Chart which tells you what symptoms you can expect from a given dosage. The chart gives you a probability of surviving any given dosage.
My numbers were for ~2.0+ Gray dosage - this gives you a survival probability of 35-40%.
Jim2BJim2B
$\begingroup$ Any effects from thermal pulse or high velocity fragments/plasma/whatever from the exploded/vaporized bomb components? $\endgroup$ – Deolater Oct 8 '15 at 22:37
$\begingroup$ The blast might actually be important. In some conditions where the ISM is denser than normal, the blast wave (described by the Sedov-Taylor solution) could be intense. $\endgroup$ – HDE 226868♦ Oct 8 '15 at 23:33
$\begingroup$ " A 1 kton nuclear bomb is borderline survivable at a range of 30 km due to the X-ray flux." Surely that's a typo. 30 km? People survived Hiroshima much closer than that. $\endgroup$ – WhatRoughBeast Oct 9 '15 at 0:35
$\begingroup$ @WhatRoughBeast, Key word: unshielded. Air provides pretty good X-ray shielding. $\endgroup$ – Mark Oct 9 '15 at 0:55
$\begingroup$ @WhatRoughBeast, Mark is correct. Atmosphere is opaque to X-ray and Gamma ray frequencies. It is the conversion of these frequencies into heat that generates the blast effect in atmospheres. $\endgroup$ – Jim2B Oct 9 '15 at 14:13
About 100 miles (160 kilometers) for no injuries
Something cool that I found a while ago is NASA's report on Nuclear Weapon Effects in Space. First thing to keep in mind:
If a nuclear weapon is exploded in a vacuum-i. e., in space-the complexion of weapon effects changes drastically:
First, in the absence of an atmosphere, blast disappears completely.
Second, thermal radiation, as usually defined, also disappears. There is no longer any air for the blast wave to heat and much higher frequency radiation is emitted from the weapon itself.
The radiation is your only real problem in space. So with a nice radiation-proof spacesuit, you could survive a nuclear blast at a ridiculously close range.
So how far would you have to be in order to be safe from radiation, assuming essentially no radiation protection from your spacesuit? According to Wikipedia, a dose of 0.1 grays (10 rads) is enough to cause radiation sickness. Let's look at one of the charts NASA included:
This is for a 20-kiloton explosion. At sea level, you'd get 10 rads from being a mile or so away from the explosion. In space? It looks like we're at about 60 rads when you're 40 miles away. To reduce that by a factor of 6, we'll need to go $\sqrt{6}\approx 2.5$ times as far away, so about 100 miles away.
Of course, you'd still survive short-term if you're closer than that, but the closer you get the worse the radiation sickness will be.
Something else to remember is that in space 100 miles is not very far - the international space station goes that far in about 20 seconds.
Rob WattsRob Watts
$\begingroup$ If I could accept two answers, I would accept yours as well. Indeed, it took me quite some time to decide which of the two answers (yours or Jim2B's) I should accept; finally I decided on Jim2Bs, because it on the whole provided me with more information. But yours is still a very close second. $\endgroup$ – celtschk Oct 10 '15 at 7:57
The Atomic Rockets site has a pretty comprehensive section on nuclear weapons, and the answers there suggest that a nuclear weapon in space isn't as much of a threat outside of short distances due to the inverse square law (much of the radiation energy is dissipated into space) and the lack of a medium to transmit the energy to the target (the main thing you are going to be hit with is a blast of x-rays, which will spoil your day if you are too close).
The modifier is if the nuclear weapon is driving some sort of amplification device.
In the 1980's, it was postulated that the energy of an exploding nuclear bomb could be converted into a laser beam of x-rays if a sufficiently long and slender rod of the correct material was placed with one end on the bomb and the other end pointing at the target. As the material was converted into plasma by the exploding bomb, there would be a point where the long line of plasma should become a lasing cavity and emit a high energy x-ray beam, with a potential range of thousands of kilometres. This was the basis of the "Excalibur" device, which took the idea to "11" by envisioning a device which looked like a sea urchin carrying dozens to hundreds of "spines", each locking onto a different target. There were many practical reasons this never got off the ground (so to speak), but one of them was the efficiency of converting the bomb's energy to laser energy was very low. This might have been resolved since then.
The other idea would be to make the bomb drive a "shaped charge". Astounding as this sounds, evidently this was experimented with and some success was achieved, with a pre scored plate being converted into a shotgun charge with pellets moving at some astounding velocity at the target (@ 70 km/sec), while using various means to shape the plasma jet from the nuclear explosion can result in a narrow jet of hot plasma moving at something like .03*c*! This was evolved from the pulse units developed for the ORION nuclear pulse drive spacecraft, and known under the name "Casaba-Howitzer" Much of the information is still classified, but you would certainly be in grave danger even in a well armoured spaceship hundreds, if not thousands of kilometres from the blast.
Tristan Klassen
ThucydidesThucydides
$\begingroup$ When you reference a website it would be nice when you would do it in form of a link so the readers do not have to look for it themselves. $\endgroup$ – Philipp Oct 9 '15 at 17:22
Although there are a lot of if's hidden in your question, a sort of worst-case number can be determined.
First, a Hiroshima blast is about 15 kt. Since 1 Mt is $4\times10^{15}$J, a Hiroshima-sized blast releases about .015 times that, or $$E=.015\times 4\times10^{15} = 6\times10^{13}\text{ J}$$ For a fission bomb, about 35% of released energy is thermal, and about 3% radiation. The amount of thermal radiation required to breach a spacesuit is unknown, but let's say something on the order of 1 MW/m2 for one second. After all, sunlight is about 1.5 $\times$103 W/m2, and a suit obviously won't have major problems with that. Assume a human body provides about 1 square meter of area (2 meters tall by 1/2 meter wide). Then the total thermal energy required will be 1 MJ. Since 35% of a bomb goes to thermal, we can write $$4\pi R^2 = .35 \times6\times10^{13} = 2.1\times10^{13}$$ and $$R = \sqrt{\frac{2.1\times10^{13}}{4\pi}} = 1.3\times10^6\text{ m}$$ 1300 km is much greater than is characteristic of terrestrial nukes but there's a good reason - the atmosphere absorbs most of it and produces blast.
Radiation is a different issue. First, about 1/10th as much energy goes into radiation as to thermal. However, since radiation will largely penetrate a suit, it might take less energy. But the fact that radiation will penetrate a suit means that some will simply pass through the body and produce no damage. Let's assume, purely as a fictional number, that 10% of radiation which hits a person will be absorbed. The unit of absorbed radiation is the Grey, which is 1 J/kg of tissue, and 5 Greys is a standard lethal dose for humans. For this set of assumptions,$$4\pi R^2 = 5\times 0.1\times .003 \times 6\times10^{13} = 9\times10^{10}$$ and $$R = \sqrt{\frac{9\times10^{10}}{4\pi}} = .84\times10^5 \text{ m}$$ So the two effects agree within less than a factor of 2, and 1000 km sounds like a nice round number for a survivable distance. Of course, that means that your suit didn't quite burn up, and you didn't necessarily die of radiation poisoning, so you might want to add a safety factor. I'd guess the 2,000 km is a much better number to use.
EDIT - In calculating radiation levels, I forgot to factor in body mass. Assuming 100 kg for mass gives a factor of 10 reduction in distance, to $$R = .84 \times 10^4 \text{m}$$
WhatRoughBeastWhatRoughBeast
$\begingroup$ I disagree about the thermal threat. So long as you are not looking at the bomb I would think the threat would be minimal at even quite short ranges--the vast majority of the energy is going to be delivered very, very quickly. There won't be time for the energy to soak in, you'll just blow off a very thin layer of the outside of your suit. It's the hot stuff that will matter. That being said, you answer doesn't pass the smell test. Did you mean m instead of km?? $\endgroup$ – Loren Pechtel Oct 9 '15 at 4:49
$\begingroup$ @LorenPechtel - ablation can only do so much. If 1 MJ isn't enough, provide your own number and do the math. And I'm using standard MKS. so m, not km. $\endgroup$ – WhatRoughBeast Oct 9 '15 at 16:01
$\begingroup$ Your math is wrong. A distance of 1000km is the distance needed for every square meter to receive 1J (not MJ!) of energy. $\endgroup$ – Rob Watts Oct 9 '15 at 16:28
$\begingroup$ You edited your message 9 hours ago but you still have the errant km values. $\endgroup$ – Loren Pechtel Oct 10 '15 at 1:41
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