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How many lags to use in the Ljung-Box test of a time series?

After an ARMA model is fit to a time series, it is common to check the residuals via the Ljung-Box portmanteau test (among other tests). The Ljung-Box test returns a p value. It has a parameter, h, which is the number of lags to be tested. Some texts recommend using h=20; others recommend using h=ln(n); most do not say what h to use.

Rather than using a single value for h, suppose that I do the Ljung-Box test for all h<50, and then pick the h which gives the minimum p value. Is that approach reasonable? What are the advantages and disadvantages? (One obvious disadvantage is increased computation time, but that is not a problem here.) Is there literature on this?

To elaborate slightly.... If the test gives p>0.05 for all h, then obviously the time series (residuals) pass the test. My question concerns how to interpret the test if p<0.05 for some values of h and not for other values.

• @user2875, I've deleted my answer. Fact is that for large $h$ the test is not reliable. So the answer really depends for which $h$, $p<0.05$. Furthermore what is exact value of $p$? If we decrease the threshold to $0.01$, does the result of the test changes? Personally in case of conflicting hypotheses I look for other indicators whether model is good or not. How well model fits? How does the model compare to alternative models? Do the alternative model have the same problems? For what other violations the test rejects the null? – mpiktas Jan 23 '11 at 19:52
• @mpiktas, The Ljung-Box test is based on a statistic whose distribution is asymptotically (as h becomes large) chi-squared. As h gets large relative to n, though, the power of the test decreases to 0. Hence the desire to choose h large enough that the distribution is close to chi-squared but small enough to have useful power. (I do not know what the risk of a false negative is, when h is small.) – user2875 Jan 24 '11 at 0:05
• @user2875, this the third time you changed the question. First you ask about the strategy of picking $h$ with smallest value, then how to interpret the test if $p<0.05$ for some values of $h$, and now what is the optimal $h$ to choose. All three questions have different answers and may even have different answers depending on the context of particular problem. – mpiktas Jan 24 '11 at 12:12
• @mpiktas, the questions are all the same, just different ways of looking at it. (As pointed out, if p>0.05 for all h, then we know how to interpret the smallest p; if we knew the optimal h--we don't--then we would not be concerned with choosing the smallest p.) – user2875 Jan 24 '11 at 18:43

The answer definitely depends on: What are actually trying to use the $Q$ test for?

The common reason is: to be more or less confident about joint statistical significance of the null hypothesis of no autocorrelation up to lag $h$ (alternatively assuming that you have something close to a weak white noise) and to build a parsimonious model, having as little number of parameters as possible.

Usually time series data has natural seasonal pattern, so the practical rule-of-thumb would be to set $h$ to twice this value. Another one is the forecasting horizon, if you use the model for forecasting needs. Finally if you find some significant departures at latter lags try to think about the corrections (could this be due to some seasonal effects, or the data was not corrected for outliers).

Rather than using a single value for h, suppose that I do the Ljung-Box test for all h<50, and then pick the h which gives the minimum p value.

It's a joint significance test, so if the choice of $h$ is data-driven, then why should I care about some small (occasional?) departures at any lag less than $h$, supposing that it is much less than $n$ of course (the power of the test you mentioned). Seeking to find a simple yet relevant model I suggest the information criteria as described below.

My question concerns how to interpret the test if $p<0.05$ for some values of $h$ and not for other values.

So it will depend on how far from the present it happens. Disadvantages of far departures: more parameters to estimate, less degrees of freedom, worse predictive power of the model.

Try to estimate the model including the MA and\or AR parts at the lag where the departure occurs AND additionally look at one of information criteria (either AIC or BIC depending on the sample size) this would bring you more insights on what model is more parsimonious. Any out-of-sample prediction exercises are also welcome here.

• +1, this is what I was trying to express but was not able to :) – mpiktas Jan 25 '11 at 14:20

Assume that we specify a simple AR(1) model, with all the usual properties,

$$y_t = \beta y_{t-1} + u_t$$

Denote the theoretical covariance of the error term as

$$\gamma_j \equiv E(u_tu_{t-j})$$

If we could observe the error term, then the sample autocorrelation of the error term is defined as

$$\tilde \rho_j \equiv \frac {\tilde \gamma_j}{\tilde \gamma_0}$$

where

$$\tilde\gamma_j \equiv \frac 1n \sum_{t=j+1}^nu_tu_{t-j},\;\;\; j=0,1,2...$$

But in practice, we do not observe the error term. So the sample autocorrelation related to the error term will be estimated using the residuals from estimation, as

$$\hat\gamma_j \equiv \frac 1n \sum_{t=j+1}^n\hat u_t\hat u_{t-j},\;\;\; j=0,1,2...$$

The Box-Pierce Q-statistic (the Ljung-Box Q is just an asymptotically neutral scaled version of it) is

$$Q_{BP} = n \sum_{j=1}^p\hat\rho^2_j = \sum_{j=1}^p[\sqrt n\hat\rho_j]^2\xrightarrow{d} \;???\;\chi^2(p)$$

Our issue is exactly whether $Q_{BP}$ can be said to have asymptotically a chi-square distribution (under the null of no-autocorellation in the error term) in this model.
For this to happen, each and everyone of $\sqrt n \hat\rho_j$ must be asymptotically standard Normal. A way to check this is to examine whether $\sqrt n \hat\rho$ has the same asymptotic distribution as $\sqrt n \tilde\rho$ (which is constructed using the true errors, and so has the desired asymptotic behavior under the null).

We have that

$$\hat u_t = y_t - \hat \beta y_{t-1} = u_t - (\hat \beta - \beta)y_{t-1}$$

where $\hat \beta$ is a consistent estimator. So

$$\hat\gamma_j \equiv \frac 1n \sum_{t=j+1}^n[u_t - (\hat \beta - \beta)y_{t-1}][u_{t-j} - (\hat \beta - \beta)y_{t-j-1}]$$

$$=\tilde \gamma _j -\frac 1n \sum_{t=j+1}^n (\hat \beta - \beta)\big[u_ty_{t-j-1} +u_{t-j}y_{t-1}\big] + \frac 1n \sum_{t=j+1}^n(\hat \beta - \beta)^2y_{t-1}y_{t-j-1}$$

The sample is assumed to be stationary and ergodic, and moments are assumed to exist up until the desired order. Since the estimator $\hat \beta$ is consistent, this is enough for the two sums to go to zero. So we conclude

$$\hat \gamma_j \xrightarrow{p} \tilde \gamma_j$$

This implies that

$$\hat \rho_j \xrightarrow{p} \tilde \rho_j \xrightarrow{p} \rho_j$$

But this does not automatically guarantee that $\sqrt n \hat \rho_j$ converges to $\sqrt n\tilde \rho_j$ (in distribution) (think that the continuous mapping theorem does not apply here because the transformation applied to the random variables depends on $n$). In order for this to happen, we need

$$\sqrt n \hat \gamma_j \xrightarrow{d} \sqrt n \tilde \gamma_j$$

(the denominator $\gamma_0$ -tilde or hat- will converge to the variance of the error term in both cases, so it is neutral to our issue).

We have

$$\sqrt n \hat \gamma_j =\sqrt n\tilde \gamma _j -\frac 1n \sum_{t=j+1}^n \sqrt n(\hat \beta - \beta)\big[u_ty_{t-j-1} +u_{t-j}y_{t-1}\big] \\+ \frac 1n \sum_{t=j+1}^n\sqrt n(\hat \beta - \beta)^2y_{t-1}y_{t-j-1}$$

So the question is : do these two sums, multiplied now by $\sqrt n$, go to zero in probability so that we will be left with $\sqrt n \hat \gamma_j =\sqrt n\tilde \gamma _j$ asymptotically?

For the second sum we have

$$\frac 1n \sum_{t=j+1}^n\sqrt n(\hat \beta - \beta)^2y_{t-1}y_{t-j-1} = \frac 1n \sum_{t=j+1}^n\big[\sqrt n(\hat \beta - \beta)][(\hat \beta - \beta)y_{t-1}y_{t-j-1}]$$

Since $[\sqrt n(\hat \beta - \beta)]$ converges to a random variable, and $\hat \beta$ is consistent, this will go to zero.

For the first sum, here too we have that $[\sqrt n(\hat \beta - \beta)]$ converges to a random variable, and so we have that

$$\frac 1n \sum_{t=j+1}^n \big[u_ty_{t-j-1} +u_{t-j}y_{t-1}\big] \xrightarrow{p} E[u_ty_{t-j-1}] + E[u_{t-j}y_{t-1}]$$

The first expected value, $E[u_ty_{t-j-1}]$ is zero by the assumptions of the standard AR(1) model. But the second expected value is not, since the dependent variable depends on past errors.

So $\sqrt n\hat \rho_j$ won't have the same asymptotic distribution as $\sqrt n\tilde \rho_j$. But the asymptotic distribution of the latter is standard Normal, which is the one leading to a chi-squared distribution when squaring the r.v.

Therefore we conclude, that in a pure time series model, the Box-Pierce Q and the Ljung-Box Q statistic cannot be said to have an asymptotic chi-square distribution, so the test loses its asymptotic justification.

This happens because the right-hand side variable (here the lag of the dependent variable) by design is not strictly exogenous to the error term, and we have found that such strict exogeneity is required for the BP/LB Q-statistic to have the postulated asymptotic distribution.

Here the right-hand-side variable is only "predetermined", and the Breusch-Pagan test is then valid. (for the full set of conditions required for an asymptotically valid test, see Hayashi 2000, p. 146-149).

• You wrote "But the second expected value is not, since the dependent variable depends on past errors." That's called strict exogeneity. I agree that it's a strong assumption, and you can build AR(p) framework without it, just by using weak exogeneity. This the reason why Breusch-Godfrey test is better in some sense: if the null is not true, then B-L loses power. B-G is based on weak exogeneity. Both tests are not good for some common econometric, applications, see e.g. this Stata's presentation, p. 4/44. – Aksakal Apr 3 '16 at 17:14
• @Aksakal Thanks for the reference. The point exactly is that without strict exogeneity, the Box-Pierce/Ljung-Box do not have an asymptotic chi-square distribution, this is what the mathematics above show. Weak exogeneity (which holds in the above model) is not enough for them. This is exactly what the presentation you link to says in p. 3/44. – Alecos Papadopoulos Apr 3 '16 at 17:39
• @AlecosPapadopoulos, an amazing post!!! Among the few best ones I have encountered here at Cross Validated. I just wish it would not disappear in this long thread and many users would find and benefit from it in the future. – Richard Hardy Apr 3 '16 at 19:08

Before you zero-in on the "right" h (which appears to be more of an opinion than a hard rule), make sure the "lag" is correctly defined.

http://www.stat.pitt.edu/stoffer/tsa2/Rissues.htm

Quoting the section below Issue 4 in the above link:

"....The p-values shown for the Ljung-Box statistic plot are incorrect because the degrees of freedom used to calculate the p-values are lag instead of lag - (p+q). That is, the procedure being used does NOT take into account the fact that the residuals are from a fitted model. And YES, at least one R core developer knows this...."

Edit (01/23/2011): Here's an article by Burns that might help:

http://lib.stat.cmu.edu/S/Spoetry/Working/ljungbox.pdf

• @bil_080, the OP does not mention R, and help page for Box.test in R mentions the correction and has an argument to allow for the correction, although you need to supply it manualy. – mpiktas Jan 22 '11 at 20:50
• @mpiktas, Oops, you're right. I assumed this was an R question. As for the second part of your comment, there are several R packages that use Ljung-Box stats. So, it's a good idea to make sure the user understands what the package's "lag" means. – bill_080 Jan 22 '11 at 22:29
• Thanks--I am using R, but the question is a general one. Just to be safe, I was doing the test with the LjungBox function in the portes package, as well as Box.test. – user2875 Jan 23 '11 at 16:16

The thread "Testing for autocorrelation: Ljung-Box versus Breusch-Godfrey" shows that the Ljung-Box test is essentially inapplicable in the case of an autoregressive model. It also shows that Breusch-Godfrey test should be used instead. That limits the relevance of your question and the answers (although the answers may include some generally good points).

• The trouble with LB test is when autoregressive models have other regressors, i.e. ARMAX not ARM models. OP explicitly states ARMA not ARMAX in the question. Hence, I think that your answer is incorrect. – Aksakal Apr 2 '16 at 21:50
• @Aksakal, I clearly see from Alecos Papadopoulos answer (and comments under it) in the above-mentioned thread that Ljung-Box test is inapplicable in both cases, i.e. pure AR/ARMA and ARX/ARMAX. Therefore, I cannot agree with you. – Richard Hardy Apr 3 '16 at 8:27
• Alecos Papadopoulos's answer is good, but incomplete. It points out to Ljung-Box test's assumption of strict exogeneity but it fails to mention that if you're fine with the assumption, then L-B test is Ok to use. B-G test, which he and I favor over L-B, relies on weak exogeneity. It's better to use tests with weaker assumptions in general, of course. However, even B-G test's assumptions are too strong in many cases. – Aksakal Apr 3 '16 at 17:27
• @Aksakal, The setting of this question is quite definite -- it considers residuals from an ARMA model. The important thing here is, L-B does not work (as shown explicitly in Alecos post in this as well as the above-cited thread) while B-G test does work. Of course, things can happen in other settings (even B-G test's assumptions are too strong in many cases) -- but that is not the concern in this thread. Also, I did not get what the assumption is in your statement if you're fine with the assumption, then L-B test is Ok to use. Is that supposed to invalidate Alecos point? – Richard Hardy Apr 3 '16 at 19:14

Escanciano and Lobato constructed a portmanteau test with automatic, data-driven lag selection based on the Pierce-Box test and its refinements (which include the Ljung-Box test).

The gist of their approach is to combine the AIC and BIC criteria --- common in the identification and estimation of ARMA models --- to select the optimal number of lags to be used. In the introduction of they suggest that, intuitively, test conducted using the BIC criterion are able to properly control for type I error and are more powerful when serial correlation is present in the first order''. Instead, tests based on AIC are more powerful against high order serial correlation. Their procedure thus choses a BIC-type lag selection in the case that autocorrelations seem to be small and present only at low order, and an AIC-type lag section otherwise.

The test is implemented in the R package vrtest (see function Auto.Q).

The two most common settings are $\min(20,T-1)$ and $\ln T$ where $T$ is the length of the series, as you correctly noted.

The first one is supposed to be from the authorative book by Box, Jenkins, and Reinsel. Time Series Analysis: Forecasting and Control. 3rd ed. Englewood Cliffs, NJ: Prentice Hall, 1994.. However, here's all they say about the lags on p.314:

It's not a strong argument or suggestion by any means, yet people keep repeating it from one place to another.

The second setting for a lag is from Tsay, R. S. Analysis of Financial Time Series. 2nd Ed. Hoboken, NJ: John Wiley & Sons, Inc., 2005, here's what he wrote on p.33:

Several values of m are often used. Simulation studies suggest that the choice of m ≈ ln(T ) provides better power performance.

This is a somewhat stronger argument, but there's no description of what kind of study was done. So, I wouldn't take it at a face value. He also warns about seasonality:

This general rule needs modification in analysis of seasonal time series for which autocorrelations with lags at multiples of the seasonality are more important.

Summarizing, if you just need to plug some lag into the test and move on, then you can use either of these setting, and that's fine, because that's what most practitioners do. We're either lazy or, more likely, don't have time for this stuff. Otherwise, you'd have to conduct your own research on the power and properties of the statistics for series that you deal with.

UPDATE.

Here's my answer to Richard Hardy's comment and his answer, which refers to another thread on CV started by him. You can see that the exposition in the accepted (by Richerd Hardy himself) answer in that thread is clearly based on ARMAX model, i.e. the model with exogenous regressors $x_t$:$$y_t = \mathbf x_t'\beta + \phi(L)y_t + u_t$$

However, OP did not indicate that he's doing ARMAX, to contrary, he explicitly mentions ARMA:

After an ARMA model is fit to a time series, it is common to check the residuals via the Ljung-Box portmanteau test

One of the first papers that pointed to a potential issue with LB test was Dezhbaksh, Hashem (1990). “The Inappropriate Use of Serial Correlation Tests in Dynamic Linear Models,” Review of Economics and Statistics, 72, 126–132. Here's the excerpt from the paper:

As you can see, he doesn't object to using LB test for pure time series models such as ARMA. See also the discussion in the manual to a standard econometrics tool EViews:

If the series represents the residuals from ARIMA estimation, the appropriate degrees of freedom should be adjusted to represent the number of autocorrelations less the number of AR and MA terms previously estimated. Note also that some care should be taken in interpreting the results of a Ljung-Box test applied to the residuals from an ARMAX specification (see Dezhbaksh, 1990, for simulation evidence on the finite sample performance of the test in this setting)

Yes, you have to be careful with ARMAX models and LB test, but you can't make a blanket statement that LB test is always wrong for all autoregressive series.

UPDATE 2

Alecos Papadopoulos's answer shows why Ljung-Box test requires strict exogeneity assumption. He doesn't show it in his post, but Breusch-Gpdfrey test (another alternative test) requires only weak exogeneity, which is better, of course. This what Greene, Econometrics, 7th ed. says on the differences between tests, p.923:

The essential difference between the Godfrey–Breusch and the Box–Pierce tests is the use of partial correlations (controlling for X and the other variables) in the former and simple correlations in the latter. Under the null hypothesis, there is no autocorrelation in εt , and no correlation between $x_t$ and $\varepsilon_s$ in any event, so the two tests are asymptotically equivalent. On the other hand, because it does not condition on $x_t$ , the Box–Pierce test is less powerful than the LM test when the null hypothesis is false, as intuition might suggest.

• I suppose that you decided to answer the question as it was bumped to the top of the active threads by my recent answer. Curiously, I argue that the test is inappropriate in the setting under consideration, making the whole thread problematic and the answers in it especially so. Do you think it is good practice to post yet another answer that ignores this problem without even mentioning it (just like all the previous answers do)? Or do you think my answer does not make sense (which would justify posting an answer like yours)? – Richard Hardy Apr 2 '16 at 14:25
• Thank you for an update! I am not an expert, but the argumentation by Alecos Papadopoulos in "Testing for autocorrelation: Ljung-Box versus Breusch-Godfrey" and in the comments under his answer suggests that Ljung-Box is indeed inapplicable on residuals from pure ARMA (as well as ARMAX) models. If the wording is confusing, check the maths there, it seems fine. I think this is a very interesting and important question, so I would really like to find agreement between all of us here. – Richard Hardy Apr 3 '16 at 8:21

... h should be as small as possible to preserve whatever power the LB test may have under the circumstances. As h increases the power drops. The LB test is a dreadfully weak test; you must have a lot of samples; n must be ~> 100 to be meaningful. Unfortunately I have never seen a better test. But perhaps one exists. Anyone know of one ?

Paul3nt

There's no correct answer to this that works in all situation for the reasons other have said it will depend on your data.

That said, after trying to figure out to reproduce a result in Stata in R I can tell you that, by default Stata implementation uses: $\mathrm{min}(\frac{n}{2}-2, 40)$. Either half the number of data points minus 2, or 40, whichever is smaller.

All defaults are wrong, of course, and this will definitely be wrong in some situations. In many situations, this might not be a bad place to start.

Let me suggest you our R package hwwntest. It has implemented Wavelet-based white noise tests that do not require any tuning parameters and have good statistical size and power.

Additionally, I have recently found "Thoughts on the Ljung-Box test" which is excellent discussion on the topic from Rob Hyndman.

Update: Considering the alternative discussion in this thread regarding ARMAX, another incentive to look at hwwntest is the availability of a theoretical power function for one of the tests against an alternative hypothesis of ARMA(p,q) model.


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# What is a phase arrow?

Feynman say's that a photon takes every path while reflecting off a mirror when going form A to B, but we only see the middle one(where incident angle = reflected angle) because all the others are cancelled out as they have longer routes and while following them their phase arrows cancel out each other.

If you can, please explain it in general terms without mathematics, as I am studying it the same way.

• A phase arrow is a complex number with unit magnitude. Like, something of the form $e^{i\theta}$. – knzhou Aug 20 '16 at 3:01
• As Larry Gopnik rightly said, trying to understand quantum mechanics without the mathematics is a doomed endeavour, and you will only get a very partial view of the subject. Past a certain (very early) point, you need the mathematics. – Emilio Pisanty Aug 20 '16 at 17:45

This question is based, as pretty much everyone knows, on the book "QED, the strange theory of light and matter", by Feynman.

My answer is based on my limited understanding of the book.

Phase Arrows has a good description of Feynman's analogy.

As well the above source, if you visit Feynman Lecture Using Phase Arrows , you can watch RPF's description between times 29:41 and 36:27 of part He draws the "arrows" diagram on the chalkboard.

Similarly, if you visit Feynman Lecture On Photons, Feynman talks about this subject between times 59:33 and 60:32.

One of the differences between classical mechanics and quantum mechanics is that classically, objects have a definite path,or trajectory, so if you kick a soccer ball, you can be pretty certain where it will end up, and the path it will take.

Whereas in q.m.you can't say that a photon will definitely either follow a path or be found at a particular place. You can only assign a probability that a photon will be found at a particular place.

So Feynman sort of took this idea to the extreme, in a mathematical sense. Mathwise,you can say the photon took every available path, but all the paths that don't confirm to classical notions cancel each other out, by destructive interference. So you are left with just the paths that take the least time.

This is sometimes known as the sum over histories approach to quantum mechanics.

• Although I didn't use feynman's book, your resources were good. – John_Nash Sep 17 '16 at 22:11

Feynman's understanding of this effect probably came from optics where it is known as Fermat's principle. The mechanism behind this principle is based on the notion of stationary phase, which is the phenomenon that light will go in that direction where the phase coming from different paths varies slowly.

I'll try to explain this without mathematics. The scenario is one where there is some field that propagates linearly through space or through some linear system, as in the case where we have linear optical components. At the output, one would then find locations where the field has a large amplitude and other locations where the amplitude is small or even zero.

If one looks at a particular output point, the field amplitude at that point is the sum of all the bits of field that arrived there via different paths. The phase of these bits is determine by the length of the path that they followed to get to the output point. If this phase value changes rapidly as a function of the varying path, then the different values will tend to cancel each other. You can think of it as a bunch of arrows pointing in all directions and you are adding these arrows to find the combined arrow. For the rapidly varying case the amplitudes (lengths of the arrows) will vary slow compared to the phase (directions of the arrows). Arrows that point in opposite directions will cancel each other. So the net result is a very small amplitude.

If, on the other hand, the bits of the fields arriving at the output point, have a slow varying phase, then the arrows would all point in more or less the same direction. So they would add up to give a larger amplitude (longer arrow).

• But if I shoot a photon at a mirror, I can be 100% sure of where it will end up and which path it will take(other paths will be cancelled,but still), how is the probability of that photon reaching that point after hitting the mirror be 100% – John_Nash Sep 17 '16 at 22:14
• The catch is, you cannot shoot a photon. You need to direct a beam of light and light beams always have a finite size. Furthermore, visible light has a very small wavelength. This makes the effect seems 100% precise. If one were able to see light with much longer wavelength the behaviour would appear less precise. – flippiefanus Sep 18 '16 at 4:05

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#### [1905.12724] Diffusion Variational Autoencoders

Variational autoencoders (VAEs) have become one of the most popular deep learning approaches to unsupervised learning and data generation. However, traditional VAEs suffer from the constraint that the latent space must distributionally match a simple prior (e.g. normal, uniform), independent of the initial data distribution. This leads to a number of issues around modeling manifold data, as there is no function with bounded Jacobian that maps a normal distribution to certain manifolds (e.g. sphere). Similarly, there are not many theoretical guarantees on the encoder and decoder created by the VAE. In this work, we propose a variational autoencoder that maps manifold valued data to its diffusion map coordinates in the latent space, resamples in a neighborhood around a given point in the latent space, and learns a decoder that maps the newly resampled points back to the manifold. The framework is built off of SpectralNet, and is capable of learning this data dependent latent space withou...

Keywords: autoencoder
Date: 2019/06/05 09:47

#### Related Entries

Deep Learningを用いた教師なし画像検査の論文調査 GAN/SVM/Autoencoderとか .pdf
Variational Autoencoder in Tensorflow - facial expression low dimensional embedding - Machine learni...

Intuitively Understanding Variational Autoencoders – Towards Data Science


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# Y Intercept

#### What’s the point?

Slope alone just gives you a direction, but that’s not enough. You need to have a starting point, or actually any point along the straight line to separate it from all the other lines going in the same direction but parallel to it. That starting point doesn’t change the slope of the line, it just moves it up or down.

#### TL;DW

• Since any point will do to define exactly which line, why not pick an easy one to calculate?In an equation like:

$\mathbf{y}\ =\ \mathbf{m}\ \ast\ \mathbf{x}\ +\ \mathbf{b}$

if you set X to 0, it’s pretty easy to compute

• Notice that the point on the X axis where X = 0 is, in fact, where the Y axis is. So… where the line you are plotting crosses the Y axis, it’s X value is 0, kind of by definition. 😉
• Another word for “crosses” is “intercepts” and because the line of the function you’re plotting “crosses” or “intercepts” the Y axis where X = 0, the Y value at that “intercept” point is called the “Y Intercept”.

### Where To:

Khan has lots of examples!


text:
College Physics (4th Edition)

The coin will start to slide $2.4~seconds$ after the turntable is turned on.
At low angular speeds, the force of static friction can provide the required centripetal force to keep the coin moving in a circle. We can find the angular speed when the force of static friction reaches its maximum possible value: $F_f = m~a_r$ $mg~\mu_s = m~\omega^2~r$ $\omega = \sqrt{\frac{g~\mu_s}{r}}$ $\omega = \sqrt{\frac{(9.80~m/s^2)(0.110)}{0.130~m}}$ $\omega = 2.88~rad/s$ We can find the time $t$ when the turntable reaches this angular speed: $\omega_f = \omega_0+\alpha~t$ $t = \frac{\omega_f - \omega_0}{\alpha}$ $t = \frac{2.88~rad/s - 0}{1.20~rad/s^2}$ $t = 2.4~s$ The coin will start to slide $2.4~seconds$ after the turntable is turned on.


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# Pouring water from the 10 liter container

The answer to the following "decanting" puzzle

Split 10L in half using 4L and 6L jugs

was

It is impossible to pour out 5 liters from 10 liter jug using 6 and 4 liter jugs

You have three rectangle prism containers: 10, 8 and 6 liters. Their cross-section is shown in the image below. All dimensions are in cm.

The 10 liter container is full of water (10 liters) others are empty.

Can you pour out 5 (+- 0.1) liters of water from it into any other container? Result should be 5 liters of water remaining in the 10 liter container and 5 liters in another container.

You have no marking or measuring tools.You can assume that any pouring you do will be done carefully with little spillage.

• We are assuming a square base? Nov 4, 2021 at 16:21
• According to your diagrams, assuming a square base, the 6 litre container can hold only 3.750 L, The 10 L 10.83 L, and the 8L, 16L. Something is wrong. Nov 4, 2021 at 16:24
• @Chris Cudmore I think it's part of the puzzle. Base is not assumed square, the volume is just prismatic, i.e base is some rectangle. It's possible, I have the dimensions. Nov 4, 2021 at 16:33
• Quick question - how can a 10L container be smaller than an 8L container? Nov 4, 2021 at 22:19
• oh so we should calculate that first? The 8L container is 10cm long in the unshown dimension, and the 6L container is 40cm? Nov 4, 2021 at 23:09

Inspired by athin's answer and DrD's response to it.

The 10 L container will spill half its contents when its edge is lowered to $$\frac{30 * 19}{\sqrt{30^2 + 19^2}} \approx 16 cm$$

This can be achieved like so:

The 8 L container's third dimension is 10 cm. Lay it on its side so it acts as a 10 cm platform. Put the 6 cm tall 6 L container on top of it. Its top edge is now 16 cm above the table. Lean the 10 L into the 6 L container so their edges meet and 5 L will flow from the 10 L container into the 6 L container.

• Assuming infinitely thin container walls, this would split as 4.9775 + 5.0224 , in case there is competition for a better answer. Nov 4, 2021 at 20:21

I'm not sure if this is fine, but

You can just tilt and pour slowly the 10 liter container, until the water surface line connects the two diagonal points.

• how do you exactly know that? Especially the bottom point? I think you are on the right track but need a convincing geometry explanation.
– DrD
Nov 4, 2021 at 13:17
• I didn't have a tool to draw a 3D, but the bottom point is trivial (?) to locate because rot13(gurl ner erpgnatyr cevfz naq lbh whfg arrq gb znxr fher gur yvar vf va cnenyyry jvgu gur jubyr rqtr.) Nov 4, 2021 at 13:20
• I still do not get it. You are pouring the water by tilting the container right? How do you know when you reached the right point?
– DrD
Nov 4, 2021 at 13:29
• @Drd When the water's surface is fully aligned with the edge of the container which is up, and on the opposite side of the one that water exits. Nov 4, 2021 at 13:42

It can be done quite nicely like this:

Use the 6L container standing on its side to measure 25/40 = 5/8 of the way up the 8L container. Then pour up to that point, which is 5L, and we're done!

• Hello @Benjamin Wang. While theortically your answer makes sense, you are essentially "eyeballing" the mark. This may not be the most accurate way. Remember no marking allowed.
– DrD
Nov 4, 2021 at 13:31
• @DrD Can you give an example of a pouring operation that doesn't involve using your eyes to visually confirm that a water level aligns with some other marker? Note that Benjamin's method doesn't require "marking" anything (e.g. with a pen): you just set the container on its side. If this answer is unacceptable, then the question itself seems poorly posed.
– Max
Nov 5, 2021 at 6:51

As all the reasonable solutions seem to be disallowed:

It is trivial to get to 4.8l, simply put 8l container inside the 6l one and fill the 6l one to the edge so it holds 4.8l.

There is no other stacking of containers that is better:

• 10L inside 6L vertically = 4L
• 10L inside 6L horizontally 1 = 2.6L
• 10L inside 6L horizontally 2 = 2.8L
• 8L inside 6L vertically = 4.8L (the best solution)
• 8L inside 6L horizontally 1 = 1.2L
• 8L inside 6L horizontally 2 = 3.6L
• 8L and 10L vertically inside 6L = 2.8L
• 8L vertically and 10L horizontally inside 6L 1 = 1.4L
• 8L vertically and 10L horizontally inside 6L 2 = 1.6L
• 8L inside 10L (if it fits, I haven't calculated): 4L
• 6L on the side inside 10L one (side that fits): 5.5L
• 6L on the side inside 10L one (side that doesn't fit, taking just surface areas): 2.8L
• 6L on the side inside 8L one (doesn't fit, taking surface areas): 2L

Yes, you can think of a sequence of moves that gets you to arbitrary point... if you have 2 extra empty unlimited drums. Otherwise you can't.

• There are a few other, better stackings. Nov 5, 2021 at 10:53
• @alecail I have now added I believe all possibilities, but they aren't better unless I miscalculated. Which option do you see? I assume that that the container put in is vertical - if you are allowed to put things at a particular angle you might be able to do better - but these solutions didn't seem allowed. Nov 5, 2021 at 12:49
• I haven't checked your calculations, but if they are correct, couldn't you do the "8L vert and 10L horiz inside 6L 1 = 1.4L" to get 1.4 L into the 6L, remove the others, pour the 1.4L into the 10L container. Then follow up with "8L inside 6L horizontally 2 = 3.6L" to get 3.6L in the 6L container, remove the 8L container and pour into the 10L container -- you should now have 1.4L + 3.6L = 5.0L in the 10L container with no extra drums. Nov 5, 2021 at 14:32
• I was thinking of the cases where the inside container is at an angle somehow, but it's unclear if this is allowed, even though it is not eyeballing, as the reference points are physical points on the container that need to be in contact. Nov 5, 2021 at 15:51
• @Vaekor Yeah but where are you putting that water in the mean time? As I said, you can reach (nearly?) any point from 0 to 10L with that if you also have 2 unlimited containers. But you can't with just these three. There are some options here that aren't real options anyway - how do you get water in 6L container in the first place if both other containers are squeezed in it? Only if you pour it in first and have it overflow out during squeezing ... but in this case you can't end up with 5+5L then. Nov 5, 2021 at 21:16
• Fill 8 liter container, 2 liters left in the 10 liter container.
• Fill 6 liter container from 8 liter container, now both 10 and 8 liter containers have 2 liters both.
• Pour 2 liters from 8 liter container to 10 liter container. Now 10 liter container has 4 liters and 6 liter container is full.
• Press 8 liter container in to the 6 liter container. This will overflow the 6 liter container and leave 0,9 liters in it. (6 liter container diameters are 25x40x6 and 8 liter container diameters are 20x10x40. When 8 liter container is in 6 liter container, it leaves the volume of 5x30x6 = 0,9 liters)
• Pour the 0,9 liters left in the 6 liter container to the 10 liter container and it will have 4,9 liters which will be in the allowed limits(5 liters +/- 0,1 liter)

This of course assumes that the walls of the containers (especially the 8 liter container) are infinitely thin.

• So $25*40-20*10=5*30$? Nov 5, 2021 at 14:04
• Step 4 is wrong . This would subtract 1.2 L , leaving 4.8 L of actual space. Because (ab)-(cd) != (a-c)*(b-d) in general. Nov 5, 2021 at 19:02


text:
# Zeller's algorithm in Python

I have implemented Zeller's algorithm in Python to calculate the day of the week for some given date. I would like to get some feedback on how I can make the code shorter and more elegant:

def input_value(input_name, prompt, min_num, max_num):
while True:
data = raw_input(prompt)
if data:
try:
input_name = int(data)
except ValueError:
print 'Invalid input...'
else:
if input_name >= min_num and input_name <= max_num:
return input_name
break
input_name = 'Please try again: '
else:
print 'Goodbye!'
break

month = 0
day = 0
year = 0
century = 0

month = input_value(month, "Please enter the month (from 1-12, where March is 1 and February is 12): ", 1, 12)
day = input_value(day, "Please enter the day (from 1-31): ", 1, 31)
year = input_value(year, "Please enter the year (from 0 - 99, eg. 88 in 1988): ", 0, 99)
century = input_value(century, "Please enter the century (from 0 - 99, eg. 19 in 1988): ", 0, 99)

A = month
B = day
C = year
D = century

W = (13*A - 1) / 5
X = C / 4
Y = D / 4
Z = W + X + Y + B + C - 2*D
R = Z % 7

birthday = "You were born on " + str(B) + "/" + str(A+2) + "/" + str(D) + str(C) + " which was a "

if R == 0:
print birthday + 'Sunday'
elif R == 1:
print birthday + 'Monday'
elif R == 2:
print birthday + 'Tuesday'
elif R == 3:
print birthday + 'Wednesday'
elif R == 4:
print birthday + 'Thursday'
elif R == 5:
print birthday + 'Friday'
elif R == 6:
print birthday + 'Saturday'

Thanks in advance for taking the time to look at this.

def input_value(input_name, prompt, min_num, max_num):

input_name is always a number - why are you calling it a 'name'? Also, there's no need to take it as an argument, since you never use its initial value in your function.

    while True:
data = raw_input(prompt)
if data:
try:
input_name = int(data)
except ValueError:
print 'Invalid input...'

Consider putting a continue here, for clarity. It isn't necessary, though.

            else:
if input_name >= min_num and input_name <= max_num:

Python understands if min_num <= input_name <= max_num:

                    return input_name
break

You don't need break here, the return makes it unreachable.

                input_name = 'Please try again: '

Assigning a string to something that's otherwise always a number is odd, and this value is never used anyway. Nix this line.

        else:
print 'Goodbye!'
break

This will end up falling off the end of the function, and so returning None. The rest of your script will continue, but eventually fail horribly. This probably isn't what you want.

month = 0
day = 0
year = 0
century = 0

Only reason to initialise these to 0 is to pass them into input_value, but, as I said above, it never uses the zero values. So, you don't need this.

month = input_value(month, "Please enter the month (from 1-12, where March is 1 and February is 12): ", 1, 12)

March = 1 is an odd system. Consider inputting it as January=1 and then transforming it with:

# Weirdify month counting
month = month - 2 if month < 3 else month + 10

day = input_value(day, "Please enter the day (from 1-31): ", 1, 31)
year = input_value(year, "Please enter the year (from 0 - 99, eg. 88 in 1988): ", 0, 99)
century = input_value(century, "Please enter the century (from 0 - 99, eg. 19 in 1988): ", 0, 99)

These lines are very long. You should wrap them per PEP8:

 year = input_value("Please enter the century (from 0 - 99, eg. 19 in 1988): ",
0, 99)

Also, you might consider inputting year and century as a single value ('1988') and separating it in the script.

A = month
B = day
C = year
D = century

Just use the long names throughout your formulae, eg X = year / 4.

W = (13*A - 1) / 5
X = C / 4
Y = D / 4
Z = W + X + Y + B + C - 2*D
R = Z % 7

birthday = "You were born on " + str(B) + "/" + str(A+2) + "/" + str(D) + str(C) + " which was a "

Use string formatting instead of concatenation: birthday = "You were born on {}/{}/{}, which was a {}{}".format(day, month+2, century, year)

if R == 0:
print birthday + 'Sunday'
elif R == 1:
print birthday + 'Monday'
elif R == 2:
print birthday + 'Tuesday'
elif R == 3:
print birthday + 'Wednesday'
elif R == 4:
print birthday + 'Thursday'
elif R == 5:
print birthday + 'Friday'
elif R == 6:
print birthday + 'Saturday'

This can go into a list:

 days = ['Sunday', 'Monday', 'Tuesday', 'Wednesday',
'Thursday', 'Friday', 'Saturday']
day_of_week = days[R]

And then that can go in the above string formatting instead of being concatenated on later.

• @Ivc thanks, this is really great advice. There are two points which I don't fully understand. I don't know how to use continue, even after looking it up on the python docs. I also don't fully understand the 'string formatting' and 'listing days of the week' point. Can you offer any more advice here? – Tom Kadwill Feb 12 '12 at 15:44
• continue is sortof like break, except that it goes back to the top of the loop instead of falling out the bottom. For string formatting, look up the documentation for str.format(), and the "String Formatting mini language". Basically it achieves the same thing as concatenating all your variables into the string, except its easier to read, and potentially more efficient (especially if your code ever gets run on, say, PyPy). For the days of the week, days[0] is 'Sunday' - it avoids the chain of if/elif, and will error instead of failing silently if your R is >6 somehow. – lvc Feb 13 '12 at 3:51


text:
# zbMATH — the first resource for mathematics

## Zhang, Qing

Compute Distance To:
 Author ID: zhang.qing Published as: Zhang, Q.; Zhang, Qing Homepage: http://www.math.uga.edu/directory/qing-zhang External Links: MGP · Wikidata · dblp
 Documents Indexed: 188 Publications since 1988, including 10 Books
all top 5

#### Co-Authors

 13 single-authored 73 Yin, Gang George 25 Sethi, Suresh P. 15 Yin, George Gang 14 Zhang, Hanqin 13 Liu, Ruihua 9 Boukas, El-Kébir 7 Liu, Yuanjin 6 Badowski, Grazyna 5 Tie, Jingzhi 5 Yin, Kewen 4 Nguyen, Duy Khuong 4 Yan, Houmin 4 Zhang, Caojin 3 Dai, Min 3 Haussmann, Ulrich G. 3 Kannan, Devika 3 Khas’minskiĭ, Rafail Zalmanovich 3 Moore, John Barratt 3 Pemy, Moustapha 3 Presman, Ernst L. 3 Song, Qingshuo 3 Taksar, Michael I. 3 Wu, Zhen 3 Yang, Hailang 3 Zhou, Xunyu 3 Zhu, Qiji Jim 2 Fleming, Wendell Helms 2 Guo, Xianping 2 Kong, Hoi Tin 2 Li, Xiaofan 2 Prager, David 2 Rushton, J. Nelson 2 Suo, Wulin 2 Wang, Jiawen 2 Wang, Leyi 2 Yang, Zhixin 2 Yu, Jie 1 Bao, Jianyang 1 Belu, A. 1 Bian, Baojun 1 Cheng, Yaosong 1 Duncan, Tyrone E. 1 Eloe, Paul W. 1 Gershon, Y. 1 Guo, Xin 1 He, Qi 1 Hou, Yumei 1 Jiang, Jiewei 1 Jiang, Lishang 1 Kuo, Kevin 1 Liu, Feilei 1 Liu, Qiaoge 1 Liu, Qinggui 1 Liu, Qinguo 1 Liu, Ruyi 1 Lu, Xianggang 1 Luu, Phong Thanh 1 Lv, Siyu 1 McEneaney, William M. 1 Miao, Shuming 1 Pasik-Duncan, Bozenna 1 Perkerson, Eric 1 Soner, Halil Mete 1 Song, Dongping 1 Tao, Ran 1 Thompson, Katherine Jenny 1 Wang, Jianwen 1 Wang, Jinwei 1 Yang, Jie 1 Yang, Jie 1 Yang, Zhou 1 Yao, David D. W. 1 Yatsuki, M. 1 Yu, Linmei 1 Zhang, Hongwei 1 Zhong, Yifei 1 Zhou, Xiaoye 1 Zhuang, Chengsan 1 Zhuang, Chungang 1 Zhuang, Ci
all top 5

#### Serials

 19 Journal of Optimization Theory and Applications 11 Journal of Mathematical Analysis and Applications 10 SIAM Journal on Control and Optimization 8 Automatica 8 IEEE Transactions on Automatic Control 6 Stochastics and Stochastics Reports 5 Applied Mathematics and Optimization 5 Stochastic Analysis and Applications 5 Mathematical Control and Related Fields 4 Dynamic Systems and Applications 4 SIAM Journal on Applied Mathematics 3 Journal of Systems Science and Complexity 2 Mathematics of Operations Research 2 Optimal Control Applications & Methods 2 The Annals of Applied Probability 2 Dynamics of Continuous, Discrete and Impulsive Systems 2 Mathematical Finance 2 Discrete and Continuous Dynamical Systems. Series B 2 Communications in Information and Systems 2 Stochastic Modelling and Applied Probability 2 Nonlinear Analysis. Theory, Methods & Applications 2 Applications of Mathematics 1 Advances in Applied Probability 1 IEEE Transactions on Information Theory 1 Journal of Multivariate Analysis 1 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 1 Quarterly of Applied Mathematics 1 Systems & Control Letters 1 Journal of Systems Science and Mathematical Sciences 1 IMA Journal of Mathematical Control and Information 1 MCSS. Mathematics of Control, Signals, and Systems 1 Journal of Applied Mathematics and Stochastic Analysis 1 IEEE Transactions on Signal Processing 1 M$$^3$$AS. Mathematical Models & Methods in Applied Sciences 1 International Journal of Computer Mathematics 1 Stochastic Processes and their Applications 1 SIAM Journal on Optimization 1 Neural, Parallel & Scientific Computations 1 Acta Mathematica Sinica. English Series 1 Probability in the Engineering and Informational Sciences 1 The ANZIAM Journal 1 Multiscale Modeling & Simulation 1 Journal of Control Theory and Applications 1 Contemporary Mathematics 1 The IMA Volumes in Mathematics and its Applications 1 Lecture Notes in Control and Information Sciences 1 International Series in Operations Research & Management Science 1 Nonlinear Analysis. Hybrid Systems 1 SIAM Journal on Financial Mathematics
all top 5

#### Fields

 103 Systems theory; control (93-XX) 76 Probability theory and stochastic processes (60-XX) 58 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 52 Operations research, mathematical programming (90-XX) 25 Calculus of variations and optimal control; optimization (49-XX) 12 Ordinary differential equations (34-XX) 9 Statistics (62-XX) 7 Numerical analysis (65-XX) 5 General and overarching topics; collections (00-XX) 2 Information and communication theory, circuits (94-XX) 1 Partial differential equations (35-XX) 1 Difference and functional equations (39-XX) 1 Operator theory (47-XX)

#### Citations contained in zbMATH

118 Publications have been cited 1,196 times in 593 Documents Cited by Year
Continuous-time Markov chains and applications. A singular perturbation approach. Zbl 0896.60039
Yin, G. George; Zhang, Qing
1998
Hierarchical decision making in stochastic manufacturing systems. Zbl 0923.90002
Sethi, S. P.; Zhang, Q.
1994
Stock trading: an optimal selling rule. Zbl 0990.91014
Zhang, Qing
2001
Closed-form solutions for perpetual American put options with regime switching. Zbl 1061.90082
Guo, Xin; Zhang, Qing
2004
Recursive algorithms for stock liquidation: a stochastic optimization approach. Zbl 1021.91022
Yin, G.; Liu, R. H.; Zhang, Q.
2002
Discrete-time Markov chains. Two-time-scale methods and applications. Zbl 1072.60002
Yin, G. George; Zhang, Qing
2005
Optimal selling rules in a regime switching model. Zbl 1365.60062
Guo, X.; Zhang, Q.
2005
Asymptotic expansions of singularly perturbed systems involving rapidly fluctuating Markov chains. Zbl 0849.34047
Khasminskii, R. Z.; Yin, G.; Zhang, Q.
1996
Trend following trading under a regime switching model. Zbl 1198.91246
Dai, M.; Zhang, Qing; Zhu, Q. J.
2010
A regime-switching model for European options. Zbl 1136.91015
Yao, David D.; Zhang, Qing; Zhou, Xun Yu
2006
Continuous-time Markov chains and applications. A two-time-scale approach. 2nd ed. Zbl 1277.60127
Yin, G. George; Zhang, Qing
2013
Zhang, Hanqin; Zhang, Qing
2008
Option pricing in a regime-switching model using the fast Fourier transform. Zbl 1140.91402
Liu, R. H.; Zhang, Q.; Yin, G.
2006
Robust production and maintenance planning in stochastic manufacturing systems. Zbl 0837.90054
Boukas, E. K.; Zhang, Q.; Yin, G.
1995
Valuation of stock loans with regime switching. Zbl 1188.91226
Zhang, Qing; Zhou, Xun Yu
2009
Asymptotic properties of a singularly perturbed Markov chain with inclusion of transient states. Zbl 1054.60078
Yin, G.; Zhang, Q.; Badowski, G.
2000
Nearly-optimal asset allocation in hybrid stock investment models. Zbl 1090.91049
Zhang, Q.; Yin, G.
2004
Discrete-time singularly perturbed Markov chains: Aggregation, occupation measures, and switching diffusion limit. Zbl 1048.60058
Yin, G.; Zhang, Q.; Badowski, G.
2003
Optimal selling rules in a regime-switching exponential Gaussian diffusion model. Zbl 1175.91079
Eloe, P.; Liu, R. H.; Yatsuki, M.; Yin, G.; Zhang, Q.
2008
Optimal stock liquidation in a regime switching model with finite time horizon. Zbl 1160.91359
Pemy, M.; Zhang, Q.
2006
Optimal production planning in a stochastic manufacturing system with long-run average cost. Zbl 0886.90084
Sethi, S. P.; Suo, W.; Taksar, M. I.; Zhang, Q.
1997
Constructing asymptotic series for probability distributions of Markov chains with weak and strong interactions. Zbl 0870.34058
Khasminskii, R. Z.; Yin, G.; Zhang, Q.
1997
Stochastic optimization methods for buying-low-and-selling-high strategies. Zbl 1163.62341
Song, Q. S.; Yin, G.; Zhang, Q.
2009
Liquidation of a large block of stock with regime switching. Zbl 1214.91106
Pemy, Moustapha; Zhang, Qing; Yin, G. George
2008
Singularly perturbed discrete-time Markov chains. Zbl 0972.60068
Yin, G.; Zhang, Q.
2000
Hybrid filtering for linear systems with non-Gaussian disturbances. Zbl 0978.93075
Zhang, Qing
2000
On nearly optimal controls of hybrid LQG problems. Zbl 1136.93466
Zhang, Qing; Yin, Gang George
1999
Hierarchical controls in stochastic manufacturing systems with machines in tandem. Zbl 0757.60101
Sethi, Suresh; Zhang, Qing; Zhou, Xun Yu
1992
An optimal pairs-trading rule. Zbl 1358.93191
Song, Qingshuo; Zhang, Qing
2013
A numerical method in optimal production and setup scheduling of stochastic manufacturing systems. Zbl 0892.90094
Yan, H.; Zhang, Q.
1997
Stock liquidation via stochastic approximation using Nasdaq daily and intra-day data. Zbl 1128.91031
Yin, G.; Zhang, Q.; Liu, F.; Liu, R. H.; Cheng, Y.
2006
A near-optimal selling rule for a two-time-scale market model. Zbl 1108.91031
Zhang, Q.; Yin, G.; Liu, R. H.
2005
Optimal control of a marketing-production system. Zbl 1017.90035
Zhang, Qing; Yin, Gang George; Boukas, El-Kebir
2001
Risk-sensitive production planning of a stochastic manufacturing system. Zbl 0915.93069
Fleming, W. H.; Zhang, Q.
1998
Optimal filtering of discrete-time hybrid systems. Zbl 0926.93064
Zhang, Qing
1999
Turnpike sets and their analysis in stochastic production planning problems. Zbl 0770.90030
Sethi, S.; Soner, H. M.; Zhang, Q.; Jiang, J.
1992
Minimax production planning in failure-prone manufacturing systems. Zbl 0839.90055
Boukas, E. K.; Yang, H.; Zhang, Q.
1995
Optimal feedback production planning in a stochastic $$N$$-machine flowshop. Zbl 0834.90068
Presman, Ernst; Sethi, Suresh; Zhang, Qing
1995
Piecewise deterministic Markov process model for flexible manufacturing systems with preventive maintenance. Zbl 0824.90071
Boukas, E. K.; Zhu, Q.; Zhang, Q.
1994
Structural properties of Markov chains with weak and strong interactions. Zbl 0913.60068
Zhang, Q.; Yin, G.
1997
Nonlinear filtering and control of a switching diffusion with small observation noise. Zbl 0911.93050
Zhang, Qing
1998
A central limit theorem for singularly perturbed nonstationary finite state Markov chains. Zbl 0855.60018
Zhang, Q.; Yin, G.
1996
Multilevel hierarchical decision making in stochastic marketing- production systems. Zbl 0826.93070
Sethi, S. P.; Zhang, Q.
1995
Large deviations for two-time-scale systems driven by nonhomogeneous Markov chains and associated optimal control problems. Zbl 1228.60083
He, Qi; Yin, George; Zhang, Qing
2011
Average-cost control of stochastic manufacturing systems. Zbl 1078.93004
Sethi, Suresh; Zhang, Hanqin; Zhang, Qing
2005
Risk-sensitive production planning of stochastic manufacturing systems: A singular perturbation approach. Zbl 0834.90072
Zhang, Qing
1995
Optimal trend following trading rules. Zbl 1336.91063
Dai, Min; Yang, Zhou; Zhang, Qing; Zhu, Qiji Jim
2016
BSDEs with regime switching: weak convergence and applications. Zbl 1306.60080
Tao, Ran; Wu, Zhen; Zhang, Qing
2013
Constrained stochastic estimation algorithms for a class of hybrid stock market models. Zbl 1050.91053
Yin, G.; Zhang, Q.; Yin, K.
2003
Nearly optimal control of singularly perturbed Markov decision processes in discrete time. Zbl 0990.90125
Liu, R. H.; Zhang, Q.; Yin, G.
2001
Asymptotic optimal controls in stochastic manufacturing systems with machine failures dependent on production rates. Zbl 0828.93065
Sethi, Suresh P.; Zhang, Qing
1994
Hierarchical production planning in dynamic stochastic manufacturing systems: Asymptotic optimality and error bounds. Zbl 0800.93062
Sethi, Suresh; Zhang, Qing
1994
An optimal trading rule under a switchable mean-reversion model. Zbl 1296.91256
Nguyen, Duy; Tie, Jingzhi; Zhang, Qing
2014
Weak convergence methods for approximation of the evaluation of path-dependent functionals. Zbl 1279.93098
Song, Qingshuo; Yin, George; Zhang, Qing
2013
Optimal decision for selling an illiquid stock. Zbl 1251.93017
Bian, Baojun; Dai, Min; Jiang, Lishang; Zhang, Qing; Zhong, Yifei
2011
An optimal trading rule of a mean-reverting asset. Zbl 1203.91323
Kong, Hoi Tin; Zhang, Qing
2010
Stock loan valuation under a regime-switching model with mean-reverting and finite maturity. Zbl 1198.91212
Prager, David; Zhang, Qing
2010
Discrete-time singularly perturbed Markov chains. Zbl 1031.60067
Yin, G.; Zhang, Q.
2003
Robust subspace clustering for multi-view data by exploiting correlation consensus. Zbl 1408.94694
Wang, Yang; Lin, Xuemin; Wu, Lin; Zhang, Wenjie; Zhang, Qing; Huang, Xiaodi
2015
Explicit solutions for an optimal stock selling problem under a Markov chain model. Zbl 1306.91134
Zhang, Q.
2014
A fluid flow model for empty container repositioning policy with a single port and stochastic demand. Zbl 1202.93177
Song, Dong-Ping; Zhang, Qing
2010
Stochastic optimization algorithms for pricing American put options under regime-switching models. Zbl 1109.91362
Yin, G.; Wang, J. W.; Zhang, Q.; Liu, Y. J.
2006
Asymptotically optimal production policies in dynamic stochastic jobshops with limited buffers. Zbl 1139.90355
Hou, Yumei; Sethi, Suresh P.; Zhang, Hanqin; Zhang, Qing
2006
Control of singularly perturbed Markov chains: A numerical study. Zbl 1054.93055
Yang, H.; Yin, G.; Yin, K.; Zhang, Q.
2003
Minimum average cost production planning in stochastic manufacturing systems. Zbl 0952.93133
Sethi, Suresh P.; Zhang, Hanqin; Zhang, Qing
1998
Hierarchical production control in a stochastic manufacturing system with long-run average cost. Zbl 0894.90074
Sethi, Suresh P.; Zhang, Hanqin; Zhang, Qing
1997
Turnpike sets in stochastic manufacturing systems with finite time horizon. Zbl 0847.90029
Zhang, Q.; Yin, G.
1994
Hierarchical controls in stochastic manufacturing systems with convex costs. Zbl 0797.90039
Sethi, S. P.; Zhang, Q.; Zhou, X. Y.
1994
Controlled partially observed diffusions with correlated noise. Zbl 0714.93064
Zhang, Qing
1990
Stochastic adaptive control with small observation noise. Zbl 0745.93081
Haussmann, U. G.; Zhang, Qing
1990
An optimal strategy for pairs trading under geometric Brownian motions. Zbl 1418.91491
Tie, Jingzhi; Zhang, Hanqin; Zhang, Qing
2018
A partitioned finite element scheme based on Gauge-Uzawa method for time-dependent MHD equations. Zbl 1388.76160
Zhang, Qing; Su, Haiyan; Feng, Xinlong
2018
Multiple failure modes reliability modeling and analysis in crack growth life based on JC method. Zbl 1426.74264
Sun, YuanTao; Liu, Chao; Zhang, Qing; Qin, XianRong
2017
Building up an illiquid stock position subject to expected fund availability: optimal controls and numerical methods. Zbl 1410.91495
Lu, Xianggang; Yin, George; Zhang, Qing; Zhang, Caojin; Guo, Xianping
2017
Asset allocation for regime-switching market models under partial observation. Zbl 1307.91170
Yu, L.; Zhang, Q.; Yin, G.
2014
Asymptotically optimal controls of hybrid linear quadratic regulators in discrete time. Zbl 1081.93528
Liu, R. H.; Zhang, Q.; Yin, G.
2002
Nearly optimal control of nonlinear Markovian systems subject to weak and strong interactions. Zbl 1011.93112
Liu, R. H.; Zhang, Q.; Yin, G.
2001
Hierarchical production control in dynamic stochastic jobshops with long-run average cost. Zbl 0992.90025
Sethi, S. P.; Zhang, H.; Zhang, Q.
2000
Singularly perturbed Markov chains: Convergence and aggregation. Zbl 0951.60034
Yin, G.; Zhang, Q.; Badowski, G.
2000
A numerical study of singularly perturbed Markov chains: Quasi-equilibrium distributions and scaled occupation measures. Zbl 0946.60074
Li, X.; Yin, G.; Yin, K.; Zhang, Q.
1999
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1996
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1993
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2001
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2001
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2001
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2001
Discrete-time dynamic systems arising from singularly perturbed Markov chains. Zbl 1042.60523
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2001
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#### Cited by 853 Authors

 82 Yin, Gang George 72 Zhang, Qing 26 Yin, George Gang 21 Boukas, El-Kébir 21 Sethi, Suresh P. 12 Wang, Leyi 12 Zhu, Chao 10 Jin, Zhuo 10 Zhang, Hanqin 8 Costa, Oswaldo Luiz V. 8 Siu, Tak Kuen 7 Kenne, Jean-Pierre 7 Liu, Ruihua 7 Liu, Yuanjin 7 Mao, Xuerong 7 Nguyen, Son Luu 7 Shi, Peng 7 Song, Qingshuo 7 Wu, Zhen 7 Yang, Hailiang 7 Yuan, Chenggui 6 Blanchini, Franco 6 Nguyen, Dung Tien 6 Xi, Fubao 5 Badowski, Grazyna 5 Gharbi, Ali 5 Özekici, Süleyman 5 Pemy, Moustapha 5 Wu, Fuke 5 Zheng, Harry H. 4 Bian, Baojun 4 Dai, Min 4 Gaitsgory, Vladimir A. 4 Ghosh, Mrinal Kanti 4 Goreac, Dan 4 Khas’minskiĭ, Rafail Zalmanovich 4 Leung, Tim 4 Li, Xiaoyue 4 Limnios, Nikolaos 4 Lu, Xianggang 4 Nguyen, Duy Khuong 4 Ocejo, Adriana 4 Petersson, Mikael 4 Tie, Jingzhi 4 Yang, Hailang 4 Yin, Gang 3 Bauso, Dario 3 Bounkhel, Messaoud 3 Çanakoğlu, Ethem 3 Elliott, Robert James 3 Fan, Kun 3 Guo, Xianping 3 Han, Zhengzhi 3 Hoang, Tuan Anh 3 Hu, Junhao 3 Il’in, Arlen Mikhailovich 3 Kan, Shaobai 3 Khaliq, Abdul Q. M. 3 Korolyuk, Volodymyr Semenovych 3 Ledoux, James 3 Liu, Jingzhen 3 Liu, RongHua 3 Liu, Yue 3 Lv, Siyu 3 Ma, Jingtang 3 Ma, Yuechao 3 Menoukeu Pamen, Olivier 3 Pesenti, Raffaele 3 Presman, Ernst L. 3 Rossomakhine, Sergey 3 Shao, Jinghai 3 Shen, Yang 3 Sun, Zhongyang 3 Suo, Wulin 3 Tadj, Lotfi 3 Tao, Ran 3 Thieullen, Michèle 3 Tran, Ky Quan 3 Wang, Rongming 3 Wang, Zheng 3 Xiong, Jie 3 Yan, Houmin 3 Yiu, Ka Fai Cedric 3 Yoshioka, Hidekazu 3 Zhang, Junfeng 3 Zhou, Xun Yu 3 Zhu, Fubo 3 Zhu, Songping 2 Altman, Eitan 2 Anabtawi, M. J. 2 Baran, Nicholas A. 2 Baten, Md. Azizul 2 Benhadid, Yacine 2 Benites, Guilherme R. A. M. 2 Bensoussan, Alain 2 Bin, Ning 2 Buckdahn, Rainer 2 Bui, Trang 2 Cai, Ning 2 Camilli, Fabio ...and 753 more Authors
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#### Cited in 151 Serials

 49 Automatica 39 Journal of Optimization Theory and Applications 27 Journal of Mathematical Analysis and Applications 26 European Journal of Operational Research 18 Stochastic Analysis and Applications 14 Stochastics 13 Quantitative Finance 12 Applied Mathematics and Optimization 12 Journal of Computational and Applied Mathematics 12 SIAM Journal on Control and Optimization 12 Stochastic Processes and their Applications 12 International Journal of Theoretical and Applied Finance 10 Computers & Mathematics with Applications 10 Nonlinear Analysis. Hybrid Systems 9 Insurance Mathematics & Economics 8 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 8 Mathematical Control and Related Fields 7 Advances in Applied Probability 7 Systems & Control Letters 7 Acta Mathematicae Applicatae Sinica. English Series 7 The Annals of Applied Probability 6 Communications in Statistics. Theory and Methods 6 International Journal of Computer Mathematics 6 Mathematical Finance 6 Methodology and Computing in Applied Probability 5 Journal of the Franklin Institute 5 Applied Mathematics and Computation 5 Optimal Control Applications & Methods 5 Operations Research Letters 5 Journal of Economic Dynamics & Control 5 Applied Mathematical Modelling 5 Abstract and Applied Analysis 5 Mathematical Methods of Operations Research 5 Nonlinear Analysis. Real World Applications 4 International Journal of Control 4 International Journal of Production Research 4 Journal of Scientific Computing 4 Annals of Operations Research 4 Mathematical Problems in Engineering 4 Journal of Systems Science and Complexity 3 International Journal of Systems Science 3 Journal of Applied Probability 3 Journal of Differential Equations 3 Statistics & Probability Letters 3 Applied Numerical Mathematics 3 Mathematical and Computer Modelling 3 International Journal of Robust and Nonlinear Control 3 Communications in Nonlinear Science and Numerical Simulation 3 Discrete and Continuous Dynamical Systems. Series B 3 Advances in Difference Equations 3 Journal of Control Theory and Applications 3 Journal of Industrial and Management Optimization 3 Mathematics and Financial Economics 3 SIAM Journal on Financial Mathematics 3 Asian Journal of Control 3 Nonlinear Analysis. Theory, Methods & Applications 3 International Journal of Systems Science. Principles and Applications of Systems and Integration 2 Mathematics of Operations Research 2 Naval Research Logistics 2 Quarterly of Applied Mathematics 2 SIAM Journal on Numerical Analysis 2 Computers & Operations Research 2 Asia-Pacific Journal of Operational Research 2 Applied Mathematics Letters 2 MCSS. Mathematics of Control, Signals, and Systems 2 Journal of Applied Mathematics and Stochastic Analysis 2 Discrete Event Dynamic Systems 2 Linear Algebra and its Applications 2 NoDEA. Nonlinear Differential Equations and Applications 2 Applied Mathematical Finance 2 Finance and Stochastics 2 European Journal of Control 2 Soft Computing 2 Discrete Dynamics in Nature and Society 2 The ANZIAM Journal 2 Scandinavian Actuarial Journal 2 Stochastic Models 2 ASTIN Bulletin 2 Asia-Pacific Financial Markets 2 Journal of Biological Dynamics 2 Frontiers of Mathematics in China 2 Set-Valued and Variational Analysis 2 Symmetry 2 Annals of Finance 1 Journal of Computational Physics 1 Journal of Mathematical Biology 1 Journal of Mathematical Physics 1 Lithuanian Mathematical Journal 1 Mathematical Biosciences 1 Mathematical Methods in the Applied Sciences 1 Metrika 1 Nonlinearity 1 Physica A 1 Bulletin of Mathematical Biology 1 Information Sciences 1 International Journal of Mathematics and Mathematical Sciences 1 Journal of Functional Analysis 1 Journal of Mathematical Economics 1 Journal of Multivariate Analysis 1 Mathematics and Computers in Simulation ...and 51 more Serials
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#### Cited in 31 Fields

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text:
## The Annals of Statistics

### Isotropy and Sphericity: Some Characterisations of the Normal Distribution

Gerard Letac

#### Abstract

Main result: $X_1, X_2, \cdots, X_n$ are independent random variables valued in Euclidean spaces $E_1, E_2, \cdots, E_n$ such that $P\lbrack X_j = 0 \rbrack = 0$ for all $j$. Denote $R = \lbrack \sum^n_{j = 1} \|X_j\|^2 \rbrack^{1/2}$. Suppose that $(R^{-1}X_1, R^{-1}X_2, \cdots, R^{-1}X_n)$ is uniformly distributed on the sphere of $\oplus^n_{j = 1} E_j$. Then the $X_j$ are normal if $n \geq 3$. The case $n = 2$ and the case of Hilbert spaces are also studied.

#### Article information

Source
Ann. Statist., Volume 9, Number 2 (1981), 408-417.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176345406

Digital Object Identifier
doi:10.1214/aos/1176345406

Mathematical Reviews number (MathSciNet)
MR606624

Zentralblatt MATH identifier
0462.62014

JSTOR


text:
# When does strict inclusion holds for the domain of subdifferential?

Recall that, given an extended real-valued function $$f: \mathbb{R}^n \to (-\infty, \infty]$$

Its effective domain is, $$\text{dom}(f) = \{x \in \mathbb{R}^n : f(x) < +\infty\}$$

The subdifferential is, $$\partial f(x) = \{v \in \mathbb{R}^n: f(x^\prime) \geq f(x) + v^\top (x^\prime - x), \forall x^\prime \in \mathbb{R}^n\}$$

and its effective domain is all the vectors where it is subdifferentiable,

$$\text{dom}(\partial f) = \{x \in \mathbb{R}^n | \partial f(x) \neq \varnothing\}.$$

Now by a theorem of Rockafellar (top of page 227, "Convex Analysis", 1970), it states,

Let $$f: \mathbb{R}^n \to (-\infty, \infty]$$ be a closed, proper, convex function, then $$\text{rint}(\text{dom}(f))) \subseteq \text{dom}(\partial f) \subseteq \text{dom}(f).$$ where $$\text{rint}$$ is the relative interior.

My question is, when does strict inclusion holds, especially for $$\text{dom}(\partial f) \subseteq \text{dom}(f)$$.

For any set I can think of that is closed, you can make a point on the boundary of that set, and there will exist some subgradient at that point. There is the possibility that this strict inequality is an equality for most cases.

• You seem to have posted this twice. Please remove one of the copies. Jan 25, 2021 at 22:51

Define $$f\colon\mathbb R\to\mathbb R$$ by letting $$f(x):=-\sqrt x$$ if $$x\ge0$$ and $$f(x):=\infty$$ if $$x<0$$. Then $$f$$ is a closed proper convex function.
However, $$f(0)=0<\infty$$, so that $$0\in\text{dom}(f)$$. On the other hand, $$f(y)=-\sqrt y for each real $$v$$ and any small enough $$y>0$$. So, $$0\notin\text{dom}(\partial f)$$.
Thus, here the set inclusion $$\text{dom}(\partial f)\subset\text{dom}(f)$$ is strict.
• I wonder if it is simpler to just make the observation that $f(x) = -\sqrt(x) + \delta_{x \geq 0}(x)$, so $\partial f(x) = -\frac{1}{2} x^{-1/2} +N_{x \geq 0}(x)$ and the first term is not defined at $x = 0$. Jan 26, 2021 at 21:53
• We can always write $\{-\frac{1}{2} x^{-1/2}\}$ as a singleton set. This is what I meant. The value in this set is not defined as $x \to 0$. Is this a simpler way to see that this inclusion is strict? Jan 26, 2021 at 22:24


text:
# How do you find the slope and intercept of -4x+2y=12?

Apr 10, 2016

Slope: $2$
y-intercept: $6$
(x-intercept: $- 3$)

#### Explanation:

slope
Given an equation in the form $A x + B y = C$ the slope is $\left(- \frac{A}{B}\right)$
So $- 4 x + 2 y = 12$ has a slope of $- \frac{- 4}{2} = 2$

Alternately, we could convert the given form
into "slope-intercept form": $y = m x + b$ with slope $m$ and y-intercept $b$
$\textcolor{w h i t e}{\text{XXX}} - 4 x + 2 y = 12$
$\textcolor{w h i t e}{\text{XXX}} \rightarrow 2 y = 4 x + 12$
$\textcolor{w h i t e}{\text{XXX}} \rightarrow y = 2 x + 6$

y-intercept
If we used the "slope-intercept form" (above) we already have the y-intercept: $6$;
otherwise we can determine the y-intercept by finding the value of $y$ when $x = 0$ in the given equation: $- 4 x + 2 y = 12$
$\textcolor{w h i t e}{\text{XXX}} - 4 \times \left(0\right) + 2 y = 12$
$\textcolor{w h i t e}{\text{XXX}} \rightarrow y = 6$

x-intercept (note sometimes when "the intercept" is asked for, what is meant is only the y-intercept)
The x-intercept can be determined by find the value of $x$ when $y = 0$ in the given equation: $- 4 x + 2 y = 12$
$\textcolor{w h i t e}{\text{XXX}} - 4 x + 2 \times \left(0\right) = 12$
$\textcolor{w h i t e}{\text{XXX}} \rightarrow x = - 3$


text:
D - Card Eater

Time Limit: 2 sec / Memory Limit: 256 MB

### 問題文

すぬけくんはカードゲームで遊ぶことにしました。 N 枚からなるカードの山があり、上から i 枚目のカードには整数 A_i が書かれています。

すぬけくんはこのカードの山に対し 0 回以上、以下の操作を行い、残ったカードに書かれた値が互いに異なるようにしたいです。最大で何枚のカードを残すことが可能か求めなさい。なお、N は奇数であり、少なくとも 1 枚のカードを残すことが可能であることが保証されます。

### 制約

• 3 ≦ N ≦ 10^{5}
• N は奇数
• 1 ≦ A_i ≦ 10^{5}
• A_i は整数

### 入力

N
A_1 A_2 A_3 ... A_{N}

### 入力例 1

5
1 2 1 3 7

### 出力例 1

3

### 入力例 2

15
1 3 5 2 1 3 2 8 8 6 2 6 11 1 1

### 出力例 2

7

Score : 400 points

### Problem Statement

Snuke has decided to play a game using cards. He has a deck consisting of N cards. On the i-th card from the top, an integer A_i is written.

He will perform the operation described below zero or more times, so that the values written on the remaining cards will be pairwise distinct. Find the maximum possible number of remaining cards. Here, N is odd, which guarantees that at least one card can be kept.

Operation: Take out three arbitrary cards from the deck. Among those three cards, eat two: one with the largest value, and another with the smallest value. Then, return the remaining one card to the deck.

### Constraints

• 3 ≦ N ≦ 10^{5}
• N is odd.
• 1 ≦ A_i ≦ 10^{5}
• A_i is an integer.

### Input

The input is given from Standard Input in the following format:

N
A_1 A_2 A_3 ... A_{N}

### Sample Input 1

5
1 2 1 3 7

### Sample Output 1

3

One optimal solution is to perform the operation once, taking out two cards with 1 and one card with 2. One card with 1 and another with 2 will be eaten, and the remaining card with 1 will be returned to deck. Then, the values written on the remaining cards in the deck will be pairwise distinct: 1, 3 and 7.

### Sample Input 2

15
1 3 5 2 1 3 2 8 8 6 2 6 11 1 1

### Sample Output 2

7


text:
# What is Algebra?

Isn’t it funny that so many people talk about algebra, do algebra, or even fear algebra but have never thought about what it actually is? Well, let’s take care of that right now. (and for the record, I’m talking about elementary algebra not abstract algebra which is a completely awesome, but a different thing).

## Back to arithmetic

Think back to when you were learning adding and subtracting, multiplying and dividing – all that stuff. When you do arithmetic, you deal with specific numbers. In algebra, we shift that perspective into something more general and  more abstract. Instead of talking about specific numbers, we will start including variables, which represent either a specific unknown number or a general number that we could allow to be anything.

## Why would variables be useful?

Kaitlyn’s mom always gives \$5 more to Kaitlyn than to her brother Alex.

This means that if Alex gets \$10, then Kaitlyn gets \$15 right? In fact, we could keep doing this forever! If the mom wins the lotto and gives Alex \$50,000, then Kaitlyn should be getting \$50,005. (Kind of weird but go with it.)

Variables can describe what is happening in general. If we let $x$ be the amount that Alex gets, then we know that $x+5$ is how much Kaitlyn will get.

Not only does this show the relationship between the two in a compact way, but it also allows a deeper analysis. Imagine if the situation was more complicated. You can’t just explain it in words to a computer so it can figure it out. Variables would have to be used!

## It’s just a different way of looking at things

In the end, think of algebra as a shift in perspective. You now have the freedom to let a symbol stand in when you don’t know the value of a number or want to know what happens when it is any number. This is a powerful ability! Algebra really is the study of expressions (and equations) with unknown values. What general rules apply? Are there certain situations where some rule is not true? These are the questions you can answer with algebra.


text:
#### Volume 11, issue 3 (2011)

 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Subscriptions Author Index To Appear Contacts ISSN (electronic): 1472-2739 ISSN (print): 1472-2747
Obtaining presentations from group actions without making choices

### Andrew Putman

Algebraic & Geometric Topology 11 (2011) 1737–1766
##### Abstract

Consider a group $G$ acting nicely on a simply-connected simplicial complex $X$. Numerous classical methods exist for using this group action to produce a presentation for $G$. For the case that $X∕G$ is $2$–connected, we give a new method that has the novelty that one does not have to identify a fundamental domain for the action. Indeed, the resulting presentation is canonical in the sense that no arbitrary choices need to be made. It can be viewed as a nonabelian analogue of a simple result in the study of equivariant homology.

##### Keywords
group presentations, group actions
##### Mathematical Subject Classification 2010
Primary: 20F05, 57M07