A Neat Trick For Determining The Integrals Of exp(x) cos x And exp(x) sin x

The standard method (typically found in first-year calculus textbooks) for determining the integrals

\int e^x \cos x \, {\rm d}x


\int e^x \sin x \, {\rm d}x

is to integrate by parts twice. If you haven’t seen the standard method, I’ll show you how to do the first one; the second one is similar. Later in the post I’ll show you the neat trick for determining both integrals at once.

It doesn’t much matter whether you let u represent the exponential function or the trigonometric function in the first integration, but you have to be consistent in the second integration. (That is, if you let u stand for the exponential function in the first integration, then also let u stand for the exponential function in the second integration. Alternatively, if you let u stand for the trigonometric function in the first integration, then also let u stand for the trigonometric function in the second integration.) Otherwise, after two integrations by parts you will end up with 0 = 0, which is true but not helpful.

So, let u = e^x and {\rm d} v = \cos x \, {\rm d}x, so that {\rm d} u = e^x \, {\rm d}x and v = \sin x. Then, calling the integral to be determined “I,” for reasons that will become clear shortly, we have

I = \int e^x \cos x \, {\rm d}x
I = uv - \int v \, {\rm d}u
I = e^x \sin x - \int e^x \sin x \, {\rm d}x

In the integral on the right side of the previous equation, integrate by parts again, letting U = e^x and {\rm d} V = \sin x \, {\rm d}x, so that {\rm d} U = e^x \, {\rm d}x and V = -\cos x. The result is

I = e^x \sin x - \left [ UV - \int V \, {\rm d}U \right ]
I = e^x \sin x - \left [ e^x \left ( -\cos x \right ) - \int e^x \left ( -\cos x \right ) \, {\rm d}x \right ]
I = e^x \sin x - \left [- e^x \cos x + \int e^x \cos x \, {\rm d}x \right ]
I = e^x \sin x + e^x \cos x - \int e^x \cos x \, {\rm d}x

We seem to be going around in circles, because the integral on the right side of the previous equation is the same as the one we started with. However, if we just replace it by its label I, the previous equation is seen to be an algebraic equation that we can solve for I. (This is the motivation for introducing the label I.) Doing this, we obtain

I = e^x \sin x + e^x \cos x - I
2I = e^x \sin x + e^x \cos x
2I = e^x \left [ \sin x + \cos x \right ]
I = \dfrac{e^x}{2} \left [ \sin x + \cos x \right ]
\int e^x \cos x \, {\rm d}x = \dfrac{e^x}{2} \left [ \sin x + \cos x \right ]

You can check the result by differentiating it to arrive at the original function. Some people enjoy this method as it seems as if we got something for nothing. We never really “finished” the integration by parts (after two iterations, we were still left with an integral), and yet the final result somehow popped out.

And of course, if you need to determine the other integral, then you have to go through the process once more, integrating by parts twice again. Try it for practice, if you wish; the result is

\int e^x \sin x \, {\rm d}x = \dfrac{e^x}{2} \left [ \sin x - \cos x \right ]

Now for the trick, which relies on you knowing about complex numbers, including Euler’s formula:

e^{ix} = \cos x + i \sin x

The strategy is to multiply the second of the original integrals by i and then add it to the first of the original integrals. It turns out that combining them in this way results in an integral that is quite easy to determine; no integration by parts four times is needed. Then we just separate the final result into a sum of real and imaginary parts; the real part is the result for the first integral and the imaginary part is the result of the second integral.

Let’s see how it works: First multiply \int e^x \sin x \, {\rm d}x by i and add it to \int e^x \cos x \, {\rm d}x and combine the integrals into one integral:

\int e^x \cos x \, {\rm d}x + i \int e^x \sin x \, {\rm d}x = \int \left [ e^x \cos x + i e^x \sin x \right ] \, {\rm d}x
\int e^x \cos x \, {\rm d}x + i \int e^x \sin x \, {\rm d}x = \int e^x \left [ \cos x + i \sin x \right ] \, {\rm d}x
\int e^x \cos x \, {\rm d}x + i \int e^x \sin x \, {\rm d}x = \int e^x \left [ e^{ix} \right ] \, {\rm d}x       (using Euler’s formula)

Now combine the exponential functions on the right side of the previous equation and antidifferentiate:

\int e^x \cos x \, {\rm d}x + i \int e^x \sin x \, {\rm d}x = \int e^{(1 + i)x} \, {\rm d}x

\int e^x \cos x \, {\rm d}x + i \int e^x \sin x \, {\rm d}x = \dfrac{1}{1 + i} e^{(1 + i)x}

The final stage is to express the right side of the previous equation as the sum of a real part and an imaginary part. Part of this process is to multiply the numerator and denominator of the fractional factor by the complex conjugate of the denominator; the other part is to separate the exponential functions on the right side of the equation:

\int e^x \cos x \, {\rm d}x + i \int e^x \sin x \, {\rm d}x = \dfrac{1}{1 + i} \times \dfrac{1 - i}{1 - i} e^x e^{ix}

\int e^x \cos x \, {\rm d}x + i \int e^x \sin x \, {\rm d}x = \dfrac{1 - i}{1 - i^2} e^x \left [ \cos x + i \sin x \right ]

\int e^x \cos x \, {\rm d}x + i \int e^x \sin x \, {\rm d}x = \dfrac{1 - i}{2} e^x \left [ \cos x + i \sin x \right ]

\int e^x \cos x \, {\rm d}x + i \int e^x \sin x \, {\rm d}x = \dfrac{e^x}{2} \left [ 1 - i \right ] \left [ \cos x + i \sin x \right ]

\int e^x \cos x \, {\rm d}x + i \int e^x \sin x \, {\rm d}x = \dfrac{e^x}{2} \left [ \cos x + i \sin x - i \cos x - i^2 \sin x \right ]

\int e^x \cos x \, {\rm d}x + i \int e^x \sin x \, {\rm d}x = \dfrac{e^x}{2} \left [ \cos x + i \sin x - i \cos x + \sin x \right ]

\int e^x \cos x \, {\rm d}x + i \int e^x \sin x \, {\rm d}x = \dfrac{e^x}{2} \left [ \sin x + \cos x + i \left ( \sin x - \cos x \right ) \right ]

\int e^x \cos x \, {\rm d}x + i \int e^x \sin x \, {\rm d}x = \dfrac{e^x}{2} \left [ \sin x + \cos x \right ] + i \dfrac{e^x}{2} \left [ \sin x - \cos x \right ]

Matching real and imaginary parts on both sides of the previous equation gives us the final results:

\int e^x \cos x \, {\rm d}x = \dfrac{e^x}{2} \left [ \sin x + \cos x \right ]

\int e^x \sin x \, {\rm d}x = \dfrac{e^x}{2} \left [ \sin x - \cos x \right ]

Conclusion: Does the trick save work? The integration step in the trick method is very easy, so we’re trading four integrations by parts plus some algebra for a simple integration and some algebra with complex numbers. It’s a trade I’d make any day. But of course, to each his own, so try both ways and decide for yourself which way you like better.

If the integrals are a bit more complex, then the savings in the trick method are even greater. For example, you might try using both methods to determine the integrals

\int e^{ax} \cos bx \, {\rm d}x


\int e^{ax} \sin bx \, {\rm d}x

Integration by parts is even more of a pain, but the trick method is hardly more difficult. If you do try them, you can check your final results against these:

\int e^{ax} \cos bx \, {\rm d}x = \dfrac{e^{ax}}{a^2 + b^2} \left [ b\sin bx + a\cos bx \right ]

\int e^{ax} \sin bx \, {\rm d}x = \dfrac{e^{ax}}{a^2 + b^2} \left [ a\sin bx - b\cos bx \right ]

I just did these two integrals with pencil and paper, and the trick method is much faster. Even if you only have to work out one of the integrals (the method is the same, you just ignore one of the final results), I think the trick method is still a time-saver and the probability of making an error is reduced, because you avoid the messy integrations by parts.

Posted in Calculus, Mathematics | Tagged , , , | 9 Comments

“No Student Left Untested,” by Diane Ravitch

Measuring teacher effectiveness by the performance of students on standardized tests is insane. New York State has just signed on to a particularly dangerous form of this insanity. Diane Ravitch has clearly explained the insanity and its destructive consequences in No Student Left Untested, in the New York Review of Books (hat-tip to Observational Epidemiology). She makes a lot of sense; so much so that the powers that be have singled her out for vicious attacks.

Here is an excerpt (the entire article is highly recommended for anyone interested in the issues; further discussion is here and here):

No high-performing nation in the world evaluates teachers by the test scores of their students; and no state or district in this nation has a successful program of this kind. The State of Tennessee and the city of Dallas have been using some type of test-score based teacher evaluation for twenty years but are not known as educational models. Across the nation, in response to the prompting of Race to the Top, states are struggling to evaluate their teachers by student test scores, but none has figured it out.

All such schemes rely on standardized tests as the ultimate measure of education. This is madness. The tests have some value in measuring basic skills and rote learning, but their overuse distorts education. No standardized test can accurately measure the quality of education. Students can be coached to guess the right answer, but learning this skill does not equate to acquiring facility in complex reasoning and analysis. It is possible to have higher test scores and worse education. The scores tell us nothing about how well students can think, how deeply they understand history or science or literature or philosophy, or how much they love to paint or dance or sing, or how well prepared they are to cast their votes carefully or to be wise jurors.

Of course, teachers should be evaluated. They should be evaluated by experienced principals and peers. No incompetent teacher should be allowed to remain in the classroom. Those who can’t teach and can’t improve should be fired. But the current frenzy of blaming teachers for low scores smacks of a witch-hunt, the search for a scapegoat, someone to blame for a faltering economy, for the growing levels of poverty, for widening income inequality.

Update: Further interesting links are provided by Japheth Wood here.

Posted in Education | Tagged , , , , , , , , , | 2 Comments

Confusing Use Of Numbers: Best-Before Dates

My wife was driving me and our children to Collingwood yesterday. I was hungry, and found a power bar in the glove box, with a best-before date of


Yesterday was 20 February 2012, and my question is: had the best-before date passed yet?

In Europe, it is typical to write dates in the form “day/month/year,” so if this were Europe, then the best-before date would have passed, and I perhaps ought not to eat the bar. (Although the best-before date would not have passed by too much, and usually the time at which the food is spoiled is beyond the best-before date, so perhaps eating it would be OK.)

In North America, the year is usually last, but the day and month could be in either order. But there are six possible permutations of the six data, so although only two of the permutations are in common use (“year/month/day” is also used, but less commonly), there is still room for confusion. For something as important as best-before dates, shouldn’t we have an unambiguous usage convention? Clarity is courtesy, but when it comes to best-before dates, clarity could also save someone from illness.

But it is difficult to reach consensus on conventions, particularly in this case with products being produced around the world. So here is my proposal for ensuring that best-before dates will never be misinterpreted:

1. Always write all four digits of the year.
2. Always use a day that is at least 13.

Then it doesn’t matter which permutation is used. The dates 13/02/2012, 02/13/2012, 2012/13/02, etc., can only be interpreted in one way. And if a packaged food is good for a year or more, then altering the day by a few to ensure that it is at least 13 will cause no problems. In my situation, this would have saved me wondering whether the best-before date was 9 February or 2 September, or even some time in 2009 (2002 seemed unlikely).

If the food product is only good for a few days after packaging (which is the case for meats, bread, and some other foods), then it is unlikely that it will travel far, and the word for the month could be safely used.

Posted in Mathematics in every-day life | Tagged , | 5 Comments

Canadian Government Shamefully Suppresses Science

The Canadian government has a sickening recent history of persecuting whistleblowing scientists who were working hard and with integrity to protect the safety of the food supply (see here and here), then abusing the legal system to deny justice, all at taxpayer expense of course. (Their record on whistleblowers in general is also terrible.) The Canadian government’s muzzling of scientists has been prominently publicized (also see the follow-up here) at the annual meeting of the AAAS in Vancouver this past week (thanks to my friend Jim Slominski for the link).

The allegation of “muzzling” came up at a session of the AAAS meeting to discuss the impact of a media protocol introduced by the Conservative government shortly after it was elected in 2008.

The protocol requires that all interview requests for scientists employed by the government must first be cleared by officials. A decision as to whether to allow the interview can take several days, which can prevent government scientists commenting on breaking news stories.

Sources say that requests are often refused and when interviews are granted, government media relations officials can and do ask for written questions to be submitted in advance and elect to sit in on the interview.

‘Orwellian’ approach

Andrew Weaver, an environmental scientist at the University of Victoria in British Columbia, described the protocol as “Orwellian”.

The protocol states: “Just as we have one department we should have one voice. Interviews sometimes present surprises to ministers and senior management. Media relations will work with staff on how best to deal with the call (an interview request from a journalist). This should include asking the programme expert to respond with approved lines.”

Professor Weaver said that information is so tightly controlled that the public is “left in the dark”.

“The only information they are given is that which the government wants, which will then allow a supporting of a particular agenda,” he said.

Shame on the Canadian government, where ideology is regularly trumping reality. (And the mayor and city council of our largest city, Toronto, is evidently cut from the same autocratic cloth.) For those of us who have never experienced life under the thumb of a KGB-like apparatus, there is apparently no need to move to North Korea or Iran. All we have to do is wait here for the control freaks who run our country to stealthily grind us down into a parallel prison.

ps. Video of the AAAS symposium “Unmuzzling Government Scientists: How to Reopen the Discourse” (along with some recent actions along these lines) is available here.

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Climate Change Denialism Highlights The Need For Public Education In Basic Science

There are many excellent newspaper and magazine columns (and internet sites and blogs) that publicize the latest research findings in science. They are important because they inform the general public about scientific findings funded by their taxes, and they communicate the excitement of scientific discovery in a way that undoubtedly inspires many young people.

However, what is also sorely needed is some sort of column that explains basic science to the lay public. I don’t mean a development of Newton’s laws of motion for beginners, but rather an understanding of how the enterprise of science works, and how to interpret what one reads in the popular media. When one reads shouting headlines that a recent study shows that Vitamin E is dangerous (without some perspective about the population of the study, the strength of the effect, the time over which the study took place, the control group, and so on), and then reads another shouting headline saying that coffee is good for you after all, really, one can’t help but have sympathy for the poor man in the street who doesn’t know what to think, and then just discounts it all because it is too difficult and troubling to think about (in the same way that the antics of some politicians cast them all into disrepute). And by the way, how did the Knicks fare last night against the Lakers? Linsanity!

This bewilderment leaves citizens particularly vulnerable to nefarious agents who actively foment confusion by loudly trumpeting complete nonsense, in the (unfortunately quite realistic) hope that doubt will be engendered in the minds of many people, which furthers their cause.

One case in point is the despicable Discovery Institute, which is completely anti-science, but tries to let on that its arguments for the existence of God are scientific. Their not-so-hidden agenda is to promote their religious views and do everything in their power (which is well-funded and quite considerable) to force their religious views to be taught in science classes. Amongst their many transgressions, their claim that evolution is “just a theory” confuses the every-day meaning of the term with its scientific meaning, a strategy that is intended to sow uncertainty in the minds of the uninformed. Citizens with a strong sense for what science is and how it works would be far less likely to be hoodwinked by such transparent fallacies.

Some striking instances of the effectiveness of this strategy are to be found in this unintentionally funny clip, showing beauty pageant contestants answering the question, “Should evolution be taught in schools?” (This spawned the hilarious parody, “Should mathematics be taught in schools?” Sometimes humour is the best teacher. More scary, for me at least, is this “news report” from The Daily Show on 26 October 2011, entitled “Science — What’s it up to?” (Here is the link for those based in the U.S.; I haven’t been able to find a link that works in Canada (not sure about the rest of the world; try the U.S. link), but a partial transcript is here.) Interviewer Aasif Mandvi gets political strategist Noelle Nikpour to admit the most ridiculous things about science, including, “Scientists are scamming the American people right and left for their own financial gain.”

Nikpour: “It’s very confusing for a child to be only taught evolution, to go home to a household where their parents say, “Well, wait a minute, God created the Earth.”
Mandvi: “What is the point of teaching children facts if it’s just going to confuse them?”
Nikpour: “It confuses the children when they go home. We as Americans, we are paying tax dollars for our children to be educated. We need to offer them every theory that’s out there. It’s all about choice. It’s all about freedom.”
Mandvi: “I mean it should be up to the American people to decide what’s true.”
Nikpour: “Absolutely! Doesn’t it make common sense?”

Another case in point is climate change denialism, which has been supported by very rich and powerful interests. A news story yesterday reports that climate denialists want to get their propaganda into schools (see also here). Nobody knows what the climate is going to be like in the next decade, the next century, or the next millennium. The system is just too complex, and nonlinear, for any kind of prediction to be meaningful. But this is exactly what is alarming; it seems clear that the system is out of equilibrium, and like a car swerving out of control, who knows where it will end up? Maybe it will end up upright and the passengers will breathe a sigh of relief at a close brush with death, but maybe the car will roll over several times, end up in a ditch and all the passengers will die. Maybe we will turn the planet into a desert, maybe we will freeze. Who knows? So the conservative action would be to plan for a car crash, and to take steps to mitigate potential catastrophes that might arise. But for the giant oil companies, and related industries and hangers-on, any sense that our actions might be causing climate change must be crushed, because mitigating action would be very bad for business. And that’s the bottom line; we must protect profits at all costs, and business must proceed as usual.

What does a scientifically educated person think about all this? Well there are many possible reasonable opinions, but categorical denial that human actions could possibly have any effect on climate change is not one of them. Weighing the evidence and taking prudent action, including funding projects that monitor weather and research historical climate changes, is essential. Just because we don’t know exactly what will happen in future does not excuse us from taking wise prophylactic action now. Doing nothing is radical and dangerous. Advocating that nothing be done using self-serving fallacious arguments, and fraudulently casting doubt on solid scientific research, is reprehensible and dangerous.

What does a person who is ignorant of the ways of science think about all this? Oh, there is a controversy, who knows who is right, nobody knows, so let’s just hope for the best and not bother any further with the issue. Those elitist scientists are confusing, and they always make me feel dumb. Pass me the chips, and turn on the flat-screen, I want to watch my Knicks. (Full disclosure: I would never think like this; I am fanatical about the Raptors.)

ps. On a brighter note, Scientists In School is a group that is taking positive action to help interest young people in science (via hands-on activities) in my locale. Undoubtedly there is some group in your area that is similarly interested in promoting public understanding of science that is worth supporting.

Posted in Environment, What is science? | Tagged , , , , , , , , , , , | 2 Comments

Both Students And Professors Need Certification, and the Elsevier Boycott

I’ve written before about the evils of grading (for example, see here and here), the main purpose of which is to make certifying students easy. Our current grading system in mathematics is counterproductive to learning (students are inhibited from engaging in essential learning activities out of the fear that is naturally induced by typical high-stakes grading systems), and there is a strong tendency for course content to be skewed towards what is easy to grade rather than what is best for students’ development. The practice of assigning partial credit allows students to accumulate marks towards a good grade without doing the intensive work necessary for mastery, to their ultimate detriment, as most of them eventually reach a level where they fail unnecessarily because their background preparation is so weak. Politicians manipulate the system to increase the number of students who reach certain grade levels, which allows them to bathe in the warm glow of counterfeit success, to the detriment of our students and our society. To make matters worse, there is intense pressure in the U.S. nowadays to hold teachers accountable by assessing teacher performance based on their students’ scores on standardized tests. This is great for the publishers who produce the tests, who are profiting handsomely, but not so great for students, teachers, and society.

After understanding that students were not responsible for setting up this dysfunctional system, it would be inappropriate to blame them for their unhealthy focus on grades. (Every teacher who ever lived can recall hearing some other teacher complain about “students nowadays, who only care about marks, not about the spirit of learning.”) Students are under enormous pressure to get high grades, because we have set up a system in which their entrance to university undergraduate programs, entrance to graduate and professional schools, scholarships — in short, their entire career success — depends on grades. It seems absolutely silly to base everything on a single number, but there you have it.

It’s worth noting that professors are subject to similar pressure for certification. Forgetting about the same competition for high grades while they were students, professors are under intense pressure to achieve high “grades” for their career success in academia. They need high “grades” to achieve tenure, to be promoted to full rank, and to receive their yearly merit increases in salary. In this case, their “grades” are composed of factors that account for their performance as teachers, their service to their department, faculty, and university community, but by far the most important factor is their research. (Despite protestations to the contrary from university administrators about how much universities value good teaching (and I’m sure they do), note that it is quite possible to become a full professor while being a consistently below-average teacher, while being a stellar teacher but a below-average researcher will never get you to the full professor rank.)

And just as rich businessmen are leading the attack on public school teachers (to the benefit of rich publishing companies), and are thereby influencing in an unhealthy way the environment and conditions of our childrens’ education, rich publishing companies are in control of the certification system for professors’ research, to the benefit of themselves, at great costs to universities, and therefore to the detriment of society.

For professors, their research contribution is graded based on the frequency and quality of their published papers, and on the number and value of their research grants. (The latter also depends on the former. Also, I know that there are other factors, such as participation in conferences, but published research is the most important factor.)

However, how on earth is a university administrator supposed to judge the value of research in a field in which he or she is not an expert? (Which is to say, practically all fields of research.)

This is where the academic journal publishers step in. They provide a service by publishing a large number of journals, with varying levels of prestige, and so a professor’s research can be judged by a non-expert by counting the number of papers published, weighting them according to the prestige of the journal, accounting for impact factors, and so on.

In days gone by, journal publishers provided services that could not be easily provided by individual professors. For example, typesetting mathematical texts used to be a very expensive task that only highly specialized typesetters could accomplish. Mathematical manuscripts were often hand-written, and turning them into professional-quality publications was a very useful step in disseminating and archiving a paper. Publishers hired copy-editors to make manuscripts more readable. The tasks of editing academic journals and refereeing papers was (and is) done on a volunteer basis by professors.

Nowadays, however, with TeX, LaTeX, and all of its friends freely available to the vast majority of mathematics and science researchers, the need for publishers is questionable. If professors are doing the work of typesetting papers, refereeing them, and editing journals anyway, what value do journal publishers add? Journal publishers have cut back drastically on copy-editing, their function as an archiver is unnecessary now that electronic means of archiving are widely available, so the only thing that they really provide is prestige. But prestige is a chimera, and can be easily accounted for in other ways.

What exacerbates the situation is that journal publishers make absolutely enormous profits, which ultimately are paid for by our taxpayers. Publishers often place the published papers behind paywalls so that the same taxpayers who paid for the research in the first place have to pay again to read the results.

Still worse is the way that publishers of academic journals shake down university libraries. Taxpayers pay for the research done at universities (yes, private universities receive some private funding, but they are still financially supported by taxpayers), and a powerful argument for open access is that taxpayers should not have to pay a second time to have access to published research. This argument gains in strength when one realizes the enormous profits made by publishers of academic journals, on the backs of volunteer labour (the same academics who referee papers submitted to journals, edit the journals, etc.).

Reed Elsevier has been judged the worst offender , and thanks to an initiative spearheaded by Tim Gowers, a boycott of Elsevier has been organized. To learn more about the boycott, which has been in the news and all over the internet for the past few weeks, see the links in the following paragraph.

Authoritative arguments in favour of open access (see here and here), and against large publishers of academic journals, are made by Tim Gowers (here and here; the latter includes an incisive open letter outlining the motivation for the boycott, signed by 34 prominent mathematicians), and Terence Tao (here and here). Other valuable resources are John Baez’s page on journal publishing reform (also see here and here, where Baez makes recommendations for introducing new systems, including Math 2.0 (a discussion forum started by Andrew Stacey and Scott Morrison), and here), and Michael Nielsen’s page on journal publishing reform (also see here for a concise summary of the Elsevier issue). Prominent bloggers who have commented on the issue include Scott Aaronson, Nassif Ghoussoub, Cathy O’Neil (here and here), and Peter Krautzberger (here and here), and Sean Carroll. The issue has even hit the mainstream, at The New York Times and The Boston Globe.

If you would like to add your signature to the protest site, The Cost of Knowledge, it is here. I just checked the site and currently 6251 mathematicians and scientists from around the world have signed to publicize their decision to boycott (in various degrees) Reed Elsevier.

This whole issue deserves serious discussion within the scientific community, and the wider public needs to become aware of it. Most universities are continually crying about how they are strapped for resources, and their libraries are blackmailed by large publishers of academic journals. Reed Elsevier is the worst offender, but not the only one.  The issue has been discussed for a long time (see some of David Mermin’s contributions, for example here, which is reprinted on pages 57–62 here; other references are listed in the open letter signed by Gowers et al linked here; a recent example is an article by Peter Olver in the September 2011 issue of the Notices of the AMS here) and it’s good to see some serious action.

It’s heart-warming to see academics working to wrest control of their research publications away from large, powerful, greedy academic publishers such as Reed Elsevier. One hopes that along with this initiative there will be movement towards a more fair and humane system for evaluating the contributions of professors. And maybe we can figure out a way to evaluate students in a more humane way while we’re at it.

Update: Cosma Shalizi makes the point a lot better and more concisely than I do here. (Thanks to Sam Alexander for the link.)

Posted in Education, Science research, Society | Tagged , , , , , , , , , , , , , , , , , , , , , , , , , , | 7 Comments

Zen Valentine

My dear friend Manolo Santiago lived in Toronto in 1996 when he wrote this poem, an alienated urban version of the proverb of Wu Li, which you can find at the bottom of this page or here.

Zen Valentine

Before Valentine’s Day: Ride subway, walk briskly, look grim.

On Valentine’s Day: Ride subway, walk briskly, carry flowers, look grim.

After Valentine’s Day: Ride subway, walk briskly, look grim.

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