They looked at an online community which moved from "easy anonymity" to "registered pseudonyms" to IDs linked to Facebook names/avatars (which they took as a proxy for "real names"). They found:
"Our results suggest that the quality of comments was highest in the middle phase. There was a great improvement after the shift from easy or disposable anonymity to what we call 'durable pseudonyms'. But instead of improving further after the shift to the real-name phase, the quality of comments actually got worse – not as bad as in the first phase, but still worse by our measure."
"What matters, it seems, is not so much whether you are commenting anonymously, but whether you are invested in your persona and accountable for its behaviour in that particular forum. There seems to be value in enabling people to speak on forums without their comments being connected, via their real names, to other contexts. The online comment management company Disqus, in a similar vein, found that comments made under conditions of durable pseudonymity were rated by other users as having the highest quality."
You might be thinking of: "One rather curious conclusion emerges, that pure mathematics is on the whole distinctly more useful than applied." I wonder what he would think if he could see the ways in which number theory, once often regarded as the purest of the branches of math, is now used in things like cryptography.
"Apology" is definitely worth reading. Some of his opinions can seem rather elitist:
"Statesmen, despise publicits, painters despise art-critics, and physiologists, physicists, or mathematicians usually have similar feelings; there is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain. Exposition, criticism, appreciation is work for second-rate minds."
At the same time, he is very honest about himself - in fact, he seems to have been suffering from depression over what he perceived as a decline in his ability to do math at the level he was accustomed to:
"If then I find myself writing, not mathematics but 'about' mathematics, it is a confession of weakness, for which I may rightly be scorned or pitied by younger and more vigorous mathematicians. I write about mathematics because, like any other mathematician who has passed sixty, I have no longer the freshness of mind, the energy, or the patience to carry on effectively with my proper job."
Or:
"A mathematician may still be competent enough at sixty, but it is useless to expect him to have original ideas."
Or more sadly, but with some serenity:
"It is plain now that my life, for what it is worth, is finished, and that nothing I can do can perceptibly increase or diminish its value. It is very diffciult to be dispassionate, but I count it a 'success'; I have had more reward and not less than was due to a man of my particular grade of ability."
"If I had been offered a life neither better nor worse when I was twenty, I would have accepted without hesitation."
The personal reflections bookend a central portion where he illustrates with several examples (e.g. Euclid's proof of the infinitude of the primes) his feelings about the "importance" of math, its "usefulness", and the distinction between pure and applied math.
It's interesting to compare "Apology" to "Littlewood's Miscellany" (I recommend the Cambridge University Press version, which contains the essay "The Mathematician's Art of Work" - ISBN 0-521-33702-X). There is more math than in "Apology" and many anecdotes. J. E. Littlewood was Hardy's long-time collaborator.
> I wonder what he would think if he could see the ways in which number theory, once often regarded as the purest of the branches of math, is now used in things like cryptography.
I would say number theory proves his statement, although perhaps not his point.
Applied math is useful for the applications that are known at the time of its creation, and it is likely that it will remain with that level of applicability in the future, although if the real world applications that it is used for fall out of favor we might find that the applied math decreases in importance, given its importance is in its applicability, and the applicability of things has an importance contingent on the importance of the thing that they are applied to.
This is of course not 100% sure, as there can also arise new applications of things in the future.
Pure math on the other hand, being not tethered to any particular application on the time of its creation, may find all sorts of applications in the future, pure math has as such infinite potential applicability waiting to be discovered and thus infinite potential usefulness, whereas applied math has limited known applicability and thus limited known usefulness.
Your concern is reasonable, and I've thought about it myself. The Wikipedia article linked downthread by int_19h notes: "Up to one percent of all those detained in immigration detention centers are nationals of the United States according to research by Jacqueline Stevens, a professor of political science at Northwestern University." There are specific cases mentioned in the article, and the case of Mark Daniel Lyttle was pretty alarming. It was written up in: https://www.newyorker.com/magazine/2013/04/29/the-deportatio...
On the other hand, you can't be detained without probable cause, and race/ethnicity alone isn't enough. For instance, the Supreme Court ruled in United States v. Brignoni-Ponce (https://case.law/caselaw/?reporter=us&volume=422&case=0873-0...): "In this case the officers relied on a single factor to justify stopping respondent’s car: the apparent Mexican ancestry of the occupants. We cannot conclude that this furnished reasonable grounds to believe that the three occupants were aliens. At best the officers had only a fleeting glimpse of the persons in the moving car, illuminated by headlights. Even if they saw enough to think that the occupants were of Mexican descent, this factor alone would justify neither a reasonable belief that they were aliens, nor a reasonable belief that the car concealed other aliens who were illegally in the country. Large numbers of native-born and naturalized citizens have the physical characteristics identified with Mexican ancestry, and even in the border area a relatively small proportion of them are aliens. The likelihood that any given person of Mexican ancestry is an alien is high enough to make Mexican appearance a relevant factor, but standing alone it does not justify stopping all Mexican-Americans to ask if they are aliens."
I second the recommendation to get a RealID. You're going to need one eventually for domestic flights, among other things. When I got mine at the DMV renewing my driver's license, they asked for a birth certificate, social security card, driver's license, and proof of (local) residency (e.g. utility bill). So why not get one and carry that as additional proof?
I'm not understanding what you're saying. The standard definition of the derivative of f at c is
f'(c) = lim_{h → 0} (f(c + h) - f(c))/h
The definition would not make sense if f wasn't defined at c (note the "f(c)" in the numerator). For instance, it can't be applied to your f(x) = (x² - 1)/(x - 1) at x = 1, because f(1) is not defined.
And it's a standard result (even stated in Calc 1 classes) that if a function is differentiable at a point, then it's continuous there. For example:
5.2 Theorem. Let f be defined on [a, b]. If f is differentiable at a point x ∈ [a, b], then f is continuous at x.
(Walter Rudin, "Principles of Mathematical Analysis", 3rd edition, p. 104)
Or:
Theorem 2.1 If f is differentiable at x = a, then f is continuous at x = a.
(Robert Smith and Roland Minton, "Calculus -Early Transcendentals", 4th edition, p. 140)
It's true that your f(x) = (x² - 1)/(x - 1) has a removable discontinuity at x = 1, since if we define g(x) = f(x) for x ≠ 1 and g(1) = 2, then g is continuous. Was this what you meant?
This is correct. You cannot have a discontinuity with any accepted definition of a derivative (and your definition is explicit about this: the value f(c) must exist). Just allowing the limits on both sides to be equal already has a mathematical definition which is that of a functional limit, the function in this case being (f(x) - flim(c))/ (x-c) where flim(c) is the value of a (different) functional limit of f(x): x->c (as f(c) doesn't exist).
and yes, by defining a new function with that hole explicitly filled in with a defined value to make it continuous is the typical prescription. It does not imply the derivative exists for the other function as the other post posits.
https://en.m.wikipedia.org/wiki/Classification_of_discontinu... is responsive and quite accessible. It notes that there doesn't have to be an undefined point for a function to be discontinuous (and that terminology often conflates the two), and matches what I recall of determining that if the limit of the derivative from both sides of the discontinuity exists and is equal, the derivative exists.
> ... there doesn't have to be an undefined point for a function to be discontinuous.
That's right. In the example f(x) = (x² - 1)/(x - 1) for x ≠ 1, if we further define f(1) = 0, the function is now defined at x = 1, but discontinuous there.
> ... if the limit of the derivative from both sides of the discontinuity exists and is equal, the derivative exists.
(You probably mean "both sides of the point", since if there's a discontinuity there the derivative can't exist.) Your point that, if the left and right-hand limits both exist and are equal, then the derivative exists (and equals their common value) is true for all limits.
Also, there's a difference between the use of the word "continuous" in calc courses and in topology. In calc courses where functions tend to take real numbers to real numbers, a function may be said to be "not continuous" at a point where it isn't defined. So f(x) = 1/(x - 2) is "not continuous at 2". But in topology, you only consider continuity for points in the domain of the function. So since the (natural) domain of f(x) = 1/(x - 2) is x ≠ 2, the function is continuous everywhere (that it's defined).
I was actually aiming for the situation where a function is defined on all reals but still discontinuous (e.g. the piecewise function in the wiki article for the removable discontinuity). So there's a discontinuity (x=1), however the function is defined everywhere.
The standard definition of a derivative c involves the assumption that f is defined at c.
However, you could also (probably) define the derivative as lim_{h->0} (f(c+h) - f(c-h))/2h, so without needing f(c) to be defined. But that's not standard.
> However, you could also (probably) define the derivative as lim_{h->0} (f(c+h) - f(c-h))/2h, so without needing f(c) to be defined. But that's not standard.
Although this gives the right answer whenever f is differentiable at c, it can wrongly think that a function is differentiable when it isn't, as for the absolute-value function at c = 0.
A study of private equity and dental practices: "Percentage Of Dentists And Dental Practices Affiliated With Private Equity Nearly Doubled, 2015-21" - https://pubmed.ncbi.nlm.nih.gov/39102603/
The abstract: "Over the course of the past twenty years, private equity (PE) has played a role in acquiring medical practices, hospitals, and nursing homes. More recently, PE has taken a greater interest in acquiring dental practices, but few data exist about the scope of PE activity within dentistry. We analyzed dentist provider data for the period 2015-21 to examine trends in PE acquisition of dental practices. The percentage of dentists affiliated with PE increased from 6.6 percent in 2015 to 12.8 percent in 2021. During this period, PE affiliation increased particularly among larger dental practices and among dental specialists such as endodontists, oral surgeons, and pediatric dentists. PE-affiliated dental practices were more likely to participate in Medicaid than practices not affiliated with PE. Future research should investigate whether PE's role in dentistry affects the affordability and quality of dental services."
Is it generally true that a dental practice that is a franchise is private-equity backed? The original article mentions Aspen Dental. If I wanted to know about (say) SmileBuilderz, how could I find out?
In https://medicalxpress.com/news/2024-07-covid-puzzle-pieces-f... the author says: "The intense scientific effort that long COVID sparked has resulted in more than 24,000 scientific publications, making it the most researched health condition in any four years of recorded human history."
Conveniently he provides the search term used to produce the "24,000" figure:
“Enhanced warming of the Arctic region relative to the rest of the globe, known as Arctic amplification, is caused by a variety of diverse factors, many of which are influenced by the Atlantic meridional overturning circulation (AMOC). Here, we quantify the role of AMOC changes in Arctic amplification throughout the twenty-first century by comparing two suites of climate model simulations under the same climate change scenario but with two different AMOC states: one with a weakened AMOC and another with a steady AMOC. We find that a weakened AMOC can reduce annual mean Arctic warming by 2 °C by the end of the century. A primary contributor to this reduction in warming is surface albedo feedback, related to a smaller sea ice loss due to AMOC slowdown. Another major contributor is the changes in ocean heat uptake. The weakened AMOC and its associated anomalous ocean heat transport divergence lead to increased ocean heat uptake and surface cooling. These two factors are inextricably linked on seasonal timescales, and their relative importance for Arctic amplification can vary by season. The weakened AMOC can also abate Arctic warming via lapse rate feedback, creating marked cooling from the surface to lower-to-mid troposphere while resulting in relatively weaker cooling in the upper troposphere. Additionally, the weakened AMOC increases the low-level cloud fraction over the North Atlantic warming hole, causing significant cooling there via shortwave (sw) cloud feedback despite the overall effect of sw cloud feedback being a slight warming of the average temperature over the Arctic.”
"Online anonymity: study found ‘stable pseudonyms’ created a more civil environment than real user names" - https://theconversation.com/online-anonymity-study-found-sta...
They looked at an online community which moved from "easy anonymity" to "registered pseudonyms" to IDs linked to Facebook names/avatars (which they took as a proxy for "real names"). They found:
"Our results suggest that the quality of comments was highest in the middle phase. There was a great improvement after the shift from easy or disposable anonymity to what we call 'durable pseudonyms'. But instead of improving further after the shift to the real-name phase, the quality of comments actually got worse – not as bad as in the first phase, but still worse by our measure."
"What matters, it seems, is not so much whether you are commenting anonymously, but whether you are invested in your persona and accountable for its behaviour in that particular forum. There seems to be value in enabling people to speak on forums without their comments being connected, via their real names, to other contexts. The online comment management company Disqus, in a similar vein, found that comments made under conditions of durable pseudonymity were rated by other users as having the highest quality."
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