There are ten and two different notes, or steps, in my people's music. The sounds go up and down such that a note that is ten and three higher than another sounds the same but higher up. We group those notes into pleasing sounds, and those groups are made up of notes that have a set number of steps between them. The notes have names, but there are fewer names than there are steps! The notes between the named notes are named by adding a changing word that says if the note is higher or lower by one step than a named note.
To write down this music, we use one or more sets of five lines drawn from side to side next to each other. The notes are drawn as spots on or between the lines in the spaces with sticks going up or down from the spots. We use more lines and some smaller spots to say how long the note should be played. Some of the steps (the ones that don't have their own names) are really in between the lines or spaces, so we use two different pictures to say if a note is really higher or lower by one step.
If you select a region with transient mode on, then `query-replace` (`M-%` by default) will only affect the region. So select the region you want the change to effect (just like in the demo video), hit `M-%`, then type the old and new text, then you can type `!` when prompted for what to do and all instances of old become new only within the region.
A camp counselor introduced a bunch of us to "Nick Danger" one summer back in the '70's, and it was instant infatuation for me. I bought all of their albums I could find and listened to them all the time. No drugs involved at all, on my side at least. Their love of language play, political stances, satire and parody, and obvious skill and love of what they did made Firesign Theater amazing.
FT is great even without drugs, but while high I found them even more hilarious, the sensory aspects of their performance were way more effective at transporting me in to the worlds they were conjuring up, and I felt I made a lot more connections in the many references they were making.
Pot has often had a similar effect for me with other comedies, but the multisensory, multi-level nature of Firesign Theater brings the whole experience to another level that's rarely been matched by anything else.
Yes, but if they did they are giving away security information: they are acknowledging that the entered email address is a legit delegated login. That's a bad thing to do, security-wise. You don't want bad guys to start trying email addresses and be able to see which ones are good email addresses.
The article's title is misleading. The mathematician has suggested an improvement or clarification to the notation used to describe a second derivative which makes it easier to manipulate formulaically.
I don't think the title is misleading given the clarity of the article. Like most titles regarding technical subjects, it needs to be carefully qualified to be understood.
But was there a flaw of calculus notation?
Bartlett introduces changes to Leibniz's notation that:
1. Allows for direct algebraic manipulation of differential relationships, for both simple and complex equations, and which extends very nicely to multivariable cases.
2. Makes understanding the relationships between higher order derivatives clearer.
3. Clearly highlights that differentials and derivatives are the two steps of differentiation.
4. Removes misunderstandings about Leibnitz's notation where the notation looks like a ratio, an interpretation that works for the first derivative, but confusingly and inconveniently does not for high order derivatives.
5. Results in equations with far fewer subscripts and superscripts. Surely a Fields Medal is in order? :)
6. Removes a now unnecessary concept from differential algebra (i.e. the Leibniz non-algebraic non-ratio).
That last one simplifies the foundations of calculus, not just notation.
Given that, it is a fix to calculus, not just its notation.
Well, I guess that's debatable. However, the way I see it, the old version was presented as a fraction, but couldn't be used as a fraction. The new version is also presented as a fraction and can be used as a fraction.
Perhaps it is due to my old-school nature, but if I wrote a fraction in school that didn't work as a fraction, that would be a mistake, would it not? Especially if it was possible to write it correctly? I couldn't go to my professor and just say, "well, actually, while it looks like a fraction, it was actually just a piece of notation that needed future clarification."
What part of it is debatable? The article clearly states that it is just a notational improvement: there was really no flaw in calculus, it was more a wart in notation. This change would not result in different answers unless the original calculations were done incorrectly.
I'd regard the current notation more like an irregular verb, or something that is spelled differently from how it is pronounced: it's inconvenient to deal with, but everyone adept at calculus knew about it and how to deal with it. This article is just describing a notational improvement that would make it easier to get right.
The fact that there was a well-known kludge to get around the problems inherent in the notation doesn't make the notation right. As the paper pointed out, the new notation didn't drop magically from the sky, it is literally the application of the quotient rule to the derivative. The reason that no one noticed this notation before, is that they didn't think to apply the quotient rule to the quotient dy/dx.
If I had a problem, and my solution involved forgetting to apply the quotient rule to a prominent quotient, it would still be wrong, even if I also came out with a set of kludges that allowed me to get right answers in the most common cases.
It seems similar to Roman numerals vs Arabic numerals. We could stick with Roman numerals, but mathematics would be pretty much at a standstill if we had to use such an unwieldy system.
A major improvement in notation is not insignificant.
With writing, music, and perhaps even software it is best to know and understand the rules and be able to use them before you start breaking them. When you do, you'll know why and do so for a purpose.
I've had the same Tom Binh Brain Bag backpack for about 15 years and I love it. Wearing it right now in fact. I also have an insert accessory for organizing papers, pens, etc. and a padded laptop protector.
There are different kinds of infinity. The integers are countable, the real numbers aren't. You can prove that if you try to map each integer to some rational number, there will be some rational numbers that are not on that list --- there are more of them. See https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument.
> You can prove that if you try to map each integer to some rational number, there will be some rational numbers that are not on that list --- there are more of them.
Did you mean reals here? There _is_ a (bijection) mapping between integers and rationals.
Yes, in theory. In practice, you'd not only remove this function but all of its calls. When somebody down the road realizes that they want this function back, they have to (A) realize it's in the VC history and (B) not only get back the function but all the calling points. That is so much of a pain and potentially error-prone that leaving this function in for a while with a comment might be the more practical approach.
On the other hand, it may no longer be called from every place where it should be. I've found that it's better to document what was wrong with the overall approach, scrape the dead code from the source base, and move on. Barnacles like these accumulate over time otherwise.
To write down this music, we use one or more sets of five lines drawn from side to side next to each other. The notes are drawn as spots on or between the lines in the spaces with sticks going up or down from the spots. We use more lines and some smaller spots to say how long the note should be played. Some of the steps (the ones that don't have their own names) are really in between the lines or spaces, so we use two different pictures to say if a note is really higher or lower by one step.