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The following sets are identical:

• { (−1)^s•f•2^e | s ∈ {0, 1}, f is the value of a 24-bit binary numeral with a radix point after the first digit, and e is an integer such that −126 ≤ e ≤ 127 }.
• { (−1)^s•f•2^e | s ∈ {0, 1}, f is the value of a 24-bit binary numeral with a radix point before the first digit, and e is an integer such that −125 ≤ e ≤ 128 }.
• { (−1)^s•f•2^e | s ∈ {0, 1}, f is the value of a 24-bit binary numeral with a radix point after the last digit, and e is an integer such that −149 ≤ e ≤ 104 }.
• { f•2^e | f is an integer such that |f| < 2²⁴, and e is an integer such that −149 ≤ e ≤ 104 }.

In other words, we may put the radix point anywhere in the significand we want, simply by adjusting the range of the exponent to compensate. Which form to use may be chosen for convenience or preference.

The third form scales the significand so it is an integer, and the fourth form incorporates the sign into the significand. This form is convenient for using number theory to analyze floating-point behavior.

IEEE 754 generally uses the first form. It refers to this as “a scientific form,” reflecting the fact that, in scientific notation, we commonly write numbers with a radix point just after the first digit, as in the mass of the Earth is about 5.9722•10²⁴ kg. In clause 3.3, IEEE 754-2008 mentions “It is also convenient for some purposes to view the significand as an integer; in which case the finite floating-point numbers are described thus:”, followed by text equivalent to the third form above except that it is generalized (the base and other parameters are arbitrary values for any floating-point format rather than the constants I used above specifically for the binary32 format).

I implemented a bootstrapping solution. I initially did standard nonparametric bootstrapping, which resamples observations, but this produces 95% CIs that look suspiciously wide — I think that this is because that form of bootstrapping fails to maintain the balance in the x-distribution (e.g. by resampling you could end up with no observations for small values of x). (It's also possible that there's just a bug in my code.)

As a second shot I switched to resampling the residuals from the initial fit and adding them to the predicted values; this is a fairly standard approach e.g. in bootstrapping time series (although I'm ignoring the possibility of autocorrelation in the residuals, which would require block bootstrapping).

Here's the basic bootstrap resampler.

If I got it right, try following:

ClassroomCoursesReadonly cannot be used to create courses, just to retrieve them.

So I found the solution:

1. Change the Scopes from Readonly to just ClassroomCourses
2. Recreate credentials
3. Upload the new file in project folder

• Classroom scopes

Tips that may help are:

Data can be posted via GET or POST. What you are describing with URLs is GET. POST is when you don't see anything in the url.

In both cases getting familiar with using your browser's developer console will help you explore how websites work. In Chrome, you can hit F12 or right click any element and select "inspect element." This is especially helpful when trying to inspect data that is passed using POST, since you can't see them in the url. Use the "network" tab while clicking around to see what the website is doing in the background.

Lastly, just play around with websites. For example, when you browse Amazon you might notice the urls look like https://www.amazon.com/Avalon-Organics-Creme-Radiant-Renewal/dp/B082G172GL/?_encoding=UTF8 but if you play around with it you notice you can delete out the title and the url still works like this: https://www.amazon.com/dp/B082G172GL

The simple thing to do is to create a B-tree index to speed up the search:

You can generate a hash to easily look up the client count using each_with_object. You can then map over the current_clients array and add the client count.

There is a difference, yes. In the code presented, one would call the constructor like so:

The sneaky thing about amortization is that real costs no longer exactly correspond to amortized costs. There's no general method for determining how operations are amortized other than you can't go bankrupt, so you could make the amortized cost of multi-pop be 5 if you wanted.

In this example, you can imagine that whenever we push an element (cost 1), we also purchase a voucher good for popping that element (cost 1, so 1 + 1 = amortized cost 2 since we did a unit of work and prepaid for another). Then we pay for the pops using vouchers only, so both pop and multi-pop are free.