I have purchased products/services off of ads, but only because I was directly searching to buy those things and they came up as sponsored results. I'm not sure if that counts.
I have never bought off of ads otherwise and find that most of the time they are not relevant to my interests. I also started using an adblocker because of other reasons (mal-ads/autoplay) even though I previously did not do so because I felt text/banner ads were quite acceptable in order to help keep the lights on.
a baire category is an old-timey synonym for a baire space. It's related to topology & analysis, not algebra (like category theory). In the modern theory we can make an analogy (but it is just an analogy!) between 'categories' of (complete non-empty) metric spaces on the one hand, and measure spaces (of positive measure) on the other, as follows:
metric spaces: measure spaces:
first category zero measure
second category positive measure
residual full measure
baire measurable
A terminology note, I think "first category"/"second category"/"residual" have largely been replaced by the clearer "meagre"/"non-meagre"/"comeagre". Or at least, I hope they've largely been so replaced, because geez the old terminology is a pain. :P
Yes. "Meagre" means small. Which is what a first-category set is, but the name "first category" doesn't tell you that (how is one supposed to remember which of first and second category is which?). And then after that, you don't need to introduce any more words. If you mean to say something is not meagre, you just say it's not meagre, same as you would with any concept. If you mean to say it's complement is meagre, you just say it's comeagre, same as with anything.
The names "first category" and "second category" tell you nothing, not even making it obvious which is which -- and moreover, they make it sound like the two are both equally useful sides of a distinction, like finite and infinite, while in reality one of these is actually directly useful and the other is just everything that isn't that (which is how most distinctions in mathematics go; you wouldn't make up a separate new word for "not compact", for instance).
And then "residual" makes it sound like there's some new thing going on, when of course it's really just being co-first-category. But co-first-category's quite the mouthful, isn't it...?
I'd say so. "non-meagre" = not meagre = positive, and "comeager" = complement of meagre = full. The only thing you have to remember is what "meagre" means.
i remember in undergrad my analysis prof saying there were two types of analysis: hard analysis (sans baire category theorem) and soft analysis (using baire category theorem). i wonder if what he actually meant was in baire spaces and not in baire spaces.
I suspect your professor was referring to 'qualitative' vs 'quantitative' approaches when referring to soft vs hard analysis. The baire category theorem is intimately related to these to approaches, and leads to three theorems (1) the uniform boundedness principle, (2) the open mapping theorem, and (3) the closed graph theorem, that establish equivalences between the qualitative and quantitative theories.
According to Tao's textbook, in practice these are not used much directly, instead one starts with quantitative bounds, then derives the qualitative corollaries. The baire category theorem tells us why those there theorems are true, and thus tells us why we can take that step from quantitative to qualitative. I think this is what your professor meant.
Referring to properties of linear operators. Examples are finiteness, surjectivity, non-negativity, monotonicity, min/max principles, contraction properties, etc.
I think it just means kindergarten to year 13. At least, Ontario (Canada) used to have a 13th year, but they phased that out 2 decades ago into just K-12 as well.
My dad went to Best Buy to replace his 10+ year old laptop last year. He actually said he wanted Intel instead of AMD because he heard AMD processors were very slow. I had to tell him the Intel N-whatever and Celerons were also very slow so that he'd stick with one of the i-whatevers.
Basically anything they sell him will be faster than his current laptop. Many of the low power i series aren't actually much better than a celeron, but they're very common in new laptops.
CPUs aren't the bottleneck though, disk I/O and I/O in general are where most "slowness" issues are promulgated from.
"Scribble is a collection of tools for creating prose documents—papers, books, library documentation, etc.—in HTML or PDF (via Latex) form. More generally, Scribble helps you write programs that are rich in textual content, whether the content is prose to be typeset or any other form of text to be generated programmatically."
You type in text and mark up using @-tags that are defined in, and can also contain, the full Racket language. It's like LaTeX with a modern language (but without the tons of available packages).
This is unlikely something you'll use in practice, but I just read about it so it came to mind.
A Braun tree is a binary tree that represents a flexible array (indexable everywhere and extendable at both ends). The gist of it is that you take your array index and read it as a binary string that tells you how to go down the tree. If your index is 1, then you're at the right place and you stop. Then you go left/right depending on whether the remainder is 0/1. That is the 1-based interpretation, and it works because 1 is in the first digit while 2 and 3 need the next digit as well.
If you want to use 0-based indices, it doesn't work out as cleanly, because 0 and 1 share the same digit, while 1 and 2 use different digits. So now you have to branch on whether your index is odd or even, and then either (i-1)/2, or (i/2)-1.
It's not a big deal either way, but it turns a simple to visualize data structure into one where you have to think about the indices.
Binary heaps are more commonly known and nicer to work with in 1-based arrays for the same reason. If arrays are 1-based, a binary heap's node's children are at 2n and 2n+1, and the node's parent is at n/2.
You shouldn't need a branch in a 0-based Braun tree though. (n-1)/2 should be correct all the time.
But why try the "PortableApps" package of a program that isn't listed on that program's homepage instead of using that program's own executable installer listed on its homepage?
Is it not possible that this "PortableApps" package broke the program?
I have never bought off of ads otherwise and find that most of the time they are not relevant to my interests. I also started using an adblocker because of other reasons (mal-ads/autoplay) even though I previously did not do so because I felt text/banner ads were quite acceptable in order to help keep the lights on.