Timeline for How to invoke constants badly
Current License: CC BY-SA 4.0
42 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 27, 2020 at 16:06 | comment | added | mathworker21 | My point is, we should always just rely on context and the reader to understand. The $o(1)$ should be clear from context. If there is a need to emphasize to the reader that $n$ is the relevant parameter shrinking/growing, then one should write $o_{n \to \infty}(1)$. I just see no point to $o(n^0)$. | |
| Dec 27, 2020 at 16:04 | comment | added | mathworker21 | @TimothyChow But the reason you're making the change is to reinforce the idea that whatever is in the o/O is the parameter growing or shrinking. My issue is not only when there are multiple variables. Suppose we have some process changing over time $t \to \infty$. As $t \to \infty$, our position $x(t)$ changes in some way. And we have two functions $f(x)$ and $g(x)$ that are functions of position $x$. If we want to say $f(x) = o(g(x))$, we'll be referring to $t \to \infty$ and not any limit of $x$. | |
| Dec 27, 2020 at 14:45 | comment | added | Timothy Chow | If there are many variables and only one is going to infinity, then I will happily adopt whatever notation you suggest. | |
| Dec 27, 2020 at 14:39 | comment | added | Timothy Chow | @mathworker21 : The fact that we use $o(n)$ means that we trust the reader to interpret the notation correctly. If we trust readers that far, then why not trust them with $o(n^0)$? It seems no more unnatural to me than $o(1)$. The trouble with $o(1)$ is that we have intentionally destroyed information irretrievably for no good reason. Note, by the way, that I am not adopting the convention that whatever is in the o/O is the parameter growing or shrinking. I am making only one change to the standard notation. | |
| Dec 27, 2020 at 14:29 | comment | added | mathworker21 | @TimothyChow No. I said $o(n^0)$ is reinforcing the false conception because it's clear the author went out of their way to emphasize that $n$ is the variable going to infinity or $0$. If $o(n^0)$ genuinely popped up naturally, that would be fine. | |
| Dec 27, 2020 at 12:51 | comment | added | Timothy Chow | @mathworker21 : So $o(n)$ is also bad, I guess, because it reinforces the same false conception? | |
| Dec 27, 2020 at 4:17 | comment | added | mathworker21 | @TimothyChow "Do you disagree, and believe that the notation $o(1)$ is strictly better than the notation $o(n^0)$?" Yes, of course. The notation $o(1)$ is easier on the eyes and the notation $o(n^0)$ reinforces the false conception that variables inside the parentheses are the relevant parameters going to infinity or $0$. | |
| Dec 27, 2020 at 4:09 | comment | added | Timothy Chow | @mathworker21 : I'm not claiming to solve all problems with my suggestion. I'm claiming only that the notation $o(n^0)$ is no worse than, and probably better than, the notation $o(1)$. Do you disagree, and believe that the notation $o(1)$ is strictly better than the notation $o(n^0)$? (Assume that no other changes are made to the standard notation.) | |
| Dec 25, 2020 at 11:34 | comment | added | mathworker21 | @TimothyChow But what if you have many variables and only one is going to infinity? Like, let's say you want to note $kn = o(kn^2)$, where $k$ is some fixed thing and $n \to \infty$. I really don't think it's wise to adopt the convention that "whatever is in the o/O is the parameter growing or shrinking". | |
| Dec 23, 2020 at 10:29 | vote | accept | Alessandro Della Corte | ||
| Dec 23, 2020 at 10:29 | vote | accept | Alessandro Della Corte | ||
| Dec 23, 2020 at 10:29 | |||||
| Dec 23, 2020 at 10:29 | vote | accept | Alessandro Della Corte | ||
| Dec 23, 2020 at 10:29 | |||||
| Dec 22, 2020 at 23:28 | comment | added | Timothy Chow | (By the way, when I say zero converts, that includes myself!) | |
| Dec 22, 2020 at 23:27 | comment | added | Timothy Chow | @LSpice : Terry Tao's suggestion includes more information because it gives the limiting value of $n$. But I doubt that many people will adopt it except in circumstances where it is really important to emphasize what the limit is. People are fundamentally lazy, after all. Who is going to write, "Comparison sorting takes $O_{n\to\infty}(n\log n)$ comparisons"? I was hoping that maybe people could be induced to switch from $o(1)$ to $o(n^0)$ since it's not all that much more work. But either because old habits die hard or because it's easier to write $1$ than $n^0$, I have zero converts so far. | |
| Dec 22, 2020 at 22:12 | vote | accept | Alessandro Della Corte | ||
| Dec 23, 2020 at 10:29 | |||||
| Dec 22, 2020 at 19:54 | comment | added | LSpice | @TimothyChow, re, isn't the information you're trying to recover precisely that provided by @TerryTao's suggestion $o_{n \to \infty}(1)$? | |
| Dec 22, 2020 at 18:03 | answer | added | pinaki | timeline score: 16 | |
| Dec 22, 2020 at 17:41 | comment | added | Alessandro Della Corte | @TerryTao I guess the writing style of mathematicians working in "hard" analysis estimates is more or less optimized for their aims - after all, while mathematics is an exact science, writing it and maybe even reading it is somewhere in the middle between an empirical science and an art. | |
| Dec 22, 2020 at 17:36 | comment | added | Terry Tao | @TimothyChow Having the asymptotic variable appear inside $o()$ can sometimes help identify the asymptotic limit but is not a reliable solution for doing so. For instance the statement $t = o(t^2)$ without context does not identify whether the asymptotic limit is $t \to \infty$ (where the statement is true) or $t \to 0$ (false). Similarly, the statement $n^k = o((n+1)^k)$ for natural numbers $n,k$ without additional context does not identify whether the asymptotic limit is $k \to \infty$ with $n$ fixed (true) or $n \to \infty$ with $k$ fixed (false). Subscripting is my recommended solution. | |
| Dec 22, 2020 at 13:33 | history | made wiki | Post Made Community Wiki by Todd Trimble | ||
| Dec 22, 2020 at 10:33 | comment | added | Sebastian Bechtel | @DeaneYang I prefer to state on which parameters a constant does not depend. I see no point in tracking the dependence on stuff that's not relevant to my problem. An incarnation of this paradigm is to state how a constant depends on certain things, like it depends on an elliptic operator only via its coefficient bounds (i.e., it does not depend on the remaining data of the operator). Of course it is then the responsibility of the user to check that one really only varies parameters that were excluded from the dependence list. | |
| Dec 22, 2020 at 9:42 | answer | added | YCor | timeline score: 8 | |
| Dec 22, 2020 at 9:34 | history | edited | YCor | CC BY-SA 4.0 |
formatting
|
| Dec 22, 2020 at 7:46 | history | became hot network question | |||
| Dec 22, 2020 at 7:23 | answer | added | Alessandro Della Corte | timeline score: 41 | |
| Dec 22, 2020 at 6:08 | vote | accept | Alessandro Della Corte | ||
| Dec 22, 2020 at 22:12 | |||||
| Dec 22, 2020 at 6:08 | vote | accept | Alessandro Della Corte | ||
| Dec 22, 2020 at 6:08 | |||||
| Dec 22, 2020 at 4:49 | comment | added | Terry Tao | @BrendanMcKay A standard phrasing in analytic number theory is "The implied constant is absolute". (There is also the further distinction of constants into "effective" and "ineffective"; what you refer to as a "numerical constant" would likely be called an "effective absolute constant" (or "effectively computable absolute constant") in analytic number theory.) | |
| Dec 22, 2020 at 3:51 | comment | added | Brendan McKay | The opposite question is how one conveys to the reader that a constant doesn't depend on anything; i.e. that it could be replaced by an explicit number. I've see "universal constant" and "numerical constant" but I don't know if everyone would get the message. | |
| Dec 22, 2020 at 2:43 | comment | added | Brendan McKay | A very similar and common sloppiness occurs with "for sufficiently small $\varepsilon>0$" when the required smallness actually depends on other parameters lurking in the background. | |
| Dec 22, 2020 at 2:20 | answer | added | R W | timeline score: 15 | |
| Dec 22, 2020 at 2:12 | comment | added | Timothy Chow | @TerryTao : Actually, I think that what bothers me is that it strikes me as an example of misguided "simplification." In the expression $o(n)$ or $O(n)$, the $n$ is there not just to indicate the rate of growth, but to indicate what variable is going to infinity. So $o(n^0)$ or $O(n^0)$ is the logical extension of that notation, but then people reflexively "simplify" the expression, perhaps without consciously realizing that they're not really simplifying, but actually destroying information. | |
| Dec 22, 2020 at 1:37 | comment | added | Terry Tao | @TimothyChow My guess what is bothering you about $o(1)$ (and possibly also $O(1)$) is that it looks like a function applied to $1$, but it really is a rather different use of the (highly overloaded) parenthesis symbol. Personally I like to use $o_{n \to \infty}(1)$ instead of $o(1)$ in order to explicitly specify the asymptotic limit involved here (and there are variants such as $o_{n \to \infty;k}(1)$ when one wants to permit the decay rate to depend on additional parameters such as $k$). | |
| Dec 22, 2020 at 1:31 | comment | added | Terry Tao | @R.vanDobbendeBruyn In any field of math, if quantifiers were introduced separately for each statement, the exposition would become very cluttered (esp. if there are a lot of nonce variables that occur in just one line). Outside of "hard" analysis, though, there are often convenient definitions that hide a lot of the quantifiers (e.g., continuity hides epsilon and delta quantifiers, a universal property in a category hides universal and existential quantifiers, etc.). But in the "fuzzy" world of estimates we often don't have this luxury (unless one uses something like nonstandard analysis). | |
| Dec 21, 2020 at 23:13 | comment | added | Timothy Chow | On a related note, the "little-oh" notation $o(1)$ has always bothered me, and for a while I tried using the notation $o(n^0)$ instead, but I couldn't persuade anyone else to adopt it. | |
| Dec 21, 2020 at 19:28 | comment | added | R. van Dobben de Bruyn | In the rest of mathematics, this problem is solved by introducing quantifiers at the right place in the statement (usually written out in words). I've never fully understood why this is not [always] used when doing estimates... | |
| Dec 21, 2020 at 19:26 | comment | added | Denis Serre | I, a nephew of Jean-Pierre Serre, am certainly guilty of invoquing such constants in my research activity. But the inequality I am proud of (Compensated Integrability) has an explicit and accurate constant. | |
| Dec 21, 2020 at 18:37 | review | Close votes | |||
| Dec 25, 2020 at 4:20 | |||||
| Dec 21, 2020 at 18:20 | comment | added | Alessandro Della Corte | Indeed I was not sure about creating a list, you can remove the tag if you want. However, I think Serre was referring to published papers. | |
| Dec 21, 2020 at 18:18 | comment | added | Michael Renardy | It is the kind of mistake students make every day, and I have certainly seen it in submitted manuscripts, which were insufficiently memorable to recall specifics. I do not see the point of creating a list here. | |
| Dec 21, 2020 at 17:37 | comment | added | Deane Yang | I can't cite any mistakes caused by this practice, but I have no doubt there have been many. When I first ran into this when studying PDEs, it made me very uncomfortable, so I always wrote explicitly what parameters $C$ did depend on. So my $C$ always looked something like $C(n,a,b,P, Q, \beta)$. If I could write an explicit formula for $C$, I often did, since the dependence of the parameter sometimes mattered, too. | |
| Dec 21, 2020 at 17:30 | history | asked | Alessandro Della Corte | CC BY-SA 4.0 |