Wikipedia:Reference desk/Mathematics#Trajectory of a projectile with air drag

I just encountered the concept of the Hamming distance for the first time. I see that it is the number of positions at which the corresponding symbols are different, and the minimum number of substitutions required to change one string into the other, and the minimum number of errors that could have transformed one string into the other.

What is the Hamming distance between 0010 and 0100? Is it two, since a digit-by-digit replacement forces us to change first to 0000 or 0110, and second to 0100? Or is it one, since each one has three 0s and a 1, and the only difference is a single error of sequence? Nyttend (talk) 00:29, 11 June 2025 (UTC)

:It's two. There are two positions at which the strings differ (the second and third positions).--Antendren (talk) 05:29, 11 June 2025 (UTC)

= June 12 =

I know they are better algorithms. But I want to solve a discrete logarithm in a finite field having a finite field of several Kb long and where the discrete logarithm solution lies into a 200bits subgroup.

The problem of such finite fields is there’s no birational equivalence to finite rings : such finite field element are polynomials. In such a case, what does it means for a finite field element to be smooth ? How do you achieve factorization into prime elements in such a case ? 2A01:E0A:ACF:90B0:0:0:A03F:E788 (talk) 21:27, 12 June 2025 (UTC)

= June 13 =

For the decimal expansion of n² to contain no digits other than 1, 4 and/or 7, what constraints are there on n? Thanks, cmɢʟee⎆τaʟκ 15:56, 13 June 2025 (UTC)

:The sequence of such $n$ is OEIS:A053898, while the sequence of their corresponding $n^\{2\}$ is OEIS:A053899. I'm searching for a pattern/constraints on $n$. GalacticShoe (talk) 17:04, 13 June 2025 (UTC)

::Thanks very much. Amazing it's already in the OEIS! cmɢʟee⎆τaʟκ 22:51, 13 June 2025 (UTC)

:::I think part of the appeal is the rarity. There are no squares containing only one digit other than the trivial single-digit squares, and the sequence of squares containing at most two digits, OEIS:A018885, seems to eventually become all multiples of powers of 10 by 1, 4, and 9 after $6661661161\; =\; 81619^\{2\}$. Squares containing exactly three distinct digits would be a natural place to continue. GalacticShoe (talk) 01:32, 15 June 2025 (UTC)

::::It is currently unknown, though, whether A018885 contains only a finite number of members without 0 tail.

::::An element of the ring of n-adic numbers (a field when n is prime) can be viewed as an infinite path on a rooted tree, in which the nodes of the path correspond to the tail segments of the element. One can do a breadth-first tree unfolding of the 10-adic numbers whose squares (or cubes etc.) use only a small finite number of digits. For example, for the case of at most two distinct digits, the square of ···1619 is ···1161, whose tail at the same length uses just two digits. So 1619 is a node in the tree, at depth level 4. For which d do the last five digits of ···d 1619 use only digits 1 and 6? Well, ···31619² = ···61161, while also ···81619² = ···61161. All other choices for d break the just-1-and-6-ness. So the tree node 1161 sprouts two branches, to nodes 31619 and 81619. Continuing, we find that ···331619² = ···161161 and ···831619² = ···161161, so 31619 sprouts branches to 331619 and 831619. Likewise, node 81619 sprouts branches to 081619 and 581619.

::::A node may fail to sprout branches. For the one-digit case, ···12² = ···44, but no ···d 12 squares to ···444, so it is a dead end. In fact, here all paths terminate except for ···00000000000. Some nodes are bingo nodes, like for the two-digit case the node 81619, since 81619² = 6661661161 in ordinary arithmetic.  ​‑‑Lambiam 08:45, 15 June 2025 (UTC)

= June 16 =

As far I understand, when it comes to finite fields, Pollard rho and Pollard’s lambda are still the best algorithm for solving discrete logarithms in a multiplicative subgroup/suborder…

Index calculus methods like ɴꜰꜱ can be made to target multiplicative subgroups, but they don’t work in multiplicative subgroups : given a finite fields with a very large characteristic which is enough larger than the target subgroup, they can have an higher complexity than using a Pollard’s method.

Therefore, if we discard quantum computers and Oracle based situations, there’s no known sub‑exponential algorithm that can take real advantage from knowledge the solution of a specific discrete logarithm is below a specific integer.

This even lead to cryptographic systems where the publicly known and used finite field’s order is relatively small for a would be sub‑exponential algorithm but the finite field itself is too large for index calculus (still used today).

Now is it because of lack of research or is there a reason in number theory that prevents creating an algorithm having at least a time and space complexity ? 2A01:E0A:ACF:90B0:0:0:A03F:E788 (talk) 23:15, 16 June 2025 (UTC)

:: There is a theorem of Victor Shoup that this is not possible. Tito Omburo (talk) 23:45, 16 June 2025 (UTC)

:::Where to find it ? 2A01:E0A:ACF:90B0:0:0:A03F:E788 (talk) 00:44, 17 June 2025 (UTC)

:::What’s the name of the article from the author ? 2A01:E0A:ACF:90B0:0:0:A03F:E788 (talk) 20:57, 18 June 2025 (UTC)

:::: I do not know the name. Shoup allegedly "proves" that in a cyclic group where multiplication is given by an oracle, it is not possible to achieve better than $O(\backslash sqrt\; p)$ time for the discrete logarithm problem. As far as I understand the question, this was it. Multiplicative groups in finite fields have more structure than this, but that structure ultimately reduces to discrete log in the multiplicative subgroups per Pohlig-Hellman. These Pohlig-Hellman subgroups are too "rarified" to expect anything more than Shoup's result. Caveat: I think Shoup is wrong, and all of the complexity estimates are bunk, but that's just me. Tito Omburo (talk) 21:12, 18 June 2025 (UTC)

:::::Here's the paper from his website: [https://www.shoup.net/papers/dlbounds1.pdf] 173.79.19.248 (talk) 01:58, 21 June 2025 (UTC)

= June 18 =

I was looking at a Circuit of the pentagonal faces of a Dodecahedron, wondering if it could be done with only passing from a side to one of the non-adjacent sides. I determined it can't since any set of three faces in such a circuit would be the ones on three consecutive sides of a fourth pentagon, which could then only enter and exit by adjacent sides.

I was looking at moving that up a level. For a given face of a Dodecahedron, designate a move from the face to one of the five adjacent faces as an Ortho move, a move from a face to an opposite face as a Para move and a move to one of the other five faces as a Meta move (see Arene substitution pattern for the naming convention) Can a circuit through the cells of a 120-Cell be done *only* with Meta moves? Can a circuit be done only with Ortho Moves? If no for either, can it be done allowing both that type of move or Para. (Obviously it can't be done with Para only has that will trace out the "equator" of the polydedron (similar to doing it on the cube or hypercube).Naraht (talk) 01:57, 18 June 2025 (UTC)

I just know it was published in 1968 by

Tables of indices and primitive roots, Royal Society Mathematical Tables, vol 9, Cambridge University Press.

But don’t know the article’s exact name nor can I afford the book… Even if I could afford it, I need it as a ᴘᴅꜰ. 2A01:E0A:ACF:90B0:0:0:A03F:E788 (talk) 20:56, 18 June 2025 (UTC)

:That volume was authored by A. E. Western and J. C. P. Miller. Part of the introduction, preceding the tables, can be viewed [https://oeis.org/A002223/a002223.pdf here].

:I don't see anyone with the surname "Stephen" who has published work on elliptic curves. Could this by any chance refer to Nelson Malcolm Stephens (1941–2024), no article on Wikipedia but author of one of the cited works in Feit–Thompson conjecture? The article [https://www.ams.org/journals/proc/2004-132-04/S0002-9939-03-07311-8/ "Primes generated by elliptic curves"] is by Graham Everest, Victor Miller and Nelson Stephens.  ​‑‑Lambiam 00:07, 19 June 2025 (UTC)

= June 20 =

For y=|x| the x=0 point is said in italian "punto angoloso". Paper and on-line dictionaries fail to translate this term. Note that I mean a generic case, not the peculiar case called cusp. In the english wiki I have not found a definition, but I have found such a point called angle, corner, sharp turn and bend. Which is the most-widely accepted term? thanks 151.29.62.127 (talk) 09:49, 20 June 2025 (UTC)

:I'm not convinced it has exactly the same meaning as you're going for, but I think vertex comes pretty close. That's the term used in the context of linear programming at least. A direct translation, "corner point" will probably be understood as well. And yes, a cusp is a special case since the tangent line moves continuously there. There are many ways that a function can be non-differentiable though, for example y=x sin (1/x) is not differentiable at 0, and I'd hesitate to call that a corner. As with many language questions, it helps to know the context, can give an example of the word being used in a sentence? --RDBury (talk) 12:10, 20 June 2025 (UTC)

::Thanks very much. I am translating italian lessons for people that know little english and almost no italian, so the context cannot help.151.29.62.127 (talk) 12:34, 20 June 2025 (UTC)

:In old calculus books, you may find two words, "cusp" for where the derivative goes to infinity, and "salient point" for where the derivative has a jump discontinuity. I think the best answer to your question as phrased is "salient point", though that excludes the cusp case. --Trovatore (talk) 20:04, 20 June 2025 (UTC)

:Anticipating a couple of possible objections. First, technically, the first derivative doesn't have a jump discontinuity at the singular point because it isn't defined there. Hopefully you can work out what I mean.

:More importantly, I'm just treating the case of the graph of a function, not more general curves. Again the extension is left as an exercise. --Trovatore (talk) 20:07, 20 June 2025 (UTC)

::I've never personally seen "salient point", and notably Richard Courant's classic Differential and Integral Calculus uses the term "corner" (although as a purely descriptive noun, rather than term of art). It's fair to say that there is no widely accepted term of art for this, and "corner" seems to be common, but even a literal translation of the Italian would be satisfactory. Tito Omburo (talk) 20:12, 20 June 2025 (UTC)

:::It's attested in {{cite book |last= Griffin|first= Frank Loxley|date= 1927|title= Mathematical Analysis Higher Course|publisher= The Riverside Press|location= Cambridge, Massachusetts|page= 390|isbn= }}. --Trovatore (talk) 20:38, 20 June 2025 (UTC)

:::File:SalientPoint.jpg

:::It happens to be PD, so I'm going to go ahead and drop in a shot of it; possibly it will give context, but also I just love these old books. --Trovatore (talk) 20:56, 20 June 2025 (UTC)

::::A book of somewhat dubious provenance, imo. Tito Omburo (talk) 21:07, 20 June 2025 (UTC)

::::: I have to say I find that comment a little reductive. It's actually an excellent book for learning certain techniques. It was my dad's college textbook and I had a great time with it in high school (or was it junior high?). It's true that it's pitched more to the engineering or physics student and less to the mathematician. I did learn some physics from it as well.

::::: This particular part of it is a little frustrating because it's not completely clear what Griffin is talking about. (It might help to know that on the previous page he's discussing curves defined by an equation $f(x,\; y)=0$, and the mysterious equation 61 he mentions is $\backslash frac\{\backslash partial\; f\}\{\backslash partial\; x\}+\backslash frac\{\backslash partial\; f\}\{\backslash partial\; y\}\backslash frac\{dy\}\{dx\}=0$.) In particular the statement that "[a] salient point is possible only when the equation involves transcendental or irrational algebraic functions of special kinds" is a bit obscure. I wondered for years what these cases would be. He does give an example in the exercises, but it's $y=5-\backslash sqrt\{x^2(x+4)\}$, which is obviously just leveraging the convention that the square-root function gives the nonnegative square root if it exists. In any case this is potentially "good trouble" as it might spur the interested student to learn about algebraic varieties.

::::: Griffin was head of the math department at Reed College, its acting president for a while, and vice president of the MAA. We probably ought to have an article about him. There's a starting point [https://www.reed.edu/president/reed_presidents/griffin.html here]. --Trovatore (talk) 00:58, 21 June 2025 (UTC)

::::The term "angular point" is also found: E. J. Townsend, G. A. Goodenough (1910). Essentials of Calculus. New York: Holt. [https://archive.org/details/cu31924000678015/page/n68/mode/1up?q=%22angular+point%22 p. 51]. Or, more recently, [https://books.google.com/books?id=QnpqBQAAQBAJ&pg=PA243&dq=%22angular+point%22&hl=en here] or [https://books.google.com/books?id=WMyZlwiQBNYC&pg=PA98&dq=%22angular+point%22&hl=en here]. Townsend was the translator of the 1902 English edition of Hilbert's Grundlagen der Geometrie.^{[https://archive.org/details/foundationsgeom00towngoog/page/n6/mode/1up]}  ​‑‑Lambiam 21:15, 20 June 2025 (UTC)

:{{ping|151.29.62.127}} I think "singular point" might be the best choice for your needs here. The only thing is it might be more general than you want, in that it at least arguably includes points of discontinuity, but if you specify that the function is continuous, that should be OK. --Trovatore (talk) 21:03, 20 June 2025 (UTC)

:: Singular point could mean anything! That's a bad choice. OP is specifically asking about corners. Tito Omburo (talk) 21:06, 20 June 2025 (UTC)

:::There is, moreover, only a singularity in the derivative.  ​‑‑Lambiam 09:16, 21 June 2025 (UTC)

::::"Singular" doesn't mean "discontinuous". A discontinuity in the derivative is absolutely a singularity, and as I said a discontinuity in the function itself could be ruled out by context. That said, it's true there are various other possibilities. However the heading asks about "non-derivable points" ("non-derivable" presumably meaning "non-differentiable"). In lots of contexts those would be precisely the singular points. --Trovatore (talk) 17:01, 21 June 2025 (UTC)

::File:AngularPoint.jpg

::Wiktionary defines the term salient point solely as: "a prominent feature or detail". This is what English readers will naturally expect the term to mean. Wiktionary lists an obsolete sense "moving by leaps or springs; jumping" for salient; apart from this being obsolete, what is moving by a leap is only the derivative.

::I have uploaded the textbook definition of angular point, which is crisp and clear. Modern texts (refs given above) using the term give equivalent definitions. The term is appropriately descriptive.  ​‑‑Lambiam 09:48, 21 June 2025 (UTC)

:::I don't know how Griffin came to give this sense in his textbook. I had assumed it was standard but perhaps it was not. Searches are confounded by a more recent sense from computer vision and machine learning. --Trovatore (talk) 17:07, 21 June 2025 (UTC)

::::Probably a calque of French point saillant.^{[https://books.google.com/books?id=5foJAAAAIAAJ&pg=PA256&dq=%22point+saillant%22&hl=en][https://books.google.com/books?id=jMsKAAAAYAAJ&pg=PA245&dq=%22point+saillant%22&hl=en][https://books.google.com/books?id=jr8KAAAAYAAJ&pg=PA326&dq=%22point+saillant%22&hl=en]}  ​‑‑Lambiam 18:28, 21 June 2025 (UTC)

= June 21 =

Is James Tanton an Australian mathematician, as stated in the article about him, or an American author, as stated of the article Encyclopedia of Mathematics (James Tanton). None of the references for either of these pages is clear. Possibly there are two people called James Tanton, one of whom is a mathematician and the other wrote an encyclopedia about mathematics. 176.108.139.1 (talk) 01:31, 21 June 2025 (UTC)

:Math genealogy knows about one [https://www.mathgenealogy.org/id.php?id=18868 James Tanton] (actually only one person with the surname Tanton). That seems to be the same person as [https://www.jamestantonmath.com/ this guy], whose CV (linked there) includes degrees from an Australian university and a book he's written called Encyclopedia of Mathematics. All this suggests there is one person, who has lived in both Australia and the US at different times. 173.79.19.248 (talk) 02:08, 21 June 2025 (UTC)

::Since graduating from the University of Adelaide in 1988, Tanton has been living in the US.^{[https://stevementz.com/earth-by-jeffrey-cohen-and-lindy-elkins-tanton-object-lessons/]} He is settled there, with a family – his spouse is planetary scientist Lindy Elkins-Tanton,^{[https://news.berkeley.edu/2025/05/06/planetary-scientist-lindy-elkins-tanton-to-head-space-sciences-laboratory/]} who was born in Ithaca, New York. While not certain, it is reasonable to assume that Tanton is now a naturalized US citizen. If so, he should be referred to as an "Australian–American mathematician", like we do for James Oxley, Terence Tao and Paula Tretkoff.  ​‑‑Lambiam 07:43, 21 June 2025 (UTC)

:::Thanks! I've updated both articles. 176.108.139.1 (talk) 18:16, 21 June 2025 (UTC)

For polynomials over the integers of degree n, one could classify them as Cantor did, by k= sum of the absolute values of the coefficients. For each k,the number of each that are factorable, the number that have Galois group C_n, D_n[or D_2n depending on convention], A_n, S_n could be counted, at least for small n and small k. If the polynomial is factorable, a "quasi" galois group could be noted as the dirct product of the galois groups of the irreducible factors.(not transitive though). So are there results anyone here know about this? Also what books are there where I can I find out about work like this? (I've heard in passing that as n->oo, most polynomials have galois group A_n, but I don't know why or where to find out.Rich (talk) 20:44, 21 June 2025 (UTC)

= June 24 =