Wikipedia:Reference desk/Archives/Mathematics/2022 January 18

= January 18 =

What's the name of the vertical bar notation?
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$$\int_a^b f(x) \, dx = F(b) - F(a) = F(x) \Bigg|_a^b$$ What's the name of the vertical bar on the right side of F(x)? - Justin545 (talk) 15:33, 18 January 2022 (UTC) Do we have an article about it? By the way, there is a similar notation without the vertical bar in integration by parts: $$\begin{align} \int_a^b u(x) v'(x) \, dx   & = \Big[u(x) v(x)\Big]_a^b - \int_a^b u'(x) v(x) \, dx\\[6pt] & = u(b) v(b) - u(a) v(a) - \int_a^b u'(x) v(x) \, dx. \end{align}$$ - Justin545 (talk) 16:25, 18 January 2022 (UTC)


 * Some texts call it an "evaluation bar", and the term is even used for the notation with square brackets. It appears not to be in common use, though; it may not be understood without context or explanation. The term "evaluation bar" is also used for the notation $$f(x)|_{x=a},$$ for which it seems more appropriate. Did you know? The vertical bar to indicate evaluation of an antiderivative at the two limits of integration was used first by French mathematician Pierre Frederic Sarrus (1798–1861) in 1823. --Lambiam 17:11, 18 January 2022 (UTC)
 * The evaluation bar is kind of unclear. For instance, the bound variable of $$g(x, y, z) \Big|_a^b$$ is unclear but $$h(x, y, z) \Big|_{y = \pi}$$ is not. Thanks for the links to give me a chance to rediscover the notation. - Justin545 (talk) 18:25, 18 January 2022 (UTC)
 * When I teach multivariable calculus I often use notations like $$g(x, y, z) \Big|_{x = a}^{x = b}$$ or similar, which resolves the ambiguity. (I don't give this symbol a name, though, and I don't think any of the textbooks I've used does, either.)  --JBL (talk) 00:42, 19 January 2022 (UTC)

cardinality of recursive set
If I define set A = {B} and set B = {A}, then does this create any sort of recursive paradox or is this perfectly fine? And is the cardinality of this set simply infinite?

Duomillia (talk) 17:57, 18 January 2022 (UTC)
 * Not a paradox, per se, but it violates the axiom of foundation, which holds in the von Neumann universe ordinarily used to interpret set-theoretic statements.
 * There are alternative set theories that use some sort of anti-foundation axiom (that redirect currently goes to one particular axiom; it should probably go to non-well-founded set theory), and which allow sets meeting your requirements. However the cardinality of the set is not infinite.  It's just one.  It has exactly one element, namely B. --Trovatore (talk) 18:04, 18 January 2022 (UTC)
 * I don't know if the axiom of foundation was designed to eliminate such strangeness, but it certainly has that effect. I'd argue as well that the definitions of A and B are circular, and so they don't actually define anything. You can certainly ask if there are sets A and B with A={B} and B={A}, but that's not a definition. --RDBury (talk) 18:31, 18 January 2022 (UTC)

Just to clarify, how do we arrive at the conclusion that the two sets violate the axiom of foundation? A = {B} and the contents of A is disjoint (B) Duomillia (talk) 00:52, 20 January 2022 (UTC)
 * If your sets A and B exist, then look at C = {A, B}. C is not empty, and yet it has no element disjoint from C. --Trovatore (talk) 01:09, 20 January 2022 (UTC)

Cuboctahedral dice (D14? D6/8?)
A cuboctahedron is a solid shape with six square and eight equilateral triangular faces. It could conceivably be used as a 14-sided die. By symmetry, the probability of rolling any one the the square faces is equal, likewise the probability of rolling any one of the triangular faces. It seems likely, however, that those probabilities differ from each other, so such a die would be more likely to land on a square (or maybe a triangular?) face. Is there a mathematical method for finding these probabilities? --Verbarson talkedits 20:56, 18 January 2022 (UTC)
 * I think it's unlikely that there's any accurate purely mathematical approach to this. It's really a physics question.  You could answer it mathematically if you had some reason, say, to think that the probability were proportional to the area of the face.  But it's not obvious to me that that should be so. --Trovatore (talk) 21:01, 18 January 2022 (UTC)
 * When did/will off-the-shelf Windows® physics software get good enough to get good odds by manual tallying without having to learn languages like R or Python or C or Fortran or whatever? Sagittarian Milky Way (talk) 22:11, 18 January 2022 (UTC)
 * Our article on the finite element method references a number of software packages; that could be a start. Note that the answer may depend on factors other than the shape of the die, so there may not be a single answer independent of the die's composition and construction. --Trovatore (talk) 22:46, 18 January 2022 (UTC)
 * If the die is cast in a vacuum, and the collisions are fully inelastic, it will never settle. One way of modeling the die's necessary loss of energy is to imagine it being cast into a layer of treacle, so that it lands in the treacle in a random orientation and instantly stops spinning, just sinking to the bottom to slowly settle there on one of its 14 faces. Given this model, what you need to equalize are the solid angles subtended by the faces, viewed from the die's centre. --Lambiam 23:50, 18 January 2022 (UTC)
 * Note to my fellow Americans: treacle is British for molasses, except, I expect, not as good. I'd be surprised if the real answer is that simple, but it's a natural enough guess in the absence of further information, I suppose. --Trovatore (talk) 05:12, 19 January 2022 (UTC)
 * Another model is to imagine the solid laying flat on a side and asking the question, for each angle, how much momentum in that direction is required to roll the die onto another side. My intuition is that the distribution of required energy by angle is more variable for the triangular sides than the square sides. This suggests that the exact probability might have a small factor distinguishing between the kinds of face, so supporting what I take to be Trovatore's contention that this isn't a problem where we can have an assumption-light exact solution. &mdash; Charles Stewart (talk) 13:47, 19 January 2022 (UTC)
 * Here's a discussion from 8 years about a version of this question, and here's a discussion from 7 years before that. I'm sure there are more if you put more effort into searching than I did. --JBL (talk) 00:48, 19 January 2022 (UTC)

I'll take it to Science and see if anyone has actually tried it out. Thanks, all.--Verbarson talkedits 23:40, 22 January 2022 (UTC)