Activity for celtschk‭

Type	On...	Excerpt	Status	Date
Question	—	Unification and generalization of limit and colimit The recent post by r reminded me of my own idea to unify and generalise limit and colimit. I also only occasionally dabble in category theory, and thus I wonder if the following is a sound construction, and if so, if it is a known concept. The idea is as follows: We have a category $\mat... (more)	—	about 1 year ago
Answer	—	A: Criterion in terms of the bases for determining whether one topology is finer than another A counterexample would be the set $S=\{0,1,2\}$ with the topologies $\mathcal T = \{\emptyset, \{0\}, \{0,1\}, S\}$and $\mathcal T' = \{\emptyset,\{0\},S\}$. Clearly $\mathcal T'$ is not finer than $\mathcal T$. Now $\mathcal T$ is generated by the basis $\mathscr B = \{\{0\},\{0,1\},S\}$ and $\ma... (more)	—	about 1 year ago
Answer	—	A: Proving $|{\bf R}^{\bf R}|=|2^{\bf R}|$ using the Schroeder-Bernstein Theorem To apply the Schröder-Bernstein theorem, we need injections in two directions. Given that the Schröder-Bernstein theorem doesn't require the axiom of choice (AC), there's a value in avoiding anything that requires AC (such as the theorem that existence of a surjection from $A$ to $B$ implies the exis... (more)	—	over 1 year ago
Question	—	LaTeX formulas not rendered correctly in summary I've noticed that the question preview of several questions on the Q&A page contains “misplaced &” with yellow background, which turns out to be a MathJax error. However opening one of them shows that the post itself is just fine. However the formula that doesn't correctly show up in the preview c... (more)	—	over 1 year ago
Question	—	How to get rid of the tag ordinals? On my latest question, which was about ordinal numbers (often just called ordinals), I originally used the new tag ordinals. Later I noticed that ordinal-numbers would be a much better name for that tag. Since my question was still the only one using the tag, I simply retagged it, assuming that the n... (more)	—	over 1 year ago
Answer	—	A: Is this formula for the minimal sum correct? I think I've solved the problem: The formula is correct. For the proof I'm using the decomposition $\alpha = \lambda + m$ and $\beta = \mu + n$ from the question. Also, I'll use the notation $A\cong B$ for “$A$ is order-equivalent to $B$”. Also note that $+$ and $\cdot$ denote the standard ordinal... (more)	—	over 1 year ago
Question	—	Is this formula for the minimal sum correct? As is well known, the addition of natural numbers can be extended to the ordinal numbers in different ways. The first way is the ordinal sum, and the second is the natural or Hessenberg sum. Now I've been thinking about other possible sums of ordinals. For that purpose I've used the following gene... (more)	—	over 1 year ago
Question	—	Does there exist a non-zero game such that the sum of three or more copies of it is zero? In combinatorial game theory, there are non-zero games $G$ with the property $G+G=0$; this is in particular true for all impartial games. Now I wonder if there also exist non-zero games such that $G+G+G=0$. I don't see an obvious reason why those shouldn't exist, but I also have no idea on how to ... (more)	—	over 1 year ago
Answer	—	A: Is there a way to encode a unique arrangement of vertices of a graph with a unique short word? I'm going to assume your points are lying on an integer grid. I'm also assuming you always have a finite number of points. Then one way to make words for your point clouds is to enumerate the grid points, for example by starting at the origin and going in a spiral. Then you can assign an unique fi... (more)	—	over 1 year ago
Answer	—	A: Why isn't MathJax table appearing? This is another instance of the general MathJax/Markdown problem noted here. If you replace all double backslashes (`\\`) with quadruple backslashes (`\\\\`), it works: $\begin{array} {|r|r|}\hline \text{If I already earned a} & \text{then I must roll another} \\\\ \hline 9 & 1 \\\\ \hline 8 & 2 ... (more)	—	over 2 years ago
Answer	—	A: Why doesn't \$ work? You need to use `\\$`, like this: \\$1. The reason is that MathJax is interpreted in the browser, while Markdown is interpreted in the server. If you write `\$`, then Markdown sees the backslash and therefore puts the dollar sign literally into the output (which it would have done anyway), with... (more)	—	over 2 years ago
Answer	—	A: How's it possible to arrange 0 objects? How can 0! = 1? How is it possible to arrange $0$ objects? Well, by doing nothing. Doing nothing is something you can do, and you can do it in exactly one way. Another way to see it is to consider that it is also the number of ways you can rearrange the objects in a row. One way to rearrange the objects is, again... (more)	—	over 2 years ago
Answer	—	A: Is Mathematical Induction truly "induction", or misnamed? In the context of Peano arithmetics, mathematical induction is actually an axiom, that is you cannot prove it from other properties of the natural numbers. It is extremely plausible that if something holds for $0$, and if it holds for $n$ it also holds for $n+1$, that it holds for all natural numbers... (more)	—	over 2 years ago
Answer	—	A: Why must percent change divide the difference by the old, NOT new, value? When you say “the value changed by 5%” what you actually mean is “the value changed by 5% of the old value” (this is by convention because that is in the vast majority of cases the quantity you are interested in). And that is why you have to divide the difference by the old value. If you want to b... (more)	—	over 2 years ago
Question	—	Does this construction always give a topological vector space? Consider an algebraic vector space $V$ over $\mathbb R$ or $\mathbb C$. Now for each possible basis $Bk = \\{ei^{(k)}|i\in I\\}$ of $V$, one can define an inner product $\langle\cdot\vert\cdot\ranglek$ by $\langle ei^{(k)}\vert ej^{(k)}\ranglek = \delta{ij}$ (for vector spaces of infinite dimensio... (more)	—	over 2 years ago
Answer	—	A: Getting backward of partial differentiation's chain rule Your main error is to treat the derivatives as fractions. From the definition, they are not fractions, they are limits of fractions. Now for total derivatives treating them like fractions generally gives correct results (at least I never have seen any case where it fails), therefore one might c... (more)	—	over 2 years ago
Question	—	MathJax doesn't render after editing comment I've just noticed that when editing a comment with MathJax, the MathJax content is not rendered after finishing the edit. Note that after a page reload, it again renders correctly. (more)	—	over 2 years ago
Answer	—	A: What is "continuous" in Math? The terms “discrete” and “continuous” are not related to cardinality, but to topology. In particular, you can have a discrete topology on a space of arbitrary cardinality. Note also that “discrete” is a property of a set in a topological space, while “continuous” is a property of functions between... (more)	—	over 2 years ago
Answer	—	A: Is $x=\int \int \ddot{x}\mathrm dx \mathrm dx$? While you can integrate twice for the same variable, your equation is not right. $\ddot x$ means deriving $x$ twice with respect to time, that is, $$\ddot x = \frac{\mathrm d^2x}{\mathrm dt^2}$$ which is very different from deriving twice with respect to $x$ (indeed, $\mathrm d^2x/dx^2=0$). To coun... (more)	—	over 2 years ago
Answer	—	A: Prove that $\int \ddot{x}(t)\mathrm dt=v_0 + \frac{F_0}{m}t$ Your calculation is not correct. What you calculated is $$\\int a\\,\mathrm d\textcolor{red}{a}$$ which is something very different from $$\\int a(t)\\,\mathrm d\textcolor{red}{t}$$ Indeed, in the specific case here, $a = F0/m$ is time independent, that is, it is a constant in respect to time. No... (more)	—	over 2 years ago
Answer	—	A: After $n - 2$ unchosen doors are opened, how does the probability of the $n - 2$ unchosen doors "shift" or "transfer" to the lone unchosen door? Maybe it's a good idea to start from a situation that's intuitive, and then gradually chance it to the Monty Hall version. So let's start with the following situation: Monty lets you choose a door. Then, after you've chosen it, he offers you to either select the chosen door, or open all the oth... (more)	—	over 2 years ago
Question	—	Editing my answer fails I just tried to edit this answer of mine, but for some reason clicking the edit button only reloads the page. Trying to open in a different tab didn't help either. In case it matters, I'm using Waterfox Classic 2021.02 (64-Bit) KDE Plasma Edition. (more)	—	over 2 years ago
Answer	—	A: How to determine area of square using Calculus in Cartesian coodinate? The correct equation for the square boundary you sketched is $$\left|x\right| + \left|y\right| = 5$$ where $\left|x\right|$ means the absolute value of $x$, that is, $$\left|x\right| = \begin{cases} x & \text{for $x\ge 0$}\\\\ -x & \text{for $x < 0$} \end{cases}$$ Now to calculate the area wi... (more)	—	over 2 years ago
Answer	—	A: differentiate under integral sign says something went wrong Assuming you copied the formula from the book correctly, your mistake is simply to assume that your book is right. As a physicist, my first instinct was to make a dimensional analysis. The formula for $I$ gives a dimensionally valid expression as long as $a$ and $b$ have the inverse unit of $x$ (s... (more)	—	almost 3 years ago
Answer	—	A: What's wrong with evaluating $n(n-1) \dots (n-[k-3])(n-[k-2])\color{red}{(n-[k-1])}$ at $k = 1$? The informal expression stands for the formal expression $$\prod{j=1}^k (n-j+1)$$ where, since it is a formal expression, you indeed can insert $k=1$, to get $$\prod{j=1}^1 (n-j+1) = (n-1+1) = n$$ as the text states. Now the author of the text obviously wanted to avoid the product notation, ei... (more)	—	almost 3 years ago
Answer	—	A: Why rational to be indifferent between two urns, when urn A has 50-50 red and white balls, but you don't know urn B's ratio? Let's assume you know that urn B has 5 balls in it. I deliberately take an odd number, because that way we know for sure that there are not exactly the same number of red and white balls in that urn. Note that since you don't know the content of the urn, you have to assign probabilities to the urn... (more)	—	almost 3 years ago
Answer	—	A: Intuitively, why's X% of Y = Y% of X? Just consider that Z is Z percent of 100. Then you get: > X percent of Y is X percent of Y percent of 100, which is Y percent of X percent of 100, which is Y percent of X. I think that X percent of Y percent of something is the same as Y percent of X percent of something is intuitive. And that ... (more)	—	about 3 years ago
Answer	—	A: In "if and only if" proofs, why's 1 direction easier to prove than the other? One thing to consider is that the equivalences are generally based on the validity of certain axioms. And then, one direction may only depend on a smaller set of axioms. Now it is possible that the additional axioms help also in the direction that doesn't need it, but it is also possible that the ea... (more)	—	about 3 years ago
Answer	—	A: Why aren't $z_1=f(xy)$ and $z_2=f(x/y)$ functions of 2 variables? You are misrepresenting what Hagen von Eitzen wrote. He did not write that $z1$ and $z2$ depend on only one variable. He wrote that $f$ is a function of only one variable. That's a massive difference. The question this refers to was whether $z1=f'(xy)$ or $z2=f'(x/y)$ are ambiguous, and the correc... (more)	—	about 3 years ago
Answer	—	A: Why should a non-commutative operation even be called "multiplication"? In most (but not all) cases where you name an operation multiplication is when you have two binary operations, and one of them distributes over the other, but not the other way round. Then the operation that distributes over the other is called multiplication and the other one addition, in analogy to... (more)	—	about 3 years ago
Question	—	Is this topology basis dependent? Consider a topological field $K$ and an algebraic(!) vector space $V$ over $K$, that is, $V$ has not (yet) a topology defined on it. I'm particularly interested in the case where $V$ has infinite dimension. Now be $V^\ast$ the algebraic dual of $V$. Define a topology on $V^\ast$ through pointwise ... (more)	—	about 3 years ago
Question	—	Does this generalization of path-connectedness also cover general connectedness? I've got the following idea to generalise path-connectedness: Given a topological space $P$ and a subspace $S$, a space $X$ is $(P,S)$-connected if every continuous function $f:S\to X$ can be extended to a continuous function $g:P\to X$. The usual path-connectedness is obtained using $P=[0,1]$ ... (more)	—	over 3 years ago
Question	—	Are there useful topologies on Cartesian products “in between” the product topology and the box topology? On the Cartesian product of topological spaces, there are two standard topologies: One is the product topology, the other is the box topology. As is well known, the box topology is generated by the product of open sets, and the product topology is generated by such products with the restriction th... (more)	—	over 3 years ago
Answer	—	A: Are 3 10% chances better than one 30% chance (when penalized by a variable for failures)? It somewhat depends on your game situation. If you have just that one weapon, need to upgrade it urgently, and are likely to have enough coins the next time you need to upgrade a weapon, then clearly the 30% option is better: It gives you a 30% probability of upgrade, while three 10% attempts give... (more)	—	over 3 years ago
Question	—	Inconsistency of answer count between main page and question page The question “Cutting the square” is listed on the main page as having one answer, however when going to that question, it says 0 answers (and indeed, no answer is shown). Something doesn't add up here. [1]: https://math.codidact.com/posts/278332 (more)	—	over 3 years ago
Question	—	$sitename not substituted in FAQ In the FAQ there appears the term "$sitename" several times. I'm pretty sure that's the name of a variable that should have been substituted here. (more)	—	over 3 years ago
Answer	—	A: Why does “unless” mean “if not”? According to the definition you quote, “unless” gives an exception to a preceding negative statement. An exception to a statement is a condition in which the statement does not apply. Therefore the statement does apply only if the condition is not fulfilled (because otherwise the exception applies, a... (more)	—	over 3 years ago
Answer	—	A: Why does the decimal expansion of $1/(10n - 1)$ have this neat pattern? I think the simplest way to see how it works is as follows: Given $x=\frac{1}{10n-1}$, it is easy to check that $\frac{x+1}{10}=nx$. Now $\frac{x+1}{10}$ means shifting the digits to the right, with an $1$ added as the first decimal. For example, in the case of $x=1/19$, we have $$\frac{x+1}{1... (more)	—	over 3 years ago
Question	—	Classification of topological points I've recently dreamed up the following idea to classify topological points, and I'm wondering if this is or relates to a known concept. Here “topological point” refers to a point together with its neighbourhood structure (the definitions below should make clear what I mean). Given a topological sp... (more)	—	over 3 years ago