Activity for Derek Elkins‭

Type	On...	Excerpt	Status	Date
Answer	—	A: Is there a $\Delta$-complex structure on the sphere with less than three $0$-simplices? If it wasn't for the right to left implication of condition 3, you could do this with a single map out of the 0-simplex. If there are 2 or fewer points in the image of a map from $\Delta^0$, then that means at least two vertices of the image of every map out of the 2-simplex must be the same. Sinc... (more)	—	16 days ago
Answer	—	A: Prove that $\forall x\in\Bbb R:\lfloor x^2\rfloor-\lfloor rx\rfloor\ge-1\iff|r|\le2$. This is definitely more complicated than it needs to be. First, we can rewrite the inequality as $$\lfloor x^2 + 1\rfloor \geq \lfloor rx \rfloor$$ Proof 1 (This is the first proof I wrote, but the second one is nicer and takes a more categorical perspective.) $\lfloor x^2 + 1 \rfloor = 1... (more)	—	30 days ago
Answer	—	A: $\sup(A\cdot B) = (\sup A)(\sup B)$ where $A$ and $B$ bounded sets of positive real numbers To further emphasize the "fake difficulty", we can show that most of this proof is completely "formal", in that it holds for very general reasons that have little to do with real numbers. See the bottom for a detailed, elementary proof that explicitly illustrates that the proof is almost entirely a m... (more)	—	4 months ago
Answer	—	A: Picture proof for expansion of $x^n−y^n$ Here's a semi-visual, semi-algebraic approach. A useful alternative perspective on (univariate) polynomials is to think of them as being represented by their sequence of coefficients – filling in zeroes for any missing monomials. With this perspective, multiplying by $z$ simply shifts the se... (more)	—	9 months ago
Answer	—	A: Intuitively, why doesn't picking unpopular integers ($> 31$) lower your probability of winning lotteries? Let's say every lottery you buy exactly one ticket and always use the same numbers. Does that lower your odds of winning versus a strategy where you always pick a random set of numbers? No. Assuming the winning numbers are uniformly distributed, your odds of winning are always directly proportiona... (more)	—	9 months ago
Answer	—	A: Should posting on Meta affect reputation? Just to put out the other option, which is the one I prefer: No, votes on meta posts should not affect main site reputation. (more)	—	10 months ago
Answer	—	A: Average distance from circle's center to a point The problem is indeed that you aren't computing the average distance. What you're computing is a median distance. Essentially, you are equally weighting points that are just slightly outside $C2$ and points that are near the boundary of $C1$, but those contribute different amounts to the average dist... (more)	—	11 months ago
Answer	—	A: how to mathematically express a relationship in which a vector can be any 3D unit vector What makes the most sense to do somewhat depends on the context of this expression. What it seems you are really trying to do is defined $\mathbf e$ as a function of $\mathbf s$. What's making it awkward is that $\mathbf e$ isn't a function of $\mathbf s$ but merely a total (multi-valued) relation... (more)	—	11 months ago
Answer	—	A: Finding a single row of Matrix after exponentiation While there are no doubt even more clever things you can do, the most straightforward approach, which will likely make a significant improvement, is to move from matrix-matrix multiplication to matrix-vector multiplication. Let $ek$ be a (column) vector with all $0$s except the $k$-th entry is $1$... (more)	—	11 months ago
Answer	—	A: On Tarski-style universes in type theory I'm a bit confused about what you're confused about for the first part. In a Russell-style universe system, we'd have a hierarchy of universes with $\mathcal Ui : \mathcal U{i+1}$, i.e. the $i$th universe is a type in the $(i+1)$th universe. As you state, for a Tarski-style universe system, $\mathcal... (more)	—	about 1 year ago
Answer	—	A: The purpose of logical frameworks in specifying type theories András Kovács answer and comments to your question on Math.SE hit some of the main points. I'll give a bit more detail and context. First, the point of logical frameworks like LF is to be a formal meta-language, and, in particular, a formal specification language. The definitions in the HoTT book ... (more)	—	about 1 year ago
Answer	—	A: The meaning of $\pm$ I think I can explain what's going on by interpreting things more formally, though it could be gotten at informally as well; it's just clearer to see formally. One way to interpret "$y = \pm x$" (and other informal notations like $y = 1,2,3$) is as $y \in \{x, -x\}$ ($y \in \{1,2,3\}$). This is lo... (more)	—	about 1 year ago
Answer	—	A: Unification and generalization of limit and colimit Let $\mathcal I$ be the category with two objects and two parallel arrows between them: $1 \rightrightarrows 2$ and $\mathcal P$ be the cone over $\mathcal I$ with apex labelled $0$. Write $\mathsf f$ and $\mathsf g$ for the two arrows $1 \to 2$. Let $\iota : \mathcal I \hookrightarrow \mathcal P$ be... (more)	—	about 1 year ago
Answer	—	A: Generalization of categorical product The $F(\pii) = id$ doesn't really fit the form of a universal property. If we drop that constraint, we can present your universal property in terms of representability $$\mathcal C(Y, X1 \timesF X2) \cong \{(f1,f2)\in\mathcal C(Y, X1)\times\mathcal C(Y, X2)\mid F(f1) = F(f2) \}$$ natural in $Y$. No... (more)	—	about 1 year ago
Answer	—	A: Proving $|{\bf R}^{\bf R}|=|2^{\bf R}|$ using the Schroeder-Bernstein Theorem By definition $\mathbb R^\mathbb R \subset \mathbf 2^{\mathbb R\times\mathbb R}$. So all we need is a surjection $\mathbb R \twoheadrightarrow \mathbb R \times \mathbb R$ of which there are plenty such as space filling curves. If you have a bijection between $\mathbf 2^\mathbb N$ and $\mathbb R$, the... (more)	—	over 1 year ago
Answer	—	A: What is the Name of Function for Probability of a Certain Sum on Random Die Rolls? I don't know some special name for exactly that sum, but it is closely related to what are known as polynomial coefficients or extended binomial coefficients. These are often written as ${n \choose j}{k+1}$ and are defined indirectly via: $$\left(\sum{i=0}^k x^i\right)^n = \sum{j=0}^{nk} {n \choose j... (more)	—	over 1 year ago
Answer	—	A: Proving that $p\mid (p+1776)$ if $p$ is a prime and $p(p+1776)$ is a perfect square You basically have it. I'll give an overly detailed rendition of an informal proof below. As you state, the assumption is that $p(p+1776) = k^2$ for some natural number $k$ with $p$ a prime. We want to show that $p\mid p + 1776$, or, equivalently, $p + 1776 = pm$ for some natural $m$. Proof:... (more)	—	over 1 year ago
Answer	—	A: Prove $e^x \ge x+1 \\\; \forall x \in \mathbb{R}$ using induction You can't do induction on the real numbers because the real numbers aren't inductively defined. That said, it's not clear what you intend "induction on reals" to mean. Before continuing, I'd like to make an aside on proof writing. First, as a matter of presentation, it's useful to make it clear wh... (more)	—	over 1 year ago
Answer	—	A: Show that $f(x) = \arctan\left(\frac{x}{x+1}\right) + \arctan\left(\frac{x+1}{x}\right) = \frac{\pi}{2}$ $\infty$ is not a number so $\arctan(\infty)$ isn't meaningful, nor does it have a reasonably standard interpretation. You can already tell something is wrong, because there's nothing in your argument that keeps it from applying for all values of $x$. One way to attempt to make an argument along the ... (more)	—	almost 2 years ago
Answer	—	A: Are vectrices useful for calculations as opposed to formalism? This is largely just a notational difference insofar as it's used in the example you give. While it often is convenient and enlightening to view a matrix as a "vector" of vectors momentarily, taking this too seriously can be problematic. A vector space expects a field) of scalars, and vectors don't f... (more)	—	almost 2 years ago
Answer	—	A: Complex numbers in 2D, quaternions in 4D, why nothing in 3D? There is a famous result) about finite-dimensional real associative division algebras. You may find that interesting, but I don't think it's what you want. It is a direct answer to the question of why we "can't" multiply "triplets of numbers" if you require that multiplication to satisfy certain prop... (more)	—	about 2 years ago
Answer	—	A: Maximize Independent Variable of Matrix Multiplication Since all the $xi$ are restricted to be non-negative, the magnitude of the vector increases if and only if the sum of the $xi$ increases. Therefore, maximizing $\sumi xi$ is equivalent to maximizing $|x|$. This sum is a linear function. Your direction constraint can be expressed as: $$\vec y \wedg... (more)	—	about 2 years ago
Answer	—	A: Are we in a "history-valley" for Topology? I'm going to focus on the "history-valley" aspect, and to jump to the upshot: No, I don't think we're in a "history-valley" with respect to topology. I more or less agree with Guilherme Gondin‭ that the significant thing in the ??? period was the foundational "crisis" (spurred on by the kind of is... (more)	—	over 2 years ago
Answer	—	A: Acceptable, usual to write $\ge 2$ pipes simultaneously? As mentioned in the answers you referenced in earlier versions of your question, $(-\mid-)$ is not standalone notation in usual probability theory notation.^[I have seen it used, e.g. in Jaynes' "Probability Theory: The Logic of Science", as standalone notation, but not in a way such that $P(A\mid B)... (more)	—	over 2 years ago
Answer	—	A: $\int dx dy dz d p_x dp_y dp_z$ Does it have any physical meaning? It's hard to answer your question specifically without the context, and obviously the physical significance of some expression depends on what the variables and operations in that expression stand for. Before considering this particular integral, I want to talk about integration generally and its not... (more)	—	over 2 years ago
Answer	—	A: Is Pythagorean theorem really valid in higher dimensional space? Consider a vector, $\mathbf v=(a,b,c)$, in $3$-space. We can project this onto the $xy$-plane, say, producing the vector $(a, b, 0)$ which we can identify with the $2$-vector, $(a, b)$. This two vector corresponds to the hypotenuse of a right triangle whose side lengths are $a$ and $b$. Therefore the... (more)	—	over 2 years ago
Answer	—	A: What're the orders for equation expressing? This question contains a lot of confusion. First, an equation is something like $f(x) = g(x)$ (i.e. there's an equality sign). Solving an equation means finding (all) values for the free variables such that both sides become equal. In the above example, this would mean finding values for $x$ such tha... (more)	—	over 2 years ago
Answer	—	A: Getting backward of partial differentiation's chain rule Traditional mathematical notation for calculus (both integral and differential) is rather incoherent. I don't think there exists a write-up providing systematic rules that would allow you to correctly and unambiguously parse this kind of notation, i.e. the kind of notation used in a typical undergrad... (more)	—	over 2 years ago
Answer	—	A: methodology of integration by parts ($e^{ax}\cos (bx+c)\mathrm dx$) I haven't seen integration by parts written that way before, and the derivation you describe seems overly complicated, albeit not for the $I$ stuff. In particular, I don't understand the purpose of the trigonometric substitutions as the integral should be solvable already without them. Maybe there wa... (more)	—	almost 3 years ago
Answer	—	A: How to derive some trigonometric formulas? Memorizing or using actual trigonometry to re-derive trigonometric formulas is a waste of time and mental resources. (This isn't to say it isn't useful to go through the trigonometric approach once and to understand how they relate.) Instead, there is just one relatively simple formula that one needs... (more)	—	almost 3 years ago
Answer	—	A: Are challenge-like questions like these considered on-scope in Mathematics CD? If your question was closed for being off-topic, then you have your answer. The way your question is worded, it sounds like the "question" is more of a prompt and the "answers" to that "question" are really just demonstrations fitting the prompt. If that's accurate, then that indeed sounds a lot l... (more)	—	almost 3 years ago
Answer	—	A: What was Justice Scalia's mathematical mistake in Penry v. Lynaugh (1989)? Consider the universal quantifier. I'll write it as: $\mathsf{forall}\ x.P(x)$ to be read as "for all $x$ in the domain $P(x)$ holds". We then have the logical identity $$(\mathsf{forall}\ x.P(x)) \land (\mathsf{forall}\ x.Q(x)) \to (\mathsf{forall}\ x. P(x) \land Q(x))$$ where $\land$ is logical con... (more)	—	almost 3 years ago
Answer	—	A: How to show if a set is simply connected? Probably one of the simpler ways of establishing this is reducing it to a case where you already know the answer. I will assume that you have proven at some point that the circle, $\mathbb S^1$, is not simply connected. Simple connectedness is a topological property so it is preserved by homeomorphis... (more)	—	almost 3 years ago
Answer	—	A: Can this site contains Physics's Math question? To start, what belongs on the Physics Codidact is up to the people active on the Physics Codidact. The question seems to have been well received there, and I suspect that it would also have been well-received here. Math questions of any sort are welcome here. However, when asking a question motiva... (more)	—	about 3 years ago
Answer	—	A: Does every divergence-free vector field arise as the curl of some vector field? tl;dr We can formulate your question more nicely with geometric algebra. As r mentioned in a comment, there is a counter-example to your question on a simply-connected but not 2-connected domain. The counter-examples provided are Green's functions for the vector derivative. Formulating the problem in... (more)	—	about 3 years ago
Answer	—	A: Why does this definition of generalized forces work? This expression is not the clearest way of writing this, but the idea is that we are defining the components of the covector $Q(q)$ on a basis of differential 1-forms $dqj$, i.e. $Q = \sum{j=1}^k Qj dqj$. This is made more clear by Wikipedia's expression of this statement: $Qj = \sum{i=1}^n \langle \... (more)	—	over 3 years ago
Question	—	Marketing Math Codidact I don't think it's controversial to say that this site is currently suffering from a lack of new questions. I believe the main problem is lack of awareness. People aren't coming to this site because they have no idea it exists. A distant second concern, in my opinion, is the value proposition rela... (more)	—	over 3 years ago
Answer	—	A: Why always rationalize a denominator? This answer combines the thoughts in both of the other (current) answers (tommi's and r's) but presents it in a more formal context. A significant portion of high school (and earlier... and a decent amount of later...) math is highly algorithmic, and, in particular, corresponds to the important id... (more)	—	over 3 years ago
Answer	—	A: Why does “unless” mean “if not”? The quotes from the books are exactly the kind of ridiculously naive and over-simplified linguistics in introductions to logic that I criticize in this blog article. Particularly for everyday natural language expressions, you will find no shortage of ways where this "translation" provides only a clum... (more)	—	over 3 years ago
Answer	—	A: Existence of a set of all sets Sure. You can simply add $\exists V.\forall x.x \in V$ as an axiom to ZF(C), and you will have such an axiomatic system. Such an addition to ZF(C) would make it inconsistent, but it would still prove the existence of a "set of all sets" (along with everything else). As Peter Taylor points out in a... (more)	—	over 3 years ago
Question	—	Can we constructively find a third element of a set $X$ satisfying $X \cong X \times X$ given two distinct elements? Consider a set $X$ such that $X \cong X\times X$. It's quite easy to prove in a classical set theory, e.g. ZF, that $X$ must be the empty set or a singleton set or an infinite set. In other words, if we additionally assume that there exists $a, b \in X$ and $a \neq b$, then we know that $X$ is an inf... (more)	—	over 3 years ago
Answer	—	A: Computational hardness of the uniform halting problem According to Wikipedia, the set of (indices of) Turing machines that compute total functions, i.e. which halt on all inputs, is a $\Pi2$ set. If we use the variation of the definition of arithmetical hierarchy which includes primitive recursive functions, then it is fairly straightforward^[If you are... (more)	—	over 3 years ago