Activity for r~~‭

Type	On...	Excerpt	Status	Date
Answer	—	A: Minimal non-standard number in non-standard models of PA Quite the opposite; in no non-standard model of Peano arithmetic is there a minimal non-standard number. Consider the formula \\(\phi(x) = \left(x = 0\right) \vee \exists y \left(x = S(y)\right)\\). The first-order induction axiom for \(\phi\) is \\[ \phi(0) \wedge \forall x \bigl(\phi(x) \Rig... (more)	—	2 months ago
Answer	—	A: Cyclical or “loop” fractals? Does the Cantor set qualify? This animation loops infinitely, and shows that you get back to where you started every time you zoom in by a factor of 3. Show animated GIF An animation of the Cantor set (more)	—	2 months ago
Answer	—	A: Strange behavior in elections and pie charts > with randomized positions in the chart Implicit in this description of your model is the notion that every location for a separator line is equally likely. This doesn't describe the actual distribution of vote totals; with random voters, lines are much less likely to appear near other lines than... (more)	—	7 months ago
Answer	—	A: What is the formula for sample standard deviation of a small sample size? The sample standard deviation (with Bessel's correction) is defined to be the first formula in your post. It doesn't ‘become’ anything else. You were possibly remembering using the sample standard deviation in an estimator for the population mean. The \(t\)-value is multiplied by the sample standa... (more)	—	7 months ago
Answer	—	A: 2 construals "of 100 patients presenting with a lump like the claimant’s in Gregg v Scott, 42 will be ‘cured’ if they are treated immediately." The author of this passage is proposing a very simple hidden variable model. This model has two variables: $A$ — whether the patient has certain unknown genes (this variable is hidden) $B$ — whether the treatment succeeds (this variable is observed) And the model proposes that $A$ influences $... (more)	—	11 months ago
Answer	—	A: The meaning of $\pm$ I think I would describe that as ‘the two claims $|x| = \pm x$’, not a single claim. It's comparable to saying ‘the two roots of $x^2 - 4$ are $x = \pm2$’ and not ‘the root of $x^2 - 4$ is $x = \pm2$’. Whether those two claims are meant to be and-ed or or-ed would depend on context. In a vacuum, I... (more)	—	about 1 year ago
Question	—	Generalization of categorical product I'm only a dabbler in category theory; this might be a basic concept just outside of my sphere of exposure. I'm looking for references to the following universal construction, a generalization of the categorical product. Given a category $\mathcal{C}$ and a functor $F$ from $\mathcal{C}$ to som... (more)	—	about 1 year ago
Answer	—	A: Example of $f:[0,1]\to\mathbf{R}$ with $\lim_{a\to 0^+}\int_a^1f(x)dx=L $ for some real number $L$ but $\int_0^1|f(x)|dx=\infty $ Since you haven't specified that such a function needs to be continuous or well-behaved in any way, it's quite easy to describe one. The integral $\int0^1 \frac1x\,dx$ diverges, so there is an infinite amount of area to work with. Measure off the section of curve with area 1 starting at $x = 1$ an... (more)	—	over 1 year ago
Answer	—	A: Finding the limit $ \lim_{x\to 0^+}e^{1/x}\sum_{n=\lfloor 1/x\rfloor}^\infty\frac{x^n}{n} $ First, let's make things simpler and do a change of variable, replacing $x$ with $\frac{1}{x}$: $$ \lim{x\to \infty}e^x\sum{n=\lfloor x\rfloor}^\infty\frac{1}{nx^n} $$ Massage into an explicit L'Hôpital indeterminate form: $$ \lim{x\to \infty}\frac{\sum{n=\lfloor x \rfloor}^\infty\frac{1}... (more)	—	over 1 year ago
Answer	—	A: What is the probability that the convex hull of $n$ randomly distributed points has $l$ of the points on its boundary? Your conjecture is incorrect. Consider the case of four points. Place three of these points randomly; these points form a triangle, and each pair of points defines a line. Consider the three half-planes bounded by each of these lines and containing the centroid of the triangle. If and only if a fo... (more)	—	over 1 year ago
Answer	—	A: Do linear and group invariant functions allowed to go inside(?) integral operators? If $f$ is a linear map (between finite-dimensional spaces), then it can be represented by multiplying by some matrix. The fact that $f$ is $\mathfrak G$-invariant presumably means that this matrix is constant with respect to the $\mathfrak g$ used as the variable of integration. It's easy to se... (more)	—	over 1 year ago
Answer	—	A: Why can't we conclude the extrema property of a function from its quadratic approximation when the discriminant is zero? You've basically walked right up to the explanation. The second-order approximation you describe, when $D = 0$, is $$g(x, y) = a\left(x + \frac{b}{2a}y\right)^2$$ Define $z = x + \frac{b}{2a}y$, and we have $g(x, y) = g(z) = az^2$. This means that $g$ describes a function that really only ha... (more)	—	over 1 year ago
Answer	—	A: organizing a library $2^{n-1}-1$ is a lower bound on the maximum, at least. For $n$ books, if you start with the ordering $n, 1, 2, \ldots, n - 1$ (I'm following your convention of 1-indexing the books), then the books could be reordered via the sequence $Sn$ defined as follows: $$ \begin{align} S1 &= () \\\\ S{n + 1... (more)	—	over 1 year ago
Answer	—	A: If $\mathbf{R}$ is thought of as a vector space over $\mathbf{Q}$, what is its dimension? The dimension of $\mathbb{R}$ as a vector space over $\mathbb{Q}$ is equal to the cardinality of $\mathbb{R}$. In general, the dimension and cardinality of any vector space $\mathbf{V}$ and the cardinality of its scalar field $\mathbf{K}$ will obey the following equation: $$ \|\mathbf{V}\| = \... (more)	—	over 1 year ago
Answer	—	A: Show that $f(x) = \arctan\left(\frac{x}{x+1}\right) + \arctan\left(\frac{x+1}{x}\right) = \frac{\pi}{2}$ You're right that division by zero is a problem if you want to use that identity. Rather than salvage that, I would recommend this identity instead: $$ \arctan a + \arctan{a^{-1}} = \frac\pi2\qquad\text{for positive $a$} $$ Proving this identity is easy: draw a $1 \times a$ rectangle with a d... (more)	—	over 1 year ago
Answer	—	A: Complex numbers in 2D, quaternions in 4D, why nothing in 3D? Intuition is a personal thing, but here are some thoughts that might be useful. (Rigorous justifications for most claims are absent, to keep this post from getting too long. I intend to restrict myself merely to pointing suggestively and waggling my eyebrows.) Don't think of the quaternion trick a... (more)	—	about 2 years ago
Answer	—	A: How can a 15 year old construe the LHS of Generalized Vandermonde's Identity, when it lacks summation limits and a summation index? Unfortunately, this is ambiguous notation. It isn't your fault; this presentation of this identity is more suited for an audience already familiar with the material than someone learning it for the first time. The answer to your question is that the summation is over all ordered length-$p$ sequenc... (more)	—	about 2 years ago
Answer	—	A: What is the parallel for edges? I would still use the term ‘edge’ for that case—as in, ‘These three arcs are the edges of a circular triangle.’ If necessary to prevent confusion, I might specify ‘curved edge’ or something similar. But while ‘edge’ in geometry often implies a strict line segment, in graph theory it simply denotes a ... (more)	—	over 2 years ago
Answer	—	A: Third kind of infinite By Cantor's diagonal argument, there can be no bijection between the power set of any set and the set itself. This is true for finite and infinite sets. So taking the power set ($\mathcal P$) of a set of some infinite cardinality always gives you a set with a strictly larger infinite cardinality. ... (more)	—	over 2 years ago
Answer	—	A: What does upper indices represent? A textbook homework problem might ask, ‘Speedy the snail creeps along at a steady pace of 60 cm per minute. How far does Speedy travel each second?’ The correct answer is, of course, one centimeter. Hopefully your teacher would also accept 10 mm or 0.01 m as equally valid answers; after all, they all... (more)	—	over 2 years ago
Answer	—	A: Can the bijection for the Lost Boarding Pass Probability Problem, be formulated or pictured? The bijection is just to swap the people sitting in the first and last seats. I feel like a ‘formula’ is more machinery than this simple concept is worth, but here, if this helps: let $P$ be the set of permutations on $\\{0\ldots n - 1\\}$, where for $p \in P$, $p(s)$ is the number of the passenge... (more)	—	over 2 years ago
Answer	—	A: transpose matrix and general matrix is completely messed up I suspect what's tripping you up here is the difference between applying a linear transformation to a vector versus applying a change of basis. Both involve matrix multiplication, but the nature of the transformation is different. First, some words about vectors. There are two ways to introduce st... (more)	—	over 2 years ago
Answer	—	A: Is Pythagorean theorem really valid in higher dimensional space? One generalization of the Pythagorean theorem to three dimensions is de Gua's theorem: > if a tetrahedron has a right-angle corner (like the corner of a cube), then the square of the area of the face opposite the right-angle corner is the sum of the squares of the areas of the other three faces ... (more)	—	over 2 years ago
Answer	—	A: What to do when there's lot of function of time? (For integration) Should I consider them as constant? No, if you're integrating with respect to time, you can't treat functions of time as constants. Remember that, by the fundamental theorem of calculus, the result of your integration must be a function that, when differentiated, yields your original integrand. So if you incorrectly concluded that \... (more)	—	over 2 years ago
Answer	—	A: What're the orders for equation expressing? Another aspect of operation precedence which Derek's otherwise excellent answer doesn't cover is operator associativity. Implicit in the PEMDAS convention is the idea that the arithmetic operators are left-associative, meaning that they should be grouped from left to right when a subexpression con... (more)	—	over 2 years ago
Answer	—	A: Is ‘How would you know to do the next step?’ always a bad question? I believe that yes, such questions are intrinsically bad. There are two ways to attempt to answer such a question. First, the answerer could try to get into the head of the author of the particular proof being examined, and figure out how they made the necessary leap. If the proof is well written,... (more)	—	over 2 years ago
Question	—	Is ‘How would you know to do the next step?’ always a bad question? We have a user who keeps posting questions of the form, ‘How would you [tortured synonym for “know”] to [do the next step in a proof]?’ Leaving aside the various other reasons that these posts are bad^[I don't want those other issues to be a distraction from the central question here. I'm only mentio... (more)	—	over 2 years ago
Answer	—	A: Getting backward of partial differentiation's chain rule The difference between a partial derivative and a total derivative is that the partial derivative measures the change in a function when only one of its arguments varies, while the total derivative measures the change in a function when all of its arguments vary. The $\frac{\partial z}{\partial x}$ n... (more)	—	over 2 years ago
Answer	—	A: Find limits of integration in polar coordinates Regardless of the coordinate system, the principle of finding the limits of integration is the same: find the minimum and maximum value of the independent coordinate that enclose the region of interest. In polar coordinates, for these two particular curves, the region of interest is best described... (more)	—	over 2 years ago
Answer	—	A: How can "information about the birth season" bring "at least one is a girl" closer to "a specific one is a girl"? ‘More specific information’, here, is informally referring to the value of the (positive) likelihood ratio: the ratio between how likely it is to get that information if a proposition is true, and how likely it is to get that information if the proposition is false. (I say ‘informally’ because, as yo... (more)	—	over 2 years ago
Answer	—	A: How does $\lim\limits_{n \to \infty} \dfrac{p[s + (1 - s)c^n]}{p[ s + (1 - s)c^n] + (1 - p)[s + (1 - s)w^n]} = p?$ > So if $n \to \infty$, then all terms above containing $c^n \to 0$. This is false. It's true that as $n \to \infty$, $c^n \to 0$ (for $0 < c < 1$). But a term like $\frac{s}{c^n}$ will diverge instead of converging to $0$. You haven't done yourself any favors by dividing everything by $c^n$... (more)	—	over 2 years ago
Answer	—	A: How to derive some trigonometric formulas? Can't use algebra, you say? Nonsense! All you need is Euler's formula $$ e^{i\theta} = \cos\theta + i\sin\theta $$ and these identities just fall out of the algebra. For example: \begin{align} \sin (\alpha + \beta) &= \frac1{2i} \left(e^{i(\alpha + \beta)} - e^{-i(\alpha + \beta)}\right)\\\\ ... (more)	—	almost 3 years ago
Answer	—	A: $\sum_{k=0}^{n} \binom{n}{k}=2^{n} \overset{?}{\iff} \sum_{k=0}^{n} \binom{2n+1}{k}=2^{2n}$ The green result can be used to prove the red one, yes. Start with the fact that $\binom{m}{k} = \binom{m}{m - k}$. If this isn't obvious to you, you can derive it either by looking at the usual formula for the binomial, or by considering that choosing the members of a subset is equivalent to choo... (more)	—	almost 3 years ago
Answer	—	A: How does the change of variable $\color{red}{r↦n−r}$ transmogrify $\sum\limits_{r=0}^n2^{n-r}\binom{n+r}{n}=2^{2n}$ into $\sum\limits_{k=0}^{n} \frac{\binom{2n-k}{n}}{2^{2n-k}}=1$? $n$ is free in the expression $\sum{r=0}^n 2^{n-r}\binom{n+r}{n}$. The only variable not free in that expression is $r$. What $\sum{r=0}^n$ means is to treat what comes next as a function of $r$, and compute the sum of that function evaluated at $0$, $1$, and so on up to $n$. If $n$ is used inside th... (more)	—	almost 3 years ago
Answer	—	A: A formal-logic formula for decimal to binary conversion If you have a non-negative integer $n$, and the base $b$ representation of $n$ is a sequence of digits, numbered from the right as $dk\dots d2d1d0$, then $di = \left\lfloor \frac{n}{b^i} \right\rfloor \bmod b$. For example, decimal 10 is binary 1010 because: $$ d3 = \left\lfloor \frac{10}{2^3}... (more)	—	almost 3 years ago
Answer	—	A: Questions with a quote/screenshot and a request to explain My approach is downvote and, if it's a first infraction, comment explaining the issue with the question. Repeat offenders get downvotes and, perhaps if this one-guy's-opinion calcifies into actual community policy, flags. I'd like us to start doing a better job of holding the line against low-effo... (more)	—	almost 3 years ago
Answer	—	A: Why $\color{red}{k\dbinom{k}{1}} \neq$ "first choose the k team members and then choose one of time to be captain"? $\(^nk\)$ is the number of ways to choose $k$ team members from $n$ players. $k$ is the number of ways to choose one captain from $k$ team members. (It isn't the number of ways to choose a captain from the original group of people, as you suggested; that would be $n$.) $k\(^nk\)$ is the product... (more)	—	almost 3 years ago
Answer	—	A: If k = 1, why $n(n-1) \dots \color{red}{(n-k+1)} = n$? You started on the right track with $n(n - 1)\ldots(n - [k - 3])(n - [k - 2])(n - [k - 1])$. What you should have noticed is that there are $k$ terms in that product. The first term subtracts 0 from $n$, the second term subtracts 1, and so on, with the $i$th term subtracting $i - 1$, all the way u... (more)	—	almost 3 years ago
Answer	—	A: Why mustn't the proportion of smokers among married people be the same as the proportion of smokers in the whole population? The key word is exactly. If I flip a fair coin, I expect about half of my flips to be heads. If I flip it twice, exactly one head is quite likely. If I flip it twenty times, exactly ten heads is not that unusual, but still a little lucky maybe. If I flip that coin one million times and get exactly 50... (more)	—	almost 3 years ago
Answer	—	A: difference between quotient rule and product rule If you work out deriving the quotient rule yourself using the exact trick you're highlighting, you can see that the quotient rule is nothing more than the product rule and the chain rule used together. If you rearrange a quotient to use the product rule, then you'll very likely be using the chain rul... (more)	—	almost 3 years ago
Answer	—	A: How can I visualize this Compound Interest Chart with indefinite integrals? Neither of those charts presents those integrals; the first illustrates annually compounded interest, and the second compares a few different compounding schedules to continuous compounding, but not in a way that highlights the integrals in the text explanation. If you wanted an illustration of th... (more)	—	about 3 years ago
Answer	—	A: How can I generalize a picture for the Mean Value Theorem to the Generalized MVT? Plotting $f$ and $g$ together is unlikely to lead to intuition about the GMVT; they're just two arbitrary curves, whereas the value from that diagram of the MVT comes from the fact that the yellow line is constructed to give intuition. The art of drawing intuitive figures is largely about figuring ou... (more)	—	about 3 years ago
Answer	—	A: Why should a non-commutative operation even be called "multiplication"? Usually (not always, but true for the examples of multiplication of real numbers and multiplication of square matrices), a binary operation that is called some variant on ‘multiplication’ is the multiplicative operation of some ring). That's the closest thing to a rigorous definition of multiplicatio... (more)	—	about 3 years ago
Answer	—	A: Given a triangle with squares on two sides, the line segments joining the centres of the squares to the midpoint of the third side are equal and perpendicular Your calculations look correct to me, but I think this is perhaps meant to be seen geometrically (apologies if this is already obvious to you and you were looking for a non-geometric explanation). Let the vertices of the triangle be $x$, $y$, $z$, starting at the vertex shared by the two squares, ... (more)	—	over 3 years ago
Answer	—	A: How can I deduce which operation removes redundacies? You have 8 people, and you have to pick 3 of them to get identical gifts. Even though the gifts are identical, let's give them labels: A, B, and C. Let's also number the people 1 to 8. The first step is to figure out how many ways there are to hand out gifts in order. You have 8 choices for gift A... (more)	—	over 3 years ago
Answer	—	A: Product of empty set of elements vs. product over empty indexing set — is there any difference? tl;dr: Having one logical choice for a notation that is consistent with the rest of one's definitions is not the same as defining that notation to mean that choice. Until one has done so, the notation has no meaning (in a strict reading of a math text). In this excerpt, Lang is defining, presumabl... (more)	—	over 3 years ago
Answer	—	A: Why always rationalize a denominator? Mathematically, absolutely nothing is wrong with fractions that don't have the smallest positive integer denominator that they could have, as far as I know. Pedagogically, the policy does help with there being a unique answer that can be checked quickly, when grading multiple homework assignments ... (more)	—	over 3 years ago