https://math.codidact.com/categories/41/feedNew Posts - Q&A - MathematicsMathematics - Codidact2022-05-17T23:22:17Zhttps://math.codidact.com/posts/286472Are vectrices useful for calculations as opposed to formalism?Fred Wamsleyhttps://math.codidact.com/users/550432022-05-17T01:22:34Z2022-05-17T23:22:17Z<p>The context here is calculations about satellite attitude control and motion. There is a frequent need to convert between reference frames, and a vectrix is a clever notation to assist.</p>
<p>...https://math.codidact.com/posts/286428$g(x)\xrightarrow{x\to\infty}\infty$ Implies $g'(x)\leq g^{1+\varepsilon}(x)$Udi Fogielhttps://math.codidact.com/users/564582022-05-10T21:34:01Z2022-05-15T19:15:28Z<p>Recently in my ordinary differential equations class we were given the following problem:</p>
<blockquote>
<p>Suppose $g:(0,\infty)\to\mathbb{R}$ is an increasing function of class $C^{1}$ suc...https://math.codidact.com/posts/286166Complex numbers in 2D, quaternions in 4D, why nothing in 3D? Fred Wamsleyhttps://math.codidact.com/users/550432022-03-27T21:22:28Z2022-05-14T07:19:45Z<p>I'm just trying to understand why quaternions are necessary. If I understand right, first off please check me on this, Hamilton kept trying to find ways to multiply triplets and found something ...https://math.codidact.com/posts/285432How to intuit p = Calvin's probability of winning each game independently = $1/2 \implies$ P(Calvin wins the match) = 1/2?DNBhttps://math.codidact.com/users/536282021-12-31T08:31:07Z2022-05-09T13:01:11Z<p>Please see the sentence beside my red line. The notion of a "sanity check" suggests that these resultant integers should be obvious, without calculation or contemplation. But why's it plain and ...https://math.codidact.com/posts/286398Matrices with rotational symmetryPeter Taylorhttps://math.codidact.com/users/363562022-05-05T22:24:34Z2022-05-05T22:46:21Z<p>I've seen a <a href="https://mathoverflow.net/a/418547">claim without proof</a> that the characteristic polynomials of matrices with rotational symmetry (i.e. $n \times n$ matrices $A$ with $A_{...https://math.codidact.com/posts/286292Why does the Dottie number $=\sqrt{1-\left(2\text I^{-1}_\frac12\left(\frac12,\frac32\right)-1\right)^2}$?Tyma Gaidashhttps://math.codidact.com/users/565582022-04-22T04:26:17Z2022-04-22T04:37:45Z<p><strong>Introduction:</strong></p>
<p>For some background information on the <a href="https://en.m.wikipedia.org/wiki/Dottie_number"><strong>Dottie Number D</strong></a>,</p>
<p>Cross posted f...https://math.codidact.com/posts/286188A discrete-time queue Markov Chain problemJoehttps://math.codidact.com/users/565152022-04-05T08:20:31Z2022-04-05T08:20:31Z<p>Hello everyone, I hope somebody can help me with this question, only for parts c and d, because I have already figured out parts a and b. Here is the question:</p>
<p>Items arrive in a queue du...https://math.codidact.com/posts/286150equilateral triangle inscribed in an ellipsemsh210https://math.codidact.com/users/81062022-03-24T19:58:14Z2022-03-25T00:12:09Z<p>A high-schooler I know was given the following problem:</p>
<blockquote>
<p>In the ellipse $x^2+3y^2=12$ is inscribed an equilateral triangle. One of the triangle's vertices is at the point $(...https://math.codidact.com/posts/286140Finding distance to parabola's focus, given some pointsmsh210https://math.codidact.com/users/81062022-03-21T20:44:16Z2022-03-21T21:21:24Z<p>A high school student I know has the following problem:</p>
<blockquote>
<p>A parabola is given by $y^2=2px$ with $p>0$. The point $D$ is on the parabola in the first quadrant at a distance...https://math.codidact.com/posts/286130How can I justify whether I'm able to apply the Gaussian elimination method to this system?dumplingshttps://math.codidact.com/users/547102022-03-20T11:01:16Z2022-03-21T12:15:58Z<p>I'm currently practicing some Gaussian elimination questions for my discrete math topic, I came across this question and I need help. How can I justify whether I'm able to apply the Gaussian eli...https://math.codidact.com/posts/286004Sample skewness consistencyGuilherme Costahttps://math.codidact.com/users/534532022-02-22T10:55:44Z2022-03-04T08:54:25Z<p>Is sample skewness a consistent estimator of skewness?</p>
<!-- g: js, mdit -->https://math.codidact.com/posts/285670Why 1. multiply the number of independent options? 2. add the number of exclusive options?DNBhttps://math.codidact.com/users/536282022-01-24T03:31:04Z2022-02-27T05:05:31Z<p>Ironically, this textbook highlights understanding over memorization, but it doesn't expatiate the two WHY's in the question title!</p>
<blockquote>
<p>When faced with a series of independent ...https://math.codidact.com/posts/285984How $ijk=\sqrt{1}$?deleted user#2022-02-18T15:34:12Z2022-02-26T13:56:37Z<p>In the <a href="https://en.wikipedia.org/wiki/Quaternion#History">Wikipedia page</a>, I can clearly see that
$$i^2=j^2=k^2=ijk=-1$$</p>
<p>But if we consider them separately</p>
<p>$$i=\sqrt{...https://math.codidact.com/posts/285977Maximize Independent Variable of Matrix MultiplicationJosh Hyatthttps://math.codidact.com/users/536292022-02-17T20:47:41Z2022-02-18T20:23:40Z<p>Let $T$ be an $m\times n$ matrix with column vectors $\vec{T_i}$:</p>
<p>$$
\vec{T_i}=\begin{bmatrix}
\vec{T_1} & ... & \vec{T_n}
\end{bmatrix}
$$</p>
<p>Let $\vec{x}$ be an unknow...https://math.codidact.com/posts/285819Is it impossible to prove Jensen's inequality by way of step functions?whybecausehttps://math.codidact.com/users/550222022-02-04T01:03:31Z2022-02-11T21:32:00Z<p>Jensen's Inequality: Let $\varphi:\Bbb R\to \Bbb R$ be convex, and $f:[0,1]\to\Bbb R$ be integrable, and suppose $\varphi\circ f$ is integrable over [0,1]. Then
$$ \varphi\left(\int_{[0,1]} f\...https://math.codidact.com/posts/285925"Pointwise equicontinuity implies uniform" implies compactwhybecausehttps://math.codidact.com/users/550222022-02-11T05:02:37Z2022-02-11T20:05:08Z<p>Suppose $(M,d)$ is a metric space such that every sequence $f_n:M\to \Bbb R$ which is pointwise equicontinuous is also uniformly equicontinuous. Does this imply that $M$ is a compact metric spa...https://math.codidact.com/posts/285788Why can more general problems — paradoxically — be easier to solve or prove?Chgg Clouhttps://math.codidact.com/users/535642022-02-01T15:37:37Z2022-02-02T14:16:10Z<p><a href="https://math.stackexchange.com/a/899131">These answers</a> list instances of the Inventor's Paradox, but doesn't expatiate WHY this can happen?</p>
<p>Here's an analogy. Let Dialect 1 ...https://math.codidact.com/posts/285679How to intuit, construe multiplicands and multiplicators $\le 10$ resulting from $\dbinom pc$, WITHOUT division or factorials?DNBhttps://math.codidact.com/users/536282022-01-25T05:46:05Z2022-02-01T10:24:37Z<p>I grok, am NOT asking about, the answers below. Rather — <strong>how can I deduce and intuit the multiplicands and multiplicators $\le 10$, resulting from simplifying $\dbinom {p \text{ people...https://math.codidact.com/posts/285756$\{f_n\}\to f$ in $X$ implies $||f||\le\liminf||f_n||$ whybecausehttps://math.codidact.com/users/550222022-01-31T02:27:52Z2022-01-31T16:28:15Z<p>Let $X$ be a normed linear space and suppose that, for each $f\in X$ there exists a bounded linear functional $ T\in X^* $ such that $T(f)=||f||$ and $||T||_*=1$. Prove that if $\{f_n\}\to f$ in...https://math.codidact.com/posts/285753Missing a solution: Are A and B always equal if A-B=0re89jhttps://math.codidact.com/users/553482022-01-30T23:10:46Z2022-01-31T09:01:14Z<p>I just came across this problem on an online learning app:</p>
<p>4x(5−x)−12(5−x)+100=100</p>
<p>I tried to solve it by subtracting 100 from each side then inferring if something minus somethi...https://math.codidact.com/posts/282608Out of 4 people, why does ways to choose a 2-person committee overcount by 2 the ways to divide the 4 into 2 teams of 2?DNBhttps://math.codidact.com/users/536282021-07-09T05:04:41Z2022-01-28T15:52:59Z<ol>
<li>
<p>Please see the sentence alongside my red line below. Why does part (a) overcount part (b) by a factor of c?</p>
</li>
<li>
<p>Scilicet, why aren't the answers to parts (a) and (b...https://math.codidact.com/posts/285671How can you "easily see that such squares [of side length $\sqrt{13}$ and $\sqrt{18}$] will not fit into the [4 × 4] grid"?DNBhttps://math.codidact.com/users/536282022-01-24T03:45:23Z2022-01-28T12:29:42Z<blockquote>
<p>Problem 2.4: How many squares of any size can be formed by connecting dots in the grid shown in Figure 2.2.</p>
</blockquote>
<p>I skip p 31, but apprise me if you want me to inc...https://math.codidact.com/posts/285680How can a 15 year old construe the LHS of Generalized Vandermonde's Identity, when it lacks summation limits and a summation index?DNBhttps://math.codidact.com/users/536282022-01-25T06:07:44Z2022-01-25T19:24:52Z<p>Paradoxically, though <a href="https://math.stackexchange.com/q/3572304">Rothe-Hagen Identity</a> (henceforth RHI)</p>
<p>$\sum\limits_{k=0}^n\frac{x}{x+kz}{x+kz \choose k}\frac{y}{y+(n-k)z}{y+...https://math.codidact.com/posts/285672To correct for overcounting, why can't we divide by the sum of the over-counts?DNBhttps://math.codidact.com/users/536282022-01-24T08:02:35Z2022-01-24T08:52:47Z<p>The book doesn't expatiate the sentence underlined in red below. 1. Why "it isn't 2 + 2"?</p>
<ol start="2">
<li>Note that $2 \times 2 = 2 + 2 = 4$! Is this a coincidence?</li>
</ol>
<blockq...https://math.codidact.com/posts/285673How can you prognosticate to rewrite the same sum backwards, then add the same sum (twice)? DNBhttps://math.codidact.com/users/536282022-01-24T08:22:11Z2022-01-24T08:23:40Z<p><a href="https://math.stackexchange.com/questions/924203/reversing-the-order-of-summation-in-sum-i-1n-i#comment1907997_924214">This comment doesn't fulfill me</a>, because it doesn't demystify t...https://math.codidact.com/posts/285443Why's Pr(the running total of a fair dice rolled repeatedly = n) = $\frac16 (p_{n-1} + p_{n-2} + p_{n-3} + p_{n-4} + p_{n-5} + p_{n-6})$?DNBhttps://math.codidact.com/users/536282022-01-01T05:01:33Z2022-01-17T09:18:39Z<p>Where did $\color{red}{p_n = \frac16 (p_{n-1} + p_{n-2} + p_{n-3} + p_{n-4} + p_{n-5} + p_{n-6})}$ spring from? The solution doesn't expatiate, and makes it appear out of the blue? Can you pleas...https://math.codidact.com/posts/285433Why's a 2-player game with "win by two" rule = Gambler's Ruin where each player starts with $2? Why N = 4?DNBhttps://math.codidact.com/users/536282021-12-31T08:41:57Z2022-01-17T08:54:19Z<ol>
<li>
<p>Why can this problem "be thought of as a gambler's ruin where each player starts out
with $2"?</p>
</li>
<li>
<p>Please see my red arrow. Why is this exponent 4? I quote op. cit....https://math.codidact.com/posts/285524How's it possible to arrange 0 objects? How can 0! = 1?DNBhttps://math.codidact.com/users/536282022-01-10T00:06:59Z2022-01-12T16:28:38Z<ol>
<li>
<p>I don't understand the following explanation. Isn't it physically impossible to "arrange 0 objects", or nothing? It's senseless to raise the concept of arrangement when you have not...https://math.codidact.com/posts/285431If C = Calvin wins the match, and $X \thicksim Bin(2, p) =$ how many of the first 2 games he wins — then why P(C|X = 1) = P(C)?DNBhttps://math.codidact.com/users/536282021-12-31T08:24:45Z2022-01-05T11:14:44Z<p>The author's solution doesn't expatiate why $\color{red}{P(C|X = 1) = P(C)}$? <a href="https://math.stackexchange.com/q/4288685">This similar question on Math Stack Exchange</a> has 0 answers, a...https://math.codidact.com/posts/285344How can you analogize mathematical induction to dominoes falling, if some domino can fail to topple?Chgg Clouhttps://math.codidact.com/users/535642021-12-21T00:33:33Z2022-01-05T10:00:41Z<p>This analogy doesn't convince me, because what if some domino (after b, the base case) fails to topple? In real life, a domino can remain standing upright if it got placed too far apart from the...https://math.codidact.com/posts/280847How can I re-write or supplant “basis point” UNambiguously?Chgg Clouhttps://math.codidact.com/users/535642021-02-18T06:29:34Z2022-01-05T07:29:51Z<p>For this question, please DON'T use or even allude to "basis point" in your rewording, because I'm trying to replace "basis point"! "basis point" befuddles me! <a href="https://www.investopedia....https://math.codidact.com/posts/285447How to deduce which ETF is redundant? a of ETF1's holdings in ETF3, a of ETF2 in ETF3, b of ETF1 are in ETF2, b of ETF2 in ETF1.Chgg Clouhttps://math.codidact.com/users/535642022-01-01T20:28:41Z2022-01-01T21:55:06Z<p>According to <a href="https://www.etfrc.com/funds/overlap.php">https://www.etfrc.com/funds/overlap.php</a>,</p>
<ol>
<li>
<p>47.2% of <a href="https://www.ftportfolios.com/retail/etf/etfsumma...https://math.codidact.com/posts/285436Are we in a "history-valley" for Topology?whybecausehttps://math.codidact.com/users/550222021-12-31T16:22:51Z2021-12-31T21:15:50Z<p>Here is my current rough timeline of Topology:</p>
<ul>
<li>Newton invents the calculus</li>
<li>People like Riemann and Cauchy make it rigorous, and by this time, we have the $\varepsilon,\d...https://math.codidact.com/posts/285430Intuitively, why does $p$ vary inversely with $P(C_3 \mid D_2)$? But directly with $P(C_2 \mid D_3)$?DNBhttps://math.codidact.com/users/536282021-12-31T07:29:04Z2021-12-31T15:05:54Z<p><strong>I'm seeking merely intuition here</strong> — NOT about the algebra that I know how to execute. Please see the fractions colored in red and orange at the bottom.</p>
<ol>
<li>$P(C_3 \...https://math.codidact.com/posts/285434"the people who move have a percentage of Democrats which is between these two values" — How does this furnish intuition for Simpson's Paradox?DNBhttps://math.codidact.com/users/536282021-12-31T08:53:03Z2021-12-31T08:55:07Z<p>Why does the phrase colored in red below ($\color{red}{\text{the people who move have a percentage of Democrats which is between these two values}}$) matter? How does it assist with intuiting t...https://math.codidact.com/posts/285429For (a variation) of the Monty Hall problem, what permits you to repaint doors 2 and 3? What permits you to replace p with $1 - p$?DNBhttps://math.codidact.com/users/536282021-12-31T07:07:53Z2021-12-31T08:34:35Z<p>Please see the text colored in red at the bottom.</p>
<ol>
<li>
<p>Why can you haughtily just — and what legitimizes you to — "[i]magine repainting doors 2 and 3, reversing which is called wh...https://math.codidact.com/posts/285015Intuitively, why can $a, b$ cycle in ${\color{red}{b}} = \frac c{\color{red}{a}} \iff {\color{red}{a}} = \frac c{\color{red}{b}}$? Chgg Clouhttps://math.codidact.com/users/535642021-11-29T17:39:29Z2021-12-31T02:19:56Z<p>I'm NOT asking about algebra behind $ab = c \iff {\color{red}{b}} = \frac c{\color{red}{a}} \iff {\color{red}{a}} = \frac c{\color{red}{b}}.$</p>
<ol>
<li>
<p>Rather, what's the intuition why...https://math.codidact.com/posts/285424$\int_{E_n} |g|^q = \left| \int_E \chi_{E_n}\cdot \text{sgn}(g)\cdot g \cdot |g|^{q-1}\cdot |g| \right|$ whybecausehttps://math.codidact.com/users/550222021-12-31T01:18:40Z2021-12-31T02:13:44Z<p>I am trying to understand why the following equation is true. Here $E$ is a measurable set and all functions are defined and measurable on it. $1<p,q,<\infty$ such that $\frac 1 p+\frac ...https://math.codidact.com/posts/282603Without calculations, how can you visualize "that half the squares are white and half are black"?DNBhttps://math.codidact.com/users/536282021-07-09T03:50:39Z2021-12-30T23:34:58Z<p>Please see the 2nd para. below alongside my red highlighted words. I can't "[i]magine rotating the chessboard 90 degrees clockwise." I can't visualize how "all the positions that had a white squ...https://math.codidact.com/posts/285364In general, does $\color{forestgreen}{P(A|M)} + \color{red}{P(B|M)} = 1$?DNBhttps://math.codidact.com/users/536282021-12-23T09:08:25Z2021-12-30T22:06:30Z<p>In this question, $\color{forestgreen}{P(A|M)} + \color{red}{P(B|M)} = 1$. But the author's solution didn't annunciate this, and doesn't unfurl how to compute $\color{red}{P(B|M)}$. Can I simply...https://math.codidact.com/posts/285365Acceptable, usual to write $\ge 2$ pipes simultaneously?DNBhttps://math.codidact.com/users/536282021-12-23T09:24:56Z2021-12-26T09:47:28Z<p><em><strong>I'm NOT asking for the solution to this exercise that's publicly accessible.</strong></em> Rather, pls see the green and red underlines. If I apply the author's green definition to t...https://math.codidact.com/posts/285371How does $P(C > D \mid C = 2) \neq P(C > D \mid C \neq 2)$ prove that B > C depends on C > D?DNBhttps://math.codidact.com/users/536282021-12-24T07:00:13Z2021-12-24T07:00:33Z<p>I grok that $\color{limegreen}{P(C > D \mid C = 2) = P(D = 1 \mid C = 6) = 1/2}$, and $\color{red}{P(C > D \mid C \neq 2) = P(C > D \mid C = 6) = 1}$. But I don't grok the last sentenc...https://math.codidact.com/posts/284280Is Pythagorean theorem really valid in higher dimensional space?deleted user#2021-09-21T16:25:27Z2021-12-22T05:38:12Z<p>I saw that someone was writing Pythagorean theorem in 3 dimensional space. The equation was :</p>
<p>$$c=\sqrt{x^2+y^2+z^2}$$
If it's really correct than Pythagorean should work in higher dime...https://math.codidact.com/posts/285342Is Mathematical Induction truly "induction", or misnamed?Chgg Clouhttps://math.codidact.com/users/535642021-12-21T00:24:22Z2021-12-21T14:06:19Z<p>By induction, I mean <a href="https://www.youtube.com/watch?v=-nn3XMoPC7s">this screenshot from this Youtube video</a></p>
<p><img alt="" src="https://i.stack.imgur.com/s8VJy.png"></p>
<p><a h...https://math.codidact.com/posts/284996How do mathematicians measure area for points?deleted user#2021-11-27T06:59:36Z2021-12-20T10:04:42Z<p>There are two points (adjacent or not adjacent) on top of an infinite plane; how to measure how much area they catch together, as a "set".</p>
<p>Because that if we put a minimal border around ...https://math.codidact.com/posts/285326Why must percent change divide the difference by the old, NOT new, value? Chgg Clouhttps://math.codidact.com/users/535642021-12-19T20:01:24Z2021-12-19T21:32:20Z<p><img alt="" src="https://www.onlinemathlearning.com/image-files/percent-change.png"></p>
<p>My 14 year old still cannot intuit why the denominator must be the old number, not the new — not even...https://math.codidact.com/posts/283400If a matrix has lots of values in a column but, not in row than, what that actually called?deleted user#2021-08-14T16:40:29Z2021-12-15T02:52:10Z<p>$\begin{bmatrix}a & b & c & d\end{bmatrix}$</p>
<p>$\begin{bmatrix}a \\ b \\ c \\ d \end{bmatrix}$</p>
<p>Which one is $1$ dimensional Matrix? I think that first matrix is $1$ dime...https://math.codidact.com/posts/284561What does upper indices represent?deleted user#2021-10-10T15:08:24Z2021-12-07T19:51:31Z<p>I saw that people were representing matrices in two ways.</p>
<ol>
<li>$$\sum_{j=1}^n a_{ij}$$</li>
</ol>
<p>It is representing a column matrix (vector actually) if we assume $i=1$.</p>
<p>...https://math.codidact.com/posts/285052What is the parallel for edges?deleted user#2021-12-02T12:38:07Z2021-12-03T08:31:16Z<p>What is the parallel to <a href="https://en.wikipedia.org/wiki/Edge_(geometry)">edge</a> in curves?</p>
<p>For example, if we have three points and three edges we get a triangle but if we defin...https://math.codidact.com/posts/284723Can we add without using addition?IDoYTSoSubHahahttps://math.codidact.com/users/541142021-10-29T18:24:39Z2021-12-01T22:16:40Z<p>Is there a formula that applies $a + b = c$ without addition. I tried many times to make such equation.</p>
<p>Here's what I have in mind, we get 2 values, call them $a$ and $b$. The objective ...