https://math.codidact.com/categories/41/feedNew Posts - Q&A - MathematicsMathematics - Codidact2021-04-09T12:17:42Zhttps://math.codidact.com/posts/280168How can I deduce which operation removes redundacies?DNBhttps://math.codidact.com/users/536282020-12-30T03:11:56Z2021-04-09T12:17:42Z<p>Please don't answer by working backwards from the answer, or by appealing to arithmetic. Act as if you're learning this for the first time.</p>
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<p>How can I deduce which operation ...https://math.codidact.com/posts/280799Is this topology basis dependent?celtschkhttps://math.codidact.com/users/80562021-02-14T19:32:06Z2021-03-30T18:08:26Z<p>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 infinit...https://math.codidact.com/posts/281063Why aren't filled black and hollow red candlesticks intuitive mathematically?Chgg Clouhttps://math.codidact.com/users/535642021-03-17T07:01:02Z2021-03-17T07:01:02Z<p>I know that color (black vs. red) compares today's close with the previous close, and hollowness (hollow vs. filled) compares today's close with today's open. But the two candlesticks, embolded ...https://math.codidact.com/posts/281010De-mystifying tricks – If $\{x_n\}$ converges, then Cesaro Mean converges.Technologically illiteratehttps://math.codidact.com/users/536962021-03-09T06:12:46Z2021-03-11T00:21:07Z<blockquote>
<p><strong>Exercise 2.3.11 (Cesaro Means).</strong> (a) Show if ${x_n}$ is a convergent sequence, then the sequences given by the averages ${\dfrac{x_1 + x_2 + ... + x_n}{n}}$ converg...https://math.codidact.com/posts/281005Intuitively, why's X% of Y = Y% of X?Chgg Clouhttps://math.codidact.com/users/535642021-03-09T05:09:37Z2021-03-09T09:27:03Z<p>I didn't spot this trick until I read <a href="https://redd.it/7yvxw4">this</a>:</p>
<blockquote>
<p>LPT: X percent of Y is equal to Y percent of X.</p>
</blockquote>
<p>I know that multipli...https://math.codidact.com/posts/281006In If and Only If Proofs, why's the proof for one direction easier than the other?Chgg Clouhttps://math.codidact.com/users/535642021-03-09T05:27:22Z2021-03-09T09:20:44Z<p><a href="https://math.stackexchange.com/q/3069488">This list</a>. <a href="https://math.stackexchange.com/q/3069488">this question</a> substantiates proofs where one direction can be proved effo...https://math.codidact.com/posts/281009Why aren't $z_1=f(xy)$ and $z_2=f(x/y)$ functions of 2 variables?Technologically illiteratehttps://math.codidact.com/users/536962021-03-09T05:48:14Z2021-03-09T07:24:02Z<p><a href="https://math.stackexchange.com/a/738461/53259">Hagen von Eitzen answered</a> that $z_1, z_2$</p>
<blockquote>
<p>depend on only one variable - there's no comma between the parentheses...https://math.codidact.com/posts/281008If a repair on a vessel costs $2 and fully repaired vessel is worth $1, don't we need know worth of broken vessel to decide whether to repair?Chgg Clouhttps://math.codidact.com/users/535642021-03-09T05:37:11Z2021-03-09T05:37:11Z<p>I divided all prices by 100,000 to simplify the question. The book beneath doesn't stipulate the current value of the vessel. But don't we need to know it, to calculate if it's worth repairing? ...https://math.codidact.com/posts/281007What kind of logic does “Logic in the Theory and Practice of Lawmaking” employ?Chgg Clouhttps://math.codidact.com/users/535642021-03-09T05:30:15Z2021-03-09T05:30:15Z<p>I stumbled on 2016 Springer book <a href="https://www.amazon.ca/Theory-Practice-Lawmaking-Michal-Araszkiewicz/dp/3319371460"><em>Logic in the Theory and Practice of Lawmaking</em></a>. I was cur...https://math.codidact.com/posts/280910Approximation of an elliptic integraltheabbiehttps://math.codidact.com/users/537872021-02-24T03:00:54Z2021-02-24T03:00:54Z<p>I used some nice guesswork to get this formula<br>
$$\int_0^{\frac{\pi}{2}}\sqrt{a^2\sin^2x+b^2\cos^2x}dx=\frac{ab\pi}{\left(a+b\right)\sin\left(\frac{a\pi}{a+b}\right)}$$<br>
The comparision ...https://math.codidact.com/posts/280851If $n = xm$ and $n \rightarrow \infty$, then $m \rightarrow \infty$?Technologically illiteratehttps://math.codidact.com/users/536962021-02-18T06:39:06Z2021-02-22T08:11:42Z<p>Did Stewart prove the titular result, also underlined in green in the image below, either by himself or as an exercise, in <em>Calculus Early Transcendentals</em> or in the normal version <em>Ca...https://math.codidact.com/posts/280842Does every divergence-free vector field arise as the curl of some vector field?Bahudari Ragamhttps://math.codidact.com/users/535162021-02-17T18:51:20Z2021-02-20T19:38:13Z<p>I was introduced to the concepts of gradient $\nabla f$, curl $\nabla \times F$ and divergence $\nabla \cdot F$ in an introductory course on calculus during my undergraduate studies. There I lea...https://math.codidact.com/posts/280848How can I visualize this Compound Interest Chart with indefinite integrals?Chgg Clouhttps://math.codidact.com/users/535642021-02-18T06:36:11Z2021-02-18T20:09:22Z<p>How can I visualize <a href="https://betterexplained.com/articles/definitions-of-e-colorized/">the integrals below</a>? Can someone draw on the chart and point to where $\int 1 , dx = x$, $\int ...https://math.codidact.com/posts/280850How to visualize division as splitting Dividend into B equal “partial groups”, then rounding up A partial groups to get a full group?Chgg Clouhttps://math.codidact.com/users/535642021-02-18T06:38:08Z2021-02-18T06:38:08Z<p>Because $10 \div \dfrac{4}{3}$ isn't an integer, <a href="https://old.reddit.com/r/learnmath/comments/ckv75r/why_multiplying_by_the_reciprocal_works_while/evurf0g/">I changed the numbers in thi...https://math.codidact.com/posts/280849How can I visualize dividing by a fraction as partial and full groups?Chgg Clouhttps://math.codidact.com/users/535642021-02-18T06:37:44Z2021-02-18T06:37:44Z<p>I can't visualize all these "full groups" and "partial groups." Can someone please picture them? Please see my questions red in-line. Please don't use the identity that $\dfr...https://math.codidact.com/posts/280847How can I re-write or supplant “basis point” unambiguously?Chgg Clouhttps://math.codidact.com/users/535642021-02-18T06:29:34Z2021-02-18T06:31:16Z<p>For this question, please don't use or even allude to "basis point" in your rewording. <a href="https://www.investopedia.com/ask/answers/what-basis-point-bps/">Basis Point (BPS) Defini...https://math.codidact.com/posts/280704Why should a non-commutative operation even be called "multiplication"?theabbiehttps://math.codidact.com/users/537872021-02-08T10:18:35Z2021-02-14T19:56:50Z<p>As per my knowledge and what was taught in school,</p>
<blockquote>
<p>$a\times b$ is $a$ times $b$ or $b$ times $a$</p>
</blockquote>
<p>Obviously this is commutative as $a$ times $b$ and $...https://math.codidact.com/posts/280663How can I generalize a picture for the Mean Value Theorem to the Generalized MVT?Technologically illiteratehttps://math.codidact.com/users/536962021-02-04T02:22:15Z2021-02-11T22:16:44Z<p>How can I transmogrify this figure for the <a href="https://math.stackexchange.com/q/296176">Generalized MVT</a>? If $f$ and $g$ are continuous
on the closed interval $[a, b]$ and differentiabl...https://math.codidact.com/posts/280758In a SARIMA model, does the order in which I difference the time series matter?anoldmanintheseahttps://math.codidact.com/users/536472021-02-10T18:11:16Z2021-02-10T18:11:16Z<p>In a SARIMA model $\text{ARIMA}(p,d,q)\times(P,D,Q)_S$, we can write it as $$\Phi_P(B^S)\phi(B)(1-B^S)^D(1-B)^d x_t= \delta +\Theta_Q(B^S)\theta(B)w_t$$ where the capital letters are the paramet...https://math.codidact.com/posts/280741What story and one-digit Natural Numbers explain Bayes' Theorem chart most simply?DNBhttps://math.codidact.com/users/536282021-02-09T23:29:08Z2021-02-10T09:42:21Z<p>Some students have sniveled that most examples of Bayes' Theorem use non-integer numbers. I want to try a Bayes' Theorem chart that uses just single digit Natural Numbers $\le 9$. To complete th...https://math.codidact.com/posts/279026Classification of topological pointsceltschkhttps://math.codidact.com/users/80562020-11-08T20:51:50Z2021-02-02T22:52:43Z<p>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 i...https://math.codidact.com/posts/280629Isn't it wrong to write that Indefinite Integral = Definite Integral with a variable in its Upper Limit?Technologically illiteratehttps://math.codidact.com/users/536962021-01-31T23:53:38Z2021-02-02T21:29:00Z<blockquote>
<p>$\int f(t) \; dt = \int_{t_0}^t f(s) \; ds \quad \text{ where $t_0$ is some convenient lower limit of integration.}$</p>
</blockquote>
<p>Isn't this wrong? Because LHS $\neq$ RH...https://math.codidact.com/posts/280653How can I visualize $\lim\limits_{x \rightarrow \pm \infty} f(x) = \lim\limits_{t \rightarrow 0^{\pm}} f(1/t)$?Technologically illiteratehttps://math.codidact.com/users/536962021-02-02T21:22:23Z2021-02-02T21:25:53Z<p>I'm not asking about <a href="https://math.stackexchange.com/a/3983012/872474">the proof</a> that I already understand. I'm longing to understand this graphically. As you can see, I added $1/t$...https://math.codidact.com/posts/280572Does this generalization of path-connectedness also cover general connectedness?celtschkhttps://math.codidact.com/users/80562021-01-24T14:06:30Z2021-01-24T14:08:17Z<p>I've got the following idea to generalise path-connectedness:</p>
<p>Given a topological space $P$ and a subspace $S$, a space $X$ is $(P,S)$-connected if every continuous function $f:S\to X$ c...https://math.codidact.com/posts/280524Are there useful topologies on Cartesian products “in between” the product topology and the box topology?celtschkhttps://math.codidact.com/users/80562021-01-20T07:44:16Z2021-01-20T08:45:21Z<p>On the Cartesian product of topological spaces, there are two standard topologies: One is the product topology, the other is the box topology.</p>
<p>As is well known, the box topology is gener...https://math.codidact.com/posts/278332Cutting the squareGrovehttps://math.codidact.com/users/534522020-10-01T21:53:10Z2021-01-18T15:15:15Z<p>I don't know how simple/diffcult this is, but it has been on my mind for some time, and I haven't managed to do anything about it, so now I'll try here to see if anyone has any input.</p>
<p>Le...https://math.codidact.com/posts/280495Why does this definition of generalized forces work?JoshuaTShttps://math.codidact.com/users/532632021-01-16T00:57:56Z2021-01-16T03:53:49Z<p>I don't understand an equation in the discussion of Lagrangian Mechanics in the book <em>Mathematical Aspects of Classical and Celestial Mechanics</em> by Arnold <em>et al.</em> (section 1.2.1)....https://math.codidact.com/posts/280460What surface do I get by attaching $g$ handles as well as $k$ crosscaps to a sphere?Bahudari Ragamhttps://math.codidact.com/users/535162021-01-13T18:25:39Z2021-01-13T19:09:42Z<p>I recently found out that there is a classification of compact connected surfaces that says that every such surface (or, $2$-manifold) is homeomorphic to either $S_g$, the sphere with $g \geq 0$...https://math.codidact.com/posts/279973In $P(X\leq q)$, what do we call $q$?anoldmanintheseahttps://math.codidact.com/users/536472020-12-17T22:34:31Z2021-01-11T03:42:17Z<p>This is a simple question, but in $P(X\leq q)=\alpha$, what do we call $q$?</p>
<p>I sometimes call them quantiles, for lack of a better word, but it's not correct, I think. My 'inspiration' co...https://math.codidact.com/posts/280310Are 3 10% chances better than one 30% chance (when penalized by a variable for failures)?dustytrashhttps://math.codidact.com/users/81702021-01-05T20:09:42Z2021-01-08T11:01:02Z<p>I'm playing a computer game in which you can spend <code>100 coins to be given a 10% chance</code> to upgrade a weapon, or spend <code>200 coins for a 20%</code> chance, or <code>300 coins for a...https://math.codidact.com/posts/280348Normal in one base and not in another?Grovehttps://math.codidact.com/users/534522021-01-07T10:55:07Z2021-01-07T10:55:07Z<p>Examples of numbers that are normal in some bases are known (Champernowne's $0.12345678910111213\ldots$ is probably the most well-known example of a number that is normal in base 10).</p>
<p>To...https://math.codidact.com/posts/280204Given 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 perpendicularBahudari Ragamhttps://math.codidact.com/users/535162020-12-31T02:06:40Z2021-01-03T14:41:34Z<p>I am reading Tristan Needham's <em>Visual Complex Analysis</em> (2012 reprint, OUP), and in $\S$1.III.3: <em>Geometry</em>, the author gives a geometric proof of the following fact, shown in Fig...https://math.codidact.com/posts/280118Product of empty set of elements vs. product over empty indexing set — is there any difference?Bahudari Ragamhttps://math.codidact.com/users/535162020-12-25T12:48:00Z2020-12-30T05:16:36Z<p>I am reading Lang's <em>Algebra</em> (3rd ed., Pearson, 2003). In $\S$I.1 <em>Monoids</em>, on page 4 the author defines the meaning of and notations for products of finitely many elements of a ...https://math.codidact.com/posts/280064Why always rationalize a denominator?msh210https://math.codidact.com/users/81062020-12-21T06:52:38Z2020-12-21T12:45:02Z<p>Schoolteachers will insist that their students present answers to problems with rational (indeed integral) denominators. Never $1/\sqrt3$, for example, but instead $\sqrt3/3$. That's also how ma...https://math.codidact.com/posts/279400Intuitively, what does the 1 mean in $\dfrac{a}{b} = \dfrac{1}{\frac{b}{a}}$?Chgg Clouhttps://math.codidact.com/users/535642020-11-20T07:21:54Z2020-12-11T01:46:58Z<p>For intuition, I instantiate variables with fruits: $a$ as apple and $b$ as berry (pick whatever berry you like).</p>
<p>In terms of apples and/or berries, what does the red $\color{red}{1}$ be...https://math.codidact.com/posts/279727Disentangling Machine Learning Theory with Cross-Validationseth.wagenmanhttps://math.codidact.com/users/535812020-12-04T20:20:56Z2020-12-04T20:20:56Z<p>Does anyone see a link between machine learning's repeated epochs of training and the concept of cross-validation in linear modeling theory?</p>
<p>This article demonstrates what I perceive to ...https://math.codidact.com/posts/279044Why does “unless” mean “if not”? PSTHhttps://math.codidact.com/users/535662020-11-10T07:30:40Z2020-11-11T00:54:46Z<p>Harry Gensler. <a href="https://www.amazon.com/Introduction-Logic-Harry-J-Gensler/dp/1138910597"><em>Introduction to Logic</em></a> (2017 3 ed). p 169.</p>
<blockquote>
<p>“Unless” is also equ...https://math.codidact.com/posts/278629Why does the decimal expansion of $1/(10n - 1)$ have this neat pattern?Bahudari Ragamhttps://math.codidact.com/users/535162020-10-19T19:07:53Z2020-11-09T19:10:15Z<p>I was playing around with the reciprocals of some positive integers and found these interesting patterns:</p>
<p>$$
\frac{1}{19} = 0.\overline{052631578947368421}
$$</p>
<p>Now, this repeating d...https://math.codidact.com/posts/278558When and how should the $Z_n^2$ statistic be used?HDE 226868https://math.codidact.com/users/363722020-10-16T20:10:54Z2020-11-02T02:28:21Z<p>I was reading a paper (<a href="https://arxiv.org/abs/2010.06638">Kuechel et al. 2020</a>, since retracted; see <a href="https://arxiv.org/abs/2010.06638v1">original version</a>) claiming a dete...https://math.codidact.com/posts/278871Proving the Equation of the Pencil of Two Intersecting CirclesTripleFaulthttps://math.codidact.com/users/535432020-10-29T11:44:33Z2020-10-29T12:53:15Z<p>Given two circles algebraically represented by the equation</p>
<p>$$
x^2 + y^2 + 2g_{1}x + 2f_{1}y + c_{1} = 0
$$
$$
x^2 + y^2 + 2g_{2}x + 2f_{2}y + c_{2} = 0
$$</p>
<p>in this post I will be t...https://math.codidact.com/posts/278141The probability distribution of rolling $n$ dice and keeping $k$ highesttommihttps://math.codidact.com/users/534072020-09-19T17:57:38Z2020-10-25T02:56:32Z<p>In many roleplaying games one rolls a handful of dice and calculates their sum. In some games there are bonus or penalty dice, so that we roll, for example, 4 dice with six sides and take the su...https://math.codidact.com/posts/278696Equation of a line given some parametrized points on it and an areamsh210https://math.codidact.com/users/81062020-10-21T13:56:43Z2020-10-21T13:56:49Z<p>My kid has this question for homework:</p>
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<p>A line passing through the point $(6,6)$ crosses the line $y=2x$ at the point $A$, and the $x$-axis at the point $B$ (the $x$-coordinat...https://math.codidact.com/posts/278647Method to calculate the area of a cyclic n-gonRazetimehttps://math.codidact.com/users/533102020-10-20T08:40:27Z2020-10-20T09:15:01Z<p>The area of a cyclic quadriateral, given the side lengths, is given as follows:
$$
\sqrt{(s-a)(s-b)(s-c)(s-d)}
$$</p>
<p>(Brahmagupta's formula)</p>
<p>Where $
s = \frac{a+b+c+d}{2}.
$</p>
<p>I ...https://math.codidact.com/posts/278545Using maximum of forecasted values to forecast maximumGuilherme Costahttps://math.codidact.com/users/534532020-10-16T08:57:55Z2020-10-18T07:06:15Z<p>I am using an algortihm to generate a daily sales forecast and have concluded that the forecast is, for pratical purposes, of good enough quality ("low" wMAPE).</p>
<p>In general, and ...https://math.codidact.com/posts/278553Which of the multiple definitions of correlation to use?Mithrandir24601https://math.codidact.com/users/81622020-10-16T16:21:29Z2020-10-17T15:32:48Z<p>In optics<sup class="footnote-ref"><a href="#fn1" id="fnref1">1</a></sup>, we have a notion of <a href="https://en.wikipedia.org/wiki/Degree_of_coherence#Degree_of_first-order_coherence">coheren...https://math.codidact.com/posts/278464Discrete data: Graphs and skewnessNezar https://math.codidact.com/users/534712020-10-11T13:49:32Z2020-10-15T10:49:47Z<p>I am studying an introductory course in statistic " Essentials of Statistics".
The author mentioned that Histograms are used to represent the frequency distribution of a contiuous data...https://math.codidact.com/posts/278430Existence of a set of all setsMathPhysicshttps://math.codidact.com/users/532052020-10-09T08:44:55Z2020-10-10T16:55:08Z<p>Suppose that we have an axiomatic set theory having the following axiom:</p>
<p><strong>The Axiom Schema of Comprehension:</strong> Let $\mathbf{P}(x)$ be a property of $x$. For any set $A$, the...https://math.codidact.com/posts/278419Can we adapt Furstenberg's proof of the infinitude of primes to other interesting subsets of the integers?HDE 226868https://math.codidact.com/users/363722020-10-08T13:41:07Z2020-10-09T01:22:39Z<p>In a textbook, I was introduced to <a href="https://en.wikipedia.org/wiki/Furstenberg%27s_proof_of_the_infinitude_of_primes">Furstenberg's topological proof of the infinitude of primes</a>, whic...https://math.codidact.com/posts/278337Similar triangles with the same areamsh210https://math.codidact.com/users/81062020-10-02T09:50:57Z2020-10-05T02:25:00Z<p>My kid was assigned this problem:</p>
<blockquote>
<p>Given</p>
<p><img src="https://math.codidact.com/uploads/N4FMPUuM5TMaqhhKXksq66mg" alt="line segments ADB, AEC, COE, and BOE;" /></p>
<p>$\o...https://math.codidact.com/posts/278331Including Association Analysis results into existing Classification ModelGuilherme Costahttps://math.codidact.com/users/534532020-10-01T21:13:15Z2020-10-01T21:13:15Z<p>I have trained a <strong>classification model</strong> that predicts (returns a score) on how likely a customer is to buy from our company.
This model <strong>does not contain any product specif...