https://math.codidact.com/categories/41/feedNew Posts - Q&A - MathematicsMathematics - Codidact2021-01-20T08:45:21Zhttps://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?Gourimanohari 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 perpendicularGourimanohari 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/280168How can I deduce which operation removes redundacies?High school studenthttps://math.codidact.com/users/536282020-12-30T03:11:56Z2021-01-02T20:42:30Z<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/280118Product of empty set of elements vs. product over empty indexing set — is there any difference?Gourimanohari 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?Gourimanohari 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/279026Classification of topological pointsceltschkhttps://math.codidact.com/users/80562020-11-08T20:51:50Z2020-11-08T23:47:31Z<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/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...https://math.codidact.com/posts/278268Asymptotics of counting integers by prime signaturePeter Taylorhttps://math.codidact.com/users/363562020-09-28T14:30:12Z2020-09-29T07:34:22Z<p>The prime counting function $\pi(n)$ which counts the number of primes up to $n$ is well-known, and it's also fairly well-known that using a well-optimised implementation of the <a href="https:/...https://math.codidact.com/posts/278280Can we constructively find a third element of a set $X$ satisfying $X \cong X \times X$ given two distinct elements?Derek Elkinshttps://math.codidact.com/users/533982020-09-29T01:50:02Z2020-09-29T01:50:02Z<p>Consider a set $X$ such that $X \cong X\times X$. It's quite easy to prove in a classical set theory, e.g. <strong>ZF</strong>, that $X$ must be the empty set or a singleton set or an infinite s...https://math.codidact.com/posts/278140Computational hardness of the uniform halting problemuser53100https://math.codidact.com/users/531002020-09-18T17:12:41Z2020-09-20T22:17:46Z<p>How hard is the decision-problem of a plain Turing Machine halting on all inputs? I'm not sure of the terminology, but by "hard" I mean its location on the arithmetical hierarchy.</p>
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