https://math.codidact.com/categories/41/feedNew Posts - Q&A - MathematicsMathematics - Codidact2024-03-26T03:02:29Zhttps://math.codidact.com/posts/291159Why does the method of separating variables work?The Amplitwisthttps://math.codidact.com/users/535162024-03-26T03:02:29Z2024-03-26T03:02:29Z<p>One of the methods to solve a partial differential equation is to use separation of variables.
For example, consider the heat equation:
$$
u_t - a^2 u_{xx} = 0, \qquad 0 < x < L,\ 0 <...https://math.codidact.com/posts/291124Does {7,4|3} have a realization on the Klein quartic?WheatWizardhttps://math.codidact.com/users/543162024-03-23T00:07:37Z2024-03-23T00:07:37Z<p>It was pointed out to me recently that the polyhedron <a href="https://www.abstract-polytopes.com/atlas/336/208/14.html">{7,4|3}</a> has the same automorphism group as the <a href="https://en.wi...https://math.codidact.com/posts/291104Find length $MP$ in trapezoidTheCodidacter, or rather ACodidacterhttps://math.codidact.com/users/609342024-03-20T11:15:58Z2024-03-20T14:20:43Z<p>Trapezoid $ABCD$ is given such that $AB\parallel CD$ and $AD=BD$. Let $M$ be the midpoint of $AB$ and $P$ be the intersection of diagonal $AC$ with the circumcircle of $\triangle BCD$. Given tha...https://math.codidact.com/posts/291056What is an example of a pathological imbedding of a(n allowed) graph into an oriented surface?The Amplitwisthttps://math.codidact.com/users/535162024-03-12T15:30:01Z2024-03-12T16:01:49Z<p>I am reading the following paper: G. A. Jones, and D. Singerman, <a href="https://doi.org/10.1112/plms/s3-37.2.273"><em>Theory of maps on orientable surfaces</em></a>, Proc. London Math. Soc. (3...https://math.codidact.com/posts/290803What are the Peano axioms?Julius H.https://math.codidact.com/users/573372024-02-14T23:50:24Z2024-03-10T08:34:35Z<p>According to <a href="https://en.m.wikipedia.org/wiki/Peano_axioms">Wikipedia</a>,</p>
<blockquote>
<p>The nine Peano axioms contain three types of statements. The first axiom asserts the exis...https://math.codidact.com/posts/291013The Fourier transform of $1/p^3$C-Thttps://math.codidact.com/users/802482024-03-07T10:06:25Z2024-03-07T10:14:20Z<p>Take the following Fourier transform conventions
$$
\tilde{f}(\mathbf{p}) =\int f(\mathbf{x})
\ e^{-i\mathbf{p}\cdot\mathbf{x}} \ d^3x
$$
$$
f(\mathbf{x}) =\int \tilde{f}(\mathbf{p})\
e^{...https://math.codidact.com/posts/291012Besov or Triebel-Lizorkin spaces versus Lorentz spacesLL 3.14https://math.codidact.com/users/802462024-03-07T09:46:41Z2024-03-07T09:47:46Z<p>At the $0$ order of derivatives of Sobolev spaces, we find Besov spaces $\dot{B}^0_{p,q}$, Triebel Lizorkin spaces $\dot{F}^0_{p,q}$ and Lorentz spaces $L^{p,q}$, with in particular if $p≥ 2$
$...https://math.codidact.com/posts/290945What is a “first-order substructural logic that has a cut-free sequent calculus”?Julius H.https://math.codidact.com/users/573372024-02-26T08:12:45Z2024-02-26T18:25:56Z<p>Is a first order sub-structural logic a synonym for a fragment of first order logic? Ie just some restriction on syntactic allowances, so we know we have a theory whose sentences are a subset of...https://math.codidact.com/posts/290943What is the significance of the K-axiom in modal logic S5?Julius H.https://math.codidact.com/users/573372024-02-26T07:08:42Z2024-02-26T07:08:42Z<p>In normal modal logic S5, the K axiom says $\square (p \rightarrow q) \rightarrow (\square p \rightarrow \square q)$.</p>
<p>First of all, is this an abuse of notation? <a href="https://en.m.wi...https://math.codidact.com/posts/290942What is a one-variable fragment of first-order logic?Julius H.https://math.codidact.com/users/573372024-02-26T06:48:12Z2024-02-26T06:48:12Z<p>It is said the one variable fragment of first order logic is FOL restricted to sentences of one variable.</p>
<p>Does this mean in the entirety of the language, there is only a single distinct ...https://math.codidact.com/posts/290865matrix inverse of $I + A$ where $A$ is skew-symmetricTrevorhttps://math.codidact.com/users/542402024-02-19T00:12:16Z2024-02-22T05:44:24Z<p>I am looking for a formula or result for
$$(I + A)^{-1}$$
where $I$ is the identity matrix and $A$ is skew-symmetric ($A^T = -A$). I have spent a lot of time looking online and through various...https://math.codidact.com/posts/290854Explain derived categories to someone without a strong background in homological algebraJulius H.https://math.codidact.com/users/573372024-02-18T02:43:58Z2024-02-19T08:25:57Z<p><a href="https://en.m.wikipedia.org/wiki/Derived_category">Wikipedia</a> has ample information about derived categories but it is too sophisticated for me to take in.</p>
<p>The derived categor...https://math.codidact.com/posts/290864What is a dual object?Julius H.https://math.codidact.com/users/573372024-02-18T15:48:46Z2024-02-19T08:25:44Z<p>According to <a href="https://en.m.wikipedia.org/wiki/Dual_object">Wikipedia</a>,</p>
<ul>
<li>
<p>A dual object in a monoidal category is analogous to the idea of a dual vector space.</p>
<...https://math.codidact.com/posts/290814is arithmetic finitely consistent?Jasonhttps://math.codidact.com/users/782162024-02-15T15:35:30Z2024-02-15T15:35:30Z<p>i have also asked this question on MSE - <a href="https://math.stackexchange.com/questions/4863426/is-arithmetic-finitely-consistent">https://math.stackexchange.com/questions/4863426/is-arithmet...https://math.codidact.com/posts/290771Minimal non-standard number in non-standard models of PAAdityahttps://math.codidact.com/users/779112024-02-12T18:11:12Z2024-02-13T07:33:50Z<p>Excuse me, if the question sounds too naive.</p>
<p>From Gödel's incompleteness theorem we know that there would be non-standard models where the Gödel sentence would be false. These models wil...https://math.codidact.com/posts/290745A program which could derive theorems given formation rules in any modal logic?Julius H.https://math.codidact.com/users/573372024-02-10T19:46:01Z2024-02-12T19:19:15Z<p>There are multiple modal logics which have different formation rules:</p>
<p><img alt="Image_alt_text" src="https://math.codidact.com/uploads/3otvo5pyjopr24hiyyxew7ymlqa3"></p>
<p>Suppose some...https://math.codidact.com/posts/290742Concrete examples of set theorists thinking independence proofs only determine provability rather than that a statement is neither true nor false?Julius H.https://math.codidact.com/users/573372024-02-10T16:15:15Z2024-02-12T09:17:56Z<p>I’m curious to know more about this quote from a paper by Joel David Hamkins.</p>
<blockquote>
<p>The pervasive independence phenomenon in set theory is described on this view as a distraction...https://math.codidact.com/posts/290728Cyclical or “loop” fractals?Julius H.https://math.codidact.com/users/573372024-02-10T03:58:19Z2024-02-10T19:26:14Z<p>I want to ask an intuitively conceived question about “fractal loops”. However, I don’t know the mathematical definition of a fractal perfectly, off the top of my head.</p>
<p>Because you can “...https://math.codidact.com/posts/290740What are the “increasingly stable consequences of the large cardinal hierarchy”?Julius H.https://math.codidact.com/users/573372024-02-10T15:35:55Z2024-02-10T15:35:55Z<p>I would like to understand the following quote, from a <a href="https://arxiv.org/abs/1108.4223">paper</a> by Joel Hamkins:</p>
<blockquote>
<p>Adherents of the universe view often point to th...https://math.codidact.com/posts/290727Defining Bayes’s theorem from scratch in ZFCJulius H.https://math.codidact.com/users/573372024-02-09T22:58:20Z2024-02-09T22:58:20Z<p>I have tons of interrelated questions which I would like resolved in order to help me answer this <a href="https://philosophy.stackexchange.com/a/108474/56485">Philosophy SE</a> question about B...https://math.codidact.com/posts/290642Fourier transform of an $L^1$ function is uniformly continuousSnoopyhttps://math.codidact.com/users/568172024-01-26T00:06:28Z2024-01-26T13:49:02Z<p>$\def\Rbb{\mathbf{R}}$$\def\Cbb{\mathbf{C}}$$\def\intw{\int_{\Rbb^n}}$If $f\in L^1(\Rbb^n)$, denote the Fourier transform of $f$ as
$$
\hat{f}(x) = \int_{\Rbb^n}f(t)e^{-2\pi x\cdot t}\ dt
$$<...https://math.codidact.com/posts/290637Are these introductory logic textbooks wrong to teach ‘unless’ = ‘or’? 8500 Wardhttps://math.codidact.com/users/773792024-01-25T19:06:06Z2024-01-26T10:16:15Z<p><a href="https://english.stackexchange.com/users/547/colin-fine">Colin Fine</a> answered that</p>
<blockquote>
<p><a href="https://english.stackexchange.com/a/296700">Unless" does not equal "o...https://math.codidact.com/posts/290599For any real number $m$, $ \left|\sum_{n=1}^{\infty}\frac{m}{n^2+m^2}\right|<\frac{\pi}{2} $Snoopyhttps://math.codidact.com/users/568172024-01-20T13:25:32Z2024-01-20T14:57:43Z<blockquote>
<p><strong>Problem.</strong> Prove that for any real number $m$,
$$
\left|\sum_{n=1}^{\infty}\frac{m}{n^2+m^2}\right|<\frac{\pi}{2}
$$</p>
</blockquote>
<hr>
<p><strong>Notes...https://math.codidact.com/posts/290567If both $\lim_{x\to\infty}f(x)$ and $\lim_{x\to\infty}f'(x)$ exist, then $\lim_{x\to\infty}f'(x)=0$.Snoopyhttps://math.codidact.com/users/568172024-01-13T22:56:11Z2024-01-17T12:52:50Z<p><strong>Question.</strong> Let $f:\mathbf{R}\to\mathbf{R}$ be a differentiable function. If both the limits $\displaystyle \lim_{x\to\infty}f(x)$ and $\displaystyle \lim_{x\to\infty}f'(x)$ exist...https://math.codidact.com/posts/290576The derivatives of a function at a boundary pointtommihttps://math.codidact.com/users/534072024-01-15T08:11:00Z2024-01-17T01:12:44Z<p>I have a function $f \colon [0, L[ \, \to \mathbb{R}$ and I want to use the derivatives of arbitrary high orders of this function at zero. The function is defined on the half-open interval $[0, ...https://math.codidact.com/posts/289823Strange behavior in elections and pie chartsTheCodidacter, or rather ACodidacterhttps://math.codidact.com/users/609342023-09-27T15:01:14Z2024-01-16T18:14:29Z<p>So, a friend asked me <strong>the probability for a candidate to get at least 50% of the total votes</strong> in an election consisting of 5 candidates (let's pretend everyone picks at random 🙂)...https://math.codidact.com/posts/290570Using convexity in the proof of Hölder’s inequalitySnoopyhttps://math.codidact.com/users/568172024-01-14T15:38:57Z2024-01-14T15:42:11Z<p>A key fact for the algebra properties of $L^p$ spaces is <a href="https://en.wikipedia.org/wiki/H%C3%B6lder%27s_inequality">Hölder’s inequality</a>:</p>
<blockquote>
<p>Let $f \in L^p$ and $g ...https://math.codidact.com/posts/290537What does it mean by saying that $C([0,1])$ is a subset of $L^\infty([0,1])$? Snoopyhttps://math.codidact.com/users/568172024-01-10T03:13:08Z2024-01-10T03:20:14Z<blockquote>
<p><strong>Question.</strong> What does it mean by saying that $C([0,1])$ is a subset of $L^\infty([0,1])$?</p>
</blockquote>
<p><strong>Notes.</strong> This is an example of questi...https://math.codidact.com/posts/290527$\liminf (a_n+b_n) = \liminf(a_n)+\liminf(b_n)$ provided that $\lim a_n$ existsSnoopyhttps://math.codidact.com/users/568172024-01-09T01:33:42Z2024-01-09T13:27:05Z<blockquote>
<p><strong>Question.</strong> Suppose $(a_n)$ and $(b_n)$ are two sequences of real numbers such that $\displaystyle \lim_{n\to\infty}a_n=a.$ Show that
$$
\liminf_{n\to\infty}(a_n+b...https://math.codidact.com/posts/290512$\sup(A\cdot B) = (\sup A)(\sup B)$ where $A$ and $B$ bounded sets of positive real numbersSnoopyhttps://math.codidact.com/users/568172024-01-07T20:48:55Z2024-01-09T01:00:26Z<blockquote>
<p><strong>Problem.</strong> Suppose $A$ and $B$ are two subsets of positive real numbers. In addition, assume that $A$ and $B$ are both bounded. Show that
$$ (\sup A)(\sup B) = \sup...https://math.codidact.com/posts/290508Why is $ \int_0^{\infty}\left|\frac{\sin x}{x}\right|\ dx=\infty$?Snoopyhttps://math.codidact.com/users/568172024-01-07T15:51:57Z2024-01-07T16:05:42Z<blockquote>
<p><strong>Question</strong>: Why is
$$
\int_0^{\infty}\left|\frac{\sin x}{x}\right|\ dx=\infty\quad ?
$$</p>
</blockquote>
<p>There are several other ways to state the fact in t...https://math.codidact.com/posts/290492Is there a $(n_3)$ configuration which is not self-dual?WheatWizardhttps://math.codidact.com/users/543162024-01-04T03:47:14Z2024-01-05T00:33:56Z<p>Coxeter points out that for a self-dual configuration $(m_c,n_d)$ it must be that $m=n$ and $c=d$, so we may abbreviate it $(m_c)$.</p>
<p>However I'm interested in the other direction of this ...https://math.codidact.com/posts/290491Picking 20 different numbers, in the same draw $\;$ vs. $\;$ picking 10 different numbers, in 2 different draws.H7Dhttps://math.codidact.com/users/757402024-01-03T21:51:29Z2024-01-04T13:16:52Z<p><a href="https://www.olg.ca/en/lottery/play-daily-keno-encore/about.html">Daily Keno</a> lets you</p>
<blockquote>
<ul>
<li><a href="https://www.olg.ca/en/lottery/play-daily-keno-encore/about...https://math.codidact.com/posts/290196Is the nth Betti number determined by orientability?WheatWizardhttps://math.codidact.com/users/543162023-11-10T04:52:07Z2023-11-30T01:20:38Z<p>I'm interested in a proof of the following claim:</p>
<blockquote>
<p>If $M$ is a connected $n$-dimensional compact manifold then the $n$th Betti number, $\beta_n(M) = 1$ if $M$ is orientable ...https://math.codidact.com/posts/290318Is it known whether most numbers are normal or notGrovehttps://math.codidact.com/users/534522023-11-29T23:42:42Z2023-11-29T23:42:42Z<p>In a <a href="https://math.stackexchange.com/questions/4816739/can-the-occurrence-of-certain-digits-be-proven-disproven-for-any-arbitrary-irra">comment over at mathematics.stackexchange.com</a>,...https://math.codidact.com/posts/288852What is special about the 11-cell and 57-cell?WheatWizardhttps://math.codidact.com/users/543162023-07-04T13:26:46Z2023-11-29T18:00:08Z<p>Reading about the 11-cell and 57-cell I find two facts implied often:</p>
<ul>
<li>They are particularly notable among the abstract regular 4-polytopes.</li>
<li>They are related to each othe...https://math.codidact.com/posts/287972How can school children intuit why over 100, D is larger? But under 100, D% is larger?Chgg Clouhttps://math.codidact.com/users/535642023-03-22T00:13:24Z2023-11-15T10:50:48Z<h4>I can prove <a href="https://www.nickkolenda.com/psychological-pricing-strategies/">the Rule of 100</a> algebraically, below. But my school kids are hankering after intuition, and a plainer ex...https://math.codidact.com/posts/288906Is there a statistical method to identify the type of outliers shown in the pictures attached to my question?Ivan Nepomnyashchikhhttps://math.codidact.com/users/610182023-07-07T20:39:56Z2023-11-14T16:23:04Z<p>I have obtained experimental data:<img alt="Image_alt_text" src="https://math.codidact.com/uploads/aif2vdb9e66zqs137owlic7b9ik3"></p>
<p>Those "horns" (leftmost and rightmost peaks) in each ima...https://math.codidact.com/posts/290157$\left(\forall \varepsilon >0: |a-b| < \varepsilon\right) \iff a=b$ vs. $\left(\forall \varepsilon > 0: a \le b + \varepsilon \right) \iff a \le b$YorkTechhttps://math.codidact.com/users/659412023-11-01T21:33:11Z2023-11-03T08:54:20Z<p>How does $\left(\forall \varepsilon >0: |a-b| < \varepsilon\right) \iff a=b$ relate to $\left(\forall \varepsilon > 0: a \le b + \varepsilon \right) \iff a \le b$? Does one equivalence ...https://math.codidact.com/posts/290025Calculation of limitKlein Morettihttps://math.codidact.com/users/659562023-10-16T17:29:16Z2023-10-16T17:29:16Z<p>How to prove
$$\lim\limits_{n\to\infty}\frac{n^{\frac{m-1}2}\sum_{k=0}^m\left(C_n^k\right)^m}{2^{nm}}=\left(\frac2\pi\right)^{\frac m2}\sqrt{\frac\pi{2m}}.$$</p>
<!-- g: js, mdit -->https://math.codidact.com/posts/289814What is the probability density function for the tau distribution?mr Tsjolderhttps://math.codidact.com/users/643002023-09-26T10:00:53Z2023-09-27T15:18:36Z<p>The <a href="https://en.wikipedia.org/wiki/Studentized_residual#Distribution">tau-distribution</a> is typically defined in terms of the <a href="https://en.wikipedia.org/wiki/Student%27s_t-distr...https://math.codidact.com/posts/289758How can I improve contrast of red and green, to prove Reverse Triangle Inequality? Tortillahttps://math.codidact.com/users/656722023-09-20T05:48:28Z2023-09-26T09:01:49Z<p>I need to improve <a href="https://math.stackexchange.com/a/774233">this original diagram</a> to the one beneath, because this diagram reappears on standardized tests <em><strong>with different ...https://math.codidact.com/posts/289757Solely by eye, how can 16 year olds visually distinguish $\color{red}{\vec{b} - \vec{r}}$ from $\color{limegreen}{|\vec{b}| - |\vec{r}|}$ ?Tortillahttps://math.codidact.com/users/656722023-09-20T05:32:39Z2023-09-22T18:23:44Z<p>Yearly, I teach 16 year olds the diagram beneath (improvement of <a href="https://math.stackexchange.com/a/774233">this original</a>) that reappears on standardized tests <em><strong>with differ...https://math.codidact.com/posts/289760While pictorializing $|x - y| < |x + y|$, how can 1 picture simultaneously prove (Reverse) △ Inequalities, $|x-y| ≤ |x|+|y|, |x|-|y| ≤ |x-y|$? Tortillahttps://math.codidact.com/users/656722023-09-20T06:21:31Z2023-09-20T12:27:27Z<p>I shall improve <a href="https://math.codidact.com/posts/289496">this post</a>, because</p>
<ul>
<li>
<p>it overlooked Triangle Inequality, $|x + y| ≤ \color{darkgoldenrod}{|x| + |y|}$. Micha...https://math.codidact.com/posts/289742Is it possible to show that a normalised random variable has zero mean and unit variance?mr Tsjolderhttps://math.codidact.com/users/643002023-09-19T15:19:37Z2023-09-19T15:20:14Z<p>Given some random variable $X_i$, is it possible to compute expectations of the normalised value, like: $$\mathbb{E}\biggl[\frac{X_i - \bar{x}}{s}\biggr],$$
where $\bar{x} = \frac{1}{N}\sum_{j=...https://math.codidact.com/posts/28808725% probability that there was a chance of avoiding injury $\quad$ vs. $\quad$ 25% chance of avoiding injuryDNBhttps://math.codidact.com/users/536282023-05-01T06:53:22Z2023-09-16T02:39:56Z<p>I ask about merely the math behind the last sentence of footnote 71 quoted below. I quote the legalistic sentences thereinbefore for context, but they may be immaterial.</p>
<h4>How does "a 25%...https://math.codidact.com/posts/289643What is the formula for sample standard deviation of a small sample size?Ivan Nepomnyashchikhhttps://math.codidact.com/users/610182023-09-08T17:01:40Z2023-09-16T02:34:00Z<p>The formula for sample standard deviation is given by:</p>
$$s = \sqrt{\frac{\sum_{i=1}^{i=N} (x_i - \bar{x})^2}{N-1}}$$<p><strong>Am I right that when the sample size is small ($N<30$), the...https://math.codidact.com/posts/289536How can I choose a point from a uniform distribution within a regular polygon?Karl Knechtelhttps://math.codidact.com/users/646562023-08-26T14:59:15Z2023-08-27T19:37:51Z<p>Suppose I have a regular polygon in the Cartesian plane, centered at the origin with circumradius $1$, aligned with one vertex at $(1, 0)$. Given a source of independent, uniformly distributed r...https://math.codidact.com/posts/286527Notation for one-sided hypothesis testingtommihttps://math.codidact.com/users/534072022-05-30T11:36:06Z2023-08-26T06:39:06Z<p>I see the following notation for one-sided hypothesis testing:</p>
<ul>
<li>$H_0$: $K = 2$</li>
<li>$H_1$: $K > 2$</li>
</ul>
<p>I would find it more natural to write:</p>
<ul>
<li>$H_...https://math.codidact.com/posts/289532Without trial and error, how can I effortlessly deduce all $n, k_i ∈ ℕ ∋ \binom n {k_1, k_2, ..., k_n} =$ given c? Chgg Clouhttps://math.codidact.com/users/535642023-08-25T10:37:06Z2023-08-26T04:37:14Z<h4>With online or computer software, for a given $c ∈ ℕ $, how can I efficiently deduce all natural numbers that $n, k_i ∈ ℕ ∋ \dbinom n { k_1, k_2, ..., k_i} = c$ ? For example below, $i = 1, \co...