https://math.codidact.com/categories/41/feedNew Posts - Q&A - MathematicsMathematics - Codidact2023-01-31T08:06:30Zhttps://math.codidact.com/posts/287819Finding a quadratic that has solutions to given valuesbrahhttps://math.codidact.com/users/589082023-01-30T20:58:04Z2023-01-31T08:06:30Z<p>Hi, i wanted to see if there is an algorithm for finding/approximating a quadratic formula that resulted in (at least one) wanted value</p>
<p>i remember finding a math.stackexchange post long ...https://math.codidact.com/posts/287419What is the probability that the convex hull of $n$ randomly distributed points has $l$ of the points on its boundary?sirichttps://math.codidact.com/users/578142022-11-18T16:23:46Z2023-01-31T08:06:14Z<p>Consider a square in which $n$ points are uniformly randomly distributed. Now consider the convex hull of these points. The "length" of the convex hull is defined as the number of points in the ...https://math.codidact.com/posts/287787Prove that 49 is the only prime square to be followed by twice a prime square and then a semiprimePeter Taylorhttps://math.codidact.com/users/363562023-01-25T15:19:06Z2023-01-25T15:20:17Z<p>Let $\tau(n)$ denote the number of divisors of $n$. OEIS sequence <a href="https://oeis.org/A309981">A309981</a> gives the smallest $k$ such that the tuple $(\tau(n), \tau(n+1), \ldots, \tau(n+k...https://math.codidact.com/posts/286961Is $f(x)=\sin(x)$ the unique function satisfying $f'(0)=1$ and $f^{(n)}(\Bbb R)\subset [-1,1]$ for all $n=0,1,\ldots$?Grovehttps://math.codidact.com/users/534522022-09-06T15:22:23Z2023-01-25T08:14:28Z<blockquote>
<p><strong>Question.</strong> Is there a function $f:\Bbb R \to \Bbb R$ with $f'(0)=1$ and $f^{(n)}(x)\in [-1,1]$ for all $n=0,1,\ldots$ and $x\in \Bbb R$, other than $f(x)=\sin(x)$?<...https://math.codidact.com/posts/287762Is there a "regular" quasi-convex function $f:\Bbb R^2 \to \Bbb R$ that is not a monotone transformation of any convex function?Pavel Kocourekhttps://math.codidact.com/users/588182023-01-23T14:58:24Z2023-01-23T14:58:24Z<h3>Question</h3>
<blockquote>
<p>Can you find an example of a differentiable quasi-convex function $f:\Bbb R^2 \to \Bbb R$ that is <em>non-degenerate</em>, but there does not exist any strictly...https://math.codidact.com/posts/287756Is there a two variable quartic polynomial with two strict local minima and no other critical point?Pavel Kocourekhttps://math.codidact.com/users/588182023-01-21T15:22:38Z2023-01-21T15:22:38Z<blockquote>
<p>Does there exist a degree 4 polynomial $p:\Bbb R^2 \to \Bbb R$ that has two strict local minima and no other critical point?</p>
</blockquote>
<p>This is the same as <a href="htt...https://math.codidact.com/posts/287625What did James Stewart mean by "the line integral reduces to an ordinary single integral in this case" ? TextKithttps://math.codidact.com/users/536962022-12-27T01:26:37Z2023-01-16T08:17:37Z<ol>
<li>Please see the question in the title, in reference to the paragraph beside my two green question marks in the image below.</li>
<li>How do you symbolize <em>"the line integral reduces to...https://math.codidact.com/posts/287720Proving $|{\bf R}^{\bf R}|=|2^{\bf R}|$ using the Schroeder-Bernstein TheoremSnoopyhttps://math.codidact.com/users/568172023-01-14T14:38:38Z2023-01-15T11:46:31Z<p>Let $A$ be the set of <em>all</em> functions from ${\bf R}$ to ${\bf R}$ and $B$ the power set of ${\bf R}$. Then $|A|=|B|$.</p>
<p>This is a well-known result in set theory. A quick search on ...https://math.codidact.com/posts/287674Is it worth it to replace Arabic numerals with Kaktovik numerals?GoldenGoldhttps://math.codidact.com/users/586302023-01-09T05:32:49Z2023-01-09T06:48:51Z<p>is there is any advantage for <a href="https://en.wikipedia.org/wiki/Kaktovik_numerals">Kaktovik numerals</a> over <a href="https://en.wikipedia.org/wiki/Arabic_numerals">Arabic numerals</a>, if...https://math.codidact.com/posts/287634Expanding the Integration problem.Anonymoushttps://math.codidact.com/users/574732022-12-28T12:43:48Z2023-01-01T02:35:57Z<p><strong>How does this work like when you expand the integration how did it result in <em>-1/4(b+a)^2</em>?</strong>
<img alt="Question on expanding the integration" src="https://math.codidact.c...https://math.codidact.com/posts/287642What is the Name of Function for Probability of a Certain Sum on Random Die Rolls?James McLellanhttps://math.codidact.com/users/567002022-12-28T21:30:29Z2022-12-29T02:30:54Z<p>Hi.</p>
<p>I'm writing a book about using statistics for roleplaying game design and am using this equation for calculating the probability of rolling a particular sum "n" on "z" throws of an "...https://math.codidact.com/posts/287410Find all integer solutions for $a_1(a_1+k)(a_1+2k)...(a_1+(2k+1)d)+1 = n^2$belohhttps://math.codidact.com/users/577962022-11-17T21:58:03Z2022-12-22T10:51:07Z<p>Find all integer solution for the equation below:
$$a_1(a_1+k)(a_1+2k)...(a_1+(2k+1)d)+1 = n^2$$</p>
<p>=$$ 1+\prod_{k=0}^{2k+1} (a_1 + kd) = n^2 $$</p>
<!-- g: js, mdit -->https://math.codidact.com/posts/287562Dividing a cuboid in fourYair Randhttps://math.codidact.com/users/531262022-12-16T03:02:31Z2022-12-19T15:26:15Z<p>Suppose we have an irregular cuboid (that is, a hexahedron with six irregular quadrilateral faces) ABCDEFGH that we wish to divide into four smaller cuboids, such that three of them each fully s...https://math.codidact.com/posts/287492Example 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 $Snoopyhttps://math.codidact.com/users/568172022-11-27T19:08:16Z2022-11-27T23:35:05Z<p>In the Wikipedia article on <a href="https://en.wikipedia.org/wiki/Improper_integral#Types_of_integrals">improper integrals</a>, the function $f(x)=\frac{\sin x}{x}$ gives an example that is imp...https://math.codidact.com/posts/287484Finding the limit $ \lim_{x\to 0^+}e^{1/x}\sum_{n=\lfloor 1/x\rfloor}^\infty\frac{x^n}{n} $Snoopyhttps://math.codidact.com/users/568172022-11-27T01:26:33Z2022-11-27T19:19:42Z<blockquote>
<p>Let $\lfloor x \rfloor$ be the maximum integer $n\le x$. Find the limit
$$
\lim_{x\to 0^+}e^{1/x}\sum_{n=\lfloor 1/x\rfloor}^\infty\frac{x^n}{n}
$$</p>
</blockquote>
<hr>
<p>...https://math.codidact.com/posts/286985Given two angles of a triangle, finding an angle formed by a medianSnoopyhttps://math.codidact.com/users/568172022-09-11T21:44:35Z2022-11-21T13:53:05Z<blockquote>
<p><strong>Problem</strong>: Suppose in $\triangle ABC$, $\angle BAC = 30^\circ$ and $\angle BCA = 15^\circ$. Suppose $BM$ is a <a href="https://en.wikipedia.org/wiki/Median_(geometry...https://math.codidact.com/posts/287269Is this formula for the minimal sum correct?celtschkhttps://math.codidact.com/users/80562022-10-16T09:13:41Z2022-11-06T11:06:36Z<p>As is well known, the addition of natural numbers can be extended to the ordinal numbers in different ways. The first way is the ordinal sum, and the second is the natural or Hessenberg sum.</p>...https://math.codidact.com/posts/287291Do linear and group invariant functions allowed to go inside(?) integral operators?sepiabrownhttps://math.codidact.com/users/572592022-10-21T09:29:08Z2022-10-25T04:32:13Z<p>I am working through the book <em>Geometric Deep Learning</em> (<a href="https://arxiv.org/abs/2104.13478">https://arxiv.org/abs/2104.13478</a>) and have hit the following formula (Chapter 3.5, ...https://math.codidact.com/posts/287188How can 3/1 ≡ 1/(1/3), when left side features merely integers, but right side features a repetend? Chgg Clouhttps://math.codidact.com/users/535642022-10-09T00:23:58Z2022-10-11T12:01:12Z<p>On one hand, I know that algebraically, $\dfrac{3}1 ≡ \dfrac{1}{\color{red}{1/3}}$.</p>
<p>On the other hand, they differ in practice, not least because $\color{red}{1/3}$ contains 3 as the rep...https://math.codidact.com/posts/287192Does there exist a non-zero game such that the sum of three or more copies of it is zero?celtschkhttps://math.codidact.com/users/80562022-10-09T09:17:14Z2022-10-09T09:17:14Z<p>In combinatorial game theory, there are non-zero games $G$ with the property $G+G=0$; this is in particular true for all impartial games.</p>
<p>Now I wonder if there also exist non-zero games ...https://math.codidact.com/posts/287160Why can't we conclude the extrema property of a function from its quadratic approximation when the discriminant is zero?Snoopyhttps://math.codidact.com/users/568172022-10-05T20:58:08Z2022-10-07T17:35:17Z<p>Suppose $f:\mathbf{R}^2\to\mathbf{R}$ is a smooth function and $P=(0,0)$ is a critical point of $f$. The <a href="https://en.wikipedia.org/wiki/Second_partial_derivative_test">second-derivative ...https://math.codidact.com/posts/287174How to calculate remaining volume of a wire spoolre89jhttps://math.codidact.com/users/553482022-10-06T23:52:49Z2022-10-07T12:40:13Z<p>There are a bunch of rolls of 3d filament at my library in various degrees of emptiness/fullness. When I pick one I need to know that it likely has enough remaining for my print job.</p>
<p>Ho...https://math.codidact.com/posts/282966What's wrong with evaluating $n(n-1) \dots (n-[k-3])(n-[k-2])\color{red}{(n-[k-1])}$ at $k = 1$? DNBhttps://math.codidact.com/users/536282021-07-30T02:00:37Z2022-10-03T08:29:04Z<p>This snag arose out of <a href="https://math.codidact.com/posts/282606">this post</a>, and <a href="https://math.codidact.com/comments/thread/3555#comment-11900">these comments by r~~</a>. In th...https://math.codidact.com/posts/280118Product of empty set of elements vs. product over empty indexing set — is there any difference?The Amplitwisthttps://math.codidact.com/users/535162020-12-25T12:48:00Z2022-10-03T08:28:02Z<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/286140Finding distance to parabola's focus, given some pointsmsh210https://math.codidact.com/users/81062022-03-21T20:44:16Z2022-10-03T08:24:52Z<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/287063Show that $\forall n \in \mathbb{Z}^{+}$, $25^n \equiv 25 \bmod{100}$.Carefree Explorerhttps://math.codidact.com/users/568512022-09-24T14:21:59Z2022-10-03T08:24:13Z<blockquote>
<p>Show that $\forall n \in \mathbb{Z}^{+}$, $25^n \equiv 25 \bmod{100}$.</p>
</blockquote>
<p>This was a simple observation I made when playing around and I came up with the follow...https://math.codidact.com/posts/287002Proving that $p\mid (p+1776)$ if $p$ is a prime and $p(p+1776)$ is a perfect squareSnoopyhttps://math.codidact.com/users/568172022-09-14T18:23:50Z2022-09-15T19:21:22Z<p><strong>Problem</strong>: Suppose $p$ is a prime number and $p(p+1776)$ is a perfect square. Prove that $p\mid (p+1776)$.</p>
<p>From the assumption of the problem, $p(p+1776)=k^2$ for some pos...https://math.codidact.com/posts/286991organizing a librarymsh210https://math.codidact.com/users/81062022-09-12T15:36:11Z2022-09-13T05:58:16Z<p>Suppose you have $n>1$ books lined up on a shelf, numbered $1$ to $n$, not in the correct order, and you wish to put them in order. Here's your method: Choose a misplaced book<sup>[1]</sup> a...https://math.codidact.com/posts/286956If $\mathbf{R}$ is thought of as a vector space over $\mathbf{Q}$, what is its dimension?Snoopyhttps://math.codidact.com/users/568172022-09-04T01:50:34Z2022-09-05T05:34:32Z<p>It is known that $\mathbf{R}$, as a vector space over the field of <em>real</em> numbers, has the dimension $1$. I know that $\mathbf{Q}$ is also a field.</p>
<p><strong>Question</strong>: If $...https://math.codidact.com/posts/286926Inhomogenous differential equations in sciencesMissMulanhttps://math.codidact.com/users/541072022-08-24T17:14:46Z2022-08-24T17:15:04Z<p>A first order inhomogenous differential equation is :</p>
<p>$$ \frac{dy}{dx}+p(x)y(x) = q(x)$$</p>
<p>with $$ q(x)\neq0$$</p>
<p>I like studying applied math but I dont like pure math so im ...https://math.codidact.com/posts/286907Prove $e^x \ge x+1 \\\; \forall x \in \mathbb{R}$ using inductionCarefree Explorerhttps://math.codidact.com/users/568512022-08-20T20:14:35Z2022-08-21T06:56:53Z<blockquote>
<p>(How) can we prove $e^x \ge x+1 \; \forall x \in \mathbb{R}$ using induction (without using the derivative of $e^x$ at any stage)? Comments on my attempt are appreciated.</p>
</bl...https://math.codidact.com/posts/286848Is there a way to encode a unique arrangement of vertices of a graph with a unique short word?Carefree Explorerhttps://math.codidact.com/users/568512022-08-11T12:52:39Z2022-08-13T09:12:57Z<p>I call graphs $G_1$ and $G_2$ <em>distinct</em> iff (i) $G_1$ has a different arrangement<sup>1</sup> of vertices than $G_2$ <strong>and</strong> (ii) $G_1$ and $G_2$ have the same number of ver...https://math.codidact.com/posts/286829Show that $f(x) = \arctan\left(\frac{x}{x+1}\right) + \arctan\left(\frac{x+1}{x}\right) = \frac{\pi}{2}$Carefree Explorerhttps://math.codidact.com/users/568512022-08-02T20:32:05Z2022-08-03T05:05:08Z<blockquote>
<p>Show that $$f(x) = \arctan\left(\frac{x}{x+1}\right) + \arctan\left(\frac{x+1}{x}\right) = \frac{\pi}{2} \quad \forall x \in (-\infty, -1)\large\cup (0, \infty)$$</p>
</blockquote...https://math.codidact.com/posts/286808What's the common ratio for this geometric sequence?General Sebast1anhttps://math.codidact.com/users/541142022-07-28T07:58:04Z2022-07-28T12:49:03Z<p>Delayed learning some math, now I'm back at it.</p>
<p>Geometric sequences. Basically a sequence where it has a common ratio.</p>
<p>An example:</p>
<pre><code>Sequence = {14, 28, 56, 112}
R...https://math.codidact.com/posts/286788The gcd of powers in a gcd domainSnoopyhttps://math.codidact.com/users/568172022-07-22T18:12:11Z2022-07-25T14:30:34Z<p><strong>Question:</strong></p>
<p>$\def\gcd{\operatorname{gcd}}$Let $R$ be a <a href="https://en.wikipedia.org/wiki/GCD_domain">gcd domain</a>. Does it always hold that $\gcd(x^m,y^m)=\gcd(x,y)...https://math.codidact.com/posts/286728Isn't "any, some, or all" redundant? Why not write just "any"?Chgg Clouhttps://math.codidact.com/users/535642022-07-16T01:02:32Z2022-07-16T08:18:24Z<p>Please see the title of this post. In the following quotations, what changes — if anything — if you replace "<strong>any, some[,] or all</strong>" with just "<code>any</code>"? Don't these autho...https://math.codidact.com/posts/286709What are the 2 arithmetic means of $x + y$ and $4x - 2y$?General Sebast1anhttps://math.codidact.com/users/541142022-07-12T07:21:05Z2022-07-12T11:44:58Z<p>I'm currently learning arithmetic sequences, and I've gotten to the means. I'm answering an activity as a test to see if what I'm doing is right.</p>
<p>Here's an example through format:</p>
<...https://math.codidact.com/posts/286612Notation for nested exponentsJRNhttps://math.codidact.com/users/529962022-06-20T01:18:16Z2022-07-07T07:41:14Z<p>An expression such as $a^{b^c}$ is usually interpreted as $a^{(b^c)}$ and not as ${(a^b)}^c$. (See, for example, the Wikipedia entry for <a href="https://en.wikipedia.org/wiki/Double_exponentia...https://math.codidact.com/posts/286150equilateral triangle inscribed in an ellipsemsh210https://math.codidact.com/users/81062022-03-24T19:58:14Z2022-07-05T16:36:36Z<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/286655Do the Faber partition polynomials have integer coefficients?Peter Taylorhttps://math.codidact.com/users/363562022-06-30T17:23:47Z2022-06-30T17:58:06Z<p>The Online Encyclopedia of Integer Sequences includes <a href="https://oeis.org/A263916">A263916: Coefficients of the Faber partition polynomials</a>. Perhaps the clearest definition given is</p...https://math.codidact.com/posts/286572Endomorphisms on an entropic structure whose pointwise product is the identity automorphism - entropic idempotent structure?Prime Moverhttps://math.codidact.com/users/566792022-06-10T10:07:24Z2022-06-10T11:20:28Z<p>Context: self-study from Warner's "Modern Algebra (1965): Exercise 16.27.</p>
<blockquote>
<p>Let $\alpha$ and $\beta$ be endomorphisms of an entropic structure $(S, \odot)$ such that $\alpha ...https://math.codidact.com/posts/281006In "if and only if" proofs, why's 1 direction easier to prove than the other? Chgg Clouhttps://math.codidact.com/users/535642021-03-09T05:27:22Z2022-06-10T09:57:37Z<p><a href="https://math.stackexchange.com/q/3069488">This list</a> on Math StackExchange instantiates (biconditional) logical equivalences where one direction can be proved swimmingly, but the oth...https://math.codidact.com/posts/286561Optimising a 3 value problem (well 2 really)KalleMPhttps://math.codidact.com/users/81142022-06-06T15:01:52Z2022-06-09T14:07:49Z<p>I have tried to find a way to optimise value selections so a range of results based on 3 values will be as evenly spaced as possible as close as possible to a geometric series.</p>
<p>I have a ...https://math.codidact.com/posts/286527Notation for one-sided hypothesis testingtommihttps://math.codidact.com/users/534072022-05-30T11:36:06Z2022-05-30T11:36:06Z<p>I see the following notation for one-sided hypothesis testing:</p>
<ul>
<li>$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/286166Complex numbers in 2D, quaternions in 4D, why nothing in 3D? Fred Wamsleyhttps://math.codidact.com/users/550432022-03-27T21:22:28Z2022-05-28T06:36:52Z<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/285984How $ijk=\sqrt{1}$?deleted user#2022-02-18T15:34:12Z2022-05-27T19:32:10Z<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/286428$g(x)\xrightarrow{x\to\infty}\infty$ Implies $g'(x)\leq g^{1+\varepsilon}(x)$Udi Fogielhttps://math.codidact.com/users/564582022-05-10T21:34:01Z2022-05-19T15:41:49Z<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/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/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_{...