https://math.codidact.com/categories/41/feed New Posts - Q&A - Mathematics Mathematics - Codidact 2023-01-31T08:06:30Z https://math.codidact.com/posts/287819 Finding a quadratic that has solutions to given values brah‭ https://math.codidact.com/users/58908 2023-01-30T20:58:04Z 2023-01-31T08:06:30Z <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/287419 What is the probability that the convex hull of $n$ randomly distributed points has $l$ of the points on its boundary? siric‭ https://math.codidact.com/users/57814 2022-11-18T16:23:46Z 2023-01-31T08:06:14Z <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/287787 Prove that 49 is the only prime square to be followed by twice a prime square and then a semiprime Peter Taylor‭ https://math.codidact.com/users/36356 2023-01-25T15:19:06Z 2023-01-25T15:20:17Z <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 $(\tau(n), \tau(n+1), \ldots, \tau(n+k... https://math.codidact.com/posts/286961 Is$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$? Grove‭ https://math.codidact.com/users/53452 2022-09-06T15:22:23Z 2023-01-25T08:14:28Z <blockquote> <p><strong>Question.</strong> Is there a function$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)$?<... https://math.codidact.com/posts/287762 Is there a "regular" quasi-convex function$f:\Bbb R^2 \to \Bbb R$that is not a monotone transformation of any convex function? Pavel Kocourek‭ https://math.codidact.com/users/58818 2023-01-23T14:58:24Z 2023-01-23T14:58:24Z <h3>Question</h3> <blockquote> <p>Can you find an example of a differentiable quasi-convex function$f:\Bbb R^2 \to \Bbb R$that is <em>non-degenerate</em>, but there does not exist any strictly... https://math.codidact.com/posts/287756 Is there a two variable quartic polynomial with two strict local minima and no other critical point? Pavel Kocourek‭ https://math.codidact.com/users/58818 2023-01-21T15:22:38Z 2023-01-21T15:22:38Z <blockquote> <p>Does there exist a degree 4 polynomial$p:\Bbb R^2 \to \Bbb R$that has two strict local minima and no other critical point?</p> </blockquote> <p>This is the same as <a href="htt... https://math.codidact.com/posts/287625 What did James Stewart mean by "the line integral reduces to an ordinary single integral in this case" ? TextKit‭ https://math.codidact.com/users/53696 2022-12-27T01:26:37Z 2023-01-16T08:17:37Z <ol> <li>Please see the question in the title, in reference to the paragraph beside my two green question marks in the image below.</li> <li>How do you symbolize <em>"the line integral reduces to... https://math.codidact.com/posts/287720 Proving$|{\bf R}^{\bf R}|=|2^{\bf R}|$using the Schroeder-Bernstein Theorem Snoopy‭ https://math.codidact.com/users/56817 2023-01-14T14:38:38Z 2023-01-15T11:46:31Z <p>Let$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|$.</p> <p>This is a well-known result in set theory. A quick search on ... https://math.codidact.com/posts/287674 Is it worth it to replace Arabic numerals with Kaktovik numerals? GoldenGold‭ https://math.codidact.com/users/58630 2023-01-09T05:32:49Z 2023-01-09T06:48:51Z <p>is there is any advantage for <a href="https://en.wikipedia.org/wiki/Kaktovik_numerals">Kaktovik numerals</a> over <a href="https://en.wikipedia.org/wiki/Arabic_numerals">Arabic numerals</a>, if... https://math.codidact.com/posts/287634 Expanding the Integration problem. Anonymous‭ https://math.codidact.com/users/57473 2022-12-28T12:43:48Z 2023-01-01T02:35:57Z <p><strong>How does this work like when you expand the integration how did it result in <em>-1/4(b+a)^2</em>?</strong> <img alt="Question on expanding the integration" src="https://math.codidact.c... https://math.codidact.com/posts/287642 What is the Name of Function for Probability of a Certain Sum on Random Die Rolls? James McLellan‭ https://math.codidact.com/users/56700 2022-12-28T21:30:29Z 2022-12-29T02:30:54Z <p>Hi.</p> <p>I'm writing a book about using statistics for roleplaying game design and am using this equation for calculating the probability of rolling a particular sum "n" on "z" throws of an "... https://math.codidact.com/posts/287410 Find all integer solutions for$a_1(a_1+k)(a_1+2k)...(a_1+(2k+1)d)+1 = n^2$beloh‭ https://math.codidact.com/users/57796 2022-11-17T21:58:03Z 2022-12-22T10:51:07Z <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/287562 Dividing a cuboid in four Yair Rand‭ https://math.codidact.com/users/53126 2022-12-16T03:02:31Z 2022-12-19T15:26:15Z <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/287492 Example 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 $Snoopy‭ https://math.codidact.com/users/56817 2022-11-27T19:08:16Z 2022-11-27T23:35:05Z <p>In the Wikipedia article on <a href="https://en.wikipedia.org/wiki/Improper_integral#Types_of_integrals">improper integrals</a>, the function$f(x)=\frac{\sin x}{x}$gives an example that is imp... https://math.codidact.com/posts/287484 Finding the limit$ \lim_{x\to 0^+}e^{1/x}\sum_{n=\lfloor 1/x\rfloor}^\infty\frac{x^n}{n} $Snoopy‭ https://math.codidact.com/users/56817 2022-11-27T01:26:33Z 2022-11-27T19:19:42Z <blockquote> <p>Let$\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/286985 Given two angles of a triangle, finding an angle formed by a median Snoopy‭ https://math.codidact.com/users/56817 2022-09-11T21:44:35Z 2022-11-21T13:53:05Z <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/287269 Is this formula for the minimal sum correct? celtschk‭ https://math.codidact.com/users/8056 2022-10-16T09:13:41Z 2022-11-06T11:06:36Z <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/287291 Do linear and group invariant functions allowed to go inside(?) integral operators? sepiabrown‭ https://math.codidact.com/users/57259 2022-10-21T09:29:08Z 2022-10-25T04:32:13Z <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/287188 How can 3/1 ≡ 1/(1/3), when left side features merely integers, but right side features a repetend? Chgg Clou‭ https://math.codidact.com/users/53564 2022-10-09T00:23:58Z 2022-10-11T12:01:12Z <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/287192 Does there exist a non-zero game such that the sum of three or more copies of it is zero? celtschk‭ https://math.codidact.com/users/8056 2022-10-09T09:17:14Z 2022-10-09T09:17:14Z <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/287160 Why can't we conclude the extrema property of a function from its quadratic approximation when the discriminant is zero? Snoopy‭ https://math.codidact.com/users/56817 2022-10-05T20:58:08Z 2022-10-07T17:35:17Z <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/287174 How to calculate remaining volume of a wire spool re89j‭ https://math.codidact.com/users/55348 2022-10-06T23:52:49Z 2022-10-07T12:40:13Z <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/282966 What's wrong with evaluating$n(n-1) \dots (n-[k-3])(n-[k-2])\color{red}{(n-[k-1])}$at$k = 1$? DNB‭ https://math.codidact.com/users/53628 2021-07-30T02:00:37Z 2022-10-03T08:29:04Z <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/280118 Product of empty set of elements vs. product over empty indexing set — is there any difference? The Amplitwist‭ https://math.codidact.com/users/53516 2020-12-25T12:48:00Z 2022-10-03T08:28:02Z <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/286140 Finding distance to parabola's focus, given some points msh210‭ https://math.codidact.com/users/8106 2022-03-21T20:44:16Z 2022-10-03T08:24:52Z <p>A high school student I know has the following problem:</p> <blockquote> <p>A parabola is given by$y^2=2px$with$p&gt;0$. The point$D$is on the parabola in the first quadrant at a distance... https://math.codidact.com/posts/287063 Show that$\forall n \in \mathbb{Z}^{+}$,$25^n \equiv 25 \bmod{100}$. Carefree Explorer‭ https://math.codidact.com/users/56851 2022-09-24T14:21:59Z 2022-10-03T08:24:13Z <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/287002 Proving that$p\mid (p+1776)$if$p$is a prime and$p(p+1776)$is a perfect square Snoopy‭ https://math.codidact.com/users/56817 2022-09-14T18:23:50Z 2022-09-15T19:21:22Z <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/286991 organizing a library msh210‭ https://math.codidact.com/users/8106 2022-09-12T15:36:11Z 2022-09-13T05:58:16Z <p>Suppose you have$n&gt;1$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></sup> a... https://math.codidact.com/posts/286956 If$\mathbf{R}$is thought of as a vector space over$\mathbf{Q}$, what is its dimension? Snoopy‭ https://math.codidact.com/users/56817 2022-09-04T01:50:34Z 2022-09-05T05:34:32Z <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/286926 Inhomogenous differential equations in sciences MissMulan‭ https://math.codidact.com/users/54107 2022-08-24T17:14:46Z 2022-08-24T17:15:04Z <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/286907 Prove $e^x \ge x+1 \\\; \forall x \in \mathbb{R}$ using induction Carefree Explorer‭ https://math.codidact.com/users/56851 2022-08-20T20:14:35Z 2022-08-21T06:56:53Z <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/286848 Is there a way to encode a unique arrangement of vertices of a graph with a unique short word? Carefree Explorer‭ https://math.codidact.com/users/56851 2022-08-11T12:52:39Z 2022-08-13T09:12:57Z <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/286829 Show that $f(x) = \arctan\left(\frac{x}{x+1}\right) + \arctan\left(\frac{x+1}{x}\right) = \frac{\pi}{2}$ Carefree Explorer‭ https://math.codidact.com/users/56851 2022-08-02T20:32:05Z 2022-08-03T05:05:08Z <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/286808 What's the common ratio for this geometric sequence? General Sebast1an‭ https://math.codidact.com/users/54114 2022-07-28T07:58:04Z 2022-07-28T12:49:03Z <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/286788 The gcd of powers in a gcd domain Snoopy‭ https://math.codidact.com/users/56817 2022-07-22T18:12:11Z 2022-07-25T14:30:34Z <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 $\gcd(x^m,y^m)=\gcd(x,y)... https://math.codidact.com/posts/286728 Isn't "any, some, or all" redundant? Why not write just "any"? Chgg Clou‭ https://math.codidact.com/users/53564 2022-07-16T01:02:32Z 2022-07-16T08:18:24Z <p>Please see the title of this post. In the following quotations, what changes — if anything — if you replace "<strong>any, some[,] or all</strong>" with just "<code>any</code>"? Don't these autho... https://math.codidact.com/posts/286709 What are the 2 arithmetic means of$x + y$and$4x - 2y$? General Sebast1an‭ https://math.codidact.com/users/54114 2022-07-12T07:21:05Z 2022-07-12T11:44:58Z <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/286612 Notation for nested exponents JRN‭ https://math.codidact.com/users/52996 2022-06-20T01:18:16Z 2022-07-07T07:41:14Z <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/286150 equilateral triangle inscribed in an ellipse msh210‭ https://math.codidact.com/users/8106 2022-03-24T19:58:14Z 2022-07-05T16:36:36Z <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/286655 Do the Faber partition polynomials have integer coefficients? Peter Taylor‭ https://math.codidact.com/users/36356 2022-06-30T17:23:47Z 2022-06-30T17:58:06Z <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/286572 Endomorphisms on an entropic structure whose pointwise product is the identity automorphism - entropic idempotent structure? Prime Mover‭ https://math.codidact.com/users/56679 2022-06-10T10:07:24Z 2022-06-10T11:20:28Z <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 $\alpha ... https://math.codidact.com/posts/281006 In "if and only if" proofs, why's 1 direction easier to prove than the other? Chgg Clou‭ https://math.codidact.com/users/53564 2021-03-09T05:27:22Z 2022-06-10T09:57:37Z <p><a href="https://math.stackexchange.com/q/3069488">This list</a> on Math StackExchange instantiates (biconditional) logical equivalences where one direction can be proved swimmingly, but the oth... https://math.codidact.com/posts/286561 Optimising a 3 value problem (well 2 really) KalleMP‭ https://math.codidact.com/users/8114 2022-06-06T15:01:52Z 2022-06-09T14:07:49Z <p>I have tried to find a way to optimise value selections so a range of results based on 3 values will be as evenly spaced as possible as close as possible to a geometric series.</p> <p>I have a ... https://math.codidact.com/posts/286527 Notation for one-sided hypothesis testing tommi‭ https://math.codidact.com/users/53407 2022-05-30T11:36:06Z 2022-05-30T11:36:06Z <p>I see the following notation for one-sided hypothesis testing:</p> <ul> <li>$H_0$:$K = 2$</li> <li>$H_1$:$K &gt; 2$</li> </ul> <p>I would find it more natural to write:</p> <ul> <li>$H_... https://math.codidact.com/posts/286166 Complex numbers in 2D, quaternions in 4D, why nothing in 3D? Fred Wamsley‭ https://math.codidact.com/users/55043 2022-03-27T21:22:28Z 2022-05-28T06:36:52Z <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/285984 How $ijk=\sqrt{1}$? deleted user # 2022-02-18T15:34:12Z 2022-05-27T19:32:10Z <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 Fogiel‭ https://math.codidact.com/users/56458 2022-05-10T21:34:01Z 2022-05-19T15:41:49Z <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/286472 Are vectrices useful for calculations as opposed to formalism? Fred Wamsley‭ https://math.codidact.com/users/55043 2022-05-17T01:22:34Z 2022-05-17T23:22:17Z <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/285432 How to intuit p = Calvin's probability of winning each game independently = $1/2 \implies$ P(Calvin wins the match) = 1/2? DNB‭ https://math.codidact.com/users/53628 2021-12-31T08:31:07Z 2022-05-09T13:01:11Z <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/286398 Matrices with rotational symmetry Peter Taylor‭ https://math.codidact.com/users/36356 2022-05-05T22:24:34Z 2022-05-05T22:46:21Z <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_{...