https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usSun, 30 Aug 2026 03:47:44 -0400https://pop.com.sg/derivative-the-tangent-line-y-x-2-4x-5-2-7.htmlhttps://pop.com.sg/derivative-the-tangent-line-y-x-2-4x-5-2-7.htmlI was just introduced to derivatives. I have a problem, Find the slope of the tangent line to the graph at the indicated point. The question was, $$y=x^2 -4x -5 ; (-2,7)$$ So the indicated ......Sun, 30 Aug 2026 07:32:41 -0400https://pop.com.sg/solving-2-2-sin-x-2-sin-x-6-0.htmlhttps://pop.com.sg/solving-2-2-sin-x-2-sin-x-6-0.htmlCan someone help me with this question. The mark scheme says its C but i have no idea why or how you get to that....Sun, 30 Aug 2026 07:30:44 -0400https://pop.com.sg/how-is-graduate-abstract-algebra-different-from-undergraduate-abstract.htmlhttps://pop.com.sg/how-is-graduate-abstract-algebra-different-from-undergraduate-abstract.htmlI am currently an undergraduate math student. (In fact, freshmen.) I know that usually abstract algebra is taught somehow late in the undergraduate course, and curious how studies of abstract alge......Sun, 30 Aug 2026 07:21:30 -0400https://pop.com.sg/characteristic-map-of-a-n-cell-in-a-cw-complex.htmlhttps://pop.com.sg/characteristic-map-of-a-n-cell-in-a-cw-complex.htmlI have a problem in understanding the purpose of the definition of a CW complex. What really would help me is to understand the following: Let $\sigma$ be a n-cell and $\Phi_\sigma:\mathbb D^n \to......Sun, 30 Aug 2026 07:11:25 -0400https://pop.com.sg/how-to-find-the-equation-of-an-ellipsoid-given-the-center-and-two-trac.htmlhttps://pop.com.sg/how-to-find-the-equation-of-an-ellipsoid-given-the-center-and-two-trac.htmlIf a center of an ellipsoid $E$ is $(1,k,0)$ and the trace of $E$ when $z=0$ is $x^2+y^2-2x-8y+16=0$ and when $x=1$ it is $-9y^2-z^2+72y=135$. Determine the equation of E So, the ellipsoid has the ......Sun, 30 Aug 2026 06:53:17 -0400https://pop.com.sg/cubing-the-expression-of-a-complex-number.htmlhttps://pop.com.sg/cubing-the-expression-of-a-complex-number.htmlCalculate the solutions to $$\left(-8-8\sqrt{3}i\right)^3$$ I would really appreciate if you could help me with this. Thanks...Sun, 30 Aug 2026 06:47:43 -0400https://pop.com.sg/if-the-lateral-surface-area-of-a-square-based-pyramid-with-equal-edges.htmlhttps://pop.com.sg/if-the-lateral-surface-area-of-a-square-based-pyramid-with-equal-edges.htmlThe lateral surface area of a square based pyramid having equal edges is $144√3$ cm^2. Find the area of base. My Attempt: let, length of base =$a$ slant height=$l$ Then, $$L.S.A. of pyramid=144......Sun, 30 Aug 2026 06:44:20 -0400https://pop.com.sg/levi-civita-and-kronecker-delta-properties.htmlhttps://pop.com.sg/levi-civita-and-kronecker-delta-properties.htmlI'm trying to grasp Levi-civita and Kronecker del notation to use when evaluating geophysical tensors, but I came across a few problems in the book I'm reading that have me stumped. 1) $\delta_{i......Sun, 30 Aug 2026 06:29:34 -0400https://pop.com.sg/what-is-the-difference-between-mapping-and-morphism.htmlhttps://pop.com.sg/what-is-the-difference-between-mapping-and-morphism.htmlI wonder if there's differences between Mapping and Morphism. Although the terms are used in different context i.e. mapping for set theory and morphism for category theory, from my understanding th......Sun, 30 Aug 2026 06:22:21 -0400https://pop.com.sg/advantage-of-first-order-logic-over-second-order-logic.htmlhttps://pop.com.sg/advantage-of-first-order-logic-over-second-order-logic.htmlAs I look over the post that has the similar question, I began to wonder: The only reason I found is that first-order logic can prove validity of some second-order logic formula/sentences, as some......Sun, 30 Aug 2026 06:16:27 -0400https://pop.com.sg/almost-sure-convergence-of-a-sequence-of-random-variables.htmlhttps://pop.com.sg/almost-sure-convergence-of-a-sequence-of-random-variables.htmlOnce again I've encountered a problem, which might not be difficult: I'm given a sequence of random variables $ (X_n) $, each with density function $g_n(x) = nx^{n-1} \textbf{1}_{(0,1]} $. I am to ...Sun, 30 Aug 2026 06:08:52 -0400https://pop.com.sg/solving-differential-equation-y-039-cos-xy.htmlhttps://pop.com.sg/solving-differential-equation-y-039-cos-xy.htmlMy question is about solving differential equation $y'=\cos(xy)$. I tried to changing variable $u=xy$ \begin{align*} u &= xy \\ \frac{du}{dx} &= y+ x\frac{dy}{dx}\\ \frac{dy}{dx} &= \fr......Sun, 30 Aug 2026 05:55:38 -0400https://pop.com.sg/on-the-definition-of-locally-connected-spaces.htmlhttps://pop.com.sg/on-the-definition-of-locally-connected-spaces.htmlI have seen two a priori different definitions of locally connected space : 1) For all point $x$, and neighborhood $V$ of $x$, there is a connected neighborhood $C$ of $x$ such that $C\subseteq V......Sun, 30 Aug 2026 05:46:12 -0400https://pop.com.sg/find-the-domain-and-range-of-the-function-f-x-y-sqrt-x-y-and-sketch-th.htmlhttps://pop.com.sg/find-the-domain-and-range-of-the-function-f-x-y-sqrt-x-y-and-sketch-th.htmlI have found the domain to be $y \geq -x$. I have found the range to be $z \geq 0$, or $[ 0, \infty )$. I'm not sure how to sketch the domain in the xy-plane. I figured it would be a straight line ...Sun, 30 Aug 2026 05:36:54 -0400https://pop.com.sg/how-to-construct-a-genus-2-surface-from-8-gon.htmlhttps://pop.com.sg/how-to-construct-a-genus-2-surface-from-8-gon.htmlI am requesting some help or reference for visualization? I am having a hard time constructing a genus 2 surface from 8-gon. May I request for some reference? Here's the construction I used from H......Sun, 30 Aug 2026 05:36:32 -0400https://pop.com.sg/working-out-the-median-of-a-beta-function.htmlhttps://pop.com.sg/working-out-the-median-of-a-beta-function.htmlI am trying to work out the median of the beta function of $\mathrm{B}(1/2,1/6)$. I have been told the answer to this is $0.9510$ but i'm unsure to get there? Is there a simple formula in order to ......Sun, 30 Aug 2026 05:27:27 -0400https://pop.com.sg/is-x-x-a-proper-or-improper-fraction.htmlhttps://pop.com.sg/is-x-x-a-proper-or-improper-fraction.htmlIf the numerator is greater than the denominator, it's improper. The other way around, it's proper. What if the numerator is the same as the denominator?...Sun, 30 Aug 2026 05:21:11 -0400https://pop.com.sg/how-to-write-k-is-equal-to-any-integer-in-symbols.htmlhttps://pop.com.sg/how-to-write-k-is-equal-to-any-integer-in-symbols.htmlHow do you state that $k$ is equal to any integer in the following? The solutions to this equation $$2\sin(3x)-1=0$$ are $$ \left\{ \begin{array}{ll} x=\dfrac{\pi}{18}+\dfrac{2\pi}{3}k\\[4......Sun, 30 Aug 2026 05:18:47 -0400https://pop.com.sg/what-is-the-isomorphic-graph-of-k4-closed.htmlhttps://pop.com.sg/what-is-the-isomorphic-graph-of-k4-closed.htmlWhat is the isomorphic graph of a k4 graph? Is every complete graph isomorphic to itself? If there are any theorems related to it, it would be highly appreciated to be pointed out here. Thanks....Sun, 30 Aug 2026 05:09:06 -0400https://pop.com.sg/expectation-of-stopping-time-w-r-t-a-brownian-motion.htmlhttps://pop.com.sg/expectation-of-stopping-time-w-r-t-a-brownian-motion.htmlHow do you take the expectation of a stopping time with respect to a Brownian motion? The specific question is: $$ \tau = \inf\{ t \ge 0: B(t) \in \{-a, b\}\} $$ I understand the optional stopping ...Sun, 30 Aug 2026 04:55:54 -0400https://pop.com.sg/partial-differential-equation-change-of-variables.htmlhttps://pop.com.sg/partial-differential-equation-change-of-variables.htmlI am given the following; $u(x,y)$ satisfes, $$2\frac{\partial ^ 2u}{\partial x^2} +3\frac{\partial ^ 2u}{\partial y^2}-7\frac{\partial ^ 2u}{\partial x \partial y}=0 $$ Use a change of variables......Sun, 30 Aug 2026 04:54:29 -0400https://pop.com.sg/how-to-take-the-third-derivative-of-this-function-without-it-getting-e.htmlhttps://pop.com.sg/how-to-take-the-third-derivative-of-this-function-without-it-getting-e.htmlI have the equation $x^2 +xy + y^3 = 1$ and I'm being asked to find $y'''(x)$ at the point where $x=1$. I calculate $f'(x) = {{-y-2x}\over{x+3y^2}}$ When I try to take the second derivative of the ...Sun, 30 Aug 2026 04:43:04 -0400https://pop.com.sg/understanding-the-definition-of-the-center-z-g-of-a-group-g.htmlhttps://pop.com.sg/understanding-the-definition-of-the-center-z-g-of-a-group-g.htmlI'm having trouble understanding the definition of "center" in group theory. My textbook says: "The center, $Z(G)$, of a group is the subset of elements in $G$ that commute with every element of......Sun, 30 Aug 2026 04:29:48 -0400https://pop.com.sg/how-do-i-tell-wolfram-alpha-that-i-want-the-fourth-derivative-of-y-in-.htmlhttps://pop.com.sg/how-do-i-tell-wolfram-alpha-that-i-want-the-fourth-derivative-of-y-in-.htmlFor example, how would I enter y^(IV) - 16y = 0? typing out fourth derivative, and putting four ' marks does not seem to work....Sun, 30 Aug 2026 04:26:07 -0400https://pop.com.sg/induction-proof-dealing-with-geometric-series-duplicate.htmlhttps://pop.com.sg/induction-proof-dealing-with-geometric-series-duplicate.html$1+r+(r^2)+...+r^n= \frac{1-r^{n+1}} {1-r}$ Any help would be appreciated in solving the geometric series....Sun, 30 Aug 2026 04:24:12 -0400https://pop.com.sg/what-does-this-notation-mean-f-u.htmlhttps://pop.com.sg/what-does-this-notation-mean-f-u.htmlIf $F$ is a function and $U$ is a set, then what does $F|U$ mean? In http://ocw.mit.edu/courses/mathematics/18-101-analysis-ii-fall-2005/lecture-notes/lecture6.pdf, the Inverse Function Theorem, ......Sun, 30 Aug 2026 04:17:46 -0400https://pop.com.sg/why-are-modular-lattices-important.htmlhttps://pop.com.sg/why-are-modular-lattices-important.htmlA lattice $(L,\leq)$ is said to be modular when $$ (\forall x,a,b\in L)\quad x \leq b \implies x \vee (a \wedge b) = (x \vee a) \wedge b, $$ where $\vee$ is the join operation, and $\wedge$ is the ......Sun, 30 Aug 2026 04:03:24 -0400https://pop.com.sg/how-to-find-the-generating-function-and-the-closed-form-for-the-genera.htmlhttps://pop.com.sg/how-to-find-the-generating-function-and-the-closed-form-for-the-genera.htmlI'm trying to find the generating function and the closed form for the generating form for this sequence: $0,1,-2,4,-8,16,-32,64...$ I've tried the following: I think it's an index shift so that......Sun, 30 Aug 2026 04:02:48 -0400https://pop.com.sg/what-is-minimum-number-of-edges-should-be-removed-from-k-6-to-get-plan.htmlhttps://pop.com.sg/what-is-minimum-number-of-edges-should-be-removed-from-k-6-to-get-plan.htmlWhat is minimum number of edges should be removed from $K_6$ to get planar graph? It is easy to show it just with picture.. but is it possible to prove it analytically?...Sun, 30 Aug 2026 04:00:21 -0400https://pop.com.sg/what-is-a-trivial-and-a-non-trivial-solution-in-terms-of-linear-algebr.htmlhttps://pop.com.sg/what-is-a-trivial-and-a-non-trivial-solution-in-terms-of-linear-algebr.htmlThe homogeneous unique solution always gives a trivial solution but trivial solution consists of zeros e.g $\{0,0,0\}$ but what if the solution is still homogeneous unique solution but not in the f......Sun, 30 Aug 2026 03:58:34 -0400https://pop.com.sg/calculating-this-riemann-sum-limit.htmlhttps://pop.com.sg/calculating-this-riemann-sum-limit.htmlCalculate the limit $$\lim_{n\to \infty} {\sum_{k=1}^{n} {\left(\frac{nk-1}{n^3}\right) \sin\frac{k}{n}}}$$ How exactly do we calculate this limit of the Riemann sum? I am never able to find what ......Sun, 30 Aug 2026 03:55:31 -0400https://pop.com.sg/coin-flip-betting-system.htmlhttps://pop.com.sg/coin-flip-betting-system.html-Read comments before posting I came up with a betting system that seems to defy logic... explain why its wrong? Here's the "logic" behind it Rule: If you flip a coin enough times(x) the number of ...Sun, 30 Aug 2026 03:53:30 -0400https://pop.com.sg/can-a-surjection-and-injection-exist-but-not-a-bijection-duplicate.htmlhttps://pop.com.sg/can-a-surjection-and-injection-exist-but-not-a-bijection-duplicate.htmlIf I there exists an injection $\phi: S_1 \to S_2$ and a surjection $\tau: S_1 \to S_2$, does there necessarily exist a bijection between sets $S_1$ and $S_2$? I'd like this to be true, but I don'......Sun, 30 Aug 2026 03:47:51 -0400https://pop.com.sg/what-is-the-riemann-sphere.htmlhttps://pop.com.sg/what-is-the-riemann-sphere.htmlReading from wikipedia I understood that Riemann Sphere is used to represent extended complex plane. But it would be great if someone could explain it in a less technical manner....Sun, 30 Aug 2026 03:42:11 -0400https://pop.com.sg/calculating-a-minimum-surface-area-of-a-box.htmlhttps://pop.com.sg/calculating-a-minimum-surface-area-of-a-box.htmlI want to calculate the minimum surface area of a (closed) box for a given volume. So let’s say I have a given volume V (e.g. V=10m^3). And I need a box where all the surface area is as minimal as ...Sun, 30 Aug 2026 03:28:00 -0400https://pop.com.sg/how-does-tree-3-grow-to-get-so-big-laymen-explanation.htmlhttps://pop.com.sg/how-does-tree-3-grow-to-get-so-big-laymen-explanation.htmlI am not a mathematician but I am interested in big numbers. I find them to be really interesting, almost god-like. I am watching a series of videos from David Metzler on YouTube. I have a basic ...Sun, 30 Aug 2026 03:19:02 -0400https://pop.com.sg/trial-solutions-for-differential-equations.htmlhttps://pop.com.sg/trial-solutions-for-differential-equations.htmlI have a question from my Differential Equations & Linear Algebra class. When you're trying to find the general solution to an nth order linear non-homogeneous differential equation, you have t......Sun, 30 Aug 2026 03:02:57 -0400https://pop.com.sg/what-is-the-difference-between-10-choose-2-and-10-choose-1-times-9-cho.htmlhttps://pop.com.sg/what-is-the-difference-between-10-choose-2-and-10-choose-1-times-9-cho.htmlI'm not able to understand the difference between these two. Don't both just give the number of ways of selecting $2$ objects from a total of $10$? Maybe the difference is that ${10 \choose 2}$ gi......Sun, 30 Aug 2026 02:54:33 -0400https://pop.com.sg/how-do-i-compute-a-double-weighted-average-in-other-words-an-average-w.htmlhttps://pop.com.sg/how-do-i-compute-a-double-weighted-average-in-other-words-an-average-w.htmlThis might be a simple question... I was wondering, how does one take a double weighted average? (honestly, don't know if that is what it is called). For example, let's say that we are working wit......Sun, 30 Aug 2026 02:49:27 -0400https://pop.com.sg/expected-number-of-coin-flips-to-get-three-heads-or-three-tails.htmlhttps://pop.com.sg/expected-number-of-coin-flips-to-get-three-heads-or-three-tails.htmlWe both play a game where we flip a coin. You win if 3 heads appear, I win if 3 tails appear. What is the expected number of flips for the game to end. The heads/tails doesn't need to be consecutive. ...Sun, 30 Aug 2026 02:47:56 -0400https://pop.com.sg/how-are-the-known-digits-of-pi-guaranteed.htmlhttps://pop.com.sg/how-are-the-known-digits-of-pi-guaranteed.htmlWhen discussing with my son a few of the many methods to calculate the digits of $\pi$ (15 yo school level), I realized that the methods I know more or less (geometric approximation, Monte Carlo and ...Sun, 30 Aug 2026 02:47:43 -0400https://pop.com.sg/proving-a-statement-using-the-definition-of-a-limit.htmlhttps://pop.com.sg/proving-a-statement-using-the-definition-of-a-limit.htmlLet $f$ be a function that is defined on a deleted neighbourhood of $x_0 ∈ \Bbb R$, such that $f(x) \ne 0$ for every $x$ on that deleted neighbourhood. Suppose that $$\lim _{x\to x_0} f(x) = \lim_{......Sun, 30 Aug 2026 02:46:56 -0400https://pop.com.sg/what-is-the-measure-of-angle-f.htmlhttps://pop.com.sg/what-is-the-measure-of-angle-f.htmlThis is an isosceles triangle with base angles being the only part filled in. I have trouble understanding that if the base angles are not like terms, then how can I solve my equation (which is angle ...Sun, 30 Aug 2026 02:45:46 -0400https://pop.com.sg/some-tips-on-3d-trig.htmlhttps://pop.com.sg/some-tips-on-3d-trig.htmlI understand this isn't a MATHS question specifically, however, sometimes I have trouble identifying when to use 3D trig, and trouble with it in general. Can some of the experienced/advanced people ...Sun, 30 Aug 2026 02:29:14 -0400https://pop.com.sg/what-is-an-indexed-set.htmlhttps://pop.com.sg/what-is-an-indexed-set.htmlI stuck on something. I am trying to understand what is said in the image I linked above. "A function I from a set Λ onto a set a is said to index the set a by Λ. The set Λ is called the index an......Sun, 30 Aug 2026 02:21:43 -0400https://pop.com.sg/question-on-the-039-hat-check-039-problem.htmlhttps://pop.com.sg/question-on-the-039-hat-check-039-problem.htmlThe famous 'Hat Check Problem' goes like this, 'n' men enter the restaurant and put their hats at the reception. Each man gets a random hat back when going back after having dinner. The goal is to ......Sun, 30 Aug 2026 02:21:34 -0400https://pop.com.sg/notation-for-elements-of-a-vector.htmlhttps://pop.com.sg/notation-for-elements-of-a-vector.htmlIf I want $x_i$ to be an arbitrary element of a vector $\vec{x}$ can I use the following notation: $x_i \in \vec{x}=[x_1\;x_2\;\cdots\;x_n]^T\in R^n$ ? And if I later want to spesify the interval o......Sun, 30 Aug 2026 02:19:25 -0400https://pop.com.sg/find-the-number-of-real-solutions-of-the-equation-x-3-1-2-2x-1-frac-1-.htmlhttps://pop.com.sg/find-the-number-of-real-solutions-of-the-equation-x-3-1-2-2x-1-frac-1-.htmlQuestion: Find the number of real solutions of the following equation: $$x^3+1=2(2x-1)^{\frac{1}{3}}$$ My Approach: Since the question was under the topic "Inverse Functions", I just tri......Sun, 30 Aug 2026 02:17:19 -0400https://pop.com.sg/relationship-between-cartesian-velocity-and-polar-velocity.htmlhttps://pop.com.sg/relationship-between-cartesian-velocity-and-polar-velocity.htmlI am trying to convert an equation from Cartesian to polar coordinates. I know for a given $x$ and $y$ Cartesian, we can get polar $r=\sqrt{x^2+y^2}$ and $\theta = \arctan(y/x)$. However, for a g......Sun, 30 Aug 2026 02:17:01 -0400https://pop.com.sg/converge-in-probability-and-big-oh-pee-1.htmlhttps://pop.com.sg/converge-in-probability-and-big-oh-pee-1.htmlI have read an econometrics textbook on "converge in distribution" and found the following sentence. \begin{equation} \text{If} \hspace{0.25 cm} Z_{N} \xrightarrow{d} Z, Z_{N} = \large{O......Sun, 30 Aug 2026 02:10:41 -0400https://pop.com.sg/how-to-define-a-vector-without-a-basis.htmlhttps://pop.com.sg/how-to-define-a-vector-without-a-basis.htmlWe know that a vector is an element of a vector space that does not depend by the basis that is chosen to represent it. But, in a finite dimensional vector space, when we want define a specific vec......Sun, 30 Aug 2026 02:09:06 -0400https://pop.com.sg/are-abelian-subgroups-normal.htmlhttps://pop.com.sg/are-abelian-subgroups-normal.htmlLet $G$ be a group, and let $H < G$ be such that all the elements of $H$ commute with each other, i.e. $H$ is abelian. Then is $H$ necessarily normal in $G$, i.e. $H \unlhd G$?...Sun, 30 Aug 2026 01:50:55 -0400https://pop.com.sg/how-to-show-an-ancillary-statistic-is-a-first-order-ancillary-statisti.htmlhttps://pop.com.sg/how-to-show-an-ancillary-statistic-is-a-first-order-ancillary-statisti.htmlA statistic $A(X)$ is first order ancillary if $\mathbb{E}_{\theta}[A(X)]$ does not depend on $\theta$ where $X \sim P_{\theta}$. Show when distribution of $A(X)$ is independent of $\theta$, $\math......Sun, 30 Aug 2026 01:50:13 -0400https://pop.com.sg/formal-definition-of-the-differential-of-a-function.htmlhttps://pop.com.sg/formal-definition-of-the-differential-of-a-function.htmlThe formal definition of the differential of a differentiable function $f: x \mapsto y=f(x)$ is that it's a two-variable function, its name is $df$ and its value is $df(x,\Delta_X) = f'(x)\cdot\Del......Sun, 30 Aug 2026 01:50:10 -0400https://pop.com.sg/prove-that-every-year-has-at-least-one-friday-the-13th.htmlhttps://pop.com.sg/prove-that-every-year-has-at-least-one-friday-the-13th.htmlEveryone knows Friday the 13th is regarded as a day of bad luck. Why does every year have at least one of this bad day? Source: 2010 ISI B Math UGB...Sun, 30 Aug 2026 01:48:20 -0400https://pop.com.sg/chain-rule-with-triple-composition.htmlhttps://pop.com.sg/chain-rule-with-triple-composition.htmlWe are supposed to apply the chain rule on the following function $f$: $$ f(x) = \sqrt{x+\sqrt{2x+\sqrt{3x}}} $$ I assumed we could rewrite this as $$ f(x) = g(h(j(x))) $$ However, I was not su......Sun, 30 Aug 2026 01:42:43 -0400https://pop.com.sg/finding-a-equation-for-a-tangent-line-to-the-curve-y-e-x-which-also-go.htmlhttps://pop.com.sg/finding-a-equation-for-a-tangent-line-to-the-curve-y-e-x-which-also-go.htmlI have an assignment Find an equation for a tangent line to the curve $y=e^x$ which also goes through the origin. However, in my formula it is asserted that the slope between at a point "p" an......Sun, 30 Aug 2026 01:37:00 -0400https://pop.com.sg/how-do-i-find-the-cumulative-distribution-function-of-a-binomial-rando.htmlhttps://pop.com.sg/how-do-i-find-the-cumulative-distribution-function-of-a-binomial-rando.htmlHow would I find the cumulative distribution function of a binomial? I know I'd have to integrate it with its given parameters but how would someone go about doing that?...Sun, 30 Aug 2026 01:34:26 -0400https://pop.com.sg/understanding-the-proof-of-sqrt-2-is-irrational-by-contradiction.htmlhttps://pop.com.sg/understanding-the-proof-of-sqrt-2-is-irrational-by-contradiction.htmlI have some difficulties in understanding the proof of "$\sqrt{2}$is irrational" by contradiction. I am reading it in 10th class(in India) Mathematics book( available online, here ) This ......Sun, 30 Aug 2026 01:33:20 -0400https://pop.com.sg/when-square-is-inscribed-how-to-identify-the-similar-triangles.htmlhttps://pop.com.sg/when-square-is-inscribed-how-to-identify-the-similar-triangles.htmlIn the picture, $ \square PQRS $ is a square inscribed in a right triangle. How can we come to conclusion that $ \triangle ABC \sim \triangle PBQ $? My understanding is that $ \triangle ABC $ and $ \...Sun, 30 Aug 2026 01:29:00 -0400https://pop.com.sg/find-adjacent-only-knowing-angle-and-opposite.htmlhttps://pop.com.sg/find-adjacent-only-knowing-angle-and-opposite.htmlCan you find the length of the adjacent side of a right triangle only knowing the length of the opposite side and the angle? If so how do you calculate it?...Sun, 30 Aug 2026 01:09:19 -0400https://pop.com.sg/backpropagation-with-linear-algebra-how.htmlhttps://pop.com.sg/backpropagation-with-linear-algebra-how.htmlI have an intresset in control theory - optimal control. But unfortunately all control theories are for linear models. That's become a very difficult issue when to implement a linear controller for a ...Sun, 30 Aug 2026 01:08:17 -0400https://pop.com.sg/basic-open-set-question.htmlhttps://pop.com.sg/basic-open-set-question.htmlIn $\mathbb{R}^2$, if I have an open set, call it $U$, and a point $u_0\in U$. Is the set $U^{'} = U - \{ u_0 \}$ is open as well? Or perhaps it is non-open & non-closed? My intuition is that it ...Sun, 30 Aug 2026 01:06:35 -0400https://pop.com.sg/calculate-the-radius-of-the-cylinder-such-that-the-tsa-is-a-max.htmlhttps://pop.com.sg/calculate-the-radius-of-the-cylinder-such-that-the-tsa-is-a-max.htmlGiven: Volume 30m^2 Diameter 6x Height H Calculate the radius of the cylinder such that the Total Surface Area is a maximum? Can someone help me understand this question and point me in the correct ...Sun, 30 Aug 2026 01:02:21 -0400https://pop.com.sg/examples-of-bijective-map-from-mathbb-r-3-rightarrow-mathbb-r.htmlhttps://pop.com.sg/examples-of-bijective-map-from-mathbb-r-3-rightarrow-mathbb-r.htmlCould any one give an example of a bijective map from $\mathbb{R}^3\rightarrow \mathbb{R}$? Thank you....Sun, 30 Aug 2026 00:58:06 -0400https://pop.com.sg/proof-of-uniqueness-of-the-bounded-linear-transformation-extended-in-t.htmlhttps://pop.com.sg/proof-of-uniqueness-of-the-bounded-linear-transformation-extended-in-t.htmlB.L.T Theorem (from Reed/Simon): Suppose $T$ is a bounded linear transformation from a normed linear space $\langle V_1, \|\cdot\|\rangle$ to a complete normed linear space $\langle V_2, \|\cdot\|\...Sun, 30 Aug 2026 00:46:07 -0400https://pop.com.sg/can-this-boolean-expression-be-simplified-any-further.htmlhttps://pop.com.sg/can-this-boolean-expression-be-simplified-any-further.htmlI have simplified a Boolean expression to $$(\lnot a \land \lnot b \land \lnot c) \lor (a \land (b \lor c)).$$ Is there any way to simplify this further, e.g. using De Morgan's or anything?...Sun, 30 Aug 2026 00:43:07 -0400https://pop.com.sg/tangentplane-of-a-hyperboloid.htmlhttps://pop.com.sg/tangentplane-of-a-hyperboloid.htmlI have a hyperboloid $H$, which is $x^2+\frac{y^2}{4}-z^2=1$. I want to show that all points $P$ on $H$, in which the tangentplane to $H$ that goes through the point $(1,4,2)$, all are in a plane.......Sun, 30 Aug 2026 00:41:02 -0400https://pop.com.sg/what-is-the-general-equation-for-rotated-ellipsoid.htmlhttps://pop.com.sg/what-is-the-general-equation-for-rotated-ellipsoid.htmlI have general equation for ellipsoid not in center: $$ \frac{(x-x_0)^2}{a^2}+\frac{(y-y_0)^2}{b^2}+\frac{(z-z_0)^2}{c^2}=1.$$ What is the equation when it's rotated based on $\alpha$(over $x$ axi......Sun, 30 Aug 2026 00:39:07 -0400https://pop.com.sg/what-is-the-derivative-of-x-y-with-respect-to-y.htmlhttps://pop.com.sg/what-is-the-derivative-of-x-y-with-respect-to-y.htmlI'm looking for the derivative of $x^y$ with respect to $y$. I have done it by taking log of both sides, how do I do it if I try to write $\log e^{x^y} = x^y$?...Sun, 30 Aug 2026 00:36:59 -0400https://pop.com.sg/how-to-calculate-average-of-sine-squared.htmlhttps://pop.com.sg/how-to-calculate-average-of-sine-squared.htmlI have problem with calculating average of sine. In my book, there is claim, that $$\langle\sin^2\omega t\rangle=\frac12$$ Now I have a question how to get this result mathematically? I know you ......Sun, 30 Aug 2026 00:36:02 -0400https://pop.com.sg/does-every-mathematical-operation-has-an-inverse-operation.htmlhttps://pop.com.sg/does-every-mathematical-operation-has-an-inverse-operation.htmlFor example, we say that the addition and subtraction are inverse operations like that does each and every mathematical operation has an inverse operation?...Sun, 30 Aug 2026 00:30:19 -0400https://pop.com.sg/what-039-s-the-differece-between-cot-x-and-arctan-x-duplicate.htmlhttps://pop.com.sg/what-039-s-the-differece-between-cot-x-and-arctan-x-duplicate.htmlI know that $\displaystyle \cot(x)=\frac{1}{\tan(x)}$ and $\space \displaystyle \arctan(x)=\tan(x)^{-1}=\frac{1}{\tan(x)}$ What is the difference between these two function? Is $\cot(x)$ the reci......Sun, 30 Aug 2026 00:24:01 -0400https://pop.com.sg/solve-the-following-equation-sqrt-x-sqrt-4x-sqrt-16x-sqrt-64x-5-sqrt-x.htmlhttps://pop.com.sg/solve-the-following-equation-sqrt-x-sqrt-4x-sqrt-16x-sqrt-64x-5-sqrt-x.htmlA past examination paper had the following question that I found interesting. I tried having a go at it but haven't come around with any solutions. How would one go about tackling it? $$\sqrt {x +......Sun, 30 Aug 2026 00:23:52 -0400https://pop.com.sg/what-is-the-fastest-way-to-multiply-and-divide-polynomials.htmlhttps://pop.com.sg/what-is-the-fastest-way-to-multiply-and-divide-polynomials.htmlI want to get faster at multiplying and dividing polynomials. Is there a way I can do it faster? For multiplying I just do it normally. But for division I have to use the box method....Sun, 30 Aug 2026 00:19:53 -0400https://pop.com.sg/how-would-you-define-twice-as-many-when-0-is-concerned.htmlhttps://pop.com.sg/how-would-you-define-twice-as-many-when-0-is-concerned.htmlConsider the statement that "$\mathscr{L}$ is defined as a language where all strings contained in it contain twice as many b's as a's." Clearly a string containing a single a will yield ......Sun, 30 Aug 2026 00:12:16 -0400https://pop.com.sg/what-are-some-applications-of-linear-approximation-in-the-real-world.htmlhttps://pop.com.sg/what-are-some-applications-of-linear-approximation-in-the-real-world.htmlWhat are examples you can give to Calculus I high school students? Here is a link to show the level linear approximations will be taught. I've found some applications by a simple google search. M......Sun, 30 Aug 2026 00:11:15 -0400https://pop.com.sg/base-and-height-of-equilateral-triangle-that-are-closest-to-integers.htmlhttps://pop.com.sg/base-and-height-of-equilateral-triangle-that-are-closest-to-integers.htmlI know that formula for the height of an equilateral triangle is height = (1/2) * √3 * base, so the height and base can never both be integers. But how could I find values that are as close as poss......Sun, 30 Aug 2026 00:08:07 -0400https://pop.com.sg/solve-sin-4x-sin-x-closed.htmlhttps://pop.com.sg/solve-sin-4x-sin-x-closed.htmlSolve $\sin(4x) = \sin(x)$ for $0 < x < 180$ and solve $\tan(4x) = \tan(x)$ for $0 < x < 2 \pi$ I think you have to use the addition formulas however I keep on getting stuck on the wo......Sun, 30 Aug 2026 00:01:16 -0400https://pop.com.sg/velocity-definite-integral.htmlhttps://pop.com.sg/velocity-definite-integral.htmlI am not sure how to approach this problem. Problem Suppose a certain object moves in a straight line with the following velocity, where $v$ is in meters per second and $t$ is in seconds: $v(t) =......Sat, 29 Aug 2026 23:56:45 -0400https://pop.com.sg/help-me-understand-the-formal-logic-notation.htmlhttps://pop.com.sg/help-me-understand-the-formal-logic-notation.htmlI'm trying to describe the predicate/proposition n>1 in formal logic notation. Apparently the answer is ∃y. y=1 $\land$ n>y. If I try to read this, it reads: There exists a y. y is 1......Sat, 29 Aug 2026 23:47:53 -0400https://pop.com.sg/why-do-you-add-1-in-counting-test-questions.htmlhttps://pop.com.sg/why-do-you-add-1-in-counting-test-questions.htmlHere's an example question from the SAT question of the day: On the last day of a one-week sale, customers numbered 149 through 201 were waited on. How many customers were waited on that day? ...Sat, 29 Aug 2026 23:37:32 -0400https://pop.com.sg/how-to-calculate-the-volume-of-an-ellipsoid-with-triple-integral.htmlhttps://pop.com.sg/how-to-calculate-the-volume-of-an-ellipsoid-with-triple-integral.htmlI'm having some troubles since this morning on an exercise. I need to find the volume of the ellipsoid defined by $\frac{x^2}{a^2} + \frac{y^2}{a^2} + \frac{z^2}{a^2} \leq 1$. So at the beginning I ...Sat, 29 Aug 2026 23:26:51 -0400https://pop.com.sg/what-is-the-probability-of-picking-a-random-number-from-1-to-100-100-t.htmlhttps://pop.com.sg/what-is-the-probability-of-picking-a-random-number-from-1-to-100-100-t.htmlSay I pick a number from 1 to 100, and repeat this process 99 more times. What is the probability that no numbers will repeat? Is it correct to say the answer is 1/100! or is there more to it?...Sat, 29 Aug 2026 23:21:34 -0400https://pop.com.sg/check-my-proof-of-0-r-q-epsilon-real-rational-duplicate.htmlhttps://pop.com.sg/check-my-proof-of-0-r-q-epsilon-real-rational-duplicate.htmlI am working on this exercise: $$ \forall \varepsilon > 0, \ \exists q \in Q \text{ where } 0 < |r - q| < \varepsilon $$ To clarify, r is a real number, q is a rational number. This is ......Sat, 29 Aug 2026 23:19:24 -0400https://pop.com.sg/disproving-a-s-s-angle-side-side-congruence-condition.htmlhttps://pop.com.sg/disproving-a-s-s-angle-side-side-congruence-condition.htmlI know that $A-S-S$$(Angle-Side-Side)$ congruence does not exist.But I cannot disprove it. Every time I draw a figure,I get two congruent triangles. My Attempt- So,we draw two lines such that $A......Sat, 29 Aug 2026 23:18:20 -0400https://pop.com.sg/converting-between-two-frames-of-reference-given-two-points.htmlhttps://pop.com.sg/converting-between-two-frames-of-reference-given-two-points.htmlI have two points ($P_1$ & $P_2$) with their coordinates given in two different frames of reference ($A$ & $B$). Given these, what I'd like to do is derive the transformation to be able to ...Sat, 29 Aug 2026 23:17:24 -0400https://pop.com.sg/euclidean-norm-of-a-vector-with-respect-to-some-basis-other-than-the-s.htmlhttps://pop.com.sg/euclidean-norm-of-a-vector-with-respect-to-some-basis-other-than-the-s.htmlWe know that if $v=a_1 e_1+\cdots+a_n e_n$ where $(e_{i})$ is the standard basis on $\mathbb{R}^{n}$ that the Euclidean norm is given by $$\|v\|^2 = a_1^2 +\cdots+a_n^2.$$ Suppose that $v$ were wri......Sat, 29 Aug 2026 23:12:56 -0400https://pop.com.sg/what-039-s-the-difference-between-list-and-sequence-as-mathematical-co.htmlhttps://pop.com.sg/what-039-s-the-difference-between-list-and-sequence-as-mathematical-co.htmlWhat distinguishes between set, multiset and list is whether the order is important or not, and whether repetitions of elements is allowed: List: order is important and repetition is allowed Multi......Sat, 29 Aug 2026 23:12:39 -0400https://pop.com.sg/cosine-similarity-vs-angular-distance.htmlhttps://pop.com.sg/cosine-similarity-vs-angular-distance.htmlWhile checking Google's Universal sentence encoder paper, I found that they mention that using a similarity based on angular distance performs better than raw cosine similarity. More specifically......Sat, 29 Aug 2026 23:04:00 -0400https://pop.com.sg/property-of-subordinate-matrix-norm-ab-leq-a-b.htmlhttps://pop.com.sg/property-of-subordinate-matrix-norm-ab-leq-a-b.htmlI do not understand why the following property for Matrix subordinate norms holds: \begin{equation} \|AB\| \leq \|A\|\|B\| \end{equation} Please explain clearly as I know that it should be shown b......Sat, 29 Aug 2026 23:02:00 -0400https://pop.com.sg/when-to-use-congruent-vs-approximately.htmlhttps://pop.com.sg/when-to-use-congruent-vs-approximately.htmlWhat is the appropriate usage of the congruent ($\cong$) vs. the approximately ($\approx$) symbols?...Sat, 29 Aug 2026 23:01:15 -0400https://pop.com.sg/thales-properties-with-similar-triangles.htmlhttps://pop.com.sg/thales-properties-with-similar-triangles.htmlLet : $\Delta ABC$ is a triangle $0<\beta<1$ We put $M$ and $N$ two points such that $AM=\beta AB$ and $AN=\beta AC$ We want to prove that $(MN)$ parallel to $(BC)$ and $MN= \beta BC$. I m......Sat, 29 Aug 2026 22:48:30 -0400https://pop.com.sg/counterclockwise-rotation-matrix.htmlhttps://pop.com.sg/counterclockwise-rotation-matrix.htmlIf I take the basis $(\vec{e_x},\vec{e_y})$ and make a rotation counterclockwise of angle $\theta$, I end up with two new vectors $(\vec{u},\vec{v})$ such that : $\vec{u} = \cos\theta \vec{e_x} + ......Sat, 29 Aug 2026 22:43:53 -0400https://pop.com.sg/riemann-sums-finding-the-lower-sum.htmlhttps://pop.com.sg/riemann-sums-finding-the-lower-sum.htmljust doing some review and I am a bit confused on how I pick the upper vs lower sum of a Riemann sum. I get for the upper sum I choose the maximum value of $f$ on $[x_{k-1},x_k]$ and the lower sum ......Sat, 29 Aug 2026 22:36:54 -0400https://pop.com.sg/how-to-calculate-matrix-rotation.htmlhttps://pop.com.sg/how-to-calculate-matrix-rotation.htmlGiven the following rotation matrix $$\left[ \begin{matrix} -1/3 & 2/3 & -2/3 \\ 2/3 & -1/3 & -2/3 \\ -2/3 & -2/3 & -1/3 \\ \end{matrix} \right]$$ ......Sat, 29 Aug 2026 22:30:20 -0400https://pop.com.sg/how-to-compute-the-gradient-of-the-norm-for-linear-least-squares.htmlhttps://pop.com.sg/how-to-compute-the-gradient-of-the-norm-for-linear-least-squares.htmlI am reading about Solution of the Linear Least Squares Problem. Given the function $$f(\theta) = \frac{1}{2} \lVert y - \Phi \theta \rVert _{2}^{2}$$ To find the minimizer we need to compute the ...Sat, 29 Aug 2026 22:26:00 -0400https://pop.com.sg/prove-the-determinant-is-the-product-of-its-diagonal-entries.htmlhttps://pop.com.sg/prove-the-determinant-is-the-product-of-its-diagonal-entries.htmlProve that the determinant of an upper triangular matrix is the product of its diagonal entries. What I have so far: We will prove this by induction for an $n$ $\times$ $n$ matrix. For the case o......Sat, 29 Aug 2026 22:20:23 -0400https://pop.com.sg/show-that-unit-circle-is-compact.htmlhttps://pop.com.sg/show-that-unit-circle-is-compact.htmlQuick question. Say we are given the unit circle $\{ (x,y)\in \mathbb{R}^2: x^2+y^2=1 \}$. Is this set compact? How can I prove that this is closed? Bounded? Do I have to take the complement of th......Sat, 29 Aug 2026 22:16:39 -0400https://pop.com.sg/derive-formula-for-number-of-cables-in-full-mesh-network.htmlhttps://pop.com.sg/derive-formula-for-number-of-cables-in-full-mesh-network.htmlI am trying to determine how they derived number of cables needed in a full mesh network According to networking books it is $\dfrac{N * (N-1)}{2}$, where N is the number of nodes. I tried drawin......Sat, 29 Aug 2026 22:14:36 -0400