https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usSun, 06 Sep 2026 17:41:15 -0400https://pop.com.sg/convex-side-of-a-spherical-mirror.htmlhttps://pop.com.sg/convex-side-of-a-spherical-mirror.htmlA convex set has the property that if you take any two points in the set and draw the line segment connecting those two points, that line segment lies entirely in the set. My textbook says tha......Sun, 06 Sep 2026 21:39:24 -0400https://pop.com.sg/bracket-of-8-tournament.htmlhttps://pop.com.sg/bracket-of-8-tournament.htmlIf I had a bracket style tournament, how many possibilities are there for a 8 team tournament. Once a team gets eliminated, they do not play a game for the 3rd place and below....Sun, 06 Sep 2026 21:35:24 -0400https://pop.com.sg/product-of-an-m-matrix-and-an-inverse-m-matrix.htmlhttps://pop.com.sg/product-of-an-m-matrix-and-an-inverse-m-matrix.htmlThis problem comes from the book "Topics in matrix analysis" by Horn and Johnson. Let $A$ and $B$ be two $M$-matrices. Show that $B^{-1}A$ is an $M$-matrix if and only if $B^{-1}A$ is a $Z$-matrix......Sun, 06 Sep 2026 21:31:29 -0400https://pop.com.sg/what-exactly-is-the-difference-between-gateaux-derivative-and-directio.htmlhttps://pop.com.sg/what-exactly-is-the-difference-between-gateaux-derivative-and-directio.htmlThe definition of the limit looks very similar between the two derivatives. It seems that directional derivative is the "amount" of the function going in the direction of a vector (arrow), wherea......Sun, 06 Sep 2026 21:26:03 -0400https://pop.com.sg/transposing-multiple-matrices.htmlhttps://pop.com.sg/transposing-multiple-matrices.htmlSo I have a question which says the following (a) Given $A= \begin{bmatrix} 3 & 0 \\ -1 & 2 \\ 1 & 1 \end{bmatrix} $, $B= \begin{bmatrix} 4 &-1 \\ 0 & 2 \\ \end{bmat......Sun, 06 Sep 2026 21:18:00 -0400https://pop.com.sg/why-does-the-taylor-series-of-frac-1-ln-x-have-a-non-infinite-radius-o.htmlhttps://pop.com.sg/why-does-the-taylor-series-of-frac-1-ln-x-have-a-non-infinite-radius-o.htmlShouldn't the taylor series of a function be equal to that function for any input value? Why does this not work for the taylor series of $\frac {1}{\ln x}$ when $|x| \gt 1$? Edit: I do mean the ......Sun, 06 Sep 2026 21:11:01 -0400https://pop.com.sg/how-do-you-find-the-probability-of-p-a-and-b.htmlhttps://pop.com.sg/how-do-you-find-the-probability-of-p-a-and-b.htmlThe probability model has a sample space of {A,B,C} with P(A) = 0.1, P(B) = 0.8, P(C) = 0.1. I found that P(B or C) = 0.9 because the probabilities add, but I cannot figure out P(A and B)....Sun, 06 Sep 2026 21:02:08 -0400https://pop.com.sg/confusion-about-topologist-039-s-whirlpool.htmlhttps://pop.com.sg/confusion-about-topologist-039-s-whirlpool.htmlI've found a solution to the title here https://static1.squarespace.com/static/5aff705c5ffd207cc87a512d/t/5c950a447817f75bd2262096/1553271370324/Topology.pdf on page $71$ But why the arc give a ...Sun, 06 Sep 2026 20:57:59 -0400https://pop.com.sg/what-does-rightarrow-mean-in-p-rightarrow-q.htmlhttps://pop.com.sg/what-does-rightarrow-mean-in-p-rightarrow-q.htmlI was looking at an exercise where it asked the following: $$\begin{array}{ccc} p&q&p\rightarrow q \\ T&T&T \\ &\ldots \end{array}$$ So, for the third column, I just put $T$ w......Sun, 06 Sep 2026 20:54:25 -0400https://pop.com.sg/up-to-date-advice-on-the-best-way-to-take-notes-maths.htmlhttps://pop.com.sg/up-to-date-advice-on-the-best-way-to-take-notes-maths.htmlI have read some old discussions about this topic and would like to get some up-to-date advice if possible. I'm going to start university next year (maths), and I know how important is to have a se......Sun, 06 Sep 2026 20:49:07 -0400https://pop.com.sg/average-proportion-for-proportions-with-different-denominators.htmlhttps://pop.com.sg/average-proportion-for-proportions-with-different-denominators.htmlSay I have an experiment in which subjects are asked to respond to some stimulus. Their responses are transcribed and coded, first as "Valid/Invalid", and then for the valid responses, "Correct/Inc......Sun, 06 Sep 2026 20:48:37 -0400https://pop.com.sg/matlab-vector-assignment-of-arbitrary-length.htmlhttps://pop.com.sg/matlab-vector-assignment-of-arbitrary-length.htmlGreetings: let's suppose I have a loop that will execute 20 times, and within that loop I'll obtain a vector of information whose length will increase, each iteration, from 2 to 20. Now suppose I h......Sun, 06 Sep 2026 20:48:22 -0400https://pop.com.sg/why-is-compactness-so-important.htmlhttps://pop.com.sg/why-is-compactness-so-important.htmlI've read many times that 'compactness' is such an extremely important and useful concept, though it's still not very apparent why. The only theorems I've seen concerning it are the Heine-Borel the......Sun, 06 Sep 2026 20:43:52 -0400https://pop.com.sg/recursion-tree-t-n-t-n-3-t-2n-3-cn.htmlhttps://pop.com.sg/recursion-tree-t-n-t-n-3-t-2n-3-cn.htmlI have a task: Explain that by using recursion tree that solution for: $T(n)=T(\frac n3)+T(\frac {2n}{3})+cn$ Where c is constance, is $\Omega(n\lg n)$ My solution: Recursion tree for $T(n)=T(\fra......Sun, 06 Sep 2026 20:26:38 -0400https://pop.com.sg/what-are-some-measures-of-connectedness-in-graphs.htmlhttps://pop.com.sg/what-are-some-measures-of-connectedness-in-graphs.htmlI am not a mathematician (I am an engineer who is working on improving his mathematics), so I apologize in advance if my question is trivial. Consider a graph of $N$ nodes, with some defined crite......Sun, 06 Sep 2026 20:20:00 -0400https://pop.com.sg/calculating-moment-generating-function-with-normal-distribution-closed.htmlhttps://pop.com.sg/calculating-moment-generating-function-with-normal-distribution-closed.htmlMy problem is to calculate $E(e^{\lambda X})$, where X has normal distribution $N(\mu, \sigma^2)$. So I have to calculate integral as follows: $\int_{-\infty}^\infty e^{\lambda x} \frac{1}{\sqrt{2\...Sun, 06 Sep 2026 20:13:36 -0400https://pop.com.sg/confused-by-limit-superior-and-limit-inferior-definition.htmlhttps://pop.com.sg/confused-by-limit-superior-and-limit-inferior-definition.htmlI'm trying to grasp the idea behind limit superior and limit inferior. Using the concept of subsequential limits, I understand that for a sequence $(x_n)$, the $\limsup x_n$ is the largest limit th......Sun, 06 Sep 2026 20:11:12 -0400https://pop.com.sg/what-039-s-math-beyond-calculus-closed.htmlhttps://pop.com.sg/what-039-s-math-beyond-calculus-closed.htmlAs a high school student I have already completed the equivalent of Calculus 2, covering advanced material such as differential equations and L'hopital's rule, so I have seen a lot of math already.......Sun, 06 Sep 2026 20:10:55 -0400https://pop.com.sg/what-is-advanced-calculus.htmlhttps://pop.com.sg/what-is-advanced-calculus.htmlI've seen "advanced calculus" as a prerequisite to some courses. What are people referring to when they say that? Is it a particular set of topics? What are some key examples of books that cover......Sun, 06 Sep 2026 20:04:18 -0400https://pop.com.sg/what-really-is-039-039-orthogonality-039-039.htmlhttps://pop.com.sg/what-really-is-039-039-orthogonality-039-039.htmlI know that we can define two vectors to be orthogonal only if they are elements of a vector space with an inner product. So, if $\vec x$ and $\vec y$ are elements of $\mathbb{R}^n$ (as a real v......Sun, 06 Sep 2026 19:59:00 -0400https://pop.com.sg/common-perpendicular-of-two-straight-lines.htmlhttps://pop.com.sg/common-perpendicular-of-two-straight-lines.htmlI have the straight lines: $$d_1: \frac{x-1}2=\frac{y-3}1=\frac{z+2}1\\[4ex] d_2: \dfrac{x-1}1=\frac{y+2}{-4}=\frac{z-9}2$$ And I have to find the common perpendicular of these lines....Sun, 06 Sep 2026 19:56:12 -0400https://pop.com.sg/must-g-be-connected.htmlhttps://pop.com.sg/must-g-be-connected.htmlLet $G$ be a graph of order $n$. If deg $u$ $+$ deg $v$ $+$ deg $w$ $\geq n-1$ for every three pairwise nonadjacent vertices $u,v$ and $w$ of $G$, must $G$ be connected? I know that if $H$ is a gr......Sun, 06 Sep 2026 19:37:46 -0400https://pop.com.sg/probability-of-getting-off-the-bus-in-drinking-game-ride-the-bus.htmlhttps://pop.com.sg/probability-of-getting-off-the-bus-in-drinking-game-ride-the-bus.htmlThis isn't so much a question, but I'm just looking for some critique on my answer for this problem to confirm/deny if it's correct. Drinking games are different everywhere, but the way I've been t......Sun, 06 Sep 2026 19:24:49 -0400https://pop.com.sg/in-probability-does-p-a-p-ab-p-ab-039.htmlhttps://pop.com.sg/in-probability-does-p-a-p-ab-p-ab-039.htmlLet A and B be events. It seems to make sense to be that the probability that A occurs is equal to the probability that A and B both occur PLUS the probability that A occurs and B does not. I have......Sun, 06 Sep 2026 19:21:23 -0400https://pop.com.sg/proof-that-the-sample-mean-is-the-best-estimator-for-the-population-me.htmlhttps://pop.com.sg/proof-that-the-sample-mean-is-the-best-estimator-for-the-population-me.htmlI've always heard that the sample mean $\overline{X}$ is "the best estimator" for the population mean $\mu$. But is that always true regardless of the population distribution? is there any proof fo......Sun, 06 Sep 2026 19:20:05 -0400https://pop.com.sg/how-is-1-cubic-decimeter-1-liter-and-1000-cubic-centimeters-equal-to-1.htmlhttps://pop.com.sg/how-is-1-cubic-decimeter-1-liter-and-1000-cubic-centimeters-equal-to-1.htmlAm I cubing these units? If so how? My thinking? decimeter =10^-1 = 0.1, centimeter 10^-2= 0.01 If 1 liter = 1 cubic decimeter, how can it also equal 1000 cubic centimeters when the two units are......Sun, 06 Sep 2026 19:09:36 -0400https://pop.com.sg/intuition-of-the-cyclic-trace-property-of-matrix-products-and-relation.htmlhttps://pop.com.sg/intuition-of-the-cyclic-trace-property-of-matrix-products-and-relation.htmlLet $A_1,\ldots,A_n$ be square matrices (not necessarily symmetric). Taking the trace of a matrix product and a cyclic permutation of the product yields: \begin{align*} \text{trace}(A_nA_{n-1}\cdot......Sun, 06 Sep 2026 19:04:37 -0400https://pop.com.sg/questions-tagged-pseudoinverse.htmlhttps://pop.com.sg/questions-tagged-pseudoinverse.htmlQ&A for people studying math at any level and professionals in related fields...Sun, 06 Sep 2026 19:02:15 -0400https://pop.com.sg/let-x-be-a-finite-set-with-lxl-6-then-the-number-of-equivalence-relati.htmlhttps://pop.com.sg/let-x-be-a-finite-set-with-lxl-6-then-the-number-of-equivalence-relati.htmlLet X be a finite set with |X| : 6. Then the number of equivalence relations on X such that each equivalence class has at least three elements in it is: (A) 10. (B) 11. (c) 20. (D) 21. My try We know ...Sun, 06 Sep 2026 18:57:07 -0400https://pop.com.sg/partial-derivative-of-f-x-with-respect-to-x-1.htmlhttps://pop.com.sg/partial-derivative-of-f-x-with-respect-to-x-1.htmlLet $f: \Bbb R^n \to R$ be a scalar field defined by $$ f(x) = \sum_{i=1}^n \sum_{j=1}^n a_{ij} x_i x_j .$$ I want to calculate $\frac{\partial f}{\partial x_1}$. I found a brute force way of ...Sun, 06 Sep 2026 18:54:57 -0400https://pop.com.sg/what-039-s-the-point-of-functions-closed.htmlhttps://pop.com.sg/what-039-s-the-point-of-functions-closed.htmlI do not understand why functions are needed in mathematics(algebra, to be exact). They are nearly the same as regular equations. Yeah, "there should be only one output for a particular input", and......Sun, 06 Sep 2026 18:31:18 -0400https://pop.com.sg/what-is-taylor-039-s-inequality-about-duplicate.htmlhttps://pop.com.sg/what-is-taylor-039-s-inequality-about-duplicate.htmlI am confused by the Taylor's Inequality formula: $$|R_n(x)| \le \frac M{(n+1)!}|x−a|^{n+1}$$ What do $R_n$, $M$, $n$, $x$, $a$, stand for i.e. what to substitute inside each variable? What purpos......Sun, 06 Sep 2026 18:30:41 -0400https://pop.com.sg/opposite-of-disjoint.htmlhttps://pop.com.sg/opposite-of-disjoint.htmlSets whose intersection is the empty set are called disjoint. What is the opposite of a disjoint set? For example the sets $\{1,2\}$ and $\{2,3\}$ satisfy this condition. I know that you can just s......Sun, 06 Sep 2026 18:28:25 -0400https://pop.com.sg/martingale-transform-question.htmlhttps://pop.com.sg/martingale-transform-question.htmlI was reading my notes and I was having trouble understanding theorem 4.3 below. I understand essentially what it is saying, but to me its simplying stating something rather intuitive? That given $......Sun, 06 Sep 2026 18:27:55 -0400https://pop.com.sg/when-the-roulette-has-hit-5-reds-why-shouldn-039-t-i-bet-to-black.htmlhttps://pop.com.sg/when-the-roulette-has-hit-5-reds-why-shouldn-039-t-i-bet-to-black.htmlFirst some context, I'm not a mathematician, not even close (as you will soon see) I do grasp some things about it but in a need to know basis, so plain english answers are appreciated (too). I ca......Sun, 06 Sep 2026 18:26:22 -0400https://pop.com.sg/calculating-the-convolution-of-a-piecewise-function.htmlhttps://pop.com.sg/calculating-the-convolution-of-a-piecewise-function.htmlLet $$f(x) = \begin{cases} \frac{1}{2}, & \text{if $\rvert x\lvert \le 1$ } \\ 0, & \text{otherwise} \end{cases}$$ I want to calculate the convolution of $f$ with itself. I am ......Sun, 06 Sep 2026 18:26:05 -0400https://pop.com.sg/discrete-measure-and-lebesgue-measurability.htmlhttps://pop.com.sg/discrete-measure-and-lebesgue-measurability.htmlAccording to wikipedia https://en.wikipedia.org/wiki/Discrete_measure a driscrite measure is defined in the following way: Let's consider a real line $\mathbb{R}$. For some (possibly finite) seq......Sun, 06 Sep 2026 18:25:19 -0400https://pop.com.sg/approximate-piecewise-constant-function-with-continuous-function.htmlhttps://pop.com.sg/approximate-piecewise-constant-function-with-continuous-function.htmlI have a function $f(t)$ that is piecewise constant: $$ f(t) = a_i \forall t\in[t_i,t_{i+1}) $$ with $n$ values $a_0, a_1, ..., a_{n-1}$, and $n+1$ values $t_0, t_1, ..., t_n$. I want to approximat......Sun, 06 Sep 2026 18:19:59 -0400https://pop.com.sg/induction-without-a-base-case-duplicate.htmlhttps://pop.com.sg/induction-without-a-base-case-duplicate.htmlI am looking for an example where you have $P(n)$ implying $P(n+1)$. However there is no base case. For which there is therefore no solution at all for the induction problem even though the inductive ...Sun, 06 Sep 2026 18:09:49 -0400https://pop.com.sg/find-cos-alpha-beta-if-alpha-beta-are-the-roots-of-the-equation-a-cos-.htmlhttps://pop.com.sg/find-cos-alpha-beta-if-alpha-beta-are-the-roots-of-the-equation-a-cos-.htmlIf $\alpha$, $\beta$ are the roots of the equation $a\cos x+b\sin x=c$, then prove that $\cos(\alpha+\beta)=\dfrac{a^2-b^2}{a^2+b^2}$ My Attempt $$ b\sin x=c-a\cos x\implies b^2(1-\cos^2x)=c^2+a^2......Sun, 06 Sep 2026 17:46:24 -0400https://pop.com.sg/integration-of-frac-1-sin-x-cos-x.htmlhttps://pop.com.sg/integration-of-frac-1-sin-x-cos-x.htmlI'm given this $\int\frac{1}{\sin x+\cos x}dx$. My attempt, $\sin x+\cos x=R\cos (x-\alpha)$ $R\cos \alpha=1$ and $R\sin \alpha=1$ $R=\sqrt{1^2+1^2}=\sqrt{2}$, $\tan\alpha=1$ $\alpha=\frac{\p......Sun, 06 Sep 2026 17:37:37 -0400https://pop.com.sg/find-the-distance-between-two-skew-lines.htmlhttps://pop.com.sg/find-the-distance-between-two-skew-lines.htmlGiven the following parametric find the distance between the two lines: $x=2+t\;,\;\; y=1+6t\;,\;\; z=2t$ and $x=1+2s\;,\;\; y=6+14s\;,\;\; z=-1+5s$ I tried using a point on the first line calle......Sun, 06 Sep 2026 17:31:39 -0400https://pop.com.sg/solve-sec-x-tan-x.htmlhttps://pop.com.sg/solve-sec-x-tan-x.htmlI have been trying to do this problem: Solve $$\sec(x)=\tan(x),\quad 0≤x<2π$$ I started by rewriting $\sec(x)$ as $\frac{1}{\cos(x)}$. I then rewrote $\tan(x)$ to get $\frac{\sin(x)}{\cos(x)}......Sun, 06 Sep 2026 17:25:57 -0400https://pop.com.sg/history-behind-the-choice-of-letters-h-and-k-for-the-vertex-of-a-parab.htmlhttps://pop.com.sg/history-behind-the-choice-of-letters-h-and-k-for-the-vertex-of-a-parab.htmlAfter failing to find a historical explanation for usage of letters $h$ and $k$ for the vertex of a parabola in most relatively recent textbooks in anglosphere, I turn to math.SE. Is there any ...Sun, 06 Sep 2026 17:20:26 -0400https://pop.com.sg/prove-n-prime-ideal-of-mathbb-z-iff-n-is-prime-or-zero.htmlhttps://pop.com.sg/prove-n-prime-ideal-of-mathbb-z-iff-n-is-prime-or-zero.htmlprove $(n)$ prime ideal of $\mathbb{Z}$ iff $n$ is prime or zero Defintions Def of prime Ideal (n) $$ ab\in (n) \implies a\in(n) \vee b\in(n) $$ Def 1] integer n is prime if $n \neq 0,\pm 1 $ and ...Sun, 06 Sep 2026 17:01:44 -0400https://pop.com.sg/is-the-classifying-space-a-fully-faithful-functor.htmlhttps://pop.com.sg/is-the-classifying-space-a-fully-faithful-functor.htmlGiven a topological group $G$, we can form its classifying space $BG$; suppose we have chosen some specific construction, say the bar construction. $B$ is a functor - given any homomorphism $G \to ......Sun, 06 Sep 2026 16:44:31 -0400https://pop.com.sg/finding-factors-of-a-complex-polynomial.htmlhttps://pop.com.sg/finding-factors-of-a-complex-polynomial.htmlLet $P(z) = z^4-z^3+z^2+2=0$. Express $P(z)$ as a product of two real quadratic factors and hence find the other two zeros. I am given that $P(1+i)=0$, but I don't know to begin finding the other ...Sun, 06 Sep 2026 16:43:13 -0400https://pop.com.sg/oracles-for-tqbf.htmlhttps://pop.com.sg/oracles-for-tqbf.htmlI've seen this question somewhere and I've been thinking about it a lot but couldnt think of an answer. Say you have oracles A and B for the TQFB (True Quantified Boolean Formula) decision problem,......Sun, 06 Sep 2026 16:36:46 -0400https://pop.com.sg/using-l-039-hopital-rule-on-a-limit-that-does-not-exist.htmlhttps://pop.com.sg/using-l-039-hopital-rule-on-a-limit-that-does-not-exist.htmlI was trying to solve an improper integral and had to evaluate the expression below with the upper limit as infinity (limit is 0) and lower limit as 0 (the limit I'm asking about here) Here's the ...Sun, 06 Sep 2026 16:36:31 -0400https://pop.com.sg/why-are-quadrants-defined-the-way-they-are.htmlhttps://pop.com.sg/why-are-quadrants-defined-the-way-they-are.htmlI was thinking about planes and things, and suddenly wondered why quadrants are defined the way they are, the first on the top-right, and so on. I wonder if this gives us any benefit, or if any ...Sun, 06 Sep 2026 16:26:51 -0400https://pop.com.sg/modulo-calculation-with-multiple-exponents-via-crt.htmlhttps://pop.com.sg/modulo-calculation-with-multiple-exponents-via-crt.htmlI'm aware that there are already a few questions like this but unfortunately I wasn't able to find an answer yet. $$ (14^{2014)^{2014}} \pmod {60} $$ So I started off by putting the modular in : ......Sun, 06 Sep 2026 16:15:44 -0400https://pop.com.sg/how-do-i-find-the-angular-acceleration-of-a-pulley-when-there-one-obje.htmlhttps://pop.com.sg/how-do-i-find-the-angular-acceleration-of-a-pulley-when-there-one-obje.htmlThe problem is as follows: The figure from below shows two blocks joined by a rope on each end goind through a pulley. The masses of each block are as follows $m_1=5\,kg$, $m_2=4\,kg$. The ...Sun, 06 Sep 2026 16:15:30 -0400https://pop.com.sg/a-tough-differential-calculus-problem.htmlhttps://pop.com.sg/a-tough-differential-calculus-problem.htmlThis is a question I've had a lot of trouble with. I HAVE solved it, however, with a lot of trouble and with an extremely ugly calculation. So I want to ask you guys (who are probably more '...Sun, 06 Sep 2026 16:13:12 -0400https://pop.com.sg/constant-of-integration-positive-or-negative.htmlhttps://pop.com.sg/constant-of-integration-positive-or-negative.htmlI was solving some equation related to mechanics ... anyways I reached this step: $\int dx=-10\int 1+\frac{20}{v-20}dv$ It's correct according to the answer key of the book. So the next step is: ......Sun, 06 Sep 2026 16:03:35 -0400https://pop.com.sg/deriving-the-normal-equation-used-to-solve-least-squares-problems.htmlhttps://pop.com.sg/deriving-the-normal-equation-used-to-solve-least-squares-problems.htmlI am studying for my linear algebra final and it was suggested to me that I learn how to derive the normal equation used in solving least squares problems. I have been looking in various places onl......Sun, 06 Sep 2026 15:59:48 -0400https://pop.com.sg/prove-that-the-gradient-of-a-unit-vector-equals-2-magnitude-of-the-vec.htmlhttps://pop.com.sg/prove-that-the-gradient-of-a-unit-vector-equals-2-magnitude-of-the-vec.htmlLet $\vec r=(x,y,z)$ Firstly find $\vec \nabla (\frac 1 r)$ where r is the magnitude of $\vec r$. I think I've done this correctly to get $-x(x^2+y^2+z^2)^{-\frac32} \hat i-y(x^2+y^2+z^2)^{-\frac32......Sun, 06 Sep 2026 15:48:56 -0400https://pop.com.sg/line-ellipsoid-intersection.htmlhttps://pop.com.sg/line-ellipsoid-intersection.htmlI am trying to determine whether a line and an ellipsoid (in general, inc. rotations) intersect. Following a similar argument to the Line-sphere intersection using vector notation I have construc......Sun, 06 Sep 2026 15:39:29 -0400https://pop.com.sg/problem-with-solving-equation-with-hyperbolic-functions-2-cosh-2x-sinh.htmlhttps://pop.com.sg/problem-with-solving-equation-with-hyperbolic-functions-2-cosh-2x-sinh.htmlI want to solve the following equation for real values of $x$, by substituting the exponential forms of the hyperbolic functions. \begin{equation} 2\cosh(2x) - \sinh(2x) = 2 \end{equation} If someone ...Sun, 06 Sep 2026 15:29:15 -0400https://pop.com.sg/is-there-any-inequality-between-2-norm-condition-number-and-frobenius-.htmlhttps://pop.com.sg/is-there-any-inequality-between-2-norm-condition-number-and-frobenius-.htmlWhat I have found in [1] Condition number inequality between Frobenius norm and 2-norm for square matrix, Consider a full rank matrix $X \in \mathbb{C}^{n \times m}$, $m=n$, then we can have, $$......Sun, 06 Sep 2026 15:11:41 -0400https://pop.com.sg/deriving-the-kinematic-equation-v-2-v-0-2-2a-x-x-0.htmlhttps://pop.com.sg/deriving-the-kinematic-equation-v-2-v-0-2-2a-x-x-0.htmlI have a question on deriving the kinematic equation $v^2=v_{0}^2+2a(x-x_{0})$ from first principles and the known kinematic equations. Is this simply differentiating, but with respect to time($t$)......Sun, 06 Sep 2026 14:59:22 -0400https://pop.com.sg/gradient-of-a-matrix.htmlhttps://pop.com.sg/gradient-of-a-matrix.htmlI was following Stephen Boyd's convex optimisation course and came across the following slide: Can somebody explain to me how the gradient was calculated for the quadratic and least-squares object......Sun, 06 Sep 2026 14:59:03 -0400https://pop.com.sg/the-volume-of-sphere-using-integrals.htmlhttps://pop.com.sg/the-volume-of-sphere-using-integrals.htmlIn spherical coordinate system I have the volume element $$dV=r^{2}\sin(\theta)\ d\theta\ d\varphi\ dr$$ I want to calculate the volume for the radius equal to $R$. I calculate the integral: $$\int......Sun, 06 Sep 2026 14:55:28 -0400https://pop.com.sg/infinimum-of-a-set.htmlhttps://pop.com.sg/infinimum-of-a-set.htmlGiven the following conditions: $x \in \Bbb R$ and $y\in (0,1)$ I was asked to prove that inf $ |x-y |=0 $ My Attempt: By the elementary properties of the modulus function , we know that $ ......Sun, 06 Sep 2026 14:47:34 -0400https://pop.com.sg/does-sin-2-x-cos-2-x-1-2-cos-2-x.htmlhttps://pop.com.sg/does-sin-2-x-cos-2-x-1-2-cos-2-x.htmlI am finishing a proof. It seems like I can use $\cos^2 + \sin^2 = 1$ to figure this out, but I just can't see how it works. So I've got two questions. Does $\sin^2 x - \cos^2 x = 1-2\cos^2 x$......Sun, 06 Sep 2026 14:44:13 -0400https://pop.com.sg/differentiating-an-inner-product.htmlhttps://pop.com.sg/differentiating-an-inner-product.htmlIf $(V, \langle \cdot, \cdot \rangle)$ is a finite-dimensional inner product space and $f,g : \mathbb{R} \longrightarrow V$ are differentiable functions, a straightforward calculation with components ...Sun, 06 Sep 2026 14:42:16 -0400https://pop.com.sg/find-the-arc-length-of-the-parametric-curve.htmlhttps://pop.com.sg/find-the-arc-length-of-the-parametric-curve.html$$x=6t-6sint$$ $$y=6-6cost$$ Find the arc length of the parametric curve $$Arc length = \int_{0}^{2\pi} \sqrt{(6-6cost)^2+(6sint)^2}dt\\ =\int_{0}^{2\pi} \sqrt{36-72cost+36cos^2t+36sin^2t}dt \\ =\......Sun, 06 Sep 2026 14:41:56 -0400https://pop.com.sg/write-an-equation-for-a-rational-function-with.htmlhttps://pop.com.sg/write-an-equation-for-a-rational-function-with.htmlWrite an equation for a rational function with: Vertical asymptotes at $x = -5$ and $x = 5$ $x$ intercepts at $x = -2$ and $x = -6$ Horizontal asymptote at $y = 6$ $y =$ ? I have $$\frac{(x+2)......Sun, 06 Sep 2026 14:33:10 -0400https://pop.com.sg/minimum-spanning-tree-edge-count.htmlhttps://pop.com.sg/minimum-spanning-tree-edge-count.htmlGiven is a weighted complete graph where every weigth is a positive ineger. Let n be the amount of vertices. I have to prove that the number of edges of a minimum spanning tree of that graph is eq......Sun, 06 Sep 2026 14:25:50 -0400https://pop.com.sg/prove-that-unit-impulse-function-integral-is-equal-to-one-duplicate.htmlhttps://pop.com.sg/prove-that-unit-impulse-function-integral-is-equal-to-one-duplicate.htmlUnit impulse function is one of the special functions which is widely used in the field of signal processing. It has nice properties that helps in some situations specially its sifting property. Bu......Sun, 06 Sep 2026 14:24:34 -0400https://pop.com.sg/order-on-eigenvalues-on-diagonal-matrix.htmlhttps://pop.com.sg/order-on-eigenvalues-on-diagonal-matrix.htmlIf the eigenvalues are say $-1$, $-1$ and $2$ for a $3$ x $3$ matrix, then when comes to the diagonal matrix, is it (from top left, to bottom right) $-1$ $-1$ $2$ or $2$ $-1$ $-1$ or $-1$ $2$ $-1$?......Sun, 06 Sep 2026 14:21:09 -0400https://pop.com.sg/example-of-a-not-recursively-enumerable-set-a-subseteq-mathbb-n.htmlhttps://pop.com.sg/example-of-a-not-recursively-enumerable-set-a-subseteq-mathbb-n.htmlCan someone give me an example if a not recursively enumerable set $A \subseteq \mathbb{N}$ ? I came up with this question, when trying to show, that there exist partial functions $f: \mathbb{N} \...Sun, 06 Sep 2026 14:18:59 -0400https://pop.com.sg/how-can-i-finish-integrating-int-sqrt-x-2-49-over-x-using-trig-substit.htmlhttps://pop.com.sg/how-can-i-finish-integrating-int-sqrt-x-2-49-over-x-using-trig-substit.html$$\int {\sqrt{x^2-49} \over x}\,dx $$ $$ x = 7\sec\theta$$ $$ dx = 7\tan\theta \sec\theta \,d\theta$$ $$\int {\sqrt{7^2\sec^2\theta - 7^2} \over 7\sec\theta}\left(7\tan\theta \sec\theta \,d\theta\r......Sun, 06 Sep 2026 14:12:17 -0400https://pop.com.sg/what-is-a-residue.htmlhttps://pop.com.sg/what-is-a-residue.htmlI've heard of residues in complex analysis, contour integration, etc. but all I really know it to be is the $c_{-1}$ term in the Laurent series for a function. Is there some sort of intuition on wh......Sun, 06 Sep 2026 13:58:47 -0400https://pop.com.sg/difference-between-parallel-and-equal-lines.htmlhttps://pop.com.sg/difference-between-parallel-and-equal-lines.htmlBasically I want to know that when did a pair of parallel lines become equal. And with the above please tell me the difference between the two...Sun, 06 Sep 2026 13:58:03 -0400https://pop.com.sg/why-can-039-t-i-use-product-rule-to-derive-x-ln-3.htmlhttps://pop.com.sg/why-can-039-t-i-use-product-rule-to-derive-x-ln-3.htmlThe product rule is defined as $$(f \cdot g)' = f' \cdot g + g' \cdot f.$$ I have the following function $u(x) = x\cdot \ln(3)$. I understand that you can derive it by implicit differentiation and......Sun, 06 Sep 2026 13:56:08 -0400https://pop.com.sg/volume-of-solid-rotated-about-y-axis.htmlhttps://pop.com.sg/volume-of-solid-rotated-about-y-axis.htmlFind the volume $V$ of the solid obtained by rotating the region bounded by the given curves about the specified line. $y = \ln{7x}$; $y = 3$ $y = 4$ $x = 0$ Rotated about the $Y$-Axis. How ......Sun, 06 Sep 2026 13:56:06 -0400https://pop.com.sg/understanding-the-musical-isomorphisms-in-vector-spaces.htmlhttps://pop.com.sg/understanding-the-musical-isomorphisms-in-vector-spaces.htmlI am trying to solidify my understanding of the muscial isomorphisms in the context of vector spaces. I believe I understand the definitions but would appreciate corrections if my understanding is ......Sun, 06 Sep 2026 13:47:18 -0400https://pop.com.sg/power-series-and-intervals-of-convergence.htmlhttps://pop.com.sg/power-series-and-intervals-of-convergence.htmlWhile teaching power series in my AP Calculus class, an interesting question came up. I gave an answer that I believe is correct, but I'm not certain why it is correct except "by definition", but ......Sun, 06 Sep 2026 13:45:42 -0400https://pop.com.sg/cross-product-of-two-sums-of-three-vectors.htmlhttps://pop.com.sg/cross-product-of-two-sums-of-three-vectors.htmlThis seems like a simple question but I couldn't find anywhere to verify Is it true that: $$(\mathbf{a}+\mathbf{b}+\mathbf{c})\times (\mathbf{d}+\mathbf{e}+\mathbf{f}) = (\mathbf{a}\times \mathbf......Sun, 06 Sep 2026 13:44:07 -0400https://pop.com.sg/solving-an-8th-degree-polynomial.htmlhttps://pop.com.sg/solving-an-8th-degree-polynomial.htmlI know that through the Abel Ruffini Theorem the general solution to a polynomial of degree five or more cannot be found explicitly. But are there are any other ways to find the roots of such a ...Sun, 06 Sep 2026 13:35:38 -0400https://pop.com.sg/how-to-find-a-basis-for-r-3-which-contains-a-basis-of-im-c.htmlhttps://pop.com.sg/how-to-find-a-basis-for-r-3-which-contains-a-basis-of-im-c.htmlFind a basis for $R^3$ which contains a basis of $im(C)$ (image of C), where $$C=\begin{pmatrix}1 & 2 & 3&4\\\ 2 & -4 & 6& -2\\ -1 & 2 & -3 &1 \end{pmatrix}$$......Sun, 06 Sep 2026 13:35:18 -0400https://pop.com.sg/how-many-faces-of-a-solid-can-one-see.htmlhttps://pop.com.sg/how-many-faces-of-a-solid-can-one-see.htmlWhat is the maximum number of faces of totally convex solid that one can "see" from a point? ...and, more importantly, how can I ask this question better? (I'm a college student with little exper......Sun, 06 Sep 2026 13:34:11 -0400https://pop.com.sg/second-order-linear-odes-and-analytic-coefficients.htmlhttps://pop.com.sg/second-order-linear-odes-and-analytic-coefficients.htmlA second order, linear, homogeneous ODE can be expressed as: $$a(x)y''(x) + b(x)y'(x) + c(x)y(x) = 0$$ I have only a basic knowledge about differential equations. However, all the texts state tha......Sun, 06 Sep 2026 13:32:52 -0400https://pop.com.sg/how-can-we-minimize-a-function-of-two-variables.htmlhttps://pop.com.sg/how-can-we-minimize-a-function-of-two-variables.htmlHere we are more interested in the method to minimize the function, rather than what the actual result is. The function I currently have, which I may need to change, is: $$b / d + (1/6 \log{(b)})......Sun, 06 Sep 2026 13:25:01 -0400https://pop.com.sg/measure-theory-semifinite-measure-folland-problem-1-14-duplicate.htmlhttps://pop.com.sg/measure-theory-semifinite-measure-folland-problem-1-14-duplicate.htmlLet $\mu$ be a semi-finite (and countably-additive) measure and $\mu(E)=\infty$. Then for any $C>0$, there exists a measurable set $F$ such that $F \subseteq E$ and $C<\mu(F)<\infty$. By ...Sun, 06 Sep 2026 13:21:05 -0400https://pop.com.sg/mathbb-z-sqrt-11-is-norm-euclidean.htmlhttps://pop.com.sg/mathbb-z-sqrt-11-is-norm-euclidean.htmlI'm trying to show that $\mathbb{Z}[\sqrt{11}]$ is Euclidean with respect to the function $a+b\sqrt{11} \mapsto|N(a+b\sqrt{11})| = | a^2 -11b^2|$ By multiplicativity, it suffices to show that $\fo......Sun, 06 Sep 2026 13:18:38 -0400https://pop.com.sg/finding-area-bounded-by-2-lines-curves-by-integration.htmlhttps://pop.com.sg/finding-area-bounded-by-2-lines-curves-by-integration.htmlWhen finding area bounded by 2 lines/curves by integration, I believe I 1st need to find the intersection point eg. $y=1-x$, $y=\sqrt{1+x}$ $1-x = \sqrt{1+x}$ ... $x(x+3)=0$ So $x=0 \text{ o......Sun, 06 Sep 2026 13:16:18 -0400https://pop.com.sg/a-good-book-for-introduction-to-mathematical-statistics.htmlhttps://pop.com.sg/a-good-book-for-introduction-to-mathematical-statistics.htmlI am trying to read this book (Introduction to Mathematical Statistics by Robert,Joseph, Allen,7th edition) I find it hard to follow. So, can anyone please recommend another book has similar conten......Sun, 06 Sep 2026 13:16:17 -0400https://pop.com.sg/the-gamma-function-and-the-pi-function.htmlhttps://pop.com.sg/the-gamma-function-and-the-pi-function.htmlI have been studying differential equation, in particular special functions. Euler's Gamma function, and Gauss's Pi function are essentially the same, differing only by an offset of one unit. for......Sun, 06 Sep 2026 13:14:02 -0400https://pop.com.sg/helmholtz-theorem.htmlhttps://pop.com.sg/helmholtz-theorem.htmlI have been told that the Helmholtz decomposition theorem says that every smooth vector field $\boldsymbol{F}$ [where I am not sure what precise assumptions are needed on $\boldsymbol{F}$] on an ...Sun, 06 Sep 2026 13:11:24 -0400https://pop.com.sg/x-coordinates-of-points-where-tangent-line-is-horizontal-or-vertical-u.htmlhttps://pop.com.sg/x-coordinates-of-points-where-tangent-line-is-horizontal-or-vertical-u.htmlFor the implicit equation $x^3 + y^3 - xy^2 = 8$ Determine the exact x-coordinates of all points where the tangent line is horizontal or vertical I figured out $\dfrac{dy}{dx} =\dfrac{y^2 - 3x^......Sun, 06 Sep 2026 13:07:58 -0400https://pop.com.sg/what-does-principal-mean-in-principal-unit-normal-vector.htmlhttps://pop.com.sg/what-does-principal-mean-in-principal-unit-normal-vector.htmlWhen working with space curves, why is it called the "Principal Unit Normal Vector"? I know there are two. So how do we know which one is the principal one? Also, is there a principal unit binor......Sun, 06 Sep 2026 12:59:01 -0400https://pop.com.sg/how-to-resolve-a-trigonometric-equation-involving-contangent.htmlhttps://pop.com.sg/how-to-resolve-a-trigonometric-equation-involving-contangent.htmlI am resolving some trigonometric equation systems. A typical system (the following one is invented) could be: $\sin(x) + \cos(y) = 1.5$ $2\sin(x) + 3\cos(y) = 1$ After getting the values, you j......Sun, 06 Sep 2026 12:38:35 -0400https://pop.com.sg/find-laplace-transform-using-unit-step-function-given-graph-of-a-perio.htmlhttps://pop.com.sg/find-laplace-transform-using-unit-step-function-given-graph-of-a-perio.htmlPlease correct my work. The textbook answer which is expressed exactly like this $1/s(1+e^{-s})$ does not match my own. Find the Laplace Transform for one period of the perpetual periodic function......Sun, 06 Sep 2026 12:33:09 -0400https://pop.com.sg/volume-of-truncated-prism.htmlhttps://pop.com.sg/volume-of-truncated-prism.htmlI needed to find the volume of what Wikipedia calls a truncated prism, which is a prism (with triangle base) that is intersected with a halfspace such that the boundary of the halfspace intersects......Sun, 06 Sep 2026 12:26:14 -0400https://pop.com.sg/proving-pascal-039-s-rule-n-choose-r-n-1-choose-r-1-n-1-choose-r-when-.htmlhttps://pop.com.sg/proving-pascal-039-s-rule-n-choose-r-n-1-choose-r-1-n-1-choose-r-when-.htmlI'm trying to prove that ${n \choose r}$ is equal to ${{n-1} \choose {r-1}}+{{n-1} \choose r}$ when $1\leq r\leq n$. I suppose I could use the counting rules in probability, perhaps combination= $......Sun, 06 Sep 2026 12:12:54 -0400https://pop.com.sg/notation-and-the-name-choice-for-meet-and-join-in-order-theory.htmlhttps://pop.com.sg/notation-and-the-name-choice-for-meet-and-join-in-order-theory.htmlI have two simple questions: From where do the names meet and join come from? I don't see any intuition between those names in context of order theory. From where does the notation come? I have to......Sun, 06 Sep 2026 12:12:00 -0400https://pop.com.sg/frames-and-co-frames.htmlhttps://pop.com.sg/frames-and-co-frames.htmlI'm trying to write a precise definition of a frame and co-frame. I have that a frame is a basis for a tangent space $T_pM$ and the co-frame is the dual basis (basis for the co-tangent space $T^{\a......Sun, 06 Sep 2026 11:58:45 -0400https://pop.com.sg/list-of-all-256-binary-combinations-of-8-digits-closed.htmlhttps://pop.com.sg/list-of-all-256-binary-combinations-of-8-digits-closed.htmli don't have any coding/java etc software. how can i view all the 256 binary combinatios from 8 digits? can someone tell me where can i find them in a list (any site/online generator etc) or copy t......Sun, 06 Sep 2026 11:51:41 -0400https://pop.com.sg/properties-of-subtraction-and-division.htmlhttps://pop.com.sg/properties-of-subtraction-and-division.htmlI'm (re)learning math and was looking for the properties of the most basic math operations. For addition and multiplication these are very easy to find (e.g. https://www.aaamath.com/g74a_px1.htm and ...Sun, 06 Sep 2026 11:50:35 -0400