https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usTue, 18 Aug 2026 19:35:55 -0400https://pop.com.sg/understanding-the-properties-and-use-of-the-laplacian-matrix-and-its-n.htmlhttps://pop.com.sg/understanding-the-properties-and-use-of-the-laplacian-matrix-and-its-n.htmlI am reading the wikipedia article on the Laplacian matrix: https://en.wikipedia.org/wiki/Laplacian_matrix I don't understand what is the particular use of it; having the diagonals as the degree an......Tue, 18 Aug 2026 23:27:55 -0400https://pop.com.sg/is-0-a-natural-number.htmlhttps://pop.com.sg/is-0-a-natural-number.htmlIs there a consensus in the mathematical community, or some accepted authority, to determine whether zero should be classified as a natural number? It seems as though formerly $0$ was considered i......Tue, 18 Aug 2026 23:21:00 -0400https://pop.com.sg/what-is-wrong-with-my-solution-int-cos-2-x-tan-3x-dx.htmlhttps://pop.com.sg/what-is-wrong-with-my-solution-int-cos-2-x-tan-3x-dx.htmlI am trying to do this problem completely on my own but I can not get a proper answer for some reason $$\begin{align} \int \cos^2 x \tan^3x dx &=\int \cos^2 x \frac{ \sin^3 x}{ \cos^3 x}dx\\ &......Tue, 18 Aug 2026 23:17:46 -0400https://pop.com.sg/what-allows-us-divide-multiply-dx-in-calculus.htmlhttps://pop.com.sg/what-allows-us-divide-multiply-dx-in-calculus.htmlI've read nearly all of the threads on this topic but none seem to answer my question or lead me in the best direction. When performing U-substituion or even in it's most basic form: $y = 2x$, $dy=......Tue, 18 Aug 2026 23:16:08 -0400https://pop.com.sg/what-is-general-term-of-the-sequence-1-2-4-4-8-8-8-8-16-16-16-16-16-16.htmlhttps://pop.com.sg/what-is-general-term-of-the-sequence-1-2-4-4-8-8-8-8-16-16-16-16-16-16.htmlI have a sequence as follows: $$1, 2, 4, 4, 8, 8, 8, 8, 16, 16, 16, 16, 16, 16, 16, 16, 32, 32 ...$$ What will be the closed form of the above sequence for the nth term? I clearly see the patter......Tue, 18 Aug 2026 23:16:02 -0400https://pop.com.sg/how-to-find-a-minimal-polynomial.htmlhttps://pop.com.sg/how-to-find-a-minimal-polynomial.htmlI need to find minimal polynomial of $\alpha = \sqrt 2 + \sqrt [3] 3 $ over $\mathbb Q$ and prove that my result is minimal polynomial. How do I do that?...Tue, 18 Aug 2026 23:11:00 -0400https://pop.com.sg/laplace-transform-of-sin-at-by-definition.htmlhttps://pop.com.sg/laplace-transform-of-sin-at-by-definition.htmlI want to find Laplace transform of $\sin(at)$ by definition $$\displaystyle \mathcal L \left\{{\sin at}\right\} = \int_0^{\to +\infty}e^{-st}\sin at \, \mathrm dt$$ After taking integration by pa......Tue, 18 Aug 2026 23:08:22 -0400https://pop.com.sg/what-is-temperature-distribution-equation.htmlhttps://pop.com.sg/what-is-temperature-distribution-equation.htmlI have $T(x,y)$ (a function of two variables $x$ and $y$) and it is the temperature distribution on a metal plate. What does temperature distribution mean? what does its equation say? if I have the ...Tue, 18 Aug 2026 23:04:52 -0400https://pop.com.sg/minimal-realization-of-a-state-space-model.htmlhttps://pop.com.sg/minimal-realization-of-a-state-space-model.htmlMinimal realization of a transfer function is about cancelling out poles against zeros. But the minimal realization of a state space model is about cancelling out non-controllable and non-observable ...Tue, 18 Aug 2026 23:03:41 -0400https://pop.com.sg/what-does-it-mean-to-solve-or-find-solutions-in-mathematics.htmlhttps://pop.com.sg/what-does-it-mean-to-solve-or-find-solutions-in-mathematics.htmlSomething that has been really confusing me lately is that this equation has four solutions $$3x(x+1)(x^2+x+2)=16x(x+1)(2x+1)$$ But what does that mean? Until now solutions to me has meant, what a......Tue, 18 Aug 2026 22:49:57 -0400https://pop.com.sg/find-a-vector-mathbf-x-whose-image-under-t-is-b.htmlhttps://pop.com.sg/find-a-vector-mathbf-x-whose-image-under-t-is-b.htmlI am having trouble with this question and how to get the answer. With $T$ defined by $T(\mathbf x)=A\mathbf x$, find a vector $x$ whose image under $T$ is $b$. $$ A = \begin{pmatrix} 1 & -3......Tue, 18 Aug 2026 22:24:16 -0400https://pop.com.sg/calculating-multivariable-limit.htmlhttps://pop.com.sg/calculating-multivariable-limit.html$$\lim_{(x,y) \to (0,0)} \frac{x^2y}{x^3 + y}$$ I tried the following, but im not sure if this is allowed: Let $$t = x^3+y$$ $$(x,y)\to (0,0)$$ $$t\to 0$$ $$y=t-x^3$$ Now we will calculate the ......Tue, 18 Aug 2026 22:20:48 -0400https://pop.com.sg/how-do-i-determine-if-a-matrix-is-in-a-span-of-two-other-matrices.htmlhttps://pop.com.sg/how-do-i-determine-if-a-matrix-is-in-a-span-of-two-other-matrices.htmlIf \begin{align} A =\begin{bmatrix} 1 & 1 \\ -1 & 1 \\ \end{bmatrix} \end{align} and \begin{align}B = \begin{bmatrix} 1 & -1 \\ 1 & 0 \\ ......Tue, 18 Aug 2026 22:13:32 -0400https://pop.com.sg/degeneracy-of-outerplanar-graphs.htmlhttps://pop.com.sg/degeneracy-of-outerplanar-graphs.htmlDoes anyone know an elegant proof to the fact that every outerplanar graph has a vertex of degree at most 2 (and hence is 2-degenerate, since every subgraph is also outerplanar). I have a proof by ...Tue, 18 Aug 2026 22:04:07 -0400https://pop.com.sg/period-vs-time-period-of-sine-wave.htmlhttps://pop.com.sg/period-vs-time-period-of-sine-wave.htmlIt's weird I'm still confused about this, but usually when we figure out the period of a sine wave from its graph, it's in radians. But the true period should be in time, like how fast we are revol......Tue, 18 Aug 2026 21:47:43 -0400https://pop.com.sg/determine-if-a-statement-is-true-or-false.htmlhttps://pop.com.sg/determine-if-a-statement-is-true-or-false.htmlI need to determine if this statement is true or false: For each non-zero integer x, there is a negative rational number y such that $x = \frac{3}{6y+4}$. If I say x is rational it can be written ......Tue, 18 Aug 2026 21:33:43 -0400https://pop.com.sg/manual-calculation-doesn-039-t-match-wolfram-alpha-why.htmlhttps://pop.com.sg/manual-calculation-doesn-039-t-match-wolfram-alpha-why.htmlSay I want to evaluate this sum: $$\sum_{x=2}^\infty \ln(x^3+1)-\ln(x^3-1)$$ We can rewrite the sum as $$\ln\left(\prod_{x=2}^\infty \frac{x^3+1}{x^3-1}\right)$$ We can split the product into two ...Tue, 18 Aug 2026 21:32:34 -0400https://pop.com.sg/simplifying-math-equation-with-factorials.htmlhttps://pop.com.sg/simplifying-math-equation-with-factorials.htmlI am trying to understand how simplified that equation: $$\frac{(x-2)!}{(x-4)!}-46=\frac{x!}{(x-1)!}$$ In my book it was solved like this: but I really don't understand how it happened? Where did ......Tue, 18 Aug 2026 21:28:43 -0400https://pop.com.sg/find-the-perimeter-of-the-trapezoid-from-the-3-side-lengths.htmlhttps://pop.com.sg/find-the-perimeter-of-the-trapezoid-from-the-3-side-lengths.htmlWhat we know about the trapezoid ABCD (AB and CD are respectively its large and small bases) is that AB=9 cm,AC=6 cm,AD=8 cm,CD=4 cm. Determine the perimeter of the trapezoid. It is quite tricky.......Tue, 18 Aug 2026 21:27:38 -0400https://pop.com.sg/there-is-any-relation-between-pi-sqrt-2-or-a-generic-polygon.htmlhttps://pop.com.sg/there-is-any-relation-between-pi-sqrt-2-or-a-generic-polygon.htmlI'm a programmer, I'm always looking for new formulas and new way of computing things, to satisfy my curiosity I would like to know if there are any formulas, or I should say equalities, that make ......Tue, 18 Aug 2026 21:24:00 -0400https://pop.com.sg/probability-of-3-people-in-a-room-of-30-having-the-same-birthday.htmlhttps://pop.com.sg/probability-of-3-people-in-a-room-of-30-having-the-same-birthday.htmlI have been looking at the birthday problem (http://en.wikipedia.org/wiki/Birthday_problem) and I am trying to figure out what the probability of 3 people sharing a birthday in a room of 30 people ......Tue, 18 Aug 2026 21:21:30 -0400https://pop.com.sg/solve-the-function-for-x-in-terms-of-y.htmlhttps://pop.com.sg/solve-the-function-for-x-in-terms-of-y.htmlEquation: $$y = \frac{x + 1}{x - 1}$$ Task: Solve for $x$ in terms of $y$. My attempt: I just dont know how to go about doing this because the variable $x$ is in both the numerator and denominator....Tue, 18 Aug 2026 21:19:39 -0400https://pop.com.sg/solve-equation-with-absolute-value-of-complex-numbers.htmlhttps://pop.com.sg/solve-equation-with-absolute-value-of-complex-numbers.htmlFind all complex numbers $z$ such that $|z| + \overline{z}=9-3i$. Thinking about using with the fact that $z=a+bi$ and $|z| = \sqrt{a^2+b^2}$ in some way. Not sure how....Tue, 18 Aug 2026 21:18:35 -0400https://pop.com.sg/how-to-perform-abstract-integration.htmlhttps://pop.com.sg/how-to-perform-abstract-integration.htmlSuppose $X$ is a Banach space. Here, page $124$, there is this sentence If $\mu$ is a finite support on $X$ we can define its barycenter $\beta(\mu)=\beta_X(\mu) \in X$ by $$\beta(\mu)=\int ......Tue, 18 Aug 2026 21:12:39 -0400https://pop.com.sg/zebra-groups-and-counting-stripes.htmlhttps://pop.com.sg/zebra-groups-and-counting-stripes.htmlHow many stripes can you paint on a $2$-group of fixed size? A group of order $2^ap^b$ is solvable, by Burnside's theorem, so its chief factors are either abelian $2$-groups or abelian $p$-groups.......Tue, 18 Aug 2026 20:53:10 -0400https://pop.com.sg/what-is-the-difference-between-shifting-functions-and-shifting-parabol.htmlhttps://pop.com.sg/what-is-the-difference-between-shifting-functions-and-shifting-parabol.htmlFor shifting a parabola to the right, do we write a "$+$" sign or "$-$" sign in the equation? Is this the same way for shifting a function, as well? Here is the equation of parabola just for the ...Tue, 18 Aug 2026 20:51:10 -0400https://pop.com.sg/limit-of-absolute-values.htmlhttps://pop.com.sg/limit-of-absolute-values.htmlHow can we solve $\lim_{h\rightarrow0}{\frac{|x+h|-|x|}{h}}$ with $x,h \in \mathbb{R}$, and similar limit problems with absolute values? Note that although this represents $\frac{d}{dx}|x|$, I w......Tue, 18 Aug 2026 20:50:10 -0400https://pop.com.sg/why-is-this-called-the-cocycle-condition.htmlhttps://pop.com.sg/why-is-this-called-the-cocycle-condition.htmlMy question is about why the condition below is called the cocycle condition. Surely it is named after some interpretation of it in cohomology. Let $C$ be a site, and let $F$ be a fibered category......Tue, 18 Aug 2026 20:38:56 -0400https://pop.com.sg/integration-involving-fractional-part-function.htmlhttps://pop.com.sg/integration-involving-fractional-part-function.htmlFind the value of $$ \int_{0}^{1}{(\{2x\}-1)(\{3x\}-1)}dx$$ where $\{x\}$ denotes the fractional part function. $\{x\}=x-[x]$ ,where $[x]$ denotes the greatest integer function. My try :: Since th......Tue, 18 Aug 2026 20:38:36 -0400https://pop.com.sg/is-a-linear-transformation-onto-or-one-to-one.htmlhttps://pop.com.sg/is-a-linear-transformation-onto-or-one-to-one.htmlThe definition of one-to-one was pretty straight forward. If the vectors are lin.indep the transformation would be one-to-one. But could it also be onto? The definition of onto was a little more ...Tue, 18 Aug 2026 20:34:37 -0400https://pop.com.sg/what-does-it-mean-for-media-to-play-at-1-5x-or-nx-speed.htmlhttps://pop.com.sg/what-does-it-mean-for-media-to-play-at-1-5x-or-nx-speed.htmlSuppose I have a video that plays for 60 minutes at normal "1x" speed. I know that if I set the video to play at "2x" speed, then it should play for 30 minutes. Now, what if the......Tue, 18 Aug 2026 20:31:20 -0400https://pop.com.sg/find-all-eigenvalues-of-a-matrix-that-holds-a-a-5.htmlhttps://pop.com.sg/find-all-eigenvalues-of-a-matrix-that-holds-a-a-5.htmlI need to find all eigenvalues of a matrix that holds $A = A^{5}$. I have no idea where to start. Any directions please?...Tue, 18 Aug 2026 20:20:48 -0400https://pop.com.sg/adjoint-of-a-differential-operator.htmlhttps://pop.com.sg/adjoint-of-a-differential-operator.htmlI am self-studying differential equations using MIT's publicly available materials. One of the recitation exercises runs as follows: Define an inner or dot product on $\mathcal{C}[a,b]$ by \......Tue, 18 Aug 2026 20:20:24 -0400https://pop.com.sg/why-is-congruence-sign-a-equiv-b-pmod-c-closed.htmlhttps://pop.com.sg/why-is-congruence-sign-a-equiv-b-pmod-c-closed.htmlWhy is congruence sign $A\equiv B\pmod{C}$? Wouldn't it be more natural to use $A \equiv_{C} B$ instead? The former notation suggests that you take B and make some operation on it (like in case of ...Tue, 18 Aug 2026 20:16:42 -0400https://pop.com.sg/vi-arnold-ode-need-help-coming-to-grips-with-early-notation-in-the-boo.htmlhttps://pop.com.sg/vi-arnold-ode-need-help-coming-to-grips-with-early-notation-in-the-boo.htmlThis seems to be a somewhat common theme among users on here trying to start reading Arnold's ODE book. I have found a few similar questions but none have sufficiently answered my problems. ...Tue, 18 Aug 2026 20:06:10 -0400https://pop.com.sg/fair-coin-and-weighted-coin.htmlhttps://pop.com.sg/fair-coin-and-weighted-coin.htmlI have a fair coin and a weighted coin which lands heads 75% of the time. I pick a coin at random and flip it 5 times and get heads 4 times and tails once. What is the probability that I picked the ...Tue, 18 Aug 2026 20:05:46 -0400https://pop.com.sg/how-can-2-100-2-98-2-99-2-97-be-written-in-terms-of-its-prime-factors.htmlhttps://pop.com.sg/how-can-2-100-2-98-2-99-2-97-be-written-in-terms-of-its-prime-factors.htmlHow can $(2^{100}-2^{98})(2^{99}-2^{97})$ be written in terms of its prime factors? I tried to expand it: $2^{199}-4^{197}+2^{195}$ What do I do next? The answer choices are: A. $2^{100}\cdot3 \......Tue, 18 Aug 2026 19:56:31 -0400https://pop.com.sg/domain-and-range-of-piecewise-function-help.htmlhttps://pop.com.sg/domain-and-range-of-piecewise-function-help.htmlI have a graph of a piecewise function below, and I am having trouble figuring out the domain of the function in interval notation. My answers are: Domain: $[-7, -1)\cup(-1, \infty)$ Range: $[-6, \...Tue, 18 Aug 2026 19:53:18 -0400https://pop.com.sg/how-to-find-sin-and-cos-of-0-pi-2-pi-without-calculator-closed.htmlhttps://pop.com.sg/how-to-find-sin-and-cos-of-0-pi-2-pi-without-calculator-closed.htmlIn my notes it shows how to calculate by using the unit circle. But I do not know why the value of sin is the y coordinate and the value of cos is the x coordinate....Tue, 18 Aug 2026 19:36:22 -0400https://pop.com.sg/degenerate-solution-in-linear-programming.htmlhttps://pop.com.sg/degenerate-solution-in-linear-programming.htmlHow can I determine if a solution in a linear programming problem is degenerate without I use any software or the graphical display of the solution; For example in the model: $$\max\{2x_1 + 4x_2\......Tue, 18 Aug 2026 19:21:47 -0400https://pop.com.sg/proof-of-bolzano-039-s-theorem.htmlhttps://pop.com.sg/proof-of-bolzano-039-s-theorem.htmlI know one proof of Bolzano's Theorem, which can be sketched as follows: Set $f$ a continuous function in $[a,b]$ such that ${f(a)<0<f(b)}$. ${A=\{x:a<x<b \text{ and } f <0\in[a,x]......Tue, 18 Aug 2026 19:19:37 -0400https://pop.com.sg/explanation-of-the-derivation-of-the-formula-for-the-sum-of-an-arithme.htmlhttps://pop.com.sg/explanation-of-the-derivation-of-the-formula-for-the-sum-of-an-arithme.htmlI am trying to understand the derivation of the formula for the sum of an arithmetic sequence of the first $n$ terms. I do not understand what rules or reasoning allow two sequences to be added in ...Tue, 18 Aug 2026 19:14:08 -0400https://pop.com.sg/polynomial-function-given-roots-and-a-point.htmlhttps://pop.com.sg/polynomial-function-given-roots-and-a-point.htmlA 3rd degree polynomial has roots at x=-2i and x=5. The y-intercept is (0,25). Write an equation for this function in factored form with real coefficients. My first hack is to say that in factored ...Tue, 18 Aug 2026 19:11:17 -0400https://pop.com.sg/how-to-find-the-transition-matrix-for-ordered-basis-of-2x2-diagonal-ma.htmlhttps://pop.com.sg/how-to-find-the-transition-matrix-for-ordered-basis-of-2x2-diagonal-ma.htmlThe problem: For the vector space of lower triangular matrices with zero trace, given ordered basis: $B=${$$ \begin{bmatrix} -5 & 0 \\ 4 & 5 \\ \end{bmatrix}......Tue, 18 Aug 2026 19:08:21 -0400https://pop.com.sg/what-does-inversion-mean.htmlhttps://pop.com.sg/what-does-inversion-mean.htmlI am in highschool taking some advanced math courses and I have some questions about terminology. There appears to be more definitions to the meaning of inversion in math than I can count. I'm hea......Tue, 18 Aug 2026 19:07:40 -0400https://pop.com.sg/why-does-a-flat-finite-type-morphism-of-irreducible-noetherian-schemes.htmlhttps://pop.com.sg/why-does-a-flat-finite-type-morphism-of-irreducible-noetherian-schemes.htmlBasically the title, I came across this statement reading some notes on dimension of fibers of flat maps. (I don't know if Noetherian actually matters for this question) I think that for maps of ...Tue, 18 Aug 2026 19:04:30 -0400https://pop.com.sg/let-f-k-be-two-fields-f-subset-k-and-suppose-f-x-g-x-in-f-x-are-relati.htmlhttps://pop.com.sg/let-f-k-be-two-fields-f-subset-k-and-suppose-f-x-g-x-in-f-x-are-relati.htmlLet $F,K$ be two fields $F \subset K$ and suppose $f(x),g(x) \in F[x]$ are relatively prime in $F[x].$ Prove they are relatively prime in $K[x].$ Suppose $f(x)$ and $g(x)$ are relatively prime in ......Tue, 18 Aug 2026 19:04:24 -0400https://pop.com.sg/four-men-hats-and-probability.htmlhttps://pop.com.sg/four-men-hats-and-probability.htmlI encountered the four men in hats puzzle for the first time today. My question is about a realisation I (think I) had while arriving at the solution, but I have no idea whether I've made a mistake ...Tue, 18 Aug 2026 19:04:10 -0400https://pop.com.sg/finding-polynomal-function-with-given-zeros-and-one-zero-is-a-square-r.htmlhttps://pop.com.sg/finding-polynomal-function-with-given-zeros-and-one-zero-is-a-square-r.htmlI've been having trouble with this problem: Find a polynomial function of minimum degree with $-1$ and $1-\sqrt{3}$ as zeros. Function must have integer coefficients. When I tried it, I got this: \...Tue, 18 Aug 2026 19:03:24 -0400https://pop.com.sg/what-is-the-domain-of-z-arcsin-dfrac-x-y.htmlhttps://pop.com.sg/what-is-the-domain-of-z-arcsin-dfrac-x-y.htmlI get that it should be $|y|>|x|$ and in the Wolfram it looks like this. But when I graph it by hand is that it should be only the "upper" part of intersection and not the "bottom" part as well,......Tue, 18 Aug 2026 19:02:19 -0400https://pop.com.sg/difference-between-variance-and-covariance.htmlhttps://pop.com.sg/difference-between-variance-and-covariance.htmlWhat exactly is the physical interpretation of variance (in terms of a data set) and the difference between variance and co-variance matrices?...Tue, 18 Aug 2026 18:54:23 -0400https://pop.com.sg/area-of-intersection-between-two-circles-duplicate.htmlhttps://pop.com.sg/area-of-intersection-between-two-circles-duplicate.htmlSuppose you have 2 circles that intersect each other in such a way that each circle passes through the other's center. What is the area between the circle(or common area) i.e. area between the cent......Tue, 18 Aug 2026 18:51:26 -0400https://pop.com.sg/how-does-one-prove-rodrigues-039-formula-for-legendre-polynomials.htmlhttps://pop.com.sg/how-does-one-prove-rodrigues-039-formula-for-legendre-polynomials.htmlI am trying to prove that $\frac{1}{n!\space2^n}\frac{d^n}{dx^n}\{(x^2-1)^n\}=P_n(x)$, where $P_n(x)$ is the Legendre Polynomial of order n. I've been told that the proof uses complex analysis, of......Tue, 18 Aug 2026 18:44:45 -0400https://pop.com.sg/what-algebraic-topology-book-to-read-after-hatcher-039-s.htmlhttps://pop.com.sg/what-algebraic-topology-book-to-read-after-hatcher-039-s.htmlI've currently finished chapter 2 of his book and done all the exercises of in chapter 0, 1 and 2. Was wondering when I finished reading this book what book do I read next in algebraic topology?...Tue, 18 Aug 2026 18:38:16 -0400https://pop.com.sg/finding-magnitude-of-a-function.htmlhttps://pop.com.sg/finding-magnitude-of-a-function.htmlI want to calculate the magnitude of the following function: $x(t)=\dfrac{i}{t}(e^{it}+1)$ I did the following to find the magnitude: $x(t)=\dfrac{i}{t}(1+\cos t+i \sin t)$ $|x(t)|=\sqrt{\dfrac{\si......Tue, 18 Aug 2026 18:29:02 -0400https://pop.com.sg/swing-twist-decomposition-for-quaternion-rotations-verification.htmlhttps://pop.com.sg/swing-twist-decomposition-for-quaternion-rotations-verification.htmlPRZEMYSLAW DOBROWOLSKI has written a paper that (I think) can be applied to swing-twist decompositions for quaternion rotations called "SWING-TWIST DECOMPOSITION IN CLIFFORD ALGEBRA" . I tried to ......Tue, 18 Aug 2026 18:15:28 -0400https://pop.com.sg/what-is-the-maximum-norm-induced-by-the-maximum-norm-for-mathbb-r-n.htmlhttps://pop.com.sg/what-is-the-maximum-norm-induced-by-the-maximum-norm-for-mathbb-r-n.htmlI'm reading a problem which uses the matrix norm induced by the maximum norm for $\mathbb{R}^n$. But I do know what it is exactly, and googling it gives different results....Tue, 18 Aug 2026 18:10:31 -0400https://pop.com.sg/how-is-the-joint-proportionality-joint-variation-equation-i-e-y-kxz-lo.htmlhttps://pop.com.sg/how-is-the-joint-proportionality-joint-variation-equation-i-e-y-kxz-lo.htmlIf $y$ is proportional to $x$, and $y$ is also proportional to $z$, then how are we able to arrive at the equation: $y = kxz$ ? My understanding so far of proportionality, which comes from this ...Tue, 18 Aug 2026 18:10:19 -0400https://pop.com.sg/does-the-series-sum-frac-1-n-ln-n-3-2-converge-or-diverge.htmlhttps://pop.com.sg/does-the-series-sum-frac-1-n-ln-n-3-2-converge-or-diverge.htmlConsider $$\sum \frac{1}{n\ \ln^{3/2}(n)}$$ The ratio test is inconclusive. The root test is inconclusive. And it seems right that $\frac{1}{n\ (\ln(n))^{3/2}}\leq\frac{1}{n}$ which diverges, but......Tue, 18 Aug 2026 18:03:03 -0400https://pop.com.sg/derivative-of-arcsin-x-in-1.htmlhttps://pop.com.sg/derivative-of-arcsin-x-in-1.htmlI need to find the derivative $\arcsin'(-1)$. I know that $(\arcsin)'(x)=\frac{1}{\sqrt{1-x^2}}$. I also know that $$f_{+}'(b)=\lim_{x \to b+}f'(x) \tag{1}$$ and that the right derivate is $+\infty......Tue, 18 Aug 2026 18:01:01 -0400https://pop.com.sg/rolling-a-fair-die.htmlhttps://pop.com.sg/rolling-a-fair-die.htmlYou and your friend play a game in which you and your friend take turns rolling a fair six-sided die and keep a running tally of the sum of the results of all rolls made. Play continues until either ...Tue, 18 Aug 2026 18:00:25 -0400https://pop.com.sg/find-the-area-common-to-the-inside-of-the-cardioid-r-1-sin-theta-and-t.htmlhttps://pop.com.sg/find-the-area-common-to-the-inside-of-the-cardioid-r-1-sin-theta-and-t.htmlFunction Plotter graph: I think the formula is $$A = \frac 1 2 \int_{\alpha}^{\beta} (\text{outer})^2 - (\text{inner})^2 d\theta$$ where $\alpha, \beta$ are where they intersect in $[0, 2\pi]$. ......Tue, 18 Aug 2026 18:00:14 -0400https://pop.com.sg/prove-sinh-x-x-for-all-x-0.htmlhttps://pop.com.sg/prove-sinh-x-x-for-all-x-0.htmlI did a proof for $\sinh x > x$ for all $x > 0$. But I am not sure if the proof is mathematically valid. I started by showing that $\frac{d}{dx} \sinh x = \cosh x$ and that the limit of $\co......Tue, 18 Aug 2026 17:57:25 -0400https://pop.com.sg/find-arcsin-frac-4-5-arccos-frac-1-sqrt-50.htmlhttps://pop.com.sg/find-arcsin-frac-4-5-arccos-frac-1-sqrt-50.htmlI can't solve this problem due to the irrational number. Used 2 methods: (1) with Pythagorean theorem I got: \begin{align} \arccos(\frac{1}{\sqrt{50}})&=\theta\\\frac{1}{\sqrt{50}}&=\cos(\t......Tue, 18 Aug 2026 17:55:25 -0400https://pop.com.sg/property-of-a-kite-inscribed-in-a-square.htmlhttps://pop.com.sg/property-of-a-kite-inscribed-in-a-square.htmlA friend of mine presented the following problem to me: Say you have a square $ABCD$ and a kite $BCEF$ (a kite is a quadrilateral object such that $\overline{BC}$ and $\overline{BF}$ have the same ......Tue, 18 Aug 2026 17:41:45 -0400https://pop.com.sg/what-distinguishes-039-family-039-vs-039-set-039-of-functions.htmlhttps://pop.com.sg/what-distinguishes-039-family-039-vs-039-set-039-of-functions.htmlSource: Stewart, James. Calculus: Early Transcendentals (6 edn 2007). [p. 50 Top:]   To understand how the expression for a function relates to its graph, it’s helpful to graph a family of ...Tue, 18 Aug 2026 17:41:18 -0400https://pop.com.sg/a-question-about-irreducible-representations-of-so-3-group.htmlhttps://pop.com.sg/a-question-about-irreducible-representations-of-so-3-group.htmlI have a perplexity about the representations of the group of 3D rotations $SO(3)$. I'm reading the following text which is describing the irreducible representations of $SO(3)$ that are associated......Tue, 18 Aug 2026 17:34:26 -0400https://pop.com.sg/evaluating-int-2x-2-1-e-x-2-dx.htmlhttps://pop.com.sg/evaluating-int-2x-2-1-e-x-2-dx.html$$\int(2x^2+1)e^{x^2}dx$$ The answer of course: $$\int(2x^2+1)e^{x^2}\,dx=xe^{x^2}+C$$ But what kind of techniques we should use with problem like this ?...Tue, 18 Aug 2026 17:18:29 -0400https://pop.com.sg/pentation-notation-how-does-it-work-duplicate.htmlhttps://pop.com.sg/pentation-notation-how-does-it-work-duplicate.htmlWhen going through with learning Grahams number, I got stuck at $$3↑↑↑3$$ Working it through, we have $$3↑3=3^3$$ $$3↑↑3=3^{3^3}=3↑(3↑3)$$ As such, it would appear to me that $$3↑↑↑3=3^{3^{3^......Tue, 18 Aug 2026 17:10:43 -0400https://pop.com.sg/is-there-a-way-to-solve-an-equation-with-both-sine-and-cosine-in-it.htmlhttps://pop.com.sg/is-there-a-way-to-solve-an-equation-with-both-sine-and-cosine-in-it.htmlI'm trying to solve a system of equations. I end up with sine and cosine in one equation and the parameter of both as the last unknown. Like this: num1 = num2 * sin(start) + num3 * cos(start) num1, ...Tue, 18 Aug 2026 17:06:23 -0400https://pop.com.sg/what-is-the-difference-between-a-unit-simplex-and-a-probability-simple.htmlhttps://pop.com.sg/what-is-the-difference-between-a-unit-simplex-and-a-probability-simple.htmlThe unit simplex is the $n$-dimensional simplex determined by the zero vector and the unit vectors, i.e., $0,e_1, \ldots,e_n\in\mathbf R^n$. It can be expressed as the set of vectors that satisfy ......Tue, 18 Aug 2026 16:57:16 -0400https://pop.com.sg/solving-ax-b-when-x-and-b-are-given.htmlhttps://pop.com.sg/solving-ax-b-when-x-and-b-are-given.htmlI am trying to study least square and linear regression and I understand the solution for $Ax = b$ when x is the unknown and the LS solution is given by $(A^TA)^{-1}A^TA$. Now, I was wondering if ...Tue, 18 Aug 2026 16:55:53 -0400https://pop.com.sg/calculate-average-wind-direction.htmlhttps://pop.com.sg/calculate-average-wind-direction.htmlWhat is the best way to average wind direction? I have found many conflicting suggestions elsewhere. Best one I saw is to average SINs of all angles in radians and take inverse SIN of the result. ......Tue, 18 Aug 2026 16:51:40 -0400https://pop.com.sg/finding-final-velocity-when-initial-velocity-distance-travelled-and-ac.htmlhttps://pop.com.sg/finding-final-velocity-when-initial-velocity-distance-travelled-and-ac.htmlI've been struggling with this, I need to find the final velocity of a car after it travels a certain distance when I know it's initial velocity and acceleration are known. Any help?...Tue, 18 Aug 2026 16:45:33 -0400https://pop.com.sg/trapezium-midsegment-theorem.htmlhttps://pop.com.sg/trapezium-midsegment-theorem.htmlA question from an ACT Math test: Please teach me a way to get the answer quickly, I must've missed some properties of trapezoid cause I can only think of making similar triangles to find the answ......Tue, 18 Aug 2026 16:35:48 -0400https://pop.com.sg/what-does-round-off-error-in-double-arithmetic-mean.htmlhttps://pop.com.sg/what-does-round-off-error-in-double-arithmetic-mean.htmlLet's say I have a function $f$ and furthermore let there also be a recursion formula for $f$. So I can evaluate $f(x)$ directly and I can evaluate $f(x)$ using the recursion formula. Now, in the ...Tue, 18 Aug 2026 16:32:15 -0400https://pop.com.sg/given-a-number-039-n-039-find-how-many-how-many-numbers-are-there-betw.htmlhttps://pop.com.sg/given-a-number-039-n-039-find-how-many-how-many-numbers-are-there-betw.htmlYou are given a number $N\le 10^{18}$. You need to find out how many numbers there exist in between $1$ to $N$, which doesn't contain the digit $'X'$ in it . Say $N = 5, X=4$ The answer is $4$. ......Tue, 18 Aug 2026 16:25:05 -0400https://pop.com.sg/given-that-the-customer-has-filed-a-claim-what-is-the-probability-that.htmlhttps://pop.com.sg/given-that-the-customer-has-filed-a-claim-what-is-the-probability-that.htmlAn automobile insurance company divides customers into three categories, good risks, medium risks, and poor risks. Assume that 70% of the customers are good risks, 20% are medium risks, and 10% are......Tue, 18 Aug 2026 16:22:15 -0400https://pop.com.sg/what-does-3-3-mean.htmlhttps://pop.com.sg/what-does-3-3-mean.htmlI was taking a test and this was a question. It does not mean $3^3$, nor is it a typo. The superscript was before the $3$ and I have no idea what it means. I tried researching but couldn't find an ......Tue, 18 Aug 2026 16:20:43 -0400https://pop.com.sg/which-of-the-following-expressions-completes-the-identity-1-sec-2-x.htmlhttps://pop.com.sg/which-of-the-following-expressions-completes-the-identity-1-sec-2-x.htmlI simplified $1 + \sec^2 x$ to $1 + 1/\cos^2 x$ then $2/\cos^2 x$ but I am stuck here....Tue, 18 Aug 2026 16:15:22 -0400https://pop.com.sg/gale-stewart-theorem-open-games-are-determined-implies-closed-games-ar.htmlhttps://pop.com.sg/gale-stewart-theorem-open-games-are-determined-implies-closed-games-ar.htmlA Gale-Stewart game $G(A)$ is played on a set $A\subseteq\mathbb N^\mathbb N$. In this game, players p0 and p1 alternately pick a natural number, forming a sequence $\alpha:=\alpha_0\alpha_1\alpha_2\...Tue, 18 Aug 2026 16:11:28 -0400https://pop.com.sg/how-to-factor-a-four-term-polynomial-without-grouping.htmlhttps://pop.com.sg/how-to-factor-a-four-term-polynomial-without-grouping.html$$2x^3 + 9x^2 +7x -6$$ This equation doesn't factor by grouping, and other than that I have no idea how to solve this problem. Will someone please help?...Tue, 18 Aug 2026 16:08:51 -0400https://pop.com.sg/rotate-graph-of-function-by-180.htmlhttps://pop.com.sg/rotate-graph-of-function-by-180.htmlsuppose that we have graph of function $$f(x)=1+x\cos(x)$$ and we should rotate it by $180$ degree,question is what is a function which describe new graph?answer is $$f(x)=x\cos(x)-1$$but i ca......Tue, 18 Aug 2026 16:08:12 -0400https://pop.com.sg/what-is-the-geodesic-between-a-point-and-a-line-geodesic-between-two-p.htmlhttps://pop.com.sg/what-is-the-geodesic-between-a-point-and-a-line-geodesic-between-two-p.htmlI found a similiar question that also asks for the distance from a point to a line but works on a sphere. Now I'm trying to figure out the length of the geodesic line d between the given geodesic......Tue, 18 Aug 2026 16:03:32 -0400https://pop.com.sg/find-the-least-positive-value-of-alpha-for-the-following-trignometric-.htmlhttps://pop.com.sg/find-the-least-positive-value-of-alpha-for-the-following-trignometric-.htmlGiven question is : $$\sin(5\alpha + \theta) = \cos(\theta - 3\alpha)$$ We are to find the least positive value of $\alpha$ for which above equation holds. The way I did is as, $$\sin5\alpha \cos......Tue, 18 Aug 2026 15:59:49 -0400https://pop.com.sg/how-to-calculate-the-area-of-a-parabolic-dish.htmlhttps://pop.com.sg/how-to-calculate-the-area-of-a-parabolic-dish.htmlI feel like this should be really easy, but I'm not sure if I'm doing it correctly so I'm going to give it a go here, and if I'm not very good at maths (I'm not) then you can hopefully correct me! ...Tue, 18 Aug 2026 15:54:20 -0400https://pop.com.sg/high-dimensional-optimization-algorithm.htmlhttps://pop.com.sg/high-dimensional-optimization-algorithm.htmlI have an optimization problem that at first sounds quite textbook. I have a convex objective function in $D$-dimensional space that is twice differentiable everywhere and has no local optima. ...Tue, 18 Aug 2026 15:50:30 -0400https://pop.com.sg/find-a-formula-for-the-area-of-the-triangle-with-vertices-in-polar-coo.htmlhttps://pop.com.sg/find-a-formula-for-the-area-of-the-triangle-with-vertices-in-polar-coo.htmlI don't know where to start with this homework problem. If anyone could help me start, or provide the solution, that'd be greatly appreciated....Tue, 18 Aug 2026 15:47:54 -0400https://pop.com.sg/defining-a-plane-using-only-2-points-and-an-axis.htmlhttps://pop.com.sg/defining-a-plane-using-only-2-points-and-an-axis.htmlSuppose you have a $3$-dimensional space in which there are 2 points ($A$ and $B$) defined (non identical). Now, you can define a line that goes through them but you cannot define a unique plane, b......Tue, 18 Aug 2026 15:43:06 -0400https://pop.com.sg/for-how-many-natural-numbers-a-is-the-expression-sqrt-frac-a-64-a-64-a.htmlhttps://pop.com.sg/for-how-many-natural-numbers-a-is-the-expression-sqrt-frac-a-64-a-64-a.htmlFor how many natural numbers $a$ is the expression $\sqrt{\frac{a+64}{a-64}}$ also a natural number? I cannot seem to do this without using trial and error....Tue, 18 Aug 2026 15:39:21 -0400https://pop.com.sg/how-to-find-the-length-of-diagonal-of-a-rhombus.htmlhttps://pop.com.sg/how-to-find-the-length-of-diagonal-of-a-rhombus.htmlsuppose,only the length of the side of a rhombus is given then how can I find the length of the diagonals?(without measuring the angle) Is there any equation where the side is related to the diagonal ...Tue, 18 Aug 2026 15:36:28 -0400https://pop.com.sg/how-do-i-solve-this-square-root-problem.htmlhttps://pop.com.sg/how-do-i-solve-this-square-root-problem.htmlI need to solve the following problem: $$\frac{\sqrt{7+\sqrt{5}}}{\sqrt{7-\sqrt{5}}}=\,?$$...Tue, 18 Aug 2026 15:31:40 -0400https://pop.com.sg/extending-the-p-adic-valuation.htmlhttps://pop.com.sg/extending-the-p-adic-valuation.htmlGiven a prime $p$, the $p$-adic valuation on the field $\mathbb{Q}$ is the map $\nu:\mathbb{Q}^*\to\mathbb{Z}$ given by $\nu(p^ka/b)=k$, where $a,b$ are prime to $p$. I want to consider extension......Tue, 18 Aug 2026 15:26:55 -0400https://pop.com.sg/how-to-find-the-n-th-term-of-the-series-1-22-333-4444-55555.htmlhttps://pop.com.sg/how-to-find-the-n-th-term-of-the-series-1-22-333-4444-55555.htmlHow to find the $n$-th term of the series $1, 22, 333, 4444, 55555..........?$ Here the $i^{\text{th}}$ term is the concatenation of $i$ for $i$ times. for ef $13$ the term = $13$ concatenated $13$......Tue, 18 Aug 2026 15:22:57 -0400https://pop.com.sg/on-when-and-when-not-to-divide-by-tan-theta.htmlhttps://pop.com.sg/on-when-and-when-not-to-divide-by-tan-theta.htmlThe textbook I'm using has this problem involving trig equations which goes $\tan(x) \sin^2(x)=2 \tan(x)$; we're trying to solve for x. The textbook advises not to divide this problem by $\tan(x)$ ......Tue, 18 Aug 2026 15:22:28 -0400https://pop.com.sg/having-a-unique-solution-for-a-rectangular-matrix.htmlhttps://pop.com.sg/having-a-unique-solution-for-a-rectangular-matrix.htmlSuppose we are given a $4\times3$ matrix $A$ and a column $b$ such that $Ax=b$ always has a unique solution. What is the reduced row echelon form of $A$? I think that since the solution is unique,......Tue, 18 Aug 2026 15:18:55 -0400https://pop.com.sg/how-or-why-are-implicit-functions-actually-functions.htmlhttps://pop.com.sg/how-or-why-are-implicit-functions-actually-functions.htmlI'm a high schooler and today my teacher taught us the implicit differentiation, in which he gave us a very brief explanation of implicit function. I didn't quite get it at that time so I decided to ...Tue, 18 Aug 2026 15:13:54 -0400https://pop.com.sg/jacobian-calculation-with-homographic-coordinates.htmlhttps://pop.com.sg/jacobian-calculation-with-homographic-coordinates.htmlI'm trying to work my way through an image processing paper (Creating Full View Panoramic Image Mosaics and Environment Maps) and am stuck on how the Jacobian is calculated, especially terms ($J_{1......Tue, 18 Aug 2026 15:03:33 -0400https://pop.com.sg/third-order-differential-equations.htmlhttps://pop.com.sg/third-order-differential-equations.htmlI am having trouble solving the third order differential equation $y'''+y'=0$ It was given to me in a quiz (which I got wrong) with boundary conditions $y(0) = 0$ $y'(0)=2$ $y(\pi)=6$ I know t......Tue, 18 Aug 2026 15:03:15 -0400https://pop.com.sg/why-do-we-need-prime-ideals-in-the-spectrum-of-a-ring.htmlhttps://pop.com.sg/why-do-we-need-prime-ideals-in-the-spectrum-of-a-ring.htmlI'm reading Atiyah Macdonald, where they introduce in the exercises of chapter one a topological space $\operatorname{Spec}(A)$ associated to a ring $A$, which is defined as $\operatorname{Spec}(A) \...Tue, 18 Aug 2026 15:01:00 -0400