https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usMon, 17 Aug 2026 20:57:53 -0400https://pop.com.sg/limits-as-x-approaches-one-from-the-left.htmlhttps://pop.com.sg/limits-as-x-approaches-one-from-the-left.htmlHow do I find $$\lim_{x \to 1^-} -\frac{1}{x-1}$$ Can someone please show me how to work it out algebraically?...Tue, 18 Aug 2026 00:57:00 -0400https://pop.com.sg/prove-that-sqrt-45-is-irrational-using-euclid-s-lemma.htmlhttps://pop.com.sg/prove-that-sqrt-45-is-irrational-using-euclid-s-lemma.htmlProve that $\sqrt{45}$ is irrational (using Euclid’s Lemma) Assume $\sqrt{45}$ is rational. By definition of rational : $\sqrt{p}=\frac{a}{b}$= $\sqrt{45}$=$\frac{a}{b}$ for some $a,b$ integers a......Tue, 18 Aug 2026 00:56:05 -0400https://pop.com.sg/how-do-i-use-interpolation-with-the-z-table.htmlhttps://pop.com.sg/how-do-i-use-interpolation-with-the-z-table.htmlMy textbook has an example of interpolation, but I am not sure how the book did it since it doesn't explain it. It says if we want $P(Z < 1.246)$ we must use interpolation and the steps given a......Tue, 18 Aug 2026 00:55:21 -0400https://pop.com.sg/find-the-value-of-sin-left-cos-1-frac-1-2-sin-1-frac-3-5-right.htmlhttps://pop.com.sg/find-the-value-of-sin-left-cos-1-frac-1-2-sin-1-frac-3-5-right.htmlFind the value of $\sin \left(\cos^{-1} \left(\dfrac {1}{2}\right) + \sin^{-1} \left(\dfrac {3}{5}\right)\right)$ My Attempt: \begin{align*} \sin \left(\cos^{-1} \left(\dfrac {1}{2}\right) + \sin......Tue, 18 Aug 2026 00:53:43 -0400https://pop.com.sg/partial-derivative-of-integral-leibniz-rule.htmlhttps://pop.com.sg/partial-derivative-of-integral-leibniz-rule.htmlThe Leibniz rule is as follows: $$\frac{d}{d\alpha} \int_{a(\alpha)}^{b(\alpha)} f(x, \alpha) dx = \frac{db(\alpha)}{d\alpha} f(b(\alpha), \alpha) - \frac{da(\alpha)}{d\alpha} f(a(\alpha), \alpha)......Tue, 18 Aug 2026 00:40:42 -0400https://pop.com.sg/intuition-behind-an-integral-identity.htmlhttps://pop.com.sg/intuition-behind-an-integral-identity.htmlA proof for the identity $$\int_{-\infty}^{\infty} f(x)\, dx=\int_{-\infty}^{\infty} f\left(x-\frac{1}{x}\right)\, dx,$$ has been asked before (for example, here), and one answer to that question ...Tue, 18 Aug 2026 00:40:35 -0400https://pop.com.sg/volume-of-tetrahedron-using-cross-and-dot-product.htmlhttps://pop.com.sg/volume-of-tetrahedron-using-cross-and-dot-product.htmlConsider the tetrahedron in the image: Prove that the volume of the tetrahedron is given by $\frac16 |a \times b \cdot c|$. I know volume of the tetrahedron is equal to the base area times height......Tue, 18 Aug 2026 00:34:01 -0400https://pop.com.sg/how-is-a-decibel-a-tenth-of-a-bel.htmlhttps://pop.com.sg/how-is-a-decibel-a-tenth-of-a-bel.htmlBy definition, one Bel is log10(P2/P1) (in the power example) and one dB is 10log10(P2/P1), so it appears to me that one dB is ten Bels, not one-tenth of a Bel. Yet, many online publications, univ......Tue, 18 Aug 2026 00:28:12 -0400https://pop.com.sg/why-does-the-higher-order-derivative-test-work.htmlhttps://pop.com.sg/why-does-the-higher-order-derivative-test-work.htmlI'm an AP Calculus BC student, so all I know about derivatives is the increasing/decreasing/ relative max/min function relation with first derivative (first derivative test), concavity (second deri......Tue, 18 Aug 2026 00:27:02 -0400https://pop.com.sg/work-to-pump-water-from-a-cylindrical-tank.htmlhttps://pop.com.sg/work-to-pump-water-from-a-cylindrical-tank.htmlWe have a cylindrical tank with radius 2m and length 10m filled with water. How much work does it tank to pump the water out of the tank from the top? My attempt at the problem goes as follow. $g......Tue, 18 Aug 2026 00:23:19 -0400https://pop.com.sg/subtracting-multiple-logarithms.htmlhttps://pop.com.sg/subtracting-multiple-logarithms.htmlI am asked to simplify $$J=\ln (x^2-16)-\ln x-\ln(x-4), \quad x>4$$ Since each logarithm's argument is non-negative, I can use $$\ln x -\ln y=\ln\frac{x}{y}$$ and obtain the correct answer $$\ln......Tue, 18 Aug 2026 00:20:02 -0400https://pop.com.sg/is-the-equator-a-line.htmlhttps://pop.com.sg/is-the-equator-a-line.htmlThe equator is a circle that goes around the earth. Does that mean that it is a line? I'm guessing that it is because it is infinite. At the same time, it's overlapping. The real question is, is a ...Tue, 18 Aug 2026 00:07:20 -0400https://pop.com.sg/what-is-an-empty-set.htmlhttps://pop.com.sg/what-is-an-empty-set.htmlWe define the term "Set" as, A set is a collection of objects. And an "Empty set" as, An empty set is a set which contains nothing. First problem I encountered: How the definition of "Emp......Tue, 18 Aug 2026 00:02:47 -0400https://pop.com.sg/dragon-curve-formula.htmlhttps://pop.com.sg/dragon-curve-formula.htmlCan someone please explain to me this formula?...Mon, 17 Aug 2026 23:51:13 -0400https://pop.com.sg/is-determinant-of-matrix-multiplied-its-transpose-always-positive.htmlhttps://pop.com.sg/is-determinant-of-matrix-multiplied-its-transpose-always-positive.htmlAssume $A$ is an arbitrary $m\times n$ real matrix. Is $\det(AA^T)$ always positive? Is it non-negative or it can have any value? Edit: It seems I have to emphasis that $m \ne n$ i.e. matrix is non-...Mon, 17 Aug 2026 23:44:23 -0400https://pop.com.sg/the-equation-of-the-line-pass-through-2-points-in-3d-space-closed.htmlhttps://pop.com.sg/the-equation-of-the-line-pass-through-2-points-in-3d-space-closed.htmlI'm looking for a line equation passes through $2$ points in a $3$-dimensional space, and use it to determine the intersection between sphere and line. Any help would be appreciated...Mon, 17 Aug 2026 23:35:35 -0400https://pop.com.sg/what-do-french-mathematicians-call-a-ray.htmlhttps://pop.com.sg/what-do-french-mathematicians-call-a-ray.htmlIn the USA we are taught at a young age that a ray is a line with one end, a compromise between a line and a line segment, something like $[0,\infty)$. Is there a word for this in french? I have ta......Mon, 17 Aug 2026 23:33:02 -0400https://pop.com.sg/the-graphs-of-f-x-y-x-2-y-2-and-g-x-y-x-2-y-2-xy-3-are-tangent-to-each.htmlhttps://pop.com.sg/the-graphs-of-f-x-y-x-2-y-2-and-g-x-y-x-2-y-2-xy-3-are-tangent-to-each.htmlWhy can we say that the graphs of $f(x,y)=x^2+y^2$ and $g(x,y)=-x^2-y^2+xy^3$ "are tangent to each other" at $(0,0)$ ? I have done the following: The tangent plane at the graph of $f(x,y)=x^2+y^2......Mon, 17 Aug 2026 23:22:31 -0400https://pop.com.sg/whats-is-the-meaning-of-the-derivative-of-a-vector-function.htmlhttps://pop.com.sg/whats-is-the-meaning-of-the-derivative-of-a-vector-function.htmlAssuming we have a continuous and well behaved vector function $$ R(t) = \langle f(t), g(t), h(t) \rangle $$ then its derivative at an arbitrary point a is $R'(a)$, which can be computed (I'm not ......Mon, 17 Aug 2026 23:21:48 -0400https://pop.com.sg/relation-between-a-convex-set-and-convex-function.htmlhttps://pop.com.sg/relation-between-a-convex-set-and-convex-function.htmlI was reading about convex sets and convex functions. I would like to know if there is a relation between convex functions and convex sets. Like, a function whose domain and range is a convex set is ...Mon, 17 Aug 2026 23:05:28 -0400https://pop.com.sg/find-the-average-velocity-and-the-instantaneous-velocity.htmlhttps://pop.com.sg/find-the-average-velocity-and-the-instantaneous-velocity.htmlFind the average velocity of an object whose position is given by: $$s(t)=\frac{13}{t+2}$$ Using the following intervals: i.[11,12] ii. [11, 11.1] iii. [11, 11.001] iv. [11, 1.00001] Then gu......Mon, 17 Aug 2026 22:58:43 -0400https://pop.com.sg/constructing-an-isomorphism.htmlhttps://pop.com.sg/constructing-an-isomorphism.htmlI need to construct an isomorphism: $T: W \to C^3$ where W is a subspace of $P_3(C)$ I got bases (for W and $C^3$ respectively) $\alpha = \{(ix + 1),\ \ (x^3 + 2x), \ (2ix^2 + ix -1)\}$ $\beta ......Mon, 17 Aug 2026 22:58:34 -0400https://pop.com.sg/trick-with-3-digit-numbers-always-get-1089.htmlhttps://pop.com.sg/trick-with-3-digit-numbers-always-get-1089.htmlWhen I was in primary school a teacher showed us the following exercise in arithmetic. Take any 3 digit number between 201 and 998 provided that the hundreds digit is bigger than the ones digit an......Mon, 17 Aug 2026 22:54:33 -0400https://pop.com.sg/how-to-calculate-moment-of-inertia-of-parabolic-spandrel.htmlhttps://pop.com.sg/how-to-calculate-moment-of-inertia-of-parabolic-spandrel.htmlI want to find the moment of inertia of this spandrel along $DD'$ $$I=\int r^2 \,\mathrm dm$$ The area of the spandrel is $\frac{3m}{ab}$ so $I=\sigma\int r^2\mathrm dA $ where $\sigma$ is aerial ...Mon, 17 Aug 2026 22:53:04 -0400https://pop.com.sg/simple-proof-show-that-4-n-1-is-divisible-by-3-guided-proof-task.htmlhttps://pop.com.sg/simple-proof-show-that-4-n-1-is-divisible-by-3-guided-proof-task.htmlFirst part of the task is just to show that $(4^n-1)$ actually is divisible by 3 for n=1,2,3,4. No problem. Second step: is to show that $(4^n -1) = (2^n-1)(2^n+1)$ No problem, just algebra. Third ...Mon, 17 Aug 2026 22:45:55 -0400https://pop.com.sg/context-sensitive-grammar-for-the-copy-language.htmlhttps://pop.com.sg/context-sensitive-grammar-for-the-copy-language.htmlI need to find a context-sensitive grammar for the copy language $$L = \{ww \; | w \in \{0,1\}^* \}$$ This is what I got so far: $\begin{eqnarray} S & \Rightarrow & \lambda \; | \; X \\ X......Mon, 17 Aug 2026 22:41:06 -0400https://pop.com.sg/continuous-projections-on-ell-1-with-norm-1.htmlhttps://pop.com.sg/continuous-projections-on-ell-1-with-norm-1.htmlI was trying to find papers and articles about non-contractive continuous projections on $\ell_1(S)$ where $S$ is an arbitrary set. If it is not studied yet, I would like to know results for the ca......Mon, 17 Aug 2026 22:37:08 -0400https://pop.com.sg/understanding-the-unit-circle.htmlhttps://pop.com.sg/understanding-the-unit-circle.htmlI'm going to need to know this (the unit circle, English) for a test, but I'm so horrible with both charts and memorization, the whole thing is failing to stick into my brain. Using non-complicated ...Mon, 17 Aug 2026 22:27:30 -0400https://pop.com.sg/a-different-type-of-infinite-power-tower-function.htmlhttps://pop.com.sg/a-different-type-of-infinite-power-tower-function.htmlThe question is about the following: Let a function be defined such that $$f_n(x) = x \uparrow \uparrow n$$ Where $n$ is a natural number Now, it is reported at many places that the function $$F_......Mon, 17 Aug 2026 22:24:32 -0400https://pop.com.sg/how-to-compute-the-similarity-transformation-matrix.htmlhttps://pop.com.sg/how-to-compute-the-similarity-transformation-matrix.htmlStuck on this question: Let $$A=\begin{pmatrix} 2&1\\ -1&-1 \end{pmatrix}$$$$B=\begin{pmatrix} -2&5\\ -1&3 \end{pmatrix}$$$$C=\begin{pmatrix} 5&2\\ 4&1 \end{pmatrix}$$ ......Mon, 17 Aug 2026 22:23:12 -0400https://pop.com.sg/a-die-is-rolled-and-a-coin-is-flipped-what-is-wrong-with-the-following.htmlhttps://pop.com.sg/a-die-is-rolled-and-a-coin-is-flipped-what-is-wrong-with-the-following.htmlA die is rolled and a coin is flipped. What is wrong with the following reasoning?: Let $A=\{1,3,6\}$ (event from die roll), let $B=\{H\}$ (event of heads) then, $A\cap B=\emptyset$ (since they ha......Mon, 17 Aug 2026 22:15:21 -0400https://pop.com.sg/how-to-calculate-carmichael-function-of-a-large-number.htmlhttps://pop.com.sg/how-to-calculate-carmichael-function-of-a-large-number.htmlHow would I go about calculating $\lambda(49392)$ by hand? I have $\lambda$ defined in a book as $$ \lambda(n) = max{\{\bar{a} | \bar{a} \in Z_n^*\}} \ \cdot \ ord_{Z_n^{*}} (\bar{a}) $$ where $......Mon, 17 Aug 2026 22:14:12 -0400https://pop.com.sg/sequence-vs-series.htmlhttps://pop.com.sg/sequence-vs-series.htmlWhat is the difference between a sequence and a series and how should they be used i.e. give examples of the usage of these terminologies in separate senarios....Mon, 17 Aug 2026 22:04:08 -0400https://pop.com.sg/regular-distributions.htmlhttps://pop.com.sg/regular-distributions.htmlI've somehow a really hard time showing that a distribution in $\mathcal{D'}(\mathbb{R^n})$ is regular. The definition seems straight forward: $ f \in \mathcal{D'}(\mathbb{R^n}) \text{ is regula......Mon, 17 Aug 2026 21:59:58 -0400https://pop.com.sg/equation-of-a-line-tangent-to-circumference.htmlhttps://pop.com.sg/equation-of-a-line-tangent-to-circumference.htmlDiscover the general equation of the tangent line to the circumference $x^2 + y^2 - 2x + 4y + 1 = 0$ by the point $(3,4)$. NO CALCULUS. by the circumference equation i discovered that $C(1, -......Mon, 17 Aug 2026 21:52:28 -0400https://pop.com.sg/calculator-similar-to-desmos-but-for-3-d.htmlhttps://pop.com.sg/calculator-similar-to-desmos-but-for-3-d.htmlIs there a calculator with functionality similar to Desmos but in $3$ dimensions? I am looking to learn about families of quadric surfaces so I am looking for a $3$D calculator with sliders....Mon, 17 Aug 2026 21:51:59 -0400https://pop.com.sg/differentiability-of-the-cantor-function.htmlhttps://pop.com.sg/differentiability-of-the-cantor-function.htmlI know that the Cantor function is differentiable a.e. but I want to prove it without using the theorem about monotonic functions. I have already proved that $f'(x) = 0$ for all $x \in [0,1] \backs......Mon, 17 Aug 2026 21:49:59 -0400https://pop.com.sg/chernoff-bound-probability-value-of-n-so-that-with-probability-999-at-.htmlhttps://pop.com.sg/chernoff-bound-probability-value-of-n-so-that-with-probability-999-at-.htmlProblem 3: Consider a biased coin with probability $p = \frac{1}{3}$ of landing heads and probability $\frac{2}{3}$ of landing tails. Suppose the coin is flipped some number $n$ of times, and let $......Mon, 17 Aug 2026 21:37:11 -0400https://pop.com.sg/standard-notation-for-writing-probability-fraction-vs-percentage.htmlhttps://pop.com.sg/standard-notation-for-writing-probability-fraction-vs-percentage.htmlI have a question about the usage of fractions and percentages in questions related to probability. I am wondering if a fraction is the same as a percentage. To better explain my question, I will p......Mon, 17 Aug 2026 21:27:08 -0400https://pop.com.sg/are-functions-of-independent-variables-also-independent.htmlhttps://pop.com.sg/are-functions-of-independent-variables-also-independent.htmlIt's a really simple question. However I didn't see it in books and I tried to find the answer on the web but failed. If I have two independent random variables, $X_1$ and $X_2$, then I define two......Mon, 17 Aug 2026 21:20:51 -0400https://pop.com.sg/proof-difference-of-absolute-values-and-absolute-value-difference-dupl.htmlhttps://pop.com.sg/proof-difference-of-absolute-values-and-absolute-value-difference-dupl.htmlprove that for all $a,b \in \mathbb{R}$, $ | |a|-|b|| ≤ |a-b|$ so what I have thought of so fat is I tried to use the definition of absolute value then getting $b-a ≤ |a|-|b| ≤ a-b.$ from there I'm ...Mon, 17 Aug 2026 21:20:39 -0400https://pop.com.sg/understanding-how-em-algorithm-actually-works-for-missing-data.htmlhttps://pop.com.sg/understanding-how-em-algorithm-actually-works-for-missing-data.htmlI am currently studying EM algorithm for handling missing data in a data set. I understand that the final goal of EM algorithm is not to impute data, but to calculate the parameters of interest. Ho......Mon, 17 Aug 2026 21:06:35 -0400https://pop.com.sg/find-ln-0-2-accurate-to-0-001.htmlhttps://pop.com.sg/find-ln-0-2-accurate-to-0-001.htmlI want to find ln(0.2) accurate to 0.001. I get ln(1+(-0.8)) to solve the question. When I find the remainder I get the below one. So I should find n for find ln(0.2). I find n=23 using a calculat......Mon, 17 Aug 2026 20:52:38 -0400https://pop.com.sg/finding-prime-factors-by-taking-the-square-root.htmlhttps://pop.com.sg/finding-prime-factors-by-taking-the-square-root.htmlI'm trying to solve the third Project Euler problem and I'd like a little help understanding a mathematical concept underlying my tentative solution. The question reads: The prime factors of 13......Mon, 17 Aug 2026 20:44:47 -0400https://pop.com.sg/derivative-of-5x-2-3.htmlhttps://pop.com.sg/derivative-of-5x-2-3.htmlHow is the derivative of $(5x-2)^3$ equal to $15(5x-2)^2$ and not $3(5x-2)^2$. According to $\frac{df}{dx} = nx^{n-1}$, it has to be $3(5x-2)^2$ right. Please explain....Mon, 17 Aug 2026 20:42:47 -0400https://pop.com.sg/arrival-probability-of-merged-poisson-process-closed.htmlhttps://pop.com.sg/arrival-probability-of-merged-poisson-process-closed.htmlIf we have two Poisson process $A$ and $B$ with arrival rate $\lambda_{1}$ and $\lambda_{2}$. Now I have question for calculating the probability of an arrival e.g., from process A in the case of m......Mon, 17 Aug 2026 20:36:13 -0400https://pop.com.sg/questions-tagged-recurrence-relations.htmlhttps://pop.com.sg/questions-tagged-recurrence-relations.htmlQ&A for people studying math at any level and professionals in related fields...Mon, 17 Aug 2026 20:27:14 -0400https://pop.com.sg/probability-of-three-events-occurring-given-correlation.htmlhttps://pop.com.sg/probability-of-three-events-occurring-given-correlation.htmlI am facing a problem that I cannot find the answer to. I have three variables, A, B and C. There are only two possibilities for each of these, A either happens or it does not, B happens or it does......Mon, 17 Aug 2026 20:17:41 -0400https://pop.com.sg/find-a-function-from-values.htmlhttps://pop.com.sg/find-a-function-from-values.htmlIs there any way to find a function, even just similar, from a set of values? I get these pairs of values from two sensors and would like to find a simple function that describes the relationship ...Mon, 17 Aug 2026 20:10:30 -0400https://pop.com.sg/laplacian-in-polar-coordinates-nabla-2f-r-f-039-039-r-frac-2-r-f-039-r.htmlhttps://pop.com.sg/laplacian-in-polar-coordinates-nabla-2f-r-f-039-039-r-frac-2-r-f-039-r.html$\vec{R}=x \hat{i} + y\hat{j} + z\hat{k}$ and $r=|\vec{R}|=\sqrt{x^2+y^2+z^2}$ Prove that $\nabla^2f(r)=f''(r)+\frac{2}{r}f'(r)$ So we need to basically show that $$\frac{\partial^2}{\parti......Mon, 17 Aug 2026 20:07:28 -0400https://pop.com.sg/how-to-find-the-matrix-of-a-linear-transformation-with-respect-to-two-.htmlhttps://pop.com.sg/how-to-find-the-matrix-of-a-linear-transformation-with-respect-to-two-.htmlI'm having some trouble understanding the process of actually finding what $[T]_\beta ^\gamma$ is, given $2$ bases $\beta$ and $\gamma$. Here's an example: Let $T$: $P_3(\mathbb{R})$ $\rightarro......Mon, 17 Aug 2026 20:00:08 -0400https://pop.com.sg/geometric-interpretation-of-the-cofactor-expansion-theorem.htmlhttps://pop.com.sg/geometric-interpretation-of-the-cofactor-expansion-theorem.htmlI find the geometric interpretation of determinants to be really intuitive - they are the "area" created by the column vectors of the matrix. Could someone give me a geometric interpretation of the ...Mon, 17 Aug 2026 19:51:03 -0400https://pop.com.sg/matrix-of-a-quadratic-form.htmlhttps://pop.com.sg/matrix-of-a-quadratic-form.htmlWhat exactly is the matrix of a quadratic form? I have seen this notation occuring in a few papers (e.g. Siegel's unreadable German papers), with particular reference to the trace of a quadratic fo......Mon, 17 Aug 2026 19:49:30 -0400https://pop.com.sg/horizontal-tank-with-hemispherical-ends-depth-to-capacity-calculation.htmlhttps://pop.com.sg/horizontal-tank-with-hemispherical-ends-depth-to-capacity-calculation.htmlI am trying to find an accurate way of calculating the capacity of an underground tank at a given depth. The tank manufacturer has provided a strapping table for the tank which tells me the capaci......Mon, 17 Aug 2026 19:46:18 -0400https://pop.com.sg/what-is-the-difference-between-the-big-o-and-big-o-star-asterisk-opera.htmlhttps://pop.com.sg/what-is-the-difference-between-the-big-o-and-big-o-star-asterisk-opera.htmlI'm doing some research on algorithms complexity and in different papers I notice both the use of the regular Big-O operator O(...) and a variant O*(...). I never saw the definition of the latter ......Mon, 17 Aug 2026 19:42:32 -0400https://pop.com.sg/about-a-divergence-theorem-proof.htmlhttps://pop.com.sg/about-a-divergence-theorem-proof.htmlI'm currently working through a proof of the divergence theorem for simple volumes. I've found some steps of the proof not formal enough, and I have not seen anything online that was of help. Here ......Mon, 17 Aug 2026 19:38:04 -0400https://pop.com.sg/what-is-the-remainder-of-65-divided-by-67.htmlhttps://pop.com.sg/what-is-the-remainder-of-65-divided-by-67.htmlWhat is the remainder of $65!$ divided by $67$? Attempt: By Wilson's theorem, we have $66! = -1\mod(67) $. $$66! = -1\mod(67) \implies 66 (65!) = -1 \mod(67)$$ and we also know that $66 = -1 \m......Mon, 17 Aug 2026 19:31:49 -0400https://pop.com.sg/proving-the-curl-of-a-gradient-is-zero.htmlhttps://pop.com.sg/proving-the-curl-of-a-gradient-is-zero.htmlI'm having trouble proving $$\nabla\times(\nabla f)=0$$ using index notation. I have started with: $$(\hat{e_i}\partial_i)\times(\hat{e_j}\partial_j f)=\partial_i\partial_jf(\hat{e_i}\times\hat{e_j......Mon, 17 Aug 2026 19:30:32 -0400https://pop.com.sg/4th-order-tensors-double-dot-product-and-inverse-computation.htmlhttps://pop.com.sg/4th-order-tensors-double-dot-product-and-inverse-computation.htmlI am currently working on a subject that involves a lot of 4th order tensors computations including double dot product and inverse of fourth order tensors. First the definitions so that we are on the ...Mon, 17 Aug 2026 19:28:31 -0400https://pop.com.sg/simpler-proof-of-an-integral-representation-of-bessel-function-of-the-.htmlhttps://pop.com.sg/simpler-proof-of-an-integral-representation-of-bessel-function-of-the-.htmlWhile doing research in electrical engineering, I derived the following integral representation of the Bessel function of the first kind: $$J_n(x)=\frac{e^{in\pi/2}}{2\pi}\int_0^{2\pi}e^{i(n\tau-x......Mon, 17 Aug 2026 19:27:11 -0400https://pop.com.sg/lim-a-b-when-lim-b-does-not-exist-closed.htmlhttps://pop.com.sg/lim-a-b-when-lim-b-does-not-exist-closed.htmlSuppose $a$ and $b$ are functions of $x$. Is it guaranteed that $$ \lim_{x \to +\infty} a + b\text{ does not exist} $$ when $$ \lim_{x \to +\infty} a = c\quad\text{and}\quad \lim_{x \to +\infty......Mon, 17 Aug 2026 19:25:13 -0400https://pop.com.sg/how-to-convert-sum-into-formula.htmlhttps://pop.com.sg/how-to-convert-sum-into-formula.htmlI am reading some combinatorics books. And here author first obtained a sum answer for a problem and then converted it to formula without explaining it. Just writing equality sign. This is that ...Mon, 17 Aug 2026 19:09:56 -0400https://pop.com.sg/how-do-i-calculate-the-area-under-a-curve-using-the-midpoints-of-recta.htmlhttps://pop.com.sg/how-do-i-calculate-the-area-under-a-curve-using-the-midpoints-of-recta.htmlI figured out how to calculate the area under the curve from the Right endpoint and Left endpoints, but I can't figure out how to calculate it using the midpoints. Especially when it says $M_3$. Il......Mon, 17 Aug 2026 19:06:56 -0400https://pop.com.sg/the-best-strategy-and-highest-possible-score-for-the-threes-game.htmlhttps://pop.com.sg/the-best-strategy-and-highest-possible-score-for-the-threes-game.html[There's still the strategy to go. A suitably robust argument that establishes what is statistically the best strategy will be accepted.] Here's my description of the game: There's a $4\times 4$ grid ...Mon, 17 Aug 2026 19:06:41 -0400https://pop.com.sg/generalized-regular-polygon-area-formula-to-any-number-of-sides.htmlhttps://pop.com.sg/generalized-regular-polygon-area-formula-to-any-number-of-sides.htmlI was looking for a way to check if the regular polygon area formula of $\frac{ns^2}{4} \cot \frac{π}{n}$ where s is the side length and n is the number of sides could be extended to fractional and ...Mon, 17 Aug 2026 19:06:16 -0400https://pop.com.sg/creating-a-steady-state-vector.htmlhttps://pop.com.sg/creating-a-steady-state-vector.htmlI'm confused on where the intuition came from to divide $w$ by the sum of its entries to find $q$. I don't really see the relation from the sum of its entries with "every solution being a multiple ......Mon, 17 Aug 2026 19:05:03 -0400https://pop.com.sg/if-n-is-a-positive-integer-then-2-n-2-2-n-2-2-2n-1.htmlhttps://pop.com.sg/if-n-is-a-positive-integer-then-2-n-2-2-n-2-2-2n-1.htmlI'm not sure why $$(-2^n)^{-2} + (2^{-n})^2=2^{-2n+1}$$ I have been going over this equation for a while now, noticing, and have successfully got quite far in the equation, finding that $$ (-2^n......Mon, 17 Aug 2026 19:03:05 -0400https://pop.com.sg/if-you-have-100-spheres-packed-into-a-sphere-shape-how-many-will-be-on.htmlhttps://pop.com.sg/if-you-have-100-spheres-packed-into-a-sphere-shape-how-many-will-be-on.htmlMy question is more about ratios. I'm wondering is there a calculator or formula I can input an X number of Spheres. If the spheres are packed in a spherical shape what is the ratio of The interior ...Mon, 17 Aug 2026 19:00:37 -0400https://pop.com.sg/wolframalpha-says-limit-exists-when-it-doesn-039-t.htmlhttps://pop.com.sg/wolframalpha-says-limit-exists-when-it-doesn-039-t.htmlI was trying to calculate the following limit: $$ \lim_{(x,y)\to (0,0)} \frac{(x^2+y^2)^2}{x^2+y^4} $$ and, feeding it into WolframAlpha, I obtain the following answer, stating the limit is $0$: ...Mon, 17 Aug 2026 19:00:18 -0400https://pop.com.sg/why-should-kids-learn-how-to-use-a-compass-and-straightedge-and-not-re.htmlhttps://pop.com.sg/why-should-kids-learn-how-to-use-a-compass-and-straightedge-and-not-re.htmlI am curious why it is necessary for people to learn how to use compasses and straightedges in geometry, and not just rely on a drawing program. I have a couple ideas, but I might be missing somet......Mon, 17 Aug 2026 18:53:07 -0400https://pop.com.sg/how-can-i-solve-cos-2-x-sin-x-1-0.htmlhttps://pop.com.sg/how-can-i-solve-cos-2-x-sin-x-1-0.htmlThe solution set of the equation $$\cos^2 x + \sin x +1 = 0$$ is? I haven't studied trigonometry, I'm kinda lost on this issue ......Mon, 17 Aug 2026 18:49:20 -0400https://pop.com.sg/given-graph-find-degree.htmlhttps://pop.com.sg/given-graph-find-degree.htmlGiven the graph I'm asked for the degree. I can see how it is odd. I can see how the $a$ value is negative. I can see the inflection point at $0$. My question is how would I know if this is degree......Mon, 17 Aug 2026 18:33:12 -0400https://pop.com.sg/example-of-bijective-function-from-1-1-to-bbb-r.htmlhttps://pop.com.sg/example-of-bijective-function-from-1-1-to-bbb-r.htmlI'm looking for a bijective function $f : (-1,1) \to \mathbb{R}$. I'm having trouble finding an example....Mon, 17 Aug 2026 18:25:21 -0400https://pop.com.sg/h-cartan-differential-calculus-query.htmlhttps://pop.com.sg/h-cartan-differential-calculus-query.htmlIn H. Cartan - Differential Calculus (1971) p. 29 he investigates differentiating a bi-linear function $f: E_1 \times E_2 \to F$ where $E_1, E_2, F$ are Banach spaces and $E_1 \times E_2$ the produ......Mon, 17 Aug 2026 18:25:13 -0400https://pop.com.sg/find-area-enclosed-by-curve.htmlhttps://pop.com.sg/find-area-enclosed-by-curve.htmlI want to find the area enclosed by the plane curve $x^{2/3}+y^{2/3}=1$. My attempt was to set $x=\cos^3t, \ y=\sin^3t$ so:$$x^{2/3}+y^{2/3}=\cos^2t+\sin^2t=1$$ Then the area is $$2A=\oint_Cxdy-yd......Mon, 17 Aug 2026 18:24:54 -0400https://pop.com.sg/could-a-square-be-a-perfect-number.htmlhttps://pop.com.sg/could-a-square-be-a-perfect-number.htmlA perfect number is the sum of its (positive) divisors (excluding itself). I am wondering if a square could be a perfect number. If it is an odd square, then, excluding itself, it has an even num......Mon, 17 Aug 2026 18:16:32 -0400https://pop.com.sg/decompose-a-simple-function-into-the-sum-of-two-simple-functions.htmlhttps://pop.com.sg/decompose-a-simple-function-into-the-sum-of-two-simple-functions.htmlLet $(\Omega, \mathscr F)$ be a measurable space and let $f,g : \Omega \to \mathbb R$ be non-negative Borel functions. Suppose that $\psi$ is a non-negative simple function, i.e. $\psi = \sum_{i=1}......Mon, 17 Aug 2026 18:07:18 -0400https://pop.com.sg/why-we-can-039-t-have-radicals-with-negative-index.htmlhttps://pop.com.sg/why-we-can-039-t-have-radicals-with-negative-index.htmlI am a high school student and we just learned about radical and radical notation. Our teacher says index of radical must be integer and greater than 2 by definition. But I can’t understand why we ......Mon, 17 Aug 2026 18:00:32 -0400https://pop.com.sg/useful-examples-of-pathological-functions.htmlhttps://pop.com.sg/useful-examples-of-pathological-functions.htmlWhat are some particularly well-known functions that exhibit pathological behavior at or near at least one value and are particularly useful as examples? For instance, if $f'(a) = b$, then $f(......Mon, 17 Aug 2026 17:56:47 -0400https://pop.com.sg/example-of-integration-by-parts-in-higher-dimension.htmlhttps://pop.com.sg/example-of-integration-by-parts-in-higher-dimension.htmlI'm looking for a concrete example of an application of integration by parts in higher dimensions. The formula I'm looking at is from here, here, and here. $\Omega$ is an open bounded subset of $\...Mon, 17 Aug 2026 17:45:10 -0400https://pop.com.sg/how-to-show-the-fundamental-group-of-torus-is-abelian-in-a-homotopic-w.htmlhttps://pop.com.sg/how-to-show-the-fundamental-group-of-torus-is-abelian-in-a-homotopic-w.htmlI know the torus is homeomorphic to $S^1 \times S^1$ and the fundamental group is $ \mathbb{Z} \times \mathbb{Z} $, but in the real case, (let the generators of the torus's fundamental group be $a$......Mon, 17 Aug 2026 17:40:23 -0400https://pop.com.sg/misunderstanding-the-taylor-remainder-theorem.htmlhttps://pop.com.sg/misunderstanding-the-taylor-remainder-theorem.htmlI believe I am misinterpreting the Taylor Remainder theorem somehow. The Taylor Remainder theorem is (taken from Briggs 3rd ed Calculus: Early Transcendentals) Let $f$ have continuous derivativ......Mon, 17 Aug 2026 17:38:08 -0400https://pop.com.sg/convergent-rearrangement-of-convergent-sequence-has-limit-in-the-seque.htmlhttps://pop.com.sg/convergent-rearrangement-of-convergent-sequence-has-limit-in-the-seque.htmlI'm working on a proof for analysis: "Let $\{x_n\}$ be a convergent sequence in $\mathbb R^d$ with limit $x$. Set $$A=\{x_1,x_2,x_3,\cdots\}\cup\{x\},$$ that is, $A$ is the set consisting of all the ...Mon, 17 Aug 2026 17:27:28 -0400https://pop.com.sg/how-to-define-linear-and-non-linear-differential-equation.htmlhttps://pop.com.sg/how-to-define-linear-and-non-linear-differential-equation.htmlI have a problem understanding how to define a linear or non-linear Differential equation. These are my answers to the questions, however, my teacher's answers are not the same as mine. Questions ......Mon, 17 Aug 2026 17:27:14 -0400https://pop.com.sg/how-to-do-capital-pi-prod-notation-on-a-ti-84-calculator.htmlhttps://pop.com.sg/how-to-do-capital-pi-prod-notation-on-a-ti-84-calculator.htmlI am trying to figure out how to input product notation into my TI-84 Plus graphing calculator. I haven't found anything so far. Is there any function I could use to do this? Maybe even a program I ...Mon, 17 Aug 2026 17:26:01 -0400https://pop.com.sg/solution-for-logistic-growth-equation-with-linearly-growing-carrying-c.htmlhttps://pop.com.sg/solution-for-logistic-growth-equation-with-linearly-growing-carrying-c.htmlDoes someone have a solution for the following equation where the carrying capacity varies linearly with time (and only time): $$\frac{\mathrm dN}{\mathrm dt}= rN\left(1-\frac{N}{K}\right)$$ $K =......Mon, 17 Aug 2026 17:22:35 -0400https://pop.com.sg/natural-number-which-can-be-expressed-as-sum-of-two-perfect-squares-in.htmlhttps://pop.com.sg/natural-number-which-can-be-expressed-as-sum-of-two-perfect-squares-in.htmlRamanujan's number is $1729$ which is the least natural number which can be expressed as the sum of two perfect cubes in two different ways. But can we find a number which can be expressed as the s......Mon, 17 Aug 2026 17:13:00 -0400https://pop.com.sg/what-is-the-difference-between-probabilistic-and-physicists-hermite-po.htmlhttps://pop.com.sg/what-is-the-difference-between-probabilistic-and-physicists-hermite-po.htmlIn the equation they differ only by a fraction of 2 and in the results one is without a constant in x^n and one has those constants. Why are they defined differently and what are their separate uses?...Mon, 17 Aug 2026 17:09:21 -0400https://pop.com.sg/is-there-a-systematic-way-to-find-the-range-of-functions.htmlhttps://pop.com.sg/is-there-a-systematic-way-to-find-the-range-of-functions.htmlI was looking for steps or a systematic way to find the range of functions. The way to find domain is quite obvious: exclude x-values that make the function undefined on the real numbers. But the......Mon, 17 Aug 2026 17:00:42 -0400https://pop.com.sg/what-does-a-b-0-mean-in-matrix-theory.htmlhttps://pop.com.sg/what-does-a-b-0-mean-in-matrix-theory.htmlThis is probably very simple but I can't seem to find an answer anywhere. I'm being asked to prove $e^{tA}e^{tB} = e^{t(A+B)}$ for all $t \in \mathbb{R}$ iff $[A,B]=0$ However, I am unsure what $......Mon, 17 Aug 2026 16:59:28 -0400https://pop.com.sg/correct-notation-to-indicate-multiplying-all-elements-within-a-set.htmlhttps://pop.com.sg/correct-notation-to-indicate-multiplying-all-elements-within-a-set.htmlIs there a correct notation to indicate multiplying all elements within a set? For example, if $M = \left\{n_0, n_1, ..., n_t\right\}$ be the set of elements where I want to multiply all the numbers ...Mon, 17 Aug 2026 16:44:27 -0400https://pop.com.sg/complete-the-square-and-write-in-standard-form-for-9y-2-6y-9-x-0.htmlhttps://pop.com.sg/complete-the-square-and-write-in-standard-form-for-9y-2-6y-9-x-0.htmlComplete the square and write in standard form: $x-a=A(y-b)^2$ $9y^2-6y-9-x=0$ I do not know how to complete the square with $4$ terms. I started off like: $$9y^2-6y-9=x$$ I don't know whether......Mon, 17 Aug 2026 16:43:42 -0400https://pop.com.sg/partial-trace-of-density-operator.htmlhttps://pop.com.sg/partial-trace-of-density-operator.htmlBefore stating my question I present my motivation: to learn more about the tensor product. Now, quantum mechanics assigns a Hilbert space to each physical system as a postulate of the theory. Su......Mon, 17 Aug 2026 16:40:27 -0400https://pop.com.sg/boolean-algebra-and-simplify-boolean-expression.htmlhttps://pop.com.sg/boolean-algebra-and-simplify-boolean-expression.htmlquestion1: from =$ab+ac+a'b'c$ , can I change it to $ab+b'c$ ? =$ab+ac+a'b'c$ =$c.!(a+a'b')+ab$ -> applied demorgan, am I right? =$c(a'.(a+b))+ab$ $c(a'a+a'b)+ab$ $a'bc+ab$ $b(a'c+a)$ ->shou......Mon, 17 Aug 2026 16:33:19 -0400https://pop.com.sg/how-do-we-define-arc-length.htmlhttps://pop.com.sg/how-do-we-define-arc-length.htmlIn trying to write a nice proof of the derivatives of $\sin(x)$ and $\cos(x)$, I encountered a serious problem, namely that I have never seen a proper definition of the notion of arc length. Based on ...Mon, 17 Aug 2026 16:31:45 -0400https://pop.com.sg/histogram-and-normal-distribution.htmlhttps://pop.com.sg/histogram-and-normal-distribution.htmlI was studying histograms and normal distribution. As far as I know, they are two different tools used for calculating probability and statistics. More specifically they help to visualize and it is......Mon, 17 Aug 2026 16:29:32 -0400https://pop.com.sg/cohomology-of-s-1.htmlhttps://pop.com.sg/cohomology-of-s-1.htmlAbove is a way to find the cohomology groups of $S^1$. But what I do not get is how they concluded that $\delta$ is surjective. Can someone explain this to me?...Mon, 17 Aug 2026 16:29:29 -0400https://pop.com.sg/when-n-is-divided-by-14-the-remainder-is-10-what-is-the-remainder-when.htmlhttps://pop.com.sg/when-n-is-divided-by-14-the-remainder-is-10-what-is-the-remainder-when.htmlI need to explain this to someone who hasn't taken a math course for 5 years. She is good with her algebra. This was my attempt: Here's how this question works. To motivate what I'll be doing, ...Mon, 17 Aug 2026 16:26:55 -0400https://pop.com.sg/proving-bi-lipschitz-function-has-an-inverse-that-is-lipschitz.htmlhttps://pop.com.sg/proving-bi-lipschitz-function-has-an-inverse-that-is-lipschitz.htmlHow could I go about proving that a bi-Lipschitz function has an inverse that is a Lipschitz function. Definition 1: bi-Lipschitz function. Given metric spaces $(X,d_X)$, $(Y,d_Y)$, a function $f......Mon, 17 Aug 2026 16:26:18 -0400https://pop.com.sg/how-many-different-number-combinations-are-there-between-1000-and-2999.htmlhttps://pop.com.sg/how-many-different-number-combinations-are-there-between-1000-and-2999.htmlThis is a question I made that I would like to solve. Here are some points: I am referring to combinations, not permutations. Therefore, whilst there are 2000 permutations of numbers between 1000 ......Mon, 17 Aug 2026 16:19:41 -0400