https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usTue, 11 Aug 2026 05:33:54 -0400https://pop.com.sg/prove-in-full-detail-that-the-set-is-a-vector-space.htmlhttps://pop.com.sg/prove-in-full-detail-that-the-set-is-a-vector-space.htmlSo I'm doing a review test and I have this problem: Prove in full detail, with the standard operations in R2, that the set {(x,2x): x is a real number} is a vector space. Attempt: Given: $(x_1, 2x......Tue, 11 Aug 2026 09:31:48 -0400https://pop.com.sg/physical-meaning-of-one-dimensional-heat-equation.htmlhttps://pop.com.sg/physical-meaning-of-one-dimensional-heat-equation.htmlConsider heat conduction in a $1$-D material. The temperature distribution $u$ after sufficiently long time can be modeled by $$-(a(x)u_x)_x = f(x), \qquad x \in [0,1]$$ where $a$ is heat conduct......Tue, 11 Aug 2026 09:25:25 -0400https://pop.com.sg/is-the-absolute-value-function-a-linear-function.htmlhttps://pop.com.sg/is-the-absolute-value-function-a-linear-function.htmlI'm inclined to say yes, as it doesn't involve exponentiation, roots, logarithmic or trigonometric functions, but I watched a video where the teacher said that the absolute value function is "clear......Tue, 11 Aug 2026 09:10:18 -0400https://pop.com.sg/maximum-number-of-horizontal-asymptotes-of-a-rational-function.htmlhttps://pop.com.sg/maximum-number-of-horizontal-asymptotes-of-a-rational-function.htmlThe maximum number of horizontal asymptotes a Rational function can have is 1 according to Thomas calculus 14th edition. The only reasoning I could come up with is: given a function f(x)/g(x), if ......Tue, 11 Aug 2026 09:04:59 -0400https://pop.com.sg/image-of-intersection-of-sects-not-equal-to-intersection-of-images-of-.htmlhttps://pop.com.sg/image-of-intersection-of-sects-not-equal-to-intersection-of-images-of-.htmlHow to disprove, if $f$ is a function, $f(A \cap B) != f(A) \cap f(B)?$...Tue, 11 Aug 2026 09:02:14 -0400https://pop.com.sg/the-math-model-of-the-slot-machines-closed.htmlhttps://pop.com.sg/the-math-model-of-the-slot-machines-closed.htmlCould you please tell me or give me the sources - what we understand by a math model of the slot machine. What does it mean that the game is well or badly designed?...Tue, 11 Aug 2026 08:58:58 -0400https://pop.com.sg/convert-boolean-expression-into-sop-and-pos.htmlhttps://pop.com.sg/convert-boolean-expression-into-sop-and-pos.htmlConvert the following expression into SOP (sum of products) and POS (product of sums) canonical forms using boolean algebra method: $(ac + b)(a + b'c) + ac$ Attempt at solution: $(ac + b)(a + b......Tue, 11 Aug 2026 08:54:43 -0400https://pop.com.sg/do-negative-dimensions-make-sense-duplicate.htmlhttps://pop.com.sg/do-negative-dimensions-make-sense-duplicate.htmlSome time ago I read in a popular physics book that in M-theory, there are some "things" which can be said to have dimension $-1$. Probably, the author was vastly exaggerating, but this left me ...Tue, 11 Aug 2026 08:43:50 -0400https://pop.com.sg/what-is-milnor-fiber.htmlhttps://pop.com.sg/what-is-milnor-fiber.htmlLet $f:\mathbb{C}^{n+1}\to\mathbb{C}$ be a map with $0\in\mathbb{C}$ a critical value. Let $X_0=f^{-1}(0)$ and $x\in X_0$, then the map $F:\frac{f}{|f|}:S_{\varepsilon,x}\setminus(S_{\varepsilon,x}......Tue, 11 Aug 2026 08:41:57 -0400https://pop.com.sg/how-can-i-figure-out-what-size-shipping-box-i-need-based-on-multiple-d.htmlhttps://pop.com.sg/how-can-i-figure-out-what-size-shipping-box-i-need-based-on-multiple-d.htmlIs there a math formula I can use to provide a guideline of what length x width x height box I would need to buy based on smaller individual boxes that I want to put inside this larger box? For ex......Tue, 11 Aug 2026 08:39:06 -0400https://pop.com.sg/dividing-100-pens-among-5-children-everyone-gets-an-odd-number-of-pens.htmlhttps://pop.com.sg/dividing-100-pens-among-5-children-everyone-gets-an-odd-number-of-pens.htmlSo I want to divide 100 pens among 5 children in a way that each of them gets an odd number of pens. I think it's not possible but what is the answer and why? Tag's may be wrong, please feel free to ...Tue, 11 Aug 2026 08:35:41 -0400https://pop.com.sg/what-is-a-tensor-product-chebyshev-grid.htmlhttps://pop.com.sg/what-is-a-tensor-product-chebyshev-grid.htmlWhat is the difference between "Chebyshev grid" and "tensor-product Chebyshev grid"? Are they defined on a 2D vector?...Tue, 11 Aug 2026 08:34:52 -0400https://pop.com.sg/why-is-a-circle-1-dimensional.htmlhttps://pop.com.sg/why-is-a-circle-1-dimensional.htmlIn the textbook I am reading, it says a dimension is the number of independent parameters needed to specify a point. In order to make a circle, you need two points to specify the $x$ and $y$ positi......Tue, 11 Aug 2026 08:33:37 -0400https://pop.com.sg/independence-of-sigma-algebras-related-to-independence-of-random-varia.htmlhttps://pop.com.sg/independence-of-sigma-algebras-related-to-independence-of-random-varia.htmlIn the lecture about probability theory, we had the following defintion for independence of a finite number of sub-sigma-algebras on the probability space $(\Omega, \mathcal{A},P)$: The sub-sigma-...Tue, 11 Aug 2026 08:33:16 -0400https://pop.com.sg/explaining-the-physical-meaning-of-an-eigenvalue-in-a-real-world-probl.htmlhttps://pop.com.sg/explaining-the-physical-meaning-of-an-eigenvalue-in-a-real-world-probl.htmlContextual Problem A PhD student in Applied Mathematics is defending his dissertation and needs to make 10 gallon keg consisting of vodka and beer to placate his thesis committee. Suppose that all ...Tue, 11 Aug 2026 08:23:09 -0400https://pop.com.sg/correct-method-of-proving-raabe-039-s-test.htmlhttps://pop.com.sg/correct-method-of-proving-raabe-039-s-test.htmlI was wondering if my method of proof for Raabe's test was valid, since it is different from the normal method used with comparing to a sequence $\frac{1}{n^{p}}$ for some p > 1. Raabe's Test (As ...Tue, 11 Aug 2026 08:10:45 -0400https://pop.com.sg/solve-int-sqrt-7x-4-dx.htmlhttps://pop.com.sg/solve-int-sqrt-7x-4-dx.htmlI need to solve the following integral $$\int \sqrt{7x + 4}\,dx$$ I did the following steps: \begin{align} \text{Let} \, u &= 7x+4 \quad \text{Let} \, du = 7 \, dx \\ \int &\sqrt{u} \, ......Tue, 11 Aug 2026 07:28:39 -0400https://pop.com.sg/find-the-maclaurin-series-for-cos-2x-using-the-series-for-sin-2x.htmlhttps://pop.com.sg/find-the-maclaurin-series-for-cos-2x-using-the-series-for-sin-2x.htmlI know that $$\sin(2x)= 2x - \frac{8x^3}{3!} + \frac{32x^5}{5!} - \frac{128x^7}{7!} + \cdots $$ $$\sin(2x)= \sum_{n=0}^\infty (-1)^n {2^{2n+1}x^{2n+1} \over (2n+1)!}$$ But I don't see how I can us......Tue, 11 Aug 2026 07:26:34 -0400https://pop.com.sg/jacobi-symbol-of-quadratic-residues.htmlhttps://pop.com.sg/jacobi-symbol-of-quadratic-residues.htmlIn my textbook there's the following definition: Let $n$ be an odd number and $n = \prod_{i=1}^s q_i^{r_i}$ its prime factorization. Then the Jacobi symbol is defined as \begin{equation*} \l......Tue, 11 Aug 2026 07:19:39 -0400https://pop.com.sg/proof-that-tree-n-where-n-3-is-finite.htmlhttps://pop.com.sg/proof-that-tree-n-where-n-3-is-finite.htmlReading online, it generally seems accepted that TREE(n) where n >= 3 is a finite number, but large enough to be incomputable and only has extremely loose lower bounds today. TREE(n) is the function ...Tue, 11 Aug 2026 07:16:00 -0400https://pop.com.sg/how-do-i-find-the-period-of-the-sine-function-y-20-sin-left-frac-5-pi-.htmlhttps://pop.com.sg/how-do-i-find-the-period-of-the-sine-function-y-20-sin-left-frac-5-pi-.htmlUsing Desmos I can see the period is $0.8$ but how do I get there? I understand that the period is $2\pi/$co-efficient of $x$ but the $-2/5$ is throwing me off....Tue, 11 Aug 2026 07:07:52 -0400https://pop.com.sg/q-rotation-matrix-preserving-properties.htmlhttps://pop.com.sg/q-rotation-matrix-preserving-properties.htmlGiven a simple rotation matrix: $$\mathbf{R}(\theta) = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \text{cos}(\theta) & \text{sin}(\theta) \\ 0 & -\text{sin}(\theta)& \text{cos}(\thet......Tue, 11 Aug 2026 07:05:38 -0400https://pop.com.sg/calculate-local-maximum-and-minimum-of-f-x-2-sin-x-5-sin-x.htmlhttps://pop.com.sg/calculate-local-maximum-and-minimum-of-f-x-2-sin-x-5-sin-x.htmlI am working on my scholarship exam practice which assumes high school or pre-university math knowledge. Could you please have a look on my approach? The minimum of the function $f(x)=(2+\sin x)......Tue, 11 Aug 2026 07:01:25 -0400https://pop.com.sg/set-with-no-elements.htmlhttps://pop.com.sg/set-with-no-elements.htmlConsider a set defined on the universe of integers. It contains a large number of subsets. These subsets include the empty set and sets which contain multiple integer elements. However our set does......Tue, 11 Aug 2026 06:58:24 -0400https://pop.com.sg/possible-n-plus-r-question.htmlhttps://pop.com.sg/possible-n-plus-r-question.htmlI am having trouble with this question on how to solve. Can someone please help? Mark and Juanita own a sandwich shop. They offer 3 kinds of bread, 5 kinds of meat, and 3 kinds of cheese. Each typ......Tue, 11 Aug 2026 06:57:16 -0400https://pop.com.sg/derivative-rules.htmlhttps://pop.com.sg/derivative-rules.htmlI have some problems when I have to derive: I don't know where I have to begin and where I have to stop. For example, if I have $$f(x)=(x^2 +1)(x^2 +3)$$ I know that I have to use the product rule ......Tue, 11 Aug 2026 06:50:14 -0400https://pop.com.sg/understanding-arctan-with-the-leibniz-series.htmlhttps://pop.com.sg/understanding-arctan-with-the-leibniz-series.htmlI am a newbie at complex math, and formulas that uses complex functions ($\sum_{i=a}^B$, $\arctan \theta$, $\tan \theta$ , etc.). I mostly understand all functions that I said in the last sentance......Tue, 11 Aug 2026 06:43:22 -0400https://pop.com.sg/sum-of-two-pascal-random-variables.htmlhttps://pop.com.sg/sum-of-two-pascal-random-variables.htmlI'm having trouble finding the PMF of a sum of pascal random variables. The problem stated is: Let $X \sim Pascal(m,p)$ and $Y \sim Pascal(l,p)$ where X and Y are independent of each other. Let $......Tue, 11 Aug 2026 06:38:11 -0400https://pop.com.sg/determine-cycle-from-adjacency-matrix.htmlhttps://pop.com.sg/determine-cycle-from-adjacency-matrix.htmlIs there a way/algorithm to determine if there is a cycle in a graph if I only have the adjacency matrix and can not visualize the graph?...Tue, 11 Aug 2026 06:36:00 -0400https://pop.com.sg/why-is-e-eulers-constant-included-in-the-formula-for-continuous-intere.htmlhttps://pop.com.sg/why-is-e-eulers-constant-included-in-the-formula-for-continuous-intere.htmlLets start with the most basic continuous interest formula and the evaluation of e: Continuous interest: $\left(1+\frac{r}{n}\right)^n$ Evaluation of e: $\lim_{x\to\infty} \left(1+\frac{1}{x}\ri......Tue, 11 Aug 2026 06:34:10 -0400https://pop.com.sg/can-the-volume-be-non-positive.htmlhttps://pop.com.sg/can-the-volume-be-non-positive.htmlI have a question please I have asked my teacher if the area could be negative she said :no never But what about the volume ? E.x If $y=x+2$ and $y=-x-2$,$x=0$ about x-axis The volume become zer......Tue, 11 Aug 2026 06:27:38 -0400https://pop.com.sg/find-the-two-sided-p-value.htmlhttps://pop.com.sg/find-the-two-sided-p-value.htmlThe book has the following text(about finding a trust zone). For $H_0 : \pi = 0.20$, for instance, the score test statistic is $Z(s) = -2.50$, which has two-sided P-value $0.012 < 0.05$, so $0.20$ ...Tue, 11 Aug 2026 06:25:42 -0400https://pop.com.sg/calculus-integration-problem-int-sin-5-x-cos-2-x-dx.htmlhttps://pop.com.sg/calculus-integration-problem-int-sin-5-x-cos-2-x-dx.htmlWhat's the integration of $$\int \sin^5 (x) \cos^2 (x)\,dx?$$...Tue, 11 Aug 2026 06:25:29 -0400https://pop.com.sg/find-the-value-of-cos-2-pi-5-using-radicals-duplicate.htmlhttps://pop.com.sg/find-the-value-of-cos-2-pi-5-using-radicals-duplicate.htmlThis is homework so if there is another example that can illustrate the technique I would happily accept that as guidance. The only thing I have been able to find is a question asking about $\cos(2......Tue, 11 Aug 2026 06:20:22 -0400https://pop.com.sg/what-does-the-relaxation-function-do-in-dijkstra-039-s-algorithm.htmlhttps://pop.com.sg/what-does-the-relaxation-function-do-in-dijkstra-039-s-algorithm.htmlI'm not sure what the relaxation function in Dijkstra algorithm does. Here is my current understanding: The relaxation function is essentially deciding which edge to choose from different alternat......Tue, 11 Aug 2026 06:02:16 -0400https://pop.com.sg/is-12-mod-13-the-same-as-1-mod-13.htmlhttps://pop.com.sg/is-12-mod-13-the-same-as-1-mod-13.htmlI'm in the process of studying for an exam and this just popped into my head. Sorry if it's a dumb question...Tue, 11 Aug 2026 05:50:28 -0400https://pop.com.sg/we-have-a-box-with-100-screws-90-good-10-defective-what-is-the-prob-th.htmlhttps://pop.com.sg/we-have-a-box-with-100-screws-90-good-10-defective-what-is-the-prob-th.htmlProbability to choose a non-defective screw for the 1st time is $\dfrac{90}{100}$ 2nd time $\dfrac{89}{99}$ 3rd time $\dfrac{88}{98}$ ... 10th time $\dfrac{81}{91}$ The final result is a ...Tue, 11 Aug 2026 05:50:26 -0400https://pop.com.sg/why-is-an-image-called-an-image.htmlhttps://pop.com.sg/why-is-an-image-called-an-image.htmlGiven a function $f : A \to B$, the image, denoted by $\operatorname{Im}f$ is the set of all $f(x)$ where $x \in A$. Why do we call this set the image? When was it first used, and what motivated it......Tue, 11 Aug 2026 05:48:02 -0400https://pop.com.sg/how-to-reverse-this-bitwise-and-xor-encoding-algorithm.htmlhttps://pop.com.sg/how-to-reverse-this-bitwise-and-xor-encoding-algorithm.htmlI have been given an "encoding" algorithm that does bitwise XOR and bitwise AND. Originally it's a C code that operates on integers with bit-shifts, but I have translated it into a simpler pseudocode ...Tue, 11 Aug 2026 05:46:56 -0400https://pop.com.sg/graph-of-this-integral-is-it-a-half-circle-or-quarter-circle.htmlhttps://pop.com.sg/graph-of-this-integral-is-it-a-half-circle-or-quarter-circle.htmlI'm reading this text: $$\int_0^1 \sqrt{1 - x^2} \cdot dx$$ Why isn't the above integral the right half of a circle? Why did they show the equation of $y^2$? Was the point of that just to show th......Tue, 11 Aug 2026 05:46:51 -0400https://pop.com.sg/a-family-has-three-children-what-is-the-probability-that-at-least-one-.htmlhttps://pop.com.sg/a-family-has-three-children-what-is-the-probability-that-at-least-one-.htmlAccording to me there are $4$ possible outcomes: $$GGG \ \ BBB \ \ BGG \ \ BBG $$ Out of these four outcomes, $3$ are favorable. So the probability should be $\frac{3}{4}$. But should you take......Tue, 11 Aug 2026 05:43:39 -0400https://pop.com.sg/show-1-sqrt-5-is-irreducible-in-mathbb-z-sqrt-5.htmlhttps://pop.com.sg/show-1-sqrt-5-is-irreducible-in-mathbb-z-sqrt-5.htmlShow $1+\sqrt{5}$ is irreducible in $\mathbb{Z}[\sqrt{5}]$. Similar questions use the norm ( see e.g. How to show $1 + \sqrt{ 5 }$ is irreducible ), but I do not know yet about such a mapping, so I......Tue, 11 Aug 2026 05:36:23 -0400https://pop.com.sg/notation-for-summation-of-factorials.htmlhttps://pop.com.sg/notation-for-summation-of-factorials.html$1!+2!+3!...$ The sum of all the factorials up to a chosen positive integer $n$. This would be expressed using sigma notation. If you can show me this, that would be enough for me, for now. If you......Tue, 11 Aug 2026 05:30:33 -0400https://pop.com.sg/law-of-total-variance-intuition.htmlhttps://pop.com.sg/law-of-total-variance-intuition.htmlIntuitively, what's the difference between 2 following terms on the right hand side of the law of total variance? $$\text{Var}(Y) = \Bbb E\left[\text{Var}\left(Y|X\right)\right] + \text{Var}\left(\......Tue, 11 Aug 2026 05:27:24 -0400https://pop.com.sg/form-of-the-particular-solution-for-odes.htmlhttps://pop.com.sg/form-of-the-particular-solution-for-odes.htmlI have been working on finding the form of the particular solution for the following differential equations, using Paul's online notes as a reference, and every time I submit my answers, they are w......Tue, 11 Aug 2026 05:26:48 -0400https://pop.com.sg/how-to-factorize-quadratic-equations-quickly.htmlhttps://pop.com.sg/how-to-factorize-quadratic-equations-quickly.htmlIt takes me more than a minute to quickly factorise this kind of quadratic expression. $$3n^2 -53n + 232$$ I need to solve them in less than $10$-$15$ seconds. Please tell me a way I can solve them....Tue, 11 Aug 2026 05:12:46 -0400https://pop.com.sg/gamma-function-identity-involving-5-4-3-4-1-4-and-pi.htmlhttps://pop.com.sg/gamma-function-identity-involving-5-4-3-4-1-4-and-pi.htmlHow can I prove this identity: $$\frac{4\pi\sqrt{2}\Gamma\Big(\frac{5}{4}\Big)}{\Gamma\Big(\frac{3}{4}\Big)}-\Gamma\Big(\frac{1}{4}\Big)^2 = 0?$$ I found it while calculating $$\prod_{i=0}^\infty\f......Tue, 11 Aug 2026 05:08:54 -0400https://pop.com.sg/schur-039-s-theorem-proof-for-ramsey-theory-misunderstanding.htmlhttps://pop.com.sg/schur-039-s-theorem-proof-for-ramsey-theory-misunderstanding.htmlSchur's Theorem in Ramsey Theory asserts that for every positive integer $r$, there is some positive integer $S(r)$ such that for every partition of the set $\{1,\ldots,S\}$ into $r$ parts, one of ......Tue, 11 Aug 2026 04:58:11 -0400https://pop.com.sg/what-exactly-goes-raising-a-number-to-a-fraction-mean.htmlhttps://pop.com.sg/what-exactly-goes-raising-a-number-to-a-fraction-mean.htmlI apologise for asking something so fundamental, but what exactly does $$2^\frac{2}{5}$$ actually mean? I get raising a whole number to another whole number $$x^y$$ means you are multiplying x with ...Tue, 11 Aug 2026 04:57:11 -0400https://pop.com.sg/domain-of-a-polynomial-function.htmlhttps://pop.com.sg/domain-of-a-polynomial-function.htmlA polynomial function is given as, $P(x)= a_nx^n+a_{n-1}x^{n-1}+.....+a_1x+a_0$ Notice the last but one term $a_1x$. This term is a simplified form of $a_{n-(n-1)}x^{n-(n-1)}$. Now let us take......Tue, 11 Aug 2026 04:57:04 -0400https://pop.com.sg/set-theory-proof-with-set-difference.htmlhttps://pop.com.sg/set-theory-proof-with-set-difference.htmlLet A, B and C be subsets of a universal set U. Prove C $\subseteq$ (A-B)$^c$ $\to$ A $\cap$ C $\subseteq$ B My attempt: Suppose C $\subseteq$ (A-B)$^c$ $\forall$x$\in$C, x$\in$(A-B)$^c$ $\for......Tue, 11 Aug 2026 04:56:43 -0400https://pop.com.sg/polynomial-coefficients-in-exponential-series-how-can-i-convert-this-i.htmlhttps://pop.com.sg/polynomial-coefficients-in-exponential-series-how-can-i-convert-this-i.htmlAssume we have the exponential-series $ \small \exp(x) = 1+ {x \over 1!} + {x^2 \over 2! } + \cdots = \sum\limits_{k=0 }^\infty {x^k \over k!} $ modified with a polynomial in the coefficients say $ \...Tue, 11 Aug 2026 04:53:13 -0400https://pop.com.sg/product-of-a-lipschitz-function-with-a-bounded-lipschitz-function-is-l.htmlhttps://pop.com.sg/product-of-a-lipschitz-function-with-a-bounded-lipschitz-function-is-l.htmlLet $f,g:A\to \mathbb R $ be Lipschitz continuous functions. Suppose $f $ is bounded. Is $f\cdot g $ Lipschitz continuous? (We can ask the question for several $A $, say $A=]0,1[$ or $A=\mathbb R$......Tue, 11 Aug 2026 04:49:45 -0400https://pop.com.sg/find-the-mathfrak-b-matrix-for-the-transformation-x-mapsto-ax-given-ma.htmlhttps://pop.com.sg/find-the-mathfrak-b-matrix-for-the-transformation-x-mapsto-ax-given-ma.htmlFind the $\mathfrak B$-matrix for the transformation $x\mapsto Ax$, where $\mathfrak B=\left\{\begin{bmatrix}2\\3\end{bmatrix},\begin{bmatrix}3\\5\end{bmatrix}\right\}$ and $A=\begin{bmatrix}1&......Tue, 11 Aug 2026 04:39:11 -0400https://pop.com.sg/desmos-sin-wave-sum-shows-different-to-code.htmlhttps://pop.com.sg/desmos-sin-wave-sum-shows-different-to-code.htmlI have followed The Coding Train's Discrete Fourier Transformation series, and added a feature which spits out the coefficients of the equation. I wrote test in the program, which looked nice. Howe......Tue, 11 Aug 2026 04:34:27 -0400https://pop.com.sg/what-is-the-remainder-of-16-is-divided-by-19.htmlhttps://pop.com.sg/what-is-the-remainder-of-16-is-divided-by-19.htmlCan anyone share me the trick to solve this problem using congruence? Thanks, Chan...Tue, 11 Aug 2026 04:25:39 -0400https://pop.com.sg/a-3-regular-graph-must-have-a-subgraph-which-is-a-cycle-of-even-length.htmlhttps://pop.com.sg/a-3-regular-graph-must-have-a-subgraph-which-is-a-cycle-of-even-length.htmlDoes a 3-regular graph necessarily contain a cycle of even length? I tried, believe me I tried, I even went and asked my professor's assistant and even he couldn't give me a straight answer... So......Tue, 11 Aug 2026 04:21:11 -0400https://pop.com.sg/how-can-i-express-the-natural-logarithm-of-a-power-series-in-terms-of-.htmlhttps://pop.com.sg/how-can-i-express-the-natural-logarithm-of-a-power-series-in-terms-of-.htmlCan I express the expression $$\ln\left( \sum_0^{\infty} a_nx^n\right)$$ in terms of another power series? Can I differentiate the term $$\ln\left( \sum_0^{\infty} a_nx^n\right)$$ and get $$\frac{\......Tue, 11 Aug 2026 04:12:45 -0400https://pop.com.sg/integrating-x-2e-x-using-feynman-039-s-trick-duplicate.htmlhttps://pop.com.sg/integrating-x-2e-x-using-feynman-039-s-trick-duplicate.htmlIn the second episode of season $8$ of "The Big Bang Theory," which aired yesterday night, it is stated that one can integrate $x^2e^{-x}$ by using Feynman's trick of differentiating under the inte......Tue, 11 Aug 2026 04:06:55 -0400https://pop.com.sg/formula-for-the-square-root-of-a-number.htmlhttps://pop.com.sg/formula-for-the-square-root-of-a-number.htmlIs there a formula for the square root of a number, that only uses addition, subtraction, multiplication, or division?...Tue, 11 Aug 2026 04:06:04 -0400https://pop.com.sg/what-are-the-most-obscure-or-advanced-mathematics-with-practical-appli.htmlhttps://pop.com.sg/what-are-the-most-obscure-or-advanced-mathematics-with-practical-appli.htmlThroughout my engineering studies there were jokes made by my professors (mostly mathematics professors) that referenced the fact that pure mathematicians strive to create mathematics with no pract......Tue, 11 Aug 2026 04:01:03 -0400https://pop.com.sg/why-cauchy-distribution-isn-039-t-exponental-family.htmlhttps://pop.com.sg/why-cauchy-distribution-isn-039-t-exponental-family.htmlWe know that the density function of standard Cauchy Distribution is $p(x)=\frac{\lambda}{\pi (\lambda^2+x^2)}$. It seems that $p(x)$ can't be written as the form of Exponential Family $$f_{X}(x;\......Tue, 11 Aug 2026 03:53:56 -0400https://pop.com.sg/krull-dimension-of-finite-rings-closed.htmlhttps://pop.com.sg/krull-dimension-of-finite-rings-closed.htmlI have the following question(s): Can the Krull dimension of a finite ring be any natural number? Thanks for the help!...Tue, 11 Aug 2026 03:52:01 -0400https://pop.com.sg/merge-sort-seems-to-take-the-same-number-of-comparisons-for-best-and-w.htmlhttps://pop.com.sg/merge-sort-seems-to-take-the-same-number-of-comparisons-for-best-and-w.htmlI am trying to clear up my conceptions of merge sort. What I cannot understand how merge sort takes less number of comparisons during best case. Let me explain, looking at the merge procedure gi......Tue, 11 Aug 2026 03:38:38 -0400https://pop.com.sg/what-is-the-difference-between-linear-transformation-and-affine-transf.htmlhttps://pop.com.sg/what-is-the-difference-between-linear-transformation-and-affine-transf.htmlRecently, I am struglling with the difference between linear transformation and affine transformation. Are they the same ? I found an interesting question on the difference between the functions. ......Tue, 11 Aug 2026 03:36:19 -0400https://pop.com.sg/how-to-define-a-triod.htmlhttps://pop.com.sg/how-to-define-a-triod.htmlI'm looking for a topological definition of a triod as a compact metric space. I have an intuitive idea of what it is (three intervals with one shared point being a boundary point of each interval)......Tue, 11 Aug 2026 03:26:00 -0400https://pop.com.sg/is-this-the-correct-way-of-solve-a-probability-question-from-a-deck-of.htmlhttps://pop.com.sg/is-this-the-correct-way-of-solve-a-probability-question-from-a-deck-of.htmlYou draw a card from a standard deck of $52$ playing cards then replace it in the deck and draw a second card. Determine the probability of drawing an eight and then heart My work: their are $4$ e......Tue, 11 Aug 2026 03:19:36 -0400https://pop.com.sg/tangent-to-the-x-1-axis.htmlhttps://pop.com.sg/tangent-to-the-x-1-axis.htmlFor a National Board Exam Review: Point (3,4) is the center of the circle tangent to the x-1 axis. What is the point of tangency? Answer is (3,0) I usually would provide an attempt but I do......Tue, 11 Aug 2026 03:15:55 -0400https://pop.com.sg/basic-properties-of-the-matrix-logarithm.htmlhttps://pop.com.sg/basic-properties-of-the-matrix-logarithm.htmlLet $|| \cdot ||$ be any submultiplicative norm on $\operatorname{Mat}_n(\mathbb C)$. For complex matrices $A$ with $||I_n - A|| < 1$, we can define the matrix logarithm $$\log(A) = \sum\limit......Tue, 11 Aug 2026 03:14:37 -0400https://pop.com.sg/pizza-topping-combination-problem.htmlhttps://pop.com.sg/pizza-topping-combination-problem.htmlI have two pizzas, I can choose up to 3 topping, there are 7 toppings to choose from. Double toppings, triple toppings of the same kind are allowed. Order doesn't matter, ham on top of chicke......Tue, 11 Aug 2026 03:06:24 -0400https://pop.com.sg/reducing-graph-3-coloring-to-10-coloring.htmlhttps://pop.com.sg/reducing-graph-3-coloring-to-10-coloring.htmlI am trying to show that the NP-Complete problem of 3-coloring a graph reduces to the problem of 10-coloring a graph.I have already shown how 10-coloring can be verified in polynomial time, and is ......Tue, 11 Aug 2026 02:56:34 -0400https://pop.com.sg/is-serge-lang-039-s-algebra-still-worth-reading.htmlhttps://pop.com.sg/is-serge-lang-039-s-algebra-still-worth-reading.htmlIs Serge Lang's famous book Algebra nowadays still worth reading, or are there other, more modern books which are better suited for an overview over all areas of algebra? EDIT: My main concern is ......Tue, 11 Aug 2026 02:54:17 -0400https://pop.com.sg/finding-the-general-solution-for-a-differential-equation.htmlhttps://pop.com.sg/finding-the-general-solution-for-a-differential-equation.htmlI'm practicing for the DE midterm, going through (supposedly) basic questions to practice. I am now stuck on this question. I am supposed to find the general equation for the given DE. I don't kno......Tue, 11 Aug 2026 02:53:07 -0400https://pop.com.sg/why-negative-factorial-doesn-039-t-exists.htmlhttps://pop.com.sg/why-negative-factorial-doesn-039-t-exists.htmlI've been told that factorials of negative numbers doesn't exists that's what I also found while trying to calculate factorial of negative $1$. But, I can see that graph of factorial $x$ is even ex......Tue, 11 Aug 2026 02:44:45 -0400https://pop.com.sg/calculating-average-with-minimum-and-maximum.htmlhttps://pop.com.sg/calculating-average-with-minimum-and-maximum.htmlMaybe it's a simple question but I have not any idea for that at the moment! How we can calculate average value for N samples with having only the minimum and maximum values?...Tue, 11 Aug 2026 02:43:24 -0400https://pop.com.sg/what-is-the-fastest-way-to-multiply-two-digit-numbers.htmlhttps://pop.com.sg/what-is-the-fastest-way-to-multiply-two-digit-numbers.htmlI been playing different math games on my Android lately (for example: Math Cruncher). I've noticed that I'm unable to quickly (under 7-8 seconds) multiply two digit numbers (i.e $ 18 * 17$). So my ...Tue, 11 Aug 2026 02:36:19 -0400https://pop.com.sg/is-symmetric-difference-xor-commutative.htmlhttps://pop.com.sg/is-symmetric-difference-xor-commutative.htmlI'm working on a problem: If A, B, C, and D are sets, does it follow that (A $\oplus$ B) $\oplus$ (C $\oplus$ D) = (A $\oplus$ C) $\oplus$ (B $\oplus$ D)? How does one approach something like thi......Tue, 11 Aug 2026 02:33:38 -0400https://pop.com.sg/bernoulli-differential-equation-of-second-order.htmlhttps://pop.com.sg/bernoulli-differential-equation-of-second-order.htmlHow one can solve a Bernoulli differential equation of second order? i.e., solve the DE \begin{align} \frac{{d^2 y}}{{dx^2 }} + p\left( x \right)\frac{{dy}}{{dx}} + q \left( x \right)y = g\left( x \...Tue, 11 Aug 2026 02:12:03 -0400https://pop.com.sg/finding-stable-distribution-of-a-stochastic-matrix.htmlhttps://pop.com.sg/finding-stable-distribution-of-a-stochastic-matrix.htmlso im currently looking at Markov diagrams and stochastic matrices and have become quite stuck on some parts of this question below and was looking for some help. Here is what I have done so far, ......Tue, 11 Aug 2026 01:57:31 -0400https://pop.com.sg/even-odd-or-neither.htmlhttps://pop.com.sg/even-odd-or-neither.htmlI would like someone to verify my solutions to the problems above? 9a. even 9b. odd 10. neither...Tue, 11 Aug 2026 01:56:31 -0400https://pop.com.sg/differences-between-homomorphisms-and-functions-closed.htmlhttps://pop.com.sg/differences-between-homomorphisms-and-functions-closed.htmlIncredibly, I haven't been able to find a good entry online comparing these two mathematical structures. Their relative familiarity doesn't make their boundaries less fuzzy. Here is a coarse attempt: ...Tue, 11 Aug 2026 01:39:32 -0400https://pop.com.sg/finding-the-volume-of-a-rectangular-based-pyramid-using-calculus.htmlhttps://pop.com.sg/finding-the-volume-of-a-rectangular-based-pyramid-using-calculus.htmlI have a maths test coming up, and I just can't seem to solve a question on finding volumes of solids (via integration). Here's the question: Find the volume of a pyramid with height h and rectang......Tue, 11 Aug 2026 01:33:22 -0400https://pop.com.sg/explain-motivation-behind-recurrence-of-100-sided-dice-problem.htmlhttps://pop.com.sg/explain-motivation-behind-recurrence-of-100-sided-dice-problem.htmlThe following question is asked in another MSE post. For completeness, I restate the question below. The following question is from a Jane Street interview. You are given a 100-sided die. After yo......Tue, 11 Aug 2026 01:30:38 -0400https://pop.com.sg/cardinality-of-cartesian-product-of-finite-sets-closed.htmlhttps://pop.com.sg/cardinality-of-cartesian-product-of-finite-sets-closed.htmlIf $a = \{1,2,3\}$ and $b = \{a,b,c\},\;$ FIND $\;n(a\times b)$ Or is it impossible to multiply these sets? What will be the answer?...Tue, 11 Aug 2026 01:29:57 -0400https://pop.com.sg/how-does-x-h-3-x-3-3x-2h-3xh-2-h-3.htmlhttps://pop.com.sg/how-does-x-h-3-x-3-3x-2h-3xh-2-h-3.htmlGood afternoon my wonderful friends! Whenever I do this equation I set it up using the difference of two cubes, which is as follows: $(a+b)^3 = (a+b)(a^2-ab+b^2) = a^3 + b^3$ Whenever I try to use ...Tue, 11 Aug 2026 01:27:31 -0400https://pop.com.sg/how-to-find-the-number-of-perfect-matchings-in-complete-graphs.htmlhttps://pop.com.sg/how-to-find-the-number-of-perfect-matchings-in-complete-graphs.htmlIn wikipedia FKT algorithm is given for planar graphs. Not anything for complete graphs. I need to find the number of perfect matchings in complete graph of six vertices....Tue, 11 Aug 2026 01:21:32 -0400https://pop.com.sg/how-to-find-the-number-of-sides-of-a-dice-with-an-unspecified-sides.htmlhttps://pop.com.sg/how-to-find-the-number-of-sides-of-a-dice-with-an-unspecified-sides.htmlPlease I need help on how to go about solving this question: A dice has $n$ sides with numbers $1,2,3,4,...,n$ marked on it. If the probability that a particular number appears at least once when the ...Tue, 11 Aug 2026 01:17:07 -0400https://pop.com.sg/wrong-result-in-wolfram-alpha-on-graph.htmlhttps://pop.com.sg/wrong-result-in-wolfram-alpha-on-graph.htmlI'm learning to code in Python, so I tried to find roots and draw a graph of a function $-(16x^2-24x+5)e^{-x}$ 1 Result i got in Python using the mat.plotlib library is this The problem is that ...Tue, 11 Aug 2026 01:10:14 -0400https://pop.com.sg/what-is-the-mathematical-point-of-straightedge-and-compass-constructio.htmlhttps://pop.com.sg/what-is-the-mathematical-point-of-straightedge-and-compass-constructio.htmlThe ancient discipline of construction by straightedge and compass is both fascinating and entertaining. But what is its significance in a mathematical sense? It is still taught in high school geom......Tue, 11 Aug 2026 01:09:48 -0400https://pop.com.sg/exterior-angle-of-a-reflex-angle.htmlhttps://pop.com.sg/exterior-angle-of-a-reflex-angle.htmlHow are exterior angles for reflex angles defined? Are they negative? This doubt came to my mind while thinking about the sum of the exterior angles of a concave polygon. I think it should be negat......Tue, 11 Aug 2026 01:09:18 -0400https://pop.com.sg/construct-the-intersection-of-a-cube-by-a-plane-through-3-points-on-it.htmlhttps://pop.com.sg/construct-the-intersection-of-a-cube-by-a-plane-through-3-points-on-it.htmlSo this is a rather old problem, but I still cannot find a pure constructive solution to it. Please, do not offer me to write a plane equation, etc. I would be grateful, if you offer a solution by ......Tue, 11 Aug 2026 01:09:09 -0400https://pop.com.sg/find-the-partial-derivative-for-e-xyz-6.htmlhttps://pop.com.sg/find-the-partial-derivative-for-e-xyz-6.htmlgiven $f(x,y,z)=e^{xyz^6}$ calculate $f_{xyz}$ $f_{x}= yz^6e^{xyz^6}$ using the chain rule and multiplying the derivative of the outisde times the inside then $f_{xy} = 5yz^6e^{xyz^6} + e^{xyz......Tue, 11 Aug 2026 01:04:52 -0400https://pop.com.sg/discrete-math-onto-1-1-functions.htmlhttps://pop.com.sg/discrete-math-onto-1-1-functions.htmlLet $S = \{3,B\}$ Give an example of a function $f: S \times S \to S$ that is onto. Give an example of a function $g: S \to S \times S$ that is 1-1. Give an example of a function $h: P(S) \to S \ti......Tue, 11 Aug 2026 01:04:22 -0400https://pop.com.sg/optimization-circle-and-square.htmlhttps://pop.com.sg/optimization-circle-and-square.htmlA piece of wire of length 60 is cut, and the resulting two pieces are formed to make a circle and a square. Where should the wire be cut to (a) minimize and (b) maximize the combined area of the ci......Tue, 11 Aug 2026 00:58:24 -0400https://pop.com.sg/prove-that-u-is-orthogonal-to-v-operatorname-proj-u-v-for-all-vectors-.htmlhttps://pop.com.sg/prove-that-u-is-orthogonal-to-v-operatorname-proj-u-v-for-all-vectors-.htmlHere's where I'm at, not sure where to go from here. Two vectors are orthogonal if their dot product is $0$. Knowing that; $$u \cdot (v - \operatorname{proj} u(v)) = 0$$ $$u \cdot \left(v - \fra......Tue, 11 Aug 2026 00:56:06 -0400https://pop.com.sg/rewriting-biconditional-statements-into-conditional-statement-without-.htmlhttps://pop.com.sg/rewriting-biconditional-statements-into-conditional-statement-without-.htmli'm currently taking discrete math and my study guide gives the problems without the answers. (very inconvenient).. but anyways i'm stuck on rewriting biconditional statements to conditional statem......Tue, 11 Aug 2026 00:54:23 -0400https://pop.com.sg/how-can-i-find-the-largest-perfect-square-in-a-really-big-number.htmlhttps://pop.com.sg/how-can-i-find-the-largest-perfect-square-in-a-really-big-number.htmlLet's say I want to find the largest number that when squared doesn't exceed 9223372036854775807. Or any other large number like that. How can I go about finding that? Is there some kind of functio......Tue, 11 Aug 2026 00:51:35 -0400https://pop.com.sg/limit-0-times-infinity-rewrite-to-find-the-limit.htmlhttps://pop.com.sg/limit-0-times-infinity-rewrite-to-find-the-limit.htmlI need some help with: $\lim_{x\to 0+} x^3\cdot e^{1/x}$. How to start? I've tried substitution $(y=1/x)$ without any luck. I would prefer not to use L'Hopitals rule and apologizes for a bad titl......Tue, 11 Aug 2026 00:46:40 -0400https://pop.com.sg/best-program-for-creating-educational-math-animations.htmlhttps://pop.com.sg/best-program-for-creating-educational-math-animations.htmlI'm looking for recommendations on what program to use for creating mathematical animations. These animations will be used in creating educational videos for high school math -- Trigonometry first,......Tue, 11 Aug 2026 00:43:08 -0400https://pop.com.sg/what-is-x-5-called.htmlhttps://pop.com.sg/what-is-x-5-called.htmlI have CS background, and I'm trying to teach fundamentals of programming to students. They seem to have difficulties in understanding the difference between x=5 and x==5. However, trying to explain, ...Tue, 11 Aug 2026 00:41:36 -0400