https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usFri, 21 Aug 2026 20:36:40 -0400https://pop.com.sg/is-the-mle-of-an-exponential-distribution-complete-and-sufficient.htmlhttps://pop.com.sg/is-the-mle-of-an-exponential-distribution-complete-and-sufficient.htmlI've just prove that the MLE of an exponential distribution is: $$ \hat \theta = \frac{n}{\sum x_i} $$ So I'm trying to show that $\hat \theta $ is complete and sufficient. For sufficiency I've ......Sat, 22 Aug 2026 00:31:27 -0400https://pop.com.sg/how-you-could-you-change-the-surface-area-formula-for-a-cylinder-to-ca.htmlhttps://pop.com.sg/how-you-could-you-change-the-surface-area-formula-for-a-cylinder-to-ca.htmlSkateboarders use half pipes for doing tricks. A half-pipe is a half cylinder. Explain how you could manipulate the surface area formula for a cylinder to calculate the curved surface area of the h......Sat, 22 Aug 2026 00:22:34 -0400https://pop.com.sg/euclidean-distance-proof.htmlhttps://pop.com.sg/euclidean-distance-proof.htmlHow can I show that the Euclidean distance satisfies the triangle inequality? Where the Euclidean distance is given by: $$d(p,q) = \sqrt{(p_1-q_1)^2 + \cdots + (p_n-q_n)^2}$$ Triangle Inequality......Sat, 22 Aug 2026 00:18:44 -0400https://pop.com.sg/shortest-distance-from-point-to-a-plane.htmlhttps://pop.com.sg/shortest-distance-from-point-to-a-plane.htmlQ: Find the shortest distance from the point $A(1,1,1)$ to the plane $2x+3y+4z=5$. I understand that we need to pick a point P on the plane such that we can form a vector PA. Can someone explain ......Sat, 22 Aug 2026 00:18:08 -0400https://pop.com.sg/how-can-i-prove-ax-bx-for-every-n-times1-column-matrix-x-implies-a-b.htmlhttps://pop.com.sg/how-can-i-prove-ax-bx-for-every-n-times1-column-matrix-x-implies-a-b.htmlLet $A$ and $B$ be matrices $n\times n$. Suppose $AX=BX$ for every $n\times 1$ column matrix $X$. How can I prove this implies $A=B$?...Sat, 22 Aug 2026 00:16:12 -0400https://pop.com.sg/lagrange-interpolating-polynomials-error-bound.htmlhttps://pop.com.sg/lagrange-interpolating-polynomials-error-bound.htmlLet $f(x) = e^{2x} - x$, $x_0 = 1$, $x_1 = 1.25$, and $x_2 = 1.6$. Construct interpolation polynomials of degree at most one and at most two to approximate $f(1.4)$, and find an error bound for the ...Sat, 22 Aug 2026 00:12:30 -0400https://pop.com.sg/unable-to-evaluate-the-integral-int-csc-left-x-frac-pi-3-right-csc-lef.htmlhttps://pop.com.sg/unable-to-evaluate-the-integral-int-csc-left-x-frac-pi-3-right-csc-lef.htmlThe Question $$\int \csc\left(x-\frac{\pi}{3}\right)\csc\left(x-\frac{\pi}{6}\right) dx $$ What I Tried- I tried dividing both numerator and denominator by $\sin \pi/6$ but couldn't get too far, w......Sat, 22 Aug 2026 00:09:36 -0400https://pop.com.sg/what-is-a-principal-minor-of-a-matrix.htmlhttps://pop.com.sg/what-is-a-principal-minor-of-a-matrix.htmlI was going through the book on operation research by Hamdy A.Taha. It referred to principal minor of a hessian matrix. Can someone explain what is meant by a principal minor? Is it different from ......Sat, 22 Aug 2026 00:07:03 -0400https://pop.com.sg/a-related-rate-problem-of-a-trough-with-a-trapezoidal-cross-section.htmlhttps://pop.com.sg/a-related-rate-problem-of-a-trough-with-a-trapezoidal-cross-section.htmlFrom the MIT exercise book (2E-7): A trough is filled with water at a rate of $1$ cubic meter per second. The trough has a trapezoidal cross section with the lower base of length half a meter and ......Fri, 21 Aug 2026 23:59:42 -0400https://pop.com.sg/the-sixth-number-system.htmlhttps://pop.com.sg/the-sixth-number-system.htmlIn high school, I learned there are 5 number systems, namely: Natural numbers ($\Bbb N$) Integers ($\Bbb Z$) Rational numbers ($\Bbb Q$) Real numbers ($\Bbb R$) Complex numbers ($\Bbb C$) I ...Fri, 21 Aug 2026 23:59:12 -0400https://pop.com.sg/discrete-math-understanding-a-proof-involving-the-definition-of-divisi.htmlhttps://pop.com.sg/discrete-math-understanding-a-proof-involving-the-definition-of-divisi.htmlIn this first course on discrete mathematics, the instructor provided this following solution to a question. The question was asked us to prove the following (the solution is provided as well): My ...Fri, 21 Aug 2026 23:56:54 -0400https://pop.com.sg/definition-and-example-of-a-partition.htmlhttps://pop.com.sg/definition-and-example-of-a-partition.htmlPartition of a Set is defined as "A collection of disjoint subsets of a given set. The union of the subsets must equal the entire original set." For example, one possible partition of $(1, 2, 3, 4......Fri, 21 Aug 2026 23:53:44 -0400https://pop.com.sg/line-integral-work-done-moving-along-a-certain-path.htmlhttps://pop.com.sg/line-integral-work-done-moving-along-a-certain-path.htmlFind the work done by the force field $F(x,y) = -xi + 6yj$ along the path $C:y = x^3$ from $(0,0)$ to $(6,216)$ I tried parameterizing C which gave me $x(t) = t$ and $y(t) = t^3$ but do I use thos......Fri, 21 Aug 2026 23:52:39 -0400https://pop.com.sg/similarity-transform-of-a-matrix-preserves-the-determinant.htmlhttps://pop.com.sg/similarity-transform-of-a-matrix-preserves-the-determinant.htmlIt is posited (in chapter $11$ of the book Numerical Recipes in C) that doing a similarity transform of a matrix $A$ preserves its eigenvalues. This is how a similarity transform is defined: $$A \...Fri, 21 Aug 2026 23:52:37 -0400https://pop.com.sg/special-derivative-function.htmlhttps://pop.com.sg/special-derivative-function.htmlI have a function $f(x)$ which I would like to have the derivative with respect to $x$. How can I get the derivative of the following function with respect to $x$? $$f(x) = \log(1-z^{e^{y^{T}x}})$$...Fri, 21 Aug 2026 23:44:29 -0400https://pop.com.sg/number-of-various-7-character-passwords-that-are-required-to-contain-a.htmlhttps://pop.com.sg/number-of-various-7-character-passwords-that-are-required-to-contain-a.htmlIn this question on Information Security, different options are given for passwords. One example states that a password contains a mixture of numbers, letters and special characters. Assuming that ...Fri, 21 Aug 2026 23:34:11 -0400https://pop.com.sg/math-in-horseshoe-puzzle.htmlhttps://pop.com.sg/math-in-horseshoe-puzzle.htmlWe know that Rubik's Cube is a good demonstration of group theory. Correspondingly, for the horseshoe puzzle as in the picture below, is there a math language for it? Does it demonstrate any math s......Fri, 21 Aug 2026 23:20:28 -0400https://pop.com.sg/what-can-be-understood-by-a-b-b-a-in-set-theory.htmlhttps://pop.com.sg/what-can-be-understood-by-a-b-b-a-in-set-theory.htmlWhat can be understood by $\rm A-B = B - A$ in set theory? What does this tell us about the characteristics of $A$ and $B$ and their relationship? I'm quite confused as I am new to set theory....Fri, 21 Aug 2026 23:09:16 -0400https://pop.com.sg/solving-a-function-where-x-mod-y-represents-the-remainder-after-divisi.htmlhttps://pop.com.sg/solving-a-function-where-x-mod-y-represents-the-remainder-after-divisi.htmlI would like help solving this practice question. For a function f(x, y) defined below, what is the value of f(775, 527)? Here, x mod y represents the remainder after division of x by y. f(x, y)......Fri, 21 Aug 2026 23:07:40 -0400https://pop.com.sg/proving-the-symmetry-of-the-ricci-tensor.htmlhttps://pop.com.sg/proving-the-symmetry-of-the-ricci-tensor.htmlConsider the Ricci tensor : $R_{\mu\nu}=\partial_{\rho}\Gamma_{\nu\mu}^{\rho} -\partial_{\nu}\Gamma_{\rho\mu}^{\rho} +\Gamma_{\rho\lambda}^{\rho}\Gamma_{\nu\mu}^{\lambda} -\Gamma_{\nu\lambda}^{\rho}\...Fri, 21 Aug 2026 23:07:21 -0400https://pop.com.sg/derivation-of-the-general-forms-of-partial-fractions.htmlhttps://pop.com.sg/derivation-of-the-general-forms-of-partial-fractions.htmlI'm learning about partial fractions, and I've been told of 3 types or "forms" that they can take (1) If the denominator of the fraction has linear factors: $${5 \over {(x - 2)(x + 3)}} \equiv {A......Fri, 21 Aug 2026 23:04:53 -0400https://pop.com.sg/how-many-solutions-does-the-system-have-why.htmlhttps://pop.com.sg/how-many-solutions-does-the-system-have-why.htmlHow can I solve this problem? Let Z be a 4 x 5 matrix. A basis for the null space of Z is {𝐛𝟏} where 𝐛𝟏 ≠ 𝟎. If the system of equations (Z^T)𝐱 = 𝐦 is consistent, how many solutions does the ...Fri, 21 Aug 2026 22:54:33 -0400https://pop.com.sg/normal-curve-1-5-standard-deviations.htmlhttps://pop.com.sg/normal-curve-1-5-standard-deviations.htmlFor a normal curve, how much of the area lies within 1.5 standard deviations of the mean? I already know about the 68–95–99.7 rule, and see that it should be between 68% and 95%. I also know that it ...Fri, 21 Aug 2026 22:43:46 -0400https://pop.com.sg/vandermonde-determinant-from-3-by-3.htmlhttps://pop.com.sg/vandermonde-determinant-from-3-by-3.htmlI am a novice at math and need help calculating the determinant of the Vandermonde matrix, which is: $V=\begin{bmatrix} 1&a_1&a_1^2&\dots&a_1^{n-1}\\ 1&a_2&a_2^2&\dots&a......Fri, 21 Aug 2026 22:40:49 -0400https://pop.com.sg/solving-a-system-of-equations-linear-algebra.htmlhttps://pop.com.sg/solving-a-system-of-equations-linear-algebra.htmlFor one of my homework problems I have the following system of equations. $$ \left\{ \begin{array}{c} x_1+2x_2-2x_3-3x_5+2x_6=-4 \\ -x_4+4x_5+3x_6=-9 \\ x_1+2x_2-5x_5+4x_6=-4 \end{array} \right......Fri, 21 Aug 2026 22:39:06 -0400https://pop.com.sg/calculate-singular-value-decomposition.htmlhttps://pop.com.sg/calculate-singular-value-decomposition.htmlI need to find the reduced singular value decomposition of this matrix. $$\begin{pmatrix} -2 & -1 & 2 \\ 2 & 1 & -2 \\ \end{pmatrix} $$ I formed $$ A^TA= \begin{pmatrix} 8 & ......Fri, 21 Aug 2026 22:37:56 -0400https://pop.com.sg/exponential-and-power-functions-through-two-points.htmlhttps://pop.com.sg/exponential-and-power-functions-through-two-points.htmlI have a problem where I'm asked to determine the constants of exponential and power functions that go through both points (5, 50) and (10, 1600). I have tried to solve them below, but would apprec......Fri, 21 Aug 2026 22:30:39 -0400https://pop.com.sg/how-can-three-vectors-be-orthogonal-to-each-other.htmlhttps://pop.com.sg/how-can-three-vectors-be-orthogonal-to-each-other.htmlIn a scenario, say that: Vectors $\mathbf{U}$, $\mathbf{V}$ and $\mathbf{W}$ are all orthogonal such that the dot product between each of these $(\mathbf{UV}\;\mathbf{VW}\;\mathbf{WU})$ is equal......Fri, 21 Aug 2026 22:25:16 -0400https://pop.com.sg/giving-a-basis-for-the-column-space-of-a.htmlhttps://pop.com.sg/giving-a-basis-for-the-column-space-of-a.htmlLet $A = \begin{bmatrix}3&3&3\\3&5&1\\-2&4&-8\\-2&-4&0\\4&9&-1\end{bmatrix}$ Give a basis for the column space of A So what I've done so far is put it in RR......Fri, 21 Aug 2026 22:19:01 -0400https://pop.com.sg/radial-and-transverse-components-of-velocity-and-acceleration.htmlhttps://pop.com.sg/radial-and-transverse-components-of-velocity-and-acceleration.htmlI need help with this problem: Find (i) the velocity $\dot r(t)$, (ii) the speed $\Vert \dot r(t)\Vert$, (iii) the acceleration $\ddot r(t)$, (iv) the radial and transverse componentes of velo......Fri, 21 Aug 2026 22:16:45 -0400https://pop.com.sg/plot-absolute-error-vs-step-size-as-a-log-log-plot-matlab.htmlhttps://pop.com.sg/plot-absolute-error-vs-step-size-as-a-log-log-plot-matlab.htmlI have approximated up to $t=2$ with step sizes $h=(1/2)^{n}$ for $n=1, ... , 10$ using a Runge-Kutta (order 4) MATLAB code. Then, I wrote down the absolute error of $y(2)$. Now, I want to plot the......Fri, 21 Aug 2026 22:15:02 -0400https://pop.com.sg/how-to-combine-manipulate-two-summations-into-one-summation-in-general.htmlhttps://pop.com.sg/how-to-combine-manipulate-two-summations-into-one-summation-in-general.htmlI always struggle to understand what I can and can't do with sums. In fact, even when convergence isn't an issue, I still get confused. What can I do about this problem? For instance, I am currently ...Fri, 21 Aug 2026 22:13:11 -0400https://pop.com.sg/generalizing-the-abel-theorem-to-higher-order-differential-equations.htmlhttps://pop.com.sg/generalizing-the-abel-theorem-to-higher-order-differential-equations.htmllet $$y''+p(t)y'+q(t)y=g(t)$$ then By the Abel theorem wronskian equals to : $$W= c e^{\int p(t)dt}$$ Now We want to generalise this to a 3rd order differential equation where : $$y'''+p_1(t)y''......Fri, 21 Aug 2026 22:03:06 -0400https://pop.com.sg/why-arctan-in-radian-mode-and-arctan-in-degrees-mode-are-different.htmlhttps://pop.com.sg/why-arctan-in-radian-mode-and-arctan-in-degrees-mode-are-different.htmlSo I've been playing around with my calculator. In degrees mode with my casio calculator, I typed, "$\arctan(1)$" and got an answer of 45. In radian mode with the same parameter, I got, "0.78539......Fri, 21 Aug 2026 22:01:17 -0400https://pop.com.sg/equation-of-a-rectangle.htmlhttps://pop.com.sg/equation-of-a-rectangle.htmlI need to graph a rectangle on the Cartesian coordinate system. Is there an equation for a rectangle? I can't find it anywhere....Fri, 21 Aug 2026 21:59:28 -0400https://pop.com.sg/is-this-the-right-way-to-prove-basic-limit-identities.htmlhttps://pop.com.sg/is-this-the-right-way-to-prove-basic-limit-identities.htmlFor example let's say I wanted to prove that: $$\lim_{x \to \infty} \frac{1}{x} = 0$$ Normally I would use epsilon-delta for such proofs but then I start out with conditions like: $|\frac{1}{x} ......Fri, 21 Aug 2026 21:56:44 -0400https://pop.com.sg/integration-of-secant-duplicate.htmlhttps://pop.com.sg/integration-of-secant-duplicate.html$$\begin{align} \int \sec x \, dx &= \int \cos x \left( \frac{1}{\cos^2x} \right) \, dx \\ &= \int \cos x \left( \frac{1}{1-\sin^2x} \right) \, dx \\ & = \int\cos x\cdot\frac{1}{1-\......Fri, 21 Aug 2026 21:52:12 -0400https://pop.com.sg/finding-a-regular-16-sided-polygon-whose-vertices-are-lattice-points.htmlhttps://pop.com.sg/finding-a-regular-16-sided-polygon-whose-vertices-are-lattice-points.htmlFind a regular equilateral and equiangular 16-sided polygon that has vertices that are lattice points. What is the minimum side length? Find the vertices of such a polygon. Find the area of the po......Fri, 21 Aug 2026 21:49:49 -0400https://pop.com.sg/given-that-g-is-the-centroid-of-delta-abc-ga-2-sqrt-3-gb-2-sqrt-2-gc-2.htmlhttps://pop.com.sg/given-that-g-is-the-centroid-of-delta-abc-ga-2-sqrt-3-gb-2-sqrt-2-gc-2.htmlGiven that $G$ is the centroid of $\Delta ABC$, $GA = 2\sqrt{3}$ , $GB = 2\sqrt{2}$, $GC = 2$ . Find $[\Delta ABC]$. What I Tried: Here is a picture :- I know the centroid divides each of the med......Fri, 21 Aug 2026 21:44:46 -0400https://pop.com.sg/prove-the-fhhf-theorem-using-as-much-abstract-non-sense-as-possible.htmlhttps://pop.com.sg/prove-the-fhhf-theorem-using-as-much-abstract-non-sense-as-possible.htmlThis is my second attempt to solve exercise 1.6H from Vakil: Assume $F$ is a covariant right-exact functor and show that we get a map $HF^i \rightarrow H^iF$. Attempt to a solution: Apply $F$ to ......Fri, 21 Aug 2026 21:33:41 -0400https://pop.com.sg/what-is-the-general-formula-for-calculating-dot-and-cross-products-in-.htmlhttps://pop.com.sg/what-is-the-general-formula-for-calculating-dot-and-cross-products-in-.htmlI was writing a C++ class for working with 3D vectors. I have written operations in the Cartesian coordinates easily, but I'm stuck and very confused at spherical coordinates. I googled my question......Fri, 21 Aug 2026 21:18:31 -0400https://pop.com.sg/taylor-series-analog-for-functionals.htmlhttps://pop.com.sg/taylor-series-analog-for-functionals.htmlFor a function $f(x)$, it is possible to write it as a taylor series centered around a point $x=a$: $$f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a){(x-a)}^{n}}{n!}=f(a)+f'(a)(x-a)+\frac{f''(a)(x-a)^{2}......Fri, 21 Aug 2026 21:11:26 -0400https://pop.com.sg/understanding-the-first-step-analysis-of-absorbing-markov-chains.htmlhttps://pop.com.sg/understanding-the-first-step-analysis-of-absorbing-markov-chains.htmlConsider a time-homogeneous Markov chain $\{X_n\}_{n=0}^\infty$ with the state space state space $S=\{0,1,2\}$ and the following transition probability matrix: \begin{pmatrix} 1 & 0 & 0 ......Fri, 21 Aug 2026 21:10:23 -0400https://pop.com.sg/how-do-you-find-a-vector-in-the-form-a-b-when-only-the-angle-and-magni.htmlhttps://pop.com.sg/how-do-you-find-a-vector-in-the-form-a-b-when-only-the-angle-and-magni.htmlHow do you find a vector in the form when only the angle and magnitude are given? Here is an example where an angle of 80 degrees is given along with a magnitude of 3....Fri, 21 Aug 2026 21:05:28 -0400https://pop.com.sg/determine-the-asymptotic-curves-of-the-catenoid.htmlhttps://pop.com.sg/determine-the-asymptotic-curves-of-the-catenoid.htmlLet be $X(u,v) = (\cosh v \cos u,\cosh v \sin u, v)$ the parametrization of catenoid. Determine the asymptotic curves of the catenoid. $\textbf{My attempt:}$ $X_u(u,v) = (-\cosh v \sin u,\cosh v \......Fri, 21 Aug 2026 20:51:07 -0400https://pop.com.sg/two-vectors-are-in-the-same-direction-if.htmlhttps://pop.com.sg/two-vectors-are-in-the-same-direction-if.htmlAre two vectors in the same direction if their dot product is greater than zero/positive? I know they are orthogonal if their dot product is 0 so they can not be in the same direction. I also read ......Fri, 21 Aug 2026 20:48:39 -0400https://pop.com.sg/what-is-growth-of-a-function-and-growth-rate-of-a-function.htmlhttps://pop.com.sg/what-is-growth-of-a-function-and-growth-rate-of-a-function.htmlI'm trying to figure out what is growth of a function and growth rate of a function means and how should I note it. As I understand growth rate of f(n) is dominant term of f(n) when n tends to in......Fri, 21 Aug 2026 20:48:37 -0400https://pop.com.sg/finding-the-possible-values-of-x.htmlhttps://pop.com.sg/finding-the-possible-values-of-x.htmlThe lengths of a triangle are 5cm, 7cm and x cm. What are the possible values of x? I don't know where to start with this question as i don't thin enough information has been given. With an ans......Fri, 21 Aug 2026 20:44:16 -0400https://pop.com.sg/square-brackets-instead-of-parens-for-functions.htmlhttps://pop.com.sg/square-brackets-instead-of-parens-for-functions.htmlWhy are sometimes square brackets used to apply parameters to functions instead of the usual round parentheses? For instance, in my probability course, they use $\text{P}[X]$ to denote the probabi......Fri, 21 Aug 2026 20:41:47 -0400https://pop.com.sg/probability-of-a-u-b-u-c.htmlhttps://pop.com.sg/probability-of-a-u-b-u-c.htmlEvent A and B are independent, A and C are mutually exclusive, B and C are independent. Given: $\Pr(A) = 1/2$ $\Pr(B) = 1/4$ $\Pr(C) = 1/8$ Find $P(A \cup B \cup C)$ ? The teacher gave the answe......Fri, 21 Aug 2026 20:19:49 -0400https://pop.com.sg/using-the-logarithm-product-rule-with-a-coefficient.htmlhttps://pop.com.sg/using-the-logarithm-product-rule-with-a-coefficient.htmlI'm trying to solve this logarithmic expression: $\left(\dfrac{\left(\log_6(27)+2\log_6(2)\right)}{\log_6\sqrt[3]{0.25}+\log_6(\frac{1}{3})}\right)^{\!2}$ But after a couple of steps, I got st......Fri, 21 Aug 2026 20:16:28 -0400https://pop.com.sg/why-is-it-enough-for-a-tangent-plane-to-only-agree-with-the-slope-of-a.htmlhttps://pop.com.sg/why-is-it-enough-for-a-tangent-plane-to-only-agree-with-the-slope-of-a.htmlThanks for reading. Short Version Why is it enough for the tangent plane to only agree with the slope of a differentiable 2-input function in the $x$ and $y$ directions at a certain point for it to......Fri, 21 Aug 2026 20:05:42 -0400https://pop.com.sg/definite-integral-area-between-two-trig-functions.htmlhttps://pop.com.sg/definite-integral-area-between-two-trig-functions.htmlI'm trying to figure out how to find the area between two trig functions. I know the procedure of integration here, finding the difference between two functions and integrating across whatever inte......Fri, 21 Aug 2026 20:04:49 -0400https://pop.com.sg/what-is-the-difference-between-arg-max-and-max.htmlhttps://pop.com.sg/what-is-the-difference-between-arg-max-and-max.htmlWhat is the exact difference between $\arg\max$ and $\max$ of a function? Is it right to say the following? $\arg\max f(x)$ is nothing but the value of $x$ for which the value of the function is ...Fri, 21 Aug 2026 20:00:19 -0400https://pop.com.sg/opposite-of-factorial-duplicate.htmlhttps://pop.com.sg/opposite-of-factorial-duplicate.htmlPossible Duplicate: Is there a way to solve for an unknown in a factorial? I was just wondering, what would be the opposite of factorial? For example, If I had $n! = 120$. How can I then show ...Fri, 21 Aug 2026 19:51:31 -0400https://pop.com.sg/cardinality-of-all-cardinalities.htmlhttps://pop.com.sg/cardinality-of-all-cardinalities.htmlLet $C = \{0, 1, 2, \ldots, \aleph_0, \aleph_1, \aleph_2, \ldots\}$. What is $\left|C\right|$? Or is it even well-defined?...Fri, 21 Aug 2026 19:50:46 -0400https://pop.com.sg/maximum-spanning-tree.htmlhttps://pop.com.sg/maximum-spanning-tree.htmlI wonder how to prove that given a Minimum Spanning Tree of a graph, the other spanning tree with the least common edge with Minimum Spanning Tree is always Maximum Spanning tree....Fri, 21 Aug 2026 19:34:07 -0400https://pop.com.sg/an-interesting-cubic-equation-x-3-3x-1-0.htmlhttps://pop.com.sg/an-interesting-cubic-equation-x-3-3x-1-0.htmlFor the cubic equation $x^3-3x-1=0$, you can check it has three real roots by calculus knowledge. Let them be $x_1<x_2<x_3$. Now prove that $$x_3^2-x_2^2=x_3-x_1.$$ It is not difficult to ch......Fri, 21 Aug 2026 19:34:03 -0400https://pop.com.sg/how-can-i-prove-that-every-group-of-n-255-elements-is-commutative.htmlhttps://pop.com.sg/how-can-i-prove-that-every-group-of-n-255-elements-is-commutative.htmlThere was previous task was same but with $N = 185$. And I prove it by showing that number of Sylow subgroups is 1 for every prime $p\mid N$. But there I have some options $N_5 \in \{1, 51\}$, $N_{......Fri, 21 Aug 2026 19:28:27 -0400https://pop.com.sg/definition-of-linear-differential-equation.htmlhttps://pop.com.sg/definition-of-linear-differential-equation.htmlI am a 13 year old self teaching myself Differential Equations from a website and a book, I came across the definition of a Linear Differential Equation but I didn't understand the definition, I lo......Fri, 21 Aug 2026 19:25:53 -0400https://pop.com.sg/does-there-exist-a-group-isomorphism-from-z-to-zxz.htmlhttps://pop.com.sg/does-there-exist-a-group-isomorphism-from-z-to-zxz.htmlI know that $\mathbb Z$ and $\mathbb{Z}\times\mathbb{Z}$ have the same cardinality because you can create a bijection between the two. The example I was taught is Cantor's pairing function, which m......Fri, 21 Aug 2026 19:16:11 -0400https://pop.com.sg/finding-a-joint-probability-density-function-given-marginal-probabilit.htmlhttps://pop.com.sg/finding-a-joint-probability-density-function-given-marginal-probabilit.htmlX and Y are independent random variables with the following PDFs: $f_{x}(x) = \begin{cases} (1/3)e^{-x/3}, & x \geq\text{ 0} \\ 0, & \text{otherwise} \end{cases}$ $f_{y}(y) = \begin{cases......Fri, 21 Aug 2026 19:13:07 -0400https://pop.com.sg/how-many-subsets-can-be-made-with-a-set-of-n-items.htmlhttps://pop.com.sg/how-many-subsets-can-be-made-with-a-set-of-n-items.htmlIf you have a set of $N$ items, how many subsets can you make? For example, for the set existing of $3$ items (Item1, Item2 and Item3) the subsets are Item1, Item2, Item3, Item2+3, Item1+2, Item1+......Fri, 21 Aug 2026 19:09:56 -0400https://pop.com.sg/understanding-the-formula-for-stereographic-projection-of-a-point.htmlhttps://pop.com.sg/understanding-the-formula-for-stereographic-projection-of-a-point.htmlI was wondering about the equation of line I can write which can help me finding the coordinates of Point $P'$ in relation with coordinates of points on the sphere that is $P$. Let the $P'(X,Y)$, ......Fri, 21 Aug 2026 19:01:08 -0400https://pop.com.sg/calculator-tool-to-find-subgroup-of-s-n-generated-by-langle-g-1-dots-g.htmlhttps://pop.com.sg/calculator-tool-to-find-subgroup-of-s-n-generated-by-langle-g-1-dots-g.htmldoes anyone know an online utility tool that computes the subgroup of the symmetric group generated by generators, for example, I want the program to tell me $$ \langle (13)(24), (13) \rangle = \{......Fri, 21 Aug 2026 18:59:58 -0400https://pop.com.sg/sketch-the-graph-of-the-integrand-function-and-use-it-to-help-evaluate.htmlhttps://pop.com.sg/sketch-the-graph-of-the-integrand-function-and-use-it-to-help-evaluate.htmlsketch the graph of the integrand function and use it to help evaluate the integral. integration from(1 , -1) |x|-1 I think I can evaluate the integration f(x) = 1/2 x^2 -x+c but how sketch the ...Fri, 21 Aug 2026 18:44:35 -0400https://pop.com.sg/how-to-rewrite-logarithmic-equation-in-exponential-form.htmlhttps://pop.com.sg/how-to-rewrite-logarithmic-equation-in-exponential-form.htmlHow would I rewrite this logarithmic equation: $\ln(37)= 3.6109$, in exponential form? -Thanks...Fri, 21 Aug 2026 18:38:00 -0400https://pop.com.sg/transformation-matrix-for-rotation-around-a-point-that-is-not-the-orig.htmlhttps://pop.com.sg/transformation-matrix-for-rotation-around-a-point-that-is-not-the-orig.htmlI need to find the matrix that rotates an arbitrary point around $\begin{bmatrix}5 \\6\end{bmatrix}$ by 35* anticlockwise. I figure I need to first move the plane to centre it at the origin, perfo......Fri, 21 Aug 2026 18:33:17 -0400https://pop.com.sg/does-such-a-natural-number-exist-that-it-would-be-divisible-by-every-o.htmlhttps://pop.com.sg/does-such-a-natural-number-exist-that-it-would-be-divisible-by-every-o.htmlI've got to prove (or disprove) the following statement: $\exists x \in \mathbb{N} \; \forall y \in \mathbb{N}: y \mid x$, which translates into "It exists such $x$ from the set of natural numbers, ...Fri, 21 Aug 2026 18:28:57 -0400https://pop.com.sg/polar-plot-r-1-2-cos-6-theta.htmlhttps://pop.com.sg/polar-plot-r-1-2-cos-6-theta.htmlI have a task for school and we need to plot a polar function with MATLAB. The function is $r = 1-2\cos(6\theta)$. I did this and I'm getting exactly the same as on Wolfram Alpha: https://www....Fri, 21 Aug 2026 18:08:26 -0400https://pop.com.sg/how-to-define-addition-through-multiplication.htmlhttps://pop.com.sg/how-to-define-addition-through-multiplication.htmlOne might define multiplication $\bullet$ on $\mathbb Z$ as follows: $\bullet: \mathbb Z\times \mathbb N\ni (a,b) \mapsto a+\cdots+a\in \mathbb Z$ where we add $b$ times. But suppose we are in a ...Fri, 21 Aug 2026 17:54:11 -0400https://pop.com.sg/when-finding-relative-minimum-maximum-what-is-the-point-of-using-the-s.htmlhttps://pop.com.sg/when-finding-relative-minimum-maximum-what-is-the-point-of-using-the-s.htmlWhy use the second derivative test over the first derivative test when finding maxima and minima if it's uncertain what $f''(x) = 0$ is? Why not just always use the first derivative since we need t......Fri, 21 Aug 2026 17:50:16 -0400https://pop.com.sg/stochastic-leibniz-rule.htmlhttps://pop.com.sg/stochastic-leibniz-rule.htmlI have come up with the following Leibniz stochastic rule and I want to check that: The result is correct; The proof is right. Statement: let $f(\cdot,t):s \rightarrow f(s,t)$, $s \in \mathbb{R}^......Fri, 21 Aug 2026 17:49:26 -0400https://pop.com.sg/how-to-get-rid-of-this-exponentiation.htmlhttps://pop.com.sg/how-to-get-rid-of-this-exponentiation.htmlI would like to remove exponentiation in this equation $y=x^t$ and use only multiplication or division. I have $x, t$ and $\ln(x)$. Is it possible?...Fri, 21 Aug 2026 17:47:13 -0400https://pop.com.sg/finding-area-of-shaded-region-on-graph.htmlhttps://pop.com.sg/finding-area-of-shaded-region-on-graph.htmlWith the three equations provided, I can find the coordinates of intersection. However, I have no idea how to find the height as there isnt any right angles. The graph is apparently a triangle, how......Fri, 21 Aug 2026 17:38:16 -0400https://pop.com.sg/perpendicular-from-incenter-of-a-triangle-to-any-side-is-equal-to-the-.htmlhttps://pop.com.sg/perpendicular-from-incenter-of-a-triangle-to-any-side-is-equal-to-the-.htmlGiven a triangle $ABC$ with incenter $I$, it is said that the perpendicular line segment from $I$ to any of the sides $AB$, $AC$, or $BC$ is equal to the radius of the incircle. (See the second pic......Fri, 21 Aug 2026 17:30:21 -0400https://pop.com.sg/how-to-solve-a-linear-system-in-matrix-form-using-laplace-transform.htmlhttps://pop.com.sg/how-to-solve-a-linear-system-in-matrix-form-using-laplace-transform.htmlHow to solve this linear system using Laplace transform? $$\mathbf X'(t)=\left[\begin{array}{r,r,r}-3&0&2\\1&-1&0\\-2&-1&0\end{array}\right]\mathbf X(t); ~~~~~~~~\mathbf X(0)=\...Fri, 21 Aug 2026 17:17:06 -0400https://pop.com.sg/probability-of-trifecta-box-bet.htmlhttps://pop.com.sg/probability-of-trifecta-box-bet.htmlIn a horse race there are 10 horses. Bob wants to make a "trifecta Box bet". A trifecta box bet is when you choose the first three horses that finish the race in ANY order. What is the probability ......Fri, 21 Aug 2026 17:13:53 -0400https://pop.com.sg/proving-theta-tan-theta-with-geometry-closed.htmlhttps://pop.com.sg/proving-theta-tan-theta-with-geometry-closed.htmlI'm looking for a simple geometrical method of proving that $$\theta < \tan\theta$$ for $0 < \theta < \frac{\pi}{2}$. I am able to prove that $\sin\theta < \theta$....Fri, 21 Aug 2026 17:11:20 -0400https://pop.com.sg/is-dijkstra-039-s-algorithm-optimal-for-unweighted-graphs.htmlhttps://pop.com.sg/is-dijkstra-039-s-algorithm-optimal-for-unweighted-graphs.htmlDijkstra's algorithm is a very good approach to the shortest path problem. But is it optimal? Are there better algorithms for unweighted graphs?...Fri, 21 Aug 2026 17:04:49 -0400https://pop.com.sg/orthonormal-basis-of-the-column-space-of-a-matrix.htmlhttps://pop.com.sg/orthonormal-basis-of-the-column-space-of-a-matrix.htmlFind an orthonormal basis of the column space of the matrix $$\begin{pmatrix}1 & -1 & 2 & 3\\ -1 & 1 & -3 & 2\\ 2 & -2 & 5 & -5 \end{pmatrix}.$$ I found the row ...Fri, 21 Aug 2026 16:48:22 -0400https://pop.com.sg/which-rational-primes-less-than-50-are-also-gaussian-primes.htmlhttps://pop.com.sg/which-rational-primes-less-than-50-are-also-gaussian-primes.htmlWhich rational primes less than 50 are also Gaussian primes? My attempt: First we need to list all of the rational prime numbers that are less than $50$ $1,2,3,5,7,11,13,17,19,23,29,31,37,39,41,4......Fri, 21 Aug 2026 16:30:07 -0400https://pop.com.sg/use-duplex-method-to-find-square-root-of-given-number.htmlhttps://pop.com.sg/use-duplex-method-to-find-square-root-of-given-number.htmlI was reading this Wikipedia page about finding square root of given number. But I did't understand the section on the "Vedic duplex method". Not because of lack of English vocabulary, but main ...Fri, 21 Aug 2026 16:29:17 -0400https://pop.com.sg/formal-definition-of-plane.htmlhttps://pop.com.sg/formal-definition-of-plane.htmlThe formal definition of plane says that: A plane is a set of points such that if any two points are taken on it, all the points lying on the line joining these two points also lie on the plane. ......Fri, 21 Aug 2026 16:24:09 -0400https://pop.com.sg/on-a-scale-of-1-to-10-how-likely-is-it-that-this-question-is-using-bin.htmlhttps://pop.com.sg/on-a-scale-of-1-to-10-how-likely-is-it-that-this-question-is-using-bin.htmlI just read this interesting xkcd strip: At first I thought it was funny, but as I got to ruminate a little over it, I was surprised to be unable to find an answer. As Karolis Juodelė pointed out,......Fri, 21 Aug 2026 16:06:25 -0400https://pop.com.sg/what-do-the-p-and-q-stand-for-in-this-quadratic-formula.htmlhttps://pop.com.sg/what-do-the-p-and-q-stand-for-in-this-quadratic-formula.htmlI study Computer Science in a German university and they use another formula to solve the quadratic equations. They call it the P-Q-Formel. I'm used to this formula. While the Germans use this on......Fri, 21 Aug 2026 15:55:43 -0400https://pop.com.sg/how-to-determine-if-a-linear-system-is-solvable.htmlhttps://pop.com.sg/how-to-determine-if-a-linear-system-is-solvable.htmlI have this problem $$\begin{array}{rcccccl} 3x &-&y &+& 2z &=& 2\\ 2x &+& y &+& z &=& -1\\ x &+& 3y & & &=&-1 \end{array......Fri, 21 Aug 2026 15:55:23 -0400https://pop.com.sg/why-does-the-symbol-for-the-multiplication-operation-change-shape.htmlhttps://pop.com.sg/why-does-the-symbol-for-the-multiplication-operation-change-shape.htmlWhy does the "$\times$" used in arithmetic change to a "$\cdot$" as we progress through education? The symbol seems to only be ambiguous because of the variable $x$; however, we wouldn't have chose......Fri, 21 Aug 2026 15:47:57 -0400https://pop.com.sg/what-is-an-algebra.htmlhttps://pop.com.sg/what-is-an-algebra.htmlIs an algebra or 'a algebra' the same thing as an algebraic structure? Or does it have a different meaning? Thanks...Fri, 21 Aug 2026 15:47:50 -0400https://pop.com.sg/what-is-the-best-way-to-solve-modular-arithmetic-equations-such-as-9x-.htmlhttps://pop.com.sg/what-is-the-best-way-to-solve-modular-arithmetic-equations-such-as-9x-.htmlWhat is the best way to solve equations like the following: $9x \equiv 33 \pmod{43}$ The only way I know would be to try all multiples of $43$ and $9$ and compare until I get $33$ for the remaind......Fri, 21 Aug 2026 15:45:24 -0400https://pop.com.sg/prove-lim-x-to3-frac-x-2-9-x-3-6-using-delta-epsilon-definition-of-lim.htmlhttps://pop.com.sg/prove-lim-x-to3-frac-x-2-9-x-3-6-using-delta-epsilon-definition-of-lim.htmlI need to prove that the $$\lim_{x\to3}\frac{x^2 - 9}{x - 3} = 6$$ using $\delta-\epsilon$ definition of limit. Now, I have started with a discussion, saying that what we want is that if $\left| x......Fri, 21 Aug 2026 15:41:27 -0400https://pop.com.sg/why-does-ln-x-ln-b-log-b-x.htmlhttps://pop.com.sg/why-does-ln-x-ln-b-log-b-x.htmlI'm doing some Java code. As far as I can tell, Java only has functions that do natural log and base $10$ log. I have a requirement to specify the base. I've seen that doing $\ln x/ \ln b$ is th......Fri, 21 Aug 2026 15:39:59 -0400https://pop.com.sg/what-are-linear-homogeneous-and-non-homoegenous-recurrence-relations.htmlhttps://pop.com.sg/what-are-linear-homogeneous-and-non-homoegenous-recurrence-relations.htmlAccording to my book, a linear homogeneous recurrence of order $k$ is expressed this way: $$A_0a_n+A_1a_{n-1}+A_2a_{n-2}+\cdots+A_ka_{n-k}=0$$ While a linear non-homogeneous recurrence of order $k$......Fri, 21 Aug 2026 15:38:31 -0400https://pop.com.sg/what-is-the-logic-behind-the-digit-sum-formula.htmlhttps://pop.com.sg/what-is-the-logic-behind-the-digit-sum-formula.htmlThe digit sum of a number is the sum of the digits in the number. Let $n$ be a natural number. Then the digit sum of $n$ is $S(n)=\displaystyle\sum_{k=0}^{\lfloor{\log_{10} n}\rfloor}\frac{1}{10^k......Fri, 21 Aug 2026 15:38:03 -0400https://pop.com.sg/boolean-expression-of-3-input-xor-gate.htmlhttps://pop.com.sg/boolean-expression-of-3-input-xor-gate.htmlhow is it equal $A$ xor $B$ xor $C = A'BC' + AB'C' + A'B'C + ABC.$ can someone explain using with boolean algebra rules....Fri, 21 Aug 2026 15:35:34 -0400https://pop.com.sg/intuition-of-the-meaning-of-homology-groups.htmlhttps://pop.com.sg/intuition-of-the-meaning-of-homology-groups.htmlI am studying homology groups and I am looking to try and develop, if possible, a little more intuition about what they actually mean. I've only been studying homology for a short while, so if poss......Fri, 21 Aug 2026 15:34:08 -0400https://pop.com.sg/show-that-each-character-of-g-which-is-zero-for-all-g-ne-1-is-an-integ.htmlhttps://pop.com.sg/show-that-each-character-of-g-which-is-zero-for-all-g-ne-1-is-an-integ.htmlThis is a question from J.P.Serre's book 'Linear representation of finite groups',section 2.4 The question: Let $G$ be a finite group. Show that each character of $G$ which is zero for all $g \ne ......Fri, 21 Aug 2026 15:27:38 -0400https://pop.com.sg/how-to-write-integral.htmlhttps://pop.com.sg/how-to-write-integral.htmlWhen solving a integral problem domain $[x_1,x_2]$, I start with writing down this: $$\int_{x_1}^{x_2} f(x) dx = F(x_2) - F(x_1).$$ Then I calculate the $F(x)$ and fill in the respected $x_1$ and $......Fri, 21 Aug 2026 15:24:29 -0400https://pop.com.sg/finding-taylor-series-of-indefinite-integral-cos-x-2-around-x-0.htmlhttps://pop.com.sg/finding-taylor-series-of-indefinite-integral-cos-x-2-around-x-0.htmlHow would I find the Taylor series around $x = 0$ for this integral? $$\int \cos(x^2)$$ My first point of confusion is if it is around $x = 0$, doesn't that make it a Maclaurin series? Would I go a......Fri, 21 Aug 2026 15:24:01 -0400https://pop.com.sg/probability-of-sampling-with-and-without-replacement.htmlhttps://pop.com.sg/probability-of-sampling-with-and-without-replacement.htmlIn sampling without replacement the probability of any fixed element in the population to be included in a random sample of size $r$ is $\frac{r}{n}$. In sampling with replacement the corresponding ...Fri, 21 Aug 2026 15:18:50 -0400