https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usFri, 21 Aug 2026 18:44:10 -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 -0400https://pop.com.sg/how-to-solve-this-elliptic-integral.htmlhttps://pop.com.sg/how-to-solve-this-elliptic-integral.htmlCan anyone explain to me how to find the integral ? $$ \int_0^1\sqrt{9x^4+4x^2+1}dx =? $$...Fri, 21 Aug 2026 15:16:13 -0400https://pop.com.sg/finding-maximum-rate-of-change-of-a-function-of-two-variables.htmlhttps://pop.com.sg/finding-maximum-rate-of-change-of-a-function-of-two-variables.htmlConsider the surface $h(x,y) = 1+2x\sqrt{y}$ (i) Find the maximum rate of change of $h$ at $(3,4)$. (ii) Find the (two) directions one could begin to move to stay level if one is standing on the ...Fri, 21 Aug 2026 15:03:18 -0400https://pop.com.sg/simplifying-cos-x-times-cos-y.htmlhttps://pop.com.sg/simplifying-cos-x-times-cos-y.htmlSo I got a problem wrong on my homework because I simplified $\cos (x) \times \cos (y)$ to be $\cos (xy)$, and my teacher wrote on my paper that it can't be simplified. I was wondering if anyone co......Fri, 21 Aug 2026 15:01:35 -0400https://pop.com.sg/algorithm-to-compute-newton-polynomial-derivative.htmlhttps://pop.com.sg/algorithm-to-compute-newton-polynomial-derivative.htmlI'm unable to find a clean solution to this problem and hope someone here can help me. Given a list of x-values: $x_0, x_1, x_2, ... x_n$ and a value $x$, I want to determine the accompanying parts......Fri, 21 Aug 2026 14:50:57 -0400https://pop.com.sg/dot-product-of-two-vectors-squared-identity.htmlhttps://pop.com.sg/dot-product-of-two-vectors-squared-identity.htmlDoes the following equation hold true in general? $$ (a^Tb )^ 2 = (a^T b b^T a) $$ Where $a, b \in R^n$ are vectors of the same size n=1, 2, ... I know it's true in n=1 and n=2 because I manua......Fri, 21 Aug 2026 14:41:37 -0400https://pop.com.sg/is-the-gradient-operator-surjective.htmlhttps://pop.com.sg/is-the-gradient-operator-surjective.htmlLet $\Omega \subset \mathbb{R}^{n}$ be open and bounded with Lipschitz boundary. Is the gradient operator $\nabla :H^{1} ( \Omega ) \rightarrow L^{2} ( \Omega )$ surjective? Here $H^{1} ( \Omega ) ......Fri, 21 Aug 2026 14:39:45 -0400https://pop.com.sg/taylor-expansion-of-sine-and-cosine.htmlhttps://pop.com.sg/taylor-expansion-of-sine-and-cosine.htmlHere: Is there a formula for sine and cosine?, one of the answers mentioned using the Taylor expansion for approximating $\sin(x)$ and $\cos(x)$, and someone commented pointing out however that this ...Fri, 21 Aug 2026 14:37:34 -0400https://pop.com.sg/solving-a-bcd-for-c-where-a-b-c-d-are-all-matrices-closed.htmlhttps://pop.com.sg/solving-a-bcd-for-c-where-a-b-c-d-are-all-matrices-closed.htmlIf $A,B,C,D$ are all matrices and $A=BCD$ (with dimensions such that all matrix multiplications are defined), how does one solve for $C$? In the particular context I'm working in, $B$ and $D$ ar......Fri, 21 Aug 2026 14:34:46 -0400https://pop.com.sg/if-the-order-in-a-set-doesn-t-matter-can-we-change-order-of-say-bbb-n.htmlhttps://pop.com.sg/if-the-order-in-a-set-doesn-t-matter-can-we-change-order-of-say-bbb-n.htmlI’m given to understand that the order of the elements of a set doesn’t matter. So can I change the order of the set of natural numbers or any set of numbers ( $\mathbb{W,Z,Q,R}$ for that matter) as ...Fri, 21 Aug 2026 14:33:12 -0400https://pop.com.sg/are-functions-considered-continuous-at-endpoints.htmlhttps://pop.com.sg/are-functions-considered-continuous-at-endpoints.htmlConsider a function $f(x)$ that has no jump, infinite, or removable discontinuities in the middle anywhere -- but maybe the domain is limited: Would the endpoint $[a,$ be considered continuous? What ...Fri, 21 Aug 2026 14:21:07 -0400https://pop.com.sg/equality-of-rows-and-columns-of-a-table-with-numbers.htmlhttps://pop.com.sg/equality-of-rows-and-columns-of-a-table-with-numbers.htmlIn a table of size $m \times n$, numbers are written so that the sum of the numbers in each row and in each column is equal to $10$. How to prove that $m = n$?...Fri, 21 Aug 2026 14:08:09 -0400https://pop.com.sg/usefulness-of-the-notion-of-hilbert-scheme-in-algebraic-geometry.htmlhttps://pop.com.sg/usefulness-of-the-notion-of-hilbert-scheme-in-algebraic-geometry.htmlCould someone tell me why and how Hilbert schemes and relative Hilbert schemes are important and useful in algebraic geometry? Could anyone give me some applications of this notion in concrete ter......Fri, 21 Aug 2026 13:45:33 -0400https://pop.com.sg/what-is-the-inverse-function-of-f-x-x-1-x-2.htmlhttps://pop.com.sg/what-is-the-inverse-function-of-f-x-x-1-x-2.htmlCan you give me a hint for how the inverse function of $f\colon (-1,1)\to \mathbb{R}\colon f(x)=\frac{x}{1-x^2}$ looks? I need to show a homeomorphism!...Fri, 21 Aug 2026 13:27:25 -0400https://pop.com.sg/using-the-alternative-formula-to-find-the-derivative-of-a-function.htmlhttps://pop.com.sg/using-the-alternative-formula-to-find-the-derivative-of-a-function.htmlI'm attempting to find the derivative of the function: $$f(x) = 4x^2+3x+5$$ Using the alternative formula: $$\frac{f(z)-f(x)}{z-x}$$ Here are my steps so far: $$\frac{4z^2+3z+5-(4x^2+3x+5)}{z-x}$$......Fri, 21 Aug 2026 13:12:58 -0400https://pop.com.sg/proving-existence.htmlhttps://pop.com.sg/proving-existence.htmlWhat does an introductory-level proof of existence look like? For example, I have done proofs involving proving uniqueness by assuming there are two arbitrary elements that have the properties I w......Fri, 21 Aug 2026 13:10:43 -0400https://pop.com.sg/how-to-measure-the-volume-of-rock.htmlhttps://pop.com.sg/how-to-measure-the-volume-of-rock.htmlI have a object which is similar to the shape of irregular rock like this I would like to find the volume of this. How to do it? If I have to find the volume, what are the things I would need. e......Fri, 21 Aug 2026 13:09:04 -0400https://pop.com.sg/how-to-prove-that-the-inverse-of-a-persymmetric-matrix-is-also-persymm.htmlhttps://pop.com.sg/how-to-prove-that-the-inverse-of-a-persymmetric-matrix-is-also-persymm.htmlAn exercise in a textbook I'm using to brush up on my linear algebra asks to prove that the inverse of a persymmetric matrix is also persymmetric. I have a colleague's old notes in front of me with a ...Fri, 21 Aug 2026 13:08:31 -0400https://pop.com.sg/is-there-a-domain-larger-than-i-e-a-supserset-of-the-complex-number-do.htmlhttps://pop.com.sg/is-there-a-domain-larger-than-i-e-a-supserset-of-the-complex-number-do.htmlI've been teaching my 10yo son some (for me, anyway) pretty advanced mathematics recently and he stumped me with a question. The background is this. In the domain of natural numbers, addition and ...Fri, 21 Aug 2026 13:06:08 -0400https://pop.com.sg/how-many-different-5-letters-word-can-be-possible-by-using-the-letters.htmlhttps://pop.com.sg/how-many-different-5-letters-word-can-be-possible-by-using-the-letters.htmlSo I took the vowels (a,e,i) permute them 3p3 and then I choose 6c2.Here I get baffled and can't figure out what should I do next. By the way multiply 3p3 and 6c2 comes up with 180 which is not the ...Fri, 21 Aug 2026 13:05:22 -0400https://pop.com.sg/where-does-feigenbaum-039-s-constant-4-6692-originate-closed.htmlhttps://pop.com.sg/where-does-feigenbaum-039-s-constant-4-6692-originate-closed.htmlFeigenbaum discovered a ratio between bifurcations that were found in all known chaotic-dynamic systems, from dripping water faucets to abstract equations on population fluctuations (as elucidated in ...Fri, 21 Aug 2026 13:03:31 -0400https://pop.com.sg/solving-sin-theta-cos-theta-1-in-the-interval-0-circ-leq-theta-leq-360.htmlhttps://pop.com.sg/solving-sin-theta-cos-theta-1-in-the-interval-0-circ-leq-theta-leq-360.htmlSolve in the interval $0^\circ\leq \theta\leq 360^\circ$ the equation $\sin \theta + \cos \theta=1$. I've got the two angles in the interval to be $0^\circ$ and $90^\circ$, it's not an answer I'm ......Fri, 21 Aug 2026 13:01:28 -0400https://pop.com.sg/definition-of-disconnected.htmlhttps://pop.com.sg/definition-of-disconnected.htmlI am confused in definition of disconnected set in topological space $X$. Which of the following definition is valid/correct? 1) a subset $A$ of $X$ is disconnected if $A=V ∪W$ where $U$ and $V$ ......Fri, 21 Aug 2026 12:59:33 -0400