https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usFri, 28 Aug 2026 17:03:11 -0400https://pop.com.sg/method-of-characteristics-for-burgers-039-equation-with-rectangular-da.htmlhttps://pop.com.sg/method-of-characteristics-for-burgers-039-equation-with-rectangular-da.htmlUsing the method of characteristics, find a solution to Burgers' equation \begin{cases} u_t+\left(\frac{u^2}{2} \right)_x =0 & \text{in }\mathbb{R}\times(0,\infty) \\ \qquad \qquad \, \, u=g......Fri, 28 Aug 2026 20:57:09 -0400https://pop.com.sg/computing-the-iterated-logarithm-log-star-by-hand.htmlhttps://pop.com.sg/computing-the-iterated-logarithm-log-star-by-hand.htmlI'm having trouble figuring out how to compute the iterated logarithm (log-star) by hand without using a calculator or programming language. I wrote out the following program in Java to check my an......Fri, 28 Aug 2026 20:54:31 -0400https://pop.com.sg/how-to-choose-between-real-and-complex-coefficients-of-fourier-series.htmlhttps://pop.com.sg/how-to-choose-between-real-and-complex-coefficients-of-fourier-series.htmlSometimes in exercises we are asked to calculate the fourier series of a function. But there are two ways to do that. If $f:\Bbb R\mapsto\Bbb R$ is $T$-periodic over $\Bbb R$ then what conditions......Fri, 28 Aug 2026 20:50:02 -0400https://pop.com.sg/why-does-two-terms-immediately-adjacent-mean-multiply.htmlhttps://pop.com.sg/why-does-two-terms-immediately-adjacent-mean-multiply.htmlI am currently teaching a GED math class. While learning about the order of operations, the students asked why does a number next to a parentheses mean multiplication? I understand the rule that two ...Fri, 28 Aug 2026 20:40:35 -0400https://pop.com.sg/what-does-the-plus-sign-contained-in-a-circle-oplus-mean-in-this-case.htmlhttps://pop.com.sg/what-does-the-plus-sign-contained-in-a-circle-oplus-mean-in-this-case.htmlThe fundamental group of the torus is isomorphic to $\mathbb{Z}\oplus\mathbb{Z}$. I know that the $\oplus$ symbol is the exclusive or symbol but I don't understand how two of the same sets are XOR......Fri, 28 Aug 2026 20:18:57 -0400https://pop.com.sg/remainder-when-2-108-is-divided-by-11-closed.htmlhttps://pop.com.sg/remainder-when-2-108-is-divided-by-11-closed.htmlWhat is the remainder obtained when $2^{108}$ is divided by $11$? I tried bringing in $11$ in the given no. such as in $(11-3)^{36}$ and then using binomial expansion...but its not helping. Any ...Fri, 28 Aug 2026 20:08:18 -0400https://pop.com.sg/complexity-analysis-of-alpha-beta-pruning-of-a-full-tree.htmlhttps://pop.com.sg/complexity-analysis-of-alpha-beta-pruning-of-a-full-tree.htmlI am trying to understand the derivation of a time complexity for an alpha-beta pruning algorithm but up till now have not found any reasonable recourse. Many recourses claim that if you take a full ...Fri, 28 Aug 2026 20:04:37 -0400https://pop.com.sg/relationship-between-groups-and-permutations.htmlhttps://pop.com.sg/relationship-between-groups-and-permutations.htmlWhat is the relationship between groups and permutations? The way I understand it, a permutation is a set of numbers that you can do things with. A group is a set of elements equipped with an opera......Fri, 28 Aug 2026 20:03:35 -0400https://pop.com.sg/combinations-probability-dividing-9-people-into-3-groups.htmlhttps://pop.com.sg/combinations-probability-dividing-9-people-into-3-groups.htmlI'm struggling to understand the procedure to solve the following problem: The group consists of 9 people, 3 men and 6 women. They are divided randomly into 3 groups (3 persons each). What's the ...Fri, 28 Aug 2026 19:53:05 -0400https://pop.com.sg/how-do-i-calculate-the-value-of-this-series.htmlhttps://pop.com.sg/how-do-i-calculate-the-value-of-this-series.htmlI want to find the value to which this series converges $$\sum_{n=0}^\infty \frac{(-1)^n}{n^2+1}$$ I tried looking at the sequence of partial sums $$S_k = \sum_{n=0}^k \frac{(-1)^n}{n^2+1}$$ and I ...Fri, 28 Aug 2026 19:42:27 -0400https://pop.com.sg/riemannian-ricci-curvature-for-round-n-sphere.htmlhttps://pop.com.sg/riemannian-ricci-curvature-for-round-n-sphere.htmlWhat is the best way to see that the Ricci scalar curvature of $(S^n(r),g_{round})$ is a constant $n(n-1)/r^2$ ? I essentially only see this value stated in the literature, but no computation assoc......Fri, 28 Aug 2026 19:41:20 -0400https://pop.com.sg/do-non-convex-platonic-solids-exist.htmlhttps://pop.com.sg/do-non-convex-platonic-solids-exist.htmlConsider a solid with the following properties - It is composed of congruent, regular polygons. At each vertex, the same number of edges and faces meet. This is the same as the requirement for the ...Fri, 28 Aug 2026 19:34:14 -0400https://pop.com.sg/motivation-behind-point-set-topology-closed.htmlhttps://pop.com.sg/motivation-behind-point-set-topology-closed.htmlWhy should I study point-set topology? What initially interested me in topology was the pop-sci rubber sheet stuff or coffee cup-donut stuff or proving fundamental theorem of algebra using curves ......Fri, 28 Aug 2026 19:28:58 -0400https://pop.com.sg/binary-decimal-primes.htmlhttps://pop.com.sg/binary-decimal-primes.htmlConverting an integer $N$ into binary and then reading the result as a base-10 number $N_2$, the prime $N=3$ gives $N_2=11$ (which is also prime) and $N=5$ gives $N_2=101$. Are $3$ and $5$ the only......Fri, 28 Aug 2026 19:19:44 -0400https://pop.com.sg/proving-that-f-n-to-z-is-a-bijection-closed.htmlhttps://pop.com.sg/proving-that-f-n-to-z-is-a-bijection-closed.htmlProve that $f: \mathbb{N} \to \mathbb{Z} $ is a bijection where the relation between $\mathbb{N}$ and $\mathbb{Z}$ is as follows: $f(n)= \frac n2$ if n is even, and $-\frac {n+1}{2}$ if n is odd ...Fri, 28 Aug 2026 19:10:59 -0400https://pop.com.sg/what-exactly-is-a-number.htmlhttps://pop.com.sg/what-exactly-is-a-number.htmlWe've just been learning about complex numbers in class, and I don't really see why they're called numbers. Originally, a number used to be a means of counting (natural numbers). Then we extend t......Fri, 28 Aug 2026 19:09:35 -0400https://pop.com.sg/chain-rule-f-t-e-t-sin2t.htmlhttps://pop.com.sg/chain-rule-f-t-e-t-sin2t.htmlI'm missing a step somewhere on this: $$f(t)=e^{t\sin2t}.$$ I know I have to use the chain rule but I'm getting tripped up. here are my steps: $\frac d{dx}(e^{t\sin2t})$ $e^{t\sin2t} \frac d{dx}(t\...Fri, 28 Aug 2026 18:58:04 -0400https://pop.com.sg/what-exactly-is-calculus.htmlhttps://pop.com.sg/what-exactly-is-calculus.htmlI've researched this topic a lot, but couldn't find a proper answer to this, and I can't wait a year to learn it at school, so my question is: What exactly is calculus? I know who invented it,......Fri, 28 Aug 2026 18:42:47 -0400https://pop.com.sg/prove-left-frac-2-5-right-frac-2-5-ln-2.htmlhttps://pop.com.sg/prove-left-frac-2-5-right-frac-2-5-ln-2.htmlInadvertently, I find this interesting inequality. But this problem have nice solution? prove that $$\ln{2}>(\dfrac{2}{5})^{\frac{2}{5}}$$ This problem have nice solution? Thank you. ago,I find ...Fri, 28 Aug 2026 18:38:03 -0400https://pop.com.sg/radius-arc-length-what-angle-from-c-p.htmlhttps://pop.com.sg/radius-arc-length-what-angle-from-c-p.htmlI know the radius and length of an arc. how do I figure out the angle from center point to the ends of the arc? ( I'm trying to figure out the length of the arc if I offset/increase radius by $2$" ......Fri, 28 Aug 2026 18:37:01 -0400https://pop.com.sg/definition-of-continuous-map.htmlhttps://pop.com.sg/definition-of-continuous-map.htmlLet $X,Y$ be topological spaces, let $f:X\longrightarrow Y$ a function. There are several (equivalent) ways to define continuity of $f$, one of these says: $f$ is continuous if $$(1)\qquad \forall A\...Fri, 28 Aug 2026 18:29:07 -0400https://pop.com.sg/finding-the-upper-and-lower-bounds-positive-and-negative-each-if-appli.htmlhttps://pop.com.sg/finding-the-upper-and-lower-bounds-positive-and-negative-each-if-appli.htmlGiven $|z^4 -8 + 3|, |z|=2 $ I need to find the positive upper and lower bounds of the above expression. My attempt: By triangle inequality, we have $||z^4|+|-8z|+3|\geq|z^{4} -8z + 3| \leq ||z^4 ......Fri, 28 Aug 2026 18:22:45 -0400https://pop.com.sg/how-to-find-the-laplace-transform-of-f-t-t-e-t.htmlhttps://pop.com.sg/how-to-find-the-laplace-transform-of-f-t-t-e-t.htmlHow to find the Laplace transform of the following function: $$f(t)=t e^t$$ $$F(s)=\int_0^\infty (te^t e^{-st})dt$$ What method do I use to find the integral?...Fri, 28 Aug 2026 18:22:28 -0400https://pop.com.sg/what-039-s-the-point-of-trigonometric-proofs-identities-in-introductor.htmlhttps://pop.com.sg/what-039-s-the-point-of-trigonometric-proofs-identities-in-introductor.htmlI remember back in high school at some point delving into worksheet after worksheet of trigonometric "identities", the vast majority of which are basically restatements of $\sin^2(x) + \cos^2(x) = ......Fri, 28 Aug 2026 18:20:11 -0400https://pop.com.sg/how-to-correctly-find-csc-of-an-angle.htmlhttps://pop.com.sg/how-to-correctly-find-csc-of-an-angle.htmlAlright, so one of my questions is CSC (angle) -5. When I plug CSC in my calculator, it says "math error." I'm using a Casio fx-300 MS, and using shift + cos, then putting an angle, such as 90....Fri, 28 Aug 2026 18:17:35 -0400https://pop.com.sg/volume-of-an-irregular-octahedron-from-edge-lengths.htmlhttps://pop.com.sg/volume-of-an-irregular-octahedron-from-edge-lengths.htmlDoes anyone know how to calculate the volume of an irregular octahedron from the lengths of the edges? The octahedron has triangular faces, but the only information are the edge lengths. Alternat......Fri, 28 Aug 2026 17:57:38 -0400https://pop.com.sg/equation-or-equality.htmlhttps://pop.com.sg/equation-or-equality.htmlIs a numerical equality (like $1+2 = 3$) an algebraic equation? In google search we have: "equation ɪˈkweɪʒ(ə)n/ noun noun: equation; plural noun: equations 1. Mathematics a statement that the va......Fri, 28 Aug 2026 17:56:47 -0400https://pop.com.sg/how-to-solve-this-simple-2d-markov-chain.htmlhttps://pop.com.sg/how-to-solve-this-simple-2d-markov-chain.htmlI have 2D Markov chain like below : I have already solved for $p_{b,0}$ and $p_{0,d}$ like this : $$p_{b,0} = \rho^bp_{0,0}$$ $$p_{0,d} = \rho^dp_{0,0}$$ But I don't know how to get $p_{b,d}$.......Fri, 28 Aug 2026 17:54:23 -0400https://pop.com.sg/what-does-039-express-in-terms-of-x-039-mean.htmlhttps://pop.com.sg/what-does-039-express-in-terms-of-x-039-mean.htmlFor the following question : $f(x) = 2x^2 + 4x $ It asks me to express the following in terms of $x$: $f(-2x)$ What does the question mean by this? Does it mean make $x$ the subject?...Fri, 28 Aug 2026 17:45:10 -0400https://pop.com.sg/approximate-the-sum-of-each-alternating-series.htmlhttps://pop.com.sg/approximate-the-sum-of-each-alternating-series.htmlApproximate the sum of each series to three decimal places. $\sum {(-1)^n {1\over {n^3}}}$ From alternating series test, this series convergence. $S\approx a_3+S_2$ $S\approx {1\over {27}} + {7\ov......Fri, 28 Aug 2026 17:30:51 -0400https://pop.com.sg/whether-multiplication-by-a-constant-matrix-alone-can-represent-every-.htmlhttps://pop.com.sg/whether-multiplication-by-a-constant-matrix-alone-can-represent-every-.htmlI read recently that every function $f : \mathbb{R}^m \to \mathbb{R}^n$ can be written as $f(\mathbf{x}) = \mathbf{Ax}$ where $\mathbf{A}$ is a matrix of constants. On the one hand this is intuitive ...Fri, 28 Aug 2026 17:23:47 -0400https://pop.com.sg/determine-relationship-between-vectors.htmlhttps://pop.com.sg/determine-relationship-between-vectors.htmlUsing the product i would like to understand if it is possible to determine relationship between two vectors,let us consider the following problem: Vectors $u$ and $v$ have length $1$. Which of......Fri, 28 Aug 2026 17:22:17 -0400https://pop.com.sg/relationship-between-smooth-manifold-and-differentiable-manifold.htmlhttps://pop.com.sg/relationship-between-smooth-manifold-and-differentiable-manifold.htmlDefine a smooth manifold to be a ringed space (a Hausdorff, second countable, $n$-manifold) that is locally isomorphic (as a ringed space) to the sheaf of smooth real valued functions on some open ......Fri, 28 Aug 2026 17:22:07 -0400https://pop.com.sg/finding-the-range-of-a-function-without-graph.htmlhttps://pop.com.sg/finding-the-range-of-a-function-without-graph.htmlI'm asked to find the range of the function : $ (-x^2 , +1) $ Taking R as a real number which represents a quantity along a line and graphing : then the range for this function from viewing the ...Fri, 28 Aug 2026 17:20:40 -0400https://pop.com.sg/book-recommendation-hard-problems-for-multivariable-calculus-w-solutio.htmlhttps://pop.com.sg/book-recommendation-hard-problems-for-multivariable-calculus-w-solutio.htmlI'm looking for a text that covers roughly what's sometimes called "Calculus III" or multivariable calculus.* But this text must satisfy certain additional criteria: (1) It must be more in-depth (......Fri, 28 Aug 2026 17:19:05 -0400https://pop.com.sg/gompertz-function-as-a-finite-product.htmlhttps://pop.com.sg/gompertz-function-as-a-finite-product.htmlI have a growth model such as: $$G(n)= (1-1/N)(1-2/N)...(1-n/N) \text{, with } n<N$$ As analytical approximation of $G(n)$, I numerically found that $1-\exp(-\exp(1.71 (1-1.05 n/\sqrt{N}))$ w......Fri, 28 Aug 2026 17:18:13 -0400https://pop.com.sg/how-many-10-letter-words-can-be-formed-using-the-26-letters-of-the-alp.htmlhttps://pop.com.sg/how-many-10-letter-words-can-be-formed-using-the-26-letters-of-the-alp.htmlHow many $10$-letter words can be formed using the $26$ letters of the alphabet if: a) Repetition is allowed b) Repetition is not allowed c) Repetition is allowed, but letters are listed ...Fri, 28 Aug 2026 17:06:15 -0400https://pop.com.sg/the-points-where-function-is-discontinuous-are-those-points-counted-co.htmlhttps://pop.com.sg/the-points-where-function-is-discontinuous-are-those-points-counted-co.htmlThe points where function is discontinuous,are those points counted/considered in the domain of the function. $(1)[x]$,greatest integer function is discontinuous at all integer points but integers ......Fri, 28 Aug 2026 17:02:43 -0400https://pop.com.sg/a-little-question-about-clairaut-039-s-theorem.htmlhttps://pop.com.sg/a-little-question-about-clairaut-039-s-theorem.htmlClairaut's Theorem Let $f$ be a function of two variables,let$(x_{0},y_{0})$ be a point and let $U$ be an open disk with center $(x_{0},y_{0})$.Assume that $f$ is defined on $U$ and its partial ...Fri, 28 Aug 2026 16:35:45 -0400https://pop.com.sg/what-is-a-jordan-cell.htmlhttps://pop.com.sg/what-is-a-jordan-cell.htmlGoogle has been surprisingly unhelpful for me. A homework problem from my algebra class asks me to Calculate p(A) where A is a Jordan cell and p is a polynomial. I'd like to attempt the problem ......Fri, 28 Aug 2026 16:32:04 -0400https://pop.com.sg/what-does-sin-theta-0-mean-here-if-tan-theta-frac-8-15-and-sin-theta-0.htmlhttps://pop.com.sg/what-does-sin-theta-0-mean-here-if-tan-theta-frac-8-15-and-sin-theta-0.htmlThe question is: If $\tan(\theta) = -\frac{8}{15}$, and $\sin(\theta) > 0$, then find $\cos(\theta)$. What I did was draw a triangle on the unit circle with sides $8, 15$ and therefore hypot......Fri, 28 Aug 2026 16:30:13 -0400https://pop.com.sg/finding-x-y-z-of-a-triangle.htmlhttps://pop.com.sg/finding-x-y-z-of-a-triangle.htmlMy younger brother was asked to do this problem and was given the problem exactly as the problem was given with no additional information. Basically, we are trying to find the values of x,y,and z ......Fri, 28 Aug 2026 16:28:19 -0400https://pop.com.sg/how-do-you-factor-x-3-8-0.htmlhttps://pop.com.sg/how-do-you-factor-x-3-8-0.htmlI factored $x^3 - 8 = 0$ and I only got $x = 2$, but the answer said it's $x=2$ and $x=-1 \pm \sqrt{3}i$? How do you get $x=-1 \pm\sqrt{3}i$?...Fri, 28 Aug 2026 16:25:39 -0400https://pop.com.sg/how-can-the-surd-sqrt-2-sqrt-3-be-expressed.htmlhttps://pop.com.sg/how-can-the-surd-sqrt-2-sqrt-3-be-expressed.htmlI was wondering how $\sqrt{2-\sqrt{3}}$ could be expressed in terms of $\frac{\sqrt{3}-1}{\sqrt{2}}$. I did try to solve both the expressions separately but none of them seemed to match. I would ...Fri, 28 Aug 2026 16:19:51 -0400https://pop.com.sg/integrate-frac-x-ln-x-sqrt-x-2-1-sqrt-x-2-1-closed.htmlhttps://pop.com.sg/integrate-frac-x-ln-x-sqrt-x-2-1-sqrt-x-2-1-closed.html$$\int \frac{x\ln\left(x+\sqrt{x^{2}+1}\right)}{\sqrt{x^{2}+1}}dx$$ I honestly have been struggling with this integral for 2 hours straight and none of my methods (including substitution and integ......Fri, 28 Aug 2026 16:19:48 -0400https://pop.com.sg/what-039-s-wrong-with-my-solution-for-equation-x-1-2-3x-1-2-10x-3-2.htmlhttps://pop.com.sg/what-039-s-wrong-with-my-solution-for-equation-x-1-2-3x-1-2-10x-3-2.htmlThe solution given in the website: Here's my solution: What am I doing wrong?...Fri, 28 Aug 2026 16:19:39 -0400https://pop.com.sg/how-to-solve-equations-when-logarithm-is-the-exponent.htmlhttps://pop.com.sg/how-to-solve-equations-when-logarithm-is-the-exponent.htmlHere is an example problem where the logarithm is expressed as an exponent. Please help me understand this concept its not properly covered in my textbook. $$ 3^{\log_3 (2k)} = 9 $$...Fri, 28 Aug 2026 16:01:27 -0400https://pop.com.sg/taking-the-area-of-isosceles-cross-sections-of-an-ellipse.htmlhttps://pop.com.sg/taking-the-area-of-isosceles-cross-sections-of-an-ellipse.htmlThe base of S is an elliptical region with boundary curve $25x^2 + 4y^2 = 100$. Cross-sections perpendicular to the x-axis are isosceles right triangles with hypotenuse in the base. I have gotten ......Fri, 28 Aug 2026 15:49:18 -0400https://pop.com.sg/prove-that-arcsin-3-5-pi-is-irrational.htmlhttps://pop.com.sg/prove-that-arcsin-3-5-pi-is-irrational.htmlI was asked to prove that on the unit circle, the rational points are dense. My idea is to first show that the point (3/5,4/5) corresponds to an irrational angle (more precisely, its ratio with res......Fri, 28 Aug 2026 15:34:35 -0400https://pop.com.sg/how-can-i-use-qr-in-matlab-to-compute-lq-decomposition.htmlhttps://pop.com.sg/how-can-i-use-qr-in-matlab-to-compute-lq-decomposition.htmlI want to compute this: $$\begin{bmatrix} U\\ Y \end{bmatrix} = \begin{bmatrix} L_{11} &0 \\ L_{21} & L_{22} \end{bmatrix}\begin{bmatrix} Q_1\\ Q_2 \end{bmatrix}$$ Is this matlab command ...Fri, 28 Aug 2026 15:23:55 -0400https://pop.com.sg/is-there-anything-special-about-intersection-points-of-vesica-pisces-c.htmlhttps://pop.com.sg/is-there-anything-special-about-intersection-points-of-vesica-pisces-c.htmlTake a triangle $\triangle ABC$ On each pair of vertices of the triangle, place a pair of vesica pisces circles with the properties that each circle has the same radius as the other circle of th......Fri, 28 Aug 2026 15:09:15 -0400https://pop.com.sg/how-to-find-the-dimensions-of-a-rectangle-given-a-the-area-and-the-dia.htmlhttps://pop.com.sg/how-to-find-the-dimensions-of-a-rectangle-given-a-the-area-and-the-dia.htmlI need to write concise formulas of these two:$\space \space \space$ $(1)$ finding the dimensions of a rectangle given the Area and the diagonal $\space \space \space$$(2)$ finding the dimensions o......Fri, 28 Aug 2026 15:01:12 -0400https://pop.com.sg/how-to-find-the-length-of-one-of-the-sides-of-a-triangle-given-the-are.htmlhttps://pop.com.sg/how-to-find-the-length-of-one-of-the-sides-of-a-triangle-given-the-are.htmlThe triangles are drawn to scale. The first triangle has side lengths of 17, 17, 16 while the second triangle has side lengths of 17,17,$k$. The triangles have the same area. Find the value of $k$ ...Fri, 28 Aug 2026 15:00:27 -0400https://pop.com.sg/what-039-s-the-intuition-behind-definition-of-chaotic-function.htmlhttps://pop.com.sg/what-039-s-the-intuition-behind-definition-of-chaotic-function.htmlI read books A First Course in Discrete Dynamical Systems by Richard A. Holmgren and An Introduction to Chaotic Dynamical Systems by Robert L. Devaney. I want to understand which concepts of "chaos......Fri, 28 Aug 2026 14:54:26 -0400https://pop.com.sg/normalizer-of-the-normalizer-of-the-sylow-p-subgroup.htmlhttps://pop.com.sg/normalizer-of-the-normalizer-of-the-sylow-p-subgroup.htmlIf $P$ is a Sylow $p$-subgroup of $G$, how do I prove that normalizer of the normalizer $P$ is same as the normalizer of $P$ ?...Fri, 28 Aug 2026 14:53:47 -0400https://pop.com.sg/can-a-5th-degree-equation-with-repeated-roots-be-solved-by-radicials.htmlhttps://pop.com.sg/can-a-5th-degree-equation-with-repeated-roots-be-solved-by-radicials.htmlI realized that generally a 5th degree equation cannot be solved by radicals, but what if we have a 5th degree equation with repeated roots, how about its solvability by radicals? Intuitively, it w......Fri, 28 Aug 2026 14:50:29 -0400https://pop.com.sg/how-to-draw-a-phase-portrait-of-a-two-dimensional-ode.htmlhttps://pop.com.sg/how-to-draw-a-phase-portrait-of-a-two-dimensional-ode.htmlIf we're given: $$\dot{x}=-x+y$$ $$\dot{y}=xy-1$$ How do I draw a phase portrait of this system? I don't understand which direction the arrows are supposed to point. This is what I got so far: I ...Fri, 28 Aug 2026 14:46:14 -0400https://pop.com.sg/finding-the-largest-singular-value-easily.htmlhttps://pop.com.sg/finding-the-largest-singular-value-easily.htmlI'm only interested in finding the largest singular value of an $m\times n$ real matrix $A$. I don't need the corresponding (left and right) singular vectors. Is there a way to do so without perf......Fri, 28 Aug 2026 14:43:04 -0400https://pop.com.sg/classification-of-pde-into-linear-nonlinear.htmlhttps://pop.com.sg/classification-of-pde-into-linear-nonlinear.htmlConsider the following second order PDEs \begin{align} u_{t} + v_{t} + x^{4}u_{xx} + v_{yy} &=0\,\, (1)\\ u_{t} + v_{t} + xu_{xx} + v_{yy} + x^{2}v_{y}&=0\,\,(2) \end{align} I want to ...Fri, 28 Aug 2026 14:41:03 -0400https://pop.com.sg/finding-the-order-of-an-element-in-the-group-of-integers-modulo-n-with.htmlhttps://pop.com.sg/finding-the-order-of-an-element-in-the-group-of-integers-modulo-n-with.htmlI'm a bit new to abstract algebra and while learning it, I've come across a somewhat tedious problem. I'm curious as to whether or not there is some modular congruency trick/number theory that let......Fri, 28 Aug 2026 14:33:08 -0400https://pop.com.sg/convert-angle-radians-to-a-heading-vector.htmlhttps://pop.com.sg/convert-angle-radians-to-a-heading-vector.htmlI have been looking everywhere trying to find out how to convert an angle in radians (expressed as -Pi to Pi) to a heading vector. The only [x,y] answer I have found is, [cos(angle), sin(angle)] , ...Fri, 28 Aug 2026 14:24:35 -0400https://pop.com.sg/user-aretor-mathematics-stack-exchange.htmlhttps://pop.com.sg/user-aretor-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Fri, 28 Aug 2026 14:23:41 -0400https://pop.com.sg/proving-the-so-called-well-ordering-principle.htmlhttps://pop.com.sg/proving-the-so-called-well-ordering-principle.htmlIs there anything wrong with the following proof? Theorem. Every non-empty subset $B \subset\mathbb{N}$ has a least member. Proof. Assume not. Then, of necessity, we'd have to have $B=\varnothing......Fri, 28 Aug 2026 14:18:20 -0400https://pop.com.sg/sample-standard-deviation-given-population-standard-deviation.htmlhttps://pop.com.sg/sample-standard-deviation-given-population-standard-deviation.htmlHow do you find the sample standard deviation when given the population standard deviation? What formula do you use? If you can make up an example that would be great....Fri, 28 Aug 2026 14:08:47 -0400https://pop.com.sg/why-are-3d-transformation-matrices-4-times-4-instead-of-3-times-3.htmlhttps://pop.com.sg/why-are-3d-transformation-matrices-4-times-4-instead-of-3-times-3.htmlBackground: Many (if not all) of the transformation matrices used in $3D$ computer graphics are $4\times 4$, including the three values for $x$, $y$ and $z$, plus an additional term which usually h......Fri, 28 Aug 2026 14:07:52 -0400https://pop.com.sg/let-t-be-the-centroid-of-a-triangle-abc-and-let-p-be-the-midpoint-of-t.htmlhttps://pop.com.sg/let-t-be-the-centroid-of-a-triangle-abc-and-let-p-be-the-midpoint-of-t.htmlQuestion : Let $T$ be the centroid of a triangle $ABC$, and let $P$ be the midpoint of the side $\overline{AC}$. The line containing the point $T$ parallel to the line $BC$ intersects the side $\ov......Fri, 28 Aug 2026 14:00:53 -0400https://pop.com.sg/how-to-express-logical-equivalence-arrow-using-pierce-039-s-arrow.htmlhttps://pop.com.sg/how-to-express-logical-equivalence-arrow-using-pierce-039-s-arrow.htmlI am really confused about this and not sure how to show $P\iff Q$ with the ↓ arrow and only the ↓ arrow. I understand that $P \iff Q$ is $P\implies Q$ and $Q\implies P$. I also know that $P\impl......Fri, 28 Aug 2026 13:45:44 -0400https://pop.com.sg/why-are-the-trig-functions-defined-by-the-counterclockwise-path-of-a-c.htmlhttps://pop.com.sg/why-are-the-trig-functions-defined-by-the-counterclockwise-path-of-a-c.htmlMy understanding is that $\cos$ is defined by the value of $x$ as you trace the graph of a circle counterclockwise, starting at the point $(1, 0)$. Similarly, $\sin$ traces the $y$ value. I underst......Fri, 28 Aug 2026 13:32:13 -0400https://pop.com.sg/find-y-function-of-cos-y-x.htmlhttps://pop.com.sg/find-y-function-of-cos-y-x.htmlLet's say that there is no limit to $x$, then I believe $y = \arccos(x) + \pi n $ because $\cos(y) = x$ for $x + \pi n , n\in \mathbb{Z}$. However, in the last step of the following calculation, it......Fri, 28 Aug 2026 13:28:50 -0400https://pop.com.sg/prove-series-frac-1-x-ln-x-diverges-duplicate.htmlhttps://pop.com.sg/prove-series-frac-1-x-ln-x-diverges-duplicate.htmlProve series $$\sum\limits_{n=2}^{\infty}\frac{1}{n\ln n}$$ doesn't converge without integral test. I really don't know any hint for $\ln n$. So I understand I have to show $$\frac{1}{n\ln n} > ......Fri, 28 Aug 2026 13:22:32 -0400https://pop.com.sg/lagrange-multipliers-with-multiple-constraints.htmlhttps://pop.com.sg/lagrange-multipliers-with-multiple-constraints.htmlI want to maximize $f(x,y,z)=x+y+z$, according to the constraints $g_1(x,y,z)=x^2-y^2-1=0$ and $g_2(x,y,z)=2x+z-1=0$ . So I get $5$ equations using lagrange multipliers solving $\nabla f = \lambda......Fri, 28 Aug 2026 13:15:05 -0400https://pop.com.sg/optimization-related-rate-searchlight.htmlhttps://pop.com.sg/optimization-related-rate-searchlight.htmlI can't seem to figure out how to set up this problem. I'm a bit confused on the relationship between the variables here; I've seen a similar problem involving a searchlight but the angle of $\thet......Fri, 28 Aug 2026 13:13:11 -0400https://pop.com.sg/center-of-mass-of-a-circle.htmlhttps://pop.com.sg/center-of-mass-of-a-circle.htmlHow would one find the center of mass of a circle? The center of mass of a rod is given by: $$\frac{1}{M}\int^{L}_{0}\rho x dx$$ So, for a sphere, it would be an area integral, such as: $$\frac{......Fri, 28 Aug 2026 13:11:14 -0400https://pop.com.sg/normalizing-eigenvectors-from-pauli-matrices.htmlhttps://pop.com.sg/normalizing-eigenvectors-from-pauli-matrices.htmlFor this example of a Pauli matrix, \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix} I found that one of its eigenvectors (for $\lambda = 1$) is \begin{bmatrix} -i \\ 1 \end{bmat......Fri, 28 Aug 2026 13:00:04 -0400https://pop.com.sg/intuition-normal-subgroup-test-fraleigh-p-141-theorem-14-13.htmlhttps://pop.com.sg/intuition-normal-subgroup-test-fraleigh-p-141-theorem-14-13.html(1.) Not querying proofs or formality. I do this in my other question. Normal Subgroup Test says H is normal in G $\iff gH{g}^{-1}\subseteq H$ for all $g \in G$. What's the intuition of this su......Fri, 28 Aug 2026 12:59:08 -0400https://pop.com.sg/is-the-division-property-of-equality-just-a-special-case-of-the-multip.htmlhttps://pop.com.sg/is-the-division-property-of-equality-just-a-special-case-of-the-multip.htmlMy textbook clearly states after the lesson on Transforming Equations: Addition and Subtraction "Notice that the subtraction property of equality is just a special case of the addition property, s......Fri, 28 Aug 2026 12:55:53 -0400https://pop.com.sg/showing-a-matrix-is-not-diagonalizable.htmlhttps://pop.com.sg/showing-a-matrix-is-not-diagonalizable.htmlFor eigenvalue 1, the eigenspace I got was $span[ 1,0,0]$, and for eigenvalue 4, the eigenspace I got was $span[ 1,-3,9]$. Do they look right? The reason the matrix is not diagonalizable is becaus......Fri, 28 Aug 2026 12:55:45 -0400https://pop.com.sg/integral-of-arcsin-x-x-2.htmlhttps://pop.com.sg/integral-of-arcsin-x-x-2.htmlI did the integration by parts and got this expression, but then I am stuck on how to take it further. I tried substituting u=1-x^2, but then I had to do a partial fraction decomposition (which I d......Fri, 28 Aug 2026 12:54:00 -0400https://pop.com.sg/complex-conjugation-continuous-function.htmlhttps://pop.com.sg/complex-conjugation-continuous-function.htmlHow can I show that complex conjugation is a continuous function? I tried looking at open sets $U$ and then the preimage. Can assume preimage is not empty so that if $z$ is in preimage then $f(z) \......Fri, 28 Aug 2026 12:51:42 -0400https://pop.com.sg/how-to-calculate-the-integral-of-exponential-functions.htmlhttps://pop.com.sg/how-to-calculate-the-integral-of-exponential-functions.htmlHaving an integral like $\int_{2}^{10}{\frac{x}{\ln x}}dx$ How does this function turns to an exponential integral of the form: $ \operatorname{Ei}(x)=-\int_{-x}^{\infty}\frac{e^{-t}}t\,dt.\,$ For ...Fri, 28 Aug 2026 12:48:56 -0400https://pop.com.sg/example-of-function-of-bounded-variation.htmlhttps://pop.com.sg/example-of-function-of-bounded-variation.htmlI want to show that $f(x) = x \sin(\frac{1}{\sqrt{x}})$ , with $f(0)=0$ on $[0,1]$ is of bounded variation. To do that I Don't want to use of showing this is continuously differentiable hence, BV. ......Fri, 28 Aug 2026 12:39:15 -0400https://pop.com.sg/given-this-transformation-matrix-how-do-i-decompose-it-into-translatio.htmlhttps://pop.com.sg/given-this-transformation-matrix-how-do-i-decompose-it-into-translatio.htmlI have this problem from my Graphics course. Given this transformation matrix: $$\begin{pmatrix} -2 &-1& 2\\ -2 &1& -1\\ 0 &0& 1\\ \end{pmatrix}$$ I need to extract ...Fri, 28 Aug 2026 12:36:14 -0400https://pop.com.sg/power-series-to-evaluate-a-definite-integral-of-f-x-frac1-1-x-7.htmlhttps://pop.com.sg/power-series-to-evaluate-a-definite-integral-of-f-x-frac1-1-x-7.htmlFind a power series expansion of the function $f(x)=\dfrac1{1+x^7}$ about $x=0$. Use it to evaluate $\int_0^1 \dfrac1{1+x^7} dx$. Your answer should be expressed in the form of numerical series (a ......Fri, 28 Aug 2026 12:34:32 -0400https://pop.com.sg/calculating-values-for-cubic-bezier-curve.htmlhttps://pop.com.sg/calculating-values-for-cubic-bezier-curve.htmlI'm trying to use cubic bezier curves for some non-linear animations in my iOS app. Let's say I'm animating position of some element on the screen. I'm using this curve from cubic-bezier.com for ...Fri, 28 Aug 2026 12:29:39 -0400https://pop.com.sg/find-s-where-s-sqrt-3-5-2-sqrt-13-sqrt-3-5-2-sqrt-13-why-am-i-getting-.htmlhttps://pop.com.sg/find-s-where-s-sqrt-3-5-2-sqrt-13-sqrt-3-5-2-sqrt-13-why-am-i-getting-.html$\large S=\sqrt[3] {5+2 \sqrt {13}}+\sqrt[3]{5-2 \sqrt {13}}$ Multiplying by conjugate: $\large S=\dfrac {-3}{\sqrt[3] {5+2 \sqrt {13}}-\sqrt[3]{5-2 \sqrt {13}}}$ From the original: $\large S-......Fri, 28 Aug 2026 12:20:20 -0400https://pop.com.sg/how-to-show-if-a-complex-function-is-analytic.htmlhttps://pop.com.sg/how-to-show-if-a-complex-function-is-analytic.htmlJust began the study of complex analysis. Let $$ f(x,y) = x^2 - y^2 + 2 i xy - x - iy. $$ I need to determine if this function is analytic. This means I have to show the partials satisfy the Cauchy-...Fri, 28 Aug 2026 12:18:21 -0400https://pop.com.sg/understanding-vector-projection.htmlhttps://pop.com.sg/understanding-vector-projection.htmlI'm learning about vector projection. I understand how to perform it, but I still can't understand what it actually means and what it gives me. Here is a common definition: Vector projection of a ...Fri, 28 Aug 2026 12:14:31 -0400https://pop.com.sg/integral-of-1-x-2.htmlhttps://pop.com.sg/integral-of-1-x-2.htmlI know that the integral of $1/x^2$ is $(-1/x +C)$ Can we split $(1/x^2)$ as $(1/x)(1/x)$ and say that the integral of $(1/x^2)$ is $\ln x\cdot\ln x$? PS: I am new to high school. And this is my fi......Fri, 28 Aug 2026 12:02:29 -0400https://pop.com.sg/is-there-any-formula-to-find-the-nth-element-in-a-sequence-where-commo.htmlhttps://pop.com.sg/is-there-any-formula-to-find-the-nth-element-in-a-sequence-where-commo.htmlTo explain my question, here is an example. Below is an AP: 2, 6, 10, 14....n Calculating the nth term in this sequence is easy because we have a formula. The common difference (d = 4) in AP is ...Fri, 28 Aug 2026 11:56:56 -0400https://pop.com.sg/is-there-a-straight-line-solution-for-this-separable-differential-equa.htmlhttps://pop.com.sg/is-there-a-straight-line-solution-for-this-separable-differential-equa.htmlThis is the question: Solve $$x^2 \frac{\text{d}y}{\text{d}x} = y^2 - 3xy -5x^2 .$$ Are there any straight line solutions? If so, state them. For the case $y(1)=0$, state the solution ...Fri, 28 Aug 2026 11:56:03 -0400https://pop.com.sg/what-is-the-difference-between-probability-density-function-and-probab.htmlhttps://pop.com.sg/what-is-the-difference-between-probability-density-function-and-probab.htmlWhats the difference between probability density function and probability distribution function?...Fri, 28 Aug 2026 11:48:02 -0400https://pop.com.sg/set-intersection-the-same-as-logical-conjunction.htmlhttps://pop.com.sg/set-intersection-the-same-as-logical-conjunction.htmlI have been exploring more about set theory beyond my textbook and I have ran into something I couldn't explain. Can you use logical conjunction/disjunction on sets and are they the same as union/...Fri, 28 Aug 2026 11:45:14 -0400https://pop.com.sg/mathematical-notation-for-distinct-numbers.htmlhttps://pop.com.sg/mathematical-notation-for-distinct-numbers.htmlHow can I represent a set of distinct numbers in shorthand mathematical notation? In particular I am trying to write that there are 4 distinct integers $a,b,c,d$. For three numbers, I could say: ......Fri, 28 Aug 2026 11:17:31 -0400https://pop.com.sg/equations-for-cubic-regression.htmlhttps://pop.com.sg/equations-for-cubic-regression.htmlSo, I'm making a simple program for drawing graphs, and I'm looking at making some simple best-fit curves using some basic regression analysis. I've happily got linear and quadratic regression work......Fri, 28 Aug 2026 11:09:24 -0400https://pop.com.sg/depressed-cubic.htmlhttps://pop.com.sg/depressed-cubic.htmlI have this string of questions, but when I look up depressed cubic, I don't quite understand. I'm not asking for all of these questions to be answered explicitly, but for some explanation on depre......Fri, 28 Aug 2026 11:05:09 -0400https://pop.com.sg/equation-for-curves-of-high-density-in-bifurcation-diagram.htmlhttps://pop.com.sg/equation-for-curves-of-high-density-in-bifurcation-diagram.htmlIs there an equation for the high density curves in the chaotic regions of the bifurcation diagram for the logistic map? I'm talking about the sinusoidal-looking dark curves in the following picture. ...Fri, 28 Aug 2026 10:55:43 -0400https://pop.com.sg/how-to-compute-confidence-interval-for-variance-with-unknown-mean-from.htmlhttps://pop.com.sg/how-to-compute-confidence-interval-for-variance-with-unknown-mean-from.htmlWhen mean is known, we note that $\frac{\bar{(X-a)^2}n}{\sigma ^2}$ has a Chi-squared distribution with $n$ degrees of freedom. However, what to do when $a$ is unknown? I can't just substitute sample ...Fri, 28 Aug 2026 10:50:25 -0400https://pop.com.sg/is-there-a-symbol-for-antiparallel.htmlhttps://pop.com.sg/is-there-a-symbol-for-antiparallel.htmlI've been doing some work where I've needed to talk about vectors that are parallel and those that are antiparallel, parallel to the negative of the other vector. Is there a symbol for this? I ......Fri, 28 Aug 2026 10:24:28 -0400https://pop.com.sg/divergence-of-series-with-n-factorial.htmlhttps://pop.com.sg/divergence-of-series-with-n-factorial.htmlI need to find the convergence or divergence of $$\sum_{n=1}^\infty \frac{n!}{3^n}$$ The first thing I did was to take the nth root. So I got: $$\lim_{n\rightarrow\infty}\big(\frac{n!}{3^n}\big)^\......Fri, 28 Aug 2026 10:12:48 -0400https://pop.com.sg/probably-an-easy-question-e-2-ln-3-without-a-calculator.htmlhttps://pop.com.sg/probably-an-easy-question-e-2-ln-3-without-a-calculator.htmlHow to evaluate $e^{2\ln(3)}$ without using a calculator? I've tried playing with $\ln$ but couldn't figure out...Fri, 28 Aug 2026 10:11:04 -0400