https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usTue, 11 Aug 2026 15:47:52 -0400https://pop.com.sg/orthogonal-position-and-velocity-vectors.htmlhttps://pop.com.sg/orthogonal-position-and-velocity-vectors.htmlIs it true that if the position and velocity vectors of a moving particle are always perpendicular the path of the particle is on a sphere? If so how do I prove it? Geometrically I believe it makes ...Tue, 11 Aug 2026 19:44:25 -0400https://pop.com.sg/lower-bound-on-tail-probabilities.htmlhttps://pop.com.sg/lower-bound-on-tail-probabilities.htmlInequalities such as Markov's and Chebyshev’s provide upper bounds on tail probabilities. Are there similar inequalities that give lower bounds in the form $P(X \geq \alpha)>\theta$?...Tue, 11 Aug 2026 19:44:08 -0400https://pop.com.sg/intuitive-and-memorable-way-to-see-n1-n1-n2-as-distinct-permutations-o.htmlhttps://pop.com.sg/intuitive-and-memorable-way-to-see-n1-n1-n2-as-distinct-permutations-o.htmlExample Problem: Count the number of distinct ways in which N objects, of which n1 are indistinguishably of one type and n2 of a second type, can be accomodated in a total of N = n1 + n2 ways. ...Tue, 11 Aug 2026 19:36:17 -0400https://pop.com.sg/eigenspace-definition.htmlhttps://pop.com.sg/eigenspace-definition.htmlThis is the definition I have for eigenspace: Let $\lambda$ be an eigenvalue of $A \in Mn(\mathbb C)$. Then {$v \in \mathbb C^n|Av= \lambda v$}$=W_\lambda$ is a subspace of $\mathbb C^n$, call......Tue, 11 Aug 2026 19:34:51 -0400https://pop.com.sg/proving-cos-4-theta-4-cos-2-theta-equiv-8-sin-4-theta-3.htmlhttps://pop.com.sg/proving-cos-4-theta-4-cos-2-theta-equiv-8-sin-4-theta-3.htmlProve the following identity: $$\cos(4\theta) - 4\cos(2\theta) \equiv 8\sin^4\theta - 3$$ How can I express $\cos(4\theta) $ in other terms?...Tue, 11 Aug 2026 19:25:37 -0400https://pop.com.sg/the-significance-and-acceptance-of-helfgott-s-proof-of-the-weak-goldba.htmlhttps://pop.com.sg/the-significance-and-acceptance-of-helfgott-s-proof-of-the-weak-goldba.htmlRecently I was browsing math Wikipedia, and found that Harald Helfgott announced the complete proof of the weak Goldbach Conjecture in 2013, a proof which has been accepted widely by the math commu......Tue, 11 Aug 2026 19:25:09 -0400https://pop.com.sg/proof-of-pythagorean-theorem-without-using-geometry-for-a-high-school-.htmlhttps://pop.com.sg/proof-of-pythagorean-theorem-without-using-geometry-for-a-high-school-.htmlThere are some proofs of Pythagoras theorem which don't even require high school maths to understand, but they all are using shapes to prove of the theorem. However, I am trying to find some proofs......Tue, 11 Aug 2026 19:21:40 -0400https://pop.com.sg/how-is-a-general-case-equivalent-to-a-special-case-and-how-showing-a-s.htmlhttps://pop.com.sg/how-is-a-general-case-equivalent-to-a-special-case-and-how-showing-a-s.html"The general theorem expressed by $\lambda a^2 = \lambda b^2 + \lambda c^2 $ is equivalent not only to the special case $a^2 = b^2 + c^2 $ but to any other special case. Therefore, if any such spe......Tue, 11 Aug 2026 19:18:19 -0400https://pop.com.sg/is-heat-equation-is-also-called-as-thermal-diffusion-equation.htmlhttps://pop.com.sg/is-heat-equation-is-also-called-as-thermal-diffusion-equation.htmlIs the following equation known as heat equation, also referred as thermal diffusion equation? Yes/No. $${\displaystyle {\frac {\partial u}{\partial t}}-\alpha \left({\frac {\partial ^{2}u}{\parti......Tue, 11 Aug 2026 19:12:24 -0400https://pop.com.sg/proof-of-yet-another-weyl-s-inequality-using-courant.htmlhttps://pop.com.sg/proof-of-yet-another-weyl-s-inequality-using-courant.htmlThe Courant-Fischer theorem states that $$\lambda_j=\max_{\dim(\mathbb{V})=j}\min_{v\in \mathbb{V},v\neq 0}\rho(v,A)=\min_{\dim(\mathbb{W})=n-j+1}\max_{w\in \mathbb{W},w\neq0}\rho(v,A)$$ where $\la......Tue, 11 Aug 2026 19:12:23 -0400https://pop.com.sg/polar-equation-of-a-circle.htmlhttps://pop.com.sg/polar-equation-of-a-circle.htmlA very long time ago in algebra/trig class we did polar equation of a circle where $r = 2a\cos\theta + 2b\sin\theta$ Now I forgot how to derive this. So I tried using the standard form of a circl......Tue, 11 Aug 2026 19:08:27 -0400https://pop.com.sg/numerically-calculate-the-second-left-hand-derivative.htmlhttps://pop.com.sg/numerically-calculate-the-second-left-hand-derivative.htmlThe Problem I have a series of measurements for which I have to calculate the first and second derivative numerically in a "live" fashion, i.e. using only previous data. This is easy for the first ...Tue, 11 Aug 2026 19:07:01 -0400https://pop.com.sg/divisibility-criteria-of-24-why-is-this.htmlhttps://pop.com.sg/divisibility-criteria-of-24-why-is-this.htmlI am currently familiar with the method of checking if a number is divisible by $2, 3, 4, 5, 6, 8, 9, 10, 11$. While Checking for divisibility for $24$ (online). I found out that the number has to ...Tue, 11 Aug 2026 18:52:47 -0400https://pop.com.sg/differential-topology-versus-differential-geometry.htmlhttps://pop.com.sg/differential-topology-versus-differential-geometry.htmlI have just finished my undergraduate studies. During last two semesters I've taken two subjects dealing with manifolds: Analysis on manifolds, containing: definition of manifold, tangent space (as ...Tue, 11 Aug 2026 18:48:31 -0400https://pop.com.sg/derivative-of-integral.htmlhttps://pop.com.sg/derivative-of-integral.htmlI'm having a little trouble with the following problem: Calculate $F'(x)$: $F(x)=\int_{1}^{x^{2}}(t-\sin^{2}t) dt$ It says we have to use substitution but I don't see why the answer can't just b......Tue, 11 Aug 2026 18:41:08 -0400https://pop.com.sg/toy-examples-for-kan-extensions.htmlhttps://pop.com.sg/toy-examples-for-kan-extensions.htmlBackground: If $\mathcal{C}$ is a cocomplete category and $f : I \to J$ is a functor between small categories, then $f^* : \mathrm{Hom}(J,\mathcal{C}) \to \mathrm{Hom}(I,\mathcal{C})$ has a left ad......Tue, 11 Aug 2026 18:38:24 -0400https://pop.com.sg/parabolic-points-and-curvature.htmlhttps://pop.com.sg/parabolic-points-and-curvature.htmlI have problems to solve this exercise: Let $p$ be a point of an oriented surface $S$ and assume that there is a neighborhood $U$ of $p$ in $S$ all points of which are parabolic. Prove that the (un......Tue, 11 Aug 2026 18:22:40 -0400https://pop.com.sg/find-the-perimeter-of-triangle-abc.htmlhttps://pop.com.sg/find-the-perimeter-of-triangle-abc.htmlThis practice engineering board exam which I've tried seems very tricky (question no.2). The inscribed circle of $\triangle ABC$ is tangent to $AB$ at $P$ . If the radius is $21$, $AP=23$ and $PB=2......Tue, 11 Aug 2026 18:21:06 -0400https://pop.com.sg/number-of-words-that-can-be-made-using-all-the-letters-of-the-word-w-i.htmlhttps://pop.com.sg/number-of-words-that-can-be-made-using-all-the-letters-of-the-word-w-i.htmlI am facing difficulty in solving the 18th Question from the above passage. Attempt: I have attempted it using principal of inclusion and exclusion. Let n be the required number of ways. $n = \...Tue, 11 Aug 2026 18:19:20 -0400https://pop.com.sg/square-of-a-differential-duplicate.htmlhttps://pop.com.sg/square-of-a-differential-duplicate.htmlJust wondering, is this valid: $$ \left(\frac{df}{dx}\right)^2=\frac{d^{2}f}{dx^{2}} $$...Tue, 11 Aug 2026 18:15:33 -0400https://pop.com.sg/expected-value-of-a-the-estimate-of-bernoulli-distribution.htmlhttps://pop.com.sg/expected-value-of-a-the-estimate-of-bernoulli-distribution.htmlI know that $E[x] = p$ for some Bernoulli random variable X with parameter $p$ where $P(X = 1) = p$ and $P(X = 0) = 1 − p$. Similarly $E[X^2] = p$ and $Var[X] = p − p^2 = p(1 − p)$. However, I'm ha......Tue, 11 Aug 2026 18:15:10 -0400https://pop.com.sg/find-from-the-first-principle-the-derivative-of-sqrt-sin-2x.htmlhttps://pop.com.sg/find-from-the-first-principle-the-derivative-of-sqrt-sin-2x.htmlFind, from the first principle, the derivative of: $$\sqrt {\sin (2x)}$$ My Attempt: $$f(x)=\sqrt {\sin (2x)}$$ $$f(x+\Delta x)=\sqrt {\sin (2x+2\Delta x)}$$ Now, $$f'(x)=\lim_{\Delta x\to 0} \dfr......Tue, 11 Aug 2026 18:06:29 -0400https://pop.com.sg/how-to-find-the-percent-variation-in-y-is-explained-by-x.htmlhttps://pop.com.sg/how-to-find-the-percent-variation-in-y-is-explained-by-x.htmlI know that the r^2 value for the data is 0.9832. Is there a way to use that value to find the percent variation in Y is explained by X? Or do I need to use the data given to me?...Tue, 11 Aug 2026 18:05:17 -0400https://pop.com.sg/list-of-common-or-useful-limits-of-sequences-and-series.htmlhttps://pop.com.sg/list-of-common-or-useful-limits-of-sequences-and-series.htmlThere are many sequences or series which come up frequently, and it's good to have a directory of the most commonly used or useful ones. I'll start out with some. Proofs are not required. $$\begin......Tue, 11 Aug 2026 18:03:51 -0400https://pop.com.sg/how-can-you-show-that-the-center-of-mass-of-a-triangle-lies-on-the-med.htmlhttps://pop.com.sg/how-can-you-show-that-the-center-of-mass-of-a-triangle-lies-on-the-med.htmlIn geometry class, it is usually first shown that the medians of a triangle intersect at a single point. Then is is explained that this point is called the centroid and that it is the balance poin......Tue, 11 Aug 2026 17:45:07 -0400https://pop.com.sg/finding-maximum-height-with-initial-velocity.htmlhttps://pop.com.sg/finding-maximum-height-with-initial-velocity.htmlAn object is projected from the surface of Earth with a velocity of one mile per second. What is the maximum height it will reach? My solution for solving this was to use the following equat......Tue, 11 Aug 2026 17:44:32 -0400https://pop.com.sg/identity-element-generating-a-cyclic-group.htmlhttps://pop.com.sg/identity-element-generating-a-cyclic-group.htmlA group is defined as a set that is: associative, contains the identity element $(e)$, and every element has an inverse under given operation $(*)$. By this definition, $\{e\}$ is a group over the ......Tue, 11 Aug 2026 17:25:02 -0400https://pop.com.sg/how-to-define-a-relation.htmlhttps://pop.com.sg/how-to-define-a-relation.htmlI was tasked with this question: Let $X = \{0,1,2,3,4\}$. Define a relation $R$ on $X$ such that $xRy$, if $x + y = 4$. I don't understand/know what syntax I should use to define this as a relat......Tue, 11 Aug 2026 17:14:32 -0400https://pop.com.sg/given-the-function-minimize-the-function-subject-to-the-constraint-lag.htmlhttps://pop.com.sg/given-the-function-minimize-the-function-subject-to-the-constraint-lag.htmlGiven the function $f(x, y, z) = (x-1)^2 + y^2 + (z+1)^2$: Minimize $f(x, y, z)$ subject to the constraint $x + y +z = 1$ I know that you need to use Lagrange's Multipliers and thus would need to......Tue, 11 Aug 2026 17:14:01 -0400https://pop.com.sg/equation-of-a-line-that-passes-halfway-between-two-points-in-other-wor.htmlhttps://pop.com.sg/equation-of-a-line-that-passes-halfway-between-two-points-in-other-wor.htmlIs there a formal proper way of finding the line between two points? By that I don't mean the line connecting the two points, I mean a line that runs the same distance away from point 1 and point ......Tue, 11 Aug 2026 16:59:07 -0400https://pop.com.sg/is-there-an-integral-that-proves-pi-333-106.htmlhttps://pop.com.sg/is-there-an-integral-that-proves-pi-333-106.htmlThe following integral, $$ \int_0^1 \frac{x^4(1-x)^4}{x^2 + 1} \mathrm{d}x = \frac{22}{7} - \pi $$ is clearly positive, which proves that $\pi < 22/7$. Is there a similar integral which prove......Tue, 11 Aug 2026 16:54:43 -0400https://pop.com.sg/curves-with-a-common-tangent-line.htmlhttps://pop.com.sg/curves-with-a-common-tangent-line.htmlQuestion Find the point where the curves $$\tag 1y = x^3 -3x + 4$$ and $$\tag 2 y = 3x^2 - 3x$$ are tangent to each other, that is, have a common tangent line. My approach Let $x = a$ and $x = b......Tue, 11 Aug 2026 16:29:20 -0400https://pop.com.sg/solving-for-given-variable-how-to-solve.htmlhttps://pop.com.sg/solving-for-given-variable-how-to-solve.htmlSo the question states: 'To convert degrees Celsius to Kelvin, the formula $K = C + 273.15$ is used. Solve this formula for $C$.' The answer I came up with was: $K - 273.15 = C$. So is this correc......Tue, 11 Aug 2026 16:17:44 -0400https://pop.com.sg/what-is-cumulative-logistic-distribution.htmlhttps://pop.com.sg/what-is-cumulative-logistic-distribution.htmlI read an article on Logistic Regression on Wikipedia: https://en.wikipedia.org/wiki/Logistic_regression. It was written that logistic function is a cumulative logistic distribution. My question......Tue, 11 Aug 2026 16:11:06 -0400https://pop.com.sg/how-do-you-derive-the-formula-of-sin-a-b-from-cos-a-b-cos-a-cos-b-sin-.htmlhttps://pop.com.sg/how-do-you-derive-the-formula-of-sin-a-b-from-cos-a-b-cos-a-cos-b-sin-.htmlI know that cos is even while sin is odd, and I know $\cos(\pi)=\sin((\pi/2)-x)$, but I still can't figure the derivation of $\sin (a+b)$ from $\cos(a+b)=\cos(a)\cos(b)-\sin(a)\sin(b)$. Could you ......Tue, 11 Aug 2026 15:38:22 -0400https://pop.com.sg/laplace-transform-to-solve-differential-equation-with-ivp-given-at-a-p.htmlhttps://pop.com.sg/laplace-transform-to-solve-differential-equation-with-ivp-given-at-a-p.htmlhow could I use Laplace Transform to solve the following differential equation: $$y''+2y'+y=0; \;\;\;\;\;y'(0)=2\;\;\;\;\;\mbox{and}\;\;\;\;\;\boldsymbol{y(1)=2}.$$ The solution may involve the ...Tue, 11 Aug 2026 15:34:12 -0400https://pop.com.sg/matrix-by-matrix-derivative-formula.htmlhttps://pop.com.sg/matrix-by-matrix-derivative-formula.htmlI need to derive $\frac{\delta(X^{T}MX)}{\delta X}$, where $X$ and $M$ are $n \times n$ matrices. I know that $\frac{\delta(AXB)}{\delta X}=B^{T} \otimes A$ but am having a hard time deriving what I ...Tue, 11 Aug 2026 15:31:07 -0400https://pop.com.sg/the-meaning-of-differentiation-of-x-with-respect-to-y.htmlhttps://pop.com.sg/the-meaning-of-differentiation-of-x-with-respect-to-y.htmlThe physical meaning of the differentiation of $x$ with respect to $y$ is the rate of change of $x$ with respect to $y$. But, I am finding it difficult in understanding the geometrical interpretati......Tue, 11 Aug 2026 15:28:43 -0400https://pop.com.sg/round-table-seating-probability.htmlhttps://pop.com.sg/round-table-seating-probability.htmlThere are $6$ people, let's call them - (a,b,c,d,e,f), to sit at a round table. The number of ways they can arrange themselves is $(6-1)! = 5! = 120$ ways. What is the probability that person 'a' ......Tue, 11 Aug 2026 15:26:42 -0400https://pop.com.sg/determinant-of-an-elementary-matrix.htmlhttps://pop.com.sg/determinant-of-an-elementary-matrix.htmlI read a very slick proof of determinant properties, in this case of the fact $\det A = \det A^T$, which says in one place It suffices to notice that for any elementary matrix $M$ we have $\det......Tue, 11 Aug 2026 15:07:13 -0400https://pop.com.sg/third-order-linear-parabolic-pde.htmlhttps://pop.com.sg/third-order-linear-parabolic-pde.htmlWhat's the best method to solve analytically an equation of the form $$f_t=f_x+af_{xx}+bf_{xxx}$$ with $a,b\in\mathbb{R}$ ?...Tue, 11 Aug 2026 14:59:52 -0400https://pop.com.sg/solving-system-of-differential-equations-using-runge-kutta-method.htmlhttps://pop.com.sg/solving-system-of-differential-equations-using-runge-kutta-method.htmlI have two differential equations: \begin{align}\frac{dx}{dt} &= 1 -(1+b)x+ax^2y\\ \frac{dy}{dt} &= bx - ax^2y\end{align} I have been asked to solve them on Python using the Runge Kutta (......Tue, 11 Aug 2026 14:52:12 -0400https://pop.com.sg/proving-that-pi-2x-2-pi-x.htmlhttps://pop.com.sg/proving-that-pi-2x-2-pi-x.htmlIn our analytic number theory class we were given the following problem as homework: prove rigorously that for large $x$ the number of primes in $(1,x]$ exceeds that in $(x,2x]$. In class we prove......Tue, 11 Aug 2026 14:46:14 -0400https://pop.com.sg/flipping-a-fraction-within-an-inequality.htmlhttps://pop.com.sg/flipping-a-fraction-within-an-inequality.htmlIs it possible to flip a faction within an inequality? Such that: $$\frac13 < x < \frac23$$ becomes, $$3 > \frac1x > \frac32$$...Tue, 11 Aug 2026 14:28:59 -0400https://pop.com.sg/tetrahedron-packing-in-cube.htmlhttps://pop.com.sg/tetrahedron-packing-in-cube.htmlI'm thinking about following solid geometry problem. Q: Suppose you have a box of "cube" shape with edge length 1. Then, How many regular tetrahedrons(with edge length 1) can be in the box? So, t......Tue, 11 Aug 2026 14:11:37 -0400https://pop.com.sg/how-can-you-visually-distinguish-between-a-parabola-and-hyperbola.htmlhttps://pop.com.sg/how-can-you-visually-distinguish-between-a-parabola-and-hyperbola.htmlI am fully aware of the mathematical definition of either and how they relate to conic sections. Still, I wonder whether there is an easy way to see that these are fundamentally different curves.......Tue, 11 Aug 2026 14:05:09 -0400https://pop.com.sg/solution-to-equations-in-3-variable-without-forming-a-cubic-polynomial.htmlhttps://pop.com.sg/solution-to-equations-in-3-variable-without-forming-a-cubic-polynomial.htmlCan we solve 3 equations:- $$a+b+c = x$$ $$abc = y$$ $$ab+bc+ac = z$$ to get the value of $a,b$ and $c$ where $x,y,z$ are constants As you can probably see I'm trying to find a solution to a cubic ...Tue, 11 Aug 2026 14:04:11 -0400https://pop.com.sg/symbol-for-kernel-and-range-of-a-linear-transformation.htmlhttps://pop.com.sg/symbol-for-kernel-and-range-of-a-linear-transformation.htmlI'm wondering what symbol is used for the kernel and the range of a linear transformation. I've seen them being written as such: $\ker(T)$ $\mathcal{R}(T)$ Are these the "correct" symbols for the ...Tue, 11 Aug 2026 13:58:40 -0400https://pop.com.sg/what-does-x-2-1-mean.htmlhttps://pop.com.sg/what-does-x-2-1-mean.htmlI am studying some inequality properties of absolute values and I bumped into some expressions like $|x-2| < 1$ that I just can't get the meaning of them. Lets say I have this expression $$ |......Tue, 11 Aug 2026 13:40:49 -0400https://pop.com.sg/is-there-a-geometric-interpretation-for-a-function-039-s-alpha-subleve.htmlhttps://pop.com.sg/is-there-a-geometric-interpretation-for-a-function-039-s-alpha-subleve.htmlIn Boyd and Vandenberghe's "Convex Optimization": The $\alpha$-sublevel set set of a function $f:\mathbb{R}^n \rightarrow \mathbb{R}$ is defined as $$C_\alpha=\{x \in \mathbf{dom} f|f(x)\leq\......Tue, 11 Aug 2026 13:40:07 -0400https://pop.com.sg/study-according-to-the-values-of-x-the-relative-position-of-line-curve.htmlhttps://pop.com.sg/study-according-to-the-values-of-x-the-relative-position-of-line-curve.htmlWe have $F(x) = x + \ln \left(\frac{1+2e^{-2x}}{1+e^{-x}}\right)$ represents curve $C$ and a line $d: y=x$ In an exercise it is required to study the relative position of these two functions, ......Tue, 11 Aug 2026 13:37:43 -0400https://pop.com.sg/finding-all-real-zeros-of-the-polynomial.htmlhttps://pop.com.sg/finding-all-real-zeros-of-the-polynomial.htmlOkay, so I need to find all real zeros in this polynomial... $$f(x) = 2x^3 + x^2 - 13x + 6$$ I know that the first step is to find the factors of 6 and 2, then see which when multiplied by the ot......Tue, 11 Aug 2026 13:36:04 -0400https://pop.com.sg/strategies-for-factoring-expressions-with-four-terms.htmlhttps://pop.com.sg/strategies-for-factoring-expressions-with-four-terms.htmlI'm trying to come up with a general strategy for factoring expressions with four terms on the basis of the symmetries of the expressions. One thought I had was the following: count up the number of ...Tue, 11 Aug 2026 13:33:22 -0400https://pop.com.sg/novikoff-039-s-proof-for-perceptron-convergence.htmlhttps://pop.com.sg/novikoff-039-s-proof-for-perceptron-convergence.htmlIn Machine Learning, the Perceptron algorithm converges on linearly separable data in a finite number of steps. One can prove that $(R/\gamma)^2$ is an upper bound for how many errors the algorithm......Tue, 11 Aug 2026 13:25:05 -0400https://pop.com.sg/simple-formula-for-a-sieries-like-1-2-5-10-20-50-100.htmlhttps://pop.com.sg/simple-formula-for-a-sieries-like-1-2-5-10-20-50-100.htmlI'm looking for a simple formula that will give a series that looks like this: $1; 2; 5; 10; 20; 50; 100; ...$ That means a function that will give this output: $f(1) = 1$ $f(2) = 2$ $f(3) = 5$ $f......Tue, 11 Aug 2026 13:20:43 -0400https://pop.com.sg/geometric-interpretation-of-eigendecomposition.htmlhttps://pop.com.sg/geometric-interpretation-of-eigendecomposition.htmlIf we have an orthogonal matrix $U$, then $U^Tx$ is essentially a rotation of the vector $x$. If we have a diagonal matrix $\Lambda$, then $\Lambda x$ is scaling the vector $x$ in each direction by......Tue, 11 Aug 2026 13:18:59 -0400https://pop.com.sg/how-to-determine-the-right-side-of-the-dice.htmlhttps://pop.com.sg/how-to-determine-the-right-side-of-the-dice.htmlsay I have a dice, with following numbering arrangements: top front right bottom back left 1 2 3 6 5 4 the dice can be rolled freely of course I think it is enough to give the ...Tue, 11 Aug 2026 12:57:09 -0400https://pop.com.sg/finding-partial-fractions-involving-complex-numbers-express-h-z-frac-2.htmlhttps://pop.com.sg/finding-partial-fractions-involving-complex-numbers-express-h-z-frac-2.htmlI'm attempting the following problem in the context of complex analysis: Express $h(z) = \dfrac{2z + 4i}{z^2 + 2z + 2}$ as partial fractions. I got $\dfrac{2z + 4i}{z^2 + 2z + 2} = \dfrac{2z + ......Tue, 11 Aug 2026 12:56:57 -0400https://pop.com.sg/cardinality-of-union-of-3-sets.htmlhttps://pop.com.sg/cardinality-of-union-of-3-sets.htmlI know there are answers to this specific question, but I'm trying to solve it through a specific line of thought which was used in a textbook on modern algebra for the case with 2 sets. It goes like ...Tue, 11 Aug 2026 12:52:28 -0400https://pop.com.sg/continuity-is-required-for-differentiability.htmlhttps://pop.com.sg/continuity-is-required-for-differentiability.htmlMy professor emphasized that: Differentiability implies continuity and Continuity is required for differentiability. Since a function like $\frac 1 x$ is differentiable but not continuous, I thou......Tue, 11 Aug 2026 12:48:10 -0400https://pop.com.sg/how-to-round-0-45-is-it-0-or-1.htmlhttps://pop.com.sg/how-to-round-0-45-is-it-0-or-1.htmlThis question is inspired by How to round 0.4999... ? Is it 0 or 1? I didn't quite understand the logic of the answer. It seems you round every decimal place no matter how far back it goes? In the ......Tue, 11 Aug 2026 12:47:14 -0400https://pop.com.sg/how-was-this-approximation-of-pi-involving-sqrt-5-arrived-at.htmlhttps://pop.com.sg/how-was-this-approximation-of-pi-involving-sqrt-5-arrived-at.htmlThe Wikipedia article for Approximations of $\pi$ contains this little gem: $$ \pi \approx \frac{63}{25}\times\frac{17 + 15\sqrt{5}}{7 + 15\sqrt{5}} $$ which is clearly in $\mathbb{Q[\sqrt{5}]}$. ...Tue, 11 Aug 2026 12:46:58 -0400https://pop.com.sg/measurable-space-vs-measure-space.htmlhttps://pop.com.sg/measurable-space-vs-measure-space.htmlFrom what I understood a Measurable Space is $(X,S)$ where $X$ is a set and $S\subset P(X)$ is a $\sigma$ algebra a Measure Space is $(X,S,\mu)$ where $X$ is a set and $S\subset P(X)$ is a $\sigma$ ...Tue, 11 Aug 2026 12:46:20 -0400https://pop.com.sg/using-conjugates-to-find-a-limit-with-a-cubic-root-lim-h-to-0-frac-sqr.htmlhttps://pop.com.sg/using-conjugates-to-find-a-limit-with-a-cubic-root-lim-h-to-0-frac-sqr.htmlI currently have $$\lim_{h\to 0}\frac{\sqrt[3]{h+1}-1}{h}$$ Now, I know that when you have square root instead of a cubic root it's easy. You just multiply by the conjugate and simplify afterwards......Tue, 11 Aug 2026 12:45:30 -0400https://pop.com.sg/039-imperfect-039-numbers.htmlhttps://pop.com.sg/039-imperfect-039-numbers.htmlA perfect number (integer) is equal to the sum of its divisors, including 1, and excluding itself. This has been around since Euclid. Recently, I noticed that at least for the initial integers, it is ...Tue, 11 Aug 2026 12:40:07 -0400https://pop.com.sg/proof-with-double-counting.htmlhttps://pop.com.sg/proof-with-double-counting.htmlUsing double counting (and without algebraic calculations), prove that $$ \sum_{k=r}^n \binom n k \binom k r = 2^{n-r} \binom{n}{n-r}. $$ This question was assigned to me on a quiz and I was not......Tue, 11 Aug 2026 12:29:16 -0400https://pop.com.sg/evans-pde-5-17.htmlhttps://pop.com.sg/evans-pde-5-17.htmlI'm working through Evans PDE $2$nd edition Chapter $5$, question $17$: Assume $F:\mathbb{R}\rightarrow\mathbb{R}$ is $C^1$ with $F'$ bounded. Suppose $U\in \mathbb{R}^n$ is bounded and $u\in W^......Tue, 11 Aug 2026 12:13:45 -0400https://pop.com.sg/eliminate-xy-term-of-conic.htmlhttps://pop.com.sg/eliminate-xy-term-of-conic.htmlFor the following problem, I am trying to eliminate xy, and I've tried numerous times to solve if with no luck. I need to find the general form of the equation rotated to eliminate the xy term. $ ......Tue, 11 Aug 2026 12:12:11 -0400https://pop.com.sg/proving-the-universal-mapping-property-for-polynomial-rings.htmlhttps://pop.com.sg/proving-the-universal-mapping-property-for-polynomial-rings.htmlI am studying from Patrick Morandi's Field and Galois Theory, and I am stuck on Problem 6 from Section 1 (on page 13). Verify the following universal mapping property for polynomial rings: ......Tue, 11 Aug 2026 12:09:12 -0400https://pop.com.sg/proof-verification-if-the-diagonals-of-a-parallelogram-are-congruent-t.htmlhttps://pop.com.sg/proof-verification-if-the-diagonals-of-a-parallelogram-are-congruent-t.htmlTo prove. If $|\overline{AC}| = |\overline{BD}|$, then $\square ABCD$ is a rectangle. $\overline{AC}=\sqrt{(c_1-a_1)^2+(d_2-0)^2}=\sqrt{(c_1-a_1)^2+d_{2}^2}$ $\overline{BD}=\sqrt{b_{1}^2+d_{2}^2}......Tue, 11 Aug 2026 12:07:36 -0400https://pop.com.sg/convolution-of-two-gaussians-is-a-gaussian.htmlhttps://pop.com.sg/convolution-of-two-gaussians-is-a-gaussian.htmlI know that the product of two Gaussians is a Gaussian, and I know that the convolution of two Gaussians is also a Gaussian. I guess I was just wondering if there's a proof out there to show that the ...Tue, 11 Aug 2026 11:58:58 -0400https://pop.com.sg/user-basanta-raj-pahari-mathematics-stack-exchange.htmlhttps://pop.com.sg/user-basanta-raj-pahari-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Tue, 11 Aug 2026 11:55:27 -0400https://pop.com.sg/intuition-behind-the-ilate-rule.htmlhttps://pop.com.sg/intuition-behind-the-ilate-rule.htmlOften I have wondered about this question, but today I had a chance to recollect it and hence I am posting it here. During high-school days one generally learns Integration and I still loving doing ...Tue, 11 Aug 2026 11:47:51 -0400https://pop.com.sg/how-is-the-gradient-of-exponential-functions-with-different-bases-in-t.htmlhttps://pop.com.sg/how-is-the-gradient-of-exponential-functions-with-different-bases-in-t.htmlI am studying exponential functions at the moment, and this table was presented in my textbook to show that for exponential functions with increasing 'bases' the gradient of the function increases.......Tue, 11 Aug 2026 11:39:34 -0400https://pop.com.sg/a-question-concerning-an-exercise-from-tao-vu.htmlhttps://pop.com.sg/a-question-concerning-an-exercise-from-tao-vu.htmlThis is the exercise 4.1.5 from Tao Vu Additive Combinatorics. $Z$ is a finite additive group with a fixed symmetric non-degenerate bilinear form $\cdot$ Define $e: \mathbb{R}/\mathbb{Z} \to \mat......Tue, 11 Aug 2026 11:30:34 -0400https://pop.com.sg/line-integral-vs-arc-length.htmlhttps://pop.com.sg/line-integral-vs-arc-length.htmlI am trying to understand when do to line integral and when to do arc length. So I know the formula for arc length varies based on $dx$ or $dy$ like so: $s=\int_a^b \sqrt{1+[f'(x)]^2} \, \mathrm{d}......Tue, 11 Aug 2026 11:12:57 -0400https://pop.com.sg/what-does-it-mean-for-a-matrix-to-be-nearly-singular.htmlhttps://pop.com.sg/what-does-it-mean-for-a-matrix-to-be-nearly-singular.htmlI am currently enrolled in numerical analysis course, and a term I have not heard of came up: nearly singular matrix. I know that a non-singular matrix is one where the column vectors are linearly ...Tue, 11 Aug 2026 11:01:03 -0400https://pop.com.sg/convert-rectangular-coordinate-3-0-to-polar-coordinate.htmlhttps://pop.com.sg/convert-rectangular-coordinate-3-0-to-polar-coordinate.htmlI'm trying to convert (-3,0) to polar coordinate. I can get r=$\sqrt {(-3)^2 +(0)^2}$ =3, but when computing for the angle $\theta$=$\tan^{-1} (\frac {0}{-3})$=0 but the answer for the a......Tue, 11 Aug 2026 11:00:44 -0400https://pop.com.sg/finding-stagnation-points.htmlhttps://pop.com.sg/finding-stagnation-points.htmlI am trying to find the stagnation point of a fluid flow from a complex potential. The complex potential is given by: $\Omega(z) = Uz + \cfrac{m}{2\pi}\ln z$. From this I found the streamfunction t......Tue, 11 Aug 2026 10:59:39 -0400https://pop.com.sg/how-to-calculate-the-surface-area-of-a-cube-given-a-diagonal.htmlhttps://pop.com.sg/how-to-calculate-the-surface-area-of-a-cube-given-a-diagonal.htmlI encountered this problem while practicing for a mathematics competition. A cube has a diagonal length of 10. What is the surface area of the cube? No Calculators Allowed. (Emphasis mine) I'......Tue, 11 Aug 2026 10:59:26 -0400https://pop.com.sg/how-do-i-get-the-square-root-of-a-complex-number.htmlhttps://pop.com.sg/how-do-i-get-the-square-root-of-a-complex-number.htmlIf I'm given a complex number (say $9 + 4i$), how do I calculate its square root?...Tue, 11 Aug 2026 10:57:10 -0400https://pop.com.sg/trigonometric-identities-like-a-sin-x-b-cos-y-cdots.htmlhttps://pop.com.sg/trigonometric-identities-like-a-sin-x-b-cos-y-cdots.htmlAre there any identities for trigonometric equations of the form: $$A\sin(x) + B\sin(y) = \cdots$$ $$A\sin(x) + B\cos(y) = \cdots$$ $$A\cos(x) + B\cos(y) = \cdots$$ I can't find any mention of them ...Tue, 11 Aug 2026 10:35:56 -0400https://pop.com.sg/platonic-solids-in-4d.htmlhttps://pop.com.sg/platonic-solids-in-4d.htmlI've noticed in this wikipedia article that each 4D figure is constructed of 3D platonic solids to get to these, just like using polygons on solids. In the hypercube, you have the cube in the cente......Tue, 11 Aug 2026 10:29:46 -0400https://pop.com.sg/find-the-cosine-of-the-angle-abc.htmlhttps://pop.com.sg/find-the-cosine-of-the-angle-abc.htmlI have problem with solving this task. I know that the answer might be A. But only with calculator by calculating the angles. can someone explain me or give me a hint to solve it. cos^-1 (5/13) =......Tue, 11 Aug 2026 10:25:27 -0400https://pop.com.sg/how-different-are-open-and-closed-intervals.htmlhttps://pop.com.sg/how-different-are-open-and-closed-intervals.htmlI am taking an online maths course and I got little confused on the concepts of open and closed intervals. In the lecture, professor says there is a big difference between $(a, b)$ and $[a, b]$. The ...Tue, 11 Aug 2026 10:24:59 -0400https://pop.com.sg/is-there-a-category-of-categories.htmlhttps://pop.com.sg/is-there-a-category-of-categories.htmlMy question is quite simple, I would like to know if we can define the category of the categories, unlike Cat which is the category of the small categories. By the way, are there any particular rea......Tue, 11 Aug 2026 10:16:49 -0400https://pop.com.sg/what-is-fog-x-f-g-x-f-x-fof-n-1-x-fog-2012-x-closed.htmlhttps://pop.com.sg/what-is-fog-x-f-g-x-f-x-fof-n-1-x-fog-2012-x-closed.htmlI found a math assuming fog(x)=f(g(x)), f"(x)=fof^(n-1)(x), fog^2012(x)=0. What do they mean actually? Please describe in a way that I understand....Tue, 11 Aug 2026 10:12:26 -0400https://pop.com.sg/int-1-sin-x-dx.htmlhttps://pop.com.sg/int-1-sin-x-dx.htmlI'm trying to find $\int 1/\sin(x)\ dx$, but I can't figure out how to do it. Also, what would be the value of $$\int\limits_0^{2\pi} \frac{1}{\sin x} dx\quad ?$$ Based on symmetry, I would try to......Tue, 11 Aug 2026 10:11:31 -0400https://pop.com.sg/winding-number-of-vector-field-on-surface.htmlhttps://pop.com.sg/winding-number-of-vector-field-on-surface.htmlI found a term "winding number of vector field with respect to another vector field" in a paper without definition. Because my paper I am reading is talking about the surface, so I don't know if I ......Tue, 11 Aug 2026 10:11:13 -0400https://pop.com.sg/average-area-of-the-shadow-of-a-convex-shape-closed.htmlhttps://pop.com.sg/average-area-of-the-shadow-of-a-convex-shape-closed.htmlWhat is the average area of the shadow of a convex shape taken over all possible orientations? If we take a sphere, its surface area is exactly 4 times the area of its shadow. How can it be genera......Tue, 11 Aug 2026 10:04:36 -0400https://pop.com.sg/ramsey-number-proof-r-3-3-3-3-leq-4-r-3-3-3-1-2.htmlhttps://pop.com.sg/ramsey-number-proof-r-3-3-3-3-leq-4-r-3-3-3-1-2.htmlI am trying to prove: $R(3,3,3,3)\leq 4(R(3,3,3)-1) + 2$ I am confused as to how one can go from a $4$ color problem to a $3$ color problem by multiplying and adding. edit: $R$ is the Ramsey ......Tue, 11 Aug 2026 10:02:57 -0400https://pop.com.sg/need-help-with-solution-for-problem-a5-from-imo-2015-closed.htmlhttps://pop.com.sg/need-help-with-solution-for-problem-a5-from-imo-2015-closed.htmlI'm going through solution for A5 from International Mathematical Olympiad 2015. Something's not right. Here's how it goes: First, we assume all functions below map integers to integers. Second, ......Tue, 11 Aug 2026 10:00:42 -0400https://pop.com.sg/find-the-general-solution-to-the-ode.htmlhttps://pop.com.sg/find-the-general-solution-to-the-ode.htmlI was asked to find the general solution to these ODEs. $$\frac{dy}{dx}=\frac{(-8x+3y-31)}{(-3x+y-11)}$$ I tried by rearranging to the form $M(x,y)+N(x,y)\frac{dy}{dx}=0$ and let $y=vx$, but what ......Tue, 11 Aug 2026 09:58:25 -0400https://pop.com.sg/how-to-prove-that-the-tangent-to-a-circle-is-perpendicular-to-the-radi.htmlhttps://pop.com.sg/how-to-prove-that-the-tangent-to-a-circle-is-perpendicular-to-the-radi.htmlI've tried drawing a parallel chord to the tangent but then how would you prove that the chord is perpendicular to the radius?...Tue, 11 Aug 2026 09:53:19 -0400https://pop.com.sg/relation-betwen-coefficients-and-roots-of-a-polynomial-duplicate.htmlhttps://pop.com.sg/relation-betwen-coefficients-and-roots-of-a-polynomial-duplicate.htmlPossible Duplicate: Create polynomial coefficients from its roots I am reading the first chapter titled Numerical Solutions Of Equations And Interpolation by K.A. Stroud (Advanced Engineering ......Tue, 11 Aug 2026 09:49:15 -0400https://pop.com.sg/determine-sqrt-1-50-cdot51-cdot52-cdot53-without-a-calculator.htmlhttps://pop.com.sg/determine-sqrt-1-50-cdot51-cdot52-cdot53-without-a-calculator.htmlI've tried a lot of things but failed to do it, I've calculated the result inside the square root which is $7027801$ using substitution and factoring but $\sqrt{7027801}$ isn't possible to simplify....Tue, 11 Aug 2026 09:49:11 -0400https://pop.com.sg/is-there-a-unique-minimal-expression-for-every-boolean-function.htmlhttps://pop.com.sg/is-there-a-unique-minimal-expression-for-every-boolean-function.htmlIs there a unique minimal expression for every boolean function? I've heard that there are some boolean expressions for which the minimal form is not unique. What are the characteristics of this k......Tue, 11 Aug 2026 09:42:22 -0400https://pop.com.sg/prove-in-full-detail-that-the-set-is-a-vector-space.htmlhttps://pop.com.sg/prove-in-full-detail-that-the-set-is-a-vector-space.htmlSo I'm doing a review test and I have this problem: Prove in full detail, with the standard operations in R2, that the set {(x,2x): x is a real number} is a vector space. Attempt: Given: $(x_1, 2x......Tue, 11 Aug 2026 09:31:48 -0400https://pop.com.sg/physical-meaning-of-one-dimensional-heat-equation.htmlhttps://pop.com.sg/physical-meaning-of-one-dimensional-heat-equation.htmlConsider heat conduction in a $1$-D material. The temperature distribution $u$ after sufficiently long time can be modeled by $$-(a(x)u_x)_x = f(x), \qquad x \in [0,1]$$ where $a$ is heat conduct......Tue, 11 Aug 2026 09:25:25 -0400https://pop.com.sg/is-the-absolute-value-function-a-linear-function.htmlhttps://pop.com.sg/is-the-absolute-value-function-a-linear-function.htmlI'm inclined to say yes, as it doesn't involve exponentiation, roots, logarithmic or trigonometric functions, but I watched a video where the teacher said that the absolute value function is "clear......Tue, 11 Aug 2026 09:10:18 -0400