https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usMon, 24 Aug 2026 21:59:17 -0400https://pop.com.sg/how-can-i-find-the-equation-for-a-reverse-exponential-curve-based-on-t.htmlhttps://pop.com.sg/how-can-i-find-the-equation-for-a-reverse-exponential-curve-based-on-t.htmlI have a curve such that as $x$ approaches infinity, $y$ approaches $2$, and as $x$ approaches negative infinity, $y$ approaches infinity. I know it contains the following approximate points: $$(......Tue, 25 Aug 2026 01:58:51 -0400https://pop.com.sg/why-is-frac-15-sqrt-4-125-sqrt-4-5-15-sqrt-5-and-not-15-sqrt-4-25.htmlhttps://pop.com.sg/why-is-frac-15-sqrt-4-125-sqrt-4-5-15-sqrt-5-and-not-15-sqrt-4-25.htmlI have an expression I am to simplify: $$\frac{15\sqrt[4]{125}}{\sqrt[4]{5}}$$ I arrived at $15\sqrt[4]{25}$. My textbook tells me that the answer is in fact $15\sqrt{5}$. Here is my thought proc......Tue, 25 Aug 2026 01:56:56 -0400https://pop.com.sg/mathematical-toys.htmlhttps://pop.com.sg/mathematical-toys.htmlAnybody know of "serious" mathematical ornaments or toys like the Gömböc, etc? Already have a rubix and abacus (that's more of a tool though)....Tue, 25 Aug 2026 01:53:55 -0400https://pop.com.sg/appropriate-notation-equiv-versus.htmlhttps://pop.com.sg/appropriate-notation-equiv-versus.htmlWith respect to assignments/definitions, when is it appropriate to use $\equiv$ as in $$M \equiv \max\{b_1, b_2, \dots, b_n\}$$ which I encountered in my analysis textbook as opposed to the "c......Tue, 25 Aug 2026 01:53:37 -0400https://pop.com.sg/how-to-prove-x-p-i-implies-x-p-n-inp-n-1.htmlhttps://pop.com.sg/how-to-prove-x-p-i-implies-x-p-n-inp-n-1.htmlHow to prove $[x,p]=i$ $\implies$ $[x,p^n]=inp^{n-1}$? I can do this using $p=i\frac{d}{dx}$, but my book hasn't introduced this yet so is there another proof without using this ? These are just l......Tue, 25 Aug 2026 01:51:26 -0400https://pop.com.sg/is-an-equilateral-triangle-the-same-as-an-equiangular-triangle-in-any-.htmlhttps://pop.com.sg/is-an-equilateral-triangle-the-same-as-an-equiangular-triangle-in-any-.htmlI have heard of both equilateral triangles and equiangular triangles. (For example, this sporcle quiz lists both.) Are these always equivalent, regardless of geometry? I know they are the same in ...Tue, 25 Aug 2026 01:49:56 -0400https://pop.com.sg/logic-coin-game.htmlhttps://pop.com.sg/logic-coin-game.html2 people are playing a game with coins. First, 10 coins are put in a circular form. Then, the players take turns removing 1 coin or 2 coins that are together at each turn. The player who takes the ......Tue, 25 Aug 2026 01:36:26 -0400https://pop.com.sg/how-to-integrate-x-1-x-1.htmlhttps://pop.com.sg/how-to-integrate-x-1-x-1.htmlI want to calculate the integral $$\int\frac{x-1}{x+1}\,\mathrm{d}x.$$ I have tried solving it by differentiating the denominator and substituting it, but I didn't get it. How else can I solve it?...Tue, 25 Aug 2026 01:33:03 -0400https://pop.com.sg/user-mathma-mathematics-stack-exchange.htmlhttps://pop.com.sg/user-mathma-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Tue, 25 Aug 2026 01:31:34 -0400https://pop.com.sg/how-to-calculate-subspace-of-a-set-of-solutions-of-matrix-ax-b.htmlhttps://pop.com.sg/how-to-calculate-subspace-of-a-set-of-solutions-of-matrix-ax-b.htmlI am looking through some old linear algebra exam papers. However i do not understand how to calculate whether a set of solutions is within a certain subspace R. This is the problem in question: I......Tue, 25 Aug 2026 01:26:14 -0400https://pop.com.sg/integrate-gaussian-function.htmlhttps://pop.com.sg/integrate-gaussian-function.htmlI am trying to integrate Gaussian distribution from -m to m to find parametar A. I have done this so far: $\int_{-m}^{m}\frac{A}{\sqrt(2\pi)\sigma}e^{-(x-m)^{2}/(2\sigma^{2})}dx=1$ after $u=\fra......Tue, 25 Aug 2026 01:25:11 -0400https://pop.com.sg/lfloor-25x-2-4-rfloor-13x-4-3.htmlhttps://pop.com.sg/lfloor-25x-2-4-rfloor-13x-4-3.htmlThis problem is designated for those who have the basic knowledge of floor brackets I know the answer will be 6. But tell me how....Tue, 25 Aug 2026 01:23:17 -0400https://pop.com.sg/mathcad-variable-undefined-even-though-i-just-defined-it.htmlhttps://pop.com.sg/mathcad-variable-undefined-even-though-i-just-defined-it.htmlHoping to run into some mathcadders: I have a worksheet which includes the following except (and error) Any help?...Tue, 25 Aug 2026 01:20:57 -0400https://pop.com.sg/when-is-an-upper-triangular-matrix-invertible.htmlhttps://pop.com.sg/when-is-an-upper-triangular-matrix-invertible.htmlI did the following (the exercise assumes $2\times 2$ matrices): $$\begin{pmatrix} {a}&{b}\\ {0}&{d} \end{pmatrix} \begin{pmatrix} {A}&{B}\\ {C}&{D} \end{pmatrix} = \begin{pmatri......Tue, 25 Aug 2026 01:19:20 -0400https://pop.com.sg/receding-horizon-vs-finite-horizon-in-planning.htmlhttps://pop.com.sg/receding-horizon-vs-finite-horizon-in-planning.htmlSo, I am reading a paper on TSP. In one of the papers it is mentioned: We plan using a fixed-horizon approach, alternating between replanning and execution until the elapsed time $t$ exceeds the ...Tue, 25 Aug 2026 01:16:35 -0400https://pop.com.sg/on-solvibility-of-heat-equation-with-time-space-reversed.htmlhttps://pop.com.sg/on-solvibility-of-heat-equation-with-time-space-reversed.htmlConsider the heat equation $$u_t=u_{xx}, (t,x)\in \mathbb{R}^2$$ with initial condition on $x=0$: \begin{cases} u(0,t)=1+\sin t\\ u_x(0,t)=\sin\left(\dfrac{\pi}{4}+t\right) \end{cases} and with per......Tue, 25 Aug 2026 01:10:48 -0400https://pop.com.sg/calculus-dx-on-top-of-fraction.htmlhttps://pop.com.sg/calculus-dx-on-top-of-fraction.htmlI'm studying for a final, and I haven't seen any mention of any problem of this form in class or in my homework. I can't figure out how to go about solving this problem: $$\int^{e^6}_{1}{\frac{dx}......Tue, 25 Aug 2026 01:09:25 -0400https://pop.com.sg/characteristics-and-mantessa.htmlhttps://pop.com.sg/characteristics-and-mantessa.htmlI've just heard about these terms. Could someone elaborate on what's their use is? And plus could you explain it using a few examples?...Tue, 25 Aug 2026 01:05:46 -0400https://pop.com.sg/value-of-cdf-at-infinity.htmlhttps://pop.com.sg/value-of-cdf-at-infinity.htmlI had a question, given a continuous CDF F, is $F(+\infty)$ essentially equal to 1?...Tue, 25 Aug 2026 01:02:08 -0400https://pop.com.sg/how-to-convert-numbers-with-related-bases-quickly.htmlhttps://pop.com.sg/how-to-convert-numbers-with-related-bases-quickly.htmlLet: $a = (1011011)_2 = (1123)_4$ There two ways to solve it: Convert the number in base 2 to base 10 then to base $4$. Consider $4 = 2^2$ and group each of two numbers in base $2$ to one in ba......Tue, 25 Aug 2026 01:00:15 -0400https://pop.com.sg/diagonalizing-quadratic-forms-linear-algebra.htmlhttps://pop.com.sg/diagonalizing-quadratic-forms-linear-algebra.htmlI have a question that reads: Diagonalize the quadratic form $A(x,y) = 3x^2 -12xy + 7y^2$ by completing the square. What is diagonalization? Is that when I should find the eigenvector matrix, ......Tue, 25 Aug 2026 00:48:47 -0400https://pop.com.sg/does-negative-infinity-squared-positive-infinity.htmlhttps://pop.com.sg/does-negative-infinity-squared-positive-infinity.htmlI googled this question and saw this answer but I wasn't satisfied....Tue, 25 Aug 2026 00:43:41 -0400https://pop.com.sg/probability-of-there-being-a-bullet-in-the-chamber.htmlhttps://pop.com.sg/probability-of-there-being-a-bullet-in-the-chamber.htmlSuppose we have three $6$-chambered guns: The first has no bullets. The second has one bullet. The third has two bullets in consecutive chambers. The cylinder advances automatically as the trigg......Tue, 25 Aug 2026 00:42:05 -0400https://pop.com.sg/what-base-is-roman-numerals.htmlhttps://pop.com.sg/what-base-is-roman-numerals.htmlWhat is the base for Roman Numerals? It starts off with unary then goes back and forth between multiples of 5 and 10....Tue, 25 Aug 2026 00:39:08 -0400https://pop.com.sg/questions-tagged-binomial-coefficients.htmlhttps://pop.com.sg/questions-tagged-binomial-coefficients.htmlQ&A for people studying math at any level and professionals in related fields...Tue, 25 Aug 2026 00:36:40 -0400https://pop.com.sg/using-implicit-differentiation-with-a-fraction.htmlhttps://pop.com.sg/using-implicit-differentiation-with-a-fraction.htmlHow do I solve this? What steps? I have been beating my head into the wall all evening. $$ x^2 + y^2 = \frac{x}{y} + 4 $$...Tue, 25 Aug 2026 00:25:30 -0400https://pop.com.sg/exponential-growth-and-decay-compound-interest.htmlhttps://pop.com.sg/exponential-growth-and-decay-compound-interest.htmlThis is the question: "If you want to have $\$75,000$ after $35$ years in your account that pays $12\%$ annual interest compounded quarterly, how much should you put in as your original investment?......Tue, 25 Aug 2026 00:17:52 -0400https://pop.com.sg/stability-of-fixed-points-for-ode-range.htmlhttps://pop.com.sg/stability-of-fixed-points-for-ode-range.htmlGiven the ordinary differential equation $$\frac{dx}{dt} = x^3 - 2x^2 + (1- \mu)x,$$ where $\mu$ is a parameter, I want to find an expression for the fixed point(s) $x^*$ as a function of $\mu$; ......Tue, 25 Aug 2026 00:16:34 -0400https://pop.com.sg/three-mutually-tangent-circles-have-centers-at-given-distances-from-ea.htmlhttps://pop.com.sg/three-mutually-tangent-circles-have-centers-at-given-distances-from-ea.htmlThree circles of different radii are tangent to each other externally. The distance between their centers are $9\ cm$, $8\ cm$, and $11\ cm$. Find the radius of each circle. Find the area in betw......Tue, 25 Aug 2026 00:07:41 -0400https://pop.com.sg/calculate-width-and-height-of-a-rectangle-given-its-diagonal-and-ratio.htmlhttps://pop.com.sg/calculate-width-and-height-of-a-rectangle-given-its-diagonal-and-ratio.htmlWell, I know, it's easy. We did it in class some time ago and I forgot it, I'm stupid because I can't figure it out: E.g. I have a 32" TV with 16:9 ratio and I want to know its width and height. ......Tue, 25 Aug 2026 00:06:57 -0400https://pop.com.sg/difference-between-indefinite-and-definite-integration-int-f-x-dx-int-.htmlhttps://pop.com.sg/difference-between-indefinite-and-definite-integration-int-f-x-dx-int-.html$\int f(x) \;dx = \int^x_0 f(x) \; dx$ I sometimes get the impression that there's meant to be a basic and important difference between definite and indefinite integration. But when I consider sim......Mon, 24 Aug 2026 23:54:35 -0400https://pop.com.sg/meaning-of-c-in-the-linear-equation-ax-by-c.htmlhttps://pop.com.sg/meaning-of-c-in-the-linear-equation-ax-by-c.htmlSo I found: $ax+by=c$ What exactly does $c$ stand for? What does it do and why is $c$ important? I googled everywhere but couldn't find an answer other than $c$ being a constant....Mon, 24 Aug 2026 23:39:24 -0400https://pop.com.sg/is-the-game-2048-always-solveable.htmlhttps://pop.com.sg/is-the-game-2048-always-solveable.htmlGames got me on math. I always want to play best. I don't know how to answer my question. My question is : How to show that the game 2048 is (always) solvable>? Is there any method other than......Mon, 24 Aug 2026 23:37:39 -0400https://pop.com.sg/what-is-the-predual-of-l-1.htmlhttps://pop.com.sg/what-is-the-predual-of-l-1.htmlIs there a nice characterization of the predual of $L^1$? So, what does the space $X$ look like, such that $X^*=L^1$, where the star denotes the dual of a Banach space. How do you start to find such ...Mon, 24 Aug 2026 23:25:21 -0400https://pop.com.sg/inductive-proof-using-fitch-software.htmlhttps://pop.com.sg/inductive-proof-using-fitch-software.htmlI am trying to prove the integer square root theorem $\forall x: \mathbb{N}, \exists y : \mathbb{N}((y^2 \leq x) \land (x < (y+1)^2))$ for $\lfloor \sqrt{x} \rfloor$. In words: for any natural ...Mon, 24 Aug 2026 23:17:56 -0400https://pop.com.sg/how-to-simulate-monty-hall-problem-in-computer.htmlhttps://pop.com.sg/how-to-simulate-monty-hall-problem-in-computer.htmlSuppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1 [but the door is not opened], and the host,......Mon, 24 Aug 2026 23:13:48 -0400https://pop.com.sg/finding-the-complementary-function-of-a-second-order-linear-differenti.htmlhttps://pop.com.sg/finding-the-complementary-function-of-a-second-order-linear-differenti.htmlI have a question stating Find the complementary function of $\frac{d^2y}{dx^2} + 4\frac{dy}{dx} + 40y = x$ So I solved the auxillary equation, $ \lambda^2 + 4\lambda +40 = 0$, which gives $\......Mon, 24 Aug 2026 23:13:45 -0400https://pop.com.sg/distribution-of-the-difference-of-two-normal-random-variables.htmlhttps://pop.com.sg/distribution-of-the-difference-of-two-normal-random-variables.htmlIf $U$ and $V$ are independent identically distributed standard normal, what is the distribution of their difference? I will present my answer here. I am hoping to know if I am right or wrong. U......Mon, 24 Aug 2026 23:08:24 -0400https://pop.com.sg/correct-way-to-calculate-numeric-derivative-in-discrete-time.htmlhttps://pop.com.sg/correct-way-to-calculate-numeric-derivative-in-discrete-time.htmlGiven a set of discrete measurements in time $x_t, t \in \{0,\Delta t, 2\Delta t,\ldots,T-\Delta t,T\}$, what is the correct way to compute the discrete derivative $\dot x_t$. Is it more correct to......Mon, 24 Aug 2026 23:07:10 -0400https://pop.com.sg/given-two-events-a-and-b-what-is-the-probability-that-at-most-one-will.htmlhttps://pop.com.sg/given-two-events-a-and-b-what-is-the-probability-that-at-most-one-will.htmlI know this is a very basic question, but my mathematics is coming out a bit wonky. Assume Events A, B are independent. Define: $Pr(A) = 1/6$ $Pr(B) = 1/4$ Let Event C = "at most one event out ......Mon, 24 Aug 2026 22:56:16 -0400https://pop.com.sg/what-is-the-correct-inter-arrival-time-distribution-in-a-poisson-proce.htmlhttps://pop.com.sg/what-is-the-correct-inter-arrival-time-distribution-in-a-poisson-proce.htmlGiven a Poisson process (e.g. radioactive decay) with rate $\lambda$, then the expression $\exp(-\lambda t)$ is the probability of observing no counts in time interval $t$. This can be interpreted (...Mon, 24 Aug 2026 22:41:36 -0400https://pop.com.sg/understanding-of-the-formal-and-intuitive-definition-of-a-limit.htmlhttps://pop.com.sg/understanding-of-the-formal-and-intuitive-definition-of-a-limit.htmlThe intuitive definition for $\lim\limits_{x \to a} f(x) = L$ is the value of $f ( x )$ can be made arbitrarily close to $L$ by making $x$ sufficiently close, but not equal to, $a$ . I can easily ...Mon, 24 Aug 2026 22:36:20 -0400https://pop.com.sg/polynomials-as-vector-spaces.htmlhttps://pop.com.sg/polynomials-as-vector-spaces.htmlCan someone please explain how polynomials are vector spaces? I don't understand this at all. Vectors are straight, so how could a polynomial be a vector. Is it just that the coefficients are vector ...Mon, 24 Aug 2026 22:34:50 -0400https://pop.com.sg/how-can-i-find-the-horizontal-asymptotes-of-the-function-y-frac-x-3-sq.htmlhttps://pop.com.sg/how-can-i-find-the-horizontal-asymptotes-of-the-function-y-frac-x-3-sq.htmlIt is not a homework question. I ask this question because it was on my test and I genuinely want to understand. I have a very minimal knowledge on the subject of limits. The question is: How can I......Mon, 24 Aug 2026 22:30:52 -0400https://pop.com.sg/laplace-transform-of-the-product-of-two-functions.htmlhttps://pop.com.sg/laplace-transform-of-the-product-of-two-functions.htmlWhat is the Laplace Transform of the product of two functions? Specifically, my function is sin(5t)*cos(5t), but I'd like to know a general principle if it's available. What's the easy way to com......Mon, 24 Aug 2026 22:22:07 -0400https://pop.com.sg/serre-duality-in-derived-category.htmlhttps://pop.com.sg/serre-duality-in-derived-category.htmlLet $X$ be a smooth projective variety over a field $k$, and $\omega_X$ its canonical bundle. Denote by $D^b(X):=D^b(\mathbf{Coh})$ the bounded derived category of coherent sheaves on $X$. Theorem......Mon, 24 Aug 2026 22:15:10 -0400https://pop.com.sg/get-the-equation-of-a-circle-when-given-3-points.htmlhttps://pop.com.sg/get-the-equation-of-a-circle-when-given-3-points.htmlGet the equation of a circle through the points $(1,1), (2,4), (5,3) $. I can solve this by simply drawing it, but is there a way of solving it (easily) without having to draw?...Mon, 24 Aug 2026 22:12:19 -0400https://pop.com.sg/recurrence-relations-question.htmlhttps://pop.com.sg/recurrence-relations-question.htmlA vending machine dispensing books of stamps accepts only one-dollar coins, $1 bills, and $5 bills. a) Find a recurrence relation for the number of ways to deposit $n$ dollars in the vending machi......Mon, 24 Aug 2026 22:09:17 -0400https://pop.com.sg/why-is-it-always-necessary-that-a-second-order-differential-equation-h.htmlhttps://pop.com.sg/why-is-it-always-necessary-that-a-second-order-differential-equation-h.htmlI'm an AL student , and i have been recently introduced to the concept of Homogeneous Second Order Linear Differential Equations.I could understand how the Characteristic Equation forms and why it ......Mon, 24 Aug 2026 22:03:50 -0400https://pop.com.sg/find-a-singular-value-decomposition-of-the-matrix.htmlhttps://pop.com.sg/find-a-singular-value-decomposition-of-the-matrix.htmlFina a singular value decomposition for $$A=\begin{bmatrix} -2 & 2 \\ -1 & 1 \\ 2 & -2\end{bmatrix}.$$ Find a (full svd)singular value decomposition for the matrix $A$. My working......Mon, 24 Aug 2026 22:02:40 -0400https://pop.com.sg/tranform-region-of-type-1-into-a-region-of-type-2.htmlhttps://pop.com.sg/tranform-region-of-type-1-into-a-region-of-type-2.htmlWe have the integral of type 1 : $\int_0^1 \left (\int_{e^x}^e f(x,y)dy\right )dx$ and we want to transform it into an integral of type 2. The region of type 1 is $D_1=\{(x,y) \mid 0\leq x\leq 1,......Mon, 24 Aug 2026 21:57:08 -0400https://pop.com.sg/if-a-plane-is-divided-by-n-lines-then-it-is-possible-to-color-the-regi.htmlhttps://pop.com.sg/if-a-plane-is-divided-by-n-lines-then-it-is-possible-to-color-the-regi.htmlI am self-studying Discrete Mathematics, and there is the following exercise. (in Portuguese) A plane is divided by many lines. Show that it is possible to color the regions formed with only two ...Mon, 24 Aug 2026 21:51:30 -0400https://pop.com.sg/how-do-you-prove-the-domain-of-a-function.htmlhttps://pop.com.sg/how-do-you-prove-the-domain-of-a-function.htmlEdit: Man, I actually had a bachelor's completed when I asked this dumb question, like even more than the terry tao dumb thing. Don't judge me! But to be fair even when I was in calculus we're always ...Mon, 24 Aug 2026 21:42:47 -0400https://pop.com.sg/how-to-simplify-frac-1-sqrt-3-3-1-frac-2-sqrt-3-3-1-to-sqrt-3-3.htmlhttps://pop.com.sg/how-to-simplify-frac-1-sqrt-3-3-1-frac-2-sqrt-3-3-1-to-sqrt-3-3.html$$\frac{1}{\sqrt[3]{3}-1} - \frac{2}{\sqrt[3]{3}+1}$$ I have simplified above to: $$\frac{3-\sqrt[3]{3}}{(\sqrt[3]{3}+1)(\sqrt[3]{3}-1)}$$ What is equal to: $$\frac{3-\sqrt[3]{3}}{\sqrt[3]{9}-1}$$ ...Mon, 24 Aug 2026 21:42:45 -0400https://pop.com.sg/how-is-it-that-tan-a-b-frac-tan-a-tan-b-1-tan-a-tan-b-for-all-angles-e.htmlhttps://pop.com.sg/how-is-it-that-tan-a-b-frac-tan-a-tan-b-1-tan-a-tan-b-for-all-angles-e.htmlHow is it that $$\tan(A +B) = \frac{\tan A + \tan B}{1-\tan A\tan B}$$ for any value of $A$, $B$? I have doubts about this since we arrive at this by dividing the numerator and denominator of $$\......Mon, 24 Aug 2026 21:40:15 -0400https://pop.com.sg/derivative-of-x-x-without-logarithmic-differentiation.htmlhttps://pop.com.sg/derivative-of-x-x-without-logarithmic-differentiation.htmlWith logarithmic differentiation, it is quite simple to compute the derivative of $x^x$: $$y=x^x$$ $$\ln {y} =x \ln{x}$$ $$\frac {1}{y} \frac {dy}{dx} = \ln{x} +1$$ $$\frac {dy}{dx} =\l......Mon, 24 Aug 2026 21:39:43 -0400https://pop.com.sg/true-definition-of-the-real-numbers.htmlhttps://pop.com.sg/true-definition-of-the-real-numbers.htmlI've found lots of resources that say this is a real number if it's not rational, but what is a real number, really? I mean what is the definition of a real number? If nothing else, anyone know o......Mon, 24 Aug 2026 21:22:14 -0400https://pop.com.sg/number-of-ways-to-draw-a-pair-in-a-poker-deck.htmlhttps://pop.com.sg/number-of-ways-to-draw-a-pair-in-a-poker-deck.htmlI am confused over calculating the number of ways in which I can select a pair out of a deck of 52 cards. This is how I go about solving the problem: Following the definition of a pair in card ga......Mon, 24 Aug 2026 21:19:07 -0400https://pop.com.sg/mathematical-operator-to-flip-a-vector.htmlhttps://pop.com.sg/mathematical-operator-to-flip-a-vector.htmlIs there a mathematical operator which flips a vector from left to right (or up to down). Say \begin{align} a = [1~ 2~ 3]\quad\text{and}\quad b = [3~ 2~ 1] \end{align} I'd like to have \begin{align......Mon, 24 Aug 2026 21:16:13 -0400https://pop.com.sg/finding-best-response-function.htmlhttps://pop.com.sg/finding-best-response-function.htmlConsider two players Player 1 choose $x_1$ Player 2 choose $x_2$ $x_1, x_2 \in [0,b]$ and $b>0$ The payoffs for players 1 and 2 are identical and equal $$Z=u(x_1+x_2)+v(2b-x_1-x_2)$$ $u$......Mon, 24 Aug 2026 21:08:01 -0400https://pop.com.sg/lambert-w-product-log-function.htmlhttps://pop.com.sg/lambert-w-product-log-function.htmlI would like to solve this equation: $$n \cdot 2^n = 15000$$ And according to WolframAlpha $$n=\frac{W(15000\log(2))}{\log(2)}, \text{ where }\log\text{ is ln}$$ Which shows that I need to use ......Mon, 24 Aug 2026 21:07:51 -0400https://pop.com.sg/what-does-expected-value-of-sum-of-two-discrete-random-variables-mean.htmlhttps://pop.com.sg/what-does-expected-value-of-sum-of-two-discrete-random-variables-mean.htmlI am confused with summing two random variables. Suppose $X$ and $Y$ are two random variables denoting how much is gained from each two games. If two games are played together, we can gain $E[X] + ......Mon, 24 Aug 2026 21:04:32 -0400https://pop.com.sg/taylor-series-third-order-approximation.htmlhttps://pop.com.sg/taylor-series-third-order-approximation.htmlThere has been this question that had been bothering me for a while and I could not find a satisfying answer on the internet or any of the books even though it is not very complex. i) Its because ......Mon, 24 Aug 2026 20:57:50 -0400https://pop.com.sg/even-digit-series-2-4-6-8-20-duplicate.htmlhttps://pop.com.sg/even-digit-series-2-4-6-8-20-duplicate.htmlI have a series of numbers whose Nth term is a number whose all of the digits are even. The series is of course divergent , but i am interested in finding a formula to find the Nth term of this ser......Mon, 24 Aug 2026 20:57:11 -0400https://pop.com.sg/calculating-an-ellipse-given-the-orbital-eccentricity-and-a-vertex.htmlhttps://pop.com.sg/calculating-an-ellipse-given-the-orbital-eccentricity-and-a-vertex.htmlI know that the formula for Eccentricity is e = c/a where c is the distance from the center to a focus and a is the distance from that focus to a vertex. I know the distance from the center of the ...Mon, 24 Aug 2026 20:50:15 -0400https://pop.com.sg/lie-algebra-of-lorentz-group-o-1-3.htmlhttps://pop.com.sg/lie-algebra-of-lorentz-group-o-1-3.htmlLet $$ O(1,3)=\{A\in GL_4(\mathbb R):A^TgA=g\} $$ where $g$ is the diagonal matrix with $1$ on the first diagonal entry, and $-1$ on the other diagonal entries. I want to show that the Lie algebra ...Mon, 24 Aug 2026 20:13:31 -0400https://pop.com.sg/construct-an-a4-paper.htmlhttps://pop.com.sg/construct-an-a4-paper.htmlThe A4 European paper format is designed such that if you cut in halve the longest side you would obtain two papers where the ratio of the longest side of each is the same as the original paper. I......Mon, 24 Aug 2026 20:07:15 -0400https://pop.com.sg/what-is-a-compact-element.htmlhttps://pop.com.sg/what-is-a-compact-element.htmlI was reading the definition of an Algebraic Lattice: "An algebraic lattice is a complete lattice L, such that every element x of L is the supremum of the compact elements below x". Then I looked f......Mon, 24 Aug 2026 20:02:41 -0400https://pop.com.sg/given-a-3-times-3-matrix-a-with-det-a-frac-1-8-find-det-3a-det-6a-1.htmlhttps://pop.com.sg/given-a-3-times-3-matrix-a-with-det-a-frac-1-8-find-det-3a-det-6a-1.htmlHow do I go about solving this? The $\det(3A)$ is not simply $3\cdot\frac{1}{8}$ correct?...Mon, 24 Aug 2026 20:01:45 -0400https://pop.com.sg/on-the-commutative-properties-of-subtraction-and-division.htmlhttps://pop.com.sg/on-the-commutative-properties-of-subtraction-and-division.htmlCan someone explain to me why subtraction and division is both commutative? The reason I believe that they are commutative are as follows: $$ 3-2 = 3+(-2) $$ $$ 3+(-2) = (-2)+(3) $$ I've read a w......Mon, 24 Aug 2026 19:55:39 -0400https://pop.com.sg/how-to-handle-optimization-problems-when-optimization-variable-is-matr.htmlhttps://pop.com.sg/how-to-handle-optimization-problems-when-optimization-variable-is-matr.htmlSuppose we have the following optimization problem $$ \min_{0\preceq M \preceq I} y^TMy $$ where $y \in \mathbb{R}^n$ and $M \in \mathbb{R}^{n \times n}$ is a positive semi-definite matrix. Notice......Mon, 24 Aug 2026 19:53:02 -0400https://pop.com.sg/what-functions-can-be-represented-as-power-series.htmlhttps://pop.com.sg/what-functions-can-be-represented-as-power-series.htmlHow do we know if a particular function can be represented as a power series? And once we have come up with a power series representation, how does one figure out its radius of convergence ?...Mon, 24 Aug 2026 19:43:11 -0400https://pop.com.sg/find-fourth-side-of-quadrilateral-given-three-sides-and-two-angles.htmlhttps://pop.com.sg/find-fourth-side-of-quadrilateral-given-three-sides-and-two-angles.htmlTo completely determine a quadrilateral, you have to have five independent pieces of information, of sides and angles. If you have five data (all outer sides and a diagonal), finding the angles is ......Mon, 24 Aug 2026 19:41:20 -0400https://pop.com.sg/relation-between-inradius-and-exradii-of-a-triangle-where-did-my-proof.htmlhttps://pop.com.sg/relation-between-inradius-and-exradii-of-a-triangle-where-did-my-proof.htmlI am given a triangle $\triangle ABC$ with side lengths $AB=c$, $AC=b$, $BC=a$. $r$ is the radius of the incircle, and $r_a,r_b,r_c$ are the radii of the excircles. The incircle divides the sides o......Mon, 24 Aug 2026 19:36:54 -0400https://pop.com.sg/how-to-find-correlation-coefficient-given-joint-distribution.htmlhttps://pop.com.sg/how-to-find-correlation-coefficient-given-joint-distribution.htmlRandom variables $X$ and $Y$ follow a joint distribution $$f(x, y) = \left\{ \begin{array}{ll} 2,& 0 < x \leq y < 1,\\ 0,&\text{otherwise}\end{array}\right. $$ Determine the correlat......Mon, 24 Aug 2026 19:22:03 -0400https://pop.com.sg/a-tourist-in-france-wants-to-visit-12-different-cities-find-the-probab.htmlhttps://pop.com.sg/a-tourist-in-france-wants-to-visit-12-different-cities-find-the-probab.htmlThe Question : A tourist in France wants to visit 12 different cities. If the route is randomly selected, what is the probability that she will visit the cities in alphabetical order? I thought ...Mon, 24 Aug 2026 19:20:16 -0400https://pop.com.sg/how-to-simplify-the-integration.htmlhttps://pop.com.sg/how-to-simplify-the-integration.htmlIf any integration is in form $$\int \frac{1}{1+x^2}dx$$ it easily follows the $\tan^{-1}x$ but How to simplify if we have $$\int \frac{1}{(1+x^2)^2}dx$$...Mon, 24 Aug 2026 19:17:33 -0400https://pop.com.sg/constructing-a-matrix-such-its-square-is-not-039-0-039-but-its-cube-is.htmlhttps://pop.com.sg/constructing-a-matrix-such-its-square-is-not-039-0-039-but-its-cube-is.htmli have been asked to construct a matrix A such that $A^2$ is not equal to '0' but, $A^3=0$. how should i proceed. i can only understand that all the eigenvalues for A , $A^2$ and $A^3$ will be 'z......Mon, 24 Aug 2026 19:16:35 -0400https://pop.com.sg/group-theory-notation.htmlhttps://pop.com.sg/group-theory-notation.htmlWhat does the notation $(G,.)$ mean in group theory? I have seen in places that $.$ implies the binary operation multiplication on group $G$. But then, why do we show an abelian group as $(G, +)$? ......Mon, 24 Aug 2026 19:10:42 -0400https://pop.com.sg/usage-of-the-word-formal-ly.htmlhttps://pop.com.sg/usage-of-the-word-formal-ly.htmlThis is weird. To me, in mathematical contexts, "formally" means something like "rigorously", i.e. the opposite of informally/heuristically. And yet, I very often read papers very the word seems t......Mon, 24 Aug 2026 18:59:47 -0400https://pop.com.sg/dfa-union-operation-how-to.htmlhttps://pop.com.sg/dfa-union-operation-how-to.htmlI'm currently looking at deterministic finite automata, and learning how to combine two DFAs using AND or OR. I think I understand how to construct the INTERSECTION (AND) of two DFAs, but I'm at a ......Mon, 24 Aug 2026 18:49:10 -0400https://pop.com.sg/which-of-the-following-statements-are-true-and-which-are-false.htmlhttps://pop.com.sg/which-of-the-following-statements-are-true-and-which-are-false.htmlLet $f$ and $g$ be continuous functions such that $f(x) ≤ g(x)$ for all $x ∈ [0, 1]$. Determine which of the following statements are true and which are false: $$ \begin{align} (a) & {}\quad \......Mon, 24 Aug 2026 18:44:39 -0400https://pop.com.sg/straightforward-logical-techniques-for-solving-matrices.htmlhttps://pop.com.sg/straightforward-logical-techniques-for-solving-matrices.htmlI'm studying for a Linear Algebra exam, and having difficulty with solving matrices in a timely manner. Our professor provided us with several sample exams to practice with, which begin with a matr......Mon, 24 Aug 2026 18:35:38 -0400https://pop.com.sg/determining-the-area-of-a-heart-curve-2d.htmlhttps://pop.com.sg/determining-the-area-of-a-heart-curve-2d.htmlI found out the area of a heart but I am not sure whether it is correct. Can someone please check whether what I have done makes any sense at all. The equation of a heart is shown below: I first ...Mon, 24 Aug 2026 18:27:39 -0400https://pop.com.sg/chromatic-number-of-a-graph-obtained-from-petersen-s-graph.htmlhttps://pop.com.sg/chromatic-number-of-a-graph-obtained-from-petersen-s-graph.htmlI am studying on Chromatic numbers of graphs and I came across Petersen’s graph. Petersen’s graph which is an undirected graph with 10 vertices and 15 edges. And it has the chromatic number 3, mea......Mon, 24 Aug 2026 18:16:21 -0400https://pop.com.sg/determining-the-derivative-of-e-5x-tan-2x.htmlhttps://pop.com.sg/determining-the-derivative-of-e-5x-tan-2x.htmlDetermine the derivative of $e^{5x}\tan(2x)$ I have no idea, what is this question mean- but what I can do is $$\begin{align*}y&=e^{5x}\\ \frac{dy}{dx}&=e^{5x}(5)\\ \frac{dy}{dx}&=5......Mon, 24 Aug 2026 18:07:38 -0400https://pop.com.sg/how-to-calculate-e-x-with-a-standard-calculator.htmlhttps://pop.com.sg/how-to-calculate-e-x-with-a-standard-calculator.htmlIs there a simple method for calculating the $e^x$ ($x\in\mathbb{R}$) with a basic add/subtract/multiply/divide calculator that converges in reasonable time, preferably without having to memorize ...Mon, 24 Aug 2026 18:05:46 -0400https://pop.com.sg/why-is-this-a-valid-proof-for-the-harmonic-series.htmlhttps://pop.com.sg/why-is-this-a-valid-proof-for-the-harmonic-series.htmlThe other day, I encountered the Harmonic series which I though is an interesting concept. There were many ways to prove that it's divergent and the one I really liked was rather simple but did the......Mon, 24 Aug 2026 18:03:57 -0400https://pop.com.sg/number-of-possible-permutations-of-three-pairs-of-socks-one-blue-one-b.htmlhttps://pop.com.sg/number-of-possible-permutations-of-three-pairs-of-socks-one-blue-one-b.htmlThere are 3 colours of socks; blue, black and white and each colour has two pairs of socks and we need to distribute them to six people and the left leg socks are distinguishable from right leg soc......Mon, 24 Aug 2026 17:59:09 -0400https://pop.com.sg/finding-the-major-and-minor-axis-vertices-for-an-ellipse-given-two-con.htmlhttps://pop.com.sg/finding-the-major-and-minor-axis-vertices-for-an-ellipse-given-two-con.htmlI've been googling, searching forums and looking in my old algebra/trig books to try to understand how to find the end points to the major and minor axis of an ellipse given the end points of two ...Mon, 24 Aug 2026 17:54:24 -0400https://pop.com.sg/how-to-calculate-this-limit-without-l-039-hopital-039-s-rule.htmlhttps://pop.com.sg/how-to-calculate-this-limit-without-l-039-hopital-039-s-rule.htmlI know that in the very small values of $x$ $$(1+x)^n ≈1+ nx$$ and I can prove it using Taylor series. But I wanted to prove it without any smell of derivative. So... In order to calculate the ...Mon, 24 Aug 2026 17:46:53 -0400https://pop.com.sg/principia-mathematica-and-axioms.htmlhttps://pop.com.sg/principia-mathematica-and-axioms.htmlI have been trying to understand how mathematicians achieve the truth through axiomatic systems. Surely, if one searches a bit more, it will get to know how axiomatic systems work, generally. After......Mon, 24 Aug 2026 17:41:49 -0400https://pop.com.sg/calculate-expected-value-in-dice-game.htmlhttps://pop.com.sg/calculate-expected-value-in-dice-game.htmlWe play a game in 2 stages: In stage one, we throw a dice until we get number 6, Let N represent the number of times played until we got 6 for the first time. In stage two, we throw N dices (each one ...Mon, 24 Aug 2026 17:37:42 -0400https://pop.com.sg/what-is-the-expected-area-of-a-polygon-whose-vertices-lie-on-a-circle.htmlhttps://pop.com.sg/what-is-the-expected-area-of-a-polygon-whose-vertices-lie-on-a-circle.htmlI came across a nice problem that I would like to share. Problem: What is expected value of the area of an $n$-gon whose vertices lie on a circle of radius $r$? The vertices are uniformly distribu......Mon, 24 Aug 2026 17:36:53 -0400https://pop.com.sg/differentiate-log-gamma-function.htmlhttps://pop.com.sg/differentiate-log-gamma-function.htmlI am working with the log of negative binomial distribution $NegBin(r,p)$. I need to differentiate the following with respect to $r$ such that at the end, I am NOT left with $r$ in factorial form. ......Mon, 24 Aug 2026 17:30:04 -0400https://pop.com.sg/writing-out-some-propositions-using-disjunctions-and-conjunctions.htmlhttps://pop.com.sg/writing-out-some-propositions-using-disjunctions-and-conjunctions.htmlSuppose that the universe of discourse of the atomic formula $P(x,y)$ is $\{1, 2, 3\}$. Write out the following propositions using disjunctions and conjunctions. a) $\exists x \forall y \ P(x,y)$ ...Mon, 24 Aug 2026 17:27:47 -0400https://pop.com.sg/calculate-the-sum-using-fourier-series-of-big-cos-frac-x-2-big.htmlhttps://pop.com.sg/calculate-the-sum-using-fourier-series-of-big-cos-frac-x-2-big.htmlI have been given task to evaluate the sum $\dfrac{(-1)^{n-1}}{4n^2-1}$ using Fourier series for function $|\cos{\frac{x}{2}}|$. I used the interval $(-\pi, +\pi)$ for evaluation of the sum and I n......Mon, 24 Aug 2026 17:19:17 -0400https://pop.com.sg/gradient-of-2-norm-squared.htmlhttps://pop.com.sg/gradient-of-2-norm-squared.htmlCould someone please provide a proof for why the gradient of the squared $2$-norm of $x$ is equal to $2x$? $$\nabla\|x\|_2^2 = 2x$$...Mon, 24 Aug 2026 17:02:39 -0400https://pop.com.sg/calculate-sin9-closed.htmlhttps://pop.com.sg/calculate-sin9-closed.htmlI found this question in the olympiad book. But I could not find the solution. The question is to calculate the following real number: $$\sin{9°}$$...Mon, 24 Aug 2026 16:55:08 -0400https://pop.com.sg/heaviside-step-function-squared.htmlhttps://pop.com.sg/heaviside-step-function-squared.htmlI have a question about the Heaviside step function $\theta(\xi)$, defined by $$\theta(\xi):=\begin{cases} 1, & \xi\geq0\\ 0, & \xi<0 \end{cases}$$ I need to evaluate the square of the ...Mon, 24 Aug 2026 16:47:44 -0400