https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usMon, 17 Aug 2026 18:39:52 -0400https://pop.com.sg/continuous-projections-on-ell-1-with-norm-1.htmlhttps://pop.com.sg/continuous-projections-on-ell-1-with-norm-1.htmlI was trying to find papers and articles about non-contractive continuous projections on $\ell_1(S)$ where $S$ is an arbitrary set. If it is not studied yet, I would like to know results for the ca......Mon, 17 Aug 2026 22:37:08 -0400https://pop.com.sg/understanding-the-unit-circle.htmlhttps://pop.com.sg/understanding-the-unit-circle.htmlI'm going to need to know this (the unit circle, English) for a test, but I'm so horrible with both charts and memorization, the whole thing is failing to stick into my brain. Using non-complicated ...Mon, 17 Aug 2026 22:27:30 -0400https://pop.com.sg/a-different-type-of-infinite-power-tower-function.htmlhttps://pop.com.sg/a-different-type-of-infinite-power-tower-function.htmlThe question is about the following: Let a function be defined such that $$f_n(x) = x \uparrow \uparrow n$$ Where $n$ is a natural number Now, it is reported at many places that the function $$F_......Mon, 17 Aug 2026 22:24:32 -0400https://pop.com.sg/how-to-compute-the-similarity-transformation-matrix.htmlhttps://pop.com.sg/how-to-compute-the-similarity-transformation-matrix.htmlStuck on this question: Let $$A=\begin{pmatrix} 2&1\\ -1&-1 \end{pmatrix}$$$$B=\begin{pmatrix} -2&5\\ -1&3 \end{pmatrix}$$$$C=\begin{pmatrix} 5&2\\ 4&1 \end{pmatrix}$$ ......Mon, 17 Aug 2026 22:23:12 -0400https://pop.com.sg/a-die-is-rolled-and-a-coin-is-flipped-what-is-wrong-with-the-following.htmlhttps://pop.com.sg/a-die-is-rolled-and-a-coin-is-flipped-what-is-wrong-with-the-following.htmlA die is rolled and a coin is flipped. What is wrong with the following reasoning?: Let $A=\{1,3,6\}$ (event from die roll), let $B=\{H\}$ (event of heads) then, $A\cap B=\emptyset$ (since they ha......Mon, 17 Aug 2026 22:15:21 -0400https://pop.com.sg/how-to-calculate-carmichael-function-of-a-large-number.htmlhttps://pop.com.sg/how-to-calculate-carmichael-function-of-a-large-number.htmlHow would I go about calculating $\lambda(49392)$ by hand? I have $\lambda$ defined in a book as $$ \lambda(n) = max{\{\bar{a} | \bar{a} \in Z_n^*\}} \ \cdot \ ord_{Z_n^{*}} (\bar{a}) $$ where $......Mon, 17 Aug 2026 22:14:12 -0400https://pop.com.sg/sequence-vs-series.htmlhttps://pop.com.sg/sequence-vs-series.htmlWhat is the difference between a sequence and a series and how should they be used i.e. give examples of the usage of these terminologies in separate senarios....Mon, 17 Aug 2026 22:04:08 -0400https://pop.com.sg/regular-distributions.htmlhttps://pop.com.sg/regular-distributions.htmlI've somehow a really hard time showing that a distribution in $\mathcal{D'}(\mathbb{R^n})$ is regular. The definition seems straight forward: $ f \in \mathcal{D'}(\mathbb{R^n}) \text{ is regula......Mon, 17 Aug 2026 21:59:58 -0400https://pop.com.sg/equation-of-a-line-tangent-to-circumference.htmlhttps://pop.com.sg/equation-of-a-line-tangent-to-circumference.htmlDiscover the general equation of the tangent line to the circumference $x^2 + y^2 - 2x + 4y + 1 = 0$ by the point $(3,4)$. NO CALCULUS. by the circumference equation i discovered that $C(1, -......Mon, 17 Aug 2026 21:52:28 -0400https://pop.com.sg/calculator-similar-to-desmos-but-for-3-d.htmlhttps://pop.com.sg/calculator-similar-to-desmos-but-for-3-d.htmlIs there a calculator with functionality similar to Desmos but in $3$ dimensions? I am looking to learn about families of quadric surfaces so I am looking for a $3$D calculator with sliders....Mon, 17 Aug 2026 21:51:59 -0400https://pop.com.sg/differentiability-of-the-cantor-function.htmlhttps://pop.com.sg/differentiability-of-the-cantor-function.htmlI know that the Cantor function is differentiable a.e. but I want to prove it without using the theorem about monotonic functions. I have already proved that $f'(x) = 0$ for all $x \in [0,1] \backs......Mon, 17 Aug 2026 21:49:59 -0400https://pop.com.sg/chernoff-bound-probability-value-of-n-so-that-with-probability-999-at-.htmlhttps://pop.com.sg/chernoff-bound-probability-value-of-n-so-that-with-probability-999-at-.htmlProblem 3: Consider a biased coin with probability $p = \frac{1}{3}$ of landing heads and probability $\frac{2}{3}$ of landing tails. Suppose the coin is flipped some number $n$ of times, and let $......Mon, 17 Aug 2026 21:37:11 -0400https://pop.com.sg/standard-notation-for-writing-probability-fraction-vs-percentage.htmlhttps://pop.com.sg/standard-notation-for-writing-probability-fraction-vs-percentage.htmlI have a question about the usage of fractions and percentages in questions related to probability. I am wondering if a fraction is the same as a percentage. To better explain my question, I will p......Mon, 17 Aug 2026 21:27:08 -0400https://pop.com.sg/are-functions-of-independent-variables-also-independent.htmlhttps://pop.com.sg/are-functions-of-independent-variables-also-independent.htmlIt's a really simple question. However I didn't see it in books and I tried to find the answer on the web but failed. If I have two independent random variables, $X_1$ and $X_2$, then I define two......Mon, 17 Aug 2026 21:20:51 -0400https://pop.com.sg/proof-difference-of-absolute-values-and-absolute-value-difference-dupl.htmlhttps://pop.com.sg/proof-difference-of-absolute-values-and-absolute-value-difference-dupl.htmlprove that for all $a,b \in \mathbb{R}$, $ | |a|-|b|| ≤ |a-b|$ so what I have thought of so fat is I tried to use the definition of absolute value then getting $b-a ≤ |a|-|b| ≤ a-b.$ from there I'm ...Mon, 17 Aug 2026 21:20:39 -0400https://pop.com.sg/understanding-how-em-algorithm-actually-works-for-missing-data.htmlhttps://pop.com.sg/understanding-how-em-algorithm-actually-works-for-missing-data.htmlI am currently studying EM algorithm for handling missing data in a data set. I understand that the final goal of EM algorithm is not to impute data, but to calculate the parameters of interest. Ho......Mon, 17 Aug 2026 21:06:35 -0400https://pop.com.sg/find-ln-0-2-accurate-to-0-001.htmlhttps://pop.com.sg/find-ln-0-2-accurate-to-0-001.htmlI want to find ln(0.2) accurate to 0.001. I get ln(1+(-0.8)) to solve the question. When I find the remainder I get the below one. So I should find n for find ln(0.2). I find n=23 using a calculat......Mon, 17 Aug 2026 20:52:38 -0400https://pop.com.sg/finding-prime-factors-by-taking-the-square-root.htmlhttps://pop.com.sg/finding-prime-factors-by-taking-the-square-root.htmlI'm trying to solve the third Project Euler problem and I'd like a little help understanding a mathematical concept underlying my tentative solution. The question reads: The prime factors of 13......Mon, 17 Aug 2026 20:44:47 -0400https://pop.com.sg/derivative-of-5x-2-3.htmlhttps://pop.com.sg/derivative-of-5x-2-3.htmlHow is the derivative of $(5x-2)^3$ equal to $15(5x-2)^2$ and not $3(5x-2)^2$. According to $\frac{df}{dx} = nx^{n-1}$, it has to be $3(5x-2)^2$ right. Please explain....Mon, 17 Aug 2026 20:42:47 -0400https://pop.com.sg/arrival-probability-of-merged-poisson-process-closed.htmlhttps://pop.com.sg/arrival-probability-of-merged-poisson-process-closed.htmlIf we have two Poisson process $A$ and $B$ with arrival rate $\lambda_{1}$ and $\lambda_{2}$. Now I have question for calculating the probability of an arrival e.g., from process A in the case of m......Mon, 17 Aug 2026 20:36:13 -0400https://pop.com.sg/questions-tagged-recurrence-relations.htmlhttps://pop.com.sg/questions-tagged-recurrence-relations.htmlQ&A for people studying math at any level and professionals in related fields...Mon, 17 Aug 2026 20:27:14 -0400https://pop.com.sg/probability-of-three-events-occurring-given-correlation.htmlhttps://pop.com.sg/probability-of-three-events-occurring-given-correlation.htmlI am facing a problem that I cannot find the answer to. I have three variables, A, B and C. There are only two possibilities for each of these, A either happens or it does not, B happens or it does......Mon, 17 Aug 2026 20:17:41 -0400https://pop.com.sg/find-a-function-from-values.htmlhttps://pop.com.sg/find-a-function-from-values.htmlIs there any way to find a function, even just similar, from a set of values? I get these pairs of values from two sensors and would like to find a simple function that describes the relationship ...Mon, 17 Aug 2026 20:10:30 -0400https://pop.com.sg/laplacian-in-polar-coordinates-nabla-2f-r-f-039-039-r-frac-2-r-f-039-r.htmlhttps://pop.com.sg/laplacian-in-polar-coordinates-nabla-2f-r-f-039-039-r-frac-2-r-f-039-r.html$\vec{R}=x \hat{i} + y\hat{j} + z\hat{k}$ and $r=|\vec{R}|=\sqrt{x^2+y^2+z^2}$ Prove that $\nabla^2f(r)=f''(r)+\frac{2}{r}f'(r)$ So we need to basically show that $$\frac{\partial^2}{\parti......Mon, 17 Aug 2026 20:07:28 -0400https://pop.com.sg/how-to-find-the-matrix-of-a-linear-transformation-with-respect-to-two-.htmlhttps://pop.com.sg/how-to-find-the-matrix-of-a-linear-transformation-with-respect-to-two-.htmlI'm having some trouble understanding the process of actually finding what $[T]_\beta ^\gamma$ is, given $2$ bases $\beta$ and $\gamma$. Here's an example: Let $T$: $P_3(\mathbb{R})$ $\rightarro......Mon, 17 Aug 2026 20:00:08 -0400https://pop.com.sg/geometric-interpretation-of-the-cofactor-expansion-theorem.htmlhttps://pop.com.sg/geometric-interpretation-of-the-cofactor-expansion-theorem.htmlI find the geometric interpretation of determinants to be really intuitive - they are the "area" created by the column vectors of the matrix. Could someone give me a geometric interpretation of the ...Mon, 17 Aug 2026 19:51:03 -0400https://pop.com.sg/matrix-of-a-quadratic-form.htmlhttps://pop.com.sg/matrix-of-a-quadratic-form.htmlWhat exactly is the matrix of a quadratic form? I have seen this notation occuring in a few papers (e.g. Siegel's unreadable German papers), with particular reference to the trace of a quadratic fo......Mon, 17 Aug 2026 19:49:30 -0400https://pop.com.sg/horizontal-tank-with-hemispherical-ends-depth-to-capacity-calculation.htmlhttps://pop.com.sg/horizontal-tank-with-hemispherical-ends-depth-to-capacity-calculation.htmlI am trying to find an accurate way of calculating the capacity of an underground tank at a given depth. The tank manufacturer has provided a strapping table for the tank which tells me the capaci......Mon, 17 Aug 2026 19:46:18 -0400https://pop.com.sg/what-is-the-difference-between-the-big-o-and-big-o-star-asterisk-opera.htmlhttps://pop.com.sg/what-is-the-difference-between-the-big-o-and-big-o-star-asterisk-opera.htmlI'm doing some research on algorithms complexity and in different papers I notice both the use of the regular Big-O operator O(...) and a variant O*(...). I never saw the definition of the latter ......Mon, 17 Aug 2026 19:42:32 -0400https://pop.com.sg/about-a-divergence-theorem-proof.htmlhttps://pop.com.sg/about-a-divergence-theorem-proof.htmlI'm currently working through a proof of the divergence theorem for simple volumes. I've found some steps of the proof not formal enough, and I have not seen anything online that was of help. Here ......Mon, 17 Aug 2026 19:38:04 -0400https://pop.com.sg/what-is-the-remainder-of-65-divided-by-67.htmlhttps://pop.com.sg/what-is-the-remainder-of-65-divided-by-67.htmlWhat is the remainder of $65!$ divided by $67$? Attempt: By Wilson's theorem, we have $66! = -1\mod(67) $. $$66! = -1\mod(67) \implies 66 (65!) = -1 \mod(67)$$ and we also know that $66 = -1 \m......Mon, 17 Aug 2026 19:31:49 -0400https://pop.com.sg/proving-the-curl-of-a-gradient-is-zero.htmlhttps://pop.com.sg/proving-the-curl-of-a-gradient-is-zero.htmlI'm having trouble proving $$\nabla\times(\nabla f)=0$$ using index notation. I have started with: $$(\hat{e_i}\partial_i)\times(\hat{e_j}\partial_j f)=\partial_i\partial_jf(\hat{e_i}\times\hat{e_j......Mon, 17 Aug 2026 19:30:32 -0400https://pop.com.sg/4th-order-tensors-double-dot-product-and-inverse-computation.htmlhttps://pop.com.sg/4th-order-tensors-double-dot-product-and-inverse-computation.htmlI am currently working on a subject that involves a lot of 4th order tensors computations including double dot product and inverse of fourth order tensors. First the definitions so that we are on the ...Mon, 17 Aug 2026 19:28:31 -0400https://pop.com.sg/simpler-proof-of-an-integral-representation-of-bessel-function-of-the-.htmlhttps://pop.com.sg/simpler-proof-of-an-integral-representation-of-bessel-function-of-the-.htmlWhile doing research in electrical engineering, I derived the following integral representation of the Bessel function of the first kind: $$J_n(x)=\frac{e^{in\pi/2}}{2\pi}\int_0^{2\pi}e^{i(n\tau-x......Mon, 17 Aug 2026 19:27:11 -0400https://pop.com.sg/lim-a-b-when-lim-b-does-not-exist-closed.htmlhttps://pop.com.sg/lim-a-b-when-lim-b-does-not-exist-closed.htmlSuppose $a$ and $b$ are functions of $x$. Is it guaranteed that $$ \lim_{x \to +\infty} a + b\text{ does not exist} $$ when $$ \lim_{x \to +\infty} a = c\quad\text{and}\quad \lim_{x \to +\infty......Mon, 17 Aug 2026 19:25:13 -0400https://pop.com.sg/how-to-convert-sum-into-formula.htmlhttps://pop.com.sg/how-to-convert-sum-into-formula.htmlI am reading some combinatorics books. And here author first obtained a sum answer for a problem and then converted it to formula without explaining it. Just writing equality sign. This is that ...Mon, 17 Aug 2026 19:09:56 -0400https://pop.com.sg/how-do-i-calculate-the-area-under-a-curve-using-the-midpoints-of-recta.htmlhttps://pop.com.sg/how-do-i-calculate-the-area-under-a-curve-using-the-midpoints-of-recta.htmlI figured out how to calculate the area under the curve from the Right endpoint and Left endpoints, but I can't figure out how to calculate it using the midpoints. Especially when it says $M_3$. Il......Mon, 17 Aug 2026 19:06:56 -0400https://pop.com.sg/the-best-strategy-and-highest-possible-score-for-the-threes-game.htmlhttps://pop.com.sg/the-best-strategy-and-highest-possible-score-for-the-threes-game.html[There's still the strategy to go. A suitably robust argument that establishes what is statistically the best strategy will be accepted.] Here's my description of the game: There's a $4\times 4$ grid ...Mon, 17 Aug 2026 19:06:41 -0400https://pop.com.sg/generalized-regular-polygon-area-formula-to-any-number-of-sides.htmlhttps://pop.com.sg/generalized-regular-polygon-area-formula-to-any-number-of-sides.htmlI was looking for a way to check if the regular polygon area formula of $\frac{ns^2}{4} \cot \frac{π}{n}$ where s is the side length and n is the number of sides could be extended to fractional and ...Mon, 17 Aug 2026 19:06:16 -0400https://pop.com.sg/creating-a-steady-state-vector.htmlhttps://pop.com.sg/creating-a-steady-state-vector.htmlI'm confused on where the intuition came from to divide $w$ by the sum of its entries to find $q$. I don't really see the relation from the sum of its entries with "every solution being a multiple ......Mon, 17 Aug 2026 19:05:03 -0400https://pop.com.sg/if-n-is-a-positive-integer-then-2-n-2-2-n-2-2-2n-1.htmlhttps://pop.com.sg/if-n-is-a-positive-integer-then-2-n-2-2-n-2-2-2n-1.htmlI'm not sure why $$(-2^n)^{-2} + (2^{-n})^2=2^{-2n+1}$$ I have been going over this equation for a while now, noticing, and have successfully got quite far in the equation, finding that $$ (-2^n......Mon, 17 Aug 2026 19:03:05 -0400https://pop.com.sg/if-you-have-100-spheres-packed-into-a-sphere-shape-how-many-will-be-on.htmlhttps://pop.com.sg/if-you-have-100-spheres-packed-into-a-sphere-shape-how-many-will-be-on.htmlMy question is more about ratios. I'm wondering is there a calculator or formula I can input an X number of Spheres. If the spheres are packed in a spherical shape what is the ratio of The interior ...Mon, 17 Aug 2026 19:00:37 -0400https://pop.com.sg/wolframalpha-says-limit-exists-when-it-doesn-039-t.htmlhttps://pop.com.sg/wolframalpha-says-limit-exists-when-it-doesn-039-t.htmlI was trying to calculate the following limit: $$ \lim_{(x,y)\to (0,0)} \frac{(x^2+y^2)^2}{x^2+y^4} $$ and, feeding it into WolframAlpha, I obtain the following answer, stating the limit is $0$: ...Mon, 17 Aug 2026 19:00:18 -0400https://pop.com.sg/why-should-kids-learn-how-to-use-a-compass-and-straightedge-and-not-re.htmlhttps://pop.com.sg/why-should-kids-learn-how-to-use-a-compass-and-straightedge-and-not-re.htmlI am curious why it is necessary for people to learn how to use compasses and straightedges in geometry, and not just rely on a drawing program. I have a couple ideas, but I might be missing somet......Mon, 17 Aug 2026 18:53:07 -0400https://pop.com.sg/how-can-i-solve-cos-2-x-sin-x-1-0.htmlhttps://pop.com.sg/how-can-i-solve-cos-2-x-sin-x-1-0.htmlThe solution set of the equation $$\cos^2 x + \sin x +1 = 0$$ is? I haven't studied trigonometry, I'm kinda lost on this issue ......Mon, 17 Aug 2026 18:49:20 -0400https://pop.com.sg/given-graph-find-degree.htmlhttps://pop.com.sg/given-graph-find-degree.htmlGiven the graph I'm asked for the degree. I can see how it is odd. I can see how the $a$ value is negative. I can see the inflection point at $0$. My question is how would I know if this is degree......Mon, 17 Aug 2026 18:33:12 -0400https://pop.com.sg/example-of-bijective-function-from-1-1-to-bbb-r.htmlhttps://pop.com.sg/example-of-bijective-function-from-1-1-to-bbb-r.htmlI'm looking for a bijective function $f : (-1,1) \to \mathbb{R}$. I'm having trouble finding an example....Mon, 17 Aug 2026 18:25:21 -0400https://pop.com.sg/h-cartan-differential-calculus-query.htmlhttps://pop.com.sg/h-cartan-differential-calculus-query.htmlIn H. Cartan - Differential Calculus (1971) p. 29 he investigates differentiating a bi-linear function $f: E_1 \times E_2 \to F$ where $E_1, E_2, F$ are Banach spaces and $E_1 \times E_2$ the produ......Mon, 17 Aug 2026 18:25:13 -0400https://pop.com.sg/find-area-enclosed-by-curve.htmlhttps://pop.com.sg/find-area-enclosed-by-curve.htmlI want to find the area enclosed by the plane curve $x^{2/3}+y^{2/3}=1$. My attempt was to set $x=\cos^3t, \ y=\sin^3t$ so:$$x^{2/3}+y^{2/3}=\cos^2t+\sin^2t=1$$ Then the area is $$2A=\oint_Cxdy-yd......Mon, 17 Aug 2026 18:24:54 -0400https://pop.com.sg/could-a-square-be-a-perfect-number.htmlhttps://pop.com.sg/could-a-square-be-a-perfect-number.htmlA perfect number is the sum of its (positive) divisors (excluding itself). I am wondering if a square could be a perfect number. If it is an odd square, then, excluding itself, it has an even num......Mon, 17 Aug 2026 18:16:32 -0400https://pop.com.sg/decompose-a-simple-function-into-the-sum-of-two-simple-functions.htmlhttps://pop.com.sg/decompose-a-simple-function-into-the-sum-of-two-simple-functions.htmlLet $(\Omega, \mathscr F)$ be a measurable space and let $f,g : \Omega \to \mathbb R$ be non-negative Borel functions. Suppose that $\psi$ is a non-negative simple function, i.e. $\psi = \sum_{i=1}......Mon, 17 Aug 2026 18:07:18 -0400https://pop.com.sg/why-we-can-039-t-have-radicals-with-negative-index.htmlhttps://pop.com.sg/why-we-can-039-t-have-radicals-with-negative-index.htmlI am a high school student and we just learned about radical and radical notation. Our teacher says index of radical must be integer and greater than 2 by definition. But I can’t understand why we ......Mon, 17 Aug 2026 18:00:32 -0400https://pop.com.sg/useful-examples-of-pathological-functions.htmlhttps://pop.com.sg/useful-examples-of-pathological-functions.htmlWhat are some particularly well-known functions that exhibit pathological behavior at or near at least one value and are particularly useful as examples? For instance, if $f'(a) = b$, then $f(......Mon, 17 Aug 2026 17:56:47 -0400https://pop.com.sg/example-of-integration-by-parts-in-higher-dimension.htmlhttps://pop.com.sg/example-of-integration-by-parts-in-higher-dimension.htmlI'm looking for a concrete example of an application of integration by parts in higher dimensions. The formula I'm looking at is from here, here, and here. $\Omega$ is an open bounded subset of $\...Mon, 17 Aug 2026 17:45:10 -0400https://pop.com.sg/how-to-show-the-fundamental-group-of-torus-is-abelian-in-a-homotopic-w.htmlhttps://pop.com.sg/how-to-show-the-fundamental-group-of-torus-is-abelian-in-a-homotopic-w.htmlI know the torus is homeomorphic to $S^1 \times S^1$ and the fundamental group is $ \mathbb{Z} \times \mathbb{Z} $, but in the real case, (let the generators of the torus's fundamental group be $a$......Mon, 17 Aug 2026 17:40:23 -0400https://pop.com.sg/misunderstanding-the-taylor-remainder-theorem.htmlhttps://pop.com.sg/misunderstanding-the-taylor-remainder-theorem.htmlI believe I am misinterpreting the Taylor Remainder theorem somehow. The Taylor Remainder theorem is (taken from Briggs 3rd ed Calculus: Early Transcendentals) Let $f$ have continuous derivativ......Mon, 17 Aug 2026 17:38:08 -0400https://pop.com.sg/convergent-rearrangement-of-convergent-sequence-has-limit-in-the-seque.htmlhttps://pop.com.sg/convergent-rearrangement-of-convergent-sequence-has-limit-in-the-seque.htmlI'm working on a proof for analysis: "Let $\{x_n\}$ be a convergent sequence in $\mathbb R^d$ with limit $x$. Set $$A=\{x_1,x_2,x_3,\cdots\}\cup\{x\},$$ that is, $A$ is the set consisting of all the ...Mon, 17 Aug 2026 17:27:28 -0400https://pop.com.sg/how-to-define-linear-and-non-linear-differential-equation.htmlhttps://pop.com.sg/how-to-define-linear-and-non-linear-differential-equation.htmlI have a problem understanding how to define a linear or non-linear Differential equation. These are my answers to the questions, however, my teacher's answers are not the same as mine. Questions ......Mon, 17 Aug 2026 17:27:14 -0400https://pop.com.sg/how-to-do-capital-pi-prod-notation-on-a-ti-84-calculator.htmlhttps://pop.com.sg/how-to-do-capital-pi-prod-notation-on-a-ti-84-calculator.htmlI am trying to figure out how to input product notation into my TI-84 Plus graphing calculator. I haven't found anything so far. Is there any function I could use to do this? Maybe even a program I ...Mon, 17 Aug 2026 17:26:01 -0400https://pop.com.sg/solution-for-logistic-growth-equation-with-linearly-growing-carrying-c.htmlhttps://pop.com.sg/solution-for-logistic-growth-equation-with-linearly-growing-carrying-c.htmlDoes someone have a solution for the following equation where the carrying capacity varies linearly with time (and only time): $$\frac{\mathrm dN}{\mathrm dt}= rN\left(1-\frac{N}{K}\right)$$ $K =......Mon, 17 Aug 2026 17:22:35 -0400https://pop.com.sg/natural-number-which-can-be-expressed-as-sum-of-two-perfect-squares-in.htmlhttps://pop.com.sg/natural-number-which-can-be-expressed-as-sum-of-two-perfect-squares-in.htmlRamanujan's number is $1729$ which is the least natural number which can be expressed as the sum of two perfect cubes in two different ways. But can we find a number which can be expressed as the s......Mon, 17 Aug 2026 17:13:00 -0400https://pop.com.sg/what-is-the-difference-between-probabilistic-and-physicists-hermite-po.htmlhttps://pop.com.sg/what-is-the-difference-between-probabilistic-and-physicists-hermite-po.htmlIn the equation they differ only by a fraction of 2 and in the results one is without a constant in x^n and one has those constants. Why are they defined differently and what are their separate uses?...Mon, 17 Aug 2026 17:09:21 -0400https://pop.com.sg/is-there-a-systematic-way-to-find-the-range-of-functions.htmlhttps://pop.com.sg/is-there-a-systematic-way-to-find-the-range-of-functions.htmlI was looking for steps or a systematic way to find the range of functions. The way to find domain is quite obvious: exclude x-values that make the function undefined on the real numbers. But the......Mon, 17 Aug 2026 17:00:42 -0400https://pop.com.sg/what-does-a-b-0-mean-in-matrix-theory.htmlhttps://pop.com.sg/what-does-a-b-0-mean-in-matrix-theory.htmlThis is probably very simple but I can't seem to find an answer anywhere. I'm being asked to prove $e^{tA}e^{tB} = e^{t(A+B)}$ for all $t \in \mathbb{R}$ iff $[A,B]=0$ However, I am unsure what $......Mon, 17 Aug 2026 16:59:28 -0400https://pop.com.sg/correct-notation-to-indicate-multiplying-all-elements-within-a-set.htmlhttps://pop.com.sg/correct-notation-to-indicate-multiplying-all-elements-within-a-set.htmlIs there a correct notation to indicate multiplying all elements within a set? For example, if $M = \left\{n_0, n_1, ..., n_t\right\}$ be the set of elements where I want to multiply all the numbers ...Mon, 17 Aug 2026 16:44:27 -0400https://pop.com.sg/complete-the-square-and-write-in-standard-form-for-9y-2-6y-9-x-0.htmlhttps://pop.com.sg/complete-the-square-and-write-in-standard-form-for-9y-2-6y-9-x-0.htmlComplete the square and write in standard form: $x-a=A(y-b)^2$ $9y^2-6y-9-x=0$ I do not know how to complete the square with $4$ terms. I started off like: $$9y^2-6y-9=x$$ I don't know whether......Mon, 17 Aug 2026 16:43:42 -0400https://pop.com.sg/partial-trace-of-density-operator.htmlhttps://pop.com.sg/partial-trace-of-density-operator.htmlBefore stating my question I present my motivation: to learn more about the tensor product. Now, quantum mechanics assigns a Hilbert space to each physical system as a postulate of the theory. Su......Mon, 17 Aug 2026 16:40:27 -0400https://pop.com.sg/boolean-algebra-and-simplify-boolean-expression.htmlhttps://pop.com.sg/boolean-algebra-and-simplify-boolean-expression.htmlquestion1: from =$ab+ac+a'b'c$ , can I change it to $ab+b'c$ ? =$ab+ac+a'b'c$ =$c.!(a+a'b')+ab$ -> applied demorgan, am I right? =$c(a'.(a+b))+ab$ $c(a'a+a'b)+ab$ $a'bc+ab$ $b(a'c+a)$ ->shou......Mon, 17 Aug 2026 16:33:19 -0400https://pop.com.sg/how-do-we-define-arc-length.htmlhttps://pop.com.sg/how-do-we-define-arc-length.htmlIn trying to write a nice proof of the derivatives of $\sin(x)$ and $\cos(x)$, I encountered a serious problem, namely that I have never seen a proper definition of the notion of arc length. Based on ...Mon, 17 Aug 2026 16:31:45 -0400https://pop.com.sg/histogram-and-normal-distribution.htmlhttps://pop.com.sg/histogram-and-normal-distribution.htmlI was studying histograms and normal distribution. As far as I know, they are two different tools used for calculating probability and statistics. More specifically they help to visualize and it is......Mon, 17 Aug 2026 16:29:32 -0400https://pop.com.sg/cohomology-of-s-1.htmlhttps://pop.com.sg/cohomology-of-s-1.htmlAbove is a way to find the cohomology groups of $S^1$. But what I do not get is how they concluded that $\delta$ is surjective. Can someone explain this to me?...Mon, 17 Aug 2026 16:29:29 -0400https://pop.com.sg/when-n-is-divided-by-14-the-remainder-is-10-what-is-the-remainder-when.htmlhttps://pop.com.sg/when-n-is-divided-by-14-the-remainder-is-10-what-is-the-remainder-when.htmlI need to explain this to someone who hasn't taken a math course for 5 years. She is good with her algebra. This was my attempt: Here's how this question works. To motivate what I'll be doing, ...Mon, 17 Aug 2026 16:26:55 -0400https://pop.com.sg/proving-bi-lipschitz-function-has-an-inverse-that-is-lipschitz.htmlhttps://pop.com.sg/proving-bi-lipschitz-function-has-an-inverse-that-is-lipschitz.htmlHow could I go about proving that a bi-Lipschitz function has an inverse that is a Lipschitz function. Definition 1: bi-Lipschitz function. Given metric spaces $(X,d_X)$, $(Y,d_Y)$, a function $f......Mon, 17 Aug 2026 16:26:18 -0400https://pop.com.sg/how-many-different-number-combinations-are-there-between-1000-and-2999.htmlhttps://pop.com.sg/how-many-different-number-combinations-are-there-between-1000-and-2999.htmlThis is a question I made that I would like to solve. Here are some points: I am referring to combinations, not permutations. Therefore, whilst there are 2000 permutations of numbers between 1000 ......Mon, 17 Aug 2026 16:19:41 -0400https://pop.com.sg/general-solution-of-system-of-equations-with-3-variables.htmlhttps://pop.com.sg/general-solution-of-system-of-equations-with-3-variables.htmlA system of equations is given as x + 4y +2z = 0 3x -2z = 4 3x -3y -4z = 5 The task is to find the general solution of the system. I wrote down the ...Mon, 17 Aug 2026 16:04:36 -0400https://pop.com.sg/calc-3-density-problem.htmlhttps://pop.com.sg/calc-3-density-problem.htmlI wrote down all the information I could for this problem but I'm having a very difficult time starting it. Let $D =$ {$(x,y): x^2 + y^2 \le a^2, y\ge 0$} be a semi-circular lamina of radius $a$ ......Mon, 17 Aug 2026 16:04:01 -0400https://pop.com.sg/conditional-probability-from-cdf.htmlhttps://pop.com.sg/conditional-probability-from-cdf.htmlCan you find a conditional probability value from a CDF? If so, how? $F(x)=[x^2-4; 0<=x<1 [ x; 1<=x<2$ how to find $P[x>1/2 | x<2]$...Mon, 17 Aug 2026 15:54:52 -0400https://pop.com.sg/does-radix-1-exists-duplicate.htmlhttps://pop.com.sg/does-radix-1-exists-duplicate.htmlIntroduction This probably sounds a bit weird to ask, but the question is, is there such a base that is base 1. One thing we know is that this base will contain one symbol only, but will this sym......Mon, 17 Aug 2026 15:52:29 -0400https://pop.com.sg/how-to-factor-the-polynomial-5x-2-2x-10.htmlhttps://pop.com.sg/how-to-factor-the-polynomial-5x-2-2x-10.htmlI want to factor this polynomial without using the quadratic formula. So, as the first step I multiply coefficient of $x^2$ with $-10$. The product obtained is $-50$. However, there are no two fact......Mon, 17 Aug 2026 15:46:42 -0400https://pop.com.sg/general-solution-of-sin-x-1-2.htmlhttps://pop.com.sg/general-solution-of-sin-x-1-2.htmlI have a doubt with regards to the general solution of $\sin(\theta) = -1/2$. I know that the principal solutions are $\frac{7\pi}{6}$ and $\frac{11\pi}{6}$. So my main question is: Should the gen......Mon, 17 Aug 2026 15:40:24 -0400https://pop.com.sg/square-of-a-graph.htmlhttps://pop.com.sg/square-of-a-graph.htmlCould someone tell me what the "square of a graph" $G^2$ is? Thanks....Mon, 17 Aug 2026 15:38:58 -0400https://pop.com.sg/probability-in-a-deck-of-cards-to-have-two-jacks-in-a-row.htmlhttps://pop.com.sg/probability-in-a-deck-of-cards-to-have-two-jacks-in-a-row.htmlIn a deck of $36$ cards ($9$ cards per color, $4$ colors) what is the probability to have $2$ jacks (or more) that follow each other?...Mon, 17 Aug 2026 15:38:39 -0400https://pop.com.sg/linear-decrease.htmlhttps://pop.com.sg/linear-decrease.htmlI have a problem of retained sales from the previous year. For instance, x client sell 200 cases of product the first month, 400 the second month, 300 the third and so on. The next year's sales are ...Mon, 17 Aug 2026 15:34:35 -0400https://pop.com.sg/poisson-kernel-for-the-half-space-and-ball.htmlhttps://pop.com.sg/poisson-kernel-for-the-half-space-and-ball.htmlI have 2 questions from Lawrence Evans' PDE book (pages 36 and 39 in the 2nd edition copy). The first is for green's function for a half space $\mathbb{R}_{+}^{n}=\{x=(x_{1},\cdots,x_{n})\in\mathbb......Mon, 17 Aug 2026 15:27:11 -0400https://pop.com.sg/choosing-h0-and-ha-in-hypothesis-testing.htmlhttps://pop.com.sg/choosing-h0-and-ha-in-hypothesis-testing.htmlThere seems to be some ambiguity or contradiction in how to correctly choose the null and alternative hypotheses, both online and in my instructor's notes. I'm trying to figure out if this stems me......Mon, 17 Aug 2026 15:22:23 -0400https://pop.com.sg/graph-entropy-what-is-it.htmlhttps://pop.com.sg/graph-entropy-what-is-it.htmlI am having trouble getting some intuition as to what graph entropy measures. The definition that I have is that given a graph $G$, $H(G) = \min_{X,Y}I(X ;Y)$, where $X$ is a uniformly random verte......Mon, 17 Aug 2026 15:14:04 -0400https://pop.com.sg/best-strategy-to-pick-a-lock-which-opens-if-at-least-two-of-its-three-.htmlhttps://pop.com.sg/best-strategy-to-pick-a-lock-which-opens-if-at-least-two-of-its-three-.htmlSuppose you want to open a lock with three digits code, the lock is a little special because it can be opened if you have two digits guessed right. To clarify, if the correct three digit is 123, then ...Mon, 17 Aug 2026 15:04:46 -0400https://pop.com.sg/lim-limits-x-to-0-frac-sin-x-x-x-2-without-l-039-hospital-039-s-rule.htmlhttps://pop.com.sg/lim-limits-x-to-0-frac-sin-x-x-x-2-without-l-039-hospital-039-s-rule.html$$\lim_{x \to 0} \frac{\sin x-x}{x^2}$$ I know it's an easy limit since inside the limit approaches $0/0$ as $x$ approaches $0$, we can use L'Hospital's twice to get $$\lim_{x \to 0} -\frac{\sin x......Mon, 17 Aug 2026 14:55:07 -0400https://pop.com.sg/sketching-the-contour-map-of-a-function-with-several-variables.htmlhttps://pop.com.sg/sketching-the-contour-map-of-a-function-with-several-variables.htmlConsider the surface given by the function $f(x,y) = \frac{4}{\sqrt{x^2+y^2-9}}$. Draw the level curves for the values $c = 1$ and $c = 2$. Be sure to label your axes to indicate the scale. Note: W......Mon, 17 Aug 2026 14:47:02 -0400https://pop.com.sg/prove-divergence-of-sum-ln-1-1-n.htmlhttps://pop.com.sg/prove-divergence-of-sum-ln-1-1-n.htmlWe have seen the harmonic series is a divergent series whose terms approach $0$. Show that $$\sum_{n = 1}^\infty \text{ln}\left(1 + \frac{1}{n}\right)$$ is another series with this property. Deno......Mon, 17 Aug 2026 14:46:09 -0400https://pop.com.sg/find-the-general-form-of-the-solution-to-the-system-of-equations-below.htmlhttps://pop.com.sg/find-the-general-form-of-the-solution-to-the-system-of-equations-below.htmlFind the general form of the solution to the system of equations below. \begin{align} 2x_1-x_2+8x_3-x_4&=0\\ -4x_1+3x_2-18x_3+x_4&=0\\ 2x_1+x_2+4x_3-3x_4&=0. \end{align} My attempt: ......Mon, 17 Aug 2026 14:36:28 -0400https://pop.com.sg/anyone-have-a-good-proof-for-the-second-part-of-ftc.htmlhttps://pop.com.sg/anyone-have-a-good-proof-for-the-second-part-of-ftc.htmlDoes anyone have a good proof for the second part of the fundamental theorem of calculus? I haven't been able to find any good videos on it so far so I'd like someone to write it down and I can throw ...Mon, 17 Aug 2026 14:08:59 -0400https://pop.com.sg/lax-wendroff-method-for-linear-advection-stability-analysis.htmlhttps://pop.com.sg/lax-wendroff-method-for-linear-advection-stability-analysis.htmlQuestion 1: Consider the wave equation $$ u_t + c(x) u_x = 0 , $$ where $x\in \Omega \subset \Bbb R$ and $c(x)$ is a function of $x$. (a) Show that the Lax-Wendroff scheme for this PDE is......Mon, 17 Aug 2026 14:02:52 -0400https://pop.com.sg/how-do-i-integrate-e-sqrt-x-using-the-substitution-rule.htmlhttps://pop.com.sg/how-do-i-integrate-e-sqrt-x-using-the-substitution-rule.htmlI've been trying to integrate that function, but it looks like I'm missing something, so can any one please show me how to integrate $e^\sqrt{x}$, in order to correct my procedure. Thanks...Mon, 17 Aug 2026 13:46:38 -0400https://pop.com.sg/given-a-quadrilateral-prove-it-is-a-square.htmlhttps://pop.com.sg/given-a-quadrilateral-prove-it-is-a-square.htmlConsider this game. You have a perfect ruler and a perfect protractor. You are also able to observe perfectly if two lines are parallel. Is it possible to prove that a given quadrilateral......Mon, 17 Aug 2026 13:42:44 -0400https://pop.com.sg/proofs-in-category-theory-monicity-of-composition-of-monic-arrows.htmlhttps://pop.com.sg/proofs-in-category-theory-monicity-of-composition-of-monic-arrows.htmlI've just started working on proofs in category theory and I'm interested in thinking about the structure of a proof in terms of this domain. This means thinking about the proof in terms of arrows ......Mon, 17 Aug 2026 13:33:16 -0400https://pop.com.sg/what-quality-of-a-distribution-describes-the-spikiness-of-its-density-.htmlhttps://pop.com.sg/what-quality-of-a-distribution-describes-the-spikiness-of-its-density-.htmlI'm a programmer, not a math guy, so please answer in English. ;) Suppose I have a multi-modal univariate distribution like: .. . .. ........... .. . ........Mon, 17 Aug 2026 13:31:57 -0400https://pop.com.sg/the-maths-required-for-an-economics-degree.htmlhttps://pop.com.sg/the-maths-required-for-an-economics-degree.htmlI have a degree in computer science and I wanted to do another degree in economics. However, my maths have been weak since high school always scoring slightly above passing rate. During my course of ...Mon, 17 Aug 2026 13:31:44 -0400https://pop.com.sg/function-of-three-variables-hessian-zero.htmlhttps://pop.com.sg/function-of-three-variables-hessian-zero.htmlGiven the function $f : \mathbb{R}^3 \to \mathbb{R}$ defined by $f(x,\,y,\,z) = (x - 1)^4 - (x - y)^4 - (y - z)^4$ it presents a unique critical point of coordinates $(1,\,1,\,1)$. Unfortunately, ......Mon, 17 Aug 2026 13:19:17 -0400https://pop.com.sg/equating-coefficients-meaning-and-example.htmlhttps://pop.com.sg/equating-coefficients-meaning-and-example.htmlI am trying to understand what "Equating the Coefficients " means I am given the following: $$ t_1 \left(t_2+\frac{1}{x+1}\right)+\frac{2 t_2}{x}=t_1^2 b_2'+t_1 \left(2 b_2 t_1'+b_1'\right)+b_1 t......Mon, 17 Aug 2026 13:18:32 -0400