https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usFri, 21 Aug 2026 09:36:48 -0400https://pop.com.sg/what-is-the-inverse-function-of-f-x-x-1-x-2.htmlhttps://pop.com.sg/what-is-the-inverse-function-of-f-x-x-1-x-2.htmlCan you give me a hint for how the inverse function of $f\colon (-1,1)\to \mathbb{R}\colon f(x)=\frac{x}{1-x^2}$ looks? I need to show a homeomorphism!...Fri, 21 Aug 2026 13:27:25 -0400https://pop.com.sg/using-the-alternative-formula-to-find-the-derivative-of-a-function.htmlhttps://pop.com.sg/using-the-alternative-formula-to-find-the-derivative-of-a-function.htmlI'm attempting to find the derivative of the function: $$f(x) = 4x^2+3x+5$$ Using the alternative formula: $$\frac{f(z)-f(x)}{z-x}$$ Here are my steps so far: $$\frac{4z^2+3z+5-(4x^2+3x+5)}{z-x}$$......Fri, 21 Aug 2026 13:12:58 -0400https://pop.com.sg/proving-existence.htmlhttps://pop.com.sg/proving-existence.htmlWhat does an introductory-level proof of existence look like? For example, I have done proofs involving proving uniqueness by assuming there are two arbitrary elements that have the properties I w......Fri, 21 Aug 2026 13:10:43 -0400https://pop.com.sg/how-to-measure-the-volume-of-rock.htmlhttps://pop.com.sg/how-to-measure-the-volume-of-rock.htmlI have a object which is similar to the shape of irregular rock like this I would like to find the volume of this. How to do it? If I have to find the volume, what are the things I would need. e......Fri, 21 Aug 2026 13:09:04 -0400https://pop.com.sg/how-to-prove-that-the-inverse-of-a-persymmetric-matrix-is-also-persymm.htmlhttps://pop.com.sg/how-to-prove-that-the-inverse-of-a-persymmetric-matrix-is-also-persymm.htmlAn exercise in a textbook I'm using to brush up on my linear algebra asks to prove that the inverse of a persymmetric matrix is also persymmetric. I have a colleague's old notes in front of me with a ...Fri, 21 Aug 2026 13:08:31 -0400https://pop.com.sg/is-there-a-domain-larger-than-i-e-a-supserset-of-the-complex-number-do.htmlhttps://pop.com.sg/is-there-a-domain-larger-than-i-e-a-supserset-of-the-complex-number-do.htmlI've been teaching my 10yo son some (for me, anyway) pretty advanced mathematics recently and he stumped me with a question. The background is this. In the domain of natural numbers, addition and ...Fri, 21 Aug 2026 13:06:08 -0400https://pop.com.sg/how-many-different-5-letters-word-can-be-possible-by-using-the-letters.htmlhttps://pop.com.sg/how-many-different-5-letters-word-can-be-possible-by-using-the-letters.htmlSo I took the vowels (a,e,i) permute them 3p3 and then I choose 6c2.Here I get baffled and can't figure out what should I do next. By the way multiply 3p3 and 6c2 comes up with 180 which is not the ...Fri, 21 Aug 2026 13:05:22 -0400https://pop.com.sg/where-does-feigenbaum-039-s-constant-4-6692-originate-closed.htmlhttps://pop.com.sg/where-does-feigenbaum-039-s-constant-4-6692-originate-closed.htmlFeigenbaum discovered a ratio between bifurcations that were found in all known chaotic-dynamic systems, from dripping water faucets to abstract equations on population fluctuations (as elucidated in ...Fri, 21 Aug 2026 13:03:31 -0400https://pop.com.sg/solving-sin-theta-cos-theta-1-in-the-interval-0-circ-leq-theta-leq-360.htmlhttps://pop.com.sg/solving-sin-theta-cos-theta-1-in-the-interval-0-circ-leq-theta-leq-360.htmlSolve in the interval $0^\circ\leq \theta\leq 360^\circ$ the equation $\sin \theta + \cos \theta=1$. I've got the two angles in the interval to be $0^\circ$ and $90^\circ$, it's not an answer I'm ......Fri, 21 Aug 2026 13:01:28 -0400https://pop.com.sg/definition-of-disconnected.htmlhttps://pop.com.sg/definition-of-disconnected.htmlI am confused in definition of disconnected set in topological space $X$. Which of the following definition is valid/correct? 1) a subset $A$ of $X$ is disconnected if $A=V ∪W$ where $U$ and $V$ ......Fri, 21 Aug 2026 12:59:33 -0400https://pop.com.sg/determine-a-matrix-knowing-its-eigenvalues-and-eigenvectors.htmlhttps://pop.com.sg/determine-a-matrix-knowing-its-eigenvalues-and-eigenvectors.htmlI read through similar questions, but I couldn't find an answer to this: How do you determine the symmetric matrix A if you know: $\lambda_1 = 1, \ eigenvector_1 = \pmatrix{1& 0&-1}^T;$ $\...Fri, 21 Aug 2026 12:55:04 -0400https://pop.com.sg/what-is-a-point.htmlhttps://pop.com.sg/what-is-a-point.htmlIn geometry, what is a point? I have seen Euclid's definition and definitions in some text books. Nowhere have I found a complete notion. And then I made a definition out from everything that I know ...Fri, 21 Aug 2026 12:44:08 -0400https://pop.com.sg/find-the-three-digit-prime-number.htmlhttps://pop.com.sg/find-the-three-digit-prime-number.htmlWhat is the largest three-digit prime each of whose digits is a prime? - I believe it is 773, but correct me if I'm wrong....Fri, 21 Aug 2026 12:41:29 -0400https://pop.com.sg/a-question-on-a-orthogonality-of-two-vectors.htmlhttps://pop.com.sg/a-question-on-a-orthogonality-of-two-vectors.htmlI am reading this monograph by Jonathan Richard Shewchuk on conjugate gradient descent method. Call two vector $a$ and $b$ A-orthonomal if $a^TAb=0$, where $A$ is a symmetric positive definite matr......Fri, 21 Aug 2026 12:35:56 -0400https://pop.com.sg/parallel-system-functioning-problem.htmlhttps://pop.com.sg/parallel-system-functioning-problem.htmlI am currently solving the following problem about conditional probability: "A parallel system functions whenever at least one of its components works. Consider a parallel system of n components, ......Fri, 21 Aug 2026 12:28:48 -0400https://pop.com.sg/is-0-infty-indeterminate.htmlhttps://pop.com.sg/is-0-infty-indeterminate.htmlIs a constant raised to the power of infinity indeterminate? I am just curious. Say, for instance, is $0^\infty$ indeterminate? Or is it only 1 raised to the infinity that is?...Fri, 21 Aug 2026 12:18:34 -0400https://pop.com.sg/alternating-series-test-question-if-it-fails-what-does-it-mean.htmlhttps://pop.com.sg/alternating-series-test-question-if-it-fails-what-does-it-mean.htmlSay I have this series: $$\sum_{n=1}^{\infty} (-1)^{n} \frac{3n-1}{2n+1}$$ so $b = \frac{3n-1}{2n+1}$ I can't tell if this is decreasing by looking at it since the numerator is smaller but the ...Fri, 21 Aug 2026 12:13:22 -0400https://pop.com.sg/is-there-a-subset-a-subseteq-mathbb-r-not-having-the-bp-baire-property.htmlhttps://pop.com.sg/is-there-a-subset-a-subseteq-mathbb-r-not-having-the-bp-baire-property.htmlLet $\mathcal{I}$ be the $\sigma$-ideal of meager sets of a topological space $X$. We say, that a set $A\subseteq X$ has the Baire property (BP) iff $A=^*U$ for some set $U\subseteq X$. Here $A=^......Fri, 21 Aug 2026 12:12:48 -0400https://pop.com.sg/how-many-dimensions-does-a-circle-have.htmlhttps://pop.com.sg/how-many-dimensions-does-a-circle-have.htmlIs a circle just a line (therefore 1 dimension) or is it a 2-dimensional object because it occupies some surface? Thanks in advance!...Fri, 21 Aug 2026 11:58:36 -0400https://pop.com.sg/use-part-1-of-the-fundamental-theorem-of-calculus-to-find-the-derivati.htmlhttps://pop.com.sg/use-part-1-of-the-fundamental-theorem-of-calculus-to-find-the-derivati.html$$y = \int_\sqrt x^{\pi/4} \theta \tan\theta \, d\theta$$ I'm using the property of definite integrals that says $\int_b^a f(x) \, dx = -\int_a^b f(x) \, dx$ and I'm getting $y'= -\sqrt x \tan\s......Fri, 21 Aug 2026 11:57:19 -0400https://pop.com.sg/can-rational-numbers-have-decimals.htmlhttps://pop.com.sg/can-rational-numbers-have-decimals.htmlI had a question in my exam paper - Which of the following is not a rational number? a) $\sqrt{25}$ b) $\sqrt{45}$ c) $\sqrt\frac{256}{225}$ d) $\frac{3}{4}$ The answer to this is b. Now, $\s......Fri, 21 Aug 2026 11:53:22 -0400https://pop.com.sg/what-039-s-the-value-of-x-in-the-following-equation.htmlhttps://pop.com.sg/what-039-s-the-value-of-x-in-the-following-equation.htmlSo this is how I approached this question, the above equations could be simplified to : $$a = \frac{4(b+c)}{b+c+4}\tag{1!}$$ $$b = \frac{10(a+c)}{a+c+10}\tag{2}$$ $$c=\frac{56(a+b)}{a+b+56}\tag{......Fri, 21 Aug 2026 11:45:02 -0400https://pop.com.sg/why-is-i-3-the-complex-number-i-equal-to-i-instead-of-i-duplicate.htmlhttps://pop.com.sg/why-is-i-3-the-complex-number-i-equal-to-i-instead-of-i-duplicate.html$$i^3=iii=\sqrt{-1}\sqrt{-1}\sqrt{-1}=\sqrt{(-1)(-1)(-1)}=\sqrt{-1}=i $$ Please take a look at the equation above. What am I doing wrong to understand $i^3 = i$, not $-i$?...Fri, 21 Aug 2026 11:35:34 -0400https://pop.com.sg/probability-of-a-dice-game-win-lose-or-draw.htmlhttps://pop.com.sg/probability-of-a-dice-game-win-lose-or-draw.html$C$ and $J$ play a game. $C$ always starts. $C$ rolls a fair dice first and wins if he throws an even number. If not, then $J$ rolls the dice. If she rolls an odd number she wins. if neither win it......Fri, 21 Aug 2026 11:30:54 -0400https://pop.com.sg/topology-without-tears.htmlhttps://pop.com.sg/topology-without-tears.htmlI have started reading the book Topology Without Tears by Sidney A. Morris. I have read the first chapter and so far it reads well. However, the name of the book is a bit deceiving and makes me thi......Fri, 21 Aug 2026 11:28:29 -0400https://pop.com.sg/what-is-the-usage-of-bernoulli-triangle.htmlhttps://pop.com.sg/what-is-the-usage-of-bernoulli-triangle.htmlI know that Pascal triangle is very usfeul in many fields of mathematics. I typed in google Bernoulli triangle and couldn't find any usage of it, even on english wikipedia there is not a lot about ......Fri, 21 Aug 2026 11:20:00 -0400https://pop.com.sg/how-to-find-a-transformation-matrix-given-coordinates-of-two-triangles.htmlhttps://pop.com.sg/how-to-find-a-transformation-matrix-given-coordinates-of-two-triangles.htmlI am an undergraduate student, and today I was given two triangles, $T_1$ (green) and $T_2$ (blue) in $R^2$: I was then asked to find the transformation matrix transforming $T_1$ to $T_2$. What I ...Fri, 21 Aug 2026 11:19:19 -0400https://pop.com.sg/function-for-a-hexagonal-2-dimensional-grid.htmlhttps://pop.com.sg/function-for-a-hexagonal-2-dimensional-grid.htmlI can make a "square" $2$D sine wave pattern with the function $f(x,y) = \cos(x) + \cos(y)$: What is the function to make a hexagonal pattern of the same kind?...Fri, 21 Aug 2026 11:19:10 -0400https://pop.com.sg/why-does-the-boundary-of-the-mandelbrot-set-contain-a-cardioid.htmlhttps://pop.com.sg/why-does-the-boundary-of-the-mandelbrot-set-contain-a-cardioid.htmlIn a comment to a previous answer it has been mentioned that the boundary of the Mandelbrot set contains the cardioid $$ c = e^{it} \, \frac{2 - e^{it}}{4} $$ but how can we prove this?...Fri, 21 Aug 2026 10:59:49 -0400https://pop.com.sg/endpoints-of-interval-where-a-functions-increasing-or-decreasing.htmlhttps://pop.com.sg/endpoints-of-interval-where-a-functions-increasing-or-decreasing.htmlWhen determining the intervals in which the graph of a function increase or decrease, some books include the ends while others do not. Which one is true or which one is more preferable?...Fri, 21 Aug 2026 10:59:18 -0400https://pop.com.sg/what-does-the-notation-pi-mean-in-probability-and-statistics.htmlhttps://pop.com.sg/what-does-the-notation-pi-mean-in-probability-and-statistics.htmlI have noticed that in the presentation of probability distribution, sometimes a certain probability is dentoed by $\pi(...)$. For example, in the wike page of Beta binomial distribution https://en....Fri, 21 Aug 2026 10:44:45 -0400https://pop.com.sg/boolean-algebra-simplification-a-039-bc-ab-039-c-abc-039-abc-closed.htmlhttps://pop.com.sg/boolean-algebra-simplification-a-039-bc-ab-039-c-abc-039-abc-closed.htmlCan't figure out how to simplify $(^\neg a)bc+a(^\neg b)c+ab(^\neg c)+abc$, I'm really bad at this......Fri, 21 Aug 2026 10:44:02 -0400https://pop.com.sg/inverse-of-a-3x3-block-matrix.htmlhttps://pop.com.sg/inverse-of-a-3x3-block-matrix.htmlI would like to get the inverse of a 3x3 (covariance) block matrix \begin{bmatrix}A&B&C\\B'&D&E\\C'&E'&F\end{bmatrix} where the prime ' indicates the transposition operato......Fri, 21 Aug 2026 10:33:28 -0400https://pop.com.sg/have-diagonalization-need-orthogonal-diagonalization.htmlhttps://pop.com.sg/have-diagonalization-need-orthogonal-diagonalization.htmlI have a symmetric matrix $A$ and I want to find an orthogonal diagonalization of $A$, i.e, matrices $B$ and $D$ such that: $$ A = B D B^{-1}$$ and $D$ is diagonal and $B^{-1} = B^T$. I use a math ...Fri, 21 Aug 2026 10:32:42 -0400https://pop.com.sg/find-7-999-999-modulo-10.htmlhttps://pop.com.sg/find-7-999-999-modulo-10.htmlI am trying to find $7^{999,999}$ modulo (10) using Euler's Theorem: If $m \in \mathbb{Z^+}, a \in \mathbb{Z}, (a,m) = 1$ then $a^{\phi(m)} \equiv 1$ (mod m). I am unsure though how to use it ......Fri, 21 Aug 2026 10:21:37 -0400https://pop.com.sg/roots-of-parabola.htmlhttps://pop.com.sg/roots-of-parabola.htmlI have a parabola with the equation $y = x^2 + 6x + 7$ and I am trying to calculate the $x$-intercept points. Here is my working so far... let $y = 0$, $x^2 + 6x + 7 = 0$ $x(x + 6) = -7$ After ......Fri, 21 Aug 2026 10:12:39 -0400https://pop.com.sg/should-i-round-a-decimal-number-of-people-up-or-down.htmlhttps://pop.com.sg/should-i-round-a-decimal-number-of-people-up-or-down.htmlThe maths question below is asking me to find the number of people. I have found the answer to be $57553.0834$, but obviously there can't be $0.0834$ of a human. The mark scheme states the answer is $...Fri, 21 Aug 2026 10:05:17 -0400https://pop.com.sg/finding-the-asymptote-root-of-a-reciprocal-function.htmlhttps://pop.com.sg/finding-the-asymptote-root-of-a-reciprocal-function.html$$y = \frac{3}{8x - 3} $$ The y-intercept is $-1$ and the vertical asymptote is $x = \frac{3}{8}$ but what would be the horizontal asymptote and the x-intercept in this case? I am asking this as ......Fri, 21 Aug 2026 09:54:17 -0400https://pop.com.sg/how-to-find-sin-1-sin-100.htmlhttps://pop.com.sg/how-to-find-sin-1-sin-100.htmlTo solve such questions we were told to infer the graph of $\sin^{-1}(\sin{x})$ but this seems too big of a number. Is there any shorter way to solve such questions?...Fri, 21 Aug 2026 09:40:02 -0400https://pop.com.sg/what-can-i-do-with-the-lifting-the-exponent-lemma.htmlhttps://pop.com.sg/what-can-i-do-with-the-lifting-the-exponent-lemma.htmlWhat can I do with LTE (lifting the exponent lemma)? If you don't know LTE lemma visit here or here...Fri, 21 Aug 2026 09:30:23 -0400https://pop.com.sg/can-an-underdetermined-system-have-a-unique-solution.htmlhttps://pop.com.sg/can-an-underdetermined-system-have-a-unique-solution.htmlIn my case, I am calling an underdetermined system as a system of linear equations where there are fewer equations than variables (unknowns). My textbook says the answer is false, however the inter......Fri, 21 Aug 2026 09:29:04 -0400https://pop.com.sg/lattice-of-subgroups-of-a4.htmlhttps://pop.com.sg/lattice-of-subgroups-of-a4.htmlI am having trouble with this problem. We just started learning it and I am unsure of how to obtain all the elements and how to form them into cyclic subgroups. The alternating group A4 consists o......Fri, 21 Aug 2026 09:23:59 -0400https://pop.com.sg/four-letter-word-contains-no-two-consecutive-equal-letters.htmlhttps://pop.com.sg/four-letter-word-contains-no-two-consecutive-equal-letters.htmlThis is taken from the book on Combinatorics, by Daniel Marcus. Request vetting: A16: Find the probability that a four-letter word that uses letters from $A,B,C,D,E$ contains (a) no repeated lette......Fri, 21 Aug 2026 09:09:56 -0400https://pop.com.sg/how-many-0-s-does-the-number-30-have-duplicate.htmlhttps://pop.com.sg/how-many-0-s-does-the-number-30-have-duplicate.htmlI want to find out the number of $0$s in the number $30!$, what should I do? Is there any trick that would work for a general question of this type, like number of $0$s in $50!$ ?...Fri, 21 Aug 2026 09:09:37 -0400https://pop.com.sg/foot-of-perpendicular-in-3-dimensions.htmlhttps://pop.com.sg/foot-of-perpendicular-in-3-dimensions.htmlLet us suppose that $Q$ is a foot of perpendicular from a point $P (2,4,3) $on the line joining the points $A(1,2,4)$ and $B(3,4,5)$; then what are the coordinates of $Q$? My try: Had this been a......Fri, 21 Aug 2026 09:08:29 -0400https://pop.com.sg/express-real-number-in-exponential-form.htmlhttps://pop.com.sg/express-real-number-in-exponential-form.htmlExpress $ z=-32 $ in exponential form. My reasoning: $ z=-32 $ is the same as $ z=-32+0i $ Exponential form should look like $ z=Re^{\theta i} $ $ R =\sqrt{(-32)^2 + 0^2} = 32 $ $ \theta = \tan^{......Fri, 21 Aug 2026 09:06:51 -0400https://pop.com.sg/why-is-compactness-in-logic-called-compactness.htmlhttps://pop.com.sg/why-is-compactness-in-logic-called-compactness.htmlIn logic, a semantics is said to be compact iff if every finite subset of a set of sentences has a model, then so to does the entire set. Most logic texts either don't explain the terminology, or ...Fri, 21 Aug 2026 08:57:27 -0400https://pop.com.sg/prove-the-theorem-of-nicomachus-by-induction.htmlhttps://pop.com.sg/prove-the-theorem-of-nicomachus-by-induction.htmlProve the theorem of Nicomachus(AD.100) by induction: $$ 1^3 = 1,\ 2^3 = 3+ 5,\ 3^3 = 7 + 9 + 11,\ 4^3 = 13 + 15 + 17 + 19,\ ... $$ My approach: from looking at the above pattern you can tell ......Fri, 21 Aug 2026 08:55:22 -0400https://pop.com.sg/get-location-of-vector-circle-intersection.htmlhttps://pop.com.sg/get-location-of-vector-circle-intersection.htmlI'm a coding guru, and I'm good with math - but only math that I know. Which isn't even at calculus level yet. So I'm hoping I can get some help here for my algorithm. Say I have a circle. I know ......Fri, 21 Aug 2026 08:43:18 -0400https://pop.com.sg/testing-for-absolute-convergence-frac-1-n-5n-1.htmlhttps://pop.com.sg/testing-for-absolute-convergence-frac-1-n-5n-1.htmlI'm trying to test the summation $\sum^\infty_{n=0}\frac{(-1)^n}{5n+1}$ for absolute convergence. By alternating series test, I can tell is is at least conditionally convergent. However, when I u......Fri, 21 Aug 2026 08:32:53 -0400https://pop.com.sg/why-does-wolfram-alpha-handle-log-and-ln-the-same.htmlhttps://pop.com.sg/why-does-wolfram-alpha-handle-log-and-ln-the-same.htmlI thought $\log(n)$ was like $10^x = n$ and $\ln(n)$ was $e^x = n$. But when I do $\ln(80)$, it gives me the answer for $\log$. Why is that?...Fri, 21 Aug 2026 08:26:10 -0400https://pop.com.sg/characteristics-in-the-theory-of-pde-039-s-what-039-s-going-on.htmlhttps://pop.com.sg/characteristics-in-the-theory-of-pde-039-s-what-039-s-going-on.htmlWhat's really going on as regards characterstics in the theory of PDE's? (Hopefully you wont have to check the links provided below, included for those who are interested really) For second order......Fri, 21 Aug 2026 08:26:07 -0400https://pop.com.sg/ordering-of-the-complex-numbers.htmlhttps://pop.com.sg/ordering-of-the-complex-numbers.htmlThe complex numbers as a whole cannot be ordered but could you order the complex numbers of the form ai where a is a real number?...Fri, 21 Aug 2026 08:20:49 -0400https://pop.com.sg/don-039-t-really-understand-the-absorption-law.htmlhttps://pop.com.sg/don-039-t-really-understand-the-absorption-law.htmlI don't really get the absorption law, specifically in this case: $$ (\lnot p \lor q) \land (\lnot r \lor q) \equiv (\lnot p \land \lnot r) \lor (\lnot p \land q) \lor (q\land \lnot r) \lor (q \l......Fri, 21 Aug 2026 08:16:17 -0400https://pop.com.sg/what-does-arbitrary-number-mean.htmlhttps://pop.com.sg/what-does-arbitrary-number-mean.htmlA FSM (Finite State Machine) can be designed to add two integers of any arbitrary length (arbitrary number of digits). Is it true ? My attempt : Arbitrary length means variable length, and there i......Fri, 21 Aug 2026 08:00:54 -0400https://pop.com.sg/solutions-of-x-2-6x-13-equiv-0-pmod-127.htmlhttps://pop.com.sg/solutions-of-x-2-6x-13-equiv-0-pmod-127.htmlI started learning number theory, specifically polynomial congruences, and need help with the following exercise. Here it is: Does the congruence $x^2-6x-13 \equiv 0 \pmod{127}$ has solutions? I ...Fri, 21 Aug 2026 08:00:11 -0400https://pop.com.sg/what-does-it-mean-for-a-matrix-to-be-non-negative-definite.htmlhttps://pop.com.sg/what-does-it-mean-for-a-matrix-to-be-non-negative-definite.htmlIn some old course notes I'm reading to touch up on statistical forecasting methods, the book often makes reference to "non-negative definite" matrices. I know what a semi-positive definite, positive ...Fri, 21 Aug 2026 07:52:03 -0400https://pop.com.sg/determining-matrix-for-relationship-reflexive-symmetric-transitive.htmlhttps://pop.com.sg/determining-matrix-for-relationship-reflexive-symmetric-transitive.htmlI have two matrices below and need to determine if R is (a) reflexive, (b) symmetric, and (c) transitive. $M_R = \begin{pmatrix} 1 & 0 & 1 & 0\\ 1 & 1 & 0 & 1 \\ 1 & 1 ......Fri, 21 Aug 2026 07:43:13 -0400https://pop.com.sg/how-can-i-find-the-measure-of-every-angle-in-a-star-polygon.htmlhttps://pop.com.sg/how-can-i-find-the-measure-of-every-angle-in-a-star-polygon.htmlI'm unfamiliar with these kinds of problems. I looked up some formulas and it says for an $(n,3)$ family of star polygons, $\theta = \frac{(1-\frac{6}{n}}{180}$ How do I get these formulas and what ...Fri, 21 Aug 2026 07:35:48 -0400https://pop.com.sg/arthur-milgram-photo.htmlhttps://pop.com.sg/arthur-milgram-photo.htmlsomeone has photo of Arthur Milgram? (the guy from Lax-Milgram Theorem) I'm curious, why I can't find one picture of Arthur Milgram on google? Anyone has one? or where can I find one? Thanks!...Fri, 21 Aug 2026 07:28:07 -0400https://pop.com.sg/cross-product-of-sides-of-a-triangle.htmlhttps://pop.com.sg/cross-product-of-sides-of-a-triangle.htmli am given with a triangle and and i read here that i can cross product of the two sides of a triangle gives it orientation. what does cross product of sides of a triangle mean. here is the formul......Fri, 21 Aug 2026 07:23:14 -0400https://pop.com.sg/given-a-steady-state-vector-is-it-possible-to-calculate-the-correspond.htmlhttps://pop.com.sg/given-a-steady-state-vector-is-it-possible-to-calculate-the-correspond.htmlKnowing that there is a probability matrix M (where all columns add to 1) which when applied to a given vector P produces the same vector P, what is the best solution to find M? I can get my head a......Fri, 21 Aug 2026 07:14:35 -0400https://pop.com.sg/3d-projection-on-a-2d-plane-weak-maths-ressources.htmlhttps://pop.com.sg/3d-projection-on-a-2d-plane-weak-maths-ressources.htmlAs the title says, i want to project 3D points with known (x, y, z) coordinates into a 2D plane with (x', y') coordinates, knowing that the x and y axes are respectively identical to the x' and y' ......Fri, 21 Aug 2026 07:12:41 -0400https://pop.com.sg/expansion-of-complex-equation.htmlhttps://pop.com.sg/expansion-of-complex-equation.htmlFind the value of $$\left(\frac{-1+\sqrt 3i}{2}\right)^{15} + \left(\frac{-1-\sqrt 3i}{2}\right)^{15}.$$ In general, how do we find the value of expansion of equation of high orders other than bi......Fri, 21 Aug 2026 07:09:16 -0400https://pop.com.sg/how-do-i-calculate-gf-2-galois-field-without-using-a-tool.htmlhttps://pop.com.sg/how-do-i-calculate-gf-2-galois-field-without-using-a-tool.htmlI am stuck and can't find anything on Google or SO. I have: 1+1+1+0 and I am asked to calculate the answer over GF(2). My question: what answer? I have no clue what to do. Can someone help me by ...Fri, 21 Aug 2026 07:04:44 -0400https://pop.com.sg/if-the-product-and-the-sum-of-two-complex-numbers-are-real-what-can-we.htmlhttps://pop.com.sg/if-the-product-and-the-sum-of-two-complex-numbers-are-real-what-can-we.htmlIf the product and the sum of two complex numbers are real, what can we say about the numbers? Prove it. Explanation will be very helpful. Actually the question asks for a proof. Hence, a proof is......Fri, 21 Aug 2026 07:00:52 -0400https://pop.com.sg/geometric-intuition-of-graph-laplacian-matrices.htmlhttps://pop.com.sg/geometric-intuition-of-graph-laplacian-matrices.htmlI am reading about Laplacian matrices for the first time and struggling to gain intuition as to why they are so useful. Could anyone provide insight as to the geometric significance of the Laplacia......Fri, 21 Aug 2026 06:52:03 -0400https://pop.com.sg/if-sin-x-cos-x-1-3-then-determine-sin-x-cos-x.htmlhttps://pop.com.sg/if-sin-x-cos-x-1-3-then-determine-sin-x-cos-x.htmlIf $$\sin(x) - \cos(x) = \frac{1}{3}$$ then determine $$\sin(x)\cos(x)$$ I know that the expected solution is squaring both sides of equation and solving it this way: \begin{gather} \sin^2(x......Fri, 21 Aug 2026 06:51:56 -0400https://pop.com.sg/graph-with-both-slant-and-horizontal-asymptotes.htmlhttps://pop.com.sg/graph-with-both-slant-and-horizontal-asymptotes.htmlIs there such a graph? A graph that increases at a decreasing rate with the graph approaching a slant asymptote as x decreases to negative infinity while the graph approaching a horizontal asymptot......Fri, 21 Aug 2026 06:51:07 -0400https://pop.com.sg/is-zfc-arithmetically-sound.htmlhttps://pop.com.sg/is-zfc-arithmetically-sound.htmlI apologize that this question is fairly philosophical and not purely mathematical. For the purposes of this question, I would like to take the point of view that that natural numbers are "rea......Fri, 21 Aug 2026 06:41:25 -0400https://pop.com.sg/how-to-solve-log-x-1-log-x-2-2.htmlhttps://pop.com.sg/how-to-solve-log-x-1-log-x-2-2.htmlI'm doing this exercise: $$\log(x - 1) + \log(x - 2) = 2$$ My steps: Step 1: $$\log(x-1)(x - 2) = 2$$ Step 2: $$(x - 1)(x - 2) = 10^2$$ Step 3: $$x^2 - 3x + 2 = 100$$ Step 4: $$x^2 = 98 ......Fri, 21 Aug 2026 06:32:46 -0400https://pop.com.sg/clearing-a-manufacturing-backlog.htmlhttps://pop.com.sg/clearing-a-manufacturing-backlog.htmlI have a simple problem that I think I may be over complicating, but I'm not certain. I work in a manufacturing environment as a process engineer and I'm setting up a calculator to determine how l......Fri, 21 Aug 2026 06:30:30 -0400https://pop.com.sg/given-two-square-matrices-a-and-b-is-ab-2-a-2b-2-true-or-false.htmlhttps://pop.com.sg/given-two-square-matrices-a-and-b-is-ab-2-a-2b-2-true-or-false.htmlI tried with a counter example and it came out that this claim is false. I took a matrix $$ A= \left[ {\begin{array}{cc} 2 & 1\\ 3 & 2\\ \end{array} } \right] $$ and a matri......Fri, 21 Aug 2026 06:29:36 -0400https://pop.com.sg/definition-of-pi-in-terms-of-cos.htmlhttps://pop.com.sg/definition-of-pi-in-terms-of-cos.htmlI have to define $\pi$ in terms of cos and to show why the definition is well-defined. I think that I haven't understood what it is asking because $\pi$ would be the angle that divided for $2$ giv......Fri, 21 Aug 2026 06:26:46 -0400https://pop.com.sg/find-max-of-total-area-of-solenoid.htmlhttps://pop.com.sg/find-max-of-total-area-of-solenoid.htmlI'm having problems findind the best way to maximize the total area of my solenoid, of course given $l$ as the costant value of the total lenght of my wire. I can't find it on the internet, so I'm ...Fri, 21 Aug 2026 06:21:53 -0400https://pop.com.sg/partial-derivative-of-x-is-quotient-rule-necessary.htmlhttps://pop.com.sg/partial-derivative-of-x-is-quotient-rule-necessary.htmlLet $$u(x,y)=\frac{x}{x^2+y^2}$$ I'm trying to determine if the given function is harmonic. I know that the 2nd partial derivative with respect to $x$ should, when added to the 2nd partial deriva......Fri, 21 Aug 2026 06:00:52 -0400https://pop.com.sg/understanding-cardano-039-s-formula.htmlhttps://pop.com.sg/understanding-cardano-039-s-formula.htmlIn deriving his formula, Cardano arrives at the equation $y^3+py+q=0$. By substituting $y=\sqrt[3]{u}+\sqrt[3]{v}$, he gets the equation $(u+v+q)+(\sqrt[3]{u}\sqrt[3]{v})(3\sqrt[3]{u} \sqrt[3]{v} +......Fri, 21 Aug 2026 05:56:28 -0400https://pop.com.sg/transforming-the-laplace-operator-from-polar-to-cartesian-coordinates.htmlhttps://pop.com.sg/transforming-the-laplace-operator-from-polar-to-cartesian-coordinates.htmlI'm trying to find the error in my logic here. Let's say we are given the Laplace operator in polar coordinates: $$ \frac{\partial^2}{\partial r^2} + \frac{1}{r}\frac{\partial}{\partial r} + \fr......Fri, 21 Aug 2026 05:38:18 -0400https://pop.com.sg/how-can-i-plot-f-x-y-x-2-y-2.htmlhttps://pop.com.sg/how-can-i-plot-f-x-y-x-2-y-2.htmlI want to plot $f(x, y) = x^2 + y^2$? I can plot functions of a single variable but I don't know how to plot multivariable function....Fri, 21 Aug 2026 05:27:42 -0400https://pop.com.sg/find-general-solution-in-radians-of-cos-x-cos-3x-cos-5x-0.htmlhttps://pop.com.sg/find-general-solution-in-radians-of-cos-x-cos-3x-cos-5x-0.htmlI study maths as a hobby. I have this problem: Find in radians the general solution of: $$\cos x + \cos 3x + \cos 5x = 0$$ I have said: $\cos 3x + \cos 5x = 2\cos\left( \frac{1}{2}(3x + 5x)\right)\......Fri, 21 Aug 2026 05:21:47 -0400https://pop.com.sg/find-the-exponential-value-of-a-number.htmlhttps://pop.com.sg/find-the-exponential-value-of-a-number.htmlI'm wondering if there is a way to add two numbers with the same base but different exponents without converting the base of the numbers. For example lets say your adding 2^2 and 2^4 the value of 2......Fri, 21 Aug 2026 05:14:11 -0400https://pop.com.sg/2-sphere-surface-coordinate-dimension.htmlhttps://pop.com.sg/2-sphere-surface-coordinate-dimension.htmlOrdinary sphere in $\mathbb{R}^3$ is two-dimensional object (2-sphere), i.e. it requires at least two coordinates to define point on a surface. As I notice, however, there is a catch. If we use ...Fri, 21 Aug 2026 05:12:45 -0400https://pop.com.sg/what-does-rigorous-proof-mean.htmlhttps://pop.com.sg/what-does-rigorous-proof-mean.htmlI have heard several times that some mathematician has given another and more rigorous for an established theorem, but I don't know what does it really mean and what differences makes it to be more '...Fri, 21 Aug 2026 05:08:11 -0400https://pop.com.sg/what-are-the-smallest-and-largest-values-of-v-w-what-are-the-smallest-.htmlhttps://pop.com.sg/what-are-the-smallest-and-largest-values-of-v-w-what-are-the-smallest-.htmlIf $||v|| = 5$ and $||w|| = 3$, what are the smallest and largest values of $||v - w||$? What are the smallest and largest values of $v \cdot w$? How can I solve these two problems? For $||v - w||......Fri, 21 Aug 2026 05:06:12 -0400https://pop.com.sg/what-is-the-difference-between-variable-argument-and-parameter.htmlhttps://pop.com.sg/what-is-the-difference-between-variable-argument-and-parameter.htmlI'm sure that these terms should be different since there exists a difference between parameter and argument in computer science but I'm not sure about their differences in math....Fri, 21 Aug 2026 05:05:54 -0400https://pop.com.sg/definition-statistics-calculations-for-evenly-spaced-numbers.htmlhttps://pop.com.sg/definition-statistics-calculations-for-evenly-spaced-numbers.htmlIt seems like this ought to have a name like "smoothness" or something similar, but smoothness relates to something entirely different, and I can't find terminology that applies to the ...Fri, 21 Aug 2026 04:49:30 -0400https://pop.com.sg/understanding-the-giry-monad.htmlhttps://pop.com.sg/understanding-the-giry-monad.htmlI would like to understand the Giry monad, which is used to reason about probability in category theory. The issue is that I'm hitting a stumbling block understanding monads in general, in the cat......Fri, 21 Aug 2026 04:23:21 -0400https://pop.com.sg/bernstein-inequality-for-non-centered-random-variables-is-there-a-coun.htmlhttps://pop.com.sg/bernstein-inequality-for-non-centered-random-variables-is-there-a-coun.htmlThe usual Bernstein inequality (see e.g. Rauhut + Fourcat, A Mathematical Introduction to Compressive Sensing, Theorem 7.30) states that if $X_1, \dots, X_m$ are independent mean zero random variab......Fri, 21 Aug 2026 03:58:58 -0400https://pop.com.sg/taylor-series-of-an-integral-function.htmlhttps://pop.com.sg/taylor-series-of-an-integral-function.htmlProblem $$I(x) = \int_{1}^x \frac{e^t - 1}{t}$$ Find $I'( \sqrt{x} )$. Solution We know that $F'(x) = f(x)$ by the fundamental theorem of calculus so $$I'(x) = \frac{e^t -1}{t}$$ And so $$I'( \......Fri, 21 Aug 2026 03:56:06 -0400https://pop.com.sg/chances-of-same-9-digit-number-repeating-in-90-000-attempts.htmlhttps://pop.com.sg/chances-of-same-9-digit-number-repeating-in-90-000-attempts.htmlI am here asking a specific question as it relates to a programming project at work. Specifically, I have a file that a user can create which contains a large number of fields. Four of these fields......Fri, 21 Aug 2026 03:53:33 -0400https://pop.com.sg/tribonacci-sequence-term.htmlhttps://pop.com.sg/tribonacci-sequence-term.htmlA tribonacci sequence is a sequence of numbers such that each term from the fourth onward is the sum of the previous three terms. The first three terms in a tribonacci sequence are called its seeds......Fri, 21 Aug 2026 03:53:02 -0400https://pop.com.sg/two-vertices-of-a-square.htmlhttps://pop.com.sg/two-vertices-of-a-square.htmlBeing (-2, 4) and B (3, -1) consecutive vertices of a square, determine the other two vertices How to solve this only algebraically? Would point-to-point distance (relative position between points......Fri, 21 Aug 2026 03:52:39 -0400https://pop.com.sg/how-to-factor-a-fourth-degree-polynomial.htmlhttps://pop.com.sg/how-to-factor-a-fourth-degree-polynomial.htmlI'm working on a math problem but I am having a hard time figuring out the method used by my textbook to make this factorization: $$x^4 + 10x^3 + 39x^2 + 70x + 50 = (x^2 + 4x + 5)(x^2 + 6x + 10)$$......Fri, 21 Aug 2026 03:48:17 -0400https://pop.com.sg/prove-that-the-sum-of-all-simple-roots-is-a-root.htmlhttps://pop.com.sg/prove-that-the-sum-of-all-simple-roots-is-a-root.htmlLet $\Delta$ be an indecomposable root system in a real inner product space $E$, and suppose that $\Phi$ is a simple system of roots in $\Delta$, with respect to an ordering of $E$. If $\Phi = \{\a......Fri, 21 Aug 2026 03:46:54 -0400https://pop.com.sg/equation-for-a-sinusoidal-wave-with-changing-frequency.htmlhttps://pop.com.sg/equation-for-a-sinusoidal-wave-with-changing-frequency.htmlI would like to find a sinusoidal wave whose period or frequency change to half (or double) with every step, someting like this But I cant find the precise coefficients for the period to decrease ......Fri, 21 Aug 2026 03:44:45 -0400https://pop.com.sg/what-are-the-names-of-numbers-in-the-binary-system.htmlhttps://pop.com.sg/what-are-the-names-of-numbers-in-the-binary-system.htmlThe names we use are very much related to the radix we use $0 - 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9$ zero - one - two - three - four - five - six - seven - eight -nine We repeat the names $21$ twenty......Fri, 21 Aug 2026 03:43:16 -0400https://pop.com.sg/what-is-the-geometric-meaning-of-the-inner-product-of-two-functions.htmlhttps://pop.com.sg/what-is-the-geometric-meaning-of-the-inner-product-of-two-functions.htmlWhen it comes to inner product I have thus far only dealt with vectors, and so the concept is very intuitive because one can easily visualize two vectors and how they get multiplied, and it is clea......Fri, 21 Aug 2026 03:40:03 -0400https://pop.com.sg/increasing-decreasing-sequence-of-events-intuition.htmlhttps://pop.com.sg/increasing-decreasing-sequence-of-events-intuition.htmlhaving trouble moving from the concept of increasing/decreasing sequence of real numbers to increasing/decreasing sequence of events. I have no problem with this concept regarding real numbers but ...Fri, 21 Aug 2026 03:39:48 -0400https://pop.com.sg/correctly-using-lotus-in-this-situation.htmlhttps://pop.com.sg/correctly-using-lotus-in-this-situation.htmlLet $X$ be a random variable with $E(X^2) < \infty$, and $Y$ be a binary random variable taking values $\{0,1\}$. The Odds Ratio (OR) between the odds of an outcome at $X=a+\epsilon$ (where $\ep......Fri, 21 Aug 2026 03:34:16 -0400https://pop.com.sg/proof-of-cauchy-039-s-theorem-for-abelian-groups.htmlhttps://pop.com.sg/proof-of-cauchy-039-s-theorem-for-abelian-groups.htmlI am reading the Dummit and Foote's proof of Cauchy's theorem for abelian groups. On the highlighted line, how do you conclude from $y\notin N$ and $y^p\in N$ that $\langle y^p \rangle \neq \langle......Fri, 21 Aug 2026 03:26:48 -0400