https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usThu, 03 Sep 2026 05:45:53 -0400https://pop.com.sg/laplace-transform-of-cosh-bt.htmlhttps://pop.com.sg/laplace-transform-of-cosh-bt.htmlSo, one of my homework assignments is to take the Laplace transform of a function such that $f(t)=\cosh{bt}$. I figured it would be equivalent to: $$\int^\infty_0{e^{-st}\frac{e^{bt}+e^{-bt}}{2}}\...Thu, 03 Sep 2026 09:42:43 -0400https://pop.com.sg/where-039-s-the-foolish-part-prove-that-0-0-2-duplicate.htmlhttps://pop.com.sg/where-039-s-the-foolish-part-prove-that-0-0-2-duplicate.htmlI've been across this on the web: \begin{align} \frac{0}{0} & = \frac{100-100}{100-100} \\ & = \frac{10^2-10^2}{10(10-10)} \\ & = \frac{(10-10)(10+10)}{(10)(10-10)} \\ & = \frac{(1......Thu, 03 Sep 2026 09:40:10 -0400https://pop.com.sg/how-to-find-length-of-vector-in-a-particular-direction.htmlhttps://pop.com.sg/how-to-find-length-of-vector-in-a-particular-direction.htmlHow do you solve this: How do I find how long the red line is?...Thu, 03 Sep 2026 09:38:42 -0400https://pop.com.sg/differential-equation-capacitor-closed.htmlhttps://pop.com.sg/differential-equation-capacitor-closed.htmlThe voltage of a capacitor can be described with the differential equation $ \frac {du} {dt} + \frac {1} {RC} u = 0$ where the voltage is u(t) at the time t. Solve the differential equation: Don't ...Thu, 03 Sep 2026 09:31:13 -0400https://pop.com.sg/alternate-translation-for-every-real-number-except-zero-has-a-multipli.htmlhttps://pop.com.sg/alternate-translation-for-every-real-number-except-zero-has-a-multipli.htmlA given text states, “Every real number except zero has a multiplicative inverse" (where mul- tiplicative inverse of a real number x is a real number y such that xy = 1). It offers the following ...Thu, 03 Sep 2026 09:15:09 -0400https://pop.com.sg/determine-slope-given-one-point-and-y-intercept.htmlhttps://pop.com.sg/determine-slope-given-one-point-and-y-intercept.htmlThe graph of the relation y = mx + 35 passes through the point (8,–77). Determine the value Of m. I understand how to find the y intercept when given slope and a point, but i can't seem to find the ...Thu, 03 Sep 2026 09:13:21 -0400https://pop.com.sg/prove-by-induction-2-n-c-n-0-c-n-1-cdots-c-n-n-duplicate.htmlhttps://pop.com.sg/prove-by-induction-2-n-c-n-0-c-n-1-cdots-c-n-n-duplicate.htmlThis is a question I came across in an old midterm and I'm not sure how to do it. Any help is appreciated. $$2^n = C(n,0) + C(n,1) + \cdots + C(n,n).$$ Prove this statement is true for all $n \ge ......Thu, 03 Sep 2026 09:12:26 -0400https://pop.com.sg/what-is-the-derivative-of-xf-x-closed.htmlhttps://pop.com.sg/what-is-the-derivative-of-xf-x-closed.htmlIf we have a function $f(x)$ what would the value of the following expression be? $$\frac{d}{dx}[xf(x)]$$...Thu, 03 Sep 2026 09:12:04 -0400https://pop.com.sg/how-to-find-bn-for-limit-comparison.htmlhttps://pop.com.sg/how-to-find-bn-for-limit-comparison.htmlI am asking how to do the Limit comparison test for this: $$\lim {An\over Bn} =L$$ You choose $B_n$ yourselve, but how do you choose it? Example: $$A_n = \frac{3n^2 + 5n + 1}{\sqrt{(n^5 + 5 )}}$......Thu, 03 Sep 2026 09:07:09 -0400https://pop.com.sg/modeling-with-markov-chains-and-one-step-analysis.htmlhttps://pop.com.sg/modeling-with-markov-chains-and-one-step-analysis.htmlI have set up the following model: Let $X_n$ be the number of heads in the $n$-th toss and $P(X_0=0)=1$. I can calculate the transition matrix $P$. Define $$ T=\min\{n\geq 0\mid X_n=5\}. $$ Then ......Thu, 03 Sep 2026 08:56:40 -0400https://pop.com.sg/how-to-solve-this-indefinite-integral.htmlhttps://pop.com.sg/how-to-solve-this-indefinite-integral.htmlI'm trying to understand how to solve this indefinite integral using the power rule. $$ \begin{eqnarray} \int \frac{x}{\sqrt{x}} dx &=& \int \frac{x^1}{x^\frac{1}{2}} dx \end{eqnarray} $$ ...Thu, 03 Sep 2026 08:54:31 -0400https://pop.com.sg/why-do-some-people-state-that-039-zero-is-not-a-number-039.htmlhttps://pop.com.sg/why-do-some-people-state-that-039-zero-is-not-a-number-039.htmlEvery now and then I read about people who wonder whether zero is a number. It never occurred to me to question this, so I checked the Wikipedia page which, when talking about the Rules of Brahmagu......Thu, 03 Sep 2026 08:40:47 -0400https://pop.com.sg/sum-of-geometric-series-formula-duplicate.htmlhttps://pop.com.sg/sum-of-geometric-series-formula-duplicate.htmlI just need the formula for the sum of geometric series when each element in the series has the value $1/2^{j+1}$, where $j = 0, 1, 2, \ldots, n$. Please help. Someone told me it is: $$S = 2 - \f......Thu, 03 Sep 2026 08:27:51 -0400https://pop.com.sg/how-far-can-one-see-over-the-ocean.htmlhttps://pop.com.sg/how-far-can-one-see-over-the-ocean.htmlSince Earth is a sphere, one has only a limited visibility radius. How far is that, actually? This Q&A was inspired by this question, about whether or not Legolas can see the 24km distant Ride......Thu, 03 Sep 2026 08:26:54 -0400https://pop.com.sg/understanding-the-bockstein-homomorphism-in-group-cohomology.htmlhttps://pop.com.sg/understanding-the-bockstein-homomorphism-in-group-cohomology.htmlLet $k$ be an algebraically closed field of characteristic $p>0$. Let $G$ be a finite group. In group cohomology, the Bockstein homomorphism is the connecting homomorphism $$\beta:H^n(G,\math......Thu, 03 Sep 2026 08:23:37 -0400https://pop.com.sg/why-is-a-circle-not-simply-connected.htmlhttps://pop.com.sg/why-is-a-circle-not-simply-connected.htmlTo be simply-connected means to be path-connected and able to continuously shrink a closed curve while remaining in the domain. According to wikipedia, a circle is not simply connected, but a dis......Thu, 03 Sep 2026 08:05:46 -0400https://pop.com.sg/how-do-i-go-about-simplifying-this-complex-radical.htmlhttps://pop.com.sg/how-do-i-go-about-simplifying-this-complex-radical.htmlI'm having a difficult time trying to simplify the radical below. When I type the radical to the left into my calculator, I arrive at the correct answer to the right. But I cannot seem to figure ou......Thu, 03 Sep 2026 08:04:12 -0400https://pop.com.sg/is-closedness-a-proper-word.htmlhttps://pop.com.sg/is-closedness-a-proper-word.htmlIn one of my papers I had to prove a list of properties of a set, say, $S=\{a,b,c\}$. Among them we have a fact that $S$ is downward closed with respect to a binary relation $R$. I found it awkward......Thu, 03 Sep 2026 08:03:58 -0400https://pop.com.sg/what-does-it-mean-when-f-a-is-defined-closed.htmlhttps://pop.com.sg/what-does-it-mean-when-f-a-is-defined-closed.htmlI am in Calculus I and my calculus jargon is lacking so please if you can explain, for lack of better terms, explain it simply....Thu, 03 Sep 2026 07:55:28 -0400https://pop.com.sg/can-a-function-have-an-x-input-with-two-y-outputs.htmlhttps://pop.com.sg/can-a-function-have-an-x-input-with-two-y-outputs.htmlFor example f(4) = 3, 2 f(5) = 4 f(6) = 10 Please forgive me if my example is atrocious, hopefully it gets my point across....Thu, 03 Sep 2026 07:52:31 -0400https://pop.com.sg/which-greek-letter-is-commonly-used-to-represent-a-count.htmlhttps://pop.com.sg/which-greek-letter-is-commonly-used-to-represent-a-count.htmlWhich Greek letter is commonly used to represent a count? For example, the Greek letter sigma ($\Sigma$) is commonly used to represent a sum....Thu, 03 Sep 2026 07:50:32 -0400https://pop.com.sg/odds-of-winning-at-minesweeper-with-perfect-play.htmlhttps://pop.com.sg/odds-of-winning-at-minesweeper-with-perfect-play.htmlHow would someone go about doing this? Assume that the first "click" will never be a bomb, and that the number of mines and the area are both known. Rather hoping there is a clever way to do this, ......Thu, 03 Sep 2026 07:45:03 -0400https://pop.com.sg/applying-difference-of-cubes-to-cube-roots.htmlhttps://pop.com.sg/applying-difference-of-cubes-to-cube-roots.htmlI am stumped as to why this application of the difference of cubes is valid... I am rationalizing the denominator. I don't understand the reasoning of why the difference of cubes formula is appli......Thu, 03 Sep 2026 07:44:01 -0400https://pop.com.sg/why-does-the-dot-product-between-two-unit-vectors-equal-the-cosine-on-.htmlhttps://pop.com.sg/why-does-the-dot-product-between-two-unit-vectors-equal-the-cosine-on-.htmlI'm confused about why does the dot product between two unit vectors equal the cosine on the angle between them Thank you....Thu, 03 Sep 2026 07:32:14 -0400https://pop.com.sg/how-can-i-check-that-a-hexagon-intersects-a-circle.htmlhttps://pop.com.sg/how-can-i-check-that-a-hexagon-intersects-a-circle.htmlIf I know the center positions and radius of a circle and hexagon, how can I determine whether are they touches or intersects each other? Update: Also the orientation of the hexagon is know:...Thu, 03 Sep 2026 07:23:34 -0400https://pop.com.sg/calculating-the-volume-of-a-hemisphere-given-only-the-height.htmlhttps://pop.com.sg/calculating-the-volume-of-a-hemisphere-given-only-the-height.htmlSo I have a math problem that I'm trying to solve with regards to hemispheres but am just confused about this part of the question If I have a hemisphere with radius $r$, then Volume $= \frac23......Thu, 03 Sep 2026 07:21:37 -0400https://pop.com.sg/how-to-intuitively-see-that-the-text-volume-of-a-pyramid-1-3-times-tex.htmlhttps://pop.com.sg/how-to-intuitively-see-that-the-text-volume-of-a-pyramid-1-3-times-tex.htmlI'm interested to know if anyone can point me to a non-calculus way of seeing that the $\text{volume of a pyramid} = \frac{1}{3}\times(\text{area of base})\times(\text{height})$. Yes, I've googled....Thu, 03 Sep 2026 07:20:56 -0400https://pop.com.sg/square-root-of-1-2-sqrt3-2.htmlhttps://pop.com.sg/square-root-of-1-2-sqrt3-2.htmlPlaying with Maple, I noticed that it gives the square root of $c = 1+\frac{\sqrt3}{2}$ as equal to $a = \frac{1}{2}+\frac{\sqrt3}{2}$. Indeed it checks out. But I got curious: how can I find that......Thu, 03 Sep 2026 07:15:50 -0400https://pop.com.sg/the-price-is-right-optimal-play.htmlhttps://pop.com.sg/the-price-is-right-optimal-play.htmlThe following situation happened on the Price is Right and I was curious about the optimal response. The rules are: A contestant rolls a wheel with 5 cent increments from 5 - 100 (20 numbers total)......Thu, 03 Sep 2026 07:13:03 -0400https://pop.com.sg/given-that-log-ab-log-ba-and-that-a-b-ne-1-and-a-ne-b-find-b-in-terms-.htmlhttps://pop.com.sg/given-that-log-ab-log-ba-and-that-a-b-ne-1-and-a-ne-b-find-b-in-terms-.htmlGiven that $\log_ba=\log_ab$, and that $a,b \ne 1$ and $a \ne b$, find $b$ in terms of $a$. I tried to solve this problem using changing the base of the first part of the equation: $$ \frac{\log......Thu, 03 Sep 2026 07:12:39 -0400https://pop.com.sg/what-are-expressions-in-mathematics.htmlhttps://pop.com.sg/what-are-expressions-in-mathematics.htmlLike algebraic expressions are logarithmic, Experimental, trigonometric, differential, etc., expressions also there? I am not referring to functions, but just expressions or equations. Are their ...Thu, 03 Sep 2026 07:08:55 -0400https://pop.com.sg/find-a-unit-vector-that-maximizes-the-directional-derivative-at-a-poin.htmlhttps://pop.com.sg/find-a-unit-vector-that-maximizes-the-directional-derivative-at-a-poin.htmlGiven the function $f(x,y)=g(2x+y), g'(3)=3$ Find the coordinates of a unit vector $v=(v_1,v_2)$ that finds the maximum directional derivative $D_ uf(1,1)$ of f in the point $(1,1)$. I know that I ...Thu, 03 Sep 2026 07:08:45 -0400https://pop.com.sg/probability-of-random-item-selection.htmlhttps://pop.com.sg/probability-of-random-item-selection.htmlIn a 15-piece batch there are 5 black balls and the 10 remaining balls are white. 3 balls are selected randomly. The batch is discarded if among the chosen ones at least one selected ball is white.......Thu, 03 Sep 2026 07:04:11 -0400https://pop.com.sg/find-the-length-of-parametric-curves-simple.htmlhttps://pop.com.sg/find-the-length-of-parametric-curves-simple.htmlso I have the following parametric curve: Now I'm trying to figure out how I should tackle this, I know the formula for finding lengths is $\int[\sqrt{g'(x)^2+f'(x)^2}]$ or when its a single ...Thu, 03 Sep 2026 07:00:25 -0400https://pop.com.sg/calculating-a-chromatic-polynomial.htmlhttps://pop.com.sg/calculating-a-chromatic-polynomial.htmlI am going through some questions in the "Bondy,Murty - Graph Theory with applications" book, and I have stumbled upon the following question: calculate the Chromatic Polynomial of the following ......Thu, 03 Sep 2026 06:56:25 -0400https://pop.com.sg/remainder-of-an-integer-divided-by-5-is-the-same-as-the-remainder-of-t.htmlhttps://pop.com.sg/remainder-of-an-integer-divided-by-5-is-the-same-as-the-remainder-of-t.htmlwe have been told in arithmetic that the remainder of an integer divided by 5 is the same as the remainder of the division of the rightmost digit by 5 how to use modular properties to prove this?...Thu, 03 Sep 2026 06:55:40 -0400https://pop.com.sg/understanding-the-difference-between-span-and-basis.htmlhttps://pop.com.sg/understanding-the-difference-between-span-and-basis.htmlI've been reading a bit around MSE and I've stumbled upon some similar questions as mine. However, most of them do not have a concrete explanation to what I'm looking for. I understand that the Sp......Thu, 03 Sep 2026 06:55:39 -0400https://pop.com.sg/proof-by-induction-sequence-of-integers.htmlhttps://pop.com.sg/proof-by-induction-sequence-of-integers.htmlSuppose a sequence of integers $a_1$, $a_2$, ... is defined as: $$a_1 = 3$$ $$a_2 = 6$$ $$a_n = 5a_{n-1} - 6a_{n-2} + 2$$ for all $n\ge3$ $\mathbf {Prove}$ $\mathbf {S(n)}$: $a_n = 1 + 2^{n-1} + ......Thu, 03 Sep 2026 06:38:00 -0400https://pop.com.sg/check-if-line-intersects-with-circles-perimeter.htmlhttps://pop.com.sg/check-if-line-intersects-with-circles-perimeter.htmlSay I have this circle here. What would be an algorithm to check if lines like the green or blue lines intersect the edge of the circle, but not the red line. So lets say if(amountOfPointsHit(l......Thu, 03 Sep 2026 06:20:04 -0400https://pop.com.sg/parameterization-of-a-part-of-a-sphere.htmlhttps://pop.com.sg/parameterization-of-a-part-of-a-sphere.htmlI have to parametrize $D=\{x^2+y^2+z^2\le 25,y\le -4\}$. I can see the I have to parametrize 2 surfaces : ($S_1$) the intersection between the plane $z=-4$ and the sphere: ($x^2+z^2\le 9$) ($S_2......Thu, 03 Sep 2026 06:12:38 -0400https://pop.com.sg/computing-the-expectation-of-the-square-of-a-random-variable-text-e-x-.htmlhttps://pop.com.sg/computing-the-expectation-of-the-square-of-a-random-variable-text-e-x-.htmlWhat is the rule for computing $ \text{E}[X^{2}] $, where $ \text{E} $ is the expectation operator and $ X $ is a random variable? Let $ S $ be a sample space, and let $ p(x) $ denote the probabil......Thu, 03 Sep 2026 06:07:16 -0400https://pop.com.sg/how-do-i-find-the-lateral-surface-area-of-an-elliptical-cone-not-a-fru.htmlhttps://pop.com.sg/how-do-i-find-the-lateral-surface-area-of-an-elliptical-cone-not-a-fru.htmlThe formula for the lateral surface area of a right circular cone is $\pi*r*l$ where $r$ is the circle's radius. But ellipse has a semi-major axis as well as a semi-minor one. Which one of them sho......Thu, 03 Sep 2026 05:49:57 -0400https://pop.com.sg/farmers-pen-part-a-part-b.htmlhttps://pop.com.sg/farmers-pen-part-a-part-b.htmlA) A rectangular pen is built with one side against a barn, 200 meters of fencing are used for the other three sides of the pen. What dimensions maximize the area of the pen? B) A rancher plans to......Thu, 03 Sep 2026 05:39:29 -0400https://pop.com.sg/how-do-you-calculate-a-fee-percentage-to-handle-a-fee-being-charged.htmlhttps://pop.com.sg/how-do-you-calculate-a-fee-percentage-to-handle-a-fee-being-charged.htmlProblem is that we get charged a 3% fee. We add this 3% fee to the invoice. When we get the amount back they charge 3% on the invoice plus on the fee we added. What formula can I use to figure out ...Thu, 03 Sep 2026 05:35:57 -0400https://pop.com.sg/what-is-the-difference-between-a-partition-and-a-subinterval-closed.htmlhttps://pop.com.sg/what-is-the-difference-between-a-partition-and-a-subinterval-closed.htmlFor example the norm of a partition is the widest subinterval...Thu, 03 Sep 2026 05:34:14 -0400https://pop.com.sg/which-pair-of-ratios-form-a-proportion.htmlhttps://pop.com.sg/which-pair-of-ratios-form-a-proportion.htmlMy daughter is stuck on this question and we're not sure what the answer is. I'm rusty with math skills, so I don't understand how to help her, or what approaches she's tried. Please help us! 10.5/......Thu, 03 Sep 2026 05:31:03 -0400https://pop.com.sg/what-is-the-condition-for-the-linear-system-to-be-consistent.htmlhttps://pop.com.sg/what-is-the-condition-for-the-linear-system-to-be-consistent.htmlconsider a linear system as follows $a_{1,1}\ x_1+a_{1,2}\ x_2+...+a_{1,n}\ x_n = b_1$ $a_{2,1}\ x_1+a_{2,2}\ x_2+...+a_{2,n}\ x_n = b_2$ ... $a_{m,1}\ x_1+a_{m,2}\ x_2+...+a_{m,n}\ x_n = b_m$ ......Thu, 03 Sep 2026 05:29:42 -0400https://pop.com.sg/differential-equation-find-the-complete-solution-closed.htmlhttps://pop.com.sg/differential-equation-find-the-complete-solution-closed.htmlHey guys I'm stuck with a Differential equation. Can anyone help me finding the complete solution to $dy/dx + 2y = 3e^x$ Any help would be greatly appreciated! :)...Thu, 03 Sep 2026 05:25:46 -0400https://pop.com.sg/what-is-the-point-of-subspaces-duplicate.htmlhttps://pop.com.sg/what-is-the-point-of-subspaces-duplicate.htmlI've been studying for my linear algebra final for the past few days, and just had this thought. I know what a subspace is; It's a subset of a vector space that contains the zero vector, and prese......Thu, 03 Sep 2026 05:05:43 -0400https://pop.com.sg/pseudo-inverse-of-product-of-matrices.htmlhttps://pop.com.sg/pseudo-inverse-of-product-of-matrices.htmlLet $A$ and $B$ are two matrices where $A \in \mathbb{R}^{m\times p}$ and $B \in \mathbb{R}^{p\times n}$ and both $A$ and $B$ are full rank matrices Now I really want to know in what cases $(AB)^......Thu, 03 Sep 2026 04:59:02 -0400https://pop.com.sg/why-is-the-magnitude-of-the-vectors-still-not-1-after-normalization-cl.htmlhttps://pop.com.sg/why-is-the-magnitude-of-the-vectors-still-not-1-after-normalization-cl.htmlI was reading that the vectors must have magnitude of $1$ to be called normalized; however, none of the problems I solved gave me a vector magnitude of $1$ after normalizing. I also thought that ta......Thu, 03 Sep 2026 04:46:12 -0400https://pop.com.sg/if-a-number-irrational-can-it-be-rational-in-some-rational-based-numbe.htmlhttps://pop.com.sg/if-a-number-irrational-can-it-be-rational-in-some-rational-based-numbe.htmlThe number $\frac{22}{7}$ is irrational in our base-$10$ system, but in, say, base-$14$, it is rational (it comes out to $3.2$ in that system). It's easy for fractions that are irrational as decim......Thu, 03 Sep 2026 04:32:54 -0400https://pop.com.sg/show-that-ln-a-b-ln-a-ln-b-when-a-frac-b-b-1.htmlhttps://pop.com.sg/show-that-ln-a-b-ln-a-ln-b-when-a-frac-b-b-1.htmlShow that $\ln(a+b) = \ln(a) + \ln(b)$ when $a = \frac{b}{b-1}$ My attempt at a solution was $$ \ln(a+b) = \ln \left(a(1+\frac{b}{a})\right) $$ and by setting $a = \frac{b}{b-1}$ we get $$ \ln ......Thu, 03 Sep 2026 04:25:12 -0400https://pop.com.sg/proof-that-sqrt-5-is-irrational.htmlhttps://pop.com.sg/proof-that-sqrt-5-is-irrational.htmlIn my textbook the following proof is given for the fact that $\sqrt{5}$ is irrational: $ x = \frac{p}{q}$ and $x^2 = 5$. We choose $p$ and $q$ so that the have no common factors, so we know that ......Thu, 03 Sep 2026 04:22:37 -0400https://pop.com.sg/how-to-find-the-sum-of-an-alternating-series.htmlhttps://pop.com.sg/how-to-find-the-sum-of-an-alternating-series.htmlFind whether the series diverges and its sum: $$\sum_{n = 1}^\infty (-1)^{n+1} \frac{3}{5^n}.$$ I found that the series converges using the Alternating Series test because the absolute value of ......Thu, 03 Sep 2026 04:12:53 -0400https://pop.com.sg/exterior-derivative-of-wedge-product-duplicate.htmlhttps://pop.com.sg/exterior-derivative-of-wedge-product-duplicate.htmlHow can I show that $$ \mathrm d(a\wedge b)=(\mathrm d\,a)\wedge b + (-1)^{q}a\wedge(\mathrm d\,b) $$ for a $q$-form $a$ and an $r$-form $b$?...Thu, 03 Sep 2026 04:11:48 -0400https://pop.com.sg/differentiate-f-x-frac-sec-x-1-tan-x.htmlhttps://pop.com.sg/differentiate-f-x-frac-sec-x-1-tan-x.htmlI don't understand line 3. After using the Quotient Rule, how do I use the Trig identities to simplify the third line? Also, what is the strategy when using trig identities, when do you know you......Thu, 03 Sep 2026 04:09:14 -0400https://pop.com.sg/definition-of-sinc-function.htmlhttps://pop.com.sg/definition-of-sinc-function.htmlI just want to make clear of the definition of sinc(x). I know there is a normalized and unnormalized definition for the sinc function. If we have unnormalized sinc then we have: $$\sin(x)/x=\text{......Thu, 03 Sep 2026 04:09:07 -0400https://pop.com.sg/base-change-for-proper-morphism-in-hartshorne.htmlhttps://pop.com.sg/base-change-for-proper-morphism-in-hartshorne.htmlAfter reading Section III.11 (The theorem on Formal functions) and III.12 (The semicontinuity Theorem), I feel that I get some kind of formalism instead of a clear picture of what is going on. So ......Thu, 03 Sep 2026 04:07:00 -0400https://pop.com.sg/proof-that-a-finer-partition-of-an-interval-has-smaller-than-or-equal-.htmlhttps://pop.com.sg/proof-that-a-finer-partition-of-an-interval-has-smaller-than-or-equal-.htmlThe question asks to prove that if a partition $Q$ of some interval $I$ is finer than partition $P$ on the same interval$(Q \supseteq P)$ then the norm of the finer partition is less than or equal......Thu, 03 Sep 2026 03:56:55 -0400https://pop.com.sg/tensor-product-of-3-vectors.htmlhttps://pop.com.sg/tensor-product-of-3-vectors.htmlI understand how to do a tensor product of 2 vectors, but I am not quite sure how to incorporate a third one. Can someone explain to me how to compute tensor product if I have 3 vectors $$u\otimes v\...Thu, 03 Sep 2026 03:56:36 -0400https://pop.com.sg/alternate-proof-of-unique-lifting-property-of-covering-spaces.htmlhttps://pop.com.sg/alternate-proof-of-unique-lifting-property-of-covering-spaces.htmlI proved one of Hatcher's propositions on my own and my proof is quite a bit different than his. The Unique Lifting Property says: Given a covering space $p:\tilde{X} \rightarrow X$ and a map $f......Thu, 03 Sep 2026 03:56:08 -0400https://pop.com.sg/discriminant-of-the-depressed-cubic.htmlhttps://pop.com.sg/discriminant-of-the-depressed-cubic.htmlHere is the question I'm currently looking at: Show that the discriminant of the equation $y^3+py+q=0$ is $-4p^3-27q^2$. I've done some research and found this, but we haven't studied Vieta's the......Thu, 03 Sep 2026 03:54:09 -0400https://pop.com.sg/writing-expression-in-terms-of-ln2-and-ln3.htmlhttps://pop.com.sg/writing-expression-in-terms-of-ln2-and-ln3.htmlExpress $\ln \sqrt[3]{32}$ in terms of $\ln2$ and/or $\ln3$ My try: $\ln\sqrt[3]{32}= \ln(32^{1/3})= \frac{1}{3} \ln32 = \frac{1}{3} \ln2^5$ This isn't correct because there's a $5$...how d......Thu, 03 Sep 2026 03:52:00 -0400https://pop.com.sg/sufficient-statistics-for-lambda-poisson-distribution.htmlhttps://pop.com.sg/sufficient-statistics-for-lambda-poisson-distribution.htmlI have the following question out of a book. I even have the solution from solutions manual that I cannot really follow either, so I thought I would ask here to see if someone could dumb it down f......Thu, 03 Sep 2026 03:35:56 -0400https://pop.com.sg/user-alon-amit-mathematics-stack-exchange.htmlhttps://pop.com.sg/user-alon-amit-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Thu, 03 Sep 2026 03:34:07 -0400https://pop.com.sg/how-to-solve-ax-xb-for-x-matrix.htmlhttps://pop.com.sg/how-to-solve-ax-xb-for-x-matrix.htmlI have two symmetric $3\times 3$ matrices $A, B$. I am interested in solving the system $$AX= XB$$ Is there a way this is usually done? The matrices are not necessarily non singular....Thu, 03 Sep 2026 03:32:22 -0400https://pop.com.sg/prove-det-ka-k-n-det-a-closed.htmlhttps://pop.com.sg/prove-det-ka-k-n-det-a-closed.htmlLet $A$ be an $n \times n$ matrix and $k$ be a scalar. Prove that $\det(kA)=k^n\det A$. I really don't know where to start. Can someone give me a hint for this proof?...Thu, 03 Sep 2026 03:23:50 -0400https://pop.com.sg/what-is-an-interpretation-of-the-matrix-exponential.htmlhttps://pop.com.sg/what-is-an-interpretation-of-the-matrix-exponential.htmlI just read about the existence of the "matrix exponential" $$e^X := \sum_{k = 0}^\infty\frac1{k!}X^k$$ Is there a simple way to interpret this? I understand the analog between real number ...Thu, 03 Sep 2026 03:22:05 -0400https://pop.com.sg/last-step-of-cardano-method-extracting-qube-roots-of-complex-numbers.htmlhttps://pop.com.sg/last-step-of-cardano-method-extracting-qube-roots-of-complex-numbers.htmlWe can derive roots of depressed cubic using Cardano method: $x=u+v$ (1) We then substitute x, group variables: $u^6+q \cdot u^3- \frac{p^3}{27} = 0$ (2) or $w^2+q \cdot w- \frac{p^3}{27} = 0$ (3)......Thu, 03 Sep 2026 03:21:44 -0400https://pop.com.sg/what-are-left-and-right-singular-vectors-in-svd.htmlhttps://pop.com.sg/what-are-left-and-right-singular-vectors-in-svd.htmlLet $USV^T$ is a singular value decomposition of matrix $A$. In the textbook "Linear Algebra and Its Applications" by D. C. Lay et. al., where SVD is introduced, it says that "the co......Thu, 03 Sep 2026 03:21:10 -0400https://pop.com.sg/the-perfect-sharing-algorithm-abbabaab.htmlhttps://pop.com.sg/the-perfect-sharing-algorithm-abbabaab.htmlLess of a question and more of an exercise, it has to do with something I found while doing some programming and being unable to find things. Basically I wanted a formula for the perfect sharing ...Thu, 03 Sep 2026 03:12:08 -0400https://pop.com.sg/the-derivation-of-the-wald-interval.htmlhttps://pop.com.sg/the-derivation-of-the-wald-interval.htmlI'm asking about the binomial proportion confidence interval, also known as the Wald interval. Recall that $$\lim_{n \to \infty}{P_p \left( -z_{1-\frac{\alpha}{2}}\frac{\sigma}{\sqrt{n}}+\bar{X_n}......Thu, 03 Sep 2026 03:10:21 -0400https://pop.com.sg/difference-between-strictly-increasing-and-increasing-functions.htmlhttps://pop.com.sg/difference-between-strictly-increasing-and-increasing-functions.htmlWhat is the precise difference between strictly increasing and increasing functions?? I see these terms being thrown around a lot My guess is that strictly increasing mean that derivative is only g......Thu, 03 Sep 2026 03:01:25 -0400https://pop.com.sg/why-does-f-ax-039-a-f-039-ax-that-is-what-effect-does-the-horizontal-d.htmlhttps://pop.com.sg/why-does-f-ax-039-a-f-039-ax-that-is-what-effect-does-the-horizontal-d.htmlToday my Calculus teacher told us that: $[f(ax)]'=a*f'(ax)$ This fact was unfortunately not proven, and I do not understand why it is so. It is clear to me that the graph of $y =f(ax)$ will be a ...Thu, 03 Sep 2026 02:44:34 -0400https://pop.com.sg/how-to-find-the-oblique-asymptote-of-root-of-a-function.htmlhttps://pop.com.sg/how-to-find-the-oblique-asymptote-of-root-of-a-function.htmlIn a test example I'm solving, the question asks to find the oblique asymptote of the following function: $f(x) = \sqrt{4x^2+x+6}$ $x$ at $+\infty$ We have only learned how to do so with rational ...Thu, 03 Sep 2026 02:43:20 -0400https://pop.com.sg/obtaining-uncertainty-in-linear-regression.htmlhttps://pop.com.sg/obtaining-uncertainty-in-linear-regression.htmlI'm trying to find the uncertainty values for a set of data points' slope and intercept. The values are: $(0.18751,0.512332), (0.17076, 0.511825), (0.23204,0.513665), (0.20878, 0.512986), (0.17172......Thu, 03 Sep 2026 02:39:30 -0400https://pop.com.sg/finding-the-inverse-of-the-function-f-defined-by-f-x-x-2-4x-for-x-geq-.htmlhttps://pop.com.sg/finding-the-inverse-of-the-function-f-defined-by-f-x-x-2-4x-for-x-geq-.htmlLet $f$ be the function defined by $$f(x)=x^2-4x$$ for $x\ge 2$. Find the inverse function indicating its domain and the range. When I try to make $x$ the subject I get $+$and $-$ two answers for......Thu, 03 Sep 2026 02:38:43 -0400https://pop.com.sg/how-to-memorize-the-sum-and-product-of-roots-for-an-n-th-degree-equati.htmlhttps://pop.com.sg/how-to-memorize-the-sum-and-product-of-roots-for-an-n-th-degree-equati.htmlFor my exams I need to know the following equations by heart: for a polynomial equation: $a_nx+a_{n-1}x^{n-1}+...+a_1x+a_0,$ the sum and product of the roots are given by $$\textrm{Sum}=-\frac{a_......Thu, 03 Sep 2026 02:38:35 -0400https://pop.com.sg/how-find-constant-speed.htmlhttps://pop.com.sg/how-find-constant-speed.htmlA car travels up a hill at a constant speed of 19 km/h and returns down the hill at a constant speed of 50 km/h. Calculate the average speed for the round trip. Am I supposed to add the numbers, ......Thu, 03 Sep 2026 02:36:38 -0400https://pop.com.sg/finding-the-area-of-an-isosceles-right-triangle-given-its-hypotenuse.htmlhttps://pop.com.sg/finding-the-area-of-an-isosceles-right-triangle-given-its-hypotenuse.htmlI was doing a problem in which I was told the lengths of all sides of an isosceles triangle and asked to find the area. I solved the problem by dividing the isosceles triangle into two equal trian......Thu, 03 Sep 2026 02:22:53 -0400https://pop.com.sg/proving-the-fundamental-theorem-of-calculus-without-mvt.htmlhttps://pop.com.sg/proving-the-fundamental-theorem-of-calculus-without-mvt.htmlIn this question Proof of the Mean Value Theorem for Integrals, someone proves the MVT for integrals using the fundamental theorem of calculus. I have seen another proof like this as well: If $f$ is ...Thu, 03 Sep 2026 01:49:20 -0400https://pop.com.sg/is-there-a-symbol-for-potential-equality.htmlhttps://pop.com.sg/is-there-a-symbol-for-potential-equality.htmlIs there a symbol for potential equality? Essentially I'd like to condense: $$ (a = b) \lor (a \ne b) $$ so that I can express the phrase "a may or may not be equal to b". Apologies if my syntax ......Thu, 03 Sep 2026 01:26:05 -0400https://pop.com.sg/what-is-a-real-world-example-of-zero-work-done-by-a-conservative-vecto.htmlhttps://pop.com.sg/what-is-a-real-world-example-of-zero-work-done-by-a-conservative-vecto.htmlI have only a high school physics background, so when I study the later parts of multivariable calculus, e.g., Greens, Gauss, and Stokes' theorems, there are some topics that I only know the mathem......Thu, 03 Sep 2026 01:21:20 -0400https://pop.com.sg/calculating-time-period-of-oscillation-of-a-mass-on-a-spring-closed.htmlhttps://pop.com.sg/calculating-time-period-of-oscillation-of-a-mass-on-a-spring-closed.htmlI have the question: "A mass of $10$ kg bounces up and down on a spring. The spring constant is $250 $ N m$^{-1}$. Calculate the time period of the oscillation." I know that time period $T =......Thu, 03 Sep 2026 01:05:43 -0400https://pop.com.sg/finding-polynomial-with-root-sqrt-2-sqrt-3-over-mathbb-q-what-is-the-d.htmlhttps://pop.com.sg/finding-polynomial-with-root-sqrt-2-sqrt-3-over-mathbb-q-what-is-the-d.htmla) In $\mathbb{R}$, $\sqrt{2}$ and $\sqrt{3}$ are algebric over $\mathbb{Q}$. Find the polynomial of degree $4$ over $\mathbb{Q}$ satisfiable by $\sqrt{2}+\sqrt{3}$ b) Wich is the degree of $\sqrt......Thu, 03 Sep 2026 01:02:35 -0400https://pop.com.sg/is-it-possible-for-a-quadratic-equation-to-have-one-rational-root-and-.htmlhttps://pop.com.sg/is-it-possible-for-a-quadratic-equation-to-have-one-rational-root-and-.htmlIs it possible for a quadratic equation to have one rational root and one irrational root? Yes, a pretty straightforward question. Is it possible?...Thu, 03 Sep 2026 00:57:04 -0400https://pop.com.sg/what-is-the-integral-of-n-th-root-of-tan-x.htmlhttps://pop.com.sg/what-is-the-integral-of-n-th-root-of-tan-x.htmlWhat will be the value of $ \int\sqrt[n]{\tan x},dx $ ? I have solved the cases for n=2 and n=3 but can't see how I can generalize it....Thu, 03 Sep 2026 00:56:43 -0400https://pop.com.sg/prove-that-x-1-p-equiv-x-p-1-mod-p-2-closed.htmlhttps://pop.com.sg/prove-that-x-1-p-equiv-x-p-1-mod-p-2-closed.htmlLet $p$ be a prime number and $x$ and integer such that $x^{2p}+x^p-1\equiv 0\mod p^2$. Prove that $(x+1)^p\equiv x^p+1 \mod p^2$. It seems to me there is no such $x$ in the first place. Any thou......Thu, 03 Sep 2026 00:54:18 -0400https://pop.com.sg/what-is-the-remainder-when-4-44-is-divided-by-44-closed.htmlhttps://pop.com.sg/what-is-the-remainder-when-4-44-is-divided-by-44-closed.htmlPlease suggest a suitable approach for this problem....Thu, 03 Sep 2026 00:41:25 -0400https://pop.com.sg/what-does-x-divides-y-mean.htmlhttps://pop.com.sg/what-does-x-divides-y-mean.htmlI need to negate the following sentence: "If for the integers $x, y, z$ we know that $x$ divides $y$ and $y$ divides $z$, then $x$ divides $z$." In this scenario, what does it mean for $x$ to "di......Thu, 03 Sep 2026 00:38:31 -0400https://pop.com.sg/negation-of-0-1.htmlhttps://pop.com.sg/negation-of-0-1.htmlI'm taking my first proof-heavy class (real analysis), and one practice problem on the first homework is to write the negation of $$0 = 1$$ My immediate thought was that it would simply be $$0 ......Thu, 03 Sep 2026 00:27:48 -0400https://pop.com.sg/finding-the-compound-interest-on-7500-dollars-at-4-per-annum-for-2-yea.htmlhttps://pop.com.sg/finding-the-compound-interest-on-7500-dollars-at-4-per-annum-for-2-yea.htmlFind compound interest on $\$7500$ at $4\%$ per annum for $2$ years, compounded annually. The choices are as follow: $\$512$, $\$552$, $\$612$, $\$622$. I tried to solve this problem by: C.I. ......Thu, 03 Sep 2026 00:14:16 -0400https://pop.com.sg/why-is-a-perfect-group-called-a-perfect-group.htmlhttps://pop.com.sg/why-is-a-perfect-group-called-a-perfect-group.htmlA group is called perfect if we have $[G,G]=G$. I was wondering in what sense is this group perfect? I've never really done anything much with perfect groups so I don't really know anything about ...Thu, 03 Sep 2026 00:09:40 -0400https://pop.com.sg/finding-the-autocorrelation-of-a-sine-wave.htmlhttps://pop.com.sg/finding-the-autocorrelation-of-a-sine-wave.htmlThe autocorrelation of sin(t) is defined as $$\displaystyle \int_{-\infty}^{\infty} \sin(t+\tau)\sin(t)d\tau$$ I've tried using the Wiener-Khinchin theorem which says that $$Corr(g,g)\...Wed, 02 Sep 2026 23:59:57 -0400https://pop.com.sg/common-area-between-2-polar-functions.htmlhttps://pop.com.sg/common-area-between-2-polar-functions.htmlFind the common area between the two polar functions: $r=\sin\theta$ and $r=\cos\theta$; however, I'm not too sure about how to set up the bounds for the integral. I know the functions have the same ...Wed, 02 Sep 2026 23:55:59 -0400https://pop.com.sg/find-the-general-solution-of-the-given-second-order-differential-equat.htmlhttps://pop.com.sg/find-the-general-solution-of-the-given-second-order-differential-equat.htmlFind the general solution of the given second order differential equation. $$4y''+y'=0$$ This was my procedure to solving this problem: $\chi(r)=4r^2+r=0$ $r(4r+1)=0$ $r=0, -\frac14$ $y_1=e^{0x}, y......Wed, 02 Sep 2026 23:50:01 -0400https://pop.com.sg/why-does-the-derivative-graph-of-a-curve-look-linear-and-not-curvy.htmlhttps://pop.com.sg/why-does-the-derivative-graph-of-a-curve-look-linear-and-not-curvy.htmlWhy is the derivative graph of $y=x^2$ linear and not some sort of curve? I know that $\frac{d}{dx}x^2=2x$, but I am not talking computing algebraically, but more conceptually, thinking in terms o......Wed, 02 Sep 2026 23:44:23 -0400https://pop.com.sg/what-does-identically-distributed-mean-closed.htmlhttps://pop.com.sg/what-does-identically-distributed-mean-closed.htmlWhen two distributions have the same variance and shape, do we call them identically distributed, regardless of their mean (as this is usually a location parameter)?...Wed, 02 Sep 2026 23:33:52 -0400https://pop.com.sg/why-must-the-pythagorean-theorem-contain-squared-values-is-there-a-rel.htmlhttps://pop.com.sg/why-must-the-pythagorean-theorem-contain-squared-values-is-there-a-rel.htmlIn a way it is self-evident that the sides of a right triangle have a special relationship to one another; visual proofs illustrate this. However, I am at a loss for how to describe this relationship ...Wed, 02 Sep 2026 23:32:12 -0400