https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usThu, 03 Sep 2026 02:37:44 -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$$ This was my procedure to solving this problem: $\chi(r)=4r^2+r=0$ $r(4r+1)=0$ $r=0, -\frac14$ $y_1=e^{0x}, y......Wed, 02 Sep 2026 23:50:01 -0400https://pop.com.sg/why-does-the-derivative-graph-of-a-curve-look-linear-and-not-curvy.htmlhttps://pop.com.sg/why-does-the-derivative-graph-of-a-curve-look-linear-and-not-curvy.htmlWhy is the derivative graph of $y=x^2$ linear and not some sort of curve? I know that $\frac{d}{dx}x^2=2x$, but I am not talking computing algebraically, but more conceptually, thinking in terms o......Wed, 02 Sep 2026 23:44:23 -0400https://pop.com.sg/what-does-identically-distributed-mean-closed.htmlhttps://pop.com.sg/what-does-identically-distributed-mean-closed.htmlWhen two distributions have the same variance and shape, do we call them identically distributed, regardless of their mean (as this is usually a location parameter)?...Wed, 02 Sep 2026 23:33:52 -0400https://pop.com.sg/why-must-the-pythagorean-theorem-contain-squared-values-is-there-a-rel.htmlhttps://pop.com.sg/why-must-the-pythagorean-theorem-contain-squared-values-is-there-a-rel.htmlIn a way it is self-evident that the sides of a right triangle have a special relationship to one another; visual proofs illustrate this. However, I am at a loss for how to describe this relationship ...Wed, 02 Sep 2026 23:32:12 -0400https://pop.com.sg/deriving-properties-of-logarithms-from-the-definition-ln-x-int-1-x-fra.htmlhttps://pop.com.sg/deriving-properties-of-logarithms-from-the-definition-ln-x-int-1-x-fra.htmlAssuming the definition of the logarithmic function as follows: $$\ln{x}=\int_1^x \frac{dt}{t}$$ I managed to derive the formulas for logarithms as we know them: $$\ln(xy)=\int_1^{xy}\frac{dt}{t}=\......Wed, 02 Sep 2026 23:24:10 -0400https://pop.com.sg/well-posed-problem.htmlhttps://pop.com.sg/well-posed-problem.htmlIn the definition of a well-posed problem it states that a problem is well posed if: 1.A solution exists. 2.The solution is unique. 3.The solution's behaviour changes continuously with the initial ...Wed, 02 Sep 2026 23:19:50 -0400https://pop.com.sg/quaternion-differentiation.htmlhttps://pop.com.sg/quaternion-differentiation.htmlI have an application that tracks an image and estimates its position and orientation. The orientation is given by a quaternion, and it is modified by an angular velocity every frame. To predict the ...Wed, 02 Sep 2026 22:55:45 -0400https://pop.com.sg/a-quick-way-for-graphing-cubic-functions.htmlhttps://pop.com.sg/a-quick-way-for-graphing-cubic-functions.htmlCubic function is $f(x) = ax^3 + bx^2 + cx + d , a\not = 0$ . I'm looking for a general way to plot it quickly . My mean is the plot it just by looking at the coefficients (i.e. $a , b ,c ,d$) . I ......Wed, 02 Sep 2026 22:52:18 -0400https://pop.com.sg/shift-numbers-into-a-different-range.htmlhttps://pop.com.sg/shift-numbers-into-a-different-range.htmlI was wondering how can I shift my data that fall between a range lets say [0, 125] to another range like [-128, 128]. Thanks for any help...Wed, 02 Sep 2026 22:46:09 -0400https://pop.com.sg/numerically-stable-extraction-of-axis-angle-from-unit-quaternion.htmlhttps://pop.com.sg/numerically-stable-extraction-of-axis-angle-from-unit-quaternion.htmlI am looking to extract an axis-angle representation from a unit quaternion. From the definition, a naive attempt might be: $ q = \begin{bmatrix} cos(\theta/2) \\ \omega_x \sin(\theta/2) \\ \omeg......Wed, 02 Sep 2026 22:43:42 -0400https://pop.com.sg/solving-cos-2x-cos-left-x-frac-pi-6-right-sin-2x-sin-left-frac-pi-6-x-.htmlhttps://pop.com.sg/solving-cos-2x-cos-left-x-frac-pi-6-right-sin-2x-sin-left-frac-pi-6-x-.htmlSolve this equation for $x\in (0 , \frac{\pi}{2})$ $$\cos(2x)\cos\left(x - \frac{\pi}{6}\right) = \sin(2x)\sin\left(\frac{\pi}{6} - x\right)$$ I gave a try using the $\cos(a-b)$ and $\sin(a-b)$ fo......Wed, 02 Sep 2026 22:41:12 -0400https://pop.com.sg/what-angle-must-two-equal-length-lines-be-at-s-t-a-line-from-the-open-.htmlhttps://pop.com.sg/what-angle-must-two-equal-length-lines-be-at-s-t-a-line-from-the-open-.htmlI've been thinking about this problem in my head for years, and finally sat down and drew it in CAD iterating until I could get an approximate solution, but the math is beyond me. Here is an image ......Wed, 02 Sep 2026 22:35:40 -0400https://pop.com.sg/find-the-angle-of-depression-given-the-following-two-right-triangle.htmlhttps://pop.com.sg/find-the-angle-of-depression-given-the-following-two-right-triangle.htmlA person in a control tower is looking down at an airplane on the runaway that is 78 yards away from the base of the tower.If this tower is 27 yards high, find the angle of depression from the towe......Wed, 02 Sep 2026 22:30:59 -0400https://pop.com.sg/finding-the-base-radius-of-the-cone.htmlhttps://pop.com.sg/finding-the-base-radius-of-the-cone.htmlA solid cone has a lateral surface area of $100\pi$ square centimeters and a total surface area of $269\pi$ square centimeters. Find the base radius of the cone. I would need to find the radius or ...Wed, 02 Sep 2026 22:26:13 -0400https://pop.com.sg/need-help-solving-an-integral-for-lagrange-remainder-proof.htmlhttps://pop.com.sg/need-help-solving-an-integral-for-lagrange-remainder-proof.htmlThis image of part of a proof for the Lagrange Remainder for Taylor's Formula. I need help solving the last integral. Can anyone explain?...Wed, 02 Sep 2026 22:23:19 -0400https://pop.com.sg/what-is-a-section-exactly.htmlhttps://pop.com.sg/what-is-a-section-exactly.htmlI have read 4 chapters of Hartshorne's Algebraic Geometry, when I go back to the beginning of scheme and the definition of a section, I am kind of confused why we call such a element "section". Le......Wed, 02 Sep 2026 22:20:03 -0400https://pop.com.sg/one-divided-by-infinity-is-not-zero-duplicate.htmlhttps://pop.com.sg/one-divided-by-infinity-is-not-zero-duplicate.htmlI know that $\frac{1}{\infty}$ is undefined. But my question is - can we say that $\frac{1}{\infty}\neq0$ ? I've got some idea how to explain that: Let's say we have a random-number generator that ...Wed, 02 Sep 2026 22:16:11 -0400https://pop.com.sg/what-is-poisson-point-process.htmlhttps://pop.com.sg/what-is-poisson-point-process.html"Points $\{A_j\}_{j\in\Phi(\lambda)}$ are assumed to be distributed according to a homogeneous PPP with intensity $\lambda$, denoted $\Phi(\lambda)=\{X_j\}$, where $X_j$ is the location of the $j$th ...Wed, 02 Sep 2026 22:14:44 -0400https://pop.com.sg/tan-x-cot-90-circ-x.htmlhttps://pop.com.sg/tan-x-cot-90-circ-x.htmlI was looking at a mark scheme for a question I was stuck on, and I came across this. You are asked to work out the value of $\tan 75^\circ$ after you've worked out $\cos 15^\circ$ and $\sin 15^\ci......Wed, 02 Sep 2026 22:14:39 -0400https://pop.com.sg/how-to-find-p-value.htmlhttps://pop.com.sg/how-to-find-p-value.htmlI am badly strugling to find P-values. I have no idea how it is coming for every problem I am doing. For example, the question below, how did they get/choose alpha 0.1 and 0.5 to calculate calculat......Wed, 02 Sep 2026 22:04:22 -0400https://pop.com.sg/how-does-multiplying-by-trigonometric-functions-in-a-matrix-transform-.htmlhttps://pop.com.sg/how-does-multiplying-by-trigonometric-functions-in-a-matrix-transform-.htmlI found this comic: But I can't understand the humor because I can't understand how trig functions affect matrix multiplication. Can someone please explain?...Wed, 02 Sep 2026 22:03:52 -0400https://pop.com.sg/how-many-nodes-before-k-clique-or-k-anti-clique.htmlhttps://pop.com.sg/how-many-nodes-before-k-clique-or-k-anti-clique.htmlI am attempting to solve some problems here. For exercise 1, the tightest result I could get is $4^k$. Is that the mininum possible bound? I am trying to either find a tight example, or find a bet......Wed, 02 Sep 2026 22:00:25 -0400https://pop.com.sg/derivative-of-sinx-with-respect-to-time-t.htmlhttps://pop.com.sg/derivative-of-sinx-with-respect-to-time-t.htmlhow does one take a time derivative of sinx it doesn't seem possible to me, shouldn't the function be in terms of time(t) to take its derivative.How can I calculate the change of the slope $w.r.t$ to ...Wed, 02 Sep 2026 22:00:16 -0400https://pop.com.sg/how-to-find-the-slope-of-a-secant-line-when-y-ln-x.htmlhttps://pop.com.sg/how-to-find-the-slope-of-a-secant-line-when-y-ln-x.htmlLet $y=\ln x$. Find the slope of the secant line between $P=(5, \ln5)$ and $Q=(9, \ln9)$. Should I just replace $\ln x$ with any of those two points? Ex: $y=\ln9$ or $y=\ln5$?? Please Help!!...Wed, 02 Sep 2026 21:53:28 -0400https://pop.com.sg/generalization-of-schwarz-reflection-principle-for-harmonic-functions.htmlhttps://pop.com.sg/generalization-of-schwarz-reflection-principle-for-harmonic-functions.htmlI recently learned about the Schwarz reflection principle and I was wondering whether any of the assumptions can be weakened. Explicitly, let $\Omega\subseteq\mathbb{C}$ be open and connected such ......Wed, 02 Sep 2026 21:39:34 -0400https://pop.com.sg/n-unlabelled-balls-in-m-labeled-buckets.htmlhttps://pop.com.sg/n-unlabelled-balls-in-m-labeled-buckets.htmlSo the question is how many ways are there to put N unlabelled balls in M labeled buckets. One way (I think) I figured it out is $M^N$ ways to put each ball, divided by $N!$ permutations of ball ...Wed, 02 Sep 2026 21:36:26 -0400https://pop.com.sg/number-combinations-multi-set.htmlhttps://pop.com.sg/number-combinations-multi-set.htmlHere is my question: Consider a multiset $\{n\cdot a, n\cdot b, 1,2,3, \ldots, n+1\}$ of size $3n+1$. Determine the number of $n$-combinations. I know from my textbook that if you have a $n$-...Wed, 02 Sep 2026 21:29:46 -0400https://pop.com.sg/why-is-the-domain-of-x-x-positive-real-numbers.htmlhttps://pop.com.sg/why-is-the-domain-of-x-x-positive-real-numbers.html(What is this function called, is it an exponential function?) Plot of $x^x$ I plotted this function and found that only positive part of it is shown. Also while computing derivative of this func......Wed, 02 Sep 2026 21:26:41 -0400https://pop.com.sg/the-sum-of-2-consecutive-numbers-is-53-closed.htmlhttps://pop.com.sg/the-sum-of-2-consecutive-numbers-is-53-closed.htmlThe sum of 2 consecutive numbers is 53. I need to find those numbers, but I'm not even sure how to set up the problem. Any suggestions?...Wed, 02 Sep 2026 21:17:52 -0400https://pop.com.sg/how-to-write-exponential-function-for-curve-that-pass-4-or-more-points.htmlhttps://pop.com.sg/how-to-write-exponential-function-for-curve-that-pass-4-or-more-points.htmlGiven 3 points, we could write exponential function for a curve that would pass those points in $y = ax^n + b$ form So I wonder that, could we write a function in exponential form that would pass 4 ...Wed, 02 Sep 2026 21:15:04 -0400https://pop.com.sg/calculate-the-probability-of-at-least-one.htmlhttps://pop.com.sg/calculate-the-probability-of-at-least-one.htmlA major electronics manufacturer has determined that when one of its televisions is sold, there is $0.08$ chance that it will need service before the warranty period expires. Suppose a retailer ......Wed, 02 Sep 2026 21:13:02 -0400https://pop.com.sg/what-is-2loglikelihood.htmlhttps://pop.com.sg/what-is-2loglikelihood.htmlI don't understand the term "loglikelihood"? I'd like to have a practical understanding of this word, and of why this is important. Besides this whenever we calculate some statistic like chisquare or ...Wed, 02 Sep 2026 20:51:20 -0400https://pop.com.sg/039-trace-trick-039-for-expectations-of-quadratic-forms.htmlhttps://pop.com.sg/039-trace-trick-039-for-expectations-of-quadratic-forms.htmlI am trying to understand the proof for the Kullback-Leibler divergence between two multivariate normal distributions. On the way, a sort of trace trick is applied for the expectation of the quadra......Wed, 02 Sep 2026 20:44:47 -0400https://pop.com.sg/what-is-a-geometry.htmlhttps://pop.com.sg/what-is-a-geometry.htmlThere is no question what topology is and what it's about: it's about topologies (= topological spaces), and that's it. There is also no question what (universal) algebra is and what it's about. (......Wed, 02 Sep 2026 20:42:59 -0400https://pop.com.sg/expected-value-with-indicator-random-variables.htmlhttps://pop.com.sg/expected-value-with-indicator-random-variables.htmlI don't understand the solution for problem 7a at http://www.ma.utexas.edu/users/geir/teaching/m362k/weeklyhw9solns.pdf. Reproduced below for reference: Suppose that A and B each randomly, and ...Wed, 02 Sep 2026 20:41:14 -0400https://pop.com.sg/proof-that-the-dimension-of-a-matrix-row-space-is-equal-to-the-dimensi.htmlhttps://pop.com.sg/proof-that-the-dimension-of-a-matrix-row-space-is-equal-to-the-dimensi.htmlI have the following theorem: Theorem 3.12. Let A be an m n matrix. Then the dimension of its row space is equal to the dimension of its column space. And the following proof is given: Proof. ...Wed, 02 Sep 2026 20:15:08 -0400https://pop.com.sg/what-is-the-essential-difference-between-ordinary-differential-equatio.htmlhttps://pop.com.sg/what-is-the-essential-difference-between-ordinary-differential-equatio.htmlPlease forgive my stupidity. So many years after my undergraduate study and so many years after dealing with various concrete ODEs and PDEs, I still cannot tell the essential difference between t......Wed, 02 Sep 2026 19:59:24 -0400https://pop.com.sg/limit-of-composite-functions.htmlhttps://pop.com.sg/limit-of-composite-functions.htmlLet $f$ and $g$ be some functions, assuming all right conditions that allow function composition, I want to prove that $$\lim_{x \to \infty} f(g(x))=f\left(\lim_{x \to \infty}g(x)\right) $$ As lo......Wed, 02 Sep 2026 19:55:41 -0400https://pop.com.sg/why-are-euclid-axioms-of-geometry-considered-039-not-sound-039.htmlhttps://pop.com.sg/why-are-euclid-axioms-of-geometry-considered-039-not-sound-039.htmlThe five postulates (axioms) are: "To draw a straight line from any point to any point." "To produce [extend] a finite straight line continuously in a straight line." "To describe a circle with ......Wed, 02 Sep 2026 19:48:23 -0400https://pop.com.sg/proving-that-sin-z-w-sin-w-cos-z-sin-z-cos-w-using-complex-exponential.htmlhttps://pop.com.sg/proving-that-sin-z-w-sin-w-cos-z-sin-z-cos-w-using-complex-exponential.htmlLet's define: $$\sin(z) = \frac{\exp(iz) - \exp(-iz)}{2i}$$ $$\cos(z) = \frac{\exp(iz) + \exp(-iz)}{2}$$ We are to prove that $$\sin(z+w)=\sin(w) \cos(z) + \sin(z)\cos(w), \forall_{z,w \in \...Wed, 02 Sep 2026 19:41:58 -0400https://pop.com.sg/is-there-a-way-to-draw-3d-graphs-on-paper-closed.htmlhttps://pop.com.sg/is-there-a-way-to-draw-3d-graphs-on-paper-closed.htmlI want to draw three dimensional graphs on paper. How would I do this though? Would I have to fold paper? Or would I have to rotate each point projected onto a 2d plane to make it suitable for my v......Wed, 02 Sep 2026 19:35:06 -0400https://pop.com.sg/question-about-existential-instantiation.htmlhttps://pop.com.sg/question-about-existential-instantiation.htmlI was having some trouble understanding existential instantiation. My textbook (Rosen - Discrete Mathematics and its Applications) states this about existential instantiation: Existential instanti......Wed, 02 Sep 2026 19:26:13 -0400