https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usWed, 26 Aug 2026 19:02:53 -0400https://pop.com.sg/bound-for-the-fourier-transform.htmlhttps://pop.com.sg/bound-for-the-fourier-transform.htmlCan someone come up with a proof for this bound of the Fourier transformation? Let $f$ be $s$ times continuously differentiable, with compact support, then there exists a real constant $c$ such th......Wed, 26 Aug 2026 22:52:12 -0400https://pop.com.sg/for-what-value-of-constant-a-is-function-continuous.htmlhttps://pop.com.sg/for-what-value-of-constant-a-is-function-continuous.htmlI know there is a similar question. I had a read through it and it didn't help me so I'm posting this one. The question is for what value(s) of the constant $a\in \mathbb R$ is $$f_a(x) = \left\{ \...Wed, 26 Aug 2026 22:50:48 -0400https://pop.com.sg/prove-this-identity-cos-x-sec-x-cos-x-sin-2x-closed.htmlhttps://pop.com.sg/prove-this-identity-cos-x-sec-x-cos-x-sin-2x-closed.htmlI have been trying to prove this identity for about 25 minutes and I can't figure it out. Can you help me please? $$\cos x(\sec x-\cos x)=\sin^2x$$...Wed, 26 Aug 2026 22:44:49 -0400https://pop.com.sg/find-the-least-significant-digit-of-2-3a-where-a-10-100.htmlhttps://pop.com.sg/find-the-least-significant-digit-of-2-3a-where-a-10-100.htmlCan anyone please tell me - What is the least significant digit in $2^{3A}$? How do we define least significant digit?...Wed, 26 Aug 2026 22:43:13 -0400https://pop.com.sg/unimodular-matrix-definition.htmlhttps://pop.com.sg/unimodular-matrix-definition.htmlI'm a bit confused. Based on Wikipedia: In mathematics, a unimodular matrix M is a square integer matrix having determinant +1, 0 or −1. Equivalently, it is an integer matrix that is invertibl......Wed, 26 Aug 2026 22:38:40 -0400https://pop.com.sg/choosing-randomly-integers-from-1-to-10.htmlhttps://pop.com.sg/choosing-randomly-integers-from-1-to-10.htmlFrom Question 5 the practice book of the GRE math subject test: Sofia and Tess will each randomly choose one of the $10$ integers from $1$ to $10$. What is the probability that neither integer cho......Wed, 26 Aug 2026 22:29:10 -0400https://pop.com.sg/rotate-a-shape-360-degrees-on-one-axis-and-not-be-the-same-as-the-star.htmlhttps://pop.com.sg/rotate-a-shape-360-degrees-on-one-axis-and-not-be-the-same-as-the-star.htmlAre there shapes in nth dimensions (in euclidian geometry), that can be rotated 360 degrees on one axis and have more than one point not in the same place? Alternatively, in non-euclidian geometry,......Wed, 26 Aug 2026 22:23:44 -0400https://pop.com.sg/summation-of-1-k-duplicate.htmlhttps://pop.com.sg/summation-of-1-k-duplicate.htmlWhat is summation of 1/k where n ranges from 1 to n. I need the general formula for the summation. I know the series tends to infinity when k tends to infinity . But upto n terms there must be a ...Wed, 26 Aug 2026 22:19:19 -0400https://pop.com.sg/sobolev-norm-in-the-definition-of-sobolev-spaces.htmlhttps://pop.com.sg/sobolev-norm-in-the-definition-of-sobolev-spaces.htmlI've seen the Sobolev space defined as: The Sobolev space $H^k(\Omega)$ is the set of all functions $u \in L_2(\Omega)$ for which the weak derivative $\partial^\alpha u \in L_2(\Omega)$ for all $|\...Wed, 26 Aug 2026 22:17:58 -0400https://pop.com.sg/conflicting-definition-of-the-hessian-matrix-does-the-order-of-the-par.htmlhttps://pop.com.sg/conflicting-definition-of-the-hessian-matrix-does-the-order-of-the-par.htmlOn Wikipedia, the Hessian matrix is defined as, https://en.wikipedia.org/wiki/Hessian_matrix $ {\displaystyle \mathbf {H} ={\begin{bmatrix}{\dfrac {\partial ^{2}f}{\partial x_{1}^{2}}}&{\dfrac {\...Wed, 26 Aug 2026 22:12:18 -0400https://pop.com.sg/how-to-visualize-triple-integrals.htmlhttps://pop.com.sg/how-to-visualize-triple-integrals.htmlI am struggling a lot with triple integrals. I can evaluate them, but I find it extremely difficult to write the triple integral. I cannot visualize them. An example question: $\iiint_E6xy\;dV$, w......Wed, 26 Aug 2026 22:00:07 -0400https://pop.com.sg/orthogonal-projection-onto-a-vector-with-matrix-transformation.htmlhttps://pop.com.sg/orthogonal-projection-onto-a-vector-with-matrix-transformation.htmlSo I have this question here which says: $a)$ Find the standard matrix of the linear operator $T:R^2\rightarrow R^2$ given by the orthogonal projection onto the vector $(1,-2)$. $b)$ Given the li......Wed, 26 Aug 2026 21:57:07 -0400https://pop.com.sg/how-to-express-rounding-down-up-to-nearest-multiplum-of-a-number.htmlhttps://pop.com.sg/how-to-express-rounding-down-up-to-nearest-multiplum-of-a-number.htmlI've written an equation in a programming language which I need to express using maths: slots = (elements_count - (elements_count % slot_size) + slot_size) / slot_size In Excel, you can also wr......Wed, 26 Aug 2026 21:48:52 -0400https://pop.com.sg/domain-and-range-of-f-x.htmlhttps://pop.com.sg/domain-and-range-of-f-x.htmlFind the domain and range of Function? $$F(x) = \frac{1}{\sqrt{25-x^2}}$$ Domanin f(x) has real value , if $$25-x^2\geqslant0$$ $$-x^2\geqslant-25$$ $$x^2\le25$$ $$-5\le x\le5$$ $$D_f= [-5,5]$$ ......Wed, 26 Aug 2026 21:46:40 -0400https://pop.com.sg/angle-between-edges-in-a-square-pyramid.htmlhttps://pop.com.sg/angle-between-edges-in-a-square-pyramid.htmlHow do you calculate the missing angle? I couldn't figure it out. This is not homework, just a math practice question that I made up to test myself....Wed, 26 Aug 2026 21:35:01 -0400https://pop.com.sg/derivative-of-1-1-x-2.htmlhttps://pop.com.sg/derivative-of-1-1-x-2.html$ \frac{\partial f}{\partial x}\dfrac{1}{(1-x)^2} = \dfrac{0 *(1-x)^2 - (-1)(1)}{(1-x)^2*(1-x)} $ But: $ \frac{\partial f}{\partial x}\dfrac{1}{(1-x)^2} \neq \dfrac{0 *(x^2-2x+1) - (2x-2)(1)}{(x^2-......Wed, 26 Aug 2026 21:33:22 -0400https://pop.com.sg/mercer-039-s-condition-and-rkhss.htmlhttps://pop.com.sg/mercer-039-s-condition-and-rkhss.htmlIn general when considering an RKHS $\mathcal{H}$ over a set $X$ one has a kernel function $k:X\times X\rightarrow\mathbb{C}$. It is then not too hard to show that its reproducing property implies a ...Wed, 26 Aug 2026 21:31:20 -0400https://pop.com.sg/instantaneous-rate-of-change-and-average-rate-of-change.htmlhttps://pop.com.sg/instantaneous-rate-of-change-and-average-rate-of-change.htmlI'm currently studying for my calculus AP exam and I'm currently stuck on a question. Unless otherwise specified, the domain of a function $f$ is assumed to be the set of all real numbers $x$ for ......Wed, 26 Aug 2026 21:27:58 -0400https://pop.com.sg/which-of-the-following-sets-of-vectors-are-linearly-independent.htmlhttps://pop.com.sg/which-of-the-following-sets-of-vectors-are-linearly-independent.htmlWhich of the following set of vectors are linearly independent? I'm a bit confused. I think the answer(s) would be A, C, and D. I'm unsure of how to actually figure out if each one is or isn't LI/LD...Wed, 26 Aug 2026 21:20:54 -0400https://pop.com.sg/applying-mean-value-theorem-to-formula.htmlhttps://pop.com.sg/applying-mean-value-theorem-to-formula.htmlAs I understand it, the mean value theorem is where $${f}'(c)=\frac{f(b)-f(a)}{b-a}$$ if $f$ is continuous on the open interval (a, b) and differentiable on the closed interval [a,b]. A proble......Wed, 26 Aug 2026 21:13:13 -0400https://pop.com.sg/dimension-of-solution-set-for-ax-b-but-the-solution-set-isn-039-t-a-ve.htmlhttps://pop.com.sg/dimension-of-solution-set-for-ax-b-but-the-solution-set-isn-039-t-a-ve.htmlThe solution set of a system of $n$ linear equations written as a matrix equation $AX=b$,$b\neq0$ cannot have zero vector in it. This means that the solution set cannot be a vector space and talking ...Wed, 26 Aug 2026 21:12:01 -0400https://pop.com.sg/assuming-a-n-is-a-convergent-sequence-prove-that-the-lim-inf-of-a-n-1-.htmlhttps://pop.com.sg/assuming-a-n-is-a-convergent-sequence-prove-that-the-lim-inf-of-a-n-1-.htmlI'm aware that you have to use the definition of a limit of a sequence, which is: $\lim\limits_{n \to \infty} a_n = L $ if for every $E > 0$, there is an $N$ such that if $n >N $then $|a_n -......Wed, 26 Aug 2026 21:04:07 -0400https://pop.com.sg/why-can-a-matrix-without-a-full-rank-not-be-invertible.htmlhttps://pop.com.sg/why-can-a-matrix-without-a-full-rank-not-be-invertible.htmlI know you could just say because the $\det = 0$. But during the introduction of determinants the professor said, obviously if two columns of the matrix are linearly dependent the matrix can't be ...Wed, 26 Aug 2026 20:58:06 -0400https://pop.com.sg/proof-of-a-2-ab-b-2-0-a-b-both-are-not-zero-both-are-real-duplicate.htmlhttps://pop.com.sg/proof-of-a-2-ab-b-2-0-a-b-both-are-not-zero-both-are-real-duplicate.htmlShow that for any reals a,b, both of which are not zero, it holds true that a^2 + ab + b^2 >0...Wed, 26 Aug 2026 20:55:07 -0400https://pop.com.sg/what-is-the-difference-between-average-and-expected-value.htmlhttps://pop.com.sg/what-is-the-difference-between-average-and-expected-value.htmlQuestion : What is the difference between Average and Expected value? I have been going through the definition of expected value on Wikipedia beneath all that jargon it seems that the expected val......Wed, 26 Aug 2026 20:54:15 -0400https://pop.com.sg/prove-that-x-is-rational.htmlhttps://pop.com.sg/prove-that-x-is-rational.htmlLet $x$ be a real number with the properties that $x^3+x$ and $x^5+x$ are rational. Prove that $x$ is rational. Denote $a=x^3+x$; $b=x^5+x$. We can multiply and add them together until we get the d......Wed, 26 Aug 2026 20:36:45 -0400https://pop.com.sg/differentiating-with-respect-to-1-x.htmlhttps://pop.com.sg/differentiating-with-respect-to-1-x.htmlI am fairly sure this is a silly question, but a Google search was insufficient to find a satisfactory answer. If I differentiate some function of $x$ with respect to $1-x$, what do I get compare......Wed, 26 Aug 2026 20:36:15 -0400https://pop.com.sg/minimal-sufficient-statistics-for-uniform-distribution-on-theta-theta.htmlhttps://pop.com.sg/minimal-sufficient-statistics-for-uniform-distribution-on-theta-theta.htmlLet $X_1,\dots,X_n$ be a sample from uniform distribution on $(-\theta,\theta)$ with parameter $\theta>0$. It is easy to show that $T(X) = (X_{(1)},X_{(n)})$ is a sufficient statistic for $\the......Wed, 26 Aug 2026 20:34:22 -0400https://pop.com.sg/partial-differential-equation-u-t-3u-xx.htmlhttps://pop.com.sg/partial-differential-equation-u-t-3u-xx.htmlSolve the partial differential equation $$u_t=3u_{xx}, u(0,t)=0, u(x,0)=\cos{x}\sin{5x}$$ Attempt: Using separation of variables, let $u(x,t)=f(x)g(t)$, so $$f(x)g'(t)=3f''(x)g(t)$$ $$\frac{g'(t)}{......Wed, 26 Aug 2026 20:26:57 -0400https://pop.com.sg/is-the-rank-of-a-matrix-the-same-of-its-transpose-if-yes-how-can-i-pro.htmlhttps://pop.com.sg/is-the-rank-of-a-matrix-the-same-of-its-transpose-if-yes-how-can-i-pro.htmlI am auditing a Linear Algebra class, and today we were taught about the rank of a matrix. The definition was given from the row point of view: "The rank of a matrix A is the number of non-zero ...Wed, 26 Aug 2026 20:26:33 -0400https://pop.com.sg/how-to-use-parametric-equations-for-line-integral.htmlhttps://pop.com.sg/how-to-use-parametric-equations-for-line-integral.htmlThe question is to find the work done in moving a particle in a force field: $$\overrightarrow F = 3x^2i+(2xy-y)j+3k$$ along the straight line from (0, 0, 0) to (2, 1, 3) So, work done $=\int \...Wed, 26 Aug 2026 20:22:25 -0400https://pop.com.sg/what-039-s-the-difference-between-theorem-lemma-and-corollary.htmlhttps://pop.com.sg/what-039-s-the-difference-between-theorem-lemma-and-corollary.htmlCan anybody explain me what is the basic difference between theorem, lemma and corollary? We have been using it for a long time but I never paid any attention. I am just curious to know....Wed, 26 Aug 2026 20:19:07 -0400https://pop.com.sg/proving-and-distributive-law-using-boolean-algebra.htmlhttps://pop.com.sg/proving-and-distributive-law-using-boolean-algebra.html$$ X(Y+Z) = (XY) + (XZ) $$ I can’t seem to derive the proper steps to prove this equation using Boolean axioms. The hint I’ve been given is using demorgans laws proofs but I still can’t seem to fi......Wed, 26 Aug 2026 20:17:02 -0400https://pop.com.sg/finding-contrapositive-of-a-universal-statement.htmlhttps://pop.com.sg/finding-contrapositive-of-a-universal-statement.htmlFor all $x$, if $x^2$ is even, then $x$ is even. The contrapositive to this statement is: For all $x$, if $x$ is odd, then $x^2$ is odd. Why do we ignore the "For all $x$" and not say ......Wed, 26 Aug 2026 20:16:58 -0400https://pop.com.sg/soft-question-definition-of-the-real-plane.htmlhttps://pop.com.sg/soft-question-definition-of-the-real-plane.htmlI have only seen the real plane to be defined as $\mathbb R \times \mathbb R$. In Euclidean geometry, the origin does not play any central role, but in this set theoretic definition, it does. Is t......Wed, 26 Aug 2026 20:13:26 -0400https://pop.com.sg/how-to-prepare-for-integral-calculus-calculus-2.htmlhttps://pop.com.sg/how-to-prepare-for-integral-calculus-calculus-2.htmlI'm majoring in computer engineering and I have Calculus 2 coming up this semester. From what I understand, Calculus 2 is the most difficult math class in the engineering path. Over the summer, I......Wed, 26 Aug 2026 20:10:41 -0400https://pop.com.sg/probability-of-drawing-a-flush-from-a-standard-deck-of-cards.htmlhttps://pop.com.sg/probability-of-drawing-a-flush-from-a-standard-deck-of-cards.htmlThe following is the problem put to me: A 5-card poker hand is dealt from a well shuffled regular 52-card playing card deck. Find the probability that the hand is a Flush (5 nonconsecutive cards e......Wed, 26 Aug 2026 19:54:52 -0400https://pop.com.sg/how-to-calculate-the-height-of-a-cone-at-particular-volume.htmlhttps://pop.com.sg/how-to-calculate-the-height-of-a-cone-at-particular-volume.htmlThe equation for the volume of a cone is $\frac{1}{3}\pi r^2h$ starting from the base. However, I wanna calculate the height of the cone for a particular volume from the tip of the cone. Could you ......Wed, 26 Aug 2026 19:31:46 -0400https://pop.com.sg/with-the-help-of-chinese-remainder-theorem-show-that-x-144-equiv-1-pmo.htmlhttps://pop.com.sg/with-the-help-of-chinese-remainder-theorem-show-that-x-144-equiv-1-pmo.htmlWith the help of Chinese remainder theorem show that $ x^{144} \equiv 1 \pmod{323}$ for all $x$ relatively prime to 323. The problem with me is that I used to use CRT when $x$ is raised to a power......Wed, 26 Aug 2026 19:25:38 -0400https://pop.com.sg/what-does-curly-curved-less-than-sign-succcurlyeq-mean.htmlhttps://pop.com.sg/what-does-curly-curved-less-than-sign-succcurlyeq-mean.htmlI am reading Boyd & Vandenberghe's Convex Optimization. The authors use curved greater than or equal to (\succcurlyeq) $$f(x^*) \succcurlyeq \alpha$$ and curved less than or equal to (\preccu......Wed, 26 Aug 2026 19:22:58 -0400https://pop.com.sg/is-it-possible-to-reverse-a-gradient-vec-nabla-operation.htmlhttps://pop.com.sg/is-it-possible-to-reverse-a-gradient-vec-nabla-operation.htmlIn calculus, the antiderivative (indefinite integral) can be considered as the reverse operation of a derivative. A gradient yields a vector. Is there a similar way of reversing gradient, as you d......Wed, 26 Aug 2026 19:15:48 -0400https://pop.com.sg/constant-slope-in-straight-line.htmlhttps://pop.com.sg/constant-slope-in-straight-line.htmlI'm now reading about the formal definition of line as the graph of the function $f(x)=f(x_0)+m(x-x_0)$ for an arbitrary $x_0$ through which the graph passes, where the constant $m$ is called slope......Wed, 26 Aug 2026 18:57:11 -0400https://pop.com.sg/height-of-pyramid.htmlhttps://pop.com.sg/height-of-pyramid.htmlKnowing the base sides and the slant heights, do we calculate the height of the pyramid by the formula $h^2=h_b^2-\left (\frac{a}{2}\right )^2$, no matter what shape the base of the pyramid has, i.e. ...Wed, 26 Aug 2026 18:47:17 -0400https://pop.com.sg/what-is-a-cardinal-basis-spline.htmlhttps://pop.com.sg/what-is-a-cardinal-basis-spline.htmlWikipedia says: the normalized cardinal B-splines tend to the Gaussian function and writes them as "Bk". Meanwhile, cnx.org Signal Reconstruction says: The basis splines Bn are shown ... ......Wed, 26 Aug 2026 18:44:09 -0400https://pop.com.sg/finding-transient-and-steady-state-solution.htmlhttps://pop.com.sg/finding-transient-and-steady-state-solution.htmlObtain the steady periodic solutin $x_{sp}(t)=Asin(\omega t+\phi)$ and the transient equation for the solution t $x''+2x'+26x=82cos(4t)$, where $x(0)=6$ & $x'(0)=0$. So I'm not sure what's being ...Wed, 26 Aug 2026 18:42:59 -0400https://pop.com.sg/limit-as-n-goes-to-infinity.htmlhttps://pop.com.sg/limit-as-n-goes-to-infinity.htmlConsider this limit: $$\lim_{n\rightarrow\infty} \left( 1+\frac{1}{n} \right) ^{n^2} = x$$ I thought the way to solve this for $x$ was to reduce it using the fact that as $n \rightarrow \infty$, $......Wed, 26 Aug 2026 18:39:58 -0400https://pop.com.sg/definition-of-left-right-exact-functors.htmlhttps://pop.com.sg/definition-of-left-right-exact-functors.htmlLet $P,Q$ be abelian categories and $F:P\to Q$ be an additive functor. Wikipedia states two definitions on left exact functors (right dually): $F$ is left exact if $0\to A\to B\to C\to 0$ is exact ...Wed, 26 Aug 2026 18:39:58 -0400https://pop.com.sg/solving-equations-with-factorials-duplicate.htmlhttps://pop.com.sg/solving-equations-with-factorials-duplicate.htmlI looked on the internet but couldn't find anything relevant, so I was hoping you could help because I have no clue where to even start with how to solve this equations: x! = 6 Obviously trial and......Wed, 26 Aug 2026 18:37:50 -0400https://pop.com.sg/how-would-you-discover-stokes-039-s-theorem.htmlhttps://pop.com.sg/how-would-you-discover-stokes-039-s-theorem.htmlLet $S$ be a smooth oriented surface in $\mathbb R^3$ with boundary $C$, and let $f: \mathbb R^3 \to \mathbb R^3$ be a continuously differentiable vector field on $\mathbb R^3$. Stokes's theorem s......Wed, 26 Aug 2026 18:13:57 -0400https://pop.com.sg/can-someone-give-me-a-deeper-understanding-of-implicit-differentiation.htmlhttps://pop.com.sg/can-someone-give-me-a-deeper-understanding-of-implicit-differentiation.htmlI'm doing calculus and I want to be an engineer so I would like to understand the essence of the logic of implicit differentials rather than just memorizing the algorithm. Yes, I could probably mem......Wed, 26 Aug 2026 18:09:26 -0400https://pop.com.sg/archimedean-spiral-symmetry-test.htmlhttps://pop.com.sg/archimedean-spiral-symmetry-test.htmlIt is usually stated in Precalculus textbooks that in polar coordinates when a relation between $r$ and $\theta$ passes a symmetry test then the curve described by that relation has that symmetry. ......Wed, 26 Aug 2026 17:58:08 -0400https://pop.com.sg/don-039-t-understand-why-this-binomial-expansion-is-not-valid-for-x-1.htmlhttps://pop.com.sg/don-039-t-understand-why-this-binomial-expansion-is-not-valid-for-x-1.htmltoday I'm studying binomial expansion and I'm a little confused about when certain expressions are valid. E.g. take this solution from my textbook: I understand that $(1-x)^{-1}$ has an infinite ...Wed, 26 Aug 2026 17:57:18 -0400https://pop.com.sg/where-does-this-probability-formula-come-from.htmlhttps://pop.com.sg/where-does-this-probability-formula-come-from.htmlFrom Introduction to Probability, 2ed (Bersekas et al), chapter 8.1 p. 417: I would like to ask where the underlined formula comes from. So far, the formulas given in that chapter are: Please he......Wed, 26 Aug 2026 17:52:38 -0400https://pop.com.sg/probability-of-balls-drawn-with-replacement.htmlhttps://pop.com.sg/probability-of-balls-drawn-with-replacement.htmlWe have two bags, Bag A has 40 red balls and 15 blue balls, Bag B has 40 blue balls and 10 red balls. One of these bags is selected at random and from it five balls are drawn at random, replacing e......Wed, 26 Aug 2026 17:45:29 -0400https://pop.com.sg/why-does-the-difference-between-the-left-hand-and-right-hand-sum-get-s.htmlhttps://pop.com.sg/why-does-the-difference-between-the-left-hand-and-right-hand-sum-get-s.htmlQ) For a given function on a given interval, the difference between the left-hand sum and right-hand sum always gets smaller as the number of subdivisions gets larger. I remember that the answer i......Wed, 26 Aug 2026 17:38:47 -0400https://pop.com.sg/solving-inhomogeneous-heat-equation.htmlhttps://pop.com.sg/solving-inhomogeneous-heat-equation.htmlSolve the inhomogeneous heat equation $u_t=c^2u_{xx} +\sin(5\pi x)$ for all $0 < x < 1, t > 0$ subject to homogeneous Dirichlet boundary conditions $u(t,0)=u(t,1)=0$ and initial condition ......Wed, 26 Aug 2026 17:36:29 -0400https://pop.com.sg/simplify-boolean-product-of-sums-function.htmlhttps://pop.com.sg/simplify-boolean-product-of-sums-function.htmlI've got a product of sums expression: F=(A'+B+C')&(A+D')(C+D') I need to show it as a sum of products and then simplify it. Right now I got: F=(A'&D')+(A&B&C)+(B&D')+(C&D......Wed, 26 Aug 2026 17:36:22 -0400https://pop.com.sg/order-of-essential-singularity.htmlhttps://pop.com.sg/order-of-essential-singularity.htmlcan we say that essential singularities has order like poles ? I mean ; we know e^(1/z) has an essential singularity at z=0 then do I need to say e^(1/z^3) has essential singularity at z=0 with or......Wed, 26 Aug 2026 17:21:30 -0400https://pop.com.sg/what-exactly-is-real-number.htmlhttps://pop.com.sg/what-exactly-is-real-number.htmlThis question may sound philosophy, but it has been bothering me for a very long time, therefore I have to ask it here. The story goes back when my first time reading Apostol's Calculus, then I had ...Wed, 26 Aug 2026 17:19:17 -0400https://pop.com.sg/how-to-find-functional-square-root-of-sin-x.htmlhttps://pop.com.sg/how-to-find-functional-square-root-of-sin-x.htmlMaybe I am overlooking something, but is there some easy way to find a function $x\to f(x)$ so that $$(f\circ f)(x) = f(f(x)) = \sin(x)$$ on some interval, say $x\in [-\pi,\pi]\subset \mathbb R$ of ...Wed, 26 Aug 2026 16:51:30 -0400https://pop.com.sg/prove-that-the-product-of-a-non-zero-rational-and-irrational-number-is.htmlhttps://pop.com.sg/prove-that-the-product-of-a-non-zero-rational-and-irrational-number-is.htmlCould you please confirm if this proof is correct? Theorem: If $q \neq 0$ is rational and $y$ is irrational, then $qy$ is irrational. Proof: Proof by contradiction, we assume that $qy$ is rational. ...Wed, 26 Aug 2026 16:44:55 -0400https://pop.com.sg/quickest-way-to-find-characteristic-polynomial-from-a-given-matrix.htmlhttps://pop.com.sg/quickest-way-to-find-characteristic-polynomial-from-a-given-matrix.htmlFind the Rational form of $$ A=\begin{pmatrix} 1&2&0&4\\ \:\:\:4&1&2&0\\ \:\:\:0&4&1&2\\ \:\:\:2&0&4&1\end{pmatrix} $$ I don't wanna the solution, ...Wed, 26 Aug 2026 16:44:26 -0400https://pop.com.sg/given-a-byte-in-2-s-complement-what-is-0x56-in-decimal-closed.htmlhttps://pop.com.sg/given-a-byte-in-2-s-complement-what-is-0x56-in-decimal-closed.htmla) -13 b) -110 c) 38 d) none of the above Im not sure on how to do this question. What I have done so far is convert the hexadecimal number into a decimal, but since it's already in 2's complement......Wed, 26 Aug 2026 16:40:46 -0400https://pop.com.sg/area-of-trapezoids-vs-area-of-parallelograms.htmlhttps://pop.com.sg/area-of-trapezoids-vs-area-of-parallelograms.htmlBy definition parallelograms are special type of trapezoids. Given a trapezoid with known sides one can calculate its area according to wikipedia. But in order to calculate the area of a parallelog......Wed, 26 Aug 2026 16:30:11 -0400https://pop.com.sg/statistics-calculating-mean-given-standard-deviation-and-percentage.htmlhttps://pop.com.sg/statistics-calculating-mean-given-standard-deviation-and-percentage.htmlThis is one of our homework problems for Statistics. I'm really stuck on this one, and it would be great if someone could show me how to solve this, or at least set up the equation. The weights of ...Wed, 26 Aug 2026 16:27:12 -0400https://pop.com.sg/how-can-i-construct-the-centroid-of-a-quadrilateral.htmlhttps://pop.com.sg/how-can-i-construct-the-centroid-of-a-quadrilateral.htmlHow can I construct the centroid of a quadrilateral? I suppose that it is the intersection between the lines that join the middles of opposite sides....Wed, 26 Aug 2026 16:25:59 -0400https://pop.com.sg/calculate-inverse-discrete-time-fourier-transform.htmlhttps://pop.com.sg/calculate-inverse-discrete-time-fourier-transform.htmlCalculate Inverse Discrete Time Fourier Transform of the following where $|a| < 1$: $$ X(e^{j\omega}) = \frac{1-a^2}{(1-ae^{-j\omega})(1-ae^{j\omega})} $$ Plugging this directly into the IDTFT ...Wed, 26 Aug 2026 16:09:18 -0400https://pop.com.sg/what-comes-before-precalculus-closed.htmlhttps://pop.com.sg/what-comes-before-precalculus-closed.htmlI studied in Europe and followed mostly European academic ways. I'm tutoring a young person , and I will to know what math class comes before pre calculus. Because soon we will be starting some ...Wed, 26 Aug 2026 16:05:56 -0400https://pop.com.sg/baby-papa-mama-big-rudin.htmlhttps://pop.com.sg/baby-papa-mama-big-rudin.htmlRecently, I was looking for the reviews of some Analysis books while encountered terms such as Baby/Papa/Mama/Big Rudin. Firstly, I thought that these are the names of a book! But it turned out that ...Wed, 26 Aug 2026 16:01:51 -0400https://pop.com.sg/relationships-between-det-a-b-and-a-b.htmlhttps://pop.com.sg/relationships-between-det-a-b-and-a-b.htmlWhen computing $\det(A+B)$ we notice that there is no relation between $\det A + \det B$. However does the $\det(A+B)$ have any relation to the matrices $A+B$ as they stand?...Wed, 26 Aug 2026 16:01:05 -0400https://pop.com.sg/what-is-the-largest-positive-n-for-which-n-3-100-is-divisible-by-n-10.htmlhttps://pop.com.sg/what-is-the-largest-positive-n-for-which-n-3-100-is-divisible-by-n-10.htmlWhat is the largest positive $n$ for which $n^3+100$ is divisible by $n+10$ I tried to factorize $n^3+100$, but $100$ is not a perfect cube. I wish it were $1000$....Wed, 26 Aug 2026 15:55:43 -0400https://pop.com.sg/prove-the-principle-of-strong-induction.htmlhttps://pop.com.sg/prove-the-principle-of-strong-induction.htmlThe following is from Analysis with an Introduction to Proof by Steven Lay Prove the principle of strong induction: Let $P(n)$ be a statement that is either true or false for each $n \in \mathbb{N}$ ...Wed, 26 Aug 2026 15:48:46 -0400https://pop.com.sg/what-is-meant-by-the-double-vertical-line-notation-here.htmlhttps://pop.com.sg/what-is-meant-by-the-double-vertical-line-notation-here.htmlWhat do the double vertical lines around $\vec i$ and $\vec j$ in this equation actually mean? $$ sim(i,j) = cos(\vec i, \vec j) = \frac{\vec i \cdot \vec j}{\lVert \vec i\rVert^2 * \lVert\vec j\r......Wed, 26 Aug 2026 15:46:52 -0400https://pop.com.sg/conditions-for-a-unique-root-of-a-fifth-degree-polynomial.htmlhttps://pop.com.sg/conditions-for-a-unique-root-of-a-fifth-degree-polynomial.htmlFifth degree polynomials cannot generally be solved analytically, but at least one solution always exists. Given the normal form $$ax^5+bx^4+cx^3+dx^2+ex+f=0,$$ is it possible to find sufficient ...Wed, 26 Aug 2026 15:43:00 -0400https://pop.com.sg/symbolic-operation-on-ti-84-calculator-closed.htmlhttps://pop.com.sg/symbolic-operation-on-ti-84-calculator-closed.htmlI want the calculator to answer with the parameter answer, not with a numeric value, I have tried to clear "M" variable with no success. Any idea? http://imgur.com/cHjfLln ( sorry I don't have ......Wed, 26 Aug 2026 15:35:36 -0400https://pop.com.sg/x-x-y-how-to-solve-for-x-duplicate.htmlhttps://pop.com.sg/x-x-y-how-to-solve-for-x-duplicate.htmlI tried looking for ways to solve this equation and came across something like Lambert's W function, which, by the way, I did not understand a bit, because I've never learned it nor do I have a dec......Wed, 26 Aug 2026 15:17:02 -0400https://pop.com.sg/possible-polyhedra-from-euler-039-s-formula.htmlhttps://pop.com.sg/possible-polyhedra-from-euler-039-s-formula.htmlI'm not very clear with the euler's formula, and I couldn't find it anywhere. I'm sorry if it is a double post. F + V - E = 2 Is the euler's formula. If the equation balances, is it polyhedra al......Wed, 26 Aug 2026 15:03:32 -0400https://pop.com.sg/the-chain-rule-and-squared-trig-functions.htmlhttps://pop.com.sg/the-chain-rule-and-squared-trig-functions.htmlI am trying to do this homework problem and I have no idea how to approach it. I have tried many methods, all resulting in failure. I went to the books website and it offers no help. I am trying to......Wed, 26 Aug 2026 14:59:04 -0400https://pop.com.sg/moment-of-inertia-around-z-axis.htmlhttps://pop.com.sg/moment-of-inertia-around-z-axis.htmlHello I am having difficulty with the following; I am wanting to find I, the moment of inertia about the z axis of the region that is bounded by the paraboloid $z=x^{2}+y^{2}$ and the $z=1$ plane,......Wed, 26 Aug 2026 14:55:40 -0400https://pop.com.sg/use-the-given-graph-f-over-the-interval-0-7-to-find-the-following.htmlhttps://pop.com.sg/use-the-given-graph-f-over-the-interval-0-7-to-find-the-following.html(a) The open intervals on which $f$ is increasing. I answered $(0, 1), (3, 5), (5,7)$ (b) The open intervals on which $f$ is concave upward. I answered $(1, 4)$ (c) The open intervals on which ......Wed, 26 Aug 2026 14:54:39 -0400https://pop.com.sg/using-l-039-hopital-039-s-rule-with-the-indeterminate-form-of-infinity.htmlhttps://pop.com.sg/using-l-039-hopital-039-s-rule-with-the-indeterminate-form-of-infinity.htmlI'm aware that with L'Hopital's rule, we're dealing with indeterminate forms of $ \lim_{x\to a} \frac{f(x)}{g(x)} $, and that includes re-writing $\lim_{x\to a} \infty - \infty$ into that format. ...Wed, 26 Aug 2026 14:51:16 -0400https://pop.com.sg/how-to-graph-the-parametric-without-calculator-x-sin-t-y-t-2.htmlhttps://pop.com.sg/how-to-graph-the-parametric-without-calculator-x-sin-t-y-t-2.htmlQuestion How to graph the parametric without calculator: $x=\sin t$ , $y=t^2$ I first make t by itself by doing the following: $\sqrt{y}=t$ and plugging into the other function to get $x=\sin(\sq......Wed, 26 Aug 2026 14:43:45 -0400https://pop.com.sg/prove-that-there-exists-a-shortest-distance-between-two-skew-lines.htmlhttps://pop.com.sg/prove-that-there-exists-a-shortest-distance-between-two-skew-lines.htmlIn every geometry textbook it states that the shortest distance between two skew lines (lines that are not co-planar) is given by the unique line that runs perpendicular to both skew lines. This is ...Wed, 26 Aug 2026 14:41:41 -0400https://pop.com.sg/finding-the-angles-of-a-triangle-using-algebraic-expressions.htmlhttps://pop.com.sg/finding-the-angles-of-a-triangle-using-algebraic-expressions.htmlI think I have solved this question but want to check if I am correct with my formula I used the theorem (rule) : the measure of the exterior angle of a triangle is equal to the sum of the remote (...Wed, 26 Aug 2026 14:40:57 -0400https://pop.com.sg/positive-definite-and-block-matrix.htmlhttps://pop.com.sg/positive-definite-and-block-matrix.htmlSuppose a $n \times n$ symmetric matrix $\mathbf{M}$ is positive definite. Its block matrix form is written as follow. \begin{align} \mathbf{M} \; = \; \begin{pmatrix} \mathbf{A} &am......Wed, 26 Aug 2026 14:40:17 -0400https://pop.com.sg/difference-between-spe-and-spne-in-game-theory.htmlhttps://pop.com.sg/difference-between-spe-and-spne-in-game-theory.htmlI am slightly confused as to what the difference between Subgame Perfect Equilibrium and Subgame Perfect Nash Equilibrium is. I was told there is a difference, but now I found many lecture notes t......Wed, 26 Aug 2026 14:34:22 -0400https://pop.com.sg/definition-singular-manifold.htmlhttps://pop.com.sg/definition-singular-manifold.htmlI'm looking for the definition of a singular manifold. I haven't found it yet. For instance, in $\mathbb{R}^4$, with $f(x,y,z,t)=xy-zt$, $f^{-1}(0)$ is a singular submanifold. I only found a few ex......Wed, 26 Aug 2026 14:20:58 -0400https://pop.com.sg/why-do-we-draw-the-xyz-coordinate-system-like-this.htmlhttps://pop.com.sg/why-do-we-draw-the-xyz-coordinate-system-like-this.htmlUsually people (including, for instance, Calculus teachers) draw the $xyz$ coordinate system in such a way that the $y$ and $z$ axes are perpendicular to each other: xyz http://pad1.whstatic.com/im......Wed, 26 Aug 2026 14:17:54 -0400https://pop.com.sg/derivation-of-normal-equation-for-linear-least-squares-in-matrix-form.htmlhttps://pop.com.sg/derivation-of-normal-equation-for-linear-least-squares-in-matrix-form.htmlThe derivation can be found on wikipedia but it's not clear how each step follows. We have $y=X\beta+\epsilon$, and want to minimize $\epsilon^2$. We write objective function as $S(\beta)=||y-X\b......Wed, 26 Aug 2026 14:15:43 -0400https://pop.com.sg/simplifying-division-of-integrals.htmlhttps://pop.com.sg/simplifying-division-of-integrals.htmlThe $\overline{x}$ coordinate of the center of mass of a plane region is calculated as $$ \overline{x} = \frac{M_y}{M} = \frac{\int_a^b xf(x) \, \textrm{d}x}{\int_a^b f(x) \, \textrm{d}x} $$ And ......Wed, 26 Aug 2026 14:11:43 -0400https://pop.com.sg/what-is-a-number.htmlhttps://pop.com.sg/what-is-a-number.htmlA dictionary I consulted said a 'number' is a 'quantity', so I looked up what quantity means and the same dictionary said it is an amount or number of some material or thing. Since quantity and nu......Wed, 26 Aug 2026 13:43:03 -0400https://pop.com.sg/how-to-find-the-number-of-x-per-second-given-the-time-elapsed.htmlhttps://pop.com.sg/how-to-find-the-number-of-x-per-second-given-the-time-elapsed.htmlI'm benchmarking a websocket server and I am very poor at math so please forgive me. I am recording the amount of messages sent, and the elapsed time: Players Connected: 2. Messages Sent: 3528, E......Wed, 26 Aug 2026 13:39:42 -0400https://pop.com.sg/justifying-why-0-0-is-indeterminate-and-1-0-is-undefined.htmlhttps://pop.com.sg/justifying-why-0-0-is-indeterminate-and-1-0-is-undefined.html$\dfrac 00=x$ $0x=0$ $x$ can be any value, therefore $\dfrac 00$ can be any value, and is indeterminate. $\dfrac 10=x$ $0x=1$ There is no such $x$ that satisfies the above, therefore $\dfrac 10$......Wed, 26 Aug 2026 13:30:14 -0400https://pop.com.sg/find-the-absolute-maximum-and-minimum-of-f-x-frac-1-x-frac-2-x-2-on-2-.htmlhttps://pop.com.sg/find-the-absolute-maximum-and-minimum-of-f-x-frac-1-x-frac-2-x-2-on-2-.htmlNote: Turns out this is a pretty tricky problem where we need to use techniques not developed in the chapter. We will go over this problem in class on Tuesday. Find the absolute maximum and minimu......Wed, 26 Aug 2026 13:30:11 -0400https://pop.com.sg/methods-of-solving-a-problem-regarding-withdrawing-and-depositing-from.htmlhttps://pop.com.sg/methods-of-solving-a-problem-regarding-withdrawing-and-depositing-from.htmlI have an imaginary bank account with an initial balance $x$. I can only withdraw $0\% < w\% < 100\%$ (of the current balance) or deposit $0\% < d\% < 100\%$ (of the current balance) at a ...Wed, 26 Aug 2026 13:30:02 -0400https://pop.com.sg/double-integral-of-graph.htmlhttps://pop.com.sg/double-integral-of-graph.htmlI am currently trying to do this question but I am having trouble getting the correct answer of 25. I am also having trouble understanding whether my limits are right and whether my method of doin......Wed, 26 Aug 2026 13:15:30 -0400https://pop.com.sg/explicitly-calculate-shape-operator-for-graph-of-f-x-y-xy.htmlhttps://pop.com.sg/explicitly-calculate-shape-operator-for-graph-of-f-x-y-xy.htmlThis seems trivial but I am stuck. To gain intuition for a bigger problem, I am trying to compute the shape operator of the graph of $f(x,y)=xy$ at the point $p=(0,0,0)$, call this surface $\Sigma$......Wed, 26 Aug 2026 12:51:37 -0400https://pop.com.sg/strong-induction-example-using-all-of-p-1-and-and-p-k-1-and-p-k-to-pro.htmlhttps://pop.com.sg/strong-induction-example-using-all-of-p-1-and-and-p-k-1-and-p-k-to-pro.htmlI'm trying to understand how to do "real" strong induction, but my textbook seems to be of no help. It defines strong induction as follows: Let $P(n)$ be a property that is defined for integers $n......Wed, 26 Aug 2026 12:45:13 -0400https://pop.com.sg/nonsingular-z-matrix-nonsingular-m-matrix.htmlhttps://pop.com.sg/nonsingular-z-matrix-nonsingular-m-matrix.htmlHere I'm considering only M-matrices that are also Z-matrices, so all the off diagonal elements are negative in all matrices I consider. If I have a Z-Matrix with real, positive eigenvalues, is it......Wed, 26 Aug 2026 12:45:11 -0400https://pop.com.sg/how-can-i-know-that-the-graph-is-symmetric-with-respect-to-the-x-axis-.htmlhttps://pop.com.sg/how-can-i-know-that-the-graph-is-symmetric-with-respect-to-the-x-axis-.htmlHow can I know that the graph is symmetric with respect to the x axis or y axis? $y=x^4-2x^2-21$ $9x^2+4y^2=36$ $y^2=x-4$...Wed, 26 Aug 2026 12:38:59 -0400