https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usWed, 19 Aug 2026 11:43:53 -0400https://pop.com.sg/in-triangle-abc-find-the-value-of-cos-a-cos-b.htmlhttps://pop.com.sg/in-triangle-abc-find-the-value-of-cos-a-cos-b.htmlThe sides of $\triangle$ABC are in Arithmetic Progression (order being $a$, $b$, $c$) and satisfy $\dfrac{2!}{1!9!}+\dfrac{2!}{3!7!}+\dfrac{1}{5!5!}=\dfrac{8^a}{(2b)!}$, Then prove that the value ......Wed, 19 Aug 2026 15:40:41 -0400https://pop.com.sg/example-of-a-real-life-graph-with-a-hole.htmlhttps://pop.com.sg/example-of-a-real-life-graph-with-a-hole.htmlAnyone ever come across a real non-textbook example of a graph with a hole in it? In Precalc, you get into graphing rational expressions, some of which reduce to a non-rational. The cancelled f......Wed, 19 Aug 2026 15:33:45 -0400https://pop.com.sg/why-is-the-inductive-hypothesis-assumed-to-be-true.htmlhttps://pop.com.sg/why-is-the-inductive-hypothesis-assumed-to-be-true.htmlThe logic of the Inductive proof seems circular, the whole proof seems to hinge on whether or not the Inductive Hypothesis is true. Sure you can show that p(k+1) is true if p(k) is true, you have ......Wed, 19 Aug 2026 15:16:24 -0400https://pop.com.sg/why-does-limit-not-exist.htmlhttps://pop.com.sg/why-does-limit-not-exist.htmlI don't understand how the limit does not exist for the composite function. The limit as x approaches -2 for g(x) is zero. So, the last step is to evaluate h(0), which is -1. Yes, there is a hole a......Wed, 19 Aug 2026 15:14:18 -0400https://pop.com.sg/difference-between-weak-or-martingale-and-strong-solutions-to-sdes.htmlhttps://pop.com.sg/difference-between-weak-or-martingale-and-strong-solutions-to-sdes.htmlHi Im fairly new to SDE theory and am struggling with the difference between a weak ( or martingale ) solution and a strong solution to an SDE : $$ d(X_{t})=b(t,X_{t})dt + \sigma(t,X_{t})dW_{t} ......Wed, 19 Aug 2026 15:08:17 -0400https://pop.com.sg/graph-x-2-y-2-z-2-4-1.htmlhttps://pop.com.sg/graph-x-2-y-2-z-2-4-1.htmlGraph $x^2+y^2+z^2/4=1$ I want to know what would be the graph of the equation...Wed, 19 Aug 2026 15:06:10 -0400https://pop.com.sg/proving-that-sin-z-2-sin-x-2-sinh-y-2.htmlhttps://pop.com.sg/proving-that-sin-z-2-sin-x-2-sinh-y-2.htmlGoal: Prove $|\sin(z)|^2 = \sin(x)^2+\sinh(y)^2$ I have \begin{align*} \sin(x+iy) &=\sin(x)\cos(iy)+\cos(x)\sin(iy)\\ &=\sin(x)\cosh(y)+i\cos(x)\sinh(y)\\ |\sin(z)|^2 &=\sin(x)^2 \cosh......Wed, 19 Aug 2026 14:56:21 -0400https://pop.com.sg/mean-and-variance-of-squared-gaussian-y-x-2-where-x-sim-mathcal-n-0-si.htmlhttps://pop.com.sg/mean-and-variance-of-squared-gaussian-y-x-2-where-x-sim-mathcal-n-0-si.htmlWhat is the mean and variance of Squared Gaussian: $Y=X^2$ where: $X\sim\mathcal{N}(0,\sigma^2)$? It is interesting to note that Gaussian R.V here is zero-mean and non-central Chi-square Distribut......Wed, 19 Aug 2026 14:51:44 -0400https://pop.com.sg/what-is-the-best-strategy-for-cookie-clicker-esque-games.htmlhttps://pop.com.sg/what-is-the-best-strategy-for-cookie-clicker-esque-games.htmlToday, I stumbled across the game Cookie Clicker, which I recommend you avoid until you have at least a few hours of time to waste. The basic idea behind the game is this: You have a large stash......Wed, 19 Aug 2026 14:48:12 -0400https://pop.com.sg/find-the-area-enclosed-by-one-loop-of-this-lemniscate.htmlhttps://pop.com.sg/find-the-area-enclosed-by-one-loop-of-this-lemniscate.htmlFind the area enclosed by one loop of the lemniscate with equation $r^2 = 81cos(2\theta)$. Choose your limits carefully. When I was solving this problem, my intuition was to take the integral from......Wed, 19 Aug 2026 14:47:33 -0400https://pop.com.sg/how-to-prove-that-the-following-function-has-exactly-two-zeroes-in-a-p.htmlhttps://pop.com.sg/how-to-prove-that-the-following-function-has-exactly-two-zeroes-in-a-p.htmlI am practicing exam questions for my own exam in complex analysis. This was one I couldn't figure out. Let $U = \mathbb{C} \setminus \{ x \in \mathbb{R} : x \leq 0 \} $ en let $\log : U \to \math......Wed, 19 Aug 2026 14:47:31 -0400https://pop.com.sg/how-to-test-whether-a-set-of-four-points-can-form-a-parallelogram.htmlhttps://pop.com.sg/how-to-test-whether-a-set-of-four-points-can-form-a-parallelogram.htmlGiven four points $(x_1,y_1)$,$(x_2,y_2)$,$(x_3,y_3)$,$(x_4,y_4)$. How can we efficiently test them to make sure whether they are vertices of a parallelogram? I think evaluating and comparing the ...Wed, 19 Aug 2026 14:46:05 -0400https://pop.com.sg/finding-area-between-two-curves-below-and-above-x-axis.htmlhttps://pop.com.sg/finding-area-between-two-curves-below-and-above-x-axis.htmlI'm trying to find the area between: $y = 2x^2 - 1$ and $y = x^2$ I have found that the intersection points are at $(-1,1)$ and $(1,1)$. But the part that confuses me is that $y = 2x^2 - 1$ goes......Wed, 19 Aug 2026 14:45:56 -0400https://pop.com.sg/direct-integration-method.htmlhttps://pop.com.sg/direct-integration-method.htmlI want to directly integrate the following equation: $\frac{d^2f}{dt^2} - af = 0$ . I am wondering whether I have used a correct method. I denote $\frac{df}{dt} = f'$. Here are my steps: $f'' - ......Wed, 19 Aug 2026 14:42:10 -0400https://pop.com.sg/using-truth-tables-verify-that-the-contrapositive-and-original-stateme.htmlhttps://pop.com.sg/using-truth-tables-verify-that-the-contrapositive-and-original-stateme.htmlHow do I approach this? I understand how to construct a truth table, such as this one: However, is this enough to prove that they are equivalent statements?...Wed, 19 Aug 2026 14:34:49 -0400https://pop.com.sg/find-probability-density-function-of-y-sigma-x-mu.htmlhttps://pop.com.sg/find-probability-density-function-of-y-sigma-x-mu.htmlX has the pdf $f_x(x)$ Consider $Y=\sigma X+\mu$ where $\sigma$ and $\mu$ are constant. Find the PDF of $Y$? I find that $\mu_Y= \mu+\sigma \mu_x$ And $\sigma_Y^2=\sigma^2\sigma_x^2$ But I cannot f......Wed, 19 Aug 2026 14:28:44 -0400https://pop.com.sg/reference-request-for-gibbs-measure.htmlhttps://pop.com.sg/reference-request-for-gibbs-measure.htmlUnderstanding the Gibbs measure seems to be essential to fulling understanding statistical mechanics. However, I don't know of any mathematically rigorous textbooks/lecture notes that talk about such ...Wed, 19 Aug 2026 14:25:14 -0400https://pop.com.sg/int-infty-infty-e-ikx-dx-equals-what.htmlhttps://pop.com.sg/int-infty-infty-e-ikx-dx-equals-what.htmlWhat would $\int\limits_{-\infty}^\infty e^{ikx}dx$ be equal to where $i$ refers to imaginary unit? What steps should I go over to solve this integral? I saw this in the Fourier transform, and am ...Wed, 19 Aug 2026 14:22:49 -0400https://pop.com.sg/topological-basis-vs-local-basis.htmlhttps://pop.com.sg/topological-basis-vs-local-basis.htmlGiven a topological space $(X, \tau)$ and a basis $B$ for $X$, every open set $A \in \tau$ can be written as $A = \cup B_\alpha$, where $B_\alpha \in B$. The definition of a local basis is supposed......Wed, 19 Aug 2026 14:19:28 -0400https://pop.com.sg/i-need-help-writing-the-expression-5-sin-2x-cos-2x-in-terms-of-sine-on.htmlhttps://pop.com.sg/i-need-help-writing-the-expression-5-sin-2x-cos-2x-in-terms-of-sine-on.htmlI need help writing the expression $5(\sin 2x- \cos 2x)$ in terms of sine only. Using the double angle identity formula, I was able to get it to $5(\sin 2x - 1 + 2\sin^2x)$, but not sure how to move ...Wed, 19 Aug 2026 14:13:47 -0400https://pop.com.sg/spivak-ch-11-theorem-7.htmlhttps://pop.com.sg/spivak-ch-11-theorem-7.htmlCan someone please explain how to supply a rigorous $\epsilon,\delta$ argument for this theorem as Spivak says ? My argument is: $f'(a)=\lim_{h\to 0} f'(\alpha_h)$ equivalent to $|f'(\alpha_h)-f'......Wed, 19 Aug 2026 14:04:30 -0400https://pop.com.sg/what-is-the-significance-of-integrating-a-function.htmlhttps://pop.com.sg/what-is-the-significance-of-integrating-a-function.htmlNow i understand how important these things can be in terms of very small changes or finding area under curves and otherwise. However, when we integrate a function such as y = x we get (x^2)/2, and......Wed, 19 Aug 2026 14:04:20 -0400https://pop.com.sg/defective-matrix-and-eigenvalues.htmlhttps://pop.com.sg/defective-matrix-and-eigenvalues.htmlA matrix $A$ is called defective if $A$ has an eigenvalue $\lambda$ of multiplicity $m > 1$ for which the associated eigenspace has a basis of fewer than $m$ vectors; that is, the dimension of the ...Wed, 19 Aug 2026 13:46:36 -0400https://pop.com.sg/how-to-find-the-center-of-a-parallelogram.htmlhttps://pop.com.sg/how-to-find-the-center-of-a-parallelogram.htmlI want to find the center of a parallelogram in order to use it in my java program. I have four coordinates of the parallelogram and I want to find the center coordinate of the parallelogram. It se......Wed, 19 Aug 2026 13:43:25 -0400https://pop.com.sg/integral-notation-confusion-x-vs-t.htmlhttps://pop.com.sg/integral-notation-confusion-x-vs-t.htmlOne of our worksheets contains the following integrals (areas) to find: $$\int_0^x 1 \, dx$$ $$\int_0^x x \, dx$$ I am skeptical. Although this is our first day learning about integrals in class......Wed, 19 Aug 2026 13:35:16 -0400https://pop.com.sg/how-do-we-find-out-angle-from-x-y-coordinates.htmlhttps://pop.com.sg/how-do-we-find-out-angle-from-x-y-coordinates.htmlI found the following sentence. To find the angle you use the arctangent function like this, angle $=\tan^{-1}\left(\frac{y}{x}\right)$. But I am curious, is this the only way to know the angle......Wed, 19 Aug 2026 13:31:40 -0400https://pop.com.sg/how-do-percentiles-work.htmlhttps://pop.com.sg/how-do-percentiles-work.htmlIf I have 5 students (A-E) that score 80%, 70%, 70%, 60% and 50% on a test what percentiles do they fall in? A - 20th percentile (80%) B - 40th percentile (70%) C - 40th percentile (70%) D - 80th ...Wed, 19 Aug 2026 13:27:46 -0400https://pop.com.sg/problem-similar-to-folland-chapter-2-problem-51.htmlhttps://pop.com.sg/problem-similar-to-folland-chapter-2-problem-51.htmlLet $(X\mathcal M,\mu)$ and $(Y,\mathcal N,v)$ be $\sigma$-finite measure spaces. If $f:X\to\Bbb R$ is $\mathcal M$-measurable, $g:Y\to\Bbb R$ is $\mathcal N$-measurable, and $h(x,y)=f(x)g(......Wed, 19 Aug 2026 13:21:39 -0400https://pop.com.sg/maclaurin-polynomial-of-tan-x.htmlhttps://pop.com.sg/maclaurin-polynomial-of-tan-x.htmlThe method used to find the Maclaurin polynomial of sin(x), cos(x), and $e^x$ requires finding several derivatives of the function. However, you can only take a couple derivatives of tan(x) before it ...Wed, 19 Aug 2026 13:17:21 -0400https://pop.com.sg/how-to-find-the-point-on-the-sphere-that-is-closest-to-a-plane.htmlhttps://pop.com.sg/how-to-find-the-point-on-the-sphere-that-is-closest-to-a-plane.htmlConsider the plane $x+2y+2z=4$, how to find the point on the sphere $x^2+y^2+z^2=1$ that is closest to the plane? I could find the distance from the plane to the origin using the formula $D=\frac{|1\...Wed, 19 Aug 2026 13:05:23 -0400https://pop.com.sg/what-is-the-reflection-across-a-parabola.htmlhttps://pop.com.sg/what-is-the-reflection-across-a-parabola.htmlReflection across a line is well familiar, reflection across a circle is the inversion, the point at a distance $d$ from the center is reflected into a point on the same ray through the center, but......Wed, 19 Aug 2026 13:02:21 -0400https://pop.com.sg/user-david-yang-mathematics-stack-exchange.htmlhttps://pop.com.sg/user-david-yang-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Wed, 19 Aug 2026 12:59:09 -0400https://pop.com.sg/what-is-so-linear-about-linear-combination.htmlhttps://pop.com.sg/what-is-so-linear-about-linear-combination.htmlWhat is linear about a linear combination of things?. In linear algebra, the "things" we are dealing with are usually vectors and the linear combination gives the span of the vectors. Or it could b......Wed, 19 Aug 2026 12:55:27 -0400https://pop.com.sg/physical-significance-of-knot-vector-in-b-spline.htmlhttps://pop.com.sg/physical-significance-of-knot-vector-in-b-spline.htmlA B-spline blending curve formulation is: $P(u)=\sum_{k=0}^np_k B_{k,d}(u)$ Given $n+1$ control points, B-spline blending functions are polynomials of degree $d-1$, $(1<d<=n+1)$. ......Wed, 19 Aug 2026 12:51:45 -0400https://pop.com.sg/product-of-cosines-prod-r-1-7-cos-left-frac-r-pi-15-right.htmlhttps://pop.com.sg/product-of-cosines-prod-r-1-7-cos-left-frac-r-pi-15-right.htmlEvaluate $$ \prod_{r=1}^{7} \cos \left({\dfrac{r\pi}{15}}\right) $$ I tried trigonometric identities of product of cosines, i.e, $$\cos\text{A}\cdot\cos\text{B} = \dfrac{1}{2}[ \cos(A+B)+\cos(A-B)......Wed, 19 Aug 2026 12:49:43 -0400https://pop.com.sg/calculate-expectation-of-a-geometric-random-variable.htmlhttps://pop.com.sg/calculate-expectation-of-a-geometric-random-variable.htmlWhen you download a file from a website, the file gets corrupted with probability 0.8. What is the expected number of downloads to get an uncorrupted file? I have no idea how to do this. I only kn......Wed, 19 Aug 2026 12:48:26 -0400https://pop.com.sg/what-is-an-intuitive-explanation-of-the-combinations-formula-duplicate.htmlhttps://pop.com.sg/what-is-an-intuitive-explanation-of-the-combinations-formula-duplicate.htmlI perfectly understand the permutations formula i.e. if you have $n$ things how many ways can you rearrange it if taken $k$ at a time (or if you have $k$ slots)? So you draw the following tree. And......Wed, 19 Aug 2026 12:43:31 -0400https://pop.com.sg/final-value-theorem-for-closed-system.htmlhttps://pop.com.sg/final-value-theorem-for-closed-system.htmlWe have a system with output given by $\frac{Y(s)}{R(s)} = \frac{F(s)G(s)}{1+F(s)G(s)}$ where $F(s)G(s) = K\frac{s+1}{s^2+s+1}$. Let $K=4$ and $R(s) = 10/s$. Using the final value theorem, calcu......Wed, 19 Aug 2026 12:36:57 -0400https://pop.com.sg/prove-that-9-n-4-n-1-is-a-multiple-of-5-for-all-n-in-mathbb-n.htmlhttps://pop.com.sg/prove-that-9-n-4-n-1-is-a-multiple-of-5-for-all-n-in-mathbb-n.htmlProve that $9^n+4^{n+1}$ is a multiple of $5$ for all $n \in \mathbb{N}$ Proof. So i'm going to prove this by induction. The first case when $n=1$ is trivial since: $$9+16=25,$$ implying $ 5 \mid......Wed, 19 Aug 2026 12:27:15 -0400https://pop.com.sg/limit-of-logarithm.htmlhttps://pop.com.sg/limit-of-logarithm.htmlI came across this relation a lot,when evaluating lim of function like natural logarithm I could find that evaluating limit for logarithm equal to evaluating logarithm for limit. Why this is true? ......Wed, 19 Aug 2026 12:17:33 -0400https://pop.com.sg/area-of-a-trapezoid-without-the-height.htmlhttps://pop.com.sg/area-of-a-trapezoid-without-the-height.htmlHow would I find the area of a non-iscoceles trapezoid and without the height? The trapezoid's bases are $30$ and $40$, and the legs $14$ and $16$. Thanks...Wed, 19 Aug 2026 12:13:08 -0400https://pop.com.sg/why-do-we-use-decimal-system-duplicate.htmlhttps://pop.com.sg/why-do-we-use-decimal-system-duplicate.htmlWhy do we use decimal system? Some people give the answer that we have 10 fingers and it is therefore convenient for us to use decimal system, but I am not satisfy with this as I think this is too......Wed, 19 Aug 2026 12:06:51 -0400https://pop.com.sg/what-is-a-binary-expansion-of-a-real-number.htmlhttps://pop.com.sg/what-is-a-binary-expansion-of-a-real-number.htmlRelated to this question I asked. I want to know what is exactly meant by a binary expansion of a number, real or natural. Can someone show me via example, how would you binary expand a real number ...Wed, 19 Aug 2026 12:04:53 -0400https://pop.com.sg/using-the-comparison-test-prove-that-the-infinite-series-of-n-2-1-n-3-.htmlhttps://pop.com.sg/using-the-comparison-test-prove-that-the-infinite-series-of-n-2-1-n-3-.htmlThe conditions of the comparison sum state that if $0\le a_n\le b_n$ and if $b_n$ converges, then $a_n$ also converges and if $a_n$ diverges, then $b_n$ also diverges. I'm not sure how to go abou......Wed, 19 Aug 2026 12:02:39 -0400https://pop.com.sg/a-spectral-theorem-for-a-skew-hermitian-complex-matrix.htmlhttps://pop.com.sg/a-spectral-theorem-for-a-skew-hermitian-complex-matrix.htmlSorry if this is a duplicate, but I tried searching on this but only found results dealing with skew symmetric real matrices. So $A\in\mathbb{C}^{n\times n}$ is called skew-Hermitian if $A^*=-A$. I'm ...Wed, 19 Aug 2026 11:58:07 -0400https://pop.com.sg/how-to-compute-two-039-s-complement-of-a-negative-number.htmlhttps://pop.com.sg/how-to-compute-two-039-s-complement-of-a-negative-number.htmlI made the search on here and on google and couldn't find anything that answered the topic title. From my bit of understanding, two's complement can be used to make a decimal number, negative. Wh......Wed, 19 Aug 2026 11:39:09 -0400https://pop.com.sg/pseudo-arclength-continuation-scheme.htmlhttps://pop.com.sg/pseudo-arclength-continuation-scheme.htmlI have implemented a simple parameter continuation scheme to find the stationary solutions of a nonlinear problem at different parameter values. However, my scheme cannot handle bifurcations - it f......Wed, 19 Aug 2026 11:36:51 -0400https://pop.com.sg/similarity-between-trapezoidal-sum-and-riemann-sums.htmlhttps://pop.com.sg/similarity-between-trapezoidal-sum-and-riemann-sums.htmlIs it accurate to say that a Trapezoidal Riemann Sum includes one more term than the corresponding Left, Right, or Midpoint Riemann Sum? Using the following notation for the latter three: Rieman......Wed, 19 Aug 2026 11:36:01 -0400https://pop.com.sg/calculate-maximum-velocity-given-accel-decel-initial-v-final-position.htmlhttps://pop.com.sg/calculate-maximum-velocity-given-accel-decel-initial-v-final-position.htmlI'm trying to calculate the maximum velocity of an object given: acceleration deceleration starting position (0) ending position initial velocity I've been trying to use formulas from: https://...Wed, 19 Aug 2026 11:34:34 -0400https://pop.com.sg/show-the-series-an-1-an-converges-given-that-series-an-converges.htmlhttps://pop.com.sg/show-the-series-an-1-an-converges-given-that-series-an-converges.htmlGiven that $0\le a_n\lt 1$ the series $\sum_{n=0}^{\infty} (a_n)$ converges. Show that the series $\sum_{n=0}^{\infty} \frac{a_n}{1-a_n}$ converges. This question is supposed to be solved from first ...Wed, 19 Aug 2026 11:12:26 -0400https://pop.com.sg/are-category-theory-and-set-theory-on-the-equal-foundational-footing.htmlhttps://pop.com.sg/are-category-theory-and-set-theory-on-the-equal-foundational-footing.htmlSet-theory is widely taken to be foundational to the rest of mathematics. So is category-theory. My question is: Are they two alternative, rival candidates for the role of a foundational theory of ...Wed, 19 Aug 2026 10:56:09 -0400https://pop.com.sg/what-do-finite-infinite-countable-not-countable-countably-infinite-mea.htmlhttps://pop.com.sg/what-do-finite-infinite-countable-not-countable-countably-infinite-mea.htmlPossible Duplicate: Different kinds of infinities? Those terms keep confusing me. Could somebody give a clear and unambiguous definition? How do we prove a set is countable, finite, ...? I kno......Wed, 19 Aug 2026 10:54:37 -0400https://pop.com.sg/is-there-a-symbol-for-difference-like-sum-for-summation.htmlhttps://pop.com.sg/is-there-a-symbol-for-difference-like-sum-for-summation.htmlIm curious, if there is a symbol for summation: $$\sum$$ then is there a symbol for difference? If not, if not how could I represent the difference of a sequence of numbers $(1-2-3-4-...)$?...Wed, 19 Aug 2026 10:44:27 -0400https://pop.com.sg/algebra-trick-for-dividing-polynomials.htmlhttps://pop.com.sg/algebra-trick-for-dividing-polynomials.htmlI saw that "once have a root, we can divide the polynomials" and somehow $\frac{3n^3 - 39n^2 + 360n + 20}{3(n-n_1)} = n^2 + (n_1 - 13)n + (n_1^2 - 13n_1 + 120)$ I'm noticing a pattern that there's......Wed, 19 Aug 2026 10:37:53 -0400https://pop.com.sg/how-does-one-calculate-all-orbits-of-an-ode-system.htmlhttps://pop.com.sg/how-does-one-calculate-all-orbits-of-an-ode-system.htmlI have the following ODE system and have been tasked to find all of it's orbits. The system is $$x' = y(x^2 + 1)$$ $$y' = x(x^2 + 1)$$ From here is where I'm lost. I found one definition of an orbit ...Wed, 19 Aug 2026 10:35:15 -0400https://pop.com.sg/how-to-calculate-curvature-of-earth-per-surface-kilometer.htmlhttps://pop.com.sg/how-to-calculate-curvature-of-earth-per-surface-kilometer.htmlI was watching a video regarding Flat Earthers giving curvatures of Earth that sounded way too big, so I decided to calculate how much the surface of Earth would drop with respect to a line perpend......Wed, 19 Aug 2026 10:35:12 -0400https://pop.com.sg/does-one-always-use-augmented-matrices-to-solve-systems-of-linear-equa.htmlhttps://pop.com.sg/does-one-always-use-augmented-matrices-to-solve-systems-of-linear-equa.htmlThe homework tag is to express that I am a student with no working knowledge of math. I know how to use elimination to solve systems of linear equations. I set up the matrix, perform row operations ...Wed, 19 Aug 2026 10:34:18 -0400https://pop.com.sg/proving-the-trace-of-a-transformation-is-independent-of-the-basis-chos.htmlhttps://pop.com.sg/proving-the-trace-of-a-transformation-is-independent-of-the-basis-chos.htmlHow would you prove the trace of a transformation from V to V (where V is finite dimensional) is independent of the basis chosen?...Wed, 19 Aug 2026 10:28:16 -0400https://pop.com.sg/why-are-certain-pde-called-elliptic-hyperbolic-or-parabolic.htmlhttps://pop.com.sg/why-are-certain-pde-called-elliptic-hyperbolic-or-parabolic.htmlWhy are the Partial Differential Equations so named? i.e, elliptical, hyperbolic, and parabolic. I do know the condition at which a general second order partial differential equation becomes these,......Wed, 19 Aug 2026 10:12:00 -0400https://pop.com.sg/how-do-i-find-a-perpendicular-line-from-an-equation-in-general-form.htmlhttps://pop.com.sg/how-do-i-find-a-perpendicular-line-from-an-equation-in-general-form.htmlSuppose I have the line $ax + bx + c = 0$. How do I find the values of $j$, $k$, and $l$ for the perpendicular line $jx + kx + l = 0$? Where the two lines cross at point $(p,q)$? I know if the lin......Wed, 19 Aug 2026 09:53:28 -0400https://pop.com.sg/projection-formula.htmlhttps://pop.com.sg/projection-formula.htmlDoes someone know if in the problem the projection of x onto U is defined like that : $x_u = \displaystyle \frac{\langle x,u\rangle}{u. u}$ $u$ Problem: Let $U,V\subset\mathbb C^n$ be two subspa......Wed, 19 Aug 2026 09:49:10 -0400https://pop.com.sg/probability-of-event-occurring-multiple-times.htmlhttps://pop.com.sg/probability-of-event-occurring-multiple-times.htmlI'm trying to calculate the probability of exactly $8$ identical independent events occurring if each has a $25\%$ chance of occurring. To simplify, if I were rolling a four-sided die $8$ times, ......Wed, 19 Aug 2026 09:44:26 -0400https://pop.com.sg/lines-rotation-problem.htmlhttps://pop.com.sg/lines-rotation-problem.htmlGiven $3$ infinite lines $L_1$, $L_2$, $L_3$ in $R^3$ ($3D$ space), each line is represented by two $3D$ points, the line $L_1$ is rotating around the axis line $L_3$ until it intersects with the l......Wed, 19 Aug 2026 09:43:30 -0400https://pop.com.sg/bounded-variation-is-bounded.htmlhttps://pop.com.sg/bounded-variation-is-bounded.htmlLet $f:[a,b]\to \mathbb{R}$ and let $D=\{x_o,x_1,...,x_n\}$ be a division of $[a,b]$. We say that $f$ is of bounded variation on $[a,b]$ if $\displaystyle \sup_{D\in \mathscr{D}} \sum_{i=1}^n ......Wed, 19 Aug 2026 09:42:00 -0400https://pop.com.sg/find-the-limit-lim-limits-x-to1-dfrac-x-1-5-1-x-1-3-1.htmlhttps://pop.com.sg/find-the-limit-lim-limits-x-to1-dfrac-x-1-5-1-x-1-3-1.htmlFind the limit of $$\lim_{x\to 1}\frac{x^{1/5}-1}{x^{1/3}-1}$$ How should I approach it? I tried to use L'Hopital's Rule but it's just keep giving me 0/0....Wed, 19 Aug 2026 09:30:11 -0400https://pop.com.sg/given-the-lenght-of-the-hypotenus-and-the-sum-of-the-legs-find-the-are.htmlhttps://pop.com.sg/given-the-lenght-of-the-hypotenus-and-the-sum-of-the-legs-find-the-are.htmlGiven: right-angled $\triangle ABC$, the length of the hypotenuse $c = 5$ and the sum of the legs $a+b = 6$. Find the area of the triangle. I tried creating the following system $a+b = 6; a^2+b^2 =......Wed, 19 Aug 2026 09:29:15 -0400https://pop.com.sg/can-someone-explain-the-abc-conjecture-to-me.htmlhttps://pop.com.sg/can-someone-explain-the-abc-conjecture-to-me.htmlI am an undergrad and I know that the conjecture may have been proven recently. But in reading about it, I am entirely confused as to what it means and why it is important. I was hoping some of you......Wed, 19 Aug 2026 09:25:33 -0400https://pop.com.sg/user-minus-one-twelfth-mathematics-stack-exchange.htmlhttps://pop.com.sg/user-minus-one-twelfth-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Wed, 19 Aug 2026 09:12:02 -0400https://pop.com.sg/maximum-principle-for-harmonic-functions.htmlhttps://pop.com.sg/maximum-principle-for-harmonic-functions.htmlI know the following classical maximum principle for harmonic functions: If $\Omega \subset \mathbb{C}$ is open and connected and $u \in C^2(\Omega)$ is harmonic, then $u$ has maximum (or ...Wed, 19 Aug 2026 09:08:07 -0400https://pop.com.sg/what-would-you-see-inside-a-spherical-mirror.htmlhttps://pop.com.sg/what-would-you-see-inside-a-spherical-mirror.htmlImage to build a huge spherical shell made of semitransparent glass, and to cover the internal part with a reflecting material. In such structure some light can enter, and an observer inside it ......Wed, 19 Aug 2026 09:04:32 -0400https://pop.com.sg/how-to-find-range-of-a-quadratic-quadratic-function-easily-without-plo.htmlhttps://pop.com.sg/how-to-find-range-of-a-quadratic-quadratic-function-easily-without-plo.htmlIs there any way to find range of a Quadratic/Quadratic function, without plotting its graph?...Wed, 19 Aug 2026 09:03:48 -0400https://pop.com.sg/algorithm-to-get-the-maximum-size-of-n-squares-that-fit-into-a-rectang.htmlhttps://pop.com.sg/algorithm-to-get-the-maximum-size-of-n-squares-that-fit-into-a-rectang.htmlI am looking for an algorithm that can return the number of size of n squares that fit into a a rectangle of a given width and height, maximizing the use of space (thus, leaving the least amount of ...Wed, 19 Aug 2026 09:03:40 -0400https://pop.com.sg/is-there-a-variant-of-the-dot-product-operation-that-returns-frac-a-1-.htmlhttps://pop.com.sg/is-there-a-variant-of-the-dot-product-operation-that-returns-frac-a-1-.htmlThe dot product of $[a_1,a_2]$ and $[b_1, b_2]$ is $a_1b_1 + a_2b_2$ (and so on for bigger vectors). What I'm wondering is if there's any definition of a function s.t. the "invdot" produc......Wed, 19 Aug 2026 08:59:02 -0400https://pop.com.sg/expected-value-of-a-poisson-process.htmlhttps://pop.com.sg/expected-value-of-a-poisson-process.htmlI've been searching for a while but I can't seem to figure out how to find the expected value of a poisson process up to an arbitrary time. Let {$N(t),t≥0$} be a Poisson process with rate $λ$. H......Wed, 19 Aug 2026 08:51:33 -0400https://pop.com.sg/coordinate-matrix-of-a-vector-in-terms-of-a-basis.htmlhttps://pop.com.sg/coordinate-matrix-of-a-vector-in-terms-of-a-basis.htmlI am taking a linear algebra class currently and working through Hoffman's textbook. One of the exercises I am unsure about is, Find the coordinate matrix of the vector $\alpha=(1,0,1)$ in the b......Wed, 19 Aug 2026 08:49:40 -0400https://pop.com.sg/derivative-of-a-square-root-with-exponential-function.htmlhttps://pop.com.sg/derivative-of-a-square-root-with-exponential-function.htmlSo I have the following function: $f(x)= \sqrt{e^{2x}}$ After applying the chain rule I sit with: $$\frac{1}{2\sqrt{e^{2x}}}2e^{2x}$$ From there I got: $$\frac{e^{2x}}{\sqrt{e^{2x}}}$$ While ......Wed, 19 Aug 2026 08:36:38 -0400https://pop.com.sg/creating-truth-tables-variable-columns.htmlhttps://pop.com.sg/creating-truth-tables-variable-columns.htmlI'm looking for the mathematically equation/algorithm to create the variable columns in a truth table. I really want to understand how this is done, so any directions will be a plus. Ok, so imagin......Wed, 19 Aug 2026 08:36:31 -0400https://pop.com.sg/combinatorics-formula-reference.htmlhttps://pop.com.sg/combinatorics-formula-reference.htmlI'm trying to organize the formulas for permutations and combinations. Let's say we sample $r$ objects from an urn which contains $n$ objects, $m$ of which are duplicates. So if the objects in the......Wed, 19 Aug 2026 08:26:40 -0400https://pop.com.sg/how-to-find-the-point-where-the-curvature-is-maximum.htmlhttps://pop.com.sg/how-to-find-the-point-where-the-curvature-is-maximum.htmlHow to find the point on the curve $y=e^x$ at which the curvature is maximum. from the graph of the above curve i think at (0,1) the curvature is maximum.Is it true? please help....Wed, 19 Aug 2026 08:24:33 -0400https://pop.com.sg/user-lucusk-mathematics-stack-exchange.htmlhttps://pop.com.sg/user-lucusk-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Wed, 19 Aug 2026 08:24:30 -0400https://pop.com.sg/is-x-1-3-differentiable-at-0.htmlhttps://pop.com.sg/is-x-1-3-differentiable-at-0.htmlIt occurred to me that functions that are "smooth" but have "infinite slope" may not be considered differentiable at the points where their slope is infinite. An easy example of this is $x^{1/3}$, ......Wed, 19 Aug 2026 08:21:38 -0400https://pop.com.sg/what-is-a-natural-isomorphism.htmlhttps://pop.com.sg/what-is-a-natural-isomorphism.htmlI came across the adjective "natural" many times in my reading and I think this has something to do with category theory. Could someone please illustrate the idea behind this adjective to me? Many ...Wed, 19 Aug 2026 08:14:41 -0400https://pop.com.sg/formal-definition-of-counterexample.htmlhttps://pop.com.sg/formal-definition-of-counterexample.htmlWhat is the preferred formal definition of “counterexample” as in: zero is a counterexample for "every integer is either positive or negative". Where in the literature is the notion of “counterexam......Wed, 19 Aug 2026 08:12:59 -0400https://pop.com.sg/projectile-motion-with-air-resistance-proportional-to-velocity-squared.htmlhttps://pop.com.sg/projectile-motion-with-air-resistance-proportional-to-velocity-squared.htmlA plane flies at a constant altitude of 1000 ft with a constant speed of 300 mph. The plane drops a relief package to a person on the ground. Assume the origin is point where the supply pack is rel......Wed, 19 Aug 2026 08:11:59 -0400https://pop.com.sg/how-to-find-average-rate-of-change.htmlhttps://pop.com.sg/how-to-find-average-rate-of-change.htmlHow would I find the average rate of change over $8$ minutes, of a person that runs at a rate of $v(t)=x\sin(x^2-7x)$ ft/min? I missed when this was taught and I have no clue on how to do it. Help......Wed, 19 Aug 2026 08:03:27 -0400https://pop.com.sg/how-to-estimate-the-range-of-a-normal-distribution-when-the-mean-and-s.htmlhttps://pop.com.sg/how-to-estimate-the-range-of-a-normal-distribution-when-the-mean-and-s.htmlFor example, how would you respond to this question? The earnings of one-hundred workers in a company are normally distributed. If the mean of this data set is 24 and the standard deviation is 4......Wed, 19 Aug 2026 08:02:10 -0400https://pop.com.sg/factoring-x-6-y-6-what-is-going-on-here.htmlhttps://pop.com.sg/factoring-x-6-y-6-what-is-going-on-here.htmlI apologize if this question already exists, but it was quite difficult to word. In my math class today, we learned how to factor a difference of two perfect cubes. One of our practice questions w......Wed, 19 Aug 2026 07:55:44 -0400https://pop.com.sg/what-does-sin-sin-x-mean.htmlhttps://pop.com.sg/what-does-sin-sin-x-mean.htmlWhat does an equation like $\sin(\sin(x))$ mean? I know it can be seen as a composite function $f(f(x))$, where $f(x)=\sin(x)$. Is there a way to simplify functions like this, and where will this b......Wed, 19 Aug 2026 07:54:48 -0400https://pop.com.sg/binomial-expansion-of-1-x-n.htmlhttps://pop.com.sg/binomial-expansion-of-1-x-n.htmlWe've been given with following binomial expansion $$(x+1)^n= 1+ nx + \frac{n(n-1)}{ 2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3\cdots$$ How can I get the formula of $(1-x)^n$...Wed, 19 Aug 2026 07:45:22 -0400https://pop.com.sg/which-versions-of-chernoff-bound-are-applied-to-binomial-distribution-.htmlhttps://pop.com.sg/which-versions-of-chernoff-bound-are-applied-to-binomial-distribution-.htmlThere are several versions of Chernoff bounds. I was wodering which versions are applied to computing the probabilities of a Binomial distribution in the following two examples, but couldn't. I have ...Wed, 19 Aug 2026 07:44:49 -0400https://pop.com.sg/parabolic-word-problem.htmlhttps://pop.com.sg/parabolic-word-problem.htmlA rectangular barge is traveling under a bridge with a parabolic archway. The barge is 60 feet tall and 80 feet wide. The bridge is 80 feet tall and 200 feet wide. If the barge must travel down the ...Wed, 19 Aug 2026 07:41:10 -0400https://pop.com.sg/why-is-the-expected-value-e-x-2-neq-e-x-2.htmlhttps://pop.com.sg/why-is-the-expected-value-e-x-2-neq-e-x-2.htmlI wish to use the Computational formula of the variance to calculate the variance of a normal-distributed function. For this, I need the expected value of $X$ as well as the one of $X^2$. Intuitive......Wed, 19 Aug 2026 07:40:51 -0400https://pop.com.sg/how-to-understand-complex-rotation.htmlhttps://pop.com.sg/how-to-understand-complex-rotation.htmlI come across A Visual, Intuitive Guide to Imaginary Numbers, and find it's difficult for me to understand this section: Let’s take a look. Suppose I’m on a boat, with a heading of 3 units East for ...Wed, 19 Aug 2026 07:38:43 -0400https://pop.com.sg/finding-the-no-of-ways-to-count-the-letters-in-an-english-alphabet.htmlhttps://pop.com.sg/finding-the-no-of-ways-to-count-the-letters-in-an-english-alphabet.htmlHow many strings of six lower case letters from the English alphabet contain a) the letter $a$? b) the letters $a$ and $b$? c) the letters $a$ and $b$ in consecutive positions with $a$ preceding......Wed, 19 Aug 2026 07:38:01 -0400https://pop.com.sg/find-an-equation-for-the-plane-that-contains-the-two-parallel-lines-l-.htmlhttps://pop.com.sg/find-an-equation-for-the-plane-that-contains-the-two-parallel-lines-l-.htmlI'm don't know where to start on this question and need some assistance....Wed, 19 Aug 2026 07:31:45 -0400https://pop.com.sg/free-lecture-notes-to-carl-bender-039-s-mathematical-physics-video-lec.htmlhttps://pop.com.sg/free-lecture-notes-to-carl-bender-039-s-mathematical-physics-video-lec.htmlI am just watching Carl Bender's Mathematical Physics video lecture course (about asymptotics and its application in physics) http://www.perimeterscholars.org/328.html which is great! Are there ......Wed, 19 Aug 2026 07:12:54 -0400https://pop.com.sg/choose-the-true-statement.htmlhttps://pop.com.sg/choose-the-true-statement.htmllet f(z) be the meromorphic function fiven by function $f(z)=\frac{z}{(1-e^z).\sin z}.$ then choose the True statement a) For every $ k$ $\in$ Z \ {0} ,$ z= k\pi$ is a simple pole b) $z=......Wed, 19 Aug 2026 07:11:47 -0400https://pop.com.sg/difference-between-arccos-and-cos-1.htmlhttps://pop.com.sg/difference-between-arccos-and-cos-1.htmlI'm trying to use inverse cosine, in the cosine rule to find an angle of a triange when you know 3 sides. I know this formula and have it written down. However I've left my calculator and I'm havin......Wed, 19 Aug 2026 07:09:14 -0400https://pop.com.sg/calculus-does-the-particle-ever-change-direction-at-what-time-interval.htmlhttps://pop.com.sg/calculus-does-the-particle-ever-change-direction-at-what-time-interval.htmlProblem: A particle moves in a straight line with position relative to the origin given by $s(t)= 2t-\frac{4}{t+1}$ cm, where $t ≥ 0$ is the time in seconds. Here's what I know: a) Find velocity ......Wed, 19 Aug 2026 07:07:37 -0400https://pop.com.sg/pi-controller-and-finding-poles-of-a-transfer-function.htmlhttps://pop.com.sg/pi-controller-and-finding-poles-of-a-transfer-function.htmlFor a Wind turbine, the transfer function is $$G(s) = \frac{4.2(s-827)(s^2-5.4s+194)}{(s+0.2)(s^2+0.1s+482)}\;.$$ Furthermore, let $K(s) = b+c/s$ be a $PI$ controller; find $$\frac{1}{1+GK},\qua......Wed, 19 Aug 2026 07:07:07 -0400