https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usSun, 09 Aug 2026 18:25:27 -0400https://pop.com.sg/find-the-average-value-of-f-over-region-d-f-x-y-6xy-d-is-the-triangle-.htmlhttps://pop.com.sg/find-the-average-value-of-f-over-region-d-f-x-y-6xy-d-is-the-triangle-.htmlI have no idea how to find the limits for this. I've tried plotting it over x but cant figure it out....Sun, 09 Aug 2026 22:16:35 -0400https://pop.com.sg/expected-value-of-function-of-exponential-random-variable.htmlhttps://pop.com.sg/expected-value-of-function-of-exponential-random-variable.htmlLet $X \sim Exp(\lambda)$ (exponential distribution) and $Y=X^a$. For what values of $a\in R$ is $E[Y]< \infty$? I computed $E[Y] = \lambda \int x^a e^{-\lambda x}dx$. How to proceed? Does the ...Sun, 09 Aug 2026 22:08:51 -0400https://pop.com.sg/equations-of-lines-tangent-to-an-ellipse.htmlhttps://pop.com.sg/equations-of-lines-tangent-to-an-ellipse.htmlDetermine the equations of the lines that are tangent to the ellipse $\displaystyle{\frac{1}{16}x^2 + \frac{1}{4}y^2 = 1}$ and pass through $(4,6)$. I know one tangent should be $x = 4$ because it......Sun, 09 Aug 2026 22:05:07 -0400https://pop.com.sg/variance-of-ax-b.htmlhttps://pop.com.sg/variance-of-ax-b.htmlHow do I show that $$\text{Var}(aX+b)=a^2\text{Var}(X)$$. Since, I am reading statistic first time, I don't have any idea how to start. Thanks for helping me....Sun, 09 Aug 2026 22:01:25 -0400https://pop.com.sg/graph-f-x-y-ln-x-y.htmlhttps://pop.com.sg/graph-f-x-y-ln-x-y.htmlI'm trying to graph $f(x,y)=\ln(x)-y$, however, I am not sure how as all of my tools are refusing to graph it. Can you please help me? Thanks...Sun, 09 Aug 2026 21:46:57 -0400https://pop.com.sg/what-is-the-name-of-the-solid-created-by-joining-two-tetrahedra-face-t.htmlhttps://pop.com.sg/what-is-the-name-of-the-solid-created-by-joining-two-tetrahedra-face-t.htmlWhen I connect the bases of two pyramids, I will receive an octahedron. I have a question about an analogical situation with two tetrahedra. What is the name of a geometrical figure, which is a ...Sun, 09 Aug 2026 21:29:29 -0400https://pop.com.sg/are-there-any-k-n-that-have-euler-trails-but-not-euler-cycles.htmlhttps://pop.com.sg/are-there-any-k-n-that-have-euler-trails-but-not-euler-cycles.htmlI know that in $K_n$, a complete graph. ALL the vertices must be of even degree for a euler cycle to exist. So n should be odd in this case if we want euler cycles. And if n is even, then $K_n$ has......Sun, 09 Aug 2026 21:22:58 -0400https://pop.com.sg/converse-of-fermat-039-s-little-theorem.htmlhttps://pop.com.sg/converse-of-fermat-039-s-little-theorem.htmlIf $a^n\equiv a \pmod n$ for all integers $a$, does this imply that $n$ is prime? I believe this is the converse of Fermat's little theorem....Sun, 09 Aug 2026 21:20:08 -0400https://pop.com.sg/is-2n-the-same-as-2-n.htmlhttps://pop.com.sg/is-2n-the-same-as-2-n.htmlI am trying to determine the convergence of a series $$\sum_{n=17}^{\infty} \frac{(n!)}{(2n)!}.$$ Using the ratio test, I have simplified $a_{n+1}/a_n$ to $$\frac{(2n!)}{2(n)!}.$$ If $(2n)!$ is th......Sun, 09 Aug 2026 21:19:23 -0400https://pop.com.sg/can-you-self-teach-yourself-mathematics-closed.htmlhttps://pop.com.sg/can-you-self-teach-yourself-mathematics-closed.htmlIf you are not a natural genius, can you teach yourself Mathematics (through textbooks) enough to make a successful mathematical career?...Sun, 09 Aug 2026 21:16:58 -0400https://pop.com.sg/independence-vs-marginal-independence.htmlhttps://pop.com.sg/independence-vs-marginal-independence.htmlI known that for give $x,y$, if we have $$p(x,y)=p(x)p(y)$$ Then we call $x,y$ are independent. For marginal independence, I found the definition here. Random variable $x$ is marginal independ......Sun, 09 Aug 2026 21:08:02 -0400https://pop.com.sg/differential-on-total-complex.htmlhttps://pop.com.sg/differential-on-total-complex.htmlGiven a bicomplex $C_{\ast,\ast}$, i.e. objects $C_{p,q}$ in an abelian category with horizontal $$d_{p,q}^h:C_{p,q}\to C_{p-1,q}$$ and vertical $$d_{p,q}^v:C_{p,q}\to C_{p,q-1}$$ differentials that ...Sun, 09 Aug 2026 21:02:32 -0400https://pop.com.sg/cis-graphs-finding-values.htmlhttps://pop.com.sg/cis-graphs-finding-values.htmlFor this question: If you guys are not familiar with the $\operatorname{cis}$ notation $$ \operatorname{cis}(x) = (\cos(x) + i\sin(x)) $$ I've tried doing this: By comparing arguments $$......Sun, 09 Aug 2026 20:52:43 -0400https://pop.com.sg/express-in-terms-of-single-vector.htmlhttps://pop.com.sg/express-in-terms-of-single-vector.htmlCan you guys give me a hint for this problem. ABCDEF is a regular hexagon. Express in terms of a single vector the sum of the vectors, $4\overrightarrow{AB}$, $2\overrightarrow{AC}$, $\overrightar......Sun, 09 Aug 2026 20:50:53 -0400https://pop.com.sg/calculate-sin-1-sin-45-circ-without-a-calculator.htmlhttps://pop.com.sg/calculate-sin-1-sin-45-circ-without-a-calculator.htmlWe know that in this triangle, $h^2=a^2+a^2$. Therefore, $$ h=\sqrt{a^2+a^2}=\sqrt{2a^2}=\sqrt{2}\:a\:. $$ As $h$ can be written as $\sqrt{2}\:a$, we now know how to write an expression for $\si......Sun, 09 Aug 2026 20:44:56 -0400https://pop.com.sg/definitions-of-baire-first-and-second-category-sets.htmlhttps://pop.com.sg/definitions-of-baire-first-and-second-category-sets.htmlFrom Planetmath A meager or Baire first category set in a topological space is one which is a countable union of nowhere dense sets. A Baire second category set is one which contains a countable u......Sun, 09 Aug 2026 20:42:18 -0400https://pop.com.sg/how-to-design-shape-a-polyhedron-to-be-nearly-spherically-symmetrical-.htmlhttps://pop.com.sg/how-to-design-shape-a-polyhedron-to-be-nearly-spherically-symmetrical-.htmlThere are only 5 platonic solids, but take a look at these images: How are these things designed? How are they shaped? It looks to me like those hexagons are all the same size and shape, and evenly ...Sun, 09 Aug 2026 20:29:13 -0400https://pop.com.sg/how-to-factorise-x-4-equations.htmlhttps://pop.com.sg/how-to-factorise-x-4-equations.htmlThis is my previous question I'm facing a problem to factorise this $64x^4+64x^3-88x^2-51x+39=0$. How to factorise $x^4$ equations?...Sun, 09 Aug 2026 20:20:07 -0400https://pop.com.sg/write-the-trigonometric-expression-as-an-algebraic-expression.htmlhttps://pop.com.sg/write-the-trigonometric-expression-as-an-algebraic-expression.htmlWrite the trigonometric expression as an algebraic expression. $6 \cos(2 \cos^{-1} x)$ Can someone explain to me how to do this? I tried it on my own after watching a youtube video from patrickjm......Sun, 09 Aug 2026 20:18:35 -0400https://pop.com.sg/find-distance-as-function-of-time.htmlhttps://pop.com.sg/find-distance-as-function-of-time.htmlA car starts from rest and accelerates in $a = \frac{2\cdot m}{3\cdot s^3}t$, After $3$ seconds, The car will be $27$ metres from beginning. Find distance as function of time. I know i have to ...Sun, 09 Aug 2026 20:13:09 -0400https://pop.com.sg/evaluate-definite-integral-using-limit-of-summations.htmlhttps://pop.com.sg/evaluate-definite-integral-using-limit-of-summations.htmlI need to evaluate a definite integral using sums. $$ \int_{-5}^{5} (x-\sqrt{25-x^2})dx $$ I start by getting de value of $\Delta x$: $$ \Delta x = \frac{5-(-5)}{n}=\frac{10}{n} $$ Then for $x......Sun, 09 Aug 2026 20:09:00 -0400https://pop.com.sg/finding-a-derivative-using-multiple-chain-rules.htmlhttps://pop.com.sg/finding-a-derivative-using-multiple-chain-rules.htmlFind the derivative of the function. $y = [x + (x + \sin^2 x)^3]^4$ I know how to use the chain rule and I found the derivative to be: $$4[x+(x+\sin^2(x))^3]^3 \cdot (1 + 3(x + \sin^2(x))^2) \cd......Sun, 09 Aug 2026 20:04:27 -0400https://pop.com.sg/some-confusion-about-what-a-function-really-is.htmlhttps://pop.com.sg/some-confusion-about-what-a-function-really-is.htmlDespite my username, my background is mostly in functional analysis where (at least to my understanding), a function $f$ is considered as a mathematical object in its own right distinctly different......Sun, 09 Aug 2026 20:00:39 -0400https://pop.com.sg/what-fraction-of-a-sphere-can-an-external-observer-see.htmlhttps://pop.com.sg/what-fraction-of-a-sphere-can-an-external-observer-see.htmlHere is a geometry problem. Let there be a ball of radius R and let's call it the Moon. Let there be an external observer: A. A is at a distance d to (the surface of) the Moon. [Edit] A is a Cy......Sun, 09 Aug 2026 19:59:33 -0400https://pop.com.sg/standard-symbol-for-to-prove-or-need-to-show-in-proofs.htmlhttps://pop.com.sg/standard-symbol-for-to-prove-or-need-to-show-in-proofs.htmlIs there a standard symbol used as shorthand for "to prove" or "need to show" in a proof? I've seen "N.T.S." but was wondering if there is anything more abstract — not bound to English....Sun, 09 Aug 2026 19:57:44 -0400https://pop.com.sg/spherical-harmonics-for-dummies.htmlhttps://pop.com.sg/spherical-harmonics-for-dummies.htmlAdding to the for dummies. The real spherical harmonics are orthonormal basis functions on the surface of a sphere. I'd like to fully understand that sentence and what it means. Still grappling ......Sun, 09 Aug 2026 19:52:43 -0400https://pop.com.sg/proof-that-a-number-can-039-t-be-both-odd-and-even.htmlhttps://pop.com.sg/proof-that-a-number-can-039-t-be-both-odd-and-even.htmlI'm trying to proof that a number can't be both odd and even at the same time. The definitions are the follows: A number is called even if there's $m\in\mathbb{Z}$ that satisfies $k = 2m$. A numbe......Sun, 09 Aug 2026 19:51:43 -0400https://pop.com.sg/where-exactly-are-complex-numbers-used-in-the-real-world.htmlhttps://pop.com.sg/where-exactly-are-complex-numbers-used-in-the-real-world.htmlI've always enjoyed solving problems in the complex numbers during my undergrad. However, I've always wondered where are they used and for what? In my domain (computer science) I've rarely seen it be ...Sun, 09 Aug 2026 19:50:52 -0400https://pop.com.sg/is-the-dirac-delta-function-really-a-function.htmlhttps://pop.com.sg/is-the-dirac-delta-function-really-a-function.htmlI am given to understand that the Dirac delta function is strictly not a function in the conventional sense and it is a "functional or a distribution". The part which I can not understand why the ......Sun, 09 Aug 2026 19:38:29 -0400https://pop.com.sg/most-efficient-solution-to-find-polynomial-congruence-for-0-mod-p.htmlhttps://pop.com.sg/most-efficient-solution-to-find-polynomial-congruence-for-0-mod-p.htmlI was given the polynomial $$f(x) = x^4 + 2x^3 + 3x^2 + x + 1$$ and told to find $$f(x) \mod 17 = 0 $$ I found the solution to be $$x = 8 + 17n$$ However, I arrived at this solution by computing al......Sun, 09 Aug 2026 19:32:24 -0400https://pop.com.sg/3d-graphing-ti-nspire-cx.htmlhttps://pop.com.sg/3d-graphing-ti-nspire-cx.htmlI'm trying to graph (x-6)^2+(y+3)^2+(z-2)^2=36. However, when I go to Graphs, Menu, View, and then 3D Graphing, I have to graph all 3D functions in terms of z. In order to graph this function, I ...Sun, 09 Aug 2026 19:31:36 -0400https://pop.com.sg/composition-of-two-reflections-is-a-rotation.htmlhttps://pop.com.sg/composition-of-two-reflections-is-a-rotation.htmlI have this problem that says: Prove that in the plane, every rotation about the origin is composition of two reflections in axis on the origin. First I have to say that this is a translation, off......Sun, 09 Aug 2026 19:31:35 -0400https://pop.com.sg/integers-with-odd-number-of-prime-factors.htmlhttps://pop.com.sg/integers-with-odd-number-of-prime-factors.htmlLet d(n) be the number of integers less then n which has an odd number of prime factors ( 2,3,5,7,8,11,12,13,17,18...). How to prove d(n)/n have a limit 1/2? Is there for all m an n such that $|n......Sun, 09 Aug 2026 19:28:00 -0400https://pop.com.sg/user-brian-cowan-ask-ubuntu.htmlhttps://pop.com.sg/user-brian-cowan-ask-ubuntu.htmlQ&A for Ubuntu users and developers...Sun, 09 Aug 2026 19:18:22 -0400https://pop.com.sg/identifying-if-constraints-are-binding-non-binding-or-redundant-visual.htmlhttps://pop.com.sg/identifying-if-constraints-are-binding-non-binding-or-redundant-visual.htmlSay you're given a linear program graphed with multiple constraints, and you're asked to identify which are binding, non-binding or redundant just by looking at the graph. A redundant constraint is ...Sun, 09 Aug 2026 19:16:06 -0400https://pop.com.sg/how-many-5-digit-numbers-are-possible-having-sum-22.htmlhttps://pop.com.sg/how-many-5-digit-numbers-are-possible-having-sum-22.htmlHow many $5$ digit numbers are possible having sum of their digits = $22$ ? How can I solve it by using Stars and Bars problem ?...Sun, 09 Aug 2026 19:04:01 -0400https://pop.com.sg/divide-matrix-using-left-division.htmlhttps://pop.com.sg/divide-matrix-using-left-division.htmlIn matlab, I defined a=[1;2;3] b=[4;5;6] both a and b are not square matrix. and execute a\b will return 2.2857 From various sources, we understand that a\b ~= inv(a)*b, but in my case, a is no......Sun, 09 Aug 2026 19:03:16 -0400https://pop.com.sg/find-an-equation-of-the-tangent-line-to-this-parametric-curve.htmlhttps://pop.com.sg/find-an-equation-of-the-tangent-line-to-this-parametric-curve.htmlI am trying to solve this problem: Find an equation of the tangent line to the curve $x=t^3 +5, y=t^2 +3$ at the point corresponding to $t=1$. Here is my work (which is wrong according the the on......Sun, 09 Aug 2026 19:00:31 -0400https://pop.com.sg/how-to-ssh-into-raspberry-pi-via-ethernet.htmlhttps://pop.com.sg/how-to-ssh-into-raspberry-pi-via-ethernet.htmlI've been trying to connect my headless raspberry pi to my Ubuntu 16.04 via the ethernet cable since a week, but have not been able to get the IP address of the connected raspberry pi. I've tried ......Sun, 09 Aug 2026 18:59:58 -0400https://pop.com.sg/expected-value-calculation-for-squared-normal-distribution.htmlhttps://pop.com.sg/expected-value-calculation-for-squared-normal-distribution.htmlI need help with the following problem. Suppose $Z=N(0,s)$ i.e. normally distributed random variable with standard deviation $\sqrt{s}$. I need to calculate $E[Z^2]$. My attempt is to do something ......Sun, 09 Aug 2026 18:43:43 -0400https://pop.com.sg/definition-of-base-formula.htmlhttps://pop.com.sg/definition-of-base-formula.htmlI'm not sure exactly how the "definition of bases" in the proof below works. I get it for base $10$, since $$642_{10}=6(10)^2+4(10)+2=642$$ but does that then extend to, for e.g, $$642_3=6(3)^2+4(......Sun, 09 Aug 2026 18:42:47 -0400https://pop.com.sg/infinite-network-of-resistors.htmlhttps://pop.com.sg/infinite-network-of-resistors.htmlIn Physic, I have to find a way to prove, with equations, that the sum of an infinite network of resistors of 1 $\Omega$ has a limit value. This question is based on the resistor's rules: In a pa......Sun, 09 Aug 2026 18:39:04 -0400https://pop.com.sg/find-two-vectors-v1-and-v2-such-that-when-added-equal-0-4-0.htmlhttps://pop.com.sg/find-two-vectors-v1-and-v2-such-that-when-added-equal-0-4-0.htmlStruggling with this question. Find two vectors $v_1$ and $v_2$ such that when added equal $(0, 4, 0)$. $v_1$ is parallel to $u(-2, 4, -2)$ and $v_2$ is perpendicular to $u$. Not sure how to start....Sun, 09 Aug 2026 18:36:22 -0400https://pop.com.sg/soft-question-how-to-pronounce-clairaut.htmlhttps://pop.com.sg/soft-question-how-to-pronounce-clairaut.htmlI'm preparing for a presentation tomorrow and I will refer to clairaut's theorem in multivariable calculus (mixed partials are equal) Can French speakers chime in as to how to correctly pronounce ...Sun, 09 Aug 2026 18:28:44 -0400https://pop.com.sg/inverse-laplace-of-an-exponential-function-exp-x-sqrt-s-h-k.htmlhttps://pop.com.sg/inverse-laplace-of-an-exponential-function-exp-x-sqrt-s-h-k.htmlI am having difficulty to figure out to use a Laplace Transform Table formula to verify a particular case. The inverse Laplace transform of $$L^{-1}\bigg[\exp{\bigg\{-x\sqrt{(s+h)/k}\bigg\}}\bigg......Sun, 09 Aug 2026 18:18:25 -0400https://pop.com.sg/in-the-basic-monty-hall-problem-why-not-the-probability-are-50-50-dupl.htmlhttps://pop.com.sg/in-the-basic-monty-hall-problem-why-not-the-probability-are-50-50-dupl.htmlWhen reading the book of (Chapman & Hall_CRC Texts in Statistical Science) Joseph K. Blitzstein-Introduction to Probability Chapter 2. I have great puzzles about the second last paragraph. To b......Sun, 09 Aug 2026 18:17:01 -0400https://pop.com.sg/covariance-times-constant-basic-rules.htmlhttps://pop.com.sg/covariance-times-constant-basic-rules.htmlI know the from the basic rule of the covariance we have: $$\text{Cov(aX,Y)=aCov(X,Y)}$$ however now i'm looking at a case that is creating me some doubt: Looking at the covariance of the same random ...Sun, 09 Aug 2026 18:16:12 -0400https://pop.com.sg/what-is-a-universal-sentence.htmlhttps://pop.com.sg/what-is-a-universal-sentence.htmlIs it just a sentence that starts with a universal quantifier? If so, isn't every sentence A equivalent to the sentence $$\forall x A$$ Where x does not appear in A?...Sun, 09 Aug 2026 18:15:14 -0400https://pop.com.sg/integrate-frac-x-2-1-x-2-2x-2-2-using-trigonometric-substitutions.htmlhttps://pop.com.sg/integrate-frac-x-2-1-x-2-2x-2-2-using-trigonometric-substitutions.htmlI've attempted to integrate the function $\frac{x^2+1}{(x^2-2x+2)^2}$ using several techniques, but none of them are solving it nicely. I know there must be a way to do this using trig substitutio......Sun, 09 Aug 2026 18:12:35 -0400https://pop.com.sg/what-is-1-to-the-power-of-a-fraction.htmlhttps://pop.com.sg/what-is-1-to-the-power-of-a-fraction.htmlIf I have any negative number (does not have to be $-1$), how would I determine that value to the power of $1/2$? I learned that $x^\frac{1}{2}$ is equal to $√x,$ so wouldn't $-1^\frac{1}{2}$ be eq......Sun, 09 Aug 2026 17:54:23 -0400https://pop.com.sg/non-linear-transformation.htmlhttps://pop.com.sg/non-linear-transformation.htmlCan someone explain to me in simple terms what a non-linear transformation is in maths? I know some single-variable calculus, but I read it has to do with multi-variable calculus, which I'm not fa......Sun, 09 Aug 2026 17:51:17 -0400https://pop.com.sg/homogeneous-elements-and-graded-algebra-any-misconception.htmlhttps://pop.com.sg/homogeneous-elements-and-graded-algebra-any-misconception.htmlElements of any factor $A_n$ of the decomposition are known as homogeneous elements of degree $n$. An ideal or other subset $\mathfrak{a} ⊂ A$ is homogeneous if every element $a ∈ \mathfrak{a}$......Sun, 09 Aug 2026 17:45:10 -0400https://pop.com.sg/what-is-the-radius-of-convergence-r-of-the-taylor-series.htmlhttps://pop.com.sg/what-is-the-radius-of-convergence-r-of-the-taylor-series.htmlThe following is a taylor series of a function centered at 8. $$\sum_{n=1}^{\infty}\frac{(-1)^n(x-8)^n}{7^n(n+8)}$$ I am trying to ind the radius of convergence of R of this series. My guess is......Sun, 09 Aug 2026 17:27:56 -0400https://pop.com.sg/export-files-from-ubuntu-wsl.htmlhttps://pop.com.sg/export-files-from-ubuntu-wsl.htmlHow can I export files from Ubuntu WSL to windows or any other directory so that I can use/edit that file in windows...Sun, 09 Aug 2026 17:19:17 -0400https://pop.com.sg/taylor-series-for-a-function-of-3-variables.htmlhttps://pop.com.sg/taylor-series-for-a-function-of-3-variables.htmlThe Taylor expansion of the function $f(x,y)$ is: \begin{equation} f(x+u,y+v) \approx f(x,y) + u \frac{\partial f (x,y)}{\partial x}+v \frac{\partial f (x,y)}{\partial y} + uv \frac{\partial^2 f (......Sun, 09 Aug 2026 17:07:33 -0400https://pop.com.sg/congruence-4x-equiv-2-pmod-6.htmlhttps://pop.com.sg/congruence-4x-equiv-2-pmod-6.htmlIs there a solution for the congruence $4x \equiv 2 \pmod 6$ ? And how can I find inverse element for $4$, when I can not use Extended Euclidean algorithm, because $6$ and $4$ are divisible by $2$. ...Sun, 09 Aug 2026 17:06:36 -0400https://pop.com.sg/how-to-calculate-an-area-enclosed-by-two-parametric-curves.htmlhttps://pop.com.sg/how-to-calculate-an-area-enclosed-by-two-parametric-curves.htmlI know the area under the curve given by parametric equations$x=f(t),y=g(t),\alpha\leq t\leq \beta$ is given by$$A=\int_{\alpha}^{\beta}g(t)f'(t)dt$$ That is in $\int_{a}^{b}ydx$ we have substitut......Sun, 09 Aug 2026 17:06:11 -0400https://pop.com.sg/how-to-say-pronounce-partial-in-the-context-of-homology.htmlhttps://pop.com.sg/how-to-say-pronounce-partial-in-the-context-of-homology.htmlI am curious what is the common way to say (in English contexts) the name of $\partial$ (the boundary homomorphism) in homology theory. I.e. when giving a talk / speaking. The boundary homomorphis......Sun, 09 Aug 2026 17:04:04 -0400https://pop.com.sg/show-that-the-set-is-compact-using-the-definition.htmlhttps://pop.com.sg/show-that-the-set-is-compact-using-the-definition.htmlThe set in question is $\{0\}\cup \{1,\frac12,\frac13,\ldots,\frac1n,\ldots\}$ (for $n\in\mathbb N$). Okay, so for a set to be compact, every open cover of it must be able to be broken down into a ...Sun, 09 Aug 2026 17:03:04 -0400https://pop.com.sg/integral-of-combination-log-and-inverse-trig-function.htmlhttps://pop.com.sg/integral-of-combination-log-and-inverse-trig-function.htmlDoes the following integral have a closed-form ?: \begin{equation} \int_{0}^{1}{\ln\left(\,x\,\right) \over 1 + x}\,\arccos\left(\,x\,\right) \,{\rm d}x \end{equation} This integral has been p......Sun, 09 Aug 2026 17:02:04 -0400https://pop.com.sg/how-do-integral-transforms-work.htmlhttps://pop.com.sg/how-do-integral-transforms-work.htmlIt's been a while since I have learnt Laplace's Transform and I am taking a look at Fourier's. But I feel I know nothing about them, just how to use in calculations. So I would like to have Any ...Sun, 09 Aug 2026 16:59:02 -0400https://pop.com.sg/av-bv-for-all-v-implies-a-b-closed.htmlhttps://pop.com.sg/av-bv-for-all-v-implies-a-b-closed.htmlIf $A$ and $B$ are two $4\times3$ matrices such that $A\mathbb{v}=B\mathbb{v}$ for all $\mathbb{v}\in\mathbb{R}^3$, then $A=B$? If it's not true, can you give me an example of it? Thank you....Sun, 09 Aug 2026 16:51:59 -0400https://pop.com.sg/euler-039-s-formula-for-complex-z.htmlhttps://pop.com.sg/euler-039-s-formula-for-complex-z.htmlGiven Euler's formula $e^{ix} = \cos x + i \sin x$ for $x \in \mathbb{R}$, how does one extend this definition for complex $z$? I.e for $z \in \mathbb{C}, $ show that $e^{iz} = \cos(z) + i\sin(z)$?......Sun, 09 Aug 2026 16:36:31 -0400https://pop.com.sg/finding-a-basis-of-an-infinite-dimensional-vector-space.htmlhttps://pop.com.sg/finding-a-basis-of-an-infinite-dimensional-vector-space.htmlThe other day, my teacher was talking infinite-dimensional vector spaces and complications that arise when trying to find a basis for those. He mentioned that it's been proven that some (or all, do......Sun, 09 Aug 2026 16:35:36 -0400https://pop.com.sg/what-does-it-mean-to-factor-over-the-real-numbers.htmlhttps://pop.com.sg/what-does-it-mean-to-factor-over-the-real-numbers.htmlI am confused on the topic of factoring over real numbers. What is the difference between normally factoring and factoring over real numbers? If anyone could explain, that would be appreciated! Tha......Sun, 09 Aug 2026 16:31:31 -0400https://pop.com.sg/integration-with-discrete-data-matlab.htmlhttps://pop.com.sg/integration-with-discrete-data-matlab.htmlI need to integrate function $\int_0^1 pur\mathrm{d}r$, where I only have discrete values for $p$,$u$ and $r$. So, if I multiply these values, would it be correct to integrate only that final value......Sun, 09 Aug 2026 16:12:28 -0400https://pop.com.sg/proof-of-the-extended-liouville-039-s-theorem.htmlhttps://pop.com.sg/proof-of-the-extended-liouville-039-s-theorem.htmlI'm trying to prove the Extended Liouville's Theorem: Let $f$ be an entire function. Assume that for some $k \in \mathbb{N}$, and sufficiently large $|z|$, we have that $|f(z)| \leq A + B |z|^k$. ...Sun, 09 Aug 2026 16:07:21 -0400https://pop.com.sg/gaussian-curvaure-of-a-surface-revolution.htmlhttps://pop.com.sg/gaussian-curvaure-of-a-surface-revolution.htmlLet $\alpha(s)=(f(s), g(s))$ be a plane curve parametrized by arc length on the $yz$-plane and assume that $f(s)>0.$ The surface revolution attained by rotating the curve parameterized by $\alpha$ ...Sun, 09 Aug 2026 15:59:32 -0400https://pop.com.sg/how-to-prove-derivative-of-logarithm-with-base-b.htmlhttps://pop.com.sg/how-to-prove-derivative-of-logarithm-with-base-b.htmlI learned how to derive a logarithm with any base. This is the formula: $$\frac{d}{dx}\log_bx=\frac{1}{x\ln b}$$ How can it be proved?...Sun, 09 Aug 2026 15:27:10 -0400https://pop.com.sg/oscillating-function-in-reference-to-limits.htmlhttps://pop.com.sg/oscillating-function-in-reference-to-limits.htmlThere is an aspect about limits that is still confusing to me. When my class was talking about cases where limits don't exist, The teacher mentioned there are 3 and exactly 3 cases where a limit is ...Sun, 09 Aug 2026 15:24:55 -0400https://pop.com.sg/which-kinds-of-rational-functions-of-one-variable-have-an-inverse-rela.htmlhttps://pop.com.sg/which-kinds-of-rational-functions-of-one-variable-have-an-inverse-rela.htmlLet's consider the rational functions whose numerator and denominator of the function term are coprime. Which kinds of rational functions of one variable have an inverse relation that contains a br......Sun, 09 Aug 2026 15:18:51 -0400https://pop.com.sg/initial-value-problem-using-laplace-transforms.htmlhttps://pop.com.sg/initial-value-problem-using-laplace-transforms.htmlSolve using Laplace Transform: $$y''(t)+2y'(t)+5y(t)=xf(t), \\ y(0)=1,y'(0)=1$$ where $x$ is a constant. Once the solution is found, evaluate the limit as $t \to\infty$. Progress: If I have ......Sun, 09 Aug 2026 15:11:23 -0400https://pop.com.sg/solve-the-differential-equation-y-039-xy-c.htmlhttps://pop.com.sg/solve-the-differential-equation-y-039-xy-c.htmlI know how to solve differential equations but I think I am getting confused because there is a constant. I began by using integrating factors but when then end up with a very complicated integral......Sun, 09 Aug 2026 15:10:53 -0400https://pop.com.sg/proof-of-cauchy-039-s-determinant-identity.htmlhttps://pop.com.sg/proof-of-cauchy-039-s-determinant-identity.htmlI am reading Matrix Analysis 2nd. Page 66 gives the proof of Caucuh's determinant identity as follow: where $(1.3.23)$ is as follow: I am very confused about these questions: How does second ide......Sun, 09 Aug 2026 14:51:05 -0400https://pop.com.sg/non-traditional-math-concepts-for-early-education.htmlhttps://pop.com.sg/non-traditional-math-concepts-for-early-education.htmlI am currently working on source material for a math-related software project with my mother, who has a PhD in Elementary Education and specializes in math education. While she has quite a strong ...Sun, 09 Aug 2026 14:49:30 -0400https://pop.com.sg/why-no-flat-metric-on-a-sphere.htmlhttps://pop.com.sg/why-no-flat-metric-on-a-sphere.htmlWhat is a simple proof that there is no flat metric on a sphere, but there is a flat metric on a torus? Ideally such a proof would clearly distinguish between the sphere and the torus. Naively one......Sun, 09 Aug 2026 14:49:12 -0400https://pop.com.sg/logwatch-email-failing-with-invalid-sender-domain.htmlhttps://pop.com.sg/logwatch-email-failing-with-invalid-sender-domain.htmlI run logwatch manually (no cron job). I use nullmailer to send emails from the server using my ISP's SMTP server. I can send test emails from command line no problem. But logwatch is using a bad......Sun, 09 Aug 2026 14:44:32 -0400https://pop.com.sg/all-tricks-to-find-eigenvalues-in-3x3-in-a-faster-way.htmlhttps://pop.com.sg/all-tricks-to-find-eigenvalues-in-3x3-in-a-faster-way.htmlI am doing a course on systems of differential equations. I have the following introductionary problem, to find repeated eigenvalues, before continuing on complex eigenvalues: Solve: $$X' = \color{......Sun, 09 Aug 2026 14:43:43 -0400https://pop.com.sg/how-can-lu-factorization-be-used-in-non-square-matrix.htmlhttps://pop.com.sg/how-can-lu-factorization-be-used-in-non-square-matrix.htmlIn my textbook, there is some information about LU factorization of square matrix $A$, but not about non-square matrix. How can LU factorization be used to factorize non-square matrix?...Sun, 09 Aug 2026 14:40:52 -0400https://pop.com.sg/basis-for-the-vector-space-p2.htmlhttps://pop.com.sg/basis-for-the-vector-space-p2.htmlI am trying to wrap my head around vector spaces of polynomials in P2. If I represent the polynomial $ ax^2 + bx + c $ with the matrix $ A = \begin{bmatrix} 1,0,0 \\ 0,1,0 \\ 0,0,1 \\ \end{bmatri......Sun, 09 Aug 2026 14:33:16 -0400https://pop.com.sg/derivation-of-the-moment-generating-function-of-the-geometric-distribu.htmlhttps://pop.com.sg/derivation-of-the-moment-generating-function-of-the-geometric-distribu.htmlLet $P(k,X)=p(1-p)^{k-1}$. When deriving the moment generating function I start off as follows: $E[e^{kt}X]=\sum\limits_{k=1}^{\infty}e^{kt}p(1-p)^{k-1}$. How I end up rearranging this is as fol......Sun, 09 Aug 2026 14:21:02 -0400https://pop.com.sg/why-are-correlation-and-standard-deviation-unitless.htmlhttps://pop.com.sg/why-are-correlation-and-standard-deviation-unitless.htmlI am sorry if this is too complicated a question to answer simply. I am interested in understanding the mathematical intution behind why standard deviation and correlation are unitless when the m......Sun, 09 Aug 2026 14:20:24 -0400https://pop.com.sg/removing-a-discontinuity.htmlhttps://pop.com.sg/removing-a-discontinuity.htmlHow would you "remove the discontinuity" of $f$ ? In other words, how would you define $f(4)$ in order to make $f$ continuous at $x=4$? $$f(x) = \dfrac{x^2-x-12}{x-4}$$...Sun, 09 Aug 2026 14:19:56 -0400https://pop.com.sg/if-p-then-q-when-p-is-false-but-we-want-to-assume-it-is-true.htmlhttps://pop.com.sg/if-p-then-q-when-p-is-false-but-we-want-to-assume-it-is-true.htmlFirst of all, I only learn "logic" in Real Analysis. So please bear with me and be welcome to define "logic" if necessary. In First-order logic, "If P then Q" can be expressed as "(not P) or Q". T......Sun, 09 Aug 2026 14:19:13 -0400https://pop.com.sg/what-is-the-significance-behind-taylor-series.htmlhttps://pop.com.sg/what-is-the-significance-behind-taylor-series.htmlWhy does taylor series have ample amount of importance in calculus? I like to know some insights behind taylor series....Sun, 09 Aug 2026 14:12:07 -0400https://pop.com.sg/what-is-the-name-of-the-shape-formed-by-a-semicircle-attached-to-a-rec.htmlhttps://pop.com.sg/what-is-the-name-of-the-shape-formed-by-a-semicircle-attached-to-a-rec.htmlI will be asking several questions like this. I have to figure out the names of the shapes of numerous parts that we make. I have looked, but the internet has failed me for several shapes. Here's ......Sun, 09 Aug 2026 13:58:41 -0400https://pop.com.sg/quadratic-function-is-given-x-2-4x-4-what-are-the-x-intercepts.htmlhttps://pop.com.sg/quadratic-function-is-given-x-2-4x-4-what-are-the-x-intercepts.htmlI am having a problem finding the x intercepts. I have found the Standard form of the Quadratic (Vertex form) -(x+2)^2 + 8 [by the way, does is matter if the 8 is in front of the minus sign or in......Sun, 09 Aug 2026 13:51:19 -0400https://pop.com.sg/why-the-divides-relation-on-the-set-of-positive-integers-antisymmetric.htmlhttps://pop.com.sg/why-the-divides-relation-on-the-set-of-positive-integers-antisymmetric.htmlI'd like to know why the divides relation on the set of positive integers antisymmetric. The book says $a|b$ and $b|a$ then $a=b$. But I think if a|b and b not divides a for example $1|2$ but not ......Sun, 09 Aug 2026 13:49:42 -0400https://pop.com.sg/is-sin-x-a-lipschitz-function.htmlhttps://pop.com.sg/is-sin-x-a-lipschitz-function.htmlI would like to know how can I prove that the function: \begin{equation} f(x)=|\sin x|, \quad x\in \mathbb{R} \end{equation} is (or isn't) Lipschitz continuous. I studied an example of the funtion......Sun, 09 Aug 2026 13:40:27 -0400https://pop.com.sg/computing-the-condition-number-of-a-matrix.htmlhttps://pop.com.sg/computing-the-condition-number-of-a-matrix.htmlGiven$$A=\begin{bmatrix}23.89&-36.48&1.432&21.65\\-36.48&54.58&-5.193&-34.45\\1.432&-5.193&-1.0717&1.937\\21.65&-34.45&1.937&20.50\end{bmatrix}.$$ I am ...Sun, 09 Aug 2026 13:39:56 -0400https://pop.com.sg/find-the-solution-of-the-recurrence-relation-fibonacci.htmlhttps://pop.com.sg/find-the-solution-of-the-recurrence-relation-fibonacci.htmlFind the solution of the recurrence relation $f_n = f_{n-1} + f_{n-2}$ with $f_0 = f_1 = 1$ I had someone show me how to solve/prove this using the linear difference equation, but I had never been ...Sun, 09 Aug 2026 13:14:34 -0400https://pop.com.sg/why-do-dummit-foote-require-an-ideal-also-be-a-subring.htmlhttps://pop.com.sg/why-do-dummit-foote-require-an-ideal-also-be-a-subring.htmlOn page 242 of Abstract Algebra, 3rd Ed., by Dummit & Foote, they write: However, in my lecture notes, there is a line that says Definition. (Ideal.) Let $R$ be a ring. A subgroup $I$ of the ...Sun, 09 Aug 2026 12:51:29 -0400https://pop.com.sg/find-derivative-of-taylor-series.htmlhttps://pop.com.sg/find-derivative-of-taylor-series.htmlHow would one differentiate a function in this form? $$f(x) = f(a)+{\frac {f'(a)}{1!}}(x-a)+{\frac {f''(a)}{2!}}(x-a)^{2}+{\frac {f'''(a)}{3!}}(x-a)^{3}$$ I'm sorry if this is something obvious, ......Sun, 09 Aug 2026 12:47:11 -0400https://pop.com.sg/is-the-percentage-symbol-a-constant.htmlhttps://pop.com.sg/is-the-percentage-symbol-a-constant.htmlIsn't the percentage symbol actually just a constant with the value $0.01$? As in $$ 15\% = 15 \times \% = 15 \times 0.01 = 0.15. $$ Isn't every unit actually just a constant? But why do we treat......Sun, 09 Aug 2026 12:45:20 -0400https://pop.com.sg/gradient-descent-in-n-dimensional-space-in-the-context-of-an-rbf-netwo.htmlhttps://pop.com.sg/gradient-descent-in-n-dimensional-space-in-the-context-of-an-rbf-netwo.htmlI am trying to implement an algorithm to perform gradient descent in a $n$-dimensional space in the context of an RBF network. My network has 5 inputs and 1 output. It has the following Gaussian ra......Sun, 09 Aug 2026 12:44:06 -0400https://pop.com.sg/proof-for-why-the-conceptual-and-computation-formulas-for-sum-of-squar.htmlhttps://pop.com.sg/proof-for-why-the-conceptual-and-computation-formulas-for-sum-of-squar.htmlThe Sum of Squared Deviates has a conceptual formula of: $SS = ∑(Xi — Mx)^2$ (where $Mx$ is the sample mean) The corresponding computational formula is: $$SS = ∑Xi^2 —\frac{(∑Xi)^2}{N}$$ How are ......Sun, 09 Aug 2026 12:35:41 -0400https://pop.com.sg/understanding-what-a-gradient-vector-is.htmlhttps://pop.com.sg/understanding-what-a-gradient-vector-is.htmlDefinition: let $A$ be an open set in a Euclidean space $X$. Let $f : A \to \mathbb R$ be a real valued differentiable function. Then,for each $a \in A$, there is a unique vector $\nabla f(a) x \in......Sun, 09 Aug 2026 12:22:16 -0400https://pop.com.sg/why-is-delta-used-to-describe-difference-between-two-entities.htmlhttps://pop.com.sg/why-is-delta-used-to-describe-difference-between-two-entities.htmlI hear a lot about extracting "delta" between two properties in my current job. I come from a User Interface programming background and I do not really have much math background. I looked up delta ......Sun, 09 Aug 2026 12:14:05 -0400https://pop.com.sg/what-is-the-fisher-information-for-a-uniform-distribution.htmlhttps://pop.com.sg/what-is-the-fisher-information-for-a-uniform-distribution.htmlIf X is U[$0$,$\theta$], then the likelihood is given by $f(X,\theta) = \dfrac{1}{\theta}\mathbb{1}\{0\leq x \leq \theta\}$. The definition of Fisher information is $I(\theta) = \mathbb{E} \left[ \......Sun, 09 Aug 2026 12:08:50 -0400https://pop.com.sg/proof-for-theorem-of-upper-and-lower-bounds-on-zeroes-of-polynomials.htmlhttps://pop.com.sg/proof-for-theorem-of-upper-and-lower-bounds-on-zeroes-of-polynomials.htmlI'm currently a high school Pre-Calculus student and my textbook presents the following theorem without proof: Let $f(x)$ be a polynomial with real coefficients and a positive leading coefficient.......Sun, 09 Aug 2026 11:56:23 -0400