https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usSun, 09 Aug 2026 13:44:13 -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 -0400https://pop.com.sg/can-an-arc-be-greater-than-360-degrees-or-less-than-0-degrees.htmlhttps://pop.com.sg/can-an-arc-be-greater-than-360-degrees-or-less-than-0-degrees.htmlCan an arc be greater than 360 degrees or less than 0 degrees? Is the (angular) arc length a signed scalar? I'm creating a Graphics and basic geometry software library. I believe an arc can be defi......Sun, 09 Aug 2026 11:54:52 -0400https://pop.com.sg/matlab-plot-vertical-line.htmlhttps://pop.com.sg/matlab-plot-vertical-line.htmlI have a system of equations, about 3 equations, that I need to plot that are dependent on 2 variables, R and z. But one of the equations is R={constant value}. Here's the code: R = 0:.1:500; z = ......Sun, 09 Aug 2026 11:52:40 -0400https://pop.com.sg/why-would-we-use-the-radius-of-a-circle-instead-of-the-diameter-when-c.htmlhttps://pop.com.sg/why-would-we-use-the-radius-of-a-circle-instead-of-the-diameter-when-c.htmlForgive me if this question is a little too strange or maybe even off. Mathematics has never been my strong point, but I definitely think it's the coolest... Anyway, I was looking into tau, pi's ......Sun, 09 Aug 2026 11:52:17 -0400https://pop.com.sg/any-way-to-easily-rip-blu-ray-disks.htmlhttps://pop.com.sg/any-way-to-easily-rip-blu-ray-disks.htmlI've seen the documentation on Blu-Ray Restricted formats here but I'm looking to see if anyone has come up with an easy solution yet? Also it would be awesome if someone knows of any type of prog......Sun, 09 Aug 2026 11:46:11 -0400https://pop.com.sg/strategy-for-7-up-7-down-game.htmlhttps://pop.com.sg/strategy-for-7-up-7-down-game.htmlThis is a dice game where the host throws the dice and the players bet on 3 numbers - numbers below 7, 7 and numbers above 7. On numbers below and above 7 the players get double their stake and on ......Sun, 09 Aug 2026 11:21:16 -0400https://pop.com.sg/ubuntu-sdk-error-039-could-not-find-or-load-the-qt-platform-plugin-xcb.htmlhttps://pop.com.sg/ubuntu-sdk-error-039-could-not-find-or-load-the-qt-platform-plugin-xcb.htmlI am trying to write my first ubuntu app using the Ubuntu SDK but keep running into an error. This error turns up both with QML and HTML5 apps. I am trying this just on the template for both project ...Sun, 09 Aug 2026 11:16:35 -0400https://pop.com.sg/how-to-state-a-recurrence-equation-for-running-time-of-a-divide-and-co.htmlhttps://pop.com.sg/how-to-state-a-recurrence-equation-for-running-time-of-a-divide-and-co.htmlI'm working on my homework and I saw this problem: A divide-and-conquer algorithm $ X $ divides any given problem instance into exactly two smaller problem instances and solve them recursively. ...Sun, 09 Aug 2026 11:14:56 -0400https://pop.com.sg/find-an-equation-of-the-plane-through-a-point-and-parallel-to-another-.htmlhttps://pop.com.sg/find-an-equation-of-the-plane-through-a-point-and-parallel-to-another-.htmlFind an equation of the plane through the point (1, -1, -1) and parallel to the plane $5{x} - {y} - {z} = 6$ I was shown an example of this problem but it was in a problem where it had to be ...Sun, 09 Aug 2026 11:14:16 -0400https://pop.com.sg/cheese-has-holes-more-cheese-more-holes-more-holes-less-cheese-more-ch.htmlhttps://pop.com.sg/cheese-has-holes-more-cheese-more-holes-more-holes-less-cheese-more-ch.htmlI found this statement on the internet and I would like to know why this is a fallacy. Cheese has holes. More cheese = more holes More holes = less cheese More cheese = less cheese Why is this fa......Sun, 09 Aug 2026 11:01:33 -0400https://pop.com.sg/calculating-the-nth-root-in-simple-calculator.htmlhttps://pop.com.sg/calculating-the-nth-root-in-simple-calculator.htmlRecently during my physics class that how to take a cube root in a simple calculator. Follow the steps given below Step1. Press the square root button 13 times Step2. subtract 1 from it Step3. ...Sun, 09 Aug 2026 10:54:37 -0400https://pop.com.sg/what-does-the-steady-state-represent-to-a-markov-chain.htmlhttps://pop.com.sg/what-does-the-steady-state-represent-to-a-markov-chain.htmlI'm a little confused as to the interpretation of the steady state in the context of a Markov chain. I know Markov chains are memoryless, in that each state only depends on its immediate predecesso......Sun, 09 Aug 2026 10:50:42 -0400https://pop.com.sg/find-the-remainder-using-fermat-039-s-little-theorem-when-5-119-is-div.htmlhttps://pop.com.sg/find-the-remainder-using-fermat-039-s-little-theorem-when-5-119-is-div.htmlFind the remainder using Fermat's little theorem when $5^{119}$ is divided by $59$. Fermat's little theorem states that if $p$ is prime and $\operatorname{gcd}(a,p)=1$,then $a^{p-1} -1$ is a multi......Sun, 09 Aug 2026 10:42:19 -0400https://pop.com.sg/max-min-inequality.htmlhttps://pop.com.sg/max-min-inequality.htmlIt is known that $\underset{x}{\max} \underset{y}{\min} f(x,y) \leq \underset{y}{\min} \underset{x}{\max} f(x,y)$ . When does equality hold in this expression?...Sun, 09 Aug 2026 10:41:10 -0400https://pop.com.sg/can-systems-of-3-linear-equations-with-3-unknowns-have-more-than-one-s.htmlhttps://pop.com.sg/can-systems-of-3-linear-equations-with-3-unknowns-have-more-than-one-s.htmlIn each part,determine whether the given vector is a solution of the linear system \begin{align} 2x-4y-z&=1\\ x-3y+z&=1\\ 3x-5y-3z&=1 \end{align} (a) $(3,1,1)$ (b) $(3,-1,1)$ (c) ......Sun, 09 Aug 2026 10:39:30 -0400https://pop.com.sg/integration-by-substitution-called-weierstrass-substitution.htmlhttps://pop.com.sg/integration-by-substitution-called-weierstrass-substitution.htmlI have seen this recent question What's wrong in my calculation of $\int_0^{3 \pi/4} \frac{\cos x}{1 + \cos x}dx$? and I have read that when I operate for integration by substitution this type ......Sun, 09 Aug 2026 10:33:52 -0400https://pop.com.sg/invertible-and-non-invertible-linear-transformation.htmlhttps://pop.com.sg/invertible-and-non-invertible-linear-transformation.htmlIf a linear transformation is represented by an non-invertible matrix $P$, then it might happen that two different vectors (points in $\mathbb{R^n}$) will be mapped to the same point. However, if the ...Sun, 09 Aug 2026 10:27:11 -0400https://pop.com.sg/noetherian-ring-not-descending-chain-condition.htmlhttps://pop.com.sg/noetherian-ring-not-descending-chain-condition.htmlAnalogous to the ascending chain condition we can define a descending chain condition: if $I_1 \supseteq I_2 \supseteq I_3 \supseteq \cdots$ is a descending chain of ideals then there exists a pos......Sun, 09 Aug 2026 10:24:52 -0400https://pop.com.sg/angles-in-a-nonagon.htmlhttps://pop.com.sg/angles-in-a-nonagon.htmlThe following figure was draw by taking midpoints on three sides of a regular nonagon. The angles are 20 degrees, 50 degrees, and 50 degrees. It's not hard to find the angles by using trigonometry......Sun, 09 Aug 2026 10:21:29 -0400https://pop.com.sg/power-regression-y-ax-b-c.htmlhttps://pop.com.sg/power-regression-y-ax-b-c.htmlI have to do reproduce a power regression but I don't have any experience in procedures like that. I read a little bit about power fit/power regression and that a formula like $y = ax^b$ is used for ...Sun, 09 Aug 2026 10:07:48 -0400https://pop.com.sg/differentiate-frac-ln-x-cos-x.htmlhttps://pop.com.sg/differentiate-frac-ln-x-cos-x.htmlPlease help me with this question. $$y= \frac{\ln(x)}{\cos(x)}$$ Just starting with calculus. Thank you....Sun, 09 Aug 2026 10:01:31 -0400https://pop.com.sg/what-does-calculating-the-inverse-of-a-matrix-mean.htmlhttps://pop.com.sg/what-does-calculating-the-inverse-of-a-matrix-mean.htmlAssume I have 3 equations, $x+2y+z=2, 3x+8y+z=12, 4y+z=2$ which could be represented in matrix form ($Ax = b$) like this: $\begin{pmatrix} 1 & 2 & 1\\ 3 & 8 & 1\\ 0 & 4 & ......Sun, 09 Aug 2026 09:58:09 -0400https://pop.com.sg/difference-between-probability-and-probability-density.htmlhttps://pop.com.sg/difference-between-probability-and-probability-density.htmlThis question is from DeGroot's "Probability and Statistics" : Unbounded p.d.f.’s. Since a value of a p.d.f.(probability density function) is a probability density, rather than a probability, ......Sun, 09 Aug 2026 09:35:36 -0400https://pop.com.sg/limit-of-x-ln-x-as-x-approaches-0.htmlhttps://pop.com.sg/limit-of-x-ln-x-as-x-approaches-0.htmlHow could I solve the limit in this form $ \frac{x}{x+1}$ using l'Hospital's rule? I know how to solve it in this way: $\frac{\ln x}{\frac{1}{x}}$ Thanks...Sun, 09 Aug 2026 09:31:36 -0400https://pop.com.sg/how-to-swap-rows-in-square-a-matrix-algebraically.htmlhttps://pop.com.sg/how-to-swap-rows-in-square-a-matrix-algebraically.htmlIs there some way to achieve swapping of rows of a 3x3 square matrix(for example exchanging rows 0 and 2) by using matrix algebra? Or is it something that cannot be done with algebra? What about row ...Sun, 09 Aug 2026 09:30:58 -0400https://pop.com.sg/find-an-equation-of-the-tangent-line-to-the-curve.htmlhttps://pop.com.sg/find-an-equation-of-the-tangent-line-to-the-curve.htmlFind an equation of the tangent line to the curve $2·(x^2+y^2)^2=25·(x^2−y^2)$ at the point $(3,1)$. Any hints?...Sun, 09 Aug 2026 09:29:54 -0400https://pop.com.sg/in-a-calculator-which-button-should-be-bigger-sinh-or-rm-asin.htmlhttps://pop.com.sg/in-a-calculator-which-button-should-be-bigger-sinh-or-rm-asin.htmlI am making a calculator app and would like to know: Which of the following functions would be used more often: $\sin()$ $\sinh()$ $\text{arsin}()$ ($\sin^{-1}$) $\text{arsinh}()$ ($\sinh^{-1}$)...Sun, 09 Aug 2026 09:29:42 -0400https://pop.com.sg/why-is-a-circle-in-a-plane-surrounded-by-6-other-circles.htmlhttps://pop.com.sg/why-is-a-circle-in-a-plane-surrounded-by-6-other-circles.htmlWhen you draw a circle in a plane you can perfectly surround it with 6 other circles of the same radius. This works for any radius. What's the significance of 6? Why not some other numbers? I'm lo......Sun, 09 Aug 2026 09:28:05 -0400https://pop.com.sg/ordinal-tetration-the-issue-of-epsilon-0-omega.htmlhttps://pop.com.sg/ordinal-tetration-the-issue-of-epsilon-0-omega.htmlSo in the past few months when trying to learn about the properties of the fixed points in ordinals as I move from $0$ to $\epsilon_{\epsilon_0}$ I noticed when moving from $\epsilon_n$ to the next......Sun, 09 Aug 2026 09:11:55 -0400https://pop.com.sg/what-is-the-exact-and-precise-definition-of-an-angle.htmlhttps://pop.com.sg/what-is-the-exact-and-precise-definition-of-an-angle.htmlOn wikipedea I found a definition of an Angle as such: "In order to measure an angle θ, a circular arc centered at the vertex of the angle is drawn, e.g. with a pair of compasses. The length o......Sun, 09 Aug 2026 09:00:44 -0400https://pop.com.sg/how-to-find-equation-of-a-graph-using-vertical-asymptote.htmlhttps://pop.com.sg/how-to-find-equation-of-a-graph-using-vertical-asymptote.htmlGraph Based on the graph, I need to find the equation. What I know: vertical asymptote x = 4, and opening at x = -4. I am struggling to find the rational function of the graph. y = 1/-x+4 is what ......Sun, 09 Aug 2026 08:58:51 -0400https://pop.com.sg/functions-linear-function-f-x-3x-5.htmlhttps://pop.com.sg/functions-linear-function-f-x-3x-5.htmlHey if i wanted to get a linear function how would i do it? Im being shown an example and the results are supposedly: x 3x+5 -2 -1 -1 2 0 5 1 8 2 11 But i cant figure out how to get it.......Sun, 09 Aug 2026 08:56:15 -0400https://pop.com.sg/calculate-the-image-and-a-basis-of-the-image-matrix.htmlhttps://pop.com.sg/calculate-the-image-and-a-basis-of-the-image-matrix.htmlWhat's the image of the matrix? What's the basis of the image? $M=\begin{pmatrix} -1 & 1 & 1\\ -2 & -3 & 6\\ 0 & -1 & 1 \end{pmatrix}$ First transposed the matrix:......Sun, 09 Aug 2026 08:55:14 -0400https://pop.com.sg/why-can-039-t-pi-be-expressed-as-a-fraction-duplicate.htmlhttps://pop.com.sg/why-can-039-t-pi-be-expressed-as-a-fraction-duplicate.htmlWhy can't $\pi$ be expressed as a fraction? If pi is the ratio of a circle's circumference to its diameter, why can't we simply take a circle, measure its circumference and diameter, and derive the ...Sun, 09 Aug 2026 08:43:52 -0400https://pop.com.sg/what-is-a-powerset-with-base-larger-than-2.htmlhttps://pop.com.sg/what-is-a-powerset-with-base-larger-than-2.htmlA powerset $P(S) $of some set $S$ can be treated as all different possible ways of partition $S$ into 2 ordered pair of disjoint subsets. And I'm curious what is the equivalence of partition $S$ into ...Sun, 09 Aug 2026 08:35:49 -0400https://pop.com.sg/inverse-of-a-vector.htmlhttps://pop.com.sg/inverse-of-a-vector.htmlHow would we calculate the inverse of a $3$-vector $v$ like we would do with the matrix? Basically the inverse of $1\times3$ (?) matrix. Would we get the negated value of the vector $-v$ or $1/v$? ......Sun, 09 Aug 2026 08:29:52 -0400https://pop.com.sg/why-is-delta-x-tending-to-be-zero-in-dy-dx-and-not-delta-y-tending-to-.htmlhttps://pop.com.sg/why-is-delta-x-tending-to-be-zero-in-dy-dx-and-not-delta-y-tending-to-.htmlWhy is $\Delta x $ and not $\Delta y $ written to be tending towards 0 in case when we give limits to form dy/dx. , $$\lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x} = \frac{dy}{dx}$$ Can we solve......Sun, 09 Aug 2026 08:24:55 -0400https://pop.com.sg/how-to-find-the-directional-derivative-of-the-following-function.htmlhttps://pop.com.sg/how-to-find-the-directional-derivative-of-the-following-function.htmlI am stuck with the following problem: Consider the function $f(x,y)=\frac{x^2}{y^2}$; where $(x,y)\in [1/2,3/2]\times[1/2,3/2]$.Then what is the derivative of $f$ at $(1,1)$along the direction ......Sun, 09 Aug 2026 08:17:51 -0400https://pop.com.sg/what-do-curly-brackets-mean-in-this-number-theory-equation.htmlhttps://pop.com.sg/what-do-curly-brackets-mean-in-this-number-theory-equation.html$$(\lfloor y\rfloor-1)(\lfloor x\rfloor-1)=\{x\}+\{y\}+1$$ From looking at the LaTeX, I can see the left-hand side symbols mean the floor of the variable, but the right-hand side doesn't give much......Sun, 09 Aug 2026 08:13:37 -0400https://pop.com.sg/what-is-the-equal-sign-with-3-lines-mean-in-wilson-039-s-theorem.htmlhttps://pop.com.sg/what-is-the-equal-sign-with-3-lines-mean-in-wilson-039-s-theorem.htmlI'm reading up on Wilson's Theorem, and see a symbol I don't know... what does an equal sign with three lines mean? I'm looking at the example table and I still can't infer what they are trying to......Sun, 09 Aug 2026 08:11:18 -0400https://pop.com.sg/what-is-the-purpose-of-port-1234-tcp-running-by-039-systemd-039-with-p.htmlhttps://pop.com.sg/what-is-the-purpose-of-port-1234-tcp-running-by-039-systemd-039-with-p.htmlWhen I used $ nmap 192.168.1.X it shows that 1234/tcp OPEN hotline is open. I am running behind firewall ufw, and there is no rule for port 1234. $ lsof -i showed that it is running by systemd with ...Sun, 09 Aug 2026 08:09:35 -0400https://pop.com.sg/a-question-about-incompressible-vector-field.htmlhttps://pop.com.sg/a-question-about-incompressible-vector-field.htmlLet $ X $ be a unit vector field on $\mathbb{R}^2$,with canonical metric $g$ and connection $\nabla$ .Show that if $$divX=0$$ then $$\nabla X=0$$ I tried: Let $X=(f,g)$,$Y=(-g,f,)$,then $X,Y$ ......Sun, 09 Aug 2026 08:07:19 -0400https://pop.com.sg/proof-about-complete-residue-system.htmlhttps://pop.com.sg/proof-about-complete-residue-system.htmlTheorem 3-5. If $a_1,a_2,\ldots,a_m$ is a complete residue system $\pmod m$ and $\gcd(k,m)=1$, then $ka_1,ka_2,\ldots,ka_m$ also is a complete residue system $\pmod m$ Proof: We show directly that ...Sun, 09 Aug 2026 08:04:54 -0400https://pop.com.sg/what-is-mathematical-structure.htmlhttps://pop.com.sg/what-is-mathematical-structure.htmlWhen we have an isomorphism, between 2 groups or vector spaces let us say, then it is said to be structure preserving. An isomorphism exists when there is at least one mutually invertible morphism ...Sun, 09 Aug 2026 07:51:02 -0400https://pop.com.sg/wine-visual-c-redistributable-for-visual-studio-2015.htmlhttps://pop.com.sg/wine-visual-c-redistributable-for-visual-studio-2015.htmlWhen I want to run Chessbase 14 I get this error output: gratis@Aurora:~$ env WINEPREFIX="/home/gratis/.wine" wine C:\\Program\ Files\\ChessBase\\CBase14\\CBase14.exe err:module:import_dll Librar......Sun, 09 Aug 2026 07:47:00 -0400https://pop.com.sg/confusion-with-showing-that-hn-n-cong-h-h-cap-n.htmlhttps://pop.com.sg/confusion-with-showing-that-hn-n-cong-h-h-cap-n.htmlSuppose $G$ is a group, $H \le G, $ and $N \triangleleft G$. Show that $$HN/N \cong H/(H \cap N)$$ Well, I know there was a similar question on math.stackexchange and I know there is a solution by ......Sun, 09 Aug 2026 07:42:17 -0400https://pop.com.sg/how-to-convert-a-right-angled-triangle-into-a-equilateral-triangle.htmlhttps://pop.com.sg/how-to-convert-a-right-angled-triangle-into-a-equilateral-triangle.htmlI want to use the Apophysis program to make a right angled sierpinski triangle into an equilateral triangle. But how can i do so? i have tried the second picture one but that is not correct....Sun, 09 Aug 2026 07:40:49 -0400https://pop.com.sg/figuring-out-price-demand-equation.htmlhttps://pop.com.sg/figuring-out-price-demand-equation.htmlWhen selling burgers at $2.50$ dollar per unit, average sales are $875$ units per night. When the price was raised to $3.20$ dollar per unit, average sales dropped to $700$ units per night. Find......Sun, 09 Aug 2026 07:35:19 -0400https://pop.com.sg/is-n-sin-n-dense-on-the-real-line.htmlhttps://pop.com.sg/is-n-sin-n-dense-on-the-real-line.htmlIs $\{n \sin n | n \in \mathbb{N}\}$ dense on the real line? If so, is $\{n^p \sin n | n \in \mathbb{N}\}$ dense for all $p>0$? This seems much harder than showing that $\sin n$ is dense on [-......Sun, 09 Aug 2026 07:33:00 -0400https://pop.com.sg/finding-poles-indicating-their-order-and-then-computing-their-residues.htmlhttps://pop.com.sg/finding-poles-indicating-their-order-and-then-computing-their-residues.htmlI really don't understand the concept behind finding poles in Complex Analysis and I can't find anything on the internet or in books that helps me grasp the concept... The following are past exam ...Sun, 09 Aug 2026 07:28:16 -0400https://pop.com.sg/prove-that-if-in-a-tetrahedron-if-two-pairs-of-opposite-edges-are-perp.htmlhttps://pop.com.sg/prove-that-if-in-a-tetrahedron-if-two-pairs-of-opposite-edges-are-perp.htmlProve that if in a tetrahedron if two pairs of opposite edges are perpendicular then the third pair is also perpendicular. Method: $\vec a+ \vec b + \vec c + \vec d +\vec e + \vec f = 0 \tag0$ ...Sun, 09 Aug 2026 07:22:37 -0400https://pop.com.sg/if-i-roll-two-fair-dice-the-probability-that-i-would-get-at-least-one-.htmlhttps://pop.com.sg/if-i-roll-two-fair-dice-the-probability-that-i-would-get-at-least-one-.html$11$ out of $36$? I got this by writing down the number of possible outcomes ($36$) and then counting how many of the pairs had a $6$ in them: $(1,6)$, $(2,6)$, $(3,6)$, $(4,6)$, $(5,6)$, $(6,6)$, ......Sun, 09 Aug 2026 07:21:17 -0400https://pop.com.sg/matrix-multiplication-notation.htmlhttps://pop.com.sg/matrix-multiplication-notation.htmlThis is from my textbook: If $A=(a_{ij})\in M_{mn}(\Bbb F), B=(b_{ij})\in M_{np}(\Bbb F)$ then $C=A\times B=(c_{ij})\in M_{mp}(\Bbb F)$. $c_{ij}=\sum_{k=1}^{n} a_{ik}b_{kj}$ where $i=1,...m,......Sun, 09 Aug 2026 07:20:37 -0400