https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usMon, 31 Aug 2026 07:37:48 -0400https://pop.com.sg/is-there-any-symbol-for-undefined-duplicate.htmlhttps://pop.com.sg/is-there-any-symbol-for-undefined-duplicate.htmlFor example we have $\frac{0}{0}$ which is undefined or we have a multiplication of $2\times2$ and $3\times3$ matrices which is also undefined. Is there any symbol for representing it?...Mon, 31 Aug 2026 11:21:51 -0400https://pop.com.sg/write-the-following-in-the-form-of-ax-b.htmlhttps://pop.com.sg/write-the-following-in-the-form-of-ax-b.htmlWrite the following system of equations in the form $AX = B$, and calculate the solution using the equation $X = A^{-1}B$. $$2x - 3y = - 1$$ $$-5x +5y = 20$$ I'm not the strongest at linear alge......Mon, 31 Aug 2026 11:20:47 -0400https://pop.com.sg/what-does-long-term-relative-frequency-mean.htmlhttps://pop.com.sg/what-does-long-term-relative-frequency-mean.htmlI'm reading an introductory statistics textbook and it mentioned this: "The probability of any outcome is the long-term relative frequency of that outcome. Probabilities are between zero and one, ...Mon, 31 Aug 2026 11:11:07 -0400https://pop.com.sg/solving-the-general-cubic-by-completing-the-cube.htmlhttps://pop.com.sg/solving-the-general-cubic-by-completing-the-cube.htmlCompleting the Square: In a quadratic of the form $x^2+ax+b=0$, we can complete the square by moving $b$ to the right hand side and finding a constant $m$ such that the left hand side becomes a pe......Mon, 31 Aug 2026 11:09:52 -0400https://pop.com.sg/what-is-the-history-behind-the-development-of-the-term-coefficient-clo.htmlhttps://pop.com.sg/what-is-the-history-behind-the-development-of-the-term-coefficient-clo.htmlWhy are coefficients called "coefficients"? For example I learned that squaring a number is called "squaring" because it actually refers to "making a square". That's how it was developed. |<-......Mon, 31 Aug 2026 10:59:07 -0400https://pop.com.sg/simplifying-nested-square-roots-sqrt-6-4-sqrt-2-sqrt-2.htmlhttps://pop.com.sg/simplifying-nested-square-roots-sqrt-6-4-sqrt-2-sqrt-2.htmlI guess I learned it many years ago at school, but I must have forgotten it. From a geometry puzzle I got to the solution $\sqrt{6-4\sqrt{2}} + \sqrt{2}$ My calculator tells me that (within its ...Mon, 31 Aug 2026 10:54:06 -0400https://pop.com.sg/using-integration-by-parts-to-evaluate-an-integrals.htmlhttps://pop.com.sg/using-integration-by-parts-to-evaluate-an-integrals.htmlCan't understand how to solve this math: use integration by parts to evaluate this integrals: $$\int x\sin(2x + 1) \,dx$$ can any one solve this so i can understand how to do this! Thanks :)...Mon, 31 Aug 2026 10:50:29 -0400https://pop.com.sg/the-square-of-an-integer-is-congruent-to-0-or-1-mod-4.htmlhttps://pop.com.sg/the-square-of-an-integer-is-congruent-to-0-or-1-mod-4.htmlThis is a question from the free Harvard online abstract algebra lectures. I'm posting my solutions here to get some feedback on them. For a fuller explanation, see this post. This problem is from ...Mon, 31 Aug 2026 10:49:13 -0400https://pop.com.sg/expectation-of-the-function-of-multiple-random-variables.htmlhttps://pop.com.sg/expectation-of-the-function-of-multiple-random-variables.htmlI am working through some basic probability stuff and have a question regarding functions of multiple variables. If I have two random variables $X,Y$ which have some joint probability distribution ......Mon, 31 Aug 2026 10:45:04 -0400https://pop.com.sg/whats-the-difference-between-logical-consequence-entailment-and-simple.htmlhttps://pop.com.sg/whats-the-difference-between-logical-consequence-entailment-and-simple.htmlAccording to Wikipedia: A logical consequence is the relationship between statements that holds true when one logically "follows from" one or more others. So, A ⊨ B B is a logical consequen......Mon, 31 Aug 2026 10:42:54 -0400https://pop.com.sg/definition-of-a-complete-metric-space.htmlhttps://pop.com.sg/definition-of-a-complete-metric-space.htmlI am having a little trouble understanding the definition of a complete set, which is the following : a metric space is said to be complete if every fundamental sequence converges in the space X. ......Mon, 31 Aug 2026 10:40:16 -0400https://pop.com.sg/what-is-the-difference-between-a-hermitian-inner-product-and-an-inner-.htmlhttps://pop.com.sg/what-is-the-difference-between-a-hermitian-inner-product-and-an-inner-.htmlAre there any difference? Or is the Hermitian inner product just a special case of an inner product on a complex space?...Mon, 31 Aug 2026 10:30:52 -0400https://pop.com.sg/can-a-directional-derivative-be-negative.htmlhttps://pop.com.sg/can-a-directional-derivative-be-negative.htmlI just calculated the directional derivative of a function and ended up with $- \frac{1}{3}$ which seems counterintuitive. Can this be possible?...Mon, 31 Aug 2026 10:28:02 -0400https://pop.com.sg/what-is-the-proper-notation-for-the-indefinite-integral-of-an-area.htmlhttps://pop.com.sg/what-is-the-proper-notation-for-the-indefinite-integral-of-an-area.htmlIm not exactly experienced with Calculus and math notation but I had question on Leibniz's notation: If we have an equation $dy=2dx$, and we wanted to find $y$, we would take the integral of both s......Mon, 31 Aug 2026 10:27:09 -0400https://pop.com.sg/uniqueness-of-a-complement-of-a-subgroup.htmlhttps://pop.com.sg/uniqueness-of-a-complement-of-a-subgroup.htmllet $A$ and $B$ be two subgroups of $G$. we say that $B$ is a complement of $A$ if : $G=AB$ $A\cap B=\{1\}$ Given a subgroup $A$ of $G$ i don't see how the complement $B$ of $A$ in $G$ is not uni......Mon, 31 Aug 2026 10:25:51 -0400https://pop.com.sg/solution-of-the-equation-below.htmlhttps://pop.com.sg/solution-of-the-equation-below.htmlThe solution of the equation $\sin 7x + \cos 2x = -2$ is/are? My Approach: For the above equation to hold true, Both $\sin 7x$ and $\cos 2x$ have to be -1. $$\sin 7x = -1$$ $$x = \frac{n\pi}{7} +......Mon, 31 Aug 2026 10:23:25 -0400https://pop.com.sg/when-does-a-linear-equation-have-infinitely-many-solutions.htmlhttps://pop.com.sg/when-does-a-linear-equation-have-infinitely-many-solutions.htmlFor which values of $k$ the equation $(k+8)x-3k = 3k(1-x)-6$ has infinitely many solutions? Since this is not a set of linear equations you can't use the augmented matrix and Gaussian elimination,......Mon, 31 Aug 2026 10:20:41 -0400https://pop.com.sg/how-can-i-find-values-that-make-a-matrix-linearly-dependent.htmlhttps://pop.com.sg/how-can-i-find-values-that-make-a-matrix-linearly-dependent.htmlLet $$A=\left(\begin{smallmatrix}2&-1&0&1\\0&0&a&3\\0&0&0&b\end{smallmatrix}\right)$$ For what choices (if any) of real numbers a and b are the rows of A linearly ...Mon, 31 Aug 2026 10:20:07 -0400https://pop.com.sg/arc-length-with-vector-valued-functions.htmlhttps://pop.com.sg/arc-length-with-vector-valued-functions.html"Consider the path of a particle in a conservative force field represented by the vector-valued function $r(t) = \langle 4(\sin t - t \cos t), 4(\sin t + t \sin t), (\frac{3}{2})t^2 \rangle$." "A)......Mon, 31 Aug 2026 10:05:44 -0400https://pop.com.sg/039-does-not-necessarily-equal-039-symbol.htmlhttps://pop.com.sg/039-does-not-necessarily-equal-039-symbol.htmlWhat symbol would I use if I wanted to express that, in the context of some binary relation $P$ implied from context, that $\exists (a,b)\in P: a\ne b$, but not to the extent that $\forall (a,b) \i......Mon, 31 Aug 2026 09:55:33 -0400https://pop.com.sg/is-the-integral-of-the-sum-really-the-sum-of-the-integrals.htmlhttps://pop.com.sg/is-the-integral-of-the-sum-really-the-sum-of-the-integrals.htmlI was asked to find the mclaurin series of $\int_0^x\frac{\arctan (t)}{t}dt$ using the known mclaurin for arctan: $\arctan(t)=\sum_{n=1}^{\infty} \frac{(-1)^{n+1}t^{2n-1}}{2n-1}$ Ok, so what I di......Mon, 31 Aug 2026 09:51:06 -0400https://pop.com.sg/how-to-solve-30-37-mod-77-without-calculator.htmlhttps://pop.com.sg/how-to-solve-30-37-mod-77-without-calculator.htmlI tried doing something like this: $$(30^2)^{15}(30^7)\mod 77$$ but it is not effective, maybe someone knows some tips and tricks to solve this ?...Mon, 31 Aug 2026 09:48:36 -0400https://pop.com.sg/solve-for-f-x-if-f-f-x-6x-f-x.htmlhttps://pop.com.sg/solve-for-f-x-if-f-f-x-6x-f-x.htmlIf $f: [0,\infty) \rightarrow [0,\infty)$ and $f(f(x))=6x-f(x)$ $f(x)>0$ $ \forall x \in (0,\infty) $ Find f(x)...Mon, 31 Aug 2026 09:32:04 -0400https://pop.com.sg/levi-civita-connection-and-riemannian-metrics.htmlhttps://pop.com.sg/levi-civita-connection-and-riemannian-metrics.htmlI'm trying to solve the following problem in preparation for an exam : (i) Let $g_0$ be a Riemannian metric and define $g=cg_0$ where $c$ is a positive constant. Prove that $g_0$ and $g$ have the ......Mon, 31 Aug 2026 09:09:06 -0400https://pop.com.sg/the-existence-of-local-maximum-and-minimum-of-a-real-valued-function.htmlhttps://pop.com.sg/the-existence-of-local-maximum-and-minimum-of-a-real-valued-function.htmlI think it's very clever to show that if $f$ has a local maximum (or minimum) at a point $x\in(a,b)$ and $f'(x)$ exists, then $f'(x)=0$, for $$a<x-\epsilon <x<x+\epsilon <b,$$ and so the ...Mon, 31 Aug 2026 08:59:36 -0400https://pop.com.sg/what-039-s-the-difference-between-the-parameter-of-interest-and-the-po.htmlhttps://pop.com.sg/what-039-s-the-difference-between-the-parameter-of-interest-and-the-po.htmlI have two problems. One is asking for the parameter of interest, and the other the population of interest: (1). A sales professional for Goetze Candy, the maker of Cow Tails, is told that the co......Mon, 31 Aug 2026 08:59:26 -0400https://pop.com.sg/what-is-the-meaning-of-omega-in-the-lambda-calculus.htmlhttps://pop.com.sg/what-is-the-meaning-of-omega-in-the-lambda-calculus.htmlAlthough I am sure the answer to my question must be trivial, I am still struggling with the exact definition of $\omega$, and consequently the definitions of $\omega_2 \dotsb \omega_n$, in the Lam......Mon, 31 Aug 2026 08:42:15 -0400https://pop.com.sg/calculating-the-inertia-of-a-real-symmetric-or-tridiagonal-matrix.htmlhttps://pop.com.sg/calculating-the-inertia-of-a-real-symmetric-or-tridiagonal-matrix.htmlI'm trying to find a quick method for evaluating the inertia of a real symmetric matrix, though I don't need to evaluate eigenvalues directly. The inertia of a matrix is a triple of the number of ...Mon, 31 Aug 2026 08:15:02 -0400https://pop.com.sg/what-is-the-practical-benefit-of-a-function-being-injective-surjective.htmlhttps://pop.com.sg/what-is-the-practical-benefit-of-a-function-being-injective-surjective.htmlI have learned that an injective function is one such that no two elements of the domain map to the same value in the codomain. Example: The function $x \mapsto x^2$ is injective when the domain is ...Mon, 31 Aug 2026 08:14:45 -0400https://pop.com.sg/asymmetric-periods-in-a-sine-curve.htmlhttps://pop.com.sg/asymmetric-periods-in-a-sine-curve.htmlI am wondering how to graph an asymmetric sine wave so that the slope of one side of the parabola is larger than the other, much in the same way as a sine graph with decreasing frequency. However I......Mon, 31 Aug 2026 08:14:18 -0400https://pop.com.sg/median-of-trapezoid-with-perpendicular-diagonals.htmlhttps://pop.com.sg/median-of-trapezoid-with-perpendicular-diagonals.htmlIn a trapezoid, the diagonals have lengths $10$ and $24$, and are perpendicular to each other. Find the length of the median?...Mon, 31 Aug 2026 08:10:51 -0400https://pop.com.sg/arccos-and-arcsin-integral-contradiction.htmlhttps://pop.com.sg/arccos-and-arcsin-integral-contradiction.htmlI am shown: $$f(x) = \arcsin x \implies f'(x) = \frac{1}{\sqrt{1-x^2}}$$ $$f(x) = \arccos x \implies f'(x) = -\frac{1}{\sqrt{1-x^2}}$$ These two derivatives can be very readily derived by a bit of ...Mon, 31 Aug 2026 08:10:49 -0400https://pop.com.sg/good-book-recommendations-on-trigonometry.htmlhttps://pop.com.sg/good-book-recommendations-on-trigonometry.htmlI need to find a good book on trigonometry, I was using trigonometry demystified but I got sad when I read this line: Now that you know how the circular functions are defined, you might wonder h......Mon, 31 Aug 2026 08:00:02 -0400https://pop.com.sg/can-a-complex-number-be-prime.htmlhttps://pop.com.sg/can-a-complex-number-be-prime.htmlI've been pondering over this question since a very long time. If a complex number can be prime then which parts of the complex number needs to be prime for the whole complex number to be prime....Mon, 31 Aug 2026 07:58:23 -0400https://pop.com.sg/omega-consistency-and-related-terms.htmlhttps://pop.com.sg/omega-consistency-and-related-terms.htmlWe know that a theory $T$ is $\omega$-inconsistent if there is a formula $\psi$ such that $T$ proves $(\exists x)\psi(x)$, and $T$ also proves $\lnot \psi(n)$ separately for each standard natural n......Mon, 31 Aug 2026 07:52:03 -0400https://pop.com.sg/topology-closure-definition-well-defined.htmlhttps://pop.com.sg/topology-closure-definition-well-defined.htmlI came upon the following definition for closure, Given a subset of a topological space $X$, the closure of $A$ is defined as the intersection of all closed sets containing $A$. How is this ...Mon, 31 Aug 2026 07:45:50 -0400https://pop.com.sg/converting-vector-in-cartesian-to-cylindrical-coordinates.htmlhttps://pop.com.sg/converting-vector-in-cartesian-to-cylindrical-coordinates.htmlThis seems like a trivial question, and I'm just not sure if I'm doing it right. I have vector in cartesian coordinate system: $\vec{N} =y\vec{a_x} −2x\vec{a_y} + y\vec{a_z}$. And I need to represe......Mon, 31 Aug 2026 07:44:09 -0400https://pop.com.sg/solving-a-triangle-given-two-sides-and-the-measure-of-the-included-ang.htmlhttps://pop.com.sg/solving-a-triangle-given-two-sides-and-the-measure-of-the-included-ang.htmlLet say you have a triangle Angle A = 41 degrees , side b = 3.41 and c = 5.83 can you use pythagoras theorem to find the side a? and how can you find Angle B and C...Mon, 31 Aug 2026 07:41:28 -0400https://pop.com.sg/definition-of-contractible-chain-complex.htmlhttps://pop.com.sg/definition-of-contractible-chain-complex.htmlA relatively simple question. A book I'm reading states "a complex homotopic to the zero complex is called contractible"... but I don't understand the statement. I know what it means for chain map......Mon, 31 Aug 2026 07:39:25 -0400https://pop.com.sg/why-does-the-graph-of-y-frac-tan-x-sin-x-cos-x-have-minima-at-multiple.htmlhttps://pop.com.sg/why-does-the-graph-of-y-frac-tan-x-sin-x-cos-x-have-minima-at-multiple.htmlI was just messing around is Desmos and plotted the graph of $$y=\frac{\frac{\tan(x)}{\sin(x)}}{\cos(x)}$$ and I realised that all the minimum points on the graph were multiples of $\pi$. Could som......Mon, 31 Aug 2026 07:28:37 -0400https://pop.com.sg/best-book-for-differential-equations.htmlhttps://pop.com.sg/best-book-for-differential-equations.htmlI know this is a subjective question, but I need some opinions on a very good book for learning differential equations. Ideally it should have a variety of problems with worked solutions and be ......Mon, 31 Aug 2026 07:28:21 -0400https://pop.com.sg/question-about-the-master-theorem-of-recurrences-no-b-term.htmlhttps://pop.com.sg/question-about-the-master-theorem-of-recurrences-no-b-term.htmlI'm using the master theorem to find the asymptotic run time of recurrences. For example, for a $T(n) = 4 T(n/5) + n^1$ I find that $T(n)$ is $\Theta(n^1)$, or, simply, constant time, via the set......Mon, 31 Aug 2026 07:27:16 -0400https://pop.com.sg/function-totally-differentiable-in-0-0.htmlhttps://pop.com.sg/function-totally-differentiable-in-0-0.htmlI asked this question here but didn't have an account yet and I can't comment yet on another question. So please forgive me for asking again. Consider the following function: $$f:\mathbb R^2\righ......Mon, 31 Aug 2026 07:13:23 -0400https://pop.com.sg/prove-that-sqrt-5-is-irrational.htmlhttps://pop.com.sg/prove-that-sqrt-5-is-irrational.htmlI have to prove that $\sqrt 5$ is irrational. Proceeding as in the proof of $\sqrt 2$, let us assume that $\sqrt 5$ is rational. This means for some distinct integers $p$ and $q$ having no common ...Mon, 31 Aug 2026 07:11:59 -0400https://pop.com.sg/what-is-the-n-th-derivative-of-coth-x.htmlhttps://pop.com.sg/what-is-the-n-th-derivative-of-coth-x.htmlI would like to know the $n$th derivative of the Hyperbolic Cotangent, i. e., $\frac{\partial^n}{\partial x^n} \coth( x )$. So far, I have only found an expression for the $n$th derivative of the ...Mon, 31 Aug 2026 07:09:23 -0400https://pop.com.sg/standard-symplectic-form.htmlhttps://pop.com.sg/standard-symplectic-form.htmlI am stuck at how to make the matrix of standard symplectic form. The given conditions are Definition Let $V$ be a vector space over $\mathbb{R}$. Then $\omega\in \Lambda^2(V)$ is called ...Mon, 31 Aug 2026 06:55:09 -0400https://pop.com.sg/what-is-the-difference-between-completeness-and-soundness-in-first-ord.htmlhttps://pop.com.sg/what-is-the-difference-between-completeness-and-soundness-in-first-ord.htmlCompleteness is defined as if given $\Sigma\models\Phi$ then $\Sigma\vdash\Phi$. Meaning if for every truth placement $Z$ in $\Sigma$ we would get $T$, then $\Phi$ also would get $T$. If the previous ...Mon, 31 Aug 2026 06:23:14 -0400https://pop.com.sg/composition-method-of-generating-random-variables.htmlhttps://pop.com.sg/composition-method-of-generating-random-variables.htmlI'm a student learning about methods for generating random variables. But I'm having a hard time finding examples or youtube tutorials on composition method and algorithm of generating random vari......Mon, 31 Aug 2026 06:11:15 -0400https://pop.com.sg/solving-a-hard-algebraic-equation.htmlhttps://pop.com.sg/solving-a-hard-algebraic-equation.htmlThe question is: Rectangular floor mats have an area of $x^2 + 2x - 15 \text{ cm}^2$. The length is $x + 5\text{ cm}$, and the width is $100\text{ cm}$. How do I find out the value of $x$, and......Mon, 31 Aug 2026 05:57:46 -0400https://pop.com.sg/geometric-series-how-to-find-r-and-n-given-the-last-term-the-first-ter.htmlhttps://pop.com.sg/geometric-series-how-to-find-r-and-n-given-the-last-term-the-first-ter.html$S_n=76$, $a=36$, last term is $16$ given that this is a geometric series how to find $r$ and number of terms I have tried to solve simultaneously but failed ....Mon, 31 Aug 2026 05:56:48 -0400https://pop.com.sg/how-to-find-perpendicular-vector-to-another-vector.htmlhttps://pop.com.sg/how-to-find-perpendicular-vector-to-another-vector.htmlHow do I find a vector perpendicular to a vector like this: $$3\mathbf{i}+4\mathbf{j}-2\mathbf{k}?$$ Could anyone explain this to me, please? I have a solution to this when I have $3\mathbf{i}+4\m......Mon, 31 Aug 2026 05:55:30 -0400https://pop.com.sg/proof-of-the-derivative-of-ln-x.htmlhttps://pop.com.sg/proof-of-the-derivative-of-ln-x.htmlI'm trying to prove that $\frac{\mathrm{d} }{\mathrm{d} x}\ln x = \frac{1}{x}$. Here's what I've got so far: $$ \begin{align} \frac{\mathrm{d}}{\mathrm{d} x}\ln x &= \lim_{h\to0} \frac{\ln(x + ......Mon, 31 Aug 2026 05:54:35 -0400https://pop.com.sg/equivalence-between-relational-algebra-statements.htmlhttps://pop.com.sg/equivalence-between-relational-algebra-statements.htmlThis is a practice question for a relational algebra question which I don't understand. Consider two relations $R(A, B)$ and $S(B,C)$. Which of the following relational algebra expressions is not ...Mon, 31 Aug 2026 05:51:55 -0400https://pop.com.sg/shouldn-039-t-we-check-for-conditionally-convergent-in-ratio-test-done.htmlhttps://pop.com.sg/shouldn-039-t-we-check-for-conditionally-convergent-in-ratio-test-done.html(By A(n) I mean the power series)I understood that we use absolute value of A(n+1)/A(n) in ratio test because A(n) isn't neccessarily a positive value. We know when there is a limit of absolute val......Mon, 31 Aug 2026 05:49:33 -0400https://pop.com.sg/difference-between-topologically-complete-space-and-complete-metric-sp.htmlhttps://pop.com.sg/difference-between-topologically-complete-space-and-complete-metric-sp.htmlDefinition: Topological space $(X,\tau)$ is called topologically complete if there is metric $d$ on $X$ which induces the topology $\tau$ of $X$ and $(X,d)$ is complete metric space. Also the foll......Mon, 31 Aug 2026 05:47:43 -0400https://pop.com.sg/if-x-and-y-are-rational-then-is-x-y-also-rational.htmlhttps://pop.com.sg/if-x-and-y-are-rational-then-is-x-y-also-rational.htmlI can think of the counter example $x = 2$ and $y = 1/2$ but how would a proof to disprove this look like?...Mon, 31 Aug 2026 05:37:46 -0400https://pop.com.sg/what-do-the-negative-and-positive-signs-next-to-the-limit-mean.htmlhttps://pop.com.sg/what-do-the-negative-and-positive-signs-next-to-the-limit-mean.htmlI’m sorry for this awkward question, but I don’t know how else I should explain it, I tried to do some research online, but couldn’t find anything. For example, the notation looks like: $$\lim_{x \......Mon, 31 Aug 2026 05:31:47 -0400https://pop.com.sg/how-to-calculate-z-score-without-standard-deviation.htmlhttps://pop.com.sg/how-to-calculate-z-score-without-standard-deviation.htmlI have been scratching my head about this for ages. I am studying intro stats for a Psychology Masters, and I can't get my head round this example problem: Example S.6.1 from https://newonlinecour......Mon, 31 Aug 2026 05:23:02 -0400https://pop.com.sg/system-in-a-6x6-matrix-of-ordinary-differential-equations.htmlhttps://pop.com.sg/system-in-a-6x6-matrix-of-ordinary-differential-equations.htmlOne must find general solution for $$y' = \left(\begin{matrix} 1&2&-1&-2&1&2\\ -1&-2&1&2&-1&-2\\ 2&4&-2&-4&2&4\\ -2&-4&2&4&......Mon, 31 Aug 2026 05:19:12 -0400https://pop.com.sg/cosets-and-quotient-groups-what-is-the-difference.htmlhttps://pop.com.sg/cosets-and-quotient-groups-what-is-the-difference.htmlSo I'm very new to group theory and I'm trying to understand the difference between a coset and a quotiënt group. To me it seems that a quotient group is always also a coset but not the other way a......Mon, 31 Aug 2026 04:56:13 -0400https://pop.com.sg/valuation-rings.htmlhttps://pop.com.sg/valuation-rings.htmlWhat's the spectrum of a valuation ring? How to describe morphisms from it to a scheme? Is it enough to set the image of generic point and of a maximal ideal and correspondent map of local rings?...Mon, 31 Aug 2026 04:53:44 -0400https://pop.com.sg/user-alvin-jin-mathematics-stack-exchange.htmlhttps://pop.com.sg/user-alvin-jin-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Mon, 31 Aug 2026 04:52:01 -0400https://pop.com.sg/is-g-x-log-x-convex-function.htmlhttps://pop.com.sg/is-g-x-log-x-convex-function.htmlThe graph of convex function is : In a book it is written that $g(x)=\log x$ is strictly convex function. So i searched for graph of $g(x)=\log x$ and found that Though it has been said that $......Mon, 31 Aug 2026 04:38:25 -0400https://pop.com.sg/binomial-theorem-and-complex-numbers.htmlhttps://pop.com.sg/binomial-theorem-and-complex-numbers.htmlUse the Binomial Theorem to calculate $z^4$ where $z$ is the complex number $2 − i$. Simplify your answer using the fact that $i^2 = −1$. Your final answer should be in the form $a + bi$ where $a$ ......Mon, 31 Aug 2026 04:32:53 -0400https://pop.com.sg/derivative-of-ln-1-beta-x-1.htmlhttps://pop.com.sg/derivative-of-ln-1-beta-x-1.htmlHow do I differentiate the term $\ln((1+\beta)^x-1)$ with respect to $x$? Is it possible to do it this way: $$\frac{1}{(1+\beta)^x-1}$$ But i get stuck if i do the normal differentiation....Mon, 31 Aug 2026 04:26:51 -0400https://pop.com.sg/which-symbol-is-used-to-represent-an-isometric-isomorphism-between-two.htmlhttps://pop.com.sg/which-symbol-is-used-to-represent-an-isometric-isomorphism-between-two.htmlI've been looking for a symbol which could be used to represent an Isometric Isomorphism between two normed spaces. Reading many books or PDF's, I noticed that authors use different symbols for it ......Mon, 31 Aug 2026 04:25:48 -0400https://pop.com.sg/fractions-with-negative-numerators-and-denominators.htmlhttps://pop.com.sg/fractions-with-negative-numerators-and-denominators.htmlI've been having this doubt in my mind.. I really don't get the concept behind fractions with negative numerators and denominators being equal to there positive form. For eg.: $\frac{-2}{-3} = \fra......Mon, 31 Aug 2026 04:21:40 -0400https://pop.com.sg/the-definition-and-meaning-of-machine-epsilon-in-matlab.htmlhttps://pop.com.sg/the-definition-and-meaning-of-machine-epsilon-in-matlab.htmlI am taking a introductory course in numerical mathematics, using MATLAB and a numerical math text that refers to MATLAB often. In the text, the machine precision is defined as: The distance $\e......Mon, 31 Aug 2026 04:13:55 -0400https://pop.com.sg/what-is-a-countable-set.htmlhttps://pop.com.sg/what-is-a-countable-set.htmlWhat is a countable set? In Wikipedia we read this definition: In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbe......Mon, 31 Aug 2026 04:08:25 -0400https://pop.com.sg/what-is-the-order-of-the-symmetry-group-of-the-cube.htmlhttps://pop.com.sg/what-is-the-order-of-the-symmetry-group-of-the-cube.htmlI figured out that there are 24 rotational symmetries, shown below. Rotation fixing: faces = 9 diagonals = 8 edges = 6 identity = 1 total = 24 Now, I don't know why the symmetry group of the 3-cub......Mon, 31 Aug 2026 04:07:15 -0400https://pop.com.sg/how-to-calculate-the-number-of-possible-combinations-with-repetitions.htmlhttps://pop.com.sg/how-to-calculate-the-number-of-possible-combinations-with-repetitions.htmlI would like to know the difference between "permutations with repetition" and "ways to choose $k$ elements from a set of $n$ elements if repetitions are allowed". For instance: In a set $S$ of $k$ ...Mon, 31 Aug 2026 03:54:31 -0400https://pop.com.sg/is-zero-to-the-power-of-an-imaginary-number-still-zero.htmlhttps://pop.com.sg/is-zero-to-the-power-of-an-imaginary-number-still-zero.htmlI just want to make sure that $0^i = 0$, but for some reason I couldn't find anything about this online. Is this true? --Background-- I'm trying to prove that some exponent is zero. I thought I'd ...Mon, 31 Aug 2026 03:50:57 -0400https://pop.com.sg/show-that-f-x-3x-2-2x-1-is-continuous-at-2.htmlhttps://pop.com.sg/show-that-f-x-3x-2-2x-1-is-continuous-at-2.htmlPlease check my proof Let$\delta ,\epsilon >0$ $$|x-2|<\delta \rightarrow |f(x)-f(2)<\epsilon $$ $$ \rightarrow |3x^{2}-2x+1-3x^{2}-2x+1|<\epsilon $$ $$ \...Mon, 31 Aug 2026 03:47:45 -0400https://pop.com.sg/solve-exponential-equation-with-exponent-variables.htmlhttps://pop.com.sg/solve-exponential-equation-with-exponent-variables.htmlOk, the question is: Solve the following for x: $2^x4^{x-1}=70$ I have just asked Wolfram Alpha, of course though, it supplies an answer without revealing its working. I started to try the pro......Mon, 31 Aug 2026 03:45:34 -0400https://pop.com.sg/how-to-calculate-norm-of-matrix-using-any-orthogonal-basis.htmlhttps://pop.com.sg/how-to-calculate-norm-of-matrix-using-any-orthogonal-basis.htmlShow that for $X \in \mathrm{M}_n(\mathbb{C})$ and any orthonormal basis $\{u_1, \ldots , u_n\}$ of $\mathbb{C}^n$, we have $$\|X\|^2=\sum_{j,k}^n|\langle u_j,Xu_k\rangle |^2.$$ My Attempt: I t......Mon, 31 Aug 2026 03:42:34 -0400https://pop.com.sg/can-a-sequence-be-called-convergent-divergent-if-it-has-finite-number-.htmlhttps://pop.com.sg/can-a-sequence-be-called-convergent-divergent-if-it-has-finite-number-.htmlNo explanation required for the question I guess....Mon, 31 Aug 2026 03:35:53 -0400https://pop.com.sg/is-there-a-good-way-to-compute-christoffel-symbols.htmlhttps://pop.com.sg/is-there-a-good-way-to-compute-christoffel-symbols.htmlLets say you have a Riemannian Manifold $(M,g)$, and you have some given chart where $g = g_{ij} dx_i dx_j$ and you wish to compute the Christoffel symbols for the Riemannian connection in this cha......Mon, 31 Aug 2026 03:35:28 -0400https://pop.com.sg/where-is-greens-theorem-used.htmlhttps://pop.com.sg/where-is-greens-theorem-used.htmlWhere is Greens theorem used? I think it's weird going from a vector field to calculating a volume on a scalar field, where do we use this kind of calculation?...Mon, 31 Aug 2026 03:31:39 -0400https://pop.com.sg/optimization-of-the-surface-area-of-a-open-rectangular-box-to-find-the.htmlhttps://pop.com.sg/optimization-of-the-surface-area-of-a-open-rectangular-box-to-find-the.htmlA rectangular storage container with an open top is to have a volume of 10 cubic meters. The length of the box is twice its width. Material for the base costs ten dollars per square meter and for the ...Mon, 31 Aug 2026 03:21:54 -0400https://pop.com.sg/total-distance-travelled-using-integrals.htmlhttps://pop.com.sg/total-distance-travelled-using-integrals.htmlFor time $t \geq 0$, the velocity of a particle moving along the $x$-axis is given by $\mathrm{v}\left(t\right) = \sin\left(t\right)$, where $t$ is measured in seconds and $\mathrm{v}\left(t\right)......Mon, 31 Aug 2026 03:14:46 -0400https://pop.com.sg/is-a-triangle-with-two-equal-angles-always-isosceles.htmlhttps://pop.com.sg/is-a-triangle-with-two-equal-angles-always-isosceles.htmlAn isosceles triangle is a triangle with two sides that are equal in length. This means that two angle will also be equal to each other. Is there any way that a triangle could have two equal angles......Mon, 31 Aug 2026 03:13:53 -0400https://pop.com.sg/p-norm-with-p-to-infinity-duplicate.htmlhttps://pop.com.sg/p-norm-with-p-to-infinity-duplicate.htmlI have to show that: for all vectors $v\in \Bbb R^n$: $\lim_{p\to \infty}||v||_p = \max_{1\le i \le n}|v_i|$ with the $||\cdot ||_p$ norm defined as $$ ||\cdot ||_p: (v_1, \dots ,v_n) \to (\sum^n_{......Mon, 31 Aug 2026 03:08:52 -0400https://pop.com.sg/how-to-integrate-sec-3-x-dx-duplicate.htmlhttps://pop.com.sg/how-to-integrate-sec-3-x-dx-duplicate.htmlPossible Duplicate: Indefinite integral of secant cubed How to integrate $\sec^3 x \, dx$? Can someone please give a method, I tried separating $\sec^3 x$ as $\sec x(\sec^2 x)$ then applying by-...Mon, 31 Aug 2026 02:59:22 -0400https://pop.com.sg/change-from-product-to-sum.htmlhttps://pop.com.sg/change-from-product-to-sum.htmlWe know that : $$a \times b = \underbrace{a + a + a + ... + a}_{\text{b times}}$$ That's how we convert from a product to a sum. So what happens if we go a little further? That is : $$\prod\limits_......Mon, 31 Aug 2026 02:57:51 -0400https://pop.com.sg/which-of-these-sentences-are-propositions-what-are-the-truth-values-of.htmlhttps://pop.com.sg/which-of-these-sentences-are-propositions-what-are-the-truth-values-of.htmlWhich of these sentences are propositions? What are the truth values of those that are propositions? a) Boston is the capital of Massachusetts. b) Miami is the capital of Florida. c) 2 + 3 = 5. d) ......Mon, 31 Aug 2026 02:53:07 -0400https://pop.com.sg/utilization-of-m-m-c-queue.htmlhttps://pop.com.sg/utilization-of-m-m-c-queue.htmlWhat is the utilization of m/m/c queue ? in some texts such as http://www.amazon.com/Queueing-Networks-Markov-Chains-Applications/dp/0471565253 said: individual server utilization $\rho =(\lambda)......Mon, 31 Aug 2026 02:49:20 -0400https://pop.com.sg/some-commutator-identities.htmlhttps://pop.com.sg/some-commutator-identities.htmlOn the way to study Lang's algebra, I cannot solve this problem. See page 69. Let G be a group and denote the commutator of x and y by & $[x,y]=xyx^{-1}y^{-1}$. I wanna prove that if $[x,y]=......Mon, 31 Aug 2026 02:45:19 -0400https://pop.com.sg/why-is-it-valid-to-multiply-both-sides-of-an-equation-by-its-complex-c.htmlhttps://pop.com.sg/why-is-it-valid-to-multiply-both-sides-of-an-equation-by-its-complex-c.htmlThis is the closest explanation I can find, although I still don't fully understand. We know what’s happening: division is subtracting an angle and shrinking the magnitude. By multiplying top ......Mon, 31 Aug 2026 02:44:13 -0400https://pop.com.sg/how-often-in-years-do-calendars-repeat-with-the-same-day-date-combinat.htmlhttps://pop.com.sg/how-often-in-years-do-calendars-repeat-with-the-same-day-date-combinat.htmlE.g. I'm using this formulas for calculating day of week (Julian calendar): \begin{align} a & = \left\lfloor\frac{14 - \text{month}}{12}\right\rfloor\\ y & = \text{year} + 4800 - a \\ m &a......Mon, 31 Aug 2026 02:25:18 -0400https://pop.com.sg/clarification-of-the-definition-of-tempered-growth.htmlhttps://pop.com.sg/clarification-of-the-definition-of-tempered-growth.htmlThis is a rather simple question, but I don't have many resources I've been able to consult in this regard (meaning that a Google search hasn't yielded any meaningful results). I just want to clari......Mon, 31 Aug 2026 02:16:58 -0400https://pop.com.sg/what-is-different-between-g-n-and-gn-n.htmlhttps://pop.com.sg/what-is-different-between-g-n-and-gn-n.htmlWhat is different between $G/N$ and $GN/N$? ($G$ denotes a group, and N does a normal subgroup of G) If you look at the element of each group above, I think $G/N$ contains element of form $gN$, ......Mon, 31 Aug 2026 02:10:56 -0400https://pop.com.sg/in-a-math-paper-what-is-a-remark.htmlhttps://pop.com.sg/in-a-math-paper-what-is-a-remark.htmlI sometimes see paragraphs labeled 'Remark.' However, papers that include remarks also include unlabeled explanatory paragraphs (i.e. all the other writing in the article) that seem to be remarks. ......Mon, 31 Aug 2026 02:06:33 -0400https://pop.com.sg/the-definition-of-distance-and-how-to-prove-the-ruler-postulate-in-euc.htmlhttps://pop.com.sg/the-definition-of-distance-and-how-to-prove-the-ruler-postulate-in-euc.htmlI have started to read some books about geometry. At the moment I've just started to read Hilbert's axioms and also some elementary books for highschool. From the basic perspective of an axiomatic ......Mon, 31 Aug 2026 02:04:52 -0400https://pop.com.sg/proof-for-symmetry-property-of-congruent-segments.htmlhttps://pop.com.sg/proof-for-symmetry-property-of-congruent-segments.htmlI am starting to learn geometrical proofs, and I have come across the Symmetry property of segment congruence (if $AB$ is congruent to $CD$, then $CD$ is congruent to $AB$). One of the exercises i......Mon, 31 Aug 2026 01:55:40 -0400https://pop.com.sg/what-is-the-difference-between-o-2-n-and-2-o-n.htmlhttps://pop.com.sg/what-is-the-difference-between-o-2-n-and-2-o-n.htmlIn the context of complexity, I have seen both $O(2^{n})$ and $2^{O(n)}$ used. Whats the difference between the two?...Mon, 31 Aug 2026 01:44:39 -0400https://pop.com.sg/what-is-the-norm-of-a-complex-number.htmlhttps://pop.com.sg/what-is-the-norm-of-a-complex-number.htmlMany articles on the web state that the norm of $z = a +bi$ is \begin{align} \sqrt{z*z^*} &= \sqrt{(a + bi)(a - bi)}\\ &= \sqrt{a^2 - abi + abi - b^2i^2}\\ &= \sqrt{a^2 - b^2\sqrt{-1......Mon, 31 Aug 2026 01:41:47 -0400https://pop.com.sg/recurrence-tree-method-t-n-3-2n-3-1-why-is-the-solution-in-log-3-2-and.htmlhttps://pop.com.sg/recurrence-tree-method-t-n-3-2n-3-1-why-is-the-solution-in-log-3-2-and.htmlI feel this is more of a question in the logarithms, but I am confused why this tree reoccurrence $$T(n) = 3T(2n/3)+1$$ solution is derived from $\log_{3/2}$ rather than $\log_{2/3}$? My thinking......Mon, 31 Aug 2026 01:30:50 -0400https://pop.com.sg/how-does-an-exponent-work-when-it-039-s-less-than-one.htmlhttps://pop.com.sg/how-does-an-exponent-work-when-it-039-s-less-than-one.htmlI'm rather familiar with exponents, I know that $y^x = y_1 \cdot y_2 \cdot y_3 .... y_x$, but what if the exponent is less than one, how would that work? I put in my computer $25^{1/2}$ anyway, ...Mon, 31 Aug 2026 01:30:08 -0400https://pop.com.sg/sweepstakes-probability.htmlhttps://pop.com.sg/sweepstakes-probability.htmlIs there a way to determine the probability of winning a particular sweepstakes with only the following information: the estimated odds of winning are $1$ in $13,000$ contestants must be owners (...Mon, 31 Aug 2026 01:16:56 -0400https://pop.com.sg/can-a-circle-have-a-negative-radius.htmlhttps://pop.com.sg/can-a-circle-have-a-negative-radius.htmlI am working on a problem that asks if a given sphere intersects the zx-plane. The equation of the sphere is $(x-2)^2+(y+6)^2+(z-4)^2=5^2$ Can someone please explain to me how the sphere does not ...Mon, 31 Aug 2026 01:12:57 -0400