https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usWed, 12 Aug 2026 23:41:14 -0400https://pop.com.sg/height-of-vertical-ropes-of-golden-gate-bridge.htmlhttps://pop.com.sg/height-of-vertical-ropes-of-golden-gate-bridge.htmlThe prompt is to find the parabola equation for the main span of Golden Gate bridge. Based on the $3$ points on the parabola: $(0,227)$; vertex $(640,75)$; and $(1280,227)$ I found the equation: $......Thu, 13 Aug 2026 03:37:27 -0400https://pop.com.sg/user-john-d-cook-mathematics-stack-exchange.htmlhttps://pop.com.sg/user-john-d-cook-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Thu, 13 Aug 2026 03:36:41 -0400https://pop.com.sg/what-is-the-best-book-for-pre-calculus.htmlhttps://pop.com.sg/what-is-the-best-book-for-pre-calculus.htmli have missed pre-calculus knowledge in my school but i was good at maths, and now i am a computer science student, i am feeling bad being bad in maths, so i am looking for the best Pre-Calculus bo......Thu, 13 Aug 2026 03:35:58 -0400https://pop.com.sg/exterior-algebra-as-quotient.htmlhttps://pop.com.sg/exterior-algebra-as-quotient.htmlGiven a vector space $W$, I understand what the tensor algebra $T(W)$ is, and I understand that the exterior algebra $\bigwedge W$ is defined as $\bigwedge W := T(W)/N$ where $N$ is the two-sided a......Thu, 13 Aug 2026 03:32:42 -0400https://pop.com.sg/two-integrating-factors-same-solution-to-differential-equation.htmlhttps://pop.com.sg/two-integrating-factors-same-solution-to-differential-equation.htmlIf we have to valid integrating factors to a differential equation we must get the same general solution using each of them although we get two new equations by multiplying the original for each of......Thu, 13 Aug 2026 03:23:39 -0400https://pop.com.sg/intuitive-understanding-of-the-operator-norm.htmlhttps://pop.com.sg/intuitive-understanding-of-the-operator-norm.htmlI understand various vector norms, but I don't understand operator norms. Specifically, norms on linear operators. Can anyone explain them?...Thu, 13 Aug 2026 03:20:15 -0400https://pop.com.sg/using-right-hand-rule-for-a-current-running-in-a-loop.htmlhttps://pop.com.sg/using-right-hand-rule-for-a-current-running-in-a-loop.htmlHello. Using the right hand rule, shouldn't the magnetic force be going into the paper, since velocity is to the left and the force caused by centripetal acceleration is downward? The answer key says ...Thu, 13 Aug 2026 03:19:14 -0400https://pop.com.sg/how-do-i-write-this-basic-recursive-formula-into-desmos.htmlhttps://pop.com.sg/how-do-i-write-this-basic-recursive-formula-into-desmos.htmlI want to graph a simple equation $f(x)$ which begins at $(0,1)$, then for every increasing $x$ integer increment, $f(x) = f(x-1) - (c * f(x-1))$. So in other words each time you go up by one $x$ ...Thu, 13 Aug 2026 03:18:26 -0400https://pop.com.sg/graph-of-the-function-2x-2-2-x-2-1.htmlhttps://pop.com.sg/graph-of-the-function-2x-2-2-x-2-1.htmlI am learning precalculus and my precalculus book gives this equation: for this graph: But when I enter that equation into some online graph tool like Symbolab ( https://www.symbolab.com/graphing-...Thu, 13 Aug 2026 03:09:19 -0400https://pop.com.sg/scalar-multiplication-of-vectors-on-left-or-right.htmlhttps://pop.com.sg/scalar-multiplication-of-vectors-on-left-or-right.htmlVectors are often written column-wise as if they were $n\times 1$ matrices: $$ \mathbf{v} := \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} $$ This notation implicitly identifies the vector $\mathbf{v......Thu, 13 Aug 2026 03:04:09 -0400https://pop.com.sg/multiplication-of-ordered-pairs.htmlhttps://pop.com.sg/multiplication-of-ordered-pairs.htmlFrom what I've seen, ordered pair multiplication is $(a,b) \times (c,d) = (ac-bd,ad+bc)$. Also, $(0,1)$ represents $i$. So if I wanted to take $(0,-1) \times (0.-1)$, that would be $= (-1,0)$, rig......Thu, 13 Aug 2026 02:53:31 -0400https://pop.com.sg/find-all-solutions-to-x-9-equiv-25-pmod-29.htmlhttps://pop.com.sg/find-all-solutions-to-x-9-equiv-25-pmod-29.htmlFind all solutions to $x^9 \equiv 25 \pmod {29}$ I referred to a table of indices in the back of my book for help but am still a bit confused, any help/hints are greatly appreciated....Thu, 13 Aug 2026 02:47:36 -0400https://pop.com.sg/approximation-of-sqrt2-using-euler-039-s-method.htmlhttps://pop.com.sg/approximation-of-sqrt2-using-euler-039-s-method.htmlConsider the differential equation $$\frac{dx}{dt}=\frac1{2x}.$$ This is a separable O.D.E. so we know how to find all of its solutions: they are of the form $$x(t)=\sqrt{t+C}$$ where $C$ is a ...Thu, 13 Aug 2026 02:39:44 -0400https://pop.com.sg/calculating-the-mean-of-a-folded-normal-distribution.htmlhttps://pop.com.sg/calculating-the-mean-of-a-folded-normal-distribution.htmlI have some data that I know is distributed according to the folded normal distribution. I am looking to calculate the mean and variance of the underyling "unfolded" normal distribution. Let's sa......Thu, 13 Aug 2026 02:38:15 -0400https://pop.com.sg/give-the-bayesian-posterior-mode.htmlhttps://pop.com.sg/give-the-bayesian-posterior-mode.htmlSuppose that $X_1, X_2, \ldots, X_n$ are IID Bernoulli random variables with success probability equal to an unknown parameter $\theta \in [0,1]$. Let $A$ and $B$ be nonnegative constants. If we i......Thu, 13 Aug 2026 02:36:11 -0400https://pop.com.sg/is-p-vs-np-problem-solved.htmlhttps://pop.com.sg/is-p-vs-np-problem-solved.htmlMany people have tried to solve the very famous problem "P vs NP" and a lot of solutions are proposed. (e.g. A. D. Plotnikov, On the Relationship between Classes P and NP). But I couldn't find any ...Thu, 13 Aug 2026 02:36:01 -0400https://pop.com.sg/factors-of-zero.htmlhttps://pop.com.sg/factors-of-zero.htmlHow many integer factors of 0 are there, and what are they? I'm just curious, but what counts as a factor of 0? My guess is that there are an infinite number of factors of 0, but is there a proof?...Thu, 13 Aug 2026 02:35:23 -0400https://pop.com.sg/how-to-prove-that-lim-x-to-pi-2-tan-x-tan2x-1.htmlhttps://pop.com.sg/how-to-prove-that-lim-x-to-pi-2-tan-x-tan2x-1.htmlHow do I prove that the limit of the following function is $1$? $\displaystyle\lim_{x\to \pi/2} (\tan x)^{\tan2x}$ I shouldn't use l'Hôpital's rule. I have proven that limit is greater than $......Thu, 13 Aug 2026 02:25:25 -0400https://pop.com.sg/what-are-the-4-fourth-roots-of-16-complex-numbers-included-closed.htmlhttps://pop.com.sg/what-are-the-4-fourth-roots-of-16-complex-numbers-included-closed.htmlI'm trying to find the 4 fourth roots of 16, so far I have -2 and 2, but I have no idea how to find the other 2 complex numbers....Thu, 13 Aug 2026 02:23:28 -0400https://pop.com.sg/rate-of-change-with-f-x-4x-2-7-on-1-b.htmlhttps://pop.com.sg/rate-of-change-with-f-x-4x-2-7-on-1-b.htmlAs part of a textbook exercise I am to find the rate of change of $f(x)=4x^2-7$ on inputs $[1,b]$. The solution provided is $4(b+1)$ and I am unable to arrive at this solution. Tried: $f(x_2)=4b......Thu, 13 Aug 2026 02:21:13 -0400https://pop.com.sg/checking-if-a-vector-field-is-conservative.htmlhttps://pop.com.sg/checking-if-a-vector-field-is-conservative.htmlI have three different vector fields and I want to check if they are conservative: $$1)\space \space\vec{f}(\vec{x}):=\frac{1}{||\vec{x}||}\vec{a}, \space \space D=\mathbb R^2 / (\vec{0}), \space ......Thu, 13 Aug 2026 02:04:20 -0400https://pop.com.sg/good-software-for-linear-integer-programming.htmlhttps://pop.com.sg/good-software-for-linear-integer-programming.htmlI never did any linear/integer programming so I am wondering the following two things What are some efficient free linear programming solvers? What are some efficient commercial linear programming ...Thu, 13 Aug 2026 01:59:29 -0400https://pop.com.sg/conditional-probability-all-heads-given-at-least-one-heads.htmlhttps://pop.com.sg/conditional-probability-all-heads-given-at-least-one-heads.htmlThree coins are flipped. If at least one of them comes up heads, then what is the probability that they all come up heads? The answer to this is 1/7, as it is the #(ways to have all heads)/#(ways t......Thu, 13 Aug 2026 01:44:59 -0400https://pop.com.sg/show-that-lim-2n-frac-1-n-1.htmlhttps://pop.com.sg/show-that-lim-2n-frac-1-n-1.htmlProve that $\lim ((2n)^\frac{1}{n}) = 1$. I have obtained the following: $$(2n)^\frac{1}{n} = 1 + k_{n}; n > 1$$ $$(2n) = (1 + k_{n})^n$$ By the Binomial Theorem: $$(1 + k_{n})^n = 1 + k_{n} + \...Thu, 13 Aug 2026 01:36:49 -0400https://pop.com.sg/why-is-the-slope-of-a-line-defined-as-the-change-in-y-over-the-change-.htmlhttps://pop.com.sg/why-is-the-slope-of-a-line-defined-as-the-change-in-y-over-the-change-.htmlI understand that this is a reasonable definition and it shows how "fast" $y$ values change corresponding to $x$ values because it's a ratio, but what I'm asking is, couldn't the slope have been de......Thu, 13 Aug 2026 01:22:53 -0400https://pop.com.sg/taylor-series-with-point-of-expansion.htmlhttps://pop.com.sg/taylor-series-with-point-of-expansion.htmlWrite the Taylor series for the function $f(x)= x^2-3x+1$ using $x=2$ as the point of expansion , i.e. write a formula for $f(2+h)$ Even though I know what Taylor Expansion is I am really confused......Thu, 13 Aug 2026 01:16:14 -0400https://pop.com.sg/if-g-is-non-abelian-show-that-aut-g-is-not-cyclic.htmlhttps://pop.com.sg/if-g-is-non-abelian-show-that-aut-g-is-not-cyclic.htmlIf $G$ is non-Abelian, show that $Aut(G)$ is not cyclic. I can see that $G$ non-Abelian $\Rightarrow$ $G / Z(G)$ is not cyclic. And $G / Z(G)$ is isomorphic to $Inn(G) \Rightarrow$ $Inn(G)$ is not ...Thu, 13 Aug 2026 01:15:38 -0400https://pop.com.sg/limits-involving-greatest-integer-function.htmlhttps://pop.com.sg/limits-involving-greatest-integer-function.htmlWhat can you say about the following limit: $$\lim_{x\rightarrow 0}\left(\left[\dfrac{100x}{\sin x}\right]+\left[\dfrac{99\sin x}{x}\right]\right)$$ where [.] represents the greatest integer funct......Thu, 13 Aug 2026 00:55:22 -0400https://pop.com.sg/is-pi-over-pi-rational.htmlhttps://pop.com.sg/is-pi-over-pi-rational.htmlA student asked this question in class today, and I wasn't sure of the answer. On the one hand, since Pi is irrational itself, Pi/Pi doesn't fit the definition of a rational number (namely a numbe......Thu, 13 Aug 2026 00:54:24 -0400https://pop.com.sg/numerically-equivalent-sets-power-set.htmlhttps://pop.com.sg/numerically-equivalent-sets-power-set.htmlWhat I am struggling with is understanding the part where we prove that the function is onto. Letting f be a element of $2^A$ means that in this particular example f could be either of the $f_1 f_......Thu, 13 Aug 2026 00:52:09 -0400https://pop.com.sg/difference-between-cauchy-theorem-and-cauchy-goursat-theorem.htmlhttps://pop.com.sg/difference-between-cauchy-theorem-and-cauchy-goursat-theorem.htmlIn some books it is given that Cauchy theorem is equivalent to Cauchy Goursat theorem.And in some other it is given to be different. In my class , our teacher has given it differently. Cauchy theorem ...Thu, 13 Aug 2026 00:42:35 -0400https://pop.com.sg/green-039-s-theorem-conditions.htmlhttps://pop.com.sg/green-039-s-theorem-conditions.htmlWhy must the following condition be satisfied before we can use Green's Theorem: L and M are functions of (x, y) defined on an open region containing D and have continuous partial derivatives there....Thu, 13 Aug 2026 00:34:52 -0400https://pop.com.sg/why-does-a-basis-of-a-vector-space-need-to-have-the-same-number-of-vec.htmlhttps://pop.com.sg/why-does-a-basis-of-a-vector-space-need-to-have-the-same-number-of-vec.htmlAs an example, in $\mathbb{R^3}$, the canonical basis is:\begin{align} B &= \left(\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix},\begin{bmatrix} ......Thu, 13 Aug 2026 00:28:42 -0400https://pop.com.sg/can-the-square-root-of-a-real-number-be-negative-duplicate.htmlhttps://pop.com.sg/can-the-square-root-of-a-real-number-be-negative-duplicate.htmlCan the square root of a real number be negative? Dealing with the questions of functions in eleventh class my maths teacher says that square root of a real number is always positive. How is it po......Thu, 13 Aug 2026 00:27:34 -0400https://pop.com.sg/the-most-powerful-test-for-variance-of-normal-distribution.htmlhttps://pop.com.sg/the-most-powerful-test-for-variance-of-normal-distribution.htmlLet $ X_1, \ldots, X_{10} $ be simple samples from normal distribution with expected value${}= 0$ and variance${}= \sigma^2 $. Write the most powerful test for $ \alpha = 0.05 $ to verify the hypo......Thu, 13 Aug 2026 00:07:11 -0400https://pop.com.sg/why-does-convex-function-mean-concave-up.htmlhttps://pop.com.sg/why-does-convex-function-mean-concave-up.htmlA function $f : \mathbb{R} \to \mathbb{R}$ is convex (or "concave up") provided that for all $x,y \in \mathbb{R}$ and $t \in [0,1]$, $$f(tx + (1-t)y) \le tf(x) + (1-t)f(y).$$ Equivalently, a line s......Thu, 13 Aug 2026 00:06:20 -0400https://pop.com.sg/a-limit-wrong-using-wolfram-alpha.htmlhttps://pop.com.sg/a-limit-wrong-using-wolfram-alpha.htmlI want to calculate the following limit: $$\displaystyle{\lim_{x \to 0} \cfrac{\displaystyle{\int_1^{x^2+1} \cfrac{e^{-t}}{t} \; dt}}{3x^2}}$$ For that, I use L'Hopital and the Fundamental Theore......Thu, 13 Aug 2026 00:05:59 -0400https://pop.com.sg/is-every-subset-of-a-set-a-set.htmlhttps://pop.com.sg/is-every-subset-of-a-set-a-set.htmlIn $\mathsf{ZF}$ we have the Power Set Axiom: if $x$ is a set, then $\mathcal P(x)$ is a set (see here for a nice listing of the axioms of $\mathsf{ZF}$). Yesterday I was explaining to some student......Thu, 13 Aug 2026 00:04:35 -0400https://pop.com.sg/how-to-find-the-period-of-the-sum-of-two-trigonometric-functions.htmlhttps://pop.com.sg/how-to-find-the-period-of-the-sum-of-two-trigonometric-functions.htmlI want to know if there exists a general method to find the period of the sum of two periodic trigonometric function. Example: $$f(x)=\cos(x/3)+\cos(x/4).$$...Thu, 13 Aug 2026 00:01:24 -0400https://pop.com.sg/what-is-usually-meant-by-logit-scale-or-log-scale.htmlhttps://pop.com.sg/what-is-usually-meant-by-logit-scale-or-log-scale.htmlThis question is more about the math terminology than about the math itself. Say we have x = logit(p). If one says "logit scale" does he mean: the scale of p, or the scale of x, i.e. the scale of ......Thu, 13 Aug 2026 00:00:15 -0400https://pop.com.sg/find-the-radius-of-convergence-r-of-the-series-and-find-the-interval-i.htmlhttps://pop.com.sg/find-the-radius-of-convergence-r-of-the-series-and-find-the-interval-i.html$$ \sum_{n=1}^{\infty} 3^n x^n / n^2.$$ I got $R=1/3$ Find the interval, I, of convergence of the series. (Enter your answer using interval notation.) I got $I= (-1/3, 1/3)$ i'm fairly certa......Thu, 13 Aug 2026 00:00:01 -0400https://pop.com.sg/let-m-and-n-be-two-integers-prove-that-mn-m-is-odd-if-and-only-if-m-is.htmlhttps://pop.com.sg/let-m-and-n-be-two-integers-prove-that-mn-m-is-odd-if-and-only-if-m-is.html---The statement: Let $m$ and $n$ be two integers. Prove that $mn+m$ is odd if and only if $m$ is odd and $n$ is even. ---Solution: First, we show that if $m$ is odd and $n$ is even, then $mn+m$ is......Wed, 12 Aug 2026 23:59:44 -0400https://pop.com.sg/cover-up-method-on-1-to-infinity.htmlhttps://pop.com.sg/cover-up-method-on-1-to-infinity.html$$\sum_{n=1}^{\infty}\frac{7}{n(n+3)}$$ I know this is convergent but I do not know how to apply "cover-up method" to solve this can I get some help pls? The answer I got is $77/18$...Wed, 12 Aug 2026 23:44:59 -0400https://pop.com.sg/evaluate-int-sin-3-x-cos-2-x-dx-duplicate.htmlhttps://pop.com.sg/evaluate-int-sin-3-x-cos-2-x-dx-duplicate.htmlI'm trying to solve $\int{\sin^3(x)\cos^2(x)}dx$. I got $-\frac{1}{2}\cos(x)+C$, but the memo says $\frac{1}{5}\cos^5(x)-\frac{1}{3}\cos^3(x)+C$ This is my working: Your help is appreciated!...Wed, 12 Aug 2026 23:40:54 -0400https://pop.com.sg/using-the-snowflake-method-to-factor-trinomials.htmlhttps://pop.com.sg/using-the-snowflake-method-to-factor-trinomials.htmlIn school, students may be taught different ways to factor the trinomial $$ax^2+bx+c$$ where $a \neq 1,0$. Possible methods include the classic Guess and Check Method, Grouping, Box Method, and the ...Wed, 12 Aug 2026 23:38:14 -0400https://pop.com.sg/eigenvalue-with-algebraic-multiplicity-1.htmlhttps://pop.com.sg/eigenvalue-with-algebraic-multiplicity-1.htmlIf I have some $\lambda_1$ as one of my eigenvalues, and it has algebraic multiplicity of $n$ with $n > 1$. When I try to get some eigenvectors associated with this $\lambda_1$, assuming that the ...Wed, 12 Aug 2026 23:27:31 -0400https://pop.com.sg/polar-to-cartesian-equation-conversion.htmlhttps://pop.com.sg/polar-to-cartesian-equation-conversion.htmlI have a polar equation defined as: $r = ae^{θ \tan m}$ where, $a$ and $m$ are constants, $θ$ is the angle between the horizontal axis from the origin $(x_c,y_c)$ to the coordinate. $e$ refers to ...Wed, 12 Aug 2026 23:27:22 -0400https://pop.com.sg/recommended-books-for-aime-usamo-preparation.htmlhttps://pop.com.sg/recommended-books-for-aime-usamo-preparation.htmlWhat books would you guys recommend to learn number theory, geometry, combinatorics, and algebra at a level appropriate for the AIME and/or USAMO? With the month of February approaching, the month ......Wed, 12 Aug 2026 23:19:59 -0400https://pop.com.sg/the-vertex-covering-of-a-graph-that-has-isolated-vertices.htmlhttps://pop.com.sg/the-vertex-covering-of-a-graph-that-has-isolated-vertices.htmlI know the theorem that in a graph G=(V,E) which has no isolated vertex, a set $C\subset V$ is a vertex covering in G if and only if the set V‐C is an independent set in G. I want to know whether t......Wed, 12 Aug 2026 23:16:33 -0400https://pop.com.sg/how-to-find-no-of-complex-root-of-any-polynomial.htmlhttps://pop.com.sg/how-to-find-no-of-complex-root-of-any-polynomial.htmlFor $f(x)=x^5+x^3-2x+1$, we know there should be $5$ roots . We can also find real roots of this polynomial. Can we conclude that: No of complex roots = No of total roots - No of real roots . D......Wed, 12 Aug 2026 22:47:08 -0400https://pop.com.sg/proving-that-the-reciprocal-of-an-irrational-is-irrational.htmlhttps://pop.com.sg/proving-that-the-reciprocal-of-an-irrational-is-irrational.htmlThe question I am working on is: Prove that if x is irrational, then 1/x is irrational. My proof differs from the one given in the answer key; but I still feel that mine is valid. Could someone ...Wed, 12 Aug 2026 22:45:22 -0400https://pop.com.sg/finding-mode-in-binomial-distribution.htmlhttps://pop.com.sg/finding-mode-in-binomial-distribution.htmlSuppose that $X$ has the Binomial distribution with parameters $n,p$ . How can I show that if $(n+1)p$ is integer then $X$ has two mode that is $(n+1)p$ or $(n+1)p-1?$...Wed, 12 Aug 2026 22:44:26 -0400https://pop.com.sg/is-there-a-solution-for-parker-039-s-square.htmlhttps://pop.com.sg/is-there-a-solution-for-parker-039-s-square.htmlFrom wiki: Mathematician Matt Parker attempted to create a 3x3 magic square using square numbers in a YouTube video on the Numberphile channel. His failed attempt is known as the Parker Square. I ......Wed, 12 Aug 2026 22:40:39 -0400https://pop.com.sg/definition-of-local-ring.htmlhttps://pop.com.sg/definition-of-local-ring.htmlI was reading some article on local rings, and it gave the following (equivalent) definitions: A ring $R$ is $local$ if it satisfies one of the following equivalent properties (so this needs provi......Wed, 12 Aug 2026 22:37:14 -0400https://pop.com.sg/making-the-mandelbrot-fractal-in-desmos-online-graphing-calculator.htmlhttps://pop.com.sg/making-the-mandelbrot-fractal-in-desmos-online-graphing-calculator.htmlI would like to make the following "animation" of the Mandelbrot Set using the Online Desmos Graphing Calculator app - as seen over here: https://www.youtube.com/watch?v=naqgsOOEHJs I co......Wed, 12 Aug 2026 22:31:53 -0400https://pop.com.sg/is-5-0-an-integer-or-decimal-number.htmlhttps://pop.com.sg/is-5-0-an-integer-or-decimal-number.htmlIs 5.0 an integer or decimal number? I was asked by one of my friends, we got both confused. I said by definition integer contains no or zero decimal part so it should be an integer. But he said t......Wed, 12 Aug 2026 22:29:13 -0400https://pop.com.sg/limits-of-integration-spherical-coordinates.htmlhttps://pop.com.sg/limits-of-integration-spherical-coordinates.htmlI have seen a lot of exercises where they solve a triple integral using spherical coordinates. But I'm confused about the limits that one should use. For example when they integrate over a sphere l......Wed, 12 Aug 2026 22:15:16 -0400https://pop.com.sg/number-of-triangles-sharing-all-vertices-but-no-sides-with-a-given-oct.htmlhttps://pop.com.sg/number-of-triangles-sharing-all-vertices-but-no-sides-with-a-given-oct.htmlThe number of triangles whose vertices are the the vertices of the vertices of an octagon but none of whose sides happen to come from the sides of the octagon. My Attempt: Let $\{A,B,C,D,E,F,G,H\}......Wed, 12 Aug 2026 22:11:08 -0400https://pop.com.sg/how-are-huffman-encoding-and-entropy-related.htmlhttps://pop.com.sg/how-are-huffman-encoding-and-entropy-related.htmlThe inherent unpredictability, or randomness, of a probability distribution can be measured by the extent to which it is possible to compress data drawn from that distribution. $$ \text{more compre......Wed, 12 Aug 2026 22:10:05 -0400https://pop.com.sg/what-does-noether-039-s-normalization-lemma-even-mean.htmlhttps://pop.com.sg/what-does-noether-039-s-normalization-lemma-even-mean.htmlNoether's Normalization lemma states that For any field $k$, and any finitely generated commutative $k$-algebra $A$, there exists a non-negative integer $d$ and algebraically independent element......Wed, 12 Aug 2026 22:02:17 -0400https://pop.com.sg/any-line-intersects-the-parabola-twice-in-projective-space.htmlhttps://pop.com.sg/any-line-intersects-the-parabola-twice-in-projective-space.htmlGiven the parabola $y-x^2=0$ in $\mathbb{C}^2$, I'm trying to show that in $\mathbb{P}^2$, any line intersects the parabola twice (up to multiplicity). This is simple enough for the "finite" ...Wed, 12 Aug 2026 22:00:16 -0400https://pop.com.sg/adding-two-subspaces.htmlhttps://pop.com.sg/adding-two-subspaces.htmlI have two subspaces: $$W_1 = \{(x, 3x) : x\in \Bbb R \}$$ and $$W_2 = \{(2x, 0): x\in \Bbb R \}$$ How do I get $W_1 + W_2$? I tried simply adding a sample vector from each, i.e. $$ (1, 3) + ......Wed, 12 Aug 2026 21:59:07 -0400https://pop.com.sg/solve-log-3-3x-2-log-9-4x-5-for-x.htmlhttps://pop.com.sg/solve-log-3-3x-2-log-9-4x-5-for-x.htmlSolve for $x$ $$ \log_3(3x + 2) = \log_9(4x + 5) $$ I changed the bases of the logs $$ \frac {\log_{10}(3x + 2)} {\log_{10}(3)} = \frac {\log_{10}(4x + 5)} {\log_{10}(9)} $$ Now I'm stuck, ......Wed, 12 Aug 2026 21:54:47 -0400https://pop.com.sg/fun-linear-algebra-problems.htmlhttps://pop.com.sg/fun-linear-algebra-problems.htmlI'm teaching a linear algebra course this term (using Lay's book) and would like some fun "challenge problems" to give the students. The problems that I am looking for should be be easy to state an......Wed, 12 Aug 2026 21:53:10 -0400https://pop.com.sg/deriving-the-hessian-from-the-limit-definition-of-the-derivative.htmlhttps://pop.com.sg/deriving-the-hessian-from-the-limit-definition-of-the-derivative.htmlCould someone possibly help me understand how I can derive the Hessian matrix of a twice-differentiable function $f$ defined on $\mathbb{R}^n$ using the limit definition of the second derivative. N......Wed, 12 Aug 2026 21:51:10 -0400https://pop.com.sg/non-isomorphism-graph-closed.htmlhttps://pop.com.sg/non-isomorphism-graph-closed.htmlIm confused what is non isomorphism graph cant post image so i upload it on tinypic Particulary with this example It is said, that this c4 graph on left side is non isomorphism graph. But this is......Wed, 12 Aug 2026 21:47:58 -0400https://pop.com.sg/to-what-refers-the-width-in-a-gaussian-function.htmlhttps://pop.com.sg/to-what-refers-the-width-in-a-gaussian-function.htmlI've plotted a dataset in SciDAVis and added the default Gaussian fit. SciDAVis used the following function: $$ f(x) = y_0 + A \cdot \frac{\sqrt{\frac{2}{\pi}}}{w} \cdot \exp{\left(-2 \cdot \left(......Wed, 12 Aug 2026 21:46:50 -0400https://pop.com.sg/mathbb-z-includes-zero-or-not.htmlhttps://pop.com.sg/mathbb-z-includes-zero-or-not.htmlDoes $\mathbb{Z}^{+}$ includes zero or not? I think that $0$ is not involved in the set of positive integers, but my book included zero in the set of positive integers in an answer....Wed, 12 Aug 2026 21:43:42 -0400https://pop.com.sg/projecting-an-angle-from-one-plane-to-another-plane.htmlhttps://pop.com.sg/projecting-an-angle-from-one-plane-to-another-plane.htmlConsider I have two intersecting planes with an angle ($\theta$). I have two intersecting vectors ($\vec a$ and $\vec b$) on one of the planes that make an angle ($\gamma$). If I project these two ...Wed, 12 Aug 2026 21:43:29 -0400https://pop.com.sg/find-function-corresponding-to-a-taylor-series.htmlhttps://pop.com.sg/find-function-corresponding-to-a-taylor-series.htmlI'm trying to find a function corresponding to the following Taylor series: $\sum_{n=2}^\infty\frac{x^n}{n(n-1)}$. I found it to converge if $-1\leq x \leq 1$, but I'm not sure how to find the func......Wed, 12 Aug 2026 21:42:33 -0400https://pop.com.sg/what-is-modern-analysis-and-abstract-analysis-closed.htmlhttps://pop.com.sg/what-is-modern-analysis-and-abstract-analysis-closed.htmlMostly I encounter textbooks named Analysis, Mathematical Analysis,... But recently, i encountered a book named A Course In Modern Analysis. Please explain the difference between Modern Analysis a......Wed, 12 Aug 2026 21:39:00 -0400https://pop.com.sg/evaluating-a-surface-integral-iint-a-ds.htmlhttps://pop.com.sg/evaluating-a-surface-integral-iint-a-ds.htmlSuppose $A=6z\hat i+(2x+y)\hat j-x\hat k$ .Evaluate $$\iint A.dS$$ Over the entire surface S of the region bounded by the cylinder $x^2+z^2=9,x=0,y=0,z=0$ and $y=8$.I split it into three surface 1.......Wed, 12 Aug 2026 21:36:19 -0400https://pop.com.sg/meaning-of-the-sign-of-a-complex-number-closed.htmlhttps://pop.com.sg/meaning-of-the-sign-of-a-complex-number-closed.htmlWhat is the meaning and the practical uses of the sign (signum function) of a complex number $z$ defined as $\frac{z}{|z|}$? Does it also extend to quaternions?...Wed, 12 Aug 2026 21:29:48 -0400https://pop.com.sg/from-natural-log-to-log-base-10.htmlhttps://pop.com.sg/from-natural-log-to-log-base-10.htmlThe constraints of this question is related to a programming problem, but I must get the math right in order for it to be applied to code. The actual problem is I need a function that evaluates to ......Wed, 12 Aug 2026 21:22:31 -0400https://pop.com.sg/is-the-support-of-a-borel-measure-measured-the-same-as-the-whole-space.htmlhttps://pop.com.sg/is-the-support-of-a-borel-measure-measured-the-same-as-the-whole-space.htmlWikipedia says Let (X, T) be a topological space. Let μ be a measure on the Borel σ-algebra on X. Then the support (or spectrum) of μ is defined to be the set of all points x in X for which ......Wed, 12 Aug 2026 21:18:53 -0400https://pop.com.sg/probability-of-the-union-of-3-events.htmlhttps://pop.com.sg/probability-of-the-union-of-3-events.htmlI need some clarification for why the probability of the union of three events is equal to the right side in the following: $$P(E\cup F\cup G)=P(E)+P(F)+P(G)-P(E\cap F)-P(E\cap G)-P(F\cap G)+P(E\c......Wed, 12 Aug 2026 21:18:36 -0400https://pop.com.sg/expected-value-of-y-ax-b.htmlhttps://pop.com.sg/expected-value-of-y-ax-b.htmlGiven a normal random variable X with parameters $\mu$ and $\sigma^2$, find the $E(Y)$ of $Y=aX + b$. So I started with $E(Y)=E(aX+b)=\frac{1}{\sqrt(2\pi)\sigma}\int_{-\infty}^{\infty}(ax+b)^{{-(......Wed, 12 Aug 2026 21:17:34 -0400https://pop.com.sg/graph-theory-properties-of-even-graph.htmlhttps://pop.com.sg/graph-theory-properties-of-even-graph.htmlThe question is that: show that each even graph can be decomposed into edge-disjoint cycles. What I think is: when drawing such graphs (randomly), I can produce two sets X and Y with vertices, in ......Wed, 12 Aug 2026 21:15:06 -0400https://pop.com.sg/the-difference-between-affine-set-and-affine-hull.htmlhttps://pop.com.sg/the-difference-between-affine-set-and-affine-hull.htmlAccording to the definition of affine hull and affine set. $$aff [C] = [\theta_1x_1+...+\theta_nx_n|x_1,...x_n \in C, \theta_1+...+\theta_n=1] $$ The data in affine hull is also in affine set. And ......Wed, 12 Aug 2026 21:06:50 -0400https://pop.com.sg/global-stability-vs-global-asymptotic-stability.htmlhttps://pop.com.sg/global-stability-vs-global-asymptotic-stability.htmlAfter trying to get my head around this for an embarrassingly long time, I think I need some help... We defined local (lyapunov) stability and asymptotic stability the following way: An equilibrium......Wed, 12 Aug 2026 21:03:38 -0400https://pop.com.sg/difference-between-the-principle-of-addition-the-principle-of-multipli.htmlhttps://pop.com.sg/difference-between-the-principle-of-addition-the-principle-of-multipli.htmlAccording to my book: 1.The fundamental principle of counting is used to count the number of possible ways in which a task can be done without actually counting manually. 2.Under the fundamental ...Wed, 12 Aug 2026 21:02:29 -0400https://pop.com.sg/choose-2-numbers-from-100-numbers.htmlhttps://pop.com.sg/choose-2-numbers-from-100-numbers.htmlIf I want to choose a set of 2 positive integers less than 100, is that a permutation or a combination? I can't understand does the order matter in this case or it does not?...Wed, 12 Aug 2026 20:59:51 -0400https://pop.com.sg/forming-a-basis-of-p3-r-from-a-set-s.htmlhttps://pop.com.sg/forming-a-basis-of-p3-r-from-a-set-s.htmlI seem to have a good understanding of spanning sets and linear independence which then becomes essential for understanding basis, but I am unsure how all this works for the field of polynomials. ......Wed, 12 Aug 2026 20:43:46 -0400https://pop.com.sg/limit-proof-of-e-x-x-n-duplicate.htmlhttps://pop.com.sg/limit-proof-of-e-x-x-n-duplicate.htmlI am wondering how to prove $$\lim_{x\to \infty} \frac{e^x}{x^n}=\infty$$ I was thinking of using L'Hospital's rule? But then not sure how to do the summation for doing L'Hospital's rule n times o......Wed, 12 Aug 2026 20:40:07 -0400https://pop.com.sg/what-is-the-intuitive-meaning-of-the-adjugate-matrix.htmlhttps://pop.com.sg/what-is-the-intuitive-meaning-of-the-adjugate-matrix.htmlThe definition of the adjugate matrix is easy to understand, but I have never seen it used for anything. What is the intuitive meaning of this matrix? Are there examples of applications which may ......Wed, 12 Aug 2026 20:39:21 -0400https://pop.com.sg/n-th-derivative-of-tanh.htmlhttps://pop.com.sg/n-th-derivative-of-tanh.htmlSuccessive derivative of $\tanh u$ can be expressed as polynomial functions of $\tanh u$: \begin{align} \frac{d}{du}\tanh u&=1-\tanh^2u\\ \frac{d^2}{du^2}\tanh u&=-2\tanh u\left(1-\tanh^2u\......Wed, 12 Aug 2026 20:38:10 -0400https://pop.com.sg/which-methods-can-be-used-to-evaluate-the-following-integral.htmlhttps://pop.com.sg/which-methods-can-be-used-to-evaluate-the-following-integral.htmlHow can I evaluate the following integral $$ \int_{0}^{\infty} x^{-1/2} \exp({-x/2})\ dx $$ I know the answer is $\sqrt{2\pi}$....Wed, 12 Aug 2026 20:35:48 -0400https://pop.com.sg/quotient-group-z-3z-equated-to-0-1-2.htmlhttps://pop.com.sg/quotient-group-z-3z-equated-to-0-1-2.htmlThis a followup question on the first answer to this post: Why the term and the concept of quotient group? In the first answer, Lahtonen says that for the quotient group Z/10Z, one can "equate 9 ......Wed, 12 Aug 2026 20:33:55 -0400https://pop.com.sg/deriving-the-power-series-for-cosine-using-basic-geometry.htmlhttps://pop.com.sg/deriving-the-power-series-for-cosine-using-basic-geometry.htmlLooking for the derivation of cosine lead to https://www.quora.com/How-do-I-calculate-cos-sine-etc-without-a-calculator and the MacLauren series. $$\cos(x)=1−\frac{x^2}{2!}+\frac{x^4}{4!}−\frac{x^......Wed, 12 Aug 2026 20:26:47 -0400https://pop.com.sg/reflexive-transitive-closure.htmlhttps://pop.com.sg/reflexive-transitive-closure.htmlThe problem I am working on is, "Show that a finite poset can be reconstructed from its covering relation. [Hint:Show that the poset is the reflexive transitive closure of its covering relation.]" I ...Wed, 12 Aug 2026 20:16:28 -0400https://pop.com.sg/prove-that-there-are-an-infinite-number-of-discontinuities-on-this-fun.htmlhttps://pop.com.sg/prove-that-there-are-an-infinite-number-of-discontinuities-on-this-fun.htmlShow the function $f:[0,1]\rightarrow\mathbb{R}$ given by $$ f(x)= \begin{cases} 1,&x=\frac{1}{n}\text{ for any positive integer $n$}\\ 0,&\text{otherwise} \end{cases} $$ has an infinite ......Wed, 12 Aug 2026 20:10:24 -0400https://pop.com.sg/explain-why-this-relation-has-a-reflexive-symmetric-antisymmetric-and-.htmlhttps://pop.com.sg/explain-why-this-relation-has-a-reflexive-symmetric-antisymmetric-and-.htmlLet S = {1, 2, 3} Let R = {(1,1),(3,3),(2,2)} So the answer is that it is reflexive, symmetric, antisymmetric, and transitive. I understand that it is reflexive, however I do not understand how it......Wed, 12 Aug 2026 19:59:40 -0400https://pop.com.sg/area-and-circumference-of-a-square.htmlhttps://pop.com.sg/area-and-circumference-of-a-square.htmlI've encountered this weird question: Show that the derivative of the area of a circle with respect to its radius is equal to its circumference. Show that this relationship holds between the area and ...Wed, 12 Aug 2026 19:56:10 -0400https://pop.com.sg/questions-tagged-legendre-polynomials.htmlhttps://pop.com.sg/questions-tagged-legendre-polynomials.htmlQ&A for people studying math at any level and professionals in related fields...Wed, 12 Aug 2026 19:52:42 -0400https://pop.com.sg/is-it-possible-to-cut-a-circle-and-make-smaller-circle.htmlhttps://pop.com.sg/is-it-possible-to-cut-a-circle-and-make-smaller-circle.htmlRecently I saw this meme: Although the pizza hasn't completely the shape of a circle, let's say we have a circle is it possible to cut a part of the circle like this in the image and make smaller ......Wed, 12 Aug 2026 19:39:39 -0400https://pop.com.sg/how-to-calculate-the-joint-probability-from-two-normal-distributions.htmlhttps://pop.com.sg/how-to-calculate-the-joint-probability-from-two-normal-distributions.htmlI have two random variables $X$ and $Y$ both normally distributed as $N(\mu, \sigma^2)$ (they have the same distribution). $X$ and $Y$ are dependent. They are defined from other random variables A......Wed, 12 Aug 2026 19:28:07 -0400https://pop.com.sg/a-nice-way-to-do-euler-039-s-method-on-a-calculator.htmlhttps://pop.com.sg/a-nice-way-to-do-euler-039-s-method-on-a-calculator.htmlAs part of the calculator paper for IB (International Baccalaureate), we may be asked to do Euler's method, for say, 10 iterations. While it is feasible to do with a calculator (slightly easier if......Wed, 12 Aug 2026 19:23:45 -0400https://pop.com.sg/why-is-l-0-norm-not-convex-closed.htmlhttps://pop.com.sg/why-is-l-0-norm-not-convex-closed.htmlI have this confusion in understanding the convexity of the $L_0$ norm. Why is $L_0$ norm not convex?...Wed, 12 Aug 2026 19:15:36 -0400https://pop.com.sg/difference-between-joint-probability-distribution-and-conditional-prob.htmlhttps://pop.com.sg/difference-between-joint-probability-distribution-and-conditional-prob.htmlCan someone explain to me the difference between joint probability distribution and conditional probability distribution?...Wed, 12 Aug 2026 19:10:27 -0400https://pop.com.sg/will-division-by-zero-be-defined-eventually-duplicate.htmlhttps://pop.com.sg/will-division-by-zero-be-defined-eventually-duplicate.htmlPossible Duplicate: Division by $0$ I've always been inclined to believe that x/0 = NaN is a placeholder for a character or constant that no one has created yet. I know assume that none of y......Wed, 12 Aug 2026 19:07:09 -0400