https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usSat, 22 Aug 2026 18:37:13 -0400https://pop.com.sg/is-the-point-at-0-0-of-the-graph-z-x-3-y-3-considered-a-saddle-point.htmlhttps://pop.com.sg/is-the-point-at-0-0-of-the-graph-z-x-3-y-3-considered-a-saddle-point.htmlIs the point at $(0,0)$ of the graph $z=x^3 + y^3$ considered a saddle point? I was given that function, and I used the second-order derivative test only to find that the $Hf(0,0) = 0$, with the ...Sat, 22 Aug 2026 22:28:54 -0400https://pop.com.sg/monic-polynomials-clarification.htmlhttps://pop.com.sg/monic-polynomials-clarification.htmlSo I know a monic polynomial is one where the coefficient of the leading term is 1, but does that mean the constant 1 is a monic polynomial too? Thanks....Sat, 22 Aug 2026 22:28:45 -0400https://pop.com.sg/drawing-vector-fields-beginner.htmlhttps://pop.com.sg/drawing-vector-fields-beginner.htmlI have $f(x,y) = xy$ which has the Gradient Vector Field F$(x,y) = (y,x)$ I believe the correct way to draw individual vectors is to map points into the formula, for example at point $(1,-3)$ we w......Sat, 22 Aug 2026 22:27:03 -0400https://pop.com.sg/difference-in-meaning-between-converges-against-and-converge-to.htmlhttps://pop.com.sg/difference-in-meaning-between-converges-against-and-converge-to.htmlMight be a silly question, but is there any difference between saying that a something (limit, series and so on) converges to a value and converges against a value?...Sat, 22 Aug 2026 22:24:23 -0400https://pop.com.sg/what-is-the-notation-for-the-empty-matrix.htmlhttps://pop.com.sg/what-is-the-notation-for-the-empty-matrix.htmlMy question: What is a notation for an empty 0x0 matrix (i.e. the matrix for the only linear map $f:\{0\}\to\{0\}$)? Is it written $()$? How can I distinguish the 0x0 matrix with for example the 0x3 ...Sat, 22 Aug 2026 22:23:39 -0400https://pop.com.sg/how-to-find-tangent-line-at-a-given-point-without-equation.htmlhttps://pop.com.sg/how-to-find-tangent-line-at-a-given-point-without-equation.htmlFind the equation of the line that is tangent to the curve at the point $(0,\sqrt{\frac{\pi}{2}})$. Given your answer in slope-intercept form. I don't know how can I get the tangent line, without a ...Sat, 22 Aug 2026 22:21:39 -0400https://pop.com.sg/special-galois-connections.htmlhttps://pop.com.sg/special-galois-connections.htmlLet $(f;g)$ is a Galois connection between two posets. Consider the special case $g\circ f = \operatorname{id}_{\operatorname{dom} f}$. What can be said about this special case of Galois connecti......Sat, 22 Aug 2026 22:19:19 -0400https://pop.com.sg/how-to-find-the-average-rate-of-change-velocity.htmlhttps://pop.com.sg/how-to-find-the-average-rate-of-change-velocity.htmlA problem asked this: The rocket club wants to model rockets straight up into the air. The height of rocket A can being modeled by the function $y=-16x^2+88x$. The function that models the heigh......Sat, 22 Aug 2026 22:16:51 -0400https://pop.com.sg/the-comparison-test-or-the-limit-comparison-test.htmlhttps://pop.com.sg/the-comparison-test-or-the-limit-comparison-test.htmlI have checked many of this site's pages yet I could not find a clear answer about how to choose between the "comparison test" OR the "limit comparison test". Because the difference between the tw......Sat, 22 Aug 2026 22:16:46 -0400https://pop.com.sg/what-is-a-bounty-how-can-i-start-one.htmlhttps://pop.com.sg/what-is-a-bounty-how-can-i-start-one.htmlQ&A for people studying math at any level and professionals in related fields...Sat, 22 Aug 2026 22:16:34 -0400https://pop.com.sg/solution-of-cubic-equations-in-terms-of-quadratic-equations.htmlhttps://pop.com.sg/solution-of-cubic-equations-in-terms-of-quadratic-equations.htmlIs there a general way in which a cubic equation of $3$rd degree can be represented by a quadratic equation of $2$nd degree such that $2$ solutions of a cubic equation is equal to $2$ solutions of ...Sat, 22 Aug 2026 22:15:57 -0400https://pop.com.sg/finding-the-inverse-distribution-for-a-standard-cauchy-distribution-is.htmlhttps://pop.com.sg/finding-the-inverse-distribution-for-a-standard-cauchy-distribution-is.htmlThe standard cauchy distribution was given as follows $f_X(x) = \frac{1}{\pi} \frac{1}{1+x^2} \;\;\;\;\; x\in \mathbb{R}$ I need to find the distribution for $Y= \frac{1}{X}$ and show it is the sam......Sat, 22 Aug 2026 22:14:43 -0400https://pop.com.sg/trig-identity-pythagorean-theorem-confusion.htmlhttps://pop.com.sg/trig-identity-pythagorean-theorem-confusion.htmlI run into a problem when I'm trying to prove how $\tan^2x+1 = \sec^2x$, and $1+\cot^2x=\csc^2x$ I understand that $\sin^2x+\cos^2x = 1$. (To my understanding 1 is the Hypotenuse, please correct m......Sat, 22 Aug 2026 22:14:13 -0400https://pop.com.sg/definitions-of-direct-product-and-of-direct-sum.htmlhttps://pop.com.sg/definitions-of-direct-product-and-of-direct-sum.htmlI was wondering if there are some general definitions for direct product and for direct sum, for example in category theory or in set theory, so that the concepts for vector spaces, Abelian groups,......Sat, 22 Aug 2026 21:45:43 -0400https://pop.com.sg/can-i-factorize-x-4-27x-without-using-the-factor-theorem.htmlhttps://pop.com.sg/can-i-factorize-x-4-27x-without-using-the-factor-theorem.htmlI am doing some self-assigned homework questions. One question asks me to factorize $x^4 + 27x$. I used the factor theorem to solve it. I observed that $f(0) = 0$ and $f(-3) = 0$, so $x$ and $(x + ......Sat, 22 Aug 2026 21:39:40 -0400https://pop.com.sg/find-general-solution-of-first-order-de.htmlhttps://pop.com.sg/find-general-solution-of-first-order-de.htmlFind general solution of $$y'+\frac{2y}{x}=\frac{3}{x^2}$$ and find solution for which $$y(2)=-1$$ I know $$xy'+\frac{r(x)y}{x}=3r(x)x$$ $$(ry')=ry'+r'y$$$$r'=\frac{r}{x},\frac{r'}{r}=\......Sat, 22 Aug 2026 21:39:02 -0400https://pop.com.sg/endomorphisms-of-a-finite-dimensional-vector-space.htmlhttps://pop.com.sg/endomorphisms-of-a-finite-dimensional-vector-space.htmlFrom Humphreys' Introduction to Lie Algebras and Representation Theory: If $V$ is a finite dimensional vector space over $F$, denote by $\text{End }V$ the set of linear transformations $V\righta......Sat, 22 Aug 2026 21:35:55 -0400https://pop.com.sg/determine-the-value-of-n.htmlhttps://pop.com.sg/determine-the-value-of-n.htmlI'm a complete n00b at math, but I'm wondering how one would go about determining the value of n in the following comparison. n * 1.5 + 12.5 = 12.5 / 2 + n I'm new to the math StackExchange, so I'm......Sat, 22 Aug 2026 21:34:46 -0400https://pop.com.sg/find-the-distribution-of-linear-combination-of-independent-random-vari.htmlhttps://pop.com.sg/find-the-distribution-of-linear-combination-of-independent-random-vari.htmlGiven independent and identically distributed random variables $X_1, X_2, \dots, X_n$, each of them has the same p.d.f $f(x) = Pr(X = x)$ on support $(a, b)$. How do I find the pdf or cdf of $Y = ......Sat, 22 Aug 2026 21:31:25 -0400https://pop.com.sg/how-many-algebras-are-there-of-finite-sized-omega.htmlhttps://pop.com.sg/how-many-algebras-are-there-of-finite-sized-omega.htmlAn algebra of $\Omega$ is a family that contains $\Omega,$ is closed under complement and finite union. If $\Omega=\{\}$, there can only be 1 algebra associated with $\Omega$: $2^{\Omega}.$ If $\......Sat, 22 Aug 2026 21:26:21 -0400https://pop.com.sg/y-2-frac-1-x-2-stretched-vertically-by-a-factor-or-3.htmlhttps://pop.com.sg/y-2-frac-1-x-2-stretched-vertically-by-a-factor-or-3.htmlAlright folks, having a little problem with this calculus question. anyone help? The factor and direction of the given function is to be stretched or compressed are given below. Give an equation f......Sat, 22 Aug 2026 21:15:35 -0400https://pop.com.sg/user-matthew-h-mathematics-stack-exchange.htmlhttps://pop.com.sg/user-matthew-h-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Sat, 22 Aug 2026 21:13:18 -0400https://pop.com.sg/compact-sets-are-closed.htmlhttps://pop.com.sg/compact-sets-are-closed.htmlI feel really ignorant in asking this question but I am really just don't understand how a compact set can be considered closed. By definition of a compact set it means that given an open cover......Sat, 22 Aug 2026 21:12:01 -0400https://pop.com.sg/lie-algebra-of-u-n-and-so-n.htmlhttps://pop.com.sg/lie-algebra-of-u-n-and-so-n.htmlU(N) and SO(N) are quite important groups in physics. I thought I would find this with an easy google search. Apparently NOT! What is the Lie algebra and Lie bracket of the two groups?...Sat, 22 Aug 2026 21:08:05 -0400https://pop.com.sg/is-there-an-easy-way-to-factor-polynomials-with-two-variables.htmlhttps://pop.com.sg/is-there-an-easy-way-to-factor-polynomials-with-two-variables.htmlOn a recent precal test, I saw a question involving the following expression: $$(x+1)^2-y^2$$ Which factored out into: $$(x+y+1)(x-y+1)$$ This wasn't very hard, considering that it was already ...Sat, 22 Aug 2026 21:07:18 -0400https://pop.com.sg/positive-slope-with-negative-values.htmlhttps://pop.com.sg/positive-slope-with-negative-values.htmlWhen I was in school I learned that slope could be found with the following: $$m=\frac{y_2-y_1}{x_2-x_1}$$ When given the coordinates $(-3, 0)$ and $(0, 5)$, plotting these points on a Cartesian pl......Sat, 22 Aug 2026 21:05:10 -0400https://pop.com.sg/how-to-rotate-the-positions-of-a-matrix-by-90-degrees.htmlhttps://pop.com.sg/how-to-rotate-the-positions-of-a-matrix-by-90-degrees.htmlI have a 5x5 matrix of values. I'm looking for a simple formula that I can use to rotate the position of the values (not the values themselves) 90 degrees within the matrix. For example, here is the ...Sat, 22 Aug 2026 20:47:36 -0400https://pop.com.sg/calculating-expected-values-of-roulette-bets.htmlhttps://pop.com.sg/calculating-expected-values-of-roulette-bets.htmlA roulette wheel has 38 numbers, 1 through 36, 0 and 00. One-half of the numbers from 1-36 are red, and the other half are black; 0 and 00 are green. A ball is rolled, and it falls into one of th......Sat, 22 Aug 2026 20:43:02 -0400https://pop.com.sg/multiplication-in-the-galois-field-gf-3-3.htmlhttps://pop.com.sg/multiplication-in-the-galois-field-gf-3-3.htmlI am trying to compute $x^3$ in the Galois field $\text{GF}(3^3)$ using the irreducible polynomial $f(x) = x^3 + 2x^2 + 1$. From the expression $x^3 = f(x) + (2x^2 +1)$ I proceed to take the modul......Sat, 22 Aug 2026 20:39:00 -0400https://pop.com.sg/a-question-about-the-definition-of-markov-measure.htmlhttps://pop.com.sg/a-question-about-the-definition-of-markov-measure.htmlI have one question about this definition Why is it enough to note equation (3.30) to see that the measure is well defined? is it already additive so you can put the measure that way? if not, what......Sat, 22 Aug 2026 20:32:08 -0400https://pop.com.sg/fourier-transform-of-sinc-function.htmlhttps://pop.com.sg/fourier-transform-of-sinc-function.htmllet us consider fourier transform of sinc function,as i know it is equal to rectangular function in frequency domain and i want to get it myself,i know there is a lot of material about this,but i w......Sat, 22 Aug 2026 20:22:30 -0400https://pop.com.sg/evaluating-the-limit-of-lim-x-to-9-frac-x-9-sqrt-x-3.htmlhttps://pop.com.sg/evaluating-the-limit-of-lim-x-to-9-frac-x-9-sqrt-x-3.htmlHow do you evaluate the limit of $$\lim_{x \to 9} \frac {x-9} {\sqrt{x} - 3}$$ by first principle? I was thinking of this way to evaluate but it seems to evaluate to $\frac {0} {0}$. $$\lim_{x \......Sat, 22 Aug 2026 20:20:17 -0400https://pop.com.sg/showing-a-binary-operation-is-well-defined.htmlhttps://pop.com.sg/showing-a-binary-operation-is-well-defined.htmlLet $G=\{x\in\mathbb{R}~|~0\le x<1\}$ and define $\star$ as follows $$x\star y=x+y-[x+y]$$ I want to show this is a well defined operation. I think it is this easy; If $x=x', y=y'$, then $$......Sat, 22 Aug 2026 20:16:48 -0400https://pop.com.sg/what-is-the-number-of-dimensions-of-a-scalar.htmlhttps://pop.com.sg/what-is-the-number-of-dimensions-of-a-scalar.htmlAm I correct that a matrix has two dimensions, and a vector has one dimension? What is the number of dimensions of a scalar? zero? Thanks....Sat, 22 Aug 2026 20:09:18 -0400https://pop.com.sg/definition-of-the-approx-symbol.htmlhttps://pop.com.sg/definition-of-the-approx-symbol.htmlTake an unending number, say e.g $π$. If we want to show $π$'s value, should we use the approximately notation or equal sign when writing: $π = 3.14...$ or $π ≈ 3.14...$ This might be a really s......Sat, 22 Aug 2026 20:06:29 -0400https://pop.com.sg/find-the-dual-of-a-linear-programming-problem.htmlhttps://pop.com.sg/find-the-dual-of-a-linear-programming-problem.htmlI have a very simple linear programming problem with the following constraints: Minimize $3x_1 + 2x_2 - 33x_3$ subject to $x_1 - 4x_2 + x_3 \leq 15$ $9x_1 + 6x_3 \leq 12$ $5x_1 + 9x_2 \geq 3$ ......Sat, 22 Aug 2026 19:58:27 -0400https://pop.com.sg/what-is-the-average-roll-on-dice-while-re-rolling-a-result-of-1.htmlhttps://pop.com.sg/what-is-the-average-roll-on-dice-while-re-rolling-a-result-of-1.htmlThe average roll on a six sided die (d6) is 3.5. If the result is a 1 you are allowed a single re-roll accepting the second result. What does the average roll increase to? And how to calculate this? ...Sat, 22 Aug 2026 19:55:25 -0400https://pop.com.sg/derivative-of-x-x-2.htmlhttps://pop.com.sg/derivative-of-x-x-2.htmlStudying past exam problems for my exam in ~$4$ weeks, and I came across this derivative as one of the questions. I actually have no idea how to solve it. $$\frac{d}{dx} (x^{x^2})$$ Using the chain ...Sat, 22 Aug 2026 19:54:35 -0400https://pop.com.sg/finding-maclaurin-series-of-integral.htmlhttps://pop.com.sg/finding-maclaurin-series-of-integral.htmlI wish to find the Maclaurin-series for the function $$L(x) = \int_0^x \cos(t^2)\space dt$$ Alternative 1 Set $f(t) = \cos(t)$. We have that $f'(t) = -\sin(t)$, $f''(t) = -\cos(t)$ and $f^{(3)}(t)......Sat, 22 Aug 2026 19:47:02 -0400https://pop.com.sg/two-definitions-of-bounded-variation-function.htmlhttps://pop.com.sg/two-definitions-of-bounded-variation-function.htmlAs far as I know, a function $f$ defined on an interval $[a, b]$ is said to be of bounded variation if $$\tag{1}V_a^b(f)=\sup\left\{\sum_{P} \lvert f(x_{j+1})-f(x_j)\rvert \ :\ P\ \text{partition ......Sat, 22 Aug 2026 19:32:16 -0400https://pop.com.sg/find-the-equivalent-value-of-radians.htmlhttps://pop.com.sg/find-the-equivalent-value-of-radians.htmlA function $f$ is called periodic with period $T$ if $f (x + T ) = f (x)$ (for all x) and $T$ is the smallest positive number with this property. The sine and cosine functions are periodic with per......Sat, 22 Aug 2026 19:28:49 -0400https://pop.com.sg/linear-algebra-how-to-interpret-vectors-and-equations-of-a-system-of-e.htmlhttps://pop.com.sg/linear-algebra-how-to-interpret-vectors-and-equations-of-a-system-of-e.htmlFrom Strang's 18.06 Linear Algebra OCW Course I realized that there is a row view and column view to interpret a matrix and the solution to Ax = b. Well the view made sense but it was hard understa......Sat, 22 Aug 2026 19:19:31 -0400https://pop.com.sg/what-does-ln-ln-x-simplify-to.htmlhttps://pop.com.sg/what-does-ln-ln-x-simplify-to.htmlWhat does $\ln(\ln(x))$ simplify to? First instinct was to write $\ln^2x$, but that looked odd, because $\ln(\ln(x))$ is not the same as $\ln(x)\ln(x)$ or $(\ln(x))^2$. Or is it ?...Sat, 22 Aug 2026 19:12:35 -0400https://pop.com.sg/ax-by-c-c-is-always-0.htmlhttps://pop.com.sg/ax-by-c-c-is-always-0.htmlFind an equation for the given line in the form $ax +by =​c$, where​ a, b, and c are integers with no factor common to all three and a ≥0. Through $(-40, 35)$; parallel to $7x + 8y = 13$ I put th......Sat, 22 Aug 2026 19:10:08 -0400https://pop.com.sg/suppose-you-draw-two-cards-from-a-deck-of-52-cards-without-replacement.htmlhttps://pop.com.sg/suppose-you-draw-two-cards-from-a-deck-of-52-cards-without-replacement.htmlSuppose you draw two cards from a deck of 52 cards without replacement. 1) What is the probability that both cards are hearts? 2) What is the probability that exactly one of the cards is hearts......Sat, 22 Aug 2026 19:07:39 -0400https://pop.com.sg/if-x-4-x-4-then-x-3-4-x-3-is.htmlhttps://pop.com.sg/if-x-4-x-4-then-x-3-4-x-3-is.htmlIf $x+(4/x)=4$ then $x^3+(4/x)^3 $is. Note: $(x+(4/x))^3=4^3$ assuming $4/x=y$ then $(x+y)^3=64$ $x^3+y^3+3xy^2+3yx^2=64$ $x^3+y^3+3xy(x+y)=64$ $x^3+y^3+3*4(4)=64$ so $x^3+y^3=16$...Sat, 22 Aug 2026 19:04:04 -0400https://pop.com.sg/how-to-prove-cos-4x-sin-4x-cos-2x-sin-2x-is-always-0.htmlhttps://pop.com.sg/how-to-prove-cos-4x-sin-4x-cos-2x-sin-2x-is-always-0.htmlSo I have a small problem here where I have to prove the following : $$\cos^4x - \sin^4x - \cos^2x + \sin^2x = 0 $$ I know that the 2nd part is always $1$, so I need to prove that the first par......Sat, 22 Aug 2026 19:01:16 -0400https://pop.com.sg/norm-of-finite-flat-morphism.htmlhttps://pop.com.sg/norm-of-finite-flat-morphism.htmlLet $X,Y$ be two schemes and $f : X \to Y$ be a finite flat map. In the following document http://www.ms.u-tokyo.ac.jp/~t-saito/talk/eepr.pdf page 20, the author talks about the norm map $$f_* : f_*\...Sat, 22 Aug 2026 18:57:45 -0400https://pop.com.sg/if-x-is-the-remainder-when-a-multiple-of-4-is-divided-by-6-and-y-is-th.htmlhttps://pop.com.sg/if-x-is-the-remainder-when-a-multiple-of-4-is-divided-by-6-and-y-is-th.htmlThe question is: if $x$ is the remainder when a multiple of $4$ is divided by $6$, and $y$ is the remainder when a multiple of $2$ is divided by $3$, what is the greatest possible value of $x+y$? ......Sat, 22 Aug 2026 18:43:45 -0400https://pop.com.sg/predator-prey-dynamical-system-find-the-vector-x-k-for-a-certain-k.htmlhttps://pop.com.sg/predator-prey-dynamical-system-find-the-vector-x-k-for-a-certain-k.htmlConsider a predator-pray model with the matrix $A= \begin{bmatrix} 0.5 & 0.6\\ -0.2 & 1.4 \end{bmatrix}$ ($A$ is the population distribution) and the initial state is $\vec{x}_0 = \begin{...Sat, 22 Aug 2026 18:29:06 -0400https://pop.com.sg/linear-regression-ssr-equation.htmlhttps://pop.com.sg/linear-regression-ssr-equation.htmlI've studied SST and the others but I saw the equation that $SSR=(n-1)(β_1)^2(S_x)^2$. I want to know how the equation is done. Thnks!!...Sat, 22 Aug 2026 18:28:40 -0400https://pop.com.sg/definition-of-an-axiom.htmlhttps://pop.com.sg/definition-of-an-axiom.htmlWhat would be some convenient and general enough definition of an axiom? If we go through mathematical theories we can observe that axioms either demand that something exists, or tell us how some ...Sat, 22 Aug 2026 18:17:22 -0400https://pop.com.sg/what-about-simon-donaldson-039-s-riemann-surfaces.htmlhttps://pop.com.sg/what-about-simon-donaldson-039-s-riemann-surfaces.htmlI am studying Riemann surfaces now. How about Donaldson's book "Riemann surfaces"? Could you recommend some references, and point out the required mathematical knowledge? Many thanks in advance!...Sat, 22 Aug 2026 18:16:10 -0400https://pop.com.sg/normal-operators-self-adjoint.htmlhttps://pop.com.sg/normal-operators-self-adjoint.htmlThis is one of my homework questions and I was hoping someone could check my proof: Prove that a normal operator with real eigenvalues is self-adjoint. So what I thought to do was, for some normal ...Sat, 22 Aug 2026 18:06:25 -0400https://pop.com.sg/how-do-i-find-the-base-and-height-of-a-triangle-when-given-only-the-hy.htmlhttps://pop.com.sg/how-do-i-find-the-base-and-height-of-a-triangle-when-given-only-the-hy.htmlMy hypotenuse is 1 and the area is .125... can someone find out how to get the base + height when just given the hypotenuse and area of a triangle? I know finding the area of a triangle is 1/2base(......Sat, 22 Aug 2026 17:54:58 -0400https://pop.com.sg/difference-between-each-every-any-etc.htmlhttps://pop.com.sg/difference-between-each-every-any-etc.htmlI'm having a bit of trouble understanding the use of these different phrases. Are there any certain rules for when which of the word is to be used? I feel like they overlap a lot, but can bear slig......Sat, 22 Aug 2026 17:47:06 -0400https://pop.com.sg/multiplicative-inverse-of-modulo-n-is-unique.htmlhttps://pop.com.sg/multiplicative-inverse-of-modulo-n-is-unique.htmlI am trying to prove this: $x,b,n\in \mathbb{Z}, \space xb\;(\mod{n}\;) = 1 \implies $ b is unique. I have tried by using proof by contradiction. That is, assume there are integers $b,c$ where the ...Sat, 22 Aug 2026 17:35:30 -0400https://pop.com.sg/calculating-the-average-speed-using-only-velocities.htmlhttps://pop.com.sg/calculating-the-average-speed-using-only-velocities.htmlFor the following question: Pedro travels by bus to school at an average speed of $40$ km/hr. He is driven home by the same route by a friend's car at an average speed of $50$ km/hr. Which ......Sat, 22 Aug 2026 17:30:42 -0400https://pop.com.sg/definition-of-a-compact-operator.htmlhttps://pop.com.sg/definition-of-a-compact-operator.htmlOperator compactness is characterized by maps the send the unit ball to relatively compact sets. Does anyone have a good justification for why we call this property compactness? The best justifica......Sat, 22 Aug 2026 17:28:15 -0400https://pop.com.sg/how-to-solve-x-times-0-n-where-n-does-not-equal-0.htmlhttps://pop.com.sg/how-to-solve-x-times-0-n-where-n-does-not-equal-0.htmlSo recently I was watching some videos on complex numbers. I was surprised to learn the uses of $i$ in different problems. For example, I never thought it would make sense that $x^2 + 1 = 0$ has any ...Sat, 22 Aug 2026 17:27:21 -0400https://pop.com.sg/proper-notation-for-distinct-sets.htmlhttps://pop.com.sg/proper-notation-for-distinct-sets.htmlConsider I have two sets that neither one is a subset of the other (for example, the set of prime numbers and the set of odd integers). Is there a specific notation that combines the meanings of bo......Sat, 22 Aug 2026 17:24:47 -0400https://pop.com.sg/when-applying-the-master-theorem-how-to-decide-on-case-1-2-or-3.htmlhttps://pop.com.sg/when-applying-the-master-theorem-how-to-decide-on-case-1-2-or-3.htmlWhen you are given a relation, such as $$T(n)=8T\Big(\frac{n}{2}\Big)+160n,$$ what are the logical steps to consider when deciding whether to apply case 1, 2, or 3? Or do you have to apply all 3 and ...Sat, 22 Aug 2026 17:17:14 -0400https://pop.com.sg/proper-definition-of-a-proper-map.htmlhttps://pop.com.sg/proper-definition-of-a-proper-map.htmlWhat is the proper definition of a proper map? Here are a few definitions I've come across: A continuous map $f: X \to Y$ is proper if preimages of compact subsets are compact. A continuous map......Sat, 22 Aug 2026 17:13:16 -0400https://pop.com.sg/is-aa-t-a-positive-definite-symmetric-matrix.htmlhttps://pop.com.sg/is-aa-t-a-positive-definite-symmetric-matrix.htmlLet $A\in M_{n\times n}(\mathbb{R})$ be a matrix. Is it true that $AA^{T}$ is positive-definite? Clearly $AA^{T}$ is symmetric. I have shown that a symmetric matrix $S\in M_{n\times n}(\mathbb{R})......Sat, 22 Aug 2026 17:06:25 -0400https://pop.com.sg/dot-product-is-an-euclidean-norm-in-euclidean-space.htmlhttps://pop.com.sg/dot-product-is-an-euclidean-norm-in-euclidean-space.htmlI might be missing something, but from what I have learned in linear algebra, for a matrix or vector multiplication to be valid you need to have number of columns in pre-multiplying vector to be th......Sat, 22 Aug 2026 16:59:08 -0400https://pop.com.sg/definition-simple-stochastic-process.htmlhttps://pop.com.sg/definition-simple-stochastic-process.htmlCan anybody explain what concretely is a simple (stochastic) process? (...and yes I tried to google it but without success)...Sat, 22 Aug 2026 16:57:55 -0400https://pop.com.sg/what-is-graphical-addition-closed.htmlhttps://pop.com.sg/what-is-graphical-addition-closed.html"Use the graphs of f and g and the method of graphical addition to sketch the graph of $f + g$". $f(x) = x, g(x) = \frac{1}{x}$...Sat, 22 Aug 2026 16:56:59 -0400https://pop.com.sg/finding-formula-for-a-cubic-polynomial.htmlhttps://pop.com.sg/finding-formula-for-a-cubic-polynomial.htmlSo I need to find a formula for a cubic polynomial with the given conditions: Local max at $x=1$ Local min at $x=3$ $Y$ intercept of $5$ $x^3$ term whose coefficient is $1$ So all I really have i......Sat, 22 Aug 2026 16:55:07 -0400https://pop.com.sg/how-can-i-calculate-the-running-time-of-my-compressor-knowing-the-cfm-.htmlhttps://pop.com.sg/how-can-i-calculate-the-running-time-of-my-compressor-knowing-the-cfm-.htmlSo, I actually have a real world application for the things my calc teacher tried to shove into my head, unfortunately it's been ten years since I took the class, and it all leaked out. That being......Sat, 22 Aug 2026 16:44:23 -0400https://pop.com.sg/is-borel-field-different-from-sigma-field.htmlhttps://pop.com.sg/is-borel-field-different-from-sigma-field.htmlMy mathematical statistics book denotes $\sigma$-field as following: Let $\Bbb B$ be the collection of subsets of $\Bbb C$ where $\Bbb C$ denotes sample space which is the collection of all possible ...Sat, 22 Aug 2026 16:39:35 -0400https://pop.com.sg/consider-the-following-system-of-linear-equations.htmlhttps://pop.com.sg/consider-the-following-system-of-linear-equations.htmlStory cut short, I have an exam in a weeks time and this is a question off a previous exam paper - I'm unsure as to how I should go about it as there are 4 variables with only three linear equation......Sat, 22 Aug 2026 16:36:15 -0400https://pop.com.sg/application-of-double-expectation-theorem.htmlhttps://pop.com.sg/application-of-double-expectation-theorem.htmlA component in system and a backup unit both have mean lifetime $\mu$. if component fails then it is in ediately replaced by backup, and there is probability $p$ that backup works. Find the expected ...Sat, 22 Aug 2026 16:34:11 -0400https://pop.com.sg/what-shape-is-a-bean.htmlhttps://pop.com.sg/what-shape-is-a-bean.htmlHow can I mathematically describe the shape of an idealised bean? (In two dimensions and in threes dimensions) At the moment I'm calling the shape I refer to an ellipse/ellipsoid on a curved major......Sat, 22 Aug 2026 16:33:49 -0400https://pop.com.sg/chain-rule-and-vector-valued-functions.htmlhttps://pop.com.sg/chain-rule-and-vector-valued-functions.htmlLet $f: R^n \to R$ be given by $f(x) = \frac{||x||^4} {1 + ||x||^2}$ . Use the chain rule to show that $f$ is differentiable at each $x \in R^n$ and compute $Df(x)$. This vector valued stuff just......Sat, 22 Aug 2026 16:31:49 -0400https://pop.com.sg/what-is-the-derivative-of-digamma-function.htmlhttps://pop.com.sg/what-is-the-derivative-of-digamma-function.htmlI want to know $(\frac{\Gamma'(\alpha)}{\Gamma(\alpha)})'$. My goal is to show $\alpha $ times this derivative of digamma is greater than 1. The background of question is to show $\bar{x}$ is not ...Sat, 22 Aug 2026 16:28:54 -0400https://pop.com.sg/is-a-spinor-an-element-of-the-spin-group-or-an-object-that-transforms-.htmlhttps://pop.com.sg/is-a-spinor-an-element-of-the-spin-group-or-an-object-that-transforms-.htmlI've been researching spinors, and I'm a bit confused by some of the terminology. In some cases, spinors seem to be presented as elements of the Spin group, whereas in others they seem to be presen......Sat, 22 Aug 2026 16:18:09 -0400https://pop.com.sg/determine-if-a-region-is-horizontally-or-vertically-simple.htmlhttps://pop.com.sg/determine-if-a-region-is-horizontally-or-vertically-simple.htmlI have an assignment problem which asks me to decide if the regions provided are horizontally or vertically simple. I am stuck on one of these which is the following: $$ A = \{(x,y)\in \Bbb R^2 | ......Sat, 22 Aug 2026 16:14:00 -0400https://pop.com.sg/little-oh-notation.htmlhttps://pop.com.sg/little-oh-notation.htmlAssume I have a function $ f:\mathbb{R}^{m}\to\mathbb{R}^n $, and let $ h\in \mathbb{R}^m $. Does the notation $ f\left(x_0+h\right)=o\left(h\right) $ (for fixed $x_0 $) mean that $$ \lim_{h\to0}\f......Sat, 22 Aug 2026 16:06:26 -0400https://pop.com.sg/what-are-the-advantages-of-studying-the-dual-problem-in-linear-program.htmlhttps://pop.com.sg/what-are-the-advantages-of-studying-the-dual-problem-in-linear-program.htmlI am studying linear programming and I came across primal-dual algorithm in Linear Programming. I understood it but I am unable to understand why there is a need to calculate a dual, if problem can......Sat, 22 Aug 2026 15:59:37 -0400https://pop.com.sg/1-2-5-12-27-what-will-be-the-formula-to-calculate-nth-term-for-this-se.htmlhttps://pop.com.sg/1-2-5-12-27-what-will-be-the-formula-to-calculate-nth-term-for-this-se.htmlHere, the difference of the digits at level 2 are in G.P., so is there any generalised formulae for calculating nth term if difference of digits at any level are in G.P....Sat, 22 Aug 2026 15:34:53 -0400https://pop.com.sg/convergence-of-student-039-s-t-distribution-to-a-standard-normal.htmlhttps://pop.com.sg/convergence-of-student-039-s-t-distribution-to-a-standard-normal.htmlI was looking at this question where it is shown that a Student's t-distribution converges to a standard normal distribution as the degrees of freedom tend to infinity.We start with the Student's t-...Sat, 22 Aug 2026 15:32:03 -0400https://pop.com.sg/number-of-4-digit-numbers-with-no-repeated-digit.htmlhttps://pop.com.sg/number-of-4-digit-numbers-with-no-repeated-digit.htmlNumber of $4$ digit numbers with no repeated digit is $4536$ $3024$ $5040$ $4823$ Well, I am very much weak in combinatorics. Please help....Sat, 22 Aug 2026 15:30:35 -0400https://pop.com.sg/calculate-order-of-magnitude-difference.htmlhttps://pop.com.sg/calculate-order-of-magnitude-difference.htmlSay if I have two Areas of size a = 6 km$^2$ and b = 0.1 km$^2$ how would I find how many orders of magnitude one is greater or less than the other. So, the way I would work this is by saying th......Sat, 22 Aug 2026 15:19:37 -0400https://pop.com.sg/user-eric-auld-mathematics-stack-exchange.htmlhttps://pop.com.sg/user-eric-auld-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Sat, 22 Aug 2026 15:13:08 -0400https://pop.com.sg/what-is-the-difference-between-a-sequence-of-functions-f-n-and-a-seque.htmlhttps://pop.com.sg/what-is-the-difference-between-a-sequence-of-functions-f-n-and-a-seque.htmlWhat is the difference between a sequence of functions $(f_n)$ and a sequence of functions $f_n(x)$? I am reading my textbook on analysis, and it seems to use 'sequence of functions' to describe bo......Sat, 22 Aug 2026 15:12:00 -0400https://pop.com.sg/number-with-finite-binary-representation-and-infinite-decimal-represen.htmlhttps://pop.com.sg/number-with-finite-binary-representation-and-infinite-decimal-represen.htmlOne can easily find numbers with finite decimal representation with infinite binary representation. (Like $0.3$ and $0.01010101..$) I assume there is an opposite case, meaning a number with finite ...Sat, 22 Aug 2026 15:11:27 -0400https://pop.com.sg/correct-interpretation-of-exponential-decay-and-decay-factor.htmlhttps://pop.com.sg/correct-interpretation-of-exponential-decay-and-decay-factor.htmlmy question is about exponential decay and its factor. English isn't my native language and therefore I'm not sure about the precise definition in my particular case. I'm reading a specific paper......Sat, 22 Aug 2026 15:10:29 -0400https://pop.com.sg/what-is-the-correct-way-to-self-learn-from-a-textbook.htmlhttps://pop.com.sg/what-is-the-correct-way-to-self-learn-from-a-textbook.htmlThey taught me just to calculate, and not to think or ask. So now I just regurgitate, and bullsh*t just to pass. They taught me how to integrate, but not the reasons why. As well, I learned to der......Sat, 22 Aug 2026 14:53:09 -0400https://pop.com.sg/is-there-a-common-symbol-for-concatenating-two-finite-sequences.htmlhttps://pop.com.sg/is-there-a-common-symbol-for-concatenating-two-finite-sequences.htmlSay we have two finite sequences $X = (x_0,...,x_n)$ and $Y = (y_0,...,y_n)$. Is there a more or less common notation for the concatenation of these sequences, like $\sum (X,Y) = (x_0,...,x_n,y_0,.......Sat, 22 Aug 2026 14:52:54 -0400https://pop.com.sg/how-would-one-evaluate-int-frac-x-sin-x-x-2-1-over-the-real-line.htmlhttps://pop.com.sg/how-would-one-evaluate-int-frac-x-sin-x-x-2-1-over-the-real-line.htmlAnother exam problem I'm looking at is to evaluate the following integral. $$ \int_{-\infty}^{\infty} \frac{x\hspace{-0.04 in}\cdot\hspace{-0.04 in}\sin(x)}{x^{\hspace{.02 in}2}+1} dx $$ This is a ...Sat, 22 Aug 2026 14:52:06 -0400https://pop.com.sg/good-book-for-an-introduction-to-differential-equations-for-engineers.htmlhttps://pop.com.sg/good-book-for-an-introduction-to-differential-equations-for-engineers.htmlI will be leading a discussion class on differential equations for engineers this coming semester and I am wondering if anyone has a book that they could recommend. The book that will be used in the ...Sat, 22 Aug 2026 14:50:01 -0400https://pop.com.sg/inverse-transformation-of-arctan2.htmlhttps://pop.com.sg/inverse-transformation-of-arctan2.htmlSimilar to this and this I'm trying to recover the original 2D field after the arctan2 operation. Here's a MWE in python: import numpy as np theta = np.random.uniform(low=-3*np.pi, high=3*np.pi, si......Sat, 22 Aug 2026 14:45:29 -0400https://pop.com.sg/when-does-a-system-of-equations-have-no-solution.htmlhttps://pop.com.sg/when-does-a-system-of-equations-have-no-solution.htmlI have performed Gaussian elimination on this matrix to reduce it to $$ \left[ \begin{array}{@{}ccc|c@{}} -3&-1&2 & 1 \\ 0& \frac{-5}{3}& \frac{10}{3} & \frac{8}{3} \\ 0&a......Sat, 22 Aug 2026 14:44:25 -0400https://pop.com.sg/maximizing-cross-sectional-area-of-trapezoid.htmlhttps://pop.com.sg/maximizing-cross-sectional-area-of-trapezoid.htmlThe task is to fold a piece of sheet metal that measures 60 cm across in such a way as to form a trapezoidal "gutter" (a trough for carrying rainwater) with the maximum possible cross-sectional are......Sat, 22 Aug 2026 14:35:15 -0400https://pop.com.sg/using-proof-by-contradiction-vs-proof-of-the-contrapositive.htmlhttps://pop.com.sg/using-proof-by-contradiction-vs-proof-of-the-contrapositive.htmlWhat is the difference between a "proof by contradiction" and "proving the contrapositive"? Intuitive, it feels like doing the exact same thing. And when I compare an exercise, one person proves by ...Sat, 22 Aug 2026 14:34:20 -0400https://pop.com.sg/ratios-of-median-mean-and-standard-deviation-iqr-in-a-normal-distribut.htmlhttps://pop.com.sg/ratios-of-median-mean-and-standard-deviation-iqr-in-a-normal-distribut.htmlI have some queries on the following question For a normal distribution, find the ratios of: (a) $\frac {\mbox{median}}{\mbox{mean}}$ (b) $\frac {\mbox{standard deviation}}{\mbox{interquartile r......Sat, 22 Aug 2026 14:33:09 -0400https://pop.com.sg/what-is-observation.htmlhttps://pop.com.sg/what-is-observation.htmlOften in mathematical writing one encounters texts like ''..we observe that this-and-that..''. Also one may find a review report basically saying ''..the paper is just a chain of observations...''.......Sat, 22 Aug 2026 14:29:53 -0400https://pop.com.sg/how-to-efficiently-compute-17-23-mod-31-by-hand.htmlhttps://pop.com.sg/how-to-efficiently-compute-17-23-mod-31-by-hand.htmlI could use that $17^{2} \equiv 10 (\mod 31)$ and express $17^{23}$ as $17^{16}.17^{4}.17^{3} = (((17^2)^2)^2)^2.(17^2)^2.17^2.17$ and take advantage of the fact that I can more easily work with po......Sat, 22 Aug 2026 14:24:11 -0400https://pop.com.sg/show-that-2222-5555-5555-2222-is-divisible-by-7-duplicate.htmlhttps://pop.com.sg/show-that-2222-5555-5555-2222-is-divisible-by-7-duplicate.htmlShow that $2222^ {5555} + 5555 ^ {2222}$ is divisible by $7$. I tried factorizing but it didnt lead to anything. Can divisibility rules be used? Any ideas please tell me....Sat, 22 Aug 2026 14:22:56 -0400https://pop.com.sg/how-to-make-a-geodesic-dome-2v.htmlhttps://pop.com.sg/how-to-make-a-geodesic-dome-2v.htmlI am trying to make a geodesic dome 2v in Solidwork just for fun. I use this dome calculator : http://www.domerama.com/calculators/2v-geodesic-dome-calculator/ The problem is once I start the ass......Sat, 22 Aug 2026 14:17:10 -0400