https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usThu, 27 Aug 2026 04:58:47 -0400https://pop.com.sg/is-there-always-a-superlinear-function-which-strongly-dominates-a-give.htmlhttps://pop.com.sg/is-there-always-a-superlinear-function-which-strongly-dominates-a-give.htmlConsider the class $\mathcal{F}$ of functions $f:[0, \infty) \to [0, \infty)$ with the following properties $(1)$ $f$ is strictly increasing with $f(0) = 0$. $(2)$ $f \in \omega(x)$ as $x \to \...Thu, 27 Aug 2026 08:57:07 -0400https://pop.com.sg/induction-problem-log-of-product-equals-sum-of-logs.htmlhttps://pop.com.sg/induction-problem-log-of-product-equals-sum-of-logs.htmlPlease help me prove by induction that $\displaystyle\forall n\in {{\mathbb{N}}^{*}}$, $\displaystyle\forall {{a}_{1}},\ldots ,{{a}_{n}}\in {\mathbb{R}}^{*}_{+}$, $\displaystyle \ln \left( \prod\...Thu, 27 Aug 2026 08:52:08 -0400https://pop.com.sg/laplace-039-s-equation-in-spherical-coordinates.htmlhttps://pop.com.sg/laplace-039-s-equation-in-spherical-coordinates.htmlThe general solution of the Laplace equation in spherical coordinates is (independant of $\phi$): $$V(r,\theta ) = \sum ^{\infty} _{l=0} \left( A_l r^l + \frac{B_l}{r^{l+1}} \right) P_l (\cos \the......Thu, 27 Aug 2026 08:48:38 -0400https://pop.com.sg/lp-relaxation-for-integer-linear-programming-ilp.htmlhttps://pop.com.sg/lp-relaxation-for-integer-linear-programming-ilp.htmlI am familiar with LP relaxation for ILP (or IP). Assume we concern with integer minimization problem, which we formalize using ILP; we then relax the ILP into LP and we say that the LP provides a ......Thu, 27 Aug 2026 08:39:23 -0400https://pop.com.sg/probability-that-two-random-numbers-are-coprime-is-frac-6-pi-2.htmlhttps://pop.com.sg/probability-that-two-random-numbers-are-coprime-is-frac-6-pi-2.htmlThis is a really natural question for which I know a stunning solution. So I admit I have a solution, however I would like to see if anybody will come up with something different. The question is ......Thu, 27 Aug 2026 08:30:13 -0400https://pop.com.sg/get-directional-vector-from-point-a-to-point-b.htmlhttps://pop.com.sg/get-directional-vector-from-point-a-to-point-b.htmlIm currently trying to figure out following vector maths problem: I have a point $A$ and a point $B$ and I want to get the directional vector (if that's incorrect, please tell me) that is needed t......Thu, 27 Aug 2026 08:29:36 -0400https://pop.com.sg/how-many-positive-integers-divide-20.htmlhttps://pop.com.sg/how-many-positive-integers-divide-20.html$20!$ Has nos that are multiples of $2,3,4$ and so on. However, the total number of integers is large. So, please help me....Thu, 27 Aug 2026 08:05:34 -0400https://pop.com.sg/intersection-of-finite-number-of-compact-sets-is-compact.htmlhttps://pop.com.sg/intersection-of-finite-number-of-compact-sets-is-compact.htmlIs the the intersection of a finite number of compact sets is compact? If not please give a counter example to demonstrate this is not true. I said that this is true because the intersection of fi......Thu, 27 Aug 2026 08:02:49 -0400https://pop.com.sg/minus-x-vs-negative-x-confusion.htmlhttps://pop.com.sg/minus-x-vs-negative-x-confusion.htmlMy teacher says that the following equation: $$-9-1(9x-6) = 3(4x+6)$$ Should be simplified at the first stage to: $$-9-9x+6 = 12x+18$$ He says that on the left side of the equation, we should multi......Thu, 27 Aug 2026 08:01:48 -0400https://pop.com.sg/what-is-the-difference-between-a-predicate-symbol-and-a-function-symbo.htmlhttps://pop.com.sg/what-is-the-difference-between-a-predicate-symbol-and-a-function-symbo.htmlThere are a lot of questions and answers on what the difference a predicate and a function in predicate logic is on this website, but there is no question/answer on what the difference between a ...Thu, 27 Aug 2026 07:46:23 -0400https://pop.com.sg/how-to-plot-a-delta-function-in-matlab.htmlhttps://pop.com.sg/how-to-plot-a-delta-function-in-matlab.htmlHow to plot this function in Matlab: $S(f)=1+\frac{1}{4}(\delta(f+2f{_1})+\delta(f-2f{_1})-\delta(f+2f{_2})+\delta(f-2f{_2}))$ Thanks for any advice!...Thu, 27 Aug 2026 07:44:25 -0400https://pop.com.sg/translate-the-following-english-sentences-into-symbolic-sentences-with.htmlhttps://pop.com.sg/translate-the-following-english-sentences-into-symbolic-sentences-with.htmlthis is my solution of my homework, is that true 100%? Please if there is any mistake tell me because my professor is so careful. Thanks. (Sorry, I don’t speak English well) i) Translate the follo......Thu, 27 Aug 2026 07:42:04 -0400https://pop.com.sg/how-ball-sort-puzzle-works.htmlhttps://pop.com.sg/how-ball-sort-puzzle-works.htmlBall sort puzzle is that offline game, there is also another form of this game called water sort puzzle. In the game there are n stacks containing different colors of balls. There are 4 balls of each ...Thu, 27 Aug 2026 07:40:01 -0400https://pop.com.sg/solve-30u-26-equiv-3-mod-7.htmlhttps://pop.com.sg/solve-30u-26-equiv-3-mod-7.htmlQuestion Solve $30u+26 \equiv 3\,\mod 7$ I know the solution is here,but i want to know what is flaw in my approach. My Approach $30u+26 \equiv 3\, \mod 7$ $30u \equiv -23\,\mod 7$ $u \equiv -23\...Thu, 27 Aug 2026 07:33:57 -0400https://pop.com.sg/how-to-find-least-positive-integer.htmlhttps://pop.com.sg/how-to-find-least-positive-integer.htmlA university exam paper gave a short question of this type- Find least positive integer $x$ such that $13|(x^2+1)$. It may be an easy problem but I am clueless that without anymore ...Thu, 27 Aug 2026 07:29:50 -0400https://pop.com.sg/how-to-find-the-total-distance-traveled-given-the-position-function.htmlhttps://pop.com.sg/how-to-find-the-total-distance-traveled-given-the-position-function.htmlA particle moves in a straight line according to the rule $x(t)=t^3-2t+5$, where $x(t)$ is given in meters and where $t$ is given in seconds. Determine the position, velocity, and acceleration of the ...Thu, 27 Aug 2026 07:27:12 -0400https://pop.com.sg/power-set-of-cartesian-product.htmlhttps://pop.com.sg/power-set-of-cartesian-product.htmlI'm trying to see the differences between a Power set of a cartisian product and the cartisian product of two power sets. I have these 2 examples: $P(\{1, 2\} \times \{3, 4\})$ $P(\{1, 2\}) \time......Thu, 27 Aug 2026 07:21:23 -0400https://pop.com.sg/why-can-a-graph-pass-through-its-horizontal-asymptote.htmlhttps://pop.com.sg/why-can-a-graph-pass-through-its-horizontal-asymptote.htmlI am confused why a "horizontal asymptote" that has been passed through by a graph can still be considered an asymptote? Can someone please explain this to me? I've read another website (http://...Thu, 27 Aug 2026 07:11:31 -0400https://pop.com.sg/what-does-open-set-mean-in-the-concept-of-a-topology.htmlhttps://pop.com.sg/what-does-open-set-mean-in-the-concept-of-a-topology.htmlGiven the following definition of topology, I am confused about the concept of "open sets". 2.2 Topological Space. We use some of the properties of open sets in the case of metric spaces......Thu, 27 Aug 2026 07:10:39 -0400https://pop.com.sg/calculating-a-double-pendulum.htmlhttps://pop.com.sg/calculating-a-double-pendulum.htmlconsider the following situation of a double pendulum. We found the moving equations as $$ \ddot{\theta_1}=-L_1\sin\theta_1 + \frac{m_2}{m_1}\cos\theta_2\sin(\theta_2-\theta_1),\\ \ddot{\theta}_2=......Thu, 27 Aug 2026 07:09:10 -0400https://pop.com.sg/fast-way-to-calculate-eigen-of-2x2-matrix-using-a-formula.htmlhttps://pop.com.sg/fast-way-to-calculate-eigen-of-2x2-matrix-using-a-formula.htmlI found this site: http://people.math.harvard.edu/~knill/teaching/math21b2004/exhibits/2dmatrices/index.html Which shows a very fast and simple way to get Eigen vectors for a 2x2 matrix. While har......Thu, 27 Aug 2026 07:07:25 -0400https://pop.com.sg/for-x-y-in-g-prove-that-x-y-e-if-xy-2-y-3x-and-yx-2-x-3y-duplicate.htmlhttps://pop.com.sg/for-x-y-in-g-prove-that-x-y-e-if-xy-2-y-3x-and-yx-2-x-3y-duplicate.htmlFor $x,y$ in G, prove that $x = y =e$ if $xy^2=y^3x $ and $yx^2=x^3y$ where $e$ is the identity element in G. I have filled pages trying to solve but unable to reach the solution. Also, Is there......Thu, 27 Aug 2026 07:06:51 -0400https://pop.com.sg/find-radius-of-circle-or-sphere-given-segment-area-or-cap-volume-and-c.htmlhttps://pop.com.sg/find-radius-of-circle-or-sphere-given-segment-area-or-cap-volume-and-c.htmlThe goal is to design a container (partial sphere) of given volume which attached to a plane via a port of a given radius. I believe this can be done as follows but the calculation is causing me ...Thu, 27 Aug 2026 07:04:09 -0400https://pop.com.sg/modularity-lifting-theorem-by-lue-pan.htmlhttps://pop.com.sg/modularity-lifting-theorem-by-lue-pan.htmlWhen I was searching for the proof of Modularity Lifting Theorem, I found the paper by Luis Dieulefait. According to this paper it seems that there is recent Modularity Lifting Theorem proved by Lu......Thu, 27 Aug 2026 06:55:55 -0400https://pop.com.sg/what-is-a-metric-matrix.htmlhttps://pop.com.sg/what-is-a-metric-matrix.htmlI'm having a bit of trouble finding a source of study to metric matrices (I'm not even sure whether this is how we call them in English). It is introduced along with vector inner products. I can't ......Thu, 27 Aug 2026 06:45:50 -0400https://pop.com.sg/suppose-we-have-an-inscribed-isosceles-triangle-in-a-circle.htmlhttps://pop.com.sg/suppose-we-have-an-inscribed-isosceles-triangle-in-a-circle.html$ABC$ is an isosceles triangle inscribed in a circle with centre $O$.Suppose the top vertex is $A$, the right vertex is $B$ and the left vertex is $C$. Also suppose $AC=AB$. Furthermore, the diamet......Thu, 27 Aug 2026 06:45:11 -0400https://pop.com.sg/precalc-factoring-x-2-10x-25-9y-2.htmlhttps://pop.com.sg/precalc-factoring-x-2-10x-25-9y-2.htmlFactor $$x^2+10x+25-9y^2$$ The solution is $$(x+5-3y)(x+5+3y)$$ I understand how to factor when there is only one variable $x$ but I am not sure how to complete this problem with the additional ...Thu, 27 Aug 2026 06:45:11 -0400https://pop.com.sg/formula-for-the-sequence-0-3-8-15-24.htmlhttps://pop.com.sg/formula-for-the-sequence-0-3-8-15-24.htmlOut of my own interest I've been practicing finding formula for for sequences and I've been having trouble finding one for the nth term for this sequence. 0,3,8,15,24 ... Clearly you add 5,7,9,1......Thu, 27 Aug 2026 06:36:55 -0400https://pop.com.sg/proof-that-every-polygon-with-an-inscribed-circle-is-convex.htmlhttps://pop.com.sg/proof-that-every-polygon-with-an-inscribed-circle-is-convex.htmlIn many elementary (and not-so-elementary) Euclidean geometry texts, a (simple) polygon is said to be tangential  if it is convex and has an inscribed circle (i.e., a circle that intersects an......Thu, 27 Aug 2026 06:35:12 -0400https://pop.com.sg/prove-the-superposition-principle-for-nonhomogeneous-equations.htmlhttps://pop.com.sg/prove-the-superposition-principle-for-nonhomogeneous-equations.htmlSuppose that $y_1$ is a solution to $Ly_1 = f(x)$ and $y_2$ is a solution to $Ly_2 = g(x)$ Show that $y = y_1 + y_2$ solves $Ly = f(x) + g(x)$. So, assuming that the linear operator $L$ is the s......Thu, 27 Aug 2026 06:35:01 -0400https://pop.com.sg/proof-of-formula-for-the-arc-length.htmlhttps://pop.com.sg/proof-of-formula-for-the-arc-length.htmlI've read the proof of formula for arc length . I wonder why the function has to have continuous derivative (According to the Stewart Calculus book). I mean in which part of the proof we used this ...Thu, 27 Aug 2026 06:34:57 -0400https://pop.com.sg/find-a-cubic-function-given-an-inflection-point-critical-point-and-fun.htmlhttps://pop.com.sg/find-a-cubic-function-given-an-inflection-point-critical-point-and-fun.htmlFind a cubic function f(x) = ax^3 + bx^2 + cx + d Given: Inflection point (0,18) Critical point x = 2 F(2) = 2 I know how to solve for the general forms of the derivatives, and to set the values ......Thu, 27 Aug 2026 06:25:43 -0400https://pop.com.sg/let-h-unlhd-g-and-a-in-g-if-the-coset-ah-has-order-3-in-g-h-and-h-10-w.htmlhttps://pop.com.sg/let-h-unlhd-g-and-a-in-g-if-the-coset-ah-has-order-3-in-g-h-and-h-10-w.htmlContext I am an undergrad student taking Abstract Algebra I, and this is a problem on a homework assignment. Problem Let $H$ be a normal subgroup of $G$, and let $a ∈ G$. If the coset $aH$ has orde......Thu, 27 Aug 2026 06:12:40 -0400https://pop.com.sg/what-is-the-derivative-of-log-x-a-where-x-is-the-base-differentiaition.htmlhttps://pop.com.sg/what-is-the-derivative-of-log-x-a-where-x-is-the-base-differentiaition.htmlI want to find out what $\frac{d}{dx}\log_x A$ is? I did this so far but I'm not sure. $y = \log_x A \Longrightarrow x^y = A$ so, $d/dx(x^y) = d/dx(A)$ [differentiating both sides w.r.t $x$] then......Thu, 27 Aug 2026 06:10:54 -0400https://pop.com.sg/differential-equations-how-does-separation-of-variables-really-work.htmlhttps://pop.com.sg/differential-equations-how-does-separation-of-variables-really-work.htmlI'm trying to brush up my differential equations knowledge for an upcoming exam. I watched a set of videos and read few articles on how to solve differential equations via some different methods. O......Thu, 27 Aug 2026 06:08:09 -0400https://pop.com.sg/properties-of-invertible-matrices.htmlhttps://pop.com.sg/properties-of-invertible-matrices.htmlShow that if $A$, $B$, and $A + B$ are invertible matrices with the same size, then $A(A^{-1}+B^{-1})B(A+B)^{-1}=I$ What does the result in the first part tell you about the matrix $(A^{-1}+......Thu, 27 Aug 2026 06:07:01 -0400https://pop.com.sg/difference-between-differentiation-and-derivatives-closed.htmlhttps://pop.com.sg/difference-between-differentiation-and-derivatives-closed.htmlWhat is the difference between differentiation and derivatives of a function ?...Thu, 27 Aug 2026 06:04:45 -0400https://pop.com.sg/origin-or-author-of-039-japanese-multiplication-method-039.htmlhttps://pop.com.sg/origin-or-author-of-039-japanese-multiplication-method-039.htmlWhat is the origin or author of the method 1 shown in the following image? Notes Also known as Japanese Multiplication Method for Kids....Thu, 27 Aug 2026 05:58:10 -0400https://pop.com.sg/why-is-cos-2-pi-3-1-2.htmlhttps://pop.com.sg/why-is-cos-2-pi-3-1-2.htmlI know the cosine function is even, but I don't understand why it works the way it does. Since this function is on the unit circle, and the rotation is reversed, wouldn't $\cos(\frac{-2\pi}{3})$ be......Thu, 27 Aug 2026 05:51:56 -0400https://pop.com.sg/finding-the-general-solution-to-x-039-ax-with-a-0-1-0-1-0-0-1-1-1.htmlhttps://pop.com.sg/finding-the-general-solution-to-x-039-ax-with-a-0-1-0-1-0-0-1-1-1.htmlThis was an exam question that I sort of guessed on. Find the general solution to $X' = AX$, where $A = \left[\begin{smallmatrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 1 & 1 & 1\end{...Thu, 27 Aug 2026 05:49:57 -0400https://pop.com.sg/algebraic-expression-in-its-most-simplified-form.htmlhttps://pop.com.sg/algebraic-expression-in-its-most-simplified-form.htmlI am trying to simplify the algebraic expression: $$\bigg(x-\dfrac{4}{(x-3)}\bigg)\div \bigg(x+\dfrac{2+6x}{(x-3)}\bigg)$$ I am having trouble though. My current thoughts are: $$=\bigg(\dfrac{x}......Thu, 27 Aug 2026 05:46:22 -0400https://pop.com.sg/how-to-express-other-logical-operations-via-pierce-039-s-arrow.htmlhttps://pop.com.sg/how-to-express-other-logical-operations-via-pierce-039-s-arrow.htmlx↑y, x⇒y, and x⇔y. So I have really given my best, but all I could do is express the conjunction, disjunction, negation, and impilcation....Thu, 27 Aug 2026 05:30:42 -0400https://pop.com.sg/prove-that-the-square-of-an-integer-a-is-congruent-to-0-or-1-modulo-4.htmlhttps://pop.com.sg/prove-that-the-square-of-an-integer-a-is-congruent-to-0-or-1-modulo-4.htmlI'm currently trying to go through Artin's Algebra textbook via a Harvard Extension Course (available on YouTube). I wanted to make sure that my proof was solid for this question. Prove that the ...Thu, 27 Aug 2026 05:29:24 -0400https://pop.com.sg/why-we-need-to-know-how-to-solve-a-quadratic.htmlhttps://pop.com.sg/why-we-need-to-know-how-to-solve-a-quadratic.htmlFive years ago I was tutoring orphans in a local hospital. One of them asked me the following question when I tried to ask him to solve a quadratic: Why do I need how to solve a quadratic? I am ......Thu, 27 Aug 2026 05:27:08 -0400https://pop.com.sg/does-an-increasing-sequence-of-reals-converge-if-the-difference-of-con.htmlhttps://pop.com.sg/does-an-increasing-sequence-of-reals-converge-if-the-difference-of-con.htmlIf $a_n$ is a sequence such that $$a_1 \leq a_2 \leq a_3 \leq \dotsb$$ and has the property that $a_{n+1}-a_n \to 0$, then can we conclude that $a_n$ is convergent? I know that without the con......Thu, 27 Aug 2026 05:22:46 -0400https://pop.com.sg/examples-of-groups-in-the-real-world.htmlhttps://pop.com.sg/examples-of-groups-in-the-real-world.htmlI'm looking for some examples of groups in the real world to show students in a liberal arts math course. For example the Rubik's cube. Keep in mind these students have only a college algebra backg......Thu, 27 Aug 2026 05:19:53 -0400https://pop.com.sg/taylor-series-expansion-about-x-3.htmlhttps://pop.com.sg/taylor-series-expansion-about-x-3.htmlFind the Taylor series expansion of $f(x)=\sqrt{6x-x^2} $ about $x = 3$. Looking through my notes, I think this could be solved with the binomial theorem. I am no longer sure binomial theorem is u......Thu, 27 Aug 2026 05:05:36 -0400https://pop.com.sg/find-x-location-using-3-known-x-y-location-using-trilateration.htmlhttps://pop.com.sg/find-x-location-using-3-known-x-y-location-using-trilateration.htmlI post this question in stackoverflow here and was advised it was best suited for here. I am trying to understand the maths behind trilateration, we have 3 access points $(\text{AP's: } 1,2,3)$ an......Thu, 27 Aug 2026 05:04:38 -0400https://pop.com.sg/compound-angle-formula.htmlhttps://pop.com.sg/compound-angle-formula.htmlI understand how to use the compound angle formula when solving $\sin(\pi/12)$. However I dont understand how I can use a compound angle formula to show $$\arctan(3)-\arctan(1/2)=\pi/4$$ Thankyou ......Thu, 27 Aug 2026 04:59:20 -0400https://pop.com.sg/define-the-subgroup-of-m2-r.htmlhttps://pop.com.sg/define-the-subgroup-of-m2-r.htmlFind a subgroup of $M_2(R)$, the group of $2 \times 2$ matrices with real entries under addition, called $H$, if for all $A,B$ in $M_2(R)$, $A-B$ is an element of subgroup $H$ implies that $\operat......Thu, 27 Aug 2026 04:55:49 -0400https://pop.com.sg/how-many-sides-can-a-polygon-have-before-it-is-considered-a-circle-clo.htmlhttps://pop.com.sg/how-many-sides-can-a-polygon-have-before-it-is-considered-a-circle-clo.htmlGood day, my family had a dinner discussion about polygons and how many sides a polygon has in relation to the angle measurement you'll get when you measure an "arc" encompassing a "side" of the po......Thu, 27 Aug 2026 04:48:50 -0400https://pop.com.sg/a-rectangle-is-scissors-congruent-to-a-square-of-the-same-area.htmlhttps://pop.com.sg/a-rectangle-is-scissors-congruent-to-a-square-of-the-same-area.htmlCan somebody explain me step 3, in the following link - here...Thu, 27 Aug 2026 04:48:36 -0400https://pop.com.sg/verifying-eigenvalues.htmlhttps://pop.com.sg/verifying-eigenvalues.htmlHow would you check whether eigenvalues $\lambda_1=8$, $\lambda_2=3$, $\lambda_3=-1$ belong to a matrix? $$ \begin{matrix} 7 & 1 & 1\\ 3 & 1 & 2 \\ 1......Thu, 27 Aug 2026 04:46:08 -0400https://pop.com.sg/formula-for-skewed-bell-curve.htmlhttps://pop.com.sg/formula-for-skewed-bell-curve.htmlI can plot a bell curve using the formula: $$ Y = \frac{1}{\sqrt{2\pi S^2} } e^{ -\frac{(X-A)^2}{(2S)^2} } $$ where A=mean and S=standard deviation. This is obviously on an X,Y plot. I want to add a ...Thu, 27 Aug 2026 04:38:20 -0400https://pop.com.sg/orthogonality-of-sine-and-cosine-integrals.htmlhttps://pop.com.sg/orthogonality-of-sine-and-cosine-integrals.htmlHow to prove that $$ \int_{t_0}^{t_0+T} \sin(m\omega t)\sin(n\omega t)\,\mathrm{d}t$$ will equal to $0$ when $m\ne n$ and $\frac{T}{2}$ when $m=n\ne 0$? Besides $$ \int_{t_0}^{t_0+T} \cos(m\omega ......Thu, 27 Aug 2026 04:30:39 -0400https://pop.com.sg/what-are-the-5-simplest-lambda-calculus-expresions.htmlhttps://pop.com.sg/what-are-the-5-simplest-lambda-calculus-expresions.htmlI'm struggling to learn lambda Calculus. I think what might really help is to see the simplest functions that you can create in lambda calculus and how they might be combined to make more complex ...Thu, 27 Aug 2026 04:23:03 -0400https://pop.com.sg/how-to-obtain-3-parameters-from-a-linearized-model.htmlhttps://pop.com.sg/how-to-obtain-3-parameters-from-a-linearized-model.htmlI have the following model: $$ q_e=\dfrac{K_s\ {C_e}^\beta}{1+a_s\ {C_e}^\beta} $$ Its linearized form is: $$ \beta\ln(C_e)=-\ln\left(\dfrac{K_s}{q_e}\right)+\ln a_s $$ The known values are $C_......Thu, 27 Aug 2026 04:20:12 -0400https://pop.com.sg/total-derivative-vs-chain-rule.htmlhttps://pop.com.sg/total-derivative-vs-chain-rule.htmlThe total derivative as I understand it would be a linear map along the lines on the wikipedia page. https://en.wikipedia.org/wiki/Total_derivative However in some instances it looks alot like a c......Thu, 27 Aug 2026 04:20:11 -0400https://pop.com.sg/parameterized-hexagonal-spiral-equation-s.htmlhttps://pop.com.sg/parameterized-hexagonal-spiral-equation-s.htmlI need to design a fully parameterized hexagonal spiral, where I can choose/change the distance between turns, the inner and outer radii etc. I understand that it will be somehow related to the ...Thu, 27 Aug 2026 04:15:28 -0400https://pop.com.sg/characterization-of-faithfully-flat-homomorphisms.htmlhttps://pop.com.sg/characterization-of-faithfully-flat-homomorphisms.htmlLet $A \to B$ be a homomorphism of commutative rings. Why are the following conditions equivalent? $A \to B$ is faithfully flat. $A \to B$ is injective, flat and $B/A$ is a flat $A$-module. This ......Thu, 27 Aug 2026 03:59:30 -0400https://pop.com.sg/how-do-you-take-the-conjugate-of-a-real-matrix.htmlhttps://pop.com.sg/how-do-you-take-the-conjugate-of-a-real-matrix.htmlI've only seen resources pointing to how to take the conjugate of a complex matrix but how do you take the conjugate of a real matrix? Let's take this matrix as an example: 0 1 1 0...Thu, 27 Aug 2026 03:40:18 -0400https://pop.com.sg/standard-deviation-with-exponential-distribution.htmlhttps://pop.com.sg/standard-deviation-with-exponential-distribution.htmlLet x denote the distance that an animal moves from its birth site to the first territorial vacancy it encounters. Suppose that x has an exponential distribution with parameter lambda = 0.01386. a......Thu, 27 Aug 2026 03:35:35 -0400https://pop.com.sg/using-unit-circle-to-explain-cos-0-1-and-sin-90-1.htmlhttps://pop.com.sg/using-unit-circle-to-explain-cos-0-1-and-sin-90-1.htmlWe have been taught $\cos(0) = 1$ and $\sin(90) = 1$. But, how do I visualize these angles on the unit circle?...Thu, 27 Aug 2026 03:20:36 -0400https://pop.com.sg/what-does-1-13-times-faster-mean.htmlhttps://pop.com.sg/what-does-1-13-times-faster-mean.htmlI'm reading High Performance JavaScript, and I think the graphs in one chapter are just plain wrong. Here is one on Google Books. The y axis is "Times faster", and it runs from -1.5 to +4.0. Now, I ...Thu, 27 Aug 2026 03:19:06 -0400https://pop.com.sg/subset-of-a-regular-language.htmlhttps://pop.com.sg/subset-of-a-regular-language.htmlI need to show that a subset of a regular language is regular or not. I think it may not be regular but I could not find a counter example. Do you have any simple example to prove that? Thanks in ...Thu, 27 Aug 2026 03:17:19 -0400https://pop.com.sg/modulus-of-continuity-properties-and-uniform-continuity.htmlhttps://pop.com.sg/modulus-of-continuity-properties-and-uniform-continuity.htmlLet $f:[a,b]\rightarrow \mathbb{R}$ bounded and $\omega(f,r)=\sup\{|f(x)-f(y)| \colon x,y \in [a,b], \ |x-y|<r\}$ (called modulus of continuity of $f$ EDIT note: the original question is in ...Thu, 27 Aug 2026 03:11:01 -0400https://pop.com.sg/0-order-logic-construction-sequence.htmlhttps://pop.com.sg/0-order-logic-construction-sequence.htmlI am asked to exhibit a construction sequence to show whether the following is a well-formed formula; Let S = {A, B, C, D} $\alpha = ((A\land C) \rightarrow (((\lnot A) \land (\lnot C)) \lor B)))......Thu, 27 Aug 2026 03:01:56 -0400https://pop.com.sg/what-does-039-039-and-039-039-mean-in-algebra.htmlhttps://pop.com.sg/what-does-039-039-and-039-039-mean-in-algebra.htmlIn the book I'm reading it says about ratios and proportions, If $a:b::c:d;$ then $ad=bc$, and $\frac{a}{b}=\frac{c}{d}$ What do ':' and '::' mean? Also, it is not stating the above as a defini......Thu, 27 Aug 2026 02:59:27 -0400https://pop.com.sg/definition-of-monotone-increasing-sets.htmlhttps://pop.com.sg/definition-of-monotone-increasing-sets.htmlI've started reading All of Statistics by Larry Wasserman with the hope of gaining understanding of machine learning fundamentals. The author gives the following definition in the first chapter: ......Thu, 27 Aug 2026 02:46:48 -0400https://pop.com.sg/square-roots-of-complex-number-in-exponential-form.htmlhttps://pop.com.sg/square-roots-of-complex-number-in-exponential-form.htmlThe complex number $z$ is defined by $z=\frac{9\sqrt{3}+9i}{\sqrt{3}-i}$. Find the square roots of $z$, giving your answers in the form $re^{i\theta}$.where $r>0$ and $-\pi < \theta \leq \pi$......Thu, 27 Aug 2026 02:46:07 -0400https://pop.com.sg/how-do-i-find-the-orthogonal-basis-for-this-plane.htmlhttps://pop.com.sg/how-do-i-find-the-orthogonal-basis-for-this-plane.htmlQuestion: $P$ is a plane through the origin given by $x + y + 2z = 0$. Find an orthogonal basis v1, v2 ∈ $P$. My answer: I'm assuming the question asks for two vectors that span this plane ......Thu, 27 Aug 2026 02:29:10 -0400https://pop.com.sg/why-is-the-pdf-of-the-sum-of-two-continuous-random-variables-the-convo.htmlhttps://pop.com.sg/why-is-the-pdf-of-the-sum-of-two-continuous-random-variables-the-convo.htmlI have taken a probability course last year but we didn't cover that notion. I do know the steps in the discrete case. finding the support of $X + Y$... calculating the the probability of each el......Thu, 27 Aug 2026 02:18:45 -0400https://pop.com.sg/how-to-show-i-1-i-duplicate.htmlhttps://pop.com.sg/how-to-show-i-1-i-duplicate.htmlHow can I show $i^{-1} = -i$, where $i$ is the imaginary unit? Here's what I've tried: $i^{-1} = (-1)^{-1/2} = \dots ?$...Thu, 27 Aug 2026 01:58:20 -0400https://pop.com.sg/how-to-find-local-maximum-and-minimum-of-function-f-n-x-n-sin-x-at-x-0.htmlhttps://pop.com.sg/how-to-find-local-maximum-and-minimum-of-function-f-n-x-n-sin-x-at-x-0.htmlHow to find local maximum and minimum of function $f_{n} = x^{n} \sin x$ at $x=0$? Where $n≥2$. I tried to find local Maxima or minima by finding the critical points, but I'm getting no critical p......Thu, 27 Aug 2026 01:54:40 -0400https://pop.com.sg/is-it-possible-to-have-a-function-differentiable-but-not-continuous-in.htmlhttps://pop.com.sg/is-it-possible-to-have-a-function-differentiable-but-not-continuous-in.htmlIs there any possible function that is not continuous but differentiable in a given interval. It sounds non-logical to me since differentiation is a special limit function in itself therefore non-...Thu, 27 Aug 2026 01:48:58 -0400https://pop.com.sg/re-learning-math-from-scratch-closed.htmlhttps://pop.com.sg/re-learning-math-from-scratch-closed.htmlTwo or three years ago I decided to re-learn math from scratch, I did some research on what to read, made a list of books. I started from a Sawyer, it opened my eyes a little bit on the fact that m......Thu, 27 Aug 2026 01:46:18 -0400https://pop.com.sg/the-pepperoni-pizza-problem.htmlhttps://pop.com.sg/the-pepperoni-pizza-problem.htmlThis problem arose in a different context at work, but I have translated it to pizza. Suppose you have a circular pizza of radius $R$. Upon this disc, $n$ pepperoni will be distributed completely ...Thu, 27 Aug 2026 01:27:41 -0400https://pop.com.sg/what-are-the-odds-of-rolling-a-1-and-a-20-in-two-d20-dice.htmlhttps://pop.com.sg/what-are-the-odds-of-rolling-a-1-and-a-20-in-two-d20-dice.htmlSo, one of these days a friend of mine rolled two d20 dice and got a $1$ in one of them and a $20$ in the other. I was thinking, what are the odds of this happening?...Thu, 27 Aug 2026 01:27:00 -0400https://pop.com.sg/unique-solution-for-ode-y-039-sqrt-1-y-2.htmlhttps://pop.com.sg/unique-solution-for-ode-y-039-sqrt-1-y-2.htmlI was given to solve the next ode: $y' = {\sqrt{1-y^2}} $ I found its solution: $y=sin(x+c)$ Now, I'm given that $y(0)=0$ and asked to show the only solution is $y=sin(x)$ in the region $(-\infty,\...Thu, 27 Aug 2026 01:23:54 -0400https://pop.com.sg/what-is-the-probability-that-a-five-card-poker-hand-has-four-aces.htmlhttps://pop.com.sg/what-is-the-probability-that-a-five-card-poker-hand-has-four-aces.htmlWhat is the probability that a five-card poker hand has four ACES? When I was solving the above stated problem, I got confused while trying different methods : Assume a normal $52$ deck of cards. ...Thu, 27 Aug 2026 01:13:10 -0400https://pop.com.sg/how-to-prove-that-a-function-is-differentiable-everywhere.htmlhttps://pop.com.sg/how-to-prove-that-a-function-is-differentiable-everywhere.htmlThe function I come up with is: $f(x)= -5x$ and I think this function can be differentiated everywhere because the domain is $\mathbb R$ right? $f'(x) = -5$ which is constant, but my question i......Thu, 27 Aug 2026 01:12:54 -0400https://pop.com.sg/partial-differential-equation-with-a-flux-term.htmlhttps://pop.com.sg/partial-differential-equation-with-a-flux-term.htmlRecall how we derived all our equations: Take an interval $[a,b]$ and consider $$\dfrac{\mathrm d}{\mathrm dt}\int_a^b(\text{quantity) d}x=\big[\text{Flux}\big]_a-\big[{\rm Flux}\big]_b.$$ For our ...Thu, 27 Aug 2026 01:08:33 -0400https://pop.com.sg/find-side-bc-of-a-triangle-given-ab-ac-and-a-relation-between-angle-a-.htmlhttps://pop.com.sg/find-side-bc-of-a-triangle-given-ab-ac-and-a-relation-between-angle-a-.htmlA question from my class: In triangle $ABC$, $3\angle A+2\angle B=180$ and $AB=10, AC=4$. So question is, what all can we comment on side $BC$. Can we find its exact length? I have a crude ...Thu, 27 Aug 2026 01:07:39 -0400https://pop.com.sg/tessellating-the-sphere.htmlhttps://pop.com.sg/tessellating-the-sphere.htmlIt is a famous result that the plane can be tessellated by regular triangles, squares, and hexagons. Which regular polygons can tessellate the sphere?...Thu, 27 Aug 2026 01:06:48 -0400https://pop.com.sg/spherical-cow-calculus-1-closed.htmlhttps://pop.com.sg/spherical-cow-calculus-1-closed.htmlCan someone please help me with the 3 calculus one word problems below, I've done all three of them several times but can never seem to arrive at the correct answer. This is greatly frustrating bec......Thu, 27 Aug 2026 01:01:48 -0400https://pop.com.sg/which-is-asymptotically-larger-lg-lg-n-or-lg-lg-n.htmlhttps://pop.com.sg/which-is-asymptotically-larger-lg-lg-n-or-lg-lg-n.htmlThis definition is extracted from "Introduction to Algorithm, 2nd Edition". The iterated logarithm function We use the notation $\lg^* n$ (read "log star of $n$") to denote the ...Thu, 27 Aug 2026 01:00:43 -0400https://pop.com.sg/how-are-the-tangential-and-normal-components-of-the-acceleration-vecto.htmlhttps://pop.com.sg/how-are-the-tangential-and-normal-components-of-the-acceleration-vecto.htmlLet acceleration be $$\overrightarrow a= a_T \overrightarrow T + a_N \overrightarrow N$$ where $\overrightarrow T$ is the unit tangent component and $\overrightarrow N$ is the unit normal component......Thu, 27 Aug 2026 00:52:53 -0400https://pop.com.sg/category-theory-homset-preserves-limits.htmlhttps://pop.com.sg/category-theory-homset-preserves-limits.htmledit I updated my question at the end, I think the claim may be false? Let $(L,\lambda)$ be a limit cone of a diagram $D$ in a category. For any object $X$ it is said that the hom functor $......Thu, 27 Aug 2026 00:48:06 -0400https://pop.com.sg/how-is-fubini-039-s-theorem-used-in-the-following-proof.htmlhttps://pop.com.sg/how-is-fubini-039-s-theorem-used-in-the-following-proof.htmlI'm having trouble to understand exactly how we are using Fubini's theorem in the following proof involving the distribution function, since it newer explicitly involves an integral with product me......Thu, 27 Aug 2026 00:47:58 -0400https://pop.com.sg/how-to-proof-that-phi-1-1-phi-1-for-standard-normal-distribution.htmlhttps://pop.com.sg/how-to-proof-that-phi-1-1-phi-1-for-standard-normal-distribution.htmlI have been studying normal distribution and frequently I see that $$\Phi(1)=1-\Phi(-1)$$ where $\Phi(x)=\int_{-\infty}^{x}\phi(u)du$ is the cdf of a standard normal variable. How one can prove eq......Thu, 27 Aug 2026 00:41:24 -0400https://pop.com.sg/what-is-equipotent-relation.htmlhttps://pop.com.sg/what-is-equipotent-relation.htmlIn Royden's book Real Analysis, page 13, he writes that "We call two sets A and b equipotent provided there is a one-to-one mapping $f$ from A onto B Equipotence defines a equivalence relation amon......Thu, 27 Aug 2026 00:17:23 -0400https://pop.com.sg/what-do-you-call-the-point-where-two-lines-meet.htmlhttps://pop.com.sg/what-do-you-call-the-point-where-two-lines-meet.htmlThis is from a third grader. His example is the point where the hands on the clock meet. It's not pivot. Or "if you start with a dot and make two lines go out from it, on straight up and one to the......Thu, 27 Aug 2026 00:16:37 -0400https://pop.com.sg/is-the-theory-of-dual-numbers-strong-enough-to-develop-real-analysis-a.htmlhttps://pop.com.sg/is-the-theory-of-dual-numbers-strong-enough-to-develop-real-analysis-a.htmlI've been interested in non-standard analysis recently. I was reading up on it and noticed the following interesting comment on the Wikipedia page about hyperreal numbers, right after giving an exa......Wed, 26 Aug 2026 23:55:08 -0400https://pop.com.sg/why-should-i-care-about-adjoint-functors.htmlhttps://pop.com.sg/why-should-i-care-about-adjoint-functors.htmlI am comfortable with the definition of adjoint functors. I have done a few exercises proving that certain pairs of functors are adjoint (tensor and hom, sheafification and forgetful, direct image ......Wed, 26 Aug 2026 23:50:41 -0400https://pop.com.sg/rank-of-an-augmented-matrix.htmlhttps://pop.com.sg/rank-of-an-augmented-matrix.htmlCan the rank of coeffecient matrix be greater than augmented matrix? Also,what is the condition for an inconsistent set of linear equations?...Wed, 26 Aug 2026 23:49:32 -0400https://pop.com.sg/the-use-of-gershgorin-circle-theorem.htmlhttps://pop.com.sg/the-use-of-gershgorin-circle-theorem.htmlIn order to estimate the eigenvalues of a real symmetric $n\times n$ matrix, I intend to use the Gershgorin Circle Theorem. Unfortunately, the examples one might find on the internet are a bit conf......Wed, 26 Aug 2026 23:42:17 -0400https://pop.com.sg/having-trouble-understanding-meaning-of-a-finite-dimensional-vector-sp.htmlhttps://pop.com.sg/having-trouble-understanding-meaning-of-a-finite-dimensional-vector-sp.htmlThe definition I have is the following: A vector space V is said to be finite-dimensional if there is a finite set of vectors in V that spans V and is said to be infinite-dimensional if no such ......Wed, 26 Aug 2026 23:31:47 -0400https://pop.com.sg/when-does-a-system-of-equations-have-infinite-unique-and-no-solutions.htmlhttps://pop.com.sg/when-does-a-system-of-equations-have-infinite-unique-and-no-solutions.htmlCan someone please tell me what a matrix looks like when there is infinite solutions, unique solution and no solutions. I have been searching the internet and I cannot find a straightforward answer......Wed, 26 Aug 2026 23:18:04 -0400https://pop.com.sg/polar-form-of-frac-1-z.htmlhttps://pop.com.sg/polar-form-of-frac-1-z.htmlWrite the following in polar form:$$\frac{1}{z}$$ $$\frac{1}{z}=\frac{1}{x+yi}$$ $$\frac{1}{x+yi}\cdot \frac{x-yi}{x-yi}=\frac{x-yi}{x^2+y^2}=\frac{x-yi}{r^2}$$ Because: $$x-yi=rcis(-\frac{\pi}......Wed, 26 Aug 2026 23:15:19 -0400https://pop.com.sg/how-to-integrate-sqrt-x-2-y-2-1-the-easy-way.htmlhttps://pop.com.sg/how-to-integrate-sqrt-x-2-y-2-1-the-easy-way.htmlI know you can change to polar coordinates but then you still have to integrate $\sqrt{1+r^2}$ which is still non-trivial. I remember there being some trigonometric substitution, possibly hyperbo......Wed, 26 Aug 2026 23:10:26 -0400