https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usTue, 11 Aug 2026 00:37:12 -0400https://pop.com.sg/desmos-sin-wave-sum-shows-different-to-code.htmlhttps://pop.com.sg/desmos-sin-wave-sum-shows-different-to-code.htmlI have followed The Coding Train's Discrete Fourier Transformation series, and added a feature which spits out the coefficients of the equation. I wrote test in the program, which looked nice. Howe......Tue, 11 Aug 2026 04:34:27 -0400https://pop.com.sg/what-is-the-remainder-of-16-is-divided-by-19.htmlhttps://pop.com.sg/what-is-the-remainder-of-16-is-divided-by-19.htmlCan anyone share me the trick to solve this problem using congruence? Thanks, Chan...Tue, 11 Aug 2026 04:25:39 -0400https://pop.com.sg/a-3-regular-graph-must-have-a-subgraph-which-is-a-cycle-of-even-length.htmlhttps://pop.com.sg/a-3-regular-graph-must-have-a-subgraph-which-is-a-cycle-of-even-length.htmlDoes a 3-regular graph necessarily contain a cycle of even length? I tried, believe me I tried, I even went and asked my professor's assistant and even he couldn't give me a straight answer... So......Tue, 11 Aug 2026 04:21:11 -0400https://pop.com.sg/how-can-i-express-the-natural-logarithm-of-a-power-series-in-terms-of-.htmlhttps://pop.com.sg/how-can-i-express-the-natural-logarithm-of-a-power-series-in-terms-of-.htmlCan I express the expression $$\ln\left( \sum_0^{\infty} a_nx^n\right)$$ in terms of another power series? Can I differentiate the term $$\ln\left( \sum_0^{\infty} a_nx^n\right)$$ and get $$\frac{\......Tue, 11 Aug 2026 04:12:45 -0400https://pop.com.sg/integrating-x-2e-x-using-feynman-039-s-trick-duplicate.htmlhttps://pop.com.sg/integrating-x-2e-x-using-feynman-039-s-trick-duplicate.htmlIn the second episode of season $8$ of "The Big Bang Theory," which aired yesterday night, it is stated that one can integrate $x^2e^{-x}$ by using Feynman's trick of differentiating under the inte......Tue, 11 Aug 2026 04:06:55 -0400https://pop.com.sg/formula-for-the-square-root-of-a-number.htmlhttps://pop.com.sg/formula-for-the-square-root-of-a-number.htmlIs there a formula for the square root of a number, that only uses addition, subtraction, multiplication, or division?...Tue, 11 Aug 2026 04:06:04 -0400https://pop.com.sg/what-are-the-most-obscure-or-advanced-mathematics-with-practical-appli.htmlhttps://pop.com.sg/what-are-the-most-obscure-or-advanced-mathematics-with-practical-appli.htmlThroughout my engineering studies there were jokes made by my professors (mostly mathematics professors) that referenced the fact that pure mathematicians strive to create mathematics with no pract......Tue, 11 Aug 2026 04:01:03 -0400https://pop.com.sg/why-cauchy-distribution-isn-039-t-exponental-family.htmlhttps://pop.com.sg/why-cauchy-distribution-isn-039-t-exponental-family.htmlWe know that the density function of standard Cauchy Distribution is $p(x)=\frac{\lambda}{\pi (\lambda^2+x^2)}$. It seems that $p(x)$ can't be written as the form of Exponential Family $$f_{X}(x;\......Tue, 11 Aug 2026 03:53:56 -0400https://pop.com.sg/krull-dimension-of-finite-rings-closed.htmlhttps://pop.com.sg/krull-dimension-of-finite-rings-closed.htmlI have the following question(s): Can the Krull dimension of a finite ring be any natural number? Thanks for the help!...Tue, 11 Aug 2026 03:52:01 -0400https://pop.com.sg/merge-sort-seems-to-take-the-same-number-of-comparisons-for-best-and-w.htmlhttps://pop.com.sg/merge-sort-seems-to-take-the-same-number-of-comparisons-for-best-and-w.htmlI am trying to clear up my conceptions of merge sort. What I cannot understand how merge sort takes less number of comparisons during best case. Let me explain, looking at the merge procedure gi......Tue, 11 Aug 2026 03:38:38 -0400https://pop.com.sg/what-is-the-difference-between-linear-transformation-and-affine-transf.htmlhttps://pop.com.sg/what-is-the-difference-between-linear-transformation-and-affine-transf.htmlRecently, I am struglling with the difference between linear transformation and affine transformation. Are they the same ? I found an interesting question on the difference between the functions. ......Tue, 11 Aug 2026 03:36:19 -0400https://pop.com.sg/how-to-define-a-triod.htmlhttps://pop.com.sg/how-to-define-a-triod.htmlI'm looking for a topological definition of a triod as a compact metric space. I have an intuitive idea of what it is (three intervals with one shared point being a boundary point of each interval)......Tue, 11 Aug 2026 03:26:00 -0400https://pop.com.sg/is-this-the-correct-way-of-solve-a-probability-question-from-a-deck-of.htmlhttps://pop.com.sg/is-this-the-correct-way-of-solve-a-probability-question-from-a-deck-of.htmlYou draw a card from a standard deck of $52$ playing cards then replace it in the deck and draw a second card. Determine the probability of drawing an eight and then heart My work: their are $4$ e......Tue, 11 Aug 2026 03:19:36 -0400https://pop.com.sg/tangent-to-the-x-1-axis.htmlhttps://pop.com.sg/tangent-to-the-x-1-axis.htmlFor a National Board Exam Review: Point (3,4) is the center of the circle tangent to the x-1 axis. What is the point of tangency? Answer is (3,0) I usually would provide an attempt but I do......Tue, 11 Aug 2026 03:15:55 -0400https://pop.com.sg/basic-properties-of-the-matrix-logarithm.htmlhttps://pop.com.sg/basic-properties-of-the-matrix-logarithm.htmlLet $|| \cdot ||$ be any submultiplicative norm on $\operatorname{Mat}_n(\mathbb C)$. For complex matrices $A$ with $||I_n - A|| < 1$, we can define the matrix logarithm $$\log(A) = \sum\limit......Tue, 11 Aug 2026 03:14:37 -0400https://pop.com.sg/pizza-topping-combination-problem.htmlhttps://pop.com.sg/pizza-topping-combination-problem.htmlI have two pizzas, I can choose up to 3 topping, there are 7 toppings to choose from. Double toppings, triple toppings of the same kind are allowed. Order doesn't matter, ham on top of chicke......Tue, 11 Aug 2026 03:06:24 -0400https://pop.com.sg/reducing-graph-3-coloring-to-10-coloring.htmlhttps://pop.com.sg/reducing-graph-3-coloring-to-10-coloring.htmlI am trying to show that the NP-Complete problem of 3-coloring a graph reduces to the problem of 10-coloring a graph.I have already shown how 10-coloring can be verified in polynomial time, and is ......Tue, 11 Aug 2026 02:56:34 -0400https://pop.com.sg/is-serge-lang-039-s-algebra-still-worth-reading.htmlhttps://pop.com.sg/is-serge-lang-039-s-algebra-still-worth-reading.htmlIs Serge Lang's famous book Algebra nowadays still worth reading, or are there other, more modern books which are better suited for an overview over all areas of algebra? EDIT: My main concern is ......Tue, 11 Aug 2026 02:54:17 -0400https://pop.com.sg/finding-the-general-solution-for-a-differential-equation.htmlhttps://pop.com.sg/finding-the-general-solution-for-a-differential-equation.htmlI'm practicing for the DE midterm, going through (supposedly) basic questions to practice. I am now stuck on this question. I am supposed to find the general equation for the given DE. I don't kno......Tue, 11 Aug 2026 02:53:07 -0400https://pop.com.sg/why-negative-factorial-doesn-039-t-exists.htmlhttps://pop.com.sg/why-negative-factorial-doesn-039-t-exists.htmlI've been told that factorials of negative numbers doesn't exists that's what I also found while trying to calculate factorial of negative $1$. But, I can see that graph of factorial $x$ is even ex......Tue, 11 Aug 2026 02:44:45 -0400https://pop.com.sg/calculating-average-with-minimum-and-maximum.htmlhttps://pop.com.sg/calculating-average-with-minimum-and-maximum.htmlMaybe it's a simple question but I have not any idea for that at the moment! How we can calculate average value for N samples with having only the minimum and maximum values?...Tue, 11 Aug 2026 02:43:24 -0400https://pop.com.sg/what-is-the-fastest-way-to-multiply-two-digit-numbers.htmlhttps://pop.com.sg/what-is-the-fastest-way-to-multiply-two-digit-numbers.htmlI been playing different math games on my Android lately (for example: Math Cruncher). I've noticed that I'm unable to quickly (under 7-8 seconds) multiply two digit numbers (i.e $ 18 * 17$). So my ...Tue, 11 Aug 2026 02:36:19 -0400https://pop.com.sg/is-symmetric-difference-xor-commutative.htmlhttps://pop.com.sg/is-symmetric-difference-xor-commutative.htmlI'm working on a problem: If A, B, C, and D are sets, does it follow that (A $\oplus$ B) $\oplus$ (C $\oplus$ D) = (A $\oplus$ C) $\oplus$ (B $\oplus$ D)? How does one approach something like thi......Tue, 11 Aug 2026 02:33:38 -0400https://pop.com.sg/bernoulli-differential-equation-of-second-order.htmlhttps://pop.com.sg/bernoulli-differential-equation-of-second-order.htmlHow one can solve a Bernoulli differential equation of second order? i.e., solve the DE \begin{align} \frac{{d^2 y}}{{dx^2 }} + p\left( x \right)\frac{{dy}}{{dx}} + q \left( x \right)y = g\left( x \...Tue, 11 Aug 2026 02:12:03 -0400https://pop.com.sg/finding-stable-distribution-of-a-stochastic-matrix.htmlhttps://pop.com.sg/finding-stable-distribution-of-a-stochastic-matrix.htmlso im currently looking at Markov diagrams and stochastic matrices and have become quite stuck on some parts of this question below and was looking for some help. Here is what I have done so far, ......Tue, 11 Aug 2026 01:57:31 -0400https://pop.com.sg/even-odd-or-neither.htmlhttps://pop.com.sg/even-odd-or-neither.htmlI would like someone to verify my solutions to the problems above? 9a. even 9b. odd 10. neither...Tue, 11 Aug 2026 01:56:31 -0400https://pop.com.sg/differences-between-homomorphisms-and-functions-closed.htmlhttps://pop.com.sg/differences-between-homomorphisms-and-functions-closed.htmlIncredibly, I haven't been able to find a good entry online comparing these two mathematical structures. Their relative familiarity doesn't make their boundaries less fuzzy. Here is a coarse attempt: ...Tue, 11 Aug 2026 01:39:32 -0400https://pop.com.sg/finding-the-volume-of-a-rectangular-based-pyramid-using-calculus.htmlhttps://pop.com.sg/finding-the-volume-of-a-rectangular-based-pyramid-using-calculus.htmlI have a maths test coming up, and I just can't seem to solve a question on finding volumes of solids (via integration). Here's the question: Find the volume of a pyramid with height h and rectang......Tue, 11 Aug 2026 01:33:22 -0400https://pop.com.sg/explain-motivation-behind-recurrence-of-100-sided-dice-problem.htmlhttps://pop.com.sg/explain-motivation-behind-recurrence-of-100-sided-dice-problem.htmlThe following question is asked in another MSE post. For completeness, I restate the question below. The following question is from a Jane Street interview. You are given a 100-sided die. After yo......Tue, 11 Aug 2026 01:30:38 -0400https://pop.com.sg/cardinality-of-cartesian-product-of-finite-sets-closed.htmlhttps://pop.com.sg/cardinality-of-cartesian-product-of-finite-sets-closed.htmlIf $a = \{1,2,3\}$ and $b = \{a,b,c\},\;$ FIND $\;n(a\times b)$ Or is it impossible to multiply these sets? What will be the answer?...Tue, 11 Aug 2026 01:29:57 -0400https://pop.com.sg/how-does-x-h-3-x-3-3x-2h-3xh-2-h-3.htmlhttps://pop.com.sg/how-does-x-h-3-x-3-3x-2h-3xh-2-h-3.htmlGood afternoon my wonderful friends! Whenever I do this equation I set it up using the difference of two cubes, which is as follows: $(a+b)^3 = (a+b)(a^2-ab+b^2) = a^3 + b^3$ Whenever I try to use ...Tue, 11 Aug 2026 01:27:31 -0400https://pop.com.sg/how-to-find-the-number-of-perfect-matchings-in-complete-graphs.htmlhttps://pop.com.sg/how-to-find-the-number-of-perfect-matchings-in-complete-graphs.htmlIn wikipedia FKT algorithm is given for planar graphs. Not anything for complete graphs. I need to find the number of perfect matchings in complete graph of six vertices....Tue, 11 Aug 2026 01:21:32 -0400https://pop.com.sg/how-to-find-the-number-of-sides-of-a-dice-with-an-unspecified-sides.htmlhttps://pop.com.sg/how-to-find-the-number-of-sides-of-a-dice-with-an-unspecified-sides.htmlPlease I need help on how to go about solving this question: A dice has $n$ sides with numbers $1,2,3,4,...,n$ marked on it. If the probability that a particular number appears at least once when the ...Tue, 11 Aug 2026 01:17:07 -0400https://pop.com.sg/wrong-result-in-wolfram-alpha-on-graph.htmlhttps://pop.com.sg/wrong-result-in-wolfram-alpha-on-graph.htmlI'm learning to code in Python, so I tried to find roots and draw a graph of a function $-(16x^2-24x+5)e^{-x}$ 1 Result i got in Python using the mat.plotlib library is this The problem is that ...Tue, 11 Aug 2026 01:10:14 -0400https://pop.com.sg/what-is-the-mathematical-point-of-straightedge-and-compass-constructio.htmlhttps://pop.com.sg/what-is-the-mathematical-point-of-straightedge-and-compass-constructio.htmlThe ancient discipline of construction by straightedge and compass is both fascinating and entertaining. But what is its significance in a mathematical sense? It is still taught in high school geom......Tue, 11 Aug 2026 01:09:48 -0400https://pop.com.sg/exterior-angle-of-a-reflex-angle.htmlhttps://pop.com.sg/exterior-angle-of-a-reflex-angle.htmlHow are exterior angles for reflex angles defined? Are they negative? This doubt came to my mind while thinking about the sum of the exterior angles of a concave polygon. I think it should be negat......Tue, 11 Aug 2026 01:09:18 -0400https://pop.com.sg/construct-the-intersection-of-a-cube-by-a-plane-through-3-points-on-it.htmlhttps://pop.com.sg/construct-the-intersection-of-a-cube-by-a-plane-through-3-points-on-it.htmlSo this is a rather old problem, but I still cannot find a pure constructive solution to it. Please, do not offer me to write a plane equation, etc. I would be grateful, if you offer a solution by ......Tue, 11 Aug 2026 01:09:09 -0400https://pop.com.sg/find-the-partial-derivative-for-e-xyz-6.htmlhttps://pop.com.sg/find-the-partial-derivative-for-e-xyz-6.htmlgiven $f(x,y,z)=e^{xyz^6}$ calculate $f_{xyz}$ $f_{x}= yz^6e^{xyz^6}$ using the chain rule and multiplying the derivative of the outisde times the inside then $f_{xy} = 5yz^6e^{xyz^6} + e^{xyz......Tue, 11 Aug 2026 01:04:52 -0400https://pop.com.sg/discrete-math-onto-1-1-functions.htmlhttps://pop.com.sg/discrete-math-onto-1-1-functions.htmlLet $S = \{3,B\}$ Give an example of a function $f: S \times S \to S$ that is onto. Give an example of a function $g: S \to S \times S$ that is 1-1. Give an example of a function $h: P(S) \to S \ti......Tue, 11 Aug 2026 01:04:22 -0400https://pop.com.sg/optimization-circle-and-square.htmlhttps://pop.com.sg/optimization-circle-and-square.htmlA piece of wire of length 60 is cut, and the resulting two pieces are formed to make a circle and a square. Where should the wire be cut to (a) minimize and (b) maximize the combined area of the ci......Tue, 11 Aug 2026 00:58:24 -0400https://pop.com.sg/prove-that-u-is-orthogonal-to-v-operatorname-proj-u-v-for-all-vectors-.htmlhttps://pop.com.sg/prove-that-u-is-orthogonal-to-v-operatorname-proj-u-v-for-all-vectors-.htmlHere's where I'm at, not sure where to go from here. Two vectors are orthogonal if their dot product is $0$. Knowing that; $$u \cdot (v - \operatorname{proj} u(v)) = 0$$ $$u \cdot \left(v - \fra......Tue, 11 Aug 2026 00:56:06 -0400https://pop.com.sg/rewriting-biconditional-statements-into-conditional-statement-without-.htmlhttps://pop.com.sg/rewriting-biconditional-statements-into-conditional-statement-without-.htmli'm currently taking discrete math and my study guide gives the problems without the answers. (very inconvenient).. but anyways i'm stuck on rewriting biconditional statements to conditional statem......Tue, 11 Aug 2026 00:54:23 -0400https://pop.com.sg/how-can-i-find-the-largest-perfect-square-in-a-really-big-number.htmlhttps://pop.com.sg/how-can-i-find-the-largest-perfect-square-in-a-really-big-number.htmlLet's say I want to find the largest number that when squared doesn't exceed 9223372036854775807. Or any other large number like that. How can I go about finding that? Is there some kind of functio......Tue, 11 Aug 2026 00:51:35 -0400https://pop.com.sg/limit-0-times-infinity-rewrite-to-find-the-limit.htmlhttps://pop.com.sg/limit-0-times-infinity-rewrite-to-find-the-limit.htmlI need some help with: $\lim_{x\to 0+} x^3\cdot e^{1/x}$. How to start? I've tried substitution $(y=1/x)$ without any luck. I would prefer not to use L'Hopitals rule and apologizes for a bad titl......Tue, 11 Aug 2026 00:46:40 -0400https://pop.com.sg/best-program-for-creating-educational-math-animations.htmlhttps://pop.com.sg/best-program-for-creating-educational-math-animations.htmlI'm looking for recommendations on what program to use for creating mathematical animations. These animations will be used in creating educational videos for high school math -- Trigonometry first,......Tue, 11 Aug 2026 00:43:08 -0400https://pop.com.sg/what-is-x-5-called.htmlhttps://pop.com.sg/what-is-x-5-called.htmlI have CS background, and I'm trying to teach fundamentals of programming to students. They seem to have difficulties in understanding the difference between x=5 and x==5. However, trying to explain, ...Tue, 11 Aug 2026 00:41:36 -0400https://pop.com.sg/cubic-differentiation-for-g-x-x-3-2x-10.htmlhttps://pop.com.sg/cubic-differentiation-for-g-x-x-3-2x-10.htmlI was told to solve for $x$ in $g(x) = x^3+2x-10$ using the cubic equation here (https://math.vanderbilt.edu/schectex/courses/cubic/). So I took $a = 1, b = 0, c = 2, d = -10$, plugged into the cu......Tue, 11 Aug 2026 00:28:48 -0400https://pop.com.sg/how-do-i-prove-that-161038-is-a-pseudoprime-to-base-2.htmlhttps://pop.com.sg/how-do-i-prove-that-161038-is-a-pseudoprime-to-base-2.htmlIf you say $$n=161038=2\cdot 73\cdot 1103$$ I know $n$ is a pseudoprime if $$2^n \equiv2 \mod n$$ I don't really know where to go from here. Can someone please help me in the right direction. Thanks....Tue, 11 Aug 2026 00:27:13 -0400https://pop.com.sg/good-book-for-self-study-of-functional-analysis.htmlhttps://pop.com.sg/good-book-for-self-study-of-functional-analysis.htmlI am a EE grad. student who has had one undergraduate course in real analysis (which was pretty much the only pure math course that I have ever done). I would like to do a self study of some basic ...Tue, 11 Aug 2026 00:23:58 -0400https://pop.com.sg/ramification-definition-confusion.htmlhttps://pop.com.sg/ramification-definition-confusion.htmlI recently out of curiosity came across this paper summarizing Tate's thesis which, in its introductory section, claims that, given a number field $K/\mathbb{Q}$, a prime ideal $\wp$ of $O_K$ is ra......Tue, 11 Aug 2026 00:17:19 -0400https://pop.com.sg/why-are-10-sided-dice-not-bipyramids.htmlhttps://pop.com.sg/why-are-10-sided-dice-not-bipyramids.htmlCommonly used $10$-sided dice are pentagonal trapezohedrons, as opposed to pentagonal bipyramids. Given that bipyramids are a more "obvious" shape for a fair die with an even number of faces, it's ...Tue, 11 Aug 2026 00:15:06 -0400https://pop.com.sg/is-there-a-name-for-this-strange-solution-to-a-quadratic-equation-invo.htmlhttps://pop.com.sg/is-there-a-name-for-this-strange-solution-to-a-quadratic-equation-invo.htmlHere's an elementary question on solving the following quadratic equation (well, it's not a quadratic until the square root is eliminated): $$\sqrt{x+5} + 1 = x$$ Upon solving the above equation ......Tue, 11 Aug 2026 00:09:30 -0400https://pop.com.sg/what-is-the-edge-set-of-a-multigraph.htmlhttps://pop.com.sg/what-is-the-edge-set-of-a-multigraph.htmlAn edge set of a graph is a set of doubletons, pairing edges. For example: has an edge set of $\{\{6,4\},\{4,5\},\{4,3\},\{5,2\},\{5,1\},\{3,2\},\{1,2\}\}$. A set, by definition, cannot have dupli......Tue, 11 Aug 2026 00:08:32 -0400https://pop.com.sg/what-is-the-derivative-of-e-i-pi.htmlhttps://pop.com.sg/what-is-the-derivative-of-e-i-pi.htmlI know that $e^{i\pi}$ =-1 the derivative of $e^{i\pi}$ is the derivative of - 1 which is 0. I guess I'm missing a rule or understood something wrong....Mon, 10 Aug 2026 23:57:53 -0400https://pop.com.sg/how-to-factorise-x-2-1.htmlhttps://pop.com.sg/how-to-factorise-x-2-1.htmlKeep in mind that I am not asking how to find the solutions to the equation $x^2 + 1 = 0$. I know that there will be imaginary answers In this case. I just want to know how to factor this using a l......Mon, 10 Aug 2026 23:54:22 -0400https://pop.com.sg/regular-matrix-and-regular-stochastic-matrix.htmlhttps://pop.com.sg/regular-matrix-and-regular-stochastic-matrix.htmlWe know that : A matrix is regular if its determinant is non zero. A stochastic matrix is regular if at a certain power all elements are positive. Question is how can I make the link between the......Mon, 10 Aug 2026 23:52:07 -0400https://pop.com.sg/chain-rule-for-change-of-variables-2nd-order-pde.htmlhttps://pop.com.sg/chain-rule-for-change-of-variables-2nd-order-pde.htmlLet $x,y$ be independent variables, and we transform into $\xi = \xi(x,y), \eta = \eta(x,y)$. By the chain rule, we clearly have: $$ \frac{\partial u}{\partial x}=\frac{\partial u}{\partial \xi} \f......Mon, 10 Aug 2026 23:44:12 -0400https://pop.com.sg/rotation-about-a-point-other-than-the-origin.htmlhttps://pop.com.sg/rotation-about-a-point-other-than-the-origin.htmlIt is challenging for me to see the rotation of image ABCD to get to A',B',C',D'. It is easy to see the translation of the prime figure to the double prime, but not so much the pre-image to the pr......Mon, 10 Aug 2026 23:44:04 -0400https://pop.com.sg/how-is-the-mode-of-an-exponential-distribution-zero.htmlhttps://pop.com.sg/how-is-the-mode-of-an-exponential-distribution-zero.htmlThe exponential distribution has a mode of $0$, which, according to Wikipedia, means that $0$ "is the value that is most likely to be sampled". This is not what I would expect, given that the expon......Mon, 10 Aug 2026 23:42:26 -0400https://pop.com.sg/derivative-using-difference-quotient.htmlhttps://pop.com.sg/derivative-using-difference-quotient.htmlHow would I find the derivative of $f(x)=2x^{2}+2x+1?$ I know that $f'(x)=4x+2$ using the power rule, but but how should I solve it using difference quotient, can someone please show the step by s......Mon, 10 Aug 2026 23:42:21 -0400https://pop.com.sg/best-algebraic-topology-book-alternative-to-allen-hatcher-free-book.htmlhttps://pop.com.sg/best-algebraic-topology-book-alternative-to-allen-hatcher-free-book.htmlAllen Hatcher seems impossible and this is set as the course text? So was wondering is there a better book than this? It's pretty cheap book compared to other books on amazon and is free online. ......Mon, 10 Aug 2026 23:35:47 -0400https://pop.com.sg/what-is-the-expected-value-of-sample-mean.htmlhttps://pop.com.sg/what-is-the-expected-value-of-sample-mean.htmlI have a simple question. $X$ is a random variable with mean $μ$, and there is a sample of size $n$: $X_1, X_2, \cdots, X_n$. Then what is the expected value of the sample mean $\overline{X}$? Th......Mon, 10 Aug 2026 23:35:41 -0400https://pop.com.sg/find-sup-and-inf-of-frac-1-n.htmlhttps://pop.com.sg/find-sup-and-inf-of-frac-1-n.htmlCan you please verify if my solution is correct. let $(a_n=\frac{1}{n})$ be a sequence of real numbers. show that $\sup(a_n)=1$ and $\inf(a_n)=0$. since $$n+1>n$$ $$\frac{1}{n+1}<\frac{1}{n......Mon, 10 Aug 2026 23:25:57 -0400https://pop.com.sg/expectation-product-of-expectations-is-expectation-of-product.htmlhttps://pop.com.sg/expectation-product-of-expectations-is-expectation-of-product.htmlI have a random variable $Z(t)$ (which represents the number of cells at time $t$). I know that $Z(t+\Delta t) = \sum_{k=1}^{\Delta t} Z^{k}(t)$ with $Z^{k}(t)$ all independent. Now it's using thi......Mon, 10 Aug 2026 23:20:59 -0400https://pop.com.sg/defining-the-smash-product.htmlhttps://pop.com.sg/defining-the-smash-product.htmlGiven a topological space $X$, in the definition I'm using (from John Lee's Introduction to Topological Manifolds), the quotient space $X/A$ formed by collapsing $A$ to a point, requires $A$ to be a ...Mon, 10 Aug 2026 22:57:52 -0400https://pop.com.sg/integral-test-error-approximation.htmlhttps://pop.com.sg/integral-test-error-approximation.htmlFind an N so that: $\sum_{n=1}^\infty \frac{1}{n^4}$ is between $\sum_{n=1}^N \frac{1}{n^4}$ and $\sum_{n=1}^N \frac{1}{n^4} + 0.005$ I am getting N = 200. Is this correct? I don't know if I am d......Mon, 10 Aug 2026 22:48:37 -0400https://pop.com.sg/curious-integrals-for-jacobi-theta-functions-int-0-1-vartheta-n-0-q-dq.htmlhttps://pop.com.sg/curious-integrals-for-jacobi-theta-functions-int-0-1-vartheta-n-0-q-dq.htmlThere are various identities for the Jacobi Theta Functions $\vartheta_n(z,q)$ on the MathWorld page and on the Wikipedia page. But I found no integral identities for these functions. Meanwhile, t......Mon, 10 Aug 2026 22:48:36 -0400https://pop.com.sg/drawing-marbles-out-of-a-bag.htmlhttps://pop.com.sg/drawing-marbles-out-of-a-bag.htmlThe infamous drawing marbles from a bag type question that I am used to seeing has became more complicated. Given a bag with 2 green marbles, 5 red marbles, and 3 yellow marbles. I then draw 2 ma......Mon, 10 Aug 2026 22:48:16 -0400https://pop.com.sg/relationship-between-complex-number-and-vectors.htmlhttps://pop.com.sg/relationship-between-complex-number-and-vectors.htmlWhat is the relation between complex numbers and vectors? I have read in some places "a complex number a 2-dimensional vector". One can easily observe that $i\cdot i=-1$ in complex multiplicatio......Mon, 10 Aug 2026 22:42:49 -0400https://pop.com.sg/calculate-the-wronskian-of-f-t-t-t-and-g-t-t-2-on-the-following-interv.htmlhttps://pop.com.sg/calculate-the-wronskian-of-f-t-t-t-and-g-t-t-2-on-the-following-interv.htmlHow to Calculate the Wronskian of $f(t)=t|t|$ and $g(t)=t^2$ on the following intervals: $(0,+\infty)$, $(-\infty, 0)$ and $0$. And then how would I show that the Wronskian of the two functions $f......Mon, 10 Aug 2026 22:41:34 -0400https://pop.com.sg/probability-of-one-pair-hand-in-poker-5-cards.htmlhttps://pop.com.sg/probability-of-one-pair-hand-in-poker-5-cards.htmlI have been working on the problem of probability of poker hands, I have been able to calculate the probability of each hand except one pair and high card hand. Here is what I have P1 P2 X1 X2 X......Mon, 10 Aug 2026 22:36:28 -0400https://pop.com.sg/how-to-integrate-displaystyle-1-e-1-x-2.htmlhttps://pop.com.sg/how-to-integrate-displaystyle-1-e-1-x-2.htmlHow to integrate $\displaystyle 1-e^{-1/x^2}$ ? as hint is given: $\displaystyle\int_{\mathbb R}e^{-x^2/2}=\sqrt{2\pi}$ If i substitute $u=\dfrac{1}{x}$, it doesn't bring anything: $\,\displaystyl......Mon, 10 Aug 2026 22:34:42 -0400https://pop.com.sg/parallel-postulate-from-playfair-039-s-axiom.htmlhttps://pop.com.sg/parallel-postulate-from-playfair-039-s-axiom.htmlParallel postulate: If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely......Mon, 10 Aug 2026 22:33:43 -0400https://pop.com.sg/what-is-the-difference-between-operatorname-arcsin-operatorname-arcsin.htmlhttps://pop.com.sg/what-is-the-difference-between-operatorname-arcsin-operatorname-arcsin.htmlThis is a homework problem I have. I don't remember learning the difference, and searching hasn't helped explain the difference between the capitalization....Mon, 10 Aug 2026 22:08:35 -0400https://pop.com.sg/definition-of-a-subfield.htmlhttps://pop.com.sg/definition-of-a-subfield.htmlGiven $\mathbb{K}$ a field, what is the definition subfield of $\mathbb{K}$? Intuitively, I think a subfield as a subset of a field which itself is a field, in other words, satisfy the axioms that a ...Mon, 10 Aug 2026 22:07:38 -0400https://pop.com.sg/questions-tagged-uniform-convergence.htmlhttps://pop.com.sg/questions-tagged-uniform-convergence.htmlQ&A for people studying math at any level and professionals in related fields...Mon, 10 Aug 2026 21:57:34 -0400https://pop.com.sg/how-to-solve-int-1-0-1-7x-1-3-dx.htmlhttps://pop.com.sg/how-to-solve-int-1-0-1-7x-1-3-dx.htmlI worked through $\int^1_0 (1+7x)^{1/3}dx$ and I got $\frac{3}{4}+C$ for the answer. However, I forgot about the exponent when I found the difference of the sides of the integral. I am retrying it......Mon, 10 Aug 2026 21:54:42 -0400https://pop.com.sg/gravitational-potential-energy-near-the-surface-of-the-earth.htmlhttps://pop.com.sg/gravitational-potential-energy-near-the-surface-of-the-earth.htmlOkay so for an object of mass m at a distance r from the Earth's centre, the GPE is $$U(r) = {{ - GMm} \over r}$$ For an object at height z above the surface (which obviously means at radius $r = {r_{...Mon, 10 Aug 2026 21:49:33 -0400https://pop.com.sg/roots-of-sum-of-two-polynomials-with-known-roots.htmlhttps://pop.com.sg/roots-of-sum-of-two-polynomials-with-known-roots.htmlI am writing a piece of software and I'm trying to avoid root finding polynomials for efficiency purposes. I have two polynomials with complex coefficients, where the roots of both polynomials are ......Mon, 10 Aug 2026 21:48:39 -0400https://pop.com.sg/finding-the-area-of-a-quadrilateral.htmlhttps://pop.com.sg/finding-the-area-of-a-quadrilateral.htmlI have a quadrilateral whose four sides are given $2341276$, $34374833$, $18278172$, $17267343$ units. How can I find out its area? What would be the area?...Mon, 10 Aug 2026 21:48:17 -0400https://pop.com.sg/derivative-of-e-x-with-respect-to-y.htmlhttps://pop.com.sg/derivative-of-e-x-with-respect-to-y.htmlI recently came across a question that asked for the derivative of $e^x$ with respect to $y$. I answered $\frac{d}{dy}e^x$ but the answer was $e^x\frac{dx}{dy}$. How is that the answer? I am confused....Mon, 10 Aug 2026 21:34:43 -0400https://pop.com.sg/how-is-3-equiv-3-bmod-5.htmlhttps://pop.com.sg/how-is-3-equiv-3-bmod-5.htmlJust tried googling but couldn't find any example, but how $3\equiv 3\bmod{5}$ Googled it...Mon, 10 Aug 2026 21:34:13 -0400https://pop.com.sg/the-row-reducing-matrix-with-complex-entries.htmlhttps://pop.com.sg/the-row-reducing-matrix-with-complex-entries.html$$\begin{pmatrix} -2i&-3&7\\ 0&-1-2i&5\\ 0&-1&1-2i \end{pmatrix} \mathbf{v} =0 \implies \begin{pmatrix} -2i&-3&7\\ 0&-1-2i&5\\ 0&0&0 \end{pmatrix} \m......Mon, 10 Aug 2026 21:31:40 -0400https://pop.com.sg/notation-of-the-second-derivative-where-does-the-d-go.htmlhttps://pop.com.sg/notation-of-the-second-derivative-where-does-the-d-go.htmlIn school I was taught that we use $\frac{du}{dx}$ as a notation for the first derivative of a function $u(x)$. I was also told that we could use the $d$ just like any variable. After some time we ......Mon, 10 Aug 2026 21:31:16 -0400https://pop.com.sg/getting-value-of-sin-x-cos-x-with-complex-exponential.htmlhttps://pop.com.sg/getting-value-of-sin-x-cos-x-with-complex-exponential.htmlI have a bit of difficulty with this. I am trying to express $\sin x + \cos x$ with complex exponential. I started by using Euler's equations. Then, I used the trigonometric substitution $\sin x =......Mon, 10 Aug 2026 21:24:09 -0400https://pop.com.sg/roots-of-equation-x-3-x-1-0.htmlhttps://pop.com.sg/roots-of-equation-x-3-x-1-0.htmlIf α , β , γ are the roots of equation $x^3 -x -1 =0$ then $$ \frac{1+\alpha}{1-\alpha} + \frac{1+\beta}{1-\beta} + \frac{1+\gamma}{1-\gamma} $$ My ...Mon, 10 Aug 2026 21:23:57 -0400https://pop.com.sg/most-general-antiderivative-involving-sec-x.htmlhttps://pop.com.sg/most-general-antiderivative-involving-sec-x.htmlI'm stumped on how to get the most general antiderivative, $F(x)$, of $f(x)=e^x+3secx(tan x + sec x)$. First, I split the equation on addition, since $\int[f(x)+g(x)]dx=\int f(x)dx+\int g(x)dx$ ......Mon, 10 Aug 2026 21:17:18 -0400https://pop.com.sg/analytical-solution-to-2d-diffusion-equation-with-a-drift-term.htmlhttps://pop.com.sg/analytical-solution-to-2d-diffusion-equation-with-a-drift-term.htmlI have been trying to solve the 2D PDE $$\frac{\partial p(x,y,t)}{\partial t}=-\frac{\partial }{\partial x}p(x,y,t)+\frac{\partial ^2}{\partial x^2}p(x,y,t)+\frac{\partial ^2}{\partial y^2}p(x,y,t)$$ ...Mon, 10 Aug 2026 21:10:37 -0400https://pop.com.sg/should-we-define-something-after-we-have-prove-that-exists-based-on-th.htmlhttps://pop.com.sg/should-we-define-something-after-we-have-prove-that-exists-based-on-th.htmlIn last days I have a hard time to understand what a definition is in mathematics. Until today I thought definition had a dual role in mathematics. Dictionary role The first role is that it mereley ...Mon, 10 Aug 2026 21:09:33 -0400https://pop.com.sg/sides-of-a-triangle-square-roots.htmlhttps://pop.com.sg/sides-of-a-triangle-square-roots.htmlThis is the exercise: Let $a,b,c\in \mathbb{R}^+$. Prove that the following propositions are equivalents: $a,b,c$ are sides of a triangle. $\sqrt{a},\sqrt{b},\sqrt{c}$ are sides of an acute triang......Mon, 10 Aug 2026 21:04:12 -0400https://pop.com.sg/how-do-i-find-all-points-of-discontinuity-of-the-function-f-x-x-2-x-3-.htmlhttps://pop.com.sg/how-do-i-find-all-points-of-discontinuity-of-the-function-f-x-x-2-x-3-.htmlBasically it is a modulus $\operatorname{fun}^c$. To begin with we'll redefine the function, i.e. $f(x)=|x+2|+|x-3|$. Taking $$ u = |x+2| = \begin{cases} -(x+2), & \text{if $x<0$,} \\ x+2, &......Mon, 10 Aug 2026 20:59:39 -0400https://pop.com.sg/why-is-triangle-in-p-p-np.htmlhttps://pop.com.sg/why-is-triangle-in-p-p-np.htmlI'm learning about $P/NP$ and my friend used an example in which he said that if you have a triangle in an undirected graph which is basically a set of three nodes in which all pairs of nodes are ...Mon, 10 Aug 2026 20:51:16 -0400https://pop.com.sg/euler-product-formula-for-riemann-zeta-function-proof.htmlhttps://pop.com.sg/euler-product-formula-for-riemann-zeta-function-proof.htmlIn class we introduced Reimann Zeta function $$ \zeta (x)=\sum_{n=1}^{+\infty} \frac{1}{n^x} $$ And we proved its domain was $D=(1,+\infty)$ Now Euler proved that $$ \zeta(x)=\prod_{p\text{ pr......Mon, 10 Aug 2026 20:51:05 -0400https://pop.com.sg/computing-the-derivative-of-a-system-of-equations-in-the-neighborhood-.htmlhttps://pop.com.sg/computing-the-derivative-of-a-system-of-equations-in-the-neighborhood-.htmlI'm solving the following problem on an old exam in real analysis. Thus, only such methods may be used. The system \begin{align*} \begin{cases} \sin(x+y)+\sin(y+z)+z=0 \\ \cos(x+y)......Mon, 10 Aug 2026 20:36:07 -0400https://pop.com.sg/can-i-assume-that-a-biologist-will-know-what-lhs-and-rhs-mean-or-what-.htmlhttps://pop.com.sg/can-i-assume-that-a-biologist-will-know-what-lhs-and-rhs-mean-or-what-.htmlI am writing a scientific article with a few mathematical equations. Can I assume that my audience will know what lhs and rhs mean?...Mon, 10 Aug 2026 20:33:50 -0400https://pop.com.sg/utility-function-plotting-indifference-curves.htmlhttps://pop.com.sg/utility-function-plotting-indifference-curves.htmlI know how to plot indifference curves; simply take the utility function and plot some level curves in $2D$. But how to plot a specific indifference curve, so all bundles on it are indifferent to a ...Mon, 10 Aug 2026 20:16:33 -0400https://pop.com.sg/laplace-equation-fourier-transform.htmlhttps://pop.com.sg/laplace-equation-fourier-transform.htmlSuppose $$ \begin{cases} u_{xx} + u_{yy} = 0, & (x,y) \in \mathbb R \times (0, \infty) \\ u_{y}(x,0) = f(x). \end{cases} $$ Show that $$u(x,y)= \frac{1}{2 \pi} \int_{-\infty}^{\infty} \ln((x-z)......Mon, 10 Aug 2026 20:13:16 -0400https://pop.com.sg/how-can-i-generate-the-binary-representation-of-any-real-number.htmlhttps://pop.com.sg/how-can-i-generate-the-binary-representation-of-any-real-number.htmlIn p. 30 of Baby Rudin, I find a reference to the fact that the binary representation of a real number implies the uncountablity of the set of real numbers. But I have two questions: Does every real ...Mon, 10 Aug 2026 20:00:35 -0400https://pop.com.sg/where-to-start-with-the-inverse-galois-problem.htmlhttps://pop.com.sg/where-to-start-with-the-inverse-galois-problem.htmlI have always been interested in the Inverse Galois Problem since I read the story of Évariste Galois. I have recently finished up courses in rings, fields, groups, and finally Galois theory. I rea......Mon, 10 Aug 2026 19:56:14 -0400https://pop.com.sg/smooth-sawtooth-wave-y-x-cos-x-cos-x-cos-x-dots.htmlhttps://pop.com.sg/smooth-sawtooth-wave-y-x-cos-x-cos-x-cos-x-dots.htmlConsider an infinite recursive function $$y(x)=\cos(x-\cos(x-\cos(x-\dots)))$$ $$y=\cos(x-y)$$ Plotting the function $y(x)$ implicitly we get a smooth sawtooth-like wave: Was this function s......Mon, 10 Aug 2026 19:56:06 -0400