https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usThu, 13 Aug 2026 19:46:43 -0400https://pop.com.sg/what-is-the-difference-of-pure-strategic-equilibria-and-nash-equilibri.htmlhttps://pop.com.sg/what-is-the-difference-of-pure-strategic-equilibria-and-nash-equilibri.htmlAre they the same thing just named differently or with minor differences?...Thu, 13 Aug 2026 23:38:09 -0400https://pop.com.sg/how-to-change-the-limits-of-a-summation-when-the-index-k-is-replaced-b.htmlhttps://pop.com.sg/how-to-change-the-limits-of-a-summation-when-the-index-k-is-replaced-b.htmlIs what I am doing below correct, please assist. $$\sum_{k=-\infty}^{-1}\frac{e^{kt}}{1-kt} = \sum_{k=1}^{\infty}\frac{e^{-{kt}}}{1-kt}$$ Is this the rule on how to "invert" the limits, and doe......Thu, 13 Aug 2026 23:23:54 -0400https://pop.com.sg/what-exactly-is-in-the-standart-atlas-of-r-n.htmlhttps://pop.com.sg/what-exactly-is-in-the-standart-atlas-of-r-n.htmlI saw a theorem saying that any atlas on a manifold can be extended to a unique maximal atlas. Concerning the standard atlas of $\mathbb{R}^n$, we saw that this was the atlas generated by the char......Thu, 13 Aug 2026 23:11:42 -0400https://pop.com.sg/can-navier-stokes-equation-be-derived-from-cauchy-momentum-equation.htmlhttps://pop.com.sg/can-navier-stokes-equation-be-derived-from-cauchy-momentum-equation.html$$\frac{{\partial (\rho {v}_x})}{\partial t}+\nabla\cdot{(\rho {v}_x\mathbf{v}) = -(\nabla P)_x + \rho g_x+\nabla\cdot\mathbf{\tau_x}}$$ This is the $x$ component of the Cauchy momentum equation, w......Thu, 13 Aug 2026 23:08:43 -0400https://pop.com.sg/let-g-be-a-group-and-a-in-z-g-prove-that-langle-a-rangle-is-a-normal-s.htmlhttps://pop.com.sg/let-g-be-a-group-and-a-in-z-g-prove-that-langle-a-rangle-is-a-normal-s.htmlLet $G$ be a group and $a\in Z(G)$, where $$ Z(G) =\lbrace x \in G\mid xg=gx~~ \forall g\in G\}.$$ Prove that $H=\langle a\rangle$ is a normal subgroup of $G$. To show that $H$ is normal in $G$, I ...Thu, 13 Aug 2026 23:00:10 -0400https://pop.com.sg/how-to-find-a-conserved-quantity-in-this-differential-equation.htmlhttps://pop.com.sg/how-to-find-a-conserved-quantity-in-this-differential-equation.htmlConsider the system: $$\ddot x = x^3 -x$$ What is the method to follow to find a conserved quantity for this system? So far what I have is: $\dot x = y$ and $\dot y = x^3 - x$ and I can find the ...Thu, 13 Aug 2026 22:58:23 -0400https://pop.com.sg/confusion-regarding-log-x-and-ln-x.htmlhttps://pop.com.sg/confusion-regarding-log-x-and-ln-x.htmlI was solving an integral and I encountered in some question $$\displaystyle \int_{2}^{4}\frac{1}{x} \, \mathrm dx$$ I know its integration is $\log(x)$. But my answer comes correct when I use $\......Thu, 13 Aug 2026 22:55:23 -0400https://pop.com.sg/tiling-a-square-problem.htmlhttps://pop.com.sg/tiling-a-square-problem.htmlQuestion: A 2018 x 2018 square is tiles with some 4 x 1 rectangles and some 2 x 2 squares. One of the 4 x 1 rectangles is removes and replaced with a 2 x 2 square. Is it still possible to tile the ...Thu, 13 Aug 2026 22:53:33 -0400https://pop.com.sg/dividing-divergent-and-convergent-sequences.htmlhttps://pop.com.sg/dividing-divergent-and-convergent-sequences.htmlLet $a_n$ be a diverging sequence and $b_n$ be a converging sequence. With that in mind does $\frac{a_n}{b_n}$ always diverge? If so, what is the proof for this?...Thu, 13 Aug 2026 22:51:21 -0400https://pop.com.sg/how-far-does-a-free-falling-object-fall-in-the-3-rd-second.htmlhttps://pop.com.sg/how-far-does-a-free-falling-object-fall-in-the-3-rd-second.htmlThe question: An object is dropped from a cliff. How far does the object fall in the 3rd second?" I calculated that a ball dropped from rest from a cliff will fall $45\text{ m}$ in $3 \text{ s}$, ...Thu, 13 Aug 2026 22:45:50 -0400https://pop.com.sg/ellipse-word-problem.htmlhttps://pop.com.sg/ellipse-word-problem.htmlThe ellipse is 5 meters across and 8 meters long with decorative fountains located at the foci. How far from the center should the fountains be located? (Rounded to the nearest hundredth). How far ......Thu, 13 Aug 2026 22:43:58 -0400https://pop.com.sg/changing-from-quadratic-formula-to-standard-form.htmlhttps://pop.com.sg/changing-from-quadratic-formula-to-standard-form.htmlThe graph of a quadratic function has $x$-intercepts $-1$ and $3$ and a range consisting of all numbers less than or equal to $4$. Determine an expression for the function. This is my problem. I k......Thu, 13 Aug 2026 22:39:45 -0400https://pop.com.sg/where-does-r-1-2-cos-theta-have-tangents.htmlhttps://pop.com.sg/where-does-r-1-2-cos-theta-have-tangents.htmlWhere does: $$r = 1 + 2\cos(\theta)$$ Have horizontal and vertical tangent lines? $x = r\cos(\theta) = \cos(\theta) + 2\cos^2(\theta)$ $y = r\sin(\theta) = \sin(\theta) + 2\sin(\theta)\cos(\the......Thu, 13 Aug 2026 22:32:28 -0400https://pop.com.sg/how-should-i-approach-finding-the-derivative-using-the-limit-process-f.htmlhttps://pop.com.sg/how-should-i-approach-finding-the-derivative-using-the-limit-process-f.htmlI am setting up the problem to find the limit as delta x goes to zero, using the definition in the correct format. No matter what I've tried, I am stuck without finding the right cancellations to ......Thu, 13 Aug 2026 22:29:18 -0400https://pop.com.sg/proof-of-the-least-upper-bound-or-completeness-axiom-corollary-greates.htmlhttps://pop.com.sg/proof-of-the-least-upper-bound-or-completeness-axiom-corollary-greates.htmlFirst off, I don't think this is a duplicate. Sorry if it is! Second this is not a homework question, so completeness is much appreciated. I am trying to prove the corollary of the least upper bound ...Thu, 13 Aug 2026 22:25:51 -0400https://pop.com.sg/how-to-show-that-the-sequence-frac-n-2-n-1-2n-2-4n-1-converges-to-frac.htmlhttps://pop.com.sg/how-to-show-that-the-sequence-frac-n-2-n-1-2n-2-4n-1-converges-to-frac.htmlGood day, I was wondering how I could prove that $$\lim_{n\to \infty}\frac{n^2+n+1}{2n^2-4n+1}=1/2$$ By definition....Thu, 13 Aug 2026 22:25:35 -0400https://pop.com.sg/proof-by-induction-cup-n-1-infty-a-n-is-countable-for-countable-a-n.htmlhttps://pop.com.sg/proof-by-induction-cup-n-1-infty-a-n-is-countable-for-countable-a-n.htmlHere is Theorem 1.5.8 in "Understanding Analysis" by S. Abbott, 2nd edition. Theorem 1.5.8 (i) If $A_1, A_2, ..., A_n$ are each countable sets, then the union $A_1\cup A_2 \cup ... \cup A_n$ is ...Thu, 13 Aug 2026 22:20:53 -0400https://pop.com.sg/evaluation-of-surface-integral-over-the-unit-disk-and-plane-in-xy-axis.htmlhttps://pop.com.sg/evaluation-of-surface-integral-over-the-unit-disk-and-plane-in-xy-axis.htmlEvaluation of $$\int_{S}\int(x^2+y^2+z^2)dS$$ Where $S:z=x+y,x^2+y^2\leq 1$ What i try: Let $g(x,y,z)=x^2+y^2+z^2$ and $-1\leq x,y\leq 1$ Let $S(x,y)=<x,y,x+y>$ Then $S_{x}(x,y)=<1,0,1......Thu, 13 Aug 2026 22:11:18 -0400https://pop.com.sg/question-about-bearings-and-trig.htmlhttps://pop.com.sg/question-about-bearings-and-trig.htmlA plane travelled for 600 km from airport A, on a bearing of 210°, before changing direction and travelling a further 250 km on a bearing of 300° to land at airport B. Calculate the three-figure be......Thu, 13 Aug 2026 22:08:05 -0400https://pop.com.sg/finding-the-turning-point-of-a-parabola.htmlhttps://pop.com.sg/finding-the-turning-point-of-a-parabola.htmlI know how to find the turning point of a parabola in most equations but I'm not sure how to solve it in this form, if anyone can help me please do! $$y=x^2+4x-5$$ Thanks...Thu, 13 Aug 2026 22:04:09 -0400https://pop.com.sg/what-is-the-difference-between-closed-form-expression-and-analytic-exp.htmlhttps://pop.com.sg/what-is-the-difference-between-closed-form-expression-and-analytic-exp.htmlWhat is the difference between closed-form expression and analytic expression? I often see them get referenced in settings where (in my opinion) they are essentially interchangeable. What is a ta......Thu, 13 Aug 2026 21:57:28 -0400https://pop.com.sg/series-expansion-for-e-x-2.htmlhttps://pop.com.sg/series-expansion-for-e-x-2.htmlI was interested in one of the topics on this website, while I noticed a notation that I haven't been able to understand. This is the original question. One of the answers to the question states th......Thu, 13 Aug 2026 21:52:22 -0400https://pop.com.sg/how-to-find-a-minimal-polynomial-field-theory.htmlhttps://pop.com.sg/how-to-find-a-minimal-polynomial-field-theory.htmlI was asked to find a minimal polynomial of $$\alpha = \frac{3\sqrt{5} - 2\sqrt{7} + \sqrt{35}}{1 - \sqrt{5} + \sqrt{7}}$$ over Q. I'm not able to find it without the help of WolframAlpha, which s......Thu, 13 Aug 2026 21:48:23 -0400https://pop.com.sg/is-a-matrix-a-with-an-eigenvalue-of-0-invertible.htmlhttps://pop.com.sg/is-a-matrix-a-with-an-eigenvalue-of-0-invertible.htmlJust wanted some input to see if my proof is satisfactory or if it needs some cleaning up. Here is what I have. Proof Suppose $A$ is square matrix and invertible and, for the sake of contradic......Thu, 13 Aug 2026 21:45:56 -0400https://pop.com.sg/why-is-the-sum-over-all-positive-integers-equal-to-1-12-duplicate.htmlhttps://pop.com.sg/why-is-the-sum-over-all-positive-integers-equal-to-1-12-duplicate.htmlRecently, sources for mathematical infotainment, for example numberphile, have given some information on the interpretation of divergent series as real numbers, for example $\sum_{i=0}^\infty i = ......Thu, 13 Aug 2026 21:37:51 -0400https://pop.com.sg/direct-sum-vs-direct-product-vs-tensor-product-closed.htmlhttps://pop.com.sg/direct-sum-vs-direct-product-vs-tensor-product-closed.htmlThere are a lot of questions like this all over the site, but I cannot find one that resolved my confusion- what are the formal definitions of direct sums, direct products, and tensor products (in ......Thu, 13 Aug 2026 21:35:01 -0400https://pop.com.sg/mississippi-problem.htmlhttps://pop.com.sg/mississippi-problem.htmlHow many arrangement of the letters in MISSISSIPPI have at least 2 adjacent S's? I was thinking that I can glue two of the S's together, so there will be 9 letters plus the special letter SS, and ......Thu, 13 Aug 2026 21:33:32 -0400https://pop.com.sg/cosine-integral.htmlhttps://pop.com.sg/cosine-integral.htmlShow that $$\int_0^x \frac{1-\cos(t)}{t}=\gamma+\ln(x)-\operatorname{Ci}(x)$$ where $$\operatorname{Ci}(x)=-\int_x^\infty \frac{\cos(t)}{t} \, dt$$ and gamma is an euler-mascheroni constant. I did......Thu, 13 Aug 2026 21:27:14 -0400https://pop.com.sg/imo-1988-problem-6.htmlhttps://pop.com.sg/imo-1988-problem-6.htmlIn 1988, IMO presented a problem, to prove that $k$ must be a square if $a^2+b^2=k(1+ab)$, for positive integers $a$, $b$ and $k$. I am wondering about the solutions, not obvious from the proof. B......Thu, 13 Aug 2026 21:25:47 -0400https://pop.com.sg/using-the-definition-of-derivative-to-find-tan-2x.htmlhttps://pop.com.sg/using-the-definition-of-derivative-to-find-tan-2x.htmlThe instructions: Use the definition of derivative to find $f'(x)$ if $f(x)=\tan^2(x)$. I've been working on this problem, trying every way I can think of. At first I tried this method: $$\lim_{h......Thu, 13 Aug 2026 21:24:15 -0400https://pop.com.sg/understanding-kleene-star-in-relation-to-regular-expressions.htmlhttps://pop.com.sg/understanding-kleene-star-in-relation-to-regular-expressions.htmlI have a homework problem for my intro computation class where I have to write a regular expression for an NFA. In order to help understand if my language is correct- I want to post some example re......Thu, 13 Aug 2026 20:59:25 -0400https://pop.com.sg/basis-of-fundamental-cycles.htmlhttps://pop.com.sg/basis-of-fundamental-cycles.htmlIn graph theory, for a connected, undirected finite graph one can easily find a set of fundamental cycles - I understand that part. This forms a basis of some linear space of all cycles of that gra......Thu, 13 Aug 2026 20:53:42 -0400https://pop.com.sg/vector-space-vs-subspace.htmlhttps://pop.com.sg/vector-space-vs-subspace.htmlCan someone explain the difference between a subspace and a vector space? I realize that a vector space has 10 axioms that define how vectors can be added and subtracted. I also realize that a subs......Thu, 13 Aug 2026 20:53:21 -0400https://pop.com.sg/finding-limits-of-rational-functions-as-x-to-infty-duplicate.htmlhttps://pop.com.sg/finding-limits-of-rational-functions-as-x-to-infty-duplicate.htmlPossible Duplicate: Finding the limit of $\frac{Q(n)}{P(n)}$ where $Q,P$ are polynomials $$\displaystyle \lim_{x\to\infty}\frac{(2x^2+1)^2}{(x-1)^2(x^2+x)}.$$ I do not know where to start o......Thu, 13 Aug 2026 20:44:56 -0400https://pop.com.sg/odds-of-being-correct-x-times-in-a-row.htmlhttps://pop.com.sg/odds-of-being-correct-x-times-in-a-row.htmlIs there a simple way to know what the chances are of being correct for a given number of opportunities? To keep this simple: I am either right or wrong with a 50/50 chance. What are the odds that......Thu, 13 Aug 2026 20:40:25 -0400https://pop.com.sg/explanation-of-the-bounded-convergence-theorem.htmlhttps://pop.com.sg/explanation-of-the-bounded-convergence-theorem.htmlTheorem (Bounded Convergence Theorem) Let $\{f_n\}$ be a sequence of measurable functions on a set of finite measure $E$. Suppose $\{f_n\}$ is uniformly pointwise bounded on $E$, that is , there i......Thu, 13 Aug 2026 20:36:57 -0400https://pop.com.sg/to-prove-that-f-n-x-frac-nx-1-n-2x-2-does-not-uniformly-converge-to-f-.htmlhttps://pop.com.sg/to-prove-that-f-n-x-frac-nx-1-n-2x-2-does-not-uniformly-converge-to-f-.htmlMy Approach. To prove the given statement it is sufficient to show that $$\exists \epsilon > 0, \exists x \in [0,1], \forall N \in \mathbb{N}, \exists n > N $$ $$ \Rightarrow | f_n(x) - f(x)|......Thu, 13 Aug 2026 20:26:42 -0400https://pop.com.sg/subtracting-exponents-with-same-base.htmlhttps://pop.com.sg/subtracting-exponents-with-same-base.htmlIf $2^4 - 2^3 = 2^3$ and $2^5 - 2^4 = 2^4$, then is below a rule of subtracting exponents with similar base and exponents which are just $1$ away from each other? $$A^e - A^{e-1} = A^{e-1}$$ I will ...Thu, 13 Aug 2026 20:25:08 -0400https://pop.com.sg/how-do-i-write-code-for-solving-partial-derivatives-numerically.htmlhttps://pop.com.sg/how-do-i-write-code-for-solving-partial-derivatives-numerically.htmlAs stated in the title. I am trying to write a function which evaluates the partial derivative at two points (a,b) for f. However, the output of the partial derivative evaluated at (0,0) is way too ...Thu, 13 Aug 2026 20:17:02 -0400https://pop.com.sg/why-frac-log-x-log-y-gives-the-same-value-as-frac-ln-x-ln-y.htmlhttps://pop.com.sg/why-frac-log-x-log-y-gives-the-same-value-as-frac-ln-x-ln-y.htmlIf $x=16384$ and $y=2$ $\ln(x)=9.704$ $\ln(y)=0.6931$ $\log(x)=4.2144$ $\log(y)=0.3010$ If we divide $\frac{\ln(x)}{\ln(y)}$ we get $14$ and same answer for $\frac{\log(x)}{\log(y)}$. So can an......Thu, 13 Aug 2026 20:11:39 -0400https://pop.com.sg/line-dividing-closed-plane-figure-into-congruent-parts.htmlhttps://pop.com.sg/line-dividing-closed-plane-figure-into-congruent-parts.htmlToday in class a student observed and asked the following: While the diagonal of a rectangle is not a line of symmetry, it does cut the rectangle into two congruent parts. This is a noteworthy ...Thu, 13 Aug 2026 20:10:51 -0400https://pop.com.sg/determine-if-a-triangle-is-right-angled-with-only-coordinates-duplicat.htmlhttps://pop.com.sg/determine-if-a-triangle-is-right-angled-with-only-coordinates-duplicat.htmlWhat is the easiest way to determine that, given 3 coords, they DON'T form a right angled triangle? EG, (0, 0, 0), (0, 1, 0), (1, 0, 0) - forms a right angled triangle (0, 0, 0), (0, 1, 0), (1, 0......Thu, 13 Aug 2026 20:03:29 -0400https://pop.com.sg/is-the-number-0-2343434343434-rational-duplicate.htmlhttps://pop.com.sg/is-the-number-0-2343434343434-rational-duplicate.htmlConsider the following number: $$x=0.23434343434\dots$$ My question is whether this number is rational or irrational, and how can I make sure that a specific number is rational if it was written......Thu, 13 Aug 2026 19:54:35 -0400https://pop.com.sg/what-is-the-optimal-value-for-x-in-a-1-1000-random-number-generator-2-.htmlhttps://pop.com.sg/what-is-the-optimal-value-for-x-in-a-1-1000-random-number-generator-2-.htmlI would like a mathematical answer to this. If I have a random number generator algorithm (let's say 1 to 1000), and I simply pick a random number over and over until I see each of the 1000 possible ...Thu, 13 Aug 2026 19:49:07 -0400https://pop.com.sg/how-many-sewings-are-there-on-a-soccer-ball.htmlhttps://pop.com.sg/how-many-sewings-are-there-on-a-soccer-ball.htmlA soccer ball is obtained by sewing $20$ hexagonal pieces of leather and $12$ pieces of leather of pentagonal shape. A sewing joins together the sides of two adjacent pieces. How many sewings ......Thu, 13 Aug 2026 19:41:56 -0400https://pop.com.sg/smoothness-of-a-function-is-independent-from-the-chart.htmlhttps://pop.com.sg/smoothness-of-a-function-is-independent-from-the-chart.htmlI would like to revamp an old question (Smooth function on a manifold not dependent on coordinate chart) as I did not understand the reply. In Loring Tu's book "An Introduction to Manifolds" I read (...Thu, 13 Aug 2026 19:39:53 -0400https://pop.com.sg/what-is-the-geometric-interpretation-of-the-transpose.htmlhttps://pop.com.sg/what-is-the-geometric-interpretation-of-the-transpose.htmlI can follow the definition of the transpose algebraically, i.e. as a reflection of a matrix across its diagonal, or in terms of dual spaces, but I lack any sort of geometric understanding of the ...Thu, 13 Aug 2026 19:32:00 -0400https://pop.com.sg/negative-area-below-x-axis-and-above-f-x.htmlhttps://pop.com.sg/negative-area-below-x-axis-and-above-f-x.htmlI was finding the area below the x-axis and above $y = x^2 - 4x$. My outcome was a negative number (fair enough, it's under the x-axis), but wolframalpha for instance gives me a positive number (I ......Thu, 13 Aug 2026 19:22:41 -0400https://pop.com.sg/a-mod-b-when-b-is-greater-than-a.htmlhttps://pop.com.sg/a-mod-b-when-b-is-greater-than-a.htmlWhy 6 % 10 = 6 As fa as I know 10 % 6 = 4, but I don't know how to handle the first one Thanks!...Thu, 13 Aug 2026 19:21:23 -0400https://pop.com.sg/permutation-test-for-one-way-anova.htmlhttps://pop.com.sg/permutation-test-for-one-way-anova.htmlI want to know what steps to do the permutation test (and the R/SAS code) for one-way ANOVA? Would you give me the specific steps? Thanks, Buddies!...Thu, 13 Aug 2026 19:16:41 -0400https://pop.com.sg/riemann-rearrangement-theorem.htmlhttps://pop.com.sg/riemann-rearrangement-theorem.htmlThe Riemann rearrangement theorem states that if $\sum\limits_{n=0} ^{+ \infty} a_n$ is conditionally convergent and $M \in \mathbb{R}$ then there exists a permutation $ \sigma (n) $ such that $\sum\...Thu, 13 Aug 2026 19:15:56 -0400https://pop.com.sg/symmetric-matrix-multiplication.htmlhttps://pop.com.sg/symmetric-matrix-multiplication.htmlLet $A$ and $B$ be symmetric matrices. Prove: $AB=BA$ $AB$ is a symmetric matrix As for 1. due to the axiom $(AB)^T=B^T A^T$ so $AB=BA$ As for 2. I did not find any axiom that can support the cla......Thu, 13 Aug 2026 19:01:14 -0400https://pop.com.sg/multiply-same-trig-functions.htmlhttps://pop.com.sg/multiply-same-trig-functions.htmlI apologize if this is a duplicate of another question in advance. I did not find any in the search. Does $\sin(x)\sin(y)=\sin(xy)$? I am wondering because I need to know if $\sin^2(44)=\sin(44)\si......Thu, 13 Aug 2026 18:53:55 -0400https://pop.com.sg/dimensions-of-singular-value-decomposition-matrices.htmlhttps://pop.com.sg/dimensions-of-singular-value-decomposition-matrices.htmlI have seen the Singular Value Decomposition (SVD) represented in two different ways and wanted to know a) if they're both correct and b) what their relationship is. In the first version a matrix ......Thu, 13 Aug 2026 18:52:36 -0400https://pop.com.sg/cardinality-of-a-power-set-is-given-be-2-n-why.htmlhttps://pop.com.sg/cardinality-of-a-power-set-is-given-be-2-n-why.htmlCardinality of power set of $A$ is determined by $2^n$, where n is the number of elements in Set $A$. How does this formula work? I was taught that there are $n$ elements in Set $A$ and each eleme......Thu, 13 Aug 2026 18:48:08 -0400https://pop.com.sg/how-to-find-vertices-of-a-rectangle-when-center-coordinates-and-angle-.htmlhttps://pop.com.sg/how-to-find-vertices-of-a-rectangle-when-center-coordinates-and-angle-.htmlNeed to find z1, z2 z3 and z4 when zc and r are given. Been trying to solve this seemingly simple geometry problem. Can't wrap my head around it. It's a rectangle with center coordinates given and......Thu, 13 Aug 2026 18:44:17 -0400https://pop.com.sg/card-drawing-with-replacement.htmlhttps://pop.com.sg/card-drawing-with-replacement.htmlIf 6 cards are drawn from a deck of cards with replacement. What is the probability that the outcome is a club at least once. I thought it would be 13/52. But I was wrong....Thu, 13 Aug 2026 18:36:44 -0400https://pop.com.sg/how-to-correctly-compate-f-n-and-g-n-when-working-through-o-n-notation.htmlhttps://pop.com.sg/how-to-correctly-compate-f-n-and-g-n-when-working-through-o-n-notation.htmlGoing through theory, missing the idea, need a bit of help. So, the initial state is: $$f(n) = O(g(n))$$ Assume that $f$ and $g$ are both nondecreasing and always bigger than 1. And, from my ...Thu, 13 Aug 2026 18:30:47 -0400https://pop.com.sg/calculate-the-area-of-the-crescent.htmlhttps://pop.com.sg/calculate-the-area-of-the-crescent.htmlI found this problem on a thread on Stack overflow where it was posted as "job interview question". Unfortunately I cannot find the question. But I saved the picture and just cannot figure it out. ......Thu, 13 Aug 2026 18:29:51 -0400https://pop.com.sg/how-to-find-the-closed-form-formula-for-this-recurrence-relation.htmlhttps://pop.com.sg/how-to-find-the-closed-form-formula-for-this-recurrence-relation.html$ x_{0} = 5 $ $ x_{n} = 2x_{n-1} + 9(5^{n-1})$ I have computed: $x_{0} = 5, x_{1} = 19, x_{2} = 83, x_{3} = 391, x_{4} = 1907$, but cannot see any pattern for the general $n^{th}$ term....Thu, 13 Aug 2026 18:21:13 -0400https://pop.com.sg/can-matrices-do-cross-product-with-vectors.htmlhttps://pop.com.sg/can-matrices-do-cross-product-with-vectors.htmlthis is a practical problem raised from my chemistry research, apologize if somewhere I used incorrect words. I'm dealing with the derivatives of a vector with respect to another vector. After some ...Thu, 13 Aug 2026 18:16:04 -0400https://pop.com.sg/why-does-a-monotonic-function-always-have-a-positive-rate-of-change.htmlhttps://pop.com.sg/why-does-a-monotonic-function-always-have-a-positive-rate-of-change.htmlBased on my book The rate of change of $f(u)$ as $u$ changes can be measured by looking at the change in $f$ between two values of $u$, divided by the change in $u$: $$\frac{Δf}{Δu} = \frac{f(......Thu, 13 Aug 2026 18:14:43 -0400https://pop.com.sg/this-statement-is-false-duplicate.htmlhttps://pop.com.sg/this-statement-is-false-duplicate.htmlIn propositional logic, a proposition is a statement that is either true or false, but not both. In a text I am reading and in many others, "this statement is false" is not considered a proposition......Thu, 13 Aug 2026 18:09:17 -0400https://pop.com.sg/the-sum-of-the-first-n-squares-1-4-9-cdots-n-2-is-frac-n-n-1-2n-1-6-du.htmlhttps://pop.com.sg/the-sum-of-the-first-n-squares-1-4-9-cdots-n-2-is-frac-n-n-1-2n-1-6-du.htmlProve that the sum of the first $n$ squares $(1 + 4 + 9 + \cdots + n^2)$ is $\frac{n(n+1)(2n+1)}{6}$. Can someone pls help and provide a solution for this and if possible explain the question...Thu, 13 Aug 2026 18:01:19 -0400https://pop.com.sg/pencils-of-circles.htmlhttps://pop.com.sg/pencils-of-circles.htmlI want to solve an exercise regarding the connection between the projective model of möbius geometry and pencils of circles. Consider the projective model of two-dimensional Möbius geometry P($R^{......Thu, 13 Aug 2026 17:59:32 -0400https://pop.com.sg/maximize-area-of-a-triangle-with-fixed-perimeter.htmlhttps://pop.com.sg/maximize-area-of-a-triangle-with-fixed-perimeter.htmlIf perimeter of a triangle is $2d$, what is the length of sides so the triangle has maximal area? I found some solution using circle and angles, but I think I have to use derivatives. I need ......Thu, 13 Aug 2026 17:55:43 -0400https://pop.com.sg/calculating-the-length-of-a-tangent-drawn-to-a-circle-from-a-named-poi.htmlhttps://pop.com.sg/calculating-the-length-of-a-tangent-drawn-to-a-circle-from-a-named-poi.htmlMy book (New Tertiary Mathematics Volume 1 Part 1, by C Plumpton and P S W Macilwaine) describes a method for calculating the length of a tangent to a circle from the point $(x_{1}, y_{1})$ outside......Thu, 13 Aug 2026 17:42:06 -0400https://pop.com.sg/applications-of-the-isomorphism-theorems.htmlhttps://pop.com.sg/applications-of-the-isomorphism-theorems.htmlIn my study of groups, rings, modules etc, I've seen the three isomorphism theorems stated and proved many times. I use the first one ( $G/\ker \phi \cong \operatorname{im} \phi$ ) very often, but ......Thu, 13 Aug 2026 17:26:43 -0400https://pop.com.sg/using-logic-laws-to-verify-the-logical-equivalences.htmlhttps://pop.com.sg/using-logic-laws-to-verify-the-logical-equivalences.htmlI need to use theorem 2.1.1 in order to verify the following: ∼((∼p ∧ q) ∨ (∼ p ∧ ∼q)) ∨ (p ∧ q) ≡ p I applied the Double Negative Law to get ((p ∧ q) ∨ (∼p ∧ ∼q)) ∨ (p ∧ q) I noticed we have (......Thu, 13 Aug 2026 17:24:42 -0400https://pop.com.sg/how-many-irrational-numbers.htmlhttps://pop.com.sg/how-many-irrational-numbers.htmlHow many irrational numbers can exist between two rational numbers ? As there are infinite numbers between two rational numbers and also there are infinite rational numbers between two rational num......Thu, 13 Aug 2026 17:23:45 -0400https://pop.com.sg/congruency-in-bow-tie-triangles.htmlhttps://pop.com.sg/congruency-in-bow-tie-triangles.htmlWe've just started congruency in my class, and we've stumbled across a question which goes like this: Prove that ∆AOB $\equiv$ ∆COD I drew up a diagram which portrays exactly the same informati......Thu, 13 Aug 2026 17:21:53 -0400https://pop.com.sg/trace-of-an-inverse-inequality-text-tr-a-1-ge-n-2-text-tr-a-1.htmlhttps://pop.com.sg/trace-of-an-inverse-inequality-text-tr-a-1-ge-n-2-text-tr-a-1.htmlLet $A \in \mathbb{R}^n$ be a positive definite matrix. Then, it is well known that \begin{align} \text{Tr}(A^{-1}) \ge n^2 \text{Tr}(A)^{-1} \end{align} The proof follows by using the fact that ......Thu, 13 Aug 2026 17:18:49 -0400https://pop.com.sg/ratio-of-circumference-to-diameter-on-a-sphere.htmlhttps://pop.com.sg/ratio-of-circumference-to-diameter-on-a-sphere.htmlI was listening to an audiobook of Einstein when they started discussing spherical geometry and how Pi was no longer the ratio of a circle's circumference to its diameter, so I set out to find the ......Thu, 13 Aug 2026 17:14:27 -0400https://pop.com.sg/how-to-define-being-inside-of-something-in-the-context-of-topology.htmlhttps://pop.com.sg/how-to-define-being-inside-of-something-in-the-context-of-topology.htmlI'm a Psychologist and Neuroscientist with interest in math and I just started reading about Topology. I have to say it's not easy to grasp the concepts without a practical example, so I'm trying to ...Thu, 13 Aug 2026 17:09:41 -0400https://pop.com.sg/function-domain-of-cubic-root-duplicate.htmlhttps://pop.com.sg/function-domain-of-cubic-root-duplicate.htmlDoes the domain of the function $y=\sqrt[3]{x^3+1}$ include $x<-1$? If yes, why is Mathematica and Wolfram Alpha not plotting that part of the function?...Thu, 13 Aug 2026 17:05:52 -0400https://pop.com.sg/why-are-horizontal-transformations-of-functions-reversed.htmlhttps://pop.com.sg/why-are-horizontal-transformations-of-functions-reversed.htmlWhile studying graph transformations I came across horizontal and vertical scale and translations of functions. I understand the ideas below. $f(x+a)$ - grouped with x, horizontal translation, inv......Thu, 13 Aug 2026 16:55:29 -0400https://pop.com.sg/how-to-convert-a-fraction-into-ternary.htmlhttps://pop.com.sg/how-to-convert-a-fraction-into-ternary.htmlIn a ternary system how is $\dfrac{1}{2}=0.\bar1$ ,$\dfrac{1}{3}=0.1=0.0\bar2$??, etc. In general, how does one write the ternary expression for a given fraction?...Thu, 13 Aug 2026 16:50:45 -0400https://pop.com.sg/is-there-a-symbol-or-abbreviation-for-coefficient-of-x.htmlhttps://pop.com.sg/is-there-a-symbol-or-abbreviation-for-coefficient-of-x.htmlDoing some binomial expansions with algebra where I need to equate different coefficients together but don't know what to write: [Coefficient of $x^3$] = _ $k=+1.5$...Thu, 13 Aug 2026 16:49:02 -0400https://pop.com.sg/why-is-this-the-basis-for-the-indiscrete-topology.htmlhttps://pop.com.sg/why-is-this-the-basis-for-the-indiscrete-topology.htmlIndiscrete topology only has $\{\varnothing, X\}$ where $X$ is the entire space. I know this is probably a dumb question. Why is the basis (according to wikipedia) for this topology just $\{X\}$ i......Thu, 13 Aug 2026 16:32:31 -0400https://pop.com.sg/questions-tagged-complex-integration.htmlhttps://pop.com.sg/questions-tagged-complex-integration.htmlQ&A for people studying math at any level and professionals in related fields...Thu, 13 Aug 2026 16:28:34 -0400https://pop.com.sg/probability-that-exactly-2-balls-are-white.htmlhttps://pop.com.sg/probability-that-exactly-2-balls-are-white.htmlWhat I did. I let $X$ be number of withdraws before $x$ white balls. We can call our succes in this case to be gettin a white ball and probability is of course $p = \frac{3}{6} = \frac{1}{2}$. So ......Thu, 13 Aug 2026 16:27:46 -0400https://pop.com.sg/if-x-2-5x-4-y-2-y-1-2y-for-all-real-values-of-y-then-what-interval-doe.htmlhttps://pop.com.sg/if-x-2-5x-4-y-2-y-1-2y-for-all-real-values-of-y-then-what-interval-doe.htmlOkay so I'm stuck on this one for a long time : If $(x^2-5x+4)(y^2+y+1)<2y$ for all real values of $y$ then what interval does $x$ belong to? My Try: Since $y$ can take any real value, I tr......Thu, 13 Aug 2026 16:06:22 -0400https://pop.com.sg/how-to-find-center-of-a-circle-from-only-an-arbitary-arc-of-that-circl.htmlhttps://pop.com.sg/how-to-find-center-of-a-circle-from-only-an-arbitary-arc-of-that-circl.htmlHow to find the center of a circle with given an arbitrary arc. we only have the arc nothing else. Is there any known equation or way to complete the circle....Thu, 13 Aug 2026 15:58:19 -0400https://pop.com.sg/proof-that-the-derivative-of-a-constant-is-zero.htmlhttps://pop.com.sg/proof-that-the-derivative-of-a-constant-is-zero.htmlI know that the derivative of a constant is zero, but the only proof that I can find is: given that $f(x) = {x}^{0}$, $$ f'(x) = \lim_{h\to 0} \frac {f(x+h) - f(x)}{h} $$ $$ f'(x) = \lim_{h\to 0} \......Thu, 13 Aug 2026 15:52:21 -0400https://pop.com.sg/difficulty-understanding-what-the-n-1-part-refers-to-in-a-basic-arithm.htmlhttps://pop.com.sg/difficulty-understanding-what-the-n-1-part-refers-to-in-a-basic-arithm.htmlI am 28 and learned no maths in school. I'm going back through and learning on my own time. I am up to arithmetic sequences. The general formula (according to Khan Academy) for a recursive arithmetic ...Thu, 13 Aug 2026 15:48:59 -0400https://pop.com.sg/integration-by-parts-twice.htmlhttps://pop.com.sg/integration-by-parts-twice.htmlUse the substition $u=4x-3$ to find $\int\frac{4x}{4x-3}dx$, giving your answer in terms of $x$ $$u=4x-3$$ $$4x=u+3$$ $$\frac{du}{dx}=4$$ $$dx=\frac{1}{4}du$$ $$\int\frac{4x}{4x-3}dx$$ $$\Rightarro......Thu, 13 Aug 2026 15:45:45 -0400https://pop.com.sg/matrix-product-notation.htmlhttps://pop.com.sg/matrix-product-notation.htmlMy lecturer has used some notation that I've never seen before: it is a (matrix) product symbol with a left-to-right arrow over the top. Does anybody know what this means? Thanks in advance. Edit......Thu, 13 Aug 2026 15:39:50 -0400https://pop.com.sg/understanding-the-rigorous-definition-of-continuity.htmlhttps://pop.com.sg/understanding-the-rigorous-definition-of-continuity.htmlI have an intuitive understanding of continuity, however, I’m struggling to understand the rigorous definition. I understand that a function is continuous when you can get infinitely close to each ...Thu, 13 Aug 2026 15:32:55 -0400https://pop.com.sg/domain-of-rational-functions.htmlhttps://pop.com.sg/domain-of-rational-functions.htmlI have the following definition for rational functions: "Let $p(x), q(x)$ be polynomials and define $D=\{x\in \mathbb{R} : q(x) \neq 0 \}$ Then, the function $$f:D \rightarrow \mathbb{R}$$ $$f......Thu, 13 Aug 2026 15:23:56 -0400https://pop.com.sg/what-are-the-components-in-the-ito-process.htmlhttps://pop.com.sg/what-are-the-components-in-the-ito-process.htmlWe know that if W(t) is a Brownian Motion, then the Ito Process is given by, X(t)= X(0) + $\int_0^t \Delta(u)dW(u)$ + $\int_0^t \Phi(u)du$ Could someone explain briefly what the two components a......Thu, 13 Aug 2026 15:23:33 -0400https://pop.com.sg/what-is-undecidability.htmlhttps://pop.com.sg/what-is-undecidability.htmlWhat does it mean that some problem is undecidable? For instance the halting problem. Does it mean that humans can never invent a new technique that always decides whether a turing machine will h......Thu, 13 Aug 2026 15:19:41 -0400https://pop.com.sg/finding-moment-of-inertia-of-a-cone-why-can-039-t-i-integrate-like-thi.htmlhttps://pop.com.sg/finding-moment-of-inertia-of-a-cone-why-can-039-t-i-integrate-like-thi.htmlI have a cone (vertex at $(x,y,z)=0$) with height $H$ and radius $a$ (radius of the base). Here is a picture: I want to find the moment of inertia around the $z$-axis using calculus. I know the in......Thu, 13 Aug 2026 15:15:04 -0400https://pop.com.sg/equivalent-systems-of-linear-equation.htmlhttps://pop.com.sg/equivalent-systems-of-linear-equation.htmlI've just begun to re-learn linear algebra because is so important, the book that I chose is naturally the Hoffman's for a lot of reason. Well, In the first chapter I'm stuck with the following, ...Thu, 13 Aug 2026 15:13:56 -0400https://pop.com.sg/real-valued-analytic-function-defined-on-a-connected-set-is-constant.htmlhttps://pop.com.sg/real-valued-analytic-function-defined-on-a-connected-set-is-constant.htmlLet $G$ be a connected set and $f : G \rightarrow \mathbb{C}$ a real valued analytic function. Prove that $f$ is constant. My idea to prove the result is to prove a subset $A \neq \varnothing$ of......Thu, 13 Aug 2026 15:13:38 -0400https://pop.com.sg/four-non-coplanar-points-in-mathbb-r-3-determine-an-unique-sphere.htmlhttps://pop.com.sg/four-non-coplanar-points-in-mathbb-r-3-determine-an-unique-sphere.htmlShow that four non-coplanar points in $\mathbb{R}^3$ determine an unique sphere. I have no idea how to solve this exercise. Thank you for your help....Thu, 13 Aug 2026 15:07:00 -0400https://pop.com.sg/finding-the-value-of-k-duplicate.htmlhttps://pop.com.sg/finding-the-value-of-k-duplicate.html$$π1:( 1+k , -3 , 6 )$$ $$π2: ( 1 , 5+k , 3k)$$ Does anyone know the values of k that would make these parallel and perpendicular, ive been trying for hours but nothing seems to work The original ...Thu, 13 Aug 2026 15:05:55 -0400https://pop.com.sg/no-mode-or-all-mode.htmlhttps://pop.com.sg/no-mode-or-all-mode.htmlLet's say I have a data set:- $${1,2,3,4,5}$$ I decide to find the mode though it would not be an appropriate measure of central tendency. But still........ Now this is an exceptional case and for ...Thu, 13 Aug 2026 15:05:45 -0400https://pop.com.sg/is-a-set-contained-in-its-cartesian-product-with-itself.htmlhttps://pop.com.sg/is-a-set-contained-in-its-cartesian-product-with-itself.htmlGiven the Cartesian product $X = A \times A$, is $A \subset X$?...Thu, 13 Aug 2026 14:38:22 -0400https://pop.com.sg/find-the-derivative-dy-dx-given-some-value-for-x.htmlhttps://pop.com.sg/find-the-derivative-dy-dx-given-some-value-for-x.htmlI know how to find the derivative of a function by itself given a function. I can use multiple rules to determine the derivative. However in a situation where I must find the derivative when the x......Thu, 13 Aug 2026 14:10:51 -0400https://pop.com.sg/how-to-prove-if-something-is-a-linear-transformation.htmlhttps://pop.com.sg/how-to-prove-if-something-is-a-linear-transformation.html(I think) My textbook says something is a linear transformation if $L(ax) = aL(x)$ $L(x+y) = L(x) + L(y)$ $L(x) = A(x)$ But a lot of sites I've been on haven't proved these 3 things, so I just w......Thu, 13 Aug 2026 13:57:13 -0400