https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usFri, 28 Aug 2026 22:38:53 -0400https://pop.com.sg/difference-between-horizontal-and-vertical-line-tests.htmlhttps://pop.com.sg/difference-between-horizontal-and-vertical-line-tests.htmlTrying to understand what the differnce is between a vertical and horizontal line test. If an equation fails the vertical line test, what does that tell you about the graph? If an equation fails the ...Sat, 29 Aug 2026 02:37:53 -0400https://pop.com.sg/integrating-sec-2-x-from-first-principles.htmlhttps://pop.com.sg/integrating-sec-2-x-from-first-principles.htmlIs it possible to solve $\int \sec^2 x ~dx$ without knowing that $\frac{d}{dx}\tan x = \sec^2 x$?...Sat, 29 Aug 2026 02:24:59 -0400https://pop.com.sg/preparations-for-reading-algebraic-number-theory-by-serge-lang.htmlhttps://pop.com.sg/preparations-for-reading-algebraic-number-theory-by-serge-lang.htmlI am eager to learn algebraic number theory. It seems that Serge Lang's Algebraic Number Theory is one of the standard introductory texts (correct me if this is an inaccurate assessment). I flipped ...Sat, 29 Aug 2026 02:24:02 -0400https://pop.com.sg/how-to-prove-the-limit-of-a-sequence-using-epsilon-n.htmlhttps://pop.com.sg/how-to-prove-the-limit-of-a-sequence-using-epsilon-n.htmlI think I have a proper understanding of the general procedure, but I'm having difficulty manipulating my inequality so that I can isolate $n$ by itself. Sadly I wasn't given many examples to model......Sat, 29 Aug 2026 02:17:56 -0400https://pop.com.sg/wolfram-alpha-how-to-show-functions-with-abs-and-sgn-as-a-piecewise-fu.htmlhttps://pop.com.sg/wolfram-alpha-how-to-show-functions-with-abs-and-sgn-as-a-piecewise-fu.htmlI was trying to obtain this result: integrate e^|x| dx The result is $$ \int e^{|x|} dx = \frac12 e^{-x} \left( (e^x-1)^2 \operatorname{sgn}(x) -2e^x +e^{2x} - 1 \right) \color{gray}{\,+ \text{...Sat, 29 Aug 2026 02:05:48 -0400https://pop.com.sg/a-nice-number-is-an-integer-ending-in-3-or-7-when-written-out-in-decim.htmlhttps://pop.com.sg/a-nice-number-is-an-integer-ending-in-3-or-7-when-written-out-in-decim.htmlMy teacher just asked me a question like this but i do not know how to start and work it out at all. Can someone help me out with that?...Sat, 29 Aug 2026 01:59:47 -0400https://pop.com.sg/what-is-meant-by-maximal-proper-factors-of-a-integer.htmlhttps://pop.com.sg/what-is-meant-by-maximal-proper-factors-of-a-integer.htmlI understand what is meant by proper factors, e.g. the proper factors of 36 are 2, 3, 4, 6, 9, 12, & 18. However I've just seen the phrase "maximal proper factors" used in the context of determ......Sat, 29 Aug 2026 01:49:15 -0400https://pop.com.sg/richardson-039-s-extrapolation-of-composite-trapezoidal-rule.htmlhttps://pop.com.sg/richardson-039-s-extrapolation-of-composite-trapezoidal-rule.htmlI have applied Richardson's formula to the composite trapezoidal rule, $I_h(f)=\frac{h}{2}(f(a)+\sum_{k=1}^{n-1}f(a+kh)+f(b))$, in an attempt to better approximate the integral $I(f)=\int_0^1 e^{-x......Sat, 29 Aug 2026 01:47:28 -0400https://pop.com.sg/the-decimal-expansion-of-the-quotient-of-two-integers.htmlhttps://pop.com.sg/the-decimal-expansion-of-the-quotient-of-two-integers.htmlIt is an exercise in a book on discrete mathematics.How to prove that in the decimal expansion of the quotient of two integers, eventually some block of digits repeats. For example: $\frac { 1 }{ 6......Sat, 29 Aug 2026 01:47:17 -0400https://pop.com.sg/how-to-know-spot-a-repeated-root-in-a-graph.htmlhttps://pop.com.sg/how-to-know-spot-a-repeated-root-in-a-graph.htmlThe diagram shows a cubic curve passing through (–1, 0), (2, 0) and (0, –8). What is the equation of the curve. So I would have said: $-8 = k(x + 1)(x-2)$ $k = 4$ $y = 4(x+1)(x-2)$ but the root at ......Sat, 29 Aug 2026 01:46:56 -0400https://pop.com.sg/clean-proof-of-baker-campbell-hausdorff-formula.htmlhttps://pop.com.sg/clean-proof-of-baker-campbell-hausdorff-formula.htmlI am thinking of the cleanest way to prove the BCH formula and I have come up with this. First, work out $e^{\lambda A}Be^{-\lambda A}$ by expanding the exponentials (sums go from $0$ to $\infty$):......Sat, 29 Aug 2026 01:46:40 -0400https://pop.com.sg/is-this-proof-of-0-a-0-correct.htmlhttps://pop.com.sg/is-this-proof-of-0-a-0-correct.htmlI need to prove that $0*a = a*0 = 0$ using only axiomatic properties for real numbers(It's a exercise from Apostol's book. My try: We can write $0$ as $(a - a)$. Hence we can rewrite the equation as ...Sat, 29 Aug 2026 01:46:04 -0400https://pop.com.sg/finding-radius-of-circle.htmlhttps://pop.com.sg/finding-radius-of-circle.htmlTaking the angle at center as $x$, I got $r \sin x =2$. But after that how to proceed?...Sat, 29 Aug 2026 01:42:04 -0400https://pop.com.sg/length-of-cross-product-vector.htmlhttps://pop.com.sg/length-of-cross-product-vector.htmlI have two vectors $u$, and $v$, I know that $\mid u \mid$ = 3 and $\mid v \mid$ = 5, and that $u\cdot v = -12$. I need to calculate the length of the vector $(3u+2v) \times (3v-u)$. Because I kno......Sat, 29 Aug 2026 01:34:09 -0400https://pop.com.sg/why-is-1-mod-3-2-closed.htmlhttps://pop.com.sg/why-is-1-mod-3-2-closed.htmlTitle says it all. Google doesn't have anything. Trying to get it to let me submit by typing more words. There is nothing really to be added but it wont let me submit without this....Sat, 29 Aug 2026 01:25:43 -0400https://pop.com.sg/what-is-an-intuitive-meaning-of-e-overline-x-and-var-overline-x.htmlhttps://pop.com.sg/what-is-an-intuitive-meaning-of-e-overline-x-and-var-overline-x.htmlLet $X$ be a random variable distributed over, for example say, the Binomial Distribution. Then $P(X)$ is the probability of getting $x$ successful trials in $n$ total trials. So I saw a notation......Sat, 29 Aug 2026 01:19:13 -0400https://pop.com.sg/solution-vector-equation-involving-cross-product.htmlhttps://pop.com.sg/solution-vector-equation-involving-cross-product.htmlI am starting to study linear algebra, and a problem appeared at the beginning of the textbook, the problem is the following: What 3-vector u satisfies $(1,1,0)\times u=(0,1,1)$. My immediate respo......Sat, 29 Aug 2026 01:12:45 -0400https://pop.com.sg/tail-of-a-divergent-p-series-with-p-1.htmlhttps://pop.com.sg/tail-of-a-divergent-p-series-with-p-1.htmlConsider a divergent $p$-series with $p<1$, for instance $\sum_{j=0}^{\infty} \frac{1}{\sqrt{j}}$. Let $R_{n}=\sum_{j=n}^{\infty} \frac{1}{\sqrt{j}}$ be the tail of this series. Obviously $R_n$ ...Sat, 29 Aug 2026 01:09:06 -0400https://pop.com.sg/riemann-sum-with-subintervals-of-unequal-width.htmlhttps://pop.com.sg/riemann-sum-with-subintervals-of-unequal-width.htmlAt the opening of a chapter on Riemann Sums and definite integrals, my book gives the following example problem: Consider the region bounded by the graph of $$f(x)=\sqrt{x}$$ and the x-axis for ......Sat, 29 Aug 2026 01:07:23 -0400https://pop.com.sg/average-value-using-triple-integrals.htmlhttps://pop.com.sg/average-value-using-triple-integrals.htmlFind the average value of the function $f(x,y,z)=xyz$ over the tetrahedron with vertices, $(0,0,0),(1,0,0), (1,1,0), \text{and }(1,1,1)$...Sat, 29 Aug 2026 01:02:24 -0400https://pop.com.sg/when-to-begin-a-new-paragraph-closed.htmlhttps://pop.com.sg/when-to-begin-a-new-paragraph-closed.htmlWhen writing a mathematical article, when should someone begin a new paragraph? Is there some specific rule or convention? And more generally, what rules are there about articles-writing?...Sat, 29 Aug 2026 00:59:35 -0400https://pop.com.sg/why-there-is-no-formula-log-a-log-b-something.htmlhttps://pop.com.sg/why-there-is-no-formula-log-a-log-b-something.htmlWhen I was studying in high school, my teacher taught me about log(a) + log(b) = log(ab) log(a) - log(b) = log(a/b) log(a) / log(b) = log$_b$(a) Then, I asked my teacher "Why there is no formul......Sat, 29 Aug 2026 00:53:36 -0400https://pop.com.sg/a-function-that-is-bounded-and-measurable-but-not-lebesgue-integrable.htmlhttps://pop.com.sg/a-function-that-is-bounded-and-measurable-but-not-lebesgue-integrable.htmlCould you give me concrete examples about "a function that is bounded and measurable but not Lebesgue integrable". Royden's textbook "Real analysis" says a bounded measurable function is said to be ...Sat, 29 Aug 2026 00:51:08 -0400https://pop.com.sg/method-for-finding-marginal-cdf.htmlhttps://pop.com.sg/method-for-finding-marginal-cdf.htmlI wish to calculate the marginal CDF of a joint probability distribution function. However, I am unsure of the bounds I am supposed to use, and wish to verify it. Suppose I have the expression: $$f......Sat, 29 Aug 2026 00:44:31 -0400https://pop.com.sg/are-matrices-rank-2-tensors.htmlhttps://pop.com.sg/are-matrices-rank-2-tensors.htmlI know that this is sometimes the case, but that some matrices are not tensors. So what is the intuitive and specific demands of a matrix to also be a tensor? Does it need to be quadratic, singular......Sat, 29 Aug 2026 00:28:47 -0400https://pop.com.sg/how-do-you-simplify-cos-4-x-by-using-power-reducing-formulae-closed.htmlhttps://pop.com.sg/how-do-you-simplify-cos-4-x-by-using-power-reducing-formulae-closed.htmlHow do you simplify $\cos^4(x)$ using the power reducing formula?...Sat, 29 Aug 2026 00:20:02 -0400https://pop.com.sg/how-to-calculate-the-lower-and-upper-bound-of-the-error-in-estimating-.htmlhttps://pop.com.sg/how-to-calculate-the-lower-and-upper-bound-of-the-error-in-estimating-.htmlI have the following problem, I am trying to estimate the conversion rate of product sales online, my data is simple, I have the number of clicks and the number of sales for each product. Sales are ...Sat, 29 Aug 2026 00:06:20 -0400https://pop.com.sg/what-does-the-inverted-v-represent-in-math.htmlhttps://pop.com.sg/what-does-the-inverted-v-represent-in-math.htmlI know that A V B represents Logical disjunction which means A OR B and the result of it is false only when both A and B are false . But I still didn't understand what an inverted V means as shown ......Fri, 28 Aug 2026 23:47:44 -0400https://pop.com.sg/why-is-glide-symmetry-its-own-type.htmlhttps://pop.com.sg/why-is-glide-symmetry-its-own-type.htmlArtin's Algebra pages 155 & 156 list the types of symmetry of a plane figure as: Reflective Rotational Translational Glide He then goes on to say "Figures such as wallpaper patterns may have ......Fri, 28 Aug 2026 23:40:57 -0400https://pop.com.sg/what-are-collinear-lines.htmlhttps://pop.com.sg/what-are-collinear-lines.htmlI know about collinear points (points lying on a same straight line). I know about collinear vector (parallel vectors). But, coming to collinear lines, I am not even able to imagine them, and I can't ...Fri, 28 Aug 2026 23:40:08 -0400https://pop.com.sg/determine-whether-the-series-is-divergent-or-convergent-and-find-its-s.htmlhttps://pop.com.sg/determine-whether-the-series-is-divergent-or-convergent-and-find-its-s.htmlAny tips on how to start this would be great. I'm aware its not geometric but the answer key indicates it diverges, however I have no idea how to physically show that with this equation. $$\sum_{n=......Fri, 28 Aug 2026 23:38:16 -0400https://pop.com.sg/what-are-all-the-elements-of-the-group-of-symmetries-of-a-regular-tetr.htmlhttps://pop.com.sg/what-are-all-the-elements-of-the-group-of-symmetries-of-a-regular-tetr.htmlI can see that why the order of the group of symmetries of a regular tetrahedron is $12$ : Roughly speaking, each time one of ${\{1,2,3,4}\}$ is on 'top' and we do to the other $3$ as we did in a r......Fri, 28 Aug 2026 23:37:32 -0400https://pop.com.sg/if-i-is-a-number-then-what-is-its-numerical-value.htmlhttps://pop.com.sg/if-i-is-a-number-then-what-is-its-numerical-value.html$ i $ is the unit imaginary part of complex number , but there is a question which it is mixed me probably i missed the definition of a number , wolfram alpha $ i $ is assumed to be a number , and ...Fri, 28 Aug 2026 23:29:49 -0400https://pop.com.sg/proofs-for-parallel-and-perpendicular-lines.htmlhttps://pop.com.sg/proofs-for-parallel-and-perpendicular-lines.htmlDoes anyone know a good proof to show that all lines parallel to a specific line AB have the same slope? Also, does anyone know a good proof to show that any line perpendicular to a specific line C......Fri, 28 Aug 2026 23:26:57 -0400https://pop.com.sg/find-the-relative-maximum-and-relative-minimum-of-the-function-f-x-49x.htmlhttps://pop.com.sg/find-the-relative-maximum-and-relative-minimum-of-the-function-f-x-49x.htmlFind the relative maximum and relative minimum of the function $f(x) = 49x + \frac{4}{x}=49x+4x^{-1}$ Solution: Step 1: Find the values of $x$ where $f'(x)=0$ and $f'(x)$ DNE. $f'(x)=49-4x^{-2}=......Fri, 28 Aug 2026 23:16:34 -0400https://pop.com.sg/numerically-solving-a-poisson-equation-with-neumann-boundary-condition.htmlhttps://pop.com.sg/numerically-solving-a-poisson-equation-with-neumann-boundary-condition.htmlThe Problem Suppose I have an equation of the form $\nabla^2 \phi(x) = f(x)$ on the interval $A \le x \le B$, where $f(x)$ is known and $\phi(x)$ is unknown. I have Neumann-type boundary condition......Fri, 28 Aug 2026 23:13:52 -0400https://pop.com.sg/understanding-how-to-prove-when-a-subset-is-a-subgroup.htmlhttps://pop.com.sg/understanding-how-to-prove-when-a-subset-is-a-subgroup.htmlLemma 3.4. Let $(G ,*)$ be a group. A nonempty subset $H$ of $G$ is a subgroup of $(G,*)$, iff, for every $a, b\in H$, $a*b^{-1}\in H$. Proof. First, suppose $H$ is a subgroup. If $b\in H$, then $b......Fri, 28 Aug 2026 23:04:53 -0400https://pop.com.sg/graph-isomorphisms-on-6-vertices-with-degree-3.htmlhttps://pop.com.sg/graph-isomorphisms-on-6-vertices-with-degree-3.htmlI want to find another graph that has 6 vertices and each has degree $3$ that is not isomorphic to these two graphs below. I know that these two graphs are isomorphic. They will all have the same d......Fri, 28 Aug 2026 22:54:33 -0400https://pop.com.sg/necessary-to-prove-the-inverse-is-invertible.htmlhttps://pop.com.sg/necessary-to-prove-the-inverse-is-invertible.htmlI am just starting out on linear algebra and I have come to a section the book that confuses me somewhat. The authour defines an invertible matrix A as: "A square matrix A is said to be invertible......Fri, 28 Aug 2026 22:53:13 -0400https://pop.com.sg/what-039-s-the-proof-that-the-euler-totient-function-is-multiplicative.htmlhttps://pop.com.sg/what-039-s-the-proof-that-the-euler-totient-function-is-multiplicative.htmlThat is, why is $\varphi(AB)=\varphi(A)\varphi(B)$, if $A$ and $B$ are two coprime positive integers? It's not just a technical trouble—I can't see why this should be, intuitively: I bellyfeel ......Fri, 28 Aug 2026 22:47:34 -0400https://pop.com.sg/maximum-modulus-principle-and-open-mapping-theorem.htmlhttps://pop.com.sg/maximum-modulus-principle-and-open-mapping-theorem.htmlThe Maximum Modulus Principal states: let $f$ be a function holomorphic on some connected open subset $D$ of the complex plane $\mathbb{C}$ and taking complex values. If $z_{0}$ is a point in $D$ s......Fri, 28 Aug 2026 22:31:18 -0400https://pop.com.sg/pursuit-curves-solution.htmlhttps://pop.com.sg/pursuit-curves-solution.htmlFor our math class we have to do some calculations with respect to pursuit curves. The chased object starts at point $(p,0)$. Chaser starts at $(0,0)$.(x,y) Speed of the chased object is $u$. Speed ...Fri, 28 Aug 2026 22:27:27 -0400https://pop.com.sg/find-the-limit-as-x-approaches-infinity-of-e-x-lnx-and-sin2x-x.htmlhttps://pop.com.sg/find-the-limit-as-x-approaches-infinity-of-e-x-lnx-and-sin2x-x.htmlI'm not sure how to evaluate the limit as $x \rightarrow \infty $ of the following: $$ \bullet \lim_{x\to\infty}e^{-x}\cdot \log x $$ $$\bullet\lim_{x\to\infty} \frac{\sin(2x)}{x}$$...Fri, 28 Aug 2026 22:22:40 -0400https://pop.com.sg/what-are-the-ways-of-proving-that-the-cantor-set-is-uncountable-apart-.htmlhttps://pop.com.sg/what-are-the-ways-of-proving-that-the-cantor-set-is-uncountable-apart-.htmlWhat are the ways of proving that the Cantor set is uncountable apart from Cantor diagonalization? Are there any based on dynamical systems?...Fri, 28 Aug 2026 22:19:54 -0400https://pop.com.sg/limit-of-sin-x-as-x-approaches-zero-from-the-left.htmlhttps://pop.com.sg/limit-of-sin-x-as-x-approaches-zero-from-the-left.htmlI'm trying to find a proof that: $$ \lim_{x\to0^-}\sin(x)=0$$ I'd like to be able to do the proof without reference to advanced theorems (mean value theorem, series, etc). I have a geometric approa......Fri, 28 Aug 2026 22:14:14 -0400https://pop.com.sg/what-is-the-difference-between-square-of-sum-and-sum-of-square.htmlhttps://pop.com.sg/what-is-the-difference-between-square-of-sum-and-sum-of-square.htmlWhat is difference between square of sum $(\sum_{i=1}^{n}x_i)^2$ and sum of square $\sum_{i=1}^{n}x_i^2$? I think square of sum is bigger than sum of square but i can not find a relation between t......Fri, 28 Aug 2026 22:13:48 -0400https://pop.com.sg/on-laplace-transform-of-periodic-functions.htmlhttps://pop.com.sg/on-laplace-transform-of-periodic-functions.htmlI recently bumped into this theorem regarding the Laplace transform of periodic function: Given a periodic function $f(t)=f(t+p)$, where $p$ is its period, then its Laplace transform is given by: ......Fri, 28 Aug 2026 22:08:09 -0400https://pop.com.sg/least-squares-estimate-b1-formula.htmlhttps://pop.com.sg/least-squares-estimate-b1-formula.htmlMy regression textbook textbook says that $\sum_{i=1}^{n} (x_i - \overline{x}) = 0$. I know that: $\sum_{i=1}^{n} (x_i - \overline{x})(y_i - \overline{y})$ $= \sum_{i=1}^{n} (x_i - \overline......Fri, 28 Aug 2026 21:58:20 -0400https://pop.com.sg/linear-ode-roots-of-characteristic-equation-having-multiplicity-1.htmlhttps://pop.com.sg/linear-ode-roots-of-characteristic-equation-having-multiplicity-1.htmlI know the method. For a linear homogeneous ordinary differential equation with a root of the characteristic polynomial in $\alpha$ has multiplicity $k$, then $y=x^me^{\alpha x}$ with $m=0,1,\cdot......Fri, 28 Aug 2026 21:55:12 -0400https://pop.com.sg/explicit-formula-for-floor-x.htmlhttps://pop.com.sg/explicit-formula-for-floor-x.htmlIn number theory we have so-called explicit formula's in terms of the Riemann zeta zero's. For instance to count the sum of the logarithms of the primes below some given integer. (second Chebyshev ...Fri, 28 Aug 2026 21:51:07 -0400https://pop.com.sg/how-to-rotate-a-polar-equation.htmlhttps://pop.com.sg/how-to-rotate-a-polar-equation.htmlTake a simple polar equation like r = θ/2 that graphs out to: But, how would I achieve a rotation of the light-grey plot in this image (roughly 135 degrees)? Is there a way to easily shift the plot?...Fri, 28 Aug 2026 21:47:34 -0400https://pop.com.sg/finding-eigenpairs-of-matrix-a.htmlhttps://pop.com.sg/finding-eigenpairs-of-matrix-a.htmlHow do you find the eigenpairs of this? Given two matrices $A=\begin{bmatrix} a& b\\b&-a\end{bmatrix}$ and $B=\begin{bmatrix} a& -b\\b&a\end{bmatrix}$ where $b \neq 0$, find all ...Fri, 28 Aug 2026 21:44:19 -0400https://pop.com.sg/if-a-is-not-relatively-prime-to-n-prove-modulo-property.htmlhttps://pop.com.sg/if-a-is-not-relatively-prime-to-n-prove-modulo-property.htmlIf $n>1$ is integer and $1\le a \le n$ is integer such that $(a,n)\neq 1$ then prove there exist integer $1 \le b <n $ such that $ab \equiv 0(mod \; n)$ I have tried everything from going to ...Fri, 28 Aug 2026 21:42:37 -0400https://pop.com.sg/expression-for-the-current-as-a-function-of-time.htmlhttps://pop.com.sg/expression-for-the-current-as-a-function-of-time.htmlWhen the electromotive force (emf) is removed from a circuit containing inductance and resistance but no capacitors, the rate of decrease of current is proportional to the current. If the initial ...Fri, 28 Aug 2026 21:33:31 -0400https://pop.com.sg/evaluate-double-integral-of-triangular-region.htmlhttps://pop.com.sg/evaluate-double-integral-of-triangular-region.htmlEvaluate $$ \int\int_D e^{y^2} dA $$ where D is the triangular region with vertices (0,0), (0,1) and (2,1) My attempts: $$ \int^{2}_0 \int^{1}_\frac{x}{2} e^{y^2} dy dx $$ or $$ \int^{0}_1\int^{......Fri, 28 Aug 2026 21:32:22 -0400https://pop.com.sg/for-which-values-of-b-c-is-the-matrix-invertible.htmlhttps://pop.com.sg/for-which-values-of-b-c-is-the-matrix-invertible.html$A =\left(\begin{array}{ccc} 0 & 1 & b \\ -1 & 0 & c \\ -b & -c & 0 \end{array}\right)$ I tried to come up with the RREF($A$), but in the final step I have the matrix: $A ......Fri, 28 Aug 2026 21:27:37 -0400https://pop.com.sg/is-there-any-deep-reason-that-23456789-is-prime-closed.htmlhttps://pop.com.sg/is-there-any-deep-reason-that-23456789-is-prime-closed.htmlI was recently coding a prime factorization function and wanted to test that it could factor large numbers reasonably well. Arbitrarily, I slid my finger across the number pad, and got 23456789 as ......Fri, 28 Aug 2026 21:26:47 -0400https://pop.com.sg/recurrent-and-transient-states.htmlhttps://pop.com.sg/recurrent-and-transient-states.htmlI know which are the recurrent states {2,4,5} and It's clear to see it drawing a diagram but how can I prove that {2,4,5} are the recurrent states? In a formal way? Also the transient states are ......Fri, 28 Aug 2026 21:22:08 -0400https://pop.com.sg/ambiguity-in-rule-for-determining-an-outlier.htmlhttps://pop.com.sg/ambiguity-in-rule-for-determining-an-outlier.htmlI have the following set of numbers, in which I need to determine if there are any outliers: 11, 14, 21, 26, 29, 33, 61. I graphed them in Desmos (see above), and visually, it appears to me that 6......Fri, 28 Aug 2026 21:12:57 -0400https://pop.com.sg/method-of-characteristics-for-burgers-039-equation-with-rectangular-da.htmlhttps://pop.com.sg/method-of-characteristics-for-burgers-039-equation-with-rectangular-da.htmlUsing the method of characteristics, find a solution to Burgers' equation \begin{cases} u_t+\left(\frac{u^2}{2} \right)_x =0 & \text{in }\mathbb{R}\times(0,\infty) \\ \qquad \qquad \, \, u=g......Fri, 28 Aug 2026 20:57:09 -0400https://pop.com.sg/computing-the-iterated-logarithm-log-star-by-hand.htmlhttps://pop.com.sg/computing-the-iterated-logarithm-log-star-by-hand.htmlI'm having trouble figuring out how to compute the iterated logarithm (log-star) by hand without using a calculator or programming language. I wrote out the following program in Java to check my an......Fri, 28 Aug 2026 20:54:31 -0400https://pop.com.sg/how-to-choose-between-real-and-complex-coefficients-of-fourier-series.htmlhttps://pop.com.sg/how-to-choose-between-real-and-complex-coefficients-of-fourier-series.htmlSometimes in exercises we are asked to calculate the fourier series of a function. But there are two ways to do that. If $f:\Bbb R\mapsto\Bbb R$ is $T$-periodic over $\Bbb R$ then what conditions......Fri, 28 Aug 2026 20:50:02 -0400https://pop.com.sg/why-does-two-terms-immediately-adjacent-mean-multiply.htmlhttps://pop.com.sg/why-does-two-terms-immediately-adjacent-mean-multiply.htmlI am currently teaching a GED math class. While learning about the order of operations, the students asked why does a number next to a parentheses mean multiplication? I understand the rule that two ...Fri, 28 Aug 2026 20:40:35 -0400https://pop.com.sg/what-does-the-plus-sign-contained-in-a-circle-oplus-mean-in-this-case.htmlhttps://pop.com.sg/what-does-the-plus-sign-contained-in-a-circle-oplus-mean-in-this-case.htmlThe fundamental group of the torus is isomorphic to $\mathbb{Z}\oplus\mathbb{Z}$. I know that the $\oplus$ symbol is the exclusive or symbol but I don't understand how two of the same sets are XOR......Fri, 28 Aug 2026 20:18:57 -0400https://pop.com.sg/remainder-when-2-108-is-divided-by-11-closed.htmlhttps://pop.com.sg/remainder-when-2-108-is-divided-by-11-closed.htmlWhat is the remainder obtained when $2^{108}$ is divided by $11$? I tried bringing in $11$ in the given no. such as in $(11-3)^{36}$ and then using binomial expansion...but its not helping. Any ...Fri, 28 Aug 2026 20:08:18 -0400https://pop.com.sg/complexity-analysis-of-alpha-beta-pruning-of-a-full-tree.htmlhttps://pop.com.sg/complexity-analysis-of-alpha-beta-pruning-of-a-full-tree.htmlI am trying to understand the derivation of a time complexity for an alpha-beta pruning algorithm but up till now have not found any reasonable recourse. Many recourses claim that if you take a full ...Fri, 28 Aug 2026 20:04:37 -0400https://pop.com.sg/relationship-between-groups-and-permutations.htmlhttps://pop.com.sg/relationship-between-groups-and-permutations.htmlWhat is the relationship between groups and permutations? The way I understand it, a permutation is a set of numbers that you can do things with. A group is a set of elements equipped with an opera......Fri, 28 Aug 2026 20:03:35 -0400https://pop.com.sg/combinations-probability-dividing-9-people-into-3-groups.htmlhttps://pop.com.sg/combinations-probability-dividing-9-people-into-3-groups.htmlI'm struggling to understand the procedure to solve the following problem: The group consists of 9 people, 3 men and 6 women. They are divided randomly into 3 groups (3 persons each). What's the ...Fri, 28 Aug 2026 19:53:05 -0400https://pop.com.sg/how-do-i-calculate-the-value-of-this-series.htmlhttps://pop.com.sg/how-do-i-calculate-the-value-of-this-series.htmlI want to find the value to which this series converges $$\sum_{n=0}^\infty \frac{(-1)^n}{n^2+1}$$ I tried looking at the sequence of partial sums $$S_k = \sum_{n=0}^k \frac{(-1)^n}{n^2+1}$$ and I ...Fri, 28 Aug 2026 19:42:27 -0400https://pop.com.sg/riemannian-ricci-curvature-for-round-n-sphere.htmlhttps://pop.com.sg/riemannian-ricci-curvature-for-round-n-sphere.htmlWhat is the best way to see that the Ricci scalar curvature of $(S^n(r),g_{round})$ is a constant $n(n-1)/r^2$ ? I essentially only see this value stated in the literature, but no computation assoc......Fri, 28 Aug 2026 19:41:20 -0400https://pop.com.sg/do-non-convex-platonic-solids-exist.htmlhttps://pop.com.sg/do-non-convex-platonic-solids-exist.htmlConsider a solid with the following properties - It is composed of congruent, regular polygons. At each vertex, the same number of edges and faces meet. This is the same as the requirement for the ...Fri, 28 Aug 2026 19:34:14 -0400https://pop.com.sg/motivation-behind-point-set-topology-closed.htmlhttps://pop.com.sg/motivation-behind-point-set-topology-closed.htmlWhy should I study point-set topology? What initially interested me in topology was the pop-sci rubber sheet stuff or coffee cup-donut stuff or proving fundamental theorem of algebra using curves ......Fri, 28 Aug 2026 19:28:58 -0400https://pop.com.sg/binary-decimal-primes.htmlhttps://pop.com.sg/binary-decimal-primes.htmlConverting an integer $N$ into binary and then reading the result as a base-10 number $N_2$, the prime $N=3$ gives $N_2=11$ (which is also prime) and $N=5$ gives $N_2=101$. Are $3$ and $5$ the only......Fri, 28 Aug 2026 19:19:44 -0400https://pop.com.sg/proving-that-f-n-to-z-is-a-bijection-closed.htmlhttps://pop.com.sg/proving-that-f-n-to-z-is-a-bijection-closed.htmlProve that $f: \mathbb{N} \to \mathbb{Z} $ is a bijection where the relation between $\mathbb{N}$ and $\mathbb{Z}$ is as follows: $f(n)= \frac n2$ if n is even, and $-\frac {n+1}{2}$ if n is odd ...Fri, 28 Aug 2026 19:10:59 -0400https://pop.com.sg/what-exactly-is-a-number.htmlhttps://pop.com.sg/what-exactly-is-a-number.htmlWe've just been learning about complex numbers in class, and I don't really see why they're called numbers. Originally, a number used to be a means of counting (natural numbers). Then we extend t......Fri, 28 Aug 2026 19:09:35 -0400https://pop.com.sg/chain-rule-f-t-e-t-sin2t.htmlhttps://pop.com.sg/chain-rule-f-t-e-t-sin2t.htmlI'm missing a step somewhere on this: $$f(t)=e^{t\sin2t}.$$ I know I have to use the chain rule but I'm getting tripped up. here are my steps: $\frac d{dx}(e^{t\sin2t})$ $e^{t\sin2t} \frac d{dx}(t\...Fri, 28 Aug 2026 18:58:04 -0400https://pop.com.sg/what-exactly-is-calculus.htmlhttps://pop.com.sg/what-exactly-is-calculus.htmlI've researched this topic a lot, but couldn't find a proper answer to this, and I can't wait a year to learn it at school, so my question is: What exactly is calculus? I know who invented it,......Fri, 28 Aug 2026 18:42:47 -0400https://pop.com.sg/prove-left-frac-2-5-right-frac-2-5-ln-2.htmlhttps://pop.com.sg/prove-left-frac-2-5-right-frac-2-5-ln-2.htmlInadvertently, I find this interesting inequality. But this problem have nice solution? prove that $$\ln{2}>(\dfrac{2}{5})^{\frac{2}{5}}$$ This problem have nice solution? Thank you. ago,I find ...Fri, 28 Aug 2026 18:38:03 -0400https://pop.com.sg/radius-arc-length-what-angle-from-c-p.htmlhttps://pop.com.sg/radius-arc-length-what-angle-from-c-p.htmlI know the radius and length of an arc. how do I figure out the angle from center point to the ends of the arc? ( I'm trying to figure out the length of the arc if I offset/increase radius by $2$" ......Fri, 28 Aug 2026 18:37:01 -0400https://pop.com.sg/definition-of-continuous-map.htmlhttps://pop.com.sg/definition-of-continuous-map.htmlLet $X,Y$ be topological spaces, let $f:X\longrightarrow Y$ a function. There are several (equivalent) ways to define continuity of $f$, one of these says: $f$ is continuous if $$(1)\qquad \forall A\...Fri, 28 Aug 2026 18:29:07 -0400https://pop.com.sg/finding-the-upper-and-lower-bounds-positive-and-negative-each-if-appli.htmlhttps://pop.com.sg/finding-the-upper-and-lower-bounds-positive-and-negative-each-if-appli.htmlGiven $|z^4 -8 + 3|, |z|=2 $ I need to find the positive upper and lower bounds of the above expression. My attempt: By triangle inequality, we have $||z^4|+|-8z|+3|\geq|z^{4} -8z + 3| \leq ||z^4 ......Fri, 28 Aug 2026 18:22:45 -0400https://pop.com.sg/how-to-find-the-laplace-transform-of-f-t-t-e-t.htmlhttps://pop.com.sg/how-to-find-the-laplace-transform-of-f-t-t-e-t.htmlHow to find the Laplace transform of the following function: $$f(t)=t e^t$$ $$F(s)=\int_0^\infty (te^t e^{-st})dt$$ What method do I use to find the integral?...Fri, 28 Aug 2026 18:22:28 -0400https://pop.com.sg/what-039-s-the-point-of-trigonometric-proofs-identities-in-introductor.htmlhttps://pop.com.sg/what-039-s-the-point-of-trigonometric-proofs-identities-in-introductor.htmlI remember back in high school at some point delving into worksheet after worksheet of trigonometric "identities", the vast majority of which are basically restatements of $\sin^2(x) + \cos^2(x) = ......Fri, 28 Aug 2026 18:20:11 -0400https://pop.com.sg/how-to-correctly-find-csc-of-an-angle.htmlhttps://pop.com.sg/how-to-correctly-find-csc-of-an-angle.htmlAlright, so one of my questions is CSC (angle) -5. When I plug CSC in my calculator, it says "math error." I'm using a Casio fx-300 MS, and using shift + cos, then putting an angle, such as 90....Fri, 28 Aug 2026 18:17:35 -0400https://pop.com.sg/volume-of-an-irregular-octahedron-from-edge-lengths.htmlhttps://pop.com.sg/volume-of-an-irregular-octahedron-from-edge-lengths.htmlDoes anyone know how to calculate the volume of an irregular octahedron from the lengths of the edges? The octahedron has triangular faces, but the only information are the edge lengths. Alternat......Fri, 28 Aug 2026 17:57:38 -0400https://pop.com.sg/equation-or-equality.htmlhttps://pop.com.sg/equation-or-equality.htmlIs a numerical equality (like $1+2 = 3$) an algebraic equation? In google search we have: "equation ɪˈkweɪʒ(ə)n/ noun noun: equation; plural noun: equations 1. Mathematics a statement that the va......Fri, 28 Aug 2026 17:56:47 -0400https://pop.com.sg/how-to-solve-this-simple-2d-markov-chain.htmlhttps://pop.com.sg/how-to-solve-this-simple-2d-markov-chain.htmlI have 2D Markov chain like below : I have already solved for $p_{b,0}$ and $p_{0,d}$ like this : $$p_{b,0} = \rho^bp_{0,0}$$ $$p_{0,d} = \rho^dp_{0,0}$$ But I don't know how to get $p_{b,d}$.......Fri, 28 Aug 2026 17:54:23 -0400https://pop.com.sg/what-does-039-express-in-terms-of-x-039-mean.htmlhttps://pop.com.sg/what-does-039-express-in-terms-of-x-039-mean.htmlFor the following question : $f(x) = 2x^2 + 4x $ It asks me to express the following in terms of $x$: $f(-2x)$ What does the question mean by this? Does it mean make $x$ the subject?...Fri, 28 Aug 2026 17:45:10 -0400https://pop.com.sg/approximate-the-sum-of-each-alternating-series.htmlhttps://pop.com.sg/approximate-the-sum-of-each-alternating-series.htmlApproximate the sum of each series to three decimal places. $\sum {(-1)^n {1\over {n^3}}}$ From alternating series test, this series convergence. $S\approx a_3+S_2$ $S\approx {1\over {27}} + {7\ov......Fri, 28 Aug 2026 17:30:51 -0400https://pop.com.sg/whether-multiplication-by-a-constant-matrix-alone-can-represent-every-.htmlhttps://pop.com.sg/whether-multiplication-by-a-constant-matrix-alone-can-represent-every-.htmlI read recently that every function $f : \mathbb{R}^m \to \mathbb{R}^n$ can be written as $f(\mathbf{x}) = \mathbf{Ax}$ where $\mathbf{A}$ is a matrix of constants. On the one hand this is intuitive ...Fri, 28 Aug 2026 17:23:47 -0400https://pop.com.sg/determine-relationship-between-vectors.htmlhttps://pop.com.sg/determine-relationship-between-vectors.htmlUsing the product i would like to understand if it is possible to determine relationship between two vectors,let us consider the following problem: Vectors $u$ and $v$ have length $1$. Which of......Fri, 28 Aug 2026 17:22:17 -0400https://pop.com.sg/relationship-between-smooth-manifold-and-differentiable-manifold.htmlhttps://pop.com.sg/relationship-between-smooth-manifold-and-differentiable-manifold.htmlDefine a smooth manifold to be a ringed space (a Hausdorff, second countable, $n$-manifold) that is locally isomorphic (as a ringed space) to the sheaf of smooth real valued functions on some open ......Fri, 28 Aug 2026 17:22:07 -0400https://pop.com.sg/finding-the-range-of-a-function-without-graph.htmlhttps://pop.com.sg/finding-the-range-of-a-function-without-graph.htmlI'm asked to find the range of the function : $ (-x^2 , +1) $ Taking R as a real number which represents a quantity along a line and graphing : then the range for this function from viewing the ...Fri, 28 Aug 2026 17:20:40 -0400https://pop.com.sg/book-recommendation-hard-problems-for-multivariable-calculus-w-solutio.htmlhttps://pop.com.sg/book-recommendation-hard-problems-for-multivariable-calculus-w-solutio.htmlI'm looking for a text that covers roughly what's sometimes called "Calculus III" or multivariable calculus.* But this text must satisfy certain additional criteria: (1) It must be more in-depth (......Fri, 28 Aug 2026 17:19:05 -0400https://pop.com.sg/gompertz-function-as-a-finite-product.htmlhttps://pop.com.sg/gompertz-function-as-a-finite-product.htmlI have a growth model such as: $$G(n)= (1-1/N)(1-2/N)...(1-n/N) \text{, with } n<N$$ As analytical approximation of $G(n)$, I numerically found that $1-\exp(-\exp(1.71 (1-1.05 n/\sqrt{N}))$ w......Fri, 28 Aug 2026 17:18:13 -0400https://pop.com.sg/how-many-10-letter-words-can-be-formed-using-the-26-letters-of-the-alp.htmlhttps://pop.com.sg/how-many-10-letter-words-can-be-formed-using-the-26-letters-of-the-alp.htmlHow many $10$-letter words can be formed using the $26$ letters of the alphabet if: a) Repetition is allowed b) Repetition is not allowed c) Repetition is allowed, but letters are listed ...Fri, 28 Aug 2026 17:06:15 -0400https://pop.com.sg/the-points-where-function-is-discontinuous-are-those-points-counted-co.htmlhttps://pop.com.sg/the-points-where-function-is-discontinuous-are-those-points-counted-co.htmlThe points where function is discontinuous,are those points counted/considered in the domain of the function. $(1)[x]$,greatest integer function is discontinuous at all integer points but integers ......Fri, 28 Aug 2026 17:02:43 -0400https://pop.com.sg/a-little-question-about-clairaut-039-s-theorem.htmlhttps://pop.com.sg/a-little-question-about-clairaut-039-s-theorem.htmlClairaut's Theorem Let $f$ be a function of two variables,let$(x_{0},y_{0})$ be a point and let $U$ be an open disk with center $(x_{0},y_{0})$.Assume that $f$ is defined on $U$ and its partial ...Fri, 28 Aug 2026 16:35:45 -0400https://pop.com.sg/what-is-a-jordan-cell.htmlhttps://pop.com.sg/what-is-a-jordan-cell.htmlGoogle has been surprisingly unhelpful for me. A homework problem from my algebra class asks me to Calculate p(A) where A is a Jordan cell and p is a polynomial. I'd like to attempt the problem ......Fri, 28 Aug 2026 16:32:04 -0400https://pop.com.sg/what-does-sin-theta-0-mean-here-if-tan-theta-frac-8-15-and-sin-theta-0.htmlhttps://pop.com.sg/what-does-sin-theta-0-mean-here-if-tan-theta-frac-8-15-and-sin-theta-0.htmlThe question is: If $\tan(\theta) = -\frac{8}{15}$, and $\sin(\theta) > 0$, then find $\cos(\theta)$. What I did was draw a triangle on the unit circle with sides $8, 15$ and therefore hypot......Fri, 28 Aug 2026 16:30:13 -0400