https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usWed, 12 Aug 2026 15:52:52 -0400https://pop.com.sg/questions-tagged-legendre-polynomials.htmlhttps://pop.com.sg/questions-tagged-legendre-polynomials.htmlQ&A for people studying math at any level and professionals in related fields...Wed, 12 Aug 2026 19:52:42 -0400https://pop.com.sg/is-it-possible-to-cut-a-circle-and-make-smaller-circle.htmlhttps://pop.com.sg/is-it-possible-to-cut-a-circle-and-make-smaller-circle.htmlRecently I saw this meme: Although the pizza hasn't completely the shape of a circle, let's say we have a circle is it possible to cut a part of the circle like this in the image and make smaller ......Wed, 12 Aug 2026 19:39:39 -0400https://pop.com.sg/how-to-calculate-the-joint-probability-from-two-normal-distributions.htmlhttps://pop.com.sg/how-to-calculate-the-joint-probability-from-two-normal-distributions.htmlI have two random variables $X$ and $Y$ both normally distributed as $N(\mu, \sigma^2)$ (they have the same distribution). $X$ and $Y$ are dependent. They are defined from other random variables A......Wed, 12 Aug 2026 19:28:07 -0400https://pop.com.sg/a-nice-way-to-do-euler-039-s-method-on-a-calculator.htmlhttps://pop.com.sg/a-nice-way-to-do-euler-039-s-method-on-a-calculator.htmlAs part of the calculator paper for IB (International Baccalaureate), we may be asked to do Euler's method, for say, 10 iterations. While it is feasible to do with a calculator (slightly easier if......Wed, 12 Aug 2026 19:23:45 -0400https://pop.com.sg/why-is-l-0-norm-not-convex-closed.htmlhttps://pop.com.sg/why-is-l-0-norm-not-convex-closed.htmlI have this confusion in understanding the convexity of the $L_0$ norm. Why is $L_0$ norm not convex?...Wed, 12 Aug 2026 19:15:36 -0400https://pop.com.sg/difference-between-joint-probability-distribution-and-conditional-prob.htmlhttps://pop.com.sg/difference-between-joint-probability-distribution-and-conditional-prob.htmlCan someone explain to me the difference between joint probability distribution and conditional probability distribution?...Wed, 12 Aug 2026 19:10:27 -0400https://pop.com.sg/will-division-by-zero-be-defined-eventually-duplicate.htmlhttps://pop.com.sg/will-division-by-zero-be-defined-eventually-duplicate.htmlPossible Duplicate: Division by $0$ I've always been inclined to believe that x/0 = NaN is a placeholder for a character or constant that no one has created yet. I know assume that none of y......Wed, 12 Aug 2026 19:07:09 -0400https://pop.com.sg/proving-a-conditional-statement-by-induction.htmlhttps://pop.com.sg/proving-a-conditional-statement-by-induction.htmlI’m a university freshman and I saw this in the lecture slides of my Discrete Mathematics module. I do not know what the name of the theorem is, so I will just copy the proof to the best of my abil......Wed, 12 Aug 2026 19:06:27 -0400https://pop.com.sg/do-equal-angles-necessarily-mean-a-polygon-is-regular.htmlhttps://pop.com.sg/do-equal-angles-necessarily-mean-a-polygon-is-regular.htmlIn a polygon, if all the sides are equal, it doesn’t necessarily mean that the polygon is regular (eg. a rhombus). Is this also true for angles? Meaning can you draw a polygon whose interior angles......Wed, 12 Aug 2026 19:05:45 -0400https://pop.com.sg/should-the-sign-be-reversed-if-i-square-both-sides-of-an-inequality.htmlhttps://pop.com.sg/should-the-sign-be-reversed-if-i-square-both-sides-of-an-inequality.htmlLet us say I have the following: $$x>y$$ Now, I want to take the square of both sides. Should it result in $$x^2>y^2$$ or $$x^2<y^2$$ I suspect there is no way to give a general answer to ...Wed, 12 Aug 2026 19:05:31 -0400https://pop.com.sg/irreducible-polynomials-of-degree-3-and-5.htmlhttps://pop.com.sg/irreducible-polynomials-of-degree-3-and-5.htmlTwo part question: 1) There are three irreducible cubic polynomials $\mathbb F_2$ Aproach: Bruteforce, there are not as much cubic polynomials, just 8 of them: $x^3$ $x^3+1$ $x^3+x$ $x^3+x+1$......Wed, 12 Aug 2026 19:02:09 -0400https://pop.com.sg/a-two-envelopes-puzzle.htmlhttps://pop.com.sg/a-two-envelopes-puzzle.htmlThis is Problem 1.25 from Tsitsiklis, Bertsekas, Introduction to Probability, 2nd edition. You are handed two envelopes, and you know that each contains a positive integer dollar amount and th......Wed, 12 Aug 2026 19:00:16 -0400https://pop.com.sg/cdf-for-negative-binomial-distribution.htmlhttps://pop.com.sg/cdf-for-negative-binomial-distribution.htmlI am trying to show that the following statement is true. $$ \sum_{x = r}^{X}\binom{x-1}{r-1}p^r(1-p)^{x-r} = \sum_{x = r}^{X}\binom{X}{x}p^x(1-p)^{X-x} $$ Where $X$ and $r$ and $p$ are constants......Wed, 12 Aug 2026 18:52:06 -0400https://pop.com.sg/if-a-graph-has-a-hamilton-path-starting-at-every-vertex-must-it-contai.htmlhttps://pop.com.sg/if-a-graph-has-a-hamilton-path-starting-at-every-vertex-must-it-contai.htmlIf a graph $G$ has at least three vertices, and has a Hamilton Path starting at every vertex, must it contain a Hamilton Circuit? I have been struggling with this problem. It seems that because......Wed, 12 Aug 2026 18:41:39 -0400https://pop.com.sg/cumulative-distribution-function-of-a-variable-with-exponential-distri.htmlhttps://pop.com.sg/cumulative-distribution-function-of-a-variable-with-exponential-distri.htmlSuppose we have a variable $X$ that has an exponential distribution with a probability density function: $f(x) = 3e^{-3x}, x >0$ Then the cumulative distribution function is: $\int_{-\infty}^......Wed, 12 Aug 2026 18:37:30 -0400https://pop.com.sg/can-a-limit-of-an-integral-be-moved-inside-the-integral.htmlhttps://pop.com.sg/can-a-limit-of-an-integral-be-moved-inside-the-integral.htmlAfter coming across this question: How to verify this limit, I have the following question: When taking the limit of an integral, is it valid to move the limit inside the integral, providing the l......Wed, 12 Aug 2026 18:30:17 -0400https://pop.com.sg/why-can-you-mix-partial-derivatives-with-ordinary-derivatives-in-the-c.htmlhttps://pop.com.sg/why-can-you-mix-partial-derivatives-with-ordinary-derivatives-in-the-c.htmlThis question is a simplified version of this previous question asked by myself. The following is a short extract from a book I am reading: If $u=(x^2+2y)^2 + 4$ and $p=x^2 + 2y$ $\space$ then ......Wed, 12 Aug 2026 18:29:05 -0400https://pop.com.sg/what-is-the-remainder-when-9-2012-is-divided-by-11.htmlhttps://pop.com.sg/what-is-the-remainder-when-9-2012-is-divided-by-11.htmlI'm trying to find the answer to the following: What is the remainder when $9^{2012}$ is divided by $11$? Apparently, you're supposed to use Fermat's Little Theorem, but I'm not sure how to use it......Wed, 12 Aug 2026 18:24:49 -0400https://pop.com.sg/conditional-probability-of-a-given-b-closed.htmlhttps://pop.com.sg/conditional-probability-of-a-given-b-closed.htmlVery basic question, but If A and B are independent events, then is the Probability of A given B has occurred = Zero or Probability of A? and why?...Wed, 12 Aug 2026 17:58:49 -0400https://pop.com.sg/finding-interest-rate-without-principal.htmlhttps://pop.com.sg/finding-interest-rate-without-principal.htmlSo, I recently sat a year ten exam and one of the questions was: Bob invested one half of his savings in an account that paid simple interest for two years and received $550$ dollars as interest......Wed, 12 Aug 2026 17:52:09 -0400https://pop.com.sg/finding-the-range-of-f-x-1-x-1-x-2.htmlhttps://pop.com.sg/finding-the-range-of-f-x-1-x-1-x-2.htmlI want to find the range of the following function $$f(x) = \frac{1}{(x-1)(x-2)} $$ Is there any way to find the range of the above function? I have found one idea. But that is too critical. Please ...Wed, 12 Aug 2026 17:49:06 -0400https://pop.com.sg/rigorous-definition-of-matrix-duplicate.htmlhttps://pop.com.sg/rigorous-definition-of-matrix-duplicate.htmlI know "Matrix" is usually defined as any array of numbers or other mathematical objects arranged in rows and columns. But the problem is that seems not really like a rigorous mathematical definiti......Wed, 12 Aug 2026 17:25:50 -0400https://pop.com.sg/conjugation-of-matrices-and-conjugation-of-complex-numbers.htmlhttps://pop.com.sg/conjugation-of-matrices-and-conjugation-of-complex-numbers.htmlAre conjugation of matrices and conjugation of complex numbers related? What I mean is that if $A$ is an $n \times n$ matrix then the conjugation of $A$ by an invertible $n \times n$ matrix $C$ is......Wed, 12 Aug 2026 17:23:37 -0400https://pop.com.sg/is-there-a-mathematical-definition-for-the-divisibility-of-rational-nu.htmlhttps://pop.com.sg/is-there-a-mathematical-definition-for-the-divisibility-of-rational-nu.htmlThe term divisibility usually refers to integer numbers only. I want to define the divisibility of a rational number $q$ by an integer number $z$ as follows: $q$ is divisible by $z$ if and only i......Wed, 12 Aug 2026 17:18:16 -0400https://pop.com.sg/how-many-different-sizes-of-infinity-are-there.htmlhttps://pop.com.sg/how-many-different-sizes-of-infinity-are-there.htmlIt's pretty straightforward to say that there is an infinite number of different sizes of infinity, but then I thought, "What size of infinity is that?" My thoughts are that the number of unique ...Wed, 12 Aug 2026 16:57:56 -0400https://pop.com.sg/user-mike-bennett-mathematics-stack-exchange.htmlhttps://pop.com.sg/user-mike-bennett-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Wed, 12 Aug 2026 16:57:37 -0400https://pop.com.sg/does-bounded-variation-imply-boundedness.htmlhttps://pop.com.sg/does-bounded-variation-imply-boundedness.htmlUsing the standard definition $$||f||_{TV} := \sup_{x_0<\cdots<x_n}\sum_{i=1}^{n} |f(x_i) - f(x_{i-1})|.$$ 1.When the domain is a bounded interval $[a,b]$, the statement holds. 2.When the ...Wed, 12 Aug 2026 16:55:22 -0400https://pop.com.sg/how-to-solve-csc-sin-1.htmlhttps://pop.com.sg/how-to-solve-csc-sin-1.htmlIs it possible to get some help on this. I'm trying to help my friend solve this but it had been over 14 years since I did this type. All I have to work with is this. Sorry if it is vague. Any help ...Wed, 12 Aug 2026 16:50:36 -0400https://pop.com.sg/find-the-eigenvalues-of-a-matrix-in-a-special-case-using-cofactors-dia.htmlhttps://pop.com.sg/find-the-eigenvalues-of-a-matrix-in-a-special-case-using-cofactors-dia.htmlI have a matrix $A_{(3x3)}$ with determinant iqual to zero, null trace and their cofactors of the diaganal such that $\Delta_{11}=-1$, $\Delta_{22}=0$, $\Delta_{33}=-1$. Some TIPS to calculate the ...Wed, 12 Aug 2026 16:43:51 -0400https://pop.com.sg/how-to-calculate-standard-deviation-for-a-series-containing-both-posit.htmlhttps://pop.com.sg/how-to-calculate-standard-deviation-for-a-series-containing-both-posit.htmlI got a stream of numbers in one of my apps to represent an electrical signal. I've observed that the signal ranges from -100 to +100. Other than that, the signal is fairly random and crosses 0 in ......Wed, 12 Aug 2026 16:43:15 -0400https://pop.com.sg/what-039-s-the-the-integral-of-tan-4x.htmlhttps://pop.com.sg/what-039-s-the-the-integral-of-tan-4x.htmlHow is $\displaystyle \int \tan(4x) dx = \cos^2 (4x)$? Shouldn't it be $\ln(\sec(4x))$? I don't understand... Please, help. Thank you....Wed, 12 Aug 2026 16:41:33 -0400https://pop.com.sg/slope-of-tangent-passing-through-point.htmlhttps://pop.com.sg/slope-of-tangent-passing-through-point.htmlGiven the equation of the curve $y=x^2+x$ and the requirement that the tangent to this curve passes through $(2, -3)$, what are the possible equations of the tangent lines? I started by differenti......Wed, 12 Aug 2026 16:39:57 -0400https://pop.com.sg/how-to-prove-the-cross-product-of-two-vectors-closed.htmlhttps://pop.com.sg/how-to-prove-the-cross-product-of-two-vectors-closed.htmlOkay, I must admit that I am lost on how to do this. I have looked up videos and tutorials about this, and they helped a little. The main thing is that my professor asked for us to solve this without ...Wed, 12 Aug 2026 16:38:39 -0400https://pop.com.sg/bedmas-where-the-order-of-addition-before-subtraction-matters.htmlhttps://pop.com.sg/bedmas-where-the-order-of-addition-before-subtraction-matters.htmlHere is a recent "tricky problem" that is making the rounds on FB: BEDMAS is explained here in a video, with this being the upshot: Everyone understands that 6 - 0 + 1 = 7, but I was challenged to ...Wed, 12 Aug 2026 16:30:49 -0400https://pop.com.sg/does-order-of-random-variables-matter-in-chain-rule-probability.htmlhttps://pop.com.sg/does-order-of-random-variables-matter-in-chain-rule-probability.htmlSuppose that we have the joint distribution of two random variables $X, Y$. Does it matter if we write it as $P(X, Y)$ or $P(Y, X)$? I would say yes if when substitute for their values we are not ...Wed, 12 Aug 2026 16:30:18 -0400https://pop.com.sg/solve-the-definite-integral-by-completing-the-square.htmlhttps://pop.com.sg/solve-the-definite-integral-by-completing-the-square.htmlThis is the original definite integral. $$\int_1^3\frac{2x-3}{\sqrt{4x-x^2}}$$ I do not know what to do after completing the square and I am stuck at that specific part after plugging back in. $$(......Wed, 12 Aug 2026 16:24:19 -0400https://pop.com.sg/evaluating-the-area-between-the-curves-r-2-sin-theta-and-r-sin-theta-c.htmlhttps://pop.com.sg/evaluating-the-area-between-the-curves-r-2-sin-theta-and-r-sin-theta-c.htmlSo the problem asked me to find the area of the region that lies inside both of the circles $$r=2\sin\theta, \quad r=\sin\theta +\cos\theta $$ I know that $r=2\sin\theta$ is $x^2+(y-1)^2=1,$but the ...Wed, 12 Aug 2026 16:10:12 -0400https://pop.com.sg/inhomogeneous-euler-cauchy-equations.htmlhttps://pop.com.sg/inhomogeneous-euler-cauchy-equations.htmlI have the following question: Given the Euler-Cauchy equation $$x^2\frac{d^2y}{dx^2}-3x\frac{dy}{dx}-5y=x^5$$ Solve the associated homogeneous equation by considering a solution of the form $y(x)......Wed, 12 Aug 2026 16:00:36 -0400https://pop.com.sg/how-to-find-impulse-response-h-t-given-an-output-of-lti-system.htmlhttps://pop.com.sg/how-to-find-impulse-response-h-t-given-an-output-of-lti-system.htmlA LTI system takes $x(t)$ as input and $y(t)$ as output. Given $$y(t)=\int_{t-5}^{t-1}2x(\tau)d\tau$$, how to find $h(t)$, the impulse response? Convolution $x(t)*h(t)=\int_{-\infty}^\infty x(\tau......Wed, 12 Aug 2026 15:50:33 -0400https://pop.com.sg/two-ways-of-defining-rank-of-a-set.htmlhttps://pop.com.sg/two-ways-of-defining-rank-of-a-set.htmlI am studying set theory, and I have some difficulties in understanding how people define the notion of rank there (I hope, specialists in logic will excuse me for this). As far as I understand, ......Wed, 12 Aug 2026 15:47:47 -0400https://pop.com.sg/when-you-flip-a-fraction-to-remove-the-negative-exponent-do-you-flip-a.htmlhttps://pop.com.sg/when-you-flip-a-fraction-to-remove-the-negative-exponent-do-you-flip-a.htmlWhen you flip a fraction to remove the negative exponent, do you flip all of what's on numerator and denominator? Or do you only do that with variables Such as: (-7a2b3c0/3a3b4c3 )-4 Will it be ...Wed, 12 Aug 2026 15:45:02 -0400https://pop.com.sg/motivating-infinite-series.htmlhttps://pop.com.sg/motivating-infinite-series.htmlWhat are some good ways to motivate the material on infinite series that appears at the end of a typical American Calculus II course? My students in this course are generally from biochemistry, co......Wed, 12 Aug 2026 15:39:42 -0400https://pop.com.sg/what-is-the-derivative-of-log-y-with-respect-to-x.htmlhttps://pop.com.sg/what-is-the-derivative-of-log-y-with-respect-to-x.htmlI was solving the following sum: We started off with the first equation and then he took log on both sides to simplify. Everything was fine here but the next step's LHS confused me, he said he used......Wed, 12 Aug 2026 15:15:34 -0400https://pop.com.sg/cutting-a-cube-with-a-plane.htmlhttps://pop.com.sg/cutting-a-cube-with-a-plane.htmlMe and a couple of friends have recently started solving Math problems in our breaks at school. We came along the Georg Mohr tournament, which is Denmark's first qualifications tournament for the I......Wed, 12 Aug 2026 15:13:22 -0400https://pop.com.sg/british-flag-theorem-generalized-and-inspired-from-british-flag-theore.htmlhttps://pop.com.sg/british-flag-theorem-generalized-and-inspired-from-british-flag-theore.htmlBritish Flag theorem: Let $P$ be a point in the plane, let $ABCD$ be a rectangle in the plane then: $$PA^2+PC^2=PB^2+PD^2$$ Generalization: Let $ABCD$ be a rectangle in a plane, Let $P$ be a poin......Wed, 12 Aug 2026 14:57:17 -0400https://pop.com.sg/is-this-a-solution-for-the-problem-a-3-b-3-c-3-has-no-nonzero-integer-.htmlhttps://pop.com.sg/is-this-a-solution-for-the-problem-a-3-b-3-c-3-has-no-nonzero-integer-.htmlIs this a solution for the problem: $\ a^3 + b^3 = c^3\ $ has no nonzero integer solutions? Suppose $\ a^3 + b^3 = c^3,\ a,b,c \in \mathbb Z^*,\ $then: $c^3 - b ^ 3 = (c - b)((c - b) ^ 2 + 3cb......Wed, 12 Aug 2026 14:56:34 -0400https://pop.com.sg/backward-stability-vs-stability.htmlhttps://pop.com.sg/backward-stability-vs-stability.htmlI'm reading a numerical analysis book, which gave me very confusing descriptions about stability and backward stability. It says that an algorithm $f$ is stable, if for any $x$, $$ \frac{\|f_b(x)......Wed, 12 Aug 2026 14:53:56 -0400https://pop.com.sg/express-using-logic-symbols.htmlhttps://pop.com.sg/express-using-logic-symbols.htmlI have to also decide whether it is true or not after I express it in logic symbols. This is what I have so far..Am I correct? and I don't know how to do a).. a) There is a smallest positive numbe......Wed, 12 Aug 2026 14:53:23 -0400https://pop.com.sg/degree-of-rational-function.htmlhttps://pop.com.sg/degree-of-rational-function.htmlThis might sound like a very trivial question but I found different answers on the web. Assume one has a rational function $$\frac{f(x)}{g(x)} ,$$ where $f(x)$ and $g(x)$ are polynomials. What i......Wed, 12 Aug 2026 14:51:21 -0400https://pop.com.sg/do-all-straight-lines-have-an-inverse-function.htmlhttps://pop.com.sg/do-all-straight-lines-have-an-inverse-function.htmlDo all straight lines have an inverse function? It would seem to make sense. All linear lines would pass the horizontal line test and thus when reflected across $y=x$ it would still be a function. ...Wed, 12 Aug 2026 14:49:55 -0400https://pop.com.sg/quaternions-problem-with-understanding-essence-of.htmlhttps://pop.com.sg/quaternions-problem-with-understanding-essence-of.htmlI can't calm down when I can't realized of principles of working something. Quaternion is that place where I spend а few days! Unfortunately, not one lectures or books what I saw or read don't expl......Wed, 12 Aug 2026 14:36:30 -0400https://pop.com.sg/trees-on-six-vertices.htmlhttps://pop.com.sg/trees-on-six-vertices.htmlShow that there are exactly six non-isomorphic trees on six vertices. ("Introduction to Combinatorics" p.183) The solutions were provided in the book. But why can we be sure that there are ONLY si......Wed, 12 Aug 2026 14:27:43 -0400https://pop.com.sg/what-is-the-difference-between-equation-and-formula.htmlhttps://pop.com.sg/what-is-the-difference-between-equation-and-formula.htmlSometimes equation and formula are used interchangeably, but I was wondering if there is a difference. For example, suppose we can calculate a car's fuel efficiency as: mpg = distance traveled in......Wed, 12 Aug 2026 14:18:18 -0400https://pop.com.sg/are-the-endpoints-of-the-domain-of-a-function-counted-as-critical-poin.htmlhttps://pop.com.sg/are-the-endpoints-of-the-domain-of-a-function-counted-as-critical-poin.htmlDo the end points of a domain come under critical points? I know we say critical point is a point where the derivative is zero or the derivative doesn't exist. For example: $$ f:[0,\pi] \to [-1,1......Wed, 12 Aug 2026 14:06:38 -0400https://pop.com.sg/cdf-for-a-continuous-random-variable.htmlhttps://pop.com.sg/cdf-for-a-continuous-random-variable.htmlLet $Y$ be a continuous random variable with probability density function$$ f_Y(x)=\begin{cases} xe^{-\frac{x^2}2};&x>0\\ 0;&x\le0\end{cases}.$$ Compute $\mathbb{E}(Y)$ and determine the ...Wed, 12 Aug 2026 14:05:00 -0400https://pop.com.sg/finding-the-radius-of-the-smallest-circle-that-can-circumscribe-an-equ.htmlhttps://pop.com.sg/finding-the-radius-of-the-smallest-circle-that-can-circumscribe-an-equ.htmlQ:A puzzle board is in the form of an equilateral triangle that has an area of $7\sqrt{3}$ if the board is placed on a circular table, what should be the min area of the table so that the whole board ...Wed, 12 Aug 2026 14:00:38 -0400https://pop.com.sg/how-to-find-a-circle-given-a-segment.htmlhttps://pop.com.sg/how-to-find-a-circle-given-a-segment.htmlIn this page, we get mathematical formulas for calculating the area of a circle segment under a few different conditions, if you already know the radius of the circle. My question is, how do you d......Wed, 12 Aug 2026 13:58:16 -0400https://pop.com.sg/continuous-differentiability-implies-lipschitz-continuity.htmlhttps://pop.com.sg/continuous-differentiability-implies-lipschitz-continuity.htmlHere's a statement in Zygmund's Measure and Integral on page 17: If $f$ has a continuous derivative on $[a,b]$, then (by the mean-value theorem) $f$ satisfies a Lipschitz condition on $[a,b]$. ......Wed, 12 Aug 2026 13:49:34 -0400https://pop.com.sg/equipotent-sets.htmlhttps://pop.com.sg/equipotent-sets.htmlWe know by definition that if a bijection between two sets A and B exists,then A and B are equivalent.The book I was reading took the function f:Z→N f(x)= -2x,for x<0 2x+1,for x=>0 w......Wed, 12 Aug 2026 13:38:42 -0400https://pop.com.sg/difference-between-direction-cosine-matrix-dcm-and-rotation-matrix.htmlhttps://pop.com.sg/difference-between-direction-cosine-matrix-dcm-and-rotation-matrix.htmlI am a bit confused about the difference between direction cosine matrix(DCM) and rotation matrix. I have searched through the literature but found no explicit explanation if they are different or ......Wed, 12 Aug 2026 13:38:37 -0400https://pop.com.sg/definition-of-suspension-topology.htmlhttps://pop.com.sg/definition-of-suspension-topology.htmlI am a bit confused with the definition of a suspension. This the definition. For a space $X$, denote $SX$ the suspension of $X$ in which this is the quotient space $$\frac{X \times I}{\sim}$$ w......Wed, 12 Aug 2026 13:37:38 -0400https://pop.com.sg/what-is-f-prime.htmlhttps://pop.com.sg/what-is-f-prime.htmlcurrently taking Measure and Integration course, which seems to have a different definition of f'. traditionally, $$f'(x)=\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$$ but in folland's book, it se......Wed, 12 Aug 2026 13:30:14 -0400https://pop.com.sg/points-in-general-position.htmlhttps://pop.com.sg/points-in-general-position.htmlI'm really confused by the definition of general position at wikipedia. I understand that the set of points/vectors in $\mathbb R^d$ is in general position iff every $(d+1)$ points are not in any ...Wed, 12 Aug 2026 13:21:33 -0400https://pop.com.sg/domain-and-range-of-a-sequence.htmlhttps://pop.com.sg/domain-and-range-of-a-sequence.htmlA sequence X is define as numbers smaller than 10 that are divisible by 2 and a natural number. What are the domain and ranges? I assume that the domain and range here is: dom X = <1,2,3,4,5,......Wed, 12 Aug 2026 13:05:34 -0400https://pop.com.sg/chain-rule-using-tree-diagram-why-does-it-work.htmlhttps://pop.com.sg/chain-rule-using-tree-diagram-why-does-it-work.htmlIn multivariable calculus, I was taught to compute the chain rule by drawing a "tree diagram" (a directed acyclic graph) representing the dependence of one variable on the others. I now want to ...Wed, 12 Aug 2026 12:52:28 -0400https://pop.com.sg/help-with-proof-of-expected-value-of-gamma-distribution.htmlhttps://pop.com.sg/help-with-proof-of-expected-value-of-gamma-distribution.htmlI am struggling with this proof of the expected value for the gamma distribution. I need help with the step indicated by the red arrow. Could someone please break it down for me. Thanks....Wed, 12 Aug 2026 12:39:09 -0400https://pop.com.sg/essentially-bounded-function-on-mathbb-r.htmlhttps://pop.com.sg/essentially-bounded-function-on-mathbb-r.htmlI don't really know a lot about measure (just finishing my undergrad) so I'm not really on good terms with this. So, let $L^\infty[a,b]$ denote the space of all essentially bounded functions on $[a......Wed, 12 Aug 2026 12:24:33 -0400https://pop.com.sg/prove-transitivity-property-congruence-mod-m.htmlhttps://pop.com.sg/prove-transitivity-property-congruence-mod-m.htmlProve transitivity property of congruence mod m. Show that if $x\equiv y \pmod m$ and $y \equiv z\pmod m$ then $x\equiv z\pmod m$ . I didn't really get the tutors explanation of this, I get what ...Wed, 12 Aug 2026 12:23:04 -0400https://pop.com.sg/how-can-i-plot-a-2d-figure-in-matlab-by-connecting-the-dots.htmlhttps://pop.com.sg/how-can-i-plot-a-2d-figure-in-matlab-by-connecting-the-dots.htmlI'm an amateur at MATLAB and I know how to do the basic stuff such as plotting functions, writing M files, functions, for loops, etc but I've be tasked to do something I have never done before. I ......Wed, 12 Aug 2026 12:22:02 -0400https://pop.com.sg/transformation-matrix-for-matrix-indices-to-cartesian-coordinates.htmlhttps://pop.com.sg/transformation-matrix-for-matrix-indices-to-cartesian-coordinates.htmlIn MatLab matrices, the indices are as follows: (1,1) (1,2) (1,3) (2,1) (2,2) (2,3) (3,1) (3,2) (3,3) This is an example 3x3 matrix. In corresponding cartesian coordinate system, the representation ...Wed, 12 Aug 2026 12:14:06 -0400https://pop.com.sg/solving-a-stratonovich-sde.htmlhttps://pop.com.sg/solving-a-stratonovich-sde.htmlI am trying to solve the following Stratonovich SDE $$dN_t=rN_tdt+\gamma N_t\circ dB_t$$ In my notes, the Stratonovich integral is defined as $$\int^t_0 N_s\circ dB_s=\int^t_0 N_sdB_s+\frac{1}{2}\l......Wed, 12 Aug 2026 12:11:38 -0400https://pop.com.sg/why-special-limit-of-euler-039-s-number-fails-at-higher-powers.htmlhttps://pop.com.sg/why-special-limit-of-euler-039-s-number-fails-at-higher-powers.htmlIn the "Calculus 6th edition" by Edwards and Penney, in the chapter of Transcendental functions, there is an interesting question about special limit that leads to the famous Euler's number 2.71828......Wed, 12 Aug 2026 12:06:58 -0400https://pop.com.sg/conversion-of-a-vector-in-a-cartesian-coordinate-system-to-a-cylindric.htmlhttps://pop.com.sg/conversion-of-a-vector-in-a-cartesian-coordinate-system-to-a-cylindric.htmlI'm having trouble converting a vector from the Cartesian coordinate system to the cylindrical coordinate system (second year vector calculus) Represent the vector $\mathbf A(x,y,z) = z\ \hat i - ......Wed, 12 Aug 2026 12:02:37 -0400https://pop.com.sg/positivity-in-algebraic-geometry.htmlhttps://pop.com.sg/positivity-in-algebraic-geometry.htmlNow I am studying vanishing theorems in algebraic geometry, and I am so curious about why the word $\textbf{Positivity}$ is used. So if somebody can give any intuition about that word, it will be ...Wed, 12 Aug 2026 11:56:34 -0400https://pop.com.sg/understanding-the-proof-to-egorov-039-s-theorem.htmlhttps://pop.com.sg/understanding-the-proof-to-egorov-039-s-theorem.htmlTheorem (Egorov). Let $\{f_n\}$ be a sequence of measurable functions converging almost everywhere on a measurable set $E$ to a function $f$. Then, given any $\delta > 0$, there exists a ...Wed, 12 Aug 2026 11:55:14 -0400https://pop.com.sg/average-smallest-prime-factors.htmlhttps://pop.com.sg/average-smallest-prime-factors.htmlI looked at the average smallest prime factor (ASPF) for the numbers up to N: $\text{ASPF}(N) = \frac{1}{N-1}\ \Sigma_{k=2}^N \text{SPF}(k)$ ASPF(100) = 13 ASPF(1,000) = 79 ASPR(10,000) = 578 ......Wed, 12 Aug 2026 11:53:40 -0400https://pop.com.sg/how-to-really-understand-the-tensor-algebra.htmlhttps://pop.com.sg/how-to-really-understand-the-tensor-algebra.htmlIf $V$ is a vector space over $F$, then we define $T^r_0(V)=V^{\otimes r}$, then we define the algebra of contravariant tensors to be $$T(V)=\bigoplus_{r=0}^\infty T^r_0(V)$$ together with the te......Wed, 12 Aug 2026 11:53:19 -0400https://pop.com.sg/derivative-of-function-raised-to-a-power-using-the-chain-rule.htmlhttps://pop.com.sg/derivative-of-function-raised-to-a-power-using-the-chain-rule.htmlHow do I find the derivative of the function $f(x)= (2x+1)^2$? I've tried doing this problem and am not fully sure that I am correct. I found the derivative to be $f'(x) = 8x+4$. Is that correct?...Wed, 12 Aug 2026 11:32:54 -0400https://pop.com.sg/lusin-spaces-issues-on-the-definition-and-on-spaces-used-in-analysis-t.htmlhttps://pop.com.sg/lusin-spaces-issues-on-the-definition-and-on-spaces-used-in-analysis-t.htmlI have two questions concerning Lusin spaces. Before the questions, the definition: Definition (Lusin Space): Given a Hausdorff space $(X, \tau)$, that space is said to be Lusin if there is a st......Wed, 12 Aug 2026 11:27:42 -0400https://pop.com.sg/how-to-find-terminal-point-coordinates-on-a-unit-circle.htmlhttps://pop.com.sg/how-to-find-terminal-point-coordinates-on-a-unit-circle.htmlHey everyone I am working on a homework assignment which covers unit circles. However I am really confused and having a lot of trouble locating terminal point coordinates. Everything I have read on......Wed, 12 Aug 2026 11:25:06 -0400https://pop.com.sg/adding-two-polar-vectors.htmlhttps://pop.com.sg/adding-two-polar-vectors.htmlIs there a way of adding two vectors in polar form without first having to convert them to cartesian or complex form?...Wed, 12 Aug 2026 11:24:12 -0400https://pop.com.sg/taylor-series-for-exponential-function.htmlhttps://pop.com.sg/taylor-series-for-exponential-function.htmlThe Taylor series for $e^x = \sum_{i=0}^\infty \frac{x^i}{i!}$. Then as $e^0 = 1$, if one evaluates the Taylor series at $x=0$ we find that $e^0 = \sum_{i=0}^\infty \frac{0^i}{i!} = \frac{0^0}{0!} + \...Wed, 12 Aug 2026 11:19:07 -0400https://pop.com.sg/how-is-boundary-of-a-set-defined-is-boundary-of-an-open-set-s-included.htmlhttps://pop.com.sg/how-is-boundary-of-a-set-defined-is-boundary-of-an-open-set-s-included.htmlHow is boundary of a set defined? Is boundary of an open set $S$, included in $S$?...Wed, 12 Aug 2026 11:18:34 -0400https://pop.com.sg/helix-around-helix-around-circle.htmlhttps://pop.com.sg/helix-around-helix-around-circle.htmlI'm trying to find the parametric equations for a helix around a helix around a circle (helix on helix on circle) That is: I would like to start with a circle, add a helix around it and a helix aro......Wed, 12 Aug 2026 11:14:48 -0400https://pop.com.sg/proof-that-every-polynomial-of-odd-degree-has-one-real-root.htmlhttps://pop.com.sg/proof-that-every-polynomial-of-odd-degree-has-one-real-root.htmlI want to prove that every real polynomial of odd degree has at least one real root, using the intermediate value theorem. Let $P(x) = x^{2n+1} + a_n x^{2n} + . . . + a_0$ for each $a_i \in \math......Wed, 12 Aug 2026 11:12:17 -0400https://pop.com.sg/liate-how-does-it-work.htmlhttps://pop.com.sg/liate-how-does-it-work.htmlWhen doing Integration By Parts, I know that using LIATE can be a useful guide most of the time. For those not familiar, LIATE is a guide to help you decide which term to differentiate and which ......Wed, 12 Aug 2026 11:11:33 -0400https://pop.com.sg/notation-what-is-the-element-wise-max-notation.htmlhttps://pop.com.sg/notation-what-is-the-element-wise-max-notation.htmlFor example I have 3 vectors. $A_1 = (1, 3, 2, 6)$ $A_2 = (6, 1, 9, 1)$ $A_3 = (3, 8, 4, 0)$ $max A_1 = 6$, $max A_2 = 9$, $max A_3 = 8$ What is the proper max notation for element-wise sit......Wed, 12 Aug 2026 11:00:10 -0400https://pop.com.sg/calculating-real-and-imaginary-part-of-a-complex-number.htmlhttps://pop.com.sg/calculating-real-and-imaginary-part-of-a-complex-number.htmlConsider the complex numbers $a = \frac{(1+i)^5}{(1-i)^3}$ and $b = e^{3-\pi i}$. How do I calculate the real and imaginary part of these numbers? What is the general approach to calculate these p......Wed, 12 Aug 2026 10:59:44 -0400https://pop.com.sg/how-to-factor-quadratic-ax-2-bx-c.htmlhttps://pop.com.sg/how-to-factor-quadratic-ax-2-bx-c.htmlHow do I shorten this? How do I have to think? $$ x^2 + x - 2$$ The answer is $$(x+2)(x-1)$$ I don't know how to get to the answer systematically. Could someone explain? Does anyone have a link......Wed, 12 Aug 2026 10:58:06 -0400https://pop.com.sg/1-to-the-power-of-infinity-why-is-it-indeterminate-duplicate.htmlhttps://pop.com.sg/1-to-the-power-of-infinity-why-is-it-indeterminate-duplicate.htmlI've been taught that $1^\infty$ is undetermined case. Why is it so? Isn't $1*1*1...=1$ whatever times you would multiply it? So if you take a limit, say $\lim_{n\to\infty} 1^n$, doesn't it converg......Wed, 12 Aug 2026 10:48:35 -0400https://pop.com.sg/triangle-inequality-theorem.htmlhttps://pop.com.sg/triangle-inequality-theorem.htmltriangle inequality theorem states Sum of any $2$ sides must be greater than $3$rd side. I am doing exercises in my text book. For example, I am checking if $10,3,5$ can form a triangle. $10+3 &g......Wed, 12 Aug 2026 10:46:42 -0400https://pop.com.sg/measure-theory-for-self-study-duplicate.htmlhttps://pop.com.sg/measure-theory-for-self-study-duplicate.htmlI have good knowledge of Elementary Real analysis. Now I'd like to study measure theory by myself (self-study). So please give me direction for where to start? Which book is good for starting? I have ...Wed, 12 Aug 2026 10:30:19 -0400https://pop.com.sg/what-does-it-mean-sequence-with-infinite-range.htmlhttps://pop.com.sg/what-does-it-mean-sequence-with-infinite-range.htmlI'm trying to understand this phrase Find a sequence with infinite range that converges only to $0$. What does it mean "sequence with infinite range"? Thanks...Wed, 12 Aug 2026 10:23:45 -0400https://pop.com.sg/what-does-this-subset-symbol-mean.htmlhttps://pop.com.sg/what-does-this-subset-symbol-mean.htmlI just came across this "subset" symbol in a PDF: $$\Omega \subsetneq T$$ I've never seen it before, and I tried looking for it via Detexify (to no avail). What does it mean?...Wed, 12 Aug 2026 10:21:30 -0400https://pop.com.sg/finding-parametric-equations-from-lines.htmlhttps://pop.com.sg/finding-parametric-equations-from-lines.htmlI have a question that reads: Find a parametric equation of each of the following lines: A. $3 x_1 + 4 x_2 = 6$ D. the line through $A=(-2,1)$ parallel to $x = (1,4) + t(3,5)$ E. ......Wed, 12 Aug 2026 10:21:05 -0400https://pop.com.sg/how-to-find-median-from-a-probability-distribution.htmlhttps://pop.com.sg/how-to-find-median-from-a-probability-distribution.htmlHaving trouble on something that should be really, really easy. I need to find the median of the following probability distribution...but according to the website I linked below...I'm doing it ...Wed, 12 Aug 2026 10:20:01 -0400https://pop.com.sg/solving-bernoulli-differential-equation.htmlhttps://pop.com.sg/solving-bernoulli-differential-equation.htmlHow to solve $$t \frac{dy}{dt} + y = t^4 y^3$$ First I divided by $t$ to get $$\frac{dy}{dt} + \frac{y}{t} = t^3 y^3$$ Then I multiplied through by $y^{-3}$ to get $$y^{-3} \frac{dy}{dt} + \frac1......Wed, 12 Aug 2026 10:17:38 -0400https://pop.com.sg/is-there-really-no-way-to-integrate-e-x-2.htmlhttps://pop.com.sg/is-there-really-no-way-to-integrate-e-x-2.htmlToday in my calculus class, we encountered the function $e^{-x^2}$, and I was told that it was not integrable. I was very surprised. Is there really no way to find the integral of $e^{-x^2}$? Grap......Wed, 12 Aug 2026 10:11:45 -0400https://pop.com.sg/how-to-find-the-volume-of-a-tetrahedron.htmlhttps://pop.com.sg/how-to-find-the-volume-of-a-tetrahedron.htmlFor this question, how do we find the volume of a tetrahedron in $\Bbb R^4$ if we have a 4 by 3 matrix? The cross product and determinant only work for square matrices. I'm not sure what to do afte......Wed, 12 Aug 2026 10:06:46 -0400https://pop.com.sg/vector-dot-product-0-for-perpendicular-vectors.htmlhttps://pop.com.sg/vector-dot-product-0-for-perpendicular-vectors.htmlI'm looking for the intuition behind this property, hopefully one that would be at a high-school level of reasoning. Looking through several of the answers here, I came onto this one, whose answers......Wed, 12 Aug 2026 09:59:40 -0400