https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usThu, 27 Aug 2026 22:01:07 -0400https://pop.com.sg/confusion-about-expectation-moment-generating-function-distributation.htmlhttps://pop.com.sg/confusion-about-expectation-moment-generating-function-distributation.html(I am currently studying a high dimensional probability course with very little background knowledge in probability theory as a whole, so I hope it's not annoying that I seem to be oblivious about ......Fri, 28 Aug 2026 02:00:43 -0400https://pop.com.sg/related-rates-of-change-cylinder-question.htmlhttps://pop.com.sg/related-rates-of-change-cylinder-question.htmlA cylindrical tank with radius 5 cm is being filled with water at rate of 3 cm^3 per min. how fast is the height of the water increasing? I dont want this question solved, but please help me corre......Fri, 28 Aug 2026 01:56:19 -0400https://pop.com.sg/coordinates-rotation-by-120-degree.htmlhttps://pop.com.sg/coordinates-rotation-by-120-degree.htmlIf I have a point on a standard grid with coordinates say: $A_1=(1000,0)$ $A_2=(707,707)$ Is there a easy way to transfer this points to $\pm 120$ degrees from the origin $(0,0)$, and kee......Fri, 28 Aug 2026 01:53:19 -0400https://pop.com.sg/difference-between-simplicial-and-singular-homology.htmlhttps://pop.com.sg/difference-between-simplicial-and-singular-homology.htmlI am having some difficulties understanding the difference between simplicial and singular homology. I am aware of the fact that they are isomorphic, i.e. the homology groups are in fact the same (......Fri, 28 Aug 2026 01:43:28 -0400https://pop.com.sg/relationship-between-simple-interest-and-arithmetic-progression.htmlhttps://pop.com.sg/relationship-between-simple-interest-and-arithmetic-progression.htmlDescribe the relationship between the mathematics of calculating simple interest, and the mathematics behind analyzing arithmetic sequences. How might simple interest be described as a problem of a......Fri, 28 Aug 2026 01:36:16 -0400https://pop.com.sg/definition-of-multidimensional-wiener-process.htmlhttps://pop.com.sg/definition-of-multidimensional-wiener-process.htmlI've been reading through Evans' excellent Introduction to Stochastic Differential Equations (used to be lecture notes available on his website a few years back, and became this book), and got some......Fri, 28 Aug 2026 01:30:10 -0400https://pop.com.sg/what-is-the-latest-verified-research-on-the-3x-1-problem-closed.htmlhttps://pop.com.sg/what-is-the-latest-verified-research-on-the-3x-1-problem-closed.htmlWikipedia : Collatz Conjecture Take any positive integer n. If n is even, divide it by $2$ to get $n / 2$. If n is odd, multiply it by $3$ and add $1$ to obtain $3n + 1$. Repeat the process (which......Fri, 28 Aug 2026 01:25:29 -0400https://pop.com.sg/understanding-the-determinant-of-an-infinite-matrix.htmlhttps://pop.com.sg/understanding-the-determinant-of-an-infinite-matrix.htmlThe fredholm determinant of an infinite matrix $A$ is defined by $$ \det(I+A) = \exp(\text{tr}(\log(I+A))). $$ I'm trying to understand the formula and how to use it. To be concrete here are my ...Fri, 28 Aug 2026 01:24:19 -0400https://pop.com.sg/what-are-differences-between-affine-space-and-vector-space.htmlhttps://pop.com.sg/what-are-differences-between-affine-space-and-vector-space.htmlI know smilar questions have been asked and I have looked at them but none of them seems to have satisfactory answer. I am reading the book a course in mathematics for student of physics vol. 1 by ......Fri, 28 Aug 2026 01:16:49 -0400https://pop.com.sg/solving-the-equation-ln-x-x.htmlhttps://pop.com.sg/solving-the-equation-ln-x-x.htmlI tried solving this equation for a long time but did not succeed. Any help is appreciated. $$\ln x=-x$$ I am not sure the tag is correct, I am not familiar with English mathematical terms. Please ...Fri, 28 Aug 2026 01:15:39 -0400https://pop.com.sg/most-general-solution-for-ode.htmlhttps://pop.com.sg/most-general-solution-for-ode.htmlI am facing some difficulties understanding the difference between a general solution and the most general solution of a 2nd order ODE for example for homogeneous constant coefficients case with 2 ......Fri, 28 Aug 2026 01:09:54 -0400https://pop.com.sg/need-help-determining-the-pairs-of-quaternions-that-anticommute.htmlhttps://pop.com.sg/need-help-determining-the-pairs-of-quaternions-that-anticommute.htmlI tried to solve another exercise and I would be grateful if someone could tell me if my answer is right. This is the exercise: Characterize the pairs $p,q \in \mathbb H$ such that $pq = -qp$. ......Fri, 28 Aug 2026 01:09:19 -0400https://pop.com.sg/deriving-the-rayleigh-distribution-from-the-gaussian.htmlhttps://pop.com.sg/deriving-the-rayleigh-distribution-from-the-gaussian.htmlTake a complex number $z=u+iv$ and call its magnitude $x$. Consider the real and imaginary parts to be Gaussian distributed: $$ f_{U}(u)=\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp\left(\frac{-u^{2}}{2\s......Fri, 28 Aug 2026 01:03:46 -0400https://pop.com.sg/solving-modulus-equation-systems.htmlhttps://pop.com.sg/solving-modulus-equation-systems.htmlI am studying for a test in discrete math and I created my own question but I cannot seem to solve it. Is it possible to solve the following equation system (without brainless testing), and if so......Fri, 28 Aug 2026 01:00:53 -0400https://pop.com.sg/why-does-0-75-times-1-25-equal-0-93-and-not-1.htmlhttps://pop.com.sg/why-does-0-75-times-1-25-equal-0-93-and-not-1.htmlIf percentage points can equal decimal points, then $1.25$ should equal $125$% when multiplying $1.25$ by any number. Therefore $125$% of $0.75$ should equal $1$ if $0.75$ is also $75$% of $1$. ......Fri, 28 Aug 2026 01:00:35 -0400https://pop.com.sg/find-the-point-of-intersection-of-the-straight-line-frac-x-1-4-frac-y-.htmlhttps://pop.com.sg/find-the-point-of-intersection-of-the-straight-line-frac-x-1-4-frac-y-.htmlFind the point of intersection of the straight line $$\frac{X+1}{4}=\frac{Y-2}{-2}=\frac{Z+6}{7}$$ and plane $3X+8Y-9Z=0$ the point of the line is $M(-1,2,-6)$ and direction vector of the line is......Fri, 28 Aug 2026 00:52:23 -0400https://pop.com.sg/find-the-value-of-c-such-that-the-system-has-a-solution-other-than-0-0.htmlhttps://pop.com.sg/find-the-value-of-c-such-that-the-system-has-a-solution-other-than-0-0.htmlI am asked to the find the value $c$ such that the system has a solution other than $(0, 0, 0)$. This is the linear system: \begin{array}{rcrcrcl} cx & & & + & 4z& = &am......Fri, 28 Aug 2026 00:46:37 -0400https://pop.com.sg/sard-039-s-theorem-proof-detail-question-regarding-partial-derivatives.htmlhttps://pop.com.sg/sard-039-s-theorem-proof-detail-question-regarding-partial-derivatives.htmlSard's Theorem: Suppose $M$ and $N$ are smooth manifolds with or without boundary and $F:M\to N$ is a smooth map. Then the set of critical values of $F$ has measure zero in $N$. In what appears to......Fri, 28 Aug 2026 00:37:41 -0400https://pop.com.sg/intuition-on-the-curl-formula.htmlhttps://pop.com.sg/intuition-on-the-curl-formula.htmlI'm trying to understand the formula for the curl: $$\operatorname{Curl}(F) = \nabla \times F $$ Why the vector product? What means the vector product of a vector and a operator? What is the mean......Fri, 28 Aug 2026 00:34:49 -0400https://pop.com.sg/equation-for-x-axis-3d-paraboloid.htmlhttps://pop.com.sg/equation-for-x-axis-3d-paraboloid.htmlI've been playing around with plenty of variants of paraboloid equations. However I couldn't come up with the equation for a x-axis parallel 3D paraboloid of revolution. For a 2D parabola the equ......Fri, 28 Aug 2026 00:34:02 -0400https://pop.com.sg/given-log2-a-log7-b-find-log-56.htmlhttps://pop.com.sg/given-log2-a-log7-b-find-log-56.htmlI don't know how to solve this. Can someone help me? How do I use the information above to help me find Log 56?...Fri, 28 Aug 2026 00:27:38 -0400https://pop.com.sg/are-there-infinitely-many-pythagorean-triples.htmlhttps://pop.com.sg/are-there-infinitely-many-pythagorean-triples.htmlI believe these questions are all asking different things, but: Are there infinitely many (integer) solutions to the pythagorean theorem? Is every positive integer part of a solution to the pytha......Fri, 28 Aug 2026 00:20:50 -0400https://pop.com.sg/find-the-self-adjoint-form-of-the-ode-x-2y-039-039-xy-039-mu-y-0.htmlhttps://pop.com.sg/find-the-self-adjoint-form-of-the-ode-x-2y-039-039-xy-039-mu-y-0.htmlFind the self-adjoint form of the ODE : $ x^2y''+xy'+\mu y=0 \ $ in the form $ \frac{d}{dx} [p_0 (x) y']+ \mu [q_0(x)]y=0 \ $ Answer: The adjoint equation is $ \frac{d^2}{dx^2}(x^2y)-\frac{......Fri, 28 Aug 2026 00:19:56 -0400https://pop.com.sg/matrix-norm-inequality-lvert-a-rvert-infty-leq-sqrt-n-lvert-a-rvert-2.htmlhttps://pop.com.sg/matrix-norm-inequality-lvert-a-rvert-infty-leq-sqrt-n-lvert-a-rvert-2.htmlSo I'm trying to prove that $\lVert A\rVert_\infty \leq \sqrt{n} \lVert A\rVert_2$. I've written the right hand side in terms of rows, but this method doesn't seem to be getting me anywhere. Wh......Fri, 28 Aug 2026 00:17:43 -0400https://pop.com.sg/can-every-irrational-number-be-written-in-terms-of-finitely-many-ratio.htmlhttps://pop.com.sg/can-every-irrational-number-be-written-in-terms-of-finitely-many-ratio.htmlConsider the irrational number $\sqrt{2}$. It can be written in terms, i.e., in a closed form expression, of two rational numbers as $2^{\frac{1}{2}}$. Does it hold in general that every irrational ...Fri, 28 Aug 2026 00:11:34 -0400https://pop.com.sg/sum-of-2-dice-rolls-is-even-probability.htmlhttps://pop.com.sg/sum-of-2-dice-rolls-is-even-probability.htmli have to calculate how probably it is, that the sum of 2 dice rolls is even. From my course i know that i have N^n Possibilities = 6^2 = 36. 6 = sides of dice and 2 = number of dice rolls Know i ...Fri, 28 Aug 2026 00:08:47 -0400https://pop.com.sg/how-do-i-negate-the-following-statement-closed.htmlhttps://pop.com.sg/how-do-i-negate-the-following-statement-closed.htmlHow do I negate the following statement? What are the steps which should be taken to negate an argument such as the one given? $$\exists y \forall x ((y > 0) \land (x < y))$$...Thu, 27 Aug 2026 23:56:35 -0400https://pop.com.sg/element-chasing-proof-that-a-setminus-c-setminus-b-setminus-c-a-setmin.htmlhttps://pop.com.sg/element-chasing-proof-that-a-setminus-c-setminus-b-setminus-c-a-setmin.htmlI would like to construct a formal proof of the following: $$(A\setminus C)\setminus(B\setminus C) = (A\setminus B)\setminus C$$ Let $a∈A$ be an arbitrary element, we will show that $a\in A \cap \...Thu, 27 Aug 2026 23:56:26 -0400https://pop.com.sg/how-to-calculate-the-dot-product-of-two-functions.htmlhttps://pop.com.sg/how-to-calculate-the-dot-product-of-two-functions.htmlHopefully someone can help me out with this. I have a question by my prof that goes like: Prove that the following function is orthogonal to each other (i.e., they form a basis in the space of square ...Thu, 27 Aug 2026 23:54:23 -0400https://pop.com.sg/why-does-i-i-equal-more-than-one-number-duplicate.htmlhttps://pop.com.sg/why-does-i-i-equal-more-than-one-number-duplicate.htmlAs I understand it, $i^i$ returns more than one real number. Why does this happen? When I raise $i$ to $i$, I'm raising a specific number to itself. So why do I get multiple values of this, as if i......Thu, 27 Aug 2026 23:53:32 -0400https://pop.com.sg/riemann-hypothesis-and-robin-039-s-inequality-implications.htmlhttps://pop.com.sg/riemann-hypothesis-and-robin-039-s-inequality-implications.htmlIs the following statement true: Let $\Bbb{A}$ be the set of all Natural numbers n, greater than or equal to 5041, for which the inequality $\displaystyle \sigma(n)<e^{\gamma}n\log\log n$ is not ...Thu, 27 Aug 2026 23:49:43 -0400https://pop.com.sg/is-integration-from-a-to-b-same-or-b-to-a-or-is-negative.htmlhttps://pop.com.sg/is-integration-from-a-to-b-same-or-b-to-a-or-is-negative.htmlThe integration is generally area under the curve in $\Bbb R^2 \to \Bbb R$ Is integration in the range from $a$ to $b$ the same as $b$ to $a$ or is it negative? If it is negative, Is it merely no......Thu, 27 Aug 2026 23:41:01 -0400https://pop.com.sg/construct-the-cross-section-of-a-cube-by-a-plane-passing-through-three.htmlhttps://pop.com.sg/construct-the-cross-section-of-a-cube-by-a-plane-passing-through-three.htmlI want to find/construct the cross section of a cube that includes the three points shown below. In other words, if a plane went through the cube such that it would slice where the points are, what......Thu, 27 Aug 2026 23:39:17 -0400https://pop.com.sg/finding-the-length-of-a-complex-vector-duplicate.htmlhttps://pop.com.sg/finding-the-length-of-a-complex-vector-duplicate.htmlLet us say the vector $u = (1 - i, 2 + 3i, 5)$. How would I find $\|u\|$? I have tried solving this through: $\sqrt{(1-i)^2 + (2+3i)^2 + 5^2}$ But I ended up needing to find the square root of a ...Thu, 27 Aug 2026 23:23:41 -0400https://pop.com.sg/calculating-probabilities-over-longer-period-of-time.htmlhttps://pop.com.sg/calculating-probabilities-over-longer-period-of-time.htmlThere's a great question/answer at: Calculating probabilities over different time intervals This is an awesome answer, but I'd like to ask a related question: What if the period goes the other dire......Thu, 27 Aug 2026 23:16:34 -0400https://pop.com.sg/proving-int-csc-2-x-dx.htmlhttps://pop.com.sg/proving-int-csc-2-x-dx.htmlAccording to the standard integral results, it is known that $\int \csc^2 x \, dx = -\cot x$ since $\frac{d}{dx} \cot x = - \csc^2 x$. However, supposing that this is not known, is there a better m......Thu, 27 Aug 2026 23:16:30 -0400https://pop.com.sg/how-to-excel-at-math.htmlhttps://pop.com.sg/how-to-excel-at-math.htmlI know this question has been asked at least a hundred times earlier but before you mark this as duplicate please read this cause I seriously need some professional help here. I'm a 15 year old ki......Thu, 27 Aug 2026 23:15:57 -0400https://pop.com.sg/find-x-as-function-of-y.htmlhttps://pop.com.sg/find-x-as-function-of-y.htmlI am trying to find $y$ as function of $x$ of: $$y=\frac{x-1}{x+1}$$ I have tried $$y=\frac{x-1}{x+1}\iff (x+1)y=x-1\iff(x+1)y+1=x$$ Which is of course wrong...Thu, 27 Aug 2026 23:15:55 -0400https://pop.com.sg/what-is-an-acyclic-connected-graph-in-graph-theory.htmlhttps://pop.com.sg/what-is-an-acyclic-connected-graph-in-graph-theory.htmlI want to know What it is and whether there is a difference in the definition when looking at undirected and directed graphs?...Thu, 27 Aug 2026 23:15:13 -0400https://pop.com.sg/about-the-definition-of-axiomatizable-theory-and-consistency.htmlhttps://pop.com.sg/about-the-definition-of-axiomatizable-theory-and-consistency.htmlDefinition: If $A$ is a theory and $B \subseteq A$ then $B$ is a set of axioms for $A$ iff 1) B is recursive and 2) $B \models C$ for all $C \in A$. We say $A$ is axiomatizable iff $A$ has a se......Thu, 27 Aug 2026 23:02:53 -0400https://pop.com.sg/trapezoidal-pyramid.htmlhttps://pop.com.sg/trapezoidal-pyramid.htmlIs the following problem correct? If it is correct, please present a solution. Given a trapezoidal pyramid ABCDM. The base $ABCD$ is trapezoid (where $AB\; ||\; CD$). The planes $ADM$ and $BCM$ ......Thu, 27 Aug 2026 22:52:04 -0400https://pop.com.sg/why-is-the-eccentricity-of-a-parabola-1.htmlhttps://pop.com.sg/why-is-the-eccentricity-of-a-parabola-1.htmlWhat I know: The standard form for a parabola is $(x-h)^2=4a(y-k)$, if the axis of symmetry of the parabola is vertical, or $(y-k)^2=4a(x-h)$, if the axis of symmetry of the parabola is horizontal......Thu, 27 Aug 2026 22:46:08 -0400https://pop.com.sg/what-is-division-sum-closed.htmlhttps://pop.com.sg/what-is-division-sum-closed.htmlWhat is the meaning of "division sum". I have seen the being used a lot but haven't found an explanation defining the term. Understanding from its usage, it seems to be another term for '...Thu, 27 Aug 2026 22:34:59 -0400https://pop.com.sg/suppose-that-x-and-y-are-irrational-but-x-y-is-rational-prove-that-x-y.htmlhttps://pop.com.sg/suppose-that-x-and-y-are-irrational-but-x-y-is-rational-prove-that-x-y.htmli was wondering if someone could check my proof $Q= \{a/b , c,d : a,c ∈ \mathbb Z , b,d ∈ N>0\}$ $a/b =x+y$ $a/b -y=x$ proof by contradiction. Let $x-y$ is rational $c/d = x-y$ sub $a/b -y......Thu, 27 Aug 2026 22:32:42 -0400https://pop.com.sg/find-the-constant-term-of-polynomial.htmlhttps://pop.com.sg/find-the-constant-term-of-polynomial.htmlThere's fifth degree polynomial, it's first coefficient equals $-7$. $$-7x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0$$ Also: $$W(1)=-2$$ $$W(2)=-4$$ $$W(3)=-6$$ $$W(4)=-8$$ $$W(5)=-10$$ Find the value of ...Thu, 27 Aug 2026 22:30:03 -0400https://pop.com.sg/power-method-matlab.htmlhttps://pop.com.sg/power-method-matlab.htmlI am trying to find the largest eigenvalue and its corresponding eigenvector by the Power method .. I did this code and I just want to display the largest eigenvalue and its eigenvector after ......Thu, 27 Aug 2026 22:22:50 -0400https://pop.com.sg/what-is-the-difference-between-a-linear-n-k-code-and-a-linear-n-k-d-co.htmlhttps://pop.com.sg/what-is-the-difference-between-a-linear-n-k-code-and-a-linear-n-k-d-co.htmlIn my lecture notes on coding theory, I have the following 2 definitions: Definition 1: Definition 2: Where: $n$ is the number of symbols in each codeword $k$ is the dimension of the vector space ...Thu, 27 Aug 2026 22:18:50 -0400https://pop.com.sg/for-any-real-function-if-f-0-0-then-f-039-0-0-closed.htmlhttps://pop.com.sg/for-any-real-function-if-f-0-0-then-f-039-0-0-closed.htmlThis seems a little too trivial but I have been facing many problems in my intro to real analysis books about functions who at first are discontinuous at zero but then "the hole is plugged" How co......Thu, 27 Aug 2026 22:06:00 -0400https://pop.com.sg/what-is-the-opposite-of-logarithmic-scale.htmlhttps://pop.com.sg/what-is-the-opposite-of-logarithmic-scale.htmlIf one wants to plot a variable whose value changes drastically, it makes sense to plot its order of magnitude. That way, the smaller values in the plot aren't squished all the way at the bottom. T......Thu, 27 Aug 2026 22:03:06 -0400https://pop.com.sg/the-class-of-all-classes-not-containing-themselves.htmlhttps://pop.com.sg/the-class-of-all-classes-not-containing-themselves.htmlIn ZF classes are used informally to resolve Russells Paradox, that is the collection of all sets that do not contain themselves does not form a set but a proper class. But doesn't the same paradox ...Thu, 27 Aug 2026 21:49:46 -0400https://pop.com.sg/mass-of-wire-using-line-integral.htmlhttps://pop.com.sg/mass-of-wire-using-line-integral.htmlIf $\rho(x, y)$ is the density of a wire (mass per unit length), then $m = \int \rho(x,y)ds$ is the mass of the wire. Find the mass of a wire having the shape of a semicircle $x = 1 + \cos(t), y = ......Thu, 27 Aug 2026 21:43:27 -0400https://pop.com.sg/what-is-the-mathematical-symbol-representation-for-the-remainder-opera.htmlhttps://pop.com.sg/what-is-the-mathematical-symbol-representation-for-the-remainder-opera.htmlIs there an equivalent in mathematical language to the modulo (or mod) function in computing?...Thu, 27 Aug 2026 21:34:29 -0400https://pop.com.sg/prove-the-riemann-lebesgue-lemma-int-b-af-x-cos-nx-dx-rightarrow-0-as-.htmlhttps://pop.com.sg/prove-the-riemann-lebesgue-lemma-int-b-af-x-cos-nx-dx-rightarrow-0-as-.htmlI would like to prove the Riemann-Lebesgue lemma, namely that $\int^b_af(x)\cos(nx)dx\rightarrow 0$ as $n\rightarrow \infty$ for any regulated function $f$. The textbook which I'm working from say......Thu, 27 Aug 2026 21:33:16 -0400https://pop.com.sg/express-frac-ln-left-frac-1-2-right-80000000-as-a-negative-exponent.htmlhttps://pop.com.sg/express-frac-ln-left-frac-1-2-right-80000000-as-a-negative-exponent.htmlI would like to express $\dfrac{\ln\left(\frac{1}{2}\right)}{80000000}$ as a negative exponent. Using Desmos online calculator I tried this variant: Why are the results different? I expected/goal ......Thu, 27 Aug 2026 21:31:52 -0400https://pop.com.sg/given-a-simple-graph-and-its-complement-prove-that-either-of-them-is-a.htmlhttps://pop.com.sg/given-a-simple-graph-and-its-complement-prove-that-either-of-them-is-a.htmlI was tasked to prove that when given 2 graphs $G$ and $\bar{G}$ (complement), at least one of them is a always a connected graph. Well, I always post my attempt at solution, but here I'm totally ......Thu, 27 Aug 2026 21:30:08 -0400https://pop.com.sg/is-empty-set-element-of-every-set-if-it-is-subset-of-every-set.htmlhttps://pop.com.sg/is-empty-set-element-of-every-set-if-it-is-subset-of-every-set.htmlThis problem is from Discrete Mathematics and its Applications My question is on 9b. I know that the sign represents an element is a member of. (from book) I know that the O with a slash across i......Thu, 27 Aug 2026 21:25:33 -0400https://pop.com.sg/pathway-to-low-dimensional-topology.htmlhttps://pop.com.sg/pathway-to-low-dimensional-topology.htmlI am currently studying Artin Algebra and Topology by Munkres. I have previously studied analysis. After reading few lecture notes on knot theory, I have developed a genuine interest in Low dimensi......Thu, 27 Aug 2026 21:15:30 -0400https://pop.com.sg/stationary-increments-definition.htmlhttps://pop.com.sg/stationary-increments-definition.htmlLet $X_t$ be a stochastic process on the reals. I was under the impression the definition of stationary increments was that $X_{t+s}-X_{s}$ has the same distribution as $X_{t}-X_{0}$ for all $s,t$. I ...Thu, 27 Aug 2026 21:03:19 -0400https://pop.com.sg/meaning-of-the-expression-mu-a-e.htmlhttps://pop.com.sg/meaning-of-the-expression-mu-a-e.htmlIn the monotone convergence theorem: Let $g_n\geq 0$ be a sequence of measurable functions, such that $g_n \uparrow g\;\; \mu \text{ a.e.},$ i.e. $g_n(\omega) \leq g_{n+1}(\omega)\;\; \mu \text{ a.......Thu, 27 Aug 2026 20:57:57 -0400https://pop.com.sg/limits-of-points-along-x-axis-of-the-2-variable-function.htmlhttps://pop.com.sg/limits-of-points-along-x-axis-of-the-2-variable-function.htmlLet $f:\Re ^2\rightarrow\Re$ be a real-valued function such that f(x,y) is defined as, $$f(x,y)= \begin{cases} xsin(1/y),& \text{if $y\neq 0$} \\[2ex] 0,& \text{if $y=0$} \end{cases}$$ The ...Thu, 27 Aug 2026 20:50:14 -0400https://pop.com.sg/product-of-two-gaussian-pdfs-is-a-gaussian-pdf-but-product-of-two-gaus.htmlhttps://pop.com.sg/product-of-two-gaussian-pdfs-is-a-gaussian-pdf-but-product-of-two-gaus.htmlThe Product of Two Gaussian Random Variables is not Gaussian distributed: Is the product of two Gaussian random variables also a Gaussian? Also Wolfram Mathworld So this is saying $X \sim N(\mu_1, \...Thu, 27 Aug 2026 20:47:00 -0400https://pop.com.sg/how-to-prove-this-is-a-rational-number.htmlhttps://pop.com.sg/how-to-prove-this-is-a-rational-number.htmlI'm not sure how to prove this is a rational number $\frac{q}{m}$, can some one show me? $$\frac{q}{m}=\frac{(\frac{1+\sqrt5}{2})^n - (\frac{1-\sqrt5}{2})^n}{\sqrt5}$$...Thu, 27 Aug 2026 20:37:36 -0400https://pop.com.sg/give-a-geometric-description-of-the-following-set-of-points.htmlhttps://pop.com.sg/give-a-geometric-description-of-the-following-set-of-points.htmlGive a geometric description of the following set of points: $x^2 +y^2 + z^2-8x+14y-18z>/= 65 $ So I completed the square and got the set to read: $(x-4)^2+(y+7)^2+(z-9)^2>/= 211 $ Howeve......Thu, 27 Aug 2026 20:30:57 -0400https://pop.com.sg/covariance-of-two-random-variables.htmlhttps://pop.com.sg/covariance-of-two-random-variables.htmlI am trying to find the covariance of two random variables but I am not having any lucky. Just for simplicity lets say that my random variables are : X = value rolled by a die Y = 1 if even 0 if ......Thu, 27 Aug 2026 20:26:13 -0400https://pop.com.sg/why-f-x-sqrt-x-is-a-function.htmlhttps://pop.com.sg/why-f-x-sqrt-x-is-a-function.htmlWhy $f(x) = \sqrt{x}$ is a function (as I found in my textbook) since for example the square root of $25$ has two different outputs ($-5,5$) and a function is defined as "A function from A to B is ......Thu, 27 Aug 2026 20:25:54 -0400https://pop.com.sg/user-al-jabra-mathematics-stack-exchange.htmlhttps://pop.com.sg/user-al-jabra-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Thu, 27 Aug 2026 19:59:52 -0400https://pop.com.sg/how-does-e-ln-x-e-ln-1-x.htmlhttps://pop.com.sg/how-does-e-ln-x-e-ln-1-x.htmlI understand the inverse of e^{x} is the natural logarithm. However I don't understand how the following expression is true: $e^{-\ln x} = e^{\ln(1/x)}$ Any assistance is appreciated....Thu, 27 Aug 2026 19:36:24 -0400https://pop.com.sg/the-volume-of-body-in-polar-coordinates.htmlhttps://pop.com.sg/the-volume-of-body-in-polar-coordinates.htmlIf we have curve $r=F(\varphi)$ and two polar radii $\varphi=\alpha$ and $\varphi=\beta$ $(\alpha<\beta)$. Then volume of body which is bounded by above polar curve and lines which is revolved a......Thu, 27 Aug 2026 19:32:43 -0400https://pop.com.sg/how-to-solve-a-system-with-more-equations-than-unkowns.htmlhttps://pop.com.sg/how-to-solve-a-system-with-more-equations-than-unkowns.htmlIn general, how do you solve a system with more equations than unknowns? I know that if I select the equations to match them with the number of unknowns, there may be zero or many solutions dependi......Thu, 27 Aug 2026 19:26:13 -0400https://pop.com.sg/to-find-an-extremal-of-the-functional-int-0-1-y-039-2-12-xy-dx.htmlhttps://pop.com.sg/to-find-an-extremal-of-the-functional-int-0-1-y-039-2-12-xy-dx.htmlI have to find extremal of following : $\int_0^1 [(y')^2 + 12 xy] dx$ with $y(0) = 0$ and $y(1) = 1$. I applied the Euler's equation $\frac{\partial F}{\partial y} - \frac{d}{dx}(\frac{\partial F}{\...Thu, 27 Aug 2026 19:18:25 -0400https://pop.com.sg/finding-parametric-equations-for-curve-of-intersection.htmlhttps://pop.com.sg/finding-parametric-equations-for-curve-of-intersection.htmlI am having difficulty finding parametric equations of the curve of intersection of $z = x^2 - y^2$ and cylinder $x^2 + y^2 = 1$ I am aware that the first equation represents a hyperboloid, and the ...Thu, 27 Aug 2026 19:16:53 -0400https://pop.com.sg/correct-interpretation-of-confidence-interval.htmlhttps://pop.com.sg/correct-interpretation-of-confidence-interval.htmlI think this question may have been asked in one form or another, but I did not find an answer that I understood or thought was satisfactory...so I apologize if it has been explained. My understan......Thu, 27 Aug 2026 19:15:19 -0400https://pop.com.sg/product-rule-for-gradient.htmlhttps://pop.com.sg/product-rule-for-gradient.htmlI have two functions scalar functions of vector $\textbf{x}$, $f(\textbf{x})$ and $g(\textbf{x})$. I know the gradient of $f(\textbf{x})$ (i.e. $\triangledown f(\textbf{x})$) and I want to find the ...Thu, 27 Aug 2026 19:08:57 -0400https://pop.com.sg/how-can-i-simplify-and-verify-the-logical-equivalence-using-these-laws.htmlhttps://pop.com.sg/how-can-i-simplify-and-verify-the-logical-equivalence-using-these-laws.html∼(p ∨∼q) ∨ (∼p ^ ~ q) ≡ ~p Please help I don't know where to start. These are the laws I need to list in each step when simplifying. Commutative laws: p ∧ q ≡ q ∧ p p ∨ q ≡ q ∨ p Associative l......Thu, 27 Aug 2026 18:55:18 -0400https://pop.com.sg/linearizing-logarithmic-function.htmlhttps://pop.com.sg/linearizing-logarithmic-function.htmlI have a given set of data points (y,x) with uncertainties. When I plot those points on a graph, the trendline appears to follow the equation y = c + a*ln(x). I want to be able to find the uncer......Thu, 27 Aug 2026 18:46:28 -0400https://pop.com.sg/what-is-this-angle-in-a-right-triangle-with-sides-of-length-5-12-and-1.htmlhttps://pop.com.sg/what-is-this-angle-in-a-right-triangle-with-sides-of-length-5-12-and-1.htmlHow do I find the missing adjacent angle to leg b in a right triangle with the following side lengths: leg a = 5, leg b = 12, and hypotenuse = 13. Thanks...Thu, 27 Aug 2026 18:45:51 -0400https://pop.com.sg/finding-remainder-using-fermat-039-s-little-theorm.htmlhttps://pop.com.sg/finding-remainder-using-fermat-039-s-little-theorm.htmlUse Fermat's Little Theorem to find the remainder of $5^{15}$ divided by $1337$. I know Fermat's Little Theorem, but failed to understand how to use it in this. output.944...Thu, 27 Aug 2026 18:45:35 -0400https://pop.com.sg/proof-of-stars-and-bars-formula.htmlhttps://pop.com.sg/proof-of-stars-and-bars-formula.htmlI am trying to prove a formula (for ways of distributing $n$ identical balls among $r$ persons when each person may get any number of balls) ${n+r-1}\choose{r-1}$. But I am not able to prove it. I ......Thu, 27 Aug 2026 18:45:01 -0400https://pop.com.sg/finding-the-parametric-form-of-a-standard-equation.htmlhttps://pop.com.sg/finding-the-parametric-form-of-a-standard-equation.htmlI need to find the parametric form of $3x - 2y + 10 = 0$. I found that the parametric form for this equation could be : \begin{align}x &= t\\ y &= 5 + \frac{3}{2} t \end{align} I did thi......Thu, 27 Aug 2026 18:39:06 -0400https://pop.com.sg/finding-domain-of-sqrt-x-12-x-9-x-4-x-1.htmlhttps://pop.com.sg/finding-domain-of-sqrt-x-12-x-9-x-4-x-1.htmlHow to find the domain of this function $\sqrt{x^{12} - x^9 + x^4 - x + 1}$? Its really hard to find all 12 roots and plot them or use tricks like "wavy curve" to check where the expression in the ...Thu, 27 Aug 2026 18:38:30 -0400https://pop.com.sg/equilibrium-probability-in-a-markov-chain-closed.htmlhttps://pop.com.sg/equilibrium-probability-in-a-markov-chain-closed.htmlcan someone please tell me what an equilibrium probability in a markov chain is and what the relation between the equlibrium probability and equilibrium distribution is? Thanks...Thu, 27 Aug 2026 18:31:17 -0400https://pop.com.sg/correct-graph-of-arccot-x.htmlhttps://pop.com.sg/correct-graph-of-arccot-x.htmlThere are 2 possible ways of choosing an interval of cot x graph for obtaining the graph of arccot x. Why do we choose the interval (0,$\pi$) instead of the first choice? This may be because the gr......Thu, 27 Aug 2026 18:29:29 -0400https://pop.com.sg/how-to-compute-a-linear-fractional-transformation-that-maps-a-circle-t.htmlhttps://pop.com.sg/how-to-compute-a-linear-fractional-transformation-that-maps-a-circle-t.htmlHow to compute a linear fractional transformation that maps a circle to a given circle? For example, let $C_1$ be the circle $|z+(2+i)|=1$ and $C_2$ be the circle $|z-5|=7$. How to find a linear ...Thu, 27 Aug 2026 18:27:55 -0400https://pop.com.sg/either-john-isn-039-t-stupid-and-he-is-lazy-or-he-is-stupid-he-039-s-s.htmlhttps://pop.com.sg/either-john-isn-039-t-stupid-and-he-is-lazy-or-he-is-stupid-he-039-s-s.htmlI'm reading How To Prove it by Daniel J. Velleman, and I came across the following example which sounds odd to me. Either John isn't stupid and he is lazy, or he is stupid. He's Stupid. Ther......Thu, 27 Aug 2026 18:20:04 -0400https://pop.com.sg/how-to-factorize-x-2-10yz-2xz-2xy-3y-2-3z-2.htmlhttps://pop.com.sg/how-to-factorize-x-2-10yz-2xz-2xy-3y-2-3z-2.htmlHow to factorize $$x^2+10yz-2xz-2xy-3y^2-3z^2$$ It is expanded and we should make them into parts and factorize each part individually. the last answer is $$(x+y-3z)(x-3y+z)$$ but how to get it ?...Thu, 27 Aug 2026 18:17:00 -0400https://pop.com.sg/is-there-a-general-formula-for-the-sum-of-a-quadratic-sequence.htmlhttps://pop.com.sg/is-there-a-general-formula-for-the-sum-of-a-quadratic-sequence.htmlI tried Googling "formula for sum of quadratic sequence", which did not give me anything useful. I just want an explicit formula for figuring out a sum for a quadratic sequence. For example, how wo......Thu, 27 Aug 2026 18:12:13 -0400https://pop.com.sg/variance-of-conditional-variance.htmlhttps://pop.com.sg/variance-of-conditional-variance.htmlIf we have two random variables X and Y which share a joint pdf and pmf (there is a discrete and a continuous scenario), how do we calculate: Var[Var[X|Y]] - I looked at the Theory of Total Varian......Thu, 27 Aug 2026 18:09:18 -0400https://pop.com.sg/proof-of-caratheodory-039-s-theorem-for-convex-sets-using-radon-039-s-.htmlhttps://pop.com.sg/proof-of-caratheodory-039-s-theorem-for-convex-sets-using-radon-039-s-.htmlI am self-studying some discrete geometry / convex analysis. Many descriptions of Caratheodory's Theorem for convex sets mention that Radon's Lemma can be used to simplify the proof, but I haven't......Thu, 27 Aug 2026 17:51:52 -0400https://pop.com.sg/finding-the-solution-set-for-an-absolute-value-equation-algebraically.htmlhttps://pop.com.sg/finding-the-solution-set-for-an-absolute-value-equation-algebraically.htmle.g. the equation $|x-2|=3$, geometrically it's easy to solve drawing the number line. I get the solution set {-1,5} which I understand. but I was looking at some notes I found online of how to so......Thu, 27 Aug 2026 17:49:02 -0400https://pop.com.sg/show-that-if-h-and-k-are-subgroups-of-one-abelian-group-g-then-g-1-hk-.htmlhttps://pop.com.sg/show-that-if-h-and-k-are-subgroups-of-one-abelian-group-g-then-g-1-hk-.htmlHere's the problem: Show that if $H$ and $K$ are subgroups of one abelian group $G$, then $G_1=\{hk\mid h\in H\text{ and } k\in K\}$ is a subgroup of $G$ I know that "abelian" means tha......Thu, 27 Aug 2026 17:41:25 -0400https://pop.com.sg/fitch-system-or-alternatives.htmlhttps://pop.com.sg/fitch-system-or-alternatives.htmlIs there a good resource for how to understand the rules of a Fitch proof system, or an alternative? I'm currently trying to get a sense or how things are proved in natural deduction, which differs......Thu, 27 Aug 2026 17:34:09 -0400https://pop.com.sg/icosahedral-symmetry-as-permutation-group.htmlhttps://pop.com.sg/icosahedral-symmetry-as-permutation-group.htmlHopefully an easy question: the icosahedral group of order 60 (orientation preserving symmetries of a regular icosahedron) is isomorphic to the alternating group on 5 points. In terms of the icosa......Thu, 27 Aug 2026 17:29:04 -0400https://pop.com.sg/when-to-use-the-contrapositive-to-prove-a-statment.htmlhttps://pop.com.sg/when-to-use-the-contrapositive-to-prove-a-statment.htmlMy question tries to address the intuition or situations when using the contrapositive to prove a mathematical statement is an adequate attempt. Whenever we have a mathematical statement of the fo......Thu, 27 Aug 2026 17:27:11 -0400https://pop.com.sg/is-my-understanding-of-antisymmetric-and-symmetric-relations-correct.htmlhttps://pop.com.sg/is-my-understanding-of-antisymmetric-and-symmetric-relations-correct.htmlSo I'm having a hard time grasping how a relation can be both antisymmetric and symmetric, or neither. Are my examples correct? symmetric & antisymmetric R ={(1,1),(2,2),(3,3)} not symmetr......Thu, 27 Aug 2026 17:18:12 -0400https://pop.com.sg/using-method-of-moments-on-binomial-distribution-closed.htmlhttps://pop.com.sg/using-method-of-moments-on-binomial-distribution-closed.htmlHi, I am stuck on this particular question. Any help? Thanks....Thu, 27 Aug 2026 17:17:09 -0400https://pop.com.sg/recurrence-relation-merge-sort.htmlhttps://pop.com.sg/recurrence-relation-merge-sort.htmlWe know the recurrence relation for normal merge sort. It is T(n) = 2T(n/2) + n. After solving it we can get T(n) = cnlogn. I would like to know the recurrence relation for K way merge sort i.e. in......Thu, 27 Aug 2026 17:11:30 -0400https://pop.com.sg/integral-from-infinity-to-infinity.htmlhttps://pop.com.sg/integral-from-infinity-to-infinity.htmlMy physics professor today wrote on the blackboard: $$ \int_{\infty}^{\infty} f(x) dx = 0 $$ for every function $f$. And the proof he gave was: $$ \int_{\infty}^{\infty} f(x) dx = \int_{\infty}^{a......Thu, 27 Aug 2026 17:11:11 -0400https://pop.com.sg/a-parallelogram-and-a-circle.htmlhttps://pop.com.sg/a-parallelogram-and-a-circle.htmlThe perimeter of a parallelogram $ABCD$ is $30$ $cm$. If $D$ and $M$ which is the midpoint of $\overline{DC}$ lie on the circle with diameter $\overline{AB}$, what is the perimeter of $ABMD$? We k......Thu, 27 Aug 2026 16:58:59 -0400https://pop.com.sg/why-is-two-the-only-even-number-that-is-prime.htmlhttps://pop.com.sg/why-is-two-the-only-even-number-that-is-prime.htmlThe other prime numbers are all odd numbers such as $5, 11, 127,$ and $37$. So, why is $2$ the only prime even number there is? Is it because it only has 1 and itself that way, even though it's ......Thu, 27 Aug 2026 16:58:54 -0400https://pop.com.sg/relaxed-menage-problem-sitting-arrangement-for-couples.htmlhttps://pop.com.sg/relaxed-menage-problem-sitting-arrangement-for-couples.htmlThe original problem is this: How many ways are there to seat n couples around a circular table such that no couple sits next to each other? For an answer, I looked at the solution here: https://...Thu, 27 Aug 2026 16:55:17 -0400