https://pop.com.sgExplosive pop stories with broad entertainment appeal.en-usSun, 16 Aug 2026 15:46:34 -0400https://pop.com.sg/proving-that-a-galois-group-gal-e-q-is-isomorphic-to-mathbb-f-p-times.htmlhttps://pop.com.sg/proving-that-a-galois-group-gal-e-q-is-isomorphic-to-mathbb-f-p-times.htmlI have seen many textbooks state this result without proof. $``$ If $E$ is the splitting field for the polynomial $f=x^p-1 \in \mathbb{Q}[X]$ where $p$ is prime, then the Galois group $Gal(E:Q)\s......Sun, 16 Aug 2026 19:42:52 -0400https://pop.com.sg/singular-value-decomposition-of-product-of-matrices.htmlhttps://pop.com.sg/singular-value-decomposition-of-product-of-matrices.htmlGiven SVD(A) and SVD(B) and B is a diagonal matrix, is there a way or method to construct SVD(AB) ?...Sun, 16 Aug 2026 19:28:29 -0400https://pop.com.sg/find-the-volume-of-the-solid-above-the-cone-z-sqrt-x-2-y-2-and-below-t.htmlhttps://pop.com.sg/find-the-volume-of-the-solid-above-the-cone-z-sqrt-x-2-y-2-and-below-t.htmlI actually have found the solution using double integral in polar coordinate. However, I am curious about whether I could find the same exact solution using triple integral in spherical coordinates......Sun, 16 Aug 2026 19:27:23 -0400https://pop.com.sg/how-to-raise-a-fraction-to-a-fractional-exponent.htmlhttps://pop.com.sg/how-to-raise-a-fraction-to-a-fractional-exponent.htmlI am trying to simplify this but I'm not sure how to approach it....Sun, 16 Aug 2026 19:19:13 -0400https://pop.com.sg/distance-point-on-ellipse-to-centre.htmlhttps://pop.com.sg/distance-point-on-ellipse-to-centre.htmlI'm trying to calculate the distance of a certain point of an ellipse to the centre of that ellipse: The blue things are known: The lengths of the horizontal major radius and vertical minor radius......Sun, 16 Aug 2026 19:17:11 -0400https://pop.com.sg/confusing-qualifiers-in-logic.htmlhttps://pop.com.sg/confusing-qualifiers-in-logic.htmlSuppose we know that there exists widgets. For any widget $X$, we have that the statements $A(X)$ and $B(X)$. Suppose it's true that for any $X$, $A(X) \not \Rightarrow B(X)$. How do I interpret th......Sun, 16 Aug 2026 19:13:32 -0400https://pop.com.sg/given-the-determinant-of-a-2-times-2-matrix-calculate-the-determinant-.htmlhttps://pop.com.sg/given-the-determinant-of-a-2-times-2-matrix-calculate-the-determinant-.htmlIf $$\det \underbrace{\begin{bmatrix} a&b\\ c&d\end{bmatrix}}_{=: A} = -3$$ calculate the determinant $$\det \underbrace{\begin{bmatrix} 2&-2&0\\ c+1&-1&2a\\ d-2&2&2......Sun, 16 Aug 2026 19:12:17 -0400https://pop.com.sg/what-point-on-the-line-y-2x-4-is-closest-to-the-origin.htmlhttps://pop.com.sg/what-point-on-the-line-y-2x-4-is-closest-to-the-origin.htmlWhat point on the line $y=2x+4$ is closest to the origin in $\mathbb{R}^2$? What is the distance from this point to the origin? I know the point closest to the origin is perpendicular to the origin, ...Sun, 16 Aug 2026 19:06:30 -0400https://pop.com.sg/level-curve-of-function-that-passes-through-given-point.htmlhttps://pop.com.sg/level-curve-of-function-that-passes-through-given-point.htmlI have a function $f(x,y)= 16-x^2-y^2$ that passes through the point $(2\sqrt(2),\sqrt(2))$ . How do I find out the equation of level curve? I dont have any idea . Pls provide hints. What I am do......Sun, 16 Aug 2026 19:05:17 -0400https://pop.com.sg/finding-the-matrix-associated-with-a-linear-map.htmlhttps://pop.com.sg/finding-the-matrix-associated-with-a-linear-map.htmlFind the matrix associated with the linear map $f:R^2 \rightarrow R^2$ defined by $f(x,y)=(3x-y,y-x)$ with respect to the ordered basis ${(1,0),(1,1)}$ Let the matrix be $A$ and let $f(x)=AX$ wher......Sun, 16 Aug 2026 19:01:43 -0400https://pop.com.sg/add-1100-0110-in-binary.htmlhttps://pop.com.sg/add-1100-0110-in-binary.htmlI'm studying encryption in class and we're doing "One-Time-Pad" encryption and I've come across this situation. I want to code $1100$, where my key is $0110$). In my book it says that the cipher is......Sun, 16 Aug 2026 18:55:49 -0400https://pop.com.sg/derivative-of-definite-integral.htmlhttps://pop.com.sg/derivative-of-definite-integral.htmlI am learning Calculus online and this problem stumped me. The solution doesn't really make sense either. I'm starting to think that the user made a typo. Here is the question to simplify: $$\f......Sun, 16 Aug 2026 18:53:10 -0400https://pop.com.sg/how-to-determine-the-values-of-the-coefficient-c-in-the-objective-func.htmlhttps://pop.com.sg/how-to-determine-the-values-of-the-coefficient-c-in-the-objective-func.htmlLet's say we have the following optimization problem : Max cx Subject to $Ax=b$ with $x\geq0$ How can we find the interval of the values for the coefficient $c_{j}$ of the objective where the o......Sun, 16 Aug 2026 18:50:20 -0400https://pop.com.sg/how-many-positive-integers-less-than-1000-are-there-which-contain-at-l.htmlhttps://pop.com.sg/how-many-positive-integers-less-than-1000-are-there-which-contain-at-l.htmlHow many positive integers less than 1000 are there which contain at least one 4 or at least one 9 (or both)? Obviously theres 100 in 400's and 900's but short of counting every other number whats......Sun, 16 Aug 2026 18:42:52 -0400https://pop.com.sg/integration-and-differentiation-of-fourier-series.htmlhttps://pop.com.sg/integration-and-differentiation-of-fourier-series.htmlI am interested in the properties of Fourier series under integration and differentiation, and I've noticed a "strange" phenomenon. Suppose I have a Fourier series which I Integrate, and suppose th......Sun, 16 Aug 2026 18:41:03 -0400https://pop.com.sg/what-is-the-modulus-of-a-number.htmlhttps://pop.com.sg/what-is-the-modulus-of-a-number.htmlWhat is the exact definition of the modulus of a number? As far as I know, it is the distance between the origin and the point associated with this number. So if $z=a+bi \in \Bbb C,|z|=OM=\sqrt{a^2......Sun, 16 Aug 2026 18:33:00 -0400https://pop.com.sg/number-of-possible-combinations-in-which-ordering-doesn-039-t-matter.htmlhttps://pop.com.sg/number-of-possible-combinations-in-which-ordering-doesn-039-t-matter.htmlI tried to figure this out the other day, but failed to arrive at a formula. It seems like a simple problem, though. Forgive me if my use of terminology sounds off, I have never had any formal ...Sun, 16 Aug 2026 18:23:17 -0400https://pop.com.sg/complex-number-how-to-find-the-angle-between-the-imaginary-axis-and-re.htmlhttps://pop.com.sg/complex-number-how-to-find-the-angle-between-the-imaginary-axis-and-re.htmlAssume I have complex number $z = a + ib$. $z$ can be represented by a polar representation as $r(\cos \theta+i\sin \theta)$, when $r$ is the absolute value of $z$, $\sqrt{a^2 + b^2}$. But how......Sun, 16 Aug 2026 18:14:51 -0400https://pop.com.sg/can-you-find-an-equation-parallel-or-perpendicular-to-a-line-when-it-i.htmlhttps://pop.com.sg/can-you-find-an-equation-parallel-or-perpendicular-to-a-line-when-it-i.htmlFor example, if I need to find the equation of the line parallel to $$2x-3y=4$$ which passes through the point $(1,-5)$ I know how to do this by putting it into slope-intercept form first to find the ...Sun, 16 Aug 2026 17:59:49 -0400https://pop.com.sg/sum-of-2-positive-irrational-numbers-is-irrational-duplicate.htmlhttps://pop.com.sg/sum-of-2-positive-irrational-numbers-is-irrational-duplicate.htmlSo, I know how the sum of 2 irractional numbers as a whole can be rational. For example, an irrational number $a$ and it's negative counterpart $-a$ have a sum of zero and so the sum is rational. But ...Sun, 16 Aug 2026 17:53:50 -0400https://pop.com.sg/what-is-the-definition-of-a-properly-immersed-arc.htmlhttps://pop.com.sg/what-is-the-definition-of-a-properly-immersed-arc.htmlI am going over a paper in which a theorem includes a condition that certain arcs are "properly-immersed geodesic arcs" in a surface $\Sigma$ with nonempty boundary. I have done a search, but I hav......Sun, 16 Aug 2026 17:39:25 -0400https://pop.com.sg/is-73-37-the-only-pair-of-reversible-primes-p-q-s-t-p-2q-1.htmlhttps://pop.com.sg/is-73-37-the-only-pair-of-reversible-primes-p-q-s-t-p-2q-1.htmlIn addition to being probably the only Sheldon Cooper prime, $73$ is a reversible prime $p$ (or emirp), such that its reverse is $q=(p+1)/2$. It is not hard to see that all other reversible primes ......Sun, 16 Aug 2026 17:35:40 -0400https://pop.com.sg/how-to-find-the-equation-of-a-line-tangent-to-a-circle-that-passes-thr.htmlhttps://pop.com.sg/how-to-find-the-equation-of-a-line-tangent-to-a-circle-that-passes-thr.htmlI am given the equation of a circle: $(x + 2)^2 + (y + 7)^2 = 25$. The radius is $5$. Center of the circle: $(-2, -7)$. Two lines tangent to this circle pass through point $(4, -3)$, which is outs......Sun, 16 Aug 2026 17:17:40 -0400https://pop.com.sg/distributive-law-for-fraction-arithmetic.htmlhttps://pop.com.sg/distributive-law-for-fraction-arithmetic.htmlI'm in seventh grade and my teacher wasn't able to explain this to me. Why is $\ \frac{c}{a+b}\neq \frac ca +\frac cb,\,$ but $\ \frac{a+b}c = \frac{a}c + \frac{b}c$? I'm sorry if this is obvious.......Sun, 16 Aug 2026 17:15:57 -0400https://pop.com.sg/prove-that-the-equation-x-10000-x-100-1-0-has-a-solution-with-0-x-1.htmlhttps://pop.com.sg/prove-that-the-equation-x-10000-x-100-1-0-has-a-solution-with-0-x-1.htmlProve that the equation $x^{10000} + x^{100} - 1 = 0$ has a solution with $0 < x < 1$. This is a homework question. I know I could probably find a solution that would complete the proof, but......Sun, 16 Aug 2026 17:13:56 -0400https://pop.com.sg/what-are-the-two-039-sides-039-of-a-decimal-number-called-duplicate.htmlhttps://pop.com.sg/what-are-the-two-039-sides-039-of-a-decimal-number-called-duplicate.htmlIs there a fancy name for the "left side" and "right side" of a decimal number? (That is, the pre-decimal part and the post-decimal part.)...Sun, 16 Aug 2026 16:52:49 -0400https://pop.com.sg/internally-tangent-circles-and-circumcircle.htmlhttps://pop.com.sg/internally-tangent-circles-and-circumcircle.htmlThe circle w is internally tangent with the circumcircle of $\triangle ABC$ at point A. $W \cap AB = K$ and $w$ intersects $BC$. The line $CL$ is tangent to $w$ at point L and $KL \cap BC = T$. Prove ...Sun, 16 Aug 2026 16:48:52 -0400https://pop.com.sg/what-does-a-dot-after-a-number-mean.htmlhttps://pop.com.sg/what-does-a-dot-after-a-number-mean.htmlSo I'm making some calculations for numerical analysis and the output I get in Wolfram or Mathematica for input like: -0.666667 (x-1) (x-0.5) (x+0.5)+0.166666 (x-1) (x+1) (x+0.5)+0.499999 (x-1) (x......Sun, 16 Aug 2026 16:48:17 -0400https://pop.com.sg/comparing-log-2-3-to-log-3-5.htmlhttps://pop.com.sg/comparing-log-2-3-to-log-3-5.htmlHaving been asked to compare $\log_2(3)$ and $\log_3(5)$, this is my proof: $$\log_2(3)>\log_3(5)$$ Then one uses the rule $\log_a(b)=\frac{1}{\log_b(a)}$, so $$\log_3(2)>\frac{1}{\log_3(5)}$$ ...Sun, 16 Aug 2026 16:42:58 -0400https://pop.com.sg/composition-of-a-rotation-and-a-traslation-is-a-rotation.htmlhttps://pop.com.sg/composition-of-a-rotation-and-a-traslation-is-a-rotation.htmlAssuming that I know that: 1) A plane isometry is either a rotation, a traslation, a reflection, or a glide reflection 2) Every isometry can be expressed as a product of reflections, 3) Rotatio......Sun, 16 Aug 2026 16:37:47 -0400https://pop.com.sg/requirements-for-a-linear-transformation.htmlhttps://pop.com.sg/requirements-for-a-linear-transformation.htmlLet's say we have a transformation: $$T: \mathbb{R}^n \rightarrow \mathbb{R}^m.$$ This is a linear transformation iff: for all $ \vec{a} , \vec{b} \in \mathbb{R}^n$ and $c \in \Bbb{R}$, $T(\vec{......Sun, 16 Aug 2026 16:35:20 -0400https://pop.com.sg/how-do-i-prove-that-arccos-x-arccos-x-pi-when-x-in-1-1-closed.htmlhttps://pop.com.sg/how-do-i-prove-that-arccos-x-arccos-x-pi-when-x-in-1-1-closed.htmlProve that $\arccos x + \arccos(-x) = \pi$ when $x \in [-1,1]$. How do I prove this? Where should I begin and what should I consider?...Sun, 16 Aug 2026 16:31:52 -0400https://pop.com.sg/chain-rule-for-partial-derivatives.htmlhttps://pop.com.sg/chain-rule-for-partial-derivatives.htmlMinton and Smith, in "Calculus" define the chain rule for full derivatives $\frac {dz} {dt}$ as it follows: Vretblad, however, in "Fourier Analysis and its Applications", mentions an "easy exercis......Sun, 16 Aug 2026 16:31:49 -0400https://pop.com.sg/solve-first-order-matrix-differential-equation.htmlhttps://pop.com.sg/solve-first-order-matrix-differential-equation.htmlIf I have a differential equation $ y'(t)=A y(t)$ where A is a constant square matrix that is not diagonalizable(although it is surely possible to calculate the eigenvalues) and no initial conditio......Sun, 16 Aug 2026 16:27:47 -0400https://pop.com.sg/slope-between-two-points-vectors-closed.htmlhttps://pop.com.sg/slope-between-two-points-vectors-closed.htmlI have two vector points, $\langle 1,3,7 \rangle$ and $\langle 23,-5,10 \rangle$. Point $1$ is at $t=13$ and Point $2$ is at $t=81$. I know how to find the slope between two vector points when $t=0......Sun, 16 Aug 2026 16:25:21 -0400https://pop.com.sg/equilateral-and-equiangular-polygon.htmlhttps://pop.com.sg/equilateral-and-equiangular-polygon.htmlCan we have an equilateral polygon $n \geq 5$, which is not equiangular? Ot does every odd n-gon which is equilateral must be equiangular? Is a construction of an equilateral but not equiangular n-......Sun, 16 Aug 2026 16:14:12 -0400https://pop.com.sg/why-is-a-symmetric-group-a-group.htmlhttps://pop.com.sg/why-is-a-symmetric-group-a-group.htmlIf the identity axiom for groups is the following $ae = ea = a$ ($e$ the identity element) and the operation for a symmetric group is function composition then how does a symmetric group pass this ......Sun, 16 Aug 2026 16:03:36 -0400https://pop.com.sg/circle-and-a-quadrilateral.htmlhttps://pop.com.sg/circle-and-a-quadrilateral.htmlQ: The polygon circumscribes the circle. Find the perimeter....Sun, 16 Aug 2026 15:54:06 -0400https://pop.com.sg/properties-of-triangles-in-non-euclidean-geometries.htmlhttps://pop.com.sg/properties-of-triangles-in-non-euclidean-geometries.htmlAs we all know, the angles in all triangles in Euclidean geometry must add up to $180^\circ$. As some of us may know, this is not true in non-Euclidean geometries; for example, on the surface of a ......Sun, 16 Aug 2026 15:53:03 -0400https://pop.com.sg/finding-the-basis-of-a-null-space.htmlhttps://pop.com.sg/finding-the-basis-of-a-null-space.htmlI am trying to understand why the method used in my linear algebra textbook to find the basis of the null space works. The textbook is 'Elementary Linear Algebra' by Anton. According to the textbo......Sun, 16 Aug 2026 15:50:34 -0400https://pop.com.sg/verification-of-my-solution-of-xy-039-2y.htmlhttps://pop.com.sg/verification-of-my-solution-of-xy-039-2y.htmlI am given: \begin{equation}xy'=2y\end{equation} Therefore, \begin{equation}y'=(2y)/x\end{equation} Solving for y, I have: $y=Cx^2$ (Originally, was $e^cx^2$, but C is arbitrary, so I simplifi......Sun, 16 Aug 2026 15:31:55 -0400https://pop.com.sg/solving-differential-equation-y-039-039-1-y-039-2.htmlhttps://pop.com.sg/solving-differential-equation-y-039-039-1-y-039-2.htmlFrom the Question $y''=1+(y')^2$ I used Reduction of Order by taking $y'=z$ to get, $$z\left(\frac{dz}{dy}\right)=1+z^2$$ Solve the equation by Separation of Variable, to get $$\frac{1}{2} \ln(1+z^......Sun, 16 Aug 2026 15:29:17 -0400https://pop.com.sg/can-any-right-triangle-be-inscribed-in-a-circle.htmlhttps://pop.com.sg/can-any-right-triangle-be-inscribed-in-a-circle.htmlCan every right triangle be inscribed in a circle? If so, we could infer: ": in any right triangle, the segment which cuts the hypotenuse in two and joins the opposite angle is equal to half the ...Sun, 16 Aug 2026 15:28:08 -0400https://pop.com.sg/how-to-construct-an-equilateral-triangle-on-2-concentric-circles.htmlhttps://pop.com.sg/how-to-construct-an-equilateral-triangle-on-2-concentric-circles.htmlConstruct an equilateral triangle with the given vertex so that the other vertices lie on the concentric circles respectively. I constructed the triangle, but I don't know how it works. How does......Sun, 16 Aug 2026 15:21:34 -0400https://pop.com.sg/divergence-of-laplacian.htmlhttps://pop.com.sg/divergence-of-laplacian.htmlIt seems to me that, in some derivations on fluid dynamic books I am reading, the identity $$\nabla \cdot (\nabla^2 u) = 0 $$ where $u$ is a vector field, is used. Does this identity exist? Is it ......Sun, 16 Aug 2026 15:10:05 -0400https://pop.com.sg/what-are-the-conditions-for-a-polygon-to-be-tessellated.htmlhttps://pop.com.sg/what-are-the-conditions-for-a-polygon-to-be-tessellated.htmlUpon one of my mathematical journey's (clicking through wikipedia), I encountered one of the most beautiful geometrical concept that I have ever encountered in my 16 and a half years on this oblate ...Sun, 16 Aug 2026 15:04:37 -0400https://pop.com.sg/log-of-softmax-function-derivative.htmlhttps://pop.com.sg/log-of-softmax-function-derivative.htmlCould someone explain how that derivative was arrived at. According to me, the derivative of $\log(\text{softmax})$ is $$ \nabla\log(\text{softmax}) = \begin{cases} 1-\text{softmax}, & \text{......Sun, 16 Aug 2026 15:03:06 -0400https://pop.com.sg/prove-eigenfunctions-corresponding-to-different-eigenvalues-are-orthog.htmlhttps://pop.com.sg/prove-eigenfunctions-corresponding-to-different-eigenvalues-are-orthog.htmlI'm considering the eigenvalue problem $L(\phi) = -\lambda \sigma(x) \phi$, subject to a given set of homogeneous boundary conditions. Assume $\sigma(x) \gt 0$ on an interval $[a,b]$. Suppose tha......Sun, 16 Aug 2026 15:02:45 -0400https://pop.com.sg/polynomial-inequality-for-imaginary-roots.htmlhttps://pop.com.sg/polynomial-inequality-for-imaginary-roots.htmlI've to solve the following polynomial inequality $$x^2 - 6x + 11 > 0$$ By using quadratic formula, I got the value of $x$ as below $$ \frac{6\pm \sqrt{-8}}{2}$$ These are imaginary roots and......Sun, 16 Aug 2026 15:01:12 -0400https://pop.com.sg/infinity-times-zero-closed.htmlhttps://pop.com.sg/infinity-times-zero-closed.htmlAssuming the multiplication property of limits I can do the following: $\lim \limits_{x \to ∞}f(a)f(b)=\lim \limits_{x \to ∞}f(a)\lim \limits_{x \to ∞}f(b)$ Why cannot do this? The second one is ...Sun, 16 Aug 2026 14:49:17 -0400https://pop.com.sg/pivot-columns-and-basic-variables.htmlhttps://pop.com.sg/pivot-columns-and-basic-variables.htmlSuppose we are solving a system of linear equation and arrive at the following (augmented) matrix: $ \begin{bmatrix} 1& 6 & 0 & 3 & 0 & 0\\ 0& 0 & 1 & -4 &......Sun, 16 Aug 2026 14:48:49 -0400https://pop.com.sg/what-is-the-difference-between-topological-and-metric-spaces.htmlhttps://pop.com.sg/what-is-the-difference-between-topological-and-metric-spaces.htmlWhat is the difference between a topological and a metric space?...Sun, 16 Aug 2026 14:45:36 -0400https://pop.com.sg/how-do-i-calculate-a-dihedral-angle-given-cartesian-coordinates.htmlhttps://pop.com.sg/how-do-i-calculate-a-dihedral-angle-given-cartesian-coordinates.htmlI have the $(x, y, z)$ coordinates for four points in space (atoms). How do I calculate the dihedral angle ($\phi$ in the below pictures from WP)? I'm somewhat hazy on my math, and am trying to ...Sun, 16 Aug 2026 14:44:15 -0400https://pop.com.sg/what-is-the-j-invariant.htmlhttps://pop.com.sg/what-is-the-j-invariant.htmlI know that it has something to do with Elliptic Curves in the Complex Plane but I don't have an intuitive sense as to what its there for. I understand it is defined in terms of multiple Einstein ......Sun, 16 Aug 2026 14:35:17 -0400https://pop.com.sg/decompose-projection-matrix-into-a-matrix-and-its-pseudoinverse.htmlhttps://pop.com.sg/decompose-projection-matrix-into-a-matrix-and-its-pseudoinverse.htmlWhile researching regression, I encountered the situation where I am trying to reconstruct a data matrix $X \in \mathbb{R}^{n\times p}$ from the $n \times n$ hat matrix $H = X(X'X)^{-1}X' = XX^{+}$ (...Sun, 16 Aug 2026 14:31:30 -0400https://pop.com.sg/expected-value-of-a-binomial-distribution.htmlhttps://pop.com.sg/expected-value-of-a-binomial-distribution.htmlIf $\mathrm P(X=k)=\binom nkp^k(1-p)^{n-k}$ for a binomial distribution, then from the definition of the expected value $$\mathrm E(X) = \sum^n_{k=0}k\mathrm P(X=k)=\sum^n_{k=0}k\binom nkp^k(1-p)^{......Sun, 16 Aug 2026 14:26:43 -0400https://pop.com.sg/finding-precise-solution-for-x-in-complicated-single-variable-equation.htmlhttps://pop.com.sg/finding-precise-solution-for-x-in-complicated-single-variable-equation.htmlI need an exact answer for $x$ in the following algebraic expression: $$ x^{4/3} - 2\sqrt{2}x^2 + 2\sqrt{2} = 0 $$ I don't need or want an approximate answer, I already know that $x≈1.20569$. So any ...Sun, 16 Aug 2026 14:23:51 -0400https://pop.com.sg/online-tools-for-doing-symbolic-mathematics.htmlhttps://pop.com.sg/online-tools-for-doing-symbolic-mathematics.htmlThis question is similar to this one, but specifically relates to resources available strictly as on-line web apps. Examples include: Wolfram Alpha Sage Notebook hicalc not sure what this is; th......Sun, 16 Aug 2026 14:18:50 -0400https://pop.com.sg/how-to-find-top-and-bottom-function-for-finding-area-between-two-curve.htmlhttps://pop.com.sg/how-to-find-top-and-bottom-function-for-finding-area-between-two-curve.htmlWhen I am trying to find the area between two curves below the $\mathbf{x}$-axis, should I always have the function furthest away from the x-axis as the function you are subtracting from, or should......Sun, 16 Aug 2026 14:18:16 -0400https://pop.com.sg/cofinite-topology.htmlhttps://pop.com.sg/cofinite-topology.htmlIf we have topological space $\mathbb{R}$ equipped with the co finite topology. If we have finite subsets in consideration they are definitely closed because open sets are of the form complement of ...Sun, 16 Aug 2026 14:10:25 -0400https://pop.com.sg/please-help-how-to-work-the-inequation-x-1-x-5-0.htmlhttps://pop.com.sg/please-help-how-to-work-the-inequation-x-1-x-5-0.htmlI am studying for college finals and I cannot solve the inequation (x-1)/(x-5)<0 using the method the professor taught us, which I have to use in the exam. This is another inequation using said ...Sun, 16 Aug 2026 14:06:06 -0400https://pop.com.sg/trace-of-a-matrix-when-to-use-what-is-trace-trick.htmlhttps://pop.com.sg/trace-of-a-matrix-when-to-use-what-is-trace-trick.htmlOn calculating log-likelihood function for some multivariate distributions, such as multivariate Normal, I see some examples where the matrices are suddenly changed to trace, even when the matrix i......Sun, 16 Aug 2026 14:00:56 -0400https://pop.com.sg/relation-between-the-roots-of-a-cubic-equation-and-the-coefficients.htmlhttps://pop.com.sg/relation-between-the-roots-of-a-cubic-equation-and-the-coefficients.html$ax^3 +bx^2 + cx + d= 0$ If the roots are $\alpha$ $\beta$ and $\gamma$, Is there any relationship between the sum of the squares of the roots and the coefficients of the quadratic equation.. In ...Sun, 16 Aug 2026 14:00:17 -0400https://pop.com.sg/generators-of-the-symmetric-group-s-3.htmlhttps://pop.com.sg/generators-of-the-symmetric-group-s-3.htmlI am trying to find the generator(s) of the Symmetric Group $S_3$ and I have attempted this via brute force by listing the permutations of $S_3$ and composing and repeating them but I have not foun......Sun, 16 Aug 2026 13:59:25 -0400https://pop.com.sg/how-is-logistic-loss-and-cross-entropy-related.htmlhttps://pop.com.sg/how-is-logistic-loss-and-cross-entropy-related.htmlI found that Kullback-Leibler loss, log-loss or cross-entropy is the same loss function. Is the logistic-loss function used in logistic regression equivalent to the cross-entropy function? If yes, ......Sun, 16 Aug 2026 13:56:53 -0400https://pop.com.sg/limit-lim-x-to0-ln-x.htmlhttps://pop.com.sg/limit-lim-x-to0-ln-x.htmlCan i ask for the limit as $x$ approaches $\lim_{x \to 0-}\ln x$? Please explain. Its because since the limit of a function only exists if the lim as $x$ approaches some number $n$ from both the po......Sun, 16 Aug 2026 13:56:01 -0400https://pop.com.sg/what-is-the-definition-of-finitely-generated.htmlhttps://pop.com.sg/what-is-the-definition-of-finitely-generated.htmlTo be specific, here is an example. Note that $(\mathbb{Z},+)$ is a group. Definition 1. (Lang) Let $G$ be a group and $S\subset G$. If $G$ is the intersection of all subgroups $H$ of $G$ containi......Sun, 16 Aug 2026 13:40:53 -0400https://pop.com.sg/how-to-solve-the-limit-lim-limits-x-to-infty-x-arctan-x-frac-x-pi-2.htmlhttps://pop.com.sg/how-to-solve-the-limit-lim-limits-x-to-infty-x-arctan-x-frac-x-pi-2.htmlNext week I have a math exam. While I was doing some exercises I came across this interesting limit: $\lim\limits_{x\to \infty} (x \arctan x - \frac{x\pi}{2})$ After struggling a lot, I decided to ...Sun, 16 Aug 2026 13:36:22 -0400https://pop.com.sg/in-a-30-60-right-triangle-the-side-opposite-the-30-degree-angle-is-hal.htmlhttps://pop.com.sg/in-a-30-60-right-triangle-the-side-opposite-the-30-degree-angle-is-hal.htmlIn a 30-60 right triangle the side opposite the 30 degree angle is half the length of the hypotenuse. A statement from the trigonometry section of Simmons' Precalculus in a nutshell. Please explain....Sun, 16 Aug 2026 13:27:06 -0400https://pop.com.sg/what-are-the-odds-are-getting-your-6-digit-birth-date-in-your-10-digit.htmlhttps://pop.com.sg/what-are-the-odds-are-getting-your-6-digit-birth-date-in-your-10-digit.htmlI was issued a cellphone number that has my 6-digit birth date in it. Does anyone know how to calculate the odds of that happening? If there are 10 billion possible 10-digit phone numbers, and a 6-......Sun, 16 Aug 2026 13:25:42 -0400https://pop.com.sg/solving-x-d-equiv-a-pmod-29-using-primitive-roots.htmlhttps://pop.com.sg/solving-x-d-equiv-a-pmod-29-using-primitive-roots.htmlCan anyone please detail the general approach to questions of the form $x^d \equiv a \pmod{29}$? For example, Wolfram Alpha states that $x^5 \equiv 8 \pmod{29}$ has one solution, $x^4 \equiv 4 \pmo......Sun, 16 Aug 2026 13:25:36 -0400https://pop.com.sg/how-to-prove-this-limit-does-not-exist.htmlhttps://pop.com.sg/how-to-prove-this-limit-does-not-exist.htmlI'm a relative novice to epsilon-delta proofs. My professor assigned this practice problem and I'm having terrible trouble understanding the answer he gave. Moreover, I can't find a good account fo......Sun, 16 Aug 2026 13:18:42 -0400https://pop.com.sg/if-z-is-a-complex-number-whose-imaginary-part-is-non-zero-and-z-1-z-is.htmlhttps://pop.com.sg/if-z-is-a-complex-number-whose-imaginary-part-is-non-zero-and-z-1-z-is.htmlTwo questions from Grade Twelve class on complex numbers If $z$ is a complex number whose imaginary part is non zero, and $z + 1/z$ is real, what is $z$? How do you solve graphically given the mod......Sun, 16 Aug 2026 13:15:41 -0400https://pop.com.sg/write-the-piecewise-function-in-terms-of-unit-step-functions.htmlhttps://pop.com.sg/write-the-piecewise-function-in-terms-of-unit-step-functions.htmlWrite the piecewise function $f(t) = \begin{cases} 2t, & 0\leq t < 3 \\ 6, & 3 \le t < 5 \\ 2t, & t \ge 5 \\ \end{cases} $ in terms of unit step ...Sun, 16 Aug 2026 13:09:43 -0400https://pop.com.sg/what-039-s-the-name-for-this-weird-shape-why-is-there-a-kite-inside-of.htmlhttps://pop.com.sg/what-039-s-the-name-for-this-weird-shape-why-is-there-a-kite-inside-of.htmlA triangle, but I drew lines in the corners going out, and did the same with the mini-shapes formed: I have no idea what to call this thing. For a square, it forms a diamond inside of it. For a h......Sun, 16 Aug 2026 13:07:45 -0400https://pop.com.sg/example-of-a-function-that-is-not-continuous-at-0-0-but-continuous-whe.htmlhttps://pop.com.sg/example-of-a-function-that-is-not-continuous-at-0-0-but-continuous-whe.htmlI am looking for a continuous function $f: \mathbb{R}^2-\{0, 0\} \to \mathbb{R}$ satisfying the following condition: For any $g, h: \mathbb{R}\to\mathbb{R}$ continuous functions with $g(0)=h(0)=0$,......Sun, 16 Aug 2026 13:05:21 -0400https://pop.com.sg/the-mean-of-a-deterministic-sequence.htmlhttps://pop.com.sg/the-mean-of-a-deterministic-sequence.htmlcould someone explain to me why the expected value of $y(n)$ is the following: $\operatorname{E}(y(n)) = f(n)$ when $y(n) = x(n) + f(n)$ and $x(n)$ has zero mean. But why is the expected value o......Sun, 16 Aug 2026 12:59:33 -0400https://pop.com.sg/hexagon-maze-algorithm.htmlhttps://pop.com.sg/hexagon-maze-algorithm.htmlCan anyone suggest a good algorithm to create structures like this? Note that what I what I am asking for is not a true maze with one start and one solution. Rather, it's for a video game, so like......Sun, 16 Aug 2026 12:52:16 -0400https://pop.com.sg/the-rate-of-change-of-distance-between-two-airplanes-closed.htmlhttps://pop.com.sg/the-rate-of-change-of-distance-between-two-airplanes-closed.htmlAn airplane passes over an airport at noon traveling $540$ mi/hr due north. At 1:00pm, another airplane passes over the same airport at the same elevation traveling due east at $580$ mi/hr. Assumin......Sun, 16 Aug 2026 12:50:41 -0400https://pop.com.sg/find-a-one-parameter-family-of-solutions.htmlhttps://pop.com.sg/find-a-one-parameter-family-of-solutions.htmlIm stuck on this problem. Am I suppose to get the anti-derivative function of the differential and then plug in the initial value?...Sun, 16 Aug 2026 12:47:31 -0400https://pop.com.sg/solve-x-x-5-5-for-x-duplicate.htmlhttps://pop.com.sg/solve-x-x-5-5-for-x-duplicate.htmlA friend who tutors high school math showed me this equation. I could not solve it. But by trial & error in Python I discovered $\sqrt[5]{5}$ is the answer. Further realized that for any intege......Sun, 16 Aug 2026 12:42:56 -0400https://pop.com.sg/pronunciation-of-the-symbol-varnothing-of-the-empty-set.htmlhttps://pop.com.sg/pronunciation-of-the-symbol-varnothing-of-the-empty-set.htmlThe symbol $\varnothing$ for the empty set was introduced by Bourbaki, inspired by the Norwegian alphabet $\varnothing.$ It has no relation with the Greek letter $\phi.$ From my schooldays, when the ...Sun, 16 Aug 2026 12:26:43 -0400https://pop.com.sg/herstein-or-herstein.htmlhttps://pop.com.sg/herstein-or-herstein.htmlI read somewhere that Herstein would prepare me nicely (abstract algebra-wise) for grad studies. I don't remember where I read it and now that I was about to buy the book I found out that there two ...Sun, 16 Aug 2026 12:23:50 -0400https://pop.com.sg/consistency-of-peano-axioms-hilbert-039-s-second-problem.htmlhttps://pop.com.sg/consistency-of-peano-axioms-hilbert-039-s-second-problem.html(Putting aside for the moment that Wikipedia might not be the best source of knowledge.) I just came across this Wikipedia paragraph on the Peano-Axioms: The vast majority of contemporary ...Sun, 16 Aug 2026 12:14:46 -0400https://pop.com.sg/fourier-transform-of-even-odd-function.htmlhttps://pop.com.sg/fourier-transform-of-even-odd-function.htmlHow can I show that the Fourier transform of an even integrable function $f\colon \mathbb{R}\to\mathbb{R}$ is even real-valued function? And the Fourier transform of an odd integrable function $f\c......Sun, 16 Aug 2026 12:04:56 -0400https://pop.com.sg/solve-sqrt-x-2-4-x-2.htmlhttps://pop.com.sg/solve-sqrt-x-2-4-x-2.htmlSolve for x $$\sqrt{x^2-4}=x-2$$ My try: $$\sqrt{(x-2)(x+2)}=x-2$$ $$\sqrt{x+2}=\sqrt{x-2}$$ $$x+2=x-2$$ $$0x=4$$ This is not correct $x={4\over 0}$ How else can I solve this equation?...Sun, 16 Aug 2026 12:00:20 -0400https://pop.com.sg/difference-between-maximum-and-minimum.htmlhttps://pop.com.sg/difference-between-maximum-and-minimum.htmlIf I have a problem such as this: We need to enclose a field with a fence. We have 500m of fencing material and a building is on one side of the field and so won’t need any fencing. Determine the ...Sun, 16 Aug 2026 11:55:49 -0400https://pop.com.sg/find-exact-value-of-tan-when-given-cos.htmlhttps://pop.com.sg/find-exact-value-of-tan-when-given-cos.htmlGiven $\cos30 = \frac{\sqrt3}{2}$ use trigonometric identities to find the exact value of $\tan\frac{\pi}{3}$ I understand that $\cos30 = \frac{\sqrt3}{2}$ from the standard trig values chart and ......Sun, 16 Aug 2026 11:53:18 -0400https://pop.com.sg/simplify-this-problem-from-spivak-039-s-calculis.htmlhttps://pop.com.sg/simplify-this-problem-from-spivak-039-s-calculis.htmlI'm sorry I don't have the book with me (don't know the problem number) since I moved, but I still remember the problem. It said something about finding $\int x^n\ dx$ by letting $p_n = \int_0^1 x......Sun, 16 Aug 2026 11:49:20 -0400https://pop.com.sg/what-039-s-the-geometrical-meaning-of-immersion.htmlhttps://pop.com.sg/what-039-s-the-geometrical-meaning-of-immersion.htmlDoes that just mean to different tangent vectors, their images are different tangent vectors?...Sun, 16 Aug 2026 11:45:21 -0400https://pop.com.sg/solve-for-x-in-e-2-ln-x-ln-x-2-x-3-1.htmlhttps://pop.com.sg/solve-for-x-in-e-2-ln-x-ln-x-2-x-3-1.htmlSo the question is to solve for x in: $$e^{[2\ln(x)-\ln(x^2+x-3)]} = 1$$ I took the natural log of both sides, and simplified. Here is what I've gotten it down to: $$2\ln(x) = \ln(x^2+x-3)$$ And ......Sun, 16 Aug 2026 11:43:21 -0400https://pop.com.sg/establishing-a-covering-relation.htmlhttps://pop.com.sg/establishing-a-covering-relation.htmlThe problem I am working on is, "What is the covering relation of the partial ordering $\{(A,B)|A⊆B\}$ on the power set of $S$, where $S=\{a, b, c\}$?" I am reading the answer key, and I can follo......Sun, 16 Aug 2026 11:41:29 -0400https://pop.com.sg/a-curiosity-how-do-we-prove-mathbb-r-is-closed-under-addition-and-mult.htmlhttps://pop.com.sg/a-curiosity-how-do-we-prove-mathbb-r-is-closed-under-addition-and-mult.htmlSo I tried looking around for this question, but I didn't find much of anything - mostly unrelated-but-similarly-worded stuff. So either I suck at Googling or whatever but I'll get to the point. S......Sun, 16 Aug 2026 11:38:11 -0400https://pop.com.sg/what-do-you-call-the-product-of-a-matrix-039-s-diagonal-elements.htmlhttps://pop.com.sg/what-do-you-call-the-product-of-a-matrix-039-s-diagonal-elements.htmlThe trace of $A$, an $N\times N$ matrix, is $\displaystyle\sum_{i=1}^N A_{ii}$. What do you call $\displaystyle\prod_{i=1}^N A_{ii}$?...Sun, 16 Aug 2026 11:32:17 -0400https://pop.com.sg/what-does-a-complex-root-signify.htmlhttps://pop.com.sg/what-does-a-complex-root-signify.htmlWhat does it tell me when I find that a polynomial has complex roots, except for the obvious fact that it crosses zero for these values? To me it seems that the existance of complex roots must hav......Sun, 16 Aug 2026 11:30:31 -0400https://pop.com.sg/multiplicative-property-of-trace.htmlhttps://pop.com.sg/multiplicative-property-of-trace.htmlLet $T_1 \in \mathcal{L}(V)$ and $T_2 \in \mathcal{L}(V)$ be positive operators. Prove that the trace of their product is non-negative i.e., tr($T_1 T_2) \geq 0$ Attempt 1: Obviously, a positive op......Sun, 16 Aug 2026 11:24:20 -0400https://pop.com.sg/why-is-the-approximation-of-hessian-j-tj-reasonable.htmlhttps://pop.com.sg/why-is-the-approximation-of-hessian-j-tj-reasonable.htmlI met this equation frequently in Guass-Newton optimizations. But I dont understand why the left and right side of the equation can be equal. Lets say the Jacobian is $2$ by $2$ and Hessian is $$\...Sun, 16 Aug 2026 11:23:41 -0400https://pop.com.sg/convergence-in-distribution-of-differences-sums-of-random-variables.htmlhttps://pop.com.sg/convergence-in-distribution-of-differences-sums-of-random-variables.htmlSuppose $X_n$ and $Y_n$ are sequences of random variables, defined on a common probability space, and such that $X_n-Y_n \to C$ converges in distribution to the constant r.v. $C\equiv c$ as $n \to ......Sun, 16 Aug 2026 11:05:31 -0400https://pop.com.sg/jordan-decomposition-of-idempotent-matrix.htmlhttps://pop.com.sg/jordan-decomposition-of-idempotent-matrix.htmlMatrix A $\in$ $\mathbb{R}^{n\times n}$ is idempotent if $A^{2} = A$. Describe the Jordan form of A. How do I do this? I am able to decompose a matrix to its Jordan form given that the matrix conta......Sun, 16 Aug 2026 10:57:11 -0400https://pop.com.sg/is-every-vector-field-the-gradient-of-a-scalar-field.htmlhttps://pop.com.sg/is-every-vector-field-the-gradient-of-a-scalar-field.htmlI was wondering whether every vector field is the gradient of a certain scalar field. If not, could you provide a counterexample? Thanks....Sun, 16 Aug 2026 10:55:55 -0400