https://liberty.ioCurated star highlights with fresh energy.en-usFri, 04 Sep 2026 09:16:56 -0400https://liberty.io/unclear-about-what-the-diagonal-relation-means.htmlhttps://liberty.io/unclear-about-what-the-diagonal-relation-means.htmlI am reading through Skornjakov's Elements of Lattice Theory (1977), and am having trouble understanding what is meant by a diagonal relation. I am also unclear about how the diagonal differs from ......Fri, 04 Sep 2026 13:15:30 -0400https://liberty.io/solve-the-ode-y-039-039-y-039-y-2.htmlhttps://liberty.io/solve-the-ode-y-039-039-y-039-y-2.htmlSolve the ODE $$ y''+ y' = y^2$$ I tried to solve, but I don't know any algorithm except for numerical methods....Fri, 04 Sep 2026 13:07:39 -0400https://liberty.io/how-to-solve-x-5-2x-6-1.htmlhttps://liberty.io/how-to-solve-x-5-2x-6-1.html$|x-5|=|2x+6|-1$. The answer is $0$ or $-12$, but how would I solve it by algebraically solving it as opposed to sketching a graph? $|x-5|=|2x+6|-1\\ (|x-5|)^2=(|2x+6|-1)^2\\ ...\\ 9x^4+204x^3+1......Fri, 04 Sep 2026 13:03:27 -0400https://liberty.io/derivative-of-an-exponentially-weighted-moving-average.htmlhttps://liberty.io/derivative-of-an-exponentially-weighted-moving-average.htmlIt has been a while since my university math courses, so let me apologize right off the bat... I'm using GSL to perform non-linear regression analysis and am mostly happy with the outcome, however......Fri, 04 Sep 2026 13:02:34 -0400https://liberty.io/what-do-trivial-and-non-trivial-solution-of-homogeneous-equations-mean.htmlhttps://liberty.io/what-do-trivial-and-non-trivial-solution-of-homogeneous-equations-mean.htmlSuppose I have system of 3 equations $$a_1x+b_1y+c_1z=0$$ $$a_2x+b_2y+c_2z=0$$ $$a_3x+b_3y+c_3z=0$$ and cofficient matrix $A=\begin{equation} \begin{pmatrix} a_1 & b_1 & c_1 \\ a_2 & b_......Fri, 04 Sep 2026 13:02:04 -0400https://liberty.io/find-the-maximum-displacement-given-trajectory.htmlhttps://liberty.io/find-the-maximum-displacement-given-trajectory.htmlI'm trying to find the maximum displacement of a object given the trajectory. The trajectory is given by the equation: $$p(t)=-11.79t^2+25.9t+4.35$$ Looking around online, the closest article I've ......Fri, 04 Sep 2026 12:57:12 -0400https://liberty.io/why-negating-universal-quantifier-gives-existential-quantifier.htmlhttps://liberty.io/why-negating-universal-quantifier-gives-existential-quantifier.htmlNegating a universal quantifier gives the existential quantifier, and vice versa: $\neg \forall x = \exists x \neg \\ \neg \exists x = \forall x \neg $ Why is this, and is there a proof for it (i......Fri, 04 Sep 2026 12:51:02 -0400https://liberty.io/range-two-039-s-complement.htmlhttps://liberty.io/range-two-039-s-complement.htmlThe most used formula's to calculate the range of numbers in a two's complement system are + $2^{n-1}-1$ for the highest number and $-2^{n-1}$ for the lowest number. The problem is that this only ......Fri, 04 Sep 2026 12:45:17 -0400https://liberty.io/dual-of-a-positive-semidefinite-cone.htmlhttps://liberty.io/dual-of-a-positive-semidefinite-cone.htmlThe PSD cone is the set of all positive semidefinite matrices. The dual is the set of all matrices $A$ such that tr($A^T X$) $\geq 0$ for all positive semidefinite matrices $X$. How to prove that......Fri, 04 Sep 2026 12:44:36 -0400https://liberty.io/finding-velocity-from-a-table.htmlhttps://liberty.io/finding-velocity-from-a-table.htmlI have a homework question I am seeking an alternative solution to. Basically, the question is... "The table provided below shows the position of a particle S, at several times, t. as the particle......Fri, 04 Sep 2026 12:42:03 -0400https://liberty.io/find-a-particular-solution-for-the-differential-equation-by-the-method.htmlhttps://liberty.io/find-a-particular-solution-for-the-differential-equation-by-the-method.htmlFind a particular solution for the differential equation by the method of undetermined coefficients. $$2y'' - 16y' + 32y = -e^{4x}$$ Also, find the general solution of this equation. The step......Fri, 04 Sep 2026 12:41:25 -0400https://liberty.io/how-well-does-probability-work-in-the-real-world.htmlhttps://liberty.io/how-well-does-probability-work-in-the-real-world.htmlWhen I was first introduced to probability at the age of about 13, me and my classmates always used to ridicule the subject in a most immature manner. We used to take a deck of well shuffled cards ......Fri, 04 Sep 2026 12:38:41 -0400https://liberty.io/what-is-the-centroid-of-a-hollow-spherical-cap.htmlhttps://liberty.io/what-is-the-centroid-of-a-hollow-spherical-cap.htmlI have a unit hollow sphere which I cut along a diameter to generate two equivalent hollow hemispheres. I place one of these hemispheres on an (x,y) plane, letting it rest on the circular planar f......Fri, 04 Sep 2026 12:32:21 -0400https://liberty.io/symbol-for-the-shape-of-a-matrix.htmlhttps://liberty.io/symbol-for-the-shape-of-a-matrix.htmlI'm writing a technical document and need to describe some mathematics involving the shape of a matrix. I realize that I can write $X_{m\times n}$ and someone can simply see the dimensions. I'm ...Fri, 04 Sep 2026 12:31:07 -0400https://liberty.io/example-of-strictly-concave-functions.htmlhttps://liberty.io/example-of-strictly-concave-functions.htmlI'm learning concave/convex functions. Wikipedia article provides us with example functions of concave or convex. However, it does not provide strictly concave function. Do you have some examples of ...Fri, 04 Sep 2026 12:15:14 -0400https://liberty.io/what-does-the-d-represent-in-the-equation-of-a-plane-ax-by-cz-d-0.htmlhttps://liberty.io/what-does-the-d-represent-in-the-equation-of-a-plane-ax-by-cz-d-0.htmlLook at the following problem: Consider the plane $\alpha$ defined by $x-2y+z+3=0$ and $A(0;0;2)$. Write an equation of the plane that is parallel to $\alpha$ and goes through A. My book ...Fri, 04 Sep 2026 12:06:06 -0400https://liberty.io/questions-about-relations-on-directed-and-possibly-cyclical-graphs.htmlhttps://liberty.io/questions-about-relations-on-directed-and-possibly-cyclical-graphs.htmlLet $(P, E)$ be a directed graph with $P$ being the set of nodes and $E \subset P^2$ the set of directed edges. It is well-known that the transitive closure of $E$ is a preorder $\leq$, which is more ...Fri, 04 Sep 2026 11:58:21 -0400https://liberty.io/finding-a-b-c-d-in-a-quartic-expression.htmlhttps://liberty.io/finding-a-b-c-d-in-a-quartic-expression.htmlLet $p(x)=x^4+ax^3+bx^2+cx+d$ where a,b,c,d are constants. If $p(1)=10$, $p(2)=20$, $p(3)=30$, compute $\frac {p(12)+p(-8)}{10}$. I have tried so far. \begin{align} a+b+c+d=&9\\8a+4b+2c+d=&amp......Fri, 04 Sep 2026 11:51:15 -0400https://liberty.io/finding-x-intercept-of-a-cubic-equation.htmlhttps://liberty.io/finding-x-intercept-of-a-cubic-equation.htmlWhat is the $x$ intercept of $y=(x-2)(x^2+25) $? To find $x$ intercept:$ 0=(x-2)(x^2+25) $ I tried $ 0=(x-2)(x+5)(x+5)$ in which the $X$ intercepts are $2,-5$ and $-5$. Is this correct?...Fri, 04 Sep 2026 11:45:29 -0400https://liberty.io/proving-the-trig-identity-sin-20-circ-cos-65-circ-cos-20-circ-sin-65-c.htmlhttps://liberty.io/proving-the-trig-identity-sin-20-circ-cos-65-circ-cos-20-circ-sin-65-c.htmlUsing the trigonometric identities I have to prove that: $$ \sin(20^\circ)\cos(65^\circ)-\cos(20^\circ)\sin(65^\circ)=-\frac{1}{\sqrt{2}} $$ I solved $$ \sin(20^\circ)\cos(65^\circ) = \frac{\s......Fri, 04 Sep 2026 11:36:06 -0400https://liberty.io/prime-divisor-of-3n-2-proof.htmlhttps://liberty.io/prime-divisor-of-3n-2-proof.htmlI have to prove that any number of the form $3n+2$ has a prime factor of the form $3m+2$. Ive started the proof I tried saying by the division algorithm the prime factor is either the form 3m,3m+1......Fri, 04 Sep 2026 11:32:36 -0400https://liberty.io/what-is-the-algebraic-closure-of-mathbb-q.htmlhttps://liberty.io/what-is-the-algebraic-closure-of-mathbb-q.htmlGiven the rational numbers $\mathbb{Q}$, I am wondering what is the algebraic closure of $\mathbb{Q}$, i.e., I need to find a field extension $\mathbb{K}/ \mathbb{Q}, $ such that for an arbitrary ...Fri, 04 Sep 2026 11:15:22 -0400https://liberty.io/why-homology-is-homotopy-invariant.htmlhttps://liberty.io/why-homology-is-homotopy-invariant.htmlI've red Hatcher's proof of homotopy invariance for singular homology. I understood how the proof works and the reason why we construct the prism operator (i.e. its a chain contraction). However, t......Fri, 04 Sep 2026 11:04:23 -0400https://liberty.io/double-integral-change-of-variables-to-polar-coordinates.htmlhttps://liberty.io/double-integral-change-of-variables-to-polar-coordinates.htmlQuick question on Polar Coordinates. When evaluating the double integral and changing variables, I'm not sure if the limits are correct. The question is as follows: Evaluate $$\int\!\!\!\int_D......Fri, 04 Sep 2026 10:32:13 -0400https://liberty.io/mean-of-gamma-distribution.htmlhttps://liberty.io/mean-of-gamma-distribution.htmlSo I was trying to prove the mean result of gamma distribution which is $\frac{\alpha}{\lambda}$. My attempt, $E(X)=\int_{0}^{\infty }x f(x)dx$ $=\int_{0}^{\infty } \frac{\lambda^{\alpha}}{\Gam......Fri, 04 Sep 2026 10:26:02 -0400https://liberty.io/visual-intuition-for-the-definition-of-asymptotically-equivalent.htmlhttps://liberty.io/visual-intuition-for-the-definition-of-asymptotically-equivalent.htmlI'm trying to intuitively grasp the following definition: The real-valued functions $f$ and $g$ are asymptotically equivalent as $x \to \infty$ if $$\lim_{x \to \infty} \dfrac{f(x)}{g(x)}=1.$$ We ......Fri, 04 Sep 2026 10:19:51 -0400https://liberty.io/what-is-a-common-symbol-for-angles-and-what-are-semantic-differences.htmlhttps://liberty.io/what-is-a-common-symbol-for-angles-and-what-are-semantic-differences.htmlI've just seen that LaTeX offers at least three symbols for denoting angles: $\text{\sphericalangle}:\sphericalangle ABC \\\text{\measuredangle}:\measuredangle ABC\\\text{\angle}: \angle ABC$ I t......Fri, 04 Sep 2026 10:18:24 -0400https://liberty.io/how-to-simplify-the-boolean-expression-a-039-bc-ab-039-c-039-a-039-b-0.htmlhttps://liberty.io/how-to-simplify-the-boolean-expression-a-039-bc-ab-039-c-039-a-039-b-0.htmlHow to simplify this Boolean expression? $$A'BC + AB'C' + A'B'C' + AB'C + AB$$ I have solved it using Kmap and found the answer to be $$A + BC + B'C'$$ I tried simplifying it using the rules but o......Fri, 04 Sep 2026 10:16:33 -0400https://liberty.io/the-need-for-the-gram-schmidt-process.htmlhttps://liberty.io/the-need-for-the-gram-schmidt-process.htmlAs far as I understood Gram–Schmidt orthogonalization starts with a set of linearly independent vectors and produces a set of mutually orthonormal vectors that spans the same space that starting ve......Fri, 04 Sep 2026 10:14:51 -0400https://liberty.io/what-is-the-purpose-of-jordan-canonical-form.htmlhttps://liberty.io/what-is-the-purpose-of-jordan-canonical-form.htmlI don't claim at all to be an expert on this topic. In many (advanced) linear algebra textbooks for undergraduates, I usually find something about the "Jordan Canonical Form" of a matrix. What is......Fri, 04 Sep 2026 10:03:05 -0400https://liberty.io/could-you-please-explain-why-the-remainder-when-the-square-of-49-is-di.htmlhttps://liberty.io/could-you-please-explain-why-the-remainder-when-the-square-of-49-is-di.htmlHere is the original question, What is the remainder when the square of 49 is divided by the square root of 49? According to UKMT, Because 49 = 7×7, it follows both that $\sqrt 49$ = 7 and that......Fri, 04 Sep 2026 10:00:10 -0400https://liberty.io/if-2-n-1-is-prime-why-must-n-be-a-power-of-2.htmlhttps://liberty.io/if-2-n-1-is-prime-why-must-n-be-a-power-of-2.htmlA little bird told me that if $2^n+1$ is prime, then $n$ is a power of $2$. I tend not to trust talking birds, so I'm trying to verify that statement independently. Suppose $n$ is not a power of $......Fri, 04 Sep 2026 10:00:05 -0400https://liberty.io/how-many-4-letter-words-can-we-make-from-bananas.htmlhttps://liberty.io/how-many-4-letter-words-can-we-make-from-bananas.htmlAs the title suggests, I want to know the value of possible permutations without repetition given the word BANANAS BA1N1A2N2A3S so for example to word A1N1A2S is the A3N1A2S and we want to exclu......Fri, 04 Sep 2026 09:57:37 -0400https://liberty.io/row-reduced-2-times3-echelon-matrices.htmlhttps://liberty.io/row-reduced-2-times3-echelon-matrices.htmlSuppose $R$ and $R'$ are two $2\times3$ row reduced echelon matrices if $RX=0$ and $R'X$ have axactly same solutions then prove that $R=R'$. My try: Let $x,y,z$ be the same solution of $R$ and......Fri, 04 Sep 2026 09:48:14 -0400https://liberty.io/pure-sub-module-in-m.htmlhttps://liberty.io/pure-sub-module-in-m.htmlWhat is the difference between a pure module $P$ and a pure module $P$ in a module $M$ ? I have a slight intuition though, what is the relationship between purity and solvability of equations....Fri, 04 Sep 2026 09:47:58 -0400https://liberty.io/help-me-understand-a-3d-graph.htmlhttps://liberty.io/help-me-understand-a-3d-graph.htmlI've just seen this graph and while it's isn't the first 3d graph I've seen, as a math "noob" I never thought how these graphs are plotted. I can draw 2d graphs on paper by marking the input and ou......Fri, 04 Sep 2026 09:43:25 -0400https://liberty.io/how-to-calculate-the-amplitude-and-phase-of-such-a-complex-expression.htmlhttps://liberty.io/how-to-calculate-the-amplitude-and-phase-of-such-a-complex-expression.htmlI got an expression like this: $a*\sin(x)+b*\cos(x)+i*c*\sin(x)+i*d*\cos(x)$. I want to calculate the amplitude and the phase of it. Can someone give me some hints? Thx very much!...Fri, 04 Sep 2026 09:42:49 -0400https://liberty.io/the-expression-sqrt-13-3-sqrt-frac-23-3-sqrt-13-3-sqrt-frac-23-3-is-wh.htmlhttps://liberty.io/the-expression-sqrt-13-3-sqrt-frac-23-3-sqrt-13-3-sqrt-frac-23-3-is-wh.html(a) A natural number, (b) A rational number but not a natural number, (c) An irrational number not exceeding 6, (d) An irrational number exceeding 6. Please help with this, i can't manage to simpl......Fri, 04 Sep 2026 09:42:20 -0400https://liberty.io/what-do-you-call-a-3d-annulus-cylindrical-shell-and-parts-of-it.htmlhttps://liberty.io/what-do-you-call-a-3d-annulus-cylindrical-shell-and-parts-of-it.htmlTake a cylinder and punch a centered circular hole through it--like a washer. What is that shape called? In 2D it's an annulus, but what's the name of its 3D equivalent? Also, what is this shape c......Fri, 04 Sep 2026 09:36:59 -0400https://liberty.io/how-to-calculate-the-distance-between-two-lines-drawn-at-separate-angl.htmlhttps://liberty.io/how-to-calculate-the-distance-between-two-lines-drawn-at-separate-angl.htmlI don't know much about math, so forgive me if this is an obvious / unnecessary question. Please feel free to point me towards any resource that might enlighten me. However I need to know this for ......Fri, 04 Sep 2026 09:22:26 -0400https://liberty.io/how-to-find-specific-percentage-in-a-normal-distribution.htmlhttps://liberty.io/how-to-find-specific-percentage-in-a-normal-distribution.htmlHere is the exercise question I'm trying to answer: A certain fast food outlet produces hamburgers with weights that are normally distributed with mean $\mu = 990g$ and standard deviation $\s......Fri, 04 Sep 2026 09:08:05 -0400https://liberty.io/traditional-marriage-gs-female-pessimality.htmlhttps://liberty.io/traditional-marriage-gs-female-pessimality.htmlIn the context of stable matchings I am currently trying to understand the Gale-Shapely Algorithm (traditional marriage algorithm). I have proved it s correctness and I have shown that its male o......Fri, 04 Sep 2026 09:06:25 -0400https://liberty.io/why-does-imread-generate-3-images-closed.htmlhttps://liberty.io/why-does-imread-generate-3-images-closed.htmlThe MATLAB help for imread says: A = imread(filename, fmt) The return value A is an array containing the image data. If the file contains a grayscale image, A is an M-by-N array. If the f......Fri, 04 Sep 2026 08:39:34 -0400https://liberty.io/how-do-i-find-percentiles-of-data-sets-even-vs-odd.htmlhttps://liberty.io/how-do-i-find-percentiles-of-data-sets-even-vs-odd.htmlGiven the following data set with an even number of values: $100, 100, 105, 113, 129, 132, 146, 152, 176, 200$ The value representing the 30th percentile, using the formula n(p/100) where n = sam......Fri, 04 Sep 2026 08:37:48 -0400https://liberty.io/finding-the-height-given-the-angle-of-elevation-and-depression.htmlhttps://liberty.io/finding-the-height-given-the-angle-of-elevation-and-depression.htmlPlease, I need help for this problem. I'm a little confused about it :( From a point A 10ft. above the water the angle of elevation of the top of a lighthouse is 46 degrees and the angle of depr......Fri, 04 Sep 2026 08:32:04 -0400https://liberty.io/find-n-of-geometric-sequence-closed.htmlhttps://liberty.io/find-n-of-geometric-sequence-closed.htmlI need to find the $n$ of a geometric sequence, this formula is to find the $n$th term, $n$th = $ar ^{n-1}$; so I need $n$ to be on the left instead of $n$th, I have all the other variables except......Fri, 04 Sep 2026 08:26:10 -0400https://liberty.io/why-is-ln-cos-x-equal-to-ln-sec-x.htmlhttps://liberty.io/why-is-ln-cos-x-equal-to-ln-sec-x.htmlWhy does the value $-\ln(|\cos(x)|)$ become $\ln(|\sec(x)|)?$ I was doing an integral and I got my final answer as that, but I don't understand how you can just send the negative sign inside and m......Fri, 04 Sep 2026 08:23:00 -0400https://liberty.io/find-the-coordinates-of-a-point-on-a-circle.htmlhttps://liberty.io/find-the-coordinates-of-a-point-on-a-circle.htmlI have a circle like so $r$, with angle $\theta$ to the $y$-axis"> Given a rotation θ and a radius r, how do I find the coordinate (x,y)? Keep in mind, this rotation could be anywhere between 0 ......Fri, 04 Sep 2026 08:16:14 -0400https://liberty.io/equation-of-a-line-as-a-determinant-equation.htmlhttps://liberty.io/equation-of-a-line-as-a-determinant-equation.htmlI've been looking for an answer in the forums and Google but couldn't find one. Given two different points in the plane, $(x_1,\,y_1)$ and $(x_2,\,y_2)$, we know that the equation of that line can......Fri, 04 Sep 2026 08:13:11 -0400https://liberty.io/how-to-express-transformations-of-exponential-functions-stretches-and-.htmlhttps://liberty.io/how-to-express-transformations-of-exponential-functions-stretches-and-.htmlWhen referring to the transformations of exponential functions, how do you refer to vertical "stretches" and "compressions" in terms of "a dilation of $z$ from the y-axis o......Fri, 04 Sep 2026 08:10:17 -0400https://liberty.io/how-to-calculate-the-moment-of-inertia-of-a-uniform-disk-about-an-axis.htmlhttps://liberty.io/how-to-calculate-the-moment-of-inertia-of-a-uniform-disk-about-an-axis.htmlConsider a uniform disk of mass m and radius R, and let's calculate the moment of inertia about an axis passing through a diameter of the disk. One easy way is to use the perpendicular axis theorem: ...Fri, 04 Sep 2026 08:01:57 -0400https://liberty.io/isomorphism-vs-equality-of-graphs.htmlhttps://liberty.io/isomorphism-vs-equality-of-graphs.htmlI have just started studying graph theory and having trouble with understanding the difference b/w isomorphism and equality of two graphs.According what I have studied so far, I am able to conclude......Fri, 04 Sep 2026 07:52:11 -0400https://liberty.io/what-is-a-co-prime.htmlhttps://liberty.io/what-is-a-co-prime.htmlI've never encountered this question in any of my math classes and it just shows up randomly in my comsci class with no further info about it. I've wiki'ed it, but can't even understand that. Could ...Fri, 04 Sep 2026 07:50:42 -0400https://liberty.io/basic-equivalence-class-discrete-math.htmlhttps://liberty.io/basic-equivalence-class-discrete-math.htmlI read through the textbook definition of the equivalence class, but still cannot clearly understand what an equivalence class is. Does anyone have a good example with a definition that can hit m......Fri, 04 Sep 2026 07:47:03 -0400https://liberty.io/what-is-the-outward-normal-vector-at-the-end-points-of-the-one-dimensi.htmlhttps://liberty.io/what-is-the-outward-normal-vector-at-the-end-points-of-the-one-dimensi.htmlIn the one-dimensional space, suppose the interval $[a, b]$ exists on the real line with boundary points $a$ and $b$. What is the outward normal vector at the point $a$ and at the point $b$? The an......Fri, 04 Sep 2026 07:43:35 -0400https://liberty.io/are-the-eigenvectors-of-a-real-symmetric-matrix-always-an-orthonormal-.htmlhttps://liberty.io/are-the-eigenvectors-of-a-real-symmetric-matrix-always-an-orthonormal-.htmlI was reading the wikipedia page for symmetric matrices, and I noticed this part: a real n×n matrix A is symmetric if and only if there is an orthonormal basis of Rn consisting of eigenvectors f......Fri, 04 Sep 2026 07:30:59 -0400https://liberty.io/how-to-pronounce-tableaux-closed.htmlhttps://liberty.io/how-to-pronounce-tableaux-closed.htmlHow do you pronounce Young tableaux? Does it sound just like its singular form?...Fri, 04 Sep 2026 07:23:53 -0400https://liberty.io/finding-cosets-and-index-in-d8.htmlhttps://liberty.io/finding-cosets-and-index-in-d8.htmla). Find all the left and right cosets of $<τ>$ in $D_8$. $<τ>$ is reflection and I know that there are two reflections about a diagonal. So I am wondering how to represent this idea a......Fri, 04 Sep 2026 07:10:48 -0400https://liberty.io/show-that-the-sum-of-2n-1-consecutive-integers-is-divisible-by-2n-1.htmlhttps://liberty.io/show-that-the-sum-of-2n-1-consecutive-integers-is-divisible-by-2n-1.htmlBase case: $n=1$. Picking $2n+1$ random numbers 5,6,7 we get $5+6+7=18$. So, $2(1)+1=3$ which indeed does divide 18. The base case holds. Let $n=k>=1$ and let $2k+1$ be true. We want to show $2......Fri, 04 Sep 2026 07:06:12 -0400https://liberty.io/zassenhaus-formula-exponents.htmlhttps://liberty.io/zassenhaus-formula-exponents.htmlfound Zassenhaus formula for noncommutative $X,Y$ $$e^{t(X+Y)}= e^{tX}~ e^{tY} ~e^{-\frac{t^2}{2} [X,Y]} ~ e^{\frac{t^3}{6}(2[Y,[X,Y]]+ [X,[X,Y]] )} ~ e^{\frac{-t^4}{24}([[[X,Y],X],X] + 3[[[X,Y],X......Fri, 04 Sep 2026 07:05:15 -0400https://liberty.io/right-triangular-pyramid.htmlhttps://liberty.io/right-triangular-pyramid.htmlI am somehow confused with this problem. It is said that this is a right triangular pyramid, but it has an equilateral triangle base? I am confused. How can I find VA? It doesn't look like it has a......Fri, 04 Sep 2026 06:53:20 -0400https://liberty.io/strangest-notation-closed.htmlhttps://liberty.io/strangest-notation-closed.htmlWhile this may be a fruitless pursuit of anecdotes, I still ask: what is the strangest (or most blatantly wrong (at least in the eyes of common notation)) mathematical notation you have ever seen?...Fri, 04 Sep 2026 06:53:19 -0400https://liberty.io/calculators-using-taylor-polynomials.htmlhttps://liberty.io/calculators-using-taylor-polynomials.htmlI've always heard that calculators (TI-84's and the like) use Taylor polynomials to approximate trigonometric/exponential/etc functions. Do any of you know this for a fact?...Fri, 04 Sep 2026 06:48:47 -0400https://liberty.io/derivative-of-y-sec-x-2-cos-x.htmlhttps://liberty.io/derivative-of-y-sec-x-2-cos-x.htmlAccording to wolfram the derivative of this is zero. This derivative though is part of a larger problem and I don't think it is possible for it to be zero. According to my own calculation the answe......Fri, 04 Sep 2026 06:46:47 -0400https://liberty.io/attempt-to-view-irrational-number-as-a-fraction.htmlhttps://liberty.io/attempt-to-view-irrational-number-as-a-fraction.htmlI am wondering if an irrational number can be represented as a fraction in this way: For example (to represent $\pi$): $$\pi= 3.14159265359...=\frac{314159265359...}{100000000000...}$$ In the fr......Fri, 04 Sep 2026 06:42:08 -0400https://liberty.io/closed-vs-compact-surface.htmlhttps://liberty.io/closed-vs-compact-surface.htmlWikipedia defines a surface to be a two-dimensional manifold, and a closed surface to be a surface that is compact and without boundary. Am I correct that this definition of "closed surface" is not ...Fri, 04 Sep 2026 06:40:22 -0400https://liberty.io/what-does-overlap-two-functions-mean.htmlhttps://liberty.io/what-does-overlap-two-functions-mean.htmlWhat does overlap mean in the case of two functions. Or atleast, what does overlap mean in the case of 2 lines or 2 curves. Thanks in advance. The complete sentence to clarify the context: ...Fri, 04 Sep 2026 06:38:31 -0400https://liberty.io/expectation-of-geometric-brownian-motion.htmlhttps://liberty.io/expectation-of-geometric-brownian-motion.htmlI was deriving the solution to the stochastic differential equation $$dX_t = \mu X_tdt + \sigma X_tdB_t$$ where $B_t$ is a brownian motion. After finding $$X_t = x_0\exp((\mu - \frac{\sigma^2}{2})t......Fri, 04 Sep 2026 06:37:24 -0400https://liberty.io/what-does-within-the-same-order-of-magnitude-convey.htmlhttps://liberty.io/what-does-within-the-same-order-of-magnitude-convey.htmlThis question originates from a quandary about the meaning of the statement that two values are within the same order of magnitude. I wonder whether there is an established usage, of (rather more ...Fri, 04 Sep 2026 06:10:17 -0400https://liberty.io/are-all-sets-ordered-sets.htmlhttps://liberty.io/are-all-sets-ordered-sets.htmlThe definition of an ordered set according to W. Rudin in his book, Principles of Mathematical Analysis is: An ordered set is a set S in which an order is defined He also defined order in his boo......Fri, 04 Sep 2026 06:08:56 -0400https://liberty.io/finding-formula-for-partial-sum-of-polynomial-terms.htmlhttps://liberty.io/finding-formula-for-partial-sum-of-polynomial-terms.htmlFor example, we know that the following is true (and can be easily derived): $\sum\limits_{x=1}^{n}x = \frac{1}{2}n(n+1)$ But, what if we wanted to find the sum of a series like this: $\sum\limi......Fri, 04 Sep 2026 06:08:18 -0400https://liberty.io/what-do-small-and-large-mean-for-sets-and-categories.htmlhttps://liberty.io/what-do-small-and-large-mean-for-sets-and-categories.htmlCategories for the Working Mathematicians says In addition to the metacategory of all sets ~ which is not a set ~ we want an actual category $Set$, the category of all small sets. We shall ...Fri, 04 Sep 2026 06:00:00 -0400https://liberty.io/why-do-we-assume-in-the-principle-of-strong-complete-induction.htmlhttps://liberty.io/why-do-we-assume-in-the-principle-of-strong-complete-induction.htmlI get induction. (i) show that the statement holds when n=1 (or some basis) (ii) show that the statement holds for a general subsequent case, $n+1$. By PMI the statement holds for all cases. Do......Fri, 04 Sep 2026 05:57:48 -0400https://liberty.io/vieta-039-s-formula-failed.htmlhttps://liberty.io/vieta-039-s-formula-failed.htmlFind the value of $p$ if $p$ and $q$ are the roots of the equation, $x^2+px+q=0, \ \ \{x,p,q\}\in\ \mathbb{R}$ By using vieta's formula for sum and product of roots, $\begin{cases} p+q=-p \......Fri, 04 Sep 2026 05:54:25 -0400https://liberty.io/what-does-mean-in-math.htmlhttps://liberty.io/what-does-mean-in-math.htmlIn a Linear Algebra textbook I am reading, the following is stated: $b\notin \operatorname{span}(A \cup \{a\})\setminus \operatorname{span}(A)$. It does so without explaining what "$\setminus$" mea......Fri, 04 Sep 2026 05:45:19 -0400https://liberty.io/can-we-compute-the-area-of-a-quadrilateral-with-one-right-angle-when-w.htmlhttps://liberty.io/can-we-compute-the-area-of-a-quadrilateral-with-one-right-angle-when-w.htmlI took an IQ test for fun recently, but I take issue with the answer to one of the questions. Here's the question: My issue is that the explanation assumes angle DC is a right angle. Given that ...Fri, 04 Sep 2026 05:39:26 -0400https://liberty.io/a-demonstration-of-lagrange-039-s-form-for-the-remainder-of-a-taylor-s.htmlhttps://liberty.io/a-demonstration-of-lagrange-039-s-form-for-the-remainder-of-a-taylor-s.htmlThe following argument for Lagrange's Form for the Remainder of a Taylor polynomial is a typical one in analysis books. Even in the case of finding the remainder when the Taylor polynomial is a li......Fri, 04 Sep 2026 05:38:49 -0400https://liberty.io/how-does-sinc-interpolation-work.htmlhttps://liberty.io/how-does-sinc-interpolation-work.htmlEvery now and then I come across mention of sinc interpolation. Trying to read up on it, I have yet to get what it's about. I have done basic DSP work, have programmed stuff using FFT (using just a ...Fri, 04 Sep 2026 05:25:31 -0400https://liberty.io/microeconomics-calculating-tax-revenue-and-tax-incidence.htmlhttps://liberty.io/microeconomics-calculating-tax-revenue-and-tax-incidence.htmlAustralian Government has imposed a tax on Beer. Assume that the tax on Beer is $20 per unit (a unit is a carton of drinks) Assume the demand and supply functions for cartons of Beers per week are:......Fri, 04 Sep 2026 05:22:46 -0400https://liberty.io/why-does-the-graph-not-show-the-vertical-asymptote.htmlhttps://liberty.io/why-does-the-graph-not-show-the-vertical-asymptote.htmlI have a rational function in which the denominator is equal to zero when $x = -1.4142135$. So: $f(x)=\dfrac{1}{x + 1.4142135}$ Then the vertical asymptote at $x = -1.4142135$ is pretty clear. ......Fri, 04 Sep 2026 05:13:12 -0400https://liberty.io/simple-proof-of-krein-smulian.htmlhttps://liberty.io/simple-proof-of-krein-smulian.htmlI have a question concerning a simple and short proof of Krein-Shmulian, the proof can be found here https://people.math.ethz.ch/~jteichma/slides_ftap.pdf on page 35/36. Here the proof: Let $X$ be a ...Fri, 04 Sep 2026 05:04:37 -0400https://liberty.io/if-f-039-2-0-and-f-039-039-2-4-what-can-you-say-about-f.htmlhttps://liberty.io/if-f-039-2-0-and-f-039-039-2-4-what-can-you-say-about-f.htmlI was doing very well in Calculus up until this point. I realize that concavity and $f'$ and $f''$ require one to really visualize what is happening with a function, but can someone please help me......Fri, 04 Sep 2026 05:04:14 -0400https://liberty.io/why-characteristic-zero-and-not-infinite-characteristic.htmlhttps://liberty.io/why-characteristic-zero-and-not-infinite-characteristic.htmlThe characteristic of a ring (with unity, say) is the smallest positive number $n$ such that $$\underbrace{1 + 1 + \cdots + 1}_{n \text{ times}} = 0,$$ provided such an $n$ exists. Otherwise, we de......Fri, 04 Sep 2026 04:57:20 -0400https://liberty.io/notation-for-compact-sets.htmlhttps://liberty.io/notation-for-compact-sets.htmlIs there some accepted notation for a set that is compact? E.g. I am currently writing "... [blah] is true if for every compact set $A \subset \mathbb{R}^n$ and ...". I could simplify my writing if ...Fri, 04 Sep 2026 04:44:39 -0400https://liberty.io/how-many-3-digit-positive-integers-are-odd-and-do-not-contain-digit-5.htmlhttps://liberty.io/how-many-3-digit-positive-integers-are-odd-and-do-not-contain-digit-5.htmlIt is a GRE question. And it has been answered here. But I still want to ask it again, just to know why I am wrong. The correct is 288. My idea is, first I get the total number of 3-digit integer......Fri, 04 Sep 2026 04:29:59 -0400https://liberty.io/a-specific-example-of-the-gns-construction.htmlhttps://liberty.io/a-specific-example-of-the-gns-construction.htmlIn an introduction to the GNS construction, I'm told that the GNS construction is a generalization of the way that $L^{\infty} (X, \mu)$ has a representation on $L^2$ where $\mu$ is a measure on $X......Fri, 04 Sep 2026 04:27:43 -0400https://liberty.io/the-transitive-closure-of-a-relation-r-on-set-a-whose-relation-matrix.htmlhttps://liberty.io/the-transitive-closure-of-a-relation-r-on-set-a-whose-relation-matrix.htmlThe transitive closure of a relation $R$ on set $A$ whose relation matrix My attempt: If I assumed node $(1, 2, 3)$, then given is: $R=\{(1, 2), (2, 3), (3, 1)\}$ The transitive closer would be......Fri, 04 Sep 2026 04:25:00 -0400https://liberty.io/proving-1-frac-1-4-frac-1-9-cdots-frac-1-n-2-leq-2-frac-1-n-for-all-n-.htmlhttps://liberty.io/proving-1-frac-1-4-frac-1-9-cdots-frac-1-n-2-leq-2-frac-1-n-for-all-n-.htmlQuestion: Let $P(n)$ be the statement that $1+\dfrac{1}{4}+\dfrac{1}{9}+\cdots +\dfrac{1}{n^2} <2- \dfrac{1}{n}$. Prove by mathematical induction. Use $P(2)$ for base case. Attempt at soluti......Fri, 04 Sep 2026 04:23:26 -0400https://liberty.io/how-to-solve-int-c-im-z-dz-where-c-is-the-unit-circle.htmlhttps://liberty.io/how-to-solve-int-c-im-z-dz-where-c-is-the-unit-circle.htmlI want to solve $\int_C Im(z) dz$ where $C$ is the unit circle. The first thing is to parameterize the curve, $C$: $$ z(t)=e^{it} $$ where $0\leq t \leq 2\pi$ Then $$ \frac {dz}{dt} = ie^{it} $$ ......Fri, 04 Sep 2026 04:23:02 -0400https://liberty.io/isometries-and-orthogonal-matrices.htmlhttps://liberty.io/isometries-and-orthogonal-matrices.htmlI know how to show that multiplying by an orthogonal matrix preserves the angle and distance between two vectors. I have seen everywhere that Orthogonal matrices are kind of related to rotations and ...Fri, 04 Sep 2026 04:19:00 -0400https://liberty.io/if-type-theories-are-all-logics.htmlhttps://liberty.io/if-type-theories-are-all-logics.htmlSo it sounds like Higher Order Logic (HOL) and Type Theory are equivalent. Then there is Intuitionistic Logic and Intuitionistic Type Theory, but I'm not sure of the connection there. I am just ...Fri, 04 Sep 2026 04:19:00 -0400https://liberty.io/universal-factoring-method-or-list-of-methods-trinomials.htmlhttps://liberty.io/universal-factoring-method-or-list-of-methods-trinomials.htmlI am a student in calculus II. I'm now failing tests solely because I cannot factor; I understand everything else. This is compounded by the fact it seems to exceedingly hard to find anything ...Fri, 04 Sep 2026 04:11:49 -0400https://liberty.io/for-which-angles-we-know-the-sin-value-algebraically-exact.htmlhttps://liberty.io/for-which-angles-we-know-the-sin-value-algebraically-exact.htmlFor example: $\sin(15^\circ) = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}$ $\sin(18^\circ) = \frac{\sqrt{5}}{4} - \frac{1}{4}$ $\sin(30^\circ) = \frac{1}{2}$ $\sin(45^\circ) = \frac{1}{\sqrt{2}}$ $\s......Fri, 04 Sep 2026 04:07:09 -0400https://liberty.io/finding-linear-dependence-relation.htmlhttps://liberty.io/finding-linear-dependence-relation.htmlFind a linear dependence relation between the following vectors: x1 = (1, -1, 2) x2 = (-3, 2, 1) x3= (1, 2, -3) x4= (2, 3, 1) I've already created a matrix and reduced and I know how to tell w......Fri, 04 Sep 2026 04:06:42 -0400https://liberty.io/how-to-simplify-sin-x-y-cos-y-cos-x-y-sin-y.htmlhttps://liberty.io/how-to-simplify-sin-x-y-cos-y-cos-x-y-sin-y.htmlthe question How to simplify $\sin(x-y)\cos(y)+\cos(x-y)\sin(y)$ my steps I tried to use trig identities on the $\sin(x-y)$ and $\cos(x-y)$ and tried to distribute the others in but it didn't wo......Fri, 04 Sep 2026 03:58:46 -0400https://liberty.io/use-rational-zero-theorem-to-find-real-zeros-of-2x-3-3x-2-x-1.htmlhttps://liberty.io/use-rational-zero-theorem-to-find-real-zeros-of-2x-3-3x-2-x-1.htmlI am to use the rational zero theorem to find the real zeros of $2x^3-3x^2-x+1$. The provided answers are $\frac{1}{2}$, $\frac{1+\sqrt{5}}{2}$ and $\frac{1-\sqrt{5}}{2}$. I was able to get $\frac{......Fri, 04 Sep 2026 03:53:55 -0400https://liberty.io/is-there-such-an-inequality-between-product-and-integral-for-functions.htmlhttps://liberty.io/is-there-such-an-inequality-between-product-and-integral-for-functions.htmlGiven a measure space $(\Omega, \mathbb{F},\mu)$ and any two measurable real-valued functions $f,g$ defined on $\Omega$, I was wondering if there is an inequality like $$ \int_\Omega |f*g| d\mu \l......Fri, 04 Sep 2026 03:42:03 -0400https://liberty.io/finding-the-standard-deviation-of-a-probability-distribution.htmlhttps://liberty.io/finding-the-standard-deviation-of-a-probability-distribution.htmlHere is the question: The time, to the nearest whole minute, that a city bus takes to go from one end of its route to the other has the probability distribution shown. As sometimes happens with ...Fri, 04 Sep 2026 03:41:02 -0400https://liberty.io/unordered-sampling-with-replacement.htmlhttps://liberty.io/unordered-sampling-with-replacement.htmlI have difficulties with understanding of how do we count this. Why can't we just divide number of ways of ordered sample with replacement $n^k$ by $k!$, i.e. to get rid of permutations since we c......Fri, 04 Sep 2026 03:38:59 -0400https://liberty.io/how-to-find-the-area-of-a-triangle-with-lengths-of-heights.htmlhttps://liberty.io/how-to-find-the-area-of-a-triangle-with-lengths-of-heights.htmlGiven the lengths of 3 heights in a triangle, I need to find its area....Fri, 04 Sep 2026 03:37:07 -0400