https://liberty.ioCurated star highlights with fresh energy.en-usSun, 23 Aug 2026 06:35:12 -0400https://liberty.io/number-of-special-ordered-set.htmlhttps://liberty.io/number-of-special-ordered-set.htmlAn ordered set is a set such that the order in which the objects appear in the set is significant. For example, (1, 2, 3) and (2, 1, 3) are two different ordered sets of integers. An ordered set of ...Sun, 23 Aug 2026 10:28:29 -0400https://liberty.io/determine-whether-the-solutions-are-stable-or-unstable.htmlhttps://liberty.io/determine-whether-the-solutions-are-stable-or-unstable.htmlDetermine whether the solutions x(t)=0 and x(t)=1 of the single scalar equation $dx/dt=-x(1-x)$ are stable or unstable. So far in the book I have just done problems like this except with dx/dt=(some ...Sun, 23 Aug 2026 10:24:55 -0400https://liberty.io/calculate-this-triple-integral-in-cylindrical-coordinates-the-result-i.htmlhttps://liberty.io/calculate-this-triple-integral-in-cylindrical-coordinates-the-result-i.htmlI want to calculate triple integral \begin{equation}\int\limits_{-1}^{1}\int\limits_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}\int\limits_{x^2+y^2}^1 2z dzdydx.\end{equation} (the surface is $z=x^2+y^2$, $0\le......Sun, 23 Aug 2026 10:19:17 -0400https://liberty.io/what-exactly-is-a-scalar-closed.htmlhttps://liberty.io/what-exactly-is-a-scalar-closed.htmlParticularly, referring to scalars I am only implying to individual numbers, for this instance, let's take the set R(real numbers). These numbers do lie of the real line, i.e. the x axis, which is ...Sun, 23 Aug 2026 10:19:09 -0400https://liberty.io/notation-for-the-first-element-in-a-set.htmlhttps://liberty.io/notation-for-the-first-element-in-a-set.htmlI'm looking for a widely used notation to describe the first element of a set. E.g.: S = {5, 7,...,123} is the set, and obviously 5 is the first element. S = {345, 123,...,33} is the set, and ...Sun, 23 Aug 2026 10:10:42 -0400https://liberty.io/is-there-some-consensus-on-the-dimensions-of-a-jacobian-matrix-and-of-.htmlhttps://liberty.io/is-there-some-consensus-on-the-dimensions-of-a-jacobian-matrix-and-of-.htmlAccording to Wikipedia, given a differentiable mapping $F: \mathbb{R}^n \to \mathbb{R}^m$, its Jacobian matrix is a $m \times n$ matrix defined as: $$ J_F=\begin{bmatrix} \dfrac{\partial y_1}{\par......Sun, 23 Aug 2026 10:07:46 -0400https://liberty.io/dividing-an-obtuse-triangle-into-acute-triangles.htmlhttps://liberty.io/dividing-an-obtuse-triangle-into-acute-triangles.htmlCan an obtuse triangle be subdivided into only acute triangles (right triangles are not allowed)? Any number of subdivisions can be made as long as all of the angles in all resulting triangles ar......Sun, 23 Aug 2026 10:00:18 -0400https://liberty.io/oblong-or-rectangle-oval-or-ellipse-why-do-we-have-different-names-for.htmlhttps://liberty.io/oblong-or-rectangle-oval-or-ellipse-why-do-we-have-different-names-for.htmlI am wondering why, sometimes, we have two different names for the same shape (for example, oblong and rectangle or oval and ellipse)? Is there one which is preferred over the other? Do they have ......Sun, 23 Aug 2026 09:59:22 -0400https://liberty.io/linear-least-squares-with-linear-inequality-constraints.htmlhttps://liberty.io/linear-least-squares-with-linear-inequality-constraints.htmlI'm trying to follow this older paper, page 19. The goal is to solve: $\min \|Ax-b\|^2 s.t. Gx \ge h$ given $A, G, b, h$ By combining the equations into a single LCP of the form: $Mz + q = w$ s.t......Sun, 23 Aug 2026 09:57:27 -0400https://liberty.io/difference-between-equilibrium-point-and-unique-equilibrium-point.htmlhttps://liberty.io/difference-between-equilibrium-point-and-unique-equilibrium-point.htmlIs there any difference between unique equilibrium point and equilibrium point? If yes, please tell me what it is and How is it used to solve a dynamical system. You can consider a dynamical syste......Sun, 23 Aug 2026 09:49:52 -0400https://liberty.io/density-of-measure.htmlhttps://liberty.io/density-of-measure.htmlLet $dx$ be the Lebesgue measure on $\mathbb R^d$. Let $u:\mathbb R^d\to{\mathbb R}\cup\{\infty\}$ is a non-negative and measurable function. The question is that, what are the conditions on $u$ so......Sun, 23 Aug 2026 09:44:24 -0400https://liberty.io/how-does-summation-index-shifting-work.htmlhttps://liberty.io/how-does-summation-index-shifting-work.htmlI'm looking over the solution to a test problem on recurrence relations, and I am stuck. I can't figure out the jump that was made. All that it says is "To simplify the algebra, do an index shift:......Sun, 23 Aug 2026 09:44:21 -0400https://liberty.io/arithmetic-overflow-and-underflowing.htmlhttps://liberty.io/arithmetic-overflow-and-underflowing.htmlI am a bit unclear about underflowing in terms of binary representation. Let's say that an unsigned 8-bit variable gets overflown from the addition of $150+150$. A signed 8-bit variable gets ...Sun, 23 Aug 2026 09:39:54 -0400https://liberty.io/proof-of-the-asa-triangle-congruence-criterium.htmlhttps://liberty.io/proof-of-the-asa-triangle-congruence-criterium.htmlSo we studied the triangle congruence criteria and we proved the ASA criterium: If two triangles have respectively congruent two angles and the side included in them, then the two triangles are con......Sun, 23 Aug 2026 09:30:13 -0400https://liberty.io/deriving-the-center-of-mass-of-a-semi-circular-disk-with-cylindrical-c.htmlhttps://liberty.io/deriving-the-center-of-mass-of-a-semi-circular-disk-with-cylindrical-c.htmlProblem: Derive the Center of Mass of a semi-circular disk of mass $M$ and radius $R$. My attempt: $$Y_{CM}=\int ydm$$ Now, $$dm=\sigma dA$$ where $\sigma$ is mass per unit area. Convertin......Sun, 23 Aug 2026 09:29:57 -0400https://liberty.io/what-is-the-magnitude-of-dot-product-of-two-vectors.htmlhttps://liberty.io/what-is-the-magnitude-of-dot-product-of-two-vectors.htmlThis was the question (or rather to prove) given by our professor: $|(a.c)| \leq |a||c|$ where $a$ and $c$ are vectors in $\mathbb{R}^n$. My friend and I solved it in two different ways and now we ......Sun, 23 Aug 2026 09:26:00 -0400https://liberty.io/why-are-homeomorphisms-important.htmlhttps://liberty.io/why-are-homeomorphisms-important.htmlI attended a guest lecture (I'm in high school) hosted by an algebraic topologist. Of course, the talk was non-rigorous, and gave a brief introduction to the subject. I learned that the goal of alg......Sun, 23 Aug 2026 09:20:09 -0400https://liberty.io/volume-of-a-pyramid-with-equilateral-triangle-as-base.htmlhttps://liberty.io/volume-of-a-pyramid-with-equilateral-triangle-as-base.htmlThe base of a right pyramid is an equilateral triangle of side 4cm each. Each slant edge is 5cm long. The volume of pyramid is For this question, answer provided by sscadda.com is $\dfrac{20......Sun, 23 Aug 2026 09:17:58 -0400https://liberty.io/how-to-show-that-the-product-of-two-irrational-numbers-may-be-irration.htmlhttps://liberty.io/how-to-show-that-the-product-of-two-irrational-numbers-may-be-irration.htmlShow that the product of two irrational numbers may be irrational. You may use any facts you know about the real numbers. All we know is that $\sqrt{2}$ is irrational and that $\sqrt{2}\cdot \sqrt......Sun, 23 Aug 2026 09:01:28 -0400https://liberty.io/egorov-039-s-theorem-on-intervals.htmlhttps://liberty.io/egorov-039-s-theorem-on-intervals.htmlFrm Folland's Real Analysis exercise 2.43 part (b): Suppose that $(X, \mathcal M, \mu)$ is a measure space, with $\mu(X) < \infty$. Let $f: X × [0,1] \to \mathbb C$ is a function such that $f......Sun, 23 Aug 2026 08:58:39 -0400https://liberty.io/intuition-degeneracy-maps.htmlhttps://liberty.io/intuition-degeneracy-maps.htmlI'm trying to get an intuitive understanding for the notion of a simplicial set. Roughly, a simplicial set consists of a set $S_n$ of $n$-simplices for each non-negative $n$, and families of face m......Sun, 23 Aug 2026 08:46:36 -0400https://liberty.io/why-are-all-circles-similar-why-is-pi-a-constant-duplicate.htmlhttps://liberty.io/why-are-all-circles-similar-why-is-pi-a-constant-duplicate.htmlI just know that I'm going to look like a crackpot, but here goes. The number $\pi$ is defined as the ratio of the circumference of a circle to its diameter. So there is an assumption here that all ...Sun, 23 Aug 2026 08:46:08 -0400https://liberty.io/product-between-a-column-vector-and-a-row-vector.htmlhttps://liberty.io/product-between-a-column-vector-and-a-row-vector.htmlI know that matrices product is correct when the number of the columns of the first matrix is equal to the number of rows of the second matrix. Why I can't do the product between a column vector a......Sun, 23 Aug 2026 08:43:03 -0400https://liberty.io/mclaurins-with-e-sin-x.htmlhttps://liberty.io/mclaurins-with-e-sin-x.htmlTo evaluate $e^{\sin(x)}$ I use the standard series $e^t$ and $\sin(t)$, combining them gives me: $e^t = 1+t+\dfrac{t^2}{2!}+\dfrac{t^3}{3!}+\dfrac{t^4}{4!}+O(t^5)$ $\sin(t) = t-\dfrac{t^3}{3!}+O......Sun, 23 Aug 2026 08:32:45 -0400https://liberty.io/origin-of-the-dot-and-cross-product.htmlhttps://liberty.io/origin-of-the-dot-and-cross-product.htmlMost questions usually just relate to what these can be used for, that's fairly obvious to me since I've been programming 3D games/simulations for a while, but I've never really understood the inner ...Sun, 23 Aug 2026 08:30:49 -0400https://liberty.io/understanding-the-proof-for-the-divergence-of-the-harmonic-series.htmlhttps://liberty.io/understanding-the-proof-for-the-divergence-of-the-harmonic-series.htmlI'm trying to understand the following proof for the divergence of the harmonic series: The parts which I can't seem to understand are: $$1. \sum_{i=0}^{n-1} \sum_{r=2^i+1}^{2^{i+1}} \frac{1}{r} ......Sun, 23 Aug 2026 08:22:36 -0400https://liberty.io/prove-sin-pi-2-1-using-taylor-series.htmlhttps://liberty.io/prove-sin-pi-2-1-using-taylor-series.htmlProve $\sin(\pi/2)=1$ using the Taylor series definition of $\sin x$, $$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots$$ It seems rather messy to substitute in $\pi/2$ for $x$. So we have $$\sin......Sun, 23 Aug 2026 08:20:32 -0400https://liberty.io/why-is-a-linear-combination-of-solutions-also-a-solution.htmlhttps://liberty.io/why-is-a-linear-combination-of-solutions-also-a-solution.htmlI'm working on DEs and I've come across some speedbumps. I'm not very familiar with linear algebra and that might be the problem. However, I'm sure that there's a non-linear algebra explanation t......Sun, 23 Aug 2026 08:06:51 -0400https://liberty.io/3d-coordinate-rotation-using-roll-pitch-yaw.htmlhttps://liberty.io/3d-coordinate-rotation-using-roll-pitch-yaw.htmlGiven an orthogonal unit vector triad (i,j,k) described in the 3-dimensional coordinate space (X,Y,Z), as shown below. I need to rotate the XYZ coordinate system such that X is collinear with i, Y......Sun, 23 Aug 2026 08:02:31 -0400https://liberty.io/area-of-octagon-constructed-in-a-square.htmlhttps://liberty.io/area-of-octagon-constructed-in-a-square.htmlThe following picture is constructed by connecting each corner of a square with the midpoint of a side from the square that is not adjacent to the corner. These lines create the following red octag......Sun, 23 Aug 2026 07:57:10 -0400https://liberty.io/must-a-complete-subgraph-be-induced.htmlhttps://liberty.io/must-a-complete-subgraph-be-induced.htmlIf we had some $K_n$ subgraph where $K_n \subseteq G$, must the complete subgraph $K_n$ be an induced subgraph from $G$? In other words, can we create a situation where we remove vertices from a si......Sun, 23 Aug 2026 07:49:50 -0400https://liberty.io/arc-length-using-integration.htmlhttps://liberty.io/arc-length-using-integration.htmlIn Thomas's calculus $10^{th}$ edition, chapter $5$, exercises $5.3$, problem $9$, I am asked to find the length of an arc with the equation: $$x=\int^y_0\sqrt{\sec^4(t)-1} dt$$ and $\displaystyl......Sun, 23 Aug 2026 07:42:25 -0400https://liberty.io/express-18x-2-12x-48-in-mathbb-q-x-as-the-product-of-a-constant-polyno.htmlhttps://liberty.io/express-18x-2-12x-48-in-mathbb-q-x-as-the-product-of-a-constant-polyno.htmlI believe I get the correct answer for the following problem but the solution says it is wrong. I have no idea why. Express $18X^2-12X+48$ in $\mathbb{Z}[X]$ and in $\mathbb{Q}[X]$ as the produc......Sun, 23 Aug 2026 07:41:32 -0400https://liberty.io/given-a-matrix-b-what-is-det-b-4.htmlhttps://liberty.io/given-a-matrix-b-what-is-det-b-4.htmlMy task is this: Compute det$(B^4)$, where $$B =\begin{pmatrix} 1 & 0 & 1\\ 1 & 1 & 2\\ 1 & 2 & 1 \end{pmatrix}$$ My work so far: It can be shown that $B = \begin{pmat......Sun, 23 Aug 2026 07:25:17 -0400https://liberty.io/finding-the-solution-set-for-a-quadratic-inequality-x-2-2-frac-7-2-x.htmlhttps://liberty.io/finding-the-solution-set-for-a-quadratic-inequality-x-2-2-frac-7-2-x.htmlIf $x^2-2<\frac{7}{2}x$ then what is the solution set for $x$? I have most of the problem done, I just don't know how to lay out my answer. The answer is supposed to be $$\boxed{-\frac12<x&l......Sun, 23 Aug 2026 07:15:22 -0400https://liberty.io/find-the-differential-equation-of-all-circles-of-radius-a-closed.htmlhttps://liberty.io/find-the-differential-equation-of-all-circles-of-radius-a-closed.htmlCan someone please post a detailed step-by-step procedure. Given the circle with a radius a, what is the differential equation of the circle....Sun, 23 Aug 2026 07:15:14 -0400https://liberty.io/fourier-cosine-expansion-of-sin-x.htmlhttps://liberty.io/fourier-cosine-expansion-of-sin-x.htmlInstructor mentioned something in class that got me thinking... Why is it possible for $\sin x$ to have a Fourier Cosine expansion in $[0, \pi]$? It seems strange to me that you're able to express......Sun, 23 Aug 2026 07:05:24 -0400https://liberty.io/a-cardinality-of-a-graph.htmlhttps://liberty.io/a-cardinality-of-a-graph.htmlIf I have graph $G=(V,E)\\$ What is the meaning of $|G|$? (The cardinality of G). I'd like to know few words about it. Thank you!...Sun, 23 Aug 2026 07:01:08 -0400https://liberty.io/why-do-4-digit-binary-numbers-have-16-solutions-and-3-digits-8-closed.htmlhttps://liberty.io/why-do-4-digit-binary-numbers-have-16-solutions-and-3-digits-8-closed.htmlIf I have a 4 digit binary number, how come there are 16 possible "numbers", and similarly, if I have a 4 digit binary number, how come there are 8 possible "numbers?"...Sun, 23 Aug 2026 06:51:25 -0400https://liberty.io/how-do-computers-draw-function-graphs.htmlhttps://liberty.io/how-do-computers-draw-function-graphs.htmlCould anyone give me a general overview of the algorithm computers use to draw graphs? I'm guessing it is either by graphing many, many points and just connecting them or by doing a general study ......Sun, 23 Aug 2026 06:48:41 -0400https://liberty.io/does-liouville-039-s-theorem-work-in-real-analysis.htmlhttps://liberty.io/does-liouville-039-s-theorem-work-in-real-analysis.htmlLiouville's Theorem states that every bounded entire function must be constant. Does it work in real analysis? Justify your answer! I asked it because Liouville's Theorem is proved by complex analy......Sun, 23 Aug 2026 06:40:50 -0400https://liberty.io/linearity-of-determinant.htmlhttps://liberty.io/linearity-of-determinant.htmlLinearity property of Determinant function: For an $n\times n$ matrix $A$, we can consider det$A$ as a function of the $n$ column vectors in $A$. We will show that if all columns except one are held ...Sun, 23 Aug 2026 06:35:15 -0400https://liberty.io/what-is-symmetric-about-set-symmetric-difference.htmlhttps://liberty.io/what-is-symmetric-about-set-symmetric-difference.htmlUsing $\Delta$ for set symmetric difference, $A \Delta B$ is all the elements in exactly one of the sets but not all of them. $A \Delta B \Delta C $ is all the elements in exactly one of the sets......Sun, 23 Aug 2026 06:33:52 -0400https://liberty.io/how-to-express-the-given-disk-d-in-polar-coordinates.htmlhttps://liberty.io/how-to-express-the-given-disk-d-in-polar-coordinates.htmlGiven $D=\{(x,y)|0\le x\le2,0\le y\le\sqrt{4-x^2}\}$ Express $D$ in polar coordinates. I have the answer as $D=\{(r,\theta)|0\le\theta\le\dfrac{\pi}{2},0\le r\le2\}$ One more example is $D=\{(x,y......Sun, 23 Aug 2026 06:15:49 -0400https://liberty.io/on-integration-when-solving-differential-equations-specifically-separa.htmlhttps://liberty.io/on-integration-when-solving-differential-equations-specifically-separa.htmlSo here is the differential equation and inititial conditions: $$x \frac{\mathrm{d}y}{\mathrm{d}x}=y(3−y) $$ and $$y(2) = 2$$ We have to find the equation $y$ in terms of $x ~~[y(x)]$ that is a so......Sun, 23 Aug 2026 06:05:20 -0400https://liberty.io/height-of-an-isosceles-trapezoid.htmlhttps://liberty.io/height-of-an-isosceles-trapezoid.htmlSuppose we have an isosceles trapezoid whose length of the associated square is $20$ and the length of the hypotenuse of the triangles is $30$. I want to determine if it is possible to find the hei......Sun, 23 Aug 2026 06:05:04 -0400https://liberty.io/learning-mathematics-using-only-audio.htmlhttps://liberty.io/learning-mathematics-using-only-audio.htmlAre there any mathematics audio books or other audio sources for learning mathematics, like for example math podcasts which really go into detail. I ask this because I make about 1 hour from my hou......Sun, 23 Aug 2026 05:59:05 -0400https://liberty.io/surface-integral-over-graph-of-a-function.htmlhttps://liberty.io/surface-integral-over-graph-of-a-function.htmlQ: Evaluate the surface integral $\iint_S \mathbf{F}\cdot d\mathbf{S}$, where $\mathbf{F}(x,y,z)= 3x^2\mathbf{i}-2xy\mathbf{j}+8\mathbf{k}$, and $S$ is the graph of the function $z=f(x,y)=2x-y$ fo......Sun, 23 Aug 2026 05:57:12 -0400https://liberty.io/help-solving-2-second-order-differential-equation.htmlhttps://liberty.io/help-solving-2-second-order-differential-equation.html$y'' + (y')^2 = y$ and $y*y''= (y')^2 - (y')^3$ I know that both of these equations can be solved by using $p(y) = y'$, but Im getting stuck in the middle, and can't seem to solve the thing aft......Sun, 23 Aug 2026 05:56:42 -0400https://liberty.io/convert-expression-to-nand-only.htmlhttps://liberty.io/convert-expression-to-nand-only.htmlI have to convert the following to NAND only $$\bar{A}\cdot\bar{B}\cdot\bar{C} + A\cdot\bar{B}\cdot C + A\cdot B\cdot \bar{C} + A \cdot B\cdot C$$ I've looked at the following website: http://en....Sun, 23 Aug 2026 05:48:23 -0400https://liberty.io/if-sin-theta-cos-theta-tan-theta-cot-theta-sec-theta-csc-theta-7-then-.htmlhttps://liberty.io/if-sin-theta-cos-theta-tan-theta-cot-theta-sec-theta-csc-theta-7-then-.html$$ 0<\theta<\pi/2$$ and $$\sin\theta+\cos\theta+\tan\theta+\cot\theta+\sec\theta+\csc\theta=7$$ then show that $\sin 2\theta$ is a root of the equation $$x^2 -44x +36=0$$ I tried to use ......Sun, 23 Aug 2026 05:41:04 -0400https://liberty.io/which-of-the-following-statements-are-true-about-finite-cyclic-groups.htmlhttps://liberty.io/which-of-the-following-statements-are-true-about-finite-cyclic-groups.html$1.$ If $G$ is a finite cyclic group, then for every subgroup $H,K$ of $G,$ we have either $H\subset K$ or $K\subset H.$ $2.$ If for every subgroup $H,K$ of a finite group $G,$ we have either $H\su......Sun, 23 Aug 2026 05:34:00 -0400https://liberty.io/da-in-polar-coordinates.htmlhttps://liberty.io/da-in-polar-coordinates.htmlI have seen a picture for $dV$ so that $dV = r^{2} \sin(\theta)\,dr\,d\theta\,d\phi$. But how can I deduce things like $dA$ and $dV$? In a simpler coordinate (not sure about the name), $dA = r \,dr......Sun, 23 Aug 2026 05:30:08 -0400https://liberty.io/are-the-domain-and-range-of-a-function-and-its-inverse-always-exactly-.htmlhttps://liberty.io/are-the-domain-and-range-of-a-function-and-its-inverse-always-exactly-.htmlThe function $ f(x)=\sqrt{x+4} $ has domain $[-4,\infty)$ and range $[0,\infty)$. Its inverse function $ f^{-1}(x)=x^2-4 $ by itself seems to have domain $\Bbb{R}$ and range $[-4,\infty)$. Howeve......Sun, 23 Aug 2026 05:26:53 -0400https://liberty.io/proving-the-inverse-of-the-invertible-matrix-theorem.htmlhttps://liberty.io/proving-the-inverse-of-the-invertible-matrix-theorem.htmlLet $A$ be an $n \times n$ invertible matrix and $v_1,v_2,...,v_m$ an element of $\mathbb{R}^{n}$. If $\{v_1,v_2,...,v_m\}$ spans $\mathbb{R}^{n}$ then $\{Av_1, Av_2,...,Av_m\}$ also spans $\mathbb......Sun, 23 Aug 2026 05:26:39 -0400https://liberty.io/diagonal-terms-of-the-inverse-of-positive-definite-matrix.htmlhttps://liberty.io/diagonal-terms-of-the-inverse-of-positive-definite-matrix.htmlI have a matrix $A = I + G$ where $G_{ii} = 0$ for all $i$, and $G_{ij} \in \{0, a\}$ with $a \in [0, 1[$, $G$ is symmetric, and such that $A$ is positive definite. We know that the inverse of $A$,......Sun, 23 Aug 2026 05:23:46 -0400https://liberty.io/arc-length-of-y-1-3-sqrt-x-x-3.htmlhttps://liberty.io/arc-length-of-y-1-3-sqrt-x-x-3.html$y = 1/3 \sqrt{x} (x-3)$ from 1-9 In my book it is actually $x = 1/3 \sqrt{y} (y-3)$ but I prefer to work with y so I just swap the two variables and I think everything should be the same. The fi......Sun, 23 Aug 2026 05:18:57 -0400https://liberty.io/what-is-the-difference-between-l-2-norm-and-ell-2-norm.htmlhttps://liberty.io/what-is-the-difference-between-l-2-norm-and-ell-2-norm.htmlI can find no precise definitions on the internet for the $L^2$ and $\ell^2$ norms. Certain websites keep switching between the two. Can someone please help me?...Sun, 23 Aug 2026 05:15:41 -0400https://liberty.io/how-to-solve-trigonometric-equations-without-a-calculator-duplicate.htmlhttps://liberty.io/how-to-solve-trigonometric-equations-without-a-calculator-duplicate.htmlHow does one solve for sine, cosine, or tangent (and inverse sine, cosine, tangent) without the use of a calculatour? For example: cosine of 131 degrees — how must one calculate this?...Sun, 23 Aug 2026 04:56:01 -0400https://liberty.io/f-x-x-1-what-does-x-1-mean.htmlhttps://liberty.io/f-x-x-1-what-does-x-1-mean.htmlThe question is, "Find the domain and the range of the real function $f$ defined by $f(x) = |x-1|$. I have no idea what $|x-1|$ means, someone help me?...Sun, 23 Aug 2026 04:51:21 -0400https://liberty.io/topological-isomorphism-vs-isometric-isomorphism.htmlhttps://liberty.io/topological-isomorphism-vs-isometric-isomorphism.htmlWe say that: $T:(X,\|\cdot\|_X)\rightarrow (Y,\|\cdot\|_Y)$ is a isometric isomorphism if it is a linear isomorphism, and it is an isometry, that is $\|T(x)\|_Y=\|x\|_X\quad \forall x\in X;$ $T:(X......Sun, 23 Aug 2026 04:49:55 -0400https://liberty.io/find-the-smallest-positive-integer-for-which-x-mod-3-2-and-x-mod-4-3.htmlhttps://liberty.io/find-the-smallest-positive-integer-for-which-x-mod-3-2-and-x-mod-4-3.htmlI came across a question on a piece of homework and my solution yields no possible solutions. Find the smallest positive integer for which x mod 3=2 and x mod 4=3 (and then find the next one) So I ...Sun, 23 Aug 2026 04:47:43 -0400https://liberty.io/related-rate-question-water-is-leaking-out-of-an-inverted-conical-tank.htmlhttps://liberty.io/related-rate-question-water-is-leaking-out-of-an-inverted-conical-tank.htmlWater is leaking out of an inverted conical tank at a rate of 9,500 cm3/min at the same time that water is being pumped into the tank at a constant rate. The tank has height 6 m and the diameter at......Sun, 23 Aug 2026 04:44:16 -0400https://liberty.io/on-provability-of-paris-harrington-theorem.htmlhttps://liberty.io/on-provability-of-paris-harrington-theorem.htmlIt is said that the Paris–Harrington theorem is true, but not provable in Peano arithmetic. I want to ask: So how do they know that it is true if it has no proof? I cannot imagine someone knows som......Sun, 23 Aug 2026 04:41:26 -0400https://liberty.io/normal-subgroups-of-s-4.htmlhttps://liberty.io/normal-subgroups-of-s-4.htmlCan anyone tell me how to find all normal subgroups of the symmetric group $S_4$? In particular are $H=\{e,(1 2)(3 4)\}$ and $K=\{e,(1 2)(3 4), (1 3)(2 4),(1 4)(2 3)\}$ normal subgroups?...Sun, 23 Aug 2026 04:37:03 -0400https://liberty.io/solving-heat-equation-using-bender-schmidt-method.htmlhttps://liberty.io/solving-heat-equation-using-bender-schmidt-method.htmlI was trying to solve this heat equation using Bender-Schmidt method $$u^{j+1} = A u^{j}+rB$$ The heat equation is $$u_{xx} = 32u_{t},0<x<1,u(x,0)=0,u(0,t)=0,u(1,t)=10+t$$taking $$h = 0.25,k......Sun, 23 Aug 2026 04:34:43 -0400https://liberty.io/reference-request-for-set-theoretic-foundations-of-geometry.htmlhttps://liberty.io/reference-request-for-set-theoretic-foundations-of-geometry.htmlMy question is, Is it possible to define geometrical concepts (say, of Euclidean Geometry) like 'point', 'striaght line' in purely set theoretic terms? So far, I could think of the following ...Sun, 23 Aug 2026 04:29:01 -0400https://liberty.io/dimension-of-mathbb-q-otimes-mathbb-z-mathbb-q-as-a-vector-space-over-.htmlhttps://liberty.io/dimension-of-mathbb-q-otimes-mathbb-z-mathbb-q-as-a-vector-space-over-.htmlThe following problem was subject of examination that was taken place in June. The document is here. Problem 1 states: The tensor product $\mathbb{Q}\otimes_{\mathbb Z}\mathbb{Q}$ is a vector sp......Sun, 23 Aug 2026 04:25:46 -0400https://liberty.io/negation-of-for-every-and-there-exist-statements-duplicate.htmlhttps://liberty.io/negation-of-for-every-and-there-exist-statements-duplicate.htmlI need to find the negations of the following statements: For every n (is an element of natural numbers) there is a prime number p such that n < p. There is n (is an element of natural numbers......Sun, 23 Aug 2026 04:15:35 -0400https://liberty.io/what-are-the-different-symbols-used-for-denoting-an-angle.htmlhttps://liberty.io/what-are-the-different-symbols-used-for-denoting-an-angle.htmlThis is not a question involving numbers and stuff, but suddenly had this basic question in mind: What are the different symbols that different people use for denoting an angle? (I mean, we usually......Sun, 23 Aug 2026 04:08:39 -0400https://liberty.io/how-do-you-compute-numerically-the-earth-mover-039-s-distance-emd.htmlhttps://liberty.io/how-do-you-compute-numerically-the-earth-mover-039-s-distance-emd.htmlI was trying to compute numerically (write a program) that calculated the EMD for two probability distribution $p_X$ and $q_X$. However, I had a hard time finding an outline of how to exactly compute ...Sun, 23 Aug 2026 03:56:45 -0400https://liberty.io/solve-the-inequality-25-x-3000-closed.htmlhttps://liberty.io/solve-the-inequality-25-x-3000-closed.htmlSolve the inequality $25^x>3000$. Note: No computer programs or calculators allowed. I tried factoring the two, I ended up with $5^{2z}>2^3\cdot3\cdot5^3$, which doesn't seem to help....Sun, 23 Aug 2026 03:52:27 -0400https://liberty.io/conditional-number-growth-of-hilbert-matrix-theoretical-vs-matlab.htmlhttps://liberty.io/conditional-number-growth-of-hilbert-matrix-theoretical-vs-matlab.htmlI need to investigate how the condition number of the Hilbert matrix grows with the size N. The Matlab command is: "cond(hilb(N),2)" Compute the condition number of the Hilbert matrices Hn ∈ R, N×......Sun, 23 Aug 2026 03:37:09 -0400https://liberty.io/intuition-behind-logarithm-inequality-1-frac1x-leq-log-x-leq-x-1.htmlhttps://liberty.io/intuition-behind-logarithm-inequality-1-frac1x-leq-log-x-leq-x-1.htmlOne of fundamental inequalities on logarithm is: $$ 1 - \frac1x \leq \log x \leq x-1 \quad\text{for all $x > 0$},$$ which you may prefer write in the form of $$ \frac{x}{1+x} \leq \log{(1+x)} \l......Sun, 23 Aug 2026 03:32:59 -0400https://liberty.io/for-f-x-x-3-2-times-e-0-35x-at-what-x-value-does-the-maximum-occur.htmlhttps://liberty.io/for-f-x-x-3-2-times-e-0-35x-at-what-x-value-does-the-maximum-occur.htmlFor $f(x) = x^{3.2}\times \mathrm{e}^{-0.35x}$, at what $x$ value does the maximum occur for minimum or maximum $f'(x)=0$ this implies $3.2 x^{2.2} \mathrm{e}^{-0.35x}-x^{3.2}\mathrm{e}^{-0.35x}\......Sun, 23 Aug 2026 03:32:34 -0400https://liberty.io/cube-roots-of-the-complex-numbers-1-i.htmlhttps://liberty.io/cube-roots-of-the-complex-numbers-1-i.htmlI cant get any good reference in my books regarding cube of complex numbers. Please help me find cube roots of the Complex number i+1??...Sun, 23 Aug 2026 03:31:39 -0400https://liberty.io/partition-of-an-interval-of-mathbb-r.htmlhttps://liberty.io/partition-of-an-interval-of-mathbb-r.htmlA partition of an inteval $[a,b]$ of $\mathbb{R}$ is generally defined as a finite sequence of the form: $$a = x_0 < x_1 < x_2 < \dots < x_n = b$$ Then, $[a,b]$ is seen as the following......Sun, 23 Aug 2026 03:26:51 -0400https://liberty.io/which-of-the-43-380-possible-nets-for-a-dodecahedron-is-the-narrowest.htmlhttps://liberty.io/which-of-the-43-380-possible-nets-for-a-dodecahedron-is-the-narrowest.htmlI want to fit multiple regular dodecahedron nets on to an infinitely long roll of paper. I want this to result in the largest possible dodecahedrons, for a roll of a given width. My hunch is that ......Sun, 23 Aug 2026 03:24:08 -0400https://liberty.io/if-det-a-2-det-a-then-a-2-in-a-a-1-a-a-1.htmlhttps://liberty.io/if-det-a-2-det-a-then-a-2-in-a-a-1-a-a-1.htmlIs the following statement always true? Let $A \in GL_n( \mathbb Z)$. If $\det(A^2)= \det(A)$, then $A^2 \in \{A, A^{-1}, -A, -A^{-1}\}$ . Note: $GL_n(\mathbb Z)$ is defined as the set of all ...Sun, 23 Aug 2026 03:18:42 -0400https://liberty.io/find-the-prime-factors-of-3-32-2-32.htmlhttps://liberty.io/find-the-prime-factors-of-3-32-2-32.htmlI'm having a go at BMO 2006/7 Q1 which states: "Find four prime numbers less than 100 which are factors of $3^{32}-2^{32}$." My working is as follows (basically just follows difference of two squa......Sun, 23 Aug 2026 03:14:42 -0400https://liberty.io/classifying-irreducible-finite-dimensional-representations-of-the-q-we.htmlhttps://liberty.io/classifying-irreducible-finite-dimensional-representations-of-the-q-we.htmlI have that $A = \langle x, y : yx = qxy \rangle$ is the $q$-Weyl algebra, with basis $x^iy^j$, $i, j \in \mathbb{Z}$. The base field is $\mathbb{C}$. I need to classify the irreducible finite-...Sun, 23 Aug 2026 03:13:54 -0400https://liberty.io/find-primitive-root-mod-17.htmlhttps://liberty.io/find-primitive-root-mod-17.htmlI have to list the quadratic residues of $17$ and find a primitive root. I have calculated that: Quadratic residues $mod(17)$ are $1,2,4,8,9,13,15,16$ How am I then meant to use this to obtain a ...Sun, 23 Aug 2026 03:13:36 -0400https://liberty.io/degree-sum-formula-closed.htmlhttps://liberty.io/degree-sum-formula-closed.htmlSuppose the G = (V,E) is a connected graph with n vertices and n-1 edges. Use the degree-sum formula for vertices to prove that G has a vertex of degree 1. I can't seem to find where to start in t......Sun, 23 Aug 2026 03:10:44 -0400https://liberty.io/what-does-the-small-number-on-top-of-the-square-root-symbol-mean.htmlhttps://liberty.io/what-does-the-small-number-on-top-of-the-square-root-symbol-mean.htmlI just came across this annotation in my school's maths compendium: The compendium is very brief and doesn't explain what this means....Sun, 23 Aug 2026 03:08:31 -0400https://liberty.io/how-to-recognize-the-language-of-a-dfa.htmlhttps://liberty.io/how-to-recognize-the-language-of-a-dfa.htmlGiven the following Deterministic Finite Automata, I have to be able to tell what language the Automata accepts.image here] As I can see, it accepts the letters a, b and c. Furthermore, my guess wo......Sun, 23 Aug 2026 02:44:20 -0400https://liberty.io/negation-of-some-logic-statement.htmlhttps://liberty.io/negation-of-some-logic-statement.htmlI need to negate the following statement : "Some integers are not odd" I have the below, where O(x) is "odd" $$ \exists x (\neg O(x)) $$ Would the negation be $$ \forall x (O(x)) $$ I'm confused, ...Sun, 23 Aug 2026 02:38:02 -0400https://liberty.io/linear-algebra-row-picture-and-column-picture-for-system-of-linear-equ.htmlhttps://liberty.io/linear-algebra-row-picture-and-column-picture-for-system-of-linear-equ.htmlI am presently going through Strang's linear algebra book in which he explains row picture and column picture for a system of linear equations as such For example: $$\begin{cases} 2x + 3y = 5 \\ ......Sun, 23 Aug 2026 02:26:45 -0400https://liberty.io/negative-variance.htmlhttps://liberty.io/negative-variance.htmlI have two independent variables $X$ and $Y$. $W=X-Y$ when $X\sim \mbox{Bernoulli}\left(1/2\right)$ and $Y\sim N(0,1)$. This puts $\operatorname{Var}(x)=1/4$ and $\operatorname{Var}(Y)=1$, but I h......Sun, 23 Aug 2026 01:54:56 -0400https://liberty.io/pitchfork-bifurcation-subcritical-and-supercritical.htmlhttps://liberty.io/pitchfork-bifurcation-subcritical-and-supercritical.htmlI am slightly confused about the terminology of a subcritical and supercritical pitchfork bifurcation. Consider the diagrams below: (both pictures from wikipedia, by Claudio Rocchini, CC by 2.5......Sun, 23 Aug 2026 01:40:24 -0400https://liberty.io/lyndon-words-over-the-binary-alphabet-that-begin-with-00.htmlhttps://liberty.io/lyndon-words-over-the-binary-alphabet-that-begin-with-00.htmlLydon words over the binary alphabet are strings of $1$s and $0$s such that all rotations of the word are strictly lexicographically later. I've been looking at Lyndon words of length $n$, which are ...Sun, 23 Aug 2026 01:34:06 -0400https://liberty.io/value-of-tan-pi-2-duplicate.htmlhttps://liberty.io/value-of-tan-pi-2-duplicate.htmlI understand that this is a very stupid question but I'm not getting the answer. At $x=\pi/2$, what is the value of $tan(x)$? Should it be $-\infty$ or $+\infty$? Text tells it to be $+\infty$. ......Sun, 23 Aug 2026 01:25:42 -0400https://liberty.io/area-of-circumscribed-circle.htmlhttps://liberty.io/area-of-circumscribed-circle.htmlI'm looking at the solutions for a 1989 ARML contest, and it has this question: If the area of a rectangle is 1/2 of the area of its circumscribed circle, compute the acute angle between the dia......Sun, 23 Aug 2026 01:22:25 -0400https://liberty.io/solve-equation-3-x-2-3-x.htmlhttps://liberty.io/solve-equation-3-x-2-3-x.htmlUsing the substitution $u=3^x$, or otherwise, solve, correct to 3 significant figures, the equation $3^x=2+3^{-x}$ My attempt: $u=2+u^{-1}$ $u-u^{-1}=2$ $ {{u^2-1}\over {u}}=2$ $\log ({{u^2......Sun, 23 Aug 2026 01:21:24 -0400https://liberty.io/show-that-limit-does-not-exist-two-variables.htmlhttps://liberty.io/show-that-limit-does-not-exist-two-variables.htmlI just started looking into multiple variable calculus and limits involving them. I'm not amazing at limits either. I want to answer this question: Show that the following limit does not exist $$\......Sun, 23 Aug 2026 01:18:48 -0400https://liberty.io/summation-of-n-squared-cubed-etc-duplicate.htmlhttps://liberty.io/summation-of-n-squared-cubed-etc-duplicate.htmlHow do you in general derive a formula for summation of n-squared, n-cubed, etc...? Clear explanation with reference would be great....Sun, 23 Aug 2026 01:15:14 -0400https://liberty.io/finding-if-a-relation-implicit-function-is-an-even-or-odd-function.htmlhttps://liberty.io/finding-if-a-relation-implicit-function-is-an-even-or-odd-function.htmlI have this relation, and if graphed on Desmos.com looks very symmetrical. $$x^{\frac{2}{3}} + y^{\frac{2}{3}} = 1 $$ It has cusps(vertical tangents on the y-axis). How can one determine if this re......Sun, 23 Aug 2026 01:07:42 -0400https://liberty.io/how-to-find-a-point-between-two-points-with-given-distance.htmlhttps://liberty.io/how-to-find-a-point-between-two-points-with-given-distance.htmlLets say I have point P1(10,10) and P2(20,20). I want to find a P3 which is on between this two points and 3 units away from P1. What is the formula to find P3 ? Known values: X1, X2, Y1 , Y2, ...Sun, 23 Aug 2026 01:04:13 -0400https://liberty.io/why-is-the-definition-of-cardinal-number-as-the-set-of-all-sets-equiva.htmlhttps://liberty.io/why-is-the-definition-of-cardinal-number-as-the-set-of-all-sets-equiva.htmlIn Fundamentals of Mathematics, Volume 1 Foundations of Mathematics: The Real Number System and Algebra, after defining set equivalence as the ability to put the elements of the related sets in one......Sun, 23 Aug 2026 01:02:13 -0400https://liberty.io/linear-dependence-of-functions.htmlhttps://liberty.io/linear-dependence-of-functions.htmlI'm in an intro to DE class. On one of the quizzes, the professor asked us to determine whether the set of functions $\ x, x-1,$ and$\ x+3$ are linearly independent or linearly dependent on $\ (- \......Sun, 23 Aug 2026 01:02:05 -0400https://liberty.io/definition-of-a-divergent-sequence.htmlhttps://liberty.io/definition-of-a-divergent-sequence.htmlI have that the definition of a convergent sequence is: ($\exists$ L $\in$ $\mathbb{R}$)($\forall$ $\epsilon$ > 0)($\exists$ N $\in$ $\mathbb{N}$)($\forall$ n $\in$ $\mathbb{N}$)[n $\ge$ N $\Right......Sun, 23 Aug 2026 01:01:07 -0400