https://liberty.ioCurated star highlights with fresh energy.en-usSun, 23 Aug 2026 16:31:33 -0400https://liberty.io/how-to-find-perimeter-of-right-angle-triangle-given-area-and-three-ang.htmlhttps://liberty.io/how-to-find-perimeter-of-right-angle-triangle-given-area-and-three-ang.htmlHello everyone, I’ve come across a question where I need to find the perimeter of a right angle triangle given its area and three sides (the only angle written in the picture is 40 degrees, but the......Sun, 23 Aug 2026 20:23:30 -0400https://liberty.io/what-039-s-another-name-for-the-integral-function.htmlhttps://liberty.io/what-039-s-another-name-for-the-integral-function.htmlI know this is certainty a basic question, but I'm wondering what you could use as an alternate name for the integral of a function. That is to say; $$\text{In } \int f(x)dx=F(x) \text{, } f(x) \te......Sun, 23 Aug 2026 20:15:05 -0400https://liberty.io/what-is-the-inverse-of-the-mbox-vec-operator.htmlhttps://liberty.io/what-is-the-inverse-of-the-mbox-vec-operator.htmlThere is a well known vectorization operator $\mbox{vec}$ in matrix analysis. I've vectorized my matrix equations, did some transformation of vectorized equations and now I want to get back to the ...Sun, 23 Aug 2026 20:09:48 -0400https://liberty.io/proof-of-squeeze-theorem-for-functions.htmlhttps://liberty.io/proof-of-squeeze-theorem-for-functions.htmlSuppose for all $x$ we know $g(x)\le f(x)\le h(x)$ and $\lim_{x\to c} g(x)=L=\lim_{x\to c} h(x)$. Does the following argument work to conclude that $\lim_{x\to c} f(x)=L$? Let $\epsilon\gt 0$ be ......Sun, 23 Aug 2026 20:08:25 -0400https://liberty.io/how-to-solve-the-differential-equation-y-039-y-1-y.htmlhttps://liberty.io/how-to-solve-the-differential-equation-y-039-y-1-y.htmlUp until now, we simply rearranged and integrated both sides, so $$y'=y(1-y)$$ $$\frac{dy}{dx}=y(1-y)$$ $$\frac{dy}{y(1-y)}=dx$$ $$\int\frac{dy}{y(1-y)}=\int dx$$ With partial fraction decompositio......Sun, 23 Aug 2026 19:45:55 -0400https://liberty.io/prove-e-ln-x-x.htmlhttps://liberty.io/prove-e-ln-x-x.htmlIs it possible to prove $e^{\ln{x}} = x$ for a student or can you only say that exponentiation is defined to be the inverse of natural logarithm and stop there?...Sun, 23 Aug 2026 19:43:36 -0400https://liberty.io/constructing-a-graph-from-a-degree-sequence.htmlhttps://liberty.io/constructing-a-graph-from-a-degree-sequence.htmlLet's say I'm given several degree sequences like $$\begin{array}{l} \{4,3,3,2,2\} \\ \{3,3,3,3\} \\ \{5,3,3,2,2,1\} \end{array}$$ I can find the number of edges using the handshaking lemma $$ \sum......Sun, 23 Aug 2026 19:25:54 -0400https://liberty.io/calculate-overlapping-area-of-two-squares.htmlhttps://liberty.io/calculate-overlapping-area-of-two-squares.htmlGiven task: Two $10 \times 10$ cm square napkins were thrown on the table, as shown in the figure. They covered an area of ​​the table equal to $172$ cm$^2$. What is their overlap area? Answer is......Sun, 23 Aug 2026 19:24:56 -0400https://liberty.io/definition-of-weak-topology.htmlhttps://liberty.io/definition-of-weak-topology.htmlThough I learned some basic theory about topological vector spaces, I always confused about the definition of weak-$*$ topology. Given $x\in X$, let $\phi_x: X^*\to \Bbb R$ denote the evaluation m......Sun, 23 Aug 2026 19:24:55 -0400https://liberty.io/how-do-i-solve-algebraic-fractions-with-two-fractions-on-one-side.htmlhttps://liberty.io/how-do-i-solve-algebraic-fractions-with-two-fractions-on-one-side.htmlHow do I solve algebraic fractions with two fractions on one side? $$\frac{a}{4}-\frac{a+2}{3}=9$$ Just like that one, what would I do? Please explain why you did what you did. I can't show worki......Sun, 23 Aug 2026 19:24:15 -0400https://liberty.io/best-way-to-find-orthonormal-vector.htmlhttps://liberty.io/best-way-to-find-orthonormal-vector.htmlWhat is the easiest method to find two vectors that make an orthonormal basis with a given vector? For example, I have a vector \begin{pmatrix} 2^{-1/2} \\ -2^{-1/2} \\ 0 \end{pmatrix} I understand ...Sun, 23 Aug 2026 19:19:42 -0400https://liberty.io/in-which-order-do-i-graph-transformations-of-functions.htmlhttps://liberty.io/in-which-order-do-i-graph-transformations-of-functions.htmlIn which order do I graph transformations of functions? The 6 function transformations are: Vertical Shifts Horizontal Shifts Reflection about the x-axis Reflection about the y-axis Vertical sh......Sun, 23 Aug 2026 19:15:12 -0400https://liberty.io/missing-statement-in-a-proof-jordan-normal-form.htmlhttps://liberty.io/missing-statement-in-a-proof-jordan-normal-form.htmlI am reading the proof for the existence of the Jordan normal form. I know the Fitting lemma. I am missing one thing though, I mean the proof in my book seems to have this particular thing missing. I ...Sun, 23 Aug 2026 19:14:44 -0400https://liberty.io/integrate-int-1-0-frac-r-3-sqrt-4-r-2-dr.htmlhttps://liberty.io/integrate-int-1-0-frac-r-3-sqrt-4-r-2-dr.html$$\int_0^1\frac{r^3}{\sqrt{4+r^2}}\ \mathrm dr$$ I have attached my work. I am stuck....Sun, 23 Aug 2026 19:12:48 -0400https://liberty.io/determine-the-values-of-x-y-and-z-using-one-equation-with-three-variab.htmlhttps://liberty.io/determine-the-values-of-x-y-and-z-using-one-equation-with-three-variab.htmlSolve for x, y, and z using one equation with three variables: $xyz = x! + y! + z!$ where factorials are defined in the usual way, namely $n! = n(n-1)...2*1$ I'm not sure where to start with this ...Sun, 23 Aug 2026 19:11:14 -0400https://liberty.io/polyhedrons-exclusively-made-out-of-even-sided-polygons.htmlhttps://liberty.io/polyhedrons-exclusively-made-out-of-even-sided-polygons.htmlI know that the cube is the only 3 d shape which falls in polyhedrons but still is composed of squares, exclusively although its a even sided shape. I have noticed that after square there is no sin......Sun, 23 Aug 2026 19:06:00 -0400https://liberty.io/if-sin-x-sin-pi-4-x-then-why-isn-039-t-x-x-pi-4.htmlhttps://liberty.io/if-sin-x-sin-pi-4-x-then-why-isn-039-t-x-x-pi-4.htmlI've been solving a question, If $\cos(x) + \sin(x)=\sqrt{2} \cos(\pi/2 - x)$ then find the value of $x$. We know that $\cos(x) + \sin(x)= \sqrt{2} \sin(\pi/4 + x)$. So, $$\sin(\pi/4 + x) = \......Sun, 23 Aug 2026 19:05:51 -0400https://liberty.io/motivation-for-spectral-graph-theory.htmlhttps://liberty.io/motivation-for-spectral-graph-theory.htmlWhy do we care about eigenvalues of graphs? Of course, any novel question in mathematics is interesting, but there is an entire discipline of mathematics devoted to studying these eigenvalues, so ......Sun, 23 Aug 2026 19:03:24 -0400https://liberty.io/for-which-values-of-x-does-this-series-converge.htmlhttps://liberty.io/for-which-values-of-x-does-this-series-converge.htmlFor which values of $x$ does the series presented below converge? $$\sum_{n=1}^{+\infty}\frac{x^n(1-x^n)}{n}$$ Neither the root test nor the ratio test is of much help - I've tried for awhile now ......Sun, 23 Aug 2026 18:58:41 -0400https://liberty.io/meaning-of-the-symbol-oplus-closed.htmlhttps://liberty.io/meaning-of-the-symbol-oplus-closed.htmlAnybody who knows/can explain what this character means?...Sun, 23 Aug 2026 18:58:25 -0400https://liberty.io/how-to-change-the-curvature-of-a-function.htmlhttps://liberty.io/how-to-change-the-curvature-of-a-function.htmlWe have learnt that the curvature of a function like $f(x)=x^2$ can be computed by its second derivative. However, a question came to my mind on the possible transformations which saves the shape o......Sun, 23 Aug 2026 18:56:14 -0400https://liberty.io/how-many-solutions-ax-0-has.htmlhttps://liberty.io/how-many-solutions-ax-0-has.htmlProblem How many solutions equation $Ax=0$ has when $A$ is defined as: $$ A = \begin{bmatrix} 2 & 3 & 5 \\ 1 & 4 & 0 \\ -4 & 2 & 1 \end{bmatrix} $$ Attemtp to solve ......Sun, 23 Aug 2026 18:54:06 -0400https://liberty.io/about-trichotomy-law-for-real-numbers.htmlhttps://liberty.io/about-trichotomy-law-for-real-numbers.htmlI must proof the "Trichotomy Law for Real Numbers": Prop. 1: let be $\Bbb{R}$ a complete ordered field, then $$\forall x,y \in \Bbb{R}(x=y \vee x < y \vee x > y)$$ Proof 1: by definition of......Sun, 23 Aug 2026 18:53:13 -0400https://liberty.io/find-the-curvature-of-a-curve-at-the-given-point.htmlhttps://liberty.io/find-the-curvature-of-a-curve-at-the-given-point.htmlEdit: I've revised this to clear up any confusion. The $T(0), N(0)$, etc were just to help me. Suppose we have a space curve $r(t)=<3t,\cos t,\sin t>$. I'm asked to find the curvature at $P......Sun, 23 Aug 2026 18:50:48 -0400https://liberty.io/if-x-y-and-z-are-sets-prove-set-related-questions.htmlhttps://liberty.io/if-x-y-and-z-are-sets-prove-set-related-questions.htmlnot Z ⊆ not Y implies Y ⊆ Z by contradiction not Z ⊆ not Y implies X ∩ Y ⊆ X ∩ Z (using part 1) So far for 1 I have if not Z is subset of not Y then all elements of not Z are in not Y which mean......Sun, 23 Aug 2026 18:49:37 -0400https://liberty.io/explanation-of-linearity-of-expectation.htmlhttps://liberty.io/explanation-of-linearity-of-expectation.htmlLinearity of expectation is the property that the expected value of the sum of random variables is equal to the sum of their individual expected values, regardless of whether they are independe......Sun, 23 Aug 2026 18:46:08 -0400https://liberty.io/subsets-in-linear-algebra.htmlhttps://liberty.io/subsets-in-linear-algebra.htmlLooking to get some help with some problems. Not quite understanding the concepts of vectors subspaces. So I need to determine which of the following are subsets in $ \Bbb R^3$ are actually subspac......Sun, 23 Aug 2026 18:35:01 -0400https://liberty.io/tensor-product-and-matrix-multiplication-distributive-properties.htmlhttps://liberty.io/tensor-product-and-matrix-multiplication-distributive-properties.htmlI am trying to find partial trace of some matrix of the form $M = (A \otimes B)\times (A^{T*} \otimes B^{T*})$ in which $\otimes$ is tensor product, $\times$ is matrix multiplication, $T*$ is conju......Sun, 23 Aug 2026 18:33:19 -0400https://liberty.io/is-there-such-a-thing-as-partial-integration.htmlhttps://liberty.io/is-there-such-a-thing-as-partial-integration.htmlRecently in my mathematics courses I was taught partial derivatives, and I wondered if the reverse exists for integrals. This may sound like a stupid question, and it probably is, but let me expla......Sun, 23 Aug 2026 18:22:09 -0400https://liberty.io/can-someone-explain-this-proof-of-the-product-property-of-square-roots.htmlhttps://liberty.io/can-someone-explain-this-proof-of-the-product-property-of-square-roots.htmlI'm studying a proof of the product property of square roots. I can follow it up to statement 8, but I can't make sense of how the last statement, 9, follows from the previous ones. I have transcri......Sun, 23 Aug 2026 18:14:12 -0400https://liberty.io/find-all-values-of-theta-such-that-cos-2-theta-1-2.htmlhttps://liberty.io/find-all-values-of-theta-such-that-cos-2-theta-1-2.htmlFind all values of $\theta$ such that $cos(2\theta)=1/2$. Give your answer in radians. I know that on the unit circle, cosine represents the $x$ axis, while sine represents the $y$ axis. From wha......Sun, 23 Aug 2026 18:11:56 -0400https://liberty.io/geometry-and-topology.htmlhttps://liberty.io/geometry-and-topology.htmlI was wondering what are the differences and relations: between geometry and topology; between differential geometry and differential topology; between algebraic geometry and algebraic topolog......Sun, 23 Aug 2026 18:02:27 -0400https://liberty.io/what-type-of-graph-is-y-1-x-2.htmlhttps://liberty.io/what-type-of-graph-is-y-1-x-2.htmlDoes this count as a reciprocal graph or am I mistaken? The graph looks like an exponential where x >= 0 although the equation clearly suggests otherwise....Sun, 23 Aug 2026 17:58:38 -0400https://liberty.io/poker-hand-probability-of-getting-4-cards-of-equal-face-value-and-1-ca.htmlhttps://liberty.io/poker-hand-probability-of-getting-4-cards-of-equal-face-value-and-1-ca.htmlA poker hand is defined as drawing 5 cards at random without replacement from a deck of 52 playing cards. Find the probability of the following poker hand: Four of a kind (4 cards of equals face va......Sun, 23 Aug 2026 17:49:45 -0400https://liberty.io/dfa-accept-all-prefix-of-every-word-in-the-regular-language-l.htmlhttps://liberty.io/dfa-accept-all-prefix-of-every-word-in-the-regular-language-l.htmlLet L be a regular language and M is a dfa of L. build a dfa that acceptable all the prefix of every word in L i think to make all the states are acceptable but the problem the new DFA willemphasized ...Sun, 23 Aug 2026 17:47:01 -0400https://liberty.io/determining-if-a-binary-string-represents-a-prime-integer.htmlhttps://liberty.io/determining-if-a-binary-string-represents-a-prime-integer.htmlLet $\Sigma = \{0,1\}$ and $w$ be the string $0011101$ over $\Sigma$. If we work out what $w$ is, $w$ is the binary representation of $57$, which is not prime. It is remarked in Introduction to ...Sun, 23 Aug 2026 17:46:10 -0400https://liberty.io/are-some-infinities-really-bigger-than-others-closed.htmlhttps://liberty.io/are-some-infinities-really-bigger-than-others-closed.htmlI'm not a mathematician, but I recently read a thread about how some infinities are bigger than others. The argument put forward was that of mapping pairs of numbers from reals to naturals. There i......Sun, 23 Aug 2026 17:38:38 -0400https://liberty.io/what-are-the-most-overpowered-theorems-in-mathematics-closed.htmlhttps://liberty.io/what-are-the-most-overpowered-theorems-in-mathematics-closed.htmlWhat are the most overpowered theorems in mathematics? By "overpowered," I mean theorems that allow disproportionately strong conclusions to be drawn from minimal / relatively simple ...Sun, 23 Aug 2026 17:38:22 -0400https://liberty.io/convolution-of-two-dimensional-gaussian-functions.htmlhttps://liberty.io/convolution-of-two-dimensional-gaussian-functions.htmlI want to calculate the sum of two probability density functions. I know that it is: $P_{U+V} (x)= (P_{U} * P_{V})(x)$ If $P_{U}$ and $P_{V}$ are gaussian functions in one dimension, i.e. $P_{U}(......Sun, 23 Aug 2026 17:30:48 -0400https://liberty.io/understanding-inverse-logic.htmlhttps://liberty.io/understanding-inverse-logic.htmlThe original statement is, "If I am in Paris, then I am in France". The inverse statement is, "If I am not in Paris, then I am not in France". If $\lnot A = \lnot B =$ True, then $\lnot A \implies......Sun, 23 Aug 2026 17:27:39 -0400https://liberty.io/moment-generating-function-for-multivariate-normal.htmlhttps://liberty.io/moment-generating-function-for-multivariate-normal.htmlLet $\underset{\sim}{X}\sim N_k(\underset{\sim}{\mu},V)$ i.e. $X$ follows $k-$variate normal distribution with the variance-covariance matrix, $V_{k\times k}$. Then, pdf of $\underset{\sim}{X}$ is ......Sun, 23 Aug 2026 17:26:31 -0400https://liberty.io/linear-transformations-and-finding-the-identity-transformation.htmlhttps://liberty.io/linear-transformations-and-finding-the-identity-transformation.htmlLet $S:R^3 \rightarrow R^2$ and $T :R^2 \rightarrow R^3$ be linear transformations. Is it possible for $S\circ T$ to be the identity linear transformation? Is it possible for $T \circ S$ to be the ...Sun, 23 Aug 2026 17:11:59 -0400https://liberty.io/prove-that-7-n-1-is-divisible-by-6-duplicate.htmlhttps://liberty.io/prove-that-7-n-1-is-divisible-by-6-duplicate.htmlI know that I have to prove this with the induction formula. If proved the first condition i.e. $n=6$ which is divisible by $6$. But I got stuck on how to proceed with the second condition i.e $k$....Sun, 23 Aug 2026 16:49:34 -0400https://liberty.io/definition-of-continuum.htmlhttps://liberty.io/definition-of-continuum.htmlIn class the teacher gave us the definition of a continuum as a metric, connected and compact space, on the book General Topology by Willard says that is a connected, compact and Hausdorff space, a......Sun, 23 Aug 2026 16:47:50 -0400https://liberty.io/calculating-a-point-that-lies-on-an-ellipse-given-an-angle.htmlhttps://liberty.io/calculating-a-point-that-lies-on-an-ellipse-given-an-angle.htmlI need to find a point (A on this diagram) given the center point of the ellipse as well as an angle. I've been melting my brain all day (as well as searching through questions here) testing out ...Sun, 23 Aug 2026 16:46:04 -0400https://liberty.io/in-a-binary-vector-x-in-0-1-k-what-does-the-k-mean.htmlhttps://liberty.io/in-a-binary-vector-x-in-0-1-k-what-does-the-k-mean.htmlIn a binary vector $ x \in \{0,1\}^{k}$ what does the $^{k}$ mean? I understand that $\in$ means 'is a possible outcome' or 'in' so x can be 0 or 1, but I'm not sure what the $^{k}$ means....Sun, 23 Aug 2026 16:30:51 -0400https://liberty.io/solving-the-heat-equation-using-the-separation-of-variables.htmlhttps://liberty.io/solving-the-heat-equation-using-the-separation-of-variables.htmlSolve the initial value problem: $$\begin{cases} u_t = ku_{xx} \ \ &\text{for} \ \ x > 0, t > 0,\\ u(x,0) = e^{-2x} \ \ &\text{for} \ \ x > 0,\\ u(0,t) = 0 \ \ &\text{for} ......Sun, 23 Aug 2026 16:26:13 -0400https://liberty.io/about-cardano-039-s-formula-and-the-cubic-equation.htmlhttps://liberty.io/about-cardano-039-s-formula-and-the-cubic-equation.htmlI've been working my way through Cardano's formula for the cubic equation and I'm getting stuck at the end. I'm clear on producing the depressed cubic, $$y^3 + Ay + B,$$ and from there, I'm clear......Sun, 23 Aug 2026 16:22:40 -0400https://liberty.io/applications-of-linear-systems-of-differential-equations.htmlhttps://liberty.io/applications-of-linear-systems-of-differential-equations.htmlDoes anyone know of an application of linear systems of DEs besides multiple spring-mass systems and parallel circuits? I'm looking for an interesting application to show my DE students and we've ...Sun, 23 Aug 2026 16:20:47 -0400https://liberty.io/closest-point-on-a-line-to-another-point.htmlhttps://liberty.io/closest-point-on-a-line-to-another-point.htmlThe problem asks to find the nearest to $P(2,2)$, on the line: $$-x -2y +3 =0$$ This is what I've tried, the normal vector, $$n = (-1,-2)$$ found an arbitrary point on the given line by settin......Sun, 23 Aug 2026 16:12:14 -0400https://liberty.io/how-are-second-order-nonlinear-ordinary-differential-equations-solved.htmlhttps://liberty.io/how-are-second-order-nonlinear-ordinary-differential-equations-solved.htmlI conceived the following second order nonlinear ordinary differential equation: $$\frac{d^2y(x)}{dx^2}=\frac{k}{(y(x))^2}$$ I can tell it's nonlinear because of the $\frac{k}{(y(x))^2}$ term and ...Sun, 23 Aug 2026 16:09:46 -0400https://liberty.io/symbol-for-the-number-of-elements-in-a-set.htmlhttps://liberty.io/symbol-for-the-number-of-elements-in-a-set.htmlI would like to express below concisely and mathmatically TP is a set of real numbers $$C = \{ x \in TP : x > threshold \}$$ c_count = len(C) Basically, in English, I want to count the number of ...Sun, 23 Aug 2026 16:09:22 -0400https://liberty.io/is-it-possible-to-calculate-sine-by-hand-duplicate.htmlhttps://liberty.io/is-it-possible-to-calculate-sine-by-hand-duplicate.htmlWithout a calculator, how can I calculate the sine of an angle, for example 32(without drawing a triangle)?...Sun, 23 Aug 2026 16:04:23 -0400https://liberty.io/stability-of-solution.htmlhttps://liberty.io/stability-of-solution.htmlI have this differential equation $x'=-x^3$ , How to study the stability and the asymtotic stability of $x=0$ ? Please help me Thank you ....Sun, 23 Aug 2026 15:52:57 -0400https://liberty.io/euler-theorem-039-s-inverse-proposition-in-number-theory.htmlhttps://liberty.io/euler-theorem-039-s-inverse-proposition-in-number-theory.htmlThe Euler theorem $\mathbf (a,n)=1 \Rightarrow a ^\phi$ = $ 1(mod n)$ ($ \phi $ means $ \phi(n)$) Inverse of the above proposition that $\mathbf a ^\phi $ = 1(mod n) $\Rightarrow $ (a,n)=1 ......Sun, 23 Aug 2026 15:51:25 -0400https://liberty.io/why-does-the-series-sum-n-1-infty-frac1n-not-converge.htmlhttps://liberty.io/why-does-the-series-sum-n-1-infty-frac1n-not-converge.htmlCan someone give a simple explanation as to why the harmonic series $$\sum_{n=1}^\infty\frac1n=\frac 1 1 + \frac 12 + \frac 13 + \cdots $$ doesn't converge, on the other hand it grows very slo......Sun, 23 Aug 2026 15:51:06 -0400https://liberty.io/why-do-the-rules-of-horizontal-asymptotes-of-rational-functions-work.htmlhttps://liberty.io/why-do-the-rules-of-horizontal-asymptotes-of-rational-functions-work.htmlNote: My current understanding is only at a college algebra level From what I've seen online, in layman terms, the rules for horizontal asymptotes are as follows: Rule 1) If the degree of the ...Sun, 23 Aug 2026 15:45:10 -0400https://liberty.io/did-i-just-discover-a-new-way-to-calculate-the-signature-of-a-matrix.htmlhttps://liberty.io/did-i-just-discover-a-new-way-to-calculate-the-signature-of-a-matrix.htmlDue to the complains for more clarity down below I've cut my post into segments. Feel free to skip right to Definitions, Algorithm & Conjecture. If this is not clear enough, then I'm afraid I c......Sun, 23 Aug 2026 15:31:22 -0400https://liberty.io/theorem-of-generalization-logic.htmlhttps://liberty.io/theorem-of-generalization-logic.htmlI'm an undergraduate student and I have take a Mathematical Logic course this semester. I just read about the theorem of generalization in constants and I have (maybe a silly one) question. Why the ...Sun, 23 Aug 2026 15:29:55 -0400https://liberty.io/transformation-of-unit-vectors-from-cartesian-coordinate-to-cylindrica.htmlhttps://liberty.io/transformation-of-unit-vectors-from-cartesian-coordinate-to-cylindrica.htmlLet $ (\hat i, \hat j, \hat k) $ be unit vectors in Cartesian coordinate and $ (\hat e_\rho, \hat e_\theta, \hat e_z)$ be on spherical coordinate. Using the relation, $$ \hat e_\rho = \frac{\frac{\...Sun, 23 Aug 2026 15:25:28 -0400https://liberty.io/reduce-specific-cubic-equation-to-quadratic.htmlhttps://liberty.io/reduce-specific-cubic-equation-to-quadratic.htmlPlease help, equation $3x-x^3=1$ has three roots. Interesting fact that $|x_3|= x_1+x_2$. Is it possible to reduce this equation to a quadratic?...Sun, 23 Aug 2026 15:24:17 -0400https://liberty.io/centre-of-mass-on-a-sphere.htmlhttps://liberty.io/centre-of-mass-on-a-sphere.htmlSuppose that $p_1,\ldots,p_n$ are points on the round sphere $S^2$ and that $p_i$ has a mass $m_i$. How would one go about computing the position $\overline p$ of their centre of mass? I guess one ...Sun, 23 Aug 2026 15:18:42 -0400https://liberty.io/branching-process-interpretation-of-giant-component.htmlhttps://liberty.io/branching-process-interpretation-of-giant-component.htmlI'm trying to understand the following intuitive interpretation of the relationship between branching processes and connected components in random graphs: What is a bit confusing to me is how to ...Sun, 23 Aug 2026 15:16:13 -0400https://liberty.io/duality-with-a-stochastic-matrix.htmlhttps://liberty.io/duality-with-a-stochastic-matrix.htmlIf I have a stochastic matrix $X$- the sum of each row is equal to $1$ and all elements are non-negative. Given this property, how can I show that: $x'X=x'$ , $x\geq 0$ Has a non-zero solution......Sun, 23 Aug 2026 15:15:12 -0400https://liberty.io/is-every-bounded-sequence-convergent-closed.htmlhttps://liberty.io/is-every-bounded-sequence-convergent-closed.htmlIt's true, that every convergent sequence is bounded, but is every bounded sequence convergent?...Sun, 23 Aug 2026 14:39:59 -0400https://liberty.io/why-are-all-prime-numbers-are-congruent-to-1-or-5-mod6-duplicate.htmlhttps://liberty.io/why-are-all-prime-numbers-are-congruent-to-1-or-5-mod6-duplicate.htmlI understand that any natural number congruent to 2 or 4 (mod6) is even and that 3 is the only prime congruent to 3(mod6). Thus, any prime, p > 3, must be congruent to either 1 or 5 (mod 6). I j......Sun, 23 Aug 2026 14:39:10 -0400https://liberty.io/is-a-sequence-of-all-the-same-numbers-monotonic.htmlhttps://liberty.io/is-a-sequence-of-all-the-same-numbers-monotonic.htmlI'm wondering based on the definition of monotonicity: A sequence where $a_n\geq a_{n+1}$ for all $n\in\mathbb{N}$ is monotonic. So given that the sequence $a_n = 3$ is all the same numbers an......Sun, 23 Aug 2026 14:30:08 -0400https://liberty.io/can-a-piece-of-a4-paper-be-folded-so-that-it-039-s-thick-enough-to-rea.htmlhttps://liberty.io/can-a-piece-of-a4-paper-be-folded-so-that-it-039-s-thick-enough-to-rea.htmlWhile procrastinating around the web I stumbled on a page that contained the image below, from cracked.com. I can't help but believe that this is false… Even though the article header says......Sun, 23 Aug 2026 14:24:03 -0400https://liberty.io/derivative-over-the-function-itself.htmlhttps://liberty.io/derivative-over-the-function-itself.htmlI am new to calculus. Let $f(X)$ be continuous and differentiable over a set of variables $X$ where $x \in X$ and $x \in (0,\infty)$. Is there any general solution or rules to solve $\frac{f'(X)}{f......Sun, 23 Aug 2026 14:23:15 -0400https://liberty.io/how-to-find-in-which-number-base-the-operation-was-done-by-looking-at-.htmlhttps://liberty.io/how-to-find-in-which-number-base-the-operation-was-done-by-looking-at-.html$$23 + 25 = 51 $$ What base is used in the above addition operation ? I have 2 methods to do this Method 1 : Through equations assume base be a $$23_a + 25_a = 51_a $$ $$2a + 3 + 2a + 5 = 5a +......Sun, 23 Aug 2026 14:12:34 -0400https://liberty.io/how-to-explain-what-it-means-to-say-a-function-is-defined-on-an-interv.htmlhttps://liberty.io/how-to-explain-what-it-means-to-say-a-function-is-defined-on-an-interv.htmlI am having difficulty in explaining the terminology "defined" to the students I am assisting. Here is the sentence: If a real-valued function $f$ is defined and continuous on the closed interval $......Sun, 23 Aug 2026 14:07:29 -0400https://liberty.io/error-of-taylor-series.htmlhttps://liberty.io/error-of-taylor-series.htmlPart of my assignment is to find the third degree Taylor Series of $\tan(x)$ about $\pi/4$ and then estimate the error of this approximation when evaluated at 0.75. Finding the series was easy eno......Sun, 23 Aug 2026 14:07:04 -0400https://liberty.io/how-to-find-the-average-value-of-y-e-x-between-x-e-and-x-2e-closed.htmlhttps://liberty.io/how-to-find-the-average-value-of-y-e-x-between-x-e-and-x-2e-closed.htmlWhat approach would be ideal in finding the average value of $y = e^x$ between $x = e$ and $x = 2e$?...Sun, 23 Aug 2026 14:04:03 -0400https://liberty.io/which-of-these-event-can-be-represented-in-either-a-venn-diagram-and-t.htmlhttps://liberty.io/which-of-these-event-can-be-represented-in-either-a-venn-diagram-and-t.htmlWhich of these events can be represented in a Venn Diagram. Independent events Dependent events Mutually exclusive events Mutually independent events Basic reasoning with example would be nice......Sun, 23 Aug 2026 14:03:10 -0400https://liberty.io/how-do-you-find-the-number-that-corresponds-to-quartile-1-when-given-a.htmlhttps://liberty.io/how-do-you-find-the-number-that-corresponds-to-quartile-1-when-given-a.htmlIf you have an odd number of data values {39, 40, 42, 44, 47, 48, 49, 51, 53}, what is the score that value that corresponds to Quartile 1? In general, I would like to know if you include the medi......Sun, 23 Aug 2026 14:01:46 -0400https://liberty.io/general-solution-to-wave-equation-algebraic-approach.htmlhttps://liberty.io/general-solution-to-wave-equation-algebraic-approach.htmlConsider the Wave Equation $$\frac{\partial^{2} u}{\partial x^{2}} - c^{2}\frac{\partial^{2} u}{\partial y^{2}} = 0$$ The algebraic approach from wikipedia is: Take $\alpha = x - cy$ and $\bet......Sun, 23 Aug 2026 13:46:51 -0400https://liberty.io/prove-that-if-x-and-y-are-sequences-such-that-x-converges-to-x-neq-0-a.htmlhttps://liberty.io/prove-that-if-x-and-y-are-sequences-such-that-x-converges-to-x-neq-0-a.htmlI want to prove that given convergent sequences $X$ and $XY$ then $Y$ converges. Call arbitrarily $z_n=(x_n)(y_n)$, as by hypothesis $\lim(x_n)=x≠0$, then it must happen that $(x_n)≠0$, thus we can ...Sun, 23 Aug 2026 13:45:59 -0400https://liberty.io/proof-by-induction-binomial-theorem.htmlhttps://liberty.io/proof-by-induction-binomial-theorem.htmlLet $n\in N$, $k\in Z$, $o\leq k \leq n$. Define $C^{n}_k$ as the coefficient of $x^{n-k}y^k$ in the expansion of $(x+y)^n$ $$(x+y)^n= \sum^{n}_{k=0} C^{n}_k x^{n-k}y^k$$ Use induction to prove......Sun, 23 Aug 2026 13:37:34 -0400https://liberty.io/cofree-construction-thought-process.htmlhttps://liberty.io/cofree-construction-thought-process.htmlLet $F: \mathbf{A} \to \mathbf{B}$ be any functor and $B$ an object of $\mathbf{B}$. We say the pair $(A,\epsilon)$, with $A$ and object of $\mathbf{A}$ and $\epsilon: FA \to B$ is a morphism of $\......Sun, 23 Aug 2026 13:34:00 -0400https://liberty.io/expressing-5-8-sqrt-40-56-4-5-8-sqrt-10-56-4-in-simplified-radical-for.htmlhttps://liberty.io/expressing-5-8-sqrt-40-56-4-5-8-sqrt-10-56-4-in-simplified-radical-for.htmlI am trying to complete the question below, but I am not sure how to simplify the radical. What I have so far is $$(5.8\sqrt{40} + 56.4) - (5.8\sqrt{10} + 56.4) \;=\; \text{the difference}$$ How do......Sun, 23 Aug 2026 13:33:57 -0400https://liberty.io/how-many-solutions-does-riemann-039-s-p-symbol-describe.htmlhttps://liberty.io/how-many-solutions-does-riemann-039-s-p-symbol-describe.htmlThe Papperitz-Riemann P-symbol $$ \tag 1 y(z) = P \left\{ \begin{matrix} z_1 & z_2 & z_3 & \; \\ \alpha_1 & \alpha_2 & \alpha_3 & z \\ \beta_1 & \beta_2 & \be......Sun, 23 Aug 2026 13:21:15 -0400https://liberty.io/improper-integral-of-sin-x-x-from-zero-to-infinity-duplicate.htmlhttps://liberty.io/improper-integral-of-sin-x-x-from-zero-to-infinity-duplicate.htmlI was having trouble with the following integral: $\int_{0}^\infty \frac{\sin(x)}{x}dx$. My question is, how does one go about evaluating this, since its existence seems fairly intuitive, while its ...Sun, 23 Aug 2026 13:13:40 -0400https://liberty.io/why-use-the-letter-k-in-the-function-transformation-formula-f-x-h-k.htmlhttps://liberty.io/why-use-the-letter-k-in-the-function-transformation-formula-f-x-h-k.htmlThis is strictly a historical "why is it the letter k rather than say v for vertical" question -- is it the initial letter of something from a specific language? Is it arbitrary? While we're at it......Sun, 23 Aug 2026 13:10:28 -0400https://liberty.io/how-to-prove-that-two-lines-in-3d-are-not-parallel-and-do-not-intersec.htmlhttps://liberty.io/how-to-prove-that-two-lines-in-3d-are-not-parallel-and-do-not-intersec.htmlProblem Given two lines: $$(l_1)=(x,y,z)=(3,-1,2)+t(1,1,0)$$ $$(l_2)=(x,y,z)=(0,5,2)+t(1,-2,1)$$ Explain why these two lines are not parallel and why they do not intersect each other. Also, find the ...Sun, 23 Aug 2026 13:10:06 -0400https://liberty.io/the-degree-of-a-polynomial-which-also-has-negative-exponents.htmlhttps://liberty.io/the-degree-of-a-polynomial-which-also-has-negative-exponents.htmlIn theory, we define the degree of a polynomial as the highest exponent it holds. However when there are negative and positive exponents are present in the function, I want to know the basis that......Sun, 23 Aug 2026 12:49:42 -0400https://liberty.io/hausdorff-topological-space.htmlhttps://liberty.io/hausdorff-topological-space.htmlWanted to explain what I think a Hausdorff is in my own words because maybe that is the root of the problem. A Hausdorff Space is one in which for every x and y in X with x does not equal y, there ...Sun, 23 Aug 2026 12:48:07 -0400https://liberty.io/what-is-arctan-9-in-terms-of-pi.htmlhttps://liberty.io/what-is-arctan-9-in-terms-of-pi.htmlWhat is the pi fraction of $\arctan (9)$? I converted it to degrees and decimal, but how is it expressed in terms of $\pi$?...Sun, 23 Aug 2026 12:46:00 -0400https://liberty.io/almost-complex-manifolds-are-orientable.htmlhttps://liberty.io/almost-complex-manifolds-are-orientable.htmlI want to verify the fact that every almost complex manifold is orientable. By definition, an almost complex manifold is an even-dimensional smooth manifold $M^{2n}$ with a complex structure,......Sun, 23 Aug 2026 12:35:15 -0400https://liberty.io/is-the-square-root-of-negative-1-equal-to-i-or-is-it-equal-to-plus-or-.htmlhttps://liberty.io/is-the-square-root-of-negative-1-equal-to-i-or-is-it-equal-to-plus-or-.htmlI didn't see a duplicate.. The motivation is you tube. Tanton lectures of which one is titled " The Complex number i is NOT the square root of negative one". Does anyone have a clue why this may......Sun, 23 Aug 2026 12:32:58 -0400https://liberty.io/user-jiayin-pan-mathematics-stack-exchange.htmlhttps://liberty.io/user-jiayin-pan-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Sun, 23 Aug 2026 12:30:56 -0400https://liberty.io/probability-of-getting-a-pair-in-texas-hold-039-em-poker.htmlhttps://liberty.io/probability-of-getting-a-pair-in-texas-hold-039-em-poker.htmlI want to know the probability of getting a pair after all 5 cards are dealt on the table and assuming that I haven't got a pocket pair. Let's assume that I hold an ace and a king and 50 cards are......Sun, 23 Aug 2026 12:16:52 -0400https://liberty.io/expected-value-of-square-of-euclidean-norm-of-a-gaussian-random-vector.htmlhttps://liberty.io/expected-value-of-square-of-euclidean-norm-of-a-gaussian-random-vector.htmlIf $X$ is a $p \times 1$ gaussian random vector with such that $X \sim \mathcal{N}(0,\Sigma)$. What is the expected value of the square of the euclidean norm i.e $E[\|AX\|_2^2]$? Here $A$ is a $n \......Sun, 23 Aug 2026 11:40:26 -0400https://liberty.io/difference-between-omitting-the-multiplication-sign-and-keeping-it-dup.htmlhttps://liberty.io/difference-between-omitting-the-multiplication-sign-and-keeping-it-dup.htmlWhat is the difference between 2*a and 2a? What is the difference between 2(3+4) and 2*(3+4)? We all know that omitting the multiplication sign still means multiplication so nothing changed! This ...Sun, 23 Aug 2026 11:33:42 -0400https://liberty.io/show-that-the-equation-represents-a-circle-and-find-the-centre.htmlhttps://liberty.io/show-that-the-equation-represents-a-circle-and-find-the-centre.htmlI have the following equation: ${x^2 + y^2 - 4x -6y + 9 = 0}$ And I am asked to show that the equation represents a circle and find the centre. In order to show that the equation represents a ci......Sun, 23 Aug 2026 11:25:08 -0400https://liberty.io/what-is-infinite-horizon-problem.htmlhttps://liberty.io/what-is-infinite-horizon-problem.htmlIn optimal control, What is infinite horizon problem? What is the difference between finite and infinite horizon? What are their real life examples (finite & infinite)?...Sun, 23 Aug 2026 11:25:01 -0400https://liberty.io/how-to-show-that-derivative-of-phi-v-with-respect-to-v-is-phi-039-v-a-.htmlhttps://liberty.io/how-to-show-that-derivative-of-phi-v-with-respect-to-v-is-phi-039-v-a-.htmlHow to show that derivative of $\phi(v)$ with respect to $v$ is $$\frac{d \phi}{d v}= \frac{a}{2}(1-\phi^2(v)),$$ where $$\phi(v) = \frac{1-\exp(-av)}{1-\exp(-av)}=\tanh(av/2).$$ What is the v......Sun, 23 Aug 2026 11:23:42 -0400https://liberty.io/how-can-i-define-an-infinitely-small-positive-value.htmlhttps://liberty.io/how-can-i-define-an-infinitely-small-positive-value.htmlHow do you define an infinitely small value that is greater than zero? $1/\infty$ is apparently invalid because you can't divide by infinity. And $0.\bar{0}1$ is invalid because you can't put a 1 a......Sun, 23 Aug 2026 11:22:22 -0400https://liberty.io/three-dimensional-random-walk.htmlhttps://liberty.io/three-dimensional-random-walk.htmlA particle starts at an origin $O$ in three-space. Thinking of point $O$ as the center of a cube 2 units on a side. One move in this walk sends the particle with equal likelihood to one of the eight ...Sun, 23 Aug 2026 11:19:04 -0400https://liberty.io/the-product-of-two-positive-definite-matrices-has-real-and-positive-ei.htmlhttps://liberty.io/the-product-of-two-positive-definite-matrices-has-real-and-positive-ei.htmlGiven two real positive definite (and therefore, symmetric) matrices $A$ and $B$, are all the eigenvalues of $AB$ real and positive? Wikipedia says $AB$ is positive definite if $A$ and $B$ are pos......Sun, 23 Aug 2026 11:14:03 -0400https://liberty.io/how-to-find-distance-between-two-different-circles.htmlhttps://liberty.io/how-to-find-distance-between-two-different-circles.htmlI am trying the find the distance between two different sized circles, both centred on the horizontal plane. I know the diameter of each circle, and the length around both circles if wrapped like a......Sun, 23 Aug 2026 11:04:00 -0400