https://liberty.ioCurated star highlights with fresh energy.en-usWed, 02 Sep 2026 09:15:37 -0400https://liberty.io/stirling-numbers-of-the-first-kind-identity.htmlhttps://liberty.io/stirling-numbers-of-the-first-kind-identity.htmlProve the following identity for the signless Stirling numbers of the first kind: $$\sum_{m=k}^n c(n,m){m \choose k}=c(n+1,k+1)$$ I'm trying to rewrite the sum as follows $$c(n,k){k \choose k}+c(n,......Wed, 02 Sep 2026 13:15:30 -0400https://liberty.io/boundedness-of-the-solution-of-the-heat-equation.htmlhttps://liberty.io/boundedness-of-the-solution-of-the-heat-equation.htmlThe general solution of the heat equation $$\left\{\begin{array}{rcl} \partial_tu-\Delta u &=& 0\\ u(x,0)&=& f \end{array} \right.$$ is given by $$u(x,t)=\int\limits_{\mathbb R^n}\P......Wed, 02 Sep 2026 13:13:03 -0400https://liberty.io/find-the-axes-of-the-ellipse.htmlhttps://liberty.io/find-the-axes-of-the-ellipse.htmlWhat are the equations of the major and minor axes of the ellipse $x^2+2y^2-2xy-1=0$. The centre of the ellipse is $(0,0)$ but the axes are tilted (with respect to $x-y$ axes). I don't know how to ......Wed, 02 Sep 2026 13:12:20 -0400https://liberty.io/techniques-to-prove-a-function-is-uniformly-continuous.htmlhttps://liberty.io/techniques-to-prove-a-function-is-uniformly-continuous.htmlSo I understand the definition of uniform continuity, but wanted some suggestions to prove that a function is or isn't uniformly continuous. I have looked ahead and have seen that if a function is ...Wed, 02 Sep 2026 13:11:13 -0400https://liberty.io/can-an-alternating-series-ever-be-absolutely-convergent.htmlhttps://liberty.io/can-an-alternating-series-ever-be-absolutely-convergent.htmlCan an alternating series EVER be absolutely convergent? I am examining practice problems in my calculus book and I haven't yet come across a case where this is so. It might be because they are s......Wed, 02 Sep 2026 13:05:05 -0400https://liberty.io/if-integral-is-zero-and-function-is-continuous-and-non-negative-then-w.htmlhttps://liberty.io/if-integral-is-zero-and-function-is-continuous-and-non-negative-then-w.htmlIf $f$ is continuous on $[a,b]$, $f(x)≥0$ on $[a,b]$ and $$\int_{a}^{b} f(x) =0$$ then prove that $f(x)=0$ for all $x \in [a,b]$. I tried with Riemann's definite integral definition but couldn't ...Wed, 02 Sep 2026 12:49:37 -0400https://liberty.io/linearize-an-exponential-looking-graph-with-log-function.htmlhttps://liberty.io/linearize-an-exponential-looking-graph-with-log-function.htmlThis may be a beginner question, but I can't quite wrap my head around logs... I have a set of data (from an experiment) which gives me an exponential-looking graph (Fig 1). I'd like to "linearize"......Wed, 02 Sep 2026 12:49:06 -0400https://liberty.io/how-to-put-9-pigs-into-4-pens-so-that-there-are-an-even-number-of-pigs.htmlhttps://liberty.io/how-to-put-9-pigs-into-4-pens-so-that-there-are-an-even-number-of-pigs.htmlPut 9 pigs into 4 pens so that there are an even number of pigs in each pen. Read this for inspiration....Wed, 02 Sep 2026 12:47:12 -0400https://liberty.io/pseudometric-and-metric.htmlhttps://liberty.io/pseudometric-and-metric.htmlPSEUDOMETRIC Any explanation i will be so thankful . a finite pseudometric on a set X is a function satisfying (M1) ,(M2),(M3) and (M2*) d(x,x)=0 What is the difference between a metric and ...Wed, 02 Sep 2026 12:46:58 -0400https://liberty.io/is-there-an-intuitive-way-of-visualising-complex-roots.htmlhttps://liberty.io/is-there-an-intuitive-way-of-visualising-complex-roots.htmlConsider the function $f(x)$ such that $f(x) = x^2-4x+13$. By considering the discriminant, it can immediately be seen that the function has no real roots, since $b^2-4ac = (-4)^2-4(13) = -36$ and ......Wed, 02 Sep 2026 12:44:19 -0400https://liberty.io/does-the-alphabet-of-the-language-of-propositional-logic-have-no-funct.htmlhttps://liberty.io/does-the-alphabet-of-the-language-of-propositional-logic-have-no-funct.htmlIn Ebbinghaus' Mathematical Logic, II.2.1 on p14 says that the alphabet of a first order language contains function symbols, relation symbols, and constants, whereas I don't see these symbols in &q......Wed, 02 Sep 2026 12:41:57 -0400https://liberty.io/what-is-the-definition-of-a-mathematical-point.htmlhttps://liberty.io/what-is-the-definition-of-a-mathematical-point.htmlA point is defined in the elements as that which have no parts, but what does this mean? There is nothing in the physical world that doesn't have an extension. You can't say a point is primitive no......Wed, 02 Sep 2026 12:35:14 -0400https://liberty.io/complete-the-squares-to-find-the-center-and-radius-of-the-circle.htmlhttps://liberty.io/complete-the-squares-to-find-the-center-and-radius-of-the-circle.htmlI've had trouble on this one problem for a couple days. Complete the square on the X and Y terms to find the center and radius of the circle. $x^2+2x+y^2-4y=-4\:\:$...Wed, 02 Sep 2026 12:31:21 -0400https://liberty.io/how-to-prove-that-the-gram-matrix-a-ta-is-symmetric-with-nonnegative-d.htmlhttps://liberty.io/how-to-prove-that-the-gram-matrix-a-ta-is-symmetric-with-nonnegative-d.htmlIf A is any real $m \times n$ matrix, then $A^TA$ is called the Gram matrix of $A$. Show that the Gram matrix of $A$ is always symmetric with nonnegative diagonal elements. I have tried several ...Wed, 02 Sep 2026 12:28:43 -0400https://liberty.io/what-is-the-relationship-between-the-leibniz-integral-rule-and-the-dom.htmlhttps://liberty.io/what-is-the-relationship-between-the-leibniz-integral-rule-and-the-dom.htmlThe Leibniz integral rule states $$ {\displaystyle {\frac {d}{dx}}\left(\int _{a}^{b}f(x,t)\,dt\right)=\int _{a}^{b}{\frac {\partial }{\partial x}}f(x,t)\,dt} $$ when the integral bounds are not a ...Wed, 02 Sep 2026 12:26:17 -0400https://liberty.io/sum-of-consecutive-numbers.htmlhttps://liberty.io/sum-of-consecutive-numbers.htmlI was wondering if there a way to figure out the number of ways to express an integer by adding up consecutive natural numbers. For example for $N=3$ there is one way to express it $1+2 = 3$ I ha......Wed, 02 Sep 2026 12:23:55 -0400https://liberty.io/prove-this-identity-tan-2x-sec-2x-tan-x-pi-4.htmlhttps://liberty.io/prove-this-identity-tan-2x-sec-2x-tan-x-pi-4.htmlI've been having a time with this problem. I tried to start with the left side but I hit a dead end quick... I then tried the right side and had a little more luck but I've hit a block. I first use......Wed, 02 Sep 2026 12:16:21 -0400https://liberty.io/how-to-039-easily-039-obtain-the-quadratic-form-of-an-inverse.htmlhttps://liberty.io/how-to-039-easily-039-obtain-the-quadratic-form-of-an-inverse.htmlHow to 'easily' obtain the quadratic form of an inverse? x = $v^TA^{-1}\,v$ $A$ is a symmetric invertible matrix of real numbers, it is also positive-definite and sparse. $v$ is a vector with few......Wed, 02 Sep 2026 12:10:52 -0400https://liberty.io/gittins-index-for-a-simple-example.htmlhttps://liberty.io/gittins-index-for-a-simple-example.htmlEverything I can find on the Gittins Index is extremely in depth and abstract with almost no examples. I have spent hours digging through academic papers, lecture notes, and Wikipedia sources. I ...Wed, 02 Sep 2026 11:58:58 -0400https://liberty.io/counter-examples-related-to-slutsky-039-s-theorem.htmlhttps://liberty.io/counter-examples-related-to-slutsky-039-s-theorem.htmlConsider some real-valued random variables $(X_n)$, $(Y_n)$, $X$ and $Y$, defined on the same probability space. What is a counterexample to the claim that $X_n\to X$ in distribution and $Y_n \......Wed, 02 Sep 2026 11:57:09 -0400https://liberty.io/dividing-both-sides-of-an-inequality-by-a-vector.htmlhttps://liberty.io/dividing-both-sides-of-an-inequality-by-a-vector.htmlAs ridiculous as the question sounds, I'm just referring to a very specific case. Suppose that I'm given two $n \times 1$ vectors $x$ and $c$, an $m \times n$ matrix $A$ and a $m \times 1$ vector $......Wed, 02 Sep 2026 11:55:12 -0400https://liberty.io/jacobian-transformation-to-find-joint-pdf-of-y-1-and-y-2.htmlhttps://liberty.io/jacobian-transformation-to-find-joint-pdf-of-y-1-and-y-2.htmlLet $X_1$ and $X_2$ have the following joint density: $$f_{X_{1},X_{2}}(x_1,x_2) = \begin{cases} {\frac{4}{\pi}e^{-(x_1^2+x_2^2)}} & \text{$x_1 \gt 0$, $x_2 \gt 0$} \\{0} & \text{...Wed, 02 Sep 2026 11:38:37 -0400https://liberty.io/how-to-know-limit-of-ln-x-approaches-negative-infinity-as-x-to-0.htmlhttps://liberty.io/how-to-know-limit-of-ln-x-approaches-negative-infinity-as-x-to-0.htmlSo $\lim_{x \to 0^+} \ln x = -\infty$. If you draw the graph then it's very obvious. But without drawing a graph, how do we know it's $-\infty$ as opposed to $+\infty$?...Wed, 02 Sep 2026 11:34:19 -0400https://liberty.io/calculating-the-perimeter-of-an-ellipse.htmlhttps://liberty.io/calculating-the-perimeter-of-an-ellipse.htmlThe hydraulic diameter is defined as $$D_h=\dfrac{4A}{P}$$ where $A$ is the area and $P$ is the perimeter. The area of an ellipse is simply $\pi ab$. The perimeter has many many different ...Wed, 02 Sep 2026 11:29:51 -0400https://liberty.io/int-0-infty-1-x-2-converge.htmlhttps://liberty.io/int-0-infty-1-x-2-converge.htmlI need to tell if $\int_{0}^{\infty}(-1)^{[x^2]}dx$ converge, diverge or absolutely converges. I managed to say it does not absolutely converges. for it's converges, diverges i tried substituting $......Wed, 02 Sep 2026 11:28:48 -0400https://liberty.io/finding-the-exact-value-of-sin-1.htmlhttps://liberty.io/finding-the-exact-value-of-sin-1.htmlI have a question that says to find the exact value of $\sin^{-1} (1/\sqrt{2})$ For the answer it shows $\frac{\pi}{4} \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ Where does that last part i......Wed, 02 Sep 2026 11:25:47 -0400https://liberty.io/does-the-coend-have-a-right-adjoint.htmlhttps://liberty.io/does-the-coend-have-a-right-adjoint.htmlLet $ C $ be a small category and $ \mathfrak{X} $ be complete and cocomplete category. The coend may be seen as a functor $ \int ^C : Fun (C\times C^{op}, \mathfrak{X} )\to \mathfrak{X} $ Indeed......Wed, 02 Sep 2026 11:23:51 -0400https://liberty.io/integration-by-substitution-fail-for-int-frac-1-1-sin-x-mathrm-dx.htmlhttps://liberty.io/integration-by-substitution-fail-for-int-frac-1-1-sin-x-mathrm-dx.htmlWe tried to do an integral with the universal trigonometric substitution $$\int \frac{1}{(1+\sin x)}\, \mathrm dx$$ Meaning, we substituted: $ t = \tan \frac{x}{2} \Rightarrow$ $$\int \frac{1}{......Wed, 02 Sep 2026 11:21:02 -0400https://liberty.io/how-to-construct-magic-squares-of-even-order.htmlhttps://liberty.io/how-to-construct-magic-squares-of-even-order.htmlCould someone kindly point me to references on constructing magic squares of even order? Does a compact formula/algorithm exist?...Wed, 02 Sep 2026 11:14:07 -0400https://liberty.io/prove-that-a-piecewise-function-f-is-differentiable-at-0.htmlhttps://liberty.io/prove-that-a-piecewise-function-f-is-differentiable-at-0.html$f(x)= e^{-\frac{1}{x}}$, $x>0$ $f(x)=-x^2$, $x\leq0$ This is a function that is defined for $x$ here above . How can I prove that the derivative of $f$ exists at $x=0$....Wed, 02 Sep 2026 11:13:52 -0400https://liberty.io/prove-variance-in-uniform-distribution-continuous.htmlhttps://liberty.io/prove-variance-in-uniform-distribution-continuous.htmlI read in wikipedia article, variance is $\frac{1}{12}(b-a)^2$ , can anyone prove or show how can I derive this?...Wed, 02 Sep 2026 11:10:50 -0400https://liberty.io/is-a-negative-number-squared-negative-duplicate.htmlhttps://liberty.io/is-a-negative-number-squared-negative-duplicate.html$-3^2 = -9$ I found this problem while doing an algebra refresher in the book The Complete Idiot's Guide to Algebra. I asked my engineer brother this problem and he got it wrong. A search on Goo......Wed, 02 Sep 2026 11:07:12 -0400https://liberty.io/zero-probability-and-impossibility.htmlhttps://liberty.io/zero-probability-and-impossibility.htmlI read a comment under this question: There are plenty of events that can occur that have zero probability. This reminds me that I have seen similar saying before elsewhere, and have never been......Wed, 02 Sep 2026 10:56:59 -0400https://liberty.io/given-coordinates-of-vertices-a-convex-quadrilateral-find-which-pairs-.htmlhttps://liberty.io/given-coordinates-of-vertices-a-convex-quadrilateral-find-which-pairs-.htmlI'm writing a program that will take in four $(x, y, z)$ points that form a quadrilateral, and produce two sets of $(x, y, z)$ points that form the two largest triangles that comprise the quadrilat......Wed, 02 Sep 2026 10:55:03 -0400https://liberty.io/finding-the-leading-digit-s-in-any-number.htmlhttps://liberty.io/finding-the-leading-digit-s-in-any-number.htmlI am somewhat perplexed by this (presumably very simple) issue. Simply, I am trying to find the leading digit(s) in any number (related question). Here is some nice python code (from this question......Wed, 02 Sep 2026 10:54:22 -0400https://liberty.io/if-5-sqrt-x-125-sqrt-x-5-1-then-x-equals-4.htmlhttps://liberty.io/if-5-sqrt-x-125-sqrt-x-5-1-then-x-equals-4.htmlFor the equation $$5\sqrt [ x ]{ 125 } =\sqrt [ x ]{ { 5 }^{ -1 } } $$ $x$ is equal to $-4$, but I'm not sure why. I've taken the right side of the equation ${ \left( \frac { 1 }{ 5 } \righ......Wed, 02 Sep 2026 10:42:23 -0400https://liberty.io/how-to-find-the-equation-of-a-parabola-with-only-one-solution-and-a-po.htmlhttps://liberty.io/how-to-find-the-equation-of-a-parabola-with-only-one-solution-and-a-po.htmlI have been given the points $(5, 0)$ and $(10, 5)$. I'm also given that the $a$ value in $ax^2 \pm bx \pm c$ is $-1$. I have been trying to work out to how to do this, but, when I graph the parabo......Wed, 02 Sep 2026 10:40:38 -0400https://liberty.io/what-is-a-direct-correlation.htmlhttps://liberty.io/what-is-a-direct-correlation.htmlI have two contrary definitions of for the direct correlation between two variables $X$ and $Y$ Their correlation coefficient is close to $1$. There is a direct causal relationship between the var......Wed, 02 Sep 2026 10:35:44 -0400https://liberty.io/ideals-in-polynomial-rings-generated-by-linear-polynomials.htmlhttps://liberty.io/ideals-in-polynomial-rings-generated-by-linear-polynomials.htmlLet $K$ be any field. Is it true that: If $f_1,\dots, f_k \in K[x_1,\dots,x_n]$ are polynomials of degree $1$, then $I=(f_1,\dots, f_k)$ is all $K[x_1,\dots,x_n]$, or, if it is a proper ideal, the......Wed, 02 Sep 2026 10:16:58 -0400https://liberty.io/minkowski-functional-being-equivalent-to-original-norm.htmlhttps://liberty.io/minkowski-functional-being-equivalent-to-original-norm.htmlDefinition1: A subset $ A$ of a normed space $E$ is absolutely convex if $ \forall x,y \in A, \quad \forall \alpha ,\beta $ s.t $|\alpha |+ |\beta| \leq 1, \quad $ $ \alpha x +\beta y \in A.$ ...Wed, 02 Sep 2026 10:16:00 -0400https://liberty.io/what-is-the-name-of-a-fully-twisted-strip.htmlhttps://liberty.io/what-is-the-name-of-a-fully-twisted-strip.htmlI know that an odd number of twists generate a Möbius strip, and that an even number of twists generate a shape with two sides. Now, as far as I could find, for the strip with an even number of twi......Wed, 02 Sep 2026 10:12:38 -0400https://liberty.io/simple-dice-game-optimal-strategy.htmlhttps://liberty.io/simple-dice-game-optimal-strategy.htmlHere's the description of a dice game which puzzles me since quite some time (the game comes from a book which offered a quite unsatisfactory solution — but then, its focus was on programming, so t......Wed, 02 Sep 2026 10:10:49 -0400https://liberty.io/the-need-for-the-formal-definition-of-a-limit-duplicate.htmlhttps://liberty.io/the-need-for-the-formal-definition-of-a-limit-duplicate.htmlThe intuitive definition for $\lim\limits_{x \to a} f(x) = L$ is the value of $f ( x )$ can be made arbitrarily close to $L$ by making $x$ sufficiently close, but not equal to, $a$. the formal ...Wed, 02 Sep 2026 10:01:13 -0400https://liberty.io/how-to-find-4th-degree-polynomial-equation-from-given-points.htmlhttps://liberty.io/how-to-find-4th-degree-polynomial-equation-from-given-points.htmlI am trying to find 4th degree polynomial equation from given points. I do not own a graphing calculator so this task is very difficult for me to solve. So far I would out what points I need. The p......Wed, 02 Sep 2026 09:54:37 -0400https://liberty.io/lebesgue-integral-basics.htmlhttps://liberty.io/lebesgue-integral-basics.htmlI'm having trouble finding a good explanation of the Lebesgue integral. As per the definition, it is the expectation of a random variable. Then how does it model the area under the curve? Let's tak......Wed, 02 Sep 2026 09:46:12 -0400https://liberty.io/why-is-pi-considered-irrational-if-it-can-be-expressed-as-ratio-of-cir.htmlhttps://liberty.io/why-is-pi-considered-irrational-if-it-can-be-expressed-as-ratio-of-cir.htmlPi = C / D (circumference / diameter) . I have read that if circumference can be expressed as an integer then diameter cannot and vice-versa, so that the ratio can never be expressed as a/b where b......Wed, 02 Sep 2026 09:41:52 -0400https://liberty.io/can-i-optimize-area-of-cylinder-with-no-givens.htmlhttps://liberty.io/can-i-optimize-area-of-cylinder-with-no-givens.htmlI have a problem which should be very easy (as the rest of them are on this worksheet) but this one has me stumped. The question reads: A metal can is in the form of a cylinder. It has a bottom ......Wed, 02 Sep 2026 09:38:54 -0400https://liberty.io/example-of-an-inverse-semigroup.htmlhttps://liberty.io/example-of-an-inverse-semigroup.htmlAn inverse semigroup $S$ is a semigroup in which for each $x\in S$ there exists a unique $y\in S$ such that $xyx=x$ and $yxy=y$. I'm trying to find an explicit example(which is not a group) of such ...Wed, 02 Sep 2026 09:33:49 -0400https://liberty.io/what-039-s-the-intuition-behind-the-co-area-formula.htmlhttps://liberty.io/what-039-s-the-intuition-behind-the-co-area-formula.htmlI mainly work in statistics and I know only basic measure theory. I was trying to understand the Co-Area formula by Federer. If $f:\mathbb{R}^M\to \mathbb{R}^N$ is a Lipschitz function with $M \ge......Wed, 02 Sep 2026 09:30:56 -0400https://liberty.io/how-to-rotate-a-vector-along-another-vector.htmlhttps://liberty.io/how-to-rotate-a-vector-along-another-vector.htmlI want to rotate a vector e.g $V = Ai + Bj + Ck$ along an another vector $U = xi + yj + zk$. The angle between $U$ and $V$ is $\theta$. How can I do that and how do I know the new value of $A$, $......Wed, 02 Sep 2026 09:20:21 -0400https://liberty.io/mean-and-variance-from-a-cumulative-distribution-function.htmlhttps://liberty.io/mean-and-variance-from-a-cumulative-distribution-function.htmlI'm trying to find the mean (expected value) and variance for the following distribution function: $F(x)=\begin{cases} 0 & \text{for } x \lt 0\\ x/4 & \text{for } 0 \le x \lt 1\\ ......Wed, 02 Sep 2026 09:14:39 -0400https://liberty.io/what-is-the-negation-of-the-implication-statement.htmlhttps://liberty.io/what-is-the-negation-of-the-implication-statement.htmlIn a course on logic and proofs the professor presented on the following lines to show an example of negation: $$ \neg (P \Rightarrow Q) \ \ \ \ \Longleftrightarrow \ \ \ \ P \wedge \neg Q $$ I ca......Wed, 02 Sep 2026 09:10:28 -0400https://liberty.io/area-between-two-curves-with-respect-to-y.htmlhttps://liberty.io/area-between-two-curves-with-respect-to-y.htmlLet $x_1$ be the red curve and $x_2$ be the blue curve. I've tried integrating $x_1-x_2$ as well as $x_2-x_1$ from $y=0$ to $y=9/2$ and both answers were marked wrong. What do I do here? I'm thinki......Wed, 02 Sep 2026 09:08:55 -0400https://liberty.io/hanson-wright-inequality-any-relation-between-the-original-quadratic-f.htmlhttps://liberty.io/hanson-wright-inequality-any-relation-between-the-original-quadratic-f.htmlHanson-Wright's inequality such as this provides a bound on the second-order chaos of the form $$P(x^\top A x -\mathbb{E}[x^\top A x] > t) $$ For example, $x\sim N(0, I_n)$ and $A$ is some $n\ti......Wed, 02 Sep 2026 09:03:52 -0400https://liberty.io/surface-integral-of-a-right-circular-cone.htmlhttps://liberty.io/surface-integral-of-a-right-circular-cone.htmlUse a surface integral to show that the surface area of a right circular cone of radius $R$ and height $h$ is $\pi R \sqrt{h^2+R^2}$. Hint -- Use the parametrization $x=r\cos\theta$, $y=r\sin\theta......Wed, 02 Sep 2026 08:58:46 -0400https://liberty.io/problems-that-are-easy-with-trigonometry-and-too-difficult-without-it.htmlhttps://liberty.io/problems-that-are-easy-with-trigonometry-and-too-difficult-without-it.htmlI was looking for examples that shows importance of trigonometry in an elementary level. Like if you wanted to surprise your high school students that how problems are made easy using trigonometry ......Wed, 02 Sep 2026 08:56:50 -0400https://liberty.io/does-euler-039-s-formula-give-e-ix-cos-x-i-sin-x.htmlhttps://liberty.io/does-euler-039-s-formula-give-e-ix-cos-x-i-sin-x.htmlDoes Eulers formula give $$e^{-ix}=\cos(x) -i\sin(x)$$ I know that $$e^{ix}=\cos(x)+i\sin(x)$$ But how does it work when we have a $-$ in front...Wed, 02 Sep 2026 08:54:40 -0400https://liberty.io/what-is-the-definition-of-a-circular-wedge.htmlhttps://liberty.io/what-is-the-definition-of-a-circular-wedge.htmlIn Ahlfors' Complex Analysis, chapter 3, section 4, the author claims that a region whose boundary consists of two circular arcs with common end points is either a "circular wedge" or its complemen......Wed, 02 Sep 2026 08:30:35 -0400https://liberty.io/how-do-i-properly-read-a-clinometer.htmlhttps://liberty.io/how-do-i-properly-read-a-clinometer.htmlIf the weight hangs down at roughly 42 degrees, would the angle be 90 degrees - 42 degrees = 48 degrees?...Wed, 02 Sep 2026 08:27:16 -0400https://liberty.io/understanding-quantifier-elimination.htmlhttps://liberty.io/understanding-quantifier-elimination.htmlI am struggling to understand Quantifier Elimination as it is treated in Hodges' "A Shorter Model Theory". The relevant definitions are : Definition: Take $K$ to be a class of $L$-struct......Wed, 02 Sep 2026 08:26:00 -0400https://liberty.io/why-is-the-method-of-prime-factorization-help-in-finding-the-lcm-dupli.htmlhttps://liberty.io/why-is-the-method-of-prime-factorization-help-in-finding-the-lcm-dupli.htmlwhy does finding the prime factors help in finding the lcm of two or more numbers. I know how to do it but why is this the case? Edit I found this answer from just thinking on my own.I don't kno......Wed, 02 Sep 2026 08:24:04 -0400https://liberty.io/definition-laplace-operator-on-h-1-0-omega.htmlhttps://liberty.io/definition-laplace-operator-on-h-1-0-omega.htmlLet $\Omega \subseteq \mathbb{R}^n$ be a bounded open set. Is the Laplace operador defined in the space $H^2(\Omega)$ as a sum of second weak derivatives? How is the Laplace operador defined in the......Wed, 02 Sep 2026 08:22:09 -0400https://liberty.io/proof-of-congruence-with-mod-3.htmlhttps://liberty.io/proof-of-congruence-with-mod-3.htmlOk so I've tried making this one work for a while and keep getting stuck. I have the following: Prove: ($x^2$ is congruent to $x\mod3$) $\iff$ ($x$ is not congruent to $2\mod3$). So I know it is bi-...Wed, 02 Sep 2026 08:19:49 -0400https://liberty.io/pigeonhole-principle-formula-using-propositonal-logic.htmlhttps://liberty.io/pigeonhole-principle-formula-using-propositonal-logic.htmlAccording to the Pigeonhole Principle, if we try to place $n+1$ pigeons in $n$ pigeonholes, then at least one pigeonhole must have two or more pigeons. For $i \in \{1, 2, \dots, n+1\}$ and $j \in \......Wed, 02 Sep 2026 08:12:48 -0400https://liberty.io/what-is-the-mathematical-term-and-symbol-for-division-without-remainde.htmlhttps://liberty.io/what-is-the-mathematical-term-and-symbol-for-division-without-remainde.htmlWhat is the mathematical term and symbol for division without remainders? Take for example 322 / 100. I know 322 / 100 is division and the result is 3.22. I know 322 % 100 is a modulo and the resul......Wed, 02 Sep 2026 08:02:00 -0400https://liberty.io/express-99-2-3-as-a-fraction-no-calculator.htmlhttps://liberty.io/express-99-2-3-as-a-fraction-no-calculator.htmlMy 9-year-old daughter is stuck on this question and normally I can help her, but I am also stuck on this! I have looked everywhere to find out how to do this but to no avail so any help/guidance is ...Wed, 02 Sep 2026 07:58:24 -0400https://liberty.io/please-verify-my-proof-sqrt-18-is-irrational.htmlhttps://liberty.io/please-verify-my-proof-sqrt-18-is-irrational.htmlThe following is one of my exam questions and I would like to know whether my proof is correct or not. Prove $\sqrt{18}$ is irrational using proof by contradiction. Following is my proof......Wed, 02 Sep 2026 07:52:34 -0400https://liberty.io/vertical-pipe-through-a-45-roof-how-to-determine-the-roof-hole-039-s-s.htmlhttps://liberty.io/vertical-pipe-through-a-45-roof-how-to-determine-the-roof-hole-039-s-s.htmlI have a tent roof that is at a 45° angle. In the tent, I have a woodstove that has a vertical stove pipe. I want to cut a hole in the roof of the tent that will be the exact shape of the protru......Wed, 02 Sep 2026 07:45:36 -0400https://liberty.io/mixtures-and-alligations-acid-concentration.htmlhttps://liberty.io/mixtures-and-alligations-acid-concentration.htmlFrom a vessel containing $1$ litre of pure acid, $100$ ml pure acid was drawn out in each of the beakers A and B. The acid in both the beakers was diluted by adding water in different proportions. ......Wed, 02 Sep 2026 07:43:55 -0400https://liberty.io/keeping-the-arc-length-constant-between-points-in-a-spiral.htmlhttps://liberty.io/keeping-the-arc-length-constant-between-points-in-a-spiral.htmlI'm making a visualization of points in a logarithmic spiral and want to keep the arc length between points (image particles) constant. I read that in an Archemedian spiral arc length is s = r * th......Wed, 02 Sep 2026 07:31:09 -0400https://liberty.io/find-max-min-volume-of-rectangular-box-using-lagrange-multipliers.htmlhttps://liberty.io/find-max-min-volume-of-rectangular-box-using-lagrange-multipliers.htmlCan anyone help me solve the problem below? This is question number 14.8.42 in the seventh edition of Stewart Calculus. Here is the problem definition: "Find the maximum and minimum volumes of a ...Wed, 02 Sep 2026 07:25:58 -0400https://liberty.io/determining-all-the-elements-of-s4.htmlhttps://liberty.io/determining-all-the-elements-of-s4.htmlWhat is an easy way to determine the elements of $S_{4}$? While going through my revision process in an attempt to stem out the nitty gritty areas that I am unsure of, I chanced upon this. I trie......Wed, 02 Sep 2026 07:25:34 -0400https://liberty.io/union-of-intersections-and-intersection-of-unions.htmlhttps://liberty.io/union-of-intersections-and-intersection-of-unions.htmlWould someone be able to help me understand the (concept of) "union of intersections" and the "intersection of unions". More specifically, how to approach (prove) the following equality of two sets......Wed, 02 Sep 2026 07:20:38 -0400https://liberty.io/a-letter-is-picked-at-random-from-the-alphabet.htmlhttps://liberty.io/a-letter-is-picked-at-random-from-the-alphabet.htmlJust a book problem I need help on: A letter is picked at random from the alphabet. Find the probability that the letter is contained in the word "house" or in the word "phone". I know this probl......Wed, 02 Sep 2026 07:17:43 -0400https://liberty.io/cos-cos-x-sin-sin-x.htmlhttps://liberty.io/cos-cos-x-sin-sin-x.htmlShow that $\forall x \in \mathbb{R}\ \cos (\cos x) > \sin (\sin x)$ I've tried the obvious thing to do: $\varphi: x \mapsto \cos(\cos x) - \sin (\sin x)$ $\varphi ' (x) = \sin (\cos x) \sin ......Wed, 02 Sep 2026 07:14:18 -0400https://liberty.io/incremental-averaging.htmlhttps://liberty.io/incremental-averaging.htmlIs there a way to incrementally calculate (or estimate) the average of a vector (a set of numbers) without knowing their count in advance? For example you have a = [4 6 3 9 4 12 4 18] and you wa......Wed, 02 Sep 2026 07:06:19 -0400https://liberty.io/find-initial-value-and-growth-in-exponential-growth-model.htmlhttps://liberty.io/find-initial-value-and-growth-in-exponential-growth-model.htmlI was given two models each with specific time($t$) and end amount($P$). $100$ after $3$ hours and $400$ after $5$ hours. Using the exponential growth model I get these $100=Pe^{3r}$ and $400=Pe^{5......Wed, 02 Sep 2026 06:58:56 -0400https://liberty.io/how-to-find-an-angle-in-range-0-360-between-2-vectors.htmlhttps://liberty.io/how-to-find-an-angle-in-range-0-360-between-2-vectors.htmlI know that the common approach in order to find an angle is to calculate the dot product between 2 vectors and then calculate arcus cos of it. But in this solution I can get an angle only in the r......Wed, 02 Sep 2026 06:57:49 -0400https://liberty.io/on-the-number-of-ways-to-draw-kissing-circles.htmlhttps://liberty.io/on-the-number-of-ways-to-draw-kissing-circles.htmlSo I was watching Numberphile with Neil Sloane of OEIS fame on the number of ways to make circles intersect. During the intro, he explicitly forbid kissing (touching, tangent) circles. This made me......Wed, 02 Sep 2026 06:44:53 -0400https://liberty.io/finding-the-domain-of-arcsin-function.htmlhttps://liberty.io/finding-the-domain-of-arcsin-function.htmlI have to find the domain of: $ f(x) = 10 \frac{2}{3\arcsin x^2}$ Assumption 1: $-1 \leq 3\arcsin x^2 \leq 1$ $-1 \leq x^2 \leq 1$ $x \in [-1;1]$ Assumption 2: $3\arcsin x^2 \neq 0$ $x^2 \ne......Wed, 02 Sep 2026 06:36:22 -0400https://liberty.io/the-intuition-behind-slant-asymptotes.htmlhttps://liberty.io/the-intuition-behind-slant-asymptotes.htmlI'm asking this question to get a better understanding of oblique asymptotes. As regards vertical asymptotes, I know that they represent the numbers that make a function undefined. An example of t......Wed, 02 Sep 2026 06:35:24 -0400https://liberty.io/find-a-recurrence-relation-for-the-number-of-sequences-of-1s-3s-and-5s.htmlhttps://liberty.io/find-a-recurrence-relation-for-the-number-of-sequences-of-1s-3s-and-5s.html(a) Find a recurrence relation for the number of sequences of $ \ 1s, \ 3s, \ and \ \ 5s \ $ whose terms sum to $ \ n $ (b) Repeat part $ \ (a) $ with the added condition that no $ \ 5 \ $ ......Wed, 02 Sep 2026 06:34:46 -0400https://liberty.io/the-integral-form-of-inner-product-closed.htmlhttps://liberty.io/the-integral-form-of-inner-product-closed.htmlI want to know how scientists know that the inner product of f and g equal to integration from $$\int_a^b f(x)g(x)\ dx.$$...Wed, 02 Sep 2026 06:31:56 -0400https://liberty.io/plotting-frac-1-ln-x.htmlhttps://liberty.io/plotting-frac-1-ln-x.htmlI need assistance in plotting the graph of $\frac{1}{\ln x}$. wolframalpha gives this. How to plot this function (both real and imaginary part) using calculus?...Wed, 02 Sep 2026 06:30:37 -0400https://liberty.io/what-is-differential-probability-closed.htmlhttps://liberty.io/what-is-differential-probability-closed.htmlI am trying to understand the following statement (taken from a context related to volumetric image rendering): Density denotes the differential probability that a ray interacts with the volumetric “...Wed, 02 Sep 2026 06:14:31 -0400https://liberty.io/to-show-that-function-is-constant-duplicate.htmlhttps://liberty.io/to-show-that-function-is-constant-duplicate.htmlLet $f$ be defined on $\mathbb{R}$ and suppose that |$f(x)$ - $f(y)$| $\leq$ $(x-y)^2$ $x,y \in\mathbb{R}$. Here I have to show that $f$ is a constant function. I think I have to show that $f'(x)$......Wed, 02 Sep 2026 06:02:42 -0400https://liberty.io/the-concept-physical-meaning-interpretations-behind-the-bessel-039-s-i.htmlhttps://liberty.io/the-concept-physical-meaning-interpretations-behind-the-bessel-039-s-i.htmlIf $\{\boldsymbol{\varphi}_1, \boldsymbol{\varphi}_2, ... \}$ is an orthonormal system in $\mathcal{H}$, then the Bessel's inequality is: $$\sum_{j=1}^{\infty} |\langle \mathbf{x},\boldsymbol{\...Wed, 02 Sep 2026 05:51:12 -0400https://liberty.io/proof-of-the-existence-of-all-phone-numbers-in-pi-duplicate.htmlhttps://liberty.io/proof-of-the-existence-of-all-phone-numbers-in-pi-duplicate.htmlBy way of motivating example, someone came up with a completely evil user interface where the user was required to select their ten digit phone number by moving a sliding window over the digits of ......Wed, 02 Sep 2026 05:48:45 -0400https://liberty.io/proving-that-the-number-of-vertices-of-odd-degree-in-any-graph-g-is-ev.htmlhttps://liberty.io/proving-that-the-number-of-vertices-of-odd-degree-in-any-graph-g-is-ev.htmlI'm having a bit of a trouble with the below question Given $G$ is an undirected graph, the degree of a vertex $v$, denoted by $\mathrm{deg}(v)$, in graph $G$ is the number of neighbors of $v$. ...Wed, 02 Sep 2026 05:45:46 -0400https://liberty.io/are-the-notations-lg-2-n-and-lg-n-2-the-same-thing.htmlhttps://liberty.io/are-the-notations-lg-2-n-and-lg-n-2-the-same-thing.htmlI have a problem that is has the notation $lg^2 n$ and I just want to verify that it actually means / is the same as $(lg n)^2$ If it is not the same please tell me how to evaluate $lg^2 n$ (always ...Wed, 02 Sep 2026 05:44:48 -0400https://liberty.io/method-to-find-colossally-abundant-numbers.htmlhttps://liberty.io/method-to-find-colossally-abundant-numbers.htmlColossally abundant numbers are positive integers $n$ for which there exists a positive exponent $\epsilon$ such that $$\frac{\sigma(n)}{n^{1+\epsilon}}>\frac{\sigma(m)}{m^{1+\epsilon}}$$ for ......Wed, 02 Sep 2026 05:31:53 -0400https://liberty.io/cardinality-of-a-set-containing-sets.htmlhttps://liberty.io/cardinality-of-a-set-containing-sets.htmlI've just started learning basic set theory and am puzzled by this question I came up with: What is the cardinality of {1, {2,3}}? Do I treat sets within sets as just one element and so the answe......Wed, 02 Sep 2026 05:27:25 -0400https://liberty.io/derivative-of-x-2-y-2-25.htmlhttps://liberty.io/derivative-of-x-2-y-2-25.htmlI can find the derivative of the above using implicit differentiation but was interested in finding it using the explicit function way. Nudged by “dx”: $y=(25-x^2)^{1/2}\\$ $dy= (25-(x+dx)^2)^{1......Wed, 02 Sep 2026 05:25:30 -0400https://liberty.io/understanding-purpose-of-dual-vector-space.htmlhttps://liberty.io/understanding-purpose-of-dual-vector-space.htmlI am a beginner in differential geometry and have questions/confusion about dual vector space. I took a look at this and this questions. But both did not resolve my question. We have standard ...Wed, 02 Sep 2026 05:21:58 -0400https://liberty.io/penrose-tile-generator.htmlhttps://liberty.io/penrose-tile-generator.htmlDoes anyone know if there's a client or web app that generates Penrose patterns which can then be converted to a tileable rectangular background image for web site? I found this http://stephencoll......Wed, 02 Sep 2026 05:15:08 -0400https://liberty.io/how-do-you-determine-if-a-matrix-is-invertible-by-investigating-the-eq.htmlhttps://liberty.io/how-do-you-determine-if-a-matrix-is-invertible-by-investigating-the-eq.htmlHow do you determine if a matrix is invertible by investigating the equation Ax = I? For a 3x3 matrix (A) with the following Row 1: 1, 0, 1 Row 2: 1, 1, 0 Row 3: 0, 1, 1 I know the identit......Wed, 02 Sep 2026 05:05:20 -0400https://liberty.io/finite-sub-cover-for-0-1.htmlhttps://liberty.io/finite-sub-cover-for-0-1.htmlWhile learning topology one learns about compact set. The standard definition is: A set $X$ is said to be compact if open cover has a finite subcover. Since $[0,1]$ is compact, if we take a open ......Wed, 02 Sep 2026 05:02:49 -0400https://liberty.io/sin-properties-why-is-this-undefined.htmlhttps://liberty.io/sin-properties-why-is-this-undefined.htmlWhy is sin(-π/2)^(1/2) undefined? sin(-π/2) is -1, so I thought the the answer would be the same, -1^(1/2)=-1? But my calculator and the answers sheet both say undefined......Wed, 02 Sep 2026 05:00:18 -0400https://liberty.io/show-that-the-polynomial-p-0-p-1-ldots-p-m-are-linearly-dependent.htmlhttps://liberty.io/show-that-the-polynomial-p-0-p-1-ldots-p-m-are-linearly-dependent.htmlLet $p_0, p_1, \ldots, p_m $ polynomials in $\mathbb{P}_m$ with the property that $p_j(1)=0\ \forall j$. Show that the polynomial $p_0, p_1, \ldots,p_m $ are linearly dependent. My approach for t......Wed, 02 Sep 2026 04:27:56 -0400https://liberty.io/how-do-you-simplify-fractions-that-have-exponents.htmlhttps://liberty.io/how-do-you-simplify-fractions-that-have-exponents.htmlI'm not sure how to simplify a fraction with a large exponent for example: $$\frac{2^{2001} \cdot 3^{2003}}{6^{2002}}$$...Wed, 02 Sep 2026 04:19:52 -0400