https://liberty.ioCurated star highlights with fresh energy.en-usWed, 02 Sep 2026 18:35:59 -0400https://liberty.io/what-angle-must-two-equal-length-lines-be-at-s-t-a-line-from-the-open-.htmlhttps://liberty.io/what-angle-must-two-equal-length-lines-be-at-s-t-a-line-from-the-open-.htmlI've been thinking about this problem in my head for years, and finally sat down and drew it in CAD iterating until I could get an approximate solution, but the math is beyond me. Here is an image ......Wed, 02 Sep 2026 22:35:40 -0400https://liberty.io/find-the-angle-of-depression-given-the-following-two-right-triangle.htmlhttps://liberty.io/find-the-angle-of-depression-given-the-following-two-right-triangle.htmlA person in a control tower is looking down at an airplane on the runaway that is 78 yards away from the base of the tower.If this tower is 27 yards high, find the angle of depression from the towe......Wed, 02 Sep 2026 22:30:59 -0400https://liberty.io/finding-the-base-radius-of-the-cone.htmlhttps://liberty.io/finding-the-base-radius-of-the-cone.htmlA solid cone has a lateral surface area of $100\pi$ square centimeters and a total surface area of $269\pi$ square centimeters. Find the base radius of the cone. I would need to find the radius or ...Wed, 02 Sep 2026 22:26:13 -0400https://liberty.io/need-help-solving-an-integral-for-lagrange-remainder-proof.htmlhttps://liberty.io/need-help-solving-an-integral-for-lagrange-remainder-proof.htmlThis image of part of a proof for the Lagrange Remainder for Taylor's Formula. I need help solving the last integral. Can anyone explain?...Wed, 02 Sep 2026 22:23:19 -0400https://liberty.io/what-is-a-section-exactly.htmlhttps://liberty.io/what-is-a-section-exactly.htmlI have read 4 chapters of Hartshorne's Algebraic Geometry, when I go back to the beginning of scheme and the definition of a section, I am kind of confused why we call such a element "section". Le......Wed, 02 Sep 2026 22:20:03 -0400https://liberty.io/one-divided-by-infinity-is-not-zero-duplicate.htmlhttps://liberty.io/one-divided-by-infinity-is-not-zero-duplicate.htmlI know that $\frac{1}{\infty}$ is undefined. But my question is - can we say that $\frac{1}{\infty}\neq0$ ? I've got some idea how to explain that: Let's say we have a random-number generator that ...Wed, 02 Sep 2026 22:16:11 -0400https://liberty.io/what-is-poisson-point-process.htmlhttps://liberty.io/what-is-poisson-point-process.html"Points $\{A_j\}_{j\in\Phi(\lambda)}$ are assumed to be distributed according to a homogeneous PPP with intensity $\lambda$, denoted $\Phi(\lambda)=\{X_j\}$, where $X_j$ is the location of the $j$th ...Wed, 02 Sep 2026 22:14:44 -0400https://liberty.io/tan-x-cot-90-circ-x.htmlhttps://liberty.io/tan-x-cot-90-circ-x.htmlI was looking at a mark scheme for a question I was stuck on, and I came across this. You are asked to work out the value of $\tan 75^\circ$ after you've worked out $\cos 15^\circ$ and $\sin 15^\ci......Wed, 02 Sep 2026 22:14:39 -0400https://liberty.io/how-to-find-p-value.htmlhttps://liberty.io/how-to-find-p-value.htmlI am badly strugling to find P-values. I have no idea how it is coming for every problem I am doing. For example, the question below, how did they get/choose alpha 0.1 and 0.5 to calculate calculat......Wed, 02 Sep 2026 22:04:22 -0400https://liberty.io/how-does-multiplying-by-trigonometric-functions-in-a-matrix-transform-.htmlhttps://liberty.io/how-does-multiplying-by-trigonometric-functions-in-a-matrix-transform-.htmlI found this comic: But I can't understand the humor because I can't understand how trig functions affect matrix multiplication. Can someone please explain?...Wed, 02 Sep 2026 22:03:52 -0400https://liberty.io/how-many-nodes-before-k-clique-or-k-anti-clique.htmlhttps://liberty.io/how-many-nodes-before-k-clique-or-k-anti-clique.htmlI am attempting to solve some problems here. For exercise 1, the tightest result I could get is $4^k$. Is that the mininum possible bound? I am trying to either find a tight example, or find a bet......Wed, 02 Sep 2026 22:00:25 -0400https://liberty.io/derivative-of-sinx-with-respect-to-time-t.htmlhttps://liberty.io/derivative-of-sinx-with-respect-to-time-t.htmlhow does one take a time derivative of sinx it doesn't seem possible to me, shouldn't the function be in terms of time(t) to take its derivative.How can I calculate the change of the slope $w.r.t$ to ...Wed, 02 Sep 2026 22:00:16 -0400https://liberty.io/how-to-find-the-slope-of-a-secant-line-when-y-ln-x.htmlhttps://liberty.io/how-to-find-the-slope-of-a-secant-line-when-y-ln-x.htmlLet $y=\ln x$. Find the slope of the secant line between $P=(5, \ln5)$ and $Q=(9, \ln9)$. Should I just replace $\ln x$ with any of those two points? Ex: $y=\ln9$ or $y=\ln5$?? Please Help!!...Wed, 02 Sep 2026 21:53:28 -0400https://liberty.io/generalization-of-schwarz-reflection-principle-for-harmonic-functions.htmlhttps://liberty.io/generalization-of-schwarz-reflection-principle-for-harmonic-functions.htmlI recently learned about the Schwarz reflection principle and I was wondering whether any of the assumptions can be weakened. Explicitly, let $\Omega\subseteq\mathbb{C}$ be open and connected such ......Wed, 02 Sep 2026 21:39:34 -0400https://liberty.io/n-unlabelled-balls-in-m-labeled-buckets.htmlhttps://liberty.io/n-unlabelled-balls-in-m-labeled-buckets.htmlSo the question is how many ways are there to put N unlabelled balls in M labeled buckets. One way (I think) I figured it out is $M^N$ ways to put each ball, divided by $N!$ permutations of ball ...Wed, 02 Sep 2026 21:36:26 -0400https://liberty.io/number-combinations-multi-set.htmlhttps://liberty.io/number-combinations-multi-set.htmlHere is my question: Consider a multiset $\{n\cdot a, n\cdot b, 1,2,3, \ldots, n+1\}$ of size $3n+1$. Determine the number of $n$-combinations. I know from my textbook that if you have a $n$-...Wed, 02 Sep 2026 21:29:46 -0400https://liberty.io/why-is-the-domain-of-x-x-positive-real-numbers.htmlhttps://liberty.io/why-is-the-domain-of-x-x-positive-real-numbers.html(What is this function called, is it an exponential function?) Plot of $x^x$ I plotted this function and found that only positive part of it is shown. Also while computing derivative of this func......Wed, 02 Sep 2026 21:26:41 -0400https://liberty.io/the-sum-of-2-consecutive-numbers-is-53-closed.htmlhttps://liberty.io/the-sum-of-2-consecutive-numbers-is-53-closed.htmlThe sum of 2 consecutive numbers is 53. I need to find those numbers, but I'm not even sure how to set up the problem. Any suggestions?...Wed, 02 Sep 2026 21:17:52 -0400https://liberty.io/how-to-write-exponential-function-for-curve-that-pass-4-or-more-points.htmlhttps://liberty.io/how-to-write-exponential-function-for-curve-that-pass-4-or-more-points.htmlGiven 3 points, we could write exponential function for a curve that would pass those points in $y = ax^n + b$ form So I wonder that, could we write a function in exponential form that would pass 4 ...Wed, 02 Sep 2026 21:15:04 -0400https://liberty.io/calculate-the-probability-of-at-least-one.htmlhttps://liberty.io/calculate-the-probability-of-at-least-one.htmlA major electronics manufacturer has determined that when one of its televisions is sold, there is $0.08$ chance that it will need service before the warranty period expires. Suppose a retailer ......Wed, 02 Sep 2026 21:13:02 -0400https://liberty.io/what-is-2loglikelihood.htmlhttps://liberty.io/what-is-2loglikelihood.htmlI don't understand the term "loglikelihood"? I'd like to have a practical understanding of this word, and of why this is important. Besides this whenever we calculate some statistic like chisquare or ...Wed, 02 Sep 2026 20:51:20 -0400https://liberty.io/039-trace-trick-039-for-expectations-of-quadratic-forms.htmlhttps://liberty.io/039-trace-trick-039-for-expectations-of-quadratic-forms.htmlI am trying to understand the proof for the Kullback-Leibler divergence between two multivariate normal distributions. On the way, a sort of trace trick is applied for the expectation of the quadra......Wed, 02 Sep 2026 20:44:47 -0400https://liberty.io/what-is-a-geometry.htmlhttps://liberty.io/what-is-a-geometry.htmlThere is no question what topology is and what it's about: it's about topologies (= topological spaces), and that's it. There is also no question what (universal) algebra is and what it's about. (......Wed, 02 Sep 2026 20:42:59 -0400https://liberty.io/expected-value-with-indicator-random-variables.htmlhttps://liberty.io/expected-value-with-indicator-random-variables.htmlI don't understand the solution for problem 7a at http://www.ma.utexas.edu/users/geir/teaching/m362k/weeklyhw9solns.pdf. Reproduced below for reference: Suppose that A and B each randomly, and ...Wed, 02 Sep 2026 20:41:14 -0400https://liberty.io/proof-that-the-dimension-of-a-matrix-row-space-is-equal-to-the-dimensi.htmlhttps://liberty.io/proof-that-the-dimension-of-a-matrix-row-space-is-equal-to-the-dimensi.htmlI have the following theorem: Theorem 3.12. Let A be an m n matrix. Then the dimension of its row space is equal to the dimension of its column space. And the following proof is given: Proof. ...Wed, 02 Sep 2026 20:15:08 -0400https://liberty.io/what-is-the-essential-difference-between-ordinary-differential-equatio.htmlhttps://liberty.io/what-is-the-essential-difference-between-ordinary-differential-equatio.htmlPlease forgive my stupidity. So many years after my undergraduate study and so many years after dealing with various concrete ODEs and PDEs, I still cannot tell the essential difference between t......Wed, 02 Sep 2026 19:59:24 -0400https://liberty.io/limit-of-composite-functions.htmlhttps://liberty.io/limit-of-composite-functions.htmlLet $f$ and $g$ be some functions, assuming all right conditions that allow function composition, I want to prove that $$\lim_{x \to \infty} f(g(x))=f\left(\lim_{x \to \infty}g(x)\right) $$ As lo......Wed, 02 Sep 2026 19:55:41 -0400https://liberty.io/why-are-euclid-axioms-of-geometry-considered-039-not-sound-039.htmlhttps://liberty.io/why-are-euclid-axioms-of-geometry-considered-039-not-sound-039.htmlThe five postulates (axioms) are: "To draw a straight line from any point to any point." "To produce [extend] a finite straight line continuously in a straight line." "To describe a circle with ......Wed, 02 Sep 2026 19:48:23 -0400https://liberty.io/proving-that-sin-z-w-sin-w-cos-z-sin-z-cos-w-using-complex-exponential.htmlhttps://liberty.io/proving-that-sin-z-w-sin-w-cos-z-sin-z-cos-w-using-complex-exponential.htmlLet's define: $$\sin(z) = \frac{\exp(iz) - \exp(-iz)}{2i}$$ $$\cos(z) = \frac{\exp(iz) + \exp(-iz)}{2}$$ We are to prove that $$\sin(z+w)=\sin(w) \cos(z) + \sin(z)\cos(w), \forall_{z,w \in \...Wed, 02 Sep 2026 19:41:58 -0400https://liberty.io/is-there-a-way-to-draw-3d-graphs-on-paper-closed.htmlhttps://liberty.io/is-there-a-way-to-draw-3d-graphs-on-paper-closed.htmlI want to draw three dimensional graphs on paper. How would I do this though? Would I have to fold paper? Or would I have to rotate each point projected onto a 2d plane to make it suitable for my v......Wed, 02 Sep 2026 19:35:06 -0400https://liberty.io/question-about-existential-instantiation.htmlhttps://liberty.io/question-about-existential-instantiation.htmlI was having some trouble understanding existential instantiation. My textbook (Rosen - Discrete Mathematics and its Applications) states this about existential instantiation: Existential instanti......Wed, 02 Sep 2026 19:26:13 -0400https://liberty.io/approximate-coth-x-around-x-0.htmlhttps://liberty.io/approximate-coth-x-around-x-0.htmlI'm trying to approximate $\coth(x)$ around $x = 0$, up to say, third order in $x$. Now obviously a simple taylor expansion doesn't work, as it diverges around $x = 0$. I'm not quite sure how to pr......Wed, 02 Sep 2026 19:18:31 -0400https://liberty.io/what-is-the-motivation-of-levy-prokhorov-metric.htmlhttps://liberty.io/what-is-the-motivation-of-levy-prokhorov-metric.htmlFrom Wikipedia Let $(M, d)$ be a metric space with its Borel sigma algebra $\mathcal{B} (M)$. Let $\mathcal{P} (M)$ denote the collection of all probability measures on the measurable space ......Wed, 02 Sep 2026 19:11:45 -0400https://liberty.io/difference-between-rician-distribution-and-gaussian-distribution.htmlhttps://liberty.io/difference-between-rician-distribution-and-gaussian-distribution.htmlcould any one please tell me the difference between Rician and Gaussian Distribution and the advantages of using one over other please.With some mathematical proof would be truly appreciated Thank ......Wed, 02 Sep 2026 19:10:02 -0400https://liberty.io/evaluating-expressions-in-modulo-15-system.htmlhttps://liberty.io/evaluating-expressions-in-modulo-15-system.htmlHi !! I tried to find information and examples to solve this modulo arithematic problem but couldn't find it.. This is my preparation questions for exam and not assignment.Could someone explain the......Wed, 02 Sep 2026 18:57:36 -0400https://liberty.io/finding-all-sides-of-a-right-triangle-from-area-and-a-angle.htmlhttps://liberty.io/finding-all-sides-of-a-right-triangle-from-area-and-a-angle.htmlThe following calculator can figure out the length of all sides of a right angle triangle using only the area and a angle. How is it doing that..? http://www.triangle-calculator.com/?what=rt&a=T%...Wed, 02 Sep 2026 18:55:20 -0400https://liberty.io/which-of-the-following-are-subspaces-of-bbb-r-3.htmlhttps://liberty.io/which-of-the-following-are-subspaces-of-bbb-r-3.htmlI marked my answers but I am incorrect. Here's my logic - A. $0$ is not a solution for this equation so not a subspace. B. ... not sure about this one - Is this a subspace of $\mathbb{R}^3$? C. ...Wed, 02 Sep 2026 18:55:03 -0400https://liberty.io/construction-of-graph-laplacian.htmlhttps://liberty.io/construction-of-graph-laplacian.htmlI have a weighted undirected graph, and all the edge-weights are non-negative. According to the definition of the graph Laplacian matrix, $L=D-W$. In literature, I found that $D$ is known as degree ...Wed, 02 Sep 2026 18:52:21 -0400https://liberty.io/the-meaning-of-unique-expression.htmlhttps://liberty.io/the-meaning-of-unique-expression.htmlPlease help me with this. Let $G$ be abelian group, and let $A_k$ be a family of subgroups of $G$. Prove that $G=\sum A_k$ (internal) if and only if every non-zero element $g\in G$ has a unique ...Wed, 02 Sep 2026 18:43:05 -0400https://liberty.io/triple-integral-application-for-mass-density.htmlhttps://liberty.io/triple-integral-application-for-mass-density.html"Find the mass of the region bounded by the xy-plane and the hemisphere $z = \sqrt{100 − x^2 − y^2}$, if the mass density of the region is given by the function $f(x,y,z) = \frac{z}{\sqrt{x^2+y^2+z......Wed, 02 Sep 2026 18:39:50 -0400https://liberty.io/showing-that-k-7-contains-at-least-4-monochromatic-triangles.htmlhttps://liberty.io/showing-that-k-7-contains-at-least-4-monochromatic-triangles.htmlA problem in my book is: Let the edges of $K_7$ be colored with the colors red and blue. Show that there are at least four subgraphs $K_3$ with all three edges the same color (monochromatic tria......Wed, 02 Sep 2026 18:39:33 -0400https://liberty.io/variance-of-a-joint-density-function.htmlhttps://liberty.io/variance-of-a-joint-density-function.htmlThe random variables $X$ and $Y$ have joint density: $$ f_{X,Y}(x,y) = \begin{cases} 2-x-y,& 0<x,y<1\\ 0,&\text{otherwise}. \end{cases} $$ My question is to find $\operatorname{Var}......Wed, 02 Sep 2026 18:35:09 -0400https://liberty.io/how-can-pi-have-infinite-number-of-digits-and-never-repeat-them.htmlhttps://liberty.io/how-can-pi-have-infinite-number-of-digits-and-never-repeat-them.htmlI am very confused about this matter, even if I searched google about this already. Please show me how this is determined and/or at least explain to me. First, I saw this "Infinite Monkey Theorem"......Wed, 02 Sep 2026 18:32:21 -0400https://liberty.io/condition-number-of-orthogonal-matrix.htmlhttps://liberty.io/condition-number-of-orthogonal-matrix.htmlLet $A\in M_n(\mathbb R)$ be an orthogonal matrix. Then: $cond (A) =1$. I tryed to use facts about the eigenvalues but is did not help. In 2-norm it is easy to prove it since $||A||_2 = \sqrt{\rho......Wed, 02 Sep 2026 18:01:55 -0400https://liberty.io/intuitive-reason-that-quantifier-order-matters.htmlhttps://liberty.io/intuitive-reason-that-quantifier-order-matters.htmlIs there some understandable rationale why $\forall x\, \exists y\, P(x,y) \not \equiv \exists y\, \forall x\, P(x,y)$? I'm looking for a sentence I can explain to students, but I am failing every ......Wed, 02 Sep 2026 18:00:43 -0400https://liberty.io/do-you-ever-say-that-the-amplitude-is-negative.htmlhttps://liberty.io/do-you-ever-say-that-the-amplitude-is-negative.htmlIf you have a trig function $f(x) =- 3\sin (2x) + 1$ then would you ever say that the amplitude is negative? I've seen it stated that it can be negative or that amplitude is a distance so it should......Wed, 02 Sep 2026 17:51:56 -0400https://liberty.io/negation-of-the-completeness-axiom.htmlhttps://liberty.io/negation-of-the-completeness-axiom.htmlThere are a lot of theoretical results in Real Analysis, Calculus, Topology, etc. that depend on or use this result below. This has been a question that has bothered me for a long time and seems to......Wed, 02 Sep 2026 17:49:28 -0400https://liberty.io/why-are-harmonic-functions-called-harmonic-functions.htmlhttps://liberty.io/why-are-harmonic-functions-called-harmonic-functions.htmlAre they related to harmonic series in any way? Or something else? Wikipedia didn't help....Wed, 02 Sep 2026 17:44:39 -0400https://liberty.io/consistency-definition.htmlhttps://liberty.io/consistency-definition.htmlMy textbook says: A deductive theory is said to be consistent if no two asserted statements of this theory contradict each other, or, in other words, if of any two contradictory sentences at least......Wed, 02 Sep 2026 17:38:40 -0400https://liberty.io/is-2x-y-0-a-plane-or-line-in-3d.htmlhttps://liberty.io/is-2x-y-0-a-plane-or-line-in-3d.htmlI am going through Strang's linear algebra video 1 (https://www.youtube.com/watch?time_continue=1131&v=ZK3O402wf1c), and he seemed to suggest that the equation: $2x - y = 0$ is a plane in 3 ...Wed, 02 Sep 2026 17:32:31 -0400https://liberty.io/interior-angles-of-irregular-quadrilateral-with-1-known-angle.htmlhttps://liberty.io/interior-angles-of-irregular-quadrilateral-with-1-known-angle.htmlI have the measurements of the four sides of an irregular polygon and I need to find out the size of each interior angle. I know the sum of the angles is 360 degrees but because it's not a regular ...Wed, 02 Sep 2026 17:06:38 -0400https://liberty.io/is-speed-a-function-of-position.htmlhttps://liberty.io/is-speed-a-function-of-position.htmlLet $x$ be a smooth function from $[0,\infty)$ to $\mathbb{R}^n$ satisfying the following differential equation $x''(t) = f(x(t))$, where $f$ is a smooth function from $\mathbb{R}^n$ to itself. The......Wed, 02 Sep 2026 17:03:21 -0400https://liberty.io/absolute-value-inequality-with-variable-on-both-sides.htmlhttps://liberty.io/absolute-value-inequality-with-variable-on-both-sides.htmlI am trying to solve the following inequality: $$|3-5x| \le x$$ I am not familiar with inequalities including one absolute value with variables on both sides. I tried to solve it as follows: ......Wed, 02 Sep 2026 17:00:10 -0400https://liberty.io/find-the-value-of-the-second-derivative-at-the-point-by-the-given-valu.htmlhttps://liberty.io/find-the-value-of-the-second-derivative-at-the-point-by-the-given-valu.htmlFind the value of $\frac{d^2y}{dx^2}$ at the point by the given value of t. $$ x=9t^2-6, y=t^5, t=1 $$ I am way off when I calculate this. As I understand it $\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\fr......Wed, 02 Sep 2026 16:46:15 -0400https://liberty.io/why-should-the-trace-of-a-3d-rotation-matrix-have-these-properties.htmlhttps://liberty.io/why-should-the-trace-of-a-3d-rotation-matrix-have-these-properties.htmlOn the Wikipedia article about Rotation Matrices (https://en.wikipedia.org/wiki/Rotation_matrix#Determining_the_angle), the article states that the trace of the matrix will be equal to 1 + 2 cos(th......Wed, 02 Sep 2026 16:46:07 -0400https://liberty.io/boxplot-skewness.htmlhttps://liberty.io/boxplot-skewness.htmlI do know there are some rules about boxes and whiskers to determine the skewness in a boxplot, but I am confused with some rules in this particular case: Keeping in mind the rules, in this boxplo......Wed, 02 Sep 2026 16:36:43 -0400https://liberty.io/5-3-6-7-are-lengths-of-consecutive-sides-of-a-hexagon-of-equal-angles-.htmlhttps://liberty.io/5-3-6-7-are-lengths-of-consecutive-sides-of-a-hexagon-of-equal-angles-.htmlIn a hexagon with equal angles, the lengths of four consecutive edges are 5, 3, 6 and 7 (in that order). Find the lengths of the remaining two edges My answer came to be 1 and 8 but i can't check my ...Wed, 02 Sep 2026 16:35:23 -0400https://liberty.io/python-difference-between-two-uses-of-np-append-closed.htmlhttps://liberty.io/python-difference-between-two-uses-of-np-append-closed.htmlI'm not sure how to explain why I want to do this but...why does this code x = np.array([1,2,3,4]) d = np.empty((0, 4)) d = np.append(d,[x],axis=0) x[0]=8 d = np.append(d,[x],axis=0) give me this ...Wed, 02 Sep 2026 16:34:41 -0400https://liberty.io/calculate-stopping-distance-from-deceleration-time-and-speed.htmlhttps://liberty.io/calculate-stopping-distance-from-deceleration-time-and-speed.htmlQuite a simple question but I'm having issues - I have the speed of an object (say 5m/s) and the time it takes the object to come to a complete stop (say 2s). I want the object to stop exactly on a ...Wed, 02 Sep 2026 16:30:58 -0400https://liberty.io/sign-of-lagrange-multiplier-with-inequality-constraints.htmlhttps://liberty.io/sign-of-lagrange-multiplier-with-inequality-constraints.htmlWhile I have read on several places that the sign of lagrange multiplier $\lambda$ is not that important I'm reading now on Patten recognition and machine learning by Bishop the following: If we w......Wed, 02 Sep 2026 16:26:58 -0400https://liberty.io/i-need-to-find-the-value-of-x-im-only-given-the-a-degree-how-would-you.htmlhttps://liberty.io/i-need-to-find-the-value-of-x-im-only-given-the-a-degree-how-would-you.htmlthis is the link to the triangle that is connected to the question. What is the value x?...Wed, 02 Sep 2026 16:25:34 -0400https://liberty.io/finding-the-volume-of-the-area-between-two-curves-when-rotated-about-t.htmlhttps://liberty.io/finding-the-volume-of-the-area-between-two-curves-when-rotated-about-t.htmlI keep going down a rabbit hole when answering this question: Find the volume of the area lying in the first quadrant and bounded by the $y$-axis, the curve $y = x^3$ and the line $y = 3x + 2$, ......Wed, 02 Sep 2026 16:20:07 -0400https://liberty.io/does-the-function-have-horizontal-or-vertical-asymptotes.htmlhttps://liberty.io/does-the-function-have-horizontal-or-vertical-asymptotes.htmlSo I'm analyzing some functions here and I need to determine whether or not they have horizontal or vertical asymptotes. The equations are: $f(x)=260$ $g(x)=1+24(0.9)^x$ $h(x)=f(x)/g(x)$ Now $......Wed, 02 Sep 2026 16:17:03 -0400https://liberty.io/classifying-linear-first-order-pde-system-elliptic-hyperbolic-or-parab.htmlhttps://liberty.io/classifying-linear-first-order-pde-system-elliptic-hyperbolic-or-parab.htmlConsider the constants $$\begin{aligned} & (\text i.)\; a_1 = b_1 = a_2 = b_2 = 1 \\ & (\text {ii}.)\; a_1 = b_2 = 1, \quad b_1 = 0, \quad a_2 = -1 \\ & (\text {iii}.)\; a_1 = b_1 = b......Wed, 02 Sep 2026 16:04:26 -0400https://liberty.io/the-significance-of-topologically-equivalence-to-a-donut.htmlhttps://liberty.io/the-significance-of-topologically-equivalence-to-a-donut.htmlA friend of mine explained how a coffee mug is topologically equivalent to a donut to me tonight. I have to say the idea is very interesting! However, I don't know topology a lot but I am wondering......Wed, 02 Sep 2026 16:02:28 -0400https://liberty.io/definition-of-convergence-of-a-spectral-sequence.htmlhttps://liberty.io/definition-of-convergence-of-a-spectral-sequence.htmlIn Lecture Notes in Algebraic Topology by Davis & Kirk, on page 241, there is written: What do $E^\infty$ and $\lim_{r\to\infty}E^r_{p,q}$ mean? If a spectral sequence is not first-quadrant an......Wed, 02 Sep 2026 15:59:34 -0400https://liberty.io/how-are-eigenvectors-eigenvalues-and-differential-equations-connected.htmlhttps://liberty.io/how-are-eigenvectors-eigenvalues-and-differential-equations-connected.htmlIn school and at university we never had eigenvalues nor differential equations, so these concepts were really giving me a hard time. Now I developed some intuition for both concepts. I learned that ...Wed, 02 Sep 2026 15:52:19 -0400https://liberty.io/find-perimeter-of-traiangle-when-you-know-two-sides.htmlhttps://liberty.io/find-perimeter-of-traiangle-when-you-know-two-sides.htmlHere's the GRE problem: The length of one side of a triangle is 12. The length of another side is 18. Which of the following could be the perimeter of the triangle? [ ] 30 [ ] 36 [X] 44 [X] 48 [ ]......Wed, 02 Sep 2026 15:52:01 -0400https://liberty.io/determine-frequency-response-from-impulse-response.htmlhttps://liberty.io/determine-frequency-response-from-impulse-response.htmlI'm studying signal processing, using MATLAB to plot filter responses. So far, I understand I can use the impulse response to apply a filter to a signal. For example, the impulse response of an $L$-...Wed, 02 Sep 2026 15:47:35 -0400https://liberty.io/find-the-formula-for-sn.htmlhttps://liberty.io/find-the-formula-for-sn.htmlFind an explicit formula for $s_n$ if $s_0,s_1,s_2....$ is a sequence satisfying the given recurrence relation and initial conditions. I'm trying to figure out how to finish the formula. $s_n = -......Wed, 02 Sep 2026 15:46:09 -0400https://liberty.io/how-to-show-gives-a-binary-operation-duplicate.htmlhttps://liberty.io/how-to-show-gives-a-binary-operation-duplicate.htmlSet $S$ be the set of all real number except -1. Define $*$ on $S$ by $$a*b=a+b+ab$$ a) Show that $*$ gives a binary operation on $S$. Answer from my lecturer was; Suppose that $a*b=a+b+ab=-1$.......Wed, 02 Sep 2026 15:45:31 -0400https://liberty.io/is-there-any-way-to-express-a-constant-matrix.htmlhttps://liberty.io/is-there-any-way-to-express-a-constant-matrix.htmlLet's say we have $P=A+V+r(A+V)$ where $r$ is an independent variable, $A$ is a variale matrix and $V$ is a constant matrix. Is there any expression that allows me to express that $V$ is an cons......Wed, 02 Sep 2026 15:45:11 -0400https://liberty.io/how-to-calculate-int-0-1-e-x-dx-using-riemann-sum.htmlhttps://liberty.io/how-to-calculate-int-0-1-e-x-dx-using-riemann-sum.htmlI need to calculate $\int_0^1 e^x \; dx$ using Riemann sum. Problem set gives a hint:"The sum is a geometric progression. You will need the limit $ \lim _{n\to \infty }n\left(e^{\frac{1}{n}}-1\r......Wed, 02 Sep 2026 15:43:01 -0400https://liberty.io/real-world-usage-of-4-dimensional-geometry-closed.htmlhttps://liberty.io/real-world-usage-of-4-dimensional-geometry-closed.html4d space and shapes, for example the hypercube, are quite strange, but is there any real world usage for 4d geometry? For example, do 4d graphs that use rules of 4d geometry exist?...Wed, 02 Sep 2026 15:39:39 -0400https://liberty.io/what-is-the-sum-of-n-1-n-2-n-k.htmlhttps://liberty.io/what-is-the-sum-of-n-1-n-2-n-k.htmlWhat is the sum of this series ? $(n-1)+(n-2)+(n-3)+...+(n-k)$ $(n-1)+(n-2)+...+3+2+1 = \frac{n(n-1)}{2}$ So how can we find the sum from $n-1$ to $n-k$ ?...Wed, 02 Sep 2026 15:34:29 -0400https://liberty.io/the-assumption-in-proof-by-induction.htmlhttps://liberty.io/the-assumption-in-proof-by-induction.htmlThe second step in proof by induction is to: Prove that if the statement is true for some integer $n=k$, where $k\ge n_0$ then it is also true for the next larger integer, $n=k+1$ My question......Wed, 02 Sep 2026 15:15:38 -0400https://liberty.io/what-is-meant-by-evenly-divisible.htmlhttps://liberty.io/what-is-meant-by-evenly-divisible.html"What is the smallest positive number that is evenly divisible by all of the numbers from 1 to 20?" Is it different from divisible?...Wed, 02 Sep 2026 15:13:59 -0400https://liberty.io/prove-all-right-angles-are-congruent.htmlhttps://liberty.io/prove-all-right-angles-are-congruent.htmlProve all right angles are congruent. I only have to prove one side to this argument, so I just need to the the other argument. So basically, if two angles are right, then they must be congruent......Wed, 02 Sep 2026 15:10:14 -0400https://liberty.io/mod-and-rem-in-maple.htmlhttps://liberty.io/mod-and-rem-in-maple.htmlI do not understand this statement "reduce the coefficient modulo 5 and take remainder"? Is the following Maple command to reduce the coefficient modulo 5 , which coefficient to reduce? rem(6*x^2......Wed, 02 Sep 2026 15:08:29 -0400https://liberty.io/how-the-recursive-structure-of-apollonian-gaskets-can-be-described-in-.htmlhttps://liberty.io/how-the-recursive-structure-of-apollonian-gaskets-can-be-described-in-.htmlThe classical Descartes-Soddy relationship between the signed curvatures $b_k$ ("b" for "bend") of 4 mutually tangent circles (Apollonian configuration): $$\sum_{k=1}^4 b_k^2=\t......Wed, 02 Sep 2026 15:06:19 -0400https://liberty.io/finding-h-so-that-a-system-is-consistent-linear-matrix.htmlhttps://liberty.io/finding-h-so-that-a-system-is-consistent-linear-matrix.htmlHow can I determine what value of $h$ would make my system consistent. The augmented matrix of my system is: $\begin{bmatrix} 1&4&-2\\ 3&h&6 \end{bmatrix}$ My first step is to add......Wed, 02 Sep 2026 15:05:26 -0400https://liberty.io/relationship-between-equicontinuity-and-total-boundedness.htmlhttps://liberty.io/relationship-between-equicontinuity-and-total-boundedness.htmlA subspace of a complete metric space is compact iff it is closed and totally bounded. Take a compact space $X$, then its space of continuous functions $C(X)$ is complete. Subspaces of $C(X)$ are ...Wed, 02 Sep 2026 15:04:35 -0400https://liberty.io/find-the-singular-points-of-the-differential-equation-x-3-x-1-y-039-03.htmlhttps://liberty.io/find-the-singular-points-of-the-differential-equation-x-3-x-1-y-039-03.htmlConsider the second order linear homogeneous equation $$a_0(x)y'' + a_1(x)y'+ a_2(x)y = 0, x \in I \tag{1}$$ Suppose that $a_0$, $a_1$ and $a_2$ are analytic at $x_0 \in I$. If $a_0(x_0) = 0$, then......Wed, 02 Sep 2026 14:58:24 -0400https://liberty.io/how-does-the-dot-product-determine-similarity.htmlhttps://liberty.io/how-does-the-dot-product-determine-similarity.htmlI want to know how the dot product can determine whether two vectors are similar? I know that the formula $$\cos(\theta) = \frac{u \cdot v }{ ||u||\,||v||}$$ means something, but don't know what....Wed, 02 Sep 2026 14:54:06 -0400https://liberty.io/projectile-motion-question-when-launched-at-an-angle.htmlhttps://liberty.io/projectile-motion-question-when-launched-at-an-angle.htmlThe question in the calculus class is If the initial velocity of a projectile is $80$ ft/s with an angle of elevation of $55^\circ$, at what height will it strike a building that is $178$ ft away f......Wed, 02 Sep 2026 14:50:15 -0400https://liberty.io/inverse-laplace-through-complex-roots.htmlhttps://liberty.io/inverse-laplace-through-complex-roots.htmlI have been asked to apply inverse laplace to this: $$ \frac{(4s+5)}{s^2 + 5s +18.5} $$ What I have done is; I found the roots of denominator which are : $$ (-5-7i)/2 $$ and $$ (-5+7i)/2 $$ Then I ...Wed, 02 Sep 2026 14:22:38 -0400https://liberty.io/what-does-spans-and-spanned-by-mean-in-the-context-of-linear-algebra.htmlhttps://liberty.io/what-does-spans-and-spanned-by-mean-in-the-context-of-linear-algebra.htmlI'm more frequently coming across phrases such as "vector $b$ spanned by $\{b_1, \dots , b_n\}$" and "$A$ spans $B$" while studying linear algebra. What do the terms "spanned by" and "spans" mean i......Wed, 02 Sep 2026 14:12:31 -0400https://liberty.io/difference-between-newton-039-s-method-and-gauss-newton-method.htmlhttps://liberty.io/difference-between-newton-039-s-method-and-gauss-newton-method.htmlI know that the Gauss-Newton method is essentially Newton's method with the modification that the Gauss-Newton method it uses the approximation $2J^TJ$ (where $J$ is the Jacobian matrix) for the He......Wed, 02 Sep 2026 14:04:04 -0400https://liberty.io/is-a-straight-line-the-shortest-distance-between-two-points.htmlhttps://liberty.io/is-a-straight-line-the-shortest-distance-between-two-points.htmlQuite simply, I heard a lot of talk about how a straight line isn't necessarily the shortest distance between two points. Is this true, and if it isn't, how would that work?...Wed, 02 Sep 2026 14:03:25 -0400https://liberty.io/how-to-find-kolmogorov-forward-equations-given-generator-matrix-q.htmlhttps://liberty.io/how-to-find-kolmogorov-forward-equations-given-generator-matrix-q.htmlI am having difficulty in forming Kolmogorov Forward Equations. I understand how the KFE is derived and that $$\frac {d}{ds} p_{ij} (s) = \sum_{k \neq j} p_{ik} (s) \lambda_{k} r_{kj} - p_{ij} (s)\...Wed, 02 Sep 2026 14:00:13 -0400https://liberty.io/concrete-fft-polynomial-multiplication-example.htmlhttps://liberty.io/concrete-fft-polynomial-multiplication-example.htmlI have read a number of explanations of the steps involved in multiplying two polynomials using fast fourier transform and am not quite getting it in practice. I was wondering if I could get some h......Wed, 02 Sep 2026 13:49:33 -0400https://liberty.io/limits-of-a-piecewise-functions.htmlhttps://liberty.io/limits-of-a-piecewise-functions.htmlI don't understand how to do this question. I've done a sketch, but wouldn't the limit as x tends to -2 from above be 2? I mean, it's on the actual point that it's tending too... However, I get the ...Wed, 02 Sep 2026 13:48:13 -0400https://liberty.io/quickest-way-to-find-coefficient-of-trinomial.htmlhttps://liberty.io/quickest-way-to-find-coefficient-of-trinomial.htmlHow can I find the coefficient of $a^3b^2$ in the expansion of: $$ (a+b+3)^5 $$ What is the quickest way to do this without doing the whole expansion?...Wed, 02 Sep 2026 13:44:27 -0400https://liberty.io/suppose-ab-ac-where-b-and-c-are-matrices-and-a-is-invertible-show-that.htmlhttps://liberty.io/suppose-ab-ac-where-b-and-c-are-matrices-and-a-is-invertible-show-that.htmlSuppose $AB=​AC$, where $B$ and $C$ are matrices, and $A$ is invertible. Show that $B$=enter image description here$C$. Is this​ true, in​ general, when $A$ is not​ invertible? What can be deduced ......Wed, 02 Sep 2026 13:41:38 -0400https://liberty.io/motivation-and-examples-for-ramification.htmlhttps://liberty.io/motivation-and-examples-for-ramification.htmlI started learning algebraic number theory, but it seems like all the sources I had are too abstract, giving me difficulty understanding the concept and tripping me up frequently. For today it is ...Wed, 02 Sep 2026 13:40:53 -0400https://liberty.io/what-is-frac-1-n-called.htmlhttps://liberty.io/what-is-frac-1-n-called.htmlI have forgotten the name of this, and Google is not helpful (I've even tried searching "1 divided by n" and the answer has not come up). What is it called when you have a number, $n$, and perform $\...Wed, 02 Sep 2026 13:37:26 -0400https://liberty.io/proof-of-the-mean-value-theorem-for-integrals.htmlhttps://liberty.io/proof-of-the-mean-value-theorem-for-integrals.htmlMy Single Variable Calc Textbook asked me to prove the Mean Value Theorem for Integrals by applying the Mean Value Theorem for Derivatives to the function $F(x)=\displaystyle\int_a^xf(t)dt$. I'm pr......Wed, 02 Sep 2026 13:30:19 -0400https://liberty.io/textbooks-for-math-contests-apart-from-aops.htmlhttps://liberty.io/textbooks-for-math-contests-apart-from-aops.htmlI am looking for textbooks on math contests that give the theory associated with the topics (such as graph theory,geometry,Trig,combinatorics,etc) before giving a large volley of problems to solve(......Wed, 02 Sep 2026 13:27:45 -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 -0400