https://track.liberty.ioFresh celebrity buzz with instant attention value.en-usSat, 05 Sep 2026 16:51:25 -0400https://track.liberty.io/compute-number-of-comparisons-in-quicksort-pivoting-on-median-or-third.htmlhttps://track.liberty.io/compute-number-of-comparisons-in-quicksort-pivoting-on-median-or-third.htmlWith a few friends we read the Algorithm Design Manual from Skiena. One of his (chapter 4) exercises asks for the number of comparisons that the quicksort algorithm does (comparing an element to the ...Sat, 05 Sep 2026 20:42:37 -0400https://track.liberty.io/finding-the-geodesic-equations-on-s-2.htmlhttps://track.liberty.io/finding-the-geodesic-equations-on-s-2.htmlThe metric on $S^2$ is given as $\bar{g} = ds^2 = d\theta^2 + \sin^2\theta d\phi^2$ where $x^1 = \theta$ and $x^2 = \phi$. The only non-zero components of the Christoffel symbols are $\Gamma^{\ 1}_......Sat, 05 Sep 2026 20:42:04 -0400https://track.liberty.io/convergence-divergence-of-the-sum-sum-n-2-infty-1-ln-n.htmlhttps://track.liberty.io/convergence-divergence-of-the-sum-sum-n-2-infty-1-ln-n.htmlIs the sum $$ \sum_{n =2}^{\infty} \frac{1}{\ln n!} $$ convergent or divergent? I have tried different methods and it doesn't work. Perhaps comparing with a divergent series will work? I'm think......Sat, 05 Sep 2026 20:37:19 -0400https://track.liberty.io/what-is-the-point-of-quadratic-residues.htmlhttps://track.liberty.io/what-is-the-point-of-quadratic-residues.htmlWhat is the most motivating way to introduce quadratic residues? Are there any real life examples of quadratic residues? Why is the Law of Quadratic Reciprocity considered as one of the most impor......Sat, 05 Sep 2026 20:35:53 -0400https://track.liberty.io/population-vs-sampling-frame-vs-sample.htmlhttps://track.liberty.io/population-vs-sampling-frame-vs-sample.htmlCould someone please explain how the sampling frame is different from population and sample? I understand that the population is all the sampling unit that match our criteria for the study. And the ...Sat, 05 Sep 2026 20:34:53 -0400https://track.liberty.io/how-to-calculate-limits-of-sequences.htmlhttps://track.liberty.io/how-to-calculate-limits-of-sequences.htmlCalculate the limits of the following sequences $$ a_n = \frac{n+4n-n^5}{n^3+3n^5-2n}$$ $$ b_n = \sqrt{n^2+4} - \sqrt{n^2+2} $$ I know the formulas to transform sequences to find out its limit bu......Sat, 05 Sep 2026 20:23:24 -0400https://track.liberty.io/understanding-zorn-039-s-lemma.htmlhttps://track.liberty.io/understanding-zorn-039-s-lemma.htmlA lot of authors assume Zorn's lemma. I am told it is not an obvious mathematical fact, but I am having problems understanding why that is. Zorn's lemma states that if every chain in a partially ...Sat, 05 Sep 2026 20:10:54 -0400https://track.liberty.io/minimum-distance-between-two-boats-travelling-in-different-velocities.htmlhttps://track.liberty.io/minimum-distance-between-two-boats-travelling-in-different-velocities.htmlQuestion: Boat $A$ was $100$ km south of boat $B$ at $9:00$ AM. If boat $A$ travels towards the north with the speed of $20$ km/h, and boat $B$ travels towards the east with the speed of $15$ km/h,......Sat, 05 Sep 2026 20:07:43 -0400https://track.liberty.io/would-it-be-fine-to-use-serge-lang-039-s-two-calculus-books-as-textboo.htmlhttps://track.liberty.io/would-it-be-fine-to-use-serge-lang-039-s-two-calculus-books-as-textboo.htmlI'm a freshman in Maths major, but the recommanded textbook(Calculus:A Complete Course by Robert A. Adams) by Prof. of Calculus course is too much expensive, well, I found there're Serge Lang's two ...Sat, 05 Sep 2026 20:00:17 -0400https://track.liberty.io/calculating-the-length-norm-of-a-vector-duplicate.htmlhttps://track.liberty.io/calculating-the-length-norm-of-a-vector-duplicate.htmlHow do you interpret this equation? I've seen far simpler equations for solving the magnitude, like the square root of all of the elements of a vector squared added to one another. What exactly is ......Sat, 05 Sep 2026 20:00:13 -0400https://track.liberty.io/maclaurin-series-of-e-x-2.htmlhttps://track.liberty.io/maclaurin-series-of-e-x-2.htmlThe question is: Find the first 3 non-zero terms in the MacLaurin series for the function: $$y =e^{-x^2}$$ I have been told to simply substitute the $-x^2$ into the standard MacLaurin series for $......Sat, 05 Sep 2026 19:45:39 -0400https://track.liberty.io/find-integral-of-absolute-values-by-splitting-integrals-int-1-4-3-2-x-.htmlhttps://track.liberty.io/find-integral-of-absolute-values-by-splitting-integrals-int-1-4-3-2-x-.htmlI have trouble splitting the integral $$\int_{-1}^{6} (5-|2-x|)\, dx$$ Tried so far: Split the 3 and the absolute value to two separate integrals. Draw absolute value graph. Integrate both. I think ...Sat, 05 Sep 2026 19:42:17 -0400https://track.liberty.io/finding-csc-theta-given-cot-theta.htmlhttps://track.liberty.io/finding-csc-theta-given-cot-theta.htmlI have the following problem: If $ \cot{C} = \frac{\sqrt{3}}{7} $, find $ \csc{C} $ From my trig identities, I know that $ \cot{\theta} = \frac{1}{\tan{\theta}} $, and $ \csc{\theta} = \frac{1}{\......Sat, 05 Sep 2026 19:34:35 -0400https://track.liberty.io/proving-that-a-symmetric-matrix-is-positive-definite-iff-all-eigenvalu.htmlhttps://track.liberty.io/proving-that-a-symmetric-matrix-is-positive-definite-iff-all-eigenvalu.htmlThis has essentially been asked before here but I guess I need 50 reputation to comment. Also, here I have some questions of my own. My Proof outline: (forward direction/Necessary direction): Call......Sat, 05 Sep 2026 19:21:33 -0400https://track.liberty.io/how-many-total-increasing-numbers-are-there.htmlhttps://track.liberty.io/how-many-total-increasing-numbers-are-there.htmlI've been asked this question and I'm close but just can't figure it out. To clarify, an "increasing number" has to have increasing digits from left to right and cannot repeat digits anywhere else......Sat, 05 Sep 2026 19:17:39 -0400https://track.liberty.io/expected-value-is-a-linear-operator-under-what-conditions-is-median-al.htmlhttps://track.liberty.io/expected-value-is-a-linear-operator-under-what-conditions-is-median-al.htmlI have always taken for granted that expected value is a linear operator. For any random variables $X$ and $Y$: $E(aX + bY) = aE(X) + bE(Y)$. Can anyone point me to a rigorous proof of this? Also, I ...Sat, 05 Sep 2026 19:17:36 -0400https://track.liberty.io/eigenvalues-of-an-antihermitian-matrix.htmlhttps://track.liberty.io/eigenvalues-of-an-antihermitian-matrix.htmlI have to prove that every eigenvalue of an antihermitian matrix is in the form of $bi $ for some $b \in R$. I already know that if A is antihermitian , it is normal , thus we can diagonalise it ......Sat, 05 Sep 2026 19:07:33 -0400https://track.liberty.io/what-is-the-state-transition-matrix-for-this-time-varying-system.htmlhttps://track.liberty.io/what-is-the-state-transition-matrix-for-this-time-varying-system.html$$\begin{align}\frac{d}{dt}x_1&=-x_1+\sin(t)x_2 \\ \frac{d}{dt}x_2&=-t^2x_2\end{align}$$ What is the state transition matrix for these differential equations? How do you get it? Please sho......Sat, 05 Sep 2026 19:02:18 -0400https://track.liberty.io/exponential-function-as-an-infinite-product.htmlhttps://track.liberty.io/exponential-function-as-an-infinite-product.htmlIs there any representation of the exponential function as an infinite product (where there is no maximal factor in the series of terms which essentially contributes)? I.e. $$\mathrm e^x=\prod_{n=......Sat, 05 Sep 2026 18:58:51 -0400https://track.liberty.io/the-simplest-way-to-calculate-area-of-a-pentagon.htmlhttps://track.liberty.io/the-simplest-way-to-calculate-area-of-a-pentagon.htmlI have a pentagon, whose all sides and angles I know. What would be the simplest way, i.e requires least calculations, to calculate its area? If possible, can I generalize your way to higher polyg......Sat, 05 Sep 2026 18:52:34 -0400https://track.liberty.io/why-can-039-t-a-negative-number-be-raised-to-a-power.htmlhttps://track.liberty.io/why-can-039-t-a-negative-number-be-raised-to-a-power.htmlI was messing around with graphs on desmos, and I found when I put in the function (n)^x, when n was negative, there would be no graph. Is there a reason for this? https://www.desmos.com/calculator/...Sat, 05 Sep 2026 18:51:05 -0400https://track.liberty.io/calculating-the-total-number-of-surjective-functions.htmlhttps://track.liberty.io/calculating-the-total-number-of-surjective-functions.htmlIt is quite easy to calculate the total number of functions from a set $X$ with $m$ elements to a set $Y$ with $n$ elements ($n^{m}$), and also the total number of injective functions ($n^{\underli......Sat, 05 Sep 2026 18:45:14 -0400https://track.liberty.io/big-o-notation-prove-that-n-2-2n-3-is-mathcal-o-n-2.htmlhttps://track.liberty.io/big-o-notation-prove-that-n-2-2n-3-is-mathcal-o-n-2.htmlI'm taking a course in Discrete Mathematics this summer, and my book doesn't offer a very good explanation of Big-O notation. I understand that if $f(x)$ is $\mathcal O(g(x))$ it means that there ......Sat, 05 Sep 2026 18:41:24 -0400https://track.liberty.io/solve-the-difference-quotient.htmlhttps://track.liberty.io/solve-the-difference-quotient.htmlSo I tried a variety of answers and they all came up incorrect. I guess I don't understand difference quotient because I feel like it's pointless and a slow way to do the problem. That being said........Sat, 05 Sep 2026 18:40:08 -0400https://track.liberty.io/can-log-x-log-y-be-reduced.htmlhttps://track.liberty.io/can-log-x-log-y-be-reduced.htmlI'm currently taking Pre-Calc and am learning about logs. I know that $\log(xy) = \log(x) + \log(y)$, but can $\log(x)\log(y)$ be reduced further?...Sat, 05 Sep 2026 18:34:11 -0400https://track.liberty.io/gram-matrix-invertible-iff-set-of-vectors-linearly-independent.htmlhttps://track.liberty.io/gram-matrix-invertible-iff-set-of-vectors-linearly-independent.htmlGiven a set of vectors $v_1 \cdots v_n$, the $n\times n$ Gram matrix $G$ is defined as $G_{i,j}=v_i \cdot v_j$ Due to symmetry in the dot product, $G$ is Hermitian. I'm trying to remember why $|G......Sat, 05 Sep 2026 18:14:10 -0400https://track.liberty.io/int-0-infty-frac-ln-x-x-2-a-2-mathrm-d-x-evaluate-integral-duplicate.htmlhttps://track.liberty.io/int-0-infty-frac-ln-x-x-2-a-2-mathrm-d-x-evaluate-integral-duplicate.htmlAnyone remember the method for this? I think this should been done on the site $$\int_0^{\infty}\frac{\ln x}{x^2+a^2}\mathrm{d}x$$...Sat, 05 Sep 2026 18:11:26 -0400https://track.liberty.io/how-many-6-letter-words-that-have-either-exactly-2-vowels-or-4-vowels-.htmlhttps://track.liberty.io/how-many-6-letter-words-that-have-either-exactly-2-vowels-or-4-vowels-.htmlI considered two cases. Case 1 (2 vowels): Pick 2 vowels $\binom{5 + 2 -1}{2}$, then pick 4 consonants $\binom{21 + 4 -1}{4}$, then order them $6!$. Case 2 (4 vowels): Pick 4 vowels $\binom{5 + ......Sat, 05 Sep 2026 18:10:47 -0400https://track.liberty.io/adding-two-ieee754-floating-point-representations-and-interpreting-the.htmlhttps://track.liberty.io/adding-two-ieee754-floating-point-representations-and-interpreting-the.htmlThis isn't for any class or homework. As part of my personal study, I'm trying to better understand the IEEE754 representation of decimal floating-point numbers in binary. I'd like to add two numbe......Sat, 05 Sep 2026 18:08:07 -0400https://track.liberty.io/damping-constant-equation.htmlhttps://track.liberty.io/damping-constant-equation.htmlI've been looking through my textbooks and I've found a number of different equations - so I wanted to confirm with you which it is. What is the equation which determines the damping constant (gam......Sat, 05 Sep 2026 18:07:26 -0400https://track.liberty.io/proof-that-cos-pi-3-1-2.htmlhttps://track.liberty.io/proof-that-cos-pi-3-1-2.htmlIs it possible to prove that $\cos(\pi/3)=1/2$ without using any trigonometric identities? To be specific, my calculus-book simply states that with an angle of $\pi/3$ on the unit-circle the point......Sat, 05 Sep 2026 18:06:07 -0400https://track.liberty.io/what-does-the-greek-e-like-symbol-mean.htmlhttps://track.liberty.io/what-does-the-greek-e-like-symbol-mean.htmlI have a simple question: what does the following mean? $$\displaystyle\sum_{k=0}^n$$...Sat, 05 Sep 2026 17:56:05 -0400https://track.liberty.io/how-to-find-the-following-limit-algebraically.htmlhttps://track.liberty.io/how-to-find-the-following-limit-algebraically.htmlI've been trying to answer this for a while and I know it's a simple question relative to most questions that are posted here. $$ \lim_{x\rightarrow -2}\: \frac{x^4+5x^3+6x^2}{x^2(x+1)-4(x+1)} $$ ......Sat, 05 Sep 2026 17:50:15 -0400https://track.liberty.io/difference-between-sin-1-x-and-frac1-sin-x-closed.htmlhttps://track.liberty.io/difference-between-sin-1-x-and-frac1-sin-x-closed.htmlHow are arcsine and cosecant different mathematically if cosecant is $\frac{1}{\sin(x)}$ and arcsine is $\sin^{-1}(x)$ which is $\frac{1}{\sin(x)}$? I have tried to find an answer before but nobody ...Sat, 05 Sep 2026 17:25:53 -0400https://track.liberty.io/inverse-standard-normal-cdf.htmlhttps://track.liberty.io/inverse-standard-normal-cdf.htmlWe are dealing with a standard normal random variable. We have $$\Phi(c) = 0.8$$ where $c$ is just some arbitrary number and $\Phi$ is just the usual notation for the CDF of a standard normal ...Sat, 05 Sep 2026 17:22:01 -0400https://track.liberty.io/what-is-the-sum-of-the-prime-numbers-up-to-a-prime-number-n-duplicate.htmlhttps://track.liberty.io/what-is-the-sum-of-the-prime-numbers-up-to-a-prime-number-n-duplicate.htmlHow to find the sum of prime numbers up to a prime number $n$, that is for example: the sum of prime numbers up to 7 is: 2+3+5+7=17. So what is the formula for finding: $$\sum_{k=0}^n p_k=????,$$ ......Sat, 05 Sep 2026 17:17:55 -0400https://track.liberty.io/gradient-descent-calculation-split-in-2-steps.htmlhttps://track.liberty.io/gradient-descent-calculation-split-in-2-steps.htmlIn a machine learning course I am taking we have an assignment with a notebook (In linear models). So far I have calculated the cost and the sigmoid in a logistic regression and the next exercise w......Sat, 05 Sep 2026 17:17:42 -0400https://track.liberty.io/name-for-1-x.htmlhttps://track.liberty.io/name-for-1-x.htmlThe multiplicative inverse of $x$ is $\frac{1}{x}$, and the additive inverse of $x$ is $-x$, is there a similar term for $(1-x)$?...Sat, 05 Sep 2026 17:12:18 -0400https://track.liberty.io/is-tan-1-for-arctan-on-my-calculator-wrong.htmlhttps://track.liberty.io/is-tan-1-for-arctan-on-my-calculator-wrong.htmlOn my calculator $\tan^{-1}$ is used to calculate the $\arctan$, but $\tan^{-1}$ actually is $\cot$. $\cot$ and $\arctan$ are not the same thing though. Am I missing something or is the labeling of......Sat, 05 Sep 2026 17:01:34 -0400https://track.liberty.io/finding-an-angle-between-two-vectors-without-a-calculator.htmlhttps://track.liberty.io/finding-an-angle-between-two-vectors-without-a-calculator.htmlI have an exercise where I have to find the angle between 2 vectors without using the calculator. I know the formula which gives you the cosine of the angle between the 2 vectors, but I don't know ......Sat, 05 Sep 2026 17:00:07 -0400https://track.liberty.io/adjoint-of-a-compact-operator-between-hilbert-space-is-compact.htmlhttps://track.liberty.io/adjoint-of-a-compact-operator-between-hilbert-space-is-compact.htmlLet's $H$ be a Hilbert space and $T:H \rightarrow H$ a compact map. Show that the adjoint operator $T^*$ is also compact. I've been thinking a lot about this problem and this is what I've done: I......Sat, 05 Sep 2026 16:46:19 -0400https://track.liberty.io/transforming-a-matrix-to-diagonal-matrix.htmlhttps://track.liberty.io/transforming-a-matrix-to-diagonal-matrix.htmlShow that the matrix $$A = \begin{bmatrix}a&h\\h&b\end{bmatrix} ,\quad a \ne b$$ is transformed to diagonal matrix $D = P^{-1}AP$, where $$P = \begin{bmatrix}\cos x& -\sin x\\\sin x&amp......Sat, 05 Sep 2026 16:42:26 -0400https://track.liberty.io/how-do-you-simplify-this-square-root-of-sum-sqrt-7-4-sqrt3.htmlhttps://track.liberty.io/how-do-you-simplify-this-square-root-of-sum-sqrt-7-4-sqrt3.htmlI came around this expression when solving a problem. $$\sqrt{7+4\sqrt{3}}$$ WolframAlpha says it equals $2+\sqrt{3}$. We can confirm it like this $$\left(2+\sqrt{3}\right)^2 \;=\; 4+4\sqrt{3} ......Sat, 05 Sep 2026 16:40:06 -0400https://track.liberty.io/what-exactly-are-fractals.htmlhttps://track.liberty.io/what-exactly-are-fractals.htmlI have always been amazed by things like the Mandelbrot set. I share the view of most that it and the Koch snowflake are absolutely beautiful. I decided to get a deeper more mathematical knowledge of ...Sat, 05 Sep 2026 16:36:55 -0400https://track.liberty.io/surface-areas-using-cylindrical-shells.htmlhttps://track.liberty.io/surface-areas-using-cylindrical-shells.htmlIf I try to find the surface area of any solid by using cylindrical slices, I'm getting wrong answer. I'm taking this as the formula, $$S = \int_a^b2\pi y dx$$ where $y$ = height ($2\pi y$ = ...Sat, 05 Sep 2026 16:33:53 -0400https://track.liberty.io/difference-between-simplex-and-simplicial-complex.htmlhttps://track.liberty.io/difference-between-simplex-and-simplicial-complex.htmlFirst I know the definition of simplex intuitively as follows, Simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. And the defintion of simplicial co......Sat, 05 Sep 2026 16:20:54 -0400https://track.liberty.io/calculating-distance-in-n-dimensions.htmlhttps://track.liberty.io/calculating-distance-in-n-dimensions.htmlDoes this method work up to any spacial dimension? (5d, 10d etc) http://freespace.virgin.net/hugo.elias/routines/r_dist.htm p.s. If I'm getting this all fundamentally wrong somehow, please say....Sat, 05 Sep 2026 16:13:22 -0400https://track.liberty.io/volumes-with-cross-sections-circle-base-and-square-cross-section.htmlhttps://track.liberty.io/volumes-with-cross-sections-circle-base-and-square-cross-section.htmlThe base of a solid is a circle centered at the origin with a radius r, and every plane section perpendicular to a diameter is a square. What is the volume of the solid? So this would usually be ea......Sat, 05 Sep 2026 16:07:39 -0400https://track.liberty.io/equation-log-2x-216-x.htmlhttps://track.liberty.io/equation-log-2x-216-x.htmlSolve the equation $\log_{2x}216=x$ where $x$ is real. My attempt, By trial and error $x=3$ as $6^3=216$ Is there any systematic way to solve it?...Sat, 05 Sep 2026 15:56:16 -0400https://track.liberty.io/improbable-vs-impossible.htmlhttps://track.liberty.io/improbable-vs-impossible.htmlI was wondering how mathematics in general or any of its sub fields e.g.statistics, probability, define the words Improbable and Impossible. I get their English meaning, that something is impossi......Sat, 05 Sep 2026 15:47:04 -0400https://track.liberty.io/finding-solution-to-a-differential-equation-given-the-initial-conditio.htmlhttps://track.liberty.io/finding-solution-to-a-differential-equation-given-the-initial-conditio.htmlGiven $dy/dx = 2y/(2x+1)$ and the initial condition $y(0)=e$ and $x>-1/2$, find the solution y. I was able to separate the variables and integrate both sides respectively. I then tried to use the ...Sat, 05 Sep 2026 15:46:19 -0400https://track.liberty.io/proof-of-exponential-function-integral.htmlhttps://track.liberty.io/proof-of-exponential-function-integral.htmlI fully apologize if this is a duplicate, but I really can’t find the answer to this online here. But a proof of the rule $$\int a^x dx= \frac{a^x}{\ln(a)}+C$$ would be really helpful. I need it ......Sat, 05 Sep 2026 15:41:59 -0400https://track.liberty.io/how-do-i-calculate-dice-probability.htmlhttps://track.liberty.io/how-do-i-calculate-dice-probability.htmlI am trying to learn how to calculate dice probability, but all I can find on the net are the odds of rolling a given number in a single roll. An example of what I am looking of is rolling a 4+ on ......Sat, 05 Sep 2026 15:34:35 -0400https://track.liberty.io/finding-eigenvectors-of-a-3x3-matrix-2.htmlhttps://track.liberty.io/finding-eigenvectors-of-a-3x3-matrix-2.htmlI am trying to find the eigenvectors of the following 3x3 matrix: \begin{bmatrix} -9 & 4 & 4 \\ -8 & 3 & 4 \\ -16 & 8 & 7 \\ \end{bmatrix} ......Sat, 05 Sep 2026 15:27:51 -0400https://track.liberty.io/sketching-the-graph-of-x-f-y.htmlhttps://track.liberty.io/sketching-the-graph-of-x-f-y.htmlIn order to sketch a graph of $x=f(y)$ when we know the graph of $y=f(x)$ We can exchange the axes of coordinates (and sketch the same graph of $y=f(x)$ on the other axis) I want use this method to ...Sat, 05 Sep 2026 15:25:57 -0400https://track.liberty.io/question-about-logic-riddle-patch-of-lily-pads-that-doubles-every-day.htmlhttps://track.liberty.io/question-about-logic-riddle-patch-of-lily-pads-that-doubles-every-day.htmlI wasn't sure if the Math SE is the right SE to post this question, but after reading this question at the Meta SE, I think this SE is the right place. In a video on YouTube about logic riddles, o......Sat, 05 Sep 2026 15:09:01 -0400https://track.liberty.io/how-to-find-the-orthogonal-complement-of-a-subspace.htmlhttps://track.liberty.io/how-to-find-the-orthogonal-complement-of-a-subspace.htmlI am trying to understand how to find the orthogonal complement of a subspace $M$ of a vector space $V$. From my understanding, $M^\perp$ is also a subspace of $V$ where all its vectors are ...Sat, 05 Sep 2026 15:04:00 -0400https://track.liberty.io/intuition-behind-convergence-and-consistency.htmlhttps://track.liberty.io/intuition-behind-convergence-and-consistency.htmlWhat is the definition of consistency? I have seen a proof that shows a finite difference scheme is consistent, where they basically plug a true solution $u(t)$ into a finite difference scheme, and ...Sat, 05 Sep 2026 14:47:34 -0400https://track.liberty.io/questions-tagged-linear-algebra.htmlhttps://track.liberty.io/questions-tagged-linear-algebra.htmlQ&A for people studying math at any level and professionals in related fields...Sat, 05 Sep 2026 14:42:09 -0400https://track.liberty.io/discrete-vs-continuous-vs-random-variables.htmlhttps://track.liberty.io/discrete-vs-continuous-vs-random-variables.htmlCan someone provide me a clear explanation (Perhaps with examples) about the difference between the following mathematical three concepts in probability: Discrete Variable. Continuous Variable. Ra......Sat, 05 Sep 2026 14:39:07 -0400https://track.liberty.io/extreme-value-theorem-proof-help.htmlhttps://track.liberty.io/extreme-value-theorem-proof-help.htmlExtreme Value Theorem: If $f$ is a continuous function on an interval [a,b], then $f$ attains its maximum and minimum values on [a,b]. Proof from my book: Since $f$ is continuous, then $f$ has ......Sat, 05 Sep 2026 14:29:39 -0400https://track.liberty.io/online-videos-on-measure-theory.htmlhttps://track.liberty.io/online-videos-on-measure-theory.htmlI am looking for lecture videos on Measure Theory. I am not able to find any. Looking forward for your response....Sat, 05 Sep 2026 14:29:34 -0400https://track.liberty.io/why-is-taylor-series-expansion-for-1-1-x-valid-only-for-x-in-1-1.htmlhttps://track.liberty.io/why-is-taylor-series-expansion-for-1-1-x-valid-only-for-x-in-1-1.htmlAfter finding an expansion of $$ \frac{1}{1-x} = 1 + x + x^2 + x^3 + \ldots$$ a quick test of various values for $x$ reveals that this expansion is not valid for $\forall x \in \mathbb{R}-\{1\}$. ......Sat, 05 Sep 2026 14:29:16 -0400https://track.liberty.io/evaluating-int-4-4-int-0-sqrt-16-x-2-int-0-16-x-2-y-2-sqrt-x-2-y-2-dz-.htmlhttps://track.liberty.io/evaluating-int-4-4-int-0-sqrt-16-x-2-int-0-16-x-2-y-2-sqrt-x-2-y-2-dz-.htmlQuestion: Evaluate the given triple integral with cylindrical coordinates: $$\int_{-4} ^4\int _0 ^{\sqrt{16-x^2}} \int _0 ^{16-x^2-y^2} \sqrt{x^2 + y^2}\,dz\,dy\,dx$$ My solution (attempt): Upon ...Sat, 05 Sep 2026 14:22:08 -0400https://track.liberty.io/what-do-eigenvalues-have-to-do-with-pictures.htmlhttps://track.liberty.io/what-do-eigenvalues-have-to-do-with-pictures.htmlI am trying to write a program that will perform OCR on a mobile phone, and I recently encountered this article : Can someone explain this to me ?...Sat, 05 Sep 2026 14:20:19 -0400https://track.liberty.io/equality-of-pairs-of-parallel-lines.htmlhttps://track.liberty.io/equality-of-pairs-of-parallel-lines.htmlIn the picture above two sets of parallel lines are given: $l, m$ and $p, q$. when can we say these pairs are the same? If the distance or altitude are same then can we say they are equal?...Sat, 05 Sep 2026 14:05:36 -0400https://track.liberty.io/how-to-represent-the-square-root-of-90-as-a-fraction.htmlhttps://track.liberty.io/how-to-represent-the-square-root-of-90-as-a-fraction.htmlHow to represent the square root of 90 as a fraction? I can usually do this with a calculator but it's not wanting to play nice. I need it to find the distance between two points but, unfortunatel......Sat, 05 Sep 2026 14:03:34 -0400https://track.liberty.io/what-is-the-invariant-in-tensors.htmlhttps://track.liberty.io/what-is-the-invariant-in-tensors.htmlIt is being said that the most important property of tensors is that they are invariant under basis changes. What exactly does stay invariant in tensors? Can you please explain this invarianc......Sat, 05 Sep 2026 14:01:18 -0400https://track.liberty.io/splitting-integral-into-two-parts.htmlhttps://track.liberty.io/splitting-integral-into-two-parts.htmlThis question might seem very simple, but I can't seem to figure it out. Suppose I have an integral over a square region. I was wondering in which case it would be incorrect to split the integral i......Sat, 05 Sep 2026 13:52:57 -0400https://track.liberty.io/inverse-matrices-for-linear-equations-with-infinite-solutions.htmlhttps://track.liberty.io/inverse-matrices-for-linear-equations-with-infinite-solutions.htmlGiven a linear system $\begin{cases}2x - 3y &= 7\\x + 5y &= -3\end{cases}$ This system can be expressed as a matrix-vector product $$\underbrace{\begin{bmatrix}2 & -3\\1 & 5\end{b......Sat, 05 Sep 2026 13:44:45 -0400https://track.liberty.io/exponential-equations-with-fractions.htmlhttps://track.liberty.io/exponential-equations-with-fractions.htmlI have had some issues with the following two equations: $$ \frac{3^{n-2}}{9^{1-n}}=9$$ $$\frac{5^{3n-3}}{25^{n-3}}=125$$ If anyone could work them out step by step that would be awesome. I k......Sat, 05 Sep 2026 13:42:33 -0400https://track.liberty.io/what-is-the-autocorrelation-of-a-squared-gaussian-process.htmlhttps://track.liberty.io/what-is-the-autocorrelation-of-a-squared-gaussian-process.htmlSuppose $ {X_t; t \in R} $ is a wss, zero mean Gaussian random process with autocorrelation function $ R_X( \tau) ; \tau \in R$ and power spectral density $S_X(\omega); \omega \in R$. If w define......Sat, 05 Sep 2026 13:42:26 -0400https://track.liberty.io/height-radius-ratio-for-maximum-volume-cylinder-of-given-surface-area.htmlhttps://track.liberty.io/height-radius-ratio-for-maximum-volume-cylinder-of-given-surface-area.htmlI am a bit confused by this problem I have encountered: A right circular cylindrical container with a closed top is to be constructed with a fixed surface area. Find the ratio of the height to the ...Sat, 05 Sep 2026 13:38:41 -0400https://track.liberty.io/function-or-relation-y-4-8x-2.htmlhttps://track.liberty.io/function-or-relation-y-4-8x-2.htmlThe question under debate: Is $y^4=8x^2$ a function or a relation? We disagree. I say, it's a relation because when I substitute in points I get two outputs for most inputs, and thus it's not a ...Sat, 05 Sep 2026 13:22:13 -0400https://track.liberty.io/why-the-domain-of-the-cube-root-function-are-all-the-real-numbers.htmlhttps://track.liberty.io/why-the-domain-of-the-cube-root-function-are-all-the-real-numbers.htmlsince it can also be written as x^(1/3) and therefore 1/(x^3) and this would not make sense for x=0 because of the division with 0. So why is 0 in the domain?...Sat, 05 Sep 2026 13:19:37 -0400https://track.liberty.io/understanding-multiplicity.htmlhttps://track.liberty.io/understanding-multiplicity.htmlin my books of algebra it talks about polynomial functions and their zeros Multiplicity and x-Intercepts If r is a zero of even multiplicity, then the graph touches the x-axis and turns arou......Sat, 05 Sep 2026 13:04:48 -0400https://track.liberty.io/why-are-fractions-with-a-negative-denominator-valid.htmlhttps://track.liberty.io/why-are-fractions-with-a-negative-denominator-valid.htmlWhenever in a fraction, there is $0$ in the denominator, the fraction becomes $\infty$ or indeterminate. But why do we consider those fractions valid that have some negative numbers in the denomina......Sat, 05 Sep 2026 12:54:39 -0400https://track.liberty.io/angular-radius-of-a-sphere.htmlhttps://track.liberty.io/angular-radius-of-a-sphere.htmlGiven a sphere with radius $r$ about a point $c$, what's the apparent angular radius $\alpha$ of that sphere from point $P$? In other words, if $\vec{o} = c - P$, what's the maximum angle another v......Sat, 05 Sep 2026 12:50:30 -0400https://track.liberty.io/how-to-compute-the-inverse-of-a-matrix-in-matlab.htmlhttps://track.liberty.io/how-to-compute-the-inverse-of-a-matrix-in-matlab.htmlI have a matrix $$X=\int_{0}^{2}\begin{bmatrix}P_1^2&-P_1P_2\\ -P_2P_1& P_2^2 \end{bmatrix}ds+\int_{0}^{1}(2-s)^{-0.5}(1-s)^{0.5}\begin{bmatrix}P_1Q_2& P_1Q_1\\-P_2Q_2& -P_2Q_1\end{...Sat, 05 Sep 2026 12:42:41 -0400https://track.liberty.io/is-the-category-of-discrete-categories-equivalent-to-the-category-of-s.htmlhttps://track.liberty.io/is-the-category-of-discrete-categories-equivalent-to-the-category-of-s.htmlA discrete category is one whose only morphisms are identity morphisms. The category of discrete categories would have as objects the discrete categories and as morphisms the functors between those ...Sat, 05 Sep 2026 12:38:59 -0400https://track.liberty.io/how-do-you-find-the-closest-square-number-to-another-number-without-us.htmlhttps://track.liberty.io/how-do-you-find-the-closest-square-number-to-another-number-without-us.htmlSay we try to find the closest square number to 26. we already know the closest square number is $25$. However, how do I calculate out 25? Because, if I try to prime factorize it like so: $\......Sat, 05 Sep 2026 12:22:53 -0400https://track.liberty.io/what-is-the-terminology-for-lemma-of-lemma.htmlhttps://track.liberty.io/what-is-the-terminology-for-lemma-of-lemma.htmlLet's say I need to prove a main theorem, to prove which I need three lemmas. Thus in writing the structure is as follows: Lemma 1 Proof Lemma 2 Proof Lemma 3 Proof Theorem Proof But if when prov......Sat, 05 Sep 2026 12:19:35 -0400https://track.liberty.io/determining-the-number-of-solutions-of-a-system-of-linear-inequalities.htmlhttps://track.liberty.io/determining-the-number-of-solutions-of-a-system-of-linear-inequalities.htmlI want to determine the number of solutions of a system of linear inequalities, and I was wondering if there was a simple way to to that. I know that linear programming is often used to check whether ...Sat, 05 Sep 2026 12:17:28 -0400https://track.liberty.io/continuous-version-of-a-poisson-r-v.htmlhttps://track.liberty.io/continuous-version-of-a-poisson-r-v.htmlI am wondering if there is a continuous version of a Poisson random variable, that has the following two features: 1) Has a CDF that agrees with the discrete Poisson distribution on the integers, ......Sat, 05 Sep 2026 12:08:17 -0400https://track.liberty.io/can-the-value-of-sin-theta-be-greater-than-1-closed.htmlhttps://track.liberty.io/can-the-value-of-sin-theta-be-greater-than-1-closed.htmlFor example, is it possible for sin theta to be 1.06 or 1.2 under any circumstances? Or is it possible for it to be exactly 1?...Sat, 05 Sep 2026 11:57:03 -0400https://track.liberty.io/why-is-a-nullary-operation-a-special-element-usually-0-or-1.htmlhttps://track.liberty.io/why-is-a-nullary-operation-a-special-element-usually-0-or-1.htmlDoes a nullary operation mean an operation not taking any argument? Then why is a nullary operation a special element, usually 0 or 1, in an algebraic structure? Thanks!...Sat, 05 Sep 2026 11:44:59 -0400https://track.liberty.io/calculating-percentage-gain-on-investment-with-regular-deposits.htmlhttps://track.liberty.io/calculating-percentage-gain-on-investment-with-regular-deposits.htmlOn 1st of January I had a total of 120 dollars invested in a scheme. I made further deposits (as per table below) and have a final figure of $620 on June 10th. How do I work out my percentage gain ......Sat, 05 Sep 2026 11:36:55 -0400https://track.liberty.io/is-my-understanding-of-quotient-rings-correct.htmlhttps://track.liberty.io/is-my-understanding-of-quotient-rings-correct.htmlAmidst all the rigorous constructions of quotient rings involving equivalence relations and ideals, I feel that I have finally grasped what a quotient ring is. I have applied this intuition to a few ...Sat, 05 Sep 2026 11:36:44 -0400https://track.liberty.io/proving-a-relation-is-anti-symmetric-and-transitive.htmlhttps://track.liberty.io/proving-a-relation-is-anti-symmetric-and-transitive.html$P$ is a binary relation. $P ⊆ \mathbb{R}^2$. $P = \{(x,y): y = |x|\}$. As I understand for relation to be transitive: $(a,b) \in P$ and $(b,c) \in P$ then $(a,c \in P)$ for this particular relati......Sat, 05 Sep 2026 11:29:56 -0400https://track.liberty.io/synthetic-division-of-polynomials-by-factor-of-the-form-ax-b.htmlhttps://track.liberty.io/synthetic-division-of-polynomials-by-factor-of-the-form-ax-b.htmlFind the quotient and remainder when $6x^4-11x^3+5x^2-7x+9$ is divided by $(2x-3)$. I expressed the divisor $(2x-3)$ as $2(x-\frac{3}{2})$ and conducted synthetic division by $\frac{3}{2}$ and obta......Sat, 05 Sep 2026 11:29:21 -0400https://track.liberty.io/how-to-tell-if-a-linear-system-is-consistent.htmlhttps://track.liberty.io/how-to-tell-if-a-linear-system-is-consistent.htmlSo I have a list of equations and have made it into REF which gives me $$\left[\begin{matrix}1 & -1 & 1 \\ 0 & 1 & -1 \\ 0 & 0 & 0\end{matrix}\middle|\begin{matrix}1 \\ 2k-......Sat, 05 Sep 2026 11:28:18 -0400https://track.liberty.io/trying-to-understand-implied-population-in-statistics.htmlhttps://track.liberty.io/trying-to-understand-implied-population-in-statistics.htmlThis is from the 12th edition of "Understandable Statistics" from Cengage Learning. Chapter 1, Section 1, Question 8: What is the average miles per gallon (mpg) for all new hybrid small c......Sat, 05 Sep 2026 11:27:15 -0400https://track.liberty.io/find-the-midpoint-between-two-points-on-the-circle.htmlhttps://track.liberty.io/find-the-midpoint-between-two-points-on-the-circle.htmlI want to place a new point in the middle of the two points which are on the circle outline (Arc). I have the coordinates $(x,y)$ of the center of the circle, the two red points and the radius of the ...Sat, 05 Sep 2026 11:21:54 -0400https://track.liberty.io/game-theory-problem-3x3-matrix-pure-and-mixed-strategies.htmlhttps://track.liberty.io/game-theory-problem-3x3-matrix-pure-and-mixed-strategies.htmlFrom the following matrix: $$ \left( \begin{array}{c|ccc} & L & C & R\\ \hline T &3,0& 1,1 &4,2\\ M &3,4& 1,2 &2,3\\ B &1,3& 0,......Sat, 05 Sep 2026 11:13:32 -0400https://track.liberty.io/how-find-equilibrant.htmlhttps://track.liberty.io/how-find-equilibrant.htmlIf $F_1$ has magnitude 30 N and $F_2$ has magnitude 40 N, what is the exact magnitude and direction of the resultant vector? I have 10 $10 \sqrt37$ and 53 degrees. The next question is What is the ...Sat, 05 Sep 2026 11:11:27 -0400https://track.liberty.io/what-is-prime-factorisation-of-polynomials.htmlhttps://track.liberty.io/what-is-prime-factorisation-of-polynomials.htmlI have the following question: Find the prime factorization in $\mathbb{Z}[x]$ of $x^3 - 1, x^4 - 1, x^6 - 1$ and $x^{12} - 1$. You will need to check the irreducibility in $\mathbb{Z}[x]$, of t......Sat, 05 Sep 2026 11:01:52 -0400https://track.liberty.io/confusion-regarding-interval-on-which-a-function-is-increasing.htmlhttps://track.liberty.io/confusion-regarding-interval-on-which-a-function-is-increasing.htmlThe question is as follows: If the function $f(x)=\cos x$ is strictly increasing on the open interval $(0,\pi)$, where will it be increasing ? The answer to this question is $[0,\pi]$. I am a li......Sat, 05 Sep 2026 10:58:36 -0400https://track.liberty.io/use-implicit-differentiation-to-find-an-equation-of-the-tangent-line-t.htmlhttps://track.liberty.io/use-implicit-differentiation-to-find-an-equation-of-the-tangent-line-t.htmlUse implicit differentiation to find an equation of the tangent line to the curve $$x^2+xy+y^2=1$$ at $(1,1)$. I am not sure how I should work this out because the given point is not on the curve....Sat, 05 Sep 2026 10:57:32 -0400https://track.liberty.io/find-the-value-of-alpha-beta-for-the-equation-cos-alpha-cos-beta-cos-a.htmlhttps://track.liberty.io/find-the-value-of-alpha-beta-for-the-equation-cos-alpha-cos-beta-cos-a.htmlFind the value of $\alpha,\beta$ for the equation $\cos\alpha \cos\beta \cos(\alpha +\beta)=-\frac{1}{8}$ $\alpha>0$ & $\beta<\frac{\pi}{2}$ I get the following step after some substit......Sat, 05 Sep 2026 10:56:34 -0400https://track.liberty.io/arctan-3-2-doesn-039-t-give-expected-result.htmlhttps://track.liberty.io/arctan-3-2-doesn-039-t-give-expected-result.htmlLet's say I want to find the angle measure (in degrees) such that $\tan(x) = -3/2$. It turns out that $x \approx 123.7$, and when I compute $\tan(123.7)$, I get $\approx -3/2$; so far so good. Ho......Sat, 05 Sep 2026 10:53:36 -0400