https://liberty.ioCurated star highlights with fresh energy.en-usSun, 06 Sep 2026 08:23:49 -0400https://liberty.io/proving-pascal-039-s-rule-n-choose-r-n-1-choose-r-1-n-1-choose-r-when-.htmlhttps://liberty.io/proving-pascal-039-s-rule-n-choose-r-n-1-choose-r-1-n-1-choose-r-when-.htmlI'm trying to prove that ${n \choose r}$ is equal to ${{n-1} \choose {r-1}}+{{n-1} \choose r}$ when $1\leq r\leq n$. I suppose I could use the counting rules in probability, perhaps combination= $......Sun, 06 Sep 2026 12:12:54 -0400https://liberty.io/notation-and-the-name-choice-for-meet-and-join-in-order-theory.htmlhttps://liberty.io/notation-and-the-name-choice-for-meet-and-join-in-order-theory.htmlI have two simple questions: From where do the names meet and join come from? I don't see any intuition between those names in context of order theory. From where does the notation come? I have to......Sun, 06 Sep 2026 12:12:00 -0400https://liberty.io/frames-and-co-frames.htmlhttps://liberty.io/frames-and-co-frames.htmlI'm trying to write a precise definition of a frame and co-frame. I have that a frame is a basis for a tangent space $T_pM$ and the co-frame is the dual basis (basis for the co-tangent space $T^{\a......Sun, 06 Sep 2026 11:58:45 -0400https://liberty.io/list-of-all-256-binary-combinations-of-8-digits-closed.htmlhttps://liberty.io/list-of-all-256-binary-combinations-of-8-digits-closed.htmli don't have any coding/java etc software. how can i view all the 256 binary combinatios from 8 digits? can someone tell me where can i find them in a list (any site/online generator etc) or copy t......Sun, 06 Sep 2026 11:51:41 -0400https://liberty.io/properties-of-subtraction-and-division.htmlhttps://liberty.io/properties-of-subtraction-and-division.htmlI'm (re)learning math and was looking for the properties of the most basic math operations. For addition and multiplication these are very easy to find (e.g. https://www.aaamath.com/g74a_px1.htm and ...Sun, 06 Sep 2026 11:50:35 -0400https://liberty.io/why-do-two-logs-with-negative-values-not-multiply-to-make-a-positive-l.htmlhttps://liberty.io/why-do-two-logs-with-negative-values-not-multiply-to-make-a-positive-l.htmlI was pondering a question while exploring logarithms, and logarithmic functions. $$ \log(-x) + \log(-x) = \log x^2 $$ Why is this considered incorrect? Seeing as: $$ \log(3) + \log(3) = \log3^2......Sun, 06 Sep 2026 11:38:29 -0400https://liberty.io/total-differential-of-a-function-f-x-y.htmlhttps://liberty.io/total-differential-of-a-function-f-x-y.htmlI found this definition of total variation of a function $F(x, y)$ \begin{eqnarray} dF(x, y) &\equiv& F(x+dx, y+dy) -F(x, y)\\ & = & [F(x+dx, y+dy) - F(x, y+dy)] + [F(x, y+dy) - F(x......Sun, 06 Sep 2026 11:32:29 -0400https://liberty.io/how-do-you-explain-what-a-hyperbolic-pde-is.htmlhttps://liberty.io/how-do-you-explain-what-a-hyperbolic-pde-is.htmlA friend of mine who just finished an introductory calculus class saw my books on fluid mechanics and asked me what a different PDEs look like. I explained elliptical and parabolic by giving exampl......Sun, 06 Sep 2026 11:28:29 -0400https://liberty.io/probability-of-an-event-happening-over-period-of-time-with-100-certain.htmlhttps://liberty.io/probability-of-an-event-happening-over-period-of-time-with-100-certain.htmlIf I have a certain event that is $100 \%$ to happen exactly once in $10$ days, how can I calculate the probability each day? I cannot be $10 \%$ per day, since by the end of $10$ days, there still......Sun, 06 Sep 2026 11:27:15 -0400https://liberty.io/is-there-a-3-dimensional-matrix-by-matrix-product.htmlhttps://liberty.io/is-there-a-3-dimensional-matrix-by-matrix-product.htmlIs it possible to multiply A[m,n,k] by B[p,q,r]? Does the regular matrix product have generalized form? I would appreciate it if you could help me to find out some tutorials online or mathematical......Sun, 06 Sep 2026 11:21:29 -0400https://liberty.io/solving-the-roots-of-redlich-kwong-equation.htmlhttps://liberty.io/solving-the-roots-of-redlich-kwong-equation.htmlGood evening everyone. After studying some equations of state, I've read about the mathematical steps formulated to model some. In the particular case of Van der Waals, where $$P=\frac{RT}{v-b}-\f......Sun, 06 Sep 2026 11:17:58 -0400https://liberty.io/why-does-the-exactness-of-a-koszul-complex-require-commutativity.htmlhttps://liberty.io/why-does-the-exactness-of-a-koszul-complex-require-commutativity.htmlMost references on Koszul complexes seem to assume that the elements $x_1,\ldots, x_n$ live in a commutative ring or are central. It appears to me that the proof also works provided the weaker assu......Sun, 06 Sep 2026 11:17:42 -0400https://liberty.io/user-michael-zieve-mathematics-stack-exchange.htmlhttps://liberty.io/user-michael-zieve-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Sun, 06 Sep 2026 11:17:33 -0400https://liberty.io/diagonalize-real-symmetric-matrix.htmlhttps://liberty.io/diagonalize-real-symmetric-matrix.htmlI believe that for any real symmetric matrix $A$, I am always be able to write in the eigen decomposition form $$A = P\Lambda P^T \tag{1}$$ whatever the rank of $A$. (see Real symmetric matrix is......Sun, 06 Sep 2026 11:13:19 -0400https://liberty.io/what-is-more-probable-one-50-percent-chance-or-ten-5-percent-chances.htmlhttps://liberty.io/what-is-more-probable-one-50-percent-chance-or-ten-5-percent-chances.htmlWhat is more probable; a single 50% chance at success, or ten 5% chances at success? If the former, how many additional 5% chances are required in order for it to be as likely to succeed using mul......Sun, 06 Sep 2026 11:09:18 -0400https://liberty.io/taking-derivatives-of-a-power-series.htmlhttps://liberty.io/taking-derivatives-of-a-power-series.htmlI've been working on understanding power series, and came across a problem asking for the derivative of a certain power series and for the derivative to be a summation with a lower limit equal to z......Sun, 06 Sep 2026 11:07:28 -0400https://liberty.io/a-simple-determinant-calculations.htmlhttps://liberty.io/a-simple-determinant-calculations.htmlI calculate the claim of the theorem below for a couple of times, always I get $\sum_{i=1}^{\infty} c_i z^{L+i}$ and not $\sum_{i=1}^{\infty} c_i z^{L+M+i}$ as the coefficient of RHS of last determ......Sun, 06 Sep 2026 11:04:38 -0400https://liberty.io/isometry-is-not-surjective.htmlhttps://liberty.io/isometry-is-not-surjective.htmlAccording to the definition I am using, an isometry is a mapping $f:X \rightarrow Y$ between two metric spaces $(X,d_{X})$ and $(Y,d_{Y})$: $$ d_{Y}(f(a),f(b)) = d_{X}(a,b) $$ for all $a,b \in X $ I ...Sun, 06 Sep 2026 11:01:23 -0400https://liberty.io/calculate-improvement-percentage-based-on-time-spent.htmlhttps://liberty.io/calculate-improvement-percentage-based-on-time-spent.htmlI am doing a task that it takes 2 hours. After some changes the same task takes 10 minutes. What formula I can use to calculate the improvement percentage between these 2 values?...Sun, 06 Sep 2026 11:01:10 -0400https://liberty.io/is-distance-between-two-graphs-defined-somehow.htmlhttps://liberty.io/is-distance-between-two-graphs-defined-somehow.htmlIf the two graphs are isomorphic, then their distance is zero. And this distance increases, if vertices or edges are added or removed to/from one of the graphs. Does this "distance" have a special......Sun, 06 Sep 2026 11:01:05 -0400https://liberty.io/limits-of-functions-and-left-hand-right-hand-limit.htmlhttps://liberty.io/limits-of-functions-and-left-hand-right-hand-limit.htmlUse the $\epsilon$,$\delta$-definition of limits to show that $\lim\limits_{x\to a} f(x)=L$ if and only if $\lim\limits_{x \to a^+} f(x)=\lim\limits_{x\to a^-}f(x)=L$ I started by defining the le......Sun, 06 Sep 2026 10:58:01 -0400https://liberty.io/teaching-myself-differential-topology-and-differential-geometry.htmlhttps://liberty.io/teaching-myself-differential-topology-and-differential-geometry.htmlI have a hazy notion of some stuff in differential geometry and a better, but still not quite rigorous understanding of basics of differential topology. I have decided to fix this lacuna once for ......Sun, 06 Sep 2026 10:51:04 -0400https://liberty.io/calculate-on-which-side-of-a-straight-line-is-a-given-point-located.htmlhttps://liberty.io/calculate-on-which-side-of-a-straight-line-is-a-given-point-located.htmlI am a programmer without really good knowledge in math. :/ So I have to write an algorithm that changes the color of pixel(dot) P to opposite if it's on left side of the straigt line in coordinate ...Sun, 06 Sep 2026 10:37:22 -0400https://liberty.io/what-is-the-size-of-a-set-of-sets-of-the-empty-set.htmlhttps://liberty.io/what-is-the-size-of-a-set-of-sets-of-the-empty-set.htmlWhat is the size of a set of sets of the empty set {{}, {{}}, {{{}}}}? I am not sure if the empty set, in this case, can be considered as a set and make the size 3 or if it is 2 or 0. thanks...Sun, 06 Sep 2026 10:25:32 -0400https://liberty.io/in-how-many-ways-can-we-arrange-4-letters-of-the-word-engine.htmlhttps://liberty.io/in-how-many-ways-can-we-arrange-4-letters-of-the-word-engine.htmlI need to know a combinatoric solution to this problem, with Generating functions in book, gives us 102. It might be a very simple problem, but Im very confused with this. Would be very nice of ......Sun, 06 Sep 2026 10:23:57 -0400https://liberty.io/solve-system-of-equations-using-calculus-linear-algebra.htmlhttps://liberty.io/solve-system-of-equations-using-calculus-linear-algebra.htmlI'm having trouble with this problem. I'm not sure how to solve it the "calculus way". My professor gave us the formula "$b = c + Dt$" to work with (y = mx + b), and then dealt with partial derivat......Sun, 06 Sep 2026 10:11:51 -0400https://liberty.io/why-is-only-a-square-matrix-invertible.htmlhttps://liberty.io/why-is-only-a-square-matrix-invertible.htmlCan anyone give a very simple proof (or explanation) as to why only square matrix can possibly be invertible?...Sun, 06 Sep 2026 10:02:55 -0400https://liberty.io/moment-generating-function-of-a-gamma-distribution.htmlhttps://liberty.io/moment-generating-function-of-a-gamma-distribution.htmlIf I have a variable $X$ that has a gamma distribution with parameters $s$ and $\lambda$, what is its momment generating function. I know that it is $\int_0^\infty e^{tx}\frac{1}{\Gamma(s)}\lambda......Sun, 06 Sep 2026 09:59:01 -0400https://liberty.io/proving-the-product-of-two-non-singular-matrices-is-also-non-singular.htmlhttps://liberty.io/proving-the-product-of-two-non-singular-matrices-is-also-non-singular.htmlI am having trouble with a proof for linear algebra. Could somebody explain to me how to prove that if $A$ and $B$ are both $n\times n$ non singular matrices, that their product $AB$ is also non si......Sun, 06 Sep 2026 09:55:32 -0400https://liberty.io/how-to-start-proving-a-times-b-times-c-ne-a-times-b-times-c.htmlhttps://liberty.io/how-to-start-proving-a-times-b-times-c-ne-a-times-b-times-c.htmlThis is a problem from Discrete Mathematics and its Applications: Explain why $A \times B \times C$ and $(A \times B) \times C$ are not the same. I understand the process behind the Cartesian pro......Sun, 06 Sep 2026 09:48:13 -0400https://liberty.io/factorization-of-an-invertible-symmetric-matrix.htmlhttps://liberty.io/factorization-of-an-invertible-symmetric-matrix.htmlGiven any invertible symmetric matrix: $A=\begin{bmatrix}a&b&c\\ b&d&e\\ c&e&f\end{bmatrix}$ over the complex number, Can be it factored as $A=T^\top T$? where $T^\top$ ......Sun, 06 Sep 2026 09:32:23 -0400https://liberty.io/what-is-the-measure-of-measuredangle-x-in-the-figure-below.htmlhttps://liberty.io/what-is-the-measure-of-measuredangle-x-in-the-figure-below.htmlFor reference: In the figure shown P and Q are points of tangency and $ \overset{\LARGE{\frown}}{BC}+\overset{\LARGE{\frown}}{FE}=130^o. $ Calculate "x". My progress: I think the starting ...Sun, 06 Sep 2026 09:32:09 -0400https://liberty.io/proving-that-a-point-is-a-global-minimum.htmlhttps://liberty.io/proving-that-a-point-is-a-global-minimum.htmlI have this cost function $$ f(x,y,z)=0.5 \left(x^2+y^2+0.1(z^2)\right)+0.55z $$ such that $$x,y,z>0$$ $$x+y+z=1$$ I was given the point $(0.5,0.5,0)$ as a local minimum. How to I go along to pr......Sun, 06 Sep 2026 09:26:03 -0400https://liberty.io/where-does-the-factor-y-2-1-come-from.htmlhttps://liberty.io/where-does-the-factor-y-2-1-come-from.htmlI keep looking and writing out the problem, but I don't know where $(y^2 + 1)$ comes from. I get $y^2(y-3)$ as the numerator instead, but I can't tell if I'm having a brain fart and am missing some......Sun, 06 Sep 2026 09:25:46 -0400https://liberty.io/how-is-the-pythagorean-theorem-related-to-the-equation-of-a-circle.htmlhttps://liberty.io/how-is-the-pythagorean-theorem-related-to-the-equation-of-a-circle.htmlMy teacher wants me to figure out how the Pythagorean Theorem and the Equation of a Circle are related. I can't figure this out because I view them as being two completely different things. I under......Sun, 06 Sep 2026 09:16:06 -0400https://liberty.io/how-do-i-write-3-6-11-18-27-38-in-set-builder-notation.htmlhttps://liberty.io/how-do-i-write-3-6-11-18-27-38-in-set-builder-notation.htmlFor the set $\{3,6,11,18,27,38,...\}$, the $(n+1)^\text{th}$ term, which I'll call $a_{n+1}$, is: $$a_{n+1} = a_n + 2n+1$$ How can I write this set in set-builder notation? My best guess doesn't seem ...Sun, 06 Sep 2026 09:07:39 -0400https://liberty.io/find-the-domain-of-the-function-it-039-s-a-square-root-polynomial.htmlhttps://liberty.io/find-the-domain-of-the-function-it-039-s-a-square-root-polynomial.htmlFind the domain of the function: $$f(x)= \sqrt{x^2 - 4x - 45}$$ I'm just guessing here; how about if I square everything and then put it in the graphing calculator? Thanks, Lauri...Sun, 06 Sep 2026 08:54:12 -0400https://liberty.io/what-is-the-difference-between-normal-and-perpendicular.htmlhttps://liberty.io/what-is-the-difference-between-normal-and-perpendicular.htmlWhat is the difference when a line is said to be normal to another and a line is said to be perpendicular to other?...Sun, 06 Sep 2026 08:52:07 -0400https://liberty.io/using-vector-identities-and-the-divergence-theorem-to-derive-simplifie.htmlhttps://liberty.io/using-vector-identities-and-the-divergence-theorem-to-derive-simplifie.htmlI am trying to follow along a review paper regarding the derivation of the Cahn-Hilliard paper. The question I have is related to the following section of the paper: I have been trying to follow a......Sun, 06 Sep 2026 08:42:20 -0400https://liberty.io/integral-power-rule-step-by-step.htmlhttps://liberty.io/integral-power-rule-step-by-step.htmlReal quick, two things: I'm sorry if my notation or terminology is incorrect, and I know what I'm asking isn't strictly necessary for my studies, but writing something out step by step helps me to ...Sun, 06 Sep 2026 08:41:16 -0400https://liberty.io/simple-application-of-prime-number-theorem.htmlhttps://liberty.io/simple-application-of-prime-number-theorem.htmlI was wondering if someone could possibly help me verify the following: For sufficiently large $n$ there is always a prime between $n- \sqrt{n}$ and $n$. I am not sure if this is true or not. If......Sun, 06 Sep 2026 08:26:51 -0400https://liberty.io/intuitive-explanation-for-why-the-15-and-290-theorems-should-be-true.htmlhttps://liberty.io/intuitive-explanation-for-why-the-15-and-290-theorems-should-be-true.htmlI recently read about the remarkable 15 and 290 theorems, the second of which states that any positive definite integral quadratic form represents all positive integers iff it represents all positive ...Sun, 06 Sep 2026 08:20:56 -0400https://liberty.io/what-039-s-the-geometrical-interpretation-of-the-magnitude-of-gradient.htmlhttps://liberty.io/what-039-s-the-geometrical-interpretation-of-the-magnitude-of-gradient.htmlIn the following picture, the author of the Field and Wave Electromagnetics shows the geometrical meaning of the direction of the gradient. That is, only by following the direction of the normal ve......Sun, 06 Sep 2026 08:19:43 -0400https://liberty.io/does-non-symmetric-positive-definite-matrix-have-positive-eigenvalues.htmlhttps://liberty.io/does-non-symmetric-positive-definite-matrix-have-positive-eigenvalues.htmlI found out that there exist positive definite matrices that are non-symmetric, and I know that symmetric positive definite matrices have positive eigenvalues. Does this hold for non-symmetric mat......Sun, 06 Sep 2026 08:18:29 -0400https://liberty.io/numerator-vs-denominator-vs-nominator.htmlhttps://liberty.io/numerator-vs-denominator-vs-nominator.htmlWhat is appropriate usage of "numerator", "denominator", and "nominator" to refer to parts of a fraction? I'm posting this question and answer here because I had little luck finding a clear answer ...Sun, 06 Sep 2026 08:18:02 -0400https://liberty.io/understanding-the-formula-for-curvature.htmlhttps://liberty.io/understanding-the-formula-for-curvature.htmlIn this khan academy article, they discuss how can define curvature as, $$ \bigg\| \frac{dT}{dS} \bigg\| = \kappa$$ In the post they write, "However, we don't want differences in the rate at ......Sun, 06 Sep 2026 08:16:03 -0400https://liberty.io/why-do-we-say-almost-surely-in-probability-theory.htmlhttps://liberty.io/why-do-we-say-almost-surely-in-probability-theory.htmlI recently asked a question and got a great answer that involved proving that "X is almost surely one of the roots of P". I know (now) that "almost surely" means "with probability 1", but I've never ...Sun, 06 Sep 2026 07:51:59 -0400https://liberty.io/define-strong-solution.htmlhttps://liberty.io/define-strong-solution.htmlLet $\Omega \subset \mathbb{R}^n$ be open bounded. We find a function $u:\overline\Omega \rightarrow R$ satisfying $-\Delta u + u = f$ in $\Omega$, $u=0$ on $\partial\Omega$ ---(*) Define strong ...Sun, 06 Sep 2026 07:49:59 -0400https://liberty.io/definite-integral-by-u-substitution-1-u-2-u-given.htmlhttps://liberty.io/definite-integral-by-u-substitution-1-u-2-u-given.html$$\int_{-3}^0 \frac{-8x}{(2x^2+3)^2}dx; u=2x^2+3$$ I need help solving this integral -- I'm completely bewildered. I've attempted it many times already and I don't know what I am doing wrong in my......Sun, 06 Sep 2026 07:43:28 -0400https://liberty.io/drawing-an-ace-card-from-a-deck-of-52-on-the-nth-try.htmlhttps://liberty.io/drawing-an-ace-card-from-a-deck-of-52-on-the-nth-try.htmlI have the following question: We draw one card at a time without replacement from the top of a shuffled standard poker deck and stop when we draw an ace. Let $X$ be the number of cards we have dr......Sun, 06 Sep 2026 07:42:22 -0400https://liberty.io/is-it-viable-to-ask-in-an-infinite-set-about-the-cardinality.htmlhttps://liberty.io/is-it-viable-to-ask-in-an-infinite-set-about-the-cardinality.htmlCan you ask given an infinite set about its cardinality? Does an infinite set have a cardinality? So, for example, what would be the cardinality of $+\infty$?...Sun, 06 Sep 2026 07:38:34 -0400https://liberty.io/computational-complexity-of-computing-the-determinant.htmlhttps://liberty.io/computational-complexity-of-computing-the-determinant.htmlThe formula for the determinant of an $n$ by $ n$ matrix given by expansion of minors involves $n!$ terms. As such, computing the determinant of a given matrix of with integer entries via expansion......Sun, 06 Sep 2026 07:34:28 -0400https://liberty.io/how-to-solve-23-2020-2020-bmod-37-please-see-the-body-of-the-question.htmlhttps://liberty.io/how-to-solve-23-2020-2020-bmod-37-please-see-the-body-of-the-question.htmlHow to solve $$23^{{2020}^{2020}} \mod 37.$$ Below given is my understanding of trying to solve the problem. From $$x^{p-1} = 1 \mod p$$ I deduce that $$23^{2020} \mod 37$$ would be $$23^{56.36+4}......Sun, 06 Sep 2026 07:32:11 -0400https://liberty.io/books-like-terence-tao-039-s-an-introduction-to-measure-theory.htmlhttps://liberty.io/books-like-terence-tao-039-s-an-introduction-to-measure-theory.htmlThis is an attempt to get a list going of books in mathematics (algebra, topology, geometry etc.) that in the spirit of Terence Tao's An Introduction to Measure Theory: According to my viewpoint, the ...Sun, 06 Sep 2026 07:29:43 -0400https://liberty.io/determine-a-in-the-system-such-that-the-system-is-consistent.htmlhttps://liberty.io/determine-a-in-the-system-such-that-the-system-is-consistent.htmlThe system is: $x_1 + 2x_2 - x_3 = 2, $ $2x_1 - x_2 + x_3 = 1 $, $-x_1 + 4x_2 -2x_3 = a $ To begin solving the system, I did Row 2 - 2(Row 1). Row 3 + Row 1, then Row 3 + Row 2. This left me with \......Sun, 06 Sep 2026 07:28:53 -0400https://liberty.io/finding-derivative-of-three-variables.htmlhttps://liberty.io/finding-derivative-of-three-variables.htmlConsider a box with dimensions x, y, and z. x is changing at a rate of 1 m/s, y at -2 m/s and z at 1 m/s. Find the rate that the volume, surface area and diagonal length ($s = \sqrt{x^2+y^2+z^2}$) ......Sun, 06 Sep 2026 07:19:14 -0400https://liberty.io/how-do-i-find-out-the-value-of-p-for-sin-p-cos-p-0.htmlhttps://liberty.io/how-do-i-find-out-the-value-of-p-for-sin-p-cos-p-0.htmlI tried doing this by putting $$\frac{ \sin p}{\sqrt2} +\frac{ \cos p}{\sqrt2} = 0 $$ which implies $$ \sin(p + π/4) = \sin 0 $$ which implies $p+ π/4 = nπ $. Now according to the question I'm solv......Sun, 06 Sep 2026 07:06:48 -0400https://liberty.io/why-does-unary-minus-operator-sometimes-take-precedence-over-exponenti.htmlhttps://liberty.io/why-does-unary-minus-operator-sometimes-take-precedence-over-exponenti.htmlHow should I evaluate 2*-2^3? Which one of these two is the correct one? 2*((-2)^3) 2*(-(2^3)) I was wondering what was the correct operator precedence. Everyone was taught at school to evaluate ...Sun, 06 Sep 2026 07:02:18 -0400https://liberty.io/an-intuitive-approach-to-the-jordan-normal-form.htmlhttps://liberty.io/an-intuitive-approach-to-the-jordan-normal-form.htmlI want to understand the meaning behind the Jordan Normal form, as I think this is crucial for a mathematician. As far as I understand this, the idea is to get the closest representation of an arbi......Sun, 06 Sep 2026 07:00:42 -0400https://liberty.io/how-many-ways-can-seven-people-sit-around-a-circular-table.htmlhttps://liberty.io/how-many-ways-can-seven-people-sit-around-a-circular-table.htmlHow many ways seven people can sit around a circular table? For first, I thought it was $7!$ (the number of ways of sitting in seven chairs), but the answer is $(7-1)!$. I don't understand how si......Sun, 06 Sep 2026 06:58:44 -0400https://liberty.io/how-to-find-the-number-of-arbitrary-constants-in-solution-of-a-given-d.htmlhttps://liberty.io/how-to-find-the-number-of-arbitrary-constants-in-solution-of-a-given-d.htmlHow to find the number of arbitrary constants in solution of a given differential equation ?...Sun, 06 Sep 2026 06:52:38 -0400https://liberty.io/redudant-condition-for-wiener-process-continuity.htmlhttps://liberty.io/redudant-condition-for-wiener-process-continuity.htmlIn my book, a Wiener process is defined as a process which has, amongst others, the conditions $W_t - W_s \sim \mathcal{N}(0,t - s), t > s,$ and that the process has a "continuous trajectory". ......Sun, 06 Sep 2026 06:51:01 -0400https://liberty.io/how-to-compute-the-volume-of-intersection-between-two-hyperspheres.htmlhttps://liberty.io/how-to-compute-the-volume-of-intersection-between-two-hyperspheres.htmlLet's say I have two n-spheres and I've no prior knowledge about the spheres (such as one of the sphere might be inside the other one) and I need to compute the volume of the intersection of the two ...Sun, 06 Sep 2026 06:44:10 -0400https://liberty.io/matrix-addition-can-a-3x2-matrix-a-2x3-matrix-be-added-together.htmlhttps://liberty.io/matrix-addition-can-a-3x2-matrix-a-2x3-matrix-be-added-together.htmlI have two matrices; $$ 1) \left( \begin{matrix} 3 & -2 \\ 1 & 2 \\ 0 & -3\\ \end{matrix} \right) $$ And $$ 2) \left( \begin{matrix} -2 & 1 & 3 \\ 4 & ......Sun, 06 Sep 2026 06:39:23 -0400https://liberty.io/how-to-calclulate-1-17-mod-60-closed.htmlhttps://liberty.io/how-to-calclulate-1-17-mod-60-closed.htmlhow can I calculate $ d = 17^{-1} (\text{mod} ~ 60) $ ? I was reading this article and then I wrote down this steps: 60 = 3 * 17 + 9 17 = 1 * 9 + 8 9 = 1 * 8 + 1 In the end (using EEA) I get x = ......Sun, 06 Sep 2026 06:31:44 -0400https://liberty.io/how-to-solve-the-inequality-of-x-2-2x-4-0.htmlhttps://liberty.io/how-to-solve-the-inequality-of-x-2-2x-4-0.htmlI'm reading Calculus with Analytic Geometry and in a problem in the first chapter (page 6) Solve the following inequalities $x^2 + 2x + 4 > 0$ Apparently, $x^2 + 2x + 4$ has no solution in real ...Sun, 06 Sep 2026 06:30:41 -0400https://liberty.io/how-to-simplify-sqrt-3-2.htmlhttps://liberty.io/how-to-simplify-sqrt-3-2.htmlMy textbook says to do the following steps to simplify the $\sqrt{3/2}$: $\sqrt{3/2}=\sqrt{((3/2)\times (2/2))} =\sqrt{6/4} = \sqrt{6}/\sqrt{4}=\sqrt{6}/2.$ However, if the fourth step is valid (1. ...Sun, 06 Sep 2026 06:21:07 -0400https://liberty.io/what-is-prime-gaps-relationship-with-number-6.htmlhttps://liberty.io/what-is-prime-gaps-relationship-with-number-6.htmlOut of the 78499 prime number under 1 million. There are 32821 prime gaps (difference between two consecutive prime numbers) of a multiple 6. A bar chart of differences and frequency of occurrence ......Sun, 06 Sep 2026 06:12:45 -0400https://liberty.io/find-the-number-of-real-values-of-x-such-that-x-2-10000-x-10000x-is-th.htmlhttps://liberty.io/find-the-number-of-real-values-of-x-such-that-x-2-10000-x-10000x-is-th.htmlFind the number of real values of $x$ such that $x^2 + 10000[x] = 10000x$ ($[]$ is the floor function) What I Tried: If $x$ is an integer then :- $$x^2 = 0$$ $$\rightarrow x = 0$$ So $x = 0$ is th......Sun, 06 Sep 2026 06:10:58 -0400https://liberty.io/why-do-we-say-pi-is-the-ratio-of-the-circumference-to-the-diameter-and.htmlhttps://liberty.io/why-do-we-say-pi-is-the-ratio-of-the-circumference-to-the-diameter-and.htmlEverywhere I see it written, $\pi$ is described as "the ratio of the circumference to the diameter". I know $\pi = C/d$, but the ratio $3.14...:1$ is $3.14$... diameters to $1$ circumference. So ...Sun, 06 Sep 2026 06:06:38 -0400https://liberty.io/cross-products-and-unit-circle.htmlhttps://liberty.io/cross-products-and-unit-circle.htmlI am first year maths student at university. I am currently studying cross products. For this question, I managed to solve the first 2 parts but am struggling with the general question at the e......Sun, 06 Sep 2026 06:00:25 -0400https://liberty.io/how-to-find-the-height-of-a-rectangle-given-its-area-and-the-differenc.htmlhttps://liberty.io/how-to-find-the-height-of-a-rectangle-given-its-area-and-the-differenc.htmlThe rectangular base is $4.50 \text{ cm}$ shorter than the height and the rectangle has a surface area of $135 \text{ cm}$. Solve the rectangle height with an equation. I know and feel that this is ...Sun, 06 Sep 2026 05:55:09 -0400https://liberty.io/velocity-increase-while-acceleration-decrease.htmlhttps://liberty.io/velocity-increase-while-acceleration-decrease.htmlI'm having trouble understanding velocity and acceleration. Is it possible for velocity to be increasing while acceleration decreases and vice versa?...Sun, 06 Sep 2026 05:46:25 -0400https://liberty.io/find-the-exact-value-of-the-trigonometric-function-sin-7-pi-6.htmlhttps://liberty.io/find-the-exact-value-of-the-trigonometric-function-sin-7-pi-6.htmlI am finding it a little difficult to solve this problem. The reference angle for $\sin 7\pi/6$ is sin 30 degrees (I think) which is sine 1/2. But that is not the answer. How do I sove this problem?...Sun, 06 Sep 2026 05:28:51 -0400https://liberty.io/integration-by-parts-when-can-you-not-use-the-table-method-why.htmlhttps://liberty.io/integration-by-parts-when-can-you-not-use-the-table-method-why.htmlI'm learning about integration by parts, primarily from Stewart's text (7th edition). In a supplemental book I have it brings up something called the Table Method. I really find this method appealing ...Sun, 06 Sep 2026 05:27:01 -0400https://liberty.io/why-do-we-have-to-calculate-1-binomcdf-for-p-x-11.htmlhttps://liberty.io/why-do-we-have-to-calculate-1-binomcdf-for-p-x-11.htmlWhy do we have to calculate 1-binomcdf for P(X>11)? Can't we just calculate binomcdf with left and right bound like the normalcdf?...Sun, 06 Sep 2026 05:26:40 -0400https://liberty.io/the-infinitely-nested-radicals-problem-and-ramanujan-039-s-wondrous-fo.htmlhttps://liberty.io/the-infinitely-nested-radicals-problem-and-ramanujan-039-s-wondrous-fo.htmlIn mathematics, a nested-radical is any expression where a radical (or root sign) is nested inside another radical, eg. $\sqrt{2 + \sqrt{3}}$. By extension, an Infinitely nested radical (aka, a Con......Sun, 06 Sep 2026 05:23:01 -0400https://liberty.io/writing-parametric-curve-in-cartesian-form-closed.htmlhttps://liberty.io/writing-parametric-curve-in-cartesian-form-closed.html$$x=4 \ln(t), \qquad y=2t$$ Write it in Cartesian form, giving $y$ explicitly in terms of $x$....Sun, 06 Sep 2026 05:22:06 -0400https://liberty.io/what-do-mean-and-what-does-it-mean-if-it-is-used-in-an-equation.htmlhttps://liberty.io/what-do-mean-and-what-does-it-mean-if-it-is-used-in-an-equation.htmlWhat do the square bracket symbols mean? Are they what I hear are "sets"? And when it is in an equation, how is it interpreted? Here is an example: $$\dfrac{dy}{dx}[2x^{2}+y(x)^{2}]=50x+2y(dy/dx)......Sun, 06 Sep 2026 05:20:53 -0400https://liberty.io/evaluate-the-angle-x-in-terms-of-angles-and-y-and-z.htmlhttps://liberty.io/evaluate-the-angle-x-in-terms-of-angles-and-y-and-z.htmlI need help with this problem and problems like this. I know that to solve this problem I have to find which angles are the same and etc, but how do I know that. How can I see that two angles are the ...Sun, 06 Sep 2026 05:11:41 -0400https://liberty.io/are-the-product-and-sum-of-two-householder-matrices-also-householder-m.htmlhttps://liberty.io/are-the-product-and-sum-of-two-householder-matrices-also-householder-m.htmlSuppose we have two Householder matrices $$H_i=I_n-2v_iv_i^T$$ where $\|v_i\|_2 = 1$ for $i \in \{1,2\}$. Is it true that $H_1 H_2$ is also a Householder matrix? Is it true that $H_1 + H_2$ is......Sun, 06 Sep 2026 05:09:35 -0400https://liberty.io/evaluate-the-indefinite-integral.htmlhttps://liberty.io/evaluate-the-indefinite-integral.html$ I = \int (x^2 + 2x)\cos(x) dx $ Integration by Parts, choose $u$: $$\begin{align} u &= \cos(x) \\ dv &= (x^2 + 2x)dx \\ du &= -\sin(x) \\ v &= \frac{1}{3}x^3 + x^2 \end{align} $$ ...Sun, 06 Sep 2026 04:53:37 -0400https://liberty.io/what-is-the-meaning-of-a-space-time-white.htmlhttps://liberty.io/what-is-the-meaning-of-a-space-time-white.htmlFor reference this is from page 27 here Consider the stochastic heat equation $$ \begin{cases} \dfrac{\partial u}{\partial t}(x,t)=\dfrac{\partial ^2 u}{\partial x^2}(x,t) +f(u(x,t))\dot{W}(x,t) &a......Sun, 06 Sep 2026 04:53:34 -0400https://liberty.io/ramanujan-nagell-theorem-proof-question.htmlhttps://liberty.io/ramanujan-nagell-theorem-proof-question.htmlI'm currently working through Stewart and Tall's Algebraic Number Theory. In particular, section 4.9 of this book provides a proof of the Ramanujan-Nagell Theorem, which states that the only integer ...Sun, 06 Sep 2026 04:53:27 -0400https://liberty.io/how-to-find-the-inverse-of-a-rational-function-when-one-or-both-of-the.htmlhttps://liberty.io/how-to-find-the-inverse-of-a-rational-function-when-one-or-both-of-the.htmlIf rational functions have the form $f(x)=\frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials, how do you find the inverse of $f(x)$ when $P(x)$ and/or $Q(x)$ are polynomials of degree $2$ or ...Sun, 06 Sep 2026 04:48:54 -0400https://liberty.io/about-graph-symmetric-with-respect-to-the-origin.htmlhttps://liberty.io/about-graph-symmetric-with-respect-to-the-origin.htmlWhy it is called a reflection about the y-axis followed by a reflection about the x-axis?...Sun, 06 Sep 2026 04:46:56 -0400https://liberty.io/how-many-factors-of-10-in-100-duplicate.htmlhttps://liberty.io/how-many-factors-of-10-in-100-duplicate.htmlPossible Duplicate: Highest power of a prime $p$ dividing $N!$ How many factors of 10 are there in $100!$ (IIT Question)? Is it 26,25,24 or any other value Please tell how you have done it...Sun, 06 Sep 2026 04:45:59 -0400https://liberty.io/geometry-prove-that-these-3-points-don-039-t-lie-on-the-same-line-clos.htmlhttps://liberty.io/geometry-prove-that-these-3-points-don-039-t-lie-on-the-same-line-clos.htmlRectangular coordinate system in the plane The Points are given - $A(5,0) B(1,3) C(-2,0)$. Prove that these $3$ points don't lie on the same line and find the surface of the triangle $ABC$....Sun, 06 Sep 2026 04:42:17 -0400https://liberty.io/combinatorial-proofs-having-a-difficult-time-understanding-how-to-writ.htmlhttps://liberty.io/combinatorial-proofs-having-a-difficult-time-understanding-how-to-writ.htmlCan someone explain how combinatorial proofs work? I've included an example questions that's been giving me a hard time. Any insight on the topic would be great. $$\sum_{k=1}^{n}k{n \choose k} =......Sun, 06 Sep 2026 04:36:37 -0400https://liberty.io/what-is-the-geometric-meaning-of-singular-matrix.htmlhttps://liberty.io/what-is-the-geometric-meaning-of-singular-matrix.htmlCould anyone help explain what is the geometric meaning of singular matrix ? What's the difference between singular and non-singular matrix ? I know the definition, but couldn't understand it very ......Sun, 06 Sep 2026 04:33:11 -0400https://liberty.io/what-is-the-correct-way-to-read-f-circ-g.htmlhttps://liberty.io/what-is-the-correct-way-to-read-f-circ-g.htmlLet $f : X \to Y$ and $g : Y \to Z$ be functions. We define the composition $g \circ f : X \to Z$ by $g \circ f(x) = g(f(x))$ for each $x \in X$. I have also heard the composition read out like th......Sun, 06 Sep 2026 04:32:23 -0400https://liberty.io/what-is-the-definition-of-a-positive-integer.htmlhttps://liberty.io/what-is-the-definition-of-a-positive-integer.htmlI am reading the book "The Number-System of Algebra (2nd edition)". At the starting of page-4 the author writes: A positive integer is a symbol for the number of things in a group of distinct ...Sun, 06 Sep 2026 04:27:08 -0400https://liberty.io/should-i-use-ds-or-dr.htmlhttps://liberty.io/should-i-use-ds-or-dr.htmlThe question states "Integrate $f(x, y, z) = x + \sqrt y -z^2$ over the patch $C = C_1 + C_2$ and they proceed to give $C_1$ and $C_2$ as $r_1(t)$ and $r_1(t)$. I've been dealing with 2 forms of line ...Sun, 06 Sep 2026 04:25:50 -0400https://liberty.io/gre-math-subject-test-closed.htmlhttps://liberty.io/gre-math-subject-test-closed.htmlI am studying for GRE Math. I am looking for specific tips. What types of questions usually come up? Does anyone know any tricks (e.g. integration tricks) that might be helpful? Which theorems are ...Sun, 06 Sep 2026 04:17:09 -0400https://liberty.io/why-is-x-5-6x-3-irreducible.htmlhttps://liberty.io/why-is-x-5-6x-3-irreducible.htmlIn my math course, it is written that since over $\mathbb{F}_5[X]$, $P$ has no root in $\mathbb{F}_{25}$, it is irreducible over $\mathbb{Q}[X]$ by Gauss Lemma. I don't really understand… If we p......Sun, 06 Sep 2026 04:14:58 -0400https://liberty.io/the-introduction-rule-for-implication.htmlhttps://liberty.io/the-introduction-rule-for-implication.htmlThe introduction rule for implication says that you can prove a statement of the form $P \implies Q$ by assuming $P$ and deducing $Q$. (Btw. I don't know why it's called an introduction rule - you ......Sun, 06 Sep 2026 04:14:02 -0400https://liberty.io/indefinite-integral-of-absolute-value-of-x-is-there-a-closed-form-solu.htmlhttps://liberty.io/indefinite-integral-of-absolute-value-of-x-is-there-a-closed-form-solu.htmlBeen searching the net for awhile and everything just comes back about doing the definite integral. So just thought to ask here. Title says it all. Is there a closed form solution for the indefin......Sun, 06 Sep 2026 04:11:30 -0400https://liberty.io/what-is-null-quantification.htmlhttps://liberty.io/what-is-null-quantification.htmlI want to know the definition of null quantification and when and why we need to use it. In a book I was learning, it says: "establish rules for null quantification that we can use when a ...Sun, 06 Sep 2026 03:58:59 -0400https://liberty.io/what-exactly-happens-when-i-do-a-bit-rotation.htmlhttps://liberty.io/what-exactly-happens-when-i-do-a-bit-rotation.htmlI'm sitting in my office and having difficulties to get to know that exactly happens, when I do a bit rotation of a binary number. An example: I have the binary number 1110. By doing bit rotatio......Sun, 06 Sep 2026 03:56:15 -0400https://liberty.io/odds-of-drawing-an-ace-on-the-first-draw-or-a-two-on-the-second.htmlhttps://liberty.io/odds-of-drawing-an-ace-on-the-first-draw-or-a-two-on-the-second.htmlI'm working on a puzzle which describes a casino game where you draw 13 cards. You win if your first card is an ace, or your second is a two, or... all the way up to the 13th draw being a king. In ...Sun, 06 Sep 2026 03:51:52 -0400