https://liberty.ioCurated star highlights with fresh energy.en-usWed, 12 Aug 2026 18:23:45 -0400https://liberty.io/limits-of-integration-spherical-coordinates.htmlhttps://liberty.io/limits-of-integration-spherical-coordinates.htmlI have seen a lot of exercises where they solve a triple integral using spherical coordinates. But I'm confused about the limits that one should use. For example when they integrate over a sphere l......Wed, 12 Aug 2026 22:15:16 -0400https://liberty.io/number-of-triangles-sharing-all-vertices-but-no-sides-with-a-given-oct.htmlhttps://liberty.io/number-of-triangles-sharing-all-vertices-but-no-sides-with-a-given-oct.htmlThe number of triangles whose vertices are the the vertices of the vertices of an octagon but none of whose sides happen to come from the sides of the octagon. My Attempt: Let $\{A,B,C,D,E,F,G,H\}......Wed, 12 Aug 2026 22:11:08 -0400https://liberty.io/how-are-huffman-encoding-and-entropy-related.htmlhttps://liberty.io/how-are-huffman-encoding-and-entropy-related.htmlThe inherent unpredictability, or randomness, of a probability distribution can be measured by the extent to which it is possible to compress data drawn from that distribution. $$ \text{more compre......Wed, 12 Aug 2026 22:10:05 -0400https://liberty.io/what-does-noether-039-s-normalization-lemma-even-mean.htmlhttps://liberty.io/what-does-noether-039-s-normalization-lemma-even-mean.htmlNoether's Normalization lemma states that For any field $k$, and any finitely generated commutative $k$-algebra $A$, there exists a non-negative integer $d$ and algebraically independent element......Wed, 12 Aug 2026 22:02:17 -0400https://liberty.io/any-line-intersects-the-parabola-twice-in-projective-space.htmlhttps://liberty.io/any-line-intersects-the-parabola-twice-in-projective-space.htmlGiven the parabola $y-x^2=0$ in $\mathbb{C}^2$, I'm trying to show that in $\mathbb{P}^2$, any line intersects the parabola twice (up to multiplicity). This is simple enough for the "finite" ...Wed, 12 Aug 2026 22:00:16 -0400https://liberty.io/adding-two-subspaces.htmlhttps://liberty.io/adding-two-subspaces.htmlI have two subspaces: $$W_1 = \{(x, 3x) : x\in \Bbb R \}$$ and $$W_2 = \{(2x, 0): x\in \Bbb R \}$$ How do I get $W_1 + W_2$? I tried simply adding a sample vector from each, i.e. $$ (1, 3) + ......Wed, 12 Aug 2026 21:59:07 -0400https://liberty.io/solve-log-3-3x-2-log-9-4x-5-for-x.htmlhttps://liberty.io/solve-log-3-3x-2-log-9-4x-5-for-x.htmlSolve for $x$ $$ \log_3(3x + 2) = \log_9(4x + 5) $$ I changed the bases of the logs $$ \frac {\log_{10}(3x + 2)} {\log_{10}(3)} = \frac {\log_{10}(4x + 5)} {\log_{10}(9)} $$ Now I'm stuck, ......Wed, 12 Aug 2026 21:54:47 -0400https://liberty.io/fun-linear-algebra-problems.htmlhttps://liberty.io/fun-linear-algebra-problems.htmlI'm teaching a linear algebra course this term (using Lay's book) and would like some fun "challenge problems" to give the students. The problems that I am looking for should be be easy to state an......Wed, 12 Aug 2026 21:53:10 -0400https://liberty.io/deriving-the-hessian-from-the-limit-definition-of-the-derivative.htmlhttps://liberty.io/deriving-the-hessian-from-the-limit-definition-of-the-derivative.htmlCould someone possibly help me understand how I can derive the Hessian matrix of a twice-differentiable function $f$ defined on $\mathbb{R}^n$ using the limit definition of the second derivative. N......Wed, 12 Aug 2026 21:51:10 -0400https://liberty.io/non-isomorphism-graph-closed.htmlhttps://liberty.io/non-isomorphism-graph-closed.htmlIm confused what is non isomorphism graph cant post image so i upload it on tinypic Particulary with this example It is said, that this c4 graph on left side is non isomorphism graph. But this is......Wed, 12 Aug 2026 21:47:58 -0400https://liberty.io/to-what-refers-the-width-in-a-gaussian-function.htmlhttps://liberty.io/to-what-refers-the-width-in-a-gaussian-function.htmlI've plotted a dataset in SciDAVis and added the default Gaussian fit. SciDAVis used the following function: $$ f(x) = y_0 + A \cdot \frac{\sqrt{\frac{2}{\pi}}}{w} \cdot \exp{\left(-2 \cdot \left(......Wed, 12 Aug 2026 21:46:50 -0400https://liberty.io/mathbb-z-includes-zero-or-not.htmlhttps://liberty.io/mathbb-z-includes-zero-or-not.htmlDoes $\mathbb{Z}^{+}$ includes zero or not? I think that $0$ is not involved in the set of positive integers, but my book included zero in the set of positive integers in an answer....Wed, 12 Aug 2026 21:43:42 -0400https://liberty.io/projecting-an-angle-from-one-plane-to-another-plane.htmlhttps://liberty.io/projecting-an-angle-from-one-plane-to-another-plane.htmlConsider I have two intersecting planes with an angle ($\theta$). I have two intersecting vectors ($\vec a$ and $\vec b$) on one of the planes that make an angle ($\gamma$). If I project these two ...Wed, 12 Aug 2026 21:43:29 -0400https://liberty.io/find-function-corresponding-to-a-taylor-series.htmlhttps://liberty.io/find-function-corresponding-to-a-taylor-series.htmlI'm trying to find a function corresponding to the following Taylor series: $\sum_{n=2}^\infty\frac{x^n}{n(n-1)}$. I found it to converge if $-1\leq x \leq 1$, but I'm not sure how to find the func......Wed, 12 Aug 2026 21:42:33 -0400https://liberty.io/what-is-modern-analysis-and-abstract-analysis-closed.htmlhttps://liberty.io/what-is-modern-analysis-and-abstract-analysis-closed.htmlMostly I encounter textbooks named Analysis, Mathematical Analysis,... But recently, i encountered a book named A Course In Modern Analysis. Please explain the difference between Modern Analysis a......Wed, 12 Aug 2026 21:39:00 -0400https://liberty.io/evaluating-a-surface-integral-iint-a-ds.htmlhttps://liberty.io/evaluating-a-surface-integral-iint-a-ds.htmlSuppose $A=6z\hat i+(2x+y)\hat j-x\hat k$ .Evaluate $$\iint A.dS$$ Over the entire surface S of the region bounded by the cylinder $x^2+z^2=9,x=0,y=0,z=0$ and $y=8$.I split it into three surface 1.......Wed, 12 Aug 2026 21:36:19 -0400https://liberty.io/meaning-of-the-sign-of-a-complex-number-closed.htmlhttps://liberty.io/meaning-of-the-sign-of-a-complex-number-closed.htmlWhat is the meaning and the practical uses of the sign (signum function) of a complex number $z$ defined as $\frac{z}{|z|}$? Does it also extend to quaternions?...Wed, 12 Aug 2026 21:29:48 -0400https://liberty.io/from-natural-log-to-log-base-10.htmlhttps://liberty.io/from-natural-log-to-log-base-10.htmlThe constraints of this question is related to a programming problem, but I must get the math right in order for it to be applied to code. The actual problem is I need a function that evaluates to ......Wed, 12 Aug 2026 21:22:31 -0400https://liberty.io/is-the-support-of-a-borel-measure-measured-the-same-as-the-whole-space.htmlhttps://liberty.io/is-the-support-of-a-borel-measure-measured-the-same-as-the-whole-space.htmlWikipedia says Let (X, T) be a topological space. Let μ be a measure on the Borel σ-algebra on X. Then the support (or spectrum) of μ is defined to be the set of all points x in X for which ......Wed, 12 Aug 2026 21:18:53 -0400https://liberty.io/probability-of-the-union-of-3-events.htmlhttps://liberty.io/probability-of-the-union-of-3-events.htmlI need some clarification for why the probability of the union of three events is equal to the right side in the following: $$P(E\cup F\cup G)=P(E)+P(F)+P(G)-P(E\cap F)-P(E\cap G)-P(F\cap G)+P(E\c......Wed, 12 Aug 2026 21:18:36 -0400https://liberty.io/expected-value-of-y-ax-b.htmlhttps://liberty.io/expected-value-of-y-ax-b.htmlGiven a normal random variable X with parameters $\mu$ and $\sigma^2$, find the $E(Y)$ of $Y=aX + b$. So I started with $E(Y)=E(aX+b)=\frac{1}{\sqrt(2\pi)\sigma}\int_{-\infty}^{\infty}(ax+b)^{{-(......Wed, 12 Aug 2026 21:17:34 -0400https://liberty.io/graph-theory-properties-of-even-graph.htmlhttps://liberty.io/graph-theory-properties-of-even-graph.htmlThe question is that: show that each even graph can be decomposed into edge-disjoint cycles. What I think is: when drawing such graphs (randomly), I can produce two sets X and Y with vertices, in ......Wed, 12 Aug 2026 21:15:06 -0400https://liberty.io/the-difference-between-affine-set-and-affine-hull.htmlhttps://liberty.io/the-difference-between-affine-set-and-affine-hull.htmlAccording to the definition of affine hull and affine set. $$aff [C] = [\theta_1x_1+...+\theta_nx_n|x_1,...x_n \in C, \theta_1+...+\theta_n=1] $$ The data in affine hull is also in affine set. And ......Wed, 12 Aug 2026 21:06:50 -0400https://liberty.io/global-stability-vs-global-asymptotic-stability.htmlhttps://liberty.io/global-stability-vs-global-asymptotic-stability.htmlAfter trying to get my head around this for an embarrassingly long time, I think I need some help... We defined local (lyapunov) stability and asymptotic stability the following way: An equilibrium......Wed, 12 Aug 2026 21:03:38 -0400https://liberty.io/difference-between-the-principle-of-addition-the-principle-of-multipli.htmlhttps://liberty.io/difference-between-the-principle-of-addition-the-principle-of-multipli.htmlAccording to my book: 1.The fundamental principle of counting is used to count the number of possible ways in which a task can be done without actually counting manually. 2.Under the fundamental ...Wed, 12 Aug 2026 21:02:29 -0400https://liberty.io/choose-2-numbers-from-100-numbers.htmlhttps://liberty.io/choose-2-numbers-from-100-numbers.htmlIf I want to choose a set of 2 positive integers less than 100, is that a permutation or a combination? I can't understand does the order matter in this case or it does not?...Wed, 12 Aug 2026 20:59:51 -0400https://liberty.io/forming-a-basis-of-p3-r-from-a-set-s.htmlhttps://liberty.io/forming-a-basis-of-p3-r-from-a-set-s.htmlI seem to have a good understanding of spanning sets and linear independence which then becomes essential for understanding basis, but I am unsure how all this works for the field of polynomials. ......Wed, 12 Aug 2026 20:43:46 -0400https://liberty.io/limit-proof-of-e-x-x-n-duplicate.htmlhttps://liberty.io/limit-proof-of-e-x-x-n-duplicate.htmlI am wondering how to prove $$\lim_{x\to \infty} \frac{e^x}{x^n}=\infty$$ I was thinking of using L'Hospital's rule? But then not sure how to do the summation for doing L'Hospital's rule n times o......Wed, 12 Aug 2026 20:40:07 -0400https://liberty.io/what-is-the-intuitive-meaning-of-the-adjugate-matrix.htmlhttps://liberty.io/what-is-the-intuitive-meaning-of-the-adjugate-matrix.htmlThe definition of the adjugate matrix is easy to understand, but I have never seen it used for anything. What is the intuitive meaning of this matrix? Are there examples of applications which may ......Wed, 12 Aug 2026 20:39:21 -0400https://liberty.io/n-th-derivative-of-tanh.htmlhttps://liberty.io/n-th-derivative-of-tanh.htmlSuccessive derivative of $\tanh u$ can be expressed as polynomial functions of $\tanh u$: \begin{align} \frac{d}{du}\tanh u&=1-\tanh^2u\\ \frac{d^2}{du^2}\tanh u&=-2\tanh u\left(1-\tanh^2u\......Wed, 12 Aug 2026 20:38:10 -0400https://liberty.io/which-methods-can-be-used-to-evaluate-the-following-integral.htmlhttps://liberty.io/which-methods-can-be-used-to-evaluate-the-following-integral.htmlHow can I evaluate the following integral $$ \int_{0}^{\infty} x^{-1/2} \exp({-x/2})\ dx $$ I know the answer is $\sqrt{2\pi}$....Wed, 12 Aug 2026 20:35:48 -0400https://liberty.io/quotient-group-z-3z-equated-to-0-1-2.htmlhttps://liberty.io/quotient-group-z-3z-equated-to-0-1-2.htmlThis a followup question on the first answer to this post: Why the term and the concept of quotient group? In the first answer, Lahtonen says that for the quotient group Z/10Z, one can "equate 9 ......Wed, 12 Aug 2026 20:33:55 -0400https://liberty.io/deriving-the-power-series-for-cosine-using-basic-geometry.htmlhttps://liberty.io/deriving-the-power-series-for-cosine-using-basic-geometry.htmlLooking for the derivation of cosine lead to https://www.quora.com/How-do-I-calculate-cos-sine-etc-without-a-calculator and the MacLauren series. $$\cos(x)=1−\frac{x^2}{2!}+\frac{x^4}{4!}−\frac{x^......Wed, 12 Aug 2026 20:26:47 -0400https://liberty.io/reflexive-transitive-closure.htmlhttps://liberty.io/reflexive-transitive-closure.htmlThe problem I am working on is, "Show that a finite poset can be reconstructed from its covering relation. [Hint:Show that the poset is the reflexive transitive closure of its covering relation.]" I ...Wed, 12 Aug 2026 20:16:28 -0400https://liberty.io/prove-that-there-are-an-infinite-number-of-discontinuities-on-this-fun.htmlhttps://liberty.io/prove-that-there-are-an-infinite-number-of-discontinuities-on-this-fun.htmlShow the function $f:[0,1]\rightarrow\mathbb{R}$ given by $$ f(x)= \begin{cases} 1,&x=\frac{1}{n}\text{ for any positive integer $n$}\\ 0,&\text{otherwise} \end{cases} $$ has an infinite ......Wed, 12 Aug 2026 20:10:24 -0400https://liberty.io/explain-why-this-relation-has-a-reflexive-symmetric-antisymmetric-and-.htmlhttps://liberty.io/explain-why-this-relation-has-a-reflexive-symmetric-antisymmetric-and-.htmlLet S = {1, 2, 3} Let R = {(1,1),(3,3),(2,2)} So the answer is that it is reflexive, symmetric, antisymmetric, and transitive. I understand that it is reflexive, however I do not understand how it......Wed, 12 Aug 2026 19:59:40 -0400https://liberty.io/area-and-circumference-of-a-square.htmlhttps://liberty.io/area-and-circumference-of-a-square.htmlI've encountered this weird question: Show that the derivative of the area of a circle with respect to its radius is equal to its circumference. Show that this relationship holds between the area and ...Wed, 12 Aug 2026 19:56:10 -0400https://liberty.io/questions-tagged-legendre-polynomials.htmlhttps://liberty.io/questions-tagged-legendre-polynomials.htmlQ&A for people studying math at any level and professionals in related fields...Wed, 12 Aug 2026 19:52:42 -0400https://liberty.io/is-it-possible-to-cut-a-circle-and-make-smaller-circle.htmlhttps://liberty.io/is-it-possible-to-cut-a-circle-and-make-smaller-circle.htmlRecently I saw this meme: Although the pizza hasn't completely the shape of a circle, let's say we have a circle is it possible to cut a part of the circle like this in the image and make smaller ......Wed, 12 Aug 2026 19:39:39 -0400https://liberty.io/how-to-calculate-the-joint-probability-from-two-normal-distributions.htmlhttps://liberty.io/how-to-calculate-the-joint-probability-from-two-normal-distributions.htmlI have two random variables $X$ and $Y$ both normally distributed as $N(\mu, \sigma^2)$ (they have the same distribution). $X$ and $Y$ are dependent. They are defined from other random variables A......Wed, 12 Aug 2026 19:28:07 -0400https://liberty.io/a-nice-way-to-do-euler-039-s-method-on-a-calculator.htmlhttps://liberty.io/a-nice-way-to-do-euler-039-s-method-on-a-calculator.htmlAs part of the calculator paper for IB (International Baccalaureate), we may be asked to do Euler's method, for say, 10 iterations. While it is feasible to do with a calculator (slightly easier if......Wed, 12 Aug 2026 19:23:45 -0400https://liberty.io/why-is-l-0-norm-not-convex-closed.htmlhttps://liberty.io/why-is-l-0-norm-not-convex-closed.htmlI have this confusion in understanding the convexity of the $L_0$ norm. Why is $L_0$ norm not convex?...Wed, 12 Aug 2026 19:15:36 -0400https://liberty.io/difference-between-joint-probability-distribution-and-conditional-prob.htmlhttps://liberty.io/difference-between-joint-probability-distribution-and-conditional-prob.htmlCan someone explain to me the difference between joint probability distribution and conditional probability distribution?...Wed, 12 Aug 2026 19:10:27 -0400https://liberty.io/will-division-by-zero-be-defined-eventually-duplicate.htmlhttps://liberty.io/will-division-by-zero-be-defined-eventually-duplicate.htmlPossible Duplicate: Division by $0$ I've always been inclined to believe that x/0 = NaN is a placeholder for a character or constant that no one has created yet. I know assume that none of y......Wed, 12 Aug 2026 19:07:09 -0400https://liberty.io/proving-a-conditional-statement-by-induction.htmlhttps://liberty.io/proving-a-conditional-statement-by-induction.htmlI’m a university freshman and I saw this in the lecture slides of my Discrete Mathematics module. I do not know what the name of the theorem is, so I will just copy the proof to the best of my abil......Wed, 12 Aug 2026 19:06:27 -0400https://liberty.io/do-equal-angles-necessarily-mean-a-polygon-is-regular.htmlhttps://liberty.io/do-equal-angles-necessarily-mean-a-polygon-is-regular.htmlIn a polygon, if all the sides are equal, it doesn’t necessarily mean that the polygon is regular (eg. a rhombus). Is this also true for angles? Meaning can you draw a polygon whose interior angles......Wed, 12 Aug 2026 19:05:45 -0400https://liberty.io/should-the-sign-be-reversed-if-i-square-both-sides-of-an-inequality.htmlhttps://liberty.io/should-the-sign-be-reversed-if-i-square-both-sides-of-an-inequality.htmlLet us say I have the following: $$x>y$$ Now, I want to take the square of both sides. Should it result in $$x^2>y^2$$ or $$x^2<y^2$$ I suspect there is no way to give a general answer to ...Wed, 12 Aug 2026 19:05:31 -0400https://liberty.io/irreducible-polynomials-of-degree-3-and-5.htmlhttps://liberty.io/irreducible-polynomials-of-degree-3-and-5.htmlTwo part question: 1) There are three irreducible cubic polynomials $\mathbb F_2$ Aproach: Bruteforce, there are not as much cubic polynomials, just 8 of them: $x^3$ $x^3+1$ $x^3+x$ $x^3+x+1$......Wed, 12 Aug 2026 19:02:09 -0400https://liberty.io/a-two-envelopes-puzzle.htmlhttps://liberty.io/a-two-envelopes-puzzle.htmlThis is Problem 1.25 from Tsitsiklis, Bertsekas, Introduction to Probability, 2nd edition. You are handed two envelopes, and you know that each contains a positive integer dollar amount and th......Wed, 12 Aug 2026 19:00:16 -0400https://liberty.io/cdf-for-negative-binomial-distribution.htmlhttps://liberty.io/cdf-for-negative-binomial-distribution.htmlI am trying to show that the following statement is true. $$ \sum_{x = r}^{X}\binom{x-1}{r-1}p^r(1-p)^{x-r} = \sum_{x = r}^{X}\binom{X}{x}p^x(1-p)^{X-x} $$ Where $X$ and $r$ and $p$ are constants......Wed, 12 Aug 2026 18:52:06 -0400https://liberty.io/if-a-graph-has-a-hamilton-path-starting-at-every-vertex-must-it-contai.htmlhttps://liberty.io/if-a-graph-has-a-hamilton-path-starting-at-every-vertex-must-it-contai.htmlIf a graph $G$ has at least three vertices, and has a Hamilton Path starting at every vertex, must it contain a Hamilton Circuit? I have been struggling with this problem. It seems that because......Wed, 12 Aug 2026 18:41:39 -0400https://liberty.io/cumulative-distribution-function-of-a-variable-with-exponential-distri.htmlhttps://liberty.io/cumulative-distribution-function-of-a-variable-with-exponential-distri.htmlSuppose we have a variable $X$ that has an exponential distribution with a probability density function: $f(x) = 3e^{-3x}, x >0$ Then the cumulative distribution function is: $\int_{-\infty}^......Wed, 12 Aug 2026 18:37:30 -0400https://liberty.io/can-a-limit-of-an-integral-be-moved-inside-the-integral.htmlhttps://liberty.io/can-a-limit-of-an-integral-be-moved-inside-the-integral.htmlAfter coming across this question: How to verify this limit, I have the following question: When taking the limit of an integral, is it valid to move the limit inside the integral, providing the l......Wed, 12 Aug 2026 18:30:17 -0400https://liberty.io/why-can-you-mix-partial-derivatives-with-ordinary-derivatives-in-the-c.htmlhttps://liberty.io/why-can-you-mix-partial-derivatives-with-ordinary-derivatives-in-the-c.htmlThis question is a simplified version of this previous question asked by myself. The following is a short extract from a book I am reading: If $u=(x^2+2y)^2 + 4$ and $p=x^2 + 2y$ $\space$ then ......Wed, 12 Aug 2026 18:29:05 -0400https://liberty.io/what-is-the-remainder-when-9-2012-is-divided-by-11.htmlhttps://liberty.io/what-is-the-remainder-when-9-2012-is-divided-by-11.htmlI'm trying to find the answer to the following: What is the remainder when $9^{2012}$ is divided by $11$? Apparently, you're supposed to use Fermat's Little Theorem, but I'm not sure how to use it......Wed, 12 Aug 2026 18:24:49 -0400https://liberty.io/conditional-probability-of-a-given-b-closed.htmlhttps://liberty.io/conditional-probability-of-a-given-b-closed.htmlVery basic question, but If A and B are independent events, then is the Probability of A given B has occurred = Zero or Probability of A? and why?...Wed, 12 Aug 2026 17:58:49 -0400https://liberty.io/finding-interest-rate-without-principal.htmlhttps://liberty.io/finding-interest-rate-without-principal.htmlSo, I recently sat a year ten exam and one of the questions was: Bob invested one half of his savings in an account that paid simple interest for two years and received $550$ dollars as interest......Wed, 12 Aug 2026 17:52:09 -0400https://liberty.io/finding-the-range-of-f-x-1-x-1-x-2.htmlhttps://liberty.io/finding-the-range-of-f-x-1-x-1-x-2.htmlI want to find the range of the following function $$f(x) = \frac{1}{(x-1)(x-2)} $$ Is there any way to find the range of the above function? I have found one idea. But that is too critical. Please ...Wed, 12 Aug 2026 17:49:06 -0400https://liberty.io/rigorous-definition-of-matrix-duplicate.htmlhttps://liberty.io/rigorous-definition-of-matrix-duplicate.htmlI know "Matrix" is usually defined as any array of numbers or other mathematical objects arranged in rows and columns. But the problem is that seems not really like a rigorous mathematical definiti......Wed, 12 Aug 2026 17:25:50 -0400https://liberty.io/conjugation-of-matrices-and-conjugation-of-complex-numbers.htmlhttps://liberty.io/conjugation-of-matrices-and-conjugation-of-complex-numbers.htmlAre conjugation of matrices and conjugation of complex numbers related? What I mean is that if $A$ is an $n \times n$ matrix then the conjugation of $A$ by an invertible $n \times n$ matrix $C$ is......Wed, 12 Aug 2026 17:23:37 -0400https://liberty.io/is-there-a-mathematical-definition-for-the-divisibility-of-rational-nu.htmlhttps://liberty.io/is-there-a-mathematical-definition-for-the-divisibility-of-rational-nu.htmlThe term divisibility usually refers to integer numbers only. I want to define the divisibility of a rational number $q$ by an integer number $z$ as follows: $q$ is divisible by $z$ if and only i......Wed, 12 Aug 2026 17:18:16 -0400https://liberty.io/how-many-different-sizes-of-infinity-are-there.htmlhttps://liberty.io/how-many-different-sizes-of-infinity-are-there.htmlIt's pretty straightforward to say that there is an infinite number of different sizes of infinity, but then I thought, "What size of infinity is that?" My thoughts are that the number of unique ...Wed, 12 Aug 2026 16:57:56 -0400https://liberty.io/user-mike-bennett-mathematics-stack-exchange.htmlhttps://liberty.io/user-mike-bennett-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Wed, 12 Aug 2026 16:57:37 -0400https://liberty.io/does-bounded-variation-imply-boundedness.htmlhttps://liberty.io/does-bounded-variation-imply-boundedness.htmlUsing the standard definition $$||f||_{TV} := \sup_{x_0<\cdots<x_n}\sum_{i=1}^{n} |f(x_i) - f(x_{i-1})|.$$ 1.When the domain is a bounded interval $[a,b]$, the statement holds. 2.When the ...Wed, 12 Aug 2026 16:55:22 -0400https://liberty.io/how-to-solve-csc-sin-1.htmlhttps://liberty.io/how-to-solve-csc-sin-1.htmlIs it possible to get some help on this. I'm trying to help my friend solve this but it had been over 14 years since I did this type. All I have to work with is this. Sorry if it is vague. Any help ...Wed, 12 Aug 2026 16:50:36 -0400https://liberty.io/find-the-eigenvalues-of-a-matrix-in-a-special-case-using-cofactors-dia.htmlhttps://liberty.io/find-the-eigenvalues-of-a-matrix-in-a-special-case-using-cofactors-dia.htmlI have a matrix $A_{(3x3)}$ with determinant iqual to zero, null trace and their cofactors of the diaganal such that $\Delta_{11}=-1$, $\Delta_{22}=0$, $\Delta_{33}=-1$. Some TIPS to calculate the ...Wed, 12 Aug 2026 16:43:51 -0400https://liberty.io/how-to-calculate-standard-deviation-for-a-series-containing-both-posit.htmlhttps://liberty.io/how-to-calculate-standard-deviation-for-a-series-containing-both-posit.htmlI got a stream of numbers in one of my apps to represent an electrical signal. I've observed that the signal ranges from -100 to +100. Other than that, the signal is fairly random and crosses 0 in ......Wed, 12 Aug 2026 16:43:15 -0400https://liberty.io/what-039-s-the-the-integral-of-tan-4x.htmlhttps://liberty.io/what-039-s-the-the-integral-of-tan-4x.htmlHow is $\displaystyle \int \tan(4x) dx = \cos^2 (4x)$? Shouldn't it be $\ln(\sec(4x))$? I don't understand... Please, help. Thank you....Wed, 12 Aug 2026 16:41:33 -0400https://liberty.io/slope-of-tangent-passing-through-point.htmlhttps://liberty.io/slope-of-tangent-passing-through-point.htmlGiven the equation of the curve $y=x^2+x$ and the requirement that the tangent to this curve passes through $(2, -3)$, what are the possible equations of the tangent lines? I started by differenti......Wed, 12 Aug 2026 16:39:57 -0400https://liberty.io/how-to-prove-the-cross-product-of-two-vectors-closed.htmlhttps://liberty.io/how-to-prove-the-cross-product-of-two-vectors-closed.htmlOkay, I must admit that I am lost on how to do this. I have looked up videos and tutorials about this, and they helped a little. The main thing is that my professor asked for us to solve this without ...Wed, 12 Aug 2026 16:38:39 -0400https://liberty.io/bedmas-where-the-order-of-addition-before-subtraction-matters.htmlhttps://liberty.io/bedmas-where-the-order-of-addition-before-subtraction-matters.htmlHere is a recent "tricky problem" that is making the rounds on FB: BEDMAS is explained here in a video, with this being the upshot: Everyone understands that 6 - 0 + 1 = 7, but I was challenged to ...Wed, 12 Aug 2026 16:30:49 -0400https://liberty.io/does-order-of-random-variables-matter-in-chain-rule-probability.htmlhttps://liberty.io/does-order-of-random-variables-matter-in-chain-rule-probability.htmlSuppose that we have the joint distribution of two random variables $X, Y$. Does it matter if we write it as $P(X, Y)$ or $P(Y, X)$? I would say yes if when substitute for their values we are not ...Wed, 12 Aug 2026 16:30:18 -0400https://liberty.io/solve-the-definite-integral-by-completing-the-square.htmlhttps://liberty.io/solve-the-definite-integral-by-completing-the-square.htmlThis is the original definite integral. $$\int_1^3\frac{2x-3}{\sqrt{4x-x^2}}$$ I do not know what to do after completing the square and I am stuck at that specific part after plugging back in. $$(......Wed, 12 Aug 2026 16:24:19 -0400https://liberty.io/evaluating-the-area-between-the-curves-r-2-sin-theta-and-r-sin-theta-c.htmlhttps://liberty.io/evaluating-the-area-between-the-curves-r-2-sin-theta-and-r-sin-theta-c.htmlSo the problem asked me to find the area of the region that lies inside both of the circles $$r=2\sin\theta, \quad r=\sin\theta +\cos\theta $$ I know that $r=2\sin\theta$ is $x^2+(y-1)^2=1,$but the ...Wed, 12 Aug 2026 16:10:12 -0400https://liberty.io/inhomogeneous-euler-cauchy-equations.htmlhttps://liberty.io/inhomogeneous-euler-cauchy-equations.htmlI have the following question: Given the Euler-Cauchy equation $$x^2\frac{d^2y}{dx^2}-3x\frac{dy}{dx}-5y=x^5$$ Solve the associated homogeneous equation by considering a solution of the form $y(x)......Wed, 12 Aug 2026 16:00:36 -0400https://liberty.io/how-to-find-impulse-response-h-t-given-an-output-of-lti-system.htmlhttps://liberty.io/how-to-find-impulse-response-h-t-given-an-output-of-lti-system.htmlA LTI system takes $x(t)$ as input and $y(t)$ as output. Given $$y(t)=\int_{t-5}^{t-1}2x(\tau)d\tau$$, how to find $h(t)$, the impulse response? Convolution $x(t)*h(t)=\int_{-\infty}^\infty x(\tau......Wed, 12 Aug 2026 15:50:33 -0400https://liberty.io/two-ways-of-defining-rank-of-a-set.htmlhttps://liberty.io/two-ways-of-defining-rank-of-a-set.htmlI am studying set theory, and I have some difficulties in understanding how people define the notion of rank there (I hope, specialists in logic will excuse me for this). As far as I understand, ......Wed, 12 Aug 2026 15:47:47 -0400https://liberty.io/when-you-flip-a-fraction-to-remove-the-negative-exponent-do-you-flip-a.htmlhttps://liberty.io/when-you-flip-a-fraction-to-remove-the-negative-exponent-do-you-flip-a.htmlWhen you flip a fraction to remove the negative exponent, do you flip all of what's on numerator and denominator? Or do you only do that with variables Such as: (-7a2b3c0/3a3b4c3 )-4 Will it be ...Wed, 12 Aug 2026 15:45:02 -0400https://liberty.io/motivating-infinite-series.htmlhttps://liberty.io/motivating-infinite-series.htmlWhat are some good ways to motivate the material on infinite series that appears at the end of a typical American Calculus II course? My students in this course are generally from biochemistry, co......Wed, 12 Aug 2026 15:39:42 -0400https://liberty.io/what-is-the-derivative-of-log-y-with-respect-to-x.htmlhttps://liberty.io/what-is-the-derivative-of-log-y-with-respect-to-x.htmlI was solving the following sum: We started off with the first equation and then he took log on both sides to simplify. Everything was fine here but the next step's LHS confused me, he said he used......Wed, 12 Aug 2026 15:15:34 -0400https://liberty.io/cutting-a-cube-with-a-plane.htmlhttps://liberty.io/cutting-a-cube-with-a-plane.htmlMe and a couple of friends have recently started solving Math problems in our breaks at school. We came along the Georg Mohr tournament, which is Denmark's first qualifications tournament for the I......Wed, 12 Aug 2026 15:13:22 -0400https://liberty.io/british-flag-theorem-generalized-and-inspired-from-british-flag-theore.htmlhttps://liberty.io/british-flag-theorem-generalized-and-inspired-from-british-flag-theore.htmlBritish Flag theorem: Let $P$ be a point in the plane, let $ABCD$ be a rectangle in the plane then: $$PA^2+PC^2=PB^2+PD^2$$ Generalization: Let $ABCD$ be a rectangle in a plane, Let $P$ be a poin......Wed, 12 Aug 2026 14:57:17 -0400https://liberty.io/is-this-a-solution-for-the-problem-a-3-b-3-c-3-has-no-nonzero-integer-.htmlhttps://liberty.io/is-this-a-solution-for-the-problem-a-3-b-3-c-3-has-no-nonzero-integer-.htmlIs this a solution for the problem: $\ a^3 + b^3 = c^3\ $ has no nonzero integer solutions? Suppose $\ a^3 + b^3 = c^3,\ a,b,c \in \mathbb Z^*,\ $then: $c^3 - b ^ 3 = (c - b)((c - b) ^ 2 + 3cb......Wed, 12 Aug 2026 14:56:34 -0400https://liberty.io/backward-stability-vs-stability.htmlhttps://liberty.io/backward-stability-vs-stability.htmlI'm reading a numerical analysis book, which gave me very confusing descriptions about stability and backward stability. It says that an algorithm $f$ is stable, if for any $x$, $$ \frac{\|f_b(x)......Wed, 12 Aug 2026 14:53:56 -0400https://liberty.io/express-using-logic-symbols.htmlhttps://liberty.io/express-using-logic-symbols.htmlI have to also decide whether it is true or not after I express it in logic symbols. This is what I have so far..Am I correct? and I don't know how to do a).. a) There is a smallest positive numbe......Wed, 12 Aug 2026 14:53:23 -0400https://liberty.io/degree-of-rational-function.htmlhttps://liberty.io/degree-of-rational-function.htmlThis might sound like a very trivial question but I found different answers on the web. Assume one has a rational function $$\frac{f(x)}{g(x)} ,$$ where $f(x)$ and $g(x)$ are polynomials. What i......Wed, 12 Aug 2026 14:51:21 -0400https://liberty.io/do-all-straight-lines-have-an-inverse-function.htmlhttps://liberty.io/do-all-straight-lines-have-an-inverse-function.htmlDo all straight lines have an inverse function? It would seem to make sense. All linear lines would pass the horizontal line test and thus when reflected across $y=x$ it would still be a function. ...Wed, 12 Aug 2026 14:49:55 -0400https://liberty.io/quaternions-problem-with-understanding-essence-of.htmlhttps://liberty.io/quaternions-problem-with-understanding-essence-of.htmlI can't calm down when I can't realized of principles of working something. Quaternion is that place where I spend а few days! Unfortunately, not one lectures or books what I saw or read don't expl......Wed, 12 Aug 2026 14:36:30 -0400https://liberty.io/trees-on-six-vertices.htmlhttps://liberty.io/trees-on-six-vertices.htmlShow that there are exactly six non-isomorphic trees on six vertices. ("Introduction to Combinatorics" p.183) The solutions were provided in the book. But why can we be sure that there are ONLY si......Wed, 12 Aug 2026 14:27:43 -0400https://liberty.io/what-is-the-difference-between-equation-and-formula.htmlhttps://liberty.io/what-is-the-difference-between-equation-and-formula.htmlSometimes equation and formula are used interchangeably, but I was wondering if there is a difference. For example, suppose we can calculate a car's fuel efficiency as: mpg = distance traveled in......Wed, 12 Aug 2026 14:18:18 -0400https://liberty.io/are-the-endpoints-of-the-domain-of-a-function-counted-as-critical-poin.htmlhttps://liberty.io/are-the-endpoints-of-the-domain-of-a-function-counted-as-critical-poin.htmlDo the end points of a domain come under critical points? I know we say critical point is a point where the derivative is zero or the derivative doesn't exist. For example: $$ f:[0,\pi] \to [-1,1......Wed, 12 Aug 2026 14:06:38 -0400https://liberty.io/cdf-for-a-continuous-random-variable.htmlhttps://liberty.io/cdf-for-a-continuous-random-variable.htmlLet $Y$ be a continuous random variable with probability density function$$ f_Y(x)=\begin{cases} xe^{-\frac{x^2}2};&x>0\\ 0;&x\le0\end{cases}.$$ Compute $\mathbb{E}(Y)$ and determine the ...Wed, 12 Aug 2026 14:05:00 -0400https://liberty.io/finding-the-radius-of-the-smallest-circle-that-can-circumscribe-an-equ.htmlhttps://liberty.io/finding-the-radius-of-the-smallest-circle-that-can-circumscribe-an-equ.htmlQ:A puzzle board is in the form of an equilateral triangle that has an area of $7\sqrt{3}$ if the board is placed on a circular table, what should be the min area of the table so that the whole board ...Wed, 12 Aug 2026 14:00:38 -0400https://liberty.io/how-to-find-a-circle-given-a-segment.htmlhttps://liberty.io/how-to-find-a-circle-given-a-segment.htmlIn this page, we get mathematical formulas for calculating the area of a circle segment under a few different conditions, if you already know the radius of the circle. My question is, how do you d......Wed, 12 Aug 2026 13:58:16 -0400https://liberty.io/continuous-differentiability-implies-lipschitz-continuity.htmlhttps://liberty.io/continuous-differentiability-implies-lipschitz-continuity.htmlHere's a statement in Zygmund's Measure and Integral on page 17: If $f$ has a continuous derivative on $[a,b]$, then (by the mean-value theorem) $f$ satisfies a Lipschitz condition on $[a,b]$. ......Wed, 12 Aug 2026 13:49:34 -0400https://liberty.io/equipotent-sets.htmlhttps://liberty.io/equipotent-sets.htmlWe know by definition that if a bijection between two sets A and B exists,then A and B are equivalent.The book I was reading took the function f:Z→N f(x)= -2x,for x<0 2x+1,for x=>0 w......Wed, 12 Aug 2026 13:38:42 -0400https://liberty.io/difference-between-direction-cosine-matrix-dcm-and-rotation-matrix.htmlhttps://liberty.io/difference-between-direction-cosine-matrix-dcm-and-rotation-matrix.htmlI am a bit confused about the difference between direction cosine matrix(DCM) and rotation matrix. I have searched through the literature but found no explicit explanation if they are different or ......Wed, 12 Aug 2026 13:38:37 -0400https://liberty.io/definition-of-suspension-topology.htmlhttps://liberty.io/definition-of-suspension-topology.htmlI am a bit confused with the definition of a suspension. This the definition. For a space $X$, denote $SX$ the suspension of $X$ in which this is the quotient space $$\frac{X \times I}{\sim}$$ w......Wed, 12 Aug 2026 13:37:38 -0400https://liberty.io/what-is-f-prime.htmlhttps://liberty.io/what-is-f-prime.htmlcurrently taking Measure and Integration course, which seems to have a different definition of f'. traditionally, $$f'(x)=\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$$ but in folland's book, it se......Wed, 12 Aug 2026 13:30:14 -0400https://liberty.io/points-in-general-position.htmlhttps://liberty.io/points-in-general-position.htmlI'm really confused by the definition of general position at wikipedia. I understand that the set of points/vectors in $\mathbb R^d$ is in general position iff every $(d+1)$ points are not in any ...Wed, 12 Aug 2026 13:21:33 -0400