https://liberty.ioCurated star highlights with fresh energy.en-usSat, 22 Aug 2026 23:32:36 -0400https://liberty.io/for-f-x-x-3-2-times-e-0-35x-at-what-x-value-does-the-maximum-occur.htmlhttps://liberty.io/for-f-x-x-3-2-times-e-0-35x-at-what-x-value-does-the-maximum-occur.htmlFor $f(x) = x^{3.2}\times \mathrm{e}^{-0.35x}$, at what $x$ value does the maximum occur for minimum or maximum $f'(x)=0$ this implies $3.2 x^{2.2} \mathrm{e}^{-0.35x}-x^{3.2}\mathrm{e}^{-0.35x}\......Sun, 23 Aug 2026 03:32:34 -0400https://liberty.io/cube-roots-of-the-complex-numbers-1-i.htmlhttps://liberty.io/cube-roots-of-the-complex-numbers-1-i.htmlI cant get any good reference in my books regarding cube of complex numbers. Please help me find cube roots of the Complex number i+1??...Sun, 23 Aug 2026 03:31:39 -0400https://liberty.io/partition-of-an-interval-of-mathbb-r.htmlhttps://liberty.io/partition-of-an-interval-of-mathbb-r.htmlA partition of an inteval $[a,b]$ of $\mathbb{R}$ is generally defined as a finite sequence of the form: $$a = x_0 < x_1 < x_2 < \dots < x_n = b$$ Then, $[a,b]$ is seen as the following......Sun, 23 Aug 2026 03:26:51 -0400https://liberty.io/which-of-the-43-380-possible-nets-for-a-dodecahedron-is-the-narrowest.htmlhttps://liberty.io/which-of-the-43-380-possible-nets-for-a-dodecahedron-is-the-narrowest.htmlI want to fit multiple regular dodecahedron nets on to an infinitely long roll of paper. I want this to result in the largest possible dodecahedrons, for a roll of a given width. My hunch is that ......Sun, 23 Aug 2026 03:24:08 -0400https://liberty.io/if-det-a-2-det-a-then-a-2-in-a-a-1-a-a-1.htmlhttps://liberty.io/if-det-a-2-det-a-then-a-2-in-a-a-1-a-a-1.htmlIs the following statement always true? Let $A \in GL_n( \mathbb Z)$. If $\det(A^2)= \det(A)$, then $A^2 \in \{A, A^{-1}, -A, -A^{-1}\}$ . Note: $GL_n(\mathbb Z)$ is defined as the set of all ...Sun, 23 Aug 2026 03:18:42 -0400https://liberty.io/find-the-prime-factors-of-3-32-2-32.htmlhttps://liberty.io/find-the-prime-factors-of-3-32-2-32.htmlI'm having a go at BMO 2006/7 Q1 which states: "Find four prime numbers less than 100 which are factors of $3^{32}-2^{32}$." My working is as follows (basically just follows difference of two squa......Sun, 23 Aug 2026 03:14:42 -0400https://liberty.io/classifying-irreducible-finite-dimensional-representations-of-the-q-we.htmlhttps://liberty.io/classifying-irreducible-finite-dimensional-representations-of-the-q-we.htmlI have that $A = \langle x, y : yx = qxy \rangle$ is the $q$-Weyl algebra, with basis $x^iy^j$, $i, j \in \mathbb{Z}$. The base field is $\mathbb{C}$. I need to classify the irreducible finite-...Sun, 23 Aug 2026 03:13:54 -0400https://liberty.io/find-primitive-root-mod-17.htmlhttps://liberty.io/find-primitive-root-mod-17.htmlI have to list the quadratic residues of $17$ and find a primitive root. I have calculated that: Quadratic residues $mod(17)$ are $1,2,4,8,9,13,15,16$ How am I then meant to use this to obtain a ...Sun, 23 Aug 2026 03:13:36 -0400https://liberty.io/degree-sum-formula-closed.htmlhttps://liberty.io/degree-sum-formula-closed.htmlSuppose the G = (V,E) is a connected graph with n vertices and n-1 edges. Use the degree-sum formula for vertices to prove that G has a vertex of degree 1. I can't seem to find where to start in t......Sun, 23 Aug 2026 03:10:44 -0400https://liberty.io/what-does-the-small-number-on-top-of-the-square-root-symbol-mean.htmlhttps://liberty.io/what-does-the-small-number-on-top-of-the-square-root-symbol-mean.htmlI just came across this annotation in my school's maths compendium: The compendium is very brief and doesn't explain what this means....Sun, 23 Aug 2026 03:08:31 -0400https://liberty.io/how-to-recognize-the-language-of-a-dfa.htmlhttps://liberty.io/how-to-recognize-the-language-of-a-dfa.htmlGiven the following Deterministic Finite Automata, I have to be able to tell what language the Automata accepts.image here] As I can see, it accepts the letters a, b and c. Furthermore, my guess wo......Sun, 23 Aug 2026 02:44:20 -0400https://liberty.io/negation-of-some-logic-statement.htmlhttps://liberty.io/negation-of-some-logic-statement.htmlI need to negate the following statement : "Some integers are not odd" I have the below, where O(x) is "odd" $$ \exists x (\neg O(x)) $$ Would the negation be $$ \forall x (O(x)) $$ I'm confused, ...Sun, 23 Aug 2026 02:38:02 -0400https://liberty.io/linear-algebra-row-picture-and-column-picture-for-system-of-linear-equ.htmlhttps://liberty.io/linear-algebra-row-picture-and-column-picture-for-system-of-linear-equ.htmlI am presently going through Strang's linear algebra book in which he explains row picture and column picture for a system of linear equations as such For example: $$\begin{cases} 2x + 3y = 5 \\ ......Sun, 23 Aug 2026 02:26:45 -0400https://liberty.io/negative-variance.htmlhttps://liberty.io/negative-variance.htmlI have two independent variables $X$ and $Y$. $W=X-Y$ when $X\sim \mbox{Bernoulli}\left(1/2\right)$ and $Y\sim N(0,1)$. This puts $\operatorname{Var}(x)=1/4$ and $\operatorname{Var}(Y)=1$, but I h......Sun, 23 Aug 2026 01:54:56 -0400https://liberty.io/pitchfork-bifurcation-subcritical-and-supercritical.htmlhttps://liberty.io/pitchfork-bifurcation-subcritical-and-supercritical.htmlI am slightly confused about the terminology of a subcritical and supercritical pitchfork bifurcation. Consider the diagrams below: (both pictures from wikipedia, by Claudio Rocchini, CC by 2.5......Sun, 23 Aug 2026 01:40:24 -0400https://liberty.io/lyndon-words-over-the-binary-alphabet-that-begin-with-00.htmlhttps://liberty.io/lyndon-words-over-the-binary-alphabet-that-begin-with-00.htmlLydon words over the binary alphabet are strings of $1$s and $0$s such that all rotations of the word are strictly lexicographically later. I've been looking at Lyndon words of length $n$, which are ...Sun, 23 Aug 2026 01:34:06 -0400https://liberty.io/value-of-tan-pi-2-duplicate.htmlhttps://liberty.io/value-of-tan-pi-2-duplicate.htmlI understand that this is a very stupid question but I'm not getting the answer. At $x=\pi/2$, what is the value of $tan(x)$? Should it be $-\infty$ or $+\infty$? Text tells it to be $+\infty$. ......Sun, 23 Aug 2026 01:25:42 -0400https://liberty.io/area-of-circumscribed-circle.htmlhttps://liberty.io/area-of-circumscribed-circle.htmlI'm looking at the solutions for a 1989 ARML contest, and it has this question: If the area of a rectangle is 1/2 of the area of its circumscribed circle, compute the acute angle between the dia......Sun, 23 Aug 2026 01:22:25 -0400https://liberty.io/solve-equation-3-x-2-3-x.htmlhttps://liberty.io/solve-equation-3-x-2-3-x.htmlUsing the substitution $u=3^x$, or otherwise, solve, correct to 3 significant figures, the equation $3^x=2+3^{-x}$ My attempt: $u=2+u^{-1}$ $u-u^{-1}=2$ $ {{u^2-1}\over {u}}=2$ $\log ({{u^2......Sun, 23 Aug 2026 01:21:24 -0400https://liberty.io/show-that-limit-does-not-exist-two-variables.htmlhttps://liberty.io/show-that-limit-does-not-exist-two-variables.htmlI just started looking into multiple variable calculus and limits involving them. I'm not amazing at limits either. I want to answer this question: Show that the following limit does not exist $$\......Sun, 23 Aug 2026 01:18:48 -0400https://liberty.io/summation-of-n-squared-cubed-etc-duplicate.htmlhttps://liberty.io/summation-of-n-squared-cubed-etc-duplicate.htmlHow do you in general derive a formula for summation of n-squared, n-cubed, etc...? Clear explanation with reference would be great....Sun, 23 Aug 2026 01:15:14 -0400https://liberty.io/finding-if-a-relation-implicit-function-is-an-even-or-odd-function.htmlhttps://liberty.io/finding-if-a-relation-implicit-function-is-an-even-or-odd-function.htmlI have this relation, and if graphed on Desmos.com looks very symmetrical. $$x^{\frac{2}{3}} + y^{\frac{2}{3}} = 1 $$ It has cusps(vertical tangents on the y-axis). How can one determine if this re......Sun, 23 Aug 2026 01:07:42 -0400https://liberty.io/how-to-find-a-point-between-two-points-with-given-distance.htmlhttps://liberty.io/how-to-find-a-point-between-two-points-with-given-distance.htmlLets say I have point P1(10,10) and P2(20,20). I want to find a P3 which is on between this two points and 3 units away from P1. What is the formula to find P3 ? Known values: X1, X2, Y1 , Y2, ...Sun, 23 Aug 2026 01:04:13 -0400https://liberty.io/why-is-the-definition-of-cardinal-number-as-the-set-of-all-sets-equiva.htmlhttps://liberty.io/why-is-the-definition-of-cardinal-number-as-the-set-of-all-sets-equiva.htmlIn Fundamentals of Mathematics, Volume 1 Foundations of Mathematics: The Real Number System and Algebra, after defining set equivalence as the ability to put the elements of the related sets in one......Sun, 23 Aug 2026 01:02:13 -0400https://liberty.io/linear-dependence-of-functions.htmlhttps://liberty.io/linear-dependence-of-functions.htmlI'm in an intro to DE class. On one of the quizzes, the professor asked us to determine whether the set of functions $\ x, x-1,$ and$\ x+3$ are linearly independent or linearly dependent on $\ (- \......Sun, 23 Aug 2026 01:02:05 -0400https://liberty.io/definition-of-a-divergent-sequence.htmlhttps://liberty.io/definition-of-a-divergent-sequence.htmlI have that the definition of a convergent sequence is: ($\exists$ L $\in$ $\mathbb{R}$)($\forall$ $\epsilon$ > 0)($\exists$ N $\in$ $\mathbb{N}$)($\forall$ n $\in$ $\mathbb{N}$)[n $\ge$ N $\Right......Sun, 23 Aug 2026 01:01:07 -0400https://liberty.io/solving-equations-that-contain-summations.htmlhttps://liberty.io/solving-equations-that-contain-summations.htmlMy previous algebra course did not go over summation at all, and now that I'm in my new course, Discrete Math for Information Technology, we have been introduced to summation. I understand summation ...Sun, 23 Aug 2026 01:00:06 -0400https://liberty.io/4-regular-planar-graph-is-unique.htmlhttps://liberty.io/4-regular-planar-graph-is-unique.htmlI'm working on a project for a class and as part of that project I (previously) decided to do the following problem from our textbook, Combinatorics and Graph Theory 2nd ed. by Harris, Hirst, & ...Sun, 23 Aug 2026 01:00:05 -0400https://liberty.io/derivation-of-fibonacci-log-n-time-sequence.htmlhttps://liberty.io/derivation-of-fibonacci-log-n-time-sequence.htmlI was trying to derive following equation to compute the nth fibonacci number in O(log(n)) time. F(2n) = (2*F(n-1) + F(n)) * F(n) which i found on wiki form the fibonacci matrix equation stated th......Sun, 23 Aug 2026 00:49:10 -0400https://liberty.io/books-on-complex-analysis-ahlfors-conway-and-lang.htmlhttps://liberty.io/books-on-complex-analysis-ahlfors-conway-and-lang.htmlTo make my question slightly different from others, I would like to know how would you rate on the complex analysis books by Ahlfors, Conway and Lang? I had a basic course on complex analysis during ...Sun, 23 Aug 2026 00:47:24 -0400https://liberty.io/regular-k-gons-that-can-tessellate-a-plane.htmlhttps://liberty.io/regular-k-gons-that-can-tessellate-a-plane.htmlWhat is the sum of all possible values of $k$ for which regular $k-$gons can tessellate a plane? This is one of the NAT problems. While I am familiar with tessellations, I don't quite get what the......Sun, 23 Aug 2026 00:41:48 -0400https://liberty.io/real-and-imaginary-part-of-1-i-sqrt-3-6.htmlhttps://liberty.io/real-and-imaginary-part-of-1-i-sqrt-3-6.htmli am a bit stuck here. As the title says i try to find out how to write complex numbers like for example$$ (1-i\sqrt{3})^6$$ in the normal representation$$ z = x + i*y$$ I already found out that ......Sun, 23 Aug 2026 00:34:48 -0400https://liberty.io/how-to-bisect-an-isosceles-trapezoid-into-two-equal-area-parts.htmlhttps://liberty.io/how-to-bisect-an-isosceles-trapezoid-into-two-equal-area-parts.htmlGiven an isosceles trapezoid: I want to draw a line parallel to the bases (that is, parallel to and in between AD and BC) such that the top half and the bottom half both have equal area. Specific......Sun, 23 Aug 2026 00:34:14 -0400https://liberty.io/show-that-n-kl-is-a-perfect-square.htmlhttps://liberty.io/show-that-n-kl-is-a-perfect-square.htmlI faced a doubt in this question while solving some maths problem. Please, solve it. A natural number $n$ is chosen strictly between two consecutive perfect squares. The smaller of these two square ...Sun, 23 Aug 2026 00:31:46 -0400https://liberty.io/what-is-an-intermediate-definition-for-a-tangent-to-a-curve.htmlhttps://liberty.io/what-is-an-intermediate-definition-for-a-tangent-to-a-curve.htmlMost students come to calculus with an intuitive sense of what a tangent line should be for a curve. It is easy enough to give a definition of a tangent to a circle that is both elementary and rigo......Sun, 23 Aug 2026 00:28:07 -0400https://liberty.io/different-definitions-of-p-points-ultrafilters.htmlhttps://liberty.io/different-definitions-of-p-points-ultrafilters.htmlRecently I have been exposed to the concept of P-Points. I have three definitions: A non-principal ultrafilter $u$ is a P-Point if: For every sequence $\left < A_n \right >_{n\in \omega}$ of ...Sun, 23 Aug 2026 00:25:48 -0400https://liberty.io/clarifying-the-definition-of-the-dual-map.htmlhttps://liberty.io/clarifying-the-definition-of-the-dual-map.htmlI would appreciate help understanding the definition of the dual map $T'$ as presented in Axler's "Linear Algebra Done Right" 3rd ed. on page 103. If $T \in \cal {L}$$(V,W)$ then the dua......Sun, 23 Aug 2026 00:22:53 -0400https://liberty.io/geometric-interpretation-of-mixed-partial-derivatives.htmlhttps://liberty.io/geometric-interpretation-of-mixed-partial-derivatives.htmlI'm looking for a geometric interpretation of this theorem: My book doesn't give any kind of explanation of it. Again, I'm not looking for a proof - I'm looking for a geometric interpretation. Th......Sun, 23 Aug 2026 00:18:52 -0400https://liberty.io/geometric-understanding-of-differential-forms.htmlhttps://liberty.io/geometric-understanding-of-differential-forms.htmlI would like to understand differential forms more intuitively. I have yet to find a book which explains how the use of the exterior product in differential forms ties into the geometrical signific......Sun, 23 Aug 2026 00:18:37 -0400https://liberty.io/given-two-adjacent-sides-of-a-rectangle-are-equivalent-prove-that-the-.htmlhttps://liberty.io/given-two-adjacent-sides-of-a-rectangle-are-equivalent-prove-that-the-.htmlIn Geometry class today, we were talking about quadrilaterals and the types of them. I was wondering that if, given a rectangle with two adjacent equal congruent sides, if that was enough to prove ......Sun, 23 Aug 2026 00:12:11 -0400https://liberty.io/when-is-a-power-series-a-geometric-series.htmlhttps://liberty.io/when-is-a-power-series-a-geometric-series.htmlA geometric series is one having a common ratio, right? Something like: $$\sum_{n=0}^{\infty} ar^n$$ And a power series is one with the form: $$\sum_{n=0}^{\infty} c_n(x-a)^n$$ Initially, I tho......Sun, 23 Aug 2026 00:04:58 -0400https://liberty.io/rolling-standard-deviations.htmlhttps://liberty.io/rolling-standard-deviations.htmlI am trying to calculate standard deviations on an array of numbers. My psuedo code looks like this: deviation = getStandardDeviation(array(32, 47, 42, 45, 80, 90)); In the above example, deviat......Sat, 22 Aug 2026 23:58:17 -0400https://liberty.io/how-does-one-interpret-the-meaning-of-a-stochastic-derivative.htmlhttps://liberty.io/how-does-one-interpret-the-meaning-of-a-stochastic-derivative.htmlMy understanding of derivatives is in the difference quotient limit sense... How does one interpret the meaning of a stochastic derivative? How can one possibly differentiate a random variable? W......Sat, 22 Aug 2026 23:47:19 -0400https://liberty.io/how-to-calculate-the-result-of-a-recursive-equation.htmlhttps://liberty.io/how-to-calculate-the-result-of-a-recursive-equation.htmlI'm looking for a tool (like Wolfram Alpha) that can calculate the result of $ B(P,N) $ Where $B(P,N)$ is a recursive function defined as follows: $ \left\{\begin{matrix} B(P,N)=\frac{-(-1)^{\fra......Sat, 22 Aug 2026 23:36:56 -0400https://liberty.io/property-of-the-2-3-4-triangle-and-its-extensions.htmlhttps://liberty.io/property-of-the-2-3-4-triangle-and-its-extensions.htmlI'm trying to answer this question but I'm fully stuck. I've tried to draw the diagram but it still is a mystery to me. Here is the problem: a) Prove that the $2$-$3$-$4$ triangle has the property......Sat, 22 Aug 2026 23:36:14 -0400https://liberty.io/why-is-determinant-a-multilinear-function.htmlhttps://liberty.io/why-is-determinant-a-multilinear-function.htmlI am trying to understand (intuitive explanation will be fine) why determinant is a multilinear function and therefore to learn how elementary row operation affect the determinant. I understand th......Sat, 22 Aug 2026 23:27:50 -0400https://liberty.io/closure-of-the-union-union-of-closures.htmlhttps://liberty.io/closure-of-the-union-union-of-closures.htmlI have seen that $\text{cl}(A\cup B)=\text{cl}(A)\cup \text{cl}(B)$. However I don't see why this is true. I can see the the right to left inclusion, but I can't see the inclusion from left to righ......Sat, 22 Aug 2026 23:13:55 -0400https://liberty.io/the-sum-of-two-odd-numbers-is-even.htmlhttps://liberty.io/the-sum-of-two-odd-numbers-is-even.htmlI know formal ways of proving this, but I am doing research on how middle grade students create arguments to support conjectures. I would like to anticipate many more solutions before I enact the t......Sat, 22 Aug 2026 23:10:31 -0400https://liberty.io/calculate-flux-through-a-surface.htmlhttps://liberty.io/calculate-flux-through-a-surface.htmlPart of the surface, S, is: $z=x^2+y^2$ above the disk $ \ x^2+y^2 = 1 \ $ oriented in the $\vec k$ direction. I need to set up an integrated integral to calculate the flux of $\vec F = yz\vec i+x......Sat, 22 Aug 2026 22:55:46 -0400https://liberty.io/need-help-visualizing-the-combinatorics-sum-product-rule.htmlhttps://liberty.io/need-help-visualizing-the-combinatorics-sum-product-rule.htmlI'm working on some assignments in my discrete math class for the sum & product combinatorics rules and I'm having trouble understanding their definitions and how they work. I'm hoping someone ......Sat, 22 Aug 2026 22:52:48 -0400https://liberty.io/is-the-point-at-0-0-of-the-graph-z-x-3-y-3-considered-a-saddle-point.htmlhttps://liberty.io/is-the-point-at-0-0-of-the-graph-z-x-3-y-3-considered-a-saddle-point.htmlIs the point at $(0,0)$ of the graph $z=x^3 + y^3$ considered a saddle point? I was given that function, and I used the second-order derivative test only to find that the $Hf(0,0) = 0$, with the ...Sat, 22 Aug 2026 22:28:54 -0400https://liberty.io/monic-polynomials-clarification.htmlhttps://liberty.io/monic-polynomials-clarification.htmlSo I know a monic polynomial is one where the coefficient of the leading term is 1, but does that mean the constant 1 is a monic polynomial too? Thanks....Sat, 22 Aug 2026 22:28:45 -0400https://liberty.io/drawing-vector-fields-beginner.htmlhttps://liberty.io/drawing-vector-fields-beginner.htmlI have $f(x,y) = xy$ which has the Gradient Vector Field F$(x,y) = (y,x)$ I believe the correct way to draw individual vectors is to map points into the formula, for example at point $(1,-3)$ we w......Sat, 22 Aug 2026 22:27:03 -0400https://liberty.io/difference-in-meaning-between-converges-against-and-converge-to.htmlhttps://liberty.io/difference-in-meaning-between-converges-against-and-converge-to.htmlMight be a silly question, but is there any difference between saying that a something (limit, series and so on) converges to a value and converges against a value?...Sat, 22 Aug 2026 22:24:23 -0400https://liberty.io/what-is-the-notation-for-the-empty-matrix.htmlhttps://liberty.io/what-is-the-notation-for-the-empty-matrix.htmlMy question: What is a notation for an empty 0x0 matrix (i.e. the matrix for the only linear map $f:\{0\}\to\{0\}$)? Is it written $()$? How can I distinguish the 0x0 matrix with for example the 0x3 ...Sat, 22 Aug 2026 22:23:39 -0400https://liberty.io/how-to-find-tangent-line-at-a-given-point-without-equation.htmlhttps://liberty.io/how-to-find-tangent-line-at-a-given-point-without-equation.htmlFind the equation of the line that is tangent to the curve at the point $(0,\sqrt{\frac{\pi}{2}})$. Given your answer in slope-intercept form. I don't know how can I get the tangent line, without a ...Sat, 22 Aug 2026 22:21:39 -0400https://liberty.io/special-galois-connections.htmlhttps://liberty.io/special-galois-connections.htmlLet $(f;g)$ is a Galois connection between two posets. Consider the special case $g\circ f = \operatorname{id}_{\operatorname{dom} f}$. What can be said about this special case of Galois connecti......Sat, 22 Aug 2026 22:19:19 -0400https://liberty.io/how-to-find-the-average-rate-of-change-velocity.htmlhttps://liberty.io/how-to-find-the-average-rate-of-change-velocity.htmlA problem asked this: The rocket club wants to model rockets straight up into the air. The height of rocket A can being modeled by the function $y=-16x^2+88x$. The function that models the heigh......Sat, 22 Aug 2026 22:16:51 -0400https://liberty.io/the-comparison-test-or-the-limit-comparison-test.htmlhttps://liberty.io/the-comparison-test-or-the-limit-comparison-test.htmlI have checked many of this site's pages yet I could not find a clear answer about how to choose between the "comparison test" OR the "limit comparison test". Because the difference between the tw......Sat, 22 Aug 2026 22:16:46 -0400https://liberty.io/what-is-a-bounty-how-can-i-start-one.htmlhttps://liberty.io/what-is-a-bounty-how-can-i-start-one.htmlQ&A for people studying math at any level and professionals in related fields...Sat, 22 Aug 2026 22:16:34 -0400https://liberty.io/solution-of-cubic-equations-in-terms-of-quadratic-equations.htmlhttps://liberty.io/solution-of-cubic-equations-in-terms-of-quadratic-equations.htmlIs there a general way in which a cubic equation of $3$rd degree can be represented by a quadratic equation of $2$nd degree such that $2$ solutions of a cubic equation is equal to $2$ solutions of ...Sat, 22 Aug 2026 22:15:57 -0400https://liberty.io/finding-the-inverse-distribution-for-a-standard-cauchy-distribution-is.htmlhttps://liberty.io/finding-the-inverse-distribution-for-a-standard-cauchy-distribution-is.htmlThe standard cauchy distribution was given as follows $f_X(x) = \frac{1}{\pi} \frac{1}{1+x^2} \;\;\;\;\; x\in \mathbb{R}$ I need to find the distribution for $Y= \frac{1}{X}$ and show it is the sam......Sat, 22 Aug 2026 22:14:43 -0400https://liberty.io/trig-identity-pythagorean-theorem-confusion.htmlhttps://liberty.io/trig-identity-pythagorean-theorem-confusion.htmlI run into a problem when I'm trying to prove how $\tan^2x+1 = \sec^2x$, and $1+\cot^2x=\csc^2x$ I understand that $\sin^2x+\cos^2x = 1$. (To my understanding 1 is the Hypotenuse, please correct m......Sat, 22 Aug 2026 22:14:13 -0400https://liberty.io/definitions-of-direct-product-and-of-direct-sum.htmlhttps://liberty.io/definitions-of-direct-product-and-of-direct-sum.htmlI was wondering if there are some general definitions for direct product and for direct sum, for example in category theory or in set theory, so that the concepts for vector spaces, Abelian groups,......Sat, 22 Aug 2026 21:45:43 -0400https://liberty.io/can-i-factorize-x-4-27x-without-using-the-factor-theorem.htmlhttps://liberty.io/can-i-factorize-x-4-27x-without-using-the-factor-theorem.htmlI am doing some self-assigned homework questions. One question asks me to factorize $x^4 + 27x$. I used the factor theorem to solve it. I observed that $f(0) = 0$ and $f(-3) = 0$, so $x$ and $(x + ......Sat, 22 Aug 2026 21:39:40 -0400https://liberty.io/find-general-solution-of-first-order-de.htmlhttps://liberty.io/find-general-solution-of-first-order-de.htmlFind general solution of $$y'+\frac{2y}{x}=\frac{3}{x^2}$$ and find solution for which $$y(2)=-1$$ I know $$xy'+\frac{r(x)y}{x}=3r(x)x$$ $$(ry')=ry'+r'y$$$$r'=\frac{r}{x},\frac{r'}{r}=\......Sat, 22 Aug 2026 21:39:02 -0400https://liberty.io/endomorphisms-of-a-finite-dimensional-vector-space.htmlhttps://liberty.io/endomorphisms-of-a-finite-dimensional-vector-space.htmlFrom Humphreys' Introduction to Lie Algebras and Representation Theory: If $V$ is a finite dimensional vector space over $F$, denote by $\text{End }V$ the set of linear transformations $V\righta......Sat, 22 Aug 2026 21:35:55 -0400https://liberty.io/determine-the-value-of-n.htmlhttps://liberty.io/determine-the-value-of-n.htmlI'm a complete n00b at math, but I'm wondering how one would go about determining the value of n in the following comparison. n * 1.5 + 12.5 = 12.5 / 2 + n I'm new to the math StackExchange, so I'm......Sat, 22 Aug 2026 21:34:46 -0400https://liberty.io/find-the-distribution-of-linear-combination-of-independent-random-vari.htmlhttps://liberty.io/find-the-distribution-of-linear-combination-of-independent-random-vari.htmlGiven independent and identically distributed random variables $X_1, X_2, \dots, X_n$, each of them has the same p.d.f $f(x) = Pr(X = x)$ on support $(a, b)$. How do I find the pdf or cdf of $Y = ......Sat, 22 Aug 2026 21:31:25 -0400https://liberty.io/how-many-algebras-are-there-of-finite-sized-omega.htmlhttps://liberty.io/how-many-algebras-are-there-of-finite-sized-omega.htmlAn algebra of $\Omega$ is a family that contains $\Omega,$ is closed under complement and finite union. If $\Omega=\{\}$, there can only be 1 algebra associated with $\Omega$: $2^{\Omega}.$ If $\......Sat, 22 Aug 2026 21:26:21 -0400https://liberty.io/y-2-frac-1-x-2-stretched-vertically-by-a-factor-or-3.htmlhttps://liberty.io/y-2-frac-1-x-2-stretched-vertically-by-a-factor-or-3.htmlAlright folks, having a little problem with this calculus question. anyone help? The factor and direction of the given function is to be stretched or compressed are given below. Give an equation f......Sat, 22 Aug 2026 21:15:35 -0400https://liberty.io/user-matthew-h-mathematics-stack-exchange.htmlhttps://liberty.io/user-matthew-h-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Sat, 22 Aug 2026 21:13:18 -0400https://liberty.io/compact-sets-are-closed.htmlhttps://liberty.io/compact-sets-are-closed.htmlI feel really ignorant in asking this question but I am really just don't understand how a compact set can be considered closed. By definition of a compact set it means that given an open cover......Sat, 22 Aug 2026 21:12:01 -0400https://liberty.io/lie-algebra-of-u-n-and-so-n.htmlhttps://liberty.io/lie-algebra-of-u-n-and-so-n.htmlU(N) and SO(N) are quite important groups in physics. I thought I would find this with an easy google search. Apparently NOT! What is the Lie algebra and Lie bracket of the two groups?...Sat, 22 Aug 2026 21:08:05 -0400https://liberty.io/is-there-an-easy-way-to-factor-polynomials-with-two-variables.htmlhttps://liberty.io/is-there-an-easy-way-to-factor-polynomials-with-two-variables.htmlOn a recent precal test, I saw a question involving the following expression: $$(x+1)^2-y^2$$ Which factored out into: $$(x+y+1)(x-y+1)$$ This wasn't very hard, considering that it was already ...Sat, 22 Aug 2026 21:07:18 -0400https://liberty.io/positive-slope-with-negative-values.htmlhttps://liberty.io/positive-slope-with-negative-values.htmlWhen I was in school I learned that slope could be found with the following: $$m=\frac{y_2-y_1}{x_2-x_1}$$ When given the coordinates $(-3, 0)$ and $(0, 5)$, plotting these points on a Cartesian pl......Sat, 22 Aug 2026 21:05:10 -0400https://liberty.io/how-to-rotate-the-positions-of-a-matrix-by-90-degrees.htmlhttps://liberty.io/how-to-rotate-the-positions-of-a-matrix-by-90-degrees.htmlI have a 5x5 matrix of values. I'm looking for a simple formula that I can use to rotate the position of the values (not the values themselves) 90 degrees within the matrix. For example, here is the ...Sat, 22 Aug 2026 20:47:36 -0400https://liberty.io/calculating-expected-values-of-roulette-bets.htmlhttps://liberty.io/calculating-expected-values-of-roulette-bets.htmlA roulette wheel has 38 numbers, 1 through 36, 0 and 00. One-half of the numbers from 1-36 are red, and the other half are black; 0 and 00 are green. A ball is rolled, and it falls into one of th......Sat, 22 Aug 2026 20:43:02 -0400https://liberty.io/multiplication-in-the-galois-field-gf-3-3.htmlhttps://liberty.io/multiplication-in-the-galois-field-gf-3-3.htmlI am trying to compute $x^3$ in the Galois field $\text{GF}(3^3)$ using the irreducible polynomial $f(x) = x^3 + 2x^2 + 1$. From the expression $x^3 = f(x) + (2x^2 +1)$ I proceed to take the modul......Sat, 22 Aug 2026 20:39:00 -0400https://liberty.io/a-question-about-the-definition-of-markov-measure.htmlhttps://liberty.io/a-question-about-the-definition-of-markov-measure.htmlI have one question about this definition Why is it enough to note equation (3.30) to see that the measure is well defined? is it already additive so you can put the measure that way? if not, what......Sat, 22 Aug 2026 20:32:08 -0400https://liberty.io/fourier-transform-of-sinc-function.htmlhttps://liberty.io/fourier-transform-of-sinc-function.htmllet us consider fourier transform of sinc function,as i know it is equal to rectangular function in frequency domain and i want to get it myself,i know there is a lot of material about this,but i w......Sat, 22 Aug 2026 20:22:30 -0400https://liberty.io/evaluating-the-limit-of-lim-x-to-9-frac-x-9-sqrt-x-3.htmlhttps://liberty.io/evaluating-the-limit-of-lim-x-to-9-frac-x-9-sqrt-x-3.htmlHow do you evaluate the limit of $$\lim_{x \to 9} \frac {x-9} {\sqrt{x} - 3}$$ by first principle? I was thinking of this way to evaluate but it seems to evaluate to $\frac {0} {0}$. $$\lim_{x \......Sat, 22 Aug 2026 20:20:17 -0400https://liberty.io/showing-a-binary-operation-is-well-defined.htmlhttps://liberty.io/showing-a-binary-operation-is-well-defined.htmlLet $G=\{x\in\mathbb{R}~|~0\le x<1\}$ and define $\star$ as follows $$x\star y=x+y-[x+y]$$ I want to show this is a well defined operation. I think it is this easy; If $x=x', y=y'$, then $$......Sat, 22 Aug 2026 20:16:48 -0400https://liberty.io/what-is-the-number-of-dimensions-of-a-scalar.htmlhttps://liberty.io/what-is-the-number-of-dimensions-of-a-scalar.htmlAm I correct that a matrix has two dimensions, and a vector has one dimension? What is the number of dimensions of a scalar? zero? Thanks....Sat, 22 Aug 2026 20:09:18 -0400https://liberty.io/definition-of-the-approx-symbol.htmlhttps://liberty.io/definition-of-the-approx-symbol.htmlTake an unending number, say e.g $π$. If we want to show $π$'s value, should we use the approximately notation or equal sign when writing: $π = 3.14...$ or $π ≈ 3.14...$ This might be a really s......Sat, 22 Aug 2026 20:06:29 -0400https://liberty.io/find-the-dual-of-a-linear-programming-problem.htmlhttps://liberty.io/find-the-dual-of-a-linear-programming-problem.htmlI have a very simple linear programming problem with the following constraints: Minimize $3x_1 + 2x_2 - 33x_3$ subject to $x_1 - 4x_2 + x_3 \leq 15$ $9x_1 + 6x_3 \leq 12$ $5x_1 + 9x_2 \geq 3$ ......Sat, 22 Aug 2026 19:58:27 -0400https://liberty.io/what-is-the-average-roll-on-dice-while-re-rolling-a-result-of-1.htmlhttps://liberty.io/what-is-the-average-roll-on-dice-while-re-rolling-a-result-of-1.htmlThe average roll on a six sided die (d6) is 3.5. If the result is a 1 you are allowed a single re-roll accepting the second result. What does the average roll increase to? And how to calculate this? ...Sat, 22 Aug 2026 19:55:25 -0400https://liberty.io/derivative-of-x-x-2.htmlhttps://liberty.io/derivative-of-x-x-2.htmlStudying past exam problems for my exam in ~$4$ weeks, and I came across this derivative as one of the questions. I actually have no idea how to solve it. $$\frac{d}{dx} (x^{x^2})$$ Using the chain ...Sat, 22 Aug 2026 19:54:35 -0400https://liberty.io/finding-maclaurin-series-of-integral.htmlhttps://liberty.io/finding-maclaurin-series-of-integral.htmlI wish to find the Maclaurin-series for the function $$L(x) = \int_0^x \cos(t^2)\space dt$$ Alternative 1 Set $f(t) = \cos(t)$. We have that $f'(t) = -\sin(t)$, $f''(t) = -\cos(t)$ and $f^{(3)}(t)......Sat, 22 Aug 2026 19:47:02 -0400https://liberty.io/two-definitions-of-bounded-variation-function.htmlhttps://liberty.io/two-definitions-of-bounded-variation-function.htmlAs far as I know, a function $f$ defined on an interval $[a, b]$ is said to be of bounded variation if $$\tag{1}V_a^b(f)=\sup\left\{\sum_{P} \lvert f(x_{j+1})-f(x_j)\rvert \ :\ P\ \text{partition ......Sat, 22 Aug 2026 19:32:16 -0400https://liberty.io/find-the-equivalent-value-of-radians.htmlhttps://liberty.io/find-the-equivalent-value-of-radians.htmlA function $f$ is called periodic with period $T$ if $f (x + T ) = f (x)$ (for all x) and $T$ is the smallest positive number with this property. The sine and cosine functions are periodic with per......Sat, 22 Aug 2026 19:28:49 -0400https://liberty.io/linear-algebra-how-to-interpret-vectors-and-equations-of-a-system-of-e.htmlhttps://liberty.io/linear-algebra-how-to-interpret-vectors-and-equations-of-a-system-of-e.htmlFrom Strang's 18.06 Linear Algebra OCW Course I realized that there is a row view and column view to interpret a matrix and the solution to Ax = b. Well the view made sense but it was hard understa......Sat, 22 Aug 2026 19:19:31 -0400https://liberty.io/what-does-ln-ln-x-simplify-to.htmlhttps://liberty.io/what-does-ln-ln-x-simplify-to.htmlWhat does $\ln(\ln(x))$ simplify to? First instinct was to write $\ln^2x$, but that looked odd, because $\ln(\ln(x))$ is not the same as $\ln(x)\ln(x)$ or $(\ln(x))^2$. Or is it ?...Sat, 22 Aug 2026 19:12:35 -0400https://liberty.io/ax-by-c-c-is-always-0.htmlhttps://liberty.io/ax-by-c-c-is-always-0.htmlFind an equation for the given line in the form $ax +by =​c$, where​ a, b, and c are integers with no factor common to all three and a ≥0. Through $(-40, 35)$; parallel to $7x + 8y = 13$ I put th......Sat, 22 Aug 2026 19:10:08 -0400https://liberty.io/suppose-you-draw-two-cards-from-a-deck-of-52-cards-without-replacement.htmlhttps://liberty.io/suppose-you-draw-two-cards-from-a-deck-of-52-cards-without-replacement.htmlSuppose you draw two cards from a deck of 52 cards without replacement. 1) What is the probability that both cards are hearts? 2) What is the probability that exactly one of the cards is hearts......Sat, 22 Aug 2026 19:07:39 -0400https://liberty.io/if-x-4-x-4-then-x-3-4-x-3-is.htmlhttps://liberty.io/if-x-4-x-4-then-x-3-4-x-3-is.htmlIf $x+(4/x)=4$ then $x^3+(4/x)^3 $is. Note: $(x+(4/x))^3=4^3$ assuming $4/x=y$ then $(x+y)^3=64$ $x^3+y^3+3xy^2+3yx^2=64$ $x^3+y^3+3xy(x+y)=64$ $x^3+y^3+3*4(4)=64$ so $x^3+y^3=16$...Sat, 22 Aug 2026 19:04:04 -0400https://liberty.io/how-to-prove-cos-4x-sin-4x-cos-2x-sin-2x-is-always-0.htmlhttps://liberty.io/how-to-prove-cos-4x-sin-4x-cos-2x-sin-2x-is-always-0.htmlSo I have a small problem here where I have to prove the following : $$\cos^4x - \sin^4x - \cos^2x + \sin^2x = 0 $$ I know that the 2nd part is always $1$, so I need to prove that the first par......Sat, 22 Aug 2026 19:01:16 -0400https://liberty.io/norm-of-finite-flat-morphism.htmlhttps://liberty.io/norm-of-finite-flat-morphism.htmlLet $X,Y$ be two schemes and $f : X \to Y$ be a finite flat map. In the following document http://www.ms.u-tokyo.ac.jp/~t-saito/talk/eepr.pdf page 20, the author talks about the norm map $$f_* : f_*\...Sat, 22 Aug 2026 18:57:45 -0400https://liberty.io/if-x-is-the-remainder-when-a-multiple-of-4-is-divided-by-6-and-y-is-th.htmlhttps://liberty.io/if-x-is-the-remainder-when-a-multiple-of-4-is-divided-by-6-and-y-is-th.htmlThe question is: if $x$ is the remainder when a multiple of $4$ is divided by $6$, and $y$ is the remainder when a multiple of $2$ is divided by $3$, what is the greatest possible value of $x+y$? ......Sat, 22 Aug 2026 18:43:45 -0400https://liberty.io/predator-prey-dynamical-system-find-the-vector-x-k-for-a-certain-k.htmlhttps://liberty.io/predator-prey-dynamical-system-find-the-vector-x-k-for-a-certain-k.htmlConsider a predator-pray model with the matrix $A= \begin{bmatrix} 0.5 & 0.6\\ -0.2 & 1.4 \end{bmatrix}$ ($A$ is the population distribution) and the initial state is $\vec{x}_0 = \begin{...Sat, 22 Aug 2026 18:29:06 -0400