https://track.liberty.ioFresh celebrity buzz with instant attention value.en-usSun, 06 Sep 2026 16:28:18 -0400https://track.liberty.io/intersection-of-normal-extensions-is-a-normal-extension.htmlhttps://track.liberty.io/intersection-of-normal-extensions-is-a-normal-extension.htmlLet's assume that we have two normal extensions $L_1$ and $L_2$ which means that $L_1=K(A_1)$ and $L_2=K(A_2)$, where $A_1$ and $A_2$ are sets of roots of proper polynomials. How can I show that ...Sun, 06 Sep 2026 20:16:56 -0400https://track.liberty.io/formula-for-infinite-sum-of-a-geometric-series-with-increasing-term.htmlhttps://track.liberty.io/formula-for-infinite-sum-of-a-geometric-series-with-increasing-term.htmlI'm looking for the Expectation of the discrete random variable X, E[X], with pmf: $$p(x)=(\frac 16)^{x+1}, x=0,1,2,3...$$ so what I tried is as follows... $$E[X]= \sum_{0}^\infty xp(x) =$$ so t......Sun, 06 Sep 2026 20:13:01 -0400https://track.liberty.io/how-do-i-find-the-minimum-force-done-by-a-person-pushing-a-box-when-tw.htmlhttps://track.liberty.io/how-do-i-find-the-minimum-force-done-by-a-person-pushing-a-box-when-tw.htmlThe problem is as follows: A box of $320\,N$ is at rest over a horizontal terrain. The coefficients of friction between the box and the terrain are $0.36$ and $0.48$. A girl is pushing the box ...Sun, 06 Sep 2026 20:08:10 -0400https://track.liberty.io/is-the-metric-tensor-relative-to-a-reference-coordinate-system.htmlhttps://track.liberty.io/is-the-metric-tensor-relative-to-a-reference-coordinate-system.htmlI am fairly new to this topic, and I don't know anything about tensors, but I have to know what a metric tensor is and this is how I have been thinking about it: 1)A metric tensor is a mathematical ...Sun, 06 Sep 2026 19:56:29 -0400https://track.liberty.io/what-is-the-fastest-way-to-find-factors-of-a-number-that-add-up-to-a-s.htmlhttps://track.liberty.io/what-is-the-fastest-way-to-find-factors-of-a-number-that-add-up-to-a-s.htmlSo I am studying for the SAT and am re-learning the grouping method for factoring. The current problem I'm on is $$7m^2 +59m+24$$ With grouping, the traditional way is to find two numbers that are ...Sun, 06 Sep 2026 19:37:22 -0400https://track.liberty.io/what-does-cayley-table-for-a-group-mathbb-z-5-cdot-5-tell-us.htmlhttps://track.liberty.io/what-does-cayley-table-for-a-group-mathbb-z-5-cdot-5-tell-us.htmlFirsly, I would like you to explain me what $\mathbb{Z}_5^*$ means. My teacher told me it is a group of units of $\mathbb{Z}_5$, but I'm not sure what a group of units is. If $\mathbb{Z}_5^*=\{0,1,......Sun, 06 Sep 2026 19:34:38 -0400https://track.liberty.io/how-do-i-find-a-unit-vector-orthogonal-to-a-line.htmlhttps://track.liberty.io/how-do-i-find-a-unit-vector-orthogonal-to-a-line.htmlI am looking at an academic paper. In one section of it I need to find a line orthogonal to the between two points. The paper says: Given a point p ∈ δΩ the normal direction is computed as f......Sun, 06 Sep 2026 19:28:56 -0400https://track.liberty.io/if-l-is-context-free-is-l-context-free-as-well.htmlhttps://track.liberty.io/if-l-is-context-free-is-l-context-free-as-well.htmlI know CFL in closed under Closure, but don't know how to prove it back. I try to construct a CFG for L* such as S->ε|AS and say since L* in a context-free language, there must exist a method to......Sun, 06 Sep 2026 19:25:32 -0400https://track.liberty.io/which-one-is-larger-8-factorial-to-the-power-of-1-8-or-9-factorial-to-.htmlhttps://track.liberty.io/which-one-is-larger-8-factorial-to-the-power-of-1-8-or-9-factorial-to-.htmlWhich one is larger? 8 factorial to the power of 1/8 or 9 factorial to the power of 1/9. Show your work without calculator....Sun, 06 Sep 2026 19:25:04 -0400https://track.liberty.io/the-subring-test.htmlhttps://track.liberty.io/the-subring-test.htmlThis is how the Wikipedia article on subring defines the subring test The subring test states that for any ring $R$, a nonempty subset of $R$ is a subring if it is closed under addition and ...Sun, 06 Sep 2026 19:12:34 -0400https://track.liberty.io/what-is-a-standard-curve.htmlhttps://track.liberty.io/what-is-a-standard-curve.htmlI know you may find definitions out there for standard curve. But, how can we define it in a way that makes it more understandable and clear of what it does?...Sun, 06 Sep 2026 19:08:50 -0400https://track.liberty.io/derivative-of-quaternions.htmlhttps://track.liberty.io/derivative-of-quaternions.htmlI am trying to calculate the Jacobian of a function that has quaternions and 3D points in it. I refer to quaternions as $q$ and 3D points as $p$ $$h_1(q)=A C(q)p $$ $$h_2(q)=q_1\otimes q \otimes q_......Sun, 06 Sep 2026 18:56:42 -0400https://track.liberty.io/do-counit-unit-adjunctions-have-the-expected-universal-properties.htmlhttps://track.liberty.io/do-counit-unit-adjunctions-have-the-expected-universal-properties.htmlLet a unit-counit adjunction between the categories $C$ and $D$ be defined as a pair of functors $F: D\rightarrow C$ and $G: C\rightarrow D$ with natural transformations $\varepsilon : FG \rightarr......Sun, 06 Sep 2026 18:52:33 -0400https://track.liberty.io/a-theorem-about-three-lines-any-two-of-them-being-coplanar.htmlhttps://track.liberty.io/a-theorem-about-three-lines-any-two-of-them-being-coplanar.htmlI am pretty convinced the following result holds in dimension 3: If three non-coplanar lines are such that any two of them are coplanar, then either all three lines are parallel, or they all ...Sun, 06 Sep 2026 18:47:29 -0400https://track.liberty.io/examples-of-some-pointwise-convergent-sequences-of-functions.htmlhttps://track.liberty.io/examples-of-some-pointwise-convergent-sequences-of-functions.htmlI have recently come across pointwise/uniformly convergent sequences of functions, and I am hoping if someone could give some examples of certain sequences of functions so that I could understand the ...Sun, 06 Sep 2026 18:36:20 -0400https://track.liberty.io/limit-question-with-bernstein-polynomial-approximating-f-x-x-2.htmlhttps://track.liberty.io/limit-question-with-bernstein-polynomial-approximating-f-x-x-2.htmlI am reading Atkinson's "An Introduction to Numerical Analysis". I am trying to verify a limit on page 198 involving the Bernstein polynomial approximating $f(x) = x^2$. The statement in the book ......Sun, 06 Sep 2026 18:35:15 -0400https://track.liberty.io/what-is-the-proper-way-to-study-more-advanced-math.htmlhttps://track.liberty.io/what-is-the-proper-way-to-study-more-advanced-math.htmlHere's what happens. I get stuck on some proof or some mathematical construction and I end up staring at the problem for hours, sometimes not making any progress. I do this because sometimes I thin......Sun, 06 Sep 2026 18:34:29 -0400https://track.liberty.io/computational-complexity-of-the-matlab-039-s-solver-for-solving-nonlin.htmlhttps://track.liberty.io/computational-complexity-of-the-matlab-039-s-solver-for-solving-nonlin.htmlI have to solve a nonlinear (trigonometric) equation (that always has only one solution) $$f(x)=0$$ where function $f: D \subset \mathbb{R} \to \mathbb{R}$ is strictly increasing. I used the MATLAB ...Sun, 06 Sep 2026 18:20:48 -0400https://track.liberty.io/finding-volume-of-a-frustum-of-a-pyramid.htmlhttps://track.liberty.io/finding-volume-of-a-frustum-of-a-pyramid.htmlI need to find the volume of a frustum of a pyramid with square base of side $b$, square top of side $a$, and height $h$(using integrals). I have no idea how to do questions like these, I only know......Sun, 06 Sep 2026 18:12:40 -0400https://track.liberty.io/proof-by-contradiction-negation-of-conjunction.htmlhttps://track.liberty.io/proof-by-contradiction-negation-of-conjunction.htmlConsider we have the following statement to prove: $P \implies Q \wedge R$. For a proof by contradiction, we assume $P \wedge (\neg Q \lor \neg R)$. How would one go about this? Typically to prov......Sun, 06 Sep 2026 18:02:24 -0400https://track.liberty.io/parallel-algorithms-for-svd.htmlhttps://track.liberty.io/parallel-algorithms-for-svd.htmlI just have completed a preliminar theoretical study of the important SVD decomposition. Now, I'm moving to numerical calculation of SVD. I would like to learn directly a parallel algorithm to ...Sun, 06 Sep 2026 17:34:42 -0400https://track.liberty.io/using-semantic-arguments-in-proving-validity.htmlhttps://track.liberty.io/using-semantic-arguments-in-proving-validity.htmlI have to show that $\forall x (\phi \to \psi) \to (\forall x \phi \to \forall x \psi)$ is valid using semantic arguments. However, what does "semantic arguments" mean? Do I use rules of inference?...Sun, 06 Sep 2026 17:26:45 -0400https://track.liberty.io/laplace-transform-involving-heaviside-step-function.htmlhttps://track.liberty.io/laplace-transform-involving-heaviside-step-function.htmlI'm trying to find the Laplace transform of $7 e^{-3t} u(t-3)$, where $u$ is the heaviside step function. However, we've never really gone through what the Laplace transform of the heaviside step ...Sun, 06 Sep 2026 17:11:44 -0400https://track.liberty.io/solution-for-the-inequality-x-1-x-2-x-3-0.htmlhttps://track.liberty.io/solution-for-the-inequality-x-1-x-2-x-3-0.htmlRecently I came across an inequality like this: $$(x-1)(x-2)(x-3)>0$$ The question was: Which solution for this in inequality is correct? $x>3$ or $x<1$ $x>3$ or $1<x<2$ $x&l......Sun, 06 Sep 2026 17:10:18 -0400https://track.liberty.io/positive-definite-and-transpose.htmlhttps://track.liberty.io/positive-definite-and-transpose.htmlWhen a matrix A has m rows and n columns (m>n), explain why $AA^{T}$ can't be positive definite. For the same matrix A, is $A^{T}A$ always positive definite? If so, explain. If not, what is the ...Sun, 06 Sep 2026 17:09:26 -0400https://track.liberty.io/graph-theory-what-is-the-definition-of-the-sorted-edge-algorithm.htmlhttps://track.liberty.io/graph-theory-what-is-the-definition-of-the-sorted-edge-algorithm.htmlI've been googling for a while and can't find a clear definition of the "sorted edge" algorithm--can anyone provide it please? A description would be helpful, but a simple statement of the algorit......Sun, 06 Sep 2026 17:09:25 -0400https://track.liberty.io/definition-of-very-ample-divisor.htmlhttps://track.liberty.io/definition-of-very-ample-divisor.htmlSorry for my bad English. Let $X$ be scheme over $Y$, and $\mathscr{L}$ be invertible sheaf on $X$. By definition of Hartshorne's algebraic geometry (p120), $\mathscr{L}$ is very ample relative to......Sun, 06 Sep 2026 17:04:36 -0400https://track.liberty.io/limit-approaching-0-of-a-rational-defined-function.htmlhttps://track.liberty.io/limit-approaching-0-of-a-rational-defined-function.htmlFor $f(x)=\begin{cases} e^{x^2}-1, & x \in \mathbb Q\\[2ex] 0, & x \not\in \mathbb Q \end{cases} $ Evaluate $\lim_{x\to 0}f(x)$. Can someone point me in the right direction? I have no ......Sun, 06 Sep 2026 16:58:45 -0400https://track.liberty.io/calculate-in-cm-the-length-of-segment-kl.htmlhttps://track.liberty.io/calculate-in-cm-the-length-of-segment-kl.htmlLet ABCD be a rectangle such that AB = $ \sqrt {2} BC $. Let E be a point on the semicircle with diameter AB, as shown in the following figure. Let K and L be the intersections of AB with ED and EC, ...Sun, 06 Sep 2026 16:53:46 -0400https://track.liberty.io/limit-comparison-test-for-checking-the-convergence-of-an-infinite-seri.htmlhttps://track.liberty.io/limit-comparison-test-for-checking-the-convergence-of-an-infinite-seri.htmlHere's the limit comparison test from an online source: $$\lim_\limits{n \to \infty} \frac{a_{n}}{b_{n}}=c$$ Where $a_{n}$ and $b_{n}$ are the general terms of two different infinite series. If $c$......Sun, 06 Sep 2026 16:41:38 -0400https://track.liberty.io/in-triangle-abc-with-ab-ac-and-angle-bac-20-circ-d-is-on-ac-with-bc-ad.htmlhttps://track.liberty.io/in-triangle-abc-with-ab-ac-and-angle-bac-20-circ-d-is-on-ac-with-bc-ad.htmlIn $\triangle ABC$ with $AB=AC$ and $\angle BAC=20^\circ$, point $D$ is on $AC$, with $BC=AD$. Find $\angle DBC$. I know the correct solution, but I'm more interested in where is the problem in my ...Sun, 06 Sep 2026 16:37:46 -0400https://track.liberty.io/use-calculus-to-maximize-the-area-of-a-triangle.htmlhttps://track.liberty.io/use-calculus-to-maximize-the-area-of-a-triangle.htmlWhat is the maximum area of a triangle that can be inscribed in a circle with radius of $5$ units. Solve using calculus. Is there a way to use $A = 0.5 \times b \times h$ and take the derivative?...Sun, 06 Sep 2026 16:30:20 -0400https://track.liberty.io/how-can-i-solve-a-triangle-knowing-its-area-one-side-and-an-opposite-a.htmlhttps://track.liberty.io/how-can-i-solve-a-triangle-knowing-its-area-one-side-and-an-opposite-a.htmlI need to calculate the missing elements of a triangle, knowing its area one side and the angle opposite the given side. The triangle is not a right angle triangle, nor is it equilateral or isoscel......Sun, 06 Sep 2026 16:24:59 -0400https://track.liberty.io/find-depth-of-a-half-filled-parabolic-cross-section.htmlhttps://track.liberty.io/find-depth-of-a-half-filled-parabolic-cross-section.htmlGiven a cross-section of an object that is parabolic in shape, how do you find the depth of the object when it is "half full". A full example given in an exam: A long trough whose cross-section......Sun, 06 Sep 2026 16:24:24 -0400https://track.liberty.io/definition-of-an-algebraic-singularity.htmlhttps://track.liberty.io/definition-of-an-algebraic-singularity.htmlI'm reading text about generating functions and how to reveal their asymptotic behaviour by means of analysing their singularities. In this context the term "algebraic singularity" or in German "...Sun, 06 Sep 2026 16:14:26 -0400https://track.liberty.io/how-to-make-a-semicircle-graph.htmlhttps://track.liberty.io/how-to-make-a-semicircle-graph.htmlWhat is the formula to make a semicircle graph that is continuous? By continuous I mean like a sine or cos graph but shaped like semicircles one after the other. Thanks...Sun, 06 Sep 2026 16:13:08 -0400https://track.liberty.io/how-to-prove-a-graph-asymmetric.htmlhttps://track.liberty.io/how-to-prove-a-graph-asymmetric.htmlFind a 3-regular graph that is has no other automorphism other than the identity. I searched and found that this means the graph is asymmetric and there is an example: the Frucht graph. But can so......Sun, 06 Sep 2026 16:13:00 -0400https://track.liberty.io/what-is-the-derivative-of-the-folium-of-descartes.htmlhttps://track.liberty.io/what-is-the-derivative-of-the-folium-of-descartes.html$$x^3+y^3=6xy\tag{1}\label{eq1}$$ $$\frac{dy}{dx}=\frac{\left(2y-x^{2}\right)}{y^{2}-2x}\tag{2}\label{eq2}$$ I embed the derivative $(2)$ into a 3D space by including variable $z$: $$z=\frac{\left(......Sun, 06 Sep 2026 16:11:35 -0400https://track.liberty.io/range-of-nonlinear-function.htmlhttps://track.liberty.io/range-of-nonlinear-function.htmlI'd like to know the range of function $$f(x)=(x^2, x^3)$$ and how to proceed in these problems. (The domain is $R$ while the codomain is $R^2$.) What is the image of interval $[0,1]$?...Sun, 06 Sep 2026 15:59:34 -0400https://track.liberty.io/solve-t-039-x-45-0-75-cdot-t-x-where-t-0-75.htmlhttps://track.liberty.io/solve-t-039-x-45-0-75-cdot-t-x-where-t-0-75.htmlI need help with homework: Let $T'(x)=45-0.75\cdot T(x)$, where $T(0)=75$. Find $T(x)$. Here is what I have tried: Let $T(x)=y.$ Rewrite the equation $$\frac{\mathrm dy}{\mathrm dx}=45-0.75y\i......Sun, 06 Sep 2026 15:58:49 -0400https://track.liberty.io/what-base-number-system-is-the-hebrew-language.htmlhttps://track.liberty.io/what-base-number-system-is-the-hebrew-language.htmlIn the hebrew bible, there number system is based off of hebrew letters. There is a single digit used going all the way up to 10, then it uses two digits... Untill it gets to 20, which is a single ......Sun, 06 Sep 2026 15:54:22 -0400https://track.liberty.io/structured-transfer-function-estimation-in-matlab.htmlhttps://track.liberty.io/structured-transfer-function-estimation-in-matlab.htmlI have some input-output data for a dynamical system (input = stimulus, output = observation). Assuming that this system is linear time-invariant, I am trying to estimate a transfer function $H(s)$......Sun, 06 Sep 2026 15:30:33 -0400https://track.liberty.io/support-of-a-measure-and-set-where-it-is-concentrated.htmlhttps://track.liberty.io/support-of-a-measure-and-set-where-it-is-concentrated.htmlI have some problems understanding the difference between the notion of support of a measure and saying that a measure is concentrated on a certain set. If I have a $\sigma$-algebra and $\mu$ a me......Sun, 06 Sep 2026 15:19:32 -0400https://track.liberty.io/how-can-i-write-the-axiom-of-specification-as-a-sentence.htmlhttps://track.liberty.io/how-can-i-write-the-axiom-of-specification-as-a-sentence.htmlI began reading Paul Halmos' "Naive Set Theory", and encountered the "Axiom of Specification". To every set $A$ and to every condition $S(x)$ there corresponds a set $B$ whose elements are exactly ...Sun, 06 Sep 2026 15:14:56 -0400https://track.liberty.io/find-series-expansion-of-1-cosx.htmlhttps://track.liberty.io/find-series-expansion-of-1-cosx.htmlFind the series expansion of 1/cosx from basic series expansions. I tried to find 1/cosx from the expansion of cosx but was unsure how to continue. When I found 1/cosx from the basic formula for f......Sun, 06 Sep 2026 15:14:52 -0400https://track.liberty.io/how-to-find-a-3x3-matrix-with-determinant-0-from-which-i-can-delete-ra.htmlhttps://track.liberty.io/how-to-find-a-3x3-matrix-with-determinant-0-from-which-i-can-delete-ra.htmlI need to find a $3 \times 3$ matrix and the determinant of this matrix has to be $0$. I also need to be able to delete randomly chosen column and row to make the determinant nonzero? Is it even ...Sun, 06 Sep 2026 15:12:37 -0400https://track.liberty.io/finding-an-unknown-value-in-a-matrix.htmlhttps://track.liberty.io/finding-an-unknown-value-in-a-matrix.htmlI've been doing lots of matrices practice today and I've come across this one which I'm finding quite tricky. $$ 𝐴 = \begin{pmatrix} 3 & 5 \\ z & -3 \end{pmatrix} $$ All I'm told is that......Sun, 06 Sep 2026 15:05:53 -0400https://track.liberty.io/mathematics-engineering-how-do-you-prove-the-power-rule.htmlhttps://track.liberty.io/mathematics-engineering-how-do-you-prove-the-power-rule.htmlConsider the real-valued function $f(x)=x^r$ where $r$ is a real number. (1) For what values of $x$ and $r$, is this function differentiable? (2) How do you prove the power rule, when $f(x)$ is ...Sun, 06 Sep 2026 15:02:26 -0400https://track.liberty.io/volume-of-irregular-solid.htmlhttps://track.liberty.io/volume-of-irregular-solid.htmlI need to calculate volume of irregular solid which is having fix $200 \times 200$ width and breadth but all four points varies in depth. I have table which gives depth at each point. How to calc......Sun, 06 Sep 2026 15:01:33 -0400https://track.liberty.io/understanding-a-sobolev-embedding-theorem.htmlhttps://track.liberty.io/understanding-a-sobolev-embedding-theorem.htmlIn my adv. Analysis course, we have studied the following Sobolev Embedding Theorem: Let $m\in\mathbb{N}$ and $s>m+d/2$. Then $$H^s(\mathbb{R}^d)\hookrightarrow C_0^m(\mathbb{R}^d)$$That is: ......Sun, 06 Sep 2026 15:00:36 -0400https://track.liberty.io/is-there-a-way-to-calculate-absurdly-high-powers-closed.htmlhttps://track.liberty.io/is-there-a-way-to-calculate-absurdly-high-powers-closed.htmlCould it be at all possible to calculate, say, $2^{250000}$, which would obviously have to be written in standard notation? It seems impossible without running a program on a supercomputer to work ......Sun, 06 Sep 2026 14:56:29 -0400https://track.liberty.io/convert-recursive-to-closed-formula.htmlhttps://track.liberty.io/convert-recursive-to-closed-formula.htmlI got a particular sequence defined by the following recursive function: $$T_n = T_{n-1} \times 2 - T_{n-10}$$ I need help converting it to a closed form so I can calculate very large values of n ...Sun, 06 Sep 2026 14:53:49 -0400https://track.liberty.io/is-there-a-sign-for-039-not-less-than-039-039-not-greater-than-039-etc.htmlhttps://track.liberty.io/is-there-a-sign-for-039-not-less-than-039-039-not-greater-than-039-etc.htmlI was wondering about this, just now, because I was trying to write something like: $880$ is not greater than $950$. I am wondering this because there is a 'not equal to': $\not=$ Not equal to is......Sun, 06 Sep 2026 14:44:40 -0400https://track.liberty.io/are-5-and-4x-algebraic-expressions.htmlhttps://track.liberty.io/are-5-and-4x-algebraic-expressions.htmlIn an elementary textbook I have, an algebraic expression is defined as The expressions in which the numbers, or variables, or both, are connected by operational signs (+, - etc.) are called ...Sun, 06 Sep 2026 14:25:56 -0400https://track.liberty.io/definition-of-upper-hemicontinuity-of-a-correspondence.htmlhttps://track.liberty.io/definition-of-upper-hemicontinuity-of-a-correspondence.htmlWhen using and examining Kakutani's fixed-point theorem, I've got a question about upper hemicontinuity. A correspondence $f:X\rightarrow2^Y$ is a point-to-set mapping. One way to define upper ...Sun, 06 Sep 2026 14:25:45 -0400https://track.liberty.io/is-it-possible-to-predict-number-of-edges-in-a-strongly-connected-dire.htmlhttps://track.liberty.io/is-it-possible-to-predict-number-of-edges-in-a-strongly-connected-dire.htmlI know that A graph that is complete and undirected with $n$ nodes will have $n(n-1)/2$ edges A graph that is complete and directed with $n$ nodes will have $n(n-1)$ edges A graph that is connec......Sun, 06 Sep 2026 14:20:15 -0400https://track.liberty.io/trigonometric-identities-with-multiplication.htmlhttps://track.liberty.io/trigonometric-identities-with-multiplication.htmlWhy aren't there Trigonometric identities with multiplication inside the function? For example for $\sin(xy)=?$....Sun, 06 Sep 2026 14:04:56 -0400https://track.liberty.io/integrating-a-function-of-tan-inverse.htmlhttps://track.liberty.io/integrating-a-function-of-tan-inverse.htmlHow would I carry out the following integration? $$\int_0^12\arctan x^2$$ I tried to substitute $x^2 = \tan\theta$ From there I wrote the integral as: $$\int_0^{\pi/4}\theta\cdot \frac{\sec^2\thet......Sun, 06 Sep 2026 13:57:50 -0400https://track.liberty.io/is-the-following-generalization-of-cauchy-schwarz-inequality-true.htmlhttps://track.liberty.io/is-the-following-generalization-of-cauchy-schwarz-inequality-true.htmlLet $\{a_{1,i}\}_{i=1}^k,\{a_{2,i}\}_{i=1}^k,\dots ,\{a_{n_,i}\}_{i=1}^k$ be real sequences. Does the following inequality hold $$(\sum_{i=1}^k a_{1,i}^2)\cdot(\sum_{i=1}^k a_{2,i}^2)\cdots(\sum_{i......Sun, 06 Sep 2026 13:56:48 -0400https://track.liberty.io/how-to-make-a-function.htmlhttps://track.liberty.io/how-to-make-a-function.htmlI dropped out of school early when I was still a teenager and now I'm trying to take my GED. I'm really close to passing but I'm still having trouble understanding some concepts. In the pre-test, ......Sun, 06 Sep 2026 13:51:14 -0400https://track.liberty.io/the-limit-of-a-product-of-functions-equals-the-product-of-the-limits-i.htmlhttps://track.liberty.io/the-limit-of-a-product-of-functions-equals-the-product-of-the-limits-i.htmlAll the proofs I've seen so for the limit of a product of functions equaling the product of the limits are based on the following: Let $f$ and $g$ be real or complex functions having the limits ......Sun, 06 Sep 2026 13:49:39 -0400https://track.liberty.io/t-n-b-and-the-curvature-and-the-equation-of-the-osculating-plane-at-a-.htmlhttps://track.liberty.io/t-n-b-and-the-curvature-and-the-equation-of-the-osculating-plane-at-a-.htmlGiven a vector valued function defined by $r(t):=(2t)i+(t^2)j+(\ln t)k$,Find the unit vectors $\vec T,\vec N,\vec B$ at the point $P(2,1,0)$,then find the curvature and the equation of the osculating ...Sun, 06 Sep 2026 13:38:49 -0400https://track.liberty.io/what-is-the-difference-between-disjoint-union-and-union.htmlhttps://track.liberty.io/what-is-the-difference-between-disjoint-union-and-union.htmlIf $S = A \cup B$, then $S$ is the collection of all points in $A$ and $B$ What about $S = A \sqcup B$?, I think disjoint union is the same as union, only $A, B$ are disjoint. So the notation is a......Sun, 06 Sep 2026 13:35:53 -0400https://track.liberty.io/distribution-of-log-x-if-x-is-uniform.htmlhttps://track.liberty.io/distribution-of-log-x-if-x-is-uniform.htmlFor $X$ and $Y$ random variables; $X$ follows the uniform distribution. (1): if $Y=-\log X$ (2): then it can be shown that $-\log X$ is distributed as $\exp(1)$ {i.e. exponential with mean 1}. W......Sun, 06 Sep 2026 13:31:27 -0400https://track.liberty.io/proving-a-relation-is-partial-ordering.htmlhttps://track.liberty.io/proving-a-relation-is-partial-ordering.htmlI have a problem proving that a very simple relation is partial ordering. It is defined explicitly (i.e. with pairs of numbers) and I have no idea how to do a formal proof for its antisymmetric pro......Sun, 06 Sep 2026 13:30:55 -0400https://track.liberty.io/multiplication-of-two-upper-triangular-matrices.htmlhttps://track.liberty.io/multiplication-of-two-upper-triangular-matrices.htmlConsider the product $C=(C_{ij})$ of two upper triangular matrices $A=(a_{ij})$ and $B=(b_{ij})$, with $n=m$ (rows=columns). Deduce the expression of $C=(C_{ij})$. I'm trying to do this proof but ......Sun, 06 Sep 2026 13:22:38 -0400https://track.liberty.io/rotation-matrix-of-rotation-around-a-point-other-than-the-origin.htmlhttps://track.liberty.io/rotation-matrix-of-rotation-around-a-point-other-than-the-origin.htmlIn homogeneous coordinates, a rotation matrix around the origin can be described as $R = \begin{bmatrix}\cos(\theta) & -\sin(\theta) & 0\\\sin(\theta) & \cos(\theta) & 0 \\ 0&0......Sun, 06 Sep 2026 13:21:57 -0400https://track.liberty.io/calculating-christoffel-symbol-of-the-second-kind-in-coordinates-where.htmlhttps://track.liberty.io/calculating-christoffel-symbol-of-the-second-kind-in-coordinates-where.htmlExercise 86,pg-73 of Pavel Grinfeld's Tensor Calculus Book: Derive the Christoffel symbols in spherical and cylindrical coordinates by the equation:$$\Gamma_{ij}^k= \frac12 Z^{km} \left[ \frac{\par......Sun, 06 Sep 2026 13:14:55 -0400https://track.liberty.io/how-to-find-time-period-of-a-discrete-time-signal.htmlhttps://track.liberty.io/how-to-find-time-period-of-a-discrete-time-signal.html𝑥[𝑛] = cos (𝑛/8− 𝜋) Getting the value of cos(0)=1. For that we get n=8𝜋 and for again getting cos(x)=1, we can put n=24𝜋. So the value is repeating itself in the interval of 16𝜋. How do i s......Sun, 06 Sep 2026 13:10:55 -0400https://track.liberty.io/implication-and-interpretation-of-banach-tarski.htmlhttps://track.liberty.io/implication-and-interpretation-of-banach-tarski.htmlAs I understand, the Banach-Tarski paradox says a ball in 3-space may be decomposed into finitely many pieces and reassembled into two balls each of the same size as the original. Despite being cal......Sun, 06 Sep 2026 13:07:42 -0400https://track.liberty.io/big-oh-notation-of-7x-2-confused.htmlhttps://track.liberty.io/big-oh-notation-of-7x-2-confused.htmlI'm supposed to figure out the Big-Oh notation of $7x^2$. Take a look at this. Now this says: Show that $7x^2$ is $O(x^3)$ When $x>7, 7x^2<x^3$, So let $C=1$ and $k=7$, we see $7x^2$ is......Sun, 06 Sep 2026 13:07:25 -0400https://track.liberty.io/solve-limit-containing-1-0-using-l-039-hopital-039-s-rule.htmlhttps://track.liberty.io/solve-limit-containing-1-0-using-l-039-hopital-039-s-rule.htmlI was assigned a problem in my calculus 2 course that I can't seem to solve. We're going over indeterminate forms solved by L'Hopital's Rule. $$ \lim_{x\to \pi^-} \left( \frac{\csc(x)+x}{\tan(\fr......Sun, 06 Sep 2026 13:06:41 -0400https://track.liberty.io/what-set-does-mathbb-w-denote.htmlhttps://track.liberty.io/what-set-does-mathbb-w-denote.htmlWhat set does $\mathbb W$ denote? I know this may horribly lack context, but I've seen multiple times on M.SE $\mathbb W$ used in some fairly elementary context I think....Sun, 06 Sep 2026 13:04:06 -0400https://track.liberty.io/question-about-right-triangle-and-sin-2theta.htmlhttps://track.liberty.io/question-about-right-triangle-and-sin-2theta.htmlThis is a pretty basic question but I just wanted clarification. I know that sin(theta) is opposite/hypotenuse regarding right triangles. But what would sin(2theta) be? would it be (opposite/hypote......Sun, 06 Sep 2026 12:57:20 -0400https://track.liberty.io/lattice-points-in-x-y-plane.htmlhttps://track.liberty.io/lattice-points-in-x-y-plane.htmlWhat are Lattice Points? Which points in x-y planes are Lattice Points? Is (m,n) a lattice point where m,n are any integers?...Sun, 06 Sep 2026 12:53:20 -0400https://track.liberty.io/why-is-l-infty-not-separable.htmlhttps://track.liberty.io/why-is-l-infty-not-separable.htmlMy functional analysis textbook says "The metric space $l^\infty$ is not separable." The metric defined between two sequences $\{a_1,a_2,a_3\dots\}$ and $\{b_1,b_2,b_3,\dots\}$ is $\sup\limits_......Sun, 06 Sep 2026 12:48:19 -0400https://track.liberty.io/why-do-people-interchange-between-int-and-sum-so-easily.htmlhttps://track.liberty.io/why-do-people-interchange-between-int-and-sum-so-easily.htmlOne of the things I found curious in many texts is how in certain cases interchange the $\sum$ operator with $\int$. What are the "terms" for such a swap? I understand that integration in the early......Sun, 06 Sep 2026 12:48:15 -0400https://track.liberty.io/n-points-can-be-equidistant-from-each-other-only-in-dimensions-ge-n-1.htmlhttps://track.liberty.io/n-points-can-be-equidistant-from-each-other-only-in-dimensions-ge-n-1.html2 points are from equal distance to each other in dimensions 1,2,3,... 3 points can be equidistant from each other in 2,3,... dimensions 4 points can be equidistant from each other only in dimen......Sun, 06 Sep 2026 12:47:11 -0400https://track.liberty.io/proving-when-gram-039-s-determinant-is-equal-to-zero-duplicate.htmlhttps://track.liberty.io/proving-when-gram-039-s-determinant-is-equal-to-zero-duplicate.htmlProve that Gram's determinant $G(x_1,\dots, x_n)=0$ if and only if $x_1, \dots, x_k$ are linearly dependent. So I know that $G(x_1,\dots, x_n)=\det \begin{vmatrix} \xi( x_1,x_1) & \xi( x_1,......Sun, 06 Sep 2026 12:35:03 -0400https://track.liberty.io/how-to-solve-function-equations.htmlhttps://track.liberty.io/how-to-solve-function-equations.htmlConsider: $$g(x)=x^{1/3}$$ $$f(g(x))=2x+3$$ Find $f(x)$ It should be fairly obvious that a solution is $f(x)=2x^3+3$ But is this solution unique? And what method should I use to solve such an ...Sun, 06 Sep 2026 12:34:53 -0400https://track.liberty.io/where-can-i-find-a-review-of-discrete-math.htmlhttps://track.liberty.io/where-can-i-find-a-review-of-discrete-math.htmlI'm looking for course notes and assignments and hopefully some example exams for Discrete Math, I'm taking a placement exam in the subject after having taken it 4 years ago....Sun, 06 Sep 2026 12:05:36 -0400https://track.liberty.io/why-historically-do-we-multiply-matrices-as-we-do.htmlhttps://track.liberty.io/why-historically-do-we-multiply-matrices-as-we-do.htmlMultiplication of matrices — taking the dot product of the $i$th row of the first matrix and the $j$th column of the second to yield the $ij$th entry of the product — is not a very intu......Sun, 06 Sep 2026 12:04:12 -0400https://track.liberty.io/proof-of-the-chain-rule-for-calculus.htmlhttps://track.liberty.io/proof-of-the-chain-rule-for-calculus.htmlI was comparing my attempt to prove the chain rule by my own and the proof given in Spivak's book but they seems to be rather different. Please tell me if I'm wrong or if I'm missing something. I r......Sun, 06 Sep 2026 12:00:57 -0400https://track.liberty.io/definition-of-compact-support.htmlhttps://track.liberty.io/definition-of-compact-support.htmlAccording to http://mathworld.wolfram.com/CompactSupport.html , A function has compact support if it is zero outside of a compact set. Alternatively, one can say that a function has compact suppo......Sun, 06 Sep 2026 12:00:56 -0400https://track.liberty.io/question-39-in-folland-039-s-real-analysis-chapter-3.htmlhttps://track.liberty.io/question-39-in-folland-039-s-real-analysis-chapter-3.htmlThe question "If {$F_j$} is a sequence of nonnegative increasing functions on $[a,b]$ such that $F(x)= \sum_1^\infty F_j(x) < \infty$" for all $x \in [a,b]$, then $F\prime(x)=\sum_1^\infty F\pri......Sun, 06 Sep 2026 11:49:45 -0400https://track.liberty.io/method-of-finding-a-p-adic-expansion-to-a-rational-number.htmlhttps://track.liberty.io/method-of-finding-a-p-adic-expansion-to-a-rational-number.htmlCould someone go though the method of finding a p-adic expansion of say $-\frac{1}{6}$ in $\mathbb{Z}_7?$...Sun, 06 Sep 2026 11:45:36 -0400https://track.liberty.io/what-does-mathbb-r-n-to-mathbb-r-m-mean-and-what-is-mathbb-r-n.htmlhttps://track.liberty.io/what-does-mathbb-r-n-to-mathbb-r-m-mean-and-what-is-mathbb-r-n.htmlWhat the does $\mathbb{R^n}$ mean? For example if something says that it is a transformation $T:\mathbb{R}^2 \rightarrow \mathbb{R}^3$. Does that mean that $\mathbb{R}^2 = 2 \times 2$ matrix? and t......Sun, 06 Sep 2026 11:43:23 -0400https://track.liberty.io/the-set-of-real-numbers-is-a-subset-of-the-set-of-complex-numbers.htmlhttps://track.liberty.io/the-set-of-real-numbers-is-a-subset-of-the-set-of-complex-numbers.htmlSo, I was taught that $\mathbb{Z}\subseteq\mathbb{Q}\subseteq\mathbb{R}$. But since the set of complex numbers is by definition $$\mathbb{C}=\{a+bi:a,b\in\mathbb{R}\},$$ doesn't this mean $\mathbb......Sun, 06 Sep 2026 11:36:23 -0400https://track.liberty.io/finding-the-equation-of-the-ellipse-by-completing-the-square.htmlhttps://track.liberty.io/finding-the-equation-of-the-ellipse-by-completing-the-square.htmlI have the following equation of an ellipse: $$5x^2+9y^2+40x=100$$ I need to put it in the form: $$\frac{(x-h)^2}{m^2} + \frac{(y-k)^2}{n^2} =1 $$ I was trying to complete the square with the ...Sun, 06 Sep 2026 11:33:35 -0400https://track.liberty.io/how-can-i-apply-absorption-law-in-this-expression.htmlhttps://track.liberty.io/how-can-i-apply-absorption-law-in-this-expression.htmlI have this expression: A + A * B * C' I know that i can use the Absorption law in this and the answer is A, but how can i apply that? Thank you....Sun, 06 Sep 2026 11:14:44 -0400https://track.liberty.io/why-measure-theory-for-probability.htmlhttps://track.liberty.io/why-measure-theory-for-probability.htmlI studied elementary probability theory. For that, density functions were enough. What is a practical necessity to develop measure theory? What is a problem that cannot be solved using elementary d......Sun, 06 Sep 2026 11:13:04 -0400https://track.liberty.io/convergence-of-the-series-involving-log-function.htmlhttps://track.liberty.io/convergence-of-the-series-involving-log-function.htmlI was asked about convergence of summation of $\log(n)/n^2$ and $\log(1+(1/n))/n^2$. I want to use only p series test. Is there any approximation for $\log(x)$?, Like sterling's approximation for ...Sun, 06 Sep 2026 11:05:53 -0400https://track.liberty.io/understanding-a-limiting-procedure.htmlhttps://track.liberty.io/understanding-a-limiting-procedure.htmlSay I have a function $f(x,y)$ that is defined for all $x \leq 0$ except at a single point $x=x_0 \neq 0.$ There exists one representation of the function in the domain $x < x_0$ and another ...Sun, 06 Sep 2026 11:02:58 -0400https://track.liberty.io/calculating-deposit-based-on-interest-and-required-withdrawal-in-futur.htmlhttps://track.liberty.io/calculating-deposit-based-on-interest-and-required-withdrawal-in-futur.htmlI am really stuck with example 2-5 and 2-6. I don't really understand example 2-6 and example 2-5 I just can't figure out...I was able to do example 2-4 which was easy... For example 2-4 I did F=P......Sun, 06 Sep 2026 10:59:04 -0400https://track.liberty.io/two-different-notations-for-inner-product.htmlhttps://track.liberty.io/two-different-notations-for-inner-product.htmlIn my linear algebra course on the faculty of mathematics, we used the following notation for the inner product: $$\langle v, w \rangle$$ On the other hand, my friends from the faculty of physics r......Sun, 06 Sep 2026 10:53:03 -0400https://track.liberty.io/give-a-big-o-estimate-for-a-function-f-x-use-a-simple-function-g-of-th.htmlhttps://track.liberty.io/give-a-big-o-estimate-for-a-function-f-x-use-a-simple-function-g-of-th.htmlTask Find a function $g(x)$ that is Big-O of $f(x)$ $f(x)$ is $\mathcal{O}(g(x))$ $f(x)=n\cdot log(n^2+1)+n^2\cdot log(n)$ I tried to follow/copy one of the examples in my book Discrete Mathematic......Sun, 06 Sep 2026 10:46:59 -0400https://track.liberty.io/solve-surface-integral-of-a-scalar-function-using-divergence-theorem.htmlhttps://track.liberty.io/solve-surface-integral-of-a-scalar-function-using-divergence-theorem.htmlUse the Divergence Theorem to evaluate: $$\iint_S (2x+2y+z^2) dS$$ where $S$ is the sphere $$x^2+y^2+z^2=1.$$ Most of the examples on the book that needs to be solved using divergence theorem are ...Sun, 06 Sep 2026 10:29:24 -0400https://track.liberty.io/how-can-i-find-the-basis-solutions-of-homogeneous-linear-ode.htmlhttps://track.liberty.io/how-can-i-find-the-basis-solutions-of-homogeneous-linear-ode.htmlSecond order linear differential equation is given below. $y''+\frac{2}{x}y'+k^2y=0,$ where $k$ is constant and $x\neq 0$ I already know that the basis are $y_1=\frac{e^{-ikx}}{x}$ and $y_2=\fra......Sun, 06 Sep 2026 10:28:46 -0400https://track.liberty.io/laplace-transform-for-solving-differential-equation.htmlhttps://track.liberty.io/laplace-transform-for-solving-differential-equation.htmlI solved the following task, but since I am new in this field I need to check if it is correct or if there is anything I am missing or doing wrong. Task : Solve differential equation using Laplace ...Sun, 06 Sep 2026 10:26:44 -0400https://track.liberty.io/user-allen-knutson-mathematics-stack-exchange.htmlhttps://track.liberty.io/user-allen-knutson-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Sun, 06 Sep 2026 10:26:08 -0400