https://track.liberty.ioFresh celebrity buzz with instant attention value.en-usSun, 06 Sep 2026 06:22:01 -0400https://track.liberty.io/how-do-i-graph-2-30-on-a-polar-grid-closed.htmlhttps://track.liberty.io/how-do-i-graph-2-30-on-a-polar-grid-closed.htmlI tried to graph the point $(-2,30)$ on a polar graph and I think I got it wrong, how should it be graphed?...Sun, 06 Sep 2026 10:12:26 -0400https://track.liberty.io/number-of-lines-to-connect-n-times-n-dots.htmlhttps://track.liberty.io/number-of-lines-to-connect-n-times-n-dots.htmlThe following is a popular riddle: Draw a $3 \times 3$ grid, and connect all the dots using only $4$ straight consecutive lines. The solution is to think outside of the box and do the followin......Sun, 06 Sep 2026 10:10:54 -0400https://track.liberty.io/uniqueness-of-hamiltonian-cycle.htmlhttps://track.liberty.io/uniqueness-of-hamiltonian-cycle.htmlLet G be a graph on $n$ vertices, where $n \geq 8$ and with minimum degree at least $n/2$. Does $G$ have at least 2 distinct Hamilton cycles? (Two Hamilton cycles are said to be distinct if they di......Sun, 06 Sep 2026 09:55:05 -0400https://track.liberty.io/solution-of-tanx-x.htmlhttps://track.liberty.io/solution-of-tanx-x.htmlHow do I find the solutions of tanx = x upto any number of decimals? (Of course, there is the graphical method but it just helps in finding the approximate value...)...Sun, 06 Sep 2026 09:53:29 -0400https://track.liberty.io/derivative-of-cost-function-for-logistic-regression.htmlhttps://track.liberty.io/derivative-of-cost-function-for-logistic-regression.htmlI am going over the lectures on Machine Learning at Coursera. I am struggling with the following. How can the partial derivative of $$J(\theta)=-\frac{1}{m}\sum_{i=1}^{m}y^{i}\log(h_\theta(x^{i}))+......Sun, 06 Sep 2026 09:50:06 -0400https://track.liberty.io/rotate-triangle-abc-around-the-origin.htmlhttps://track.liberty.io/rotate-triangle-abc-around-the-origin.htmlHow do you get from triangle ABC (blue) to triangle ABC (red) The instructions are to rotate it $270$ degrees I am trying to help a friend but forgot how to do this? Is it using a formula?...Sun, 06 Sep 2026 09:48:54 -0400https://track.liberty.io/what-is-the-opposite-of-a-cross-term.htmlhttps://track.liberty.io/what-is-the-opposite-of-a-cross-term.htmlWhen we multiply out $(x + y)(x + y)$, we refer to the two $xy$ terms as "cross terms". Is there a corresponding term for the $x^2$ and $y^2$ terms?...Sun, 06 Sep 2026 09:46:02 -0400https://track.liberty.io/what-is-the-difference-between-a-sound-argument-and-a-valid-argument.htmlhttps://track.liberty.io/what-is-the-difference-between-a-sound-argument-and-a-valid-argument.htmlIn my notes, these are the definitions of a valid argument An argument form is valid if and only if whenever the premises are all true, then conclusion is true. An argument is valid if its arg......Sun, 06 Sep 2026 09:44:17 -0400https://track.liberty.io/how-do-i-write-e-x-2-as-a-summation.htmlhttps://track.liberty.io/how-do-i-write-e-x-2-as-a-summation.htmlI am new to power series. I know how to write $e^x$ as a summation, but i do not know how that helps me....Sun, 06 Sep 2026 09:35:22 -0400https://track.liberty.io/how-to-calculate-critical-value-in-hypothesis-testing.htmlhttps://track.liberty.io/how-to-calculate-critical-value-in-hypothesis-testing.htmlYou wish to test the following claim ($H_a$) at a significance level of alpha=0.10. $H_o: \mu=70.4$ $H_a: \mu<70.4$ You believe the population is normally distributed and you know the standard ...Sun, 06 Sep 2026 09:31:48 -0400https://track.liberty.io/elementary-product-rule-for-ito-formula.htmlhttps://track.liberty.io/elementary-product-rule-for-ito-formula.htmlIn "Baxter, Rennie" book, there is an explanation of product rule for Ito formula. They apply Ito formula (without any details) to $$\frac{1}{2}((X_t + Y_t)^2 - X_t^2 - Y_t^2) = X_tY_t$$ and ob......Sun, 06 Sep 2026 09:14:39 -0400https://track.liberty.io/integrate-int-sqrt-frac-x-3-3-x-11-dx-closed.htmlhttps://track.liberty.io/integrate-int-sqrt-frac-x-3-3-x-11-dx-closed.html$$ \int \sqrt{\frac{x^3-3}{x^{11}}}\ dx $$ It seems like substitution could not do it. Is there another way?...Sun, 06 Sep 2026 09:14:37 -0400https://track.liberty.io/condition-of-the-mean-value-theorem.htmlhttps://track.liberty.io/condition-of-the-mean-value-theorem.htmlThe usual formulation of the mean value theorem in a real analysis course is something like this: Let $f\colon [a,b] \to \mathbb{R}$ be continous on $[a,b]$ and differentiable on $]a,b[$. Then the......Sun, 06 Sep 2026 09:12:43 -0400https://track.liberty.io/how-to-find-the-limit-of-cos-2x-x-as-x-approaches-zero-duplicate.htmlhttps://track.liberty.io/how-to-find-the-limit-of-cos-2x-x-as-x-approaches-zero-duplicate.htmlBasicly the title. I know it's easy but I can't figure it out. [Edit] Not a duplicate has a 2x instead of x....Sun, 06 Sep 2026 09:11:28 -0400https://track.liberty.io/find-from-first-principle-the-derivative-of.htmlhttps://track.liberty.io/find-from-first-principle-the-derivative-of.htmlFind, from first principle, the derivative of: $$\log (ax+b)$$ My Attempt: $$f(x)=\log (ax+b)$$ $$f(x+\Delta x)=\log (ax+a\Delta x+b)$$ Now, $$f'(x)=\lim_{\Delta x\to 0} \dfrac {f(x+\Delta x)-f(x)}{\...Sun, 06 Sep 2026 09:10:59 -0400https://track.liberty.io/why-two-line-coincide-in-the-product-of-reflections.htmlhttps://track.liberty.io/why-two-line-coincide-in-the-product-of-reflections.htmlExample. Let $l$ and $m$ be two different lines intersecting at a point $P$. Show that $R_m R_l =R_{p, \theta}$ where $B$ is twice the directed angle $\alpha$ from $l$ to $m$. Solution. Let $h$ be......Sun, 06 Sep 2026 09:07:16 -0400https://track.liberty.io/how-do-you-divide-an-inequality-by-another-inequality.htmlhttps://track.liberty.io/how-do-you-divide-an-inequality-by-another-inequality.htmlThe question I'm having trouble with is follows: Suppose you have already proved the proposition that, “If $a$ and $b$ are nonnegative real numbers, then $\frac{a + b}{2} ≥ \sqrt{ab}$.” a. Explai......Sun, 06 Sep 2026 09:01:14 -0400https://track.liberty.io/the-rotation-group-so-3-may-be-mapped-to-a-2-sphere-by-sending-a-rotat.htmlhttps://track.liberty.io/the-rotation-group-so-3-may-be-mapped-to-a-2-sphere-by-sending-a-rotat.htmlThe rotation group $SO(3)$ may be mapped to a $2$-sphere by sending a rotation matrix to its first column. How can I describe the fibres of the map?...Sun, 06 Sep 2026 09:00:41 -0400https://track.liberty.io/how-do-you-prove-a-number-is-prime.htmlhttps://track.liberty.io/how-do-you-prove-a-number-is-prime.htmlI am a software engineer but try to keep up with maths as I really enjoyed the subject at school. I just saw a great TED talk: Why I fell in love with monster prime numbers The talk states that the ...Sun, 06 Sep 2026 08:30:55 -0400https://track.liberty.io/proving-that-the-composition-of-relations-are-equal-to-each-other.htmlhttps://track.liberty.io/proving-that-the-composition-of-relations-are-equal-to-each-other.htmlPlease excuse my English if it's not understandable, exercises are translated so I don't know all the English terms in math. So I'm doing exercises regarding relations and compositions, and one of ......Sun, 06 Sep 2026 08:21:13 -0400https://track.liberty.io/finding-the-error-of-a-taylor-series.htmlhttps://track.liberty.io/finding-the-error-of-a-taylor-series.htmlWhen calculating the error of a Taylor series, the formula is as follows: $$R_n(x)=\frac{f^{(n+1)}(z)(x-c)^{(n+1)}}{(n+1)!}$$ $z$ is the maximum value of the expression on the interval between $x$......Sun, 06 Sep 2026 08:15:00 -0400https://track.liberty.io/two-sides-and-angle-between-them-triangle-question.htmlhttps://track.liberty.io/two-sides-and-angle-between-them-triangle-question.htmlis it possible to find the third side of a triangle if you know the lengths of the other two and the angle between the known sides? the triangle is not equilateral. we're using the kinect camera a......Sun, 06 Sep 2026 08:14:25 -0400https://track.liberty.io/what-is-a-coordinate-function-x-i-of-a-manifold-given-a-chart-u-x.htmlhttps://track.liberty.io/what-is-a-coordinate-function-x-i-of-a-manifold-given-a-chart-u-x.htmlI am trying to understand the notes here: http://unapologetic.wordpress.com/2011/04/13/cotangent-vectors-differentials-and-the-cotangent-bundle/. Specifically, this sentence: If we have local ...Sun, 06 Sep 2026 08:13:23 -0400https://track.liberty.io/if-e-euler-039-s-constant-1-frac-1-n-n-as-n-approaches-infinity-why-is.htmlhttps://track.liberty.io/if-e-euler-039-s-constant-1-frac-1-n-n-as-n-approaches-infinity-why-is.htmlIf $e$ (Euler's constant) $= (1+\frac{1}{n})^n$ as $n$ approaches infinity, why is $e^x$ not equal to $e$ if x equals any number, real or not real. Let r = any real number I think that $\left(1+\......Sun, 06 Sep 2026 08:05:53 -0400https://track.liberty.io/local-max-min-critical-points-of-integral.htmlhttps://track.liberty.io/local-max-min-critical-points-of-integral.htmlGiven $$f(t) = \int_0^t \frac{x^2+14x+45}{1+\cos^2(x)}dx $$ I need to find the local max of f(t). Well here using the fundamental theorem of calculus, I know I can just replace the $x$ with $t$. B......Sun, 06 Sep 2026 08:02:00 -0400https://track.liberty.io/show-that-mathbb-r-when-equipped-with-the-standard-topology-is-not-com.htmlhttps://track.liberty.io/show-that-mathbb-r-when-equipped-with-the-standard-topology-is-not-com.htmlProof: Let $\mathscr{U}$ := {$U_n$}$_{n \geq 1}$ = {$(n-1,n+1)$}$_{n \geq 1}$ be an open cover of $\mathbb{R}$. Consider finitely many of them, say: $U_{n_1} \ldots U_{n_k}$. Setting $N := \max(n_1 \...Sun, 06 Sep 2026 07:55:05 -0400https://track.liberty.io/is-there-a-correct-mathematical-term-for-the-inverse-of-the-slope.htmlhttps://track.liberty.io/is-there-a-correct-mathematical-term-for-the-inverse-of-the-slope.htmlLinear graphs all have a slope that can be calculated by deviding the progress on the y-axis by the progress on the x-axis. Is there a correct term to refer to the inverse of the slope, that means ......Sun, 06 Sep 2026 07:48:16 -0400https://track.liberty.io/what-is-the-affine-connection-and-what-is-the-intuition-behind-for-aff.htmlhttps://track.liberty.io/what-is-the-affine-connection-and-what-is-the-intuition-behind-for-aff.htmlHere is the definition of affine connection, as appears in Milnor's book Morse Theory. DEFINITION. An affine connection at a point $p \in \text{M}$ is a function which assigns to each tangent vect......Sun, 06 Sep 2026 07:46:51 -0400https://track.liberty.io/polynomial-division-modulo-5-gcd-of-two-polynomials.htmlhttps://track.liberty.io/polynomial-division-modulo-5-gcd-of-two-polynomials.htmlI have to find the gcd $h$ of $f$ and $g$ in $\mathbb{F}_5[X]$. $f=X^9+X^8+X^7+X^6+X^5+X^4+X^3+X^2+X+1,$ $g=X^4+X-2.$ Here is my polynomial division: My solution is $3X^3-X^2+3X+1$ but this is ......Sun, 06 Sep 2026 07:42:25 -0400https://track.liberty.io/prove-that-an-isometry-preserves-straight-lines-using-intuitive-geomet.htmlhttps://track.liberty.io/prove-that-an-isometry-preserves-straight-lines-using-intuitive-geomet.htmlThis is an exercise of the book "Basic Mathematics" by Serge Lang, p.145. I've been working on this proof for a few days and I can't seem to make it more coherent than this. Would appreciate some h......Sun, 06 Sep 2026 07:38:03 -0400https://track.liberty.io/what-makes-the-cauchy-principal-value-the-correct-value-for-a-integral.htmlhttps://track.liberty.io/what-makes-the-cauchy-principal-value-the-correct-value-for-a-integral.htmlI haven't been able to find a good answer to this searching around online. There is a related old question here, but it never received much attention. Suppose I have some physical property that I ...Sun, 06 Sep 2026 07:36:54 -0400https://track.liberty.io/calculating-expected-value-and-variance-of-a-probability-density-funct.htmlhttps://track.liberty.io/calculating-expected-value-and-variance-of-a-probability-density-funct.htmlIf I was given a probability density function: $$f(y) = \left\{\begin{array}{ll}\frac{3y^2(4-y)}{64} & \textrm{for } 0 \leq y \leq 4\\ 0 & \textrm{elsewhere} \end{array}\right.......Sun, 06 Sep 2026 07:26:38 -0400https://track.liberty.io/prove-taylor-expansion-with-mean-value-theorem.htmlhttps://track.liberty.io/prove-taylor-expansion-with-mean-value-theorem.htmlhttp://vapour-trail.blogspot.com/2006/03/brief-explanation-of-taylor-series-via.html provides an intuitive derivation of Taylor expansions from the mean value theorem that confuses me. The derivat......Sun, 06 Sep 2026 07:18:28 -0400https://track.liberty.io/how-to-find-the-exact-value-of-the-cosine-of-50-degree-angle.htmlhttps://track.liberty.io/how-to-find-the-exact-value-of-the-cosine-of-50-degree-angle.htmlI want to know the exact value of $\cos 50^\circ$. Actually I have already tried lot of times to solve but I cannot find the exact value of $\cos 50^\circ$....Sun, 06 Sep 2026 07:07:44 -0400https://track.liberty.io/calculating-the-coordinates-of-end-terminal-point-of-an-arc-from-known.htmlhttps://track.liberty.io/calculating-the-coordinates-of-end-terminal-point-of-an-arc-from-known.htmlI have searched the site and found similar questions of mine but not exaclty I needed. I have an arc. I know the coordinates of the arc center (P0), r (radius), coordinates of starting point of th......Sun, 06 Sep 2026 06:55:05 -0400https://track.liberty.io/more-comprehensive-general-topology-039-s-book-than-engelking.htmlhttps://track.liberty.io/more-comprehensive-general-topology-039-s-book-than-engelking.htmlOther than Engelking General Topology, I also come across other graduate general topology text such as Dugundji and Kelley, which I also find them interesting. However, I find Engelking's book more ...Sun, 06 Sep 2026 06:54:52 -0400https://track.liberty.io/solve-for-x-in-matrix-equation.htmlhttps://track.liberty.io/solve-for-x-in-matrix-equation.htmlHow can I solve for $X$ in this matrix equation? $$\begin{bmatrix}-3&-8\\-9&5\end{bmatrix} X + \begin{bmatrix}4&-7\\3&-2\end{bmatrix} = \begin{bmatrix}5&8\\-1&-1\end{bmatri......Sun, 06 Sep 2026 06:53:21 -0400https://track.liberty.io/why-is-this-answer-wrong-point-of-intersection-between-parabola-and-li.htmlhttps://track.liberty.io/why-is-this-answer-wrong-point-of-intersection-between-parabola-and-li.htmlQuestion: Use the discriminant to determine the number of points of intersection of the line $y=3x+5$ and the quadratic functions $f(x)=3x^2-2x-4$. Solve to find the points of intersection. U......Sun, 06 Sep 2026 06:49:43 -0400https://track.liberty.io/how-to-solve-5x5-grid-with-16-diagonals.htmlhttps://track.liberty.io/how-to-solve-5x5-grid-with-16-diagonals.htmlI have a grid 5x5 and I have to fill 16 little squares with a diagonal Rules You cannot place more than 1 diagonal in each square The diagonals cannot touch each other (example bellow) Click her......Sun, 06 Sep 2026 06:44:30 -0400https://track.liberty.io/how-to-calculate-the-transformation-matrix-if-the-coordinate-system-tr.htmlhttps://track.liberty.io/how-to-calculate-the-transformation-matrix-if-the-coordinate-system-tr.htmlI was trying to calculate the new coordinates of the 4 corners of the rectangle in the image, but I think I only formed the rotation matrix. what is the correct transformation matrix for this opera......Sun, 06 Sep 2026 06:44:03 -0400https://track.liberty.io/solutions-manual-for-analysis-on-manifolds.htmlhttps://track.liberty.io/solutions-manual-for-analysis-on-manifolds.htmlA few months ago,I wanted to learn something fundmental about manifolds. From highly recommend , I decided to choice Analysis on Manifolds by James R.Munkres as my self-learning textbook.Until now ,I ...Sun, 06 Sep 2026 06:33:15 -0400https://track.liberty.io/properties-of-a-non-invertible-square-matrix.htmlhttps://track.liberty.io/properties-of-a-non-invertible-square-matrix.htmlI am wondering what properties does $A$, an $n\times n$ non invertible matrix, have. An obvious one is $\det(A)=0$ But, I am not sure about other intuitions that I am having. For example: $rank(......Sun, 06 Sep 2026 06:31:30 -0400https://track.liberty.io/how-can-i-determine-the-value-of-angle-x-in-this-figure.htmlhttps://track.liberty.io/how-can-i-determine-the-value-of-angle-x-in-this-figure.html[![I used geometry but Did not find the answer. ][1]][1] **I used Geometry but not avail ** [1]: https://i.stack.imgur.com/Ejxoz.jpg N:B The fiqure is a square. Here is what I tred - When using ...Sun, 06 Sep 2026 06:30:43 -0400https://track.liberty.io/find-if-three-points-in-3-dimensional-space-are-collinear.htmlhttps://track.liberty.io/find-if-three-points-in-3-dimensional-space-are-collinear.htmlFind if the points joining $A=(6,7,1), B=(2,-3,1)$ and $C=(4,-5,0)$ are collinear. How to determine collinearity in three dimensions? In two dimensions, one can compare the slopes of segments $AB$......Sun, 06 Sep 2026 06:29:44 -0400https://track.liberty.io/geometry-proof-convex-quadrilateral.htmlhttps://track.liberty.io/geometry-proof-convex-quadrilateral.htmlA quadrilateral $ABCD$ is formed from four distinct points (called the vertices), no three of which are collinear, and from the segments $AB$, $CB$, $CD$, and $DA$ (called the sides), which have no ...Sun, 06 Sep 2026 06:29:22 -0400https://track.liberty.io/how-do-you-validate-a-pmf-closed.htmlhttps://track.liberty.io/how-do-you-validate-a-pmf-closed.html$$P\left(X=x\right)=\left\{ \begin{array}{rl} 0.02 & X=0\\ 0.03 & X=1\\ 0.2 & X\in\left\{ 2,\,5\right\} \\ 0.25 & X\in\left\{ 3,\,4\right\} \\ c & X=6 \end{array}\right.$$ I ha......Sun, 06 Sep 2026 06:28:34 -0400https://track.liberty.io/why-there-is-no-sign-of-logic-symbols-in-mathematical-texts.htmlhttps://track.liberty.io/why-there-is-no-sign-of-logic-symbols-in-mathematical-texts.htmlEither in undergraduate or graduate textbooks on Mathematics (Real/Complex Analysis, General Topology, Differential Geometry, ...), I never saw symbols $\Rightarrow$, $\iff$, $\forall$, $\exists$, ......Sun, 06 Sep 2026 06:28:10 -0400https://track.liberty.io/maximising-largest-integer-on-a-list-given-mean-median-and-mode.htmlhttps://track.liberty.io/maximising-largest-integer-on-a-list-given-mean-median-and-mode.htmlA list of $10$ positive integers has a mean of $11$, a median of $10$ and a unique mode of $7$. What is the largest possible value of an integer in the list? In an ascending (albeit not strictly ...Sun, 06 Sep 2026 06:24:24 -0400https://track.liberty.io/find-the-first-three-non-zero-terms-of-the-taylor-series-of-f.htmlhttps://track.liberty.io/find-the-first-three-non-zero-terms-of-the-taylor-series-of-f.htmlConsider the function: $$f(x) = \frac{1}{x^2+10}$$ Find the first three non-zero terms of the Taylor series of...Sun, 06 Sep 2026 06:22:43 -0400https://track.liberty.io/how-to-calculator-plus-minus-quadratic.htmlhttps://track.liberty.io/how-to-calculator-plus-minus-quadratic.htmlThe book says there should be a quadratic formula, but I don't understand how to solve it. I have: $\pm 6.0 = \sqrt(\frac{36 - 4(4.0)( \pm 7.04)}{14.2})$ I don't understand how to input this into a ...Sun, 06 Sep 2026 06:00:47 -0400https://track.liberty.io/how-to-calculate-sin-65-without-a-calculator.htmlhttps://track.liberty.io/how-to-calculate-sin-65-without-a-calculator.htmlI know about the sum and difference formula but I can't think of two values which will be able to use for sin(65). Therefore, I come to the question: How to calculate sin(65) without a calculator....Sun, 06 Sep 2026 06:00:03 -0400https://track.liberty.io/q-quadratic-division-how-to-divide-two-quadratics.htmlhttps://track.liberty.io/q-quadratic-division-how-to-divide-two-quadratics.htmlMy studies into graphs and models following examples from Khan Academy has helped me on my goal to learn how to chart and model via the quadratic formula. However, while I have been successful in ...Sun, 06 Sep 2026 05:59:14 -0400https://track.liberty.io/what-039-s-the-difference-between-a-proof-and-a-derivation.htmlhttps://track.liberty.io/what-039-s-the-difference-between-a-proof-and-a-derivation.htmlI did a BSc in Theoretical Physics, meaning that a lot of my time was spent deriving equations, making hand-wavy arguments, and arriving at solutions with a distinct lack of rigour. I'm now doing......Sun, 06 Sep 2026 05:57:23 -0400https://track.liberty.io/why-does-r-cos-theta-produce-a-circle.htmlhttps://track.liberty.io/why-does-r-cos-theta-produce-a-circle.htmlI am trying to do a double integral over the following region in polar coordinates: I know that the limits of integration are: $$\theta = -\frac{\pi}{2} \quad \to \quad \theta = \frac{\pi}{2} \\ ......Sun, 06 Sep 2026 05:50:16 -0400https://track.liberty.io/finding-all-solutions-to-an-inequality-equation.htmlhttps://track.liberty.io/finding-all-solutions-to-an-inequality-equation.htmlI have the following inequality that I need to find all solutions of: $2x^3-8x > 5x^2-20$ My guess is that you would have to turn this into a polynomial equation and let the right hand side ......Sun, 06 Sep 2026 05:49:20 -0400https://track.liberty.io/what-is-the-domain-of-a-division-of-functions.htmlhttps://track.liberty.io/what-is-the-domain-of-a-division-of-functions.htmlThis question is about real functions of real variables. I think that, in general, if the domain of some function $f(x)$ is A, and the domain of another function $g(x)$ is B, then the domain of $(......Sun, 06 Sep 2026 05:32:59 -0400https://track.liberty.io/finding-deceleration-and-initial-speed-given-time-and-distance.htmlhttps://track.liberty.io/finding-deceleration-and-initial-speed-given-time-and-distance.html(I'm sorry if this is a bad way to frame a math question) Let's say a car is moving at 40m/s and a brake is applied, the car decelerates at 10m/s thus stops after 4 seconds, traveled 100m after the ...Sun, 06 Sep 2026 05:23:42 -0400https://track.liberty.io/lim-x-to0-sin-2x-2x-x-3.htmlhttps://track.liberty.io/lim-x-to0-sin-2x-2x-x-3.htmlEvaluate $$ \lim_{x\to0} \frac{\sin(2x)-2x}{x^3} $$ I thought that I could use L'Hopital's rule to get to an answer of $-1$ but according to the answer manual that isn't correct. What am I doing w......Sun, 06 Sep 2026 05:21:26 -0400https://track.liberty.io/is-there-any-difference-between-empty-graph-and-null-graph.htmlhttps://track.liberty.io/is-there-any-difference-between-empty-graph-and-null-graph.htmlI was going through one of the links Is the empty graph connected? here I found that it is mentioned that empty graph is a graph with no edges while null graph is a graph with no vertices but I ...Sun, 06 Sep 2026 05:18:27 -0400https://track.liberty.io/how-to-prove-operatorname-tr-ab-operatorname-tr-ba.htmlhttps://track.liberty.io/how-to-prove-operatorname-tr-ab-operatorname-tr-ba.htmlthere is a similar thread here Coordinate-free proof of $\operatorname{Tr}(AB)=\operatorname{Tr}(BA)$?, but I'm only looking for a simple linear algebra proof....Sun, 06 Sep 2026 05:08:57 -0400https://track.liberty.io/evaluating-limits-algebraically.htmlhttps://track.liberty.io/evaluating-limits-algebraically.htmlI bought the eighth edition of Stewart Calculus (metric version) and I'm up to the section about limits. It's been pretty easy so far, but I've come across a class of limit problems that don't seem ...Sun, 06 Sep 2026 04:52:52 -0400https://track.liberty.io/what-is-the-definition-of-tensor-contraction.htmlhttps://track.liberty.io/what-is-the-definition-of-tensor-contraction.htmlAccording to Wikipedia's page on tensor contraction: In general, a tensor of type $(m,n)$ (with $m \geq 1$ and $n \geq 1$) is an element of the vector space $V \otimes \ldots \otimes V \otimes V......Sun, 06 Sep 2026 04:41:54 -0400https://track.liberty.io/eigenfunction-fourier-series-relationship.htmlhttps://track.liberty.io/eigenfunction-fourier-series-relationship.htmlI'm having difficulty formulating a proof detailing that every eigenfunction of $D^2$ is either a constant or of the form $a$cos$(nx)$ + $b$sin($nx$) for some value $n$. I understand that the Fou......Sun, 06 Sep 2026 04:35:26 -0400https://track.liberty.io/property-of-dirac-delta-function-in-mathbb-r-n.htmlhttps://track.liberty.io/property-of-dirac-delta-function-in-mathbb-r-n.htmlHow does one prove the following identity? $$\int _Vf(\pmb{r})\delta (g(\pmb{r})) \, d\pmb{r}=\int_S \frac{f(\pmb{r})}{\left|\operatorname{grad} g(\pmb{r})\right|} \, d\sigma$$ where $S$ is the sur......Sun, 06 Sep 2026 04:34:30 -0400https://track.liberty.io/is-there-a-way-to-find-all-roots-of-a-polynomial-equation.htmlhttps://track.liberty.io/is-there-a-way-to-find-all-roots-of-a-polynomial-equation.htmlIs there a way to find all roots of a polynomial equation? Lets say $$x^5+ax^4+bx^3+cx^2+dx+e=0$$ how to find its all roots?...Sun, 06 Sep 2026 04:30:35 -0400https://track.liberty.io/what-is-the-mathematical-definition-of-index-set.htmlhttps://track.liberty.io/what-is-the-mathematical-definition-of-index-set.htmlI find some descriptions http://en.wikipedia.org/wiki/Index_set and http://mathworld.wolfram.com/IndexSet.html . But can't find any definition....Sun, 06 Sep 2026 04:27:20 -0400https://track.liberty.io/how-to-prove-that-derivatives-have-the-intermediate-value-property.htmlhttps://track.liberty.io/how-to-prove-that-derivatives-have-the-intermediate-value-property.htmlI'm reading a book which gives this theorem without proof: If a and b are any two points in an interval on which ƒ is differentiable, then ƒ' takes on every value between ƒ'(a) and ƒ'(b). As ......Sun, 06 Sep 2026 04:25:37 -0400https://track.liberty.io/what-is-a-wavenumber.htmlhttps://track.liberty.io/what-is-a-wavenumber.htmlI have been struggling to understand the concept of wavenumber for a number of years. Is a wavenumber we talk about in the context of wave propagation in physics (say ocean waves, or acoustics) sa......Sun, 06 Sep 2026 04:16:05 -0400https://track.liberty.io/testing-for-associativity-using-the-multiplication-table-cayley-table-.htmlhttps://track.liberty.io/testing-for-associativity-using-the-multiplication-table-cayley-table-.htmlI seem to recall that there is a relatively easy method for determining the associativity of an operation by using its Cayley table. What is it?...Sun, 06 Sep 2026 04:12:40 -0400https://track.liberty.io/feynman-integration-problem.htmlhttps://track.liberty.io/feynman-integration-problem.html$$ I = \frac{\pi^2}{8} - \int_0^1 \frac{\arctan(x)}{\sqrt{1-x^2}} \,dx $$ Evaluate $I$ $$ I = \frac{\pi^2}{8} - \int_0^1 \frac{\arctan(x)}{\sqrt{1-x^2}} \,dx$$ $$f(a) = \int_0^1 \frac{\arctan(ax......Sun, 06 Sep 2026 03:57:44 -0400https://track.liberty.io/9x-multiplication-table-trick-how-does-it-work.htmlhttps://track.liberty.io/9x-multiplication-table-trick-how-does-it-work.htmlThere is a trick that is taught to children to help them remember their 9 times tables. As illustrated below, the digits of the multiplication problem in question can be determined by: Splitting ......Sun, 06 Sep 2026 03:56:46 -0400https://track.liberty.io/user-quinn-greicius-mathematics-stack-exchange.htmlhttps://track.liberty.io/user-quinn-greicius-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Sun, 06 Sep 2026 03:54:17 -0400https://track.liberty.io/less-than-or-equal-sign.htmlhttps://track.liberty.io/less-than-or-equal-sign.htmlIf I know for two numbers a and b that $${a < b }$$ Then is it correct to say that $$ a \leq b $$ I know that the second statement is true as long as the first one is. It seems OK as it is ...Sun, 06 Sep 2026 03:51:05 -0400https://track.liberty.io/would-going-first-in-egg-russian-roulette-increase-or-decrease-your-ch.htmlhttps://track.liberty.io/would-going-first-in-egg-russian-roulette-increase-or-decrease-your-ch.htmlEgg Russian Roulette: 6 eggs, 5 of which are hard boiled & 1 of which is raw. Players (1v1) each choose eggs in a turn based system whereby the egg is smashed against the forehead. The f......Sun, 06 Sep 2026 03:49:44 -0400https://track.liberty.io/why-average-value-of-a-function-is-not-calculated-by-using-the-formula.htmlhttps://track.liberty.io/why-average-value-of-a-function-is-not-calculated-by-using-the-formula.htmlI know the formula for calculating the average value of a function as $$ \frac{1}{b-a}\int_a^b f(x)\,dx$$ But in elementry level maths and physics problems we generally use a very simple approach to ...Sun, 06 Sep 2026 03:38:12 -0400https://track.liberty.io/what-is-the-meaning-of-the-symbol-pitchfork.htmlhttps://track.liberty.io/what-is-the-meaning-of-the-symbol-pitchfork.htmlI have seen this symbol in the formulation of this question. There, it is said: Let $f: X \to Y$ be a smooth map with $f \pitchfork Z$. I was googling, but I haven't found any answers....Sun, 06 Sep 2026 03:37:33 -0400https://track.liberty.io/what-does-the-definition-of-curvature-mean.htmlhttps://track.liberty.io/what-does-the-definition-of-curvature-mean.htmlFirst question, am I right in saying that curvature measure how quickly the direction of a cruve changes? Also, we have been given the "definition": $$T'(t) = \kappa(t) | \gamma'(t) | U(t)$$ wh......Sun, 06 Sep 2026 03:30:08 -0400https://track.liberty.io/is-a-non-repeating-and-non-terminating-decimal-always-an-irrational.htmlhttps://track.liberty.io/is-a-non-repeating-and-non-terminating-decimal-always-an-irrational.htmlWe can build $\frac{1}{33}$ like this, $.030303$ $\cdots$ ($03$ repeats). $.0303$ $\cdots$ tends to $\frac{1}{33}$. So,I was wondering this: In the decimal representation, if we start writing the ......Sun, 06 Sep 2026 03:21:55 -0400https://track.liberty.io/proof-that-eigenvalues-are-the-diagonal-entries-of-the-upper-triangula.htmlhttps://track.liberty.io/proof-that-eigenvalues-are-the-diagonal-entries-of-the-upper-triangula.htmlThis is 5.18 from Axler's Linear Algebra Done Right: Theorem: Suppose $T \in L(V)$ has an upper-triangular matrix with respect to some basis of $V$. Then the eigenvalues of $T$ consist precisely o......Sun, 06 Sep 2026 03:21:32 -0400https://track.liberty.io/block-diagonal-matrix-diagonalizable.htmlhttps://track.liberty.io/block-diagonal-matrix-diagonalizable.htmlI am trying to prove that: The matrix $C = \left(\begin{smallmatrix}A& 0\\0 & B\end{smallmatrix}\right)$ is diagonalizable, if only if $A$ and $B$ are diagonalizable. If $A\in GL(\mathbb{......Sun, 06 Sep 2026 03:16:48 -0400https://track.liberty.io/an-infinite-union-of-closed-sets-is-a-closed-set.htmlhttps://track.liberty.io/an-infinite-union-of-closed-sets-is-a-closed-set.htmlQuestion: {$B_n$}$ \in \Bbb R$ is a family of closed sets. Prove that $\cup _{n=1}^\infty B_n$ is not necessarily a closed set. What I thought: Using a counterexample: If I say that each $B_i$ is ......Sun, 06 Sep 2026 03:14:04 -0400https://track.liberty.io/learning-projective-geometry.htmlhttps://track.liberty.io/learning-projective-geometry.htmlMy ultimate goal is to learn some algebraic geometry with the more concrete immediate goal of understanding things like how $\mathbb R\mathrm P^2$ is embedded in $\mathbb C\mathrm P^2$ or how $\mat......Sun, 06 Sep 2026 03:07:48 -0400https://track.liberty.io/absolute-maximum-and-minimum-values-of-a-function-over-a-region.htmlhttps://track.liberty.io/absolute-maximum-and-minimum-values-of-a-function-over-a-region.htmlThe prompt is to find the absolute maximum and minimum values of the function $f(x, y) = x^2 + 2x - y^2$ on the upper half of a unit disc, $D =\{(x, y): x^2 + y^2 \leq 1, y \geq 0 \}$ I began by f......Sun, 06 Sep 2026 03:00:26 -0400https://track.liberty.io/help-with-proving-that-the-transpose-of-the-product-of-any-number-of-m.htmlhttps://track.liberty.io/help-with-proving-that-the-transpose-of-the-product-of-any-number-of-m.htmlSpecifically I am trying to show that (An)T = (AT)n where A is an mxm square matrix and n is a positive integer. This is where I'm stuck: To prove the theorem I would like to show that ((An)T)ij =......Sun, 06 Sep 2026 02:55:15 -0400https://track.liberty.io/calculate-matrix-a-from-null-space-basis-of-a-4i.htmlhttps://track.liberty.io/calculate-matrix-a-from-null-space-basis-of-a-4i.htmlHow to find a matrix $A$ when you are given some parameters and the basis for the null space? The problem I've been scratching my head over is this. The basis for the null space of $A-4I$ is $$\......Sun, 06 Sep 2026 02:54:24 -0400https://track.liberty.io/methods-to-prove-frac-21n-4-14n-3-is-irreducible-for-every-natural-num.htmlhttps://track.liberty.io/methods-to-prove-frac-21n-4-14n-3-is-irreducible-for-every-natural-num.htmlProve $\frac{21n + 4}{14n + 3}$ is irreducible for every natural number $n$. I was thinking of taking a number-theory based approach. Can you suggest the following method Calculus/Number theory ......Sun, 06 Sep 2026 02:31:52 -0400https://track.liberty.io/why-is-the-function-f-x-x-3-pmod-10-periodic-with-this-strange-propert.htmlhttps://track.liberty.io/why-is-the-function-f-x-x-3-pmod-10-periodic-with-this-strange-propert.htmlI've noticed that the function $f(x)=x^3 \pmod{10}$ is periodic. For example, listing mod(x^3,10) we get: 0,1,8,7,4,5,6,3,2,9,0,1,8,7,4,5,6,3,2,9,0,1,8,7,4,5,6,3,2,9,0,1,8,7,4,5,6,3,2,9,0,1,8,7,4......Sun, 06 Sep 2026 02:29:45 -0400https://track.liberty.io/summation-symbol-with-just-one-letter-below-it-how-to-verbally-state.htmlhttps://track.liberty.io/summation-symbol-with-just-one-letter-below-it-how-to-verbally-state.htmlI'm used to seeing summation capital sigmas with a "start" at bottom and a "stop" at top and would say the sum from i to 100 or whatever. But in this case, there is just a y at bottom. How would I ...Sun, 06 Sep 2026 02:29:30 -0400https://track.liberty.io/what-trig-identity-would-help-solve-2-cos-2x-3-cos-x.htmlhttps://track.liberty.io/what-trig-identity-would-help-solve-2-cos-2x-3-cos-x.htmlI need help with a homework question that has me puzzled. I need to solve the following equation: $$2 + \cos(2x) = 3\cos(x)$$ I don't see a good trig identity to apply. I tried $\cos(2x) = 2\cos^......Sun, 06 Sep 2026 02:25:00 -0400https://track.liberty.io/cos-4x-sin-2x-in-terms-of-first-power-of-cosine.htmlhttps://track.liberty.io/cos-4x-sin-2x-in-terms-of-first-power-of-cosine.htmlI believe that I have his correct but if someone could check it and see that'd be great. Here's a pic!...Sun, 06 Sep 2026 02:21:14 -0400https://track.liberty.io/simplifying-a-square-root-fraction.htmlhttps://track.liberty.io/simplifying-a-square-root-fraction.htmlSimplify the following $$\frac{\sqrt{3}}{\sqrt{2}(\sqrt{6} - \sqrt{3})}$$ Apparently the answer is $\frac{1}{2} (2 + \sqrt{2})$ but can't for the life of me see how to get it. Any help is massiv......Sun, 06 Sep 2026 02:13:15 -0400https://track.liberty.io/related-rates-how-fast-is-the-water-leaking-from-a-conical-shaped-tank.htmlhttps://track.liberty.io/related-rates-how-fast-is-the-water-leaking-from-a-conical-shaped-tank.htmlWater is poured at the rate of 8 cubic feet per minute into a conical-shaped tank, 20 ft deep and 10 ft in diameter at the top. If the tank has a leak in the bottom and the water level is rising at......Sun, 06 Sep 2026 02:12:14 -0400https://track.liberty.io/why-do-we-care-about-dual-spaces.htmlhttps://track.liberty.io/why-do-we-care-about-dual-spaces.htmlWhen I first took linear algebra, we never learned about dual spaces. Today in lecture we discussed them and I understand what they are, but I don't really understand why we want to study them within ...Sun, 06 Sep 2026 02:11:19 -0400https://track.liberty.io/definition-of-concave-set.htmlhttps://track.liberty.io/definition-of-concave-set.htmlWe know that a set is convex if the straight line joining any two points of the set lies completely in the set. Or, mathematically, a set $X$ is convex if $$x_1, x_2 \in X, 0 \leq \lambda \leq 1 \...Sun, 06 Sep 2026 02:08:11 -0400https://track.liberty.io/multivariable-polynomial-long-division-calculator.htmlhttps://track.liberty.io/multivariable-polynomial-long-division-calculator.htmlIs there code/online calculator that divides polynomials in multiple variables? The answers are always other polynomials, but they have many variables....Sun, 06 Sep 2026 02:05:48 -0400https://track.liberty.io/what-are-these-bracketing-symbols-and-what-do-they-mean.htmlhttps://track.liberty.io/what-are-these-bracketing-symbols-and-what-do-they-mean.htmlWhat do the matching "L" shapes (near .5 and 20) mean in this forumla? The document where I found this formula can be found here: http://www.cs.cmu.edu/~hcwong/Pdfs/icip02.ps...Sun, 06 Sep 2026 01:54:20 -0400https://track.liberty.io/x-1-x-2-iid-standard-normal-with-polar-coordinates-r-and-p-are-r-and-p.htmlhttps://track.liberty.io/x-1-x-2-iid-standard-normal-with-polar-coordinates-r-and-p-are-r-and-p.htmlI have two scalar random variables $X_1$ and $X_2$ and they are both independent and both have standard normal distribution $N(0,1)$. I am letting $r$ and $p$ be the polar coordinates of the point ......Sun, 06 Sep 2026 01:49:48 -0400https://track.liberty.io/why-don-039-t-parabolas-have-asymptotes.htmlhttps://track.liberty.io/why-don-039-t-parabolas-have-asymptotes.htmlAs given on the Wikipedia page, an asymptote is a line which becomes the tangent of the curve as the $x$ or $y$ cordinates of the curve tends to infinity. Hyperbola has asymptotes but parabolas ......Sun, 06 Sep 2026 01:48:43 -0400https://track.liberty.io/finding-the-diagonal-of-a-rhombus.htmlhttps://track.liberty.io/finding-the-diagonal-of-a-rhombus.htmlI am trying to show that the diagonals of a RHOMBUS intersect each other at 90 degrees However I need to find the length of the diagonal without using Trig. ratios. Any ideas how i could find t......Sun, 06 Sep 2026 01:43:32 -0400https://track.liberty.io/what-is-the-most-general-definition-of-a-graph.htmlhttps://track.liberty.io/what-is-the-most-general-definition-of-a-graph.htmlI would like to give the most general definition of a graph. Based on what I read about graphs, I came up with the following definition: Let us introduce some symbols. They are (a) $V$:: a non-em......Sun, 06 Sep 2026 01:42:42 -0400