https://track.liberty.ioFresh celebrity buzz with instant attention value.en-usSat, 05 Sep 2026 11:37:28 -0400https://track.liberty.io/how-do-i-calculate-dice-probability.htmlhttps://track.liberty.io/how-do-i-calculate-dice-probability.htmlI am trying to learn how to calculate dice probability, but all I can find on the net are the odds of rolling a given number in a single roll. An example of what I am looking of is rolling a 4+ on ......Sat, 05 Sep 2026 15:34:35 -0400https://track.liberty.io/finding-eigenvectors-of-a-3x3-matrix-2.htmlhttps://track.liberty.io/finding-eigenvectors-of-a-3x3-matrix-2.htmlI am trying to find the eigenvectors of the following 3x3 matrix: \begin{bmatrix} -9 & 4 & 4 \\ -8 & 3 & 4 \\ -16 & 8 & 7 \\ \end{bmatrix} ......Sat, 05 Sep 2026 15:27:51 -0400https://track.liberty.io/sketching-the-graph-of-x-f-y.htmlhttps://track.liberty.io/sketching-the-graph-of-x-f-y.htmlIn order to sketch a graph of $x=f(y)$ when we know the graph of $y=f(x)$ We can exchange the axes of coordinates (and sketch the same graph of $y=f(x)$ on the other axis) I want use this method to ...Sat, 05 Sep 2026 15:25:57 -0400https://track.liberty.io/question-about-logic-riddle-patch-of-lily-pads-that-doubles-every-day.htmlhttps://track.liberty.io/question-about-logic-riddle-patch-of-lily-pads-that-doubles-every-day.htmlI wasn't sure if the Math SE is the right SE to post this question, but after reading this question at the Meta SE, I think this SE is the right place. In a video on YouTube about logic riddles, o......Sat, 05 Sep 2026 15:09:01 -0400https://track.liberty.io/how-to-find-the-orthogonal-complement-of-a-subspace.htmlhttps://track.liberty.io/how-to-find-the-orthogonal-complement-of-a-subspace.htmlI am trying to understand how to find the orthogonal complement of a subspace $M$ of a vector space $V$. From my understanding, $M^\perp$ is also a subspace of $V$ where all its vectors are ...Sat, 05 Sep 2026 15:04:00 -0400https://track.liberty.io/intuition-behind-convergence-and-consistency.htmlhttps://track.liberty.io/intuition-behind-convergence-and-consistency.htmlWhat is the definition of consistency? I have seen a proof that shows a finite difference scheme is consistent, where they basically plug a true solution $u(t)$ into a finite difference scheme, and ...Sat, 05 Sep 2026 14:47:34 -0400https://track.liberty.io/questions-tagged-linear-algebra.htmlhttps://track.liberty.io/questions-tagged-linear-algebra.htmlQ&A for people studying math at any level and professionals in related fields...Sat, 05 Sep 2026 14:42:09 -0400https://track.liberty.io/discrete-vs-continuous-vs-random-variables.htmlhttps://track.liberty.io/discrete-vs-continuous-vs-random-variables.htmlCan someone provide me a clear explanation (Perhaps with examples) about the difference between the following mathematical three concepts in probability: Discrete Variable. Continuous Variable. Ra......Sat, 05 Sep 2026 14:39:07 -0400https://track.liberty.io/extreme-value-theorem-proof-help.htmlhttps://track.liberty.io/extreme-value-theorem-proof-help.htmlExtreme Value Theorem: If $f$ is a continuous function on an interval [a,b], then $f$ attains its maximum and minimum values on [a,b]. Proof from my book: Since $f$ is continuous, then $f$ has ......Sat, 05 Sep 2026 14:29:39 -0400https://track.liberty.io/online-videos-on-measure-theory.htmlhttps://track.liberty.io/online-videos-on-measure-theory.htmlI am looking for lecture videos on Measure Theory. I am not able to find any. Looking forward for your response....Sat, 05 Sep 2026 14:29:34 -0400https://track.liberty.io/why-is-taylor-series-expansion-for-1-1-x-valid-only-for-x-in-1-1.htmlhttps://track.liberty.io/why-is-taylor-series-expansion-for-1-1-x-valid-only-for-x-in-1-1.htmlAfter finding an expansion of $$ \frac{1}{1-x} = 1 + x + x^2 + x^3 + \ldots$$ a quick test of various values for $x$ reveals that this expansion is not valid for $\forall x \in \mathbb{R}-\{1\}$. ......Sat, 05 Sep 2026 14:29:16 -0400https://track.liberty.io/evaluating-int-4-4-int-0-sqrt-16-x-2-int-0-16-x-2-y-2-sqrt-x-2-y-2-dz-.htmlhttps://track.liberty.io/evaluating-int-4-4-int-0-sqrt-16-x-2-int-0-16-x-2-y-2-sqrt-x-2-y-2-dz-.htmlQuestion: Evaluate the given triple integral with cylindrical coordinates: $$\int_{-4} ^4\int _0 ^{\sqrt{16-x^2}} \int _0 ^{16-x^2-y^2} \sqrt{x^2 + y^2}\,dz\,dy\,dx$$ My solution (attempt): Upon ...Sat, 05 Sep 2026 14:22:08 -0400https://track.liberty.io/what-do-eigenvalues-have-to-do-with-pictures.htmlhttps://track.liberty.io/what-do-eigenvalues-have-to-do-with-pictures.htmlI am trying to write a program that will perform OCR on a mobile phone, and I recently encountered this article : Can someone explain this to me ?...Sat, 05 Sep 2026 14:20:19 -0400https://track.liberty.io/equality-of-pairs-of-parallel-lines.htmlhttps://track.liberty.io/equality-of-pairs-of-parallel-lines.htmlIn the picture above two sets of parallel lines are given: $l, m$ and $p, q$. when can we say these pairs are the same? If the distance or altitude are same then can we say they are equal?...Sat, 05 Sep 2026 14:05:36 -0400https://track.liberty.io/how-to-represent-the-square-root-of-90-as-a-fraction.htmlhttps://track.liberty.io/how-to-represent-the-square-root-of-90-as-a-fraction.htmlHow to represent the square root of 90 as a fraction? I can usually do this with a calculator but it's not wanting to play nice. I need it to find the distance between two points but, unfortunatel......Sat, 05 Sep 2026 14:03:34 -0400https://track.liberty.io/what-is-the-invariant-in-tensors.htmlhttps://track.liberty.io/what-is-the-invariant-in-tensors.htmlIt is being said that the most important property of tensors is that they are invariant under basis changes. What exactly does stay invariant in tensors? Can you please explain this invarianc......Sat, 05 Sep 2026 14:01:18 -0400https://track.liberty.io/splitting-integral-into-two-parts.htmlhttps://track.liberty.io/splitting-integral-into-two-parts.htmlThis question might seem very simple, but I can't seem to figure it out. Suppose I have an integral over a square region. I was wondering in which case it would be incorrect to split the integral i......Sat, 05 Sep 2026 13:52:57 -0400https://track.liberty.io/inverse-matrices-for-linear-equations-with-infinite-solutions.htmlhttps://track.liberty.io/inverse-matrices-for-linear-equations-with-infinite-solutions.htmlGiven a linear system $\begin{cases}2x - 3y &= 7\\x + 5y &= -3\end{cases}$ This system can be expressed as a matrix-vector product $$\underbrace{\begin{bmatrix}2 & -3\\1 & 5\end{b......Sat, 05 Sep 2026 13:44:45 -0400https://track.liberty.io/exponential-equations-with-fractions.htmlhttps://track.liberty.io/exponential-equations-with-fractions.htmlI have had some issues with the following two equations: $$ \frac{3^{n-2}}{9^{1-n}}=9$$ $$\frac{5^{3n-3}}{25^{n-3}}=125$$ If anyone could work them out step by step that would be awesome. I k......Sat, 05 Sep 2026 13:42:33 -0400https://track.liberty.io/what-is-the-autocorrelation-of-a-squared-gaussian-process.htmlhttps://track.liberty.io/what-is-the-autocorrelation-of-a-squared-gaussian-process.htmlSuppose $ {X_t; t \in R} $ is a wss, zero mean Gaussian random process with autocorrelation function $ R_X( \tau) ; \tau \in R$ and power spectral density $S_X(\omega); \omega \in R$. If w define......Sat, 05 Sep 2026 13:42:26 -0400https://track.liberty.io/height-radius-ratio-for-maximum-volume-cylinder-of-given-surface-area.htmlhttps://track.liberty.io/height-radius-ratio-for-maximum-volume-cylinder-of-given-surface-area.htmlI am a bit confused by this problem I have encountered: A right circular cylindrical container with a closed top is to be constructed with a fixed surface area. Find the ratio of the height to the ...Sat, 05 Sep 2026 13:38:41 -0400https://track.liberty.io/function-or-relation-y-4-8x-2.htmlhttps://track.liberty.io/function-or-relation-y-4-8x-2.htmlThe question under debate: Is $y^4=8x^2$ a function or a relation? We disagree. I say, it's a relation because when I substitute in points I get two outputs for most inputs, and thus it's not a ...Sat, 05 Sep 2026 13:22:13 -0400https://track.liberty.io/why-the-domain-of-the-cube-root-function-are-all-the-real-numbers.htmlhttps://track.liberty.io/why-the-domain-of-the-cube-root-function-are-all-the-real-numbers.htmlsince it can also be written as x^(1/3) and therefore 1/(x^3) and this would not make sense for x=0 because of the division with 0. So why is 0 in the domain?...Sat, 05 Sep 2026 13:19:37 -0400https://track.liberty.io/understanding-multiplicity.htmlhttps://track.liberty.io/understanding-multiplicity.htmlin my books of algebra it talks about polynomial functions and their zeros Multiplicity and x-Intercepts If r is a zero of even multiplicity, then the graph touches the x-axis and turns arou......Sat, 05 Sep 2026 13:04:48 -0400https://track.liberty.io/why-are-fractions-with-a-negative-denominator-valid.htmlhttps://track.liberty.io/why-are-fractions-with-a-negative-denominator-valid.htmlWhenever in a fraction, there is $0$ in the denominator, the fraction becomes $\infty$ or indeterminate. But why do we consider those fractions valid that have some negative numbers in the denomina......Sat, 05 Sep 2026 12:54:39 -0400https://track.liberty.io/angular-radius-of-a-sphere.htmlhttps://track.liberty.io/angular-radius-of-a-sphere.htmlGiven a sphere with radius $r$ about a point $c$, what's the apparent angular radius $\alpha$ of that sphere from point $P$? In other words, if $\vec{o} = c - P$, what's the maximum angle another v......Sat, 05 Sep 2026 12:50:30 -0400https://track.liberty.io/how-to-compute-the-inverse-of-a-matrix-in-matlab.htmlhttps://track.liberty.io/how-to-compute-the-inverse-of-a-matrix-in-matlab.htmlI have a matrix $$X=\int_{0}^{2}\begin{bmatrix}P_1^2&-P_1P_2\\ -P_2P_1& P_2^2 \end{bmatrix}ds+\int_{0}^{1}(2-s)^{-0.5}(1-s)^{0.5}\begin{bmatrix}P_1Q_2& P_1Q_1\\-P_2Q_2& -P_2Q_1\end{...Sat, 05 Sep 2026 12:42:41 -0400https://track.liberty.io/is-the-category-of-discrete-categories-equivalent-to-the-category-of-s.htmlhttps://track.liberty.io/is-the-category-of-discrete-categories-equivalent-to-the-category-of-s.htmlA discrete category is one whose only morphisms are identity morphisms. The category of discrete categories would have as objects the discrete categories and as morphisms the functors between those ...Sat, 05 Sep 2026 12:38:59 -0400https://track.liberty.io/how-do-you-find-the-closest-square-number-to-another-number-without-us.htmlhttps://track.liberty.io/how-do-you-find-the-closest-square-number-to-another-number-without-us.htmlSay we try to find the closest square number to 26. we already know the closest square number is $25$. However, how do I calculate out 25? Because, if I try to prime factorize it like so: $\......Sat, 05 Sep 2026 12:22:53 -0400https://track.liberty.io/what-is-the-terminology-for-lemma-of-lemma.htmlhttps://track.liberty.io/what-is-the-terminology-for-lemma-of-lemma.htmlLet's say I need to prove a main theorem, to prove which I need three lemmas. Thus in writing the structure is as follows: Lemma 1 Proof Lemma 2 Proof Lemma 3 Proof Theorem Proof But if when prov......Sat, 05 Sep 2026 12:19:35 -0400https://track.liberty.io/determining-the-number-of-solutions-of-a-system-of-linear-inequalities.htmlhttps://track.liberty.io/determining-the-number-of-solutions-of-a-system-of-linear-inequalities.htmlI want to determine the number of solutions of a system of linear inequalities, and I was wondering if there was a simple way to to that. I know that linear programming is often used to check whether ...Sat, 05 Sep 2026 12:17:28 -0400https://track.liberty.io/continuous-version-of-a-poisson-r-v.htmlhttps://track.liberty.io/continuous-version-of-a-poisson-r-v.htmlI am wondering if there is a continuous version of a Poisson random variable, that has the following two features: 1) Has a CDF that agrees with the discrete Poisson distribution on the integers, ......Sat, 05 Sep 2026 12:08:17 -0400https://track.liberty.io/can-the-value-of-sin-theta-be-greater-than-1-closed.htmlhttps://track.liberty.io/can-the-value-of-sin-theta-be-greater-than-1-closed.htmlFor example, is it possible for sin theta to be 1.06 or 1.2 under any circumstances? Or is it possible for it to be exactly 1?...Sat, 05 Sep 2026 11:57:03 -0400https://track.liberty.io/why-is-a-nullary-operation-a-special-element-usually-0-or-1.htmlhttps://track.liberty.io/why-is-a-nullary-operation-a-special-element-usually-0-or-1.htmlDoes a nullary operation mean an operation not taking any argument? Then why is a nullary operation a special element, usually 0 or 1, in an algebraic structure? Thanks!...Sat, 05 Sep 2026 11:44:59 -0400https://track.liberty.io/calculating-percentage-gain-on-investment-with-regular-deposits.htmlhttps://track.liberty.io/calculating-percentage-gain-on-investment-with-regular-deposits.htmlOn 1st of January I had a total of 120 dollars invested in a scheme. I made further deposits (as per table below) and have a final figure of $620 on June 10th. How do I work out my percentage gain ......Sat, 05 Sep 2026 11:36:55 -0400https://track.liberty.io/is-my-understanding-of-quotient-rings-correct.htmlhttps://track.liberty.io/is-my-understanding-of-quotient-rings-correct.htmlAmidst all the rigorous constructions of quotient rings involving equivalence relations and ideals, I feel that I have finally grasped what a quotient ring is. I have applied this intuition to a few ...Sat, 05 Sep 2026 11:36:44 -0400https://track.liberty.io/proving-a-relation-is-anti-symmetric-and-transitive.htmlhttps://track.liberty.io/proving-a-relation-is-anti-symmetric-and-transitive.html$P$ is a binary relation. $P ⊆ \mathbb{R}^2$. $P = \{(x,y): y = |x|\}$. As I understand for relation to be transitive: $(a,b) \in P$ and $(b,c) \in P$ then $(a,c \in P)$ for this particular relati......Sat, 05 Sep 2026 11:29:56 -0400https://track.liberty.io/synthetic-division-of-polynomials-by-factor-of-the-form-ax-b.htmlhttps://track.liberty.io/synthetic-division-of-polynomials-by-factor-of-the-form-ax-b.htmlFind the quotient and remainder when $6x^4-11x^3+5x^2-7x+9$ is divided by $(2x-3)$. I expressed the divisor $(2x-3)$ as $2(x-\frac{3}{2})$ and conducted synthetic division by $\frac{3}{2}$ and obta......Sat, 05 Sep 2026 11:29:21 -0400https://track.liberty.io/how-to-tell-if-a-linear-system-is-consistent.htmlhttps://track.liberty.io/how-to-tell-if-a-linear-system-is-consistent.htmlSo I have a list of equations and have made it into REF which gives me $$\left[\begin{matrix}1 & -1 & 1 \\ 0 & 1 & -1 \\ 0 & 0 & 0\end{matrix}\middle|\begin{matrix}1 \\ 2k-......Sat, 05 Sep 2026 11:28:18 -0400https://track.liberty.io/trying-to-understand-implied-population-in-statistics.htmlhttps://track.liberty.io/trying-to-understand-implied-population-in-statistics.htmlThis is from the 12th edition of "Understandable Statistics" from Cengage Learning. Chapter 1, Section 1, Question 8: What is the average miles per gallon (mpg) for all new hybrid small c......Sat, 05 Sep 2026 11:27:15 -0400https://track.liberty.io/find-the-midpoint-between-two-points-on-the-circle.htmlhttps://track.liberty.io/find-the-midpoint-between-two-points-on-the-circle.htmlI want to place a new point in the middle of the two points which are on the circle outline (Arc). I have the coordinates $(x,y)$ of the center of the circle, the two red points and the radius of the ...Sat, 05 Sep 2026 11:21:54 -0400https://track.liberty.io/game-theory-problem-3x3-matrix-pure-and-mixed-strategies.htmlhttps://track.liberty.io/game-theory-problem-3x3-matrix-pure-and-mixed-strategies.htmlFrom the following matrix: $$ \left( \begin{array}{c|ccc} & L & C & R\\ \hline T &3,0& 1,1 &4,2\\ M &3,4& 1,2 &2,3\\ B &1,3& 0,......Sat, 05 Sep 2026 11:13:32 -0400https://track.liberty.io/how-find-equilibrant.htmlhttps://track.liberty.io/how-find-equilibrant.htmlIf $F_1$ has magnitude 30 N and $F_2$ has magnitude 40 N, what is the exact magnitude and direction of the resultant vector? I have 10 $10 \sqrt37$ and 53 degrees. The next question is What is the ...Sat, 05 Sep 2026 11:11:27 -0400https://track.liberty.io/what-is-prime-factorisation-of-polynomials.htmlhttps://track.liberty.io/what-is-prime-factorisation-of-polynomials.htmlI have the following question: Find the prime factorization in $\mathbb{Z}[x]$ of $x^3 - 1, x^4 - 1, x^6 - 1$ and $x^{12} - 1$. You will need to check the irreducibility in $\mathbb{Z}[x]$, of t......Sat, 05 Sep 2026 11:01:52 -0400https://track.liberty.io/confusion-regarding-interval-on-which-a-function-is-increasing.htmlhttps://track.liberty.io/confusion-regarding-interval-on-which-a-function-is-increasing.htmlThe question is as follows: If the function $f(x)=\cos x$ is strictly increasing on the open interval $(0,\pi)$, where will it be increasing ? The answer to this question is $[0,\pi]$. I am a li......Sat, 05 Sep 2026 10:58:36 -0400https://track.liberty.io/use-implicit-differentiation-to-find-an-equation-of-the-tangent-line-t.htmlhttps://track.liberty.io/use-implicit-differentiation-to-find-an-equation-of-the-tangent-line-t.htmlUse implicit differentiation to find an equation of the tangent line to the curve $$x^2+xy+y^2=1$$ at $(1,1)$. I am not sure how I should work this out because the given point is not on the curve....Sat, 05 Sep 2026 10:57:32 -0400https://track.liberty.io/find-the-value-of-alpha-beta-for-the-equation-cos-alpha-cos-beta-cos-a.htmlhttps://track.liberty.io/find-the-value-of-alpha-beta-for-the-equation-cos-alpha-cos-beta-cos-a.htmlFind the value of $\alpha,\beta$ for the equation $\cos\alpha \cos\beta \cos(\alpha +\beta)=-\frac{1}{8}$ $\alpha>0$ & $\beta<\frac{\pi}{2}$ I get the following step after some substit......Sat, 05 Sep 2026 10:56:34 -0400https://track.liberty.io/arctan-3-2-doesn-039-t-give-expected-result.htmlhttps://track.liberty.io/arctan-3-2-doesn-039-t-give-expected-result.htmlLet's say I want to find the angle measure (in degrees) such that $\tan(x) = -3/2$. It turns out that $x \approx 123.7$, and when I compute $\tan(123.7)$, I get $\approx -3/2$; so far so good. Ho......Sat, 05 Sep 2026 10:53:36 -0400https://track.liberty.io/what-is-the-order-big-o-of-this-polynomial.htmlhttps://track.liberty.io/what-is-the-order-big-o-of-this-polynomial.html$$\frac{2n^{14} + 7 n^8 - 3}{3n^8 - n^4 + 3}$$ If this division is $p(n)$, I have to write $p(n) = O(n^k)$ I guess the answer is $n^6$, but how can i solve it step by step?...Sat, 05 Sep 2026 10:50:21 -0400https://track.liberty.io/the-number-of-ways-to-fill-a-3-by-3-grid.htmlhttps://track.liberty.io/the-number-of-ways-to-fill-a-3-by-3-grid.htmlI am currently studying the problem of combination. And when I am doing an exercise, I saw the following question: There is a 3 x 3 grid, and for each cells in the grid, two players take turn to f......Sat, 05 Sep 2026 10:49:57 -0400https://track.liberty.io/what-is-steepness-what-is-flatness.htmlhttps://track.liberty.io/what-is-steepness-what-is-flatness.htmlI have 3 graphs rather like $$y = \frac{1}{x}$$ and I am supposed to describe which one is steepest and which one is flattest. This is for an Econ class, so I'm not sure the terminology being used is ...Sat, 05 Sep 2026 10:37:30 -0400https://track.liberty.io/how-to-calculate-sum-of-a-sigma-notation-for-sum-expression-with-expon.htmlhttps://track.liberty.io/how-to-calculate-sum-of-a-sigma-notation-for-sum-expression-with-expon.htmlSo I have got this practice problem $$\sum_{k=1}^n \frac{(-1)^k\cdot2^{2k}}{3}$$ Now I am fairly new to the sigma notation for the sum of elements. I know the basic stuff like the sum of the arithm......Sat, 05 Sep 2026 10:37:09 -0400https://track.liberty.io/absolute-value-inequality-domain-restrictions.htmlhttps://track.liberty.io/absolute-value-inequality-domain-restrictions.htmlI'm almost embarrassed to post this elementary question, but I beg your indulgence. For the inequality $$-3 \left| x-4 \right| +2 > 6 + 2x$$ a graphical examination immediately shows no soluti......Sat, 05 Sep 2026 10:33:14 -0400https://track.liberty.io/how-does-one-calculate-the-area-of-a-rectangle-using-a-single-integral.htmlhttps://track.liberty.io/how-does-one-calculate-the-area-of-a-rectangle-using-a-single-integral.htmlI tried to ask this in a different way and did not correctly explain myself. I am ok integrating the line $y = x$ , let us say from $0$ to $2$ using calculus. If I want to get the square I can eas......Sat, 05 Sep 2026 10:31:59 -0400https://track.liberty.io/function-for-converting-mph-to-min-mile.htmlhttps://track.liberty.io/function-for-converting-mph-to-min-mile.htmlanyone have a good function for converting mph to min/mile. I can see that graph starting at say 60min/mile for 1 mpg decreases by half for every doubling of input, I.e. 2mph -> 30min/mile. 4pmh ->......Sat, 05 Sep 2026 10:31:31 -0400https://track.liberty.io/what-does-this-limit-statement-say-in-plain-english.htmlhttps://track.liberty.io/what-does-this-limit-statement-say-in-plain-english.htmlLet $ f(x) = x^2$ What is $\displaystyle\lim_{x \to 1}f(3)$ What is this statement saying in plain english? Is it "What is $f(3)$ approaching as $x$ approaches $1$"?...Sat, 05 Sep 2026 10:25:31 -0400https://track.liberty.io/the-expectation-of-the-half-normal-distribution.htmlhttps://track.liberty.io/the-expectation-of-the-half-normal-distribution.htmlFor the density function below, I need to find $E(X)$ and $E(X^2)$. For $E(X)$, I did the following steps and got the answer of $-2/\sqrt{2\pi}$. However, this is incorrect as the correct answer is $\...Sat, 05 Sep 2026 10:21:23 -0400https://track.liberty.io/system-of-equations-calculator.htmlhttps://track.liberty.io/system-of-equations-calculator.htmlI have a problem that is very simple in concept but very difficult to actually solve. I need to find the best fit polynomial of degree at most 4 for f(x)=ln(x) on the interval [1,4]. So far I have ...Sat, 05 Sep 2026 10:17:10 -0400https://track.liberty.io/propositional-logic-proving-contingency-without-truthtable.htmlhttps://track.liberty.io/propositional-logic-proving-contingency-without-truthtable.htmlHow can one proof that a proposition is a contingency? With a truth table it is easy (show that there is at least 1 true and 1 false outcome). But with rewriting propositions using logical equival......Sat, 05 Sep 2026 10:08:34 -0400https://track.liberty.io/what-are-the-required-backgrounds-of-robin-hartshorne-039-s-algebraic-.htmlhttps://track.liberty.io/what-are-the-required-backgrounds-of-robin-hartshorne-039-s-algebraic-.htmlIt seems that Robin Hartshorne's Algebraic Geometry is the place where a whole generation of fresh minds have successfully learned about the modern AG. But is it possible for someone who is out of ......Sat, 05 Sep 2026 10:07:03 -0400https://track.liberty.io/simple-way-to-evaluate-the-double-integration-of-sqrt-x-2-y-2-1-limits.htmlhttps://track.liberty.io/simple-way-to-evaluate-the-double-integration-of-sqrt-x-2-y-2-1-limits.htmlTo evaluate the integral, I performed the integration with respect to x and put the limits of x. I got this result: $$\sqrt{y^2 + 2}/2 + (y^2 + 1) \ln(1+ \sqrt{y^2 + 2})/2 - (y^2 + 1)/2 \ln(\sqrt{......Sat, 05 Sep 2026 10:00:15 -0400https://track.liberty.io/are-rectangles-are-trapezium.htmlhttps://track.liberty.io/are-rectangles-are-trapezium.htmlRectangle by definition given as the quadrilateral in which opposite sides are equal and each angle is equal to 90 degree Where as in case of trapezium any one pair of opposite sides is parallel. ......Sat, 05 Sep 2026 09:57:56 -0400https://track.liberty.io/find-dy-dx-for-an-integral.htmlhttps://track.liberty.io/find-dy-dx-for-an-integral.htmlCan someone walk me through how to find dy/dx (one of the problems I'm reviewing in my Calculus book): $$\int_{1/x}^{2} t\sqrt{t-4} dt $$ I know I need my (x) value to be in the numerator so I ca......Sat, 05 Sep 2026 09:54:27 -0400https://track.liberty.io/subtraction-of-elements-in-sets.htmlhttps://track.liberty.io/subtraction-of-elements-in-sets.htmlLet A be a set of finite elements. $A=\{1,2,3,4,5\}$ If I want to remove one element and show I removed one element, how should I do? Pseudo mathematical notation: $A - \{2\} = \{1,3,4,5\}$ Th......Sat, 05 Sep 2026 09:54:05 -0400https://track.liberty.io/is-there-a-formula-to-calculate-the-euclidean-distance-of-two-matrices.htmlhttps://track.liberty.io/is-there-a-formula-to-calculate-the-euclidean-distance-of-two-matrices.htmlWiki gives this formula ${\displaystyle {\begin{aligned}d(\mathbf {p} ,\mathbf {q} )=d(\mathbf {q} ,\mathbf {p} )&={\sqrt {(q_{1}-p_{1})^{2}+(q_{2}-p_{2})^{2}+\cdots +(q_{n}-p_{n})^{2}}}\\[8p......Sat, 05 Sep 2026 09:49:03 -0400https://track.liberty.io/isomorphic-quotient-rings-of-polynomial-ring-and-hilbert-functions.htmlhttps://track.liberty.io/isomorphic-quotient-rings-of-polynomial-ring-and-hilbert-functions.htmlLet $k$ be a field, $R=k[x_1,\cdots,x_n]$ and $I,J$ homogeneous ideals of $R$. Denote by $H_I(s), H_J(s)$ the Hilbert functions of $I,J$ respectively. If $R/I, R/J$ are isomorphic as graded rings, ......Sat, 05 Sep 2026 09:48:19 -0400https://track.liberty.io/proof-of-orthogonal-matrix-property-a-1-a-t.htmlhttps://track.liberty.io/proof-of-orthogonal-matrix-property-a-1-a-t.htmlI have prooved this orthogonal property. Please correct it or show your version of the proof if I am wrong: $A^{-1} = A^t$ $A^{-1} \times A = A^t \times A$ $I = I$ I appreciate your answer....Sat, 05 Sep 2026 09:47:23 -0400https://track.liberty.io/basis-of-trigonometric-functions.htmlhttps://track.liberty.io/basis-of-trigonometric-functions.htmlThe problem is as follows: Let $C[a,b]$ be the vector space (over $\mathbb{R}$) of continuous functions defined on $[a,b]$, $a < b$. For a fixed $n \in \mathbb{Z}^+$, consider the following s......Sat, 05 Sep 2026 09:36:21 -0400https://track.liberty.io/what-does-this-expected-value-notation-mean.htmlhttps://track.liberty.io/what-does-this-expected-value-notation-mean.htmlFrom: Learning from Data, section 2.3.1 - Bias and Variance: Let $f : X \rightarrow Y$, and let $D=\{(x,y=f(x)) : x \in A \subseteq X\}$ where each $x \in A$ is chosen independently with distribut......Sat, 05 Sep 2026 09:36:15 -0400https://track.liberty.io/what-is-the-remainder-of-19-1999-divided-by-25.htmlhttps://track.liberty.io/what-is-the-remainder-of-19-1999-divided-by-25.htmlHow to solve this problem without using Euler's Theorem?n Any help is highly appreciated! What is the remainder of $19^{1999}$ divided by 25?...Sat, 05 Sep 2026 09:35:40 -0400https://track.liberty.io/n-circles-and-1-straight-line-divide-into-regions.htmlhttps://track.liberty.io/n-circles-and-1-straight-line-divide-into-regions.htmlOriginal question: There are n circles and 1 straight line of a plane such that you can divide the plane into at most 44 parts. Find n. So I have no idea how to do this question, the word ‘plan......Sat, 05 Sep 2026 09:29:33 -0400https://track.liberty.io/why-is-the-frobenius-norm-of-a-matrix-greater-than-or-equal-to-the-spe.htmlhttps://track.liberty.io/why-is-the-frobenius-norm-of-a-matrix-greater-than-or-equal-to-the-spe.htmlHow can one prove that $ \|A\|_2 \le \|A\|_F $ without using $ \|A\|_2^2 := \lambda_{\max}(A^TA) $? It makes sense that the $2$-norm would be less than or equal to the Frobenius norm but I don't k......Sat, 05 Sep 2026 09:26:13 -0400https://track.liberty.io/log-power-rule-problem.htmlhttps://track.liberty.io/log-power-rule-problem.htmlAccording to many parts of the Internet, this log rule is used. log(a^b) = b*log(a) The proof is: Now let's say I want to use the rule in a Cartesian plane. I currently have the function........Sat, 05 Sep 2026 09:26:10 -0400https://track.liberty.io/completing-the-square-with-an-a-greater-than-1.htmlhttps://track.liberty.io/completing-the-square-with-an-a-greater-than-1.htmlI need some assistance with a specific problem where the equation given is $-3x^2-3x+9=0$ I have divided everything by $-3$ to get $x^2+x-3=0$ Then I move the $3$ to the other side $x^2+x=3$ ......Sat, 05 Sep 2026 09:24:59 -0400https://track.liberty.io/cardinality-of-the-union-of-two-sets.htmlhttps://track.liberty.io/cardinality-of-the-union-of-two-sets.htmlI am having trouble attempting to prove the inequality $|X\cup Y| \le |X|+|Y|$. Here is my intuitive argument when we take the union of $X\cup Y$ if there are repeated elements then they are not c......Sat, 05 Sep 2026 09:13:43 -0400https://track.liberty.io/alternate-definition-of-prime-number.htmlhttps://track.liberty.io/alternate-definition-of-prime-number.htmlI know the definition of prime number when dealing with integers, but I can't understand why the following definition also works: A prime is a quantity $p$ such that whenever $p$ is a factor of ......Sat, 05 Sep 2026 09:00:27 -0400https://track.liberty.io/system-of-equations-with-exponents.htmlhttps://track.liberty.io/system-of-equations-with-exponents.htmlWhat are the steps to solving a system of equations when $x$ and $y$ are exponents? Here is the problem to solve for $x$ and $y$. $$8^{3x}=4^{2y}$$ $$x-y=5$$...Sat, 05 Sep 2026 08:59:48 -0400https://track.liberty.io/what-happens-if-you-follow-the-sun.htmlhttps://track.liberty.io/what-happens-if-you-follow-the-sun.htmlTravelling around for quite a while and sometimes, well, just following the sun, today the question occurred to me: What happens if you really do this? So let's say some point is moving along the ......Sat, 05 Sep 2026 08:54:59 -0400https://track.liberty.io/trigonometric-identity-via-complex-exponential.htmlhttps://track.liberty.io/trigonometric-identity-via-complex-exponential.htmlNoting that $$\text{Re}[z_1z_2] = \text{Re}[z_1]\text{Re}[z_2]-\text{Im}[z_1]\text{Im}[z_2],$$ how can $$\cos(\alpha+\beta) = \text{Re}\left[e^{j(\alpha+\beta)}\right]$$ be expressed? Give ......Sat, 05 Sep 2026 08:29:16 -0400https://track.liberty.io/why-is-n-0-1-duplicate.htmlhttps://track.liberty.io/why-is-n-0-1-duplicate.htmlWhy is any number to the zeroeth power equal to 1? I would think it would be equal to zero, since nothing multiplied by nothing is, well, I would think 0. But it is 1? Examples: $(-5)^0 = 1$; $0......Sat, 05 Sep 2026 08:14:04 -0400https://track.liberty.io/best-way-to-denote-some-trigonometric-functions-tg-vs-tan-ctg-vs-cot.htmlhttps://track.liberty.io/best-way-to-denote-some-trigonometric-functions-tg-vs-tan-ctg-vs-cot.htmlWhat is the best way to denote tangent and other trigonometric functions: tg or tan, ctg or cot. What notation is commonly used and standardized?...Sat, 05 Sep 2026 08:09:18 -0400https://track.liberty.io/what-is-the-difference-between-a-function-and-a-curve.htmlhttps://track.liberty.io/what-is-the-difference-between-a-function-and-a-curve.htmlAre all curves functions? To my current knowledge a function gives only one output for a given input so x->f(x) , but for example a circle which is a curve is "made" out of two functions, so genera......Sat, 05 Sep 2026 08:07:50 -0400https://track.liberty.io/is-first-order-logic-syntax-context-free.htmlhttps://track.liberty.io/is-first-order-logic-syntax-context-free.htmlIf I were to build a first order logic parser could I do it using a context free parser generator (like bison)? I found some articles where First Order Logic Syntax is described with EBNF notation, ...Sat, 05 Sep 2026 08:04:11 -0400https://track.liberty.io/how-does-adding-big-o-notations-works.htmlhttps://track.liberty.io/how-does-adding-big-o-notations-works.htmlcan someone please explain how adding big O works. i.e. $O(n^3)+O(n) = O(n^3)$ why does the answer turn out this way? is it because $O(n^3)$ dominates the whole expression thus the answer is sti......Sat, 05 Sep 2026 08:00:53 -0400https://track.liberty.io/do-perfect-square-trinomials-only-have-one-root.htmlhttps://track.liberty.io/do-perfect-square-trinomials-only-have-one-root.htmlI apologize for the basic question, but I'm just now learning of perfect square trinomials in my math class. Google hasn't provided any relevant answers. Throughout all of the examples I have been......Sat, 05 Sep 2026 07:59:23 -0400https://track.liberty.io/real-life-applications-of-topology.htmlhttps://track.liberty.io/real-life-applications-of-topology.htmlThe other day I and my friend were having an argument. He was saying that there is no real life application of Topology at all whatsoever. I want to disprove him, so posting the question here. Wha......Sat, 05 Sep 2026 07:48:09 -0400https://track.liberty.io/limit-of-0-n-as-n-approaches-infinity.htmlhttps://track.liberty.io/limit-of-0-n-as-n-approaches-infinity.htmlIt's been a while I've been thinking about this question: How does a number change? How 1 reaches 2 for example? Aren't there infinite numbers between these two? This may have more to do with the w......Sat, 05 Sep 2026 07:42:23 -0400https://track.liberty.io/what-is-the-scale-factor-of-the-two-similar-triangles.htmlhttps://track.liberty.io/what-is-the-scale-factor-of-the-two-similar-triangles.htmlHere's the question: A triangle has side lengths 7 cm, 10 cm and 15 cm. A similar triangle to it has 25 times the area of the first triangle. Find the length scale factor between the two triangles.......Sat, 05 Sep 2026 07:40:32 -0400https://track.liberty.io/differential-equation-to-difference-equation.htmlhttps://track.liberty.io/differential-equation-to-difference-equation.htmlI have the following equation : $$\frac{dx}{dt} = -5(x-2)$$ $$\frac{dy}{dt} = 0$$ How do I change this differential equation to a difference equation ? Do I use Euler forward method ? I remember ...Sat, 05 Sep 2026 07:37:49 -0400https://track.liberty.io/advantage-of-l-stability-compared-to-a-stability.htmlhttps://track.liberty.io/advantage-of-l-stability-compared-to-a-stability.htmlAs we know L-stability is a special kind of A-stability in numerics. The point here is the rational function goes to zero as the variable goes to infinity. But why does it help to improve the numer......Sat, 05 Sep 2026 07:34:30 -0400https://track.liberty.io/should-a-linear-function-always-fix-the-origin-duplicate.htmlhttps://track.liberty.io/should-a-linear-function-always-fix-the-origin-duplicate.htmlI became very confused about linear functions after reading this question What is the difference between linear and affine function In the comments it says that $F(x)=2*x+4$ is NOT a linear funct......Sat, 05 Sep 2026 07:33:19 -0400https://track.liberty.io/what-is-the-amalg-symbol.htmlhttps://track.liberty.io/what-is-the-amalg-symbol.htmlI saw this formula: $A \amalg B$, where $A$ and $B$ are sets. I searched for the name of the symbol, "amalg", but haven't found a definition. What is the meaning of this symbol?...Sat, 05 Sep 2026 07:20:25 -0400https://track.liberty.io/why-are-there-two-different-forms-of-the-characteristic-polynomial.htmlhttps://track.liberty.io/why-are-there-two-different-forms-of-the-characteristic-polynomial.htmlExcuse me for my ignorance towards this topic. Unfortunately my professor waited to the last of day class to cover this important chapter. He merely told us that the Characteristic Polynomial equ......Sat, 05 Sep 2026 07:12:46 -0400https://track.liberty.io/what-is-the-number-j.htmlhttps://track.liberty.io/what-is-the-number-j.htmlSo, I've been hearing that $e$ and $i$ are not the only numbers out there that are represented and known by a single letter in the English alphabet. Apparently, there is also the number $j$. What i......Sat, 05 Sep 2026 07:12:40 -0400https://track.liberty.io/calculating-probability-when-range-of-values-is-given.htmlhttps://track.liberty.io/calculating-probability-when-range-of-values-is-given.htmlAssume the number of pounds of peanuts consumed in a year is equally likely to assume any value (including fractions) between 20 and 100 pounds. What is the probability that someone consumes less ...Sat, 05 Sep 2026 06:41:04 -0400https://track.liberty.io/how-to-use-atan2.htmlhttps://track.liberty.io/how-to-use-atan2.htmlI've looked at Wikipedia and don't really understand what I'm reading or how to implement atan2(). I was hoping someone could point me to a better resource or give me an example of how to use atan......Sat, 05 Sep 2026 06:40:50 -0400https://track.liberty.io/if-we-have-had-n-random-number-generations-between-1-and-100-what-is-t.htmlhttps://track.liberty.io/if-we-have-had-n-random-number-generations-between-1-and-100-what-is-t.htmlIf we have had n random number generations between 1 and 100, where n is a number less than 100, what is the probability of the same number coming up twice? After 2 generations, I'm guessing there......Sat, 05 Sep 2026 06:40:15 -0400https://track.liberty.io/find-the-number-of-seven-letter-words-that-use-letters-from-the-set-al.htmlhttps://track.liberty.io/find-the-number-of-seven-letter-words-that-use-letters-from-the-set-al.htmlFind the number of seven-letter words that use letters from the set $\{\alpha,\beta,\gamma,\delta, \epsilon\}$ and contain at least one each of $\alpha$, $\beta$, and $\gamma$. My attempt: using ...Sat, 05 Sep 2026 06:39:36 -0400https://track.liberty.io/epimorphism-and-monomorphism-explained-without-math.htmlhttps://track.liberty.io/epimorphism-and-monomorphism-explained-without-math.htmlI'm trying to understand category theory to increase my coding skills and epimorphism and monomorphism aren't clear to me. Unfortunately, my last formal education was when I was 12 due to circumsta......Sat, 05 Sep 2026 06:38:02 -0400https://track.liberty.io/how-do-i-find-the-average-acceleration-and-instantaneous-acceleration-.htmlhttps://track.liberty.io/how-do-i-find-the-average-acceleration-and-instantaneous-acceleration-.htmlI have completed parts $a$, $b$, and $c$ of the attached problem. However, I am having trouble with parts $d$ and $e$. Can anyone check to see if I am on the right track and how do I solve for part......Sat, 05 Sep 2026 06:29:31 -0400