https://track.liberty.ioFresh celebrity buzz with instant attention value.en-usSat, 05 Sep 2026 08:35:12 -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 -0400https://track.liberty.io/finding-eigenvectors-with-repeated-eigenvalues.htmlhttps://track.liberty.io/finding-eigenvectors-with-repeated-eigenvalues.htmlI have a matrix $A = \left(\begin{matrix} -5 & -6 & 3\\3 & 4 & -3\\0 & 0 & -2\end{matrix}\right)$ for which I am trying to find the Eigenvalues and Eigenvectors. In this cas......Sat, 05 Sep 2026 06:26:35 -0400https://track.liberty.io/construct-two-circles-tangent-to-each-other-and-to-a-line-and-a-circle.htmlhttps://track.liberty.io/construct-two-circles-tangent-to-each-other-and-to-a-line-and-a-circle.htmlI saw a question that was nearly the same as this, but I couldn't understand the answers. Assume that everything that seems to be tangent should be tangent, and that everything that appears to be a ...Sat, 05 Sep 2026 06:16:56 -0400https://track.liberty.io/term-document-vs-document-term-matrix-closed.htmlhttps://track.liberty.io/term-document-vs-document-term-matrix-closed.htmlI am not sure if this is the right website to ask this question but I cant figure out where else to get the answer so, please, dont be mad :-) As my bachelor thesis/project, I am trying to constru......Sat, 05 Sep 2026 06:13:12 -0400https://track.liberty.io/convergent-sequence-of-real-numbers.htmlhttps://track.liberty.io/convergent-sequence-of-real-numbers.htmlSuppose that a_n is a convergent sequence of real numbers such that lim a_n = a > 0. Then there exists a K belongs to N such that a_n > 4a/5 holds for all n is greater than or equal to K. Is it tru......Sat, 05 Sep 2026 06:04:01 -0400https://track.liberty.io/are-cumulative-distribution-function-for-a-continuous-random-variable-.htmlhttps://track.liberty.io/are-cumulative-distribution-function-for-a-continuous-random-variable-.htmlI was reading the book "probability random variables and stochastic processes by Athanasios Papoulis - Third Edition" where by in the properties of distribution functions on Page 69 of the book the ...Sat, 05 Sep 2026 05:56:34 -0400https://track.liberty.io/assigning-60-apples-to-4-baskets-so-that-each-basket-contains-15-apple.htmlhttps://track.liberty.io/assigning-60-apples-to-4-baskets-so-that-each-basket-contains-15-apple.htmlAmong $60$ apples collected from an apple tree, there are three bad apples. The apples are randomly placed into four baskets so that each basket contains $15$ apples. (i)Compute the probability tha......Sat, 05 Sep 2026 05:54:15 -0400https://track.liberty.io/gauss-seidel-linear-system-solver-cases-when-it-is-unstable.htmlhttps://track.liberty.io/gauss-seidel-linear-system-solver-cases-when-it-is-unstable.htmlI have a Gauss-Seidel linear system solver that has always been able to solve many kinds of linear systems. However, it struggles when solving this very simple system: $$ 22.7986\,x_1 + 4981.2903 \......Sat, 05 Sep 2026 05:53:12 -0400https://track.liberty.io/derivative-of-x-arctan-x.htmlhttps://track.liberty.io/derivative-of-x-arctan-x.htmlI have the following derivative: $$\frac d{dx}x\arctan x$$ I'm unsure of how to solve this derivative; obviously it's a product rule, however is it in a similar vein to finding the derivative of ar......Sat, 05 Sep 2026 05:38:38 -0400https://track.liberty.io/how-to-tell-if-a-set-of-vectors-spans-r4.htmlhttps://track.liberty.io/how-to-tell-if-a-set-of-vectors-spans-r4.htmlFirst year linear algebra. Question is Exercise 5.1.3.b from "Linear Algebra w Applications, K. Nicholson" Determine if the given vectors span $R^4$: {$(1,3,-5,0), (-2,1,0,0), (0,2,1,-1), (1,-4,5......Sat, 05 Sep 2026 05:35:31 -0400https://track.liberty.io/find-all-numbers-n-such-that-phi-n-20.htmlhttps://track.liberty.io/find-all-numbers-n-such-that-phi-n-20.htmlI'm trying to find all values of $n$ that satisfy euler's phi function in the title. I start by knowing $n$ cannot be a power of $2$ since $\phi (n)$ is not. Then, $n$ is divisible by an odd prime......Sat, 05 Sep 2026 05:34:55 -0400https://track.liberty.io/what-039-s-the-definition-of-line-segments-in-the-context-of-euclidean.htmlhttps://track.liberty.io/what-039-s-the-definition-of-line-segments-in-the-context-of-euclidean.htmlLet $\mathbb R^3$ be the set ${\{(r_1,r_2,r_3):r_1,r_2,r_3 \in \mathbb R}\}$. Let $d:\mathbb R^3 \times \mathbb R^3 \rightarrow \mathbb R$ be a function defined by $d(x,y)=\sqrt {(x_1-y_1)^2+(x_2-......Sat, 05 Sep 2026 05:34:14 -0400https://track.liberty.io/the-choice-of-scalar-factors-in-the-proof-of-the-schwarz-inequality.htmlhttps://track.liberty.io/the-choice-of-scalar-factors-in-the-proof-of-the-schwarz-inequality.htmlIn this proof for the Schwarz Inequality, they seemingly arbitrarily choose $r = w\cdot w$ and $s =-(v\cdot w)$. Why did they make these selections? I don't understand where these values for $r$ an......Sat, 05 Sep 2026 05:34:07 -0400https://track.liberty.io/how-to-use-weierstrass-test.htmlhttps://track.liberty.io/how-to-use-weierstrass-test.htmlHow can I use Weierstrass test to determine a set $I$ on which the series $\displaystyle \sum_{n=1}^{\infty} nx^n(1-x)^n$ converges uniformly? Weierstrass test: Let $f_n : I \to \mathbb{R}$ be a ...Sat, 05 Sep 2026 05:30:42 -0400https://track.liberty.io/why-are-there-nine-regular-polyhedra.htmlhttps://track.liberty.io/why-are-there-nine-regular-polyhedra.htmlContext for question: The study of polyhedra and more generally of polytopes has never been particularly focused on rigor, and many references for results are often either non-existent, or impossi......Sat, 05 Sep 2026 05:25:57 -0400https://track.liberty.io/how-many-permutations-can-a-18-digit-number-have.htmlhttps://track.liberty.io/how-many-permutations-can-a-18-digit-number-have.htmlHow can I work out how many permutations a 18 digit number may have that using digits 0-9? I found this (How many permutations are there if you have n+1 items, where the extra item can be repeated?) ...Sat, 05 Sep 2026 05:23:17 -0400https://track.liberty.io/how-to-understand-e-xy-intuitively.htmlhttps://track.liberty.io/how-to-understand-e-xy-intuitively.htmlI have no trouble understanding $\displaystyle E(X)=\int xf(x)\,dx $ and $\displaystyle E(Y)=\int y f(y)\,dy$ As each $x$ multiplies the corresponding $f(x)$ and we take the integral of it to cal......Sat, 05 Sep 2026 05:18:42 -0400https://track.liberty.io/calculation-of-the-covariance-of-gaussian-mixtures.htmlhttps://track.liberty.io/calculation-of-the-covariance-of-gaussian-mixtures.htmlI have a Gaussian mixture model, given by: $$ X \sim \sum_{i = 1}^M \alpha_i N_p(\mu_i, C_i) $$ such that $\sum_{i=1}^M\alpha_i =1 $. Is there a way I can compute the overall covariance matrix if $......Sat, 05 Sep 2026 05:18:16 -0400https://track.liberty.io/how-to-find-the-range-of-a-function-without-a-graph.htmlhttps://track.liberty.io/how-to-find-the-range-of-a-function-without-a-graph.htmlOkay so I know how to find the domain of the function $f(x) = \frac{3x^2}{x^2-1}$, which is $x\neq 1$ and $x\neq -1$, but I'm totally confused on how to find the range without using a graph....Sat, 05 Sep 2026 05:07:45 -0400https://track.liberty.io/why-do-limits-results-change-with-algebraic-manipulation.htmlhttps://track.liberty.io/why-do-limits-results-change-with-algebraic-manipulation.htmlI just started studying calculus by myself, and I am in limits, but I don't seem to understand why the result changes when I use algebra. I understand how to do it. But why is the result different ......Sat, 05 Sep 2026 05:06:29 -0400https://track.liberty.io/converting-between-explicit-implicit-and-parametric-function.htmlhttps://track.liberty.io/converting-between-explicit-implicit-and-parametric-function.htmlGiven an explicit function $y = \sin(x) + \cos(x)$, how to convert it to the respective parametric functions $x = f_1(t)$, $y = f_2(t)$? Given parametric functions $x = \sin(t)$ and $y = \cos(t)$, ......Sat, 05 Sep 2026 04:56:52 -0400https://track.liberty.io/help-on-surjection-injection-and-bijection-duplicate.htmlhttps://track.liberty.io/help-on-surjection-injection-and-bijection-duplicate.htmlI am a undergraduate majoring in CS. In preparation for a discrete mathematics exam coming up next week, I am looking through problems I got wrong on the homework. A concept I don't understand are ...Sat, 05 Sep 2026 04:55:23 -0400https://track.liberty.io/equation-of-plane-passing-through-3-points.htmlhttps://track.liberty.io/equation-of-plane-passing-through-3-points.htmlGiven three points $a, b, c\in \mathbb R^3$, how do we find the equation of the plane passing through these three points? Original Question : Let $p : \mathbb{R} \to \mathbb{R}^3$ be a twice ...Sat, 05 Sep 2026 04:53:12 -0400https://track.liberty.io/the-cartesian-product-of-two-regular-graphs-is-regular.htmlhttps://track.liberty.io/the-cartesian-product-of-two-regular-graphs-is-regular.htmlConsider two regular graphs $G_1$ and $G_2$ of degrees $d_1$ and $d_2$, respectively. I would like to prove that the Cartesian product $G_1 \square G_2$ is regular of degree $d_1 + d_2$. Currently I ...Sat, 05 Sep 2026 04:49:26 -0400https://track.liberty.io/find-how-many-positive-divisors-a-number-has-what-would-you-do.htmlhttps://track.liberty.io/find-how-many-positive-divisors-a-number-has-what-would-you-do.htmlRecently I learned an amazing thing. Suppose you are given a number and you have to find how many positive divisors it has. What would you do ? Solution: Suppose you select $12$. It has $1,2,3,4,6......Sat, 05 Sep 2026 04:35:08 -0400https://track.liberty.io/calculator-for-ax-b-where-a-is-singular.htmlhttps://track.liberty.io/calculator-for-ax-b-where-a-is-singular.htmlI have found many online calculators that can calculate $Ax=b$ only when $A$ isn't singular. However in most of my exercises, $A$ is singular and I want to ensure that I have found the right soluti......Sat, 05 Sep 2026 04:28:28 -0400https://track.liberty.io/limitations-of-marc-32-machine-numbers.htmlhttps://track.liberty.io/limitations-of-marc-32-machine-numbers.htmlI am in doubt whether I understand Marc-32 machine numbers limitations correctly, however, before I try to explain my questions, I will just shortly explain how one find Marc-32 numbers (to ensure ......Sat, 05 Sep 2026 04:24:19 -0400https://track.liberty.io/understanding-dot-and-cross-product.htmlhttps://track.liberty.io/understanding-dot-and-cross-product.htmlWhat purposes do the Dot and Cross products serve? Do you have any clear examples of when you would use them?...Sat, 05 Sep 2026 04:24:05 -0400https://track.liberty.io/what-are-the-components-of-a-function.htmlhttps://track.liberty.io/what-are-the-components-of-a-function.htmlCan someone tell me what is meant by The action of f on input x is written out in component form is $f(x) = (f_1 (x), \dots, f_m(x) )$ What is the component of a function? So if $m = 2$, and say $......Sat, 05 Sep 2026 04:23:56 -0400