Recent Questions - Mathematics Stack Exchange

---

• Can f(x) = 1/x be expressed in quadratic form for x > 0
• example of Grade of module
• On inequality involving logarithm
• Total variation fo a complex measure
• Spectral radius and maximum degree of a graph
• Transverse Foliation to the Flow of a Differential Equation on a Tangent Bundle?
• What is the algebraic rule applied to get $5\left(3^{n-1}-2^{n}\right)-6\left(3^{n-2}-2^{n-1}\right)=(15-6) \times 3^{n-2}-(10-6) \times 2^{n-1}$?
• Modeling body temperature in a continuous framework
• Find all possible values of abc, for given $a^2+b^2=c^2$ and $a+b+c=k$
• a question about formal definition of limit
• Intuition behind $\frac{a}{b} \equiv k \pmod{p} $
• Elementary Probability: Permutation and combinations question
• Combined pseudometric, triangle inequality, proof
• Simplifying $ \sum_{k=1}^n\binom n{k-1}\frac{x^{k+n}y^{n-k+1}}k. $
• Finding zeroes for function $f(t) = e^{k(t-1)} -t$ for $k> 0$ analytically
• Quadratic forms: Existence of $(x,y)\in \mathbb{Z}^2 \setminus \{0\}$ such that $P(x,y) < 2\sqrt{\lvert \det(P) \rvert}$
• Criteria for $3 \times 3$ matrix to positive definite
• The maximum of $W_t-t$ with respect to $t$
• How many ways are there to arrange $3$ men, $3$ women, $3$ children in a circle such that they alternate clockwise in this order: man, woman, child?
• I am trying to generate random matrix with based on some condition, How many matrices the can be generated?
• How to prove $2n+1$ is odd for $n \in N$?
• PHI function to list relative primes
• Intuition for fibers over a point in $\operatorname{Spec}(A)$
• How to prove: If $A$ and $B$ are normal matrices and AB = BA, $A+B$ and $AB$ are also normal, and $A$ and $B$ are simultaneously diagonalizable.
• Can the geodesic equation be used to solve the Brachistochrone Problem?
• Unital commutative Banach algebra $A$, $A/\operatorname{radical}(A)$ has no quasinilpotent elements
• How to prove that $ \bar e_2 = e_1 \sin (w) + e_2 \cos(w)$ where $w$ is the angle between $e_1$ and $e_2$
• Poisson Distribution Problem relating to birthdays
• Proof that every repeating decimal is rational
• Why is negative times negative = positive?

Can f(x) = 1/x be expressed in quadratic form for x > 0 Posted: 16 May 2022 11:25 AM PDT We can consider 1/x an continuous function when x > 0. Chart of 1/x where x > 0 This curve appears to has some form of inflection at (1,1). I could see transforming the equation in a manner to produce a curve with a minimum at (1,1). Is there any way to express 1/x where x > 0 as a quadratic function? Or do misunderstand something about the properties of 1/x?

example of Grade of module Posted: 16 May 2022 11:22 AM PDT Let $R$ be a commutative Noetherian ring, $I$ be a proper ideal of $R$ and $M$ is finitely generated $R$-module. I want to find an example of module $M$ such that $$grade(I,M)>grade(I,R)$$ thank you

On inequality involving logarithm Posted: 16 May 2022 11:22 AM PDT I have been researching in order to do a paperwork about inequalities for my university and I ran into this inequality I have been trying to understand without results. it says. For every $t\geq 1$ we have that \begin{equation} \log(t)\leq \frac{(t+1)(t^3-1)}{3t(t^2+1)} \end{equation}

Total variation fo a complex measure Posted: 16 May 2022 11:18 AM PDT In Folland's text he states, The total variation of a complex measure $\nu$ is the positive measure $|\nu|$ determined by the property that if $d\nu = f d\mu$ where $\mu$ is a positive measure, then $d|\nu| = |f|d\mu$. To see that this is well defind, we observe first that every $\nu$ is of the form $fd\mu$ for some finite measure $\mu$ and some $L^1(\mu)$; indeed we can take $\mu = |\nu_r| + |\nu_i|$ and use Theorem 3.12 to obtain $f$. Here $\nu_r$ and $\nu_i$ represent the real and positive portions of the complex measure $\nu$. The theorem he references is the Lebesgue-Radon-Nikodym Theorem for complex measures: Theorem 3.12 If $\nu$ is a complex measure and $\mu$ is a $\sigma$-finite positive measure on $(X, \mathcal{M})$, there exists a complex measure $\lambda$ and an $f \in L^1(\mu)$ such that $\lambda \perp \mu$ and $d\nu = d\lambda + f d\mu$. What I am unclear on is his claim that $\nu = \int f d\mu$. To my understanding, Theorem 3.12 states we should instead have $\nu = \lambda + \int fd\,\mu$, for some complex measure $\lambda$. Why is the $\lambda$ omitted here?

Spectral radius and maximum degree of a graph Posted: 16 May 2022 11:16 AM PDT How can I show that the spectral radius of a graph G is less than or equal to its maximum degree?

Transverse Foliation to the Flow of a Differential Equation on a Tangent Bundle? Posted: 16 May 2022 11:10 AM PDT Let $Q^n$ be a closed manifold, $M = TQ$ its tangent bundle, $\xi$ be a differential equation on $M$ that satisfies the "canonical flip on $TTQ$" (a "second-order differential equation on $Q$"), but suppose $\xi$ is the zero vector at some point(s) of $M$. If we define $p \sim q$ if there is a flow line $\Phi_t(g)$ of $\xi$ with $\Phi_{t_1}(g) = p$ and $\Phi_{t_2}(g) = q$, do the equivalence classes form a 1-dimensional foliation $\mathscr{F}$ of $M$? If the equivalence classes form a 1-dimensional foliation of $M$, when is there a codimension-1 foliation $\mathscr{G}$ transverse to $\mathscr{F}$? If there is a codimension-1 foliation $\mathscr{G}$ of $M$ transverse to $\mathscr{F}$, if $L_g$ is any leaf of $\mathscr{G}$, is $\{K_g\ |\ g \in L_g\} = \mathscr{F}$, where each $K_g = \{p \in M\ |\ \exists\ t \in \mathbb{R} \text{ with } \Phi_t(g) = p\}$? Thanks in advance.

What is the algebraic rule applied to get $5\left(3^{n-1}-2^{n}\right)-6\left(3^{n-2}-2^{n-1}\right)=(15-6) \times 3^{n-2}-(10-6) \times 2^{n-1}$? Posted: 16 May 2022 11:21 AM PDT Apperently the following equivalence holds: $$5\left(3^{n-1}-2^{n}\right)-6\left(3^{n-2}-2^{n-1}\right)=(15-6) \times 3^{n-2}-(10-6) \times 2^{n-1}$$ What algebraic rule is applied to get from the LHS to the RHS?

Modeling body temperature in a continuous framework Posted: 16 May 2022 11:11 AM PDT I am reviewing for an exam and was reviewing last year's exam. Since our professor doesn't want to solve it in class, I come here to see if someone is so kind to solve it. The problem has to be modeled with differential equations. Attached is the problem: at 23:08 hours alerting that a lifeless body has been found inside a cold room of the President Hotel. Two patrol cars are immediately dispatched to the hotel. Upon arrival of the first car at 23:11, one of the agents inspects the scene, confirms the death of the victim and takes his temperature, which is 12.40ºC. After 30 seconds, the second patrol car arrives and another officer takes the body temperature again, which is 11.85ºC. The officer in charge confirms that the victim is L. Palmer, one of his employees. The detective in charge of the case observes that access to the cold room is controlled by a magnetic lock and asks for the access log to the cold room, which each employee can only enter by swiping his or her identification card. The manager provides the police with the access list shown in the attached table. At what time do you think L. Palmer could have died, taking into account that the cold room is at approximately 0°C? Who would be your prime suspect? Justify your answers.

Find all possible values of abc, for given $a^2+b^2=c^2$ and $a+b+c=k$ Posted: 16 May 2022 11:21 AM PDT We have a system of two equations: $a^2+b^2=c^2$ and $a+b+c=k$ Where $a,b,c,k \in R$ Find all possible values of $abc$ That is defined for given conditions in real numbers(without complex ones) Something like this : $abc \in [H,B]$ or any other range What I did Rearranged system of equations to get following $a^2+b^2=(k-a-b)^2$ Opened brackets and get $2 a b - 2 a k - 2 b k + k^2=0$ $k^2=2ak+2bk-2ab$ And if $k \in R$, so $k^2 \in [0,+\infty]$ So $2ak+2bk-2ab \in [0,+\infty]$ $2ak+2bk-2ab \ge 0 $ $ak+bk-ab \ge 0 $ $k(a+b) \ge ab $ $ck(a+b) \ge abc $ And I don't really know what to do next. I kinda got abc in some conditions but cannot come up with solution yet. I even created graph to visualize how it looks like https://www.desmos.com/calculator/ati9wsixnq

a question about formal definition of limit Posted: 16 May 2022 11:08 AM PDT i tried to understand the formal definition of limts, which is the function $f$ approaches the limit $l$ near $a$ means; for every $\epsilon >0 $ there is some $\theta$ such that ,for all $x$,if $0<|x-a|<\theta$, then $|f(x)-l|<\epsilon$ my problem is that i cannot understand why $f(x)>l-\epsilon$ for all $\epsilon$.intuitively if we take any fixed $x$ such that $0<|x-a|<\theta$ we will get $f(x)>l-\epsilon$ just for some $\epsilon$,because we can choose $\epsilon$ very close to $l$ in which $f(x)<l-\epsilon$ . I really tried to convince myself of this definition, but I could not, because when I look at the curve, I find that it is not true for all $\epsilon$. Can someone explain more?

Intuition behind $\frac{a}{b} \equiv k \pmod{p} $ Posted: 16 May 2022 11:22 AM PDT I am working with $p$-adic numbers at the moment and am having some trouble with a basic fact. I know that for $\frac{a}{b}\in\mathbb{Q}$ there is a solution $k\in\mathbb{Z}$ to $\frac{a}{b} \equiv k\pmod{p^n}$ iff $p^n$ does not divide b. The condition that $p^n$ doesn't divide b makes sense to me as otherwise $b\equiv 0\pmod{p^n}$ and thus there would not be an inverse. I know that i can view $\frac{a}{b} = a*b^{-1}$ but I still don't find it very intuitive to think of the equivalence class of a fraction in modular arithmetic. Can somebody please explain to me in simple terms the connection between fractions and proving that they have an equivalence class in Z modulo a prime number? From my understanding there would need to be a canonical homomorphism $\mathbb{Z}_{(p)} \mapsto \mathbb{Z}/ p^n$. Is there such a map and how do you define it and proof its existence and well definedness? Edit: And as a bonus question: How is $\frac{a}{b} \equiv k\pmod{p^n} \iff a \equiv bk\pmod{p^n}$ ? It's probably very basic but I cant really wrap my head around it, as in general you can't always say $ab \equiv c \pmod{p}\iff a \equiv b^{-1}c \pmod{p^n}$

Elementary Probability: Permutation and combinations question Posted: 16 May 2022 11:03 AM PDT What is the probability of picking a permutation of the letters NUMBER which starts and ends with a vowel. I thought that the there are two situations: when the u is in front and when the e is in front. When the e is in front, there are 4! different combinations and the same for when the u is in front. I therefore concluded that the answer must be (4!*2)/6! = 1/15 but my book says 1/45. Thanks in advance

Combined pseudometric, triangle inequality, proof Posted: 16 May 2022 11:08 AM PDT Is there any proof that the pseudometric $$ \left\Vert x-y\right\Vert _{2}\sin\frac{\angle(x,y)}{2}, $$ for any $x,y,z\in [0,\pi]$ holds the triangle inequality $$ \left\Vert x-z\right\Vert _{2}\sin\frac{\angle(x,z)}{2}\leq\left\Vert x-y\right\Vert _{2}\sin\frac{\angle(x,y)}{2}+\left\Vert y-z\right\Vert _{2}\sin\frac{\angle(y,z)}{2}. $$ Initially, it can be rewritten to the form $$ \left\Vert x-z\right\Vert _{2}\sin\frac{\arccos\left\langle n_{x},n_{z}\right\rangle }{2}\leq\left\Vert x-y\right\Vert _{2}\sin\frac{\arccos\left\langle n_{x},n_{y}\right\rangle }{2}+\left\Vert y-z\right\Vert \sin\frac{\arccos\left\langle n_{y},n_{z}\right\rangle }{2}, $$ since $\left\Vert n\right\Vert =1$. Taking into account that $\sin\frac{\arccos x}{2}=\sqrt{\frac{1-x}{2}}$, is \begin{align*} \left\Vert x-z\right\Vert _{2}\sqrt{\frac{1-\cos\arccos\left\langle n_{x},n_{z}\right\rangle }{2}} & \leq\left\Vert x-y\right\Vert _{2}\sqrt{\frac{1-\cos\arccos\left\langle n_{x},n_{y}\right\rangle }{2}}+\left\Vert y-z\right\Vert \sqrt{\frac{1-\cos\arccos\left\langle n_{y},n_{z}\right\rangle }{2}},\\ \left\Vert x-z\right\Vert _{2}\sqrt{\frac{1-\left\langle n_{x},n_{z}\right\rangle }{2}} & \leq\left\Vert x-y\right\Vert _{2}\sqrt{\frac{1-\left\langle n_{x},n_{y}\right\rangle }{2}}+\left\Vert y-z\right\Vert \sqrt{\frac{1-\left\langle n_{y},n_{z}\right\rangle }{2}},\\ \left\Vert x-z\right\Vert _{2}\sqrt{\frac{1-\cos\omega_{xz}}{2}} & \leq\left\Vert x-y\right\Vert _{2}\sqrt{\frac{1-\cos\omega_{xy}}{2}}+\left\Vert y-z\right\Vert \sqrt{\frac{1-\cos\omega_{yz}}{2}},\\ \left\Vert x-z\right\Vert _{2}\sin\frac{\omega_{xz}}{2} & \leq\left\Vert x-y\right\Vert _{2}\sin\frac{\omega_{xy}}{2}+\left\Vert y-z\right\Vert \sin\frac{\omega_{yz}}{2}. \end{align*} which represents the triangle inequlity... Is the proof correct or not? Thanks for your help.

Simplifying $ \sum_{k=1}^n\binom n{k-1}\frac{x^{k+n}y^{n-k+1}}k. $ Posted: 16 May 2022 11:22 AM PDT I want to simplify this binomial expression: $$ \sum_{k=1}^n\binom n{k-1}\frac{x^{k+n}y^{n-k+1}}k. $$ I tried to simplify it but it's pretty hard . So if someone can help with a hint or solution.

Finding zeroes for function $f(t) = e^{k(t-1)} -t$ for $k> 0$ analytically Posted: 16 May 2022 11:05 AM PDT I tried using Lambert W function the following way $$e^{k(t-1)} -t=0$$ $$e^{k(t-1)}=t$$ $$-ke^{-k} = -kte^{-kt}$$ $$W(-ke^{-k}) = W(-kte^{-kt})$$ $$-k = -kt \implies t = 1$$ but this only gives me one trivial root, graphing the function clearly shows that this function has a root $t_0$ where $0\leq t_0< 1$ as well, is there any analytical way I can reach that point? I need a general form of the root in terms of $k$ so I can apply a limit on it, any kind of help would be appreciated.

Quadratic forms: Existence of $(x,y)\in \mathbb{Z}^2 \setminus \{0\}$ such that $P(x,y) < 2\sqrt{\lvert \det(P) \rvert}$ Posted: 16 May 2022 11:23 AM PDT I am stuck on the following exercise: Show that for any non-degenerate quadratic form $P$ over $\mathbb{R}$, that is either indefinite or positive definite, exists an integer point $(x,y) \in \mathbb{Z}^2 \setminus \{0\}$ such that $$0 < P(x,y) < 2\sqrt{\lvert \det(P) \rvert}.$$ From the lecture I know about Minkowski's Theorem, but I do not see how to use this here. Could you please help me? EDIT: The part with positive definite forms was already solved here: Minkowski's convex body theorem and binary quadratic forms So we are only left with the case of indefinite forms.

Criteria for $3 \times 3$ matrix to positive definite Posted: 16 May 2022 11:07 AM PDT Here it is said that a $2\times 2$ matrix $A$ is positive definite if and only if $tr(A) >0$ and $det(A)>0$. This will not work if $A$ is $3\times 3$. But is there any way to enforce the positive definiteness of the matrix $A$ via the trace and determinant of $A$, if $A$ is of size $3\times 3$?

The maximum of $W_t-t$ with respect to $t$ Posted: 16 May 2022 11:20 AM PDT I'm asked to calculate $$ P\left(\sup_{t\geq 0} W_t-t-1\geq 0\right)$$ Where $W_t$ is the Brownian motion. I tried to calculate it as the way I calculated the distribution of $\sup_{0\leq s\leq t} W_s$ by considering the stopping time $\tau_b=\inf\{t>0,W_t-t=b\}$ for $b>0$. But, here I can't say $\tau_b$ is a stopping time with finite expection, which is important in calculating the distribution of $\sup_{0\leq s\leq t} W_s$. So, I don't know how to do it now.

How many ways are there to arrange $3$ men, $3$ women, $3$ children in a circle such that they alternate clockwise in this order: man, woman, child? Posted: 16 May 2022 11:28 AM PDT Problem How many ways are there to arrange $3$ men, $3$ women, $3$ children in a circle such that they alternate clockwise in this order: man, woman, child? The seats are not numbered. My Opinion I wish to clarify this working Arrange $3$ men in a circle takes $(3-1)! = 2$ ways Arrange $3$ women next to men takes $3! = 6$ ways Arrange $3$ children next to women takes $3! = 6$ ways Total $= 2 \cdot 6 \cdot 6 = 72$ ways Please Comment if my approach is correct or not.

I am trying to generate random matrix with based on some condition, How many matrices the can be generated? Posted: 16 May 2022 11:16 AM PDT I am trying to generate random matrices based on the following conditions. There will be 3X3 matrices. The first column can have 1 to 10, the second column can have 11 to 20 and the third column can have 21 to 30. The numbers are not allowed to repeat in the matrix. Example : 2 | 11 | 21 8 | 13 | 28 9 | 14 | 23 I wrote the logic to generate the matrix. Multiple matrices can have the same number but the distribution will be different for each matrix. I want to find out the maximum number of matrices that can be generated based on the above condition? Can anybody help me to do the math?

How to prove $2n+1$ is odd for $n \in N$? Posted: 16 May 2022 11:08 AM PDT Usually, it seems that if $x\in N$, then "$x$ is odd" is translated by definition as $\exists y\in N: x=2 y+1$, but can we prove this? Given: $\forall a\in N:[Even(a)\iff \exists b\in N: a=2 b]$ $\forall a:[Odd(a)\iff \neg Even(a)]$ Can we actually prove: $\forall a\in N: Odd(2a+1)$ Any help or suggestions would be appreciated. Follow-up Thanks all for your suggestions. Using them, I was finally able to formally prove the above. See Corollary at https://dcproof.com/EvenNextOdd.htm (127 lines, not for the faint of heart!)

PHI function to list relative primes Posted: 16 May 2022 11:04 AM PDT I am using a website called dcode to input numbers into the PHI function, and then receive an output of numbers relatively prime with my input. The website, unfortunately, limits output to just 500 relatively prime numbers. For my testing this is too low, preferably I would like a complete list. Could anyone recommend a desktop software which does not have such limitations? It can be for either Windows or Linux. I know about Matlab, but was hoping there was already a program which does what I want, without me having to learn a new scripting language.

Intuition for fibers over a point in $\operatorname{Spec}(A)$ Posted: 16 May 2022 11:06 AM PDT This is a rather soft question, but I would like to see what I get. I'm currently reading/working through Atiyah Macdonald, and I just did exercise 21 of Chapter 3. I won't repeat the entire exercise here as it is rather long, but the point is to get to the identification of $(f^*)^{-1}(\mathfrak{p})$ with $\operatorname{Spec}(k(\mathfrak{p}) \otimes_A B)$, where $f : A \to B$ is a homomorphism of commutative rings, $\mathfrak{p}$ is a prime ideal of $A$, and $k(\mathfrak{p})$ is the residue fireld of $A_\mathfrak{p}$. My question is: Can someone give geometric intuition about what the fiber over $\mathfrak{p}$ "is", or why we would expect it to be homeomorphic to $\operatorname{Spec}(k(\mathfrak{p}) \otimes_A B)$? I don't know much about schemes, but I have a solid understanding of the basics in classical algebraic geometry (projective varieties, nullstellensatz, etc.), so I would love to get some intuition that makes sense from that perspective. Thanks!

How to prove: If $A$ and $B$ are normal matrices and AB = BA, $A+B$ and $AB$ are also normal, and $A$ and $B$ are simultaneously diagonalizable. Posted: 16 May 2022 11:02 AM PDT If $A$ and $B$ are normal matrices and they commute $(AB=BA)$, then: $A+B $ is normal $AB $ is normal $A$ and $B$ are simultaneously diagonalizable: there is a unitary matrix $U$ such that both $U^*AU$ and $U^*BU$ are diagonal. How can I prove the above statements? I found the proposition above on Wikipedia, https://en.wikipedia.org/wiki/Normal_matrix#Consequences. I could have used it to prove another theorem, but I could not prove the proposition itself.

Can the geodesic equation be used to solve the Brachistochrone Problem? Posted: 16 May 2022 11:17 AM PDT Assume the initial condition is that a point mass starts at height $y_0$. After descending to height $y < y_0$, we know that its speed will be $v = \sqrt{2mg(y_0 - y)}$. Thus, the displacement element can be written as $ds^2 = dx^2 + dy^2 = v^2 dt^2$, so that we have $dt^2 = \frac{1}{2mg(y_0 - y)} (dx^2 + dy^2)$. We are trying to minimize the functional $\int \sqrt{dt^2}$. This looks just like the geodesic problem (from general relativity) where the metric is $g_{\mu \nu} = \frac{1}{2mg}\begin{pmatrix} \frac{1}{y_0 - y} & 0 \\ 0 & \frac{1}{y_0 - y}\end{pmatrix}$. The geodesic equation reads, $$\frac{d^2 x^\mu}{dt^2} + \frac{1}{2}\Gamma^{\mu}_{\alpha\beta} \frac{dx^\alpha}{dt} \frac{dx^\beta}{dt} = 0.$$ I believe that with this metric, we have the Christoffel symbols $$\Gamma^x_{xy} = \Gamma^x_{yx} = \frac{1}{2(y_0-y)}$$ $$\Gamma^y_{yy} = \frac{1}{2(y_0-y)}$$ $$\Gamma^y_{xx} = \frac{-1}{2(y_0-y)}$$ (note the negative sign) and all else are zero. This would give the equations $$\frac{d^2 x}{dt^2} + \frac{1}{2(y_0-y)}\frac{dx}{dt}\frac{dy}{dt} = 0$$ $$\frac{d^2 y}{dt^2} + \frac{1}{2(y_0-y)}\left(\frac{dy}{dt}\right)^2 - \frac{1}{2(y_0-y)}\left(\frac{dx}{dt}\right)^2= 0$$ Is this all correct so far, and can this be continued into a useful solution to the problem? I am also concerned that I might not have the right derivative in the geodesic equation - I think it should be derivatives with respect to $t$ but I could imagine being wrong and it should be with $s$ (related by $ds = vdt \implies \frac{d}{dt} = v \frac{d}{ds}$.

Unital commutative Banach algebra $A$, $A/\operatorname{radical}(A)$ has no quasinilpotent elements Posted: 16 May 2022 11:23 AM PDT I am trying to show that for a unital commutative Banach algebra $A$, $A/\operatorname{radical}(A)$ has no quasinilpotent elements (where $\operatorname{rad}(A)=\{x\in A: x \,\text{quasinilpotent}\}$). I know that $\operatorname{rad}(A)$ is a closed ideal, and that the quotient map is continuous, so I'd want to do something like this: Let x not be quasinilpotent, so $\lim_{n\rightarrow \infty} \|x^n\|^{1/n} =\lambda \neq 0$. Let $\pi(x)=\overline{x}$, where $\pi(x):A\rightarrow A/\operatorname{rad}(A)$ is the canonical quotient map. Suppose that $\|\overline{x}^n\|^{1/n}=0$. Then it's spectrum $\sigma(\overline{x})=0$, so $\overline{x}$ is not invertible in $A/\operatorname{rad}(A)$, and thus generates a proper ideal in $A/\operatorname{rad}(A)$. So then $\pi^{-1}(\overline{x})$ generates a proper ideal in A containing $\operatorname{rad}(A)$. From here, if $\operatorname{rad}(A)$ is maximal I think I'd have a contradiction, but I don't know if that's true. If not, does anyone have another strategy I could try?

How to prove that $ \bar e_2 = e_1 \sin (w) + e_2 \cos(w)$ where $w$ is the angle between $e_1$ and $e_2$ Posted: 16 May 2022 11:19 AM PDT In the book of Linear Algebra by Werner Greub, at page 201, it is asked that Consider an oriented inner product space of dimension 2. Given two positive orthonormal bases $e_1, e_2$ and $\bar e_1, \bar e_2$, prove that $$\begin{align*}\bar e_1 &= e_1 \cos(w) - e_2 \sin(w) \\ \bar e_2 &= e_1 \sin (w) + e_2 \cos(w)\end{align*}$$ where $w$ is the oriented angle between $e_1$ and $\bar e_1.$ Note: $$\cos(w) = \frac{(e_1, \bar e_1)}{|e_1|\cdot|\bar e_1|}, \quad \sin(w) = \frac{\Delta(e_1, \bar e_1)}{|e_1| \cdot |\bar e_1|}$$ where $\Delta$ is the oriented determinant function. I have proved the first statement easily, but to prove the second, I have argued that the coefficient matrix of this system has to be a orthogonal matrix and proved the second part in that way. So my question is that how can we prove the second part differently ?

Poisson Distribution Problem relating to birthdays Posted: 16 May 2022 11:02 AM PDT Consider a random collection of n individuals. When approximating the probability that no 3 of these individuals share the same birthday, a better Poisson approximation than that obtained in the text (at least for values of n between 80 and 90) is obtained by letting $E_i$ be the event that there are at least 3 birthdays on day i, i=1,......,365. (a) Find P($E_i$). (b) Give an approximation for the probability that no three individuals share the same birthday. (c) Evaluate the preceding when n=88 (which can beshown to be the smallest value of n for which theprobability exceeds 0.5). Any help is greatly appreciated!

Proof that every repeating decimal is rational Posted: 16 May 2022 11:18 AM PDT Wikipedia claims that every repeating decimal represents a rational number. According to the following definition, how can we prove that fact? Definition: A number is rational if it can be written as $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$.

Why is negative times negative = positive? Posted: 16 May 2022 11:06 AM PDT Someone recently asked me why a negative $\times$ a negative is positive, and why a negative $\times$ a positive is negative, etc. I went ahead and gave them a proof by contradiction like so: Assume $(-x) \cdot (-y) = -xy$ Then divide both sides by $(-x)$ and you get $(-y) = y$ Since we have a contradiction, then our first assumption must be incorrect. I'm guessing I did something wrong here. Since the conclusion of $(-x) \cdot (-y) = (xy)$ is hard to derive from what I wrote. Is there a better way to explain this? Is my proof incorrect? Also, what would be an intuitive way to explain the negation concept, if there is one?

You are subscribed to email updates from Recent Questions - Mathematics Stack Exchange. To stop receiving these emails, you may unsubscribe now.	Email delivery powered by Google
Google, 1600 Amphitheatre Parkway, Mountain View, CA 94043, United States