Recent Questions - Mathematics Stack Exchange |
- Weak form of cahn hilliard equation
- Help with proof: if G is a group with neutral e, $g \in G$, g^n = e$ and $g^{n/p} \neq e \forall$ prime $p$ such that $p|n$, then order of g is n
- What is $\operatorname{Cov}[X+Y]$ when $X$ and $Y$ are NOT independent?
- Does the expression $\frac{1}{(1-hL)}$ have an upper bound?
- The question about finding a basis for a linear subspace of a vector space
- The zeroth moment of a random variable X
- 1. Prove or disprove that $\log_{10}{\sqrt{n}} = Θ(\lg{n^{5}})$ 2. Prove or disprove that $4\log_{10}{n}=O(\sqrt{n}\lg{n^{3}})$.
- Given that m^2/3 * (3/y^3)^m * (y/2)^m+2 = (n/y^2), find m and n
- What is the probability of selecting some item i, x times before some item j is ever selected, when there are n items?
- Set of discontinuities is given by the set and closure problem
- How do I find the derivative of a position vector and displacement, $d\textbf{r}(\theta)$ and $dr(\theta)$, when finding the unit tangent?
- Dedekind's Theorem
- Getting an explicit formula for a recurrence relation
- The convergence of a bounded sequence with a specific property
- Solutions to Cours d'Algèbra by Godement
- Prove that $f(x)=a_nx^n+a_{n−1}x^{n−1}+⋅⋅⋅+a_1x+a_0, x^n=O(f(x))$
- If limit exists, is that function continuous?
- Exponents with power 2019 mod 144
- Bound on the rank of the composition map
- Triangle inequality in an ellipse
- Compute volume over a solid cone $\int_K (x^2+y^2)dV$
- how vector space look like after applying weyl unitarian trick on finite group representation
- Chinese Remainder Theorem without coprime moduli [duplicate]
- A converging theorem about kernel function
- Parameterization of an ellipse by arc length
- Probability of hitting time conditional on the last step of Simple random walk.
- A family $\mathscr{F}\subset \operatorname{Aut}(G)$ is compact
- A Golden Angle Conjecture
- Finding functions $g$ such that the Fourier transform of $g(x+c)g(x)$ does not vanish
- How do I find the minimum force done by a person pushing a box when two coefficients of frictions are given?
| Weak form of cahn hilliard equation Posted: 23 Oct 2021 08:53 PM PDT I am trying to write the weak formulation of Cahn-hilliard equation as under. $\frac{\partial \phi}{\partial t} = \nabla .\{\phi(1-\phi) [\phi(1-\phi)\nabla \mu - \nabla (A(\phi)q)]\}$ $\frac{\partial q}{\partial t} = - \frac{q}{T(\phi)} -A(\phi)\nabla . [[\phi(1-\phi)\nabla \mu - \nabla (A(\phi)q)]]$ For the first equation, if w is the test function then, $(\phi_t,w)+(\phi^2(1-\phi)^2\nabla \mu, \nabla w)-(\phi(1-\phi)\nabla A(\phi)q,\nabla w)=([\phi(1-\phi)\nabla \mu - \nabla (A(\phi)q)],w)_{boundary}$ Kindly guide me, where I am committing mistake  |
| Posted: 23 Oct 2021 08:51 PM PDT If $g^n = e$, I know that the order of g divides $n$. I suppose there exists $q \in \mathbb{Z}, q|n, q \neq n$, such that order of g is q. Then $g^q = e$. Let $n = p_1p_2...p_k$. Then, for some $p_i$, $n/p_i = kq, k \in \mathbb{Z}$. For that $p_i$, $g^{n/p_i}=g^{kq}$, and as $g^q = e$, then $g^{kq}=g^{n/p_i} = e$. And then by counter-reciprocal, if $g^{n/p} \neq e \forall$ prime $p$ such that $p|n$, then order of g is n. The part of the proof I'm not sure about, is when I state that if $q|n$ and $n = p_1p_2...p_k$, then for some $p_i$, $n/p_i = kq, k \in \mathbb{Z}$. Is that statement true? If it is, how can I prove it? Also, what's the notation for order of g? Thanks!  |
| What is $\operatorname{Cov}[X+Y]$ when $X$ and $Y$ are NOT independent? Posted: 23 Oct 2021 08:49 PM PDT I have spent 2 days trying to search a formula that can help me obtain the covariance of (X+Y) especially when X and Y are independent with no luck. I could find for Variance(X+Y) however nothing for covariance. Nor could I find a way to obtain Covariance from Variance. To the questions, say we have a model $Z = aX + Y$ Where $X$ has a multivariate dist. with mean $\mu_X$ and $\sigma_Y$ And we were tasked with finding the mean and covariance for $Z$ For the mean, I just added the individual means $\mu_Z = a*\mu_X + \mu_Y$ However, I am very much stuck for $\operatorname{Cov}[aX + Y]$ . Is there a specific formula that I am missing? Or a property to calculate covariance of addition of 2 dependant variables?  |
| Does the expression $\frac{1}{(1-hL)}$ have an upper bound? Posted: 23 Oct 2021 08:46 PM PDT Considering that $1+hL\leq e^{Lh}$, is there an expression that bounds $\frac{1}{(1-hL)}$ from above in a similar fassion? If not, what are some other identities I can try?  |
| The question about finding a basis for a linear subspace of a vector space Posted: 23 Oct 2021 08:33 PM PDT Let's say there's a subspace U={(x1,x2,x3)|x1 = 2*(x2)} It is a linear subspace over R^3. Then how can I find the basis for this subspace? My idea is to firstly find a list of vectors in U such as (v1,v2...) and make sure that they can span U, then I start to check the redundancy of this list(linear independence). If the list is not redundant anymore(this list is linearly independent), then it is the basis for U. My question is, how can I find a list of vectors in U that can span U (find a spanning list for U), or my idea might be wrong, could somebody help me and give some ideas?  |
| The zeroth moment of a random variable X Posted: 23 Oct 2021 08:27 PM PDT In the book it says: When the mean of E(X) is finite, it is convenient to consider X^(0)=X-E(X) instead of X because its first moment is equal to zero. We shall say X^(0) is obtained from X by centering. Perhaps I am overthinking it, but I don't understand what the zeroth moment would even mean? To my understanding, the first moment (X^(1)) is equal to E(X). But I'm confused by the part that says "because its first moment is equal to zero." Is it saying the first moment if the zeroth moment?  |
| Posted: 23 Oct 2021 08:28 PM PDT
For the first question, I divide $\lg{n^{5}}$ on both side and find that $\frac{\log_{10}{\sqrt{n}}}{\lg{n^{5}}} = \frac{\frac{1}{2}\log_{10}{n}}{5\log_{2}{n}} = \frac{1}{10}\frac{\log_{n}{2}}{\log_{n}{10}} = \frac{1}{10}\log_{10}{2}$, which is approximately equal to 0.03. Hence I can find $c_{1} = 0.02,c_{2} = 0.1$ such that $c_{1}\lg{n^{5}} \leq \log_{10}{\sqrt{n}} \leq c_{2}\lg{n^{5}}, \forall n \geq 1$. And I don't know whether this proof if right? For the second question, I tried to follow the similar logic as the first question, but after dividing $\sqrt{n}\lg{n^{3}}$ on both side, I got stuck. Trying to compute the derivative of $\frac{4^{\log_{10}{n}}}{\sqrt{n}\lg{n^{3}}}$ and find the asymptotic properties is just too time consuming and doesn't necessarily lead to the proof or disproof, so can anyone help me on this?  |
| Given that m^2/3 * (3/y^3)^m * (y/2)^m+2 = (n/y^2), find m and n Posted: 23 Oct 2021 08:25 PM PDT I don't exactly know how to do this, but I suppose you need to delete y somehow  |
| Posted: 23 Oct 2021 08:19 PM PDT Story is I am making races from a set of 220 cars. I "randomly" select a subset of cars to race and tabulate. I notice that some cars have appeared 6-7 times while others have never appeared. How likely is that? Is it truly random? I understand the base of this. In a coin you can get 7 in a row with a probability of [2 / (2^7)]. But once I go to 3 items, I get lost. Thanks  |
| Set of discontinuities is given by the set and closure problem Posted: 23 Oct 2021 08:37 PM PDT let $f:[a,b] \rightarrow \mathbb{C}$ be any function. Let $D$ be the set of discontinuity point of $f$. Let $D(\epsilon)=\{x \mid \exists y_n \rightarrow x; \limsup|f(y_n)-f(x)|\geq \epsilon\}$. Show $D=\bigcup\limits_{n=1}^{\infty}D(\frac{1}{n})$ and $\overline{D(\epsilon)} \subset D(\frac{\epsilon}{2})$. Since $f$ is continuous at $x$ if and only if $y_n \rightarrow x \implies f(y_n) \rightarrow y$, $f$ is discontinuous at $x$ if $y_n \rightarrow x$ and $\limsup|f(y_n)-f(x)|\geq \epsilon$ for some $\epsilon$. So for each point at which $f$ is discontinuous, choose an $\epsilon>0$ and $y_n$ satisfying the inequality. Let $C$ be the set of all such $\epsilon$. Since $\inf C \geq 0$ we see that $D=\bigcup\limits_{n=1}^{\infty}D(\frac{1}{n})$ since $\bigcup\limits_{n=1}^{\infty}D(\frac{1}{n})=\{x \mid \exists y_n \rightarrow x; \limsup|f(y_n)-f(x)|\geq 0\}$. Let $x \in \overline{D(\epsilon)}$. Choose a sequence $\{y_n\} \subset D(\epsilon)$ such that $y_n \rightarrow x$. Then for each $y_n$ there is a $z_{kn} \rightarrow y_n$ as $k \to \infty$ with $\limsup\limits_{k \to \infty}|f(z_{kn})-f(y_n)|\geq \epsilon$. Then $\limsup\limits_{n \to \infty}(\lim\sup\limits_{k \to \infty}z_{kn}) \rightarrow x$ as $n \to \infty$. Then for fixed $n$, $$\epsilon \leq \limsup\limits_{k \to \infty}|f(z_{kn})-f(x)+f(x)-f(y_n)|\leq \\ \limsup\limits_{k \to \infty} |f(z_{kn})-f(x)|+\limsup\limits_{k \to \infty}|f(y_n)-f(x)|$$ Now if I can somehow show that $\limsup|f(y_n)-f(x)|<\frac{\epsilon}{2}$, the result will follow, however I think there might be some issues with the setup. Can I next take the $\limsup$ over $n$ and would this imply $\limsup|f(y_n)-f(x)|\geq \epsilon$. But what about the $\frac{\epsilon}{2}$? Actually I realize there is a major issue with this, since I need to address whether I am taking the $\limsup$ over $k$ or $n$ in the string of inequalities. I know it would be a major issue if $z_{kn}$ does not converge to $x$ as $k \to \infty$ for some $n$ large. If this weren't true, my method of proof wouldn't work since I think it is implied I am taking the $\limsup$ over $k$ by the first inequality in the string. So I realize this is definitely going along the wrong route. Any suggestions on how to start the closure portion correctly?Also, is there any technical detail I am missing for the first part?  |
| Posted: 23 Oct 2021 08:27 PM PDT I have a particle moving along an elliptical path, such that the position, $\textbf{r}(\theta(t))$, is a function of the angle, $\theta(t)$, that is defined to be the angle between the horizontal axis and the radial unit vector, $\hat{e}_r$. Thus, \begin{equation} \textbf{r}(\theta(t))=r(\theta(t))\,\hat{e}_r. \end{equation} The particle only rotates around the unit binormal direction, $\hat{e}_b$, (equivalently in this example, the z-direction, $\hat{k}$) and I believe it does not twist around the unit tangent direction, $\hat{e}_t$, because the ellipse is on the 2D $x_tx_n$-plane. Since it is on the 2D plane, $\hat{e}_b$ doesn't change directions, so I believe the angular velocity of the $\{\hat{e}_t,\hat{e}_n,\hat{e}_b\}$ basis is (with no $\hat{e}_t$ component for the angular velocity), \begin{equation} \omega_{tnb}=\dot{\theta}(t)\,\hat{e}_b. \end{equation} I say "I believe" since I am not 100% sure I am thinking of the angular velocity for the $\{\hat{e}_t,\hat{e}_n,\hat{e}_b\}$ basis in 2D correctly, and don't want to make a definitive statement on something I am not sure of, but these are the assumptions I am working off of. I need to find the expressions for the $\hat{e}_t$ and $\hat{e}_n$ unit vectors in terms of the polar unit vectors, $\hat{e}_r$ and $\hat{e}_\theta$. So far, I have the position vector as mentioned before of, $\textbf{r}(\theta(t))=r(\theta(t))\hat{e}_r$, and for the velocity vector, \begin{equation} \dot{\textbf{r}}(\theta(t))=\dot{r}(\theta(t))\,\hat{e}_r+r\dot{\theta}(t)\,\hat{e}_\theta. \end{equation} So to find $\hat{e}_t$, I have to do \begin{equation} \hat{e}_t=\frac{d\textbf{r}(\theta)}{ds(\theta)}, \end{equation} where \begin{equation} ds(\theta) = \lVert d\textbf{r}(\theta)\rVert. \end{equation} This leads to my question: how do I find $d\textbf{r}(\theta)$ and $dr(\theta)$? These are my two thoughts, and my question can then be more specifically, which thought process, if any, is correct for finding $d\textbf{r}(\theta)$?: 1. \begin{equation} \frac{d\textbf{r}(\theta)}{ds(\theta)}=\frac{d\textbf{r}(\theta)/dt}{ds(\theta)/dt}=\frac{\dot{\textbf{r}}(\theta)}{\lVert\dot{\textbf{r}}(\theta)\rVert}, \end{equation} where I use the full expression for the velocity of the mass. The reason I am unsure of this thought process is because $d\textbf{r}(\theta)$ did not include $dt$ in the original $\hat{e}_t$ equation, and the expession for $\hat{e}_t$ becomes rather long (which isn't unusual, so not too much of a problem). I am also unsure if the rightmost side of the above equation is written correctly, since I made the assumption that $ds(\theta)/dt=||\dot{\textbf{r}}||$. The reason I am thinking of this method is because I think multiplying by $1=\frac{d/dt}{d/dt}$ still results in a true statement, and I already know the velocity and speed of the particle. 2. \begin{equation} \frac{d\textbf{r}(\theta)}{ds(\theta)} = \frac{d\textbf{r}/d\theta}{dr/d\theta} = \frac{\hat{e}_r\,dr/d\theta+r\,d\hat{e}_r/d\theta}{dr/d\theta}, \end{equation} where I only take the derivative with respect to $\theta$. My concern with this method is that I do not know how to find $d\hat{e}_r/d\theta$, and I am not sure if the unit vectors are functions of $\theta$, or are only functions of time. The secondary problem, if the unit vectors are not a function of $\theta$, is that $\hat{e}_t$ would be in the $\hat{e}_r$ direction based on the way I wrote it, which is clearly not correct, in my opinion. The reason I am thinking of this method is because I am not sure if I am supposed to include the time derivative, and this method does not include that. Are either of these approaches correct, and if not, how should I be thinking about this problem?  |
| Posted: 23 Oct 2021 08:00 PM PDT Use Dedekind's Theorem to factorise the following principal ideals in the ring of integers of the following fields. a) $Q(√3): ⟨2⟩,⟨3⟩,⟨5⟩,⟨30⟩$ b) $Q( ^3√2): ⟨7⟩, ⟨29⟩, ⟨31⟩$ Here is what I understand about the theorem Let $[K:Q]=n$ with the ring integers $O_k=Z[ø]$ Given the minimal polynomial of $√3$, we have $x^2-3$. And for $^3√2$, we have $x^3-2$. a) $p=2$; $x^2-3=x^2mod2$. By Dedekind, $⟨2⟩ = ⟨2, √3⟩$ $p=3$; $x^2-3=x^2-1mod3$. Of course, we know $x^2-1=(x+1)(x-1)$. Hence $⟨3⟩ = ⟨3, 1+√3⟩•⟨3, 1-√3⟩$ $p=5$; $x^2-3=x^3-2+10mod5=(x+2)(x^2-2x+4)mod5$ $⟨5⟩ = ⟨5, 1+i⟩•⟨5, i^2-2i+4⟩$ The first three numbers generated were prime numbers. However, 30 is not. I'm thinking of factoring $30$ to $2, 3$, and $5$, I could use three of those prime numbers and it would give me a set of generated numbers by Dedekind. I am having trouble with applying Dedekind on part b). They're a little trickier than part a for I'm dealing with the cube root of 3.  |
| Getting an explicit formula for a recurrence relation Posted: 23 Oct 2021 08:40 PM PDT As you can maybe infer from the above pictures, I am trying to find the complexity of searching an element in a sorted matrix recursively (binary search). Suppose I search for the element $K$ in the matrix $A$.
Our barometer operation will be the number of times a comparison is made. Moreover, if we are working with a $ n \times m $ matrix, then the size of the instance will be $ N = \max (n, m) $. Assume $C(n)$ is the number of time the barometer operation is counted for I think the recurrence relation is $$C(N) = 1 + C(\text{size first sub matrix}) + C(\text{size second sub matrix}) + C(\text{size third sub matrix})$$ or $$C(N) = \begin{cases} 1 & \quad \text{si } A(\lceil \frac{n}{2} \rceil, \lceil \frac{m}{2} \rceil) = K\\ 1 + C(\max(\lceil \frac{n}{2} \rceil, m - \lceil \frac{m}{2} \rceil - 1)) + C(\max(n - \lceil \frac{n}{2} \rceil - 1, m - \lceil \frac{m}{2} \rceil - 1)) + C(\max(m - \lceil \frac{n}{2} \rceil - 1, \lceil \frac{m}{2} \rceil)) & \quad \text{si } A(\lceil \frac{n}{2} \rceil, \lceil \frac{m}{2} \rceil) < K\\ 1 + C(\max(\lceil \frac{n}{2} \rceil - 1, \lceil \frac{m}{2} \rceil - 1)) + C(\max( \lceil \frac{n}{2} \rceil - 1, m - \lceil \frac{m}{2} \rceil)) + C(\max(n - \lceil \frac{n}{2} \rceil, \lceil \frac{m}{2} \rceil-1)) & \quad \text{si } A(\lceil \frac{n}{2} \rceil, \lceil \frac{m}{2} \rceil) > K\\ \end{cases} $$ This is strangely complex. How can I get an explicit formula for this recurrence relation? It seems the complexity of this algorithm is in $\Theta(\log(n))$, but I have just no clue how to solve it.  |
| The convergence of a bounded sequence with a specific property Posted: 23 Oct 2021 08:19 PM PDT
I tried to show that the series associated with this sequence to converge and tried many other things like Ratio test but none of this worked... It would be a big help if anyone gives me a hint.  |
| Solutions to Cours d'Algèbra by Godement Posted: 23 Oct 2021 08:28 PM PDT I am a beginner at algebra. I came across the Cours d'Algèbra by Godement in the library, which is the most impressive book for me. However, there are a lot of exercises in the book, among which there are plenty of hard questions. How can I find solutions to it for self-study?Thank you!  |
| Prove that $f(x)=a_nx^n+a_{n−1}x^{n−1}+⋅⋅⋅+a_1x+a_0, x^n=O(f(x))$ Posted: 23 Oct 2021 08:29 PM PDT I want to prove that if $f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_1x+a_0$ and $a_n\neq0,$ $f(x)=\Theta(x^{n})$. I have known that $f(x)=O(x^{n})$, but I don't know how to prove $x^{n}=O(f(x))$. Please offer me some hints. Thanks.  |
| If limit exists, is that function continuous? Posted: 23 Oct 2021 08:00 PM PDT If I have a function $f$ and the limit exists at $a$ and $\lim_{x \to a} f(x) = \infty $ or $\lim_{x \to a} f(x) = k $ where $k$ is a constant, does that mean that the function is continuous on some interval in the domain of $f$? I think this is true, but I am not sure. Because for example, if $\lim_{x \to a} f(x) = \infty $, if I graph function that fulfill this, there is at least some interval in its domain where it is continuous.  |
| Exponents with power 2019 mod 144 Posted: 23 Oct 2021 08:44 PM PDT Problem: What is the remainder when $11^{2019}+13^{2019}$ is divided by $144$? I saw that $$11^{2019}+13^{2019} \equiv (-1)^{2019}+(1)^{2019} \equiv 0 \pmod{12},$$ but I don't now how to proceed from here. Can you help me?  |
| Bound on the rank of the composition map Posted: 23 Oct 2021 08:23 PM PDT I am trying to solve the below problem.
The only thing I know for sure I can do is use the rank-nullity theorem. I'm interested in the rank of $\phi \circ \psi$, so I can say: \begin{align*} n = \dim V = \mathrm{rank}(\phi \circ \psi) + \dim \mathrm{ker}(\phi \circ \psi). \end{align*} So $\mathrm{rank}(\phi \circ \psi) = n - \dim\mathrm{ker}(\phi \circ \psi)$, and the problem is reduced to showing that \begin{align*} n - \dim\mathrm{ker}(\phi \circ \psi) \geq \mathrm{rank}(\phi) + \mathrm{rank}(\psi) - n. \end{align*} My next instinct is to apply rank-nullity to $\phi$ and $\psi$ individually, but there's no way around having to compare the dimension of the kernel of the composition map to the rank of the individual maps. It seems to me that, because the kernel of the composition map is a subspace of the domain $\psi$, it must have dimension $\leq n$. Am I on the right track, or is some better way to compare these?  |
| Triangle inequality in an ellipse Posted: 23 Oct 2021 08:42 PM PDT In the standard ellipse of variable axes dimensions $( a, b, c= \sqrt{a^2-b^2}), $
what is the geometrical significance of triangle inequality difference for variable $(2a-2c)$ in the first and third cases? The second case result is known.. in the ongoing question in-circle diameter. It was ensured to be geometrically constructible and answer interesting... hope forms my part of problem solving attempt.  |
| Compute volume over a solid cone $\int_K (x^2+y^2)dV$ Posted: 23 Oct 2021 07:54 PM PDT I have a few exercises on computing the volume of a solid cone, however the inequalities introduced in the exercises aren't something I have faced with before. So I would really just appreciate some hints on how to tackle exercises where inequalities for $z$ are given as so $\sqrt{x^2+y^2}\le z\le1$. For example, compute the integral $\int_K (x^2+y^2)dV$ where $K$ is the solid cone. I can see the integral bounds for $z$ as $\int_{\sqrt{x^2+y^2}}^1dz$ Althlugh for $y$ I have assumed this $\sqrt{x^2+y^2} \le 1 \implies y \le \sqrt{1-x^2}$ Such that $\int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}dy$ as for the integral bounds of $x$ $\int_{-1}^1$ Which gives me $$\int_{-1}^1\int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}\int_{\sqrt{x^2+y^2}}^1(x^2+y^2)dzdydx$$  |
| how vector space look like after applying weyl unitarian trick on finite group representation Posted: 23 Oct 2021 08:45 PM PDT I have a question about following theorem.
To prove this, we make a new inner product from original one by $$ \langle v,w \rangle := \sum_{g\in G} \bigl(\rho(g)v,\rho(g)w\bigr), $$ where $(v,w)$ is original inner product, and then check that this is invariant under linear transformation by representation of group. because this new inner product is invariant, our representation is considered to be unitary in this new vector space. on the other hand dimension of this new vector space is same as original one, these two vector space are isomorphic and hence our representation is equivalent to some unitary representation. I was curious how this new vector space looks like. On original space these equidistant points forms circle and any reversible linear transformations should transform this to some rotated ellipse. but what I actually observed was ellipse with some dimple.
Am I missing some thing? Also, how can I obtain linear transformation that leads original vector space to new vector space with this new inner product. Thanks. Code I used to make a plot is following.  |
| Chinese Remainder Theorem without coprime moduli [duplicate] Posted: 23 Oct 2021 08:03 PM PDT I have two equations $$ x = 2 \mod 10$$ $$ x = 7 \mod 15$$ I know I need to break this down as $10$ and $15$ are not coprime Could I have $$ x = 2\mod 10$$ $$ x = 7\mod 3$$ and $$x = 7\mod 5$$ Also is it then accurate to say $x=7\mod3=1\mod3$?  |
| A converging theorem about kernel function Posted: 23 Oct 2021 08:45 PM PDT Let $X$ be a smooth paracompact manifold and $E,F$ be two complex vector bundle on $X$ with hermitian inner product. Let $D:C^\infty(X,E)\to C^\infty(X,F)$ be an elliptic differential operator of order $m$, $\bar{D}=hDh^{-1}$ and $h:C^\infty(X,E)\to C^\infty(X,F)$ is the antilinear isomorphism defined by the metric on $E$. I'm confused to prove the following theorem: (i)The kernel $p(x,x)$ of the projection $P$ onto the space $\mathcal{H}(D)$ of $L^2$ soultion of the elliptic equation $Du=0$ is $C^\infty$. (ii) Moreover, if $\phi_n(x)$ is an orthonormal base of $\mathcal{H}(D)$ and $\bar{\phi}_n=h\phi_n$ the corresponding base of $\mathcal{H}(D)$, we have $$ p(x,y)=\sum_{n=1}^\infty \phi_n(x)\bar{\phi}_n(y) $$ the sequence converging uniformly on compact sets of $X\times X$, together with all its derivatives. (iii)Additionally, we have $$ \dim \mathcal{H}(D)=\int_X \sum_n \| \phi_n(x) \|^2 $$ Could you please give me some help with the details? Thank you very much!  |
| Parameterization of an ellipse by arc length Posted: 23 Oct 2021 08:31 PM PDT I'm trying to parameterize the ellipse $x= a\cos(\vartheta)$ , $y= b\sin(\vartheta)$ in terms of the arc length $S$ but I don't know how to do it. Supposing that $\gamma:[a,b]\to \mathbb{R}$ is a smooth curve with $\gamma'(t)\neq 0$ for $t\in [a,b]$ , I know that $s(t)$= $\int_{a}^{t}\left\| \gamma'(\psi)\right\|d\psi$ for $t\in [a,b]$ then I find the inverse funtion of $s$. How can I find a closed formula for this specific parameterization? Can anybody give a hint, please?  |
| Probability of hitting time conditional on the last step of Simple random walk. Posted: 23 Oct 2021 08:14 PM PDT I have a trouble proving the below result: Let $\left(S_{n}\right)_{n \geq 0}$ be a simple random walk defined by $p(1)=p$ and $p(-1)=q$ where $p+q=1$ Let $k$ be a positive integer. Prove: $$ P_{0}\left(T_{0}^{\prime}=2 k \mid S_{2 k}=0\right)=\frac{1}{2 k-1} $$ where $T_{0}^{\prime}=\inf \left\{n \geq 1: S_{n}=0\right\}$. This is what I have done so far: Using the definition of conditional probability, I found that: $$P_{0}\left(T_{0}^{\prime}=2 k \mid S_{2 k}=0\right)= \frac{P_{0}(T_{0}^{\prime}=2 k, S_{2 k}=0)}{P_{0}(S_{2 k}=0)}$$ $P_{0}(T_{0}^{\prime}=2 k, S_{2 k}=0)$ means that $S_{2k}$ is the first time it hits zero. Therefore using Catalan numbers, this is the same as $$ P_{0}\left(S_{1}>0, \ldots, S_{2 k-1}>0, S_{2 k}=0\right)=\frac{1}{2 k-1}\left(\begin{array}{c} 2 k-1 \\ k-1 \end{array}\right) p^{k} q^{k} $$ And $P_{0}(S_{2 k})=0$ is $(\begin{array}{c} 2 k \\ k \end{array}) p^{k} q^{k}$. Thus: $$\frac{P_{0}(T_{0}^{\prime}=2 k, S_{2 k}=0)}{P_{0}(S_{2 k}=0)} = \frac{ \frac{1}{2 k-1}\left(\begin{array}{c} 2 k-1 \\ k-1 \end{array}\right) p^{k} q^{k} }{ \left(\begin{array}{c} 2 k \\ k \end{array}\right) p^{k} q^{k}}= \frac{1}{2 k-1} * \frac{1}{2} $$ Which is wrong as I am suppose to get $$ \frac{1}{2 k-1} $$ I will be so grateful for any help!  |
| A family $\mathscr{F}\subset \operatorname{Aut}(G)$ is compact Posted: 23 Oct 2021 08:18 PM PDT Let $G\subset\mathbb{C}$ be any bounded region, and let $w\in G$. Let $\mathscr{F}$ be the set $\{f\in \operatorname{Aut}(G):f(w)=w\}$ ; where $\operatorname{Aut}(G)$ is the group of all biholomorphic maps on G. Then we have to show that $\mathscr{F}$ is compact. My idea : Let $\{f_n\}_{n=1}^{\infty}$ be a sequence in $\mathscr{F}$. Since G is bounded region, $\overline{G}$ is compact in $\mathbb{C}$, and the diameter of $G$ is finite.. Thus there exists $M>0$ such that $\lvert f(z) \rvert \leq M$ for every $z\in G$ and for every $f\in \mathscr{F}$. Hence $\mathscr{F}$ is a locally uniformly bounded family. So, by Montel's theorem, there exists a subsequence $\{f_{n_k}\}_{k=1}^{\infty}$ of $\{f_n\}_{n=1}^{\infty}$ which converges uniformly to $f$ on every compact subset of $G$. Since $\mathscr{H}(G)$, the family of all holomorphic functions on $G$, is a complete metric space, we say that $f$ is also holomorphic on $G$. It is also clear that $f(w)=w$. Now I am stuck. How do I show that $f$ is infact a member of $\mathscr{F}$ ? If this thing can be shown then we can say, by sequential compactness, that the family $\mathscr{F}$ is indeed compact.  |
| Posted: 23 Oct 2021 08:41 PM PDT Update: this conjecture is broken, for n=7. See final note. This conjecture is related to the process of phyllotaxis in plants, and understanding why nature would choose to iterate the Golden Angle for this process, after a sufficiently long period of evolution. Essentially, I want to know if the Golden Angle is ideal for distributing plant structures around a 2-d radial point of origin at low iterations. For instance, if you want to distribute three seeds around a circle, iterating the Golden Angle three times will place a seed in each of the thirds of a circle starting from the angle origin. You want to add another seed, so you simply iterate another golden angle, placing the four seeds in each quarter of a circle, and so on. My conjecture is That for every integer $n > 0$, and for each integer $i$ in $[1...n]$ then $i$ maps to one unique integer $j$ from $[1...n]$ where $\frac {i-1} n<(j\theta)\mod 1<\frac {i}n$ is true, when $\theta = \frac{3 - \sqrt 5}2$ (acute golden angle) or $\theta = \frac{\sqrt 5 - 1}2$ (obtuse golden angle) In other words, if every value $j$ from $[1...n]$ is iterated, $j\theta $ will determine each of the $n$ regular sectors of the circle only once. note: The angle being discussed here is in units of turns of a circle, and has nothing to do with $\pi$, other than considering this angle to be the distance traveled on the circumference a circle with $r=\frac{1}{2\pi}$. note: Equality in some of the ordered comparisons could be used, but the assumption is that $\theta$ is some irrational number that can never be equal to a rational number, because this would obviously fail when the denominator of rational $\theta$ is a multiple of $n$. Q1: Is there an easy proof of this conjecture, or derivation from another proof? Q2: Is there another value of $\theta$ that this conjecture is also true? Q3: Making a model where some specific number of seeds/structures is geometrically more likely than other adjacent numbers, could evolution chose a slightly different iterative angle that does a better job? Notes: One simple verification of this conjecture for n=10 is to observe the mod 1 residues of 10 iterations of the obtuse Golden Angle on a calculator will cycle through all 10 digits: (and for the most arithmetically nimble of mind, you can also use this list to verify for n=[1..9] as well. ) The Equi-distribution theorem is important to this conjecture, which states: that the sequence a, 2a, 3a, ... mod 1 is uniformly distributed on the circle R/Z, when a is an irrational number The Three-Gap theorem, formerly the Steinhaus conjecture, proved in 1950, is possibly important to this, which states: if one places n points on a circle, at angles of θ, 2θ, 3θ ... from the starting point, then there will be at most three distinct distances between pairs of points in adjacent positions around the circle. When there are three distances, the largest of the three always equals the sum of the other two. Intriguing is Hurwitz' Irrational Number theorem, which considers the golden ratio 'bad' in terms of tightening up Lagrange's Irrational number theorem for rational approximations of irrational numbers. I have not yet read the proof. Final Note: This conjecture is quite broken, as I have discovered. I have since learned to use python code to test conjectures, so hopefully not embarrass myself in the future. I ran used some python code to check how a distribution converges from the 1/n distribution around the circle for increasing values of n. It seems that an iteratively added golden angle will diverge from a sector distribution of $1/n$, and apparently will never converge. My conjecture had assumed that this was $x/n$ where $x<1$. It truth, it turns out that it will diverge, the maximum divergence at up to $n=1000$ is $\frac {2.6894...}n$ at $n=10000$, the maximum divergence is $\frac{3.894...}n$. I think this can be shown by applying the 3-gap theorem, and I think that this maximum divergence has something to do with an insufficient rate of reduction of the maximum gap angle, as n increases. I made another test, calculating the sum $\sum _{k=1}^{n}e^{k\theta i}$ where $\theta$ is the acute golden angle in radians, that is, $\theta = \frac {2\pi (3-\sqrt 5)}2$. This would represent the center of gravity for $n$ iteratively added golden angle points around the unit circle in the complex plane. At $n=1$, the absolute value of this sum would obviously be $1$, but I thought it should converge to zero as $n$ increases, and it does, as the sum gets very close to (0,0) for many values of n, as the equ-distribution theorem would imply this. However, I was surprised to discover that the absolute value of the sum can exceed $1$ for some values of $n$, and for the values of $n$ up to 10000, I occasionally get a new maximums: In conclusion, I am a bit perplexed. Obviously, irrational angles are not easily tamed. The phyllotaxis process seems to be very tuned to using this angle, and I really want to know why it does so.  |
| Finding functions $g$ such that the Fourier transform of $g(x+c)g(x)$ does not vanish Posted: 23 Oct 2021 08:09 PM PDT Suppose that $f$ is a function of the form $$ f(x) = g(x+c)g(x), $$ i.e. $f$ is a product of $g$ with a shifted $g$. If $g$ is a Gaussian, so is $f$ for every $c \in \mathbb R$. In particular, the Fourier transform of $f$ does not vanish. Does there exist other functions with this property or even a class of functions $g$ such that the Fourier transform of $x \mapsto g(x+c)g(x)$ does not vanish for every $c$?  |
| Posted: 23 Oct 2021 08:07 PM PDT The problem is as follows:
The alternatives given in my book are as follows: $\begin{array}{ll} 1.&40\,N\\ 2.&200\,N\\ 3.&240\,N\\ 4.&300\,N\\ \end{array}$ In this problem I'm confused at why? they are giving two coefficients of friction?. I'm assuming one is static friction and the other is kinetic friction. But exactly which is which?. I think that the biggest one must be the static friction. I attempted my solution as included in the diagram from above. If I were to use this I attempted to use the equation: $F\cos 37^{\circ}- \mu \left(W+F \sin 37^{\circ}\right)=0$ $F\left(\frac{4}{5}\right)-\frac{48}{100}\left(320+\frac{3}{5}F\right)=0$ $F\left(\frac{4}{5}\right)=\frac{48}{100}\left(320+\frac{3}{5}F\right)$ $\frac{4F}{5}=\frac{48}{100}\left(\frac{1600+3F}{5}\right)$ $F=300\,N$ Which would be the option four. But would this be the minimum force?. Or instead?. $F\left(\frac{4}{5}\right)=\frac{36}{100}\left(320+\frac{3}{5}F\right)$ Which by doing all the calculations would give: $F\approx 197.26\,N$ Can somebody help me to establish exactly what's the meaning of those two coefficients of friction and are the vectors okay?.  |
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