Recent Questions - Mathematics Stack Exchange |
- Is it possible to make a connection between the surface area of the tetrahedron's circumsphere and its faces' areas?
- I know the matter, but don't know how to solve it applied on this problem I found...
- Find the normalizing constant for $\exp\left(\theta\sum_{i=1}^{n-1}x_ix_{i+1}\right)$
- Constructing a confidence interval question
- Two correlated AR(1) series
- Sine cosine function approximation
- The expected log value of exponential distribution
- $\|\mathbf{r}'\|$ in terms of Frenet-Serret Frame variables
- finite presentation $\mathcal{O}_X$ module over an integral scheme
- nuances of optimization with constraints problem
- Extension of an Hilbertian basis
- Is there a collection of prime residue classes whose union covers all integers?
- Permutations/Combinations and Multiset
- Spectral radius of
- When a simple algebraic ring extension of a UFD is a UFD
- Could use some tips on a Functional Analysis Problem
- Reduction of t-s TSP to TSP
- can you solve the part a to g?
- Prove or disprove $SO(4n) \supseteq \frac{(Sp(1)\times Sp(n))}{\mathbb{Z}_2}$?
- Understanding minimum distance from a point to the curve ${x^3} - 8x - 2y = 0$ from the sum-of-square relaxation perspective?
- How come nobody noticed there was a hole big enough to drive a truck through in Prove SST=SSE+SSR (answer1) [closed]
- Combinatorics- The number of ways to arrange marbles
- Possible values for continuous function?
- Finding the number of automorphisms in $G_{\Bbb{Q}(\sqrt[4]{2},i),\Bbb{Q}}$?
- Piecewise function notation and applications
- Two absolute values satisfy $|x|_1=|x|_2^t$ iff they satisfy $c_1|x|_1\leq |x|_2\leq c_2|x|_1$.
- Properly drawing a Penrose tiling using the pentagrid method
- What is the last non-zero digit of $(\dots((2018\underset{! \text{ occurs }1009\text{ times}}{\underbrace{!)!)!\dots)!}}$?
- Integers $1,2,...,n$ are placed in a way that each value is either bigger or smaller than all preceding values. In how many ways this can be done?
- The matrix notation of signum?
| Posted: 01 May 2021 07:54 PM PDT Can you somehow conclude that the surface area of the tetrahedron's circumsphere is greater than or equal to the area of its faces? I tried to go from the 2D version (i.e. the diameter of the circumcircle of a triangle is always greater than or equal to the triangle's edges), but is kinda stuck of how to make the proper connection. In the 2D version, the proof is rather straightforward by drawing a diameter that intersects with one of the vertices and using the observation that the hypotenuses are always the largest edge of a square triangle. Can we make use of that argument in 3D?  |
| I know the matter, but don't know how to solve it applied on this problem I found... Posted: 01 May 2021 07:52 PM PDT That's the problem: Assume the size P of a population grows following the differential equation \frac{dP(t)}{dt} = bP(t) The rate of growth b is the difference between the birth rate and the mortality rate. The Malthusian model supposes that this rate is constant. Of course we know that does not always happen: factors such as famines, outbreaks of disease or advances in medicine do influence these rates. When modeling some process mathematically it is important to recognize what our assumptions are and when they no longer hold. In this problem and the next we will look at one particular event that would result in a violation of Malthus' assumption that the growth rate b is constant over time: the occurrence of famines. Experimental data suggests that food production F grows linearly over time: F(t) = \F_0 + at We will now make two assumptions:
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| Find the normalizing constant for $\exp\left(\theta\sum_{i=1}^{n-1}x_ix_{i+1}\right)$ Posted: 01 May 2021 07:56 PM PDT Let $X_i,i=1,...,n$ be random variables assuming the values $1$ and $-1$. Suppose that the joint distribution of $X_1,...,X_n$ is given by $$\Pr(X_i=x_i,i=1,...,n)\propto \exp\left(\theta\sum_{i=1}^{n-1}x_ix_{i+1}\right):=h$$ for $x_i\in\{1, -1\}$, where $\theta\in\mathbb R$. (a) Find the normalizing constant $Z_n(\theta)=\sum_{x_1,...,x_n}h$ Ans: If n=2, we have the normalizing constant to be $e^{\theta1(1)}+e^{\theta(-1)}+e^{\theta(-1)}+e^{\theta1(1)}$ If n=3 the normalizing constant is $e^{\theta(2)}+e^{\theta(-2)}+e^0+e^0+e^0+e^0+e^{\theta(-2)}+e^{\theta(2)}$ I couldn't list all the combinations for n=4. Is there a combinatorial trick? I don't think it's supposed to be particularly difficult, so maybe I'm calculating it wrong. Also, part (c) says, is $X_t$ a markov chain? If so, determine its transition matrix. I have not yet wrapped my head around what shape or form this markov chain is hidden in.  |
| Constructing a confidence interval question Posted: 01 May 2021 07:48 PM PDT I'm trying to solve this question but I'm having trouble figuring how to solve it or even start I know that the final answer is (739.8028 ml, 746.1972 ml) but I'd really appreciate any help on how to get there? Thank you!  |
| Posted: 01 May 2021 07:46 PM PDT I have two AR(1) series that are correlated. $$X_{t,1} = \rho_1X_{t-1,1} + e_{t,1}$$ $$X_{t,2} = \rho_2X_{t-1,2} + e_{t,2}$$ and $corr(X_{t,1}, X_{t,2}) = \rho.$ I want to generate at each time $t$ a certain number of simulated values of each series. So far I did this: However, my $X_{t,1}$ and $X_{t,2}$ series are decreasing and I have a feeling that I am doing something wrong. I wanted to use arima.sim function, however, could not figure out how to generate at each given time 1000 values. Any helps is appreciated. Thanks.  |
| Sine cosine function approximation Posted: 01 May 2021 07:46 PM PDT I have originally posted this question in electrical stackexchange. I am just interested to know the derivation of the sin phi approximation  |
| The expected log value of exponential distribution Posted: 01 May 2021 07:43 PM PDT Suppose $X$ is exponentially distributed with the rate parameter $\lambda$. What is the expected value of $\log X$ and $\log\Gamma(X)$?  |
| $\|\mathbf{r}'\|$ in terms of Frenet-Serret Frame variables Posted: 01 May 2021 07:36 PM PDT On Wikipedia I find the equation: $$\frac{d}{dt} \begin{bmatrix} \mathbf{T}\\ \mathbf{N}\\ \mathbf{B} \end{bmatrix} = \|\mathbf{r}'(t)\| \begin{bmatrix} 0&\kappa&0\\ -\kappa&0&\tau\\ 0&-\tau&0 \end{bmatrix} \begin{bmatrix} \mathbf{T}\\ \mathbf{N}\\ \mathbf{B} \end{bmatrix}. $$ I was wandering if it was possible to replace $\|\mathbf{r}'(t)\|$ in terms of $\mathbf{T},\mathbf{N},\mathbf{B},\kappa,\tau$. My attempt was to to expand the equation for $\frac{d}{dt}\mathbf{T}$ and take the absolute value on both sides. However, then we get $\|\mathbf{r}'(t)\|$ in terms of the absolute value of the derivative of the tangent which is not ideal in my case.  |
| finite presentation $\mathcal{O}_X$ module over an integral scheme Posted: 01 May 2021 07:31 PM PDT Let $X$ be an integral scheme and let $\mathcal F$ be an $\mathcal O_X$-module of finite presentation. Show that there exists an open dense subset $U$ of $X$ and an integer $n ≥ 0$ such that $\mathcal F|_U\cong \mathcal O^n_X|_U .$ Since $\mathcal F$ is finite presentation, there exists an open set $Y=\{x\in X|\mathcal F_x\cong \mathcal O_{X,x}^n\}$ for $n\ge 0$. And $X$ is irreducible, the $n$ is unique. Consider the kernel $K$ of the map $\mathcal O^n_X|_U\rightarrow\mathcal F|_U\rightarrow0.$ If take an affine open neighborhood Spec$A$$\subset U$ of $x$, then for all $p_x\in$ Spec $A$, $K_{p_x}=0$? Qestion: Is this correct? How can I show that there is an open set $U$ s.t. $\mathcal O^n_X|_U\cong\mathcal F|_U.$  |
| nuances of optimization with constraints problem Posted: 01 May 2021 07:30 PM PDT Consider an optimization problem with constraints: $F(x,y)$ subject to $g(x,y)=0$. Assume F is smooth function, $g$ is a simple smooth curve. Instead of a standard Lagrange multipliers approach, do the following:
There are few things omitted above
If $g$ is a closed curve, $\tilde g(y)$ is a multivalued function. For example if $g$ is a unit circle, $\tilde g(y)=\pm\sqrt{1-y^2}$. Furthermore $\tilde g' = \mp\frac{y}{\sqrt{1-y^2}}$ has a singularity at $y=\pm 1$. Note that this singularity appeared at the points of connection between the two parts of of $g$. This singularity may not reflect in $f'$, for example if $F(x,y)=F(x^2,y)$. Since the singularities found only in the $\tilde g'$ (everything else assumed smooth), may it happen that these points be actually min/max points of the original problem? The points of connection of the two parts of $g$ are somewhat fictitious. Do we always need to pay a special attention to those points( i.e. check the value of $F$)? Do we need to do it only when the singularities reflects in $f'$, although it is still a problem of the choice of the parameterization, since the curve assumed smooth. Perhaps these points will never be a min/max under described (or more narrowed) scenarios, i.e. those values would never be of interest if we would use the Lagrange multipliers approach.  |
| Extension of an Hilbertian basis Posted: 01 May 2021 07:23 PM PDT The picture below is taken from this paper: http://real.mtak.hu/22877/. The authors claim that the basis of $H^2(\Omega) \cap H^1(\Omega)$ denoted by $\lbrace w_i \rbrace _{i \geq 1}$ can be extended to be a basis of $L^2(\Omega;H^1(0,1))$. I don't see how it can be possible. In my thinking, we have to multiply $w_i(x)$ by $h_i(x)$ where $h_i(x)=\frac{cos(i \pi x )}{i\pi}$ is a basis of $H^1(0,1)$. Is this write?. Thank you.
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| Is there a collection of prime residue classes whose union covers all integers? Posted: 01 May 2021 07:39 PM PDT Is it possible to create a collection {$a_1$(mod $p_1$), $a_2$(mod $p_2$), $a_3$(mod $p_3$) ...} where $p_i$ is a prime, such that their union covers all Integers? The size of the collection can be infinite. Intuitively it seems impossible but I'm not sure how to prove it.  |
| Permutations/Combinations and Multiset Posted: 01 May 2021 07:30 PM PDT I've recently been studying about combinations and permutations and am confused about the different techniques to use to solve different problems. For example, question 1 v.s. question 2: Qn 1: The number of way to distribute 26 indistinguishable pens to 7 different students so that each student will have at least one pen Qn 2: The number of way to distribute 3 balls to 5 boxes For question 1, I first distributed 7 pens to the 7 different students. Then I applied the multiset technique for the remaining 19 pens. I used the bar and crosses technique and got the answer of (19+7-1 choose 7-1). I tried using the same technique for the second question but it didnt seem to work which made me confused about when to use the multiset formula (n+r-1, r-1) and when to just use the normal permutation/combination equations. Thank you for helping!  |
| Posted: 01 May 2021 07:54 PM PDT Consider square matrices $A$ and $B$, both of which are inverses of M-matrices, and let $\mathcal{D}(v)$ be the diagonal matrix with entries $d_{ii} = v_i$. I want to find $$\max_v \rho \left( (\mathcal{D}(v))^{-1} A (\mathcal{D}(v)) B \right).$$ If $B=I$ then of course the spectral radius would be invariant to $v$, since $\rho \left( (\mathcal{D}(v))^{-1} A (\mathcal{D}(v)) \right) = \rho(A)$. How does this change when $B \neq I$?  |
| When a simple algebraic ring extension of a UFD is a UFD Posted: 01 May 2021 07:53 PM PDT Let $k$ be a field and let $R$ be a UFD, which is a $k$-algebra. Let $w$ be an algebraic element over $R$, namely, there exists a polynomial $f(T) \in R[T]$ such that $f(w)=0$. Denote $S=R[w]$. Examples: (1) $R=k[x^2]$, $w=x^3$, $f(T)=T^2-x^2x^2x^2$, $S=k[x^2,x^3]$. (2) $R=\mathbb{Z}$, $w=\frac{1}{2}$, $f(T)=2T-1$, $S=\mathbb{Z}[\frac{1}{2}]$. (3) $R=\mathbb{Q}$, $w=\sqrt{2}$, $f(T)=T^2-2$, $S=\mathbb{Q}[\sqrt{2}]$. (4) $R=\mathbb{R}$, $w=i$, $f(T)=T^2+1$, $S=\mathbb{R}[i]=\mathbb{C}$. If I am not wrong, $S$ is a UFD + $w$ is prime or invertible in $S$, in examples (2),(3),(4). In contrast, $S$ is not a UFD and $w$ is not a prime ($x^3$ divides $x^2x^2x^2$, but $x^3$ does not divide $x^2$) nor invertible in example (1).
Remark: $w \in S$ is a prime element in $S$ iff $(w)$ is a prime ideal in $S$ iff $S/(w)$ is an integral domain. Thank you very much!  |
| Could use some tips on a Functional Analysis Problem Posted: 01 May 2021 07:31 PM PDT could use some help answering a question: Let $A \in \mathscr{B}\left(\ell^{1}\right)$ be defined by $(A f)(m)=\sum_{n}\left(2^{-m}\left(1-2^{-n}\right) f(n)\right)$. Prove that $A\left(\right.$ ball $\left.\ell^{1}\right)$ is not closed. Let $\tau \in \ell^{1}$ be defined by $\tau(n)=2^{-n} .$ Then --this is where I'm stuck-- $\tau \in$ $\left(\operatorname{cl} A\left(\right.\right.$ ball $\left.\left.\ell^{1}\right)\right) \backslash A\left(\right.$ ball $\left.\ell^{1}\right) .$ Thus $A\left(\right.$ ball $\left.\ell^{1}\right)$ is not closed.  |
| Posted: 01 May 2021 07:46 PM PDT Define t-s TSP to a path on an undirected, weighted (non-negative) graph that starts at $s$, visits every node exactly once and ends at $t$. The goal is to find a reduction to regular TSP - A path that starts at some node, visits every node exactly once and ends at the start node. I've come up with the following approach and looking for some feedback on whether this holds. Let $G$ be the input graph to t-s TSP. Construct $G^{'}$ by converting
Now, if there is a solution to t-s TSP on $G$, there is a solution to TSP on $G^{'}$ by following the same path as in t-s TSP on $G$, and appending the edge (t, s). If there is a solution to TSP on $G^{'}$, then there is a t-s TSP solution on $G$ by following the path of TSP on $G^{'}$ and excluding the edge - (t, s). Thus, the reduction holds.  |
| can you solve the part a to g? Posted: 01 May 2021 07:37 PM PDT [QUESTION REUPLOADED], PLEASE TRY TO GIVE DETAILED EXPLAINATION, I HAVE SOLVED THIS BUT NEED TO MATCH MY ANSWERS. This is my first time using this platform, so uploaded a screenshot.  |
| Prove or disprove $SO(4n) \supseteq \frac{(Sp(1)\times Sp(n))}{\mathbb{Z}_2}$? Posted: 01 May 2021 07:38 PM PDT I suspect that this is true $$ \boxed{SO(4n) \supseteq \frac{(Sp(1)\times Sp(n))}{\mathbb{Z}_2}.} $$ How do we prove it?
How to prove that general $n$, it is all true?It is better even to show the explicit embedding. p.s. Here I use the symplectic group $Sp(1)=SU(2)$ and $Sp(2)=Spin(5)$. We can see that a generic $SU(2)$ group can be represented by a rank-2 unitary matrix satisfies $$V^\dagger V =\mathbb{I}.$$ Then we can write such a complex $V = \begin{pmatrix} a & b\\ - b^* & a~* \end{pmatrix}$. It can be checked that it obeys $Sp(1)$ condition $$ V^T \begin{pmatrix} 0& 1\\ -1 & 0 \end{pmatrix} V = \begin{pmatrix} 0& 1\\ -1 & 0 \end{pmatrix}. $$  |
| Posted: 01 May 2021 07:41 PM PDT From example 7.4 (page 22/28) of the paper Semidefinite programming relaxations for semialgebraic problems, the author is trying to find the distance from the point $\left( {{x_0},{y_0}} \right) = \left( {1,1} \right)$ to the curve $C\left( {x,y} \right): = {x^3} - 8x - 2y = 0$ by reformulating it in to an SOS optimization problem $\begin{gathered} {\text{Maximize}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\gamma ^2} \hfill \\ \text{s.t.}\,\,\,\,\,\,\,\,\,\,\,{\left( {x - 1} \right)^2} + {\left( {y - 1} \right)^2} - {\gamma ^2} + \left( {\alpha + \beta x} \right)\left( {{x^3} - 8x - 2y} \right)\,\,{\text{is}}\,{\text{SOS}} \hfill \\ \end{gathered}$ 1/ Why does SOS reformulation required us to add the extra term $\left( {\alpha + \beta x} \right)$ ? 2/ The author said that $\gamma ,\alpha ,\beta$ could be found by Semidefinite programming (SDP) but how can we rewrite this problem as an SDP ? 3/ When the bound $\gamma$ is not sharp due to small tolerance setting for the SDP solver is there any way to iteratively refined of improve this bound ? Thank you very much !  |
| Posted: 01 May 2021 07:56 PM PDT Try beta1 = 0 and see if the proof holds water (as the author is dividing by Beta1. This occurs when the author solves for x(i).(Actually it is easy enough to fix this aspect of the proof, but at least it should have been mentioned in the proof.)  |
| Combinatorics- The number of ways to arrange marbles Posted: 01 May 2021 07:58 PM PDT There are R red marbles, G green marbles and B blue marbles $(R \leq G \leq B)$ Count the number of ways to arrange them in a straight line so that the two marbles next to each other are of different colors. I can't solve the general problem. But if $B \geq R+G+2$ the answer is $0$.  |
| Possible values for continuous function? Posted: 01 May 2021 07:31 PM PDT Suppose $f$ is continuous on $[2,6]$, and the only solutions of the equation $f(x)=7$ are $x=2$ and $x=5$. If $f(3)=9$, then one of the following CANNOT be the value of $f(4)$ A) 9 I found this on the exam book and it says the answer is 5. But how??? How on earth could we justify that? I tried so hard that finally, I'm asking it here. This was my futile approach: Write Any insight is appreciated!  |
| Finding the number of automorphisms in $G_{\Bbb{Q}(\sqrt[4]{2},i),\Bbb{Q}}$? Posted: 01 May 2021 07:56 PM PDT I am trying to solve the following problem:
I think we must count all the $\Bbb{Q}$-automorphisms $\sigma, \tau$ with the given conditions, is that correct? I tried to write the expression for the elements in $K$: $$a+b\sqrt[4]{2}+c\sqrt[4]{2}^2+d\sqrt[4]{2}^3+ei+fi\sqrt[4]{2}+gi\sqrt[4]{2}^2+hi\sqrt[4]{2}^3$$ And check for the automorphisms but this looks a bit unwieldy. Is there a more practical way to do it? I suspect there is because the $\sigma$ and $\tau$ looks a bit specific and I have found before that $K=\Bbb{Q}(\sqrt[4]{2},i)$ and $2,4$ are the degrees of the extensions. I have also seen other similar questions to mine but they don't seem to address exactly what I am looking for.  |
| Piecewise function notation and applications Posted: 01 May 2021 07:32 PM PDT I would like to define a piecewise function to imitate some physical phenomenon, which, for example, occurs once in a year (at June for a whole month long). Say, frequency/domain (integers) of the func is representing months, and January is at 0.  |
| Two absolute values satisfy $|x|_1=|x|_2^t$ iff they satisfy $c_1|x|_1\leq |x|_2\leq c_2|x|_1$. Posted: 01 May 2021 07:51 PM PDT Let $k$ be a field and $|\cdot|_1$, $|\cdot|_2$ be two absolute values on $k$. Consider the following propositions: (1) There exists $t>0$ such that $|x|_1=|x|_2^t$ for all $x\in k$; (2) There exists $c_1,c_2>0$ such that $c_1|x|_1\leq |x|_2\leq c_2|x|_1$ for all $x\in k$. I know how to prove that (1) holds if and only if both absolute values induce the same topology. But two norms induce the same topology iff (2) holds. In other words, these propositions are equivalent. I really think there should exist a simpler proof of this fact (with just some algebraic manipulations) but I don't know how. I appreciate any ideas!  |
| Properly drawing a Penrose tiling using the pentagrid method Posted: 01 May 2021 07:29 PM PDT As part of my work, I create tools for artists to make various types of patterns for artistic purposes. I am trying to make a tool to create a Penrose tiling and I would like to use the pentagrid method of generating it, as it seems like the easiest way to allow the user to arbitrarily scale and translate the plane and still generate a nice random aperiodic tiling that easily stretches to infinity in any direction. Whereas using something like inflation or deflation becomes problematic when the user decides to scale the tile size down or translate by a large amount in any direction. Given that, I've found several references on the how to generate a tiling from a pentagrid, such as these:
They've been incredibly helpful, but I feel like I'm missing a step. I'm able to generate the pentagrid:
But when I attempt to generate the tiles, I end up with tiles that are separated:
I'm working in Swift and I'm generating the vertexes of each tile by doing the following at each intersection point: The Is this the expected result? Some of the references I've read have made vague statements about needing to move the tiles together (without suggesting how to do that), while others have seemed to indicate that they'll all fall into the correct positions. If they do need to be moved together, is there a particular algorithm to do that? If they don't need to be moved together, then what have I missed or misunderstood?  |
| Posted: 01 May 2021 07:24 PM PDT Without using computer programs, can we find the last non-zero digit of $(\dots((2018\underset{! \text{ occurs }1009\text{ times}}{\underbrace{!)!)!\dots)!}}$? What I know is that the last non-zero digit of $2018!$ is $4$, but I do not know what to do with that $4$. Is it useful that $!$ occurs $1009$ times where $1009$ is half of $2018$? If that is useful, then what if $1009$ was another value, say $1234$? Any help will be appreciated. THANKS!  |
| Posted: 01 May 2021 07:25 PM PDT Firstly I should mention an example of this patter which is also given in the book that is for $n=5$ $3,2,4,1,5$ is valid whereas $3,2,5,1,4$ is not.  |
| The matrix notation of signum? Posted: 01 May 2021 07:59 PM PDT The following question on a notation might look trivial but I am really not sure how to deal with it. If I have a variable $x$, I could write out: $$x=|x|\;\text {sgn} (x)$$ a notation that helps me with an operator for the signs that could point to $-1$, $0$ or $+1$. But then I have a matrix $\bf X$ with elements $x_{i,j}$ while the equation above holds for each element $x_{i,j}$, simply $$x_{i,j}=|x_{i,j}|\;\text {sgn} (x_{i,j})$$ How does the matrix notation for the equation above look like, in terms of a matrix of $\bf X$ (and not individual elements)?  |
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