All Questions

0
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0answers
2 views

Approximate distribution of sum of squared standardized Poisson variables

Suppose that $X_1, ..., X_n$ are independent and identically distributed Poisson($\lambda$) random variables. What is a good approximating distribution for $\sum_{i = 1}^{200} \frac{(X_i - \lambda)^2}...
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0answers
8 views

In general $d \phi (X)$ is not defined where $X$ is a vector field

Let $\phi:M\to N$ be a smooth map between manifolds and $X$ a vector field on $M$. Moreover, suppose that $d\phi$ is injective at every point. I want to show that in general $d\phi (X)$ is not defined ...
1
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0answers
3 views

Algorithm/approach for regridding 3D coordinates

I have a question concerning surfaces of 3-dimensional structures, specifically if there is a way to interpolate or ‘regrid’ points/coordinates on a surface generated from the structure. I ask here on ...
0
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0answers
3 views

An equation defined by norm

Let $f$ be an Eisenstein polynomial of degree $n$ and the prime $p$. $\alpha$ is a root of $f$. Let $\mathbb{Q}(\alpha)=K$, Prove that for any $\gamma\in O_K$, there exist $a\in \mathbb{Z}$, such that ...
0
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0answers
9 views

Locus of circles centers

Let $\Delta$ a line and $A$ a point not lying on $\Delta$. Find the locus of all centers of circles passing trough $A$ and tangent to $\Delta$. This is an old french book exercise.
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0answers
3 views

Hodge theory: $\Delta \alpha = 0$ iff $d\alpha = d^* \alpha = 0$ on a noncompact manifold?

Let $M$ be a Riemannian manifold (connected, oriented). One can define the co-differential $d^* : \Omega^k(M, \mathbb{R}) \to \Omega^{k-1}(M, \mathbb{R})$ even if $M$ is not compact (for example use ...
0
votes
1answer
24 views

Counting method: how many elements have the following set.

my problem is the following: I have $n$ identical items and $m$ enumerated boxes with unlimited capacity. I have to put all my items in the boxes such that all the boxes have at least one item. If $n&...
-1
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0answers
6 views

Finding Cartesian coordinates of remaining vertices of triangle, given angle from y-axis with point A and a vertex

I have an isosceles triangle ABC, where the height h and angle at point A are known. The Cartesian coordinates of point A are also known. How to find B(x2,y2) and C(x3,y3)? https://i.stack.imgur.com/...
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0answers
5 views

General formula for decomposition $n^{th}$ power of $V_1$ rep of $SU(2)$

If $V_n$ is the $n+1$ dimensional irreducible complex rep of $SU(2)$ I'd like to find closed form formula for: $$ V_1^{\otimes n}$$ Using tensor distributivity over direct sum and Clebsh formula I'...
1
vote
0answers
11 views

Is there a product formula for the Bernouilli numbers?

There are many well-known formulas available for the Bernouilli numbers, many of these can be found on the Wikipedia page. However, these are all in the form of infinite sums. I have not found any of ...
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0answers
6 views

Conserved quantities for a group action?

I was reading this wiki about moment maps, and they say that moment maps are [...]used to construct conserved quantities for the action What exactly does it mean to be a conserved quantity for an ...
3
votes
0answers
30 views

The art of proof summarizing. Are there known rules, or is it a purely common sense matter?

When a proof is long and difficult , it can be really nice vis-à-vis the reader to give a summary or an outline of the deduction before beginning hard work. Are there known rules to give a good ...
0
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0answers
4 views

Example of a bilinear form of abelian groups

Let $X$ and $Y$ be abelian groups. Then, a $Y$-valued bilinear form on $X$ is a $\mathbb{Z}$-module homomorphism $$\alpha: X \otimes_{\mathbb{Z}} X \rightarrow Y$$ How does this relate to the standard ...
0
votes
3answers
26 views

For what values of $x$ is the $\sum_{n=1}^\infty \frac{(7x)^n}{n!}$ convergent?

I was wondering for which values of $x$ the following series converges: $$\sum_{n=1}^\infty \frac{(7x)^n}{n!}.$$ I applied the ratio test to get $$\lim_{n\to \infty} \frac{7x}{n+1}$$ So then $7|x|&...
0
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0answers
29 views

Enigma having to do with bricks

A professor told us this one but I can't remember much of it, I hope you can help me find it. This is what I remember, you have bricks, each brick has only (or at least) a side which length is an ...
2
votes
1answer
29 views

Why do Linear Transformations Take a circle to an ellipse

Short Version: How can it be intuitively shown that non-singular 2D linear transformations take circles to ellipses? (Also, its probably important to state I'd prefer an explanation that doesn't use ...
0
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0answers
3 views

Triangulating a 3d point by 2 camera matrices

can you give me a hint how to solve this issue? Two cameras are placed with a 1m gap in x-direction next to each other (with focal length 1). The rotation matrices Ri and coordinates of the camera ...
1
vote
1answer
20 views

What does the symbol $\delta$ mean on this page?

On the Wikipedia page on Arithmetic Functions, the section Relations Among The Functions makes frequent references to a variable $\delta$ (or is it a function? Some other kind of value?). It's ...
2
votes
2answers
23 views

Does $2^{O(n)} $ mean $O(2^n)$? If not, what does $2^{O(n)} $ mean?

In this "tutorial", in the end, they say exponential asymptotic notation is $2^{O(n)} $. Is $2^{O(n)} $ the same thing as $O(2^n)$? Is there any reason to notate it that way? According to one of the ...
0
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0answers
13 views

Solid angle: Must a region subtending a solid angle be (simply) connected?

Although answers to the question "What is a Solid Angle?" explain that the shape of the area subtending a solid angle doesn't matter, my question is if the region has to be simply connected (no holes)....
0
votes
1answer
22 views

Show that the order of an element g is well-defined

Suppose $G$ is a group and let $g∈G$, explain why the order of $g$ is well-defined, while the definition of the order is the following: The smallest positive r such that $g^r=e$, if no such r is ...
0
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0answers
6 views

Edexcel A Level Statistics and Mechanics Year 1/AS Ex. 2E p.33 Q8 mean/standard deviation

I agree with the answer to (a). The problem I have is that if the answer to (c) is correct, then the mean would be (4+17)/2 = 10.5kn, which is different from the value 8.1kn from part (a). So my ...
0
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0answers
6 views

Given density function p(x;θ) = (1+θx)/2. Find the generalized likelihood ratio test for the hypotheses: H0: θ<=0; H1: θ>0

For this question, the range of x is from -1 to 1. I just don't know how to compute the MLE for θ in this single observation case. And I haven't seen a similar problem before. Since in this case, the ...
0
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0answers
2 views

Quantifying positining and alignement of two line segments

For simplicity we can consider a 2D case: where we have line segments of the same length $l,$ and we suppose we have $n$ of them with their centre coordinates and orientations randomly assigned. For ...
0
votes
1answer
14 views

Uniform convergence of $\sum -\frac{sin(nx)}{log(n)^2}$ over a closed and bounded subinterval of $[0,2 \pi]$

As mentioned in the title, I want to study the uniform convergence of the following series. $\sum -\frac{sin(nx)}{log(n)^2}$ for $x\in[a,b]$ where $0<a<b<2\pi$ I try to apply M-test but I am ...
0
votes
0answers
24 views

Group theory and Schur’s lemma

I am a little bit confused by consequence of Schur’s lemma: “irreducible representation of any abelian group must be of dimension one”. The proof is quite simple but i have some doubts: for exemple if ...
0
votes
1answer
13 views

How to find limits for $\theta$ for Gaussian Integrals

I'm an engineering student who's been tasked with a maths problem slightly outside my field of study. It involves the evaluating the Gaussian integral $$\int_{0}^{\infty} e^{-x^2} dx$$ After doing ...
0
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0answers
9 views

Projection of a circle via Matlab

I am currently stuck at this exercise - I dont know where to begin, which forumals should I use :/ You have to implement the projection of a circle as viewed by a camera in 3D with different angles ...
-1
votes
3answers
15 views

How to solve DE where the derivative is in the denominator

I want to solve the following DE of first order. $\frac{u}{\sqrt{( u')^2+1}}=c$ where c is constant. The problem that i have is that i don't know how to handle the DE if the derivative is in the ...
0
votes
0answers
8 views

minimal $KC$-topological space

If $P$ is a topological property, then a space $(X, \tau)$ is said to be minimal $P$ (respectively, maximal) if $(X, \tau)$ has property $P$ but no topology on $X$ which is strictly smaller (...
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0answers
6 views

Basic feasible solution on convex sets

Let $P = \{x \in R^n | Ax \geq b \}$. Suppose that at a particular basic feasible solution, there are $k$ active constraints with $k>n$. Is it true that there exists exactly $C(k,n)$ bases that ...
0
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0answers
11 views

Quick simple question - doesn't y=secx have a horizontal asymptote at y<1 [on hold]

doesn't y= sec(x) have a horizontal asymptote at y<1 I know there are asymptotes at π/2 + nπ, onwards thank you.
3
votes
2answers
25 views

Finding all continuous function which maps any sequence in geometric progression to another geometric progression

Find all continuous functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for any geometric progression $x_n$ the sequence $f(x_n)$ is also a geometric progression. I tried first by taking constant ...
0
votes
1answer
10 views

What is a region of type 3 with regards to Green's Theorem?

I understand that a region of type 1 is where two curves are connected by two vertical lines and that a region of type 2 is where two curves are connected by two horizontal lines. But what is a region ...
1
vote
0answers
14 views

Sobolev space identification

Let $\Omega$ be a open subset of $\mathbb{R}^{d}$. It is well known that $L^2(\Omega\times (0,T))$ can be identified with $L^2(0,T;L^2(\Omega))$. Now, let us consider $$H^{1}(0,T;H_{0}^{1}(\Omega))=\{...
0
votes
1answer
18 views

topological $KC$ space

The spaces are called $KC$-spaces in which every compact subset is closed. let $(X,\tau_1)$ be a $KC$-topological space, is there a $KC$ topology space $(X, \tau_2)$, so that $ \tau_ 1 \subset \...
0
votes
0answers
7 views

Approximation of extremal problem

Let $J$ be linear functional on some functional space $U$, $X_n \subset U$ -- some functional sets, $n = 1,2, ...$ such that $X_n \to X_0 \subset U $. And we have sequence of extremal problems: $\...
1
vote
1answer
21 views

For a function with mirror- and translational symmetry how can I find the domain where the function has a strict local minimum point?

Let $\ f:X \rightarrow {\rm I\!R}$ be a function and $X=\{(x,y)\in {\rm I\!R}^2\}$. The explicit expression of this function is unknown, but it can be assumed to be smooth and continuous. It is know ...
1
vote
1answer
13 views

Showing W=ku+kv has a unique module structure over kG

Let $G=<x> \times <y>$ where $|x|=|y|=p$ and so $|G|=p^2$. Let $k$ be a field of characteristic $p$. Let $W$ be the $k$-span of $v$ and $u$. We wish to show the module structure given by $...
0
votes
0answers
7 views

deriving an implicit Runge Kutta method from its Butcher tableau

I would like to use the Gauss–Legendre method of order four (which is a particular Runge Kutta method) to solve numerically an ode but I find only it's Butcher Tableau and I fail to derive the method ....
0
votes
3answers
25 views

Return 0 for negative numbers using a short list of functions [duplicate]

I'm using a CAD software that allows you to calculate a value using a small set of built in functions. I have the formula complete for the input (call it 'x') of this formula, I just need to return ...
2
votes
1answer
52 views

Find : $\int_0^{\pi/2}\frac{x\ln(\cos x)}{\sin^2 x}dx$

Find integral without use harmonic series: $$\int_0^{\pi/2}\frac{x\ln(\cos x)}{\sin^2 x}dx$$ I don't have any ideas for this type integration without using harmonic series. I will happy to see ...
0
votes
0answers
12 views

A vector calculus formula

Let $A, B$ be vector fields in $\mathbb R^3$. We have $$ \text{curl}\bigl((A\cdot \nabla)B\bigr)=(A\cdot \nabla)\text{curl}B -((\text{curl}A)\cdot \nabla)B+R(A,B). $$ I know that $R(A,A)=0$ and I ...
1
vote
0answers
24 views

Proving that $f$ is analytic. [duplicate]

The problem is: Let $f: \mathbb{R} \to \mathbb{R}$ be a $C^\infty$ function with non negative derivatives at every point. Prove that $f$ is analytic. I know that if the derivatives are bounded by $f^...
0
votes
0answers
20 views

Summation and Product sum of series

what is the answer of $$\sum_{m=1}^B\Pi_{n=1}^A \frac{m+n}{mn}=?$$ What I tried $$\sum_{m=1}^B\Pi_{n=1}^A (\frac{1}{m} +\frac{1}{n})$$ $$=\Pi_{n=1}^A \sum_{m=1}^B(\frac{1}{m} +\frac{1}{n})$$ $$=\...
0
votes
1answer
17 views

a question about strong law for renewal process

Let $\{N(t); t > 0\}$ be a renewal counting process with inter-renewal rv s $\{X_n; n \geq 1\}$. Then, $lim_{t\rightarrow \infty} N(t) = \infty$ with probability 1. Proof: We only need to show $P\...
0
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0answers
10 views

Standard form for the Characteristic Matrix/Polynomial

I'm currently taking Linear Algebra and Differential Equations, and in talking about eigenvalues of a matrix, both professors have given the same information: for some square n x n matrix A, the ...
0
votes
0answers
8 views

n-singular chains are result of a group action?

I´m beginning in the study of Singular Homology using Kosniowski's text. Since it seems that the n-chains group $S_n(X)$ inherites the group structure of $\mathbb{Z}$, we can consider that, in ...
0
votes
0answers
9 views

Tensor products of modules over non-commutative rings

I've been learning about tensor products over modules, but where the ring acting on the module is commutative. When $R$ is non-commutative, we consider a right $R$-module $M$ and a left $R$-module $N$...

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