0
votes
0answers
4 views

If two groups act on a set in the same way then are the two groups related?

Let $G_1, G_2$ be two groups that act on a set $S$ on the left, such that for all $g \in G_1$ there's $g' \in G_2$ such that $g\cdot s = g'\cdot s$ for all $s \in S$. Define $h : G_1 \to G_2$, $h(g) ...
0
votes
0answers
3 views

System of inequalities. Points of intersection?

$x^2+y^2<=81$ $y<x$ Is this correct? My answer: (-9sqrt(2)/2,-9sqrt(2)/2), (9sqrt(2)/2,9sqrt(2)/2)
0
votes
0answers
4 views

Prove Linear Dependence in T: V -> W

Problem: "Let $V$ and $W$ be vector spaces and let $T:V \rightarrow W$ be a linear transformation. Prove that, if $\{v_1, v_2, v_3\}$ is a set of three linearly dependent vectors in $V$, then the set ...
0
votes
0answers
4 views

Prove that equation satisfies Laplace's Eq.

I'm trying prove that the equation $T(x,y)$ satisfies Laplace's Equation where $T(x,y)$ is given as $T(x,y) = -Im\Omega(z)$ where $\Omega = 1/\omega$ and we are told to use the substitution $z = ...
0
votes
0answers
5 views

How to show A={q belongs to X ; p belongs to X, d(p,q)>delta, delta>0} is closed?

first of all, i used "belongs to" to mean that p is an arbitrary element of X i tried it directly and take A's complement and show it is open but failed to both way i know when union boundary of ...
0
votes
3answers
15 views

How to calculate $f(n)=1\binom{n}{1}+2\binom{n}{2}+3\binom{n}{3}…n\binom{n}{n}$

$$f(n)=1\binom{n}{1}+2\binom{n}{2}+3\binom{n}{3}.....n\binom{n}{n}$$ Is there a formula for this?
0
votes
0answers
2 views

Prevent Maple to evaluate before simplify a function

I have been trying to find a domain of $f(x)=\frac{x}{\frac{(x+2)}{(x-3)}}$ using different kind of software ( its clear the domain of this function is $\mathbb{R}\backslash \{-2,3\}$ ). When I tried ...
2
votes
0answers
11 views

What is this sequence of polynomials?

i was trying to calculate the probability of something and i came upon them. i needed to know what this was equal to: $$p_n(x)=\sum_{k_n=k_{n-1}}^{x}....\sum_{k_3=k_2}^{x} \sum_{k_2=k_1}^{x} ...
0
votes
0answers
3 views

How to compute the unique MLE from an Exponential Family of Distributions?

Let $$ f(x;\theta)=\frac{2e^{\theta x}cos(\theta \pi/x)}{\pi cosh(x)}, x\in{\mathbb{R}} $$ be a family of densities and which is clearly exponential family. Then what is the Maximum Likelihood ...
0
votes
0answers
5 views

How to simultaneously search multiple finite sequences in OEIS?

Suppose I have several finite sequences and I want to know if they show up in a family of sequences or any one particular sequence or on rows of an array on OEIS; is there such a search option.
0
votes
0answers
8 views

Fourier Transform of rational function

So I have this function: $$f(t)=\frac{1}{(1-it)^{n+1}}$$ And I have the Fourier Transform defined as $$\hat{f}(\lambda)=\frac{1}{\sqrt{2\pi}}\int_\mathbb{R}f(t)e^{-\lambda.it}dt $$ Now my ...
0
votes
1answer
11 views

If $a\equiv b [p^k]$ then $a^p \equiv b^p [p^{k+1}]$

Can anyone explain the steps to this proof? I'm really lost/ If $k\geq 1$ and $a\equiv b[p^k]$ then $a^p \equiv b^p [p^{k+1}]$ Proof: Since $a= b + qp^k$ for some $q\in \mathbb{Z}$ we have $a^p = ...
0
votes
0answers
7 views

Show that there is a matris only $ A $ such that $ \varphi (t) = e ^ {tA} $.

Let $ \varphi(t)$ of a matrix $n \times n$ functions $C^1$. If $\varphi(0)=I$ (identity) and $\varphi(t + s) = \varphi (t) + \varphi (s)$ for all $ t, s \in \Re $, show that there is a matris only $ A ...
0
votes
1answer
12 views

Was I wrong to omit angles in the solution set for this multiple angle problem?

I may have missed this in my precalculus course, but why was I wrong to omit angles that did not have a positive value for cosine? I didn't include $\frac{3\pi}{4},\frac{7\pi}{12},\frac{5\pi}{4}$ ...
1
vote
0answers
6 views

What is some prerequisite to study nonlinear programming?

What is some prerequisite to study nonlinear programming? I already know calculus and linear programming is two perquisite, what else?
0
votes
0answers
7 views

Is $L:=\mathbb{Q}_{2}[\sqrt{1+\sqrt{-2}}]$ Galois extension of $\mathbb{Q}_{2}$? completely ramified?

I think it is because normal: The minimal polynomial is $f(x)=(x^{2}-1)^{2}+2$ which has roots $1,\sqrt{-2}, \sqrt{1+\sqrt{-2}},\sqrt{-2}\sqrt{1+\sqrt{-2}}$ and those are all contained in L. ...
0
votes
2answers
17 views

Proof by induction sum $2^j = 2^{n+1} - 1$

I am trying to solve a previous test for an exam, and there are no solutions. The problem I am trying to solve is If $n$ is a natural number, then $1 + 2 + 2^2 + 2+3 + ... + 2^n = 2^{n+1} -1$ ...
0
votes
0answers
12 views

Number of necessary stickers to complete a sticker album

I have the following problem, and I was hoping you guys could help me solve it: Consider a sticker album that has $t$ uniquely collectable stickers (that accounts for the universe of collectable ...
0
votes
0answers
6 views

Find the flux of the vector field across the boundary of the cube

Find the flux of the vector $F=e^{xy} \hat{i} +e^{yz} \hat{j} +z \hat{k}$ across the boundary of $[0,1] \times [0,1] \times [0,1]$. Can someone tell me the setup of this problem?
0
votes
1answer
9 views

Showing that no non-identity element of $G/F_g$ has finite order where $G$ abelian, $F_g$ the set of elements of G that have finite order

Let $G$ be an abelian group and $F_g$ the set of elements of $G$ that have finite order. Show $F\trianglelefteq G$ and no non identity element of $G/F_g$ has finite order. $G$ is abelian $\implies ...
0
votes
0answers
3 views

Find the flux of the vector field across the closed surface of the three-dimensional region E

Let E be the part of the ball $x^2+y^2+z^2 \leq 1$ with $z \geq 0$. Find the flux of the vector field $F=2x^2y \hat{i} +2yz \hat{j}-z^2 \hat{k}$ across the closed surface of the three-dimensional ...
0
votes
0answers
13 views

This is a lemma about function spaces. I'm not really understanding it however. Can someone try explaining it to me?

Lemma: let X be a set (Y, d) in the metric space, fn, f is in Y^X then fn->f iff fn->f uniformly
0
votes
1answer
11 views

maximizing a quadratic objective subject to sum constraint

Is it possible to solve the following problem $$ \max \,{x^ \top }x \\[0.1in] st: \sum\limits_{i = 1}^n {{x_i} = 1} \\[0.1in] 0 \le x_i \le 1 $$ Intuitively it happens when any one of the n x's is 1 ...
0
votes
1answer
15 views

Explanation for example of flow generated by vector field

The text I am reading has an example for flow in a section titled "Flows and Lie derivatives." Below is the example: Let $M = \mathbb{R}^2$, and let $X((x,y)) = -y \partial/\partial x + x ...
1
vote
1answer
18 views

Asymptotic result about analytic number theory

I don't know if there is any done work done about ehis matter, and I don't have access to research news. I'm interested in this question (I haen't tried to answer it myself, but it seems very ...
1
vote
5answers
58 views

$\sum\limits_{n=1}^\infty n(\frac{1}{2})^{n}$

I am trying to find the expected value of the number of even numbers rolled before the first odd number when rolling a fair die until an odd number comes up. I arrived at $\sum\limits_{n=1}^\infty ...
0
votes
1answer
13 views

Graph Theory Question Related to Domination number.

Let G be a graph whose diameter is at least 3. Prove that the domination number of the complement of G is at most 2. I know that since the diameter of G is at least 3, the diameter of the complement ...
0
votes
0answers
7 views

transpose of integral operator and covariance of its image

Define the linear operator $O_T: \mathbb{R}^n \to L_2[0,T] $ \begin{equation} O_Tx = Ce^{At} x, \end{equation} where $t \in [0, T]$, $ C$ and $A$ are matrices with compatible dimentions, and $x \in ...
0
votes
0answers
8 views

Dual quaternion inverse

Is it true that for every dual quaternion $Q$ I can find it's inverse such that $QQ^{-1} = 1?$ Using the usual definition $Q^{-1}=\frac{Q^{*}}{||Q||^2}$ doesn't work for me, since the dual part ...
0
votes
1answer
29 views

Set notation of $S^1 \times S^1$

This is a simple question, but should this be written as: $\hspace{120pt}S^1 \times S^1 = \{(z_1,z_2)\in\mathbb{C}\times\mathbb{C}:|z_1|=|z_2|=1\}$
1
vote
2answers
22 views

Polynomials in Linear Algebra

Let $n$ be a positive integer and $\mathbb{F}$ be a field. Suppose $A$ $\in$ $M_{n\times n}(\mathbb{F})$ and $P$ is and invertible matrix, such that $P \in M_{n\times n}(\mathbb{F})$. If $f$ is any ...
1
vote
0answers
13 views

Parking Lot Optimization Problem — How To Find the Minimal Path In A Periodic Set?

When I was a commuter student, I would park in a very large parking that that had a set of stairs in a corner that I had to climb. In general, I had to park far away from this corner in an almost full ...
1
vote
1answer
39 views

Triple integration problem

Prove that: $$\iiint\limits_B \frac{1}{x} dx\,dy\,dz=\frac{8-4\sqrt2}{3}$$ where $$B=\{(x,y,z):1\leqslant x \leqslant e^z, y\geqslant z, y^2+x^2\leqslant 4\}.$$ I used Mathematica's regionplot3D to ...
3
votes
2answers
45 views

Why can't we have an $y$ such that $xy\equiv 1\; (mod\; n)$ when $n$ is not prime?

I'm reading Avner's Fearless Symmetry: Here he says that we can only have the cancelation law if the modulus is prime: I got curious with the statement and then I kept reading the chapter: ...
0
votes
0answers
16 views

Equivalence of Cohomology groups

Suppose $n=i+j,$ with $n, i,j$ positive integers. Let $I^k$ denote the $k$-dimensional unit square. It is claimed (in Hatcher's Algebraic Topology text) that $H_i(\mathbb{R}^n, \mathbb{R}^n \setminus ...
0
votes
2answers
19 views

Continuous function such that image of closed set is not closed

Could anyone give me an example of a continuous function with domain $\mathbb{R}$ such that the image of a closed set is not closed? I can't seem to think of one but my friend insists that one exists. ...
2
votes
0answers
14 views

Zero-distortion map projection

Is it possible to take a limit of map projections (from a sphere to a plane) with ever-smaller distortion factors to get some kind of dendritic limit projection that has zero distortion everywhere? My ...
3
votes
1answer
34 views

A quick way to estimate eigenvector/eigenvalue of a matrix

Is there a quick way to give a raw estimation of an eigenvector/eigenvalue of a matrix? By "quick" I mean some method which can be computed without a computer or paper and pencil...something you could ...
0
votes
1answer
9 views

Sylow p-subgroup: Understanding a proof

I don't understand the part of the followng proof that I underlined with red. I don't see why $PN/N \cong P/(P \cap N)$ implies $PN/N$ is a p-subgroup of $G/N$.
1
vote
0answers
2 views

Understanding PAC (probably approximately correct) bounds on the realizable case (and finite hypothesis class)

I was trying to understand PAC bounds on the realizable case (i.e. when there is some perfect $h^* \in \mathcal{H}$ and its generalization error is zero). Notation: Training data: $$S_n$$ Training ...
0
votes
0answers
12 views

How to find the discrete probability vector given a transition probability matrix?

I have a transition probability matrix for the above Discrete Time Markov Chain I want to find the 'discrete probability vector' of this state space. My understanding is that the discrete ...
2
votes
1answer
23 views

Let $p(t)$ be a polynomial in $\mathbb{R}$. Defining $p_0(t) = p(t)$

Let $p(t)$ be a polynomial in $\mathbb{R}$. Defining $p_0(t) = p(t), p_1(t) = 1 + \int_0^t p_0(s)ds, ... , p_k = 1+ \int_0^t p_{k-1}(s)ds.$ Prove that $p_k(t)$ converges uniformly on each compact ...
0
votes
0answers
8 views

Calculating $\arg\min_x (1-\Phi(x;\mu_1,\sigma_1^2)+\Phi(x;\mu_2,\sigma_2^2))$

I would like to find $x$ satisfying the following expression: $$\arg \min_x R(x,\mu_1,\mu_2,\sigma^2_1,\sigma^2_2)$$ where $$R(x,\mu_1,\mu_2,\sigma^2_1,\sigma^2_2) ...
3
votes
0answers
21 views

Is there a known method for finding extremely huge squarefree numbers?

People often compete to beat the record for largest known prime (it is currently $2^{57,885,161}-1$). There are also big money prizes for finding explicit prime numbers exceeding specific magnitudes. ...
1
vote
1answer
25 views

Prove that $E(\mathbb{C})^{\text{tor}} \cong \mathbb{Q}/\mathbb{Z} \times \mathbb{Q}/\mathbb{Z}$.

Let $E$ be an elliptic curve over $\mathbb{C}$. We know that $E(\mathbb{C}) \cong \mathbb{C}/L$ (this is a group isomorphism) for some lattice $L \subset \mathbb{C}$. Using this fact prove that ...
0
votes
0answers
14 views

Why is $\hat{x}$ in the linear regression equation $A^TA\hat{x} = A^Tb$ part of $C(A^T)$

When finding the best fit line for a number of points, we use $A^TA\hat{x} = A^Tb$ where we solve for $\hat{x}$. I understand that the projection $p=A\hat{x}$ is part of the column-space of $A$ and ...
7
votes
3answers
67 views

How to understand “Union of balls centered at rational numbers is way less than $\mathbb{R}$

A few month ago I had to prove $\lambda(\mathbb{Q}) = 0$ (where $\lambda$ is the one-dimensional Lebesgue measure). The idea: Let $\varepsilon \gt 0, r_n := \frac{\varepsilon}{2^n}$ and $\mathbb{Q} = ...
0
votes
0answers
6 views

Zooming views relative to point

I'm making a viewer for a fractal generated by convergence to a root of some polynomial using a root finding algorithm. (Examble fractal) in the complex plane. I then made an interactive viewer, ...
0
votes
1answer
11 views

Exponential generating function for number of 10 length sequences built from the alphabet, with some restrictions

I've got the following homework question. If anybody could possibly point me in the right direction, that would be great: Suppose X is a sequence with 10 terms built from 26 letters {a, b, c, ..., ...
0
votes
0answers
48 views

How to solve this integral ($\int _{\frac{\pi }{6}}^{\frac{\pi }{4}}\sqrt{1-\tan ^2\left(x\right)}dx$)

$$\int _{\pi/6}^{\pi/4}\sqrt{1-\tan ^2\left(x\right)}dx$$ Hey, can you help me to solve this integral please? Thanks.

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