1
vote
1answer
17 views

What does it mean a transitive permutation?

let $X$ be a finite set. Let G be a group. What is the meaning of $G$ is a transitive permutation on set $X$?
2
votes
1answer
25 views

Eigenvalue equation and separable solutions

Recently, studying Quantum Mechanics I found a doubt regarding separable solutions and eigenvalue equations for differential operators. Suppose we are considering some space $\mathcal{H}$ of functions ...
1
vote
0answers
29 views

Group theory: Intuition as to what a group is [duplicate]

In group theory the group is an algebraic structure consisting of a set which has elements associated with definite finitiary operations. Can an intuitive explanation be provided as to what this ...
2
votes
0answers
21 views

Generating subsets with 1 common element

I have a number $n$ and a set $S$ of $n(n-1)/2$ elements : $ \{1, 2, \ldots, n(n-1)/2\}$ I'm looking for an algorithm to generate $n$ distinct subsets of $S$, each having $n-1$ elements, with the ...
1
vote
1answer
18 views

Distance between a point $X$ and a line $2x-y = 1$

I am asked to state a condition for a point $X$ such that the point $X$ is a distance of 10 units from the line $2x-y = 1$, and then find the Cartesian equation of the set of all points $X$ that are a ...
0
votes
0answers
4 views

What is the parametrization of the set of points in $\mathbb{R}^2$ with $L^p$-(semi)norm $1$ for any $p$?

I'm looking for a curve $t_p: [0,L] \rightarrow \mathbb{R}^2$ that describes the set $T_p = \{ (x,y) \in \mathbb{R}^2 : |x|^p + |y|^p = 1\}.$
-1
votes
0answers
9 views

Mean and Gaussian curvature - normalization to interval $[0, 1]$

I can compute curvature of the $2.5D$ surface. Problem is, I need the results scaled in interval $[0,1]$ (or $[-1,1]$). Is it possible to compute this directly or I need to compute all curvatures of ...
-2
votes
1answer
22 views

Evaluate a double integral over a region $R$

Let $R$ be the refion enclosed by $x^2+4y^2\ge 1$ and $x^2+y^2\le 1$. Calculate $$\iint_R \lvert xy\rvert\,dxdy$$ I think the answer is $0$ because the area in the positive quadrants cuts the ...
0
votes
0answers
15 views

Is this a special probability distribution?

Does the distribution function: $\frac{1}{\theta}e^\frac{-y}{\theta} $ Have a special name? If not, how can I find the variance? I keep running into a dead end when I try.
0
votes
0answers
11 views

Finding the autocorrelation of $X(t)$ and $Y(t)$ from the autocorrelation and pseudocorrelation of $Z(t) = X(t)+i Y(t)$

Consider $Z(t) = X(t) + iY(t)$, $i$ being imaginary. Knowing that $$ r(t_1,t_2) = e^{i(t_1 - t_2) - (t_1+t_2)^{2}} \quad\quad\text{and}\quad\quad \mathrm{pseudo}-r(t_1, t_2) = 0 $$ how can one ...
0
votes
0answers
9 views

Representation of left ideals of the matrix ring [duplicate]

We know that $J$ is an ideal of the full matrix ring $S=M_n(R)$ if and only if $J$ is the ring of all $n\times n$ matrices over $I$ for some ideal $I$ of the ring $R$ with identity. Now, my question ...
1
vote
0answers
16 views

Normal Approximation to the Binomial

I need to solve this problem using Normal Approximation to the Binomial Distribution to check if the value is similar to the one that I found using the Binomial distribution. Question: What is the ...
1
vote
0answers
13 views

“Chemists triple point” in percolation theory

This is a vague question asking about the existence of a mathematical object, instead of properties of a well defined one. I am sorry if this is not the correct forum. I know if you have a random ...
1
vote
3answers
38 views

The line $2x-y=5$ turns about a point…

The line $2x-y=5$ turns about a point on it, whose ordinate and abscissae are equal, through an angle of $45°$, in anti clockwise direction. Find the equation of line in the new position. My attempt ...
2
votes
3answers
60 views

Dealing with integrals of the form $\int{e^x(f(x)+f'(x))}dx$

Integrals of the form $$\int{e^x(f(x)+f'(x))}dx$$ are very common. And I have seen this form appearing in several exam papers.But the problem I face with this particular type of integral is finding ...
-1
votes
1answer
32 views

Image of a linear transformation

Let $T : V \to W$ be a linear transformation. If $A$ is a subspace of $V$, show that its image, $$ T(A) = \left\{ T(x) \in W \mid x \in A \right\}, $$ is a subspace of $W$. I have no idea how ...
0
votes
0answers
11 views

Analytical solution of parabolic equation

What is the analytical expression of the solution of the following 1D parabolic equation ? $$\dfrac{∂f}{∂t}+V\dfrac{∂f}{∂x}=abe^{at}$$ where $t, x$ – independent variable (time and space position ...
0
votes
0answers
5 views

Getting the shape (or bounding tail estimates) of a probability distribution from its generating function

My original motivation was the question: in the digits of $\pi$, where do we first encounter $10$ consecutive identical digits? (The answer is that there are $10$ consecutive $6$s at position ...
-2
votes
1answer
12 views

To find dimension of $N(A) \cap R(B)$ over R

To find dimension of $N(A) \cap R(B)$ over R A = $\begin{bmatrix} 1 & 2 & 0 \\ -1 & 5 & 2 \end{bmatrix}$ B=$\begin{bmatrix} 1 & 2 \\ -1 & 0 \\ 3 & 1 \end{bmatrix}$ i ...
0
votes
1answer
20 views

Show the convergence of a sequence using a convergent sequence in a metric space

Let $X$ a metric space with metric $d$ and let $P \subset X$ not empty. Let $f :X \to \mathbb{R}$ a function and define $f(x) = \inf\{d(x,y): y \in P\}$. Let $(x_n)_{n \in \mathbb{N}}$ a convergent ...
3
votes
1answer
10 views

Linear independent set of functionals makes certain map surjective

Let $V$ be a finite $n$-dimensional vector space over a field $K$ and $\{\lambda_{1},\ldots, \lambda_{n}\}$ be a linearly independent set of functionals Show that the linear map $$\Lambda:V\to K^n$$ ...
2
votes
1answer
19 views

The Levi-Civita Connection on the Hyperbolic Plane

In this question here, I asked about computing the Levi-Civita connection matrix on the Hyperbolic Plane, defined as $\mathbb{H}^2=\{x+iy\in\mathbb{C}\ \vert\ y>0\}$ equipped with the metric $g = ...
0
votes
1answer
25 views

Way to calculate cumulative sum of a sequence m times without summing all elements m times.

Suppose we have a sequence 2 1 3 1 Now , I want calculate it's cumulative sum m times and determine the element at position x in the sequence. Lets's say I want to perform cumulative sum operations ...
-1
votes
2answers
30 views

A question an normal to the circle

The equation of the normal to the circle $(x-1)^2+(y-2)^2=4$ which is at a maximum distance from the point $(-1,-1)$ is (A) $x+2y=5$ (B) $2x+y=4$ (C) $3x+2y=7$ (D) $2x+3y=8$ Since its a ...
1
vote
4answers
59 views

How Many Times The Clock Hands Make an angle Theta?

You might have encountered such a question where $\theta=90^o$ but this a little different and is causing a little problem. I approached this problem in the following matter and solved for $\theta= ...
0
votes
1answer
30 views

prove properties of upper and lower integrals [on hold]

How to prove these properties of upper integrals? I am trying to use limit rules and/or a sequential approach but cannot figure it out
0
votes
0answers
26 views

Finding closed form expression for a multiple sum.

Let $n_1$, $n_2$ and $m$ be non-negative integers and let $\theta_1$ and $\theta_2$ be real numbers subject to $\frac{\theta_1}{\theta_2} = 1+m$. We consider a following multiple sum: \begin{eqnarray} ...
-1
votes
1answer
31 views

Doubt about associative property of a group (Abstract Algebra).

I am new to abstract algebra and I have a doubt about the associative property. Suppose a set is given, such as $G=\{0,1,2,3,4\}$ under $\pmod{5}$ addition operation and we have to check whether $G$ ...
0
votes
0answers
16 views

$G$ be a finite group and $f \in Aut (G)$ such that $f^3$ is identity and $f$ has unique fixed point , then any $p$-Sylow subgroup is normal?

Let $G$ be a finite group and $f \in Aut (G)$ such that $f^3$ is identity and $f(x)=x \implies x=e$ ; then is it true that for every prime $p$ dividing $|G|$ , there is exactly one $p$-Sylow subgroup ...
3
votes
4answers
60 views

Solve the equation $7t+[2t] =52 $ ,where $[x]$ denotes the floor function for $x$.

Solve the equation $7t+[2t] =52 $ ,where $[x]$ denotes the floor function for $x$. My effort Using the fact that for any number $x$ we have that $x=[x]+\{x\}$ (where $\{x\}$ is the fractional ...
1
vote
1answer
21 views

Existence of a sequence related to the convergence of a series

Trying to prove an exercise, I arrived at the following question: Let $\{a_j\}\subseteq\mathbb{R}^+$ be a monotone increasing sequence with limit $+\infty$. Suppose that there is a $D>0$ such that ...
1
vote
1answer
27 views

Show that function has maximum in a given interval

This is a question from my exam in Calculus I: Problem 4 Prove or disprove: [...] a) The function $f(x) = \left(\sin(x) + \sqrt{\log(1+x^2)}\right)^3 e^{\cos(x) - 1}$ has a maximum ...
0
votes
3answers
93 views

When $\operatorname{im}(A) = \ker(A)$

Consider the following true/ false qustion: There exists a $2 \times 2$ matrix $A$ such that $\operatorname{im}(A) = \ker(A)$. I know that this is true, but I am not sure how to show it. If $A$ ...
0
votes
1answer
20 views

Prove $(H \times 1)(1\times K)= H\times K$ where $H,K$ are groups.

Prove $(H \times 1)(1\times K)= H\times K$ where $H,K$ are groups. Suppose $x=ab,a\in H\times 1,b\in 1\times K$ Then $x=(h,1)(1,k)$ where $h\in H,k\in K$ Hence $x=(h,k)\in H\times K$ Let ...
1
vote
0answers
13 views

Prove that if $0\le p_n \lt 1$ and $S:=\sum p_n \lt 1$, then $\Pi (1-p_n) \ge 1-S$. [duplicate]

Prove that if $0\le p_n \lt 1$ and $S:=\sum p_n \lt 1$, then $\Pi (1-p_n) \ge 1-S$. I'm having real trouble proving this inequality. I'd greatly appreciate any help.
2
votes
0answers
16 views

Find $f \in C^1(U,\mathbb{R})$ which satisfiyes the following differential-form

I am quiet new to differential forms so I am not sure if my solution to the following problem is correct. The Problem: given: $U \subset \mathbb{R}^3$ and $U$ is an open set $$ V \in C^1(U, ...
0
votes
0answers
28 views

Why is the unit disc not a topological surface? [duplicate]

I am trying to prove that the unit disc $D^2$ is not a topological manifold. Clearly it is Hausdorff and second countable, so I think I should show that it is not locally Euclidean. The following is ...
-1
votes
0answers
26 views

Prove that every simple function is Riemann integrable.

Prove that every simple function on $[a,b]$ is Riemann-integrable. I understand why this is true, but I am not sure how to go about proving it. Does it have to do with the upper and lower integrals ...
0
votes
0answers
16 views

Proof that $\operatorname{End}(V) \rightarrow Gl_n(K), F \mapsto M_A^A(F)$ denotes a group-isomoprhism.

Definition: Let $A$ be a Basis of $V$, $V$ a $K$ - Vectorspace. $M_A^A(F) = \Phi_A \circ F \circ \Phi_A^{-1} $, where $\Phi_A$ denotes the following function: $n := \dim V, \{x_1,\ldots,x_n\} = A$ ...
0
votes
0answers
12 views

Solving $f(x) \leq 10 f(kx) + 10kg(x)$ for $f, g$ nonnegative on $(0, 1]$

Suppose we are given two nonnegative functions $f$ and $g$ on $(0,1]$ that satisfy $f(x) \leq x^{-1/2}$ and $$f(x) \leq 10 f(kx) + 10kg(x)$$ for all $k$ sufficiently large. Is it possible to reduce ...
1
vote
0answers
9 views

The induced map $\phi: \mathbb{C}^n \to \mathbb{C}^m$ in the construction of toric varieties

Let $\Sigma(1)$ denote the set of one dimensional cones in a fan $\Sigma$. The corresponding vectors in the lattice are denoted $(v_1, \ldots, v_n)$ and to each $v_i$ we associate a homogeneous ...
-1
votes
2answers
42 views

To check differentiability of multivariable function at origin

To check function is differentiable or not $$f(x,y) = \left\lbrace\begin{array}{cr} \frac{xy}{|x|} ,& x \neq 0\\ 0 , & x=0 \end{array}\right. $$ I think due to modulus ...
1
vote
1answer
23 views

Anticommuting matrices and their eigenvalues

Let $A,B\in \mathcal{M}_n(\mathbb{C})$. It is known that if $AB=BA$ and $\lambda_1, \lambda_2, \dots, \lambda_n $ are the eigenvalues of $A$ and $\beta_1, \beta_2, \dots, \beta_n$ are the ...
1
vote
0answers
15 views

Property of a sequence being an enumeration of the rationals.

Let $(r_n)$ be an enumeration of the rationals and $x\in\mathbb{R}$. Is it possible to find out whether the set $\left\{n\in\mathbb{N}:\left|x-r_n\right|<\frac{1}{2^n}\right\}$ is finite or ...
0
votes
0answers
12 views

Exercise: ODE with specified initial condition.

a)Consider the following ODE: $y'=4t \sqrt y$ $y(0)=1$ and find the unique solution. b)What is required for a numerical method to solve the problem exactly? c)Now consider the modified problem: ...
0
votes
1answer
32 views

A metric between functions on $\mathbb{R}^2$

I want to measure the distance between functions $f$ and $g$ (not necessarily continuous) on a bounded subset $M\subset\mathbb{R}^2$. I assume $f$ and $g$ are locally integrable and bounded on $M$. ...
0
votes
1answer
22 views

why are the Bisection and Newton Method for finding roots complementary to each other?

my lecture note states that the bisection and newton method for finding roots are most of the time complementary to each other but I can not figure out why. I have basic understanding of both of the ...
3
votes
0answers
21 views

Problem on Stein and Shakarchi's Real Analysis Chapter 5 regarding characteristic polynomial and elliptic

I was wondering if someone can help me with a problem in Stein and Shakarchi's Real Analysis Chapter 5. Consider the linear partial differential operator $$L = \sum_{|\alpha|\leq n} a_\alpha ...
1
vote
0answers
6 views

Centered Poisson, Scaled Poisson, Transformed Poisson

Given $y_1,y_2,\ldots,y_N$ with $y\sim \operatorname{Poisson}(\lambda)$. The question is, what is the distribution of $y_i-\bar{y}$ and $\frac{y_i-\bar{y}}{\bar{y}}$, where $\bar{y}=\sum_1^N y_i/N$. ...
0
votes
0answers
23 views

Categorically deducding measurability of sections

Two lemmas which are often proved in elementary measure theory courses are that sections of measurable sets are measurable, and sections of measurable functions are measurable. Note $E_x= \left\{y\in ...

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