All Questions

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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 $X$?
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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 ...
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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 ...
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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 ...
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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 ...
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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 $T_p = \{ (x,y) \in \mathbb{R}^2 : |x|^p + |y|^p = 1\}.$
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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 ...
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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 ...
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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.
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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 ...
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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 ...
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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 ...
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“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 ...
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 ...
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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 ...
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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $$\Lambda:V\to K^n$$ ...
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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
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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} ...
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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 $G$ ...
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$G$ 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 ...
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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 ...
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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 ...
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 ...
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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 $A$ ...
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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 ...
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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.
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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 $$V \in C^1(U, ... 0answers 28 views Why is the unit disc not a topological surface? [duplicate] I am trying to prove that the unit disc D^2 is not a topological manifold. Clearly it is Hausdorff and second countable, so I think I should show that it is not locally Euclidean. The following is ... 0answers 26 views Prove that every simple function is Riemann integrable. Prove that every simple function on [a,b] is Riemann-integrable. I understand why this is true, but I am not sure how to go about proving it. Does it have to do with the upper and lower integrals ... 0answers 16 views Proof that \operatorname{End}(V) \rightarrow Gl_n(K), F \mapsto M_A^A(F) denotes a group-isomoprhism. Definition: Let A be a Basis of V, V a K - Vectorspace. M_A^A(F) = \Phi_A \circ F \circ \Phi_A^{-1} , where \Phi_A denotes the following function: n := \dim V, \{x_1,\ldots,x_n\} = A ... 0answers 12 views Solving f(x) \leq 10 f(kx) + 10kg(x) for f, g nonnegative on (0, 1] Suppose we are given two nonnegative functions f and g on (0,1] that satisfy f(x) \leq x^{-1/2} and$$f(x) \leq 10 f(kx) + 10kg(x)$$for all k sufficiently large. Is it possible to reduce ... 0answers 9 views The induced map \phi: \mathbb{C}^n \to \mathbb{C}^m in the construction of toric varieties Let \Sigma(1) denote the set of one dimensional cones in a fan \Sigma. The corresponding vectors in the lattice are denoted (v_1, \ldots, v_n) and to each v_i we associate a homogeneous ... 2answers 42 views To check differentiability of multivariable function at origin To check function is differentiable or not$$f(x,y) = \left\lbrace\begin{array}{cr} \frac{xy}{|x|} ,& x \neq 0\\ 0 , & x=0 \end{array}\right. $$I think due to modulus ... 1answer 23 views Anticommuting matrices and their eigenvalues Let A,B\in \mathcal{M}_n(\mathbb{C}). It is known that if AB=BA and \lambda_1, \lambda_2, \dots, \lambda_n  are the eigenvalues of A and \beta_1, \beta_2, \dots, \beta_n are the ... 0answers 15 views Property of a sequence being an enumeration of the rationals. Let (r_n) be an enumeration of the rationals and x\in\mathbb{R}. Is it possible to find out whether the set \left\{n\in\mathbb{N}:\left|x-r_n\right|<\frac{1}{2^n}\right\} is finite or ... 0answers 12 views Exercise: ODE with specified initial condition. a)Consider the following ODE: y'=4t \sqrt y y(0)=1 and find the unique solution. b)What is required for a numerical method to solve the problem exactly? c)Now consider the modified problem: ... 1answer 32 views A metric between functions on \mathbb{R}^2 I want to measure the distance between functions f and g (not necessarily continuous) on a bounded subset M\subset\mathbb{R}^2. I assume f and g are locally integrable and bounded on M. ... 1answer 22 views why are the Bisection and Newton Method for finding roots complementary to each other? my lecture note states that the bisection and newton method for finding roots are most of the time complementary to each other but I can not figure out why. I have basic understanding of both of the ... 0answers 21 views Problem on Stein and Shakarchi's Real Analysis Chapter 5 regarding characteristic polynomial and elliptic I was wondering if someone can help me with a problem in Stein and Shakarchi's Real Analysis Chapter 5. Consider the linear partial differential operator$$L = \sum_{|\alpha|\leq n} a_\alpha ...
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 $\bar{y}=\sum_1^N y_i/N$. ...