0
votes
2answers
132 views

Calculate angle in triangle having 2 points and two lines

I have 2 points $B$ and $P$ and need to calculate angle $\alpha$ (maybe also I will need point $C$ and $E$) How can I do this. I know that I can calculate point $D$ it's $(\frac{1}{2}(x_P-x_B), ...
2
votes
2answers
37 views

Simplify this expression that came from integration

I was doing a calculation and arrived at a term $\left[P_{l-1}(\cos(\theta)) -P_{l+1}(\cos(\theta))\right]_{0}^{\pi}$(So this is the result of an integration). Does anybody of you know how to simplify ...
3
votes
1answer
112 views

Show that map is conformal

I want to show that the map $\phi(r,\theta) = r^\lambda (\cos(\lambda \theta), \sin(\lambda \theta))$, where $\lambda \in \mathbb{C}$, is conformal on the slit plane $\{(r,\theta)| r > 0, -\pi < ...
2
votes
1answer
62 views

How many combination exist in this situation?

There are $h$ groups of different elements each has a different size: $$ \begin{matrix} \text{Group 1}: ~~~~a_{11}& a_{12} & ... & a_{1k_1} & \\ \text{Group 2}:~~~~a_{21}& a_{22} ...
2
votes
3answers
72 views

Notation for definition and equivalence

I would like some clarification about the usage/meaning of $:=$ and $\equiv$. I have been using $A := B$ to denote "Let $A$ be defined as $B$." This is akin to assignment ...
4
votes
1answer
208 views

When are the following multiple improper integrals convergent?

This is a question from a past exam. For which $p, q\in \mathbb R$ do the following improper integrals converge? ...
6
votes
0answers
75 views

Generalization of Kummer isomorphism?

Let $p$ be a prime number and denote by $\mathbb{F}_p(1)$ the one dimensional vector space over $\mathbb{F}_p$ endowed with an action of $G:=Gal(\bar{\mathbb{Q}}_p / \mathbb{Q}_p)$ via the mod $p$ ...
0
votes
1answer
33 views

Cumulative Normal Distribution.

Let $X_1,\ldots,X_n$ be a random sample from $f(X;\theta)=\phi_{\theta,25}$, that is, $X_1,\ldots,X_n$ be normally distributed with mean $\theta$ and variance $25$. I am not understanding how ...
1
vote
4answers
122 views

Simplifying compound fraction: $\frac{3}{\sqrt{5}/5}$

I'm trying to simplify the following: $$\frac{3}{\ \frac{\sqrt{5}}{5} \ }.$$ I know it is a very simple question but I am stuck. I followed through some instructions on Wolfram which suggests that I ...
3
votes
1answer
55 views

A question on divisible groups

Let $p$ be a prime and $H=\prod_{n=1}^{\infty}\mathbb Z(p^{n})$ ($\mathbb Z(p^{n})$ is the finite cyclic group of order $p^{n}$). Is $H/t(H)$ divisible ($t(H)$ denotes the maximal torsion subgroup of ...
4
votes
1answer
41 views

Finding appropriate function

In order to obtain an estimate I'm wondering if there is a positive function $f:\mathbb{R}\to \mathbb{R}_+$ such that $f(x) < |x|$ for $|x|$ large enough (say $|x|\ge C_0>0$) so that the ...
0
votes
1answer
62 views

$E+F=E\oplus F \Leftarrow \bigcap$ of their bases $=∅$

I'm trying to understand this theorem: I will traduce it literally from my lecture notes: Given n≥1 subspaces $E_1,E_2,...,E_n$ of a vector space V and considering the subspace ...
1
vote
2answers
123 views

Calculate integrals $\int_{{\pi \over 4}}^{\arctan {1 \over 2}} {{{\sqrt {{\mathop{\rm cos}\nolimits u} } } \over {\sqrt {\sin u} }}du} $

This is my homework. And it's really a challenge for me. Can anyone solve this. $$\large\int\limits_{{\pi \over 4}}^{\arctan {1 \over 2}} {{\sqrt{\cos u} \over {\sqrt {\sin u} }}du} $$
0
votes
1answer
82 views

inequality by taking cases

The setup is the following: We have a sequence of r.v. $X_n$ and $X$. We define $Y_n:=\frac{1}{2}(X_n+X)$. Moreover $F$ is a strict convex Function such that $$ \lim\sup P[F(Y_n)\le ...
9
votes
2answers
608 views

Convolution intuition: clarifying Terence Tao's “blurring”/“fuzz” interpretation

On this math.MO post, "What is convolution intuitively?", Terence Tao's answer (in the case where one function is a bump function) involves "blurring" and "fuzz." Could someone clarify his ...
1
vote
2answers
65 views

Sequence problem, find root

the equation $x^3-5x+1=0$ has a root in $(0,1)$. Using a proper sequence for which $$|a(n+1)-a(n)|\le c|(a(n)-a(n-1)|$$ with $0<c<1$ , find the root with an approximation of $10^{-4}$.
2
votes
1answer
365 views

Is there formula for this squared geometric (?) progression?

Is there non-recursive formula for the following sequence: $$a_1=\frac12,$$ $$a_n=\frac12a_{n-1}^2+\frac12$$ If there is, how do you suggest I can determine it?
1
vote
1answer
401 views

Asymptotics of maxima of i.i.d. chi-square random variables

How to find the following: Let $X_1$, $X_2$, $X_3$,..., $X_n$, be i.i.d with chi-square distribution with one-degree of freedom. Find $a_n$ and $b_n$ such that $ a_n(\max_i X_i - b_n)$ converges in ...
1
vote
1answer
65 views

Solving logarithmic equation

I'm having trouble solving this equation. I know there is a solution as my graphics calculator can solve it, but I want to see the steps on how to get the answer. The mathematical equation is: ...
4
votes
1answer
116 views

Convexity of Exponential Composite Function

$f:\mathbb{R}_{+}^M\rightarrow\mathbb{R}_+$ is a convex analytic function. For $\mathbf{x}\in\mathbb{R}_{+}^M$ and $y\in\mathbb{R}_{+}$, consider the function ...
8
votes
1answer
405 views

reference for operator algebra

I am taking a course on operator algebra this semester. My instructor has suggested a reference "Kadinson and Ringrose." Are there any other good/standard references for this subject that I can look ...
2
votes
0answers
87 views

Systems of parameters for a $K$-algebra

I don't know how to solve the next problem: If we have two systems of parameters, $\{x_1,\ldots,x_n\}$ and $\{y_1,\ldots,y_n\}$ for a finitely generated $K$-algebra $A$ which is also an integral ...
10
votes
1answer
319 views

Tarski Monster group with prime $3$ or $5$

Is there any Tarski Monster Groups for the prime $3$ or $5$? I know that there is no Tarski Monster Groups with prime $2$, but I don't know for the prime $3$ and $5$.
0
votes
1answer
3k views

relationship between circumference and revolution

i would like to clarify two things by this problem:first what is relationship between circumference and revolution and also revolution and distant traveled by round object.let us consider following ...
2
votes
1answer
126 views

Confused by Kolmogorov's Strong law and Borel-Cantelli lemma

Notation: Let $X_n$ be i.i.d. random variables with mean $\mathrm{E}[X_n]=\mu$ and variance $\mathrm{E}[(X_n-\mu)^2]=\sigma^2$. Denote the sample average as $\bar{x}_m = \frac{1}{m}\sum_{n=1}^m X_n$. ...
2
votes
2answers
257 views

Theory set problem, determining the min and max number of elements from $(B \cup A) \bigtriangleup (C \cap A)$

Given that $$ |A| = 5 \\ |B| = 6 \\ |C| = 7 \\ |\Omega| = 10 \\ A \subseteq (B \cup C) $$ Determine the min and max numbers of elements from $$(B \cup A) \bigtriangleup (C \cap A)$$ I tried to solve ...
2
votes
1answer
83 views

Linear map induced by bilinear maps

Suppose $f:X\times Y\rightarrow Z$ and $g:X\times Y\rightarrow W$ are bilinear maps in the category of vector spaces (say, real). Define the null space $N_1(f) :=\{x \in X: f(x,y)=0 \ \forall\, y\in ...
6
votes
1answer
106 views

Ackermann function and $f_\omega$

The Wikipedia page of Ackermann function states that Ackermann function is "roughly comparable" to $f_\omega$ in fast-growing hierarchy. Is there some standard way to make the "roughly comparable" ...
1
vote
2answers
169 views

Rational Solution of a System of Linear Equations [duplicate]

I am having a little trouble with this problem - Let $A$ be a $m\times n$ matrix and $v$ be a $n\times 1$ matrix, both of which only has rational entries. It is known that the equation $Ax=v$ has ...
2
votes
2answers
574 views

Why is Lebesgue integral first defined for step function and then for larger classes of functions?

Most books on Lebesgue integration define the concept first for step functions (and simple functions) and later on extend the definition to Lebesgue measurable functions. Why is this approach ...
1
vote
1answer
589 views

Metric of the flat torus

I am studying the flat torus $T^n=\mathbb{R}^n/\mathbb{Z}^n$. I am interested in the metric and the connection used. Unfortunately, in the books I am reading those things aren't defined. Does anyone ...
3
votes
2answers
538 views

Uncountable basis and separability

We know that a Hilbert space is separable if and only if it has a countable orthonormal basis. What I want to ask is If a Hilbert space has an uncountable orthonormal basis, does it mean that it is ...
0
votes
0answers
467 views

Kolmogorov continuity criterion

Kolmogorov's continuity criterion states that, suppose $X=\{X_t,0\leq t\leq1\}$ is a real-valued stochastic process, and if there are constants $a,b,K>0$ such that for all $s,t\in[0,1]$ we have ...
5
votes
3answers
464 views

How to be good at angles and trigonometry

I am Computer Science Engineer and loved algebra side of Mathematics. But when it comes to trigonometry and angles and triangles, I do not understand anything since college time. And till now also ...
3
votes
0answers
120 views

Regular schemes and base change

Suppose $X$ and $Y$ are regular schemes (i.e. all local rings are regular) and flat over some base $S$. Assume further that $Y$ has relative dimension $0$ over $S$. Does it follow that $X \times_S Y$ ...
6
votes
1answer
336 views

To prove a vector bundle $\xi$ is orientable $\iff w_1(\xi)=0$

I went about it this way: $\xi$ an $n$-rank vector bundle over a topological space $M$ is orientable. $\iff$ its top exterior power $\bigwedge^n\xi$ is trivial $\iff$ the 1-frame bundle ...
1
vote
1answer
146 views

About Mean Value Property of Harmonic Function

I know the question may seem foolish to you but I am not quite sure how to show it in a decent way. My problem is to show that for bounded Borel measurable $f:\mathbb{D}^2\to\mathbb{R}$, (D1) is ...
1
vote
2answers
286 views

Can there be a perfect square whose digits consist of exactly 4 ones, 4 twos and 4 zeros in any order? [duplicate]

Is there any perfect square whose digits consist of exactly 4 ones, 4 twos and 4 zeros in any order?
1
vote
1answer
134 views

Induced orientation on boundary

I am trying to understand how the orientation is induced on the boundary of a differentiable manifold with boundary. Here is what I have worked out so far: Let $M$ be a differentiable manifold and ...
2
votes
1answer
445 views

If $P$ is a regular transition probability matrix then $P^{n^2}$ has no zero element

A transition probability matrix $P\in M_{n\times n}$ is regular if for some $k$ the matrix $P^k$ has all of its elements strictly positive. I read that this can be tested by using the following ...
-1
votes
1answer
219 views

Conditions for $v \otimes v$ to be positive semidefinite for complex $v$

I have a complex-symmetric matrix (in the sense $A=A^{T}$ not $A=A^{H}$), which is required to be positive semi-definite in the following sense (sometimes referred to as positive real): $ \Re(x^{*} A ...
0
votes
1answer
62 views

Complex matrix with a single Eigenvalue

Im not sure about a question, and need your help. Is a complex matrix with a single Eigenvalue necessarily diagonalizable? I'm thinking that the answer is true because the opposite case does happen ...
0
votes
2answers
78 views

How to use binomial theorem

How to use binomial theorem,From (1) to get (2)? $$\begin{align*}\left(1+\frac{1}{n}\right)^n\tag{1}\end{align*}$$ ...
0
votes
1answer
57 views

Derive a formula to solve a specific task

I have a specific problem. I have 8 different variables a, b, c, d, e, f, g, h. Each of these variables has a score out of 5, where 1 is bad and 5 is good. So a max score of 45 and a min of 0. Of ...
5
votes
2answers
468 views

Continuous extension of a function

Can anybody help me with this problem? Justify whether the following statement is true or false: Every continuous function on $\Bbb Q\cap [0,1]$ can be extended to a continuous function on ...
1
vote
0answers
119 views

Bound on the angle between a vector and a subspace

Suppose you have three complex vectors $x_1$, $x_2$, and $x_3$. Define $a = \angle(x_1,x_2)$, $b = \angle(x_1,x_3)$. My question is about $c = \angle(x_1, span(x_2,x_3))$, the angle between the vector ...
0
votes
1answer
289 views

Isn't $(0)$ a prime ideal in a field?

I have read in multiple places that a field $K$ has a Krull dimension of $0$. How is this true? Isn't $(0)\subset K$ a prime ideal in $K$? Obviously $K$ is an integral domain. Thanks in advance!
1
vote
1answer
52 views

Question on boundedness

Page number 479 in partial differential equation by Evans book how to say that the derivative of I is bounded on bounded sets
1
vote
1answer
616 views

find correlation coefficient of $f(x,y)=2$ for $0<x \leq y<1$

Find the correlation coefficient for the random variables $X$ and $Y$ having joint density $f(x,y)=2$ for $0 < x \leq y<1$. Seem like a simple problem but I'm stuck. Since $Corr(X,Y) = ...
2
votes
0answers
128 views

Probability that none has to wait for change

Suppose $2n$ customers stand in line at a box office, $n$ with $5$-dollar bills and $n$ with $10$-dollar bills. Suppose each ticket costs $5$ dollars, and the box office has no money initially. What ...

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