1
vote
0answers
62 views

Unramified extension of a imaginary quadratic field.

Let $K$ be an imaginary quadratic number field over $\Bbb{Q}$. Let $K(\sqrt{a})$ be an extension with $a$ an integer. Let $d$ be discriminant of field $K$. Then how to show that $K(\sqrt{a})$ is ...
1
vote
1answer
40 views

Complex solutions to trig functions

We consider: $$2\sin{z} + \cos{z} = i\sin{z}$$ Where $z \in \mathbb{C}$. We are to prove the solutions are given by $$z = \left(n\pi - \frac{\pi}{8}\right) - \frac{1}{4}i\ln{2}$$ For $n \in ...
2
votes
0answers
106 views

Riemann's explicit formula for $\pi(x)$

Riemann's explicit formula $J(x)=\mathrm{Li}(x)-\sum_{\Im\varrho>0}\left(\mathrm{Li}(x^\varrho)+\mathrm{Li}(x^{1-\varrho})\right)+\int_x^\infty\frac{\mathrm{d}t}{t(t^2-1)\log t}-\log2,$ where ...
1
vote
1answer
46 views

Shortest distance between point and line (vectors)

If I am given position vector $a$ and a line $r = βb$, how do I prove that the shortest distance is$ |a- ((a.b)/b^2 )b|$? I understand that somewhere the dot product is = $0$ but am unable to prove ...
4
votes
1answer
53 views

primitive element for a Hilbert class field

I am trying to solve the following problem which I found in a book. Find a primitive element for the Hilbert class field for $\Bbb{Q}(\sqrt{-17})$? Any hints..
0
votes
1answer
279 views

(Near) complete euclidean geometry theorems and postulates list

I‘ve been looking for a euclidean geometry book filled with as many theorems and axioms as possible, even better if it‘s as condensed as possible (say, proofs given separately in another book, or not ...
1
vote
1answer
11 views

balls have empty boundary with regard to the $p$-adic norm

Let $p$ be prime, $a\in\mathbb{Q}$ and $r\geq0$. How can I show that the closed ball $D(a,r)$ in $(\mathbb{Q},|\cdot|_p)$ must have an empty boundary (with regard to the topology induced by the ...
1
vote
1answer
141 views

Show that a class of holomorphic functions is a normal family.

Let $F$ be the class of all holomorphic functions on the unit disc $D$ satisfying $$\int_0^{2\pi}|f(re^{i\theta})|d\theta\le1$$ for each $r$ in $(0,1)$. Prove that $F$ is a normal family in the unit ...
0
votes
2answers
55 views

Onesided limit of integral

Let a function $f$ from 0 to $\infty$ be continuous and define $F$ by $$ F(x) = \frac{1}{x} \int_0^x f(t)dt $$ How can I show that $\lim_{x\to 0^+}F(x) = f(0)$
1
vote
1answer
31 views

Determining if a function is real anaytic at the point $a$?

Is there a method, other then using refer to the Taylor series to determine if a real function is analytic at the point $a$. If so please, if possible, could you give a source.
6
votes
2answers
269 views

“Bizarre” continued fraction of Ramanujan! But where's the proof?

$$\frac{e^\pi-1}{e^\pi+1}=\cfrac\pi{2+\cfrac{\pi^2}{6+\cfrac{\pi^2}{10+\cfrac{\pi^2}{14+...}}}}$$ "Bizarre" continued fraction of Ramanujan! But where's the proof? i have no training in continued ...
1
vote
1answer
25 views

Circle function solution in integers

I there possible to get all solustions of circle function as integer pairs? I have problem in image proccessing. I need get every pixel on circle from fixed center and and fixed radius. Pixels are ...
5
votes
1answer
196 views

What is the algebraic tangent cone really?

Let $A$ be a (commutative unital) ring, let $\mathfrak{a} \subseteq A$ be an ideal, and let $B = A / \mathfrak{a}$. Then we have a descending filtration $$\cdots \subseteq \mathfrak{a}^3 \subseteq ...
3
votes
3answers
85 views

Which quadrant is the “first quadrant”?

In the coordinate plane split into four quadrants by the $x$- and $y$-axes, I learned (educated in a public school in the U.S.) that the "first quadrant" was the one with both $x$ and $y$ positive, ...
2
votes
2answers
439 views

confusion about cosets and quotient space

The set of cosets $V/U = \{ v+U: u \in V \}$ with operations $$(v+U) + (w+U) = v+w+U$$ $$a(v+U) = av+U$$ (which is well defined) is a vector space called Quotient Space. I am having a difficult ...
0
votes
1answer
55 views

A question on Mortiz Pasch works in foundation of mathematics

I am reading "Introduction to the Foundation of Mathematics" ,by R L.Wilder(2nd Ed.), where Mortiz Pasch's works are described in paragraph 1.5. There is a quotation in that paragraph- " For if,on ...
1
vote
1answer
30 views

a question about normal matrices

Let $A$ be a normal matrix and $\lambda$ a scalar. Show that $A-\lambda I$ is also a normal matrix. $A$ is a normal matrix then there is a unitary diagonalization of $A$ over $\mathbb{C}$. ...
0
votes
1answer
87 views

Projective coordinates for point at infinity on elliptic curve

What is the unique characteristic of the projective coordinates of a point at infinity? I am specifically looking for a characteristic on (short) weierstrass curves. I know that the point at infinity ...
0
votes
2answers
55 views

Statistical Mechanics of interacting Particles. Quantized Fields. Solving Integral?

Hi everyone How we can analytically without using a software solve below integral . Chapter 11 of Pathria (edition 1). and x is dimensionless.
2
votes
2answers
734 views

The integral of dv

I'm solving the homogeneous differential equation $$\frac{dy}{dx} = \dfrac{x + y}{x}$$ After substituting $y = vx$ and $\dfrac{dy}{dx} = v + x\cdot \dfrac{dv}{dx}$ (Product rule of $y = vx$), we ...
1
vote
1answer
26 views

Show T being prime element in $ F_{2}(T) $

Show that $X^4+TX^2+T$ is irreducible in $ F_{2}(T) $ Using Eisenstein with T as a prime element this proof is simple. Can I proof that T is prime any easier than in the folowing: Theorem 1: K is ...
0
votes
4answers
103 views

The union of linear subspaces is not necessarily a subspace

$W$ and $U$ are subspaces of vector space $V$. Show that $U\cup W$ is not necessarily a subspace of $V$. I know that one counter-example is enough to show that. What example could fit here?
0
votes
2answers
61 views

Confusion in combinatorics

Question (1) The number of different ways in which $10$ telegrams can be distributed to 2 message boys is ____? The answer as per the book is $2^{10}$. But, I think answer should be $10^{2}$. If ...
0
votes
1answer
45 views

Let f(z) be analytic function on a disk D s.t. f(1)=1 then which is not correct

The answer of this question is (A) but not getting a proper way to solve it. Can max mod theorem be used to solve it.Is there any concrete method to solve such type of question. Please help....
1
vote
1answer
408 views

How to choose a convergence test for a given infinite series?

I have a general question when it comes to deciding if an infinite series is convergent or divergent. The tests im familiar with are ; Ratio test, Direct comparison test, Limit comparison test, Root ...
6
votes
0answers
151 views

estimating a particular analytic function on a bounded sector.

Let $f(z)$ be an analytic function on $C^+=\{\Re z>0\}$, and we have the following (weaker) estimates $$ |f(re^{i\theta},a)|\leq C (r\cos\theta)^{-n}, ...
3
votes
1answer
84 views

Integral inequality given the bounds for derivative

Let $f:[0,1]\rightarrow \mathbb{R}$ be a continuously differentiable function such that $$\int_0^1 f(x) \, dx=0$$ and $m \leq f'(x) \leq M$ on $(0,1)$. Prove that $$\frac{m}{12} \leq \int_0^1 xf(x) \, ...
1
vote
1answer
72 views

$IJ$ is the set of nilpotent elements

Let $R$ be a commutative ring with identity which is Noetherian. Let $V(A)$ denote the set of all prime ideals of $R$ containing the ideal $A$. Suppose that $V(0) = V(I) \cup V(J)$ and $V(I) \cap V(J) ...
1
vote
0answers
117 views

A maximization problem

I'm trying to find the maximum value of the function $f(x,y)=(ax+by)^p+x^p$ subject to the constraint $x^p+y^p=1$. Here, $a,b$ and $p$ are constants with $a,b>0$ and $p>1$, and $x,y>0$. I ...
0
votes
1answer
46 views

Convergence of series for specific values of $\lambda$.

Let $\lambda$ be a positive real number. For which values of $\lambda$ does the following series converge? $$\sum_{n=1}^\infty \frac{n^{-\lambda}}{1+\lambda^{-n}}$$ I can see that the series ...
0
votes
1answer
42 views

Can I define a measure $\mu$ only through the values it takes in the elements of the generator of its $\sigma$-algebra?

In other words, Let $X$ be a set and $\mathcal{B}$ a family of pairwise disjoint subsets of $X$ such that $\bigcup \mathcal{B} = X$. Denote by $\sigma(\mathcal{B})$ the smallest $\sigma$-algebra ...
-1
votes
1answer
57 views

The existence of concatenation functions in Godel Numbering?

I know that there are many schema of Gödel Numbering, and each has its own method of Concatenation, n★m. But is there a general proof that shows 'For every Gödel Numbering scheme there exists a ...
0
votes
1answer
19 views

Prove that $\sum_{x\in I}\delta_x$ diverges if $\delta_x>0$ and $I$ is more then countable.

Take $F:\Bbb R\to\Bbb R$ s.t. $F$ is continous from the right, not decreasing, $\lim_{x\to-\infty}F(x)=0$ and $\lim_{x\to+\infty}F(x)=1$. Call $I:=\{x\in\Bbb R\;:\;F\;\; \mbox{is not continous ...
3
votes
1answer
77 views

Showing irrationality of $\zeta(k)$ for some $k$ without calculating the value.

For $s\in (1,\infty)$ let $\zeta(s):=\sum_{n=1}^\infty \dfrac 1{n^s}$. Is there a way to show that $\zeta(2k)$ is irrational for some integer $k\geq 1$ without finding explicit formulae?
0
votes
3answers
73 views

Graphic of the probability distribution function : How does it works?

Here is the graphic of the probability distribution function for a random variable $X$. How can I find the distribution of $Y=-X$? By definition the distribution function for a random variable is ...
0
votes
1answer
42 views

Value of given series $\sum_{n=1}^{n=\infty}\frac{\cos{\frac{n\pi}{3}}}{n+1}$ and $\sum_{n=1}^{n=\infty}\frac{\sin{\frac{n\pi}{3}}}{n+1}$

I was doing a problem,after some calculation it all came down to the value of series $$\sum_{n=1}^{n=\infty}\frac{\cos{\frac{n\pi}{3}}}{n+1}$$ and $$ ...
13
votes
2answers
134 views

Six of a kind .

$$\begin{align} ...
0
votes
1answer
108 views

Given a set of 2D vertices, how to create a minimum-area polygon which contains all the given vertices?

Not sure whether this question belongs here or mathoverflow. You can assume that all the vertices are unique. The given vertices can be the vertices of the polygon, thus they do NOT have to be inside ...
0
votes
0answers
50 views

Show that $\frac{1}{x}$ is not integrable on $[0,1]$ - why we use fundamental theorem of calculus?

I found the followin proof that $f(x)=\frac{1}{x}$ is not Lebesgue integrable on $[0,1]$: $$\int_0^1\frac{1}{x}dx=\ln1-\ln 0=-\infty$$ But how can one employ here fundamental theorem of calculus?? ...
3
votes
1answer
113 views

How can I convert $A =$ {$c > 1$ : there exists a natural number $m$ … to mathematical notation?

In this post A = {c > 1 : there exists a natural number m, such that for every n > m, there is a prime between n and cn}. Now I don't want to discuss on the problem itself, but only on how to write ...
3
votes
4answers
287 views

How to solve $\tan x =\sin(x+45^{\circ})$?

How do I solve $\tan x = \sin(x +45^{\circ})$? This is how far I have come: $\sqrt{2}\sin x = \sin x\cdot\cos x + \cos^2 x$
5
votes
1answer
105 views

Algebraic closure of the rational inside a quotient of product of finite fields

I'm trying to solve the following exercise: " Consider the ring $R = \prod_{p} \mathbb{F}_p$, where $p$ runs over all prime numbers and $\mathbb{F}_p$ is a field with $p$ elements. Show that there ...
1
vote
0answers
110 views

What is the value of the integral $ \int_{-\infty}^{\infty}\frac{\sin(2x)}{x^3}dx$?

I tried to evaluate the integral $$ \int\limits_{-\infty}^{\infty}\frac{\sin(2x)}{x^3}dx$$ using residues but the answer comes out to be a negative value, $-2 \pi$, which seems strange. Any help on ...
0
votes
1answer
124 views

Uniform continuity 2 variable function

Let $f:\Bbb{R^2}\rightarrow\Bbb{R}$ and two real parameters $a,b$ such that $$f (\mathbf{x,y})= \begin{cases}a(x^2+y^2),&(x,y)\in B_{d_{2}}(0;2)\\ \frac{b}{\sqrt{x^2+y^2}},&(x,y)\in \Bbb{R^2}- ...
1
vote
1answer
335 views

Difference between local, global and maximum solutions of a differential equation?

Unfortunately I can't find a good answer in 3 different books I have, and I dont get the difference, nor could I find any explanation online for this. Can someone be so kind to explain it to me or ...
3
votes
1answer
323 views

Euler Vs. Diderot

I'm reading The Music of the Primes by Marcus Du Sautoy and I came across a page with the following excerpt about Leonhard Euler: "The role of the court mathematician is perfectly illustrated by a ...
2
votes
1answer
94 views

A question on prime density

Let A = {c > 1 : there exists a natural number m, such that for every n > m, there is a prime between n and cn}. Bertrand's postulate says that A contains 2. My question is : Is inf A = 1 ? If not, ...
2
votes
0answers
125 views

Least Impossible Subset Sum

Given a set A which contains natural numbers from 1 to N. Also given another set B which contains p natural numbers between 1 to N. We have to find out the least sum of subset which is not possible in ...
2
votes
1answer
68 views

The difference of two IID random variables (exponential)

So I'm self-studying probability distributions a bit, and have run into several trouble figuring out the sum and difference and product (etc) of distributions. Let $X_1$and $X_2$ be independent and ...
1
vote
1answer
68 views

Integral of a simple function

The definition of a simple function is that let ($\Omega$,F, $\mu$) be a measure space and for let $\Omega$ be written as disjoint union of $A_i$'s where $i=0,1,..,n$ . A function $f$ from $\Omega$ to ...

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