0
votes
1answer
18 views

General Form of a Open Set in the Product Topology in a Countably Infinite Product.

Suppose $\{X_n\}_{n\in\Bbb N^+}$ is a family of topological spaces. I understand that a typical basis element of the product topology has the form $$\prod_{n=1}^k U_n\times\prod_{n=k+1}^\infty ...
0
votes
1answer
18 views

positiv Martingale process

I would to like to prove that the process: $$e^{\int_{0}^{T}\theta _{s}dW_{s}-\frac{1}{2}\int_{0}^{T}\theta _{s}^2ds}$$ is a martingale which is positiv and has a mean=1 $$\theta is continuous ...
2
votes
2answers
18 views

Composing Linear Transformations

Hello and thank you in advance; The problem: "Let V be a vector space and T a linear operator $T:V\rightarrow V $, show that $$[T^m]_B =[T]_B^m$$ Where $B$ is a basis(any) of $V$ and $T^m=T\circ T ...
1
vote
1answer
25 views

If $f \in L^2 \cap C_c$ then $\sum_{n \in \mathbb{Z}}|\hat{f}(\xi+2\pi n)|^2 = a_0+…a_n \cos(2 \pi n \xi)$

Let $f \in L^2 \cap C_c$ , then I want to show that $$g(\xi):=\sum_{n \in \mathbb{Z}}|\hat{f}(\xi+2\pi n)|^2 = a_0 + 2 \sum_{n=1}^{N}c_n \cos(2\pi n \xi) $$ for some $N \in \mathbb{N}.$ Does ...
0
votes
1answer
12 views

Prove that $B^n$ is diagonalisable for all $n=2,3,\dots$ and that every eigenvalue of $B^2$ is the square of some eigenvalue of $B$.

I would like to ask you for some help in the following problem: Suppose that a matrix $B$ is diagonalisable over $\mathbb{C}$. Prove that $B^n$ is diagonalisable for all $n=2,3,\dots$ and that ...
0
votes
1answer
36 views

A measure theory question-1 [on hold]

Let $ (\Omega, \mathcal U, P)$ be a measure space and any events $A_1, A_2, A_3 \in \mathcal{U}$ And $ B$ is defined as event of occurrence of at least one of these three events. First I need to ...
1
vote
2answers
22 views

direct sum of vectors

$$U = \{(x,y,z,t) \in \mathbb{R}^4 | x + 5y + 4z + t = 0 , y + 2z + t = 0 \} $$ $$W = \{(x,y,z,t) \in \mathbb{R}^4 | x + z + 3t = 0, 2x-3y-4z+3t = 0\} $$ $U \oplus W = \mathbb{R}^4$? This is my ...
2
votes
0answers
14 views

Computation of Ext as a cohomologies of certain complex

Let $R$ be a ring and $K^\bullet$ be a complex of $R$-modules such that $K^\bullet$ has only one nontrivial cohomology $H^0(K^\bullet)=M$. Suppose that $R$-module $N$ is such that ...
-2
votes
1answer
20 views

Cumulative distribution function of the ratio of the maximum and minimum of two random variables [on hold]

Let $X_{1}$ and $X_{2}$ be independent, absolutely continuous random variables, each uniformly distributed between 0 and 1. I want to find the cumulative distribution function of the random variable ...
3
votes
1answer
48 views

Fields of characteristic $p$ that are isomorphic as vector spaces, but not as fields

Give an example of a field $\mathbb{K}$ of characteristic $p > 0$ and elements $a,b \in \bar{\mathbb{K}}$, such that $\mathbb{K}(a)$ and $\mathbb{K}(b)$ are isomorphic as vector spaces over ...
1
vote
1answer
15 views

Construction of addition and multiplication table for GF(4)

I am dealing with finite fields and somehow got stuck. The construction of a prime field $GF(p), p \in \mathbb{P}$ is pretty easy because every operation is modulo p. In other words $GF(p)$ contains ...
0
votes
2answers
62 views

How to find the greatest prime number that is smaller than $x$? [duplicate]

I want to find the greatest prime number that is smaller than $x$, where $ x \in N$. I wonder that is there any formula or algorithm to find a prime ?
3
votes
0answers
33 views

Fourier transform exercise

I am currently dealing with a problem in functional analysis where I want to show the following. Let $\phi_{k+1}(x):=\sqrt{2} \sum_{n \in \mathbb{Z}} h(n) \phi_k(2x-n)$ and $\phi_0= \chi_{[0,1]},$ if ...
5
votes
3answers
61 views

Find the Galois group of the polynomial $x^{4} + x - 1$ over the finite field $\mathbb{F}_{3}$

Find the Galois group of the polynomial $x^{4} + x - 1$ over the finite field $\mathbb{F}_{3}$. I'm particurly struggling with what would be the approach to finding the splitting field of this ...
0
votes
1answer
19 views

Show that $a^{p+1\over 4}$ solves the equation $x^2 ≡ a \pmod p$.

Let $p \equiv3 \pmod 4$ be a prime number, and let $1 \le a\le p − 1$ be a quadratic residue. Show that $a^{p+1\over 4}$ solves the equation $x^2 ≡ a \pmod p$. I know that if $(a,n)=1$ and $p\ge ...
-2
votes
0answers
21 views

Example of projection sequence on Hilbert space with strong limit P [on hold]

Let $P_n$ be a sequence of projections on a Hilbert space $H$ with strong limit $P$. Suppose that $P_n(H)$ is infinite dimensional. Show that $P(H)$ may be finite dimensional.
0
votes
1answer
24 views

Probability of last cheese

I hope that someone could help me with understanding the exercise. In a cycle shaped house there are n chambers. In this house there is a mouse and each chamber has cheese except the room where the ...
-1
votes
0answers
22 views

In how many ways can five different keys be put in a flat leather key case? [on hold]

In how many ways can five different keys be put in a flat leather key case?
0
votes
0answers
19 views

Roadmap to $p$-adic numbers: where a self-learner should look for references

TL;DR at the end of the question. I’m currently trying to learn as much as possible about p-adic numbers. I’m not sure what is the most fascinating part of the theory, but the use of the adjective ...
1
vote
3answers
42 views

Proof of sum in an inequality

I was having hard time solving this one, any help will be greatly appreciated. prove that: $$ {39\over e^2}\le\sum_{n=1}^\infty {4n^2-1\over e^n}-{3\over e}\le{54\over e^2} $$
-2
votes
4answers
32 views

Proof $log_{r} a = log_r s \cdot log_s a $ [on hold]

Do you know any proof of this logarithms property: $log_{r} a = log_r s \cdot log_s a $
1
vote
1answer
24 views

RSA fixed point

What is the number of RSA fixed points, in other words how many $m$ are there such that $$m^e\equiv m \pmod{n}$$ where $n=pq$, for primes $p,q$. I know that the answer is ...
0
votes
0answers
8 views

Error estimation - spline interpolation

I got a question regarding error estimation and spline interpolation. I got a parabola shaped graph that I've used spline interpolation on to get more accurate data. I've used a much smaller step on ...
2
votes
0answers
20 views

Integral of Brownian Motion with respect to an independent Brownian motion

I have this seemingly simple problem which I haven't been able to solve. I have two standard Brownian motions, $B$ and $W$, on the same probability space and under the same filtration (I am not so ...
-2
votes
0answers
9 views

Calculate the matrix of a lineal aplication with some information [on hold]

‪If f:(Z7)3 à(Z7)3 is the only lineal application with Ker(f)= and V2={(1,0,1), (1,1,0)} = L[(1,0,1),(1,1,0)] where V2 is the subspace associated to the proper value 2. Calculate the matrix ...
-1
votes
0answers
8 views

Proving existence of unique maximal subfields of Galois extensions with particular properties

A question I am working on asks the following: Let $K / \mathbb{Q}$ be a Galois extension. Prove that there exists a unique maximal subfield $F$ of $K$ such that $F / \mathbb{Q}$ is Galois with ...
1
vote
2answers
72 views

derivative of $\ln(4)$

what is the derivative of $\ln(4)$? I am trying to find the derivative of this equation: $h(x)=\ln(\frac{x^3\cdot e^x}{4})$ by rules of logs I simplified the $h(x)$ to the following: ...
0
votes
0answers
30 views

The degree-genus formula cannot be applied to singular curves in $\mathbb{P}_2$?

(The degree-genus formula) The Euler number $\chi$ and genus $g$ of a nonsingular projective curve of degree $d$ in $\mathbb{P}_2$ are given by$$\chi = d(3-d)$$and$$g = {1\over2}(d-1)(d-2).$$ My ...
0
votes
1answer
24 views

Analytic paths through converging sequences in the complex space.

Assume we have a Cauchy sequence $\{\vec{a_i}:i\in\mathbb{N}\}$ converging to $\vec{0}$ in $\mathbb{C}^n$ such that $|\vec{a_i}|<|\vec{a_j}|$ whenever $i>j$. Can we find an analytic path ...
-1
votes
2answers
26 views

A probability theory question [on hold]

let X be a rondom variable and coonsider a non-negative function $g: \Bbb R \to \Bbb R^+$ Please help me sshowing this following statement; for $r\in \Bbb R^+ $, $$P(g(X)\gt r) ...
0
votes
1answer
16 views

Isomorphism between vector spaces of linear transformations

Let $V,W$ vector spaces over the field $F$,and let $U: V\rightarrow W$ an isomorphism between them. Prove that the linear transformation $\mathcal{U}:\mathcal{L}(V,V)\rightarrow ...
-2
votes
1answer
19 views

How can I find Laurent Series and the region of convergence for $z/((z+1)(z+2))$ for$ z= -1$? [on hold]

How can I find Laurent Series and the region of convergence for $z = -1$ of $$ \frac{z}{(z+1)(z+2)} $$
-2
votes
0answers
37 views

What mathematician you would have liked to know? [on hold]

I know this is a classic forum question but, to be honest, I would like know your opinions... yours opinions, the opinions of the people that participate in mathexchange. I will ask to the moderators ...
0
votes
0answers
24 views

On the rearrangement of an infinite series of real numbers. [duplicate]

A chapter of a text book ended with. If we rearrange infinitely terms of a series that converges only conditionally, we may get results that are far different from the original series ...
3
votes
1answer
41 views

Folland, “Real Analysis”, Chapter 5.3, Exercise 36.

Folland, "Real Analysis", Chapter 5.3, Exercise 36: Let $\mathcal{X}$ be a separable Banach space and let $\mu$ be counting measure on $\mathbf{N}$. Suppose that $\left\{x_n\right\}_1^\infty$ ...
1
vote
0answers
21 views

Which definition of a neighborhood is more standard? [duplicate]

I came across the following two definitions of a neighborhood in a topological space $X$. Definition: A set $N\subset X$ is a neighborhood of $x\in X$ if $N$ contains a open set in $X$ which ...
0
votes
0answers
17 views

Let f be a analytic map that sends the annulus A(0,1,2) to the unit disk such that $|z|=1,|z|=2$, Furthermore f is not constant. Prove:

Let $f$ be a analytic map that sends the annulus $A(0,1,2)$ to the unit disk such that $|z|=1,|z|=2$ get mapped to the points $|f(z)| = 1$. Furthermore f is not constant. Prove: 1) $f$ has at least ...
1
vote
1answer
19 views

Can a compact set of $\mathbb{R}$ have some properties and not being convex

The question is related to this one On a condition when bounded sets in R n is convex ?. Suppose that $n > 1 $ and that $C \subset \mathbb{R}^n$ is a compact (closed and bounded) set having a ...
0
votes
0answers
23 views

Tensor independence

Let $(e_{i})$ be a basis in $V$, $( \epsilon_{i} )$ - basis in $V^{*}$ so that $\epsilon_{i} (e_{j})= \delta_{i}^{j}$ (Kronecker delta, $\epsilon_{i} (e_{j}) = 1 \Leftrightarrow i=j$, otherwise it's ...
-2
votes
3answers
37 views

$\lim _{ x->\infty }{ [(x+2)\arctan(x+2) } -(x)\arctan(x)]$ [on hold]

What would be the best way to find $\lim _{ x->\infty }{ [(x+2)\arctan(x+2) } -(x)\arctan(x)]$ ?
0
votes
2answers
32 views

Error in a Maclaurin series

I'm having trouble figuring out what I have to do with this question. "Using Taylor's theorem, determine the largest positive real value $r$ for which we can guarantee that the Maclaurin polynomial ...
1
vote
1answer
33 views

Subspaces that undo Products

I have been working on Munkre's homework sets, and I have come across the following phenomenon: Let $\mathbb{R}_\ell$ be the lower limit topology on the real numbers. If you consider a line as a ...
2
votes
2answers
188 views

Can Zorn's Lemma be 'inverted' like this:?

Let $R$ be a (commutative) ring not equal to $0$. I want to show that the set of prime ideals of $R$ has a minimal element w.r.t. inclusion. This may be a wholeheartedly wrong attempt, but I thought ...
0
votes
1answer
13 views

use laplace transform to solve the given integral equation

use Laplace transform to solve the given integral equation I don't know how start because it differences on other Laplace question I see before
0
votes
1answer
8 views

What is the formulae to draw a straight between the given ratio?

when $X_{min}=50, Y_{min}= 1.0$ when $X_{max} > 50, Y_{max}= 1.5$, where $X_{max}$ varies from $51, 52, 53, \ldots$ What is the value of $Y$ at any given point fo $X$? If $X_{min}$, $X$ & ...
1
vote
1answer
29 views

What will be the equation of side $BC$.

The equation of two equal sides $AB$ and $AC$ of an isosceles triangle $ABC$ are $x+y=5$ and $7x-y=3$ respectively . What will be the equation of the side $BC$ if the area of the triangle ...
-4
votes
1answer
31 views

In how many ways can two chocolate chip, three raisin, and one peanut butter cookie be distributed to six children? [on hold]

A mother has six cookies, two chocolate chip, three raisin, and one peanut butter. In how many distinct ways can she pass them out to six children so that each gets one? Assume that those of the ...
-2
votes
1answer
16 views

Determine the Equation of the Locus. [on hold]

P is a point that is twice as far from the point(0;5) as it is from the line y=2. And a another question. P is the point that is twice as far from the line y=1 as from the point (2;4) Determine ...
1
vote
0answers
22 views

Why do the vectors perpendicular to [1, 1, 1] and [1, 2, 3] fall on a line, as opposed to a plane?

And what's the intuition here? This is question 6(c) in pset 1.2, Strang's Linear Algebgra, 4th Ed.
1
vote
0answers
11 views

Iterative methods monotonically decreasing of the residual

For a question on Iterative Methods I have to show that the 2-norm of the residual is monotonically decreasing. We are given the following formula: $r^{(k+1)} = r^{(k)} - \alpha^{(k)} A z^{(k)}$ where ...

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