# All Questions

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### 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.
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### 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 ...
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### 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?
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### 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 ...
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### Proof of sum in an inequality

I was having hard time solving this one, any help will be greatly appreciated. prove that: $${39\over e^2}\le\sum_{n=1}^\infty {4n^2-1\over e^n}-{3\over e}\le{54\over e^2}$$
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### Proof $log_{r} a = log_r s \cdot log_s a$ [on hold]

Do you know any proof of this logarithms property: $log_{r} a = log_r s \cdot log_s a$
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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: ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $\left\{x_n\right\}_1^\infty$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### $\lim _{ x->\infty }{ [(x+2)\arctan(x+2) } -(x)\arctan(x)]$ [on hold]

What would be the best way to find $\lim _{ x->\infty }{ [(x+2)\arctan(x+2) } -(x)\arctan(x)]$ ?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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
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### 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 $X_{min}$, $X$ & ...
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### 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 ...
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### 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 ...
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### 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 ...
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 ...