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23 views

Is the quotient ring, a field?

I'm working with the quotient ring F7[x]/(x^3 - a) and I want to show the values of a that make it a field. Does anyone know how to do this?
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existence of inverse

Suppose $M = I - Q$ where $I$ is the identity matrix and $Q$ is a general matrix where all of its row sums are no more than $1$ and all entries of $Q$ are nonnegative, I have no idea about how to ...
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Prove |A⋃B| = |A| + |B - A|

I'm not really sure how to start this. I considered proving that there is a bijection between A⋃B and A⋃(B-A) but I'm struggling finding a 1-1 function f: A⋃B -> A⋃(B-A) so that A⋃B <= A⋃(B-A) or ...
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Moving a rocket between two points on a straight line, when to rotate from prograde to retrograde?

Imagine you have a rocket and you want to move it from point a to point b. The flight plan is as follows: Fire the rocket ...
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Truth, Proof and Axiomatic Systems

I still struggle mighty with basic conceptions of truth and proof. For example: The Continuum Hypothesis (CH) is either true or false, i.e. either CH or ~CH holds. Now, Goedel and Cohen proved that ...
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Section to Skew-Symmetrization Map

Let $A$ be an $n\times n$ matrix skew-symmetric matrix. Define the map $\mathbb{R}^{d^2}\to Skew_d$ by $$B\mapsto B^{\top} - B.$$ Does this map have a continuous right inverse?
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Are infinite subsets of the real field definable by a single formula?

Consider the structure $(\mathbb{R},+,*,0,1,<)$. We adjoin to it a subset $S$ of $\mathbb{R}$. Is it possible to give a single formula $F$ in the expanded language such that $F$ is true precisely ...
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Bijective Pairing given one set

****Disclaimer this is a homework help question not sure if that's a problem but I'll try to format my question in a way so I'm not just trying to get the answer but get some understanding of what it ...
43 views

Goal of the form $\forall x P(x)$ and universal generalization

When we're proving a statement with a goal of the form $\forall x P(x)$, we usually begin our proof by expanding the universal quantifier to the entire formula. Afterwards, we proceed with the common ...
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Foundations in Silverman's “Arithmetic of Elliptic Curves”

I am currently self-studying elliptic curves using Silverman's AEC. I find his treatment of the background on varieties quite sloppy , and have so far kept going back and forth between AEC and Chapter ...
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How does factoring a quadratic equation relate to the parabola on the graph and to common sense?

I just learned how to factor quadratic equations (at least, how to factor them when they're simple and easy; I don't know how to factor when a isn't equal to 1). I'm fine at doing this, and I ...
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Shadow rushed by a disk in space

A disk of radius 1 is centered at the point $A(0,1,2)$ and is parallel to the plane $xOy$. A source of light is placed at the point $P(0,1,4)$. Characterize analytically the shadow and the disk rushed ...
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About measures acting on measurable functions.

If $\phi : X \rightarrow \mathbb{R}$ is a measurable function and $\mu$ is a measure on $X$ then what is $\mu(\phi)"$? Is this a notation for some function? I came across this notation for the ...
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Solving the time-independent Shrödinger equation

The time-independent Shrödinger equation is $$\dfrac{-\hbar^2}{2m} \nabla^2 \psi + V \psi = E \psi.$$ I am then told that the general solution for a uniform potential can be written as the sum of ...
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Find the maximum number of scientists can refute this work

250 scientists listen to a speech that contains 5 theories. Half of the scientists do not agree with each of the theories. If a scientist does not agree with 3 or more theories, he refutes this work....
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Radius of Convergence of Laurent Series

How does one prove that if a function as $n$ poles, and domain the complex plane except the poles, then the radius of convergence of the Laurent series at each pole is the distance to the nearest pole....
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Is $H_m - H_n$ a surjection onto $\mathbb{Q}^+$?

I was wondering whether, for each rational $q$, we may always write $$q = \sum_{k=a}^b \frac 1k$$ For some positive integers $a \leq b$. I get the feeling that this is not true (although an ...
22 views

About continuous local martingales, question on Le-Gall's book

Background Hello, I'm working on question 4.24 on Le-Gall's Brownian motion(...) and I would ask you to check if my ideas are correct. The question is as follows: $(M_t)$ is a cont. local ...
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Help with Poisson Stochastic Process

Cars pass along the road in accordance with the Poisson process of intensity $\lambda$ . A pedestrian crosses the road at time $W$ as soon as he sees that there will be no cars during time $T$ (...
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Show that the series $\sum_{n=1}^\infty \sin \left( \frac{x}{n^2} \right)$ does not converge uniformly

I asked this question about a week ago but I am little bit unsure about the way to solve it so I hope it is ok if I ask again about some things I do not fully understand. I have to show that the ...
698 views

What are the differences between Heat equations and Poisson Equations?

Am fairly new into heat equations and wanted to have some clarifications. What are the distinguishing features between the heat equation and the Poisson equation?
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The best (or worst?) counterexample to Schwarz theorem

The well known theorem of Schwarz asserts the following: suppose that $f:U \to \mathbb{R}$ where $U \subset \mathbb{R}^n$ is $C^k$ function and pick some sequence $(j_1,...,j_k)$ of length $k$ where ...