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1
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2answers
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?
-1
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2answers
40 views

If $G/H_1\cong H_2$ and $G/H_2\cong H_1$, can we conclude that $G\cong H_1\times H_2$ if $H_1, H_2\neq \{\operatorname{id}_G\}$?

Consider a (discrete or continuous) group $G$ and two normal/invariant subgroups $H_1$ and $H_2$ (possibly invariant/normal subgroups). Provided that $H_1,H_2\neq \{\operatorname{id}_G\}$, if $G/H_1\...
0
votes
0answers
4 views

Interior set, boundary set, clousure set, limit set and isolated set

I have to define the Interior set of $A$ , boundary set of $A$, clousure set of $A$, limit set of $A$ and isolated set of $A$ denoted by $(A^o,\partial A,\overline A, A',A^{s})$: If $A$=$\{3,4,6,8\}$ ...
-2
votes
0answers
15 views

P adic convergent series

Find a 7-adically convergent series x = $\sum$ $a_n$ with $a_n$ ∈ Z(7), such that $x^2 ≡2 mod7^r$ for all r>0.
1
vote
1answer
24 views

What to study after Riemann-Roch Theorem

I am a few days away from finishing my first course on Algebraic Geometry which started with Varieties (upto Bezout's Theorem and group structure on cubic elliptic curve) and then went on to Schemes (...
-1
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2answers
19 views

Working out the number of solutions…

How would I go about figuring out the number of solutions in $x^2=-1211 ($mod $2020)$. I wrote out the prime factorisation of $2020$ which is $2^2 \times 5 \times 101 $but I sort of found myself ...
0
votes
1answer
18 views

Proving a collection of open sets is a basis for a topology of $X$

${\bf exercise:}$ Let $X$ be a topological space. Suppose $\mathscr{C}$ is a collection of open sets of $X$ such that for each open set $U$ of $X$ and each $x \in U$, there is an element $C \in \...
0
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0answers
28 views

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 ...
0
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1answer
23 views

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 ...
0
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0answers
8 views

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 ...
1
vote
2answers
17 views

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 ...
2
votes
1answer
29 views

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?
2
votes
1answer
10 views

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 ...
0
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0answers
4 views

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 ...
1
vote
0answers
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 ...
1
vote
0answers
8 views

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 ...
0
votes
1answer
23 views

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 ...
0
votes
0answers
7 views

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 ...
-1
votes
1answer
19 views

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 ...
0
votes
0answers
5 views

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 ...
0
votes
0answers
16 views

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....
2
votes
1answer
119 views

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....
-1
votes
0answers
9 views

Collinear points — are they coplanar by definition?

Say I have three collinear points, $A$, $B$, and $C$. Must they as a result all lie in the same plane, and are there any counterexamples? Sorry if this isn't a good question -- first time here $\ddot\...
0
votes
0answers
7 views

A module annihilated by a maximal ideal is semisimple

I want to show that if $M$ is a module over a commutative ring $R$ that is annihilated by a maximal ideal $I$ of $R$, then $M$ is a semisimple $R$-module. What I have in mind is the following: if $M$ ...
0
votes
0answers
16 views

Convexity of $f(x) = \sum\limits_{i=1}^n \sqrt{x_i}$

Suppose we have some $x = (x_1, x_2, .. x_n)$ such that $x_i \geq 0$ and $\sum x_i = 1$. Let $f(x) = \sum\limits_{i=1}^n \sqrt{x_i}$. Is $f$ a convex function of $x$? It appears to be the case from ...
0
votes
0answers
12 views

Is stopped integral of predictable process predictable?

Assume that $H$ is a predictable process that is locally bounded with localizing sequence $(\tau_n)_n$. And assume that $\langle M, M \rangle$ is a increasing, right-continuous, predictable process. ...
0
votes
0answers
15 views

Closed marginal distributions for parametric family with random parameters

I'm struggling to find any info on this problem Given a parametric family of probability distributions $f(x;\theta)$ and a random element on the set of parameters $\Theta$, for which ...
1
vote
4answers
46 views

A closed discrete set

Let $V$ be a normed vector space. Let $(b_n)\subseteq V, b_n \to b\in V.$ Show that $B := \{b,b_1,b_2\dots\}$ is closed. I know that if $b_n\to b,$ then $b_n$ is Cauchy. That is, $\forall \...
2
votes
0answers
16 views

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 ...
3
votes
1answer
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 ...
-1
votes
1answer
13 views

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 $T$ (...
0
votes
2answers
55 views

Prove $g'(x) = f(x + b) - f(x+a)$

Let $f\colon \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function. $a, b \in \mathbb{R}$ and $a<b$. Let $g\colon \mathbb{R} \rightarrow \mathbb{R}$ be defined as $g(x) = \int_{a}^{b}f(...
5
votes
4answers
1k views

Shortest distance between two points via calculus of variations

This problem might be trivial but when solving it using calculus of variations it's not so stupid. Suppose we have a fixed boundary condition $f(a) = f(b) = 0$ and we want to find the shortest ...
0
votes
1answer
33 views

Probability of event in Monte Carlo roulette wheel problem

A Monte Carlo roulette wheel has $18$ red numbers, $18$ black numbers and one green number. The probability of the ball NOT landing on a black number is ? My work : $18+18=36$ $\frac{18}{36}$ = ...
0
votes
1answer
17 views

Convergence of a stochastic process

Suppose that $\{X_n(t),t\in\mathbb{R}\}_{n=0,1,...}$ is a collection of stochastic processes, i.e., for any fixed $n$, we have and stochastic process $\{X_n(t),t\in\mathbb{R}\}$. Assume that for any ...
1
vote
2answers
28 views

Does multiplying by $i=\sqrt{-1}$ count as multiplying by a constant for the purposes of linear independence?

E.g. are $i \sin(x)$ and $\sin(x)$ linearly independent or linearly dependent? For context, I'm trying to reconcile the $e^{ix} = \cos(x) + i\sin(x)$ formula with the generalized solution to a ...
2
votes
2answers
35 views

Simplifying $\int\cos^2(x)\sin(2x)dx$ via the optimal substitution

I was just tutoring and a student's question was: Make a trig substitution to make evaluating $\int \cos^2(x)\sin(2x)dx$ simpler. So yeah, the question is only asking for what trigonometric ...
0
votes
0answers
7 views

When is the region between two Lipschitz graphs a Lipschitz domain?

For $f,g : \mathbb{R}^d \to \mathbb{R}$ Lipschitz with $L_1$ and $L_2$ norms, let $\Omega= \{ (a,b) \in \mathbb{R^{n+1}} : f(a) \leq b \leq g(a) \}$. I was wondering under which (hopefully mild) ...
0
votes
0answers
24 views

Taylor serie for even function. Proof

We let $f:\mathbb{R}\to\mathbb{R}$ be infinitely often differentiable function and we let the Taylor series be: $$\displaystyle\sum_{n=0}^{\infty}\left(\left(\frac{f^{n}(0)}{n!}\right)x^n\right) $$Let ...
0
votes
0answers
12 views

Combinatorial number system alternate proof

The Combinatorial number system is the theorem that every positive integers can be expressed as the combination of a using a unique sequence of of a strictly decreasing sequence $c_k > ... c_2 > ...
2
votes
1answer
37 views

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 ...
5
votes
3answers
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?
0
votes
0answers
22 views

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 ...
0
votes
0answers
6 views

Angle between two midpoints equals angle between point at perpendicular from top to base to other midpoint

Let, X, Y, Z be the midpoints of the sides AB, BC, CA of the triangle ABC. Let P be defined on BC so that ∠CPZ = ∠YXZ. Prove that AP is perpendicular to BC. This question is from a book I'm reading ...
1
vote
0answers
17 views

Plot $u(x,t)$ for a string of length $10$

Consider a taut string of length $10$ with wave speed $c = 1$ (in suitable units), with a fixed end at $x = −5$ and a free end at $x = 5$. The deflection of the string is denoted $u(t, x)$ for $−5 ...
-1
votes
0answers
15 views

question about martingales

Let $\xi_{i}$ - i.i.d., $\mathbb{P}(\xi_{i} = 1) = \mathbb{P}(\xi_{i} = -2) = \frac{1}{2}$; $X_{i} = \sum_{j=1}^{i} \xi_{j}$. How to find the $\mathbb{P}(\exists n \geq 0: X_{i} = 1)$? Give me some ...
0
votes
1answer
39 views

Stabilizing controls in linear quadratic regulator

I am studying a linear quadratic control problem with discounting. For $\gamma \in (0,1)$, $Q \succeq 0$ and $R \succ 0$ and linear dynamics $s_{t+1}=As_t + B a_t$, let the total cost starting in ...
1
vote
2answers
1k views

Solution to the second order differential equation of $LCR$ series circuit.

When applying Kirchoff's voltage law to a $\text{LCR}$ series circuit. The following differential equation pops up.. $$E\sin(Kt)-\dfrac{q}{C}-R\dfrac{dq}{dt}-L\dfrac{d^2q}{dt^2}=0$$ Where $E$, $K$,...
2
votes
2answers
56 views

Evaluate $\int_{-1}^{1} [ \frac{2}{3} x^3 + \frac{2}{3}(2-x^2)^{3/2}] dx $

Evaluate $$\int_{-1}^{1} \left[ \frac{2}{3} x^3 + \frac{2}{3}(2-x^2)^{3/2}\right] dx $$ My attempt :$$ \frac{2}{3} \left[\frac{x^4}{4}\right]_{x=-1}^{x=1} + \frac{2}{3} \left[\frac{(2-x^2)^\frac{-1}{...
0
votes
2answers
47 views

prove that for every integrable function $f(x)$ exists step function $h(x)$ such $ \intop_{a}^{b}|f\left(x\right)-h\left(x\right)|dx<\varepsilon $

If a function $h(x)$ satisfies: exists partition $ P=\left\{ a_{0},a_{1},...,a_{n}\right\} $ of the interval [a,b], such that $ h $ is constant in the segment $(a_{k-1},a_{k}) $ for any $1\leq k\...

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