# All Questions

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### Sum of squares at integer points for $L^2$ function

Let $f\in L^2(\mathbb{R})$ be a continuous function such that $f(x)\rightarrow 0$ as $x\rightarrow\pm\infty$. Is it true that $\sum_{n=1}^\infty |f(n)|^2$ is finite? If the continuity and going to ...
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### Salvage of a given propostion

Consider the statement: For all integers $r$, $s$, and $a$, and natural numbers $m$, if $ra \equiv sa \pmod m$ then $r\equiv s \pmod m$. I have found this statement to be false by the counterexample ...
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### Give an example of a subgroup of order $9$ in $\mathbb{Z}_3 + \mathbb{Z}_3 +\mathbb{Z}_3$.

I'm struggling to find an example for this.
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### Continuous $L^2$ function has finite sum at integer points?

Let $f\in L^2(\mathbb{R})$ be a continuous function. Is it true that $\sum_{n=1}^\infty |f(n)|^2$ is finite? If the continuity condition is dropped, the statement is not true, because $f(n)$ could ...
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### Sum of Gaussian Variables may not Gaussian

I am currently trying to understand the following three points which we discussed in lectures recently: We say that $X=(X_1,\ldots,X_d)$ is $d$-dimensional multivariate Gaussian distributed if ...
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### Computing the derivative of square root of a matrix

maybe this is an idiot question, however I could not figure out how to solve it. Let $X =M_n(\mathbb{R})$ be the space of $n \times n$ matrix over the reals, then there exists two open neighborhoods ...
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### Real analysis question about inequalities? [duplicate]

To preface, I asked this question yesterday and got a response, but I didn't understand part of their answer and they haven't replied to my comment asking for help. Let $g, h: [−2,1] → [3,5]$ and ...
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### Universal covering Spaces Drawings

I just have trouble drawing universal covers, how can I draw the universal covers of the following spaces: $X$ is the union of a circle with a projective plane $\mathbb{P}^2$ identified along a ...
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### Probability of 2 points uniformly distributed over unit square

If either point is above or on the y=1/4 line, or below or on the y=-1/4 line, then the two points are definitely on the same side of the x-axis. For the other possible points I know I need to ...
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### Sum of squares finite for two sequences implies sum of products finite?

Let $a_1,a_2,\ldots$ and $b_1,b_2,\ldots$ be sequences of real numbers such that $\sum_{i=-\infty}^\infty a_i^2$ and $\sum_{i=-\infty}^\infty b_i^2$ are both finite. Is it true that ...
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### Help with Stokes problem

Well, I hope this is a stokes problem. Im honestly a bit lost on this so please help me out! Suppose I have a simple closed curve, C, in the plane w/ counterclockwise direction. I need to calculate ...
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### Evaluate the integral

I struggle to solve this problem. I am not really sure what do after let $u=2 x$. $$\int e^{2x}+ 2\sqrt{x}$$
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Is $\mathbb{Z}_7$ an example of a commutative ring with no zero divisors? I came to this conclusion because $7$ is prime. Can someone tell me if I am correct?
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### Need clarity with the maximum modulus principle of analytic functions

I was reading on the maximum modulus principle and I stumbled upon a Theorem: If a function $f$ is analytic and not constant in a given domain $D$, then $|f(z)|$ has no maximum value in $D$. That ...
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### Algebraic Equation vs Algebraic Function

By definition, a function given by $y=f(x)$ is algebraic if it can be expressed in the form $$p_n(x)y^n+p_{n-1}(x)y^{n-1}+⋯+p_1(x)y+p_0(x)=0$$ where $p_i(x)$ is an at-most $i$ degree polynomial and ...
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### Notation for Subspaces

Is there a proper notation for denoting subspaces? For example, if $U$ is a subspace of some vector space $V$. I would usually just write "the subspace $U \subseteq V$" but I'm wondering is there is a ...
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### how do i represent the square root of 90 as a fraction?

I can usually do this with a calculator but it's not wanting to play nice. I need it to find the distance between two points but, unfortunately I am unable to do so because it asks for it in the form ...
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### h(xy) = h(x)h(y) Proof

Let $h$ be a differentiable function where $h(xy) = h(x)h(y), \forall x>0$. Let $h(1)=1$ be an initial condition. Prove $\exists c$ such that $h(x)=x^c$. I've tried differentiating both sides of ...
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### Confidence interval hormone replacement

Problems from Chapter 7 beginning page 282: 4ab, 7, 10, 13 Confidence intervals for the Hormone Replacement Therapy Trial: 8506 women received hormone replacement therapy, 8102 received a placebo. ...
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### A characterization of nonincreasing functions

Let $f(x)$ be continuous in the interval $I := (0,1).$ Define $$D_+f(x_0) := \liminf_{h \to 0^+} \frac{f(x_0+h)-f(x_0)}{h}.$$ Put $$S:= \{x \in I: D_+ f(x) < 0\}.$$ Suppose that the set ...
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### Algorithm for determining whether two real quadratic numbers are equivalent under a modular transformation

Let $\alpha \in \mathbb{C}$ be an algebraic number. If the minimal plynomial of $\alpha$ over $\mathbb{Q}$ has degree $2$, we say $\alpha$ is a quadratic number. Then $\alpha$ is a root of a unique ...
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### characterization of characteristic functions (Bochner Theorem Proof?) Simple case.

Prove the following theorem: Let $\phi: \Bbb R \to \Bbb C$. $\phi$ is the characteristic function of a real random variable $X:\Omega \to \Bbb R$ if and only if $\phi(0)=1$ $\phi$ is uniformly ...
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### A formal procedure to successfully create a tree over $2^k$ vertices

I have a graph $G$ with $2^k$ vertices and initially zero edges. I am trying to successfully adding edges to end up with a tree with $2^k-1$ edges. Each time I add $2^{k-i}$ edges for $i={1,2,..,k}$. ...
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### Prove that $\sqrt{a^2-ab+b^2}+\sqrt{a^2-ac+c^2\vphantom{b^2}}\ge\sqrt{b^2+bc+c^2}$

Let $a$, $b$, and $c$ be positive real numbers. Prove that $$\sqrt{a^2-ab+b^2}+\sqrt{a^2-ac+c^2\vphantom {b^2}}\ge\sqrt{b^2+bc+c^2}.$$ When does equality occur?
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### Boolean rings, subring of the ring its elements come from?

If $B=\{x \in C:x=x^2\}$ then is the boolean ring $B$ a subring of the ring $C$?
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### Prove $\frac{\sum_{i=1}^{n}a_iX_i}{\sum_{i=1}^{n}a_i}\overset{a.s.}{\rightarrow}0$?

Suppose $X_i$ are independent r.v.s. such that $P(|X_i|≤1)=1$ and $\mathrm{E}\left[X_i\right]=0$. Let $a_i$ be positive constants such that $a_i≤1$ and $\sum_{i=1}^{\infty}a_i=\infty$. How to prove ...
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### finding the number of sub fields such that $(K : Q) = 2$

Consider the polynomial $f(x) = x^5 - 4x + 2$. Let $L$ be the complex splitting field of $f(x)$ over $\mathbb{Q}$. I want to find the number of subfields $K$ of $L$ such that $(K : \mathbb{Q}) = 2$. ...
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How did this: $2(1-\sin^2x)=1+\sin x$ Become this: $2\sin^2x+\sin x-1=0$ Wouldn't it be: $2 -2\sin^2x-1+\sin x=0 ?$
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### Taylor series for $\log(1+x)$ and its convergence

I know the taylor series of $\log(1+x)$. However I don't understand how to find the convergence for $x>1$ and divergence if $x<1$.
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### The mapping $(r,\theta)\mapsto(r\cos\theta,r\sin\theta)$ on $[0,1]\times [0,2\pi]$ [on hold]

Any help would be highly appreciated Consider the polar coordinate map $(x,y)=\varphi(r,\theta)=(r\cos\theta,r\sin\theta)$ defined on $\Bbb R^2$, and its behavior on the set ...
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### Alternating Series

A professor of mine gave me this problem and asked me to figure it out. I can not seem to figure it out. Express ln($2/3$) as an alternating series and use alternating series estimates to find ...
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### Limiting values for logistic function

Given the logistic function (map) $x_{n+1} = r\cdot{xn}\cdot (1 - {xn})$and an initial value $x_{0} = 0.4$ When r = 0.5, i worked out $x_{1}=0.12$ and $x_{2}=0.0528$ How do i work out the ...
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### Computing the Fourier transform of a function not in $L^1$?

The Fourier transform is defined on $L^2$ by a density argument. It doesn't seem like it's constructive. So how would one go about computing the Fourier transform of a function in $L^2$ but not $L^1$? ...
Consider the Poisson’s equation with Robin’s boundary conditions as follows \begin{array}{ll} −\Delta u = f, &\text{in $U$,}\\ \frac{\partial u}{\partial \nu}+u=g, &\text{on $\partial U$,} ...