1
vote
2answers
18 views

winding number partial fractions

I need to compute the winding numbers for a particular contour integration (which I understand and is not relevant for this question). However, before doing so I need to use partial fractions to get ...
0
votes
2answers
23 views

Expected Value of the product of an indicator R.V. and continuous R.V.

I have an indicator random variable $X \in \{0,1\}$ and a continuous random variable $Y$. I am looking for $E[XY]$. Intuitively, it seems $E[XY] = P(X=1)E[Y|X=1]$, but am having a hard time ...
0
votes
0answers
30 views

If $f(x)\le 1$ implies $f(x)\le 1/2$ then $f(x+\delta)\le 1$?

Let $f(x)$ be a nonnegative continuous function of $x\in [0,K)$ with $f(0)\le 1/2$, and satisfies "$f(x)\le 1$ implies $f(x)\le 1/2$". Let $x_0\in[0,K)$ (so that $f(x_0)\le 1$ implies $f(x_0)\le ...
0
votes
2answers
24 views

Proving no vector potential for gravitation field defined on all of $\mathbb{R}^3 -$ origin

Let: $$F=\frac{x,y,z}{(x^2+y^2+z^2)^{3/2}}$$ Show that there is no vector potential for F which is defined on all of $\mathbb{R}^3 - \text{origin}$ I can find a vector potential which is not ...
0
votes
1answer
27 views

Prove that $X$ is finite.

Let $\mathcal T$ be the finite-closed topology on a set $X$. If $\mathcal T$ is also the discrete topology, prove that the set $X$ is finite. My attempt: We know that $\emptyset$ and $X$ are in ...
2
votes
0answers
15 views

The matrix square root is not differentiable on the boundary of the manifold of positive semi-definite matrices?

$\newcommand{\psym}{\operatorname{P}_{\ge 0}}$ $\newcommand{\Sig }{\Sigma}$ Let $\psym$ denote the subset of symmetric positive semi-definite matrices. Let $S:\psym \setminus \{0\} \to \psym ...
0
votes
1answer
37 views

Deduce the Pythagorean Theorem [duplicate]

Let V be an inner product space,and suppose that $x$ and $y $ are orthogonal vectors in V.We also know $\left\lVert x+y \right\rVert^2=\left\lVert x \right\rVert^2+\left\lVert y \right\rVert^2$.My ...
0
votes
0answers
15 views

Write out the explicit Kolmogorov forward differential equation

Let $(X_t)$ be a continuous-time Markov process with two states, as shown below. Assume that there are two positive numbers $a$ and $b$ such that for all times $t\geq 0$ and $h>0$, $P(X_{t+h} = 2 ...
0
votes
3answers
46 views

Is this a vector field?

One of the example questions we've been given in lectures is to plot the vector field given by the first order equation $y' = y - y^{3}$. The solution we were given looks like this: However, I ...
1
vote
1answer
11 views

conformal mapping of the region outside two circles

Two circles lie outside one another except for common point of tangency. How to map the region exterior to both circles (including the point of inifity) onto an infinite strip by one to one comformal ...
1
vote
0answers
25 views

Definable over $(\mathbb{R}, <, +, \cdot, 0, 1)$

I have to solve the following task and got some problems with it: a) Be $n\in\mathbb{Z}$. Is $\{n\}$ definable over $(\mathbb{R}, <, +, \cdot, 0, 1)$ b) Be $q\in\mathbb{Q}$. Is $\{q\}$ definable ...
1
vote
1answer
18 views

Invariant dimension property of a ring $R$ which admits a homomorphism to a division ring $D$

I know that if the homomorphism is surjective, then the ring has invariant dimension property. But if the homomorphism is not surjective, is it still true that $R^n$ equivalent to $R^m$ as $R-$modules ...
-2
votes
1answer
39 views

Sovle $I=\int_{0}^{\infty}\frac{\ln^2x}{1+x^4}dx$. Using function Euler. Help me please.. [on hold]

Solve $$I=\int_{0}^{\infty}\frac{\ln^2x}{1+x^4}dx$$ Using function Euler. Help me please..
0
votes
1answer
24 views

How to find the slope of a line when you only have a point and an angle?

A line passing through $(4,7)$ makes an angle of $45$ degrees with the $y$-axis. How do I find the slope of this line?
1
vote
2answers
40 views

solve for $\lim_{n \to \infty} \frac{(3n)!(1/27)^n}{(n!)^3}$

I believe the $\lim_{n \to \infty} \frac{(3n)!(1/27)^n}{(n!)^3}$ -> 0. But I am not sure if my reasoning is correct. Because there is a higher power in the denomination that the numerator, the ...
0
votes
1answer
18 views

If $f\in S(\mathbb R^n)$ (schwarz space), why $f\in L^p(\mathbb R^n)$?

Let $$\mathcal S(\mathbb R^n)=\left\{f\in \mathcal C^\infty (\mathbb R^n)\mid \forall N\in\mathbb N,\forall \alpha \in\mathbb N^n, \sup_{x\in\mathbb R^n}|(1+|x|^N)\partial _x^\alpha f(x)|<\infty ...
1
vote
1answer
12 views

Method for calculating minimum number of transmissions?

(This is a real issue I face.) I have $42$ files I want to transmit. I tried sending them in a single archive but four of them had issues, and as a result the entire archive was rejected. I do know ...
2
votes
2answers
33 views

continuous (on 3, 4 and 5) f is constant, if $f(x+2)+f(4x)=f(2x+1)+f(2x+2),\forall x\in\mathbb{R}$

Let $f:\mathbb{R}\to\mathbb{R}$ be a function that is continuous on 3, 4 and 5, such that $f(x+2)+f(4x)=f(2x+1)+f(2x+2),\forall x\in\mathbb{R}$. Show that f is constant. I don't know what to do ...
2
votes
0answers
21 views

Tauberian theorem when limit is zero

Let $h \geq 0$ be a non-negative increasing function with Laplace transform $H$. Let $\rho \geq 0$ be a constant. A simple Tauberian theorem says that the following two statements are equivalent: I. ...
0
votes
2answers
19 views

Linear Algebra: Spanning Sets, Closures, Subspaces

So I need help on three of my questions and I don't have any idea where to begin for any of these three so any help would be appreciated. Determine whether the set of all solutions to the ...
0
votes
0answers
16 views

Is Joint CDF of two random variables with a Joint PDF always continuous? [on hold]

I know it must be continuous with respect to each variable, but does it have to be continuous?
0
votes
1answer
15 views

Rectangular to polar conversion

I am trying to write this fraction in polar form (4+10i)/(24i-5) . I am having trouble to get the angle of the polar conversion. I know that in order to get the angle I need to write ...
0
votes
0answers
15 views

The harmonic functions are smooth

Let $u \in C^2(\Omega)$ a harmonic function, then $u \in C^{\infty}(\Omega)$. Here $\Omega$ is a domain of $\mathbb{R}^{n}$. Step 1 for proof: $u \equiv u^{\epsilon}$ in $\Omega_{\epsilon}$, for ...
3
votes
2answers
24 views

A relation $R$ is defined on $\mathbb{Z}$ by $aRb$ if and only if $2a + 2b\equiv 0\pmod 4$. Prove that $R$ is an equivalence relation.

A relation $R$ is defined on $\mathbb{Z}$ by $aRb$ if and only if $2a + 2b\equiv 0\pmod 4$. Prove that $R$ is an equivalence relation. My method: Let $a \in \mathbb{Z}$ be given. So, for any $a \in ...
0
votes
0answers
25 views

Continuously differentiable functions

Let $f, g,$ be $ C^2$ functions $\mathbb{R} \rightarrow \mathbb{R}$, $ F: \mathbb{R}^2 \rightarrow \mathbb{R}, F(x,y) = f(x+g(y))$ Check that $(D_1F)(D_{12}F)=(D_2F)(D_{11}F)$ I know how to ...
0
votes
3answers
27 views

An increasing smooth map $f:(0,1)\rightarrow(0,1)$ which does not extend to any smooth function on a larger domain

Although I'm not sure it's related, I have found a smooth map $f:(0,1)\rightarrow(0,1)$ which does not extend to any continuous function on a larger domain, namely ...
2
votes
0answers
19 views

Elementary properties of closures

Prove this lemma (Elementary properties of closures). Let $X$ and $Y$ be arbitrary subsets of $\mathbb R$. Then $X\subseteq \bar X$, $\overline{X\cup Y}$= $\bar X\cup \bar Y$, and $\overline {X\cap ...
2
votes
1answer
27 views

Is there a systematic way of finding the factorization over the closure of $\mathbb{Z}_2$ of $p(x) = x^{32} - x$?

And similar polynomials of the form $x^{p^n} - x$. I know that the degrees of the irreducible monic polynomials that factorize $x^{32} - x$ will have degree $d \vert 5 = 1, 5$. Also, I know that $x$ ...
0
votes
1answer
31 views

Is $f$ surjective and injective?

Let $f$ be a function from $f:\mathbb{N}\to \mathbb{Z}$ given by $f(a) = (-1)^aa$. Decide whether $f$ is injective and whether it is surjective. Which function do I start with in determining this? ...
0
votes
1answer
12 views

Basis of an eigenspace with complex eigenvalues

I understand how to find basis of an eigenspace. But currently it confuses me when I get complex eigenvalues. I have this matrix: $A=\begin{pmatrix} 1 & 1 & -1 \\ 0 & 1 & 0 \\ 1 & ...
0
votes
0answers
19 views

Closed forms in ${\mathbb C}^n$ are exact

Given a closed form $\omega$ over $\mathbb{C}^n$ with continuously differentiable (but not necessarily holomorphic) coefficients, how would one show that it is exact? More formally, we start with an ...
2
votes
2answers
31 views

In $\mathbb C^2$, Show that $\langle x,y\rangle=xAy*$ is an inner product.

$A= \left[ \begin{array}{cc} 1&i\\ -i&2 \end{array} \right] $ I've shown that (a) $\langle x,y+z\rangle=\langle x,y\rangle+\langle x,z\rangle$. (b) $\langle ...
0
votes
2answers
26 views

Help solve a Limit Question?

See this . What he's meant that "in particular"? where the $|g(x)|<|M|+1$ formula from? How deduced? What is the meaning of it?
1
vote
2answers
50 views

I am facing hard in complex analysis and so can't understand why the following steps happened?

If $z,z+\delta z$ are two points in complex plane and to differentiate a complex function $f(z)$ we do, $$\frac{df}{dz}=\lim_{\delta z \to 0} \frac{f(z+\delta z)-f(z)}{\delta z}$$ for $\delta ...
0
votes
0answers
7 views

What is the importance of Leray Schauder non linear alternative

I'm questioning the importance of the Leray-Schaulder alternative, isn't more simple to use directly the Schauder fixed point theorem. Especially that, to prove the alternative we use the fixed ...
1
vote
1answer
27 views

Why is $\frac{1}{x^{1/p} (\ln(x)^2+1)}$ in $L^1$ but not in $L^p$ for any $p>1$

From a practice qualifying exam, the goal is to find a function $f \geq 0$ on $(0,\infty))$ that $f \in L^p(0,\infty)$ iff $p=1$. One function suggested was: $$\frac{1}{x^{1/p} (\ln(x)^2+1)}$$ So ...
3
votes
0answers
23 views

Why do we know that , besides the known idoneal numbers , there is at most one more?

Here https://en.wikipedia.org/wiki/Idoneal_number the definition of an idoneal number is given : A number $n$ is idoneal if there are no integers $a,b,c$ with $0<a<b<c$ and $n=ab+ac+bc$ A ...
1
vote
0answers
15 views

How to show Plancherel's Theorem for Fourier Transform implies $L^2$ Transform Convergence.

The Plancherel Theorem for the Fourier transform $\hat{f}(s)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(t)e^{-ist}dt$ on $\mathbb{R}$ states that $$ ...
1
vote
0answers
15 views

Definable over $(\mathbb{R}, +, \cdot)$

I have the following task and I am not so sure about my solution: a) Is $\{0\}$ definable over $(\mathbb{R}, +, \cdot)$? b) Is $\{1\}$ definable over $(\mathbb{R}, +, \cdot)$? c) Is $<$ ...
0
votes
1answer
16 views

Exchangeability and independence of random variables

I have a question on the relation between exchangeability and independence between random variables. Consider the random vectors $$u_1:= \begin{pmatrix} \epsilon_{1}\\ \epsilon_2\\ \epsilon_3 ...
0
votes
0answers
15 views

Necessary and sufficient condition for lebesgue measurability

I'm trying to determine whether the following statement is true or not: If $E\subseteq \mathbb{R}$ is a set with $\lambda^{*}(E)$ finite ($\lambda^{*}(E)$ is the lebesgue exterior measure of $E$), ...
-1
votes
0answers
9 views

Divide the angle between two lines into angles and create lines for each angle

I have 5 points. (p1x,piy), (p2x,p2y), (p3x,p3y), (p4x,p4y) and (p5x,p5y) Line 1 is (p1-p2) and line 2 is (p1-p3). I want to divide the angle between these two lines and calculate lines from p1 ...
0
votes
2answers
16 views

Finding Maxima and Minima Values when the second derivative is a constant

I am given the following quadratic over the closed interval $[0,3]$ $f(x) = x-x^2$ I'm asked to find the value inside that interval that is the largest value and smallest value. I easily see that if ...
0
votes
3answers
51 views

Determine: $13^{-1} \pmod {67}$

Determine: $13^{-1} \pmod {67}$ I'm not sure how to deal with the negative one here as it inverts the integer? Any help would be appreciated!
0
votes
0answers
12 views

Good derivation of Romberg's method?

I'm confused about the Romberg's method even after viewing numerous explanations of it. I understand the "high-level" view of what's happening, but not how it's derived/proven. Can anyone recommend a ...
1
vote
0answers
30 views

Grasshoppers jumping on circle

Ten points $A_1,\dots,A_{10}$ are marked in clockwise order on a circle, so that $A_iA_{i+5}$ forms a diameter for all $1\leq i\leq 5$. Initially, a grasshopper is at each point. Every minute, one ...
1
vote
2answers
47 views

Do there exist geometric figures with multiple centers of symmetry?

Do there exist geometric figures with multiple centers of symmetry? Under what conditions does the center symmetry lie inside a geometric figure? Recall the definition of a center of symmetry: A ...
1
vote
0answers
10 views

Does a certain type of connected subset exist in Euclidean spaces having an arbitrarily high dimension

Let $\mathbb R$ be the set of all real numbers and for each positive integer $n$, let $f_n$ be a mapping of $\mathbb R^n$ into $\mathbb R$. For each positive integer $n$, does there exist an $f_n$ ...
1
vote
1answer
20 views

Functions over a $C$ vector space with geometric importance. (How to find the basis?)

Searching through our suggested exercises of linear and abstract algebra for solving, I found the following exercise. The reason I am posting this, is that because we haven't went through complex ...
0
votes
2answers
12 views

If we have a real line and $X$ is any subset of it, let $Y$ be a set such that $X\subseteq Y \subseteq \bar{X}$

If we have a real line and $X$ is any subset of it, let $Y$ be a set such that $X\subseteq Y \subseteq \bar{X}$. Prove that $\bar{X}=\bar{Y}$ My Attempt We can use definition and show that $\bar{X} ...

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