1
vote
1answer
348 views

Infinite tensor product definition

My question is short and popular: how to define an infinite tensor product of modules over a ring? So, there is an infinite set $I$ and $A$-modules $M_i$. I should understand what $\otimes_{i\in I} ...
2
votes
1answer
125 views

Proof $d(x)=\text{GCD}(f(x),g(x))$

Question1: Proof: If $d(x)|f(x),d(x)|g(x)$, $d(x)$ is an combination of $f(x)$ and $g(x)$, then $d(x)$ is $\text{GCD}(f(x),g(x))$. My proof 1 $d(x)=u(x)f(x)+v(x)g(x)$. ...
3
votes
1answer
100 views

Automorphisms between two tuples of the same type

Fix some first order structure in some fixed language. If tuple $b$ is the image of tuple $a$ under some automorphism, then they have the same type. Is the converse true? That is, for any fixed tuples ...
1
vote
1answer
63 views

Question about the behaviour of $f(z)=e^{1\over z-\sin z}$ at $z = 0$

Let $f$ be given by $f(z)=e^{1\over z-\sin z}$. Then at $z = 0$, $f$: a) has a pole; b) has an essential singularity; c) has a removable singularity; d) is analytic. as $z\to0\...
0
votes
1answer
77 views

$f$ analytic such that $f (z)$ has only essential singularity

Let $f$ be analytic such that $1/f(z)$ has only essential singularity. Then which of the following hold? a) $f$ must be a polynomial. b) $f$ cannot be a polynomial. c) $f (1/z)$ must have a pole. ...
5
votes
2answers
138 views

nonlinear pde equation

I want to solve the problem: find some non-trivial particular solution of nonlinear PDE. Are the any methods for this? I understand that there is no general method to find general solution, but.. One ...
1
vote
1answer
297 views

Calculate the yield rate on transaction

Problem:- 'A' is able to borrow $1000 from 'B' for one year at 8% effective and at the same time lend it to 'C' for one year at 10% effective. what is 'A's yield rate on this transaction? My answer:-...
1
vote
1answer
81 views

properties of block matrix

$X=\begin{pmatrix}A&B\\C&D\end{pmatrix}$, $2n\times 2n$ matrix Then a) A, B,C,D are nilpotent ⇔ X is nilpotent b)If X is diagonalisable so is A,B,C,D. c)min polynomial of X divides the ...
0
votes
1answer
103 views

Limit to zero of two variables

Given functions $f:\mathbb{R}\rightarrow\mathbb{R},g:\mathbb{R}^2\rightarrow\mathbb{R}$, suppose that the values $$A=\lim_{h\rightarrow 0}\left(f(h)\lim_{i\rightarrow 0}g(h,i)\right)$$ and $$B=\lim_{h\...
2
votes
1answer
71 views

Remainders of compactifications are images of the Stone-Čech remainder.

I need to show that if $\gamma X$ is a compactification of $X$, then $\gamma X\setminus X$ is the continuous image of $\beta X\setminus X$. I know that there exists a continuous function from $\beta ...
2
votes
1answer
684 views

Proving principle of the Iterated Suprema

Let $X$ and $Y$ be nonempty sets and let $h : X\times Y \to R$ have bounded range in $\mathbb{R}$. Let $F: X \to\mathbb{R}$ and $G : y \to \mathbb{R}$ be defined by $F(x):=\sup\{h(x,y) : y\in Y\}$, ...
1
vote
1answer
107 views

what is no. of positive eigen value of symmetric matrix A with some given relationship

Suppose A is a 3*3 symmetric matrix s.t. $$\begin{pmatrix} x & y & 1 \\ \end{pmatrix} A \begin{pmatrix} x \\ y\\ 1\end{pmatrix} = xy -1 $$ let p be the no. of positive eigen value of ...
-1
votes
2answers
165 views

related rates (calculus) questions [closed]

The minute and hour hands of the GPO clock are 2m and 1.5m long respectively. How fast are their ends approaching at (a) 2 o'clock (b) 6 o'clock? A fuel storage tank is in the shape of a right ...
0
votes
1answer
79 views

construct two rational numbers

Can one explain me a bit or more about how to construct the two rational numbers? From 《Principles of Mathematical Analysis》page 2 in proving $\sqrt{2}$ is not a rational number. $p>0$ $$q=p-\...
3
votes
3answers
83 views

$\lim_{x\rightarrow 0^+}x^x$

How can I calculate $\lim_{x\rightarrow 0^+}x^x$? I can only write it in the form $e^{x\ln x}$. I would like to use L'Hospital rule somehow, but I can't write it in form of fractions.
12
votes
3answers
626 views

Motivation for Jordan Canonical Form

I took linear algebra and understood the proof that linear operators on a vector space over an algebraically closed field have a Jordan Canonical Form. Why should I care about this theorem? I ...
1
vote
0answers
540 views

Understanding Sum of product and complete sum of product

I have a pair of problems, the first two of my homework, and I'm already unclear on how finding SOP and CSOP for them work. The first: E=xy(1+z)y' It seems like this just reduces to 0, since 1+z ...
5
votes
3answers
841 views

Does the product of positive definite matrices have a positive trace

Let $ A_{1}, A_{2}, \ldots, A_{m}$ be a real symmetric semi-positive definite matrices, I want to know whether $ tr (A_{1} \cdot A_{2} \cdots A_{m} ) \geq 0$ ? When $m=2$, it seems a rather ...
3
votes
1answer
86 views

Which of these statements about $2 \times 2$ real matrices $A$ such that $A^5 = I$ are true?

There exists a $2 \times 2$ real matrix $A$ such that $A^5 = I$. a) $A$ must be identity. b) $A$ must be similar to an element of $SO(2)$ . c) $A$ must be diagonalisable. I have ...
2
votes
1answer
78 views

Explicit expression of Galois group of $x^8-5$ over $\mathbb{Q}$

Let $F$ be the splitting field of $x^8-5$, we have $F=\mathbb{Q}(5^{1/8},i,\sqrt{2})$. I need to calculate the galois group Gal$(F/\mathbb{Q})$. I know the three generators are \begin{align*} &\...
0
votes
1answer
2k views

Graph Probability Density Function in Matlab

I am having a lot of trouble trying to graph this function in matlab and trying to get it to look like it should. I need to generate 200 random samples of a uniform random variable X on the range (0,1)...
1
vote
0answers
60 views

Which of these statements about multivariable limits are true?

Let $f :\mathbb{R}^2 \to \mathbb{R}$ be a map. Then a) $\lim\limits_{x\to 0} \lim\limits_{y\to 0} f (x, y)$ exists implies $\lim\limits_{(x,y)\to 0} f (x, y)$ exists. b) $\lim\limits_{(x,y)\...
1
vote
2answers
338 views

Parametrization of $x^2+y^2=z^2$

How can we show that any point on $x^2+y^2=z^2$ can be written in the form (z $\cos(\theta)$, z $\sin(\theta)$, z) for some $\theta$? Here is how I tried to approach it: $$(z \cos(\theta))^2+(z \sin(\...
6
votes
2answers
647 views

Maximal ideals in rings of polynomials [duplicate]

Let $k$ be a field and $D = k[X_1, . . . , X_n]$ the polynomial ring in $n$ variables over $k$. Show that: a) Every maximal ideal of $D$ is generated by $n$ elements. b) If $R$ is ring and $\...
1
vote
1answer
344 views

If $AB=BA=0$ and $A+B$ is invertible, is $A-B$ invertible?

Let $A$ and $B$ be two $n\times n$ matrices such that $AB=BA=0$ and $A+B$ is invertible. Is it true that $A-B$ is also invertible?
4
votes
1answer
182 views

Prove that $A$ has Lebesgue measure $0$.

Suppose $G$ is a connected open set of $\mathbb{C}^n$. Prove that: (1). If $f \in$ PSH(G) and $f \not \equiv \infty$ then $A=\{z \in G: f(z)=-\infty\}$ has Lebesgue measure $0$. (2). If $f \in \...
1
vote
1answer
339 views

The set of all nilpotent matrices is closed. Is it connected?

Prove that the set of all $n\times n$ nilpotent matrices is a closed set when considered as a subset of $\mathbb{R}^{n^2}$ and the metric is the usual euclidean metric. Also if it is connected as a ...
1
vote
0answers
324 views

Expected Value as an Integral without using Fubini's Thm

For a positive random variable $X$, show that $ \mathbb{E}(X) = \int_{0}^{\infty} \mathbb{P}(X > t ) \ dt $. This question has been asked before, and a solution is given here: Integral of CDF ...
4
votes
1answer
81 views

Commuting squares in abelian categories

Here $A,B,C$ and $D$ are all objects in an Abelian category. $\require{AMScd} \begin{CD} A @> >> B @> >> C;\\ @VVV @VVV @VVV\\ D @> >>E @> >> F; \end{CD} $ The ...
4
votes
4answers
179 views

Showing that a transformation $T:\mathbb R^3 \to \mathbb R^2$ is linear

OK, I am trying to prove the following transformation is linear, and find the basis for $\ker(T)$ and Im$(T)$ (also denoted in our textbook by $N(T)$ and $R(T)$ ). Then we're suposed to find the ...
4
votes
0answers
599 views

Show that if $S$ is a subspace of a vector space $V$, then $\dim (S) \leq \dim (V)$

Show that if $S$ is a subspace of a vector space $V$, then $\dim(S) \leq \dim(V)$. Furthermore, if $\dim(S)= \dim(V) < \infty$ thn $S=V$. Give an example to show that the finiteness is required in ...
1
vote
3answers
69 views

Understanding Equivalence and Relations

Can someone please explain these answers? I have reviewed the slides and read about properties of equality but I still don't understand how to apply it to these sets. For each the following relations ...
1
vote
1answer
74 views

Why doesn't sign (appear to) change in inequality?

Given the equation $\frac{1}{x}\gt -1$, I would assume one would work it as $$\frac{1}{x}\gt-1$$ $$x\cdot\frac{1}{x}\gt-1\cdot x$$ $$1 \gt -x$$ $$-1\cdot 1 \gt -x \cdot -1$$ $$-1 \lt x $$ which is ...
1
vote
1answer
78 views

When is gradient flow an isometry?

$M$ is a Riemannian manifold,$f$ is a function on $M$. Under what conditions is the gradient flow $F(t)$ of $f$ an isometry for $t>0$?
5
votes
1answer
1k views

What is the difference between a ball and a neighbourhood?

I am presently reading chapter two of Rudin, Principles of Mathematical Analysis (ed. 3). He provides the following definitions: Definition: If $\boldsymbol{x} \in \mathbb{R} ^ k$ and $r > 0$, the ...
1
vote
1answer
550 views

Normal Distribution Probability

At a large publishing company, the mean age of proofreaders is 36.2 years, and the standard deviation is 3.7 years. Assume the variables are normally distributed. a. If a proofreader in the company ...
2
votes
1answer
100 views

Gradient flow of harmonic function is measure preserving?

Let $M$ be a manifold with $Ric \ge - \left( {n - 1} \right)$ and $f:{B_R}\left( p \right) \to R$ is a lipschitz and harmonic function. In a paper, it says "As the gradient flow ${\Phi _t}$ of $f$ is ...
1
vote
0answers
17 views

Double Angle Identities [duplicate]

Given that $\sin(2x)=\frac 5 {13}$ and $0^\circ < x < 45^\circ$, I need to find $\cos(x)$ and $\sin(x)$. If I work it out, I get $$\begin{gather} \frac 5 {13} = 2 \sin(x) \cos(x) \\ \frac 5 {26}...
8
votes
1answer
94 views

For any three vectors $x,y,z\in\mathbb{R}^d$, we have $ \|y-z\|\cdot\|x\|\leq\|x-y\|\cdot\|z\|+\|z-x\|\cdot\|y\|$

Does anyone know a proof of the following problem? Problem: Show that for any three vectors ${\bf x}, {\bf y}, {\bf z}\in \mathbb{R}^d$ the following holds, $$ \|{\bf y} - {\bf z}\|\cdot \|{\bf x}\| \...
2
votes
1answer
51 views

Does $n \log\left(\cos\left(\frac{\pi\,n!}{n^2}\right )\right ) \neq 0 \implies n = p$?

Let $p$ denote a prime $> 3$. Take any $\text{odd }n\geq3,\;n \in \mathbb{N}$. How could one (dis)prove that: $$n \log\left(\cos\left(\frac{\pi\,n!}{n^2}\right )\right ) \neq 0 \implies n = p$$ ...
6
votes
1answer
328 views

$k$-cells: Why $a_i < b_i$ instead of $a_i \le b_i$

In Rudin, The Principles of Mathematical Analysis, there is the following definition: Definition: If $a_i < b_i$ for $i=1,2,...,k$, the set of all points, $ \boldsymbol{x} = ( x_1, x_2, ..., ...
1
vote
1answer
59 views

Calculate the value of x, y and z coordinates

System of the equations: Equation 1: $$-360 = -6x + x^2 - 40y + y^2 - 20z + z^2$$ Equation 2: $$-600 = -40x + x^2 - 30y + y^2 - 100z + z^2$$ Equation 3: $$59.85 = -10x + x^2 - 4y + y^2 - 10z + z^...
2
votes
1answer
54 views

If $(M_{\lambda})$ be a chain of closed affine subspaces of $X$ $\Longrightarrow$ $\displaystyle \bigcap_{\lambda\in L} M_{\lambda}\neq \emptyset \;$?

Let $X$ be a Banach space Let $(M_{\lambda})_{\lambda \in L}$ be a chain of closed affine subspaces of $X$ We can say that $$\displaystyle \bigcap_{\lambda\in L} M_{\lambda}\neq \emptyset \;\;?$$ ...
1
vote
3answers
147 views

Doubt with bounds and integrand of $\int_{0}^{2\pi}\int_{0}^{\pi/2}\int_{0}^{2}\rho^2\sin{\phi}d\rho d\phi d\theta$

Question as follows. Find the volume of the solid enclosed between the spheres $x^2+y^2+z^2=4$ and $x^2+y^2+z^2=4z \Leftrightarrow x^2+y^2+(z-2)^2=4$. I constructed the following integral and after ...
0
votes
1answer
1k views

Range of Uniform Distribution

I'm trying to compute the density for the range $R_n$ for samples of a random variable $X$ that are uniformly distributed on the interval $(a,b)$. We define the range as $$ R_n = X_{(n)} - X_{(1)}, $...
3
votes
2answers
78 views

Can $\mathbb{R}$ be obtained as union of disjoint translations of a dense subset?

Doing some homework I had to find out maximal subgroups of $\mathbb{R}$ and my first approach was that subgroups of $\mathbb{R}$ are discrete or dense and, of course, a maximal subgroup $G$ can't be ...
6
votes
4answers
310 views

Greatest integer $n$ where $n \lt (\sqrt5 +\sqrt7)^6$

I'm really not sure how to do this. I factored out a power of $3$ and squared so that I have $2^3 (6+\sqrt{35})^3 \gt n$ , and I know that if I can prove that $12^3-1 \le (6+\sqrt{35})^3 \lt 12^3$ I ...
2
votes
1answer
330 views

Making exponent of $a^x$ object of the function

Is it possible to make a variable the subject of a formula when it is an exponent in the equation? For example: $$y=a^x\quad a\;\text{is constant}$$ For example, let the constant $a = 5.$ $$ \begin{...
-1
votes
2answers
421 views

Related rates question - wheat running from a hole at a constant rate to make a cone

Wheat runs from a hole in a silo at a constant rate and forms a conical heap whose base radius is treble the height. If after 1 minute, the height of the heap is 20 cm, find the rate at which the ...
3
votes
3answers
187 views

Show $X=\left\{x \in [0,1]: x \neq \frac1n\text{ for any }n \in \Bbb N\right\}$ is neither compact nor connected

I am stuck on the following question: Let $X=\{x \in [0,1]: x \neq \frac1n: n \in \Bbb N\}$ be given the subspace topology. Then I have to prove that $X$ is neither compact nor connected. Can ...

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