4
votes
2answers
132 views

Conservative Measures under a group action (reference request)

I was reading a paper and the author define the concept of conservative measure: Let $(X,\mathcal{B})$ a measurable space and $G$ a group that acts on $X$ by $$G\times X:(g,x)\mapsto T_g(x)$$ where ...
1
vote
0answers
54 views

Inverse with respect to a given circle

Determine the inverse with respect to a given circle $g:\mathbb{R}^{2} \to \mathbb{R}^{+}, g(x,y)=x^{2}+y^{2}$. I have looked around for non geometric derivations without finding any of value. ...
0
votes
0answers
37 views

How to find union and intersection of events?

I have sample space of experiment $S=\left\{x|-\infty<x<\infty\right\}$. I consider events $$A_i=\left\{x \;\middle|\;\frac{1}{2^{i-1}}\le x<\frac{3}{2^i}\right\};i=1,2...$$ And I want to ...
3
votes
5answers
103 views

When $a\to \infty$, $\sqrt{a^2+4}$ behaves as $a+\frac{2}{a}$?

What does it mean that $\,\,f(a)=\sqrt{a^2+4}\,\,$ behaves as $\,a+\dfrac{2}{a},\,$ as $a\to \infty$? How can this be justified? Thanks.
0
votes
1answer
67 views

Normed space and convex hull of closed subset

Let $(V, ||\cdot||)$ be a normed space. If $ C\subseteq V$ is a closed set we do not know if $ch(C)$ is closed or not. The professor provided this example that as of now I'm not getting: Consider the ...
0
votes
1answer
45 views

Bounded quantifier and it's meaning

It's explained in Velleman's how to prove book that $\exists x \in AP(x)$ means that there is at least one value of x in the set A such that P(x) is true. Then ...
0
votes
1answer
31 views

Finding pooled variance

Find the variance of $S^2_p$ under the conditions; $\bar{x_1}, \bar{x_2}, s_1, s_2$ are the means and standard deviations of independent random samples of sizes $n_1$ and $n_2$ from normal populations ...
3
votes
1answer
107 views

Isomorphism of ring localized twice - Atiyah Macdonald Exercise 3.3

I studied AM before studying universal properties. When I solved the following exercise, I had a tedious solution that involved dealing with elements. Let $ A $ be a ring with multiplicatively ...
3
votes
1answer
202 views

Hartshorne II prop 6.6

I'm having a really hard time understanding the proof of this proposition. $X$ is a noetherian integral separated scheme that is regular in codimension 1. We consider $X\times \mathbb{A}^1$ and the ...
0
votes
1answer
29 views

Calculating the limit of the “$\dfrac{volume}{area}$” ratio for a 2D function

Let's assume that we have a well behaving, continuous function $f(x,y)$ defined on $\mathbb{R^2}$. The double integral $\int_{x_0}^{x_1}\int_{y_0}^{y_1}f(x,y)dxdy$ gives the volume of the space ...
0
votes
2answers
28 views

Non-trivial centers, abelian towers, Lang

I am currently reading a proof in Lang's algebra that says that: because $G$ is a finite $p$-group, with non-trivial center "we have an abelian tower for $G/Z$ by induction, we can lift this abelian ...
1
vote
5answers
68 views

For small $x$, one has $\ln(1+x)=x$?

What does it mean that for small $x$, one has $\ln(1+x)=x$? How can you explain this thing ? Thanks in advance for your reply.
2
votes
3answers
147 views

Estimating the series: $\sum_{k=0}^{\infty} \frac{k^a b^k}{k!}$

Any idea on how to estimate the following series: $$\sum_{k=0}^{\infty} \frac{k^a b^k}{k!}$$ where $a$ and $b$ are constant values. Greatly appreciate any respond.
1
vote
1answer
181 views

Evaluate and find the principal value of $(-1+i)^ {2-i}$

Can anyone please help me evaluate and find the principal value of $(-1+i)^{2-i}$ I got up to $=e^{2-i}(ln(-1+i))$ $=e^{(2-i)(1/2 ln(2)+i(3pi/4))}$
0
votes
1answer
335 views

physical meaning of heat equation

consider the heat equation $u_t=a(t)u_{xx}+f(x,t)$, $0<x<L$, $0<t<T$ subject to the initial condition $u(x,0)=g(x)$ and boundary conditions $u(1,t)=0,$ $u_x(0,t)+hu(0,t)=0$ where ...
0
votes
1answer
76 views

Fixed Point Theorem in finite dimensional Euclidean space

A fixed point theorem says that: "any continuous mapping of $\mathbb{R}^n$ into a bounded subset of $\mathbb{R}^n$ has a fixed point". So consider $f: \mathbb{R}^n \rightarrow X \subset ...
1
vote
2answers
28 views

How to integrate $(x-1)^4/(x^2 )$?

How to integrate $\frac {(x-1)^4} {x^2 }$ ? I really tried hard but don't know how to start please guide me to just start thanks in advance
4
votes
2answers
3k views

What are the formal names of operands and results for basic operations?

I'm trying to mentally summarize the names of the operands for basic operations. I've got this so far: Addition: Augend + Addend = Sum. Subtraction: Minuend - Subtrahend = Difference. ...
0
votes
1answer
26 views

Finding a confidence interval

Given Distribution: $f(R) =\begin{cases} \frac{2}{\theta^2}(\theta-R), & \text{for } 0 <R<\theta \\[2ex] 0, & \text{elsewhere} \end{cases}$ Question: Find $c$ so that ...
1
vote
4answers
3k views

How do you find the number of multiples of a given range of numbers?

I know this sounds a bit stupid but this question always confounds me. Say that you are given a range of numbers like $20$-$300$. And it asks you to find how many multiples of $5$ are given in that ...
3
votes
1answer
99 views

Convex optimization: interpretation of the dual variable

Let us consider the convex optimization problem $$ \tag{P} \underset{x\in\mathbb R^n}{\sf minimize} ~~ f(x) ~+~ g({\bf L}x) $$ where ${\bf L}\in\mathbb R^{m\times n}$. Using the convex conjugate, ...
0
votes
1answer
18 views

Is an image of a function f for Dom(f) always equal to Rg(f)?

Let's look at the definition of an image of some function $f: A\rightarrow B$. An image of a function $f$ for some subset $X\subseteq A$ is $\{b\in B \text{ | }f(x)=b \text{ for some } x \in X\}$. ...
-2
votes
1answer
675 views

Diagonalizing a block matrix

Suppose the matrices $A,B,C,D$ are $M\times M$, $M\times N$, $N\times M$ and $N\times N$, respectively. Then I give you the following block matrix: $$ G = \begin{pmatrix} A & B \\ C & D ...
2
votes
1answer
121 views

Multiplication operator on $L^2$ and spectral theorem.

Let's consider the multiplication operator by the independent variable in $L^2(\mu)$, where $\mu$ is a borel regular measure on $\mathbb{C}$: $Mf(z)=zf(z)$. I want to show that if $\phi$ is a borel ...
1
vote
1answer
75 views

Find conditional probability $\mathbb{P}(X \le x | \max(X,Y)) $

Let $X,Y$ be iid such that $X\sim F>0$ and $Y \sim F>0$ ($X$ and $Y$ have the same probability distribution). Find $\mathbb{P}(X \le x | \max(X,Y)) $. I know that $\max(X,Y) \sim F^2$. I ...
-1
votes
1answer
37 views

Cumulative Distribution Function applied to exponential variables

Let P be a program composed by two sub-programs that have execution time of T1 and T2 distributed with exponential law of parameters u1 and u2. I have to calculate the Cumulative Distribution ...
2
votes
0answers
91 views

When can I leave the absolute value from Chebyshev's inequality?

I have a positive random variable which distribution is unkown, but its mean is $10$. I have to find an estimation of its variance, given, that $Pr(X\geq9$)=0.9980 I thought of Chebyshev's ...
0
votes
1answer
52 views

Is this vector identity accurate?

Does this identity hold true for vectors $A$, $B$ and the gradient operator? $(\nabla \cdot A)B = (A\cdot \nabla)B + (B\cdot \nabla)A$
1
vote
2answers
124 views

Linear Algebra - Prove $AB=BA$

Let $A$ and $B$ be any $n \times n$ defined over the real numbers. Assume that $A^2+AB+2I=0$. Prove $AB=BA$ My solution (Not full) I didn't managed to get so far. $A(A+B)=-2I$ ...
1
vote
0answers
13 views

Proving $Corr(\hat{e}_{ij}, \hat{e}_{jk}) = \frac{-1}{n_i-1}$ for $ j \neq k$

For the model of a single factor experiment: $y_{ij}= \mu + \alpha_i + e_{ij}$, $(1 \leq i \leq a, 1 \leq j \leq n_i)$, where a = the number of treatments, $n_i$ = the number of experimental units ...
2
votes
2answers
53 views

Probability Question about Full Houses

So I've figured out the probability of getting a full house. I want to show that P(getting a full house | my first card is the 9h) is the same. Essentially, I want to show that getting a full house ...
0
votes
1answer
46 views

Smallest irreducible periodic Markov chain

What would be the smallest periodic Markov chain? We're studying periodic Markov chains in my probability course. I'm just trying to picture the smallest possible one but I can't seem to come up ...
3
votes
1answer
80 views

A base of topology

Consider a space of smooth functions $C^{\infty}[a,b]$ and a set $$\tau=\left\{B(f,\varepsilon_0,\varepsilon_1...\varepsilon_r):f\in C^{\infty}[a,b],r\in\mathbb{N}\right\} $$ where $f$ is arbitrary ...
1
vote
1answer
79 views

The Car's cost is 9,00,000 and it is inclusive of local tax of 2%. How to find the original value of car?

The Car's cost is 9,00,000 and it is inclusive of local tax of 2%. How to find the original value of car? As far as calculation goes I calculated it as below 2% of 9,00,000 is 18,000 Final ...
2
votes
2answers
194 views

Proof that $[v, Tv, T²v, … , T^n v]$ is a basis for $V$ ($dim(V)=n$)?

Let $T:V\to V$ be a linear map from a finite dimensional vector space over a field $F$ to itself. Assume $[v,Tv,T²v,...]$ spans $V$ for some $v \in V$. Don't know at all how to prove that $[v, Tv, ...
1
vote
0answers
131 views

Calculating the probability of letter assignment

We have 10 letters written to 10 different friends and the 10 addressed envelops. The letters are put into the envelops at random, that is, all 10! assignment are equally likely. (a) What is the ...
2
votes
1answer
188 views

How many ways to place n distingusishable balls into m distinguishable bins of size s?

Let there be $n$ distinguishable balls and $m$ distinguishable bins, each bin of size $s$, that is, we cannot place more than $s$ balls into it. How many possibilites are there to place the balls into ...
0
votes
1answer
37 views

Given a function f that is continious, differentiable. Show that there is a c in the interval so that f'(c)=1.

I have a question regarding an exam question that I just had. There was a function f: [0,2]$\rightarrow \mathbb{R}$ that was continious on [0,2], differentiable on (0,2). Besides that f(0)=f(1)=0 ...
0
votes
1answer
49 views

Limit equivalence

"Let $f:A\subset\Bbb{R}^n\to\Bbb{R}$ be a function and denote $\Bbb{x}=(x_1,\dots,x_n)$ and $\Bbb{p}=(p_1,\dots,p_n)$. Show the following equivalence: ...
0
votes
2answers
52 views

combinatorics: even numbers

There are given 6 numbers: 1,2,4,6,7,8. I need to find out how many 5 number combinations are there ($A_6^5$=720) and how many of those combinations are even numbers. The numbers can't recur. The book ...
2
votes
1answer
42 views

Difference between contradiction and paradox?

In multivalued logic one can distinguish at contradictions (of the type $P\wedge\neg P$) and paradoxes (of type $P\leftrightarrow \neg P$). How about in mathematics? Does the appearance of ...
1
vote
0answers
29 views

gcd of product of exponents of prime factors and product of prime factors

Let $n = \prod\limits_i p_i^{k_i}$. I want to express $$ \gcd(\prod\limits_i k_i, \prod\limits_i p_i) $$ as an arithmetic function (i.e get rid of gcd). Is that possible? Thanks!
1
vote
2answers
142 views

Configuration of five or more mutually equidistant points in space.

How is it proved that there is no configuration of five or more mutually equidistant points in $R^3$? Is it done by induction? I'm stuck. Help would be appreciated. Well, surely equilateral ...
2
votes
1answer
55 views

On the limit of $S(x)=\sum\limits_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^x}$ when $x\to 0^+$

Sorry I haven't any ideas. Maybe it equals ${1 \over 2}$. $$\lim_{x\to0^+}\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^x}$$
0
votes
1answer
29 views

Does the specific bias random coins determines whether functions of these are independent or not?

Consider 3 independent r.v.s $X_1$, $X_2$, $X_3$ that represent the outcomes of three (independent) fair coin tosses. Let 1 denote heads and 0 denote tails. Let two new random variable be defined ...
1
vote
0answers
41 views

Prove identities-p norm

A $p-$ norm of $\mathbb{Q}_p$ is a function $||_p: \mathbb{Q}_p \to \mathbb{R}$ $$x \neq 0, x=p^{w_p(x)}u \mapsto p^{-w_p(x)}$$ $$\text{ For } x=0 \Leftrightarrow w_p(x)=\infty \\ p^{-\infty}:=0$$ ...
4
votes
1answer
67 views

Show that $\Gamma \cup \{\neg \phi\}$ is satisfiable if and only if $\Gamma\not \models \phi$

Let $\Gamma$ be a set of formulas and $\phi$ be a formula. Show that $\Gamma \cup \{\neg \phi\}$ is satisfiable if and only if $\Gamma\not \models \phi$. This seemed pretty obvious but I wanted ...
2
votes
2answers
72 views

Can $f:\Bbb{N}\rightarrow\Bbb{N}$ return an empty set?

I have a function $f:\Bbb{N}\rightarrow\Bbb{N}$. An empty set is not a member of $\Bbb{N}$. Can $f$ still return an empty set for some arguments $x\in\Bbb{N}$?
0
votes
0answers
112 views

Quotient Field of an Integral Domain

The question is: Let $n$ be a positive integer and let $D$ be the subset of all rational numbers of the form $$\frac{a}{n^k}$$ with $a \in \mathbb{Z}$ and $k$ any positive integer. Show that $D$ ...
1
vote
1answer
45 views

Bayesian Network/ Number of parameters

Please consider the following Bayesian Network out of $Graphical Models in Applied Multivariate Statistics" by Joe Whittaker: Now the factorization property says that the joint probability ...

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