instruction stringlengths 12 30k |
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In a boolean algebra, 0 (the lattice's bottom) is the identity element for the join operation $\lor$, and 1 (the lattice's top) is the identity element for the meet operation $\land$. For an element in the boolean algebra, its inverse/complement element for $\lor$ is wrt 1 and its inverse/complement element for $\land$... |
$\displaystyle\lim_{\epsilon \rightarrow 0} \int \frac{1}{x^{1+\epsilon}} \mathrm{d}x $
How should I go about evaluating this integral?
Does this integral converge to $log_e x $ or to something else?
|
These operators are written in different forms in Cartesian,
cylindrical and spherical coordinates. For instance, in spherical
coordinate system, one has
$$\nabla \cdot \overrightarrow{F}=\frac{1}{r^{2}}\frac{\partial }{\partial r}
\left( r^{2}F_{r}\right) +\frac{1}{r\sin \theta }\frac{\partial }{\partial
\theta... |
Do Vector Calculus Cartesian coordinates identities with Div, Grad, Curl hold in cylindrical and spherical coordinates? |
Suppose I have a binary vector $v$ that is copied twice by two separate machines $M$ and $N$, resulting in two new vectors $x$ and $y$.
Both machines are faulty in the sense that they both might fail to copy each character correctly. More formally, $M$ has an associated vector $e_M$ of length $|v|$, where $(e_M)_i$... |
In MK (Morse-Kelley) set theory life is easy: $\forall X\forall y\left(y\in\bigcap X\leftrightarrow\forall x\left(x\in X\rightarrow y\in x\right)\right)$. If $X=\left\\{\right\\}$ then $\bigcap X=U$, where $U$ is the universal class. So the (unary) intersection of the empty set is the class that contains all sets as el... |
The only other way I can think of is to use specification, which is really the only way you know that such a set exists. Given a universe $U$, define $\bigcap X$ to be the unique $B$ satisfying the following formula:
$$\forall x(x\in B\Leftrightarrow(x\in U\wedge \exists y(y\in X\wedge x\in y)))$$
Then $\bigcap\... |
Hey guys, I have a couple things I want to prove. I'm pretty sure a proof by induction is the best route for these.
----------
First, I need to show that $5^n < n!$ from some $n_{0} > 0$. I'm choosing $n_{0} = 12$ since that's the smallest positive integer where $n! > 5^n$.
So: $P(k) : 5^k < k!$
1. Show... |
While proving that some normed spaces were complete, two questions came to my mind. They relate the topological and the metric structures induced by a norm.
1) Is it possible to find two equivalent norms $\|\cdot\|_1$ and $\|\cdot\|_2$ on a vector space $V \ $ such that $(V \ ,\|\cdot\|_1)$ is complete and $(V \ ,\|... |
These operators are written in different forms in Cartesian,
cylindrical and spherical coordinates. For instance, in spherical
coordinate system, one has
$$\nabla \cdot \overrightarrow{F}=\frac{1}{r^{2}}\frac{\partial }{\partial r}
\left( r^{2}F_{r}\right) +\frac{1}{r\sin \theta }\frac{\partial }{\partial
\theta... |
If you don't require trapezoids of the same width, you can get a closed form by taking a partition of $[0,n]$ into intervals with endpoints $n,nq,nq^2,\dots nq^k$ for $q<1$ and $k$ large (or infinite), or by using $(i/n)^2, i=0 \dots n$ as the endpoints. The first approach leads to the sum of a geometric series and t... |
If you don't require trapezoids of the same width, you can get a closed form by taking a partition of $[0,n]$ into intervals with endpoints $n,nq,nq^2,\dots nq^k$ for $q<1$ and $k$ large (or infinite), or by using $(i/n)^2, i=0 \dots n$ as the endpoints. The first approach leads to the sum of a geometric series and t... |
I have an inequality:
$n \ge C\log^k n \times (A\log(n^r\log^l n) + B)$
I want to turn it into an inquality
$n > F(A,B,C,k,l,r)$
that implies the first inequality, while making it $F$ as tight as possible.
How can I do that? |
how do I transform an inequality on $n$? |
I am going to assume that you meant to state that this was a faithful representation, meaning that the map $G \to GL_n(\mathbb{C})$ has no kernel. Otherwise, as Akhil says, the problem is dramatically underconstrained.
An infinite class of examples is the semidriect product of $S_5$ acting on $(\mathbb{Z}/N)^4$, for... |
While proving that some normed spaces were complete, two questions came to my mind. They relate the topological and the metric structures induced by a norm.
1) Is it possible to find two equivalent norms $\|\cdot\|_1$ and $\|\cdot\|_2$ on a vector space $V \ $ such that $(V \ ,\|\cdot\|_1)$ is complete and $(V \ ,\|... |
Suppose I'm given a very slowly converging sequence $\sum a_k$. In the literature the Levin's U- transfomation is mentioned as a good universal technique for convergence acceleration.
I'm have difficult understanding how this method actually works and what the transformation actually is. Can anyone help?
After ... |
Consider the function $Pt(n)$. It tells us how many Pythagorean Triples there are (below $n$) when any argument $n \in \mathbb{N}$ is plugged in. Is there an exact formula that describes $Pt(n)$ ?
Max
P.S. I presume the limit of $Pt(n)/n$ is zero as $n$ tends to infinity. Can this presumption be verified with a p... |
I am of, and I would like to retain a, mindset that mathematics does not have to have numbers as the central object of interest. With that in mind, I have done a fair amount of self-study on topics in undergrad modern algebra, looking for examples to show this is the case.
For objects like groups, it is often stated... |
Are there broad or powerful theorems of rings that do not involve the familiar numerical operations (+) and (*) in some fundamental way? |
Going closer to the foundations of mathematics than most ever need or are comfortable with, we can also write down the classic model of $\mathbb{N_0}$ in ZF. This is of course going to be _very_ bird's eye view and rather devoid of details.
"There exists a set, $I$, such that $\emptyset \in I$ and $x \in I \Rightarr... |
Consider the function $Pt(n)$. It tells us how many primitive Pythagorean Triples there are (below $n$) when any argument $n \in \mathbb{N}$ is plugged in. Is there an 'exact formula'; e.g. an elementary function of even a combination of known special functions like the Gamma and Error Function, that describes $Pt(n)$ ... |
Suppose I have a variable $s$ that has a geometric distribution with success parameter x. So the probability of success on trial $s$ is
$p_s = (1 - x)^{s - 1} x$,
Consider the following function of $s$
$V(s) = \delta^s$,
where $\delta < 1 $. Then the expected value of this function is given by
$\su... |
Consider the function $Pt(n)$. It tells us how many primitive Pythagorean Triples there are (below $n$) when any argument $n \in \mathbb{N}$ is plugged in. Is there an 'exact formula'; i.e. an elementary function of even a combination of known special functions like the Gamma and Error Function, that describes $Pt(n)$ ... |
I have a homework assignment to find the two smallest probable primes with 12 digits, where a probable prime is defined as a number such that a^p-1 = 1 (mod p), where a = 2,3,5,7.
I'm obviously not asking for this to be done for me, but I'm having trouble figuring out a way to do anything other than to brute-force ... |
I'm trying to find a factorization for $f(z) = z - c$ in inner and outer functions.
So, first I try to find for what $c$, $f$ is cyclic for the unilateral shift $M_z$. There is a theorem that states that if $A$ is a bounded operator on a Hilbert space $H$ then $x \in H$ is cyclic for $A$ iff zero is the only element... |
I'm trying to find a factorization for $f(z) = z - c$ in inner and outer functions.
So, first I try to find for what $c$, $f$ is cyclic for the unilateral shift $M_z$. There is a theorem that states that if $A$ is a bounded operator on a Hilbert space $H$ then $x \in H$ is cyclic for $A$ iff zero is the only element... |
Let $f : {\cal S}_+^n \mapsto \mathbb{R}$ be a function defined as
$f(Q) := {\rm tr} WQ^{-1}W + {\rm tr} Q$
where $W \in {\cal S}_{+}^{n}$ is a symmetric positive definite matrix.
How to compute a subdifferential of $f$ at a point $Q \in {\cal S}_+^n$ for which $f(Q) < \infty$?
Note that $f(Q) = \infty$ f... |
How to compute the subdifferential of a variational represention of the trace norm? |
I'm trying to find a factorization for $f(z) = z - c$ in inner and outer functions.
So, first I try to find for what $c$, $f$ is cyclic for the unilateral shift $M_z$. There is a theorem that states that if $A$ is a bounded operator on a Hilbert space $H$ then $x \in H$ is cyclic for $A$ iff zero is the only element... |
I'm trying to find a factorization for $f(z) = z - c$ in inner and outer functions.
So, first I try to find for what $c$, $f$ is cyclic for the unilateral shift $M_z$. There is a theorem that states that if $A$ is a bounded operator on a Hilbert space $H$ then $x \in H$ is cyclic for $A$ iff zero is the only element... |
I'm trying to find a factorization for $f(z) = z - c$ in inner and outer functions.
So, first I try to find for what $c$, $f$ is cyclic for the unilateral shift $M_z$. There is a theorem that states that if $A$ is a bounded operator on a Hilbert space $H$ then $x \in H$ is cyclic for $A$ iff zero is the only element... |
can KL divergence converge to 0, but variance of log odds not converge to 0? |
We know that the Mandelbrot set is derived from the iterations of z^2 + c.
Do anyone know something about magnet Mandelbrot? I found it in the software UltraFractal, and it is much more beautiful than the original Mandelbrot, in my opinion.
Do you know anything about it? What is it iterating?
EDIT: Here's a pi... |
If x is an element in a standard convex linear optimization set constrained by Ax = b, x>=0
Then how can I prove d is a feasible direction only if Ad=0 and di>=0 for every i where xi=0? |
If $x$ is an element in a standard convex linear optimization set constrained by $Ax = b, x \geq 0$, then how can I prove $d$ is a feasible direction only if $Ad=0$ and $di \geq 0$ for every $i$ where $xi=0$? |
Reffered here by http://mathoverflow.net/questions/41750/determining-n-in-sigma-x0n
I'm not entirely sure if this question falls under MathOverflow but neither of my Calculus AP teachers in high school could help me with this:
Given $\Sigma_{x=0}^n {f(x)\over2}$ and the output of the summation, how would you fin... |
How do you prove $|\sqrt[3]{x}$ - $\sqrt[3]{y}|$ $\leq$ $\sqrt[3]{|x-y|}$?
|
I can't seem to understand Generalized Set Operations. Can someone explain it for me? For example, in Example 1.12, suppose `k = 3` then set `A(of k) = {3, 4, 5, 6 ...}`, then how is `{3,4,5 ...} = {1,2,3 ...}`
![alt text][1]
[1]: https://i.stack.imgur.com/NvXFE.jpg |
I'm facing a practical problem where I've calculated a formula that, with the help of some programming, can bring me to my final answer. However, the numbers involved are so big that it takes ages to compute, and I think it might not be necessary. I have the following formula,
$\sum\limits^{k}_{i=m}(N-i)^{k-i}(\fra... |
California issues license plates in numeric order (if we turn the letters into numbers). I have fun noticing the latest plate I have seen. I am interested in what you can derive from a series of these observations. I understand that sampling from $\\{1,2,3...n\\}$ the only useful data is the highest value you have s... |
How do you prove $|\sqrt[3]{x} - \sqrt[3]{y}|$ $\leq$ $\sqrt[3]{|x-y|}$?
|
I've been trying to work out the best way to understand why Fourier series converge, and it's a little embarrassing but I don't even know a rigorous proof. Can someone please help put me on the right track to thinking about these issue's in the proper way? I am especially interested any geometric ways to think about ... |
**HINT** $\ \ \rm bd = ac\ \Rightarrow\ (a,b)\:(b,c)\ =\ b\: (a,b,c,d)\ $ via GCD distributive, associative laws.
See also [my post][1] on Euler's four number theorem (Vierzahlensatz), Riesz interpolation, Schreier refinement.
[1]:http://groups.google.com/groups?selm=y8z7k9leoku.fsf%40nestle.ai.mit.edu |
Is there a simple exlamation of what the Laplace transformation do exactly and how they work? Reading my math book has left me in a foggy haze of proofs that I don't completely understand. I'm looking for an explanation on lamans terms so that I understand what it is doing as I make these seemingly "magical" transform... |
This is probably a silly question, but a couple of people that I have talked to have had different responses.
Does $C_0$ denote the set of continuous functions with compact support or the set of continuous functions which vanish at infinity? |
Going closer to the foundations of mathematics than most ever need or are comfortable with, we can also write down the classic model of $\mathbb{N_0}$ in ZF. This is of course going to be _very_ bird's eye view and rather devoid of details.
"There exists a set, $I$, such that $\emptyset \in I$ and $x \in I \Rightarr... |
Think of $t$ and $r$ as two independent variables.
- Suppose $E$ be a function of $r$ and $V$ be a function of $(t,r)$ such that both go to $0$ at $r=0$.
- There exists a positive function $M(r)$ such that $M(0)=0$ and $V(t,r) = -\frac{M(r)}{R(t,r)}$ where $R$ is another positive function such that $R(0,r)=r$... |
Consider a candy bag that contains **N=100** candies. There are only two types of candy in the bag. Say the caramel candy and the chocolate candy. Nothing more is known about the contents of the bag.
Now, you are going to draw (randomly) one candy at a time from the bag until the first caramel appear. Suppose that t... |
Seems like such polyhedra are called _not necessarily closed_ (NNC) and are usually represented as closed polyhedra with additional dimension $\varepsilon$: every strict inequality $a_{i1}x_1 + \ldots + a_{in}x_n > c_i$ is replaced by $a_{i1}x_1 + \ldots + a_{in}x_n - \varepsilon \geq c_i$ and two additional inequaliti... |
We know that the Mandelbrot set is derived from the iterations of z^2 + c.
Do anyone know something about magnet Mandelbrot? I found it in the software UltraFractal, and it is much more beautiful than the original Mandelbrot, in my opinion.
Do you know anything about it? What is it iterating?
EDIT: Here's a pi... |
I don't know about a geometric interpretation, but here is a brief sketch of a proof. First we need to be precise about what we mean by "convergence." In the naive sense, Fourier series *don't* always converge - that is, pointwise. (If you change the value of a function at a single point, the Fourier series remains ... |
Let G be a group with generators and relations.
I know that in general it is difficult to determine what a group is from its generators and relations. I am interested in learning about techniques for figuring out the order of a group from the given information.
For example, I know that if the number of generators... |
The question you are asking here is the classical question of [inferential statistics][1]: "Given the outcome of an experiment, what can be said about the underlying probability distribution?"
You could, for example, give an [estimator][2] for the unknown quantity "number of caramels" (called $a$ from here on). The ... |
The question you are asking here is the classical question of [inferential statistics][1]: "Given the outcome of an experiment, what can be said about the underlying probability distribution?"
You could, for example, give an [estimator][2] for the unknown quantity "number of caramels" (called $a$ from here on). The ... |
Show that a number is not prime? |
Let $p(x)$ be a fixed distribution.
Let $A, C > 0$ be constants.
Let $\epsilon > 0$. Can we find an example of a distribution $q_{\epsilon}$ such that $\mathrm{KL}(p||q_{\epsilon}) < \epsilon$, but $E[(\log (p(x)/q(x)))^2] \ge A * \epsilon^C$?
If $p$ and $q_{\epsilon}$ would be a coin toss or a geometric distrib... |
Given $n$ smooth real functions $f_1, f_2, \dots, f_n$, define a composite function like this:
$$f(x) = \max(f_1(x), f_2(x), \dots, f_n(x)) - \min(f_1(x), f_2(x), \dots, f_n(x))$$
Is it possible to say anything useful in general about the shape of this function?
Intuitively, it seems like $f$ will be at least ... |
In [previous question][1], I asked how one would simplify the following equation for the case where the variables are very big:
$\sum\limits^{k}_{i=m}(N-i)^{k-i}(\frac{1}{N})^k\frac{k!}{(k-i)!i!} \leq a$
This answer was basically to use an approximation like Stirling's formula. Having implemented this with some c... |
The standard approach for showing $\int \sec \theta d \theta = \ln |\sec \theta + \tan \theta| + C$ is to multiply by $\frac{\sec \theta + \tan \theta}{\sec \theta + \tan \theta}$ and then do a substitution with $u = \sec \theta + \tan \theta$. I like the fact that this trick leads to a fast and clean derivation, but... |
In [previous question][1], I asked how one would simplify the following equation for the case where the variables are very big:
$\sum\limits^{k}_{i=m}(N-i)^{k-i}(\frac{1}{N})^k\frac{k!}{(k-i)!i!} \leq a$
This answer was basically to use an approximation like Stirling's formula. Having implemented this with some c... |
Using the definitions $$sec \theta = 1/\cos \theta \quad \text{and} \quad cos \theta = (\exp(i \theta) + \exp(-i \theta))/2$$ gives $$\int \sec \theta d \theta = \int \frac{2 d \theta}{\exp(i \theta) + \exp(-i \theta)}.$$ The only insight needed is to find the substitution $u = \exp( i \theta )$ (what else is there ... |
**HINT** $\ \ \:\rm bd = ac\ \Rightarrow\ (a,b)\:(b,c)\ =\ b\: (a,b,c,d)\ $ via GCD distributive, associative laws.
Alternatively, if you know $\rm\ [a,b]\: (a,b) = ab\ $ then
we have $\rm\ \ a,b\:|\:ac\ \:\Rightarrow\ \: [a,b]\:|\:ac\ \:\Rightarrow\:\ ab\:|(a,b)\:ac\ \Rightarrow\ b\:|\:(a,b)\:c$
Doing the sam... |
**HINT** $\ \ \ \rm bd = ac\ \:\Rightarrow\:\ (a,b)\:(b,c)\: =\ b\: (a,b,c,d)\ $ via GCD distributive, associative laws.
Alternatively, if you know $\rm\ [a,b]\: (a,b)\: =\: ab\ $ then by this LCM*GCD law
we have $\rm\ \ \: a,b\:|\:ac\ \:\Rightarrow\ \: [a,b]\:|\:ac\ \ \Rightarrow\:\ ab\:|(a,b)\:ac\ \Rightarrow\ ... |
I have the following sequences:
a = [0, 1, 2, 3, 4, 5]
b = [1, 2, 4, 7, 13, 26]
c = [1, 1, 2, 4, 7, 13]
I'm trying to reverse-engineer an undocumented API and believe that the values in `a` can be used to determine the corresponding values in `b` and `c`. For instance, we could say that third elemen... |
Using stereoscopic cameras, I track a certain object through it's path in space. For every frame, I compute it's pose in 3D. I can represent it's pose by either a translation vector + rotation matrix or a 6 element parameter vector (X, Y, Z, roll, pitch, yaw).
The problem is that the output generated by the cameras ... |
Let $G = \langle x , y \ | \ x^{12}y=yx^{18} \rangle$. I want to continue a discussion [here](http://math.stackexchange.com/questions/6376/normal-closure-in-groups) on the normal closure of $y$ in $G$.
How can we determine a group presentation for $N$? |
**HINT** $\ \ \ \rm bd = ac\ \:\Rightarrow\:\ (a,b)\:(b,c)\: =\ b\: (a,b,c,d)\ $ via GCD distributive, associative laws.
Alternatively, if you know $\rm\ [a,b]\: (a,b)\: =\: ab\ $ then by this LCM*GCD law
we have $\rm\ \ \: a,b\:|\:ac\ \:\Rightarrow\ \: [a,b]\:|\:ac\ \ \Rightarrow\:\ ab\:|(a,b)\:ac\ \Rightarrow\ ... |
Is there an algorithm out there that will show you how to fit the most rectangles into a circle? |
**HINT** $\ \ \ \rm bd = ac\ \:\Rightarrow\:\ (a,b)\:(b,c)\: =\ b\: (a,b,c,d)\ $ via GCD distributive, associative laws.
Alternatively, if you know $\rm\ [a,b]\: (a,b)\: =\: ab\ $ then using this LCM*GCD law
we have $\rm\ \ \: a,b\:|\:ac\ \:\Rightarrow\ \: [a,b]\:|\:ac\ \ \Rightarrow\:\ ab\:|(a,b)\:ac\ \Rightarro... |
Think of $t$ and $r$ as two independent variables.
- Suppose $E$ be a function of $r$ and $V$ be a function of $(t,r)$ such that both go to $0$ at $r=0$.
- There exists a positive function $M(r)$ such that $M(0)=0$ and $V(t,r) = -\frac{M(r)}{R(t,r)}$ where $R$ is another positive function such that $R(0,r)=r$... |
Suppose I have a variable $s$ that has a geometric distribution with success parameter x. So the probability of success on trial $s$ is
$p_s = (1 - x)^{s - 1} x$,
Consider the following function of $s$
$V(s) = \delta^s$,
where $\delta < 1 $. Then the expected value of this function is given by
$\su... |
Suppose that $\mathcal{E}$ is a well-pointed elementary topos, that $X$ and $Y$ are objects of $\mathcal{E}$, and that $F$ is a function which maps global elements $p: 1 \to X$ to global elements $F(p): 1 \to Y$ (here $1$ is the terminal object of $\mathcal{E}$). Does there exist a (necessarily unique) arrow $f: X \to ... |
Do functions defined on global elements give rise to arrows in a well-pointed topos? |
I'm playing around with a collection of subsets of $\mathbb{R}^n$ let's call them $X_i$. What I want to know is, is the following condition sufficient for some $X_i$ being measurable?
Almost all elements of $X_i$ are surrounded by a neighbourhood contained in $X_i$
This is motivated by the idea that non measurabl... |
Sufficient condition for being measurable? |
This may be a really basic question but the typical atan2 method isn't working when I write it in a program.
I need a camera object to face a certain target in 3D space.
I need to calculate the 3 Euler angles (yaw, pitch, roll) for the camera.
![alt text][1]
How can I do this?
[1]: https://i.stack.i... |
This sounds like homework. Can you translate the words into an equation? Think about what happens when the light ray strikes the mirror. |
I read that Smale recast classical mechanics in terms of symplectic geometry. I know a bit about classical mechanics but nothing about symplectic geometry.
Are there any writings from Smale on this which I could read?
And also what other books or texts are there which cover this?
Thank you! |
what are some *lower* bounds (if they exist) on $w^t A w$ in terms of $||w||_2^2$ for $A$ an matrix with all positive values? The lower bound can depend on $A$ and $||w||_2^2$. $w$ is $k \times 1$. Note when $A = I$ then we exactly get the squared norm of $w$.
If there are no such values, what in the case that $A$ i... |
what are simple lower bounds on w^t A w in terms of ||w||_2^2? |
Using the definitions $$\sec \theta = 1/\cos \theta \quad \text{and} \quad \cos \theta = (\exp(i \theta) + \exp(-i \theta))/2$$ gives $$\int \sec \theta d \theta = \int \frac{2 d \theta}{\exp(i \theta) + \exp(-i \theta)}.$$ The only insight needed is to find the substitution $u = \exp( i \theta )$ (what else is ther... |
If I have a bounded set F in N dimensional space and another set G where every element g in G has h'g=c and also must exist in F. H is a vector in the N dimensional space and c is any constant 1x1 matrix (scalar). h is a vector of appropriate dimension.
How can I prove every extreme point of G lies on the boundary o... |
I've been trying to simplify this equation:
Y = (¬AB + ¬C)(¬A + ¬D)
.. into this equation.
Y = ¬A¬C + ¬C¬D + ¬AB
Unfortunately I keep going in circles with expanding and minimizing the booleans. Any tips or advice? Thanks.
|
The following is a question from the sample GRE Mathematics Subject Test
found on the ETS website:
Let $M$ be a $5\times 5$ real matrix.
Exactly four of the following five conditions on $M$ are equivalent
to each other.
Which of the five conditions is equivalent to NONE of the other four?
(A) For any two di... |
**Note:** [This question has been posted on StackOverflow][1]. I have moved it here because:
1. I am curious about the answer
2. The OP has not shown any interest in moving it himself
---
In the Communications of the ACM, [August 2008 "Puzzled" column](http://delivery.acm.org/10.1145/1790000/1787260/p128-wink... |
Originally I was just going to ask the problem on my practice math contest, which is asking how many ways there are to write a nine-digit number containing each digit 1-9 so that the first digit is twice as big as the second and the last two digits are odd. (e.g. 846527913).
I understood that in this case, the numbe... |
Nice! The above proof proves that *any* configuration of 10 dots can be covered. What you have here is an example of the [probabilistic method](http://en.wikipedia.org/wiki/Probabilistic_method), which uses probability but gives a certain (not a probabilistic) conclusion (an example of [probabilistic proofs of non-prob... |
You're on the right track. I'd break the problem up into four cases: where the number starts with 21, 42, 63 or 84.
In the case where you start with 21, you must have 21(EEEOO)OO in your notation. (There are four even digits, so three of them must be in the middle part of the number.)
So we can ask the questions... |
It's a well known fact in probability theory that if $\lbrace X_i\rbrace$ is a family of random variables, $\phi : [0,\infty) \to [0,\infty)$ satisfies $$\lim_{x \to \infty} \frac{\phi(x)}{x} = \infty \quad (*) $$ and $\sup_i E\phi(|X_i|) < \infty$, then the family $\lbrace X_i\rbrace$ is uniformly integrable. I want ... |
Let $T:X \times Y \to \mathbb{R}$ be a continuous bilinear operator defined on normed linear spaces $X$ and $Y$ s.t.
$T(\alpha x + \beta y,z) = \alpha T(x,z) + \beta T(y,z)$) and $T(x,y) = T(y,x)$.
Does there exist a constant $C$ s.t. $||T(x,y)|| \leq C$ $||x||$ $||y|| \forall x,y$?
I know that the result i... |
I'm trying to wrap my brain around adjoint functors. One of the examples I've seen is ${IntLE} = (\mathbb{Z}, ≤)$ and $RealLE = (\mathbb{R}, ≤)$. Where the ceiling functor ${ceil} : {RealLE} \rightarrow {IntLE}$ is left adjoint to the inclusion functor ${incl} : {IntLE} \rightarrow {RealLE}$. I want to check that the f... |
I'm trying to wrap my brain around adjoint functors. One of the examples I've seen is the categories $IntLE = (\mathbb{Z}, ≤)$ and $RealLE = (\mathbb{R}, ≤)$, where the ceiling functor $ceil : RealLE \rightarrow IntLE$ is left adjoint to the inclusion functor $incl : IntLE \rightarrow RealLE$. I want to check that the ... |
First of all, i'll confess i'm no math geek. I'm from Stackoverflow, but this question seemed more apt here, so i decided to ask you guys :)
Now, i know noone has discovered (or ever will) a Polynomial that generates Prime Numbers. But i've read about Curve Fitting (or Polynomial Fitting) so i was wondering if there... |
What is the difference complete metric space and a closed set?
Can a set be closed but not complete? |
I'm trying to wrap my brain around adjoint functors. One of the examples I've seen is the categories $\bf IntLE \bf = (\mathbb{Z}, ≤)$ and $\bf RealLE \bf = (\mathbb{R}, ≤)$, where the ceiling functor $ceil : \bf RealLE \rightarrow IntLE$ is left adjoint to the inclusion functor $incl : \bf IntLE \rightarrow RealLE$. I... |
Let $g_{ij}$ be the components of a symmetric rank-2 positive definite tensor (metric on a Riemannian manifold). Let $\{C^i\}$ and $\{ \beta ^i \}$ be components of a vector field on it, the former of which is a functions of a variable $t$ and let $N$ and $M$ be two positive constants. Now let the following inequality ... |
In measure theory there are three fundamentally related theorems about exchanging limits and integrals: Fatou's lemma, Lebesgue's Dominated Convergence Theorem, and Monotone Convergence Theorem. It is difficult to prove any of these from scratch, but once you have one, the others are easier.
My question is, for thos... |
I prefer the direction: Dominated convergence theorem -> Beppo-Levi (monotone convergence) -> Fatou.
This direction requires more machinery (like Egoroff and the absolute continuity of the Lebesgue integral) but it is in my opinion more tidy. A book that uses this approach is [Bogachev - Measure Theory](http://books... |
I'm playing around with a collection of subsets of $\mathbb{R}^n$ let's call them $X_i$. What I want to know is, is the following condition sufficient for some $X_i$ being measurable?
Almost all elements of $X_i$ are surrounded by a neighbourhood contained in $X_i$
This is motivated by the idea that non measurabl... |
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