What’s the order type of the set of all integer polynomials, where $f \leq g \leftrightarrow \exists x \in \mathbb{Z} : \forall y \in \mathbb{Z} : f(y) \leq g(y)$?
Are the set of integer polynomials with non-negative coefficients order-isomorphic to the set of ordinals below $\omega^\omega$?
Also, what’s the relation between addition and multiplication of these polynomials and Hessenberg addition and multiplication of their corresponding ordinals?
For an inner product space $W = \{A \in M_{n \times n}(\mathbb{R}) \mid tr(A)=0\}$,
with inner product $\langle A,B\rangle:=tr(A^TB)=\sum_{i=1}^n\sum_{j=1}^na_{ij}b_{ij}$,
find $W^{\perp}$.
I know that:
$dim(W) = n^2 -1$, and that
$W \oplus W^{\perp} = M$
But I'm not sure of how to proce...
If someone has a PhD in some math or math related area (say CS theory or electrical engineering), is it possible for this person to teach an undergrad course, like say Intro Complex Analysis, at an university without asking for re-enumeration?
The motivation for not asking for re-enumeration is t...
In this question we are concerned only with positive integers $\mathbb N$ and other finitary objects that can be encoded using integers. A term function means a total computable function $\mathbb N^n\to\mathbb N$ with one or multiple arguments. Letters $m,n,k$ range over $\mathbb N$. Letters $f,...
Anyway, I have a very general question. I'm looking for some examples in math where suppose something (anything) that depends on a parameter $n$ has different behaviors/values/limits/etc. when $n$ is a large but finite number, and when $n \to \infty$.
Any cute examples?
Some behavior/property/etc. (anything) changes in a non-trivial manner when the limit $n \to \infty$ is taken.
@Avantgarde It's probably hard to do this in the way that you'd want, because limiting behavior/asymptotic behavior pretty much exactly deal with "n large enough". I guess a cheeky example might be $(-1)^n$---for all finite $n$, this is meaningful, but the limit doesn't exist.
@UserX H is of order 6. Z4 x Z6/H is of order 24/6 = 4, no?
@Avantgarde There are notions of scaled limits in probability, where you can view continuous random variables as limit (in distribution) of the form $\lim_{n \to \infty} n^{-1/\alpha} X_n$ where $\alpha$ is some real constant and $X_n$ are discrete distributions.
As a simple example, consider $X_n \sim \text{Geo}(\lambda/n)$ distributed geometrically, so $\Bbb P(X_n = m)$ is the probability that a coin with success probability $\lambda/n$ returns head after $m$ tosses.
Then $X_n/n$ converges in distribution to the exponential distribution $\text{Exp}(\lambda)$ which is a continuous distribution, very different as a random variable than the $X_n$'s.
One way to interpret it as "you have a coin with very low success probability, of order 1/n, but since that'd take very long time to return a head usually, of order n, you scale it by n"
Then there's the classical example of the central limit theorem, where no matter whatever random variable $X_0$ you start with, as long as it has finite second moment, taking iid copies $X_1, X_2, X_3, \cdots \sim X_0$ and taking the scaled limit $\lim_{n \to \infty} n^{-1/2}(X_1 + \cdots + X_n)$ one always ends up (in dsitrbution) at a normal distribution.
This is a sort of "second order ergodic theorem", which sort of says that this scaled mixing process converges to the normal distribution (in an appropriate space of distributions) regardless of the initial condition
As another example, random walks on $\Bbb Z$ are mathematically very easy to describe as just the discrete-time process $(S_n)_{n \geq 1}$ where $S_n = \sum_{1 \leq i \leq n} X_i$, $X_i$ being iid Bernoulli variables, taking $1$ or $-1$ with probability $1/2$
One can obtained Brownian motions on $\Bbb R$, which are usually complicated to describe properly, as scaled limits of random walks on $\Bbb Z$. Namely, take $B_t = \lim_{n \to \infty} n^{-1/2} S_{\lfloor nt \rfloor}$ (this exists pointwise for $t$ by CLT). Then $(B_t)_{t \in [0, 1]}$ is in fact a Brownian motion on $[0, 1]$.
Ok, so here's some context.
Solving regular equations we might have something like this:
$2 + x = 5$, solving for $x$ we get 3. We might even have an equation like $x + y = 5$ where there are multiple solutions.
But what's in common with all these equations is that the process, or the algorith...
These equations are part of a more general class of functional equations. In the context of Banach spaces, where the unknown operators are bounded operators, things can get very complicated:
I have been working on a problem in Quantum Mechanics and I have encountered a equation as given below.
$$\frac{d\hat A(t)}{dt} = \hat F(t)\hat A(t)$$
Where ^ denotes it is an operator
How will this differential equation be solved? Will the usual rules for linear homogeneous first order diffe...
Suppose X is a convergent square matrix with real elements, Xt is its transpose, and M is some other real matrix with same dimensions. Does a sum from i = 1 to infinity where the general term is of the form X^i M Xt^i have a name or might there be some related results? I think this sum should converge for arbitrary real M, any chance the result could be expressed in closed form in terms of M?
Let $\Gamma$ be some group, and let $\ell^2(\Gamma) := \{f : \Gamma \to \Bbb{C} \mid \sum_{\gamma \in \Gamma} |f(\gamma)|^2 < \infty\}$. Given $\gamma \in \Gamma$, define $u_\gamma : \ell^2(\Gamma) \to \ell^2(\Gamma)$ by $u_\gamma(f)(\eta) := f(\gamma^{-1} \eta)$. I am reading an expository paper in which in claims that $u_\gamma$ is a unitary operator and that $\gamma \mapsto u_\gamma$ defines a unitary representation (called the left regular representation).
However, this seems to be a mistake. Shouldn't $u_\gamma$ actually be $u_\gamma (f)(\eta) := f(\gamma \eta)$?
Defined the former way, I wasn't able to show that $\gamma \mapsto u_\gamma$ is a homomorphism.
Let $\Gamma$ be some group, and let $$\ell^2(\Gamma) := \{f : \Gamma \to \Bbb{C} \mid \sum_{\gamma \in \Gamma} |f(\gamma)|^2 < \infty\}.$$ Given $\gamma \in \Gamma$, define $u_\gamma : \ell^2(\Gamma) \to \ell^2(\Gamma)$ by $u_\gamma(f)(\eta) := f(\gamma^{-1} \eta)$. I am reading an expository pap...
Can we say that a function, which is continuous at all points in which it exists, is continuous even thought it doesn't exist at all points in the real line?
Take a function $f$. In the definition of $f$, there are two sets: the domain and the codomain.
Sometimes we write it as $f\colon A\to B$, where $A$ is the domain, and $B$ is the codomain.
A function $f\colon A\to B$ is continuous if it is continuous at x for all x in A.
The square root function is continuous on its domain $[0,\infty)$.
But it is not defined elsewhere.
So to answer your question, for a function from R to R to be continuous, it must be both defined AND continuous at each x in R. However, you can have functions from a subset of R to R which are continuous, defined similarly as I mentioned above.
Does anyone know what techniques might be useful for dealing with/simplifying a recursively-defined coordinate replacement with an integral? Specifically, if it helps, f(x)=x + integral(sin(f(x)/f(x)^2) from 1 to x, where f(x) k=0 is f(x)=x? Using Weierstrass t-replacement I end up with f(x)=f(x), which is correct but not helpful. And I don't know what to do with the endless stream of gamma functions I get when I do a definite integral and induction.
Part of learning math is learning how to be pedantic. Both -1 and 1 have the property that $x^2=1$. So, you have to restrict the codomain for the square root to be a function.
Otherwise it is a relation (if I am using that term correctly).
See, to answer accurately, a person really needs to know how the function is defined. A particular definition of the square root function might not actually be continuous depending on that.
I can show there is a sylow subgroup of order 9. To show that it is normal, I need to show that it is the only one. Currently, I know that the number of sylow subgroups is either 1 or 4.
Because they're subgroups, they must contain the identity. So, there are 8 elements not of order 1 in those subgroups. If there are 4 such subgroups, then you have 32 such elements, no?
Okay, so in the case of 36, can you rule out that two 3-subgroups have a non-trivial intersection, but the collection of all four 3-subgroups has a trivial intersection?
@BalarkaSen, Well, I visited to the TIFR website to check out the selection procedure for Masters in Mathematics. Apparently they want only the EXCEPTIONALLY TALENTED students. I failed even before I applied for the entrance.
@Rithaniel I don't think the 3-core argument works, yeah. The intersection of two 3-Sylows has to be of order at least 3. But then it's normalizer contains the 3-Sylows as well, because groups of order 9 are abelian. That should be big enough to carry out the argument.
@BalarkaSen I was thinking the other day what is actually a polynomial equation instead of thinking about it as formal sum. I wanted to understand what it represents.
Given a field k. Polynomial equations gives us a way to detect the complexity of the underlying field by seeing how geometrical data are arranged.
I don't know man, they're the most natural thing in the world. Quadratic equations historically came from word problems involving finding out the number of people in a game when you know the number of times they have mutually shook hands, etc
Italian dudes tried to do cubics and quartics as a natural generalization
Then nobody was able to solve the quintics, hence the development of field theory, to prove it cannot be "solved"
I think sheaf is just a way to detect algebraic data locally. I guess a sheaf of a sheaf your putting some kinda of topology on your sheaf and maybe detecting different way of arranging algebraic data over your object ?
@AlessandroCodenotti If $F : H \to H$ is Fredholm, $\ker F \subset H$ is finite-dimensional, so $H = (\ker F)^\perp \oplus \ker F$, where $F$ kills the second factor and is injective on the first factor. $F : (\ker F)^\perp \to \text{range}\, F$ is bijection, so invert it to get $G : \text{range} F \to (\ker F)^\perp$ and I think range is a closed subspace of $H$, so you can extend that to $G : H \to H$.
Uh
I guess you just set $G$ to be zero on $(\text{range} F)^\perp$, that's finite dimensional too (cokernel)
I like how every question with Balarka starts as someone asking him something he knows nothing about but he just figures out the whole theory on the spot
Then $GF = I - K_1$ where matrix of $K_1$ is just a bunch of $1$'s on the $\ker F$ block and $FG = I - K_2$ where matrix of $K_2$ is a bunch of $1$'s on the $(\text{range} \, F)^\perp$ block
For an inner product space $W = \{A \in M_{n \times n}(\mathbb{R}) \mid tr(A)=0\}$,
with inner product $\langle A,B\rangle:=tr(A^TB)=\sum_{i=1}^n\sum_{j=1}^na_{ij}b_{ij}$,
find $W^{\perp}$.
I know that:
$dim(W) = n^2 -1$, and that
$W \oplus W^{\perp} = M$
But I'm not sure of how to proce...
Prove that if $d$ divides $n$, then $2^d -1$ divides $2^n -1$.
Use the identity $x^k -1 = (x-1)*(x^{k-1} + x^{k-2} + \cdots + x +1)$
Ok, so here's some context.
Solving regular equations we might have something like this:
$2 + x = 5$, solving for $x$ we get 3. We might even have an equation like $x + y = 5$ where there are multiple solutions.
But what's in common with all these equations is that the process, or the algorith...
I have been working on a problem in Quantum Mechanics and I have encountered a equation as given below.
$$\frac{d\hat A(t)}{dt} = \hat F(t)\hat A(t)$$
Where ^ denotes it is an operator
How will this differential equation be solved? Will the usual rules for linear homogeneous first order diffe...
