Hey, guys, I posted a question about Integral Transforms, but no one answered that. The thing is: can I create a Integral Transform? Can I set K(s,t) the way that I want? Does any of you have a good book recommendation about ITs? I do know the applications, but wanted to try things out.... And also I have no one to talk about this, so here I am. Any help would be great
@topologicalmagician No. The discrete topology comes from a metric that takes on two values. What happens if you try three values? Can you make up a metric?
Would the statement "For every field $\mathbb{F}$, the only solution to $3x = x$ is $x = 0$" be FALSE since in a field with only 2 elements such as $\mathbb{F} = \{0 ,1\}$, $x$ simply cannot be equal to $0$ since every element has to be distinct?
because if we have a field with two elements only, if one of them is equal to the other, then it contradicts the fact we have 2 elements in it, isn't it?
Yes, it will be much clearer indeed! So let me just make sure I understand the reason x is also 1.Would it be a solution because in a field with 2 elements such as $\mathbb{F} = \{0,1\}$, 1 + 1 = 0 and 1 corresponds to the multiplicative identity, thus x = 1? Or am I missing something?
Nice, so that gets you to arrive at the classification of the types of x. But is there an easy, naive way of checking if something is the solution of an equation?
@Abwatts Remember that the crucial point in both your examples is that if $ab=0$ and $a,b\in F$ ($F$ a field — or more generally an integral domain) then either $a=0$ or $b=0$. So if $2=0$ in $F$, then $x$ can be arbitrary (and the field could have lots more than 2 elements). If not, then you must have $x=0$.
@TedShifrin So, in my example, since we're given the equation $3x = x$ and are told that for every field, the only solution would be $x=0$, could we just do the following to prove this statement is false? $3x = x \iff 2x = 0 \iff x(1+1)=0 \iff x(1+0) = 0$ so, $x=0$ or $1+0=0$ and by the additive identity axiom this becomes $1=0$. Thus we get $x=0$ or $x=1$?
I'm not entirely sure I fully comprehend how we can show this statement is false..
$1+1+1 = 1 \iff 1+1 = 0$ which is also true in fields with 2 elements?
So to find all the solutions if we're allowed to work in all fields, we basically need to factor and solve for $x$ and check (by plugging the result back in) if this actually results in something that we know is true?
Can anyone help me learning a concept, actually I have an equilateral triangle does there exist a specific method to directly find the coordinate of ortho Centre of triangle, like we have in right angle triangle that ortho Centre lies on the mid point of hypotense.
Hi, I have a question about derivative of exponent in power. I want to find the limit of the following: $(2/3)^n$ as n goes to infinity, it equals to 0. But I still want to prove this fact. How to proceed?
operation paper clip went very well I'm guessing that might be one reason
well results wise anyway naturally it was a human rights violatey kind of nightmare depends how you look at it really
anyway if I can get out a proper proof for this ive got a very solid proof for the tpc I mean I think there are already lots of proofs but anyway this is the last bit really
$${\Biggl\{ \frac{2n^2\pm9}{9}-\Bigl\lfloor\frac{2n^2\pm9}{9} \Bigr\rfloor:n \in \mathbb N }\Biggr\} = {\Biggl\{0,\frac{2}{9},\frac{5}{9},\frac{8}{9}}\Biggr\}$$
Integrals where the integrand contains an absolute value can be very hard or impossible to express in closed form.
Computing the zero’s of the integrand might help.
But what if the zero’s have no closed form or there are infinitely many ?
We can express any single zero by a contour integral but...
when people knew you are a philosopher, they actually treat you as a philosopher
all jokes dropped and even if the question seemed to have no real consequences or in some cases, mathematical consequences, a fruitful discussion follows
I also have a seminar on local class field theory and my talk for algebra is right at the end so I’ll have some experience with local fields by then lol
he's an algebraist, so he probably won't do the L-functions approach to LCFT in ANT2, Lubin-Tate-theory is covered in the seminar, so among the major modern approaches to CFT that leaves only group cohomology
Also, Böckle mentioned that instead of splitting the Vorträge, he might end up splitting the seminar entirely and offering a seminar on Darstellungstheorie and another on Galoiskohomologie
@Balarka there's a NT research seminar here next term that has some interesting topology in it: Dehn twists, mapping class groups, intersection cohomology... though I won't give one of the topological talks
actually the seminar uses a crazy amount of cohomology theories: singular, smooth De Rham, algebraic de Rham, étale, crystalline, intersection homology
if a dating profile only shows half a face split down the y axis is it a block worthy offense to ask if she smelt toast when she had the stroke? American cartoons corrupted my mind for starters, and it's not like it was offensive, I mean it was merely an expression of needing to see the other side of the axis before consensual intercourse could be considered
The definition is as I told you. I have no clue about that paper, but if the authors use some non-standard definition, then it's their job to specify that.
@TedShifrin are you asking to demonstrate that because we know the inverse of $A$ exists, there must exist a real scalar constant that is the gradient of linear proportion of $x$ to $b$, and $b$ being in $\mathbb R^n$, concurrently making $x$ also in $\mathbb R^n$?
as in given the knowledge that the inverse exists, $A \cdot A^{-1}$ must be $I$, and this is suffice to show that $x$ must belong to the same domain as $b$?
i mean otherwise im probably going to spend the next decade as the last, all about $\mathbb N$ and considering matrices above and beyond being handy not really something I should worry about. really who wants to talk about dimensions of space anyone that does for too long becomes mentally ill
I just don't see how everyone is able to keep physics and algebra separate in that respect it's so much simpler to observe and relate number patterns and ignore physics tbh
Every time I log on this chatroom I always get intimidated and humbled by the amount of stuff I don't know. most often you guys talk about things that may as well be arab for me.
The basic idea seems to be the following: The setup is you have a parametrized family of distributions $\mathscr{F}_\theta$ on $\Bbb R^n$, $\theta \in \Theta \subset \Bbb R^k$, then given data $(x_1, \cdots, x_n)$, you want to ask what is the $\theta$ of the distribution from which this came. This is the goal of statistics, summarized. So in the Bayesian world they assume $\theta$ is also some random quantity, so has some "prior distribution"
Assume the pdf of $\mathscr{F}_\theta$ exists and is $f_\theta$, and assume $\theta$ follows pdf $\pi$, the prior distribution.
Then you setup the thing as follows: $X$ be the random variable whose realization is $(x_1, \cdots, x_n)$. Essentially, we are saying $X \sim f_{\widehat{\theta}}$ for some $\widehat{\theta} \in \Theta$, and we want to find this given that we know the distribution of the conditional $X|\theta$.
The knowledge is $X|\theta$ and $\theta$ are independently distributed. So their joint pdf is $f_{X|\theta}(x|\theta)\pi(\theta)$. We want the pdf of $\theta|X$, the posterior distribution, so we can evaluate it near $(x_1, \cdots, x_n)$ to say what is the most likely $\widehat{\theta}$ from which it may come from.
That is why it's called Bayesian, given we know $X|\theta$, we want to infer about $\theta|X$
Which says if the pdf of $\theta|X$ is $\pi(\theta|X)$ then $\pi(\theta|X=x) = f_{X|\theta}(x|\theta) \pi(\theta)/\int f_{X|\eta}(x|\eta) \pi(\eta) d\eta$.
If $X$ is a noncompact LCH space (locally compact, Hausdorff) then its one point compactification is $X^*=X\cup \{\infty\}$ with topology $\mathcal{T^*}$ given by $U \in \mathcal{T^*}$ iff either
a) $U \subset X$ is open, or
b) if $\infty \in U$ then $U^c \subset X$ is compact
What whould be t...
Integrals where the integrand contains an absolute value can be very hard or impossible to express in closed form.
Computing the zero’s of the integrand might help.
But what if the zero’s have no closed form or there are infinitely many ?
We can express any single zero by a contour integral but...
Is it possible to estimate the capacity of a neural network model? If so, what are the techniques involved?
I have a solution in laplace images:
\begin{align}
&p_f(x,s) = - \frac{1}{s}\frac{b}{a} \frac{1}{\sqrt{\frac{s}{a}+ \frac{b}{a} \frac{\sqrt{ s}}{1+c\sqrt{s}}}} e^{-x \sqrt{\frac{s}{a}+ \frac{b}{a} \frac{\sqrt{ s}}{1+c\sqrt{s}}}} \label{pressure_frac_sol:laplace2c}\\
&p_r(x,y,s) = - \frac{1}{s}...
If $X$ is a noncompact LCH space (locally compact, Hausdorff) then its one point compactification is $X^*=X\cup \{\infty\}$ with topology $\mathcal{T^*}$ given by $U \in \mathcal{T^*}$ iff either
a) $U \subset X$ is open, or
b) if $\infty \in U$ then $U^c \subset X$ is compact
What whould be t...
