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I'm working through an exercise in Hartschorne concerning direct limits of sheaves on noetherian topological spaces. It seems natural to use the the fact that every open subset of a noetherian topological space is the quasi-compact in its induced topology. I'm some trouble proving what I need when I only use this prope... |
Are there non-noetherian topological spaces in which every open subset is quasi-compact? |
In school and at university we never had eigenvalues nor differential equations, so these concepts were really giving me a hard time. Now I developed some intuition for both concepts.
I learned that both are connected in some way, i.e. there is an eigenvector/eigenvalue approach to solve differential equations. Unf... |
How are eigenvectors/eigenvalues and differential equations connected? |
What is $x$ in closed form if $2x-\sin2x=\pi/2$, $x$ in the first quadrant? |
Rubin [1] gives an algorithm for counting Hamiltonian paths.
Mathematica has tools for working with Hamiltonian paths, but the naive
Length[HamiltonianCycles[G, All]]
gives an out-of-memory exception for this particular graph.
[1] Frank Rubin, <a href="http://portal.acm.org/citation.cfm?id=321854">A sea... |
To get a graphical impression of the solution have a look here:
http://www.wolframalpha.com/input/?i=solve+2+x+-+sin+(2+x)+%3D+Pi/2+for+x
Edit: That WA doesn't give a closed form is at least a hint that there might be none (very probable) |
I was reviewing quantum factorization and am slightly unclear on a classical detail of order-finding.
Given a (suitably nice) periodic function $f$ with unknown period $r$ and a power of two $N > r^2$, the quantum subroutine yields (with bounded probability) a result of the form $\lfloor jr/N \rceil$ for $1 \le j \... |
What is the length of a continued fraction expansion of a rational number? |
This is a newbie question, but I would be grateful for any reference that you could give.
let $f(x) \in \mathbb{A}$, where $x \in \mathbb{A}$
Is there a symbol to indicate the repeated application of $f$ unto its results, in a manner similar to $\sum$ and $\prod$?
Thank you! |
Specifically, if you have a topos with natural numbers object (assume choice if you want to), what familiar theorems don't hold? I've heard that showing the Dedekind reals are the same as the Cauchy reals is one. Where in the arguments is well-pointedness used? It seems hard to find examples of this.
Edit: Ah, I for... |
Let $T$ denote the period of time until Jill finishes painting $3$ walls. Then, $T$ has gamma density $f_T (t) = t^2 {\rm e}^{-t}/2$, $t > 0$. Viewing the number of walls painted by Jane up to time $t$ as a Poisson process (with unit rate), the question amounts to calculate
$$
\int_0^\infty {{\rm P}(N_t < 2|T = t)f... |
Specifically, if you have a topos with natural numbers object (assume choice if you want to), what familiar theorems don't hold? I've heard that showing the Dedekind reals are the same as the Cauchy reals is one. Where in the arguments is well-pointedness used? It seems hard to find examples of this. |
In _Classical Fourier Analysis_ by Loukas Grafakos we have in Proposition 2.3.25 the following definition for $\mathcal{S}_\infty(\mathbf{R}^n)$, namely that these are all the Schwartz functions $\phi$ such that for all multi-indices $\alpha$ we have that
$$\int_{\mathbf{R}^n} x^\alpha \phi(x) \, dx = 0.$$
Now I'... |
i'm a CompSci university student and i want some help for my math questions.
Therefore i am looking for a good discussion board for **math homework questions**.
I don't know any (good), because i come from an *non-native-english-speaking* country.
any ideas ?
> **note** : I know that i could post my questions h... |
Which discussion board is good for homework questions ? |
I'm reading Hatcher's notes on spectral sequences and he mentions that steenrod squares commute with the coboundary operator for pairs (X,A) which would then explain why these operations commute with the transgression. It says it's because
that coboundary operator can be defined in terms of suspension and we know ste... |
In _Classical Fourier Analysis_ by Loukas Grafakos we have in Proposition 2.3.25 the following definition for $\mathcal{S}_\infty(\mathbf{R}^n)$, namely that these are all the Schwartz functions $\phi$ such that for all multi-indices $\alpha$ we have that
$$\int_{\mathbf{R}^n} x^\alpha \phi(x) \, dx = 0.$$
Now I'... |
As $x\to\infty$, $\exp(-1/x^2)\to1$ so this isn't in the Schwartz
space. Why not multiply this by $\exp(-x)$ to get it decaying to
$0$ at infinity rapidly enough? |
While reading about Moufang loops in the book "An introduction to Quasigroups and their Representations" by Smith, I've encountered the following statement:
>The set $ S^7 $ of nonzero octonions of norm 1 forms a Moufang loop under multiplication. Geometrically, this set is a 7-sphere.
While I understand why this s... |
How about this algorithm? At each moment your decision is which spell to cast next. Having decided, if the cool down time for that one has expired, then go ahead and cast it. If not, wait until it has expired, then cast it. To make the decision, rank the spells in damage/time where time is from now to completion (i... |
This is only a first approximation to your answer, but I think it's a good start.
Let $x$ and $y$ be vectors in $\mathbb{R}^n$ and assume $x\cdot y = 0$. Now, I'm going to add an additional condition that neither $x$ nor $y$ is $0$. To give away the punchline, it will turn out there's a nice description of <i>thes... |
An empty set contains nothing. A power set of some set $A$, $\mathscr{P}(A)$ always contains something, at least $A$ itself, thus $\mathscr{P}(A)$ is not empty.
The reason I decided to add my two cents is to point out to the frequent source of confusion which is associating empty set with zero, and then associatin... |
**From WIKI:** In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression obtained in the context of limits. Limits involving algebraic operations are often performed by replacing subexpressions by their limits; if the expression obtained after this substitution does not... |
(I think this will work, but I'm not terribly familiar with LP. I think it's a worthwhile idea at least.)
Let's rephrase to: what's the maximal damage that I can cause in, say, 100 seconds? The constraints then become:
1. Can't spend more than 100 seconds casting
2. Can't cast a single spell more than $\frac{1... |
so I dont know if this is really what the website is meant for so let me know. I have a midterm on abstract algebra (intro) and I've been doing practice problems and I just need to know whether my answers are correct and if my way of doing them is right.
1) Does the following have a solution: $x^2 \equiv 1 \pmod{3... |
### Background###
Recall that an integral scheme $X$ is a scheme which is both irreducible and reduced; equivalently, its ring of functions is an integral domain on every open subset.
Given any point $p$, there is a local ring $R_p$ at $p$, which is given by localizing the ring of functions $R_U$ on any affine neig... |
Can an integral scheme have closed points of both positive and zero characteristic? |
so I dont know if this is really what the website is meant for so let me know. I have a midterm on abstract algebra (intro) and I've been doing practice problems and I just need to know whether my answers are correct and if my way of doing them is right.
1) Does the following have a solution: $x^2 \equiv 1 \pmod{3... |
For 1) you have just rewritten the equation backwards, so have not proved anything. You can use the same approach as 3)-square each number mod 3 and see if any square to 1
For 2) why do you know 4n+3 cannot be the sum of two squares? You need to list the numbers mod 4 that are squares, then find that no pair of th... |
Let's rephrase to: what's the maximal damage that I can cause in, say, 100 seconds? The constraints then become:
1. Can't spend more than 100 seconds casting
2. Can't cast a single spell more than $\frac{100}{cast\ time + cool\ time}$ times
Let $n_i$ be the number of times I cast spell i. I want to maximize $\... |
If $X,Y$ and $Z$ have identical distributions and are independent, and can assume $0$ or $1$, what is $\text{Var}(XYZ)$? So this is either $0$ or $1$. It's $1$ with probability $1/8$ and $0$ with probability $7/8$. This means that $\text{Var}(XYZ) = 7/16$? Or is this not right? How does this use the definition of varia... |
Let $G$ be a (finite?) group. By definition, the Eilenberg-MacLane space $K(G,1)$ is a CW complex such that $\pi_1(K(G,1)) = G$ while the higher homotopy groups are zero. One can consider the singular cohomology of $K(G,1)$, and it is a theorem that this is isomorphic to the group cohomologies $H^*(G, \mathbb{Z})$. Acc... |
Why is the cohomology of a $K(G,1)$ group cohomology? |
Let's rephrase to: what's the maximal damage that I can cause in, say, 100 seconds? The constraints then become:
1. Can't spend more than 100 seconds casting
2. Can't cast a single spell more than $\frac{100}{cast\ time + cool\ time}$ times
Let $n_i$ be the number of times I cast spell i. I want to maximize $\... |
Let $T$ be a distribution, $f$ be smooth. It is known that
$T(f(x - \cdot) ) \in C ^ \infty$ .
On the other hand, for $\phi$ a test function
$( T \ast f ) (\phi) = T( f( - \cdot ) \ast \phi ) = T( \int f( x - \cdot ) \phi( \cdot ) )$,
because the convolution is still a distribution. However, with regard to ... |
Let $\bar{\mathcal{P}}$ denote the closed, convex polytope with vertices at the positive rational points $(b_{1}, \dots, 0), \dots, (0, \dots, b_{n})$. Define the Ehrhart quasi-polynomial $L_{\mathcal{P}}(t) = |t \bar{\mathcal{P}} \cap \mathbb{Z}^{n}|$, which has the form: $\sum c_{k}(t) \ t^{k}$ with periodic function... |
When are Ehrhart Quasi-polynomials Polynomials? |
Let $\bar{\mathcal{P}}$ denote the closed, convex polytope with vertices at the positive rational points $(b_{1}, \dots, 0), \dots, (0, \dots, b_{n})$. Define the Ehrhart quasi-polynomial $L_{\mathcal{P}}(t) = |t \bar{\mathcal{P}} \cap \mathbb{Z}^{n}|$, which has the form: $\sum c_{k}(t) \ t^{k}$ with periodic function... |
Let $\bar{\mathcal{P}}$ denote the closed, convex polytope with vertices at the positive rational points $(b_{1}, \dots, 0), \dots, (0, \dots, b_{n})$. Define the Ehrhart quasi-polynomial $L_{\mathcal{P}}(t) = |t \bar{\mathcal{P}} \cap \mathbb{Z}^{n}|$, which has the form: $\sum c_{k}(t) \ t^{k}$ with periodic function... |
I'm approximating an integral with only exponentials. i.e., it is equal to
$\displaystyle \int_{-\pi}^\pi{\frac{\displaystyle\sum_{j=a}^b{c_j e^{i\cdot d_j \cdot t}}}{\displaystyle\sum_{k=a}^b{r_k e^{i \cdot s_k \cdot t}}} dt}$
where $c_j, d_j, r_k, s_k \in \mathbb{Z}$
In other words, it's a fraction of exponen... |
Is this integration approximation method known/used? |
so we have points A, B, C on 2d plane (x, y). How having such points coordinates calculate triangle square? |
How to calculate square of triangle having its points 2d coordinates? |
so I dont know if this is really what the website is meant for so let me know. I have a midterm on abstract algebra (intro) and I've been doing practice problems and I just need to know whether my answers are correct and if my way of doing them is right.
1) Does the following have a solution: $x^2 \equiv 1 \pmod{3... |
For (1), you confused the condition for a congruence of the form $ax\equiv b\pmod{n}$ to have a solution with the congruence you have here. This congruence is not a linear congruence, so the condition does not apply.
No, that does not even begin to work; it's completely different kinds of congruences (linear vs. qua... |
I think the intuitive point here is captured in the idea that between -n and n there are only finitely many integers and infinitely many rationals. So over any finite range (and intuitively therefore over the entire range because it is uniform) the density of integers in the rationals is zero. The effort is to carry ... |
For (1), you confused the condition for a congruence of the form $ax\equiv b\pmod{n}$ to have a solution with the congruence you have here. This congruence is not a linear congruence, so the condition does not apply.
No, that does not even begin to work; it's completely different kinds of congruences (linear vs. qua... |
Suppose $r_i$, $1 \le i \le n$, and $c_j$, $1 \le j \le m$, are nonnegative integers. When does there exist an $n \times m$ matrix in $\text{Mat}_{n \times m} (\mathbb{Z}^+)$, i.e. nonnegative entries, such that $r_i$ is the sum of the entries in its $i$th row and $c_j$ is the sum of the entries in its $j$th columns? |
Why not use a real "function machine," which each of your students should have -- a scientific calculator? After all, most, if not all the functions in your course will be numerical examples.
For a function of one variable, use any of the trigonometric, squaring, cubing, square roots, or log functions. Use square r... |
in a game of contract bridge, partner and me together have 11 cards in a suite. The remaining 2 cards in the same suite can be distributed amongst our opponents as either 1-1 or 2-0.
What is the probability that it will be distributed as 1-1 and what is the probability it will be distributed as 2-0?
Once the abo... |
The area of a pyramid with a square base with side length $L$ and height $h$ is calculated as follows:
1. In $\mathbb{R}^3$ place the pyramid upright with one side flush with the z-axis so that the corners of the base are at $(0,0,\frac{1}{L})$, $(0,0,-\frac{1}{L})$, $(L,0,\frac{1}{L})$ and $(L,0,-\frac{1}{L})$.
!... |
How to explain why this method of slicing a solid to find volume fails? |
Suppose that $\mathcal{B}$ is a base for a topology on a space $X$. Is there a nice way of thinking about how we can modify $\mathcal{B}$ (for instance, to simplify computations) without changing the topology it generates? It seems non-trivial to compute the topology generated by a base, but maybe some "small enough" c... |
When does a subbase of a base generate the same topology? |
The area of a pyramid with a square base with side length $L$ and height $h$ is calculated as follows:
1. In $\mathbb{R}^3$ place the pyramid upright with one side flush with the z-axis so that the corners of the base are at $(0,0,\frac{1}{L})$, $(0,0,-\frac{1}{L})$, $(L,0,\frac{1}{L})$ and $(L,0,-\frac{1}{L})$.
!... |
Currently I am reading
Logic an Structure by Dirk van Dalen (2008).
As I am missing some basics I try to find related lectures on youtube.
I frequently watch MIT, Stanford, and University of Nottinham lectures on youtube.
www.youtube.com/user/MIT
www.youtube.com/user/StanfordUniversity
www.youtube.com/user/... |
Are you aware that integral of this type can be computed exactly by means of the residue theorem? The price to pay is that you have to find the roots of the equation $\sum_a^b r_k z^{s_k}=0$ in the unit disk. (You may assume all $s_k\ge0$.) |
I have a set and its cardinality. How can I calculate the cardinality of its complementary set? |
If you haven't been using ["The Art and Craft of Problem Solving" by Zeitz][1], you should. Like, right at this moment. :)
For neat problems... you might want to look into ["Which Way Did The Bicycle Go?" by Konhauser, Wagon and Velleman][2].
[1]: http://www.amazon.com/Art-Craft-Problem-Solving/dp/0471789011/... |
If in a $\bigtriangleup$ ABC, $\cot ( \frac{A}{2} ) = \frac{b+c}{a} $, then $\bigtriangleup$ ABC is of which [type][1] ?
[1]: http://en.wikipedia.org/wiki/Triangle#By_relative_lengths_of_sides |
If in a $\displaystyle\bigtriangleup$ ABC, $\displaystyle\cot \biggl( \frac{A}{2} \biggr) = \frac{b+c}{a} $, then $\displaystyle\bigtriangleup$ ABC is of which [type][1] ?
[1]: http://en.wikipedia.org/wiki/Triangle#By_relative_lengths_of_sides |
We have points A, B, C. On 2d plane. How having points coordinates (x, y) calculate square of triangle formed by them? |
How to calculate square of triangle, having its points 2d coordinates? |
We have points A, B, C. On 2d plane. How having points coordinates (x, y) calculate area of triangle formed by them? |
How to calculate area of triangle, having its points 2d coordinates? |
This is extracted from my module :
In a $\displaystyle\bigtriangleup ABC$,
$$\displaystyle \frac{\sin (A) + \sin(B)}{\sin(a) -\sin(B)} = \frac{k \cdot(\sin (A) + \sin(B))}{k \cdot(\sin(a) -\sin(B))} = \frac{a+b}{a-b}$$
where $A,B,C$ are the angles and $a,b,c$ are the opposite sides,I can't figure out how this s... |
This is extracted from my module :
In a $\displaystyle\bigtriangleup ABC$,
$$\displaystyle \frac{\sin (A) + \sin(B)}{\sin(A) -\sin(B)} = \frac{k \cdot(\sin (A) + \sin(B))}{k \cdot(\sin(A) -\sin(B))} = \frac{a+b}{a-b}$$
where $A,B,C$ are the angles and $a,b,c$ are the opposite sides,I can't figure out how this s... |
$X_{t}$ is a stochastic process. Maybe you mean a natural filtration?
The sigma algebra generated by $X_{t}$ is $\sigma(X_{t})=\{\emptyset,[0;1],t,[0;t)\cup (t;1]\}$.
The natural filtration $\mathcal{F}_{t}$ is the minimal sigma algebra which contains all $\sigma(X_{u})$ for $u\in [0;t]$.
So $\mathcal{F}_{t}$ ... |
Geometrically, this is not too hard to see. Presumably, you understand why
$$\frac{\sin(A) + \sin(B)}{\sin(A) - \sin(B)} = \frac{k(\sin(A) + \sin(B))}{k(\sin(A) - \sin(B))}$$
Let k be the length of the normal from C onto AB. What is then k sin(A) and k sin(B)? |
I claim that it is commonly believed that mathematical objects are genuinely static: there is no "Platonic" time in which they do genuinely evolve.
Nevertheless time has its place in mathematics:
1. An endomorphism of a set (seen as a set of states of a system) into itself can be seen as evolution of the system ... |
<strike>I claim that it is commonly believed that</strike> Mathematical objects can be seen as genuinely static, with no "Platonic" time in which they do genuinely evolve.
Nevertheless time has its place in mathematics:
1. An endomorphism of a set (seen as a set of states of a system) into itself can be seen as ... |
(1). I was wondering about what are the relation and differences between formal and informal logic?
What topics does each of them have?
For example, topics such as Meaning and Definition, Syllogistic Logic, Inductive Logic, Probabilistic/Statistical Reasoning, Deductive Logic
Is symbolic logic same as formal log... |
At the high school level, I like *[Thinking Physics][1]* by Lewis Carroll Epstein very much. It's entertaining and illustrated by cartoons, and it delves deeply into the right and wrong answers to basic physics problems. An example: Can you float a battleship in a bathtub?
[1]: http://www.amazon.com/Thinking-Ph... |
I would like to add that books that cultivate interest and critical thinking do not have to be primarily about math (or physics). It is very important to offer young students a broad view of the culture of mathematics and science in general. To this end I recommend Feynman's [*Surely You're Joking, Mr. Feynman*](http... |
Let $f$ be a distribution on $\mathbf{R}^n$ such that
$$\frac{\partial f}{\partial x_i} = 0 \text{ for $i = 1, \ldots, n$.}$$
Then how to prove that $f$ is a constant? I had this exercise in a class last year but I couldn't find how to do the induction step (for $n = 1$ it is clear of course). |
In 1D projective geometry,
i want to compute the 2x2 Homography matrix H (in hom. coords), given 3 pairs of corresponding points.
i.e. i want to find H such that:
[h11 h12; h21 h22] * [0 a a+b; 1 1 1] = [0 a' a'+b'; 1 1 1]
(Matlab matrix notation)
However, i've got 6 equations here and only 3... |
How to compute a 2x2 Homography out of 3 corresponding points? |
<strike>I claim that it is commonly believed that</strike> Mathematical objects can be seen as genuinely static, with no "Platonic" time in which they do genuinely evolve.
Nevertheless time has its place in mathematics:
1. An endomorphism of a set (seen as a set of states of a system) into itself can be seen as ... |
I have a set and its cardinality. How can I calculate the cardinality of its complementary set?
**EDIT**: Gotta apologize. I was talking about well-defined sets with a fairly low positive integer as cardinality. Qwirk comes pretty close when talking about subsets and a given superset.
So see it this way. I have a... |
In 1D projective geometry,
I want to compute the 2x2 Homography matrix $H$ (in homogeneous coordinates), given 3 pairs of corresponding points.
i.e. I want to find H such that:
$$\left(\begin{array}{cc}
h_{11} & h_{12}\\
h_{21} & h_{22}\end{array}\right)\left(\begin{array}{ccc}
0 & a & a+b\\
1 & 1 & 1
\end{a... |
How are operations such as the sum, the product, the quotient, exponentiation, etc. of random variables solved or approached?
<a href="http://math.stackexchange.com/questions/1416/algebra-of-random-variables">This question</a> addresses a similar problem but starts one step further: knowing how to find the distribut... |
Let $f$ be a distribution on $\mathbf{R}^n$ (in the Schwartz sense) such that
$$\frac{\partial f}{\partial x_i} = 0 \text{ for $i = 1, \ldots, n$.}$$
Then how to prove that $f$ is a constant? I had this exercise in a class last year but I couldn't find how to do the induction step (for $n = 1$ it is clear of course... |
Are there necessary and sufficient conditions for the existence of a circuit, or a disjoint set of circuits, that passes through each vertex once in a directed graph? |
I believe the result should be in the form
$$\left(\begin{array}{ccc}
0 & k_2a' & k_3(a'+b')\\
k_1 & k_2 & k_3
\end{array}\right)$$
with $k_1,k_2,k_3$ nonzero, to account for the homogeneous coordinates. Remember that the projective point $[x:y]$ is the same as the point $[kx:ky]$. So the middle answer doesn't hav... |
I've come up with the following variation of a logarithm, intended to work on values that can be 0, or can grow exponentially from zero in either positive or negative direction.
$$myLog(x) =
\begin{cases}
\log(x+1), & \mbox{if }x \geq 0 \\
-\log(-x+1), & \mbox{if }x \lt 0
\end{cases}
$$
Is there a name for... |
Name for logarithm variation that works on non-positive values? |
What is the difference between:
1) $\frac{\partial (\textbf{x}^{T}A)}{\partial \textbf{x}}$
and
2) $\frac{\partial (A\textbf{x})}{\partial \textbf{x}}$
where A is a nxn matrix and x is a n sized column vector.
Using the definition of a Jacobian on wikipedia (http://en.wikipedia.org/wiki/Jacobian_matrix_a... |
I've come up with the following variation of a logarithm, intended to work on values that can be 0, or can grow exponentially from zero in either positive or negative direction.
$$myLog(x) =
\begin{cases}
\log(x+1), & \mbox{if }x \geq 0 \\
-\log(-x+1), & \mbox{if }x \lt 0
\end{cases}
$$
Is there a name for... |
(1). I was wondering about what are the relation and differences between formal and informal logic?
What topics does each of them have?
For example, topics such as Meaning and Definition, Syllogistic Logic, Inductive Logic, Probabilistic/Statistical Reasoning, Deductive Logic
Is symbolic logic same as formal log... |
Unfortunately I'm not versed in philosophical logic, so let's start with some "different categories of logic".
<h3> Scope </h3>
The first relevant category of logic is <i>scope</i>, what is the logic talking about. <b>Propositional logic</b> talks about nonparametric propositions. A typical rule (modus ponens) allo... |
Observe this formulas:
$sin^2 u = \frac{1 - cos 2u}{2}$
$cos^2 u = \frac{1 + cos 2u}{2}$
Then:
$\int\frac{1}{1-\cos (2u)}\mathrm{d}u = - \frac{cot (u)}{2}$
Using of this formulas and partial integration formula you can easily get last part [of the answer][1].
May be this will help you.
... |
A manifold M is simply connected if for every pair of 1-cubes c1,c2: [0,1] -> M with
c1(0) = c1(1) = c2(0) = c2(1) = t
there is a 2-cube b such that
1) b1,0 = c1 and b1,1 = c2
2) for all p in [0,1] b2,0(p) = b2,1(p) = t
My question is, if I'm given a manifold M,
for example choose M = S^2 = {(x,y,z)... |
Given a countable family of semi-norms $p_i$, we can define a metric
$d(f,g) = \sum \limits_{i=0}^{\infty} \frac{ p_i(f-g) }{ 1 + p_i(f-g) }$
We have the locally convex topology induced by the semi-norms as above, as well as the topology induced by the metric.
How does the proof work to show their equality?
... |
Let $p:E\rightarrow B$ be a Serre fibration of path connected spaces with fiber $F$. Are the connecting homomorphisms $\partial:\pi_{n+1}(B)\rightarrow \pi_{n}(F)$ in the long exact sequence of $p$ induced by a **continuous** map $\Omega B\rightarrow F$? |
Why is $\frac{k \cdot(\sin (A) + \sin(B))}{k \cdot(\sin(A) -\sin(B))} = \frac{a+b}{a-b}$ valid in a triangle? |
Given a countable family of semi-norms $p_i$, we can define a metric
$d(f,g) = \sum \limits_{i=0}^{\infty} 2^{-i} \frac{ p_i(f-g) }{ 1 + p_i(f-g) }$
We have the locally convex topology induced by the semi-norms as above, as well as the topology induced by the metric.
How does the proof work to show their equal... |
What do the finite-dimensional continuous complex representations of the additive group $\mathbb{Q}$ with the usual topology look like? With the discrete topology? Which representations are indecomposable? Irreducible?
The only ones I can think of are of the form $t \mapsto e^{tA}$ for some $A \in \mathcal{M}_n(\... |
Representation theory of the additive group of the rationals? |
Let $T$ be a distribution, $f$ be smooth. It is known that
$T(f(x - \cdot) ) \in C ^ \infty$ .
On the other hand, for $\phi$ a test function
$( T \ast f ) (\phi) = T( f( - \cdot ) \ast \phi ) = T( \int f( x - \cdot ) \phi( \cdot ) )$,
because the convolution is still a distribution. However, with regard to ... |
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