TITLE: How to prove by induction that $10^n = 3m+1$
QUESTION [2 upvotes]: Let $n∈ℕ_0$ and $m∈ℕ_0$. Show that $10^n = 3m+1$.
You can clearly see that for $n=0$, $m=0$ and $n=1$, $m=3$ and $n=2$, $m=33$ ... $n$ is the number of how many times a 3 is added to $m$ like $n=5$, then $m$ has to be $33333$ (five three's). But how to show that by induction? I always have problem to show something in a correct mathematical way...

REPLY [3 votes]: You can prove it also like this (without induction)
$$10^k-1 = (10-1)(\underbrace{10^{k-1}+10^{k-2}+...+10+1}_m)=9m$$
so you easyl get a stronger result, that is $$10^k=9m+1$$

REPLY [0 votes]: To proceed by induction we have

base case: $n=0 \implies 10^0=1=3\cdot 0+1$
induction step: assume $10^n=3m+1$ true (Ind. Hyp.) we want to show that $10^{n+1}=3s+1$

then we have
$$10^{n+1}=10\cdot 10^n \stackrel{Ind. Hyp.}= 10 \cdot (3m+1)=$$$$=30m+10=30m+9+1=3(10m+3)+1=3s+1$$


TITLE: Euler - Operator Splitting for $y'+y=3e^2t$, $y(0)=1$
QUESTION [1 upvotes]: $y'+y=3e^2t$, $y(0)=1$
How can we solve this equation with operator splitting methods in matlab by using forward Euler method?

REPLY [1 votes]: $${ y }^{ \prime  }+y=3{ e }^{ 2t }\\ \\ { y }^{ \prime  }+y=0\\ { y }^{ \prime  }=-y\\ \int { \frac { dy }{ y }  } =-\int { dt } \\ \ln { y } =-t+C\\ y=C{ e }^{ -t }\\ y=C\left( t \right) { e }^{ -t }\\ { y }^{ \prime  }={ C }^{ \prime  }\left( t \right) { e }^{ -t }-C\left( t \right) { e }^{ -t }\\ { C }^{ \prime  }\left( t \right) { e }^{ -t }-C\left( t \right) { e }^{ -t }+C\left( t \right) { e }^{ -t }=3{ e }^{ 2t }\\ { C }^{ \prime  }\left( t \right) { e }^{ -t }=3{ e }^{ 2t }\\ { C }^{ \prime  }\left( t \right) =3{ e }^{ 3t }\\ C\left( t \right) =\int { 3{ e }^{ 3t } } dt={ e }^{ 3t }+{ C }_{ 1 }\\ y={ e }^{ -t }\left( { e }^{ 3t }+{ C }_{ 1 } \right) $$
and consider the condition that $y\left( 0 \right) =1$ $$y\left( 0 \right) =1+C=1\\ C=0$$ so the final answer is

$$y={ e }^{ 2t } $$


TITLE: Class group of $\mathbb{P}^2\setminus Y$ is isomorphic to $\mathbb{Z}/d\mathbb{Z}$
QUESTION [0 upvotes]: I'm trying to understand why the following claim is true (this is example $6.5.1$ from Hartshorne's Algebraic Geometry):

If $Y\subset \mathbb{P}^2$ is an irreducible curve of degree $d$, then $\text{Cl}(\mathbb{P}^2\setminus Y)\simeq \mathbb{Z}/d\mathbb{Z}$.

The author simply argues that this is a direct application of proposition $6.4$ (which basically says $\text{Cl}(\mathbb{P}^2)=\mathbb{Z}$) and proposition $6.5$, which says there is an exact sequence:
$$\mathbb{Z}\to \underbrace{\text{Cl}(\mathbb{P}^2)}_{\mathbb{Z}}\to\text{Cl}(\mathbb{P}^2\setminus Y)\to 0$$
where the first map is $1\mapsto [Y]$ and the second is $\sum_i n_i[Z_i]\mapsto \sum_i n_i[U\cap Z_i]$ with $U:=\mathbb{P}^2\setminus Y$.
Since the map $\mathbb{Z}\to\text{Cl}(\mathbb{P}^2\setminus Y)$ is surjective, we have that $\text{Cl}(\mathbb{P}^2\setminus Y)$ is cyclic. But how do we make $d$ show up?

REPLY [2 votes]: The class group of $\mathbb P^2$ is generated by the class of a line, call it $L$. In this setting, $[Y] = dL$, so the (in fact injective) map on the left includes $\mathbb Z$ into $\operatorname{Cl}(\mathbb P^2)$ as $d\mathbb Z$.


text:
In this paper, we define \ourcodes, which are codes defined on the Hermitian curve
 with small locality, high availability, and rate bounded below by a constant.
They are the evaluation code of polynomials whose restrictions to lines, intersected with the Hermitian curve, are all low-degree.  We study these codes as a first example of curve-lifted codes.  We establish the lower bound on the rate via a counting argument applied to certain ``good'' monomials.

We conclude with a few open questions.  First, it is an interesting question to completely characterize the ``good'' monomials for \ourcodes; determining their number would pin down the rate of these codes.
Second, it is interesting to explore other constructions of curve-lifted codes.  We view one of the main contributions of this work as introducing this paradigm for code constructions, and it is our hope that our construction and analysis of \ourcodes\ may serve as a prototype for  the construction and analysis of
other families of curve-lifted codes.

\section*{Acknowledgements} The authors thank AIM for hosting this collaboration through its SQuaREs program.

config: arxiv


TITLE: What are some specific operations for matrices, like is dot product for vectors?
QUESTION [0 upvotes]: I read this question and the associated answer (What is the dot product between a vector of matrices?), and it wasn't what I am looking for.
I want to know if there are some special operations designed for bi-dimensional or multi-dimensional vectors ?
When I refer to dot product, I don't mean the equivalent for matrices, but rather an operation that deals with matrices and computes something.

REPLY [3 votes]: There are many specific operations for matrices, usually given as a bilinear product from $M_n(K)\times M_n(K)\rightarrow M_n(K)$, like the matrix product or the Lie bracket $(X,Y)\mapsto [X,Y]=XY-YX$. Another operation is the Kronecker product from
$M_{m,n}(K)\otimes M_{p,q}(K)\rightarrow M_{mp,nq}(K)$.


TITLE: Trouble solving polynomial equation with exponent
QUESTION [2 upvotes]: I'm having trouble solving this equation.It looks simple, but I just can't find the answer.Can someone help me?
$$9x^4-13x^2+4 = 0$$

REPLY [2 votes]: Let $t = x^2$ then we have $9t^2 - 13t + 4=0$
This factorises to give $(9t-4)(t-1)=0$ so our solutions are $t = \frac{4}{9}$ or $t=1$.
So $x = \pm\frac{2}{3}$ or $x=\pm1$.


TITLE: Prove Surjectivity of map iff there exists an inverse of the map
QUESTION [2 upvotes]: To prove the problem below I am having some trouble finding an inverse to prove $\Rightarrow$.  Is there a theorem I can use to prove the existence of the identity or is the inverse the correct approach?
Thanks in advance

Let $T\in \mathcal{L}(V,W)$, with $dimV<\infty$, $dimW<\infty$
Prove that $T$ is surjective iff $\exists$ $S\in \mathcal{L}(W,V)$ such that $T\circ S=I_w$

REPLY [4 votes]: First of all I think that the title is a little misleading. In fact the theorem does not say that $T$ is surjective iff exists the inverse, it says instead iff exists the right inverse. I'll make an example. If you have a linear application $F:\mathbb R^3\rightarrow\mathbb R^2$ such that $F(x,y,z)=(x,y)$ you have for sure a surjective application but you won't find it's inverse. In fact that is valid only if the two spaces have the same (finite) dimension (that was just in case you didn't know, if you wrote that title in order to simplify it I'm sorry). That said, let's demonstrate the $(\Leftarrow)$ part: take $x \in W$ then we have that $S(x)=y \in V$, so $T(y)=T(S(x))=I_W(x)=x$.
Now the $(\Rightarrow)$ part $: \forall y \in W, \exists x \in V: T(x)=y $. Now I define $S:W \rightarrow V$ such that $S(y)=x$, if I take $y \in W, \rightarrow T(S(y))=T(x)=y \Rightarrow T \circ S=I_w$ (note that $S$ is linear). Hope it helps.


TITLE: Quadric isomorphic to affine space
QUESTION [1 upvotes]: Let $K$ be a field and $X$ be irreducible in $\mathbb{A}^{n+1}_K$. Prove, that $X$ is birationally isomorphic to $\mathbb{A}^{n}_K$ if and only if $X$ contains a point over $K$.
Actually, I can't prove the converse statement: if X is birationally isomorphis to nK then it contains a point over K. The direct statement is almost trivial.

REPLY [4 votes]: (Note: in the question statement, you don't specify that X is an irreducible quadric hypersurface.)
It's interesting that you found the converse harder. If two varieties $X$ and $Y$ are birational, then there are actually open dense subsets $U \subset X$ and $V \subset Y$ such that $U$ and $V$ are actually isomorphic (see for instance p.188 of http://math.stanford.edu/~vakil/216blog/FOAGjun1113public.pdf), any certainly any open dense subset of affine space contains $K$-rational points.
You imply that you already understand the other direction, but for completeness let me say that it's the usual trick of picking a rational point and projecting along it.


TITLE: How can I apply Planck's law to calculate the photon energy density of a warm room?
QUESTION [2 upvotes]: I am trying to calculate the rate at which photons of any particular frequency will pass through a volume in a room illuminated by black body radiation only.
I've found a couple of starting points but now I'm stuck putting it together.
I could start with
$B_\lambda(\lambda, T) =\frac{2 hc^2}{\lambda^5}\frac{1}{ e^{\frac{hc}{\lambda k_\mathrm{B}T}} - 1}$ and pretend all the radiation is coming from one spot on the wall, then multiply it by the effective solid angle to the volume and the volume itself, but I'm not sure if this is reasonable.
Also since I'm counting photons it seems I want to take a definite integral over the relevant frequency range. In which case, setting $x=\lambda$, $k=2kc^2$, and $q=hc/(k_BT)$ I'll end up with this less-than-elegant answer:

I suspect there's an easier way. Is there a simple formula for the energy density of empty space when the volume under examination is "far" from the walls?
To restate the problem: take a small volume of space within a room whose walls are emitting black body radiation at a known temperature; the size and shape of the room is known (we can assume any shape which is convenient, if that matters) -- how many photons per second with wavelength $\lambda_{min}<\lambda<\lambda_{max}$ are passing through this volume?

REPLY [0 votes]: Just take the volume as your system and integrate the Plank distribution over the wavelengths/frequencies you are interested in (tip: don't use the wave-length but the angular frequency version, it is more concise). This works, because the Planck law describes an equilibrium setting, whether your volume is in equilibrium with the walls or the surrounding vacuum does not matter.
Then the average number of photons in the volume can easily be given, by the well known formulas for the number of photons in a cavity.
(This procedure obviously gets invalid if the volume becomes so small that a significant number of the photons have a wavelength larger than the sides of the box).