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We have already seen the counterexample of rotation by $90$ for $k = $;
this was the same counterexample we gave to the assertion that all linear maps
have eigenvalues.
The representations of $_d(k)$
To give an example of the kind of progress already possible, we prove:
[Representations of $_d(k)$]
thm:rep_1mat
Let $... | An Infinitely Large Napkin | napkin | general | advanced | Representation Theory | Representations of algebras | 23_rep-alg.md | 3 | 1,205 | |
Representations of algebras
ch:representations_of_algebras
In the 19th century, the word ``group'' hadn't been invented yet;
all work was done with subsets of $(n)$ or $S_n$.
Only much later was the abstract definition of a group was given,
an abstract set $G$ which was an object in its own right.
While this abstracti... | An Infinitely Large Napkin | napkin | general | advanced | Representation Theory | Representations of algebras | 23_rep-alg.md | 4 | 1,564 | |
More generally, given a vector space $V$ over any field $k$,
there is an obvious representation of $A = (V)$
by $a v = (a)(v) = a(v)$ (since $a (V)$).
From the matrix perspective: if $A = (V)$,
then we can just represent $A$ as matrices over $V$.
There are other representations of $A = _2()$.
A silly example is the r... | An Infinitely Large Napkin | napkin | general | advanced | Representation Theory | Representations of algebras | 23_rep-alg.md | 5 | 1,532 | |
Let $V$ be a representation of $A$.
A subrepresentation $W V$ is a subspace $W$
with the property that for any $a A$ and $w W$,
$a w W$.
In other words, this subspace is invariant under actions by $A$.
Thus for example if $V = W_1 W_2$ for representations $W_1$, $W_2$
then $W_1$ and $W_2$ are subrepresentations of $V$... | An Infinitely Large Napkin | napkin | general | advanced | Representation Theory | Representations of algebras | 23_rep-alg.md | 6 | 1,519 | |
We have already seen the counterexample of rotation by $90$ for $k = $;
this was the same counterexample we gave to the assertion that all linear maps
have eigenvalues.
The representations of $_d(k)$
To give an example of the kind of progress already possible, we prove:
[Representations of $_d(k)$]
thm:rep_1mat
Let $... | An Infinitely Large Napkin | napkin | general | advanced | Representation Theory | Representations of algebras | 23_rep-alg.md | 7 | 1,205 | |
Semisimple algebras
ch:semisimple_algebras
In what follows, assume the field $k$ is algebraically closed.
Fix an algebra $A$ and suppose
you want to study its representations.
We have a ``direct sum'' operation already.
So, much like we pay special attention to prime numbers,
we're motivated to study irreducible repre... | An Infinitely Large Napkin | napkin | general | advanced | Representation Theory | Semisimple algebras | 24_semisimple.md | 0 | 1,567 | |
[Jacobson density theorem]
Let $(V_1, _1)$, , $(V_r, _r)$ be pairwise nonisomorphic
finite-dimensional irreps of $A$.
Then there is a surjective map of vector spaces
\[ _i=1^r _i A _i=1^r (V_i). \]
The right way to think about this theorem is that
Density is the ``Chinese remainder theorem''
for finite-dimensional ir... | An Infinitely Large Napkin | napkin | general | advanced | Representation Theory | Semisimple algebras | 24_semisimple.md | 1 | 1,539 | |
Note that unlike the case where $A$ is a PID,
$k^ d_i$ is not isomorphic to a quotient of the ring $_d_i(k)$.
In fact, if we combine the above result with
the density theorem (and cor:finiteness), we obtain:
[Sum of squares formula]
For a finite-dimensional algebra $A$ we have
\[ _i (V_i)^2 A \]
where the $V_i$ are t... | An Infinitely Large Napkin | napkin | general | advanced | Representation Theory | Semisimple algebras | 24_semisimple.md | 2 | 1,564 | |
Pick any $v V$, then the subspace
spanned by elements $g v$ for $v V$
is $G$-invariant;
this is a finite-dimensional subspace,
so it must equal all of $V$.
prob:dten_irrep
Determine all the complex irreps of $D_10$.
There are only two one-dimensional ones
(corresponding to the only two homomorphisms $D_10 ^$).
So the... | An Infinitely Large Napkin | napkin | general | advanced | Representation Theory | Semisimple algebras | 24_semisimple.md | 3 | 326 | |
Semisimple algebras
ch:semisimple_algebras
In what follows, assume the field $k$ is algebraically closed.
Fix an algebra $A$ and suppose
you want to study its representations.
We have a ``direct sum'' operation already.
So, much like we pay special attention to prime numbers,
we're motivated to study irreducible repre... | An Infinitely Large Napkin | napkin | general | advanced | Representation Theory | Semisimple algebras | 24_semisimple.md | 4 | 1,567 | |
[Jacobson density theorem]
Let $(V_1, _1)$, , $(V_r, _r)$ be pairwise nonisomorphic
finite-dimensional irreps of $A$.
Then there is a surjective map of vector spaces
\[ _i=1^r _i A _i=1^r (V_i). \]
The right way to think about this theorem is that
Density is the ``Chinese remainder theorem''
for finite-dimensional ir... | An Infinitely Large Napkin | napkin | general | advanced | Representation Theory | Semisimple algebras | 24_semisimple.md | 5 | 1,539 | |
Note that unlike the case where $A$ is a PID,
$k^ d_i$ is not isomorphic to a quotient of the ring $_d_i(k)$.
In fact, if we combine the above result with
the density theorem (and cor:finiteness), we obtain:
[Sum of squares formula]
For a finite-dimensional algebra $A$ we have
\[ _i (V_i)^2 A \]
where the $V_i$ are t... | An Infinitely Large Napkin | napkin | general | advanced | Representation Theory | Semisimple algebras | 24_semisimple.md | 6 | 1,564 | |
Pick any $v V$, then the subspace
spanned by elements $g v$ for $v V$
is $G$-invariant;
this is a finite-dimensional subspace,
so it must equal all of $V$.
prob:dten_irrep
Determine all the complex irreps of $D_10$.
There are only two one-dimensional ones
(corresponding to the only two homomorphisms $D_10 ^$).
So the... | An Infinitely Large Napkin | napkin | general | advanced | Representation Theory | Semisimple algebras | 24_semisimple.md | 7 | 326 | |
Characters
ch:characters
Characters are basically the best thing ever.
To every representation $V$ of $A$ we will attach a
so-called character $_V A k$.
It will turn out that the characters of finite-dimensional irreps of $V$
will determine the representation $V$ completely.
Thus a finite-dimensional irrep is just spec... | An Infinitely Large Napkin | napkin | general | advanced | Representation Theory | Characters | 25_characters.md | 0 | 1,536 | |
Let $(G)$ denote the set of functions $(G) $
viewed as a vector space over $$.
We endow it with the inner form
\[
< f_1, f_2 > =
1|G|
_g G f_1(g) f_2(g).
\]
This is the same ``dot product'' that we mentioned at the beginning,
when we looked at the character table of $S_3$.
We now aim to prove the following orthogonali... | An Infinitely Large Napkin | napkin | general | advanced | Representation Theory | Characters | 25_characters.md | 1 | 1,477 | |
Now, the remaining three representations have dimensions
$d_1$, $d_2$, $d_3$ with
\[ d_1^2 + d_2^2 + d_3^2 = 4! - 2 = 22 \]
which has only $(d_1, d_2, d_3) = (2,3,3)$ and permutations.
Now, we can take the $_0$ representation
\[ \ (w,x,y,z) w+x+y+z=0 \ \]
with basis $(1,0,0,-1)$, $(0,1,0,-1)$ and $(0,0,1,-1)$.
This can... | An Infinitely Large Napkin | napkin | general | advanced | Representation Theory | Characters | 25_characters.md | 2 | 905 | |
Characters
ch:characters
Characters are basically the best thing ever.
To every representation $V$ of $A$ we will attach a
so-called character $_V A k$.
It will turn out that the characters of finite-dimensional irreps of $V$
will determine the representation $V$ completely.
Thus a finite-dimensional irrep is just spec... | An Infinitely Large Napkin | napkin | general | advanced | Representation Theory | Characters | 25_characters.md | 3 | 1,536 | |
Let $(G)$ denote the set of functions $(G) $
viewed as a vector space over $$.
We endow it with the inner form
\[
< f_1, f_2 > =
1|G|
_g G f_1(g) f_2(g).
\]
This is the same ``dot product'' that we mentioned at the beginning,
when we looked at the character table of $S_3$.
We now aim to prove the following orthogonali... | An Infinitely Large Napkin | napkin | general | advanced | Representation Theory | Characters | 25_characters.md | 4 | 1,477 | |
Now, the remaining three representations have dimensions
$d_1$, $d_2$, $d_3$ with
\[ d_1^2 + d_2^2 + d_3^2 = 4! - 2 = 22 \]
which has only $(d_1, d_2, d_3) = (2,3,3)$ and permutations.
Now, we can take the $_0$ representation
\[ \ (w,x,y,z) w+x+y+z=0 \ \]
with basis $(1,0,0,-1)$, $(0,1,0,-1)$ and $(0,0,1,-1)$.
This can... | An Infinitely Large Napkin | napkin | general | advanced | Representation Theory | Characters | 25_characters.md | 5 | 905 | |
With all this setup, we now take the time to develop some nice results which are of independent interest.
With all this setup, we now take the time to develop some nice results which are of independent interest.
Let $V$ be a complex irrep of a finite group $G$. Then $\dim V$ divides $|G|$.
The proof of this will req... | An Infinitely Large Napkin | napkin | general | advanced | Representation Theory | Some applications | 26_applications.md | 0 | 638 | |
We now prove a group-theoretic result. This is the famous poster child for representation theory (in the same way that RSA is the poster child of number theory) because the result is purely group theoretic.
Recall that a group is **simple** if it has no normal subgroups. In fact, we will prove:
Let $G$ be a nonabelia... | An Infinitely Large Napkin | napkin | general | advanced | Representation Theory | Some applications | Burnside's theorem | 26_applications.md | 1 | 821 |
We finish with the following result, the problem that started the branch of representation theory. Given a finite group $G$, we create $n$ variables $\{x_g\}_{g \in G}$, and an $n \times n$ matrix $M_G$ whose $(g,h)$th entry is $x_{gh}$.
(a) If $G = \mathbb{Z}/2\mathbb{Z} = \left< T \mid T^2 = 1\right>$ then the matri... | An Infinitely Large Napkin | napkin | general | advanced | Representation Theory | Some applications | Frobenius determinant | 26_applications.md | 2 | 936 |
Quantum states and measurements
ch:quantum_states_and_measurements
In this chapter we'll explain how to set up quantum states using
linear algebra. This will allow me to talk about quantum circuits
in the next chapter, which will set the stage for Shor's algorithm.
I won't do very much physics (read: none at all).
Tha... | An Infinitely Large Napkin | napkin | general | advanced | Quantum Algorithms | Quantum states and measurements | 27_vectors.md | 0 | 1,470 | |
Suppose for simplicity that we observe $$
with $T$ and obtain an eigenvalue $$,
and that $i_T$ is the only eigenvector with this eigenvalue.
Then, the state $$ collapses to just the state
$c_i i_T$: all the other information is destroyed.
(In fact, we may as well say it collapses to $i_T$,
since again constant factors ... | An Infinitely Large Napkin | napkin | general | advanced | Quantum Algorithms | Quantum states and measurements | 27_vectors.md | 1 | 1,469 | |
(We could have used other bases, like $_A 0_B$ and
$_A 0_B$ for the first eigenspace, but it doesn't matter.)
Expanding $$ in the four-element basis, we find that
we'll get the first eigenspace with probability
\[ |i10|^2
+ |210|^2 = . \]
and the second eigenspace with probability $$ as well.
(Note how the coefficients... | An Infinitely Large Napkin | napkin | general | advanced | Quantum Algorithms | Quantum states and measurements | 27_vectors.md | 2 | 1,137 | |
Quantum states and measurements
ch:quantum_states_and_measurements
In this chapter we'll explain how to set up quantum states using
linear algebra. This will allow me to talk about quantum circuits
in the next chapter, which will set the stage for Shor's algorithm.
I won't do very much physics (read: none at all).
Tha... | An Infinitely Large Napkin | napkin | general | advanced | Quantum Algorithms | Quantum states and measurements | 27_vectors.md | 3 | 1,470 | |
Suppose for simplicity that we observe $$
with $T$ and obtain an eigenvalue $$,
and that $i_T$ is the only eigenvector with this eigenvalue.
Then, the state $$ collapses to just the state
$c_i i_T$: all the other information is destroyed.
(In fact, we may as well say it collapses to $i_T$,
since again constant factors ... | An Infinitely Large Napkin | napkin | general | advanced | Quantum Algorithms | Quantum states and measurements | 27_vectors.md | 4 | 1,469 | |
(We could have used other bases, like $_A 0_B$ and
$_A 0_B$ for the first eigenspace, but it doesn't matter.)
Expanding $$ in the four-element basis, we find that
we'll get the first eigenspace with probability
\[ |i10|^2
+ |210|^2 = . \]
and the second eigenspace with probability $$ as well.
(Note how the coefficients... | An Infinitely Large Napkin | napkin | general | advanced | Quantum Algorithms | Quantum states and measurements | 27_vectors.md | 5 | 1,137 | |
Quantum circuits
ch:quantum_circuits
Now that we've discussed qubits, we can talk about how to use them in circuits.
The key change --- and the reason that quantum circuits can do things that
classical circuits cannot --- is the fact that we are allowing
linear combinations of $0$ and $1$.
Classical logic gates
In cla... | An Infinitely Large Napkin | napkin | general | advanced | Quantum Algorithms | Quantum circuits | 28_circuits.md | 0 | 1,574 | |
Quantum logic gates
In quantum mechanics, since we can have linear combinations of basis
elements, our logic gates will instead consist of linear maps.
Moreover, in quantum computation, gates are always reversible,
which was why we took the time in the previous section to show
that we can still simulate any function wh... | An Infinitely Large Napkin | napkin | general | advanced | Quantum Algorithms | Quantum circuits | 28_circuits.md | 1 | 1,538 | |
[Deutsch-Jozsa]
The Deutsch-Jozsa problem can be determined in a quantum circuit
with only a single call to the black box.
For concreteness, we do the case $n=1$ explicitly;
the general case is contained in prob:deutsch_jozsa.
We claim that the necessary circuit is
\[
@C=1em @R=0.7em
0 & H & 1U_f & H & \\
1 & H & U_f ... | An Infinitely Large Napkin | napkin | general | advanced | Quantum Algorithms | Quantum circuits | 28_circuits.md | 2 | 1,532 | |
Quantum circuits
ch:quantum_circuits
Now that we've discussed qubits, we can talk about how to use them in circuits.
The key change --- and the reason that quantum circuits can do things that
classical circuits cannot --- is the fact that we are allowing
linear combinations of $0$ and $1$.
Classical logic gates
In cla... | An Infinitely Large Napkin | napkin | general | advanced | Quantum Algorithms | Quantum circuits | 28_circuits.md | 3 | 1,574 | |
Quantum logic gates
In quantum mechanics, since we can have linear combinations of basis
elements, our logic gates will instead consist of linear maps.
Moreover, in quantum computation, gates are always reversible,
which was why we took the time in the previous section to show
that we can still simulate any function wh... | An Infinitely Large Napkin | napkin | general | advanced | Quantum Algorithms | Quantum circuits | 28_circuits.md | 4 | 1,538 | |
[Deutsch-Jozsa]
The Deutsch-Jozsa problem can be determined in a quantum circuit
with only a single call to the black box.
For concreteness, we do the case $n=1$ explicitly;
the general case is contained in prob:deutsch_jozsa.
We claim that the necessary circuit is
\[
@C=1em @R=0.7em
0 & H & 1U_f & H & \\
1 & H & U_f ... | An Infinitely Large Napkin | napkin | general | advanced | Quantum Algorithms | Quantum circuits | 28_circuits.md | 5 | 1,532 | |
OK, now for Shor's Algorithm: how to factor $M = pq$ in $O\left( (\log M)^2 \right)$ time.
This is arguably the reason agencies such as the US's National Security Agency have been diverting millions of dollars toward quantum computing.
OK, now for Shor's Algorithm: how to factor $M = pq$ in $O\left( (\log M)^2 \right... | An Infinitely Large Napkin | napkin | general | advanced | Quantum Algorithms | Shor's algorithm | 29_shor.md | 0 | 946 | |
The "crux move" in Shor's algorithm is the so-called quantum Fourier transform. The Fourier transform is used to extract *periodicity* in data, and it turns out the quantum analogue is a lot faster than the classical one.
Let me throw the definition at you first. Let $N$ be a positive integer, and let $\omega_N = \exp... | An Infinitely Large Napkin | napkin | general | advanced | Quantum Algorithms | Shor's algorithm | The classical (inverse) Fourier transform | 29_shor.md | 1 | 810 |
Note that to compute a Fourier transform, we need to multiply an $N \times N$ matrix with an $N$-vector, so this takes $O(N^2)$ multiplications. However, we are about to show that with a quantum computer, one can do this using $O( (\log N)^2 )$ quantum gates when $N = 2^n$, on a system with $n$ qubits.
First, some mor... | An Infinitely Large Napkin | napkin | general | advanced | Quantum Algorithms | Shor's algorithm | The quantum Fourier transform | 29_shor.md | 2 | 1,211 |
The quantum Fourier transform is the key piece of Shor's algorithm. Now that we have it, we can solve the factoring problem.
Let $p,q > 3$ be odd primes, and assume $p \neq q$. The main idea is to turn factoring an integer $M = pq$ into a problem about finding the order of $x \pmod M$; the latter is a "periodicity" pr... | An Infinitely Large Napkin | napkin | general | advanced | Quantum Algorithms | Shor's algorithm | Shor's algorithm | 29_shor.md | 3 | 1,372 |
The quantum Fourier transform is the key piece of Shor's algorithm. Now that we have it, we can solve the factoring problem.
Let $p,q > 3$ be odd primes, and assume $p \neq q$. The main idea is to turn factoring an integer $M = pq$ into a problem about finding the order of $x \pmod M$; the latter is a "periodicity" pr... | An Infinitely Large Napkin | napkin | general | advanced | Quantum Algorithms | Shor's algorithm | Shor's algorithm | 29_shor.md | 4 | 1,372 |
Limits and series
ch:calc_limits
Now that we have developed the theory
of metric (and topological) spaces well,
we give a three-chapter sequence which
briskly covers the theory of single-variable calculus.
Much of the work has secretly already been done,
For example, if $x_n$ and $y_n$ are real sequences
with $_n x_n ... | An Infinitely Large Napkin | napkin | general | advanced | Category Theory | Limits in categories (TO DO) | 30_limits.md | 0 | 1,441 | |
size(10cm);
draw( (-8,0)--(8,0), Arrows );
label("$ R$", (8,0), dir(-90));
draw( (3.1,1)--(3.1,-1), red );
draw( (2.7,1.5)--(2.7,-1.5), deepgreen );
label("$M$", (3.1,-1), dir(-45), red);
label("$M-12$", (2.7,1.5), dir(90), deepgreen);
dot("$a_1$", (-5, 0), dir(-90), blue);
dot("$a_2$", (-1, 0), dir(-90), blue);
dot("$... | An Infinitely Large Napkin | napkin | general | advanced | Category Theory | Limits in categories (TO DO) | 30_limits.md | 1 | 1,532 | |
Consider a sequence $a_1$, of real numbers.
The series $_k a_k$ converges to a limit $L$ if
the sequence of ``partial sums''
s_1 &= a_1 \\
s_2 &= a_1 + a_2 \\
s_3 &= a_1 + a_2 + a_3 \\
&= \\
s_n &= a_1 + + a_n
converges to the limit $L$.
Otherwise it diverges.
[Writing divergence as $+$]
It is customary, if all the ... | An Infinitely Large Napkin | napkin | general | advanced | Category Theory | Limits in categories (TO DO) | 30_limits.md | 2 | 1,564 | |
The terms in the first line sum up to within $$ of $L$,
and the terms in the second line have sum at most $$
in absolute value, so the total $b_1 + + b_M$
is within $ + = $ of $L$.
In particular, when you have nonnegative terms, the world is great:
Nonnegative series can be rearranged at will.
And the good news is t... | An Infinitely Large Napkin | napkin | general | advanced | Category Theory | Limits in categories (TO DO) | 30_limits.md | 3 | 1,570 | |
Iff the sequence is convergent!
[Comparison test]
Let $ a_n$ and $ b_n$ be two series.
Assume $ b_n$ is absolutely convergent,
and $|a_n| |b_n|$ for all integers $n$.
Prove that $_n a_n$ is absolutely convergent.
[Geometric series]
prob:geometric
Let $-1 < r < 1$ be a real number.
Show that the series
\[ 1 + r + r^2 ... | An Infinitely Large Napkin | napkin | general | advanced | Category Theory | Limits in categories (TO DO) | 30_limits.md | 4 | 1,062 | |
Limits and series
ch:calc_limits
Now that we have developed the theory
of metric (and topological) spaces well,
we give a three-chapter sequence which
briskly covers the theory of single-variable calculus.
Much of the work has secretly already been done,
For example, if $x_n$ and $y_n$ are real sequences
with $_n x_n ... | An Infinitely Large Napkin | napkin | general | advanced | Category Theory | Limits in categories (TO DO) | 30_limits.md | 5 | 1,441 | |
size(10cm);
draw( (-8,0)--(8,0), Arrows );
label("$ R$", (8,0), dir(-90));
draw( (3.1,1)--(3.1,-1), red );
draw( (2.7,1.5)--(2.7,-1.5), deepgreen );
label("$M$", (3.1,-1), dir(-45), red);
label("$M-12$", (2.7,1.5), dir(90), deepgreen);
dot("$a_1$", (-5, 0), dir(-90), blue);
dot("$a_2$", (-1, 0), dir(-90), blue);
dot("$... | An Infinitely Large Napkin | napkin | general | advanced | Category Theory | Limits in categories (TO DO) | 30_limits.md | 6 | 1,532 | |
Consider a sequence $a_1$, of real numbers.
The series $_k a_k$ converges to a limit $L$ if
the sequence of ``partial sums''
s_1 &= a_1 \\
s_2 &= a_1 + a_2 \\
s_3 &= a_1 + a_2 + a_3 \\
&= \\
s_n &= a_1 + + a_n
converges to the limit $L$.
Otherwise it diverges.
[Writing divergence as $+$]
It is customary, if all the ... | An Infinitely Large Napkin | napkin | general | advanced | Category Theory | Limits in categories (TO DO) | 30_limits.md | 7 | 1,564 | |
The terms in the first line sum up to within $$ of $L$,
and the terms in the second line have sum at most $$
in absolute value, so the total $b_1 + + b_M$
is within $ + = $ of $L$.
In particular, when you have nonnegative terms, the world is great:
Nonnegative series can be rearranged at will.
And the good news is t... | An Infinitely Large Napkin | napkin | general | advanced | Category Theory | Limits in categories (TO DO) | 30_limits.md | 8 | 1,570 | |
Iff the sequence is convergent!
[Comparison test]
Let $ a_n$ and $ b_n$ be two series.
Assume $ b_n$ is absolutely convergent,
and $|a_n| |b_n|$ for all integers $n$.
Prove that $_n a_n$ is absolutely convergent.
[Geometric series]
prob:geometric
Let $-1 < r < 1$ be a real number.
Show that the series
\[ 1 + r + r^2 ... | An Infinitely Large Napkin | napkin | general | advanced | Category Theory | Limits in categories (TO DO) | 30_limits.md | 9 | 1,062 | |
This is a bonus chapter meant for those who have also read about **rings and fields**: it's a nice tidbit at the intersection of algebra and analysis.
In this chapter, we are going to redo most of the previous chapter with the absolute value $\left\lvert - \right\rvert$ replaced by the $p$-adic one. This will give us ... | An Infinitely Large Napkin | napkin | general | advanced | Calculus 101 | P Adic | 31_p-adic.md | 0 | 544 | |
Before really telling you what $\mathbb{Z}_p$ and $\mathbb{Q}_p$ are, let me tell you what you might expect them to do.
In elementary/olympiad number theory, we're already well-familiar with the following two ideas:
- Taking modulo a prime $p$ or prime power $p^e$, and
- Looking at the exponent $\nu_p$.
Let me expan... | An Infinitely Large Napkin | napkin | general | advanced | Calculus 101 | P Adic | Motivation | 31_p-adic.md | 1 | 888 |
Before really telling you what $\mathbb{Z}_p$ and $\mathbb{Q}_p$ are, let me tell you what you might expect them to do.
In elementary/olympiad number theory, we're already well-familiar with the following two ideas:
- Taking modulo a prime $p$ or prime power $p^e$, and
- Looking at the exponent $\nu_p$.
Let me expan... | An Infinitely Large Napkin | napkin | general | advanced | Calculus 101 | P Adic | Motivation | 31_p-adic.md | 2 | 888 |
We now construct $\mathbb{Z}_p$ and $\mathbb{Q}_p$. I promised earlier that a $p$-adic integer will let you look at "all residues modulo $p^e$" at once. This definition will formalize this.
We now construct $\mathbb{Z}_p$ and $\mathbb{Q}_p$. I promised earlier that a $p$-adic integer will let you look at "all residues... | An Infinitely Large Napkin | napkin | general | advanced | Calculus 101 | P Adic | Algebraic perspective | 31_p-adic.md | 3 | 932 |
Here is another way to think about $p$-adic integers using "base $p$". As in the example earlier, every usual integer can be written in base $p$, for example $$50 = \overline{1212}_3 = 2 \cdot 3^0 + 1 \cdot 3^1 + 2 \cdot 3^2 + 1 \cdot 3^3.$$ More generally, given any $x = (x_1, \dots) \in \mathbb{Z}_p$, we can write do... | An Infinitely Large Napkin | napkin | general | advanced | Calculus 101 | P Adic | Base $p$ expansion | 31_p-adic.md | 4 | 871 |
Here is one way in which your intuition from generating functions carries over:
The number $x \in \mathbb{Z}_p$ is invertible if and only if $x_1 \neq 0$. In symbols, $$x \in \mathbb{Z}_p^\times \iff x \not\equiv 0 \pmod p.$$
Contrast this with the corresponding statement for $K[ [ X ] ]$: a generating function $F \i... | An Infinitely Large Napkin | napkin | general | advanced | Calculus 101 | P Adic | Constructing $\mathbb{Q}_p$ | 31_p-adic.md | 5 | 1,130 |
Up until now we've been thinking about things mostly algebraically, but moving forward it will be helpful to start using the language of analysis. Usually, two real numbers are considered "close" if they are close on the number of line, but for $p$-adic purposes we only care about modulo $p^e$ information. So, we'll in... | An Infinitely Large Napkin | napkin | general | advanced | Calculus 101 | P Adic | Definition | 31_p-adic.md | 6 | 836 |
In this way, $\mathbb{Q}_p$ and $\mathbb{Z}_p$ becomes a metric space with metric given by $\left\lvert x-y \right\rvert_p$.
Suppose $f \colon \mathbb{Z}_p \to \mathbb{Q}_p$ is continuous and $f(n) = (-1)^n$ for every $n \in \mathbb{Z}_{\ge 0}$. Prove that $p = 2$.
In fact, these spaces satisfy a stronger form of the... | An Infinitely Large Napkin | napkin | general | advanced | Calculus 101 | P Adic | Ultrametric space | 31_p-adic.md | 7 | 780 |
Let's finally state the $p$-adic analog of the geometric series formula.
Let $x \in \mathbb{Z}_p$ with $\left\lvert x \right\rvert_p < 1$. Then $$\frac{1}{1-x} = 1 + x + x^2 + x^3 + \dots.$$
*Proof.* Note that the partial sums satisfy $1 + x + x^2 + \dots + x^n = \frac{1-x^n}{1-x}$, and $x^n \to 0$ as $n \to \infty$ ... | An Infinitely Large Napkin | napkin | general | advanced | Calculus 101 | P Adic | More fun with geometric series | 31_p-adic.md | 8 | 264 |
Note that the definition of $\left\lvert \bullet \right\rvert_p$ could have been given for $\mathbb{Q}$ as well; we didn't need $\mathbb{Q}_p$ to introduce it (after all, we have $\nu_p$ in olympiads already). The big important theorem I must state now is:
The space $\mathbb{Q}_p$ is the completion of $\mathbb{Q}$ wit... | An Infinitely Large Napkin | napkin | general | advanced | Calculus 101 | P Adic | Completeness | 31_p-adic.md | 9 | 360 |
Let me justify why this definition is philosophically nice. Suppose you are an ancient Greek mathematician who is given:
> **Problem for Ancient Greeks.** Estimate the value of the sum $$S = \frac{1}{1^2} + \frac{1}{2^2} + \dots + \frac{1}{10000^2}$$ to within $0.001$.
The sum $S$ consists entirely of rational number... | An Infinitely Large Napkin | napkin | general | advanced | Calculus 101 | P Adic | Philosophical notes | 31_p-adic.md | 10 | 1,352 |
One of the big surprises of $p$-adic analysis is that:
We can basically describe all continuous functions $\mathbb{Z}_p \to \mathbb{Q}_p$.
They are given by a basis of functions $$\binom xn := \frac{x(x-1) \dots (x-(n-1))}{n!}$$ in the following way.
Let $f \colon \mathbb{Z}_p \to \mathbb{Q}_p$ be continuous, and de... | An Infinitely Large Napkin | napkin | general | advanced | Calculus 101 | P Adic | Mahler coefficients | 31_p-adic.md | 11 | 1,438 |
I suspect most of you have seen this before, but:
Let $U$ be an open subset[^1] of $\mathbb{R}$ and let $f \colon U \to \mathbb{R}$ be a function. Let $p \in U$. We say $f$ is **differentiable** at $p$ if the limit[^2] $$\lim_{h \to 0} \frac{f(p+h) - f(p)}{h}$$ exists. If so, we denote its value by $f'(p)$ and refer t... | An Infinitely Large Napkin | napkin | general | advanced | Calculus 101 | Differentiation | Definition | 32_differentiate.md | 0 | 1,029 |
I suspect most of you have seen this before, but:
Let $U$ be an open subset[^1] of $\mathbb{R}$ and let $f \colon U \to \mathbb{R}$ be a function. Let $p \in U$. We say $f$ is **differentiable** at $p$ if the limit[^2] $$\lim_{h \to 0} \frac{f(p+h) - f(p)}{h}$$ exists. If so, we denote its value by $f'(p)$ and refer t... | An Infinitely Large Napkin | napkin | general | advanced | Calculus 101 | Differentiation | Definition | 32_differentiate.md | 1 | 1,029 |
Same old, right? Sum rule, all that jazz.
In what follows $f$ and $g$ are differentiable functions, and $U$, $V$ are open subsets of $\mathbb{R}$.
- (Sum rule) If $f,g \colon U \to \mathbb{R}$ then then $(f+g)'(x) = f'(x) + g'(x)$.
- (Product rule) If $f,g \colon U \to \mathbb{R}$ then then $(f \cdot g)'(x) = f'(x) g... | An Infinitely Large Napkin | napkin | general | advanced | Calculus 101 | Differentiation | How to compute them | 32_differentiate.md | 2 | 1,461 |
You may remember from high school that one classical use of calculus was to extract the minimum or maximum values of functions. We will give a rigorous description of how to do this here.
Let $f \colon U \to \mathbb{R}$ be a function. A **local maximum** is a point $p \in U$ such that there exists an open neighborhood... | An Infinitely Large Napkin | napkin | general | advanced | Calculus 101 | Differentiation | Local (and global) maximums | 32_differentiate.md | 3 | 1,215 |
One corollary of the work in the previous section is Rolle's theorem.
Suppose $f \colon [a,b] \to \mathbb{R}$ is a continuous function, which is differentiable on the open interval $(a,b)$, such that $f(a) = f(b)$. Then there is a point $c \in (a,b)$ such that $f'(c) = 0$.
*Proof.* Assume $f$ is nonconstant (otherwis... | An Infinitely Large Napkin | napkin | general | advanced | Calculus 101 | Differentiation | Rolle and friends | 32_differentiate.md | 4 | 1,329 |
Let $f \colon U \to \mathbb{R}$ be differentiable, thus giving us a function $f' \colon U \to \mathbb{R}$. If our initial function was nice enough, then we can take the derivative again, giving a function $f'' \colon U \to \mathbb{R}$, and so on. In general, after taking the derivative $n$ times, we denote the resultin... | An Infinitely Large Napkin | napkin | general | advanced | Calculus 101 | Differentiation | Smooth functions | 32_differentiate.md | 5 | 902 |
Let $f \colon U \to \mathbb{R}$ be differentiable, thus giving us a function $f' \colon U \to \mathbb{R}$. If our initial function was nice enough, then we can take the derivative again, giving a function $f'' \colon U \to \mathbb{R}$, and so on. In general, after taking the derivative $n$ times, we denote the resultin... | An Infinitely Large Napkin | napkin | general | advanced | Calculus 101 | Differentiation | Smooth functions | 32_differentiate.md | 6 | 902 |
Power series and Taylor series
ch:power_series_taylor_series
Polynomials are very well-behaved functions,
and are studied extensively for that reason.
From an analytic perspective, for example, they are smooth,
and their derivatives are easy to compute.
In this chapter we will study power series,
which are literally `... | An Infinitely Large Napkin | napkin | general | advanced | Calculus 101 | Power series and Taylor series | 33_taylor.md | 0 | 1,536 | |
[Differentiation works term by term]
Let $_n 0 a_n z^n$ be a power series with radius of convergence $R > 0$,
and consider the corresponding function
\[ f (-R,R)
f(x) = _n 0 a_n x^n. \]
Then all the derivatives of $f$ exist and are given by power series
f'(x) &= _n 1 n a_n x^n-1 \\
f''(x) &= _n 2 n(n-1) a_n x^n-2 \\
&... | An Infinitely Large Napkin | napkin | general | advanced | Calculus 101 | Power series and Taylor series | 33_taylor.md | 1 | 1,563 | |
In particular, we can now even define complex exponentials,
giving us a function \[ \]
since the power series still has $R = $.
More generally if $a > 0$ and $z $
we may still define \[ a^z (z a). \]
(We still require the base $a$ to be a positive real
so that $ a$ is defined, though.
So this $i^i$ issue is still there... | An Infinitely Large Napkin | napkin | general | advanced | Calculus 101 | Power series and Taylor series | 33_taylor.md | 2 | 610 | |
Power series and Taylor series
ch:power_series_taylor_series
Polynomials are very well-behaved functions,
and are studied extensively for that reason.
From an analytic perspective, for example, they are smooth,
and their derivatives are easy to compute.
In this chapter we will study power series,
which are literally `... | An Infinitely Large Napkin | napkin | general | advanced | Calculus 101 | Power series and Taylor series | 33_taylor.md | 3 | 1,536 | |
[Differentiation works term by term]
Let $_n 0 a_n z^n$ be a power series with radius of convergence $R > 0$,
and consider the corresponding function
\[ f (-R,R)
f(x) = _n 0 a_n x^n. \]
Then all the derivatives of $f$ exist and are given by power series
f'(x) &= _n 1 n a_n x^n-1 \\
f''(x) &= _n 2 n(n-1) a_n x^n-2 \\
&... | An Infinitely Large Napkin | napkin | general | advanced | Calculus 101 | Power series and Taylor series | 33_taylor.md | 4 | 1,563 | |
In particular, we can now even define complex exponentials,
giving us a function \[ \]
since the power series still has $R = $.
More generally if $a > 0$ and $z $
we may still define \[ a^z (z a). \]
(We still require the base $a$ to be a positive real
so that $ a$ is defined, though.
So this $i^i$ issue is still there... | An Infinitely Large Napkin | napkin | general | advanced | Calculus 101 | Power series and Taylor series | 33_taylor.md | 5 | 610 | |
> "Trying to Riemann integrate discontinuous functions is kind of outdated."\
> --- Dennis Gaitsgory,
We will go ahead and define the Riemann integral, but we won't do very much with it. The reason is that the Lebesgue integral is basically better, so we will define it, check the fundamental theorem of calculus (or r... | An Infinitely Large Napkin | napkin | general | advanced | Calculus 101 | Riemann integrals | 34_integrate.md | 0 | 922 | |
Let $S$ be a subset (or subspace) of a topological space $X$. Then we say that $S$ is **dense** if every open subset of $X$ contains a point of $S$.
(a) $\mathbb{Q}$ is dense in $\mathbb{R}$.
(b) In general, any metric space $M$ is dense in its completion $\overline M$.
Dense sets lend themselves to having functions... | An Infinitely Large Napkin | napkin | general | advanced | Calculus 101 | Riemann integrals | Dense sets and extension | 34_integrate.md | 1 | 1,358 |
Extensions will allow us to define the Riemann integral. I need to introduce a bit of notation so bear with me.
Let $[a,b]$ be a closed interval.
- We let $C^0([a,b])$ denote the set of continuous functions on $[a,b] \to \mathbb{R}$.
- We let $R([a,b])$ denote the set of **rectangle functions** on $[a,b] \to \mathbb{... | An Infinitely Large Napkin | napkin | general | advanced | Calculus 101 | Riemann integrals | Defining the Riemann integral | 34_integrate.md | 2 | 1,054 |
The above definition might seem fantastical, overcomplicated, hilarious, or terrible, depending on your taste. But if you unravel it, it's really the picture you are used to. What we have done is taking every continuous function $f \colon [a,b] \to \mathbb{R}$ and showed that it can be approximated by a rectangle funct... | An Infinitely Large Napkin | napkin | general | advanced | Calculus 101 | Riemann integrals | Meshes | 34_integrate.md | 3 | 606 |
*Proof.* The right-hand side corresponds to the areas of some rectangle functions $g_1$, $g_2$, ... with increasingly narrow rectangles. As in the proof , as the meshes of those rectangles approaches zero, by uniform continuity, we have $d(f, g_n) \to 0$ as well. Thus by continuity in the diagram of , we get $\lim_n \S... | An Infinitely Large Napkin | napkin | general | advanced | Calculus 101 | Riemann integrals | Meshes | 34_integrate.md | 4 | 1,518 |
Holomorphic functions
ch:holomorphic_functions
Throughout this chapter, we denote by $U$ an open subset of the complex plane,
and by $$ an open subset which is also simply connected.
The main references for this chapter were ref:dartmouth,ref:bak_ca.
The nicest functions on earth
In high school you were told how to di... | An Infinitely Large Napkin | napkin | general | advanced | Complex Analysis | Holomorphic functions | 35_holomorphic.md | 0 | 1,558 | |
import graph;
graph.xaxis("Re", -1, 1, grey, NoTicks, Arrows);
graph.yaxis("Im", -1, 1, grey, NoTicks, Arrows);
If a function $f U $ is complex differentiable
at all the points in its domain it is called holomorphic.
In the special case of a holomorphic function with domain $U = $,
we call the function entire.Sorry, I... | An Infinitely Large Napkin | napkin | general | advanced | Complex Analysis | Holomorphic functions | 35_holomorphic.md | 1 | 1,509 | |
Obviously $_ (z) \; dz = 0$.
But heaven knows what $_ (z) \; dz$ is supposed to equal.
We can compute it now just out of non-laziness.
If you like, you are welcome to compute it yourself (it's a little annoying but not hard).
If I myself didn't mess up, it is
\[ _ (z) \; dz = - _ (z) \; dz
= - _0^ (i (t)) ie^it \; dt =... | An Infinitely Large Napkin | napkin | general | advanced | Complex Analysis | Holomorphic functions | 35_holomorphic.md | 2 | 1,488 | |
where we've used thm:central_cauchy_computation.
Thus, all we have to do is show that
\[ __ f(z)-f(a)z-a \; dz = 0. \]
For this we can basically use the weakest bound possible, the so-called $ML$ lemma
which I'll cite without proof:
it says ``bound the function everywhere by its maximum''.
[$ML$ estimation lemma]
Let ... | An Infinitely Large Napkin | napkin | general | advanced | Complex Analysis | Holomorphic functions | 35_holomorphic.md | 3 | 1,543 | |
(For arbitrary loops, it gets a bit more difficult, however. What does ``inside $$'' mean?)
Phrasing like this, it isn't that difficult. You may want to look at $f(z) = 1z$ a bit and
try to figure out how the proof follows before continue reading.
For simplicity, I will prove the statement for $$ being a rectangle, l... | An Infinitely Large Napkin | napkin | general | advanced | Complex Analysis | Holomorphic functions | 35_holomorphic.md | 4 | 1,565 | |
Let $f $ be an entire function.
Suppose that $ f(z) < 1000$ for all complex numbers $z$.
Prove that $f$ is a constant function.
Look at the Taylor series of $f$,
and use Cauchy's differentiation formula to
show that each of the larger coefficients must be zero.
% %
% It's true more generally that if
% $ f(z) < A+B z ... | An Infinitely Large Napkin | napkin | general | advanced | Complex Analysis | Holomorphic functions | 35_holomorphic.md | 5 | 1,227 | |
Holomorphic functions
ch:holomorphic_functions
Throughout this chapter, we denote by $U$ an open subset of the complex plane,
and by $$ an open subset which is also simply connected.
The main references for this chapter were ref:dartmouth,ref:bak_ca.
The nicest functions on earth
In high school you were told how to di... | An Infinitely Large Napkin | napkin | general | advanced | Complex Analysis | Holomorphic functions | 35_holomorphic.md | 6 | 1,558 | |
import graph;
graph.xaxis("Re", -1, 1, grey, NoTicks, Arrows);
graph.yaxis("Im", -1, 1, grey, NoTicks, Arrows);
If a function $f U $ is complex differentiable
at all the points in its domain it is called holomorphic.
In the special case of a holomorphic function with domain $U = $,
we call the function entire.Sorry, I... | An Infinitely Large Napkin | napkin | general | advanced | Complex Analysis | Holomorphic functions | 35_holomorphic.md | 7 | 1,509 | |
Obviously $_ (z) \; dz = 0$.
But heaven knows what $_ (z) \; dz$ is supposed to equal.
We can compute it now just out of non-laziness.
If you like, you are welcome to compute it yourself (it's a little annoying but not hard).
If I myself didn't mess up, it is
\[ _ (z) \; dz = - _ (z) \; dz
= - _0^ (i (t)) ie^it \; dt =... | An Infinitely Large Napkin | napkin | general | advanced | Complex Analysis | Holomorphic functions | 35_holomorphic.md | 8 | 1,488 | |
where we've used thm:central_cauchy_computation.
Thus, all we have to do is show that
\[ __ f(z)-f(a)z-a \; dz = 0. \]
For this we can basically use the weakest bound possible, the so-called $ML$ lemma
which I'll cite without proof:
it says ``bound the function everywhere by its maximum''.
[$ML$ estimation lemma]
Let ... | An Infinitely Large Napkin | napkin | general | advanced | Complex Analysis | Holomorphic functions | 35_holomorphic.md | 9 | 1,543 | |
(For arbitrary loops, it gets a bit more difficult, however. What does ``inside $$'' mean?)
Phrasing like this, it isn't that difficult. You may want to look at $f(z) = 1z$ a bit and
try to figure out how the proof follows before continue reading.
For simplicity, I will prove the statement for $$ being a rectangle, l... | An Infinitely Large Napkin | napkin | general | advanced | Complex Analysis | Holomorphic functions | 35_holomorphic.md | 10 | 1,565 | |
Let $f $ be an entire function.
Suppose that $ f(z) < 1000$ for all complex numbers $z$.
Prove that $f$ is a constant function.
Look at the Taylor series of $f$,
and use Cauchy's differentiation formula to
show that each of the larger coefficients must be zero.
% %
% It's true more generally that if
% $ f(z) < A+B z ... | An Infinitely Large Napkin | napkin | general | advanced | Complex Analysis | Holomorphic functions | 35_holomorphic.md | 11 | 1,227 | |
Meromorphic functions
ch:meromorphic_fn
The second nicest functions on earth
If holomorphic functions are like polynomials,
then meromorphic functions are like rational functions.
Basically, a meromorphic function is a function of the form
$ A(z)B(z) $
where $A , B U $ are holomorphic and $B$ is not zero.
The most impo... | An Infinitely Large Napkin | napkin | general | advanced | Complex Analysis | Meromorphic functions | 36_meromorphic.md | 0 | 1,453 | |
The order of a pole tells you how ``bad'' the pole is.
The order of a pole is the ``opposite'' concept of the multiplicity of a zero.
If $f$ has a pole at zero, then its Laurent series near $z=0$ might look something like
\[ f(z) = 1z^5 + 8z^3 - 2z^2 + 4z + 9 - 3z + 8z^2 + \]
and so $f$ has a pole of order five.
By ana... | An Infinitely Large Napkin | napkin | general | advanced | Complex Analysis | Meromorphic functions | 36_meromorphic.md | 1 | 1,446 | |
Let the poles with nonzero winding number be $p_1, , p_k$
(the others do not affect the sum).To show
that there must be finitely many such poles: recall that all our contours $ [a,b] $
are in fact bounded, so there is some big closed disk $D$ which contains all of $$.
The poles outside $D$ thus have winding number zero... | An Infinitely Large Napkin | napkin | general | advanced | Complex Analysis | Meromorphic functions | 36_meromorphic.md | 2 | 1,547 | |
Digression: the Argument Principle viewed geometrically
There is another, more geometric, way to understand the Argument Principle.
Assume a function $f$ is holomorphic on a connected open set $U$ containing $0$,
and possibly has a zero or a
pole at $0$. Let $ [0, 2 ] U$ be some curve contained in $U$, such that $0$ ... | An Infinitely Large Napkin | napkin | general | advanced | Complex Analysis | Meromorphic functions | 36_meromorphic.md | 3 | 1,415 | |
Meromorphic functions
ch:meromorphic_fn
The second nicest functions on earth
If holomorphic functions are like polynomials,
then meromorphic functions are like rational functions.
Basically, a meromorphic function is a function of the form
$ A(z)B(z) $
where $A , B U $ are holomorphic and $B$ is not zero.
The most impo... | An Infinitely Large Napkin | napkin | general | advanced | Complex Analysis | Meromorphic functions | 36_meromorphic.md | 4 | 1,453 | |
The order of a pole tells you how ``bad'' the pole is.
The order of a pole is the ``opposite'' concept of the multiplicity of a zero.
If $f$ has a pole at zero, then its Laurent series near $z=0$ might look something like
\[ f(z) = 1z^5 + 8z^3 - 2z^2 + 4z + 9 - 3z + 8z^2 + \]
and so $f$ has a pole of order five.
By ana... | An Infinitely Large Napkin | napkin | general | advanced | Complex Analysis | Meromorphic functions | 36_meromorphic.md | 5 | 1,446 | |
Let the poles with nonzero winding number be $p_1, , p_k$
(the others do not affect the sum).To show
that there must be finitely many such poles: recall that all our contours $ [a,b] $
are in fact bounded, so there is some big closed disk $D$ which contains all of $$.
The poles outside $D$ thus have winding number zero... | An Infinitely Large Napkin | napkin | general | advanced | Complex Analysis | Meromorphic functions | 36_meromorphic.md | 6 | 1,547 | |
Digression: the Argument Principle viewed geometrically
There is another, more geometric, way to understand the Argument Principle.
Assume a function $f$ is holomorphic on a connected open set $U$ containing $0$,
and possibly has a zero or a
pole at $0$. Let $ [0, 2 ] U$ be some curve contained in $U$, such that $0$ ... | An Infinitely Large Napkin | napkin | general | advanced | Complex Analysis | Meromorphic functions | 36_meromorphic.md | 7 | 1,415 | |
Holomorphic square roots and logarithms
ch:complex_log
In this chapter we'll make sense of a holomorphic square root and logarithm.
The main results are thm:nth_root, thm:holomorphic_log,
cor:nonvanishing, and cor:principal.
If you like, you can read just these four results, and skip the discussion of how they came to ... | An Infinitely Large Napkin | napkin | general | advanced | Complex Analysis | Holomorphic square roots and logarithms | 37_log.md | 0 | 1,423 |
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