Tag: theory

ct.category theory – Category morphisms and structure homomorphisms

Different types of homomorphisms (e. g. group hom.s, vector space hom.s, algebra hom.s) seem to be able to be generalized using two strategies.

  • Either describe the homomorphism class as a class preserving a specific structure (structure-preserving hom.s, the ,,logician’s way”).
  • Or see them as morphisms within a category (the ,,category theorist’s” way).

Are these two approaches always interchangeable or is one more potent than the other? Or do they have each their one expressive abilities not shared by the other one?

Asking this question on math.SE some time ago did not yield results definitely answering my questions, therefore I ask it here.

nt.number theory – Discrete logarithm and the sequence $a(n)=(g^n bmod p)^{p-1} bmod p^2$

Let $p$ be prime and $g,n$ integers.

Define $a(n)=(g^n bmod p)^{p-1} bmod p^2$

Some properties of $a(n)$:

  1. $a(n)$ is periodic with period divisor of $p-1$.
  2. The multiplicative order of $a(n)$ modulo $p^2$ is $p$.
  3. Let $D(n)$ be the discrete logarithm of $a(n)$, i.e.
    given $p$, $g$, $a(n)$ we have $g^{D(n)} mod p^2=a(n)$.
    We can efficiently compute $D(n)=k(p-1)$ via p-adic logarithms.
  4. Let $g=2$. Experimentally with high probability
    we have $D(n) bmod p=D(n+1)+1 bmod p$.

Q1 Are there other functional relations between $g$, $n$, $a(n)$, $D(n)$?

Q2 For $g=2$, when do we have $D(n) bmod p=D(n+1)+1 bmod p$?

Q3 What is the intuition for efficiently computing $D(n)$
for period divisor of $p-1$?

sagemath code follows, one can run it in a browser:

def seqanp2(p,g,n):
    """
    a(n)=(g^n mod p)^(p-1) mod p^2
    """
    try:  g=lift(g)
    except:  pass
    r1=lift((Integers(p)(g))**n)
    K2=Integers(p**2)
    res=K2(r1)**((p-1))
    return res

def solveseqan(p,g,a):
    """
    g^res =a(n)  mod p^2
    """
    try:  g=lift(g)
    except:  pass
    try:  a=lift(a)
    except:  pass
    K=Qp(p,2)
    t=lift(K(a).log()/K(g).log() )
    res=(p-1)*(p-t%p)
    return res

set_random_seed(1)

p=next_prime(10**20);
K2=Integers(p**2);
g=K2(2) 
n0=randint(2,p-2)
r1=seqanp2(p,g,n0);r2=seqanp2(p,g,n0+1);
s1=solveseqan(p,g,r1);s2=solveseqan(p,g,r2)

print(g**s1==r1,g**s2==r2,seqanp2(p,g,n0)==seqanp2(p,g,n0+p-1)) #True True True

analytic number theory – A question on regularization of sum of expressions involving primes:

We know the following:

$$gamma=lim_{ntoinfty }left(sum_{k=1}^nfrac{1}{k}-ln(n)right)$$

This could be the good candidate for regularized sum of $left(sum_{k=1}^{infty}frac{1}{k}right)$.

Also we know the following:

$$-gamma=lim_{ntoinfty }left(sum_{pleq n}frac{ln(p)}{p-1}-ln(n)right)$$

I want to ask does this analogously mean that $-gamma$ is regularized value of $(sum_{p}frac{ln(p)}{p-1})$?

Also I wanted to ask similar question:

What is the regularized value of the following sum in above sense?

$$sum_{p} frac{1}{sqrt{cp}-1}$$

Here $c$ is a constant

analytic number theory – What’s the average order of the reduction of a section of an elliptic curve

Suppose $E$ is an elliptic curve over $mathbb Q$ and $x in E(mathbb Q)$ is not torsion. We can reduce $x pmod p$ for a prime $p$ of good reduction and it will have some order $n_p$ in the group $E(mathbb F_p)$. Has there been any work on the asympotitcs of $n_p$ as $p to infty$?

More generally, suppose $x,y in E(mathbb Q)$ are two linearly independent sections and let them generate subgroups $G_x(p),G_y(p) subset E(mathbb F_p)$ for a prime of good reduction. Have the asymptotics of $G_x(p)cap G_y(p)$ been studied?

This question seems tangentially related.

rt.representation theory – The product of $Z(mathfrak{g})$-finite functions is also $Z(mathfrak{g})$-finite?

Let $G$ be a classical group defined over $mathbb{Q}$.

Let $mathfrak{g}$ be the Lie algebra of $G(mathbb{R})$ and $U(mathfrak{g}_{mathbb{C}})$ its universal enveloping algebra of $mathfrak{g}_{mathbb{C}}$.

Let $Z(mathfrak{g})$ be the center of $U(mathfrak{g}_{mathbb{C}})$. We regard the elements of $U(mathfrak{g}_{mathbb{C}})$ as differential operators on $C^{infty}(G)$, the space of smooth functions on $G(mathbb{R})$, acting by right infinitesimal translation.

Let $f,g in C^{infty}(G)$ be $Z(mathfrak{g})$-finite. (I.e. $<zcdot f | zin Z(mathfrak{g})>, <zcdot g | zin Z(mathfrak{g})>$ are finite dimensional vector space.)

Then I am wondering whether $f cdot g$ is also $Z(mathfrak{g})$-finite.

Any comments are appreciated!

queueing theory – Why is a polling-system with zero switch-over-times and exhaustive service-discipline unstable, if the workload is less than 1?

I’m asking for a short proof (or the idea of the proof), why a discrete polling system (finite number of stations) with zero switch-over-times and exhaustive service-discipline is unstable, if the workload $rho:=sum_{i=1}^Klambda_i b_i$ is less than 1.
To be more precisely, the polling system consists of $K$ M/GI/1-Queues, which share one server. $lambda_i$ is the intensity of the arrival-process and $b_i$ the mean service-time of station $i$.

complexity theory – Constant-time adding an element?

Is a computer with infinite memory and infinite word size a Truing machine equivalent (in the sense that polynomial time remains polynomial time and exponential time remains exponential time) if we allow constant-time linked link element insertion (at the beginning of the list)?

I doubt this because element insertion requires memory allocation and allocation is usually not a constant-time operation.

information theory – Why is “CNOT” gate the only non trivial for two input bits?

Just yesterday, I found the theory about quantum computing and I am studying by myself.
While trying to understand Toffoli gate on wiki (https://en.wikipedia.org/wiki/Toffoli_gate),
I faced the sentence ‘CNOT’ gate is the only non trivial for two input bits
like 00 -> 00, 01 -> 01, 10 -> 11, 11-> 10. At this point,

Question 1

the question popped up that why not 00 -> 01, 01 -> 00, 10 -> 10, 11 -> 11. I think this matrix is presented by
$$
begin{bmatrix}
0 & 1 & 0 & 0 \
1 & 0 & 0 & 0\
0 & 0 & 1 & 0\
0 & 0 & 0 & 1\
end{bmatrix}
quad
$$

is different with $$
begin{bmatrix}
1 & 0 & 0 & 0 \
0 & 1 & 0 & 0\
0 & 0 & 0 & 1\
0 & 0 & 1 & 0\
end{bmatrix}
quad
$$

and is also unitary.

Question 2.

Is the order of the basis matter whenever to present the operation as a matrix? If then, what is the rule?

Question 3.

I am studying with this lecture note- https://homes.cs.washington.edu/~oskin/quantum-notes.pdf
Page 12 of the note, $$ frac{1}{sqrt{2}}(a |0 rangle (|00rangle +|11rangle)+b |1 rangle (|00 rangle + |11 rangle))= frac{1}{sqrt{2}} begin{bmatrix}
a \
0 \
0 \
a \
b \
0 \
0 \
b \
end{bmatrix}
quad
$$

but I think $$ |0 rangle in mathbb{C}^2 $$ and $$ |00 rangle, |11 rangle in mathbb{C}^4$$ so the product of two vector is non-sense. Should I consider it as a tensor product of the two vectors? If we consider it as a tensor product, then it’s okay and the order of basis vectors looks important..

complexity theory – Show that NL ⊆ P

$textsf{PATH}$ is in $textsf{NL}$, because to solve it, you just need to keep in memory the current vertex you are in, and guess (non-deterministicaly) the next one on the path until you reach your destination. Since you keep the current vertex $v$, numbered from $0$ to $|V| – 1$, you need a memory space corresponding to the binary encoding of $v$, which is at most $1 + log_2(|V| – 1)$. You also need to keep the potential adjacent vertex of $v$, next in the path.

All in all, a Turing Machine solving this problem would only need $O(log |V|)$ additionnal space memory (the memory of the graph and of the starting vertex and the destination vertex of the path you are guessing are not considered in the memory used, because they are part of the input).

$textsf{PATH}$ is $textsf{NL}$-hard, because to solve any $textsf{NL}$ problem, you have to determine if there exists a sequence of possible transitions from the initial configuration to an accepting configuration in the Turing Machine of the problem. If you consider a graph of the possible configurations (where there exists an edge from a configuration $alpha$ to a configuration $beta$ if and only if one can go from $alpha$ to $beta$ in one transition in the Turing Machine), then solving the $textsf{NL}$ problem is the same as solving $textsf{PATH}$ in the graph of possible configurations.

You then need to prove that the graph of configurations can be constructed in logarithmic additionnal space. This can be done, because if a non-deterministic Turing Machine works in space $s(n)$, then the number of possible configurations is $2^{O(s(n))}$. Considering the binary encoding of those configurations, one can determine if there exists an edge between two configurations in deterministic space $O(s(n))$.

Now, since $textsf{PATH}$ is solvable in polynomial time (with a graph traversal algorithm, for example), that means that any $textsf{NL}$ problem is solvable in polynomial time (via the $textsf{NP}$-completude of $textsf{PATH}$), so $textsf{NL}subseteq textsf{P}$.

game theory – Algorithm for winning solitaire

Is there an algorithm that could check whether it is possible to win in the current situation in solitaire? That is, can we remove all cards from the table in a certain number of moves?

Description of the game:

Two decks of cards (36 pieces each) are connected and shuffled, and then they are laid out in 8 piles of 9 cards each. Possible work only with the top cards in each of the piles, namely, transfer one or several consecutive cards of a lower value to a card of a larger one (suits are not taken into account). If there are 9 cards in one of the piles in the correct order (ace-king-queen-jack-10-9-8-7-6), then this set is removed. If all the cards are removed from one pile, it can still be used and any card can be placed on top