I know that dynamic programming is used to solve in “pseudo-polynomial time” some NP problems, like the knapsack. If P = NP, would it mean that every problem that we solve with dynamic programming would have a more efficient (polynomial) way to solve? Or are there problems that even if P = NP, we would still use dynamic programming

# Tag: theory

## operator theory – String Concatenation Comparision

Let $A$ and $B$ be strings consisting of small latin alphabets.

We will say $A<B$ iff $AB$ is lexicography smaller than $BA$. ($AB$ is string concatenation of $A$ and then $B$)

Can we show that if $A<B$ and $B<C$, then $A<C$ (or show that it is wrong) ?

## gr.group theory – Rack cohomology as derived functor cohomology

Let $X$ be a rack and $A$ be an $X$-module. By this paper, p. 33, we can associate a cochain complex $C^bullet(X,A)$ to the pair $(X,A)$. This complex is explicitly defined by a differential $d$. I wonder if the cohomology $H^bullet(X,A)$ of the complex has an interpretation as derived functor cohomology. What functor from $X$-modules to $X$-modules do we have to derive? And how to show then the equivalence of the two definitions? I think the analogy to group cohomology is not very helpful, or can we somehow define the invariants of an $X$-module and make it fit?

## nt.number theory – Polynomial bijections $mathbb{Z}^2tomathbb{Z}^2$ are linear

Let $G$ be the group of bijections $phi:mathbb{Z}^2tomathbb{Z}^2$ such that $phi(x, y)=(P(x, y), Q(x, y))$ for some polynomials $P, Qin mathbb{Z}(x, y)$.

Do affine transformations and maps of the form $(x, y)to (x+R(y), y)$ where $Rin mathbb{Z}(y)$ generate $G$?

## 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 the average of $n_p$ for $p < X$ as $X to infty$?

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 the average 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!

## complexity theory – Why can’t MIP be simulated in PSPACE

The proof that $IP subseteq PSPACE$ is done by considering by game tree of the interactions between the prover and the verifier; this tree is polynomial in depth and each message is polynomial in length, so the tree can be explored in a DFS like fashion to simulate an optimal prover in PSPACE.

Why can’t the same be done for MIP? Proofs that $MIP subseteq NEXP$ usually show that the multiple provers can be simulated by non-deterministically guessing a strategy for each, but why can’t you just simulate the tree of interactions as in the single-prover case?

## probability theory – Definition of expectation for non-real-valued random variables

I have checked many sources for the definition of expectation for non-real-valued random variables. I am interested on the **conditions of the image space** that guarantee the existence of an expectation operator. For example, I have seen definitions of expectation for complex-valued and real-vector-valued random variables, and these work “component by component”.

**Question.** I am wondering if there is a general definition of expected value that asks for some conditions about the image space of the random variables and captures all image spaces for which expected values make sense.

So far I have only concluded that the image space must necessarily support *addition* and *scalar multiplication* to be able to define expectation for random variables with finite set of possible outcomes, i.e. $sum_{i=1}^k P(X=x_i) x_i$. Some other conditions that come to my mind are being a measure space, a metric space and being complete, but I am not sure how are they necessary and if they are sufficient.

## 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.

## gr.group theory – How many finitely-generated-by-elements-of-finite-order-groups are there?

I do not know where this question is on the trivial to intractable spectrum.

Consider the set of isomorphism classes of groups finitely generated by elements of finite order. What is the cardinality of this set?

The motivation here is that the dual of every such group gives a quantum permutation group. Finite graphs, which is to say pairs $Gamma=(V,E)$, where $1leq|V|<infty$ and $Esubset mathcal{P}(V)$ consists of two elements subsets, have quantum automorphism groups which are also quantum permutation groups.

Frucht’s theorem says that every finite group is the automorphism group of a finite graph, but surely a quantum Frucht could not hold if there were uncountably many finitely-generated-by-elements-of-finite-order groups.