## finite automata – NFA designing for strings starting with \$01\$

Construct an NFA with set of all strings that start with $$10$$.

The solution provided to me is But my question is what if the automaton receives an input $$0$$ at the starting? Also there is no option for $$q_1$$ to transit after receiving $$1$$. So I think the solution should be Please correct me if I am wrong.

## For distributions \$P\$ and \$Q\$ w/ finite KL, and \$epsilon > 0\$, is there a sequence \$R_i\$ such that \$KL(P || R_1) + dots + KL(R_n || Q) < epsilon\$?

Let $$P$$ and $$Q$$ be two probability distributions defined over the same space, with $$KL(P || Q) < infty$$. For $$epsilon > 0$$, is it possible to choose an $$n in mathbb{N}$$ and a sequence of distributions $$R_1, dots, R_n$$ such that $$KL(P || R_1) + dots + KL(R_n || Q) < epsilon$$?

Due to the strict convexity of the KL divergence in either one of its arguments (fixing the other), for each $$i$$ we have (assuming $$R_i neq R_{i+1}$$ and $$lambda in (0, 1)$$) $$KL(R_i || lambda R_i + (1-lambda)R_{i+1}) + KL(lambda R_i + (1-lambda)R_{i+1} || R_{i+1}) < (1 – lambda) KL(R_i || R_{i+1}) + lambda KL(R_i || R_{i+1}) = KL(R_i || R_{i+1}).$$
Therefore it would seem we can always just insert $$lambda R_i + (1-lambda)R_{i+1}$$ into the sequence of distributions, between $$i$$ and $$i+1$$, and push down the sum of the KL divergences. But maybe there are diminishing returns from repeating this procedure and there is some limit to how low you can go.

## Problem with initial conditions for numerically solving the Sine-Gordon equation using finite differences method (in MatLab).

We are trying to solve numerically a kink solution in matlab using the sine-gordon equation, therefore followed the steps according to the paper (1) listed below. The sine gordon equation is a non-linear pde and the numerical solution was clearly explained in the paper (1). However, the problem is that if we use the initial conditions f(x) and g(x) (page 6 of the paper), the solution will have a disturbance and is not as clean as figure 5.4a (p. 6) from the paper it self. To solve this we ‘doubled’ the size of the soliton by replacing ‘x’ in the initial conditions for ‘x/2’. Therefore my question: What is the explanition that dividing x by 2 instead of taking just x justifies? Perhaps we made a mistake in the numerical model?

Source

1. The paper used: https://www.uni-muenster.de/imperia/md/content/physik_tp/lectures/ws2016-2017/num_methods_i/sinegordon.pdf
2. There are a few references from source 1 to chapter 4: https://www.uni-muenster.de/imperia/md/content/physik_tp/lectures/ws2016-2017/num_methods_i/wave.pdf

Matlab code

The changes of the inital conditions are cleary commented in the matlab code.

```clear all; clc; close all;
%% Define variables
L = 40; %Choosing a bigger L can make it look nicer, also change it in the function at the bottom of the code
delta_x = 0.05;
delta_t = 0.01;
c = 0.2;
t_end = 90;
T = t_end/delta_t;
M = L/delta_x;
N = M+1;
x = linspace(-L,L,M+1);
t = linspace(0,t_end,T+1);
%end
%% Calculate first time row

u_0 = transpose(f(x));
gamma_1 = transpose(g(x));
beta_1 = sin(u_0);
alpha = delta_t/delta_x;

a = 1-alpha^2;
b = alpha^2/2;
c = b;
A = diag(a*ones(1,N)) + diag(b*ones(N-1,1),1) + diag(c*ones(1,N-1),-1);
A(1,2) = A(1,2) + b;
A(N,M) = A(N,M) + b;
B = 2*A;

%% Calculate second time row
u_1 = delta_t*gamma_1 + A*u_0 - (delta_t^2/2)*beta_1;
u = zeros(N,T);
u(:,1) = u_0;
u(:,2) = u_1;

%% Calculate the full matrix
beta = zeros(N,T);
beta(:,1) = beta_1;

for j = 2:(T)
beta(:,j) = sin(u(:,j));
u(:,j+1) = -u(:,j-1) + B*u(:,j) - (delta_t^2)*beta(:,j);
end

%figure
mesh(x(1:10:end),t(1:10:end),transpose(u(1:10:end,1:10:end)))
xlabel('x')
ylabel('t')
zlabel('u')

for n = 1:T-1
for l = 1:N-1
e(l,n) = (1/2*((u(l,n+1)-u(l,n))/delta_t)^2+1/2*((u(l+1,n+1)-u(l,n+1))/delta_x)*((u(l+1,n)-u(l,n))/delta_x)+(1-cos(u(l,n+1)+1-cos(u(l,n))))/2);
end
E(n) = delta_x*sum(e(:,n));

end

%% Define the functions
% Single soliton
function f = f(x)
c = 0.2; %This can be varied to create a nicer picture, also change it in the 'g'function
L = 20; %This can be varied to create a nicer picture, also change it in the 'g'function
f = 4*atan(exp((x/2)/sqrt(1-c^2))); % the '+L/2' can also be changed to have a different starting point for the soliton, don't forget to also change it in the 'g'function
end
function g = g(x)
c = 0.2;
L = 20;
g = -2*c/sqrt(1-c^2)*sech((x/2)/sqrt(1-c^2));
end```

## Finite Automaton to Turing Machine Example

I cannot seem to find an example of an NFA and Turing Machine that both accept the same languages. I am trying to understand how to convert an NFA to its equivalent Turing Machine and I think a simple example would help a lot!

## Subgroup rank of finite simple groups

Definition: The subgroup rank of a finite group G is the minimal natural number n such that every subgroup of G can be generated by n elements (or fewer).

This invariant has been studied extensively for various families of groups. I am interested in the family of finite simple groups and I have been unable to find and relevant information in the literature.

Question 1: Are there only finitely-many finite simple, non-abelian groups G of a given subgroup rank n?

Some relatively straight-forward comments and reductions:

It is not too difficult to show that there are only finitely-many alternating groups of subgroup rank at most n (by explicitly constructing elementary-abelian subgroups of a certain subgroup rank). There are also only finitely-many sporadic groups, according to the classification. These observations reduce the above question to finite simple groups of Lie-type.

Question 2: Are there only finitely-many finite simple groups G of Lie-type with given subgroup rank n?

It is again not too difficult to show that the “field rank” of G is bounded from above by a function of n (by looking at the natural homomorphism from the field to the root subgroups). It is also possible to show that the Lie-rank of G is bounded from above by a function of n. These observations further reduce question 2 to bounding the defining characteristic of the simple group of Lie-type by some function that depends only on the subgroup rank n. Unfortunately, I do not have any good intuition to determine whether the latter statement is true or not.

I hope both questions have a positive answer because that would give us a nice property about the FSG. But I suspect we can prove the answers to be “no” by simply making some judicious choice for the Lie-type, field-rank, and Lie-rank and by then looking at the structure of the Sylow-subgroups of G, as the characteristic goes through the different primes.

## proof explanation – Cardinality of a finite product of sets

My lecture notes state that for a set $$S$$, we have $$|S times S| = |S|$$. Some reading on this topic here suggests that this requires the axiom of choice, which implies to me that the assumption of $$S$$ being infinite is necessary. This makes sense, as I can come up with a counterexample for finite $$S$$. If $$S = {1,2,3}$$, then $$|S times S| = 3 cdot 3 = 9 > |S| = 3$$.

Am I correct that $$S$$ must be infinite for this result to hold? Further, is the canonical proof an explicit bijection or using Shroder-Bernstein? Every method of trying to find a bijection led to problems with either injectivity or surjectivity.

## matrices – Can all finite dimensional non commutative algebras be embedded into matrix rings?

Suppose I have a finite (non-)commutative ring $$R/k$$ (over a field $$k$$ of char $$0$$) with a linear “trace” function $$t: R to k$$. Can I find square matrices $$A_1,dots,A_n$$ (of some dimension $$r$$) so that I have an embedding $$f: R to M_r(k)$$ compatible with the trace functions on both sides?

One restriction I can see for the trace function on $$R$$ is that it should be invariant under cyclic permutations : $$t(a_1a_2dots a_n) = t(a_2dots a_na_1)$$. Is this the only restriction?

## Finite self-maps exist on rigid CY3s

Let $$X$$ be a smooth projective rigid Calabi-Yau threefold. Does there exist a finite map $$Xto X$$ of degree>1?

## ct.category theory – Filtered 2-colimits commute with finite 2-limits

Is there an explicit proof anywhere in the literature that filtered 2-colimits commute with finite 2-limits (all meant in the weak bicategorical sense) in the 2-category of groupoids? I have only been able to find a handful of papers about filtered 2-colimits, notably Dubuc and Street’s A construction of 2-filtered bicolimits of categories, and have not found this result in any of them.

Of course one could try to deduce this from $$infty$$-categorical results, but the latter proofs are also often sketchy, and in the case of groupoids it ought to be possible to give a concrete constructive proof.

## Convert the Finite Automata (FSA) into its equivalent regular expression, using stepwise minimization

I was doing an assignment of Theory of automata but while doing this question I am stuck there is no such state that can be eliminated even from transition table. I am very confused and stuck please help me out 