How do I find a regular expression for this language, and how do I define a finite state machine that recognizes words in the language (input alphabet, states, start state, state transition table, and accept states). Including a state digraph for the FSM for this language, L: For alphabet $\{0,1...

Let $A$ and $B$ be two bugs lying on two distinct points $a_0, b_0$ on a fixed circle. They start to walk along the circle in the same direction such that at time $t$ their coordinates $a_t, b_t$ satisfy $G(a_t,b_t)=0,$ where $G$ is a given implicit function. If $L_t$ is the line segment joi...

Let $M,N$ be abstract smooth manifolds of dimension $m,n$ respectively , $f:M \to N$ be a smooth map such that for some open set $U $ in $M$ , $df_p:T_pM \to T_{f(p)}N$ has rank $k$ for every $p\in U$ , if $q\in f(M)$ be such that $ f^{-1}\{q\} \subseteq U$ then is it true that $f^{-1}\{q\}$ is...

Let $R$ be a field , let us adjoin $i,j,k$ to $R$ and write $$R[i,j,k]:=\{a+bi+cj+dk :a,b,c,d \in R \}$$ where $i,j,k$ satisfies $a+bi+cj+dk=0 $ iff $a=b=c=d=0$, $i,j,k$ commutes with every element of $R$, $i^2=j^2=k^2=-1$ and $ij=k , jk=i , ki=j$. Then is it true that $R[i,j,k]$ is a divisi...