Maximally transitive semigroups of $n\times n$ matrices
We prove that, in both real and complex cases, there exists a pair of matrices that generates a dense subsemigroup of the set of $n\times n$ matrices.
math.DS↗
arXiv subjects
Mohammad Javaheri
Publications and source records attributed to Mohammad Javaheri.
Dense 2-generator subsemigroups of $2\times 2$ matrices
We show that the semigroup of real linear fractional transformations on a proper subinterval of the real line does not admit any 2-generator dense subsemigroups, and then we construct a 3-parameter family of examples of 3-generator dense subsemigroups. We also construct an explicit example of a 2-generator dense subsemigroup of $2\times 2$ real matrices. In the complex case, we prove the existence of uncountably many 2-generator dense subsemigroups.
math.NT↗
A Collatz-type conjecture on the set of rational numbers
Define $θ(x)=(x-1)/3$ if $x\geq 1$, and $θ(x)=2x/(1-x)$ if $x<1$. We conjecture that the orbit of every positive rational number ends in 0. In particular, there does not exist any positive rational fixed point for a map in the semigroup $Ω$ generated by the maps $3x+1$ and $x/(x+2)$. In this paper, we prove that the asymptotic density of the set of elements in $Ω$ that have rational fixed points is zero.
math.NT↗
Semigroups of matrices with dense orbits
We give examples of $n \times n$ matrices $A$ and $B$ over the filed $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ such that for almost every column vector $x \in \mathbb{K}^n$, the orbit of $x$ under the action of the semigroup generated by $A$ and $B$ is dense in $\mathbb{K}^n$.
math.DS↗
Semigroups of real functions with dense orbits
Let ${\mathcal F}_I=\{f:I \to I| f(x)= (Ax+B)/(Cx+D); AD-BC \neq 0 \}$, where $I$ is an interval. For $x\in I$, let $Ω_x$ be the orbit of $x$ under the action of the semigroup of functions generated by $f,g \in {\mathcal F}_I$. Our main result in this paper is to describe all $f,g \in {\mathcal F}_I$ such that $Ω_x$ is dense in $I$ for all $x$.
math.DS↗
Convergence of Ricci flow on $\mathbb{R}^2$ to flat space
We prove that, starting at an initial metric $g(0)=e^{2u_0}(dx^2+dy^2)$ on $\mathbb{R}^2$ with bounded scalar curvature and bounded $u_0$, the Ricci flow $\partial_t g(t)=-R_{g(t)}g(t)$ converges to a flat metric on $\mathbb{R}^2$.
math.DG↗
Harmonic functions via restricted mean-value theorems
Let $f$ be a function on a bounded domain $Ω\subseteq \mathbb{R}^n$ and $δ$ be a positive function on $Ω$ such that $B(x,δ(x))\subseteq Ω$. Let $σ(f)(x)$ be the average of $f$ over the ball $B(x,δ(x))$. The restricted mean-value theorems discuss the conditions on $f,δ,$ and $Ω$ under which $σ(f)=f$ implies that $f$ is harmonic. In this paper, we study the stability of harmonic functions with respect to the map $σ$. One expects that, in general, the sequence $σ^n(f)$ converges to a harmonic function. Among our results, we show that if $Ω$ is strongly convex (respectively $C^{2,α}$-smooth for some $α\in [0,1]$), the function $δ(x)$ is continuous, and $f\in C^0(\bar Ω)$ (respectively, $f\in C^{2,α}(\bar Ω)$), then $σ^n(f)$ converges to a harmonic function uniformly on $\bar Ω$.
math.AP↗
Line crossing problem for biased monotonic random walks in the plane
In this paper, we study the problem of finding the probability that the two-dimensional (biased) monotonic random walk crosses the line $y=αx+d$, where $α,d \geq 0$. A $β$-biased monotonic random walk moves from $(a,b)$ to $(a+1,b)$ or $(a,b+1)$ with probabilities $1/(β+ 1)$ and $β/(β+ 1)$, respectively. Among our results, we show that if $β\geq \lceil α\rceil$, then the $β$-biased monotonic random walk, starting from the origin, crosses the line $y=αx+d$ for all $d\geq 0$ with probability 1.
math.PR↗