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Register a new userMeasurements of magnetization on the Sierpi{n}ski carpet
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Phase transition of the classical Ising model on the Sierpi{n}ski carpet, which has the fractal dimension $log_3^{~} 8 approx 1.8927$, is studied by an adapted variant of the higher-order tensor renormalization group method. The second-order phase transition is observed at the critical temperature $T_{rm c}^{~} = 1.4783(1)$. Position dependence of local functions is studied by means of impurity tensors, which are inserted at different locations on the fractal lattice. The critical exponent $beta$ associated with the local magnetization varies by two orders of magnitude, depending on lattice locations, whereas $T_{rm c}^{~}$ is not affected.
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The feedback vertex number $tau(G)$ of a graph $G$ is the minimum number of vertices that can be deleted from $G$ such that the resultant graph does not contain a cycle. We show that $tau(S_p^n)=p^{n-1}(p-2)$ for the Sierpi{n}ski graph $S_p^n$ with $pgeq 2$ and $ngeq 1$. The generalized Sierpi{n}ski triangle graph $hat{S_p^n}$ is obtained by contracting all non-clique edges from the Sierpi{n}ski graph $S_p^{n+1}$. We prove that $tau(hat{S}_3^n)=frac {3^n+1} 2=frac{|V(hat{S}_3^n)|} 3$, and give an upper bound for $tau(hat{S}_p^n)$ for the case when $pgeq 4$.
In this paper, we investigate the existence of Sierpi{n}ski numbers and Riesel numbers as binomial coefficients. We show that for any odd positive integer $r$, there exist infinitely many Sierpi{n}ski numbers and Riesel numbers of the form $binom{k}{r}$. Let $S(x)$ be the number of positive integers $r$ satisfying $1leq rleq x$ for which $binom{k}{r}$ is a Sierpi{n}ski number for infinitely many $k$. We further show that the value $S(x)/x$ gets arbitrarily close to 1 as $x$ tends to infinity. Generalizations to base $a$-Sierpi{n}ski numbers and base $a$-Riesel numbers are also considered. In particular, we prove that there exist infinitely many positive integers $r$ such that $binom{k}{r}$ is simultaneously a base $a$-Sierpi{n}ski and base $a$-Riesel number for infinitely many $k$.
The Fokker--Planck equation describes the evolution of a probability distribution towards equilibrium--the flow parameter is the equilibration time. Assuming the distribution remains normalizable for all times, it is equivalent to an open hierarchy of equations for the moments. Ways of closing this hierarchy have been proposed; ways of explicitly solving the hierarchy equations have received much less attention. In this paper we show that much insight can be gained by mapping the Fokker--Planck equation to a Schrodinger equation, where Plancks constant is identified with the diffusion coefficient.
We study the ground state energy E_G(n) of N classical n-vector spins with the hamiltonian H = - sum_{i>j} J_ij S_i.S_j where S_i and S_j are n-vectors and the coupling constants J_ij are arbitrary. We prove that E_G(n) is independent of n for all n > n_{max}(N) = floor((sqrt(8N+1)-1) / 2) . We show that this bound is the best possible. We also derive an upper bound for E_G(m) in terms of E_G(n), for m<n. We obtain an upper bound on the frustration in the system, as measured by F(n), which is defined to be (sum_{i>j} |J_ij| + E_G(n)) / (sum_{i>j} |J_ij|). We describe a procedure for constructing a set of J_ijs such that an arbitrary given state, {S_i}, is the ground state.
The quantum O(N) model in the infinite $N$ limit is a paradigm for symmetry-breaking. Qualitatively, its phase diagram is an excellent guide to the equilibrium physics for more realistic values of $N$ in varying spatial dimensions ($d>1$). Here we investigate the physics of this model out of equilibrium, specifically its response to global quenches starting in the disordered phase. If the model were to exhibit equilibration, the late time state could be inferred from the finite temperature phase diagram. In the infinite $N$ limit, we show that not only does the model not lead to equilibration on account of an infinite number of conserved quantities, it also does emph{not} relax to a generalized Gibbs ensemble consistent with these conserved quantities. Nevertheless, we emph{still} find that the late time states following quenches bear strong signatures of the equilibrium phase diagram. Notably, we find that the model exhibits coarsening to a non-equilibrium critical state only in dimensions $d>2$, that is, if the equilibrium phase diagram contains an ordered phase at non-zero temperatures.
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