Termination of Linear Programs with Nonlinear Constraints

Tiwari proved that termination of linear programs (loops with linear loop conditions and updates) over the reals is decidable through Jordan forms and eigenvectors computation. Braverman proved that it is also decidable over the integers. In this paper, we consider the termination of loops with polynomial loop conditions and linear updates over the reals and integers. First, we prove that the termination of such loops over the integers is undecidable. Second, with an assumption, we provide an complete algorithm to decide the termination of a class of such programs over the reals. Our method is similar to that of Tiwari in spirit but uses different techniques. Finally, we conjecture that the termination of linear programs with polynomial loop conditions over the reals is undecidable in general by %constructing a loop and reducing the problem to another decision problem related to number theory and ergodic theory, which we guess undecidable.Comment: 17pages, 0 figure

On approximation of Markov binomial distributions

For a Markov chain $\mathbf{X}=\{X_i,i=1,2,...,n\}$ with the state space $\{0,1\}$, the random variable $S:=\sum_{i=1}^nX_i$ is said to follow a Markov binomial distribution. The exact distribution of $S$, denoted $\mathcal{L}S$, is very computationally intensive for large $n$ (see Gabriel [Biometrika 46 (1959) 454--460] and Bhat and Lal [Adv. in Appl. Probab. 20 (1988) 677--680]) and this paper concerns suitable approximate distributions for $\mathcal{L}S$ when $\mathbf{X}$ is stationary. We conclude that the negative binomial and binomial distributions are appropriate approximations for $\mathcal{L}S$ when $\operatorname {Var}S$ is greater than and less than $\mathbb{E}S$, respectively. Also, due to the unique structure of the distribution, we are able to derive explicit error estimates for these approximations.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ194 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Global α\alpha-decay study based on the mass table of the relativistic continuum Hartree-Bogoliubov theory

The $\alpha$-decay energies ($Q_\alpha$) are systematically investigated with the nuclear masses for $10 \leq Z \leq 120$ isotopes obtained by the relativistic continuum Hartree-Bogoliubov (RCHB) theory with the covariant density functional PC-PK1, and compared with available experimental values. It is found that the $\alpha$-decay energies deduced from the RCHB results present similar pattern as those from available experiments. Owing to the large predicted $Q_\alpha$ values ($\geq$ 4 MeV), many undiscovered heavy nuclei in the proton-rich side and super-heavy nuclei may have large possibilities for $\alpha$-decay. The influence of nuclear shell structure on $\alpha$-decay energies is also analysed.Comment: 7 pages, 4 figures. arXiv admin note: text overlap with arXiv:1309.3987 by other author