Sequences of binary irreducible polynomials

In this paper we construct an infinite sequence of binary irreducible polynomials starting from any irreducible polynomial f_0 \in \F_2 [x]. If $f_0$ is of degree $n = 2^l \cdot m$, where $m$ is odd and $l$ is a non-negative integer, after an initial finite sequence of polynomials $f_0, f_1, ..., f_{s}$ with $s \leq l+3$, the degree of $f_{i+1}$ is twice the degree of $f_i$ for any $i \geq s$.Comment: 7 pages, minor adjustment

Non-Existence of Stabilizing Policies for the Critical Push-Pull Network and Generalizations

The push-pull queueing network is a simple example in which servers either serve jobs or generate new arrivals. It was previously conjectured that there is no policy that makes the network positive recurrent (stable) in the critical case. We settle this conjecture and devise a general sufficient condition for non-stabilizability of queueing networks which is based on a linear martingale and further applies to generalizations of the push-pull network.Comment: 14 pages, 3 figure

Risk-sensitive optimal control for Markov decision processes with monotone cost

The existence of an optimal feedback law is established for the risk-sensitive optimal control problem with denumerable state space. The main assumptions imposed are irreducibility and anear monotonicity condition on the one-step cost function. A solution can be found constructively using either value iteration or policy iteration under suitable conditions on initial feedback law

Asymptotic entanglement in 1D quantum walks with a time-dependent coined

Discrete-time quantum walk evolve by a unitary operator which involves two operators a conditional shift in position space and a coin operator. This operator entangles the coin and position degrees of freedom of the walker. In this paper, we investigate the asymptotic behavior of the coin position entanglement (CPE) for an inhomogeneous quantum walk which determined by two orthogonal matrices in one-dimensional lattice. Free parameters of coin operator together provide many conditions under which a measurement perform on the coin state yield the value of entanglement on the resulting position quantum state. We study the problem analytically for all values that two free parameters of coin operator can take and the conditions under which entanglement becomes maximal are sought.Comment: 23 pages, 4 figures, accepted for publication in IJMPB. arXiv admin note: text overlap with arXiv:1001.5326 by other author

Geometric ergodicity in a weighted sobolev space

For a discrete-time Markov chain $\{X(t)\}$ evolving on $\Re^\ell$ with transition kernel $P$, natural, general conditions are developed under which the following are established: 1. The transition kernel $P$ has a purely discrete spectrum, when viewed as a linear operator on a weighted Sobolev space $L_\infty^{v,1}$ of functions with norm, $$\|f\|_{v,1} = \sup_{x \in \Re^\ell} \frac{1}{v(x)} \max \{|f(x)|, |\partial_1 f(x)|,\ldots,|\partial_\ell f(x)|\},$$ where $v\colon \Re^\ell \to [1,\infty)$ is a Lyapunov function and $\partial_i:=\partial/\partial x_i$. 2. The Markov chain is geometrically ergodic in $L_\infty^{v,1}$: There is a unique invariant probability measure $\pi$ and constants $B<\infty$ and $\delta>0$ such that, for each $f\in L_\infty^{v,1}$, any initial condition $X(0)=x$, and all $t\geq 0$: $$\Big| \text{E}_x[f(X(t))] - \pi(f)\Big| \le Be^{-\delta t}v(x),\quad \|\nabla \text{E}_x[f(X(t))] \|_2 \le Be^{-\delta t} v(x),$$ where $\pi(f)=\int fd\pi$. 3. For any function $f\in L_\infty^{v,1}$ there is a function $h\in L_\infty^{v,1}$ solving Poisson's equation: $$h-Ph = f-\pi(f).$$ Part of the analysis is based on an operator-theoretic treatment of the sensitivity process that appears in the theory of Lyapunov exponents

A Random Search Framework for Convergence Analysis of Distributed Beamforming with Feedback

The focus of this work is on the analysis of transmit beamforming schemes with a low-rate feedback link in wireless sensor/relay networks, where nodes in the network need to implement beamforming in a distributed manner. Specifically, the problem of distributed phase alignment is considered, where neither the transmitters nor the receiver has perfect channel state information, but there is a low-rate feedback link from the receiver to the transmitters. In this setting, a framework is proposed for systematically analyzing the performance of distributed beamforming schemes. To illustrate the advantage of this framework, a simple adaptive distributed beamforming scheme that was recently proposed by Mudambai et al. is studied. Two important properties for the received signal magnitude function are derived. Using these properties and the systematic framework, it is shown that the adaptive distributed beamforming scheme converges both in probability and in mean. Furthermore, it is established that the time required for the adaptive scheme to converge in mean scales linearly with respect to the number of sensor/relay nodes.Comment: 8 pages, 3 figures, presented partially at ITA '08 and PSU School of Info. Theory '0

Markov Chain Monte Carlo Method without Detailed Balance

We present a specific algorithm that generally satisfies the balance condition without imposing the detailed balance in the Markov chain Monte Carlo. In our algorithm, the average rejection rate is minimized, and even reduced to zero in many relevant cases. The absence of the detailed balance also introduces a net stochastic flow in a configuration space, which further boosts up the convergence. We demonstrate that the autocorrelation time of the Potts model becomes more than 6 times shorter than that by the conventional Metropolis algorithm. Based on the same concept, a bounce-free worm algorithm for generic quantum spin models is formulated as well.Comment: 5 pages, 5 figure