Ergodicity, Decisions, and Partial Information

In the simplest sequential decision problem for an ergodic stochastic process X, at each time n a decision u_n is made as a function of past observations X_0,...,X_{n-1}, and a loss l(u_n,X_n) is incurred. In this setting, it is known that one may choose (under a mild integrability assumption) a decision strategy whose pathwise time-average loss is asymptotically smaller than that of any other strategy. The corresponding problem in the case of partial information proves to be much more delicate, however: if the process X is not observable, but decisions must be based on the observation of a different process Y, the existence of pathwise optimal strategies is not guaranteed. The aim of this paper is to exhibit connections between pathwise optimal strategies and notions from ergodic theory. The sequential decision problem is developed in the general setting of an ergodic dynamical system (\Omega,B,P,T) with partial information Y\subseteq B. The existence of pathwise optimal strategies grounded in two basic properties: the conditional ergodic theory of the dynamical system, and the complexity of the loss function. When the loss function is not too complex, a general sufficient condition for the existence of pathwise optimal strategies is that the dynamical system is a conditional K-automorphism relative to the past observations \bigvee_n T^n Y. If the conditional ergodicity assumption is strengthened, the complexity assumption can be weakened. Several examples demonstrate the interplay between complexity and ergodicity, which does not arise in the case of full information. Our results also yield a decision-theoretic characterization of weak mixing in ergodic theory, and establish pathwise optimality of ergodic nonlinear filters.Comment: 45 page

On the exchange of intersection and supremum of sigma-fields in filtering theory

We construct a stationary Markov process with trivial tail sigma-field and a nondegenerate observation process such that the corresponding nonlinear filtering process is not uniquely ergodic. This settles in the negative a conjecture of the author in the ergodic theory of nonlinear filters arising from an erroneous proof in the classic paper of H. Kunita (1971), wherein an exchange of intersection and supremum of sigma-fields is taken for granted.Comment: 20 page

The politics of time: Political entropy, settler colonialism and urban ruination in Hebron / Al-Khalil, Palestine

This paper explores the relationships between settler colonialism and temporal regimes of urban ruination in the Palestinian city of Hebron/Al-Khalil. The city is divided into two sectors: H1, controlled by the Palestinian National Authority, and H2, roughly 20% of the city, including 33,000 Palestinians and 700 Jewish settlers, living under direct Israeli military occupation. Drawing on fieldwork in H2, we identify several temporal formations shaping H2: the settler-messianic time that spatializes the past as a platform to facilitate colonial expansion; contemporary military control that seeks to destabilize Palestinian homes, their everyday lives and ontological security in the present; and the Palestinian effort to recalibrate the rhythm of elimination and recover the future. The paper develops the concept of political entropy: a temporal technology of control that builds on the process of natural decay. Israel uses the power of time and entropy to let Palestinian assets decay slowly while prohibiting proper renovation and maintenance. The mundane violence of political entropy tends to remain unseen, as the harm is done slowly, allegedly naturally. Facing the settler colonial temporalities that maintain a state of undecidedness, where deferral itself becomes a weapon which lets entropy operate without interruption, Hebron\u27s Palestinian inhabitants aim at slowing down the effects of time by applying friction to the process of ruination, as well as by their insistence on futurability. By identifying the temporalities trapping the city\u27s Palestinian population, our paper thus frames Hebron\u27s H2 as space that is simultaneously ruined and reclaimed

Optimal Design of Robust Combinatorial Mechanisms for Substitutable Goods

In this paper we consider multidimensional mechanism design problem for selling discrete substitutable items to a group of buyers. Previous work on this problem mostly focus on stochastic description of valuations used by the seller. However, in certain applications, no prior information regarding buyers' preferences is known. To address this issue, we consider uncertain valuations and formulate the problem in a robust optimization framework: the objective is to minimize the maximum regret. For a special case of revenue-maximizing pricing problem we present a solution method based on mixed-integer linear programming formulation

Sliding mode control of quantum systems

This paper proposes a new robust control method for quantum systems with uncertainties involving sliding mode control (SMC). Sliding mode control is a widely used approach in classical control theory and industrial applications. We show that SMC is also a useful method for robust control of quantum systems. In this paper, we define two specific classes of sliding modes (i.e., eigenstates and state subspaces) and propose two novel methods combining unitary control and periodic projective measurements for the design of quantum sliding mode control systems. Two examples including a two-level system and a three-level system are presented to demonstrate the proposed SMC method. One of main features of the proposed method is that the designed control laws can guarantee desired control performance in the presence of uncertainties in the system Hamiltonian. This sliding mode control approach provides a useful control theoretic tool for robust quantum information processing with uncertainties.Comment: 18 pages, 4 figure

Feedback control of spin systems

The feedback stabilization problem for ensembles of coupled spin 1/2 systems is discussed from a control theoretic perspective. The noninvasive nature of the bulk measurement allows for a fully unitary and deterministic closed loop. The Lyapunov-based feedback design presented does not require spins that are selectively addressable. With this method, it is possible to obtain control inputs also for difficult tasks, like suppressing undesired couplings in identical spin systems.Comment: 16 pages, 15 figure

Automorphisms of graphs of cyclic splittings of free groups

We prove that any isometry of the graph of cyclic splittings of a finitely generated free group $F_N$ of rank $N\ge 3$ is induced by an outer automorphism of $F_N$. The same statement also applies to the graphs of maximally-cyclic splittings, and of very small splittings.Comment: 22 pages, 5 figures. Small modifications. To appear in Geometriae Dedicat

Tur\'an numbers for Ks,tK_{s,t}-free graphs: topological obstructions and algebraic constructions

We show that every hypersurface in $\R^s\times \R^s$ contains a large grid, i.e., the set of the form $S\times T$, with $S,T\subset \R^s$. We use this to deduce that the known constructions of extremal $K_{2,2}$-free and $K_{3,3}$-free graphs cannot be generalized to a similar construction of $K_{s,s}$-free graphs for any $s\geq 4$. We also give new constructions of extremal $K_{s,t}$-free graphs for large $t$.Comment: Fixed a small mistake in the application of Proposition

The impact of low erythrocyte density in human blood on the fitness and energetic reserves of the African malaria vector Anopheles gambiae

Background Anaemia is a common health problem in the developing world. This condition is characterized by a reduction in erythrocyte density, primarily from malnutrition and/or infectious diseases such as malaria. As red blood cells are the primary source of protein for haematophagous mosquitoes, any reduction could impede the ability of mosquito vectors to transmit malaria by influencing their fitness or that of the parasites they transmit. The aim of this study was to determine the impact of differences in the density of red blood cells in human blood on malaria vector (Anopheles gambiae sensu stricto) fitness. The hypotheses tested are that mosquito vector energetic reserves and fitness are negatively influenced by reductions in the red cell density of host human blood meals commensurate with those expected from severe anaemia. Methods Mosquitoes (An. gambiae s.s.) were offered blood meals of different packed cell volume(PCV) of human blood consistent with those arising from severe anaemia (15%) and normalPCV (50%). Associations between mosquito energetic reserves (lipid, glucose and glycogen)and fitness measures (reproduction and survival) and blood meal PCV were investigated. Results The amount of protein that malaria vectors acquired from blood feeding (indexed by haematin excretion) was significantly reduced at low blood PCV. However, mosquitoes feeding on blood of low PCV had the same oviposition rates as those feeding on blood of normal PCV, and showed an increase in egg production of around 15%. The long-term survival of An. gambiae s.s was reduced after feeding on low PCV blood, but PCV had no significant impact on the proportion of mosquitoes surviving through the minimal period required to develop and transmit malaria parasites (estimated as 14 days post-blood feeding). The impact of blood PCV on the energetic reserves of mosquitoes was relatively minor. Conclusions These results suggest that feeding on human hosts whose PCV has been depleted due to severe anaemia does not significantly reduce the fitness or transmission potential of malaria vectors, and indicates that mosquitoes may be able exploit resources for reproduction more efficiently from blood of low rather than normal PCV