periodic points of holomorphic maps via lefschetz numbers in this paper we study the set of periods of holomorphic maps on compact manifolds using the periodic lefschetz numbers introduced by dold and llibre which can be computed from the homology class of the map we show that these numbers contain information about the existence of periodic points of a given period and if we assume the map to be transversal then they give us the exact number of such periodic orbits we apply this result to the complex projective space of dimension n and to some special type of hopf surfaces partially characterizing their set of periods in the first case we also show that any holomorphic map of cp n of degree greater than one has infinitely many distinct periodic orbits hence generalizing a theorem of fornaess and sibony we then characterize the set of periods of a holomorphic map on the riemann sphere hence giving an alternative proof of baker s theorem

asymptotically optimal anomaly detection via sequential testing sequential detection of independent anomalous processes among k processes is considered at each time only m 1 leq m leq k processes can be observed and the observations from each chosen process follow two different distributions depending on whether the process is normal or abnormal each anomalous process incurs a cost per unit time until its anomaly is identified and fixed switching across processes and state declarations are allowed at all times while decisions are based on all past observations and actions the objective is a sequential search strategy that minimizes the total expected cost incurred by all the processes during the detection process under reliability constraints we develop index type algorithms for the case with both known observation distributions and the case when the observation distributions have unknown parameters we show that the proposed algorithms are asymptotically optimal in terms of minimizing the total expected cost as the error constraints approach zero simulation results demonstrate strong performance in the finite regime

active sequential hypothesis testing consider a decision maker who is responsible to dynamically collect observations so as to enhance his information about an underlying phenomena of interest in a speedy manner while accounting for the penalty of wrong declaration due to the sequential nature of the problem the decision maker relies on his current information state to adaptively select the most informative sensing action among the available ones in this paper using results in dynamic programming lower bounds for the optimal total cost are established the lower bounds characterize the fundamental limits on the maximum achievable information acquisition rate and the optimal reliability moreover upper bounds are obtained via an analysis of two heuristic policies for dynamic selection of actions it is shown that the first proposed heuristic achieves asymptotic optimality where the notion of asymptotic optimality due to chernoff implies that the relative difference between the total cost achieved by the proposed policy and the optimal total cost approaches zero as the penalty of wrong declaration hence the number of collected samples increases the second heuristic is shown to achieve asymptotic optimality only in a limited setting such as the problem of a noisy dynamic search however by considering the dependency on the number of hypotheses under a technical condition this second heuristic is shown to achieve a nonzero information acquisition rate establishing a lower bound for the maximum achievable rate and error exponent in the case of a noisy dynamic search with size independent noise the obtained nonzero rate and error exponent are shown to be maximum

fixed point indices of iterated smooth maps in arbitrary dimension abstract let f be a smooth self map of r m when m is an arbitrary natural number we give a complete description of possible sequences of indices of iterations of f at an isolated fixed point answering in affirmative the chow mallet paret and yorke conjecture posed in s n chow j mallet parret j a yorke a periodic point index which is a bifurcation invariant in geometric dynamics rio de janeiro 1981 in lecture notes in math vol 1007 springer berlin 1983 pp 109 131

fixed point indices and periodic points of holomorphic mappings let n be the ball x 1 in the complex vector space mathbb c n let f delta n rightarrow mathbb c n be a holomorphic mapping and let m be a positive integer assume that the origin 0 0 dots 0 is an isolated fixed point of both f and the mth iteration f m of f then for each factor m of m the origin is again an isolated fixed point of f m and the fixed point index mu f m 0 of f m at the origin is well defined and so is the local dold s index invent math 74 3 419 435 1983 at the origin p m f 0 sum tau subset p m 1 tau mu f m tau 0 where p m is the set of all primes dividing m the sum extends over all subsets of p m is the cardinal number of and m tau m prod p in tau p 1 p m f 0 can be interpreted to be the number of periodic points of period m of f overlapped at the origin any holomorphic mapping f 1 delta n rightarrow mathbb c n sufficiently close to f has exactly p m f 0 distinct periodic points of period m near the origin provided that all the fixed points of f 1 m near the origin are simple note that f itself has no periodic point of period m near the origin if m 1 according to shub and sullivan s work topology 13 189 191 1974 a necessary condition so that p m f 0 0 is that the linear part of f at the origin has a periodic point of period m the goal of this paper is to prove that this condition is sufficient as well for holomorphic mappings

active classification for pomdps a kalman like state estimator author s zois ds levorato m mitra u abstract 1991 2012 ieee the problem of state tracking with active observation control is considered for a system modeled by a discrete time finite state markov chain observed through conditionally gaussian measurement vectors the measurement model statistics are shaped by the underlying state and an exogenous control input which influence the observations quality exploiting an innovations approach an approximate minimum mean squared error mmse filter is derived to estimate the markov chain system state to optimize the control strategy the associated mean squared error is used as an optimization criterion in a partially observable markov decision process formulation a stochastic dynamic programming algorithm is proposed to solve for the optimal solution to enhance the quality of system state estimates approximate mmse smoothing estimators are also derived finally the performance of the proposed framework is illustrated on the problem of physical activity detection in wireless body sensing networks the power of the proposed framework lies within its ability to accommodate a broad spectrum of active classification applications including sensor management for object classification and tracking estimation of sparse signals and radar scheduling

fixed point indices of iterated maps on determine toutes les relations universelles qui existent entre les indices de point fixe i f n n 1 2 des iterees f n d une application f v y ou y est un enr vcy est un sous ensemble ouvert

structured threshold policies for dynamic sensor scheduling a partially observed markov decision process approach we consider the optimal sensor scheduling problem formulated as a partially observed markov decision process pomdp due to operational constraints at each time instant the scheduler can dynamically select one out of a finite number of sensors and record a noisy measurement of an underlying markov chain the aim is to compute the optimal measurement scheduling policy so as to minimize a cost function comprising of estimation errors and measurement costs the formulation results in a nonstandard pomdp that is nonlinear in the information state we give sufficient conditions on the cost function dynamics of the markov chain and observation probabilities so that the optimal scheduling policy has a threshold structure with respect to a monotone likelihood ratio mlr ordering as a result the computational complexity of implementing the optimal scheduling policy is inexpensive we then present stochastic approximation algorithms for estimating the best linear mlr order threshold policy

generalized pattern search methods for a class of nonsmooth optimization problems with structure we propose a generalized pattern search gps method to solve a class of nonsmooth minimization problems where the set of nondifferentiability is included in the union of known hyperplanes and therefore is highly structured both unconstrained and linearly constrained problems are considered at each iteration the set of poll directions is enforced to conform to the geometry of both the nondifferentiability set and the boundary of the feasible region near the current iterate this is the key issue to guarantee the convergence of certain subsequences of iterates to points which satisfy first order optimality conditions numerical experiments on some classical problems validate the method

asymptotic efficiency of a sequential multihypothesis test a sequential multihypothesis test known as the m ary sequential probability ratio test msprt is generalized to account for nonuniform decision costs bounds on error probabilities and asymptotic expressions for the stopping time and error probabilities are given a key result of this correspondence is a proof that the generalized msprt is asymptotically efficient

periodic points for area preserving birational maps of surfaces it is a basic problem to count the number of periodic points of a surface mapping since the growth rate of this number as the period tends to infinity is an important dynamical invariant however this problem becomes difficult when the map admits curves of periodic points in this situation we give a precise estimate of the number of isolated periodic points for an area preserving birational map of a projective complex surface

error exponents for finite hypothesis channel identification we consider the problem of designing optimal probing signals for finite hypothesis testing equivalently we cast the problem as the design of optimal channel input sequences for identifying a discrete channel under observation from a finite set of known channels the optimality criterion that we employ is the exponent of the bayesian probability of error in our study we consider a feedforward scenario where there is no feedback from the channel output to the signal selector at the channel input and a feedback scenario where the past channel outputs are revealed to the signal selector in the feedforward scenario only the type of the input sequence matters and our main result is an expression for the error exponent in terms of the limiting distribution of the input sequence in the feedback case we show that when discriminating between two channels the optimal scheme in the first scenario is simultaneously the optimal time invariant markov feedback policy of any order

a stochastic sensor selection scheme for sequential hypothesis testing with multiple sensors we study the problem of binary sequential hypothesis testing using multiple sensors with associated observation costs an off line randomized sensor selection strategy in which a sensor is chosen at every time step with a given probability is considered the objective of this work is to find a sequential detection rule and a sensor selection probability vector such that the expected total observation cost is minimized subject to constraints on reliability and sensor usage first the sequential probability ratio test is shown to be the optimal sequential detection rule in this framework as well efficient algorithms for obtaining the optimal sensor selection probability vector are then derived in particular a special class of problems in which the algorithm has complexity that is linear in the number of sensors is identified an upper bound for the average sensor usage to estimate the error incurred due to wald s approximations is also presented this bound can be used to set a safety margin for guaranteed satisfaction of the constraints on the sensor usage

periods for transversal maps via lefschetz numbers for periodic points let f m m be a c1 map on a c differentiable manifold the map f is called transversal if for all m e n the graph of fm intersects transversally the diagonal of m x m at each point x x such that x is a fixed point of fm we study the set of periods of f by using the lefschetz numbers for periodic points we focus our study on transversal maps defined on compact manifolds such that their rational homology is ho z q hi q e q and hk t 0 for k 0 1

controlled sensing for multihypothesis testing the problem of multiple hypothesis testing with observation control is considered in both fixed sample size and sequential settings in the fixed sample size setting for binary hypothesis testing the optimal exponent for the maximal error probability corresponds to the maximum chernoff information over the choice of controls and a pure stationary open loop control policy is asymptotically optimal within the larger class of all causal control policies for multihypothesis testing in the fixed sample size setting lower and upper bounds on the optimal error exponent are derived it is also shown through an example with three hypotheses that the optimal causal control policy can be strictly better than the optimal open loop control policy in the sequential setting a test based on earlier work by chernoff for binary hypothesis testing is shown to be first order asymptotically optimal for multihypothesis testing in a strong sense using the notion of decision making risk in place of the overall probability of error another test is also designed to meet hard risk constrains while retaining asymptotic optimality the role of past information and randomization in designing optimal control policies is discussed

optimal index policies for anomaly localization in resource constrained cyber systems abstract the problem of anomaly localization in a resource constrained cyber system is considered each anomalous component of the system incurs a cost per unit time until its anomaly is identified and fixed different anomalous components may incur different costs depending on their criticality to the system due to resource constraints only one component can be probed at each given time the observations from a probed component are realizations drawn from two different distributions depending on whether the component is normal or anomalous the objective is a probing strategy that minimizes the total expected cost incurred by all the components during the detection process under reliability constraints we consider both independent and exclusive models in the former each component can be abnormal with a certain probability independent of other components in the latter one and only one component is abnormal we develop optimal index policies under both models the proposed index policies apply to a more general case where a subset more than one of the components can be probed simultaneously the problem under study also finds applications in spectrum scanning in cognitive radio networks and event detection in sensor networks

multihypothesis sequential probability ratio tests ii accurate asymptotic expansions for the expected sample size for pt i see ibid vol 45 p 2448 61 1999 we proved in pt i that two specific constructions of multihypothesis sequential tests which we refer to as multihypothesis sequential probability ratio tests msprts are asymptotically optimal as the decision risks or error probabilities go to zero the msprts asymptotically minimize not only the expected sample size but also any positive moment of the stopping time distribution under very general statistical models for the observations in this paper based on nonlinear renewal theory we find accurate asymptotic approximations up to a vanishing term for the expected sample size that take into account the overshoot over the boundaries of decision statistics the approximations are derived for the scenario where the hypotheses are simple the observations are independent and identically distributed i i d according to one of the underlying distributions and the decision risks go to zero simulation results for practical examples show that these approximations are fairly accurate not only for large but also for moderate sample sizes the asymptotic results given here complete the analysis initiated by baum and veeravalli 1994 where first order asymptotics were obtained for the expected sample size under a specific restriction on the kullback leibler distances between the hypotheses

discrimination of two channels by adaptive methods and its application to quantum system the optimal exponential error rate for adaptive discrimination of two channels is discussed in this problem adaptive choice of input signal is allowed this problem is discussed in various settings it is proved that adaptive choice does not improve the exponential error rate in these settings these results are applied to quantum state discrimination

convex stochastic dominance in bayesian localization filtering and controlled sensing pomdps this paper provides conditions on the observation probability distribution in bayesian localization and optimal filtering so that the conditional mean estimate satisfies convex stochastic dominance convex dominance allows us to compare the unconditional mean square error between two optimal bayesian state estimators over arbitrary time horizons instead of using brute force monte carlo computations the proof uses two key ideas from microeconomics namely integral precision dominance and aggregation of single crossing the convex dominance result is then used to give sufficient conditions so that the optimal policy of a controlled sensing two state partially observed markov decision process pomdp is lower bounded by a myopic policy numerical examples are presented where the shannon capacity of the observation distribution using one sensor dominates that of another and convex dominance holds but blackwell dominance does not hold these illustrate the usefulness of the main result in localization filtering and controlled sensing applications

a sequential procedure for multihypothesis testing the sequential testing of more than two hypotheses has important applications in direct sequence spread spectrum signal acquisition multiple resolution element radar and other areas a useful sequential test which we term the msprt is studied in this paper the test is shown to be a generalization of the sequential probability ratio test under bayesian assumptions it is argued that the msprt approximates the much more complicated optimal test when error probabilities are small and expected stopping times are large bounds on error probabilities are derived and asymptotic expressions for the stopping time and error probabilities are given a design procedure is presented for determining the parameters of the msprt two examples involving gaussian densities are included and comparisons are made between simulation results and asymptotic expressions comparisons with bayesian fixed sample size tests are also made and it is found that the msprt requires two to three times fewer samples on average

multihypothesis sequential probability ratio tests i asymptotic optimality the problem of sequential testing of multiple hypotheses is considered and two candidate sequential test procedures are studied both tests are multihypothesis versions of the binary sequential probability ratio test sprt and are referred to as msprts the first test is motivated by bayesian optimality arguments while the second corresponds to a generalized likelihood ratio test it is shown that both msprts are asymptotically optimal relative not only to the expected sample size but also to any positive moment of the stopping time distribution when the error probabilities or more generally risks associated with incorrect decisions are small the results are first derived for the discrete time case of independent and identically distributed i i d observations and simple hypotheses they are then extended to general possibly continuous time statistical models that may include correlated and nonhomogeneous observation processes it also demonstrated that the results can be extended to hypothesis testing problems with nuisance parameters where the composite hypotheses due to nuisance parameters can be reduced to simple ones by using the principle of invariance these results provide a complete generalization of the results given by veeravalli and baum see ibid vol 41 p 1994 97 1995 where it was shown that the quasi bayesian msprt is asymptotically efficient with respect to the expected sample size for i i d observations