control design of uncertain quantum systems with fuzzy estimators an approach of control design using fuzzy estimators fes is proposed for quantum systems with uncertainties two types of quantum control problems are considered 1 control of a pure state quantum system in the presence of uncertainties and 2 control design of quantum systems with initial mixed states and uncertainties for the first type of tasks a partial feedback control scheme with an fe is presented to design controllers in this scheme an fe is trained to estimate the quantum state for feedback control of a quantum system and controlled projective measurement is used to assist in controlling the system for the second type of quantum control tasks a probabilistic fuzzy estimator pfe is trained to estimate the quantum state for control design of a quantum system with an initial mixed state and a corresponding control algorithm is proposed to design a control law that drives the system from the mixed state to a target pure state two examples of two spin 1 2 systems are also presented and analyzed to demonstrate the process of control design and potential applications of the proposed approach

averaged controllability of parameter dependent conservative semigroups abstract we consider the problem of averaged controllability for parameter depending either in a discrete or continuous fashion control systems the aim being to find a control independent of the unknown parameters so that the average of the states is controlled we do it in the context of conservative models both in an abstract setting and also analysing the specific examples of the wave and schrodinger equations our first result is of perturbative nature assuming the averaging probability measure to be a small parameter dependent perturbation in a sense that we make precise of an atomic measure given by a dirac mass corresponding to a specific realisation of the system we show that the averaged controllability property is achieved whenever the system corresponding to the support of the dirac is controllable similar tools can be employed to obtain averaged versions of the so called ingham inequalities particular attention is devoted to the 1d wave equation in which the time periodicity of solutions can be exploited to obtain more precise results provided the parameters involved satisfy diophantine conditions ensuring the lack of resonances

the sample average approximation method for stochastic discrete optimization in this paper we study a monte carlo simulation based approach to stochastic discrete optimization problems the basic idea of such methods is that a random sample is generated and the expected value function is approximated by the corresponding sample average function the obtained sample average optimization problem is solved and the procedure is repeated several times until a stopping criterion is satisfied we discuss convergence rates stopping rules and computational complexity of this procedure and present a numerical example for the stochastic knapsack problem

cooling schedules for optimal annealing a monte carlo optimization technique called simulated annealing is a descent algorithm modified by random ascent moves in order to escape local minima which are not global minima the level of randomization is determined by a control parameter t called temperature which tends to zero according to a deterministic cooling schedule we give a simple necessary and sufficient condition on the cooling schedule for the algorithm state to converge in probability to the set of globally minimum cost states in the special case that the cooling schedule has parametric form tt c log1 t the condition for convergence is that c be greater than or equal to the depth suitably defined of the deepest local minimum which is not a global minimum state

diffusions for global optimizations we seek a global minimum of u 0 1 n to r the solution to d dt x t nabla u x t will find local minima the solution to dx t nabla u x t dt sqrt 2t dw t where w is standard n dimensional brownian motion and the boundaries are reflecting will concentrate near the global minima of u at least when temperature t is small the equilibrium distribution for x t is gibbs with density pi t x alpha exp u x t this suggests setting t t t downarrow 0 to find the global minima of u we give conditions on u x and t t such that the solution to dx t nabla u x t dt sqrt 2t dw t converges weakly to a distribution concentrated on the global minima of u

inequalities and bounds for generalized complete elliptic integrals computable bounds for the generalized complete elliptic integrals of the first and second kind are obtained also bounds for some combinations and products for integrals under discussion are established it has been proven that both families of integrals are logarithmically convex as functions of the first parameter this property has been employed to obtain several inequalities involving integrals in question

further results on sampled data design for robust control of a single qubit this paper presents further results on the robust control method for qubit systems in dong et al 2013 based on the properties of an antisymmetric system an alternative method is presented to analyse and exclude singularity intervals in the proof of partial original results for the case of amplitude damping decoherence a larger sampling period is presented when the upper bound of the probability of failure is small enough for the case of phase damping decoherence a larger sampling period is given when the lower bound of the target coherence is large enough furthermore we provide improved sampling periods for amplitude damping decoherence and phase damping decoherence without the above prior constraints

free endpoint optimal control of inhomogeneous bilinear ensemble systems optimal control of bilinear systems has been a well studied subject in the areas of mathematical and computational optimal control however effective methods for solving emerging optimal control problems involving an ensemble of deterministic or stochastic bilinear systems are underdeveloped these burgeoning problems arise in diverse applications from quantum control and molecular imaging to neuroscience in this work we develop an iterative method to find optimal controls for an inhomogeneous bilinear ensemble system with free endpoint conditions the central idea is to represent the bilinear ensemble system at each iteration as a time varying linear ensemble system and then solve it in an iterative manner we analyze convergence of the iterative procedure and discuss optimality of the convergent solutions the method is directly applicable to solve the same class of optimal control problems involving a stochastic bilinear ensemble system driven by independent additive noise processes we demonstrate the robustness and applicability of the developed iterative method through practical control designs in neuroscience and quantum control

ensemble control of bloch equations in this article we study a class of control problems which involves controlling a continuum of dynamical systems with different values of parameters characterizing the system dynamics by using the same control signal we call such problems control of ensembles the motivation for looking into these problems comes from the manipulation of an ensemble of nuclear spins in nuclear magnetic resonance nmr spectroscopy and imaging mri with dispersions in natural frequencies and the strengths of the applied radio frequency rf field from the standpoint of mathematical control theory the challenge is to simultaneously steer a continuum of systems between points of interest with the same control input this raises some new and unexplored questions about controllability of such systems we show that controllability of an ensemble can be understood by the study of the algebra of polynomials defined by the noncommuting vector fields that govern the system dynamics a systematic study of these systems has immediate applications to broad areas of control of ensembles of quantum systems as arising in coherent spectroscopy and quantum information processing the new mathematical structures appearing in such problems are excellent motivation for new developments in control theory

ramanujan s theories of elliptic functions to alternative bases in his famous paper 3 10 pp 23 39 ramanujan offers several beautiful series representations for 1 pi he first states three formulas one of which is frac 4 pi sum limits mathbf n 0 infty frac left 6 mathbf n 1 right left frac 1 2 right 3 n left mathbf n right 3 4 n where a o 1 and for each positive integer n left mathbf a right n mathbf a left mathbf a 1 right left mathbf a 2 right left mathbf a mathbf n 1 right

simulation based discrete optimization of stochastic discrete event systems subject to non closed form constraints this technical note addresses the discrete optimization of stochastic discrete event systems for which both the performance function and the constraint function are not known but can be evaluated by simulation and the solution space is either finite or unbounded our method is based on random search in a neighborhood structure called the most promising area proposed in and a moving observation area the simulation budget is allocated dynamically to promising solutions simulation based constraints are taken into account in an augmented performance function via an increasing penalty factor we prove that under some assumptions the algorithm converges with probability 1 to a set of true local optimal solutions these assumptions are restrictive and difficult to verify but we hope that the encouraging numerical results would motivate future research exploiting ideas of this technical note

complexity of graph self assembly in accretive systems and self destructible systems self assembly is a process in which small objects autonomously associate with each other to form larger complexes it is ubiquitous in biological constructions at the cellular and molecular scale and has also been identified by nanoscientists as a fundamental method for building nano scale structures recent years have seen convergent interest and efforts in studying self assembly from mathematicians computer scientists physicists chemists and biologists however most complexity theoretical studies of self assembly utilize mathematical models with two limitations 1 only attraction while no repulsion is studied 2 only assembled structures of two dimensional square grids are studied in this paper we study the complexity of the assemblies resulting from the cooperative effect of repulsion and attraction in a more general setting of graphs this allows for the study of a more general class of self assembled structures than the previous tiling model we define two novel assembly models namely the accretive graph assembly model and the self destructible graph assembly model and identify a fundamental problem in them the sequential construction of a given graph we refer to it as the accretive graph assembly problem agap and the self destructible graph assembly problem dgap in the respective models our main results are i agap is np complete even if the maximum degree of the graph is restricted to 4 or the graph is restricted to be planar with maximum degree 5 ii counting the number of sequential assembly orderings that result in a target graph agap is p complete and iii dgap is pspace complete even if the maximum degree of the graph is restricted to 6 this is the first pspace complete result in self assembly we also extend the accretive graph assembly model to a stochastic model and prove that determining the probability of a given assembly in this model is p complete

minimum time frictionless atom cooling in harmonic traps frictionless atom cooling in harmonic traps is formulated as a time optimal control problem and a synthesis of optimal controlled trajectories is obtained

time optimal simultaneous control of two level quantum systems in this paper we solve the problem of simultaneously driving in minimum time to arbitrary final conditions n two level quantum systems subject to independent controls the solution of this problem is obtained via an explicit description of the reachable set of the associated control system on s u 2 the treatment generalizes previous results on the time optimal control of two level quantum systems and suggests that similar techniques could be used to solve the minimum time control problem for a larger class of right invariant systems on lie groups

nested partitions method for global optimization we propose a new randomized method for solving global optimization problems this method the nested partitions np method systematically partitions the feasible region and concentrates the search in regions that are the most promising the most promising region is selected in each iteration based on information obtained from random sampling of the entire feasible region and local search the method hence combines global and local search we first develop the method for discrete problems and then show that the method can be extended to continuous global optimization the method is shown to converge with probability one to a global optimum in finite time in addition we provide bounds on the expected number of iterations required for convergence and we suggest two stopping criteria numerical examples are also presented to demonstrate the effectiveness of the method

penalty function with memory for discrete optimization via simulation with stochastic constraints we consider a discrete optimization via simulation dovs problem with stochastic constraints on secondary performance measures in which both objective and secondary performance measures need to be estimated by stochastic simulation to solve the problem we develop a new method called the penalty function with memory pfm it is similar to an existing penalty type method which consists of a penalty parameter and a measure of violation of constraints in a sense that it converts a dovs problem with constraints into a series of unconstrained problems however pfm uses a different penalty parameter called a penalty sequence determined by the past history of feasibility checks on a solution specifically assuming a minimization problem a penalty sequence diverges to infinity for any infeasible solution but converges to zero for any feasible solution under certain conditions as a result a dovs algorithm combined with pfm performs well even when an optimal feasible solution is a boundary solution with o

stochastic discrete optimization in this paper a stochastic search method is proposed for finding a global solution to the stochastic discrete optimization problem in which the objective function must be estimated by monte carlo simulation although there are many practical problems of this type in the fields of manufacturing engineering operations research and management science there have not been any nonheuristic methods proposed for such discrete problems with stochastic infrastructure the proposed method is very simple yet it finds a global optimum solution the method exploits the randomness of monte carlo simulation and generates a sequence of solution estimates this generated sequence turns out to be a nonstationary markov chain and it is shown under mild conditions that the markov chain is strongly ergodic and that the probability that the current solution estimate is global optimum converges to one furthermore the speed of convergence is also analyzed

a method for discrete stochastic optimization this paper addresses the problem of optimizing a function over a finite or countably infinite set of alternatives in situations where this objective function cannot be evaluated exactly but has to be estimated or measured a special focus is on situations where simulation is used to evaluate the objective function we present two versions of a new iterative method for solving such discrete stochastic optimization problems in each iteration of the proposed method a neighbor of the current alternative is selected and estimates of the objective function evaluated at the current and neighboring alternatives are compared the alternative that has a better observed function value becomes the next current alternative we show how one version of the proposed method can be used to solve discrete optimization problems where the objective function is evaluated using transient or steady state simulation and we show how the other version can be applied to solve a special class of discrete stochastic optimizati

control and synchronization of neuron ensembles synchronization of oscillations is a phenomenon prevalent in natural social and engineering systems controlling synchronization of oscillating systems is motivated by a wide range of applications from surgical treatment of neurological diseases to the design of neurocomputers in this paper we study the control of an ensemble of uncoupled neuron oscillators described by phase models we examine controllability of such a neuron ensemble for various phase models and furthermore study the related optimal control problems in particular by employing pontryagin s maximum principle we analytically derive optimal controls for spiking single and two neuron systems and analyze the applicability of the latter to an ensemble system finally we present a robust computational method for optimal control of spiking neurons based on pseudospectral approximations the methodology developed here is universal to the control of general nonlinear phase oscillators

sensor placement for calibration of spatially varying model parameters abstract this paper presents a sensor placement optimization framework for the calibration of spatially varying model parameters to account for the randomness of the calibration parameters over space and across specimens the spatially varying parameter is represented as a random field based on this representation bayesian calibration of spatially varying parameter is investigated to reduce the required computational effort during bayesian calibration the original computer simulation model is substituted with kriging surrogate models based on the singular value decomposition svd of the model response and the karhunen loeve expansion kle of the spatially varying parameters a sensor placement optimization problem is then formulated based on the bayesian calibration to maximize the expected information gain measured by the expected kullback leibler k l divergence the optimization problem needs to evaluate the expected k l divergence repeatedly which requires repeated calibration of the spatially varying parameter and this significantly increases the computational effort of solving the optimization problem to overcome this challenge an approximation for the posterior distribution is employed within the optimization problem to facilitate the identification of the optimal sensor locations using the simulated annealing algorithm a heat transfer problem with spatially varying thermal conductivity is used to demonstrate the effectiveness of the proposed method

convexity with respect to holder mean involving zero balanced hypergeometric functions in this note we investigate the convexity of zero balanced hypergeometric functions with respect to holder mean

planar formulae and their uses we define the set of planar boolean formulae and then show that the set of true quantified planar formulae is polynomial space complete and that the set of satisfiable planar formulae is np complete using these results we are able to provide simple and nearly uniform proofs of np completeness for planar node cover planar hamiltonian circuit and line geometric connected dominating set and of polynomial space completeness for planar generalized geography the np completeness of planar node cover and planar hamiltonian circuit and line were first proved elsewhere m r garey and d s johnson the rectilinear steiner tree is np complete siam j appl math 32 1977 pp 826 834 and m r garey d s johnson and r e tarjan the planar hamilton circuit problem is np complete siam j comp 5 1976 pp 704 714

functional inequalities for the quotients of hypergeometric functions letf a b c x be the gaussian hypergeometric series and for 0 r 1 let formula g d anderson m k vamanamurthy and m vuorinen raised recently the following problem for whicha b 0 1 does formula hold for allr s 0 1 they also proved this inequality fora b 1 2 the main purpose of this paper is to give an answer to this problem fora b 1 and to find a lower bound for the summ r m s fora 0 2 andb 0 2 a

retrospective optimization of mixed integer stochastic systems using dynamic simplex linear interpolation abstract we propose a family of retrospective optimization ro algorithms for optimizing stochastic systems with both integer and continuous decision variables the algorithms are continuous search procedures embedded in a ro framework using dynamic simplex interpolation rodsi by decreasing dimensions corresponding to the continuous variables of simplex the retrospective solutions become closer to an optimizer of the objective function we present convergence results of rodsi algorithms for stochastic convex systems numerical results show that a simple implementation of rodsi algorithms significantly outperforms some random search algorithms such as genetic algorithm ga and particle swarm optimization pso

log convexity and log concavity of hypergeometric like functions abstract we find sufficient conditions for log convexity and log concavity for the functions of the forms a f k a k x k a f k a k x k and a f k x k a k the most useful examples of such functions are generalized hypergeometric functions in particular we generalize the turan inequality for the confluent hypergeometric function recently proved by barnard gordy and richards and log convexity results for the same function recently proved by baricz besides we establish a reverse inequality which complements naturally the inequality of barnard gordy and richards similar results are established for the gauss and the generalized hypergeometric functions a conjecture about monotonicity of a quotient of products of confluent hypergeometric functions is made

a simulated annealing algorithm with constant temperature for discrete stochastic optimization we present a modification of the simulated annealing algorithm designed for solving discrete stochastic optimization problems like the original simulated annealing algorithm our method has the hill climbing feature so it can find global optimal solutions to discrete stochastic optimization problems with many local solutions however our method differs from the original simulated annealing algorithm in that it uses a constant rather than decreasing temperature we consider two approaches for estimating the optimal solution the first approach uses the number of visits the algorithm makes to the different states divided by a normalizer to estimate the optimal solution the second approach uses the state that has the best average estimated objective function value as estimate of the optimal solution we show that both variants of our method are guaranteed to converge almost surely to the set of global optimal solutions and discuss how our work applies in the discrete deterministic optimization setting we also show how both variants can be applied for solving discrete optimization problems when the objective function values are estimated using either transient or steady state simulation finally we include some encouraging numerical results documenting the behavior of the two variants of our algorithm when applied for solving two versions of a particular discrete stochastic optimization problem and compare their performance with that of other variants of the simulated annealing algorithm designed for solving discrete stochastic optimization problems

linear constructions for dna codes in this paper we translate in terms of coding theory constraints that are used in designing dna codes for use in dna computing or as bar codes in chemical libraries we propose new constructions for dna codes satisfying either a reverse complement constraint a gc content constraint or both that are derived from additive and linear codes over four letter alphabets we focus in particular on codes over gf 4 and we construct new dna codes that are in many cases better sometimes far better than previously known codes we provide updated tables up to length 20 that include these codes as well as new codes constructed using a combination of lexicographic techniques and stochastic search

simulated annealing to cool or not abstract the optimization algorithm simulated annealing is considered presented is a class of problem instances and neighbor selection matrices for which no monotone decreasing temperature sequence is optimal with respect to certain natural performance criteria an algorithm called threshold random search is introduced and use is made of the fact that simulated annealing is a randomized version of threshold random search with deterministic thresholds

global random optimization by simultaneous perturbation stochastic approximation we examine the theoretical and numerical global convergence properties of a certain ldquogradient freerdquo stochastic approximation algorithm called the ldquosimultaneous perturbation stochastic approximation spsa rdquo that has performed well in complex optimization problems we establish two theorems on the global convergence of spsa the first involving the well known method of injected noise the second theorem establishes conditions under which ldquobasicrdquo spsa without injected noise can achieve convergence in probability to a global optimum a result with important practical benefits

a global search method for discrete stochastic optimization this paper is concerned with the problem of optimizing the performance of a stochastic system over a finite set of alternatives in situations where the performance of the system cannot be evaluated analytically but must be estimated or measured for instance through simulation we present two variants of a new method for solving such discrete stochastic optimization problems this new method uses global search to look for the optimal solution it generates a sequence taking values in the set of feasible alternatives where each new element of the sequence is generated by comparing the current element with another candidate alternative and letting the next element of the sequence be the one of the current and candidate alternatives that appears to yield better performance for both versions of the proposed method the element of the set of feasible alternatives that the generated sequence visits most often is shown to converge almost surely to a globally optimal solution of the underlying optimization pr

a modification of the stochastic ruler method for discrete stochastic optimization abstract we propose a modified stochastic ruler method for finding a global optimal solution to a discrete optimization problem in which the objective function cannot be evaluated analytically but has to be estimated or measured our method generates a markov chain sequence taking values in the feasible set of the underlying discrete optimization problem it uses the number of visits this sequence makes to the different states to estimate the optimal solution we show that our method is guaranteed to converge almost surely a s to the set of global optimal solutions then we show how our method can be used for solving discrete optimization problems where the objective function values are estimated using either transient or steady state simulation finally we provide some numerical results to check the validity of our method and compare its performance with that of the original stochastic ruler method

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

a two step strategy for stabilizing control of quantum systems with uncertainties as a matter of course designed control laws are to be acted on actual systems however most designed control laws are proven to be effective when performed on ideal models of the systems as opposed to the actual systems themselves one fundamental problem is how to ensure the satisfactory performance of the designed control laws when they are performed on actual systems focusing on the state stabilization of quantum systems a two step strategy is proposed to solve this problem a feedback strategy and the open loop control technique of dynamical decoupling are combined therein to deal with the inevitable differences between an actual system and its model by taking advantage of the distinct quantum characteristics the measurement induced state transfer and the tensor product structure with the structure characterizing the coupling between a quantum system and its environment specifically in the first step a measurement based feedback control strategy is selected according to a model of the actual system in the second step by identifying the differences between the actual system and the established model as decoherence noise a specific control procedure is designed through dynamical decoupling this procedure allows the realization of the chosen control strategy so as to deal with the differences

a modified hooke and jeeves algorithm with likelihood ratio performance extrapolation for simulation optimization the hooke and jeeves algorithm hj is a pattern search procedure widely used to optimize non linear functions that are not necessarily continuous or differentiable the algorithm performs repeatedly two types of search routines an exploratory search and a pattern search the hj algorithm requires deterministic evaluation of the function being optimized in this paper we consider situations where the objective function is stochastic and can be evaluated only through monte carlo simulation to overcome the problem of expensive use of function evaluations for monte carlo simulation a likelihood ratio performance extrapolation lrpe technique is used we extrapolate the performance measure for different values of the decision parameters while simulating a single sample path from the underlying system our modified hooke and jeeves algorithm uses a likelihood ratio performance extrapolation for simulation optimization computational results are provided to demonstrate the performance of the proposed modified hj algorithm

stabilization of an arbitrary profile for an ensemble of half spin systems we consider the feedback stabilization of a variable profile for an ensemble of non interacting half spins described by the bloch equations we propose an explicit feedback law that stabilizes asymptotically the system around a given arbitrary target profile the convergence proof is done when the target profile is entirely in the south hemisphere or in the north hemisphere of the bloch sphere the convergence holds for initial conditions in a h1 neighbourhood of this target profile this convergence is shown for the weak h1 topology the proof relies on an adaptation of the lasalle invariance principle to infinite dimensional systems numerical simulations illustrate the efficiency of these feedback laws even for initial conditions far from the target profile

regime switching stochastic approximation algorithms with application to adaptive discrete stochastic optimization this work is devoted to a class of stochastic approximation problems with regime switching modulated by a discrete time markov chain our motivation stems from using stochastic recursive algorithms for tracking markovian parameters such as those in spreading code optimization in cdma code division multiple access wireless communication the algorithm uses constant step size to update the increments of a sequence of occupation measures it is proved that least squares estimates of the tracking errors can be developed assume that the adaptation rate is of the same order of magnitude as that of the time varying parameter which is more difficult to deal with than that of slower parameter variations due to the time varying characteristics and markovian jumps the usual stochastic approximation sa techniques cannot be carried over in the analysis by a combined use of the sa method and two time scale markov chains asymptotic properties of the algorithm are obtained which are distinct from the usual sa

controllability issues for continuous spectrum systems and ensemble controllability of bloch equations we study the controllability of the bloch equation for an ensemble of non interactinghalf spins inastaticmagneticfield withdispersioninthelarmorfrequency this system may be seen as a prototype for infinite dimensional bilinear systems with continuous spectrum whose controllability is not well understood we provide several mathematical answers with discrimination between approximate and exact controlla bility and between finite time or infinite time controllability this system is not exactly controllable in finite time t with bounded controls in l 2 0 t but it is approximately controllable in l in finite time with unbounded controls in l 0 moreover we propose explicit controls realizing the asymptotic exact controllability to a uniform state of spin 1 2o r 1 2

ensemble control of finite dimensional time varying linear systems in this article we investigate the problem of simultaneously steering an uncountable family of finite dimensional time varying linear systems with the same control signal this class of control problems motivates further research in the new subject of control theory called ensemble control a notion coming from the study of complex spin dynamics in nuclear magnetic resonance spectroscopy and imaging we derive the necessary and sufficient controllability conditions and an accompanying analytical optimal control law for an ensemble of finite dimensional time varying linear systems we show that ensemble controllability is in connection with singular values of the operator characterizing the system dynamics in addition the problem of optimal ensemble control of harmonic oscillators is studied to demonstrate the controllability results we show that the optimal solutions are pertinent to the study of time frequency limited signals and prolate spheroidal wave functions a systematic study of ensemble control systems has immediate applications to dynamical systems with parameter uncertainty as well as to wide ranging areas such as neuroscience and quantum control the work in ensemble control will foster further developments in mathematical control and systems theory

stabilization for an ensemble of half spin systems feedback stabilization of an ensemble of non interacting half spins described by the bloch equations is considered this system may be seen as an interesting example for infinite dimensional systems with continuous spectra we propose an explicit feedback law that stabilizes asymptotically the system around a uniform state of spin 1 2 or 1 2 the proof of the convergence is done locally around the equilibrium in the h 1 topology this local convergence is shown to be a weak asymptotic convergence for the h 1 topology and thus a strong convergence for the c 0 topology the proof relies on an adaptation of the lasalle invariance principle to infinite dimensional systems numerical simulations illustrate the efficiency of these feedback laws even for initial conditions far from the equilibrium

adaptive mimo antenna selection via discrete stochastic optimization recently it has been shown that it is possible to improve the performance of multiple input multiple output mimo systems by employing a larger number of antennas than actually used and selecting the optimal subset based on the channel state information existing antenna selection algorithms assume perfect channel knowledge and optimize criteria such as shannon capacity or various bounds on error rate this paper examines mimo antenna selection algorithms where the set of possible solutions is large and only a noisy estimate of the channel is available in the same spirit as traditional adaptive filtering algorithms we propose simulation based discrete stochastic optimization algorithms to adaptively select a better antenna subset using criteria such as maximum mutual information bounds on error rate etc these discrete stochastic approximation algorithms are ideally suited to minimize the error rate since computing a closed form expression for the error rate is intractable we also consider scenarios of time varying channels for which the antenna selection algorithms can track the time varying optimal antenna configuration we present several numerical examples to show the fast convergence of these algorithms under various performance criteria and also demonstrate their tracking capabilities

generalized convexity and inequalities involving special functions abstract in this paper the authors show a relation between the generalized convexity and super sub multiplicative property and discuss some generalized convexity and inequalities involving the gaussian hypergeometric function the generalized distortion function k a x and the generalized grotzsch function a r

concavity of the complete elliptic integrals of the second kind with respect to holder means abstract in this paper we establish a necessary and sufficient condition for the concavity of the complete elliptic integrals of the second kind with respect to holder means

the 4 way deterministic tiling problem is undecidable it is shown that the infinite tiling problem by wang tiles is undecidable even if the given tile set is deterministic by all four corners i e a tile is uniquely determined by the colors of any two adjacent edges the reduction is done from the turing machine halting problem and uses the aperiodic tile set of kari and papasoglu

sharp landen transformation inequalities for hypergeometric functions with applications abstract the authors present sharp landen transformation inequalities for the hypergeometric functions f 1 2 a b a b x and f 1 2 a b a b 1 2 x by showing the monotonicity properties of certain combinations defined in terms of one of these two hypergeometric functions and linear or rational functions thus giving complete solutions of the problem on extending the well known landen transformation identities for the complete elliptic integrals of the first kind to these two hypergeometric functions and substantially improving the related known results as applications of these results sharp landen transformation inequalities are obtained for the generalized grotzsch ring functions and the modular functions which appear in ramanujan s modular equations some other properties of hypergeometric functions and several properties of the ramanujan constant are obtained too

natural actor critic algorithms we present four new reinforcement learning algorithms based on actor critic natural gradient and function approximation ideas and we provide their convergence proofs actor critic reinforcement learning methods are online approximations to policy iteration in which the value function parameters are estimated using temporal difference learning and the policy parameters are updated by stochastic gradient descent methods based on policy gradients in this way are of special interest because of their compatibility with function approximation methods which are needed to handle large or infinite state spaces the use of temporal difference learning in this way is of special interest because in many applications it dramatically reduces the variance of the gradient estimates the use of the natural gradient is of interest because it can produce better conditioned parameterizations and has been shown to further reduce variance in some cases our results extend prior two timescale convergence results for actor critic methods by konda and tsitsiklis by using temporal difference learning in the actor and by incorporating natural gradients our results extend prior empirical studies of natural actor critic methods by peters vijayakumar and schaal by providing the first convergence proofs and the first fully incremental algorithms

ensemble observability of bloch equations with unknown population density abstract we introduce in the paper a novel observability problem for a large population in the limit a continuum ensemble of nonholonomic control systems with unknown population density we address the problem by focussing on a prototype namely the ensemble of bloch equations which is known for its use of describing the evolution of the bulk magnetization of a collective of non interacting nuclear spins in a static field modulated by a radio frequency rf field the dynamics of the equations are structurally identical but show variations in larmor dispersion and rf inhomogeneity we assume that the initial state of any individual system i e individual bloch equation is unknown and moreover the population density of these individual systems is also unknown furthermore we assume that at any time there is only one scalar measurement output at our disposal the measurement output integrates a certain observation function common to all individual systems over the continuum ensemble the observability problem we pose in the paper is thus the following whether one is able to use the common control input i e the rf field and the single measurement output to estimate both the initial states of the individual systems and the population density amongst other things we establish a sufficient condition for the ensemble system to be observable we show that if the common observation function is any harmonic homogeneous polynomial of positive degree then the ensemble system is observable the proof of the result utilizes tools from representation theory of lie algebras although the results established in the paper are for the specific ensemble of bloch equations the approach developed along the analysis can be generalized to investigate observability of other ensemble systems with single integrated measurement outputs

a simulated annealing technique for multi objective simulation optimization in this paper we present a simulated annealing algorithm for solving multi objective simulation optimization problems the algorithm is based on the idea of simulated annealing with constant temperature and uses a rule for accepting a candidate solution that depends on the individual estimated objective function values the algorithm is shown to converge almost surely to an optimal solution it is applied to a multi objective inventory problem the numerical results show that the algorithm converges rapidly

stochastic global optimization using tangent minorants for lipschitz functions abstract this paper deals with global stochastic optimization where the decision variable belongs to a compact subset x of r the objective function is the mathematical expectation of a partial bivariate lipschitz function f x depending on a decision variable x and a random variable whose probability distribution depends on x in the first part of the present paper we propose a branch and bound algorithm based on tangent minorants that provides a global minimum in the second part we consider the case where the function f is discontinuous we show the manner we correct that discontinuity without modifying the global minimum of e f x we also illustrate how to extend this framework to multidimensional stochastic optimization problems by using the alienor method then we validate the proposed method by applying it to some test functions and compare it to known algorithms

functional inequalities involving special functions ii this paper is motivated by an open problem of anderson vamanamurthy and vuorinen we prove a chain of inequalities for hypergeometric functions generalized and normalized bessel functions kummer functions and for some general power series some particular cases and refinements are given

balancing exploitation and exploration in discrete optimization via simulation through a gaussian process based search random search algorithms are often used to solve discrete optimization via simulation dovs problems the most critical component of a random search algorithm is the sampling distribution that is used to guide the allocation of the search effort a good sampling distribution can balance the trade off between the effort used in searching around the current best solution which is called exploitation and the effort used in searching largely unknown regions which is called exploration however most of the random search algorithms for dovs problems have difficulties in balancing this trade off in a seamless way in this paper we propose a new scheme that derives a sampling distribution from a fast fitted gaussian process based on previously evaluated solutions we show that the sampling distribution has the desired properties and can automatically balance the exploitation and exploration trade off furthermore we integrate this sampling distribution into a random research algorithm called a gaussian process based search gps and show that the gps algorithm has the desired global convergence as the simulation effort goes to infinity we illustrate the properties of the algorithm through a number of numerical experiments

concavity with respect to holder means involving the generalized grotzsch function the generalized grotzsch function has numerous applications in geometric function theory and analytic number theory and its properties have been investigated by many authors in this paper we study the concavity of the generalized grotzsch function with respect to holder means

a new approach to discrete stochastic optimization problems solving a discrete stochastic optimization problem involves two efforts one is to sample and search the design space and the other is to estimate the performance values of the sampled designs when the amount of computational resources is limited we need to know how to balance these two efforts in order to obtain the best result in this paper two performance measures which quantify the marginal contribution of the searching process as well as the performance evaluation process are proposed using these two measures we recommend a framework that can dynamically allocate the computational resources to the search process and the performance evaluation process two algorithms based on the proposed framework are tested on several scenarios and the results produced are very promising

structural decomposition for quantum two level systems an input output model of a two level quantum system in the heisenberg picture is of bilinear form with constant system matrices which allows the introduction of the concepts of controllability and observability in analogy with those of quantum linear systems by means of the notions of controllability and observability coordinate transformations which are rotation matrices can be constructed explicitly that transform an input output model to a new one the new input output model enables us to investigate many interesting properties of the two level quantum system such as steady state solutions to the lindblad master equation quantum decoherence free df subspaces quantum non demolition qnd variables and the realization of quantum back action evading bae measurements the physical system in wang and wiseman 2001 is re studied to illustrate the results presented in this paper

control in the spaces of ensembles of points we study the controlled dynamics of the ensembles of points of a riemannian manifold m parameterized ensemble of points of m is the image of a continuous map gamma theta to m where the

stochastic comparison algorithm for discrete optimization with estimation in this paper we study a class of discrete optimization problems where the objective function for a given configuration can be expressed as the expectation of a random variable in such problems only samples of the random variables are available for the optimization process an iterative algorithm called the stochastic comparison sc algorithm is developed the convergence of the sc algorithm is established based on an examination of the quasi stationary probabilities of a time inhomogeneous markov chain we also present some numerical experiments

sleeping experts and bandits approach to constrained markov decision processes this communique presents simple simulation based algorithms for obtaining an approximately optimal policy in a given finite set in large finite constrained markov decision processes the algorithms are adapted from playing strategies for sleeping experts and bandits problem and their computational complexities are independent of state and action space sizes if the given policy set is relatively small we establish convergence of their expected performances to the value of an optimal policy and convergence rates and also almost sure convergence to an optimal policy with an exponential rate for the algorithm adapted within the context of sleeping experts

generalized convexity and inequalities abstract let r 0 and let m be the family of all mean values of two numbers in r some examples are the arithmetic geometric and harmonic means given m 1 m 2 m we say that a function f r r is m 1 m 2 convex if f m 1 x y m 2 f x f y for all x y r the usual convexity is the special case when both mean values are arithmetic means we study the dependence of m 1 m 2 convexity on m 1 and m 2 and give sufficient conditions for m 1 m 2 convexity of functions defined by maclaurin series the criteria involve the maclaurin coefficients our results yield a class of new inequalities for several special functions such as the gaussian hypergeometric function and a generalized bessel function

notes on sliding mode control of two level quantum systems in dong d and petersen i r 2012 sliding mode control of two level quantum systems automatica 48 725 735 a sliding mode control approach has been proposed for two level quantum systems to deal with bounded uncertainties in the system hamiltonian this paper further extends these results in two directions one extension is to consider the effect of uncertainties during the process of driving the system s state back to the sliding mode domain from outside and we propose two approaches to accomplish this control task the other extension generalizes the previous design approach to consider uncertainties described as perturbations in the free hamiltonian

asymptotic ensemble stabilizability of the bloch equation in this paper we are concerned with the stabilizability to an equilibrium point of an ensemble of non interacting half spins we assume that the spins are immersed in a static magnetic field with dispersion in the larmor frequency and are controlled by a time varying transverse field our goal is to steer the whole ensemble to the uniform down position two cases are addressed for a finite ensemble of spins we provide a control function in feedback form that asymptotically stabilizes the ensemble in the down position generically with respect to the initial condition for an ensemble containing a countable number of spins we construct a sequence of control functions such that the sequence of the corresponding solutions pointwise converges asymptotically in time to the target state generically with respect to the initial conditions the control functions proposed are uniformly bounded and continuous

optimal ensemble control of stochastic time varying linear systems abstract we consider the optimal guidance of an ensemble of independent structurally identical finite dimensional stochastic linear systems with variation in system parameters between initial and target states of interest by applying a common control function without the use of feedback our exploration of such ensemble control systems is motivated by practical control design problems in which variation in system parameters and stochastic effects must be compensated for when state feedback is unavailable such as in pulse design for nuclear magnetic resonance spectroscopy and imaging in this paper we extend the notion of ensemble control to stochastic linear systems with additive noise and jumps which we model using white gaussian noise and poisson counters respectively and investigate the optimal steering problem in our main result we prove that the minimum norm solution to a fredholm integral equation of the first kind provides the optimal control that simultaneously minimizes the mean square error mse and the error in the mean of the terminal state the optimal controls are generated numerically for several example ensemble control problems and monte carlo simulations are used to illustrate their performance this work has immediate applications to the control of dynamical systems with parameter dispersion or uncertainty that are subject to additive noise which are of interest in quantum control neuroscience and sensorless robotic manipulation

on moduli of rings and quadrilaterals algorithms and experiments moduli of rings and quadrilaterals are frequently applied in geometric function theory see e g the handbook by kuhnau handbook of complex analysis geometric function theory vols 1 and 2 north holland amsterdam 2005 yet their exact values are known only in a few special cases previously the class of planar domains with polygonal boundary has been studied by many authors from the point of view of numerical computation we present here a new hp fem algorithm for the computation of moduli of rings and quadrilaterals and compare its accuracy and performance with previously known methods such as the schwarz christoffel toolbox of driscoll and trefethen we also demonstrate that the hp fem algorithm applies to the case of nonpolygonal boundary and report results with concrete error bounds

constraint ordinal optimization ordinal optimization has been developed as the solution to single performance measurement problem however in reality multiple performance measures are required to solve optimization problems constraint optimization was developed so that single measure could be optimized while the remaining performance measures were used as constraints in this paper we will build a new framework and quantification methods for dealing with the constraint optimization problem we call the framework constraint ordinal optimization

a decentralized approach to discrete optimization via simulation application to network flow we study a new class of decentralized algorithms for discrete optimization via simulation which is inspired by the fictitious play algorithm applied to games with identical interests in this approach each component of the solution vector of the optimization model is artificially assumed to have a corresponding player and the interaction of these players in simulation allows for exploration of the solution space and for some problems ultimately results in the identification of the optimal solution our algorithms also allow for correlation in players decision making a key feature when simulation output is shared by multiple decision makers we first establish convergence under finite sampling to equilibrium solutions in addition in the context of discrete network flow models we prove that if the underlying link cost functions are convex then our algorithms converge almost surely to an optimal solution

complexities for generalized models of self assembly in this paper we study the complexity of self assembly under models that are natural generalizations of the tile self assembly model in particular we extend rothemund and winfree s study of the tile complexity of tile self assembly proceedings of the 32nd annual acm symposium on theory of computing portland or 2000 pp 459 468 they provided a lower bound of omega frac log n log log n on the tile complexity of assembling an n times n square for almost all n adleman et al proceedings of the 33rd annual acm symposium on theory of computing heraklion greece 2001 pp 740 748 gave a construction which achieves this bound we consider whether the tile complexity for self assembly can be reduced through several natural generalizations of the model one of our results is a tile set of size o sqrt log n which assembles an n times n square in a model which allows flexible glue strength between nonequal glues this result is matched for almost all n by a lower bound dictated by kolmogorov complexity for three other generalizations we show that the omega frac log n log log n lower bound applies to n times n squares at the same time we demonstrate that there are some other shapes for which these generalizations allow reduced tile sets specifically for thin rectangles with length n and width k we provide a tighter lower bound of omega frac n 1 k k for the standard model yet we also give a construction which achieves o frac log n log log n complexity in a model in which the temperature of the tile system is adjusted during assembly we also investigate the problem of verifying whether a given tile system uniquely assembles into a given shape we show that this problem is np hard for three of the generalized models

evolutionary policy iteration under a sampling regime for stochastic combinatorial optimization this article modifies the evolutionary policy selection algorithm of chang et al which was designed for use in infinite horizon markov decision processes mdps with a large action space to a discrete stochastic optimization problem in an algorithm called evolutionary policy iteration monte carlo epi mc epi mc allows epi to be used in a stochastic combinatorial optimization setting with a finite action space and a noisy cost value function by introducing a sampling schedule convergence of epi mc to the optimal action is proven and experimental results are given

inequalities for the generalized elliptic integrals with respect to holder means abstract in this paper the authors prove some inequalities for the generalized elliptic integrals with respect to holder means

moments of ramanujanʼs generalized elliptic integrals and extensions of catalanʼs constant abstract we undertake a thorough investigation of the moments of ramanujan s alternative elliptic integrals and of related hypergeometric functions along the way we are able to give some surprising closed forms for catalan related constants and various new hypergeometric identities

analysis of quantum linear systems response to multi photon states the purpose of this paper is to present a mathematical framework for analyzing the response of quantum linear systems driven by multi photon states both the factorizable namely no correlation among the photons in the channel and unfactorizable multi photon states are treated pulse information of a multi photon input state is represented in terms of tensor and the response of quantum linear systems to multi photon input states is characterized by tensor operations analytic forms of output correlation functions and output states are derived the proposed framework is applicable no matter whether the underlying quantum dynamic system is passive or active examples from the physics literature are used to illustrate the results presented

characterization of the probabilistic traveling salesman problem we show that stochastic annealing can be successfully applied to gain new results on the probabilistic traveling salesman problem the probabilistic traveling salesman must decide on an a priori order in which to visit n cities randomly distributed over a unit square before learning that some cities can be omitted we find the optimized average length of the pruned tour follows e l over bar pruned rootnp 0 872 0 105p f np where p is the probability of a city needing to be visited and f np 1 as np infinity the average length of the a priori tour before omitting any cities is found to follow e l a priori rootn pbeta p where beta p 1 1 25 0 82 ln p is measured for 0 05less than or equal topless than or equal to0 6 scaling arguments and indirect measurements suggest that beta p tends towards a constant for p 0 03 our stochastic annealing algorithm is based on limited sampling of the pruned tour lengths exploiting the sampling error to provide the analog of thermal fluctuations in simulated thermal annealing the method has general application to the optimization of functions whose cost to evaluate rises with the precision required

the k hbox p problem on tensor products of lie groups and time optimal control of n quantum bits with a bounded field we use properties of sub riemannian k hbox p problems and the optimal synthesis on su 2 to solve the time optimal control of n noninteracting two level quantum systems qubits controlled in parallel with a control field bounded in l 2 norm every qubit is controlled independently but the total energy of the controls used is bounded for this system we describe the reachable set at every time moreover we compute the diameter that is the worst case minimum time the mathematical problem is motivated by the control of two level systems with different resonance frequencies where the control fields are obtained from a single signal having bounded energy

ensemble control of time invariant linear systems with linear parameter variation in this paper we study the control of a class of time invariant linear ensemble systems whose natural dynamics are linear in the system parameter this class of ensemble control systems arises from practical engineering and physical applications such as transport of quantum particles and control of uncertain harmonic systems we establish explicit algebraic criteria to examine controllability of such ensemble systems our derivation is based on the notion of polynomial approximation where the elements of the reachable set of the ensemble system are represented in polynomials of the system parameter and used to approximate the desired state of interest in addition we highlight the role of the spectra of the system matrices play in the determination of ensemble controllability finally illustrative examples and numerical simulations for optimal control of this class of linear ensemble systems are presented to demonstrate the theoretical results

dipole codes attractively encode glue functions dipole words are sequences of magnetic dipoles in which alike elements repel and opposite elements attract magnetic dipoles contrast with more general sets of bonding types called glues in which pairwise bonding strength is specified by a glue function we prove that every glue function g has a set of dipole words called a dipole code that attractively encodes g the pairwise attractions positive or non positive bond strength between the words are identical to those of g moreover we give such word sets of asymptotically optimal length similar results are obtained for a commonly used subclass of glue functions

uniform ensemble controllability for one parameter families of time invariant linear systems abstract in this paper we derive necessary as well as sufficient conditions for approximate controllability of parameter dependent linear systems in the supremum norm using tools from complex approximation theory we prove the existence of parameter independent open loop controls that steer the zero initial state of an ensemble of linear systems uniformly to a prescribed family of terminal states new necessary conditions for uniform ensemble controllability of single input systems are derived our results extend earlier ones of li for ensemble controllability of linear systems