on a definition of random sequences with respect to conditional probability we study a universal martin lof test with respect to a computable probability on a product space then we define random sequences with respect to a conditional probability by using a section of the set of random points of product space we show that 1 our definition is consistent with fubini s theorem and 2 it is equivalent to the relative notion of randomness under a condition this is an extension of lambalgen s theorem 1987 to a correlated probability

kolmogorov complexity and the recursion theorem several classes of diagonally nonrecursive dnr functions are characterized in terms of kolmogorov complexity in particular a set of natural numbers a can wtt compute a dnr function iff there is a nontrivial recursive lower bound on the kolmogorov complexity of the initial segments of a furthermore a can turing compute a dnr function iff there is a nontrivial a recursive lower bound on the kolmogorov complexity of the initial segments of a a is pa complete that is a can compute a 0 1 valued dnr function iff a can compute a function f such that f n is a string of length n and maximal c complexity among the strings of length n a t k iff a can compute a function f such that f n is a string of length n and maximal h complexity among the strings of length n further characterizations for these classes are given the existence of a dnr function in a turing degree is equivalent to the failure of the recursion theorem for this degree thus the provided results characterize those turing degrees in terms of kolmogorov complexity which no longer permit the usage of the recursion theorem

stochastic controllability and minimum energy control of systems with multiple delays in control in recent years various controllability problems for different types of linear dynamical systems have been considered in many publications and monographs

space time transmit code and receive filter design for colocated mimo radar this paper deals with the design of multiple input multiple output radar space time transmit code sttc and space time receive filter strf to enhance moving targets detection in the presence of signal dependent interferences an iterative procedure whose convergence is analytically proved is devised to maximize the signal to interference plus noise ratio sinr accounting for both a similarity constraint and a constant modulus requirement on the probing waveform each iteration of the algorithm involves the solution of hidden convex problems specifically both a convex problem whose solution is provided in closed form and a set of fractional programming problems that can be globally solved in polynomial time via the dinkelback s procedure are settled the computational complexity is linear with the number of iterations and polynomial with the sizes of the sttc and the strf in particular the proposed technique provides a monotonic sinr improvement without limitations on the size of the similarity constraint and ensures convergence to a stationary point filling these important gaps in the open literature besides the reported results highlight that the new devised procedure outperforms both in the optimized sinr value and the computational complexity than the available counterparts

process and truth table characterisations of randomness this paper uses quick process machines to provide characterisations of computable randomness schnorr randomness and weak randomness the quick process machine is a type of process machine first considered in work of levin and zvonkin a new technique for building process machines and quick process machines is presented this technique is similar to the kc theorem for prefix free machines using this technique a method of translating computable martingales to quick process machines is given this translation forms the basis for these new randomness characterisations quick process machines are also used to provide characterisations of computable randomness schnorr randomness and weak randomness in terms of truth table reducibility

approximate controllability of fractional nonlocal delay semilinear systems in hilbert spaces we study the existence and approximate controllability of a class of fractional nonlocal delay semilinear differential systems in a hilbert space the results are obtained by using semigroup theory fractional calculus and schauder s fixed point theorem multi delay controls and a fractional nonlocal condition are introduced furthermore we present an appropriate set of sufficient conditions for the considered fractional nonlocal multi delay control system to be approximately controllable an example to illustrate the abstract results is given

approximate controllability of semi linear neutral integro differential systems with finite delay abstract this paper considers the approximate controllability of semilinear neutral integrodifferential systems with finite delay in hilbert space since the considered equation admits nonlinear terms involving spatial derivatives the fraction power theory and norm are used to discuss the problem so that the established results can be applied to them the results are obtained by using the sadovskii fixed point theorem and the theory of analytic resolvent operator an example is presented to illustrate the applications of the obtained results

a computational model for metric spaces for every metric space x we define a continuous poset bx such that x is homeomorphic to the set of maximal elements of bx with the relative scott topology the poset bx is a dcpo iff x is complete and continuous iff x is separable the computational model bx is used to give domain theoretic proofs of banach s fixed point theorem and of two classical results of hutchinson on a complete metric space every hyperbolic iterated function system has a unique non empty compact attractor and every iterated function system with probabilities has a unique invariant measure with bounded support we also show that the probabilistic power domain of bx provides an continuous computational model for measure theory on a separable complete metric space x

computability on the probability measures on the borel sets of the unit interval while computability theory on many countable sets is well established and for computability on the real numbers several mutually non equivalent definitions are applied for most other uncountable sets in particular for measures no generally accepted computability concepts at all ha ve been available until now in this contribution we introduce computability on the set m of probability measures on the borel subsets of the unit interval 0 1 its main purpose is to demonstrate that this concept of computability is not merely an ad hoc definition but has very natural properties although the definitions and many results can of course be transferred to more general spaces of measures we restrict our attention to m in order to keep the technical details simple and concentrate on the central ideas in particular we show that simple obvious reqirements exclude a number of similar definitions that the definition leads to the expected computability results that there are other natural definitions inducing the same computability theory and that the theory is embedded smoothly into classical measure theory as background we consider tte type 2 theory of effectivity kw84 kw85 which provides a frame for very realistic computability definitions in this approach computability is defined on finite and infinite sequences of symbols explicitly by turing machines and on other sets by means of notations and representations canonical representations are derived from information structures wei97 we introduce a standard representation delta m subseteq sum omega to m via some natural information structure defined by a subbase the atomic properties of some topology on m and a standard notation of while several modifications of m suggesting themselves at first glance violate simple and obvious requirements m has several very natural properties and hence should induce an important computability theory many interesting functions on measures turn out to be computable in particular linear combination integration of continuous functions and any transformation defined by a computable iterated function system with probabilities some other natural representations of m are introduced among them a cauchy representation associated with the hutchinson metric and proved to be equivalent to m as a corollary the final topology of m is the well known weak topology on m

when does randomness come from randomness a result of shen says that if f 2 n 2 n is an almost everywhere computable measure preserving transformation and y 2 n is martin lof random then there is a martin lof random x 2 n such that f x y answering a question of bienvenu and porter we show that this property holds for computable randomness but not schnorr randomness these results combined with other known results imply that the set of martin lof randoms is the largest subset of 2 n satisfying this property and also satisfying randomness conservation if f 2 n 2 n is an almost everywhere computable measure preserving map and if x 2 n is random then f x is random

the descriptive complexity of brownian motion abstract a continuous function x on the unit interval is a generic brownian motion when every probabilistic event which holds almost surely with respect to the wiener measure is reflected in x provided that the event has a suitably effective description we show that a generic one dimensional brownian motion can be computed from an infinite binary string which is complex in the sense of kolmogorov chaitin conversely one can construct a kolmogorov chaitin random string from the values at the rational numbers of a generic brownian motion in this way we construct a recursive isomorphism between encoded versions of generic brownian motions and kolmogorov chaitin random reals

computability on subsets of euclidean space i closed and compact subsets in this paper we introduce and compare computability concepts on the set of closed subsets of euclidean space we use the language and framework of type 2 theory of effectivity tte which supplies a concise language for distinguishing a variety of effectivity properties and which admits highly effective versions of classical theorems in particular type 2 theory of effectivity allows to separate topological from computational aspects of effectivity we consider three different computability concepts on the set of closed subsets each of which is characterized by several representations which are proved to be equivalent the three induced types of computable closed sets have already been considered by many authors however under different and partly inconsistent names our characterizations show that they can be regarded as straightforward generalizations of the r e co r e and recursive subsets of natural numbers therefore we suggest to call them the recursively enumerable the co recursively enumerable and the recursive closed subsets of euclidean space open subsets obtain the dual names we extend the investigation by introducing several natural representations of the compact subsets of euclidean space and proving equivalences the paper extends and generalizes earlier definitions adds new ones and compares them in a single framework the resultant canonical computability concepts induce computability of objects as well as computability of operators on the space of closed and compact subsets

on controllability and near controllability of discrete time inhomogeneous bilinear systems without drift in this paper controllability and near controllability of discrete time inhomogeneous bilinear systems without drift are studied necessary and sufficient conditions for controllability and near controllability of the systems are respectively presented in particular algorithms for computing the control inputs to achieve the transition of a given pair of states for the controllable systems and nearly controllable systems are also provided examples are shown to demonstrate the conceptions and results of the paper

controllability of non linear impulsive stochastic systems in this article we consider the finite dimensional dynamical control systems described by non linear impulsive stochastic differential equations sufficient conditions for the complete and approximate controllability of non linear impulsive stochastic systems are formulated and proved under the natural assumption that the corresponding linear system is appropriately controllable

kolmogorov s contributions to information theory and algorithmic complexity briefly information theory says a random object x p x has complexity entropy h xp x log p x with the attendant interpretation that h bits are sufficient to describe x on the average algorithmic complexity says an object x has a complexity k x equal to the length of the shortest binary program that describes x it is a beautiful fact that these ideas are much the same in fact it is roughly true that ek x h moreover if we let pu x pr u prints x be the probability that a given computer u prints x when given a random program it can be shown that log 1 pu x k x for all x thus establishing a vital link between the universal probability measure pu and the universal complexity k more on this later the relationship of these ideas to probability theory was summed up in kolmogorov s 1983 paper which was based on his 1970 talk in nice perhaps only

a coordinate descent framework to design low psl isl sequences this paper is focused on the design of phase sequences with good aperiodic autocorrelation properties in terms of peak sidelobe level and integrated sidelobe level the problem is formulated as a biobjective pareto optimization forcing either a continuous or a discrete phase constraint at the design stage an iterative procedure based on the coordinate descent method is introduced to deal with the resulting optimization problems that are nonconvex and np hard in general each iteration of the devised method requires the solution of a nonconvex min max problem it is handled either through a novel bisection or an fft based method respectively for the continuous and the discrete phase constraint additionally a heuristic approach to initialize the procedures employing the l p norm minimization technique is proposed simulation results illustrate that the proposed methodologies can outperform some counterparts providing sequences with good autocorrelation features especially in the discrete phase binary case

controllability of the impulsive functional bbm equation with nonlinear term involving spatial derivative abstract for a broad class of control systems modeling real life problems delays and impulses are natural properties that do not modify their behavior so we conjecture that under certain conditions delays and abrupt changes as perturbations of a system do not modify properties such as controllability in this regard we prove the approximate controllability of the impulsive functional benjamin bona mahony bbm equation with nonlinear term involving spatial derivative this can be achieved using the technique avoiding fixed point theorems employed by a e bashirov et al in this case the delay helps us to prove the approximate controllability by pulling back the control solution to a fixed curve in a short time interval

designing unimodular codes via quadratic optimization the np hard problem of optimizing a quadratic form over the unimodular vector set arises in radar code design scenarios as well as other active sensing and communication applications to tackle this problem which we call unimodular quadratic program uqp several computational approaches are devised and studied power method like iterations are introduced for local optimization of uqp furthermore a monotonically error bound improving technique merit is proposed to obtain the global optimum or a local optimum of uqp with good sub optimality guarantees the provided sub optimality guarantees are case dependent and may outperform the 4 approximation guarantee of semi definite relaxation several numerical examples are presented to illustrate the performance of the proposed method the examples show that for several cases including rank deficient matrices the proposed methods can solve uqps efficiently in the sense of sub optimality guarantee and computational time

on concepts of controllability for deterministic and stochastic systems the new necessary and sufficient conditions which are formulated in terms of convergence of a certain sequence of operators involving the resolvent of the negative of the controllability operator are found for deterministic linear stationary control systems to be completely and approximately controllable respectively these conditions are applied to study the s controllability a property of attaining an arbitrarily small neighborhood of each point in the state space with a probability arbitrarily close to one and c controllability the s controllability fortified with some uniformity of stochastic systems it is shown that the s controllability the c controllability of a partially observable linear stationary control system with an additive gaussian white noise disturbance on all the intervals 0 t for t 0 is equivalent to the approximate complete controllability of its deterministic part on all the intervals 0 t for t 0

effective symbolic dynamics random points statistical behavior complexity and entropy we consider the dynamical behavior of martin lof random points in dynamical systems over metric spaces with a computable dynamics and a computable invariant measure we use computable partitions to define a sort of effective symbolic model for the dynamics through this construction we prove that such points have typical statistical behavior the behavior which is typical in the birkhoff ergodic theorem and are recurrent we introduce and compare some notions of complexity for orbits in dynamical systems and prove i that the complexity of the orbits of random points equals the kolmogorov sinai entropy of the system ii that the supremum of the complexity of orbits equals the topological entropy

non stochastic infinite and finite sequences abstract combining outcomes of coin tossing and transducer algorithms it is possible to generate with probability close to 1 very pathological sequences for which computable probabilistic forecasting is impossible these sequences are not random with respect to any reasonable probability distribution a natural consequence from the definition of such sequences is that each simple measure of the set of all such sequences is equal to 0 it was kolmogorov s and levin s idea to estimate the probability of generating of such sequences in the combinations of probabilistic and algorithmic processes 8 14 21 we collect several results in this direction for infinite sequences and asymptotic results for finite sequences including estimation of space and time of losing randomness for time bounded forecasting systems a correction to 22

computability of probability measures and martin lof randomness over metric spaces in this paper we investigate algorithmic randomness on more general spaces than the cantor space namely computable metric spaces to do this we first develop a unified framework allowing computations with probability measures we show that any computable metric space with a computable probability measure is isomorphic to the cantor space in a computable and measure theoretic sense we show that any computable metric space admits a universal uniform randomness test without further assumption

design of phase codes for radar performance optimization with a similarity constraint this paper deals with the design of coded waveforms which optimize radar performances in the presence of colored gaussian disturbance we focus on the class of phase coded pulse trains and determine the radar code which approximately maximizes the detection performance under a similarity constraint with a prefixed radar code this is tantamount to forcing a similarity between the ambiguity functions of the devised waveform and of the pulse train encoded with the prefixed sequence we consider the cases of both continuous and finite phase alphabet and formulate the code design in terms of a nonconvex np hard quadratic optimization problem in order to approximate the optimal solutions we propose techniques with polynomial computational complexity based on the method of semidefinite program sdp relaxation and randomization moreover we also derive approximation bounds yielding a ldquomeasure of goodnessrdquo of the devised algorithms at the analysis stage we assess the performance of the new encoding techniques both in terms of detection performance and ambiguity function under different choices for the similarity parameter we also show that the new algorithms achieve an accurate approximation of the optimal solution with a modest number of randomizations

mimo radar waveform design with constant modulus and similarity constraints we consider the problem of waveform design for multiple input multiple output mimo radar in the presence of signal dependent interference embedded in white gaussian disturbance we present two sequential optimization procedures to maximize the signal to interference plus noise ratio sinr accounting for a constant modulus constraint as well as a similarity constraint involving a known radar waveform with some desired properties e g in terms of pulse compression and ambiguity the presented sequential optimization algorithms based on a relaxation method yield solutions with good accuracy their computational complexity is linear in the number of iterations and trials in the randomized procedure and polynomial in the receive filter length finally we evaluate the proposed techniques by considering their sinr performance beam pattern as well as pulse compression property via numerical simulations

randomness for non computable measures different approaches have been taken to defining randomness for non computable probability measures we will explain the approach of reimann and slaman along with the uniform test approach first introduced by levin and also used by gacs hoyrup and rojas we will show that these approaches are fundamentally equivalent having clarified what it means to be random for a non computable probability measure we turn our attention to levin s neutral measures for which all sequences are random we show that every pa degree computes a neutral measure we also show that a neutral measure has no least turing degree representation and explain why the framework of the continuous degrees a substructure of the enumeration degrees studied by miller can be used to determine the computational complexity of neutral measures this allows us to show that the turing ideals below neutral measures are exactly the scott ideals since x 2 is an atom of a neutral measure if and only if it is computable from every representation of we have a complete understanding of the possible sets of atoms of a neutral measure one simple consequence is that every neutral measure has a martin lof random atom 1 defining randomness let x be an element of cantor space and a borel probability measure on cantor space what should it mean for x to be random with respect to in the case that is the lebesgue measure then the theory of randomness is well developed for recent treatises on the subject the reader is referred to downey and hirschfeldt and nies 2 13 in fact if is a computable measure then early work of levin showed that randomness can be seen as essentially a variant on randomness for lebesgue measure 10 this leaves the question of how to define randomness if is non computable we will show that the two approaches that have previously been used to define randomness for non computable are equivalent later in theorem 4 12 we will provide another characterization of randomness using the enumeration degrees last compilation september 8 2011 last time the following date was changed november 22 2010 2010 mathematics subject classification primary 03d32 secondary 68q30 03d30 the second author was supported by the national science foundation under grants dms 0945187 and dms 0946325 the latter being part of a focused research group in algorithmic randomness

successive qcqp refinement for mimo radar waveform design under practical constraints the authors address the problem of designing a waveform for multiple input multiple output mimo radar under the important practical constraints of constant modulus and waveform similarity incorporating these constraints in an analytically tractable manner is a longstanding open challenge this is due to the fact that the optimization problem that results from signal to interference plus noise ratio sinr maximization subject to these constraints is a hard non convex problem the authors develop a new analytical approach that involves solving a sequence of convex quadratically constrained quadratic programing qcqp problems which they prove converges to a sub optimal solution because an improvement in sinr results via solving each problem in the sequence they call the method successive qcqp refinement sqr furthermore the proposed sqr method can be easily extended to incorporate emerging requirements of spectral coexistence as shown briefly in this paper the authors evaluate sqr against other candidate techniques with respect to sinr performance beam pattern and pulse compression properties in a variety of scenarios results show that sqr outperforms state of the art methods that also employ constant modulus and or similarity constraints while being computationally less burdensome

partial approximate controllability of nonlocal fractional evolution equations via approximating method abstract in this paper we study partial approximate controllability of semilinear nonlocal fractional evolution equations in hilbert spaces by using fractional calculus variational approach and approximating technique we give the approximate problem of the control system and get the compactness of approximate solution set then new sufficient conditions for the partial approximate controllability of the control system are obtained when the compactness conditions or lipschitz conditions for the nonlocal function are not required finally we apply our abstract results to the partial approximate controllability of the semilinear heat equation and delay equation

randomness and differentiability we characterize some major algorithmic randomness notions via differentiability of effective functions 1 we show that a real number z 0 1 is computably random if and only if every nondecreasing computable function 0 1 r is differentiable at z 2 we prove that a real number z 0 1 is weakly 2 random if and only if every almost everywhere differentiable computable function 0 1 r is differentiable at z 3 recasting results of the constructivist demuth from 1975 in classical language we show that a real z is martin lof random if and only if every computable function of bounded variation is differentiable at z and similarly for absolutely continuous functions we also use the analytic methods to show that computable randomness of a real is base invariant and to derive preservation results for randomness notions

strong reductions in effective randomness we study generalizations of demuth s theorem which states that the image of a martin lof random real under a tt reduction is either computable or turing equivalent to a martin lof random real we show that demuth s theorem holds for schnorr randomness and computable randomness answering a question of franklin but that it cannot be strengthened by replacing the turing equivalence in the statement of the theorem with wtt equivalence we also provide some additional results about the turing and tt degrees of reals that are random with respect to some computable measure

tensor decompositions and applications this survey provides an overview of higher order tensor decompositions their applications and available software a tensor is a multidimensional or n way array decompositions of higher order tensors i e n way arrays with n geq 3 have applications in psycho metrics chemometrics signal processing numerical linear algebra computer vision numerical analysis data mining neuroscience graph analysis and elsewhere two particular tensor decompositions can be considered to be higher order extensions of the matrix singular value decomposition candecomp parafac cp decomposes a tensor as a sum of rank one tensors and the tucker decomposition is a higher order form of principal component analysis there are many other tensor decompositions including indscal parafac2 candelinc dedicom and paratuck2 as well as nonnegative variants of all of the above the n way toolbox tensor toolbox and multilinear engine are examples of software packages for working with tensors

a constructive version of birkhoff s ergodic theorem for martin lof random points we prove the effective version of email protected s ergodic theorem for martin lof random points and effectively open sets improving the results previously obtained in this direction in particular those of vyugin nandakumar and hoyrup rojas the proof consists of two steps first we prove a generalization of email protected s theorem which is a particular case of effective ergodic theorem a trajectory of a computable ergodic mapping that starts from a random point cannot remain inside an effectively open set of measure less than 1 second we show that the full statement of the effective ergodic theorem can be reduced to this special case both steps use the statement of classical ergodic theorem but not its usual classical proof therefore we get a new simple proof of the effective ergodic theorem with weaker assumptions than before this result was recently obtained independently by franklin greenberg miller and ng

approximate controllability of semilinear functional equations in hilbert spaces in this paper approximate and complete controllability for semilinear functional differential systems is studied in hilbert spaces sufficient conditions are established for each of these types of controllability the results address the limitation that linear systems in infinite dimensional spaces with compact semigroup cannot be completely controllable the conditions are obtained by using the schauder fixed point theorem when the semigroup is compact and the banach fixed point theorem when the semigroup is not compact

schnorr randomness for noncomputable measures abstract this paper explores a novel definition of schnorr randomness for noncomputable measures we say x is uniformly schnorr random if t x for all lower semicomputable functions t x such that t x d x is computable we prove a number of theorems demonstrating that this is the correct definition which enjoys many of the same properties as martin lof randomness for noncomputable measures nonetheless a number of our proofs significantly differ from the martin lof case requiring new ideas from computable analysis

lowness for the class of schnorr random reals we answer a question of ambos spies and kucera in the affirmative they asked whether when a real is low for schnorr randomness it is already low for schnorr tests

partial controllability concepts the main results in theory of controllability are formulated for deterministic or stochastic control systems given in a standard form i e given as a first order differential equation driven by an infinitesimal generator of strongly continuous semigroup in an abstract hilbert or banach space and disturbed by a deterministic function or by a white noise process at the same time some deterministic or stochastic linear systems can be written in a standard form if the state space is enlarged respectively the ordinary controllability conditions for them are too strong since they assume extended state space it is reasonable to introduce partial controllability concepts which assume original state space in this paper we study necessary and sufficient conditions of partial controllability for deterministic and stochastic linear control systems given in a standard form and their implications to particular cases

random elements in effective topological spaces with measure following a suggestion of zvonkin and levin we generalize martin lof s definition of infinite random sequences over a finite alphabet via randomness tests to effective topological spaces with a measure we show that under weak computability conditions there is a universal randomness test we prove a theorem on randomness preserving functions which corrects and extends a result by schnorr and apply it to a number of examples in particular we show that a real number is random if and only if it has a random b ary representation for any b 2 we show that many computable continuously differentiable real functions preserve randomness especially all computable analytic functions which are not constant on any open subset of their domain preserve randomness finally we introduce a new randomness concept for subsets of natural numbers which we characterize in terms of random sequences surprisingly it turns out that there are infinite co r e random sets

on the empirical validity of the bayesian method the ideas in kolmogorov s programme for algorithmc substantiation of applications of probability make it possible to define a measure of disagreement between the probability distribution representing the attitude of a coherent individual towards a random experiment and the outcome of the experiment when there is agreement we say that the probability distribution is empirically valid we prove quantitatively that formulae of bayesian statistics transform empirically valid probability distributions into other empirically valid distributions

maximum block improvement and polynomial optimization in this paper we propose an efficient method for solving the spherically constrained homogeneous polynomial optimization problem the new approach has the following three main ingredients first we establish a block coordinate descent type search method for nonlinear optimization with the novelty being that we accept only a block update that achieves the maximum improvement hence the name of our new search method maximum block improvement mbi convergence of the sequence produced by the mbi method to a stationary point is proved second we establish that maximizing a homogeneous polynomial over a sphere is equivalent to its tensor relaxation problem thus we can maximize a homogeneous polynomial function over a sphere by its tensor relaxation via the mbi approach third we propose a scheme to reach a kkt point of the polynomial optimization provided that a stationary solution for the relaxed tensor problem is available numerical experiments have shown that our new method works very efficiently for a majority of the test instances that we have experimented with the method finds the global optimal solution at a low computational cost

a constructive borel cantelli lemma constructing orbits with required statistical properties in the general context of computable metric spaces and computable measures we prove a kind of constructive borel cantelli lemma given a sequence constructive in some way of sets a i with effectively summable measures there are computable points which are not contained in infinitely many a i as a consequence of this we obtain the existence of computable points which follow the typical statistical behavior of a dynamical system they satisfy the birkhoff theorem for a large class of systems having computable invariant measure and a certain logarithmic speed of convergence of birkhoff averages over lipschitz observables this is applied to uniformly hyperbolic systems piecewise expanding maps systems on the interval with an indifferent fixed point and it directly implies the existence of computable numbers which are normal with respect to any base

stochastic maximum principle in the pontryagin s form for wide band noise driven systems the control problems for wide band noise driven systems have the following specific feature a wide band noise is given by its autocovariance function at the same time different wide band noises may have the same autocovariance function therefore it is important to obtain the results which are invariant under the change of wide band noises corresponding to the same autocovariance function in this paper one such result is obtained a stochastic maximum principle in the pontryagin s form is proved for a control system driven by a state dependent wide band noise the result is demonstrated on an example of controlled stochastic differential equation of risky asset price

on binary probing signals and instrumental variables receivers for radar the so called merit factor approach mfa to radar binary sequence design has led to several theoretical contributions in fairly diverse research areas including information theory computer science combinatorial optimization and analytical number theory however the mfa which basically aims at minimizing the clutter effect on radar performance implicitly assumes the use of a least squares ls receiver that is optimal only when there is no clutter this problem can be eliminated by using a more general optimal instrumental variables iv receiver in lieu of the ls receiver the iv receiver can reject clutter more efficiently than the ls receiver additionally the binary sequence design problem associated with the iv approach has an interesting form

computational complexity of real functions in this chapter we introduce the formal definitions of computational complexity of real functions we first review the definitions of computable real numbers and computable real functions and their basic properties then the oracle turing machine is introduced as the formal model for computing real functions this model allows us to define the complexity measures for computing real functions in a natural way the class of polynomial time computable real functions is then defined and several characterizations of this class will be given other complexity classes of real numbers and real functions such as np real functions and log space computable real functions will be defined in later chapters

probabilistic computability and choice we study the computational power of randomized computations on infinite objects such as real numbers in particular we introduce the concept of a las vegas computable multi valued function which is a function that can be computed on a probabilistic turing machine that receives a random binary sequence as auxiliary input the machine can take advantage of this random sequence but it always has to produce a correct result or to stop the computation after finite time if the random advice is not successful with positive probability the random advice has to be successful we characterize the class of las vegas computable functions in the weihrauch lattice with the help of probabilistic choice principles and weak weak k nig s lemma among other things we prove an independent choice theorem that implies that las vegas computable functions are closed under composition in a case study we show that nash equilibria are las vegas computable while zeros of continuous functions with sign changes cannot be computed on las vegas machines however we show that the latter problem admits randomized algorithms with weaker failure recognition mechanisms the last mentioned results can be interpreted such that the intermediate value theorem is reducible to the jump of weak weak k nig s lemma but not to weak weak k nig s lemma itself these examples also demonstrate that las vegas computable functions form a proper superclass of the class of computable functions and a proper subclass of the class of non deterministically computable functions we also study the impact of specific lower bounds on the success probabilities which leads to a strict hierarchy of classes in particular the classical technique of probability amplification fails for computations on infinite objects we also investigate the dependency on the underlying probability space besides cantor space we study the natural numbers the euclidean space and baire space

relative entropy based waveform design for mimo radar detection in the presence of clutter and interference this paper considers the waveform design problem for multiple input multiple output mimo radars in order to improve the detection performance of the systems we assume that target echoes are embedded in signal dependent clutter as well as colored interference considering the optimal neyman pearson detector obtaining waveforms which maximize the detection probability for a fixed value of the probability of false alarm is intractable therefore we employ relative entropy associated with the detection problem as the figure of merit for the waveform design we devise an iterative method based on minorization maximization mm technique to tackle the nonconvex design problem this method also includes a novel trick for replacing a nonconvex constraint set associated with an equivalent form of the design problem with a convex one iteratively the proposed method increases the design metric monotonically and is guaranteed to converge we extend the devised method for using in design with peak to average power ratio par and similarity constraints the method can be applied to both statistical and colocated mimo radars several numerical examples are included to demonstrate the effectiveness of the proposed method

new algorithms for designing unimodular sequences with good correlation properties unimodular i e constant modulus sequences with good autocorrelation properties are useful in several areas including communications and radar the integrated sidelobe level isl of the correlation function is often used to express the goodness of the correlation properties of a given sequence in this paper we present several cyclic algorithms for the local minimization of isl related metrics these cyclic algorithms can be initialized with a good existing sequence such as a golomb sequence a frank sequence or even a pseudo random sequence to illustrate the performance of the proposed algorithms we present a number of examples including the design of sequences that have virtually zero autocorrelation sidelobes in a specified lag interval and of long sequences that could hardly be handled by means of other algorithms previously suggested in the literature

uniform test of algorithmic randomness over a general space the algorithmic theory of randomness is well developed when the underlying space is the set of finite or infinite sequences and the underlying probability distribution is the uniform distribution or a computable distribution these restrictions seem artificial some progress has been made to extend the theory to arbitrary bernoulli distributions by martin lof and to arbitrary distributions by levin we recall the main ideas and problems of levin s theory and report further progress in the same framework the issues are the following allow non compact spaces like the space of continuous functions underlying the brownian motion the uniform test deficiency of randomness dp x depending both on the outcome x and the measure p should be defined in a general and natural way see which of the old results survive existence of universal tests conservation of randomness expression of tests in terms of description complexity existence of a universal measure expression of mutual information as deficiency of independence the negative of the new randomness test is shown to be a generalization of complexity in continuous spaces we show that the addition theorem survives the paper s main contribution is introducing an appropriate framework for studying these questions and related ones like statistics for a general family of distributions

algorithmic randomness and monotone complexity on product space we study algorithmic randomness and monotone complexity on product of the set of infinite binary sequences we explore the following problems monotone complexity on product space lambalgen s theorem for correlated probability classification of random sets by likelihood ratio tests decomposition of complexity and independence and bayesian statistics for individual random sequences formerly lambalgen s theorem for correlated probability is shown under a uniform computability assumption in h takahashi inform compt 2008 in this paper we show the theorem without the assumption

dynamics of a generic brownian motion recursive aspects we study the local fluctuations of brownian motions which are represented by infinite binary strings which are random in the sense of kolmogorov chaitin we show how the dynamical properties of such a brownian motion at a point depend on its recursive properties

lowness properties and randomness abstract the set a is low for martin lof randomness if each random set is already random relative to a a is k trivial if the prefix complexity k of each initial segment of a is minimal namely n k a n k n o 1 we show that these classes coincide this answers a question of ambos spies and kucera in p cholak s lempp m lerman r shore eds computability theory and its applications current trends and open problems american mathematical society providence ri 2000 each low for martin lof random set is 2 0 our class induces a natural intermediate 3 0 ideal in the r e turing degrees which generates the whole class under downward closure answering a further question in p cholak s lempp m lerman r shore eds computability theory and its applications current trends and open problems american mathematical society providence ri 2000 we prove that each low for computably random set is computable

on positive realness negative imaginariness and h control of state space symmetric systems this paper studies positive realness and negative imaginariness of state space symmetric systems first a necessary and sufficient condition is provided for a state space symmetric system to be respectively positive real and negative imaginary then an explicit formulation for the infimum of h norm under a mixed h and negative imaginary performance and an explicit solution for the associated optimal static output feedback control gain are developed based on symmetric negative imaginary theorem finally three examples are provided to illustrate the main results of the paper

complex quadratic optimization and semidefinite programming in this paper we study the approximation algorithms for a class of discrete quadratic optimization problems in the hermitian complex form a special case of the problem that we study corresponds to the max 3 cut model used in a recent paper of goemans and williamson j comput system sci 68 2004 pp 442 470 we first develop a closed form formula to compute the probability of a complex valued normally distributed bivariate random vector to be in a given angular region this formula allows us to compute the expected value of a randomized with a specific rounding rule solution based on the optimal solution of the complex semidefinite programming relaxation problem in particular we present an m 2 1 cos frac 2 pi m 8 pi approximation algorithm and then study the limit of that model in which the problem remains np hard we show that if the objective is to maximize a positive semidefinite hermitian form then the randomization rounding procedure guarantees a worst case performance ratio of pi 4 approx 0 7854 which is better than the ratio of 2 pi approx 0 6366 for its counterpart in the real case due to nesterov furthermore if the objective matrix is real valued positive semidefinite with nonpositive off diagonal elements then the performance ratio improves to 0 9349

effective dimension of points visited by brownian motion we consider the individual points on a martin lof random path of brownian motion we show that 1 khintchine s law of the iterated logarithm holds at almost all points and 2 there exist points besides the trivial example of the origin having effective dimension 1 the proof of 1 shows that for almost all times t the path f is martin lof random relative to t and so the effective dimension of t f t is 2

effective properties of sets and functions in metric spaces with computability structure we consider an abstract metric space with a computability structure and an effective separating set in this article we also introduce an effectively compact space the computability of real valued functions on such a space is defined it is shown that some of typical propositions in a metric space namely baire category theorem tietze s extension theorem and decomposition of unity can be effectivized it is also proved that computable functions are dense in continuous functions

approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces various sufficient conditions for approximate controllability of linear evolution systems in abstract spaces have been obtained but approximate controllability of semilinear control systems usually requires some complicated and limited assumptions in this paper we show the approximate controllability of the abstract semilinear deterministic and stochastic control systems under the natural assumption that the associated linear control system is approximately controllable the results are obtained using new properties of symmetric operators which are proved in this paper compact semigroups the schauder fixed point theorem and or the contraction mapping principle

on partial s controllability of semilinear partially observable systems in this paper the partial s controllability of semilinear partially observable control systems is investigated the essence of study is the presence of partial observations we define a weakened partial s controllability concept for partially observable systems a sufficient condition for this controllability concept is proved the method of proof differs from the traditional proofs by fixed point theorems the obtained result is demonstrated with examples

cognitive radar based sequence design via sinr maximization the ambiguity function plays an important role in radar systems in fact many radar design problems can be interpreted from the perspective of persuing desired ambiguity functions to adapt to various application scenes in this paper we consider designing a radar sequence subject to a peak to average power ratio par constraint to maximize the signal to interference plus noise ratio which can also be interpreted as designing a sequence with a desired ambiguity function from an optimization point of view this is equivalent to optimizing a complex quartic function with the par constraint an efficient algorithm based on the general majorization minimization mm method is developed to solve this problem with guaranteed convergence to a stationary point under some mild conditions in addition the unit modulus constraint as a special case is considered and another algorithm is proposed which is the combination of the general mm and the coordinate descent method numerical experiments show that the proposed algorithms can shape a desired ambiguity function based on the prior information and the performance is much better compared with the existing methods

partial controllability of stochastic linear systems the controllability concepts for linear stochastic differential equations driven by different kinds of noise processes can be reduced to the partial controllability concepts for the same systems driven by correlated white noises based on this fact in this article we study the conditions of exact and approximate controllability for linear stochastic control systems under various kinds of noise processes including correlated white noises as well as coloured wide band and shifted white noises it is proved that such systems are never exactly controllable while their approximate controllability is equivalent to the approximate controllability of the associated linear deterministic systems at all past time moments

on controllability conception for stochastic systems the controllability notions for partially observed stochastic systems are defined their relation with complete and approximate controllabilities is shown in particular it is proven that the approximate controllability condition is necessary and the complete controllability condition is sufficient for the partially observed linear gaussian control system to attain the arbitrarily small neighborhood of each point in the state space with probability arbitrarily closely to one

ambiguity function shaping for cognitive radar via complex quartic optimization in this paper we propose a cognitive approach to design phase only modulated waveforms sharing a desired range doppler response the idea is to minimize the average value of the ambiguity function of the transmitted signal over some range doppler bins which are identified exploiting a plurality of knowledge sources from a technical point of view this is tantamount to optimizing a real and homogeneous complex quartic order polynomial with a constant modulus constraint on each optimization variable after proving some interesting properties of the considered problem we devise a polynomial time waveform optimization procedure based on the maximum block improvement mbi method and the theory of conjugate partial symmetric conjugate super symmetric fourth order tensors at the analysis stage we assess the performance of the proposed technique showing its capability to properly shape the range doppler response of the transmitted waveform

quadratic optimization with similarity constraint for unimodular sequence synthesis this paper considers unimodular sequence synthesis under similarity constraint for both the continuous and discrete phase cases a computationally efficient iterative algorithm for the continuous phase case ia cpc is proposed to sequentially optimize the quadratic objective function the quadratic optimization problem is turned into multiple one dimensional optimization problems with closed form solutions for the discrete phase case we present an iterative block optimization algorithm specifically we partition the design variables into k blocks and then we sequentially optimize each block via exhaustive search while fixing the remaining k 1 blocks finally we evaluate the computational costs and performance gains of the proposed algorithms in comparison with power method like and semidefinite relaxation related techniques

computability on computable metric spaces in previous papers e g weihrauch 1987 type 2 theory of effectivity tte has been shown to be a very general and powerful computer oriented theory for studying effectivity in mathematics in this contribution computability on certain computable separable metric spaces is studied in detail by applying the framework of tte computationally admissible representations of metric spaces are introduced after showing that four different effective representations are computationally equivalent for extending computability to the set of continuous functions several effective namings which are related to definitions of continuity are introduced and compared the definitions are used to prove effective g extension theorems for continuous functions and an effective g characterization of the domains of strongly continous functions which generalize the known properties of real functions kreitz 1984

comparing notions of randomness it is an open problem in the area of effective algorithmic randomness whether kolmogorov loveland randomness coincides with martin lof randomness joe miller and andre nies suggested some variations of kolmogorov loveland randomness to approach this problem and to provide a partial solution we show that their proposed notion of injective randomness is still weaker than martin lof randomness since in this proof some of the ideas we use are clearer we also show the weaker theorem that permutation randomness is weaker than martin lof randomness

computability on subsets of metric spaces the notions recursively enumerable and recursive are the basic notions of effectivity in classical recursion theory in computable analysis these notions are generalized to closed subsets of euclidean space using their metric distance functions we study a further generalization of these concepts to subsets of computable metric spaces it appears that different characterizations which coincide in case of euclidean space lead to different notions in the general case however under certain additional conditions such as completeness and effective local compactness the situation is similar to the euclidean case we present all results in the framework of type 2 theory of effectivity which allows to express effectivity properties in a very uniform way instead of comparing properties of single subsets we compare corresponding representations of the hyperspace of closed subsets such representations do not only induce a concept of computability for single subsets but they even yield a concept of computability for operations on hyperspaces such as union intersection etc we complete our investigation by studying the special situation of compact subsets

every sequence is reducible to a random one every infinite sequence is turing reducible to an infinite sequence which is random in the sense of martin lof

computational design of sequences with good correlation properties in this paper we introduce a computational framework based on an iterative twisted approximation itrox and a set of associated algorithms for various sequence design problems the proposed computational framework can be used to obtain sequences or complementary sets of sequences possessing good periodic or aperiodic correlation properties and in an extended form to construct zero or low correlation zone sequences furthermore as constrained e g finite alphabets are of interest in many applications we introduce a modified version of our general framework that can be useful in these cases several applications of itrox are studied and numerical examples focusing on the construction of real valued and binary sequences are provided to illustrate the performance of itrox for each application

mathematical metaphysics of randomness abstract we consider various mathematical refinements of the notion of randomness of an infinite sequence and relations between them on the base of the game approach a new possible refinement of the notion of randomness is introduced unpredictability the case of finite sequences is also considered

effective metric spaces and representations of the reals based on standard notions of classical recursion theory a natural model of approximate computability for partial functions between effective metric spaces is presented it generalizes the ko friedman approach to computability of real functions by means of oracle turing machines follows the main ideas of weihrauch s type 2 theory of effectivity but it avoids the explicit use of representations the topological arithmetical hierarchy is introduced and shown to be strict if the underlying space contains an effectively discrete sequence the domains of computable functions are exactly the 2 sets of this hierarchy if the space admits a finitary stratification finally this framework is used to investigate and characterize the standard representations of the real numbers they are just those functions from the name space onto the reals which have both computable extensions and inversions that are computable as relations

on tables of random numbers the set theoretic axioms of the calculus of probability in formulating which i had the opportunity of playing some part kolmogorov 1950 had solved the majority of formal difficulties in the construction of a mathematical apparatus which is useful for a very large number of applications of probabilistic methods so successfully that the problem of finding the basis of real applications of the results of the mathematical theory of probability became rather secondary to many investigators i have already expressed the view see kolmogorov 1950 chapter i that the basis for the applicability of the results of the mathematical theory of probabi lity to real random phenomena must depend on some form of the frequency concept of probability the unavoidable nature of which has been established by von mises in a spirited manner however for a long time i had the following views 1 the frequency concept based on the notion of limiting frequency as the number of trials increases to infinity does not contribute anything to substantiate the applicability of the results of probability theory to real practical problems where we have always to deal with a finite number of trials 2 the frequency concept applied to a large but finite number of trials does not admit a rigorous formal exposition within the framework of pure mathematics accordingly i have sometimes put forward the frequency concept which involves the conscious use of certain not rigorously formal ideas about practical reliability approximate stability of the frequency in a long series of trials without the precise definition of the series which are sufficiently large etc see foundation of the theory of probability chapter i and for more details great soviet encyclo paedia section on probability and mathematika iou metod i znachenye chapter on probability theory i still maintain the first of the two theses mentioned above as regards the second however i have come to realise that the concept of random distribution of a property in a large finite population can have a strict formal mathematical exposition in fact we can show that in sufficiently large populations the distribution of the pro perty may be such that the frequency of its occurrence will be almost the same for all sufficiently large sub populations when the law of choosing these is sufficiently simple such a conception in its full development requires the introduction of a measure of the complexity of the algorithm i propose to discuss this question in another article in the present article however i shall use the fact that there cannot be a very large number of simple algorithms 369 3