recurrence for discrete time unitary evolutions we consider quantum dynamical systems specified by a unitary operator u and an initial state vector phi in each step the unitary is followed by a projective measurement checking whether the system has returned to the initial state we call the system recurrent if this eventually happens with probability one we show that recurrence is equivalent to the absence of an absolutely continuous part from the spectral measure of u with respect to phi we also show that in the recurrent case the expected first return time is an integer or infinite for which we give a topological interpretation a key role in our theory is played by the first arrival amplitudes which turn out to be the complex conjugated taylor coefficients of the schur function of the spectral measure on the one hand this provides a direct dynamical interpretation of these coefficients on the other hand it links our definition of first return times to a large body of mathematical literature

open quantum random walks ergodicity hitting times gambler s ruin and potential theory in this work we study certain aspects of open quantum random walks oqrws a class of quantum channels described by attal et al j stat phys 147 832 852 2012 as a first objective we consider processes which are nonhomogeneous in time i e at each time step a possibly distinct evolution kernel inspired by a spectral technique described by saloff coste and zuniga stoch proc appl 117 961 979 2007 we define a notion of ergodicity for finite nonhomogeneous quantum markov chains and describe a criterion for ergodicity of such objects in terms of singular values as a second objective and based on a quantum trajectory approach we study a notion of hitting time for oqrws and we see that many constructions are variations of well known classical probability results with the density matrix degree of freedom on each site giving rise to systems which are seen to be nonclassical in this way we are able to examine open quantum versions of the gambler s ruin birth and death chain and a basic theorem on potential theory

existence and nonexistence results of an optimal control problem by using relaxed control relaxed controls have proved to be very useful in studying the existence of optimal controls in optimal control theory many positive results have been obtained in the literature however negative results have also made their rare appearances the optimal control problem considered in this paper looks quite simple yet by treating such a problem we can get interesting results substantiating our idea as to whether an optimal control exists or not in our opinion the method used in the paper can be applied to more generalized cases

equilibrium conditions for young measures we explore variations of young measures in order to establish equilibrium conditions for minimizers of generalized variational principles where young measures enter the minimization problem the slicing measure decomposition is extremely useful in rendering equilibrium conditions tractable they usually yield information on the support of generalized minimizers it also enables us to describe new equivalent variational principles where these equilibrium constraints are taken into account the only case where computations can be made explicit is the scalar one dimensional case

open quantum random walks on the half line the karlin mcgregor formula path counting and foster s theorem in this work we consider open quantum random walks on the non negative integers by considering orthogonal matrix polynomials we are able to describe transition probability expressions for classes of walks via a matrix version of the karlin mcgregor formula we focus on absorbing boundary conditions and for simpler classes of examples we consider path counting and the corresponding combinatorial tools a non commutative version of the gambler s ruin is studied by obtaining the probability of reaching a certain fortune and the mean time to reach a fortune or ruin in terms of generating functions in the case of the hadamard coin a counting technique for boundary restricted paths in a lattice is also presented we discuss an open quantum version of foster s theorem for the expected return time together with applications

the distance between two convex sets abstract in this paper we explore the duality relations that characterize least norm problems the paper starts by presenting a new minimum norm duality mnd theorem one that considers the distance between two convex sets roughly speaking the new theorem says that the shortest distance between the two sets is equal to the maximal separation between the sets where the term separation refers to the distance between a pair of parallel hyperplanes that separates the two sets the second part of the paper brings several examples of applications the examples teach valuable lessons about the role of duality in least norm problems and reveal new features of these problems one lesson exposes the polar decomposition which characterizes the solution of an inconsistent system of linear inequalities another lesson reveals the close links between the mnd theorem theorems of the alternatives steepest descent directions and constructive optimality conditions

some perspectives on the eigenvalue problem this expository paper explores the relationships among a number of algorithms for solving eigenvalue problems including the power method subspace iteration the qr algorithm and the arnoldi and symmetric lanczos algorithms the symmetric lanczos algorithm is shown to be identical to the three term recursion stieltjes procedure for computing orthogonal polynomials with respect to a measure on the real line the connection between measures on the line and symmetric tridiagonal jacobi matrices is investigated if such a matrix is transformed by a step of the qr algorithm there is a corresponding transformation in the measure the tridiagonal matrices are also exploited for the construction of gaussian quadrature formulas for measures on the line the developments on the real line are replicated with suitable modifications on the unit circle via lanczos like procedures for unitary operators the best known procedure of this type is the recursion of szego for computing orthogonal polynomials on the

homogeneous open quantum random walks on a lattice we study open quantum random walks oqrws for which the underlying graph is a lattice and the generators of the walk are homogeneous in space using the results recently obtained in carbone and pautrat ann henri poincare 2015 we study the quantum trajectory associated with the oqrw which is described by a position process and a state process we obtain a central limit theorem and a large deviation principle for the position process we study in detail the case of homogeneous oqrws on the lattice mathbb z d with internal space mathfrak h mathbb c 2

block iterative projection methods for parallel computation of solutions to convex feasibility problems abstract an iterative method is proposed for solving convex feasibility problems each iteration is a convex combination of projections onto the given convex sets where the weights of the combination may vary from step to step it is shown that any sequence of iterations generated by the algorithm converges if the intersection of the given family of convex sets is nonempty and that the limit point of the sequence belongs to this intersection under mild conditions on the sequence of weight functions special cases are block iterative processes where in each iterative step a certain subfamily of the given family of convex sets is used in particular a block iterative version of the agmon motzkin schoenberg relaxation method for solving systems of linear inequalities is derived such processes lend themselves to parallel implementation and will be useful in various areas of applications including image reconstruction from projections image restoration and other fully discretized inversion problems

new existence results for optimal controls in the absence of convexity the importance of extremality a new approach to existence without convexity is presented instead of applying a weierstrass tonelli type existence result to a convexified i e relaxed form of the control problem which is standard it uses a new extremum principle this principle guarantees the existence of an optimal relaxed control function that is a convex combination of relaxations of ordinary control functions the same linear relationship is maintained for the corresponding trajectories and this is of much benefit in the final phase which consists of a deconvexification by a global subgradient argument and lyapunov s theorem the extremum principle is based on extreme point features of the relaxed control problem it can be stated in an abstract form and extends the classical extremum principle of bauer the new approach applies both to optimal control problems of suitable type dynamics linear in the state variable and to abstract variational problems without dynamics well known in economics with a different interpretation it can also give bang bang existence results several new applications serve to illustrate the power of the approach

on some necessary conditions of optimality for a nonlocal variational principle the aim of this work is to deduce optimality conditions for a nonlocal variational principle in the one dimensional scalar case we consider the relaxation of the problem in terms of young measures this relaxation is a new problem where we perform variations to derive optimality conditions and those conditions yield explicit information about minimizers in the homogeneous case it also provides a method capable of finding minimizers for some specific problems

a computational solution of the inverse problem in radiation therapy treatment planning radiation therapy concerns the delivery of the proper dose of radiation to a tumor volume without causing irreparable damage to healthy tissue and critical organs the forward problem refers to calculating the dose distribution that would be delivered to a measured patient cross section when the radiation field generated by the beam sources is specified the inverse problem refers to calculating the radiation field that will provide a specified dose distribution in the patient the forward and inverse problems of radiation therapy treatment planning are first formulated in their continuous versions and the point is made that in this field of application the inverse problem calls for the inversion of an operator for which no analytic closed form mathematical representation exists to attack the inverse problem under such circumstances a discretized model is set up in which both patient section and radiation field are finely discretized this leads to a linear feasibility problem which is solved by a relaxation method the paper includes full details of this new approach with discussion of technical aspects such as dose apportionment software development and experimental results consequences and limitations as well as a precise comparison with other methodologies are also discussed

a quantum dynamical approach to matrix khrushchev s formulas khrushchev s formula is the cornerstone of the so called khrushchev theory a body of results which has revolutionized the theory of orthogonal polynomials on the unit circle this formula can be understood as a factorization of the schur function for an orthogonal polynomial modification of a measure on the unit circle no such formula is known in the case of matrix valued measures this constitutes the main obstacle to generalize khrushchev theory to the matrix valued setting which we overcome in this paper it was recently discovered that orthogonal polynomials on the unit circle and their matrix valued versions play a significant role in the study of quantum walks the quantum mechanical analogue of random walks in particular schur functions turn out to be the mathematical tool which best codify the return properties of a discrete time quantum system a topic in which khrushchev s formula has profound and surprising implications we will show that this connection between schur functions and quantum walks is behind a simple proof of khrushchev s formula via quantum diagrammatic techniques for cmv matrices this does not merely give a quantum meaning to a known mathematical result since the diagrammatic proof also works for matrix valued measures actually this path counting approach is so fruitful that it provides different matrix generalizations of khrushchev s formula some of them new even in the case of scalar measures furthermore the path counting approach allows us to identify the properties of cmv matrices which are responsible for khrushchev s formula on the one hand this helps to formalize and unify the diagrammatic proofs using simple operator theory tools on the other hand this is the origin of our main result which extends khrushchev s formula beyond the cmv case as a factorization rule for schur functions related to general unitary operators 2015 wiley periodicals inc

comment on an existence result for a noncoercive nonconvex variational problem an essential improvement is given of a recent existence result of crasta g crasta siam j control optim 38 1999 pp 237 253 for a nonconvex noncoercive variational problem whose integrand does not depend on the state variable this is shown to follow by the methods based on fatou s lemma in several dimensions of e j balder j math anal appl 101 1984 pp 527 539 the associated euler lagrange inclusions follow from well known optimality conditions for such problems v m alekseev v m tichimirov and s v fomin optimal control consultants bureau new york 1987 v i arkin and v l levin uspekhi mat nauk 27 1972 pp 21 77

jl unitary factorization and the schur algorithm for nevanlinna functions in an indefinite setting abstract we introduce a schur transformation for generalized nevanlinna functions and show that it can be used in obtaining the unique minimal factorization of a class of rational j l unitary 2 2 matrix functions into elementary factors from the same class

on ws convergence of product measures a number of fundamental results centered around extensions of prohorov s theorem is proven for the ws topology for measures on a product space these results contribute to the foundations of stochastic decision theory they also subsume the principal results of young measure theory which only considers product measures with a fixed common marginal specializations yield the criterion for relative ws compactness of scha l 1975 the refined characterizations of ws convergence of galda ano and truffert 1997 1998 and a new version of fatou s lemma in several dimensions in a separate nonsequential development a generalization is given of the relative ws compactness criterion of jacod and ma min 1981 new applications are given to the existence of optimal equilibrium distributions over player action pairs in game theory and the existence of most optimistic scenarios in stochastic decision theory

schur agler and herglotz agler classes of functions positive kernel decompositions and transfer function realizations abstract we discuss transfer function realization for multivariable holomorphic functions mapping the unit polydisk or the right polyhalfplane into the operator analogue of either the unit disk or the right halfplane schur herglotz functions over either the unit polydisk or the right polyhalfplane which satisfy the appropriate stronger contractive positive real part condition for the values of these functions on commutative tuples of strict contractions strictly accretive operators schur agler herglotz agler functions over either the unit polydisk or the right polyhalfplane as originally shown by agler the first case polydisk to disk can be solved via unitary extensions of a partially defined isometry constructed in a canonical way from a kernel decomposition for the function the lurking isometry method we show how a geometric reformulation of the lurking isometry method embedding of a given isotropic subspace of a kre n space into a lagrangian subspace the lurking isotropic subspace method can be used to handle the second two cases polydisk to halfplane and polyhalfplane to disk as well as the last case polyhalfplane to halfplane if an additional growth condition at is imposed for the general fourth case we show how a linear fractional transformation change of variable can be used to arrive at the appropriate symmetrized nonhomogeneous bessmertny long resolvent realization we also indicate how this last result recovers the classical integral representation formula for scalar valued holomorphic functions mapping the right halfplane into itself

parallel application of block iterative methods in medical imaging and radiation therapy some row action algorithms which exploit special objective function and constraints structure have proven advantageous for solving huge and sparse feasibility or optimization problems recently developed block iterative versions of such special purpose methods enable parallel computation when the underlying problem is appropriately decomposed this opens the door for parallel computation in image reconstruction problems of computerized tomography and in the inverse problem of radiation therapy treatment planning all in their fully discretized modelling approach since there is more than one way of deriving block iterative versions of any row action method the choice has to be made with reference to the underlying real world problem

a nonlocal anisotropic model for phase transitions where u is a scalar density function on a domain of r and takes values in 1 1 w is a positive double well potential which vanishes at 1 and j is a positive possibly anisotropic interaction potential which vanishes at infinity see paragraph 1 2 for precise definitions the scalar function u represents the macroscopic density profile of a system which has two equilibrium pure phases described by the profiles u 1 and u 1 the integral w u at the right side of 1 1 forces a minimizer of f to take values close to 1 and 1 phase separation while the double integral represents an interaction energy which penalizes the spatial inhomogeneity of the system surface tension in equilibrium statistical mechanics functionals of the form 1 1 arise as free energies of continuum limits of ising spin systems on lattices in this setting u plays the role of a macroscopic magnetization density and j is a ferromagnetic kac potential see for instance 2 and references therein we underline the analogy with the more familiar gradient theory for phase transition proposed in 9 where the free energy of the system is of the form

nevanlinna representations in several variables abstract we generalize to several variables the classical theorem of nevanlinna that characterizes the cauchy transforms of positive measures on the real line we show that for the loewner class a large class of analytic functions that have non negative imaginary part on the upper polyhalf plane there are representation formulae in terms of densely defined self adjoint operators on a hilbert space we find four different representation formulae and we show that every function in the loewner class has one of the four representations corresponding precisely to four different growth conditions at infinity

new sequential compactness results for spaces of scalarly integrable functions abstract a new general relative sequential compactness criterion for scalarly integrable functions is presented the novel feature of this result is the fact that it deals with a strong form of pointwise cesaro convergence almost everywhere which is stronger than the usual types of weak convergence applications of the main result include extensions of a classical weak compactness criterion for abstract l1 spaces and a recent relative sequential compactness criterion of prohorov type for transition probabilites the main result can also be seen as an abstract version of a famous theorem of komlos which is also essential for its proof

open quantum random walks a new model of quantum random walks is introduced on lattices as well as on finite graphs these quantum random walks take into account the behavior of open quantum systems they are the exact quantum analogues of classical markov chains we explore the quantum trajectory point of view on these quantum random walks that is we show that measuring the position of the particle after each time step gives rise to a classical markov chain on the lattice times the state space of the particle this quantum trajectory is a simulation of the master equation of the quantum random walk the physical pertinence of such quantum random walks and the way they can be concretely realized is discussed differences and connections with the already well known quantum random walks such as the hadamard random walk are established

some geometric properties of the set of generalized young functionals this paper studies geometric properties in particular extreme points and rays of various generalizations of young measures applications of the knowledge of extreme points are illustrated on existence results for optimal control problems and on various convergence results for young measures by using the choquet theory

operator monotone functions and lowner functions of several variables we prove generalizations of lowner s results on matrix monotone functions to several variables we give a characterization of when a function of d variables is locally monotone on d tuples of commuting self adjoint n by n matrices we prove a generalization to several variables of nevanlinna s theorem describing analytic functions that map the upper half plane to itself and satisfy a growth condition we use this to characterize all rational functions of two variables that are operator monotone

stability results for doubly nonlinear differential inclusions by variational convergence we present a stability result for a wide class of doubly nonlinear equations featuring general maximal monotone operators and possibly nonconvex and nonsmooth energy functionals the limit analysis consists in the reformulation of the differential evolution as a scalar energy conservation equation with the aid of the so called fitzpatrick theory for the representation of monotone operators in particular our result applies to the vanishing viscosity approximation of rate independent systems

optimal design and relaxation of variational problems iii abstract this paper has attempted a synthesis of three subjects that have developed separately optimal design homogenization and relaxation in the calculus of variations the underlying theme is that of oscillations they occur with certain definite patterns in minimizing sequences for the variational problem the same patterns recur in nearly optimal designs and in the coefficients of the equation being homogenized each problem is resolved by determining the structure of those oscillations and above all their macroscopic averages it is those averages that solve the relaxed problem they also determine the optimal design and the solution of the homogenized equation we emphasize also the role of optimization the homogenized equations and composite materials of interest here come not from arbitrary mixtures but from those that are extremal since they correspond to minimizing sequences for the variational problem the effective properties of these composites attain the optimal bounds for given materials in fixed proportions

a generalization of schur functions applications to nevanlinna functions orthogonal polynomials random walks and unitary and open quantum walks abstract recent work on return properties of quantum walks the quantum analogue of random walks has identified their generating functions of first returns as schur functions this is connected with a representation of schur functions in terms of the operators governing the evolution of quantum walks i e the unitary operators on hilbert spaces in this paper we propose a generalization of schur functions by extending the above operator representation to arbitrary closed operators on banach spaces such generalized schur functions meet the formal structure of first return generating functions thus we call them fr functions we derive some general properties of fr functions among them a simple relation with an operator version of stieltjes functions which generalizes the renewal equation already known for random and quantum walks we also prove that fr functions satisfy splitting properties which extend useful factorizations of schur functions when specialized to self adjoint operators on hilbert spaces we show that fr functions become nevanlinna functions this allows us to obtain properties of nevanlinna functions which as far as we know seem to be new the fr function structure leads to a new operator representation of nevanlinna functions in terms of self adjoint operators whose spectral measures provide also new integral representations of such functions this allows us to characterize each nevanlinna function by a measure on the real line which we refer to as the measure of the nevanlinna function in contrast to standard operator and integral representations of nevanlinna functions these new ones are exact analogues of those already known for schur functions the above results are also the source of a very simple schur algorithm for nevanlinna functions based on interpolations at points on the real line which we refer to as the schur algorithm on the real line the paper is completed with several applications of fr functions to orthogonal polynomials and random and quantum walks which illustrate their wide interest an analogue for orthogonal polynomials on the real line of the khrushchev formula for orthogonal polynomials on the unit circle and the use of fr functions to study recurrence in random walks quantum walks and open quantum walks these applications provide numerous explicit examples of fr functions clarifying the meaning of these functions as first return generating functions and their splittings which become recurrence splitting rules they also show that these new tools despite being extensions of very classical ones play an important role in the study of physical problems of a highly topical nature

matrix valued szegő polynomials and quantum random walks we consider quantum random walks qrw on the integers a subject that has been considered in the last few years in the framework of quantum computation we show how the theory of cmv matrices gives a natural tool to study these processes and to give results that are analogous to those that karlin and mcgregor developed to study classical birth and death processes using orthogonal polynomials on the real line in perfect analogy with the classical case the study of qrws on the set of nonnegative integers can be handled using scalar valued laurent polynomials and a scalar valued measure on the circle in the case of classical or quantum random walks on the integers one needs to allow for matrix valued versions of these notions we show how our tools yield results in the well known case of the hadamard walk but we go beyond this translation invariant model to analyze examples that are hard to analyze using other methods more precisely we consider qrws on the set of nonnegative integers the analysis of these cases leads to phenomena that are absent in the case of qrws on the integers even if one restricts oneself to a constant coin this is illustrated here by studying recurrence properties of the walk but the same method can be used for other purposes the presentation here aims at being self contained but we refrain from trying to give an introduction to quantum random walks a subject well surveyed in the literature we quote for two excellent reviews see 1 9 see also the recent notes 20 2009 wiley periodicals inc

quantum recurrence of a subspace and operator valued schur functions a notion of monitored recurrence for discrete time quantum processes was recently introduced in grunbaum et al commun math phys 2 320 543 569 2013 taking the initial state as an absorbing one we extend this notion of monitored recurrence to absorbing subspaces of arbitrary finite dimension the generating function approach leads to a connection with the well known theory of operator valued schur functions this is the cornerstone of a spectral characterization of subspace recurrence that generalizes some of the main results in grunbaum et al commun math phys 2 320 543 569 2013 the spectral decomposition of the unitary step operator driving the evolution yields a spectral measure which we project onto the subspace to obtain a new spectral measure that is purely singular iff the subspace is recurrent and consists of a pure point spectrum with a finite number of masses precisely when all states in the subspace have a finite expected return time this notion of subspace recurrence also links the concept of expected return time to an aharonov anandan phase that in contrast to the case of state recurrence can be non integer even more surprising is the fact that averaging such geometrical phases over the absorbing subspace yields an integer with a topological meaning so that the averaged expected return time is always a rational number moreover state recurrence can occasionally give higher return probabilities than subspace recurrence a fact that reveals once more the counter intuitive behavior of quantum systems all these phenomena are illustrated with explicit examples including as a natural application the analysis of site recurrence for coined walks

nonexistence and existence results of an optimal control problem governed by a class of multisolution semilinear elliptic equations abstract an optimal control problem governed by a class of multisolution semilinear elliptic equations is studied in this paper it is pointed out that the control domain and cost functional are nonconvex by analyzing the maximum condition for an optimal relaxed control one nonexistence and some existence results of an optimal pair are obtained the idea of using relaxed controls is mainly derived from the paper lou 2007 by lou where a linear and well posed controlled system was considered the present paper is mainly devoted to discussing the case not contained in lou 2007

five diagonal matrices and zeros of orthogonal polynomials on the unit circle it is shown that monic orthogonal polynomials on the unit circle are the characteristic polynomials of certain five diagonal matrices depending on the schur parameters this result is achieved through the study of orthogonal laurent polynomials on the unit circle more precisely it is a consequence of the five term recurrence relation obtained for these orthogonal laurent polynomials and the one to one correspondence established between them and the orthogonal polynomials on the unit circle as an application some results relating the behaviour of the zeros of orthogonal polynomials and the location of schur parameters are obtained

recurrence and polya number of quantum walks we analyze the recurrence probability polya number for d dimensional unbiased quantum walks a sufficient condition for a quantum walk to be recurrent is derived as a by product we find a simple criterion for localization of quantum walks in contrast with classical walks where the polya number is characteristic for the given dimension the recurrence probability of a quantum walk depends in general on the topology of the walk choice of the coin and the initial state this allows us to change the character of the quantum walk from recurrent to transient by altering the initial state

on a class of quantum channels open random walks and recurrence we study a particular class of trace preserving completely positive maps called pq channels for which classical and quantum evolutions are isolated in a certain sense by combining open quantum random walks with a notion of recurrence we are able to describe criteria for recurrence of the walk related to this class of channels positive recurrence for open walks is also discussed in this context

characterization of generalized gradient young measures generated by sequences in w1 1 and bv generalized young measures as introduced by diperna and majda commun math phys 108 667 689 1987 provide a quantitative tool for studying the one point statistics of oscillation and concentration in sequences of functions in this work after developing a functional analytic framework for such measures including a compactness theorem and results on the generation of such young measures by l1 bounded sequences or even by sequences of bounded radon measures we turn to investigation of those young measures that are generated by bounded sequences of w1 1 gradients or bv derivatives we provide several techniques to manipulate such measures including shifting averaging and approximation by piecewise homogeneous young measures and then establish the main new result of this work the duality characterization of the set of bv or w1 1 gradient young measures in terms of jensen type inequalities for quasiconvex functions with linear growth at infinity this result is the natural generalization of the kinderlehrer pedregal theorem arch ration mech anal 115 329 365 1991 j geom anal 4 59 90 1994 for classical young measures to the w1 1 and bv case and contains its version for weakly converging sequences in w1 1 as a special case finally we give an application to a new lower semicontinuity theorem in bv

row action methods for huge and sparse systems and their applications this paper brings together and discusses theory and applications of methods identified and labelled as row action methods for linear feasibility problems find x in bf r n such that ax leqq b linearly constrained optimization problems minimize f x subject to ax leqq b and some interval convex programming problems minimize f x subject to c leqq ax leqq b the main feature of row action methods is that they are iterative procedures which without making any changes to the original matrix a use the rows of a one row at a time such methods are important and have demonstrated effectiveness for problems with large or huge matrices which do not enjoy any detectable or usable structural pattern apart from a high degree of sparaseness fields of application where row action methods are used in various ways include image reconstruction from projection operations research and game theory learning theory pattern recognition and transportation theory a row action method for the

a general realization theorem for matrix valued herglotz nevanlinna functions abstract new special types of stationary conservative impedance and scattering systems the so called non canonical systems involving triplets of hilbert spaces and projection operators are considered it is established that every matrix valued herglotz nevanlinna function of the form v z q lz r 1 t z t 1 t 2 d t can be realized as a transfer function of such a new type of conservative impedance system in this case it is shown that the realization can be chosen such that the main and the projection operators of the realizing system satisfy a certain commutativity condition if and only if l 0 it is also shown that v z with an additional condition namely l is invertible or l 0 can be realized as a linear fractional transformation of the transfer function of a non canonical scattering f system in particular this means that every scalar herglotz nevanlinna function can be realized in the above sense moreover the classical livsic systems brodskii livsic operator colligations can be derived from f systems as a special case when f i and the spectral measure d t is compactly supported the realization theorems proved in this paper are strongly connected with and complement the recent results by ball and staffans

fatou s lemma for multifunctions with unbounded values for multifunctions having unbounded sets as values we give fatou type inclusions for the kuratowski limes superior in finite as well as infinite dimensions these are derived from similar known fatou type inequalities for single valued multifunctions i e ordinary functions that is from balder s unifying fatou lemma in case the image set is finite dimensional and from an update of related results by balder in the infinite dimensional case for this extension from the single valued to the multivalued situation a lemma due to hess used to prove earlier fatou type inclusions is of critical importance also an asymptotic correction term introduced here plays an important role the two man results thus obtained subsume and extend an entire class of comparable fatou lemmas

consequences of denseness of dirac young measures 1 introductiona fundamental property of the set of dirac young measures is itsdenseness in the set of all young measures when the underlying measurespace is nonatomic e g see 11 theorem iv 2 6 6 in what would wx appear to be its most general form this result was stated for a completelyregular suslin space of control points 3 moreover a simple observation wxstemming directly from the proof of the result states that such densenessof the dirac young measures is maintained under a finite number ofcertain integral functional inequalities 3 corollary 3 for this reason awxrecent characterization of

radiation theraphy treatment planning an l2 approach we formulate the radiation theraphy treatment planning rttp problem as constrained optimization in l 2 this is accomplished by first describing the dose operator as a bounded operator from treatment space into patient space both suitable l 2 spaces within this context we examine two problems of interest in rttp first we take the case where the sources of radiation are fixed a priori and provide algorithms which lead to the optimal choices of beam profiles for these sources secondly we look at the problem of simultaneously choosing the sources and beam profiles and also suggest an algorithm for this case

extreme points of moment sets the extreme points of sets of probability measures determined by a finite number of generalized moment conditions are characterized together with an integral representation theorem the characterization is used to optimize affine functionals

disordered quantum walks in one lattice dimension we study a spin 12 particle moving on a one dimensional lattice subject to disorder induced by a random space dependent quantum coin the discrete time evolution is given by a family of random unitary quantum walk operators where the shift operation is assumed to be deterministic each coin is an independent identically distributed random variable with values in the group of two dimensional unitary matrices we derive sufficient conditions on the probability distribution of the coins such that the system exhibits dynamical localization put differently the tunneling probability between two lattice sites decays rapidly for almost all choices of random coins and after arbitrary many time steps with increasing distance our findings imply that this effect takes place if the coin is chosen at random from the haar measure or some measure continuous with respect to it but also for a class of discrete probability measures which support consists of two coins one of them being the hadamard coin

rational cayley inner herglotz agler functions positive kernel decompositions and transfer function realizations abstract the bessmertny class consists of rational matrix valued functions of d complex variables representable as the schur complement of a block of a linear pencil a z z 1 a 1 z d a d whose coefficients a k are positive semidefinite matrices we show that it coincides with the subclass of rational functions in the herglotz agler class over the right poly halfplane which are homogeneous of degree one and which are cayley inner the latter means that such a function is holomorphic on the right poly halfplane and takes skew hermitian matrix values on i r d or equivalently is the double cayley transform over the variables and over the matrix values of an inner function on the unit polydisk using agler knese s characterization of rational inner schur agler functions on the polydisk extended now to the matrix valued case and applying appropriate cayley transformations we obtain characterizations of matrix valued rational cayley inner herglotz agler functions both in the setting of the polydisk and of the right poly halfplane in terms of transfer function realizations and in terms of positive kernel decompositions in particular we extend bessmertny s representation to rational cayley inner herglotz agler functions on the right poly halfplane where a linear pencil a z is now in the form a z a 0 z 1 a 1 z d a d with a 0 skew hermitian and the other coefficients a k positive semidefinite matrices

optimality conditions for mass design problems and applications to thin plates we derive necessary and sufficient optimality conditions for a quite large class of structural design problems which can be formulated as follows under a given load and a total volume constraint minimize a suitable notion of compliance among all admissible mass distributions represented by positive measures with prescribed integral mean as a special case we focus attention on the optimization of thin plates we detail the corresponding optimality conditions and we show how they can be handled in order to determine analytically some optimal plates

on linear infeasibility arising in intensity modulated radiation therapy inverse planning abstract intensity modulated radiation therapy imrt gives rise to systems of linear inequalities representing the effects of radiation on the irradiated body these systems are often infeasible in which case one settles for an approximate solution such as an relaxation meaning that no more than percent of the inequalities are violated by no more than percent for real world imrt problems there is a feasible relaxation for sufficiently large 0 however large values of these parameters may be unacceptable medically the relaxation problem is combinatorial and for given values of the parameters can be solved exactly by mixed integer programming mip but this may be impractical because of problem size and the need for repeated solutions as the treatment progresses as a practical alternative to the mip approach we present a heuristic non combinatorial method for finding an approximate relaxation the method solves a linear program for each pair of values of the parameters and progresses through successively increasing values until an acceptable solution is found or is determined non existent the method is fast and reliable since it consists of solving a sequence of linear progrms

extreme points of sets of randomized strategies in constrained optimization and control problems this paper concerns the existence and characterization of optimal randomized strategies for some constrained optimization and control problems we first present a characterization of the extreme points of a set of randomized strategies that satisfy n moment like constraints conditions are given under which those extreme points are randomizations of at most n 1 deterministic strategies this result is then applied to obtain the existence and characterization of optimal strategies for a class of deterministic allocation like optimization problems and their young relaxations similar results are obtained for constrained markov control processes in borel spaces

a general approach to lower semicontinuity and lower closure in optimal control theory a self contained approach to lower semicontinuity and lower closure evolves from an extension of relaxed control theory which is based on a central relative weak compactness criterion called tightness and relaxation in all but one variable two lower closure results for outer integral functionals with variable abstract time domain are developed the first of these has a convexity condition for the integrand and generalizes all similar results in the literature the second lower closure result is of a new kind among other things it implies a quite general version of fatou s lemma in several dimensions