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

certain properties of a music type imaging functional in inverse scattering from an open sound hard arc abstract this paper concerns a mathematical formulation of the well known multiple signal classification music type imaging functional in the inverse scattering problem by an open sound hard arc based on the physical factorization of the so called multi static response msr matrix and the structure of left singular vectors linked to the non zero singular values of the msr matrix we construct a relationship between the imaging functional and the bessel functions of order 0 1 and 2 of the first kind we then expound certain properties of music and present numerical results for several differently chosen smooth arcs

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

a direct algorithm for ultrasound imaging of internal corrosion we develop a direct noniterative algorithm to address the inverse problem of identifying a collection of disjoint internal corrosive parts of small hausdorff measures in pipelines from exterior ultrasound boundary measurements the method is based on an asymptotic expansion of the effect of the corrosion in terms of the size of the corrosive parts we numerically test the validity of the asymptotic formula at high frequencies we also propose a simple procedure to remove high frequency instabilities in our inversion procedure we illustrate our main findings with a variety of computational examples

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

knots links and 4 manifolds in this paper we investigate the relationship between isotopy classes of knots and links in s and the di eomorphism types of homeomorphic smooth 4 manifolds as a corollary of this initial investigation we begin to uncover the surprisingly rich structure of di eomorphism types of manifolds homeomorphic to the k3 surface in order to state our theorems we need to view the seiberg witten invariant of a smooth 4 manifold as a multivariable laurent polynomial to do this recall that the seiberg witten invariant of a smooth closed oriented 4 manifold x with b2 x 1 is an integer valued function which is de ned on the set of spinc structures over x cf w km ko1 t1 in case h1 x z has no 2 torsion which will be the situation in this paper there is a natural identi cation of the spinc structures of x with the characteristic elements of h2 x z in this case we view the seiberg witten invariant as

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

music imaging and electromagnetic inverse scattering of multiple scattering small anisotropic spheres the foldy lax equation is used to derive a multiple scattering model for the multiple scattering small anisotropic spheres by this model if the number of the non zero singular values of the multistatic response msr matrix is smaller than the number of the antennas the range space of the msr matrix is found to be spanned by the background green s function vectors corresponding to the and components of the electric and magnetic dipoles induced in each scatterer which indicates that the multiple signal classification music method could be implemented to obtain the locations of the scatterers after estimating the positions of the scatterers a non iterative analytical method is proposed for retrieving the polarization strength tensors as well as the orientations of the principle axes of each scatterer two numerical simulations show that the music method and the non iterative method are efficacious for the nonlinear inverse scattering problem of determining the locations and polarization strength tensors of multiple scattering small anisotropic spheres such methods could also be applied to the inversion of small isotropic spheres or extended to the inversion of small bianisotropic spheres

a comparison theorem for gromov witten invariants in the symplectic category abstract we exploit the geometric approach to the virtual fundamental class due to fukaya ono and li tian to compare gromov witten invariants of a symplectic manifold and a symplectic submanifold whenever all constrained stable maps to the former are contained in the latter to first order various special cases of the comparison theorem in this paper have long been used in the algebraic category some of them have also appeared in the symplectic setting combined with the inherent flexibility of the symplectic category the main theorem leads to a confirmation of pandharipande s gopakumar vafa prediction for gw invariants of fano classes in 6 dimensional symplectic manifolds the proof of the main theorem uses deformations of the cauchy riemann equation that respect the submanifold and carleman similarity principle for solutions of perturbed cauchy riemann equations in a forthcoming paper we apply a similar approach to relative gromov witten invariants and the absolute relative correspondence in genus 0

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

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

analysis of music type imaging functional for single thin electromagnetic inhomogeneity in limited view inverse scattering problem this study analyzes the well known multiple signal classification music algorithm to identify unknown support of thin penetrable electromagnetic inhomogeneity from scattered field data collected within the so called multi static response matrix in limited view inverse scattering problems the mathematical theories of music are partially discovered e g in the full view problem for an unknown target of dielectric contrast or a perfectly conducting crack with the dirichlet boundary condition transverse magnetic tm polarization and so on hence we perform further research to analyze the music type imaging functional and to certify some well known but theoretically unexplained phenomena for this purpose we establish a relationship between the music imaging functional and an infinite series of bessel functions of integer order of the first kind this relationship is based on the rigorous asymptotic expansion formula in the existence of a thin inhomogeneity with a smooth supporting curve various results of numerical simulation are presented in order to support the identified structure of music although a priori information of the target is needed we suggest a least condition of range of incident and observation directions to apply music in the limited view problem

electromagnetic music type imaging of perfectly conducting arc like cracks at single frequency we propose a non iterative music multiple signal classification type algorithm for the time harmonic electromagnetic imaging of one or more perfectly conducting arc like cracks found within a homogeneous space r 2 the algorithm is based on a factorization of the multi static response msr matrix collected in the far field at a single nonzero frequency in either transverse magnetic tm mode dirichlet boundary condition or transverse electric te mode neumann boundary condition followed by the calculation of a music cost functional expected to exhibit peaks along the crack curves each half a wavelength numerical experimentation from exact noiseless and noisy data shows that this is indeed the case and that the proposed algorithm behaves in robust manner with better results in the tm mode than in the te mode for which one would have to estimate the normal to the crack to get the most optimal results

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

direct imaging of small scatterers using reduced time dependent data we introduce qualitative methods for locating small objects using time dependent acoustic near field waves these methods have reduced data collection requirements compared to typical qualitative imaging techniques in particular we only collect scattered field data in a small region surrounding the location from which an incident field was transmitted the new methods are partially theoretically justified and numerical simulations demonstrate their efficacy we show that these reduced data techniques give comparable results to methods which require full multistatic data and that these time dependent methods require less scattered field data than their time harmonic analogs

asymptotic properties of music type imaging in two dimensional inverse scattering from thin electromagnetic inclusions the main purpose of this paper is to study the structure of the well known noniterative multiple signal classification music algorithm for identifying the shape of extended electromagnetic inclusions of small thickness located in the two dimensional homogeneous space we construct a relationship between a music type imaging functional for thin inclusions and bessel functions of integer order of the first kind our construction is based on the structure of left singular vectors of a collected multi static response matrix whose elements are a measured far field pattern and an asymptotic expansion formula of the existence of thin inclusions some numerical examples are shown to support the constructed structure of music

imaging schemes for perfectly conducting cracks we consider the problem of locating perfectly conducting cracks and estimating their geometric features from multistatic response matrix measurements at a single or multiple frequencies a main objective is to design specific crack detection rules and to analyze their receiver operating characteristics and the associated signal to noise ratios in this paper we introduce an analytic framework that uses asymptotic expansions which are uniform with respect to the wavelength to crack size ratio in combination with a hypothesis test based formulation to construct specific procedures for detection of perfectly conducting cracks a central ingredient in our approach is the use of random matrix theory to characterize the signal space associated with the multistatic response matrix measurements we present numerical experiments to illustrate some of our main findings

direct sampling method for identifying magnetic inhomogeneities in limited aperture inverse scattering problem abstract the direct sampling method dsm in limited aperture inverse scattering problems for transverse electric te polarization is considered based on the asymptotic expansion formula in the presence of small targets we demonstrate that the indicator function of dsm can be represented by an infinite series of bessel functions of integer order and correspondingly examine various properties of dsm simulation results with noisy data are created to support theoretical results based on the identified structure of the indicator function we design a new indicator function of dsm analyze its mathematical structure and generate simulation results to demonstrate its effectiveness and improvement

the symplectic sum formula for gromov witten invariants in the symplectic category there is a connect sum operation that glues symplectic manifolds by identifying neighborhoods of embedded codimension two submanifolds this paper establishes a formula for the gromov witten invariants of a symplectic sum z x y in terms of the relative gw invariants of x and y several applications to enumerative geometry are given gromov witten invariants are counts of holomorphic maps into symplectic manifolds to define them on a symplectic manifold x w one introduces an almost complex structure j compatible with the symplectic form w and

topology of surface diagrams of smooth 4 manifolds surface diagrams are a new way to specify any smooth closed orientable 4 manifold by an orientable surface decorated with simple closed curves these curves are cyclically indexed and each curve has a unique transverse intersection with the next the aim of this paper is to announce a uniqueness theorem for these objects within a fixed homotopy class that turns out to be similar to the reidemeister singer theorem for heegaard splittings of 3 manifolds

relative gromov witten invariants we define relative gromov witten invariants of a symplectic manifold relative to a codimension two symplectic submanifold these invariants are the key ingredients in the symplectic sum formula of ip4 the main step is the construction of a compact space of v stable maps simple special cases include the hurwitz numbers for algebraic curves and the enumerative invariants of caporaso and harris

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

music algorithm for location searching of dielectric anomalies from s parameters using microwave imaging abstract motivated by the biomedical engineering used in early stage breast cancer detection we investigated the use of multiple signal classification music algorithm for location searching of small anomalies using s parameters we considered the application of music to functional imaging where a small number of dipole antennas are used our approach is based on the application of born approximation or physical factorization we analyzed cases in which the anomaly is respectively small and large in relation to the wavelength and the structure of the left singular vectors is linked to the nonzero singular values of a multi static response msr matrix whose elements are the s parameters using simulations we demonstrated the strengths and weaknesses of the music algorithm in detecting both small and extended anomalies

sw gr from the seiberg witten equations to pseudo holomorphic curves the purpose of this article is to explain how pseudo holomorphic curves in a symplectic 4 manifold can be constructed from solutions to the seiberg witten equations as such the main theorem proved here theorem 1 3 is an existence theorem for pseudo holomorphic curves this article thus provides a proof of roughly half of the main theorem in the announcement t1 that theorem theorem 4 1 asserts an equivalence between the seiberg witten invariants for a symplectic manifold and a certain gromov invariant which counts with signs the number of pseudoholomorphic curves in a given homology class the seiberg witten invariants were introduced to mathematicians by witten w based on his joint work with nat seiberg sw1 sw2 a description of these invariants is given in section 1 see also km1 t1 suffice it to say here that when x is a compact oriented 4 dimensional manifold with

a structure theorem for the gromov witten invariants of kahler surfaces we prove a structure theorem for the gromov witten invariants of compact kahler surfaces with geometric genus pg 0 under the technical assumption that there is a canonical divisor that is a disjoint union of smooth components the theorem shows that the gw invariants are universal functions determined by the genus of this canonical divisor components and the holomorphic euler characteristic of the surface we compute special cases of these universal functions much of the work on the gromov witten invariants of kahler surfaces has focused on rational and ruled surfaces which have geometric genus pg 0 this paper focuses on surfaces with pg 0 a class that includes most elliptic surfaces and most surfaces of general type in this context we prove a general structure theorem that shows with one technical assumption how the gw invariants are completely determined by the local geometry of a generic canonical divisor the structure theorem is a consequence of a simple fact the image localization lemma of section 3 given a kahler surface x and a canonical divisor d kx this lemma shows that the complex structure j on x can be perturbed to a non integrable almost complex structure jd with the property the image of all jd holomorphic maps lies in the support of d this immediately gives some striking vanishing theorems for the gw invariants of kahler surfaces see section 3 more importantly it implies that the gromov witten invariant of x for genus g and n marked points is a sum gwg n x a gw loc g n dk ak over the connected components dk of d of local invariants that count the contribution of maps whose image lies in or after perturbing to a generic moduli space near dk these local invariants have not been previously defined the proof of their existence relies on using non integrable structures and geometric analysis techniques our structure theorem characterizes the local invariants by expressing them in terms of usual gw invariants of certain standard surfaces for this we make the mild assumption that one can deform the kahler structure on x and choose a canonical divisor d so that all components of d are smooth with this assumption the restriction of kx to each component dk of multiplicity mk is the normal bundle nk to dk and nk k kdk partially supported by the n s f

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

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

standard surfaces and nodal curves in symplectic 4 manifolds continuing the program of ds and u1 we introduce refinements of the donaldson smith standard surface count which are designed to count nodal pseudoholomorphic curves and curves with a prescribed decomposition into reducible components in cases where a corresponding analogue of the gromov taubes invariant is easy to define our invariants agree with those analogues we also prove a vanishing result for some of the invariants that count nodal curves

3 dimensional seiberg witten invariants and non kahlerian geometry the paper is concerned with seiberg witten invariants of closed 3 manifolds and their relation to the 4 dimensional invariants the main points are 1 the introduction of parameter dependent seiberg witten invariants for 3 manifolds with b1 0 these invariants depend on a complicated chamber structure defined by a differential geometric brill noether locus in the parameter space 2 a comparison theorem which shows that the seiberg witten invariants of 3 manifolds m with b 1 m 1 are simply the restrictions of the 4 dimensional invariants of the product s 1 m to the set of pull back spinc structures as an immediate consequence one gets the wall crossing formula for the invariants of 3 dimensional manifolds with b1 1 from the 4 dimensional wall crossing formula for manifolds with b 1 3 a novel method to compute the seiberg witten invariants of 3 manifolds m using non kahlerian complex geometry on s 1 m provided this product has such a structure which is compatible with a product metric and orientation induced from m this method relies essentially on a kobayashi hitchin correspondence which is new in the non kahler case 4 a complete description of those 3 manifolds to which our method applies these are riemannian manifolds which admit an oriented riemannian foliation by geodesics such manifolds are always diffeomorphic to seifert manifolds but the foliation has not to be seifert it can have non closed leaves 5 explicit computations of seiberg witten invariants for the 3 sphere s 3 using non seifert riemannian foliations we show that the chamber structure induced by the brill noether locus is very complicated there exist countably many dif ferent chambers and the seiberg witten invariant takes on all integer values

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

asymptotic imaging of perfectly conducting cracks in this paper we consider cracks with dirichlet boundary conditions we first derive an asymptotic expansion of the boundary perturbations that are due to the presence of a small crack based on this formula we design a noniterative approach for locating a collection of small cracks in order to do so we construct a response matrix from the boundary measurements the location and the length of the crack are estimated respectively from the projection onto the noise space and the first significant singular value of the response matrix indeed the direction of the crack is estimated from the second singular vector we then consider an extended crack with dirichlet boundary conditions we rigorously derive an asymptotic expansion for the boundary perturbations that are due to a shape deformation of the crack to reconstruct an extended crack from many boundary measurements we develop two methods for obtaining a good guess several numerical experiments show how the proposed techniques for imaging small cracks as well as those for obtaining good initial guesses toward reconstructing an extended crack behave

the limited aperture problem of inverse acoustic scattering dirichlet boundary conditions let d be a bounded simply connected domain in the plane and let f theta k underset raise0 3em hbox smash scriptscriptstyle thicksim alpha l be the far field pattern arising from the scattering of a time harmonic acoustic plane wave u l underset raise0 3em hbox smash scriptscriptstyle thicksim x exp ik underset raise0 3em hbox smash scriptscriptstyle thicksim x cdot underset raise0 3em hbox smash scriptscriptstyle thicksim alpha l where underset raise0 3em hbox smash scriptscriptstyle thicksim alpha l is a unit vector and k is the wave number and the time harmonic factor e i omega t has been factored out it is assumed in addition that the total field satisfies homogeneous dirichlet boundary conditions on partial d in this paper a method is presented for recovering partial d given the far field patterns f theta k underset raise0 3em hbox smash scriptscriptstyle thicksim alpha l

multi frequency subspace migration for imaging of perfectly conducting arc like cracks in full and limited view inverse scattering problems multi frequency subspace migration imaging techniques are usually adopted for the non iterative imaging of unknown electromagnetic targets such as cracks in concrete walls or bridges and anti personnel mines in the ground in the inverse scattering problems it is confirmed that this technique is very fast effective robust and can not only be applied to full but also to limited view inverse problems if a suitable number of incidents and corresponding scattered fields are applied and collected however in many works the application of such techniques is heuristic with the motivation of such heuristic application this study analyzes the structure of the imaging functional employed in the subspace migration imaging technique in two dimensional full and limited view inverse scattering problems when the unknown targets are arbitrary shaped arc like perfectly conducting cracks located in the two dimensional homogeneous space in contrast to the statistical approach based on statistical hypothesis testing our approach is based on the fact that the subspace migration imaging functional can be expressed by a linear combination of the bessel functions of integer order of the first kind this is based on the structure of the multi static response msr matrix collected in the far field at nonzero frequency in either transverse magnetic tm mode dirichlet boundary condition or transverse electric te mode neumann boundary condition the investigation of the expression of imaging functionals gives us certain properties of subspace migration and explains why multi frequency enhances imaging resolution in particular we carefully analyze the subspace migration and confirm some properties of imaging when a small number of incident fields are applied consequently we introduce a weighted multi frequency imaging functional and confirm that it is an improved version of subspace migration in tm mode various results of numerical simulations performed on the far field data affected by large amounts of random noise are similar to the analytical results derived in this study and they provide a direction for future studies

reduced open gromov witten invariants on k3 surfaces and multiple cover formula abstract in the paper we study the wall crossing phenomenon of reduced open gromov witten invariants on k3 surfaces with rigid special lagrangian boundary condition as a corollary we derived the multiple cover formula for the reduced open gromov witten invariants which is an open analog of the gopakumar vafa conjecture of genus zero

the homotopy lie algebra of symplectomorphism groups of 3 fold blow ups of the projective plane by a result of kedra and pinsonnault we know that the topology of groups of symplectomorphisms of symplectic 4 manifolds is complicated in general however in all known very specific examples the rational cohomology rings of symplectomorphism groups are finitely generated in this paper we compute the rational homotopy lie algebra of symplectomorphism groups of the 3 point blow up of the projective plane with an arbitrary symplectic form and show that in some cases depending on the sizes of the blow ups it is infinite dimensional moreover we explain how the topology is generated by the toric structures one can put on the manifold our method involve the study of the space of almost complex structures compatible with the symplectic structure and it depends on the inflation technique of lalonde mcduff

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

enumerative geometry of calabi yau 4 folds gromov witten theory is used to define an enumerative geometry of curves in calabi yau 4 folds the main technique is to find exact solutions to moving multiple cover integrals the resulting invariants are analogous to the bps counts of gopakumar and vafa for calabi yau 3 folds we conjecture the 4 fold invariants to be integers and expect a sheaf theoretic explanation

gromov s compactness theorem for pseudo holomorphic curves we give a complete proof for gromov s compactness theorem for pseudo holomorphic curves both in the case of closed curves and curves with boundary

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

the topology of the space of symplectic balls in rational 4 manifolds we study in this paper the rational homotopy type of the space of symplectic embeddings of the standard ball b4 c r4 into 4 dimensional rational symplectic manifolds where c r2 is the capacity of the standard ball of radius r we compute the rational homotopy groups of that space when the 4 manifold has the form m s2 s2 0 0 where 0 is the area form on the sphere with total area 1 and belongs to the interval 1 2 we show that when is 1 this space retracts to the space of symplectic frames for any value of c however for any given 1 2 the rational homotopy type of that space changes as c crosses the critical parameter 1 which is the difference of areas between the two s2 factors we prove moreover that the full homotopy type of that space changes only at that value that is that the restriction map between these spaces is a homotopy equivalence as long as these values of c remain either below or above that critical value the same methods apply to all other values of and other rational 4 manifolds as well the methods rely on two different tools the study of the action of symplectic groups on the stratified space of almost complex structures developed by gromov abreu and mcduff and the analysis of the relations between the group corresponding to a manifold m the group corresponding to its blow up m and the space of symplectic embedded balls in m

music type imaging of perfectly conducting cracks in limited view inverse scattering problems abstract although standard multiple signal classification music algorithm has been considered a promising non iterative imaging technique for cracks its application is restricted to the full view inverse scattering problems many experimental results revealed that music can be applied to the imaging of perfectly conducting cracks with small length in limited view inverse scattering problem on the other hand music is not applicable for imaging of extended cracks in limited view problems but the reason behind this restricted application has not been theoretically investigated this gave an impetus for this study to attempt to identify the structure of music type algorithm that appears in the limited view inverse scattering problems by establishing a relationship between music and bessel functions of integer order of the first kind some numerical experiments are illustrated to support the identified structure of music

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

a phase and space coherent direct imaging method a direct imaging algorithm for point and extended targets is presented the algorithm is based on a physical factorization of the response matrix of a transducer array the factorization is used to transform a passive target problem to an active source problem and to extract principal components tones in a phase consistent way the multitone imaging function can superpose multiple tones spatial diversity aperture of the array and frequencies bandwidth of the probing signal based on phase coherence the method is a direct imaging algorithm that is simple and efficient since no forward solver or iteration is needed robustness of the algorithm with respect to noise is demonstrated via numerical examples