homotopical stable ranks for banach algebras abstract the connected stable rank and the general stable rank are homotopy invariants for banach algebras whereas the bass stable rank and the topological stable rank should be thought of as dimensional invariants this paper studies the two homotopical stable ranks viz their general properties as well as specific examples and computations the picture that emerges is that of a strong affinity between the homotopical stable ranks and a marked contrast with the dimensional ones

spectral determinant of schrodinger operators on graphs we study the spectral properties of the operator v x on a graph is the laplacian and v x is some potential defined on the graph in particular we derive an expression for the spectral determinant that generalizes one previously obtained for the laplacian operator

variational properties and orbital stability of standing waves for nls equation on a star graph we study standing waves for a model of nonlinear schr odinger equation on a graph the graph is obtained joining n halflines at a vertex i e it is a star graph at the vertex an interaction occurs due to a boundary condition of delta type with strength alpha leqslant 0 which includes the free or kirchhoff case alpha 0 there imposed the nonlinearity is of focusing power type the dynamics is given by an equation of the form i frac d dt psi t h psi t psi t 2 mu psi t where h is the s a operator which generates the linear schr odinger dynamics on the graph in the hilbert space l 2 gg we show the existence of several families of standing waves solutions of the form phi t e i omega t psi omega where psi omega is a vector amplitude parametrized by a frequency omega they exist for every sign of the coupling at the vertex attractive alpha 0 and for every omega frac alpha 2 n 2 the number of the families depends on the number n of the edges of the graph we moreover determine the ground state which is the standing wave the amplitude of which minimizes the action on the natural nehari constraint and order the various families of standing waves according to their increasing action thus determining a nonlinear spectrum of the problem finally we show that the ground state is orbitally stable for every allowed omega if the nonlinearity is subcritical or critical for supercritical nonlinearities there exists an omega frac alpha 2 n 2 such that the ground state is orbitally stable for frac alpha 2 n 2 omega omega action minimization requires technical subtleties in particular an adaptation of decreasing rearrangements on a star graph is developed with a generalization of the p olya szeg h o inequality

note on basic features of large time behaviour of heat kernels large time behaviour of heat semigroups and more gener ally of positive selfadjoint semigroups is studied convergence of the semigroup to the ground state and of averaged logarithms of kernels to the ground state energy is shown in the general framework of positivity improving selfadjoint semigroups this framework includes laplacians on manifolds metric graphs and discrete graphs

a trust region algorithm for heterogeneous multiobjective optimization this paper presents a trust region method for multiobjective heterogeneous optimization problems one of the objective functions is an expensive black box function given for example by a time co

multi objective retrospective optimization using stochastic zigzag search abstract we propose a new retrospective optimization ro method for multi objective simulation optimization moso problems ro algorithms generate a sequence of sample path sp problems and solve these sp problems iteratively using a nonlinear optimizer in this study a stochastic zigzag search algorithm is chosen in the ro framework to solve sp problems the key idea of zigzag search is searching around the pareto front by applying an efficient local search procedure using the gradients of the objective functions many continuous moso problems have smooth objective functions and their non dominated objective function values form a smooth surface in the image space this fact motivates developing the zigzag search method embedded in ro for such relatively well posed moso problems a numerical implementation of this method multi objective retrospective optimization using zigzag search morozs is presented particularly for continuous bi objective simulation optimization boso problems with well connected pareto optimal solutions morozs is designed for boso problems in which a simulation oracle returns both objective function values and gradients due to the local nature of zigzag search morozs can only guarantee the asymptotic convergence to local pareto optimality the efficiency of morozs is studied using three boso problems with noisy objective functions and is compared to that of genetic algorithms based nsga ii and a recently developed method mo compass

existence of a tree of stieltjes strings corresponding to two given spectra transversal vibrations of a plane rooted tree of stieltjes strings are considered with dirichlet boundary conditions at each pendant vertex continuity and kirchhoff conditions are imposed at each interior vertex except the root the dirichlet problem is that with the dirichlet condition at the root and the neumann problem is that with continuity and kirchhoff conditions at the root it is shown that strict interlacing is a sufficient condition for two sequences of real numbers to be the spectra of neumann and dirichlet problems generated by stieltjes string recurrent relations on any prescribed tree

the stable rank of topological algebras and a problem of r g swan abstract various notions of stable ranks are studied for topological algebras some partial answers to r g swan s problem have two banach or good frechet algebras as in the density theorem in k theory the same stable rank are obtained for example a frechet dense subalgebra a of a c algebra b closed under c functional calculus of self adjoint elements has the same bass stable rank as b

distribution theory for discontinuous test functions and differential operators with generalized coefficients investigation of the differential operators with the generalized coefficients having singular support on a disjoint set of points requires the consideration of the distribution theory with the set of discontinuous test functions such a distribution theory for test functions having discontinuity at one point is developed a four parameter family of schrodinger operators formed by the operators with singular potential singular metrics and singular gauge field is considered it is proved that this family of singular interactions describes all possible selfadjoint extensions of the second derivative operator defined on the functions vanishing in a neighbourhood of the point approximation by operators with smooth coefficients is discussed c 1996 academic press inc

zeta functions of quantum graphs in this paper we construct zeta functions of quantum graphs using a contour integral technique based on the argument principle we start by considering the special case of the star graph with neumann matching conditions at the center of the star we then extend the technique to allow any matching conditions at the center for which the laplace operator is self adjoint and finally obtain an expression for the zeta function of any graph with general vertex matching conditions in the process it is convenient to work with new forms for the secular equation of a quantum graph that extend the well known secular equation of the neumann star graph in the second half of this paper we apply the zeta function to obtain new results for the spectral determinant vacuum energy and heat kernel coefficients of quantum graphs these have all been topics of current research in their own right and in each case this unified approach significantly expands results in the literature

rapidly decreasing functions in reduced algebras of groups let r be a group we associate to any length function l on r the space h r of rapidly decreasing functions on r with respect to l which coincides with the space of smooth functions on the k dimensional torus when r zk we say that r has property rd if there exists a length function l on r such that h r is contained in the reduced c algebra c r of r we study the stability of property rd with respect to some constructions of groups such as subgroups over groups of finite index semidirect and amalgamated products finally we show that the following groups have property rd 1 finitely generated groups of polynomial growth 2 discrete cocompact subgroups of the group of all isometries of any hyperbolic space

relatively spectral morphisms and applications to k theory abstract spectral morphisms between banach algebras are useful for comparing their k theory and their noncommutative dimensions as expressed by various notions of stable ranks in practice one often encounters situations where the spectral information is only known over a dense subalgebra we investigate such relatively spectral morphisms we prove a relative version of the density theorem regarding isomorphism in k theory we also solve swan s problem for the connected stable rank in fact for an entire hierarchy of higher connected stable ranks that we introduce

scattering matrices and weyl functions for a scattering system a a 0 consisting of self adjoint extensions a and a 0 of a symmetric operator a with finite deficiency indices the scattering matrix s and a spectral shift function are calculated in terms of the weyl function associated with a boundary triplet for a and a simple proof of the krein birman formula is given the results are applied to singular sturm liouville operators with scalar and matrix potentials to dirac operators and to schrodinger operators with point interactions

approximation by point potentials in a magnetic field we discuss magnetic schrodinger operators perturbed by measures from the generalized kato class using an explicit krein like formula for their resolvent we prove that these operators can be approximated in the strong resolvent sense by magnetic schrodinger operators with point potentials since the spectral problem of the latter operators is solvable one in fact gets an alternative way to calculate discrete spectra we illustrate it by numerical calculations in the case when the potential is supported by a circle

on the number of negative eigenvalues of the laplacian on a metric graph the number of negative eigenvalues of self adjoint laplacians on metric graphs is calculated in terms of the boundary conditions and the underlying geometric structure this extends and complements earlier results by kostrykin and schrader 2006 contemp math 415 201 25

spectral statistics for the dirac operator on graphs we determine conditions for the quantization of graphs using the dirac operator for both two and four component spinors according to the bohigas giannoni schmit conjecture for such systems with time reversal symmetry the energy level statistics are expected in the semiclassical limit to correspond to those of random matrices from the gaussian symplectic ensemble this is confirmed by numerical investigation the scattering matrix used to formulate the quantization condition is found to be independent of the type of spinor we derive an exact trace formula for the spectrum and use this to investigate the form factor in the diagonal approximation

dirac operators and the calculation of the connes metric on arbitrary infinite graphs as an outgrowth of our investigation of non regular spaces within the context of quantum gravity and non commutative geometry we develop a graph hilbert space framework on arbitrary infinite graphs and use it to study spectral properties of graph laplacians and graph dirac operators we define a spectral triplet sharing most of the properties of what connes calls a spectral triple with the help of this scheme we derive an explicit expression for the connes distance function on general directed or undirected graphs we derive a series of a priori estimates and calculate it for a variety of examples of graphs as a possibly interesting side we show that the natural setting for approaching such problems may be the framework of non linear programming or optimization we compare our results arrived at within our particular framework with those of other authors and show that the seeming differences depend on the use of different graph geometries and or dirac operators

the berry keating operator on l 2 mathbb r _ gt rm d x and on compact quantum graphs with general self adjoint realizations the berry keating operator berry and keating 1999 siam rev 41 236 governing the schrodinger dynamics is discussed in the hilbert space and on compact quantum graphs it has been proved that the spectrum of hbk defined on is purely continuous and thus this quantization of hbk cannot yield the hypothetical hilbert polya operator possessing as eigenvalues the nontrivial zeros of the riemann zeta function a complete classification of all self adjoint extensions of hbk acting on compact quantum graphs is given together with the corresponding secular equation in form of a determinant whose zeros determine the discrete spectrum of hbk in addition an exact trace formula and the weyl asymptotics of the eigenvalue counting function are derived furthermore we introduce the squared berry keating operator which is a special case of the black scholes operator used in financial theory of option pricing again all self adjoint extensions the corresponding secular equation the trace formula and the weyl asymptotics are derived for h2bk on compact quantum graphs while the spectra of both hbk and h2bk on any compact quantum graph are discrete their weyl asymptotics demonstrate that neither hbk nor h2bk can yield as eigenvalues the nontrivial riemann zeros some simple examples are worked out in detail

quantum graphs ii some spectral properties of quantum and combinatorial graphs the paper deals with some spectral properties of mostly infinite quantum and combinatorial graphs quantum graphs have been intensively studied lately due to their numerous applications to mesoscopic physics nanotechnology optics and other areas a schnol type theorem is proven that allows one to detect that a point belongs to the spectrum when a generalized eigenfunction with an sub exponential growth integral estimate is available a theorem on spectral gap opening for decorated quantum graphs is established its analogue is known for the combinatorial case it is also shown that if a periodic combinatorial or quantum graph has a point spectrum it is generated by compactly supported eigenfunctions scars

k theorie bivariante pour les algebres de banach et conjecture de baum connes je suis parvenu dans ma these a construire une k theorie bivariante pour les algebres de banach cela m a permis de demontrer la conjecture de baum connes pour les groupes de lie semi simples et les groupes reductifs p adiques ainsi que pour les sous groupes discrets de type fini de sl 3 f avec f une extension finie de q p et pour les sous groupes discrets cocompacts de sp n 1 sl 3 r et sl 3 c j ai aussi demontre une conjecture analogue a la conjecture de baum connes pour les algebres l 1 et plus generalement toutes les bonnes completions des sous groupes fermes des groupes de lie semi simples et des groupes reductifs p adiques

integrable models from twisted half loop algebras this paper is devoted to the construction of new integrable quantum mechanical models based on certain subalgebras of the half loop algebra of various results about these subalgebras are proven by presenting them in the notation of the st petersburg school these results are then used to demonstrate the integrability and find the symmetries of two types of physical system twisted gaudin magnets and calogero type models of particles on several half lines meeting at a point

asymptotic behavior of jost functions near resonance points for wigner von neumann type potentials in this work are considered radial schrodinger operators v r e where v r a sin br r w r with w r bounded w r o r 2 at infinity a b real the asymptotic behavior of the jost function and the scattering matrix near the resonance point e0 b2 4 are studied if a b then this point may be an eigenvalue embedded in the continuous spectrum the leading behavior of the jost function for all values of a and b were determined somewhat surprisingly situations were found where the jost function becomes singular as e e0 even if e0 is an embedded eigenvalue moreover it was found that the scattering matrix is always discontinuous at e0 except in a few special cases it is also shown that the asymptotics for the jost function and the scattering matrix hold under weaker assumptions on w r in particular an angular momentum term l l 1 r 2 may be incorporated into w r the results were also applied to a whole line problem with a potential v x such that v x 0 for x 0 and v x of wigner von neumann

inverse spectral problem for quantum graphs the inverse spectral problem for the laplace operator on a finite metric graph is investigated it is shown that this problem has a unique solution for graphs with rationally independent edges and without vertices having valence 2 to prove the result a trace formula connecting the spectrum of the laplace operator with the set of periodic orbits for the metric graph is established

characteristic functions of quantum graphs in this paper we give a reduction formula for the characteristic functions of the sturm liouville boundary value problems defined on a tree we also discuss the multiplicity of the eigenvalues and interlacing properties between two spectral sets associated with different boundary conditions

high energy analysis and levinson s theorem for the selfadjoint matrix schrodinger operator on the half line the matrix schrodinger equation with a selfadjoint matrix potential is considered on the half line with the general selfadjoint boundary condition at the origin when the matrix potential is integrable the high energy asymptotics are established for the related jost matrix the inverse of the jost matrix and the scattering matrix under the additional assumption that the matrix potential has a first moment levinson s theorem is derived relating the number of bound states to the change in the argument of the determinant of the scattering matrix

the free dirac operator on compact and noncompact graphs the free dirac operator defined on composite one dimensional structures consisting of finitely many half lines and intervals is investigated the influence of the connection points between the constituents is modeled by transition conditions for the wave functions or equivalently by different self adjoint extensions of the dirac operator general relations between the parameters of the extensions and the eigenvalues resp the scattering coefficients are derived and then applied to the cases of a bundle of half lines a point defect a branching line and an eye shaped structure

four parameter point interaction in 1d quantum systems the author constructs a four parameter point interaction for a nonrelativistic particle moving on a line as the limit of a short range interaction with a range tending toward zero for particular choices of the parameters one can obtain a delta interaction or the so called delta interaction the hamiltonian corresponding to the four parameter point interaction is shown to correspond to the four parameter self adjoint hamiltonian of the free particle moving on the line with the origin excluded

on agmon metrics and exponential localization for quantum graphs we investigate the rate of decrease at infinity of eigenfunctions of quantum graphs by using agmon s method to prove l2 and l infty bounds on the product of an eigenfunction with the exponential of a certain metric a generic result applicable to all graphs is that the exponential rate of decay is controlled by an adaptation of the standard estimates for a line which are of classical liouville green wkb form examples reveal that this estimate can be the best possible but that a more rapid rate of decay is typical when the graph has additional structure in order to understand this fact we present two alternative estimates under more restrictive assumptions on the graph structure that pertain to a more rapid decay one of these depends on how the eigenfunction is distributed along a particular chosen path while the other applies to an average of the eigenfunction over edges at a given distance from the root point

spectral functions for the schrodinger operator on r with a singular potential in this article we analyze the spectral zeta function the heat kernel and the resolvent of the operator d2 dr2 r2 r2 over the interval 0 for 1 4 depending on the self adjoint extension chosen nonstandard properties of the zeta function and of asymptotic properties of the heat kernel and resolvent are observed in particular for the zeta function nonstandard locations of poles as well as logarithmic branch cuts at s k k n0 do occur this implies that the small t asymptotic expansion of the heat kernel can have nonstandard powers as well as terms such as tk ln t l 1 for k l n0 the corresponding statements for the resolvent are also shown furthermore we evaluate the zeta determinant of the operator for all values of and any self adjoint extension

quantum graphs with singular two particle interactions we construct quantum models of two particles on a compact metric graph with singular two particle interactions the hamiltonians are self adjoint realizations of laplacians acting on functions defined on pairs of edges in such a way that the interaction is provided by boundary conditions in order to find such hamiltonians closed and semi bounded quadratic forms are constructed from which the associated self adjoint operators are extracted we provide a general characterization of such operators and furthermore produce certain classes of examples we then consider identical particles and project to the bosonic and fermionic subspaces finally we show that the operators possess purely discrete spectra and that the eigenvalues are distributed following an appropriate weyl asymptotic law

dispersion for the schrodinger equation on networks in this paper we consider the schrodinger equation on a network formed by a tree with the last generation of edges formed by infinite strips we give an explicit description of the solution of the linear schrodinger equation with constant coefficients this allows us to prove dispersive estimates which in turn are useful for solving the nonlinear schrodinger equation the proof extends also to the laminar case of positive step function coefficients having a finite number of discontinuities

scattering theory on graphs we consider the scattering theory for the schrodinger operator dx2 v x on the graphs made of one dimensional wires connected to external leads we derive two expressions for the scattering matrix on arbitrary graphs one involves matrices that couple arcs oriented bonds the other involves matrices that couple vertices we discuss a simple way to tune the coupling between the graph and the leads the efficiency of the formalism is demonstrated with a few known examples

index theorems for quantum graphs in geometric analysis an index theorem relates the difference of the numbers of solutions of two differential equations to the topological structure of the manifold or bundle concerned sometimes using the heat kernels of two higher order differential operators as an intermediary in this paper the case of quantum graphs is addressed a quantum graph is a graph considered as a singular one dimensional variety and equipped with a second order differential hamiltonian h a laplacian with suitable conditions at vertices for the case of scale invariant vertex conditions i e conditions that do not mix the values of functions and of their derivatives the constant term of the heat kernel expansion is shown to be proportional to the trace of the internal scattering matrix of the graph this observation is placed into the index theory context by factoring the laplacian into two first order operators h a a and relating the constant term to the index of a an independent consideration provides an index formula for any differential operator on a finite quantum graph in terms of the vertex conditions it is also found that the algebraic multiplicity of 0 as a root of the secular determinant of h is the sum of the nullities of a and a

on δ like potential scattering on star graphs we discuss the potential scattering on the noncompact star graph the schrodinger operator with the short range potential localized in a neighborhood of the graph vertex is considered we study the asymptotic behavior of the corresponding scattering matrix in the zero range limit it has been known for a long time that in dimension 1 there is no non trivial hamiltonian with the distributional potential i e the potential acts as a totally reflecting wall several authors have in recent years studied the scattering properties of the regularizing potentials e 2q x e approximating the first derivative of the dirac delta function a non zero transmission through the regularized potential has been shown to exist as e 0 we extend these results to star graphs with the point interaction which is an analog of the potential on the line we prove that generically such a potential on the graph is opaque we also show that there exists a countable set of resonant intensities for which a partial transmission through the potential occurs this set of resonances is referred to as the resonant set and is determined as the spectrum of an auxiliary sturm liouville problem associated with q on the graph

can one hear the shape of a graph we show that the spectrum of the schroperator on a finite metric graph determines uniquely the connectivity matrix and the bond lengths provided that the lengths are non commensurate and the connectivity is simple no parallel bonds between vertices and no loops connecting a vertex to itself that is one can hear the shape of the graph we also consider a related inversion problem a compact graph can be converted into a scattering system by attaching to its vertices leads to infinity we show that the scattering phase determines uniquely the compact part of the graph under similar conditions as above

comment on factorization of scattering matrices due to partitioning of potentials in one dimensional schrodinger type equations j math phys 37 5897 1996

a factorization of the scattering matrix for the schrodinger equation and for the wave equation in one dimension for the one dimensional schrodinger equation with a potential decaying at both ends of the real axis the scattering matrix of the potential is given explicitly in terms of the scattering matrices corresponding to the fragments of this potential a similar result also holds for the wave equation in a nonhomogeneous nondispersive medium where the wave speed has the same asymptotics at both ends of the real line

on multiplicity of a quantum graph spectrum the sturm liouville equations on the edges of a metric connected graph together with the boundary and matching conditions at the vertices generate a spectral problem for a self adjoint operator it is shown that if the graph is not cyclically connected then the maximal multiplicity of an eigenvalue of such an operator is gt ptin where is the cyclomatic number of the graph and gt and ptin are the number of edges and the number of interior vertices respectively for the tree obtained by contracting all the cycles of the graph into vertices if the graph is cyclically connected then the maximal multiplicity of an eigenvalue is 1

appendix resonances on a simple graph we study the scattering problem on a graph consisting of a line with a finite length appendix the two parts are coupled through boundary conditions depending on three parameters the motion on the line is free while the appendix supports a potential the appendix bound states give rise to a ladder of resonances we construct the resolvent and solve the corresponding pole condition for a weak coupling in general the condition only admits an analytic solution in particular cases we find the pole positions numerically for a linear potential and show that the poles eventually return to the real axis when the coupling strength increases

quantum graphs as holonomic constraints we consider the dynamics on a quantum graph as the limit of the dynamics generated by a one particle hamiltonian in r2 with a potential having a deep strict minimum on the graph when the width of the well shrinks to zero for a generic graph we prove convergence outside the vertices to the free dynamics on the edges for a simple model of a graph with two edges and one vertex we prove convergence of the dynamics to the one generated by the laplacian with dirichlet boundary conditions in the vertex

green function approach for general quantum graphs we present a recursive prescription to calculate the exact green function for general quantum graphs for the closed case the expression for the poles of g which gives the individual eigenstates has the structure of a semiclassical formula where the sum over the periodic orbit is already performed as applications we discuss eigenstate localization for a three arm closed star and filter like mechanisms for transmission throughout an open trident graph

the number of eigenvalues of the matrix schrodinger operator on the half line with general boundary conditions we prove a bound of the bargmann birman schwinger type on the number of eigenvalues of the matrix schrodinger operator on the half line with the most general self adjoint boundary condition at the origin and with selfadjoint matrix potentials that are integrable and have a finite first moment

compact group automorphisms addition formulas and fuglede kadison determinants for a countable amenable group and an element f in the integral group ring z being invertible in the group von neumann algebra of we show that the entropy of the shift action of on the pontryagin dual of the quotient of z by its left ideal generated by f is the logarithm of the fuglede kadison determinant of f for the proof we establish an p version of rufus bowen s denition of topological entropy addition formulas for group extensions of countable amenable group actions and an approximation formula for the fuglede kadison determinant of f in terms of the determinants of perturbations of the compressions of f

on the number of states bound by one dimensional finite periodic potentials bound states and zero energy resonances of one dimensional finite periodic potentials are investigated by means of levinson s theorem for finite range potentials supporting no bound states a lower bound for the reduced time delay at threshold is derived

recovering a quantum graph spectrum from vertex data we study the question to what extent spectral information on a schrodinger operator on a finite compact metric graph subject to standard or type matching conditions can be recovered from a corresponding titchmarsh weyl function on the boundary of the graph in contrast to the case of ordinary or partial differential operators the knowledge of the titchmarsh weyl function is in general not sufficient for recovering the complete spectrum of the operator or the potentials on the edges however it is shown that those eigenvalues with sufficiently high depending on the cyclomatic number of the graph multiplicities can be recovered moreover we prove that under certain additional conditions the titchmarsh weyl function even contains the full spectral information

uniform existence of the integrated density of states for random schrodinger operators on metric graphs over zd abstract we consider ergodic random schrodinger operators on the metric graph z d with random potentials and random boundary conditions taking values in a finite set we show that normalized finite volume eigenvalue counting functions converge to a limit uniformly in the energy variable this limit the integrated density of states can be expressed by a closed shubin pastur type trace formula it supports the spectrum and its points of discontinuity are characterized by existence of compactly supported eigenfunctions among other examples we discuss random magnetic fields and percolation models

dimension and stable rank in the k theory of c algebras in topological k theory which can be viewed as the algebraic side of the theory of vector bundles some of the interesting properties which one investigates are for example the conditions under which bundles must possess trivial direct summands or the extent to which the cancellation property for direct whitney sums of bundles holds such properties turn out to be controlled in part by the dimension of the base space and results describing the nature of this control are among what are frequently called stability results 17 during the past few years there has been a sudden flowering of a theory for c algebras see surveys 35 36 37 since c algebras are profitably thought of as non commutative locally compact spaces with the finitely generated projective modules being the appropriate generalization of vector bundles according to swan s theorem 40 it would be natural to look for stability results for c algebras but until now the theory has been focussed on the c groups of c algebras and there has been little discussion of stability properties presumably in part for lack of an appropriate concept of dimension for c algebras one of the main objectives of this paper is to introduce a concept of dimension for c algebras which directly generalizes the classical concept of dimension for compact spaces 16 20 22 and to develop some techniques for calculating it notably for c algebras which are obtained as crossed product algebras for an action of the group of integers in algebraic k theory which is the generalization of topological c theory to rings a substantial number of stability properties have been discovered 2 39 42 48 within that theory the concept which has played a role most analogous to that of dimension is the bass stable rank but despite its many successes the bass stable rank has often been difficult to calculate even for relatively uncomplicated rings another main objective of the present paper is to make accessible for application to c algebras some of the stability results from algebraic c theory this comes about from the fact that the concept of dimension which we introduce dominates the bass stable rank i was led to seek stability results for c algebras by my desire to determine whether the cancellation property for projective modules holds for the irrational rotation c algebras discussed in 30 which as elliott 11 has suggested can appropriately be called 2 dimensional non commutative tori in more concrete terms the question is whether two projections in such an algebra which have the same trace must be unitarily equivalent the present paper provides the general theory which is needed for the affirmative answer which i obtained in a subsequent paper i will derive the

factorization of scattering matrices due to partitioning of potentials in one dimensional schrodinger type equations the one dimensional schrodinger equation and two of its generalizations are considered as they arise in quantum mechanics wave propagation in a nonhomogeneous medium and wave propagation in a nonconservative medium where energy may be absorbed or generated generically the zero energy transmission coefficient vanishes when the potential is nontrivial but in the exceptional case this coefficient is nonzero resulting in tunneling through the potential it is shown that any nontrivial exceptional potential can always be fragmented into two generic pieces furthermore any nontrivial potential generic or exceptional can be fragmented into generic pieces in infinitely many ways the results remain valid when dirac delta functions are included in the potential and other coefficients are added to the schrodinger equation for such schrodinger equations factorization formulas are obtained that relate the scattering matrices of the fragments to the scattering matrix of the full problem

hermitian unitary matrices with modular permutation symmetry abstract we study hermitian unitary matrices s c n n with the following property there exist r 0 and t 0 such that the entries of s satisfy s j j r and s j k t for all j k 1 n j k we derive necessary conditions on the ratio d r t and show that these conditions are very restrictive except for the case when n is even and the sum of the diagonal elements of s is zero examples of families of matrices s are constructed for d belonging to certain intervals the case of real matrices s is examined in more detail it is demonstrated that a real s can exist only for d n 2 1 or for n even and n 2 d 1 mod 2 we provide a detailed description of the structure of real s with d n 4 3 2 and derive a sufficient and necessary condition of its existence in terms of the existence of certain symmetric v k designs we prove that there exists no real s with d n 6 1 n 4 3 2 a parametrization of hermitian unitary matrices is also proposed and its generalization to general unitary matrices is given at the end of the paper the role of the studied matrices in quantum mechanics on graphs is briefly explained

finite propagation speed for solutions of the wave equation on metric graphs abstract we provide a class of self adjoint laplace operators on metric graphs with the property that the solutions of the associated wave equation satisfy the finite propagation speed property the proof uses energy methods which are adaptations of corresponding methods for smooth manifolds

boundary conditions at the derivative of a delta function the author derives the joining conditions on psi at a point where the potential is the nth derivative of a dirac delta function

reducible boundary conditions in coupled channels we study hamiltonians with point interactions in spaces of vector valued functions using some information from the theory of quantum graphs we describe a class of the operators which can be reduced to the direct sum of several one dimensional problems it shown that such a reduction is closely connected with the invariance under channel permutations examples are provided by some model interactions in particular the so called and the kirchhoff couplings

direct multisearch for multiobjective optimization in practical applications of optimization it is common to have several conflicting objective functions to optimize frequently these functions are subject to noise or can be of black box type preventing the use of derivative based techniques we propose a novel multiobjective derivative free methodology calling it direct multisearch dms which does not aggregate any of the objective functions our framework is inspired by the search poll paradigm of direct search methods of directional type and uses the concept of pareto dominance to maintain a list of nondominated points from which the new iterates or poll centers are chosen the aim of our method is to generate as many points in the pareto front as possible from the polling procedure itself while keeping the whole framework general enough to accommodate other disseminating strategies in particular when using the here also optional search step dms generalizes to multiobjective optimization moo all direct search methods of directional type

zeta functions of the dirac operator on quantum graphs we construct spectral zeta functions for the dirac operator on metric graphs we start with the case of a rose graph a graph with a single vertex where every edge is a loop the technique is then developed to cover any finite graph with general energy independent matching conditions at the vertices the regularized spectral determinant of the dirac operator is also obtained as the derivative of the zeta function at a special value in each case the zeta function is formulated using a contour integral method which extends results obtained for laplace and schrodinger operators on graphs

on the inverse scattering problem on branching graphs the inverse scattering problem on branching graphs is studied the definition of the schrodinger operator on such graphs is discussed the operator is defined with real potentials with finite first momentum and using special boundary conditions connecting values of the functions at the vertices it is shown that in general the scattering matrix does not determine the topology of the graph the potentials on the edges and the boundary conditions uniquely

kirchhoff s rule for quantum wires we formulate and discuss one particle quantum scattering theory on an arbitrary finite graph with n open ends and where we define the hamiltonian to be minus the laplace operator with general boundary conditions at the vertices this results in a scattering theory with n channels the corresponding on shell s matrix formed by the reflection and transmission amplitudes for incoming plane waves of energy e 0 is given explicitly in terms of the boundary conditions and the lengths of the internal lines it is shown to be unitary which may be viewed as the quantum version of kirchhoff s law we exhibit covariance and symmetry properties it is symmetric if the boundary conditions are real also there is a duality transformation on the set of boundary conditions and the lengths of the internal lines such that the low energy behaviour of one theory gives the high energy behaviour of the transformed theory finally we provide a composition rule by which the on shell s matrix of a graph is factorizable in terms of the s matrices of its subgraphs all proofs use only known facts from the theory of self adjoint extensions standard linear algebra complex function theory and elementary arguments from the theory of hermitian symplectic forms

asymptotically isospectral quantum graphs and generalised trigonometric polynomials abstract the theory of almost periodic functions is used to investigate spectral properties of schrodinger operators on metric graphs also known as quantum graphs in particular we prove that two schrodinger operators may have asymptotically close spectra if and only if the corresponding reference laplacians are isospectral our result implies that a schrodinger operator is isospectral to the standard laplacian on a may be different metric graph only if the potential is identically equal to zero

an analogue of the thom isomorphism for crossed products of a c algebra by an action of r abstract in this paper we show the existence and uniqueness of a natural isomorphism oj of kj a with kj 1 a r j z 2 where a r is a c dynamical r system k is the functor of topological k theory and a r is the crossed product of a by the action of r the pimsner voiculescu exact sequence is obtained as a corollary we show that given an invariant trace on a with dual trace gt one has gto 1 u 1 2 i u u for any unitary u in the domain of the derivation of a associated to the action finally we show that the crossed product of c s3 continuous functions on the 3 sphere by a minimal diffeomorphism is a simple c algebra with no nontrivial idempotent

on the calculation of a membership function for the solution of a fuzzy linear optimization problem in the present paper the fuzzy linear optimization problem with fuzzy coefficients in the objective function is considered recent concepts of fuzzy solution to the fuzzy optimization problem based on the level cut and the set of pareto optimal solutions of a multiobjective optimization problem are applied chanas and kuchta suggested one approach to determine the membership function values of fuzzy optimal solutions of the fuzzy optimization problem which is based on calculating the sum of lengths of certain intervals the purpose of this paper is to determine a method for realizing this idea we derive explicit formulas for the bounds of these intervals in the case of triangular fuzzy numbers and show that only one interval needs to be considered

a numerical method for constructing the pareto front of multi objective optimization problems in this paper a new numerical method is presented for constructing an approximation of the pareto front of multi objective optimization problems this method is based on the well known scalarization approach by pascoletti and serafini the proposed method is applied to four test problems that illustrate specific difficulties encountered in multi objective optimization problems such as nonconvex disjoint and local pareto fronts the effectiveness of the proposed method is demonstrated by comparing it with the nsga ii algorithm and the normal constraint method

a mesh adaptive direct search algorithm for multiobjective optimization abstract this work studies multiobjective optimization mop of nonsmooth functions subject to general constraints we first present definitions and optimality conditions as well as some single objective formulations of mop parameterized with respect to some reference point in the space of objective functions next we propose a new algorithm called m ulti m ads multiobjective mesh adaptive direct search for mop m ulti m ads generates an approximation of the pareto front by solving a series of single objective formulations of mop generated using the nbi natural boundary intersection framework these single objective problems are solved using the m ads mesh adaptive direct search algorithm for constrained nonsmooth optimization the pareto front approximation is shown to satisfy some first order necessary optimality conditions based on the clarke calculus m ulti m ads is then tested on problems from the literature with different pareto front landscapes and on a styrene production process simulation problem from chemical engineering

hermitian symplectic geometry and extension theory here we give brief account of hermitian symplectic spaces showing that they are intimately connected to symmetric as well as self adjoint extensions of a symmetric operator furthermore we find an explicit parametrization of the lagrange grassmannian in terms of the unitary matrices this allows us to explicitly describe all self adjoint boundary conditions for the schrodinger operator on the graph in terms of a unitary matrix we show that the asymptotics of the scattering matrix can be expressed simply in terms of this unitary matrix

scattering on graphs and one dimensional approximations to n dimensional schrodinger operators in the present article we develop the spectral analysis of schrodinger operators on lattice type graphs for the basic example of a cubic periodic graph the problem is reduced to the spectral analysis of certain regular differential operators on a fundamental star like subgraph with a selfadjoint condition at the central node and quasiperiodic conditions at the boundary vertices using an explicit expression for the resolvent of lattice type operator we develop in the second section appropriate lippmann schwinger techniques for the perturbed periodic operator and construct the corresponding scattering matrix it serves as a base for the approximation of the multi dimensional schrodinger operator by a one dimensional operator on the graph in the third section of the paper for given n dimensional schrodinger operators with rapidly decreasing potential we construct a lattice type operator on a cubic graph embedded into rn and show that the original n dimensional scattering problem can be approximated in a p

scattering solutions in networks of thin fibers small diameter asymptotics small diameter asymptotics is obtained for scattering solutions in a network of thin fibers the asymptotics is expressed in terms of solutions of related problems on the limiting quantum graph we calculate the lagrangian gluing conditions at vertices v in gamma for the problems on the limiting graph if the frequency of the incident wave is above the bottom of the absolutely continuous spectrum the gluing conditions are formulated in terms of the scattering data for each individual junction of the network

two body scattering on a graph and application to simple nanoelectronic devices the two body system on a graph with one junction is considered the effective three body scattering problem turns out to be exactly solvable for pointwise interactions additional degrees of freedom corresponding to a dynamics of some structure e g an atomic cluster located in the junction point of common contact of three thin electrodes are considered these degrees of freedom bring effective energy dependent interaction into the effective schrodinger equation in the scattering channel the wave function of the system is constructed in the explicit form using the extension theory methods the obtained results are applied to the qualitative description of a simple three electrode nanoelectronic device the perturbation theory approach based on the analysis of the liouville equation is suggested for calculation of the conductivity for such a device in terms of the obtained wave function

convergence of spectra of mesoscopic systems collapsing onto a graph abstract let m be a finite graph in the plane and let me be a domain that looks like the e fattened graph m exact conditions on the domain are given it is shown that the spectrum of the neumann laplacian on me converges when e 0 to the spectrum of an ode problem on m the presence of an electromagnetic field is also allowed considerations of this kind arise naturally in mesoscopic physics and other areas of physics and chemistry the results of the paper extend the ones previously obtained by j rubinstein and m schatzman

classifying the general family of 1d point interactions a scattering approach we propose a classification scheme for the complete family of 1d point interactions to do so we first review the solutions of the wavefunctions and green functions of the problem second we derive the exact time dependent propagators in such a way that we can write the expressions for the k s in a very compact form as they should such expressions do reduce to the known results in the literature according to the potential parameter values then we analyse in general terms how the different point interactions scatter off arbitrary initially localized wave packets finally we show that the physical features associated with the scattering process can be used to establish a classification procedure moreover these physical characteristics are directly related to the potential parameters leading to the many formulae for the k s as an application we present numerical calculations for gaussian wave packets

multiobjective optimization through a series of single objective formulations this work deals with bound constrained multiobjective optimization mop of nonsmooth functions for problems where the structure of the objective functions either cannot be exploited or are absent typical situations arise when the functions are computed as the result of a computer simulation we first present definitions and optimality conditions as well as two families of single objective formulations of mop next we propose a new algorithm called for the biobjective optimization bop problem i e mop with two objective functions the property that pareto points may be ordered in bop and not in mop is exploited by our algorithm generates an approximation of the pareto front by solving a series of single objective formulations of bop these single objective problems are solved using the recent mesh adaptive direct search algorithm for nonsmooth optimization the pareto front approximation is shown to satisfy some first order necessary optimality conditions based on the clarke calculus finally is tested on problems from the literature designed to illustrate specific difficulties encountered in biobjective optimization such as a nonconvex or disjoint pareto front local pareto fronts or a nonuniform pareto front

an adaptive strategy on the error of the objective functions for uncertainty based derivative free optimization in this work a strategy is developed to deal with the error affecting the objective functions in uncertainty based optimization we refer to the problems where the objective functions are the statistics of a quantity of interest computed by an uncertainty quantification technique that propagates some uncertainties of the input variables through the system under consideration in real problems the statistics are computed by a numerical method and therefore they are affected by a certain level of error depending on the chosen accuracy the errors on the objective function can be interpreted with the abstraction of a bounding box around the nominal estimation in the objective functions space in addition in some cases the uncertainty quantification methods providing the objective functions also supply the possibility of adaptive refinement to reduce the error bounding box the novel method relies on the exchange of information between the outer loop based on the optimization algorithm and the inner uncertainty quantification loop in particular in the inner uncertainty quantification loop a control is performed to decide whether a refinement of the bounding box for the current design is appropriate or not in single objective problems the current bounding box is compared to the current optimal design in multi objective problems the decision is based on the comparison of the error bounding box of the current design and the current pareto front with this strategy fewer computations are made for clearly dominated solutions and an accurate estimate of the objective function is provided for the interesting non dominated solutions the results presented in this work prove that the proposed method improves the efficiency of the global loop while preserving the accuracy of the final pareto front the proposed method improves the efficiency of the uncertainty based optimization by reducing the number of objective evaluations the method relies on the presence of a derivative free optimization method and an adaptive uncertainty quantification method the proposed method is conceived for multi objective problems but it applies to single objective optimization as well results prove that the proposed method saves computations on the dominated solutions and predicts with accuracy the pareto front

inverse scattering on matrices with boundary conditions we describe inverse scattering for the matrix schrodinger operator with general self adjoint boundary conditions at the origin using the marchenko equation our approach allows the recovery of the potential as well as the boundary conditions it is easily specialized to inverse scattering on star shaped graphs with boundary conditions at the node

variational problems on multiply connected thin strips i basic estimates and convergence of the laplacian spectrum let m be a planar embedded graph whose arcs meet transversally at the vertices let be a strip shaped domain around m of width except in a neighborhood of the singular points assume that the boundary of is smooth we define comparison operators between functions on and on m and we derive energy estimates for the compared functions we define a laplace operator on m which is in a certain sense the limit of the laplace operator on with neumann boundary conditions in particular we show that the p th eigenvalue of the laplacian on converges to the p th eigenvalue of the laplacian on m as tends to 0 a similar result holds for the magnetic schrodinger operator