the induction coefficient of induced representation theory its algebra and an analytic expression the induction coefficient is defined as the element of a matrix reducing the induced representation into irreducible constituents the authors give the algebra of these coefficients and a closed analytic expression in terms of matrix elements of coset representatives the consequences of this choice are derived and the importance of some of these results for the racah wigner coupling algebra of the symmetric and unitary groups noted

the racah wigner approach to harmonic oscillations of the cluster of nodes of a cube the racah wigner approach is applied to demonstrate various group theoretic labelling schemes for modes of oscillations and elasticity parameters of the cluster of nodes of a cube it is shown that there exists a fundamental basis in the configuration space closed under the action of the group oh the geometric symmetry group of the equilibrium it provides a transparent interpretation of all basic invariants the fibre bundle and tensor product structure of invariant bilinear forms over the configuration space yields three classification schemes for elasticity parameters in a way that resembles racah recoupling of several angular momenta

corrections to the kalman yakubovich popov lemma for pritchard salamon systems abstract in this note we correct the result in the paper the kalman yakubovich popov lemma for pritchard salamon systems 3 there was a gap in the proof which can be bridged but only by assuming that the system is exactly controllable

an optimal adaptive wavelet method for first order system least squares in this paper it is shown that any well posed 2nd order pde can be reformulated as a well posed first order least squares system this system will be solved by an adaptive wavelet solver in optimal computational complexity the applications that are considered are second order elliptic pdes with general inhomogeneous boundary conditions and the stationary navier stokes equations

h adaptive least squares finite element methods for the 2d stokes equations of any order with optimal convergence rates abstract the adaptive least squares finite element method ls fem for the stokes equations has recently been based on alternative error estimators in bringmann and carstensen 2016 for the lowest order case since the first order div ls fem measures the flux errors in h div the data resolution error measures the l 2 norm of the right hand side f minus the piecewise polynomial approximation k f without a mesh size factor this enforces separate marking with an overall abstract theory carstensen and rabus 2016 this paper contributes a a discussion of the scaling of the ls fem b optimal rates of the adaptive h version of any order k and c comparing numerical results in 2d with all details on inhomogeneous dirichlet data covered by the analysis

tensor fields on crystals a new method is presented to determine the irreducible representations of the space group of a crystal contained in the representation whose basis functions are the components of a tensor field defined on the atoms of a crystal this reducible representation is the direct product of a tensor representation dependent only on the tensor and a permutation representation dependent only on how the atoms permute under elements of the space group the permutation representation is first separately reduced prior to the reduction of the direct product the permutation representation is shown to be an induced representation and its reduction is facilitated using the theory of induced representations examples and tables of results of applying this method are given in the case of a polar vector tensor field applicable to lattice vibrational problems and crystals as the diamond structure of space group symmetry 07h

symmetry properties of direct sum systems an interpretation of double coset decompositions of groups is given in terms of the construction of systems from similar disjoint subsystems a natural labelling scheme is developed for the direct product states of systems of this type providing an extension of the concept of unique labelling an application of the decoupling transformation to finite direct sum systems is investigated

controllability properties of perturbed distributed parameter systems abstract we analyse the action of perturbations on controllable distributed parameter systems we show that the most common controllability properties are not preserved under the action of perturbations whose norm is smaller than any prescribed tolerance v 0 in particular we show that small perturbations of the values of the coefficients of a system with delays can destroy the property of modal controllability finally a positive result which is important for application to the stabilization problem is presented

existence and comparison theorems for algebraic riccati equations for continuous and discrete time systems abstract we discuss two comparison theorems for algebraic riccati equations of the form xbr 1 b x x a br 1 c a br 1 c x q c r 1 c 0 simultaneously we give sufficient conditions to obtain the existence of the maximal hermitian solution of a riccati equation from the existence of a hermitian solution of a second riccati equation further similar results are given for discrete algebraic riccati equations

unique labelling schemes for local modes in lattices two procedures are described which together specify a unique point group labelling scheme for the irreducible components of local distortions in crystals in the first procedure the group operators are factorised into a permutation part and a local atomic point group operation the second depends on dividing the complete shells of atoms into subshells which are invariant under subgroups of the site permutation group

comparison theorems for infinite dimensional riccati equations abstract comparison theorems are proved for the algebraic riccati equation in the framework of infinite dimensional hilbert spaces these theorems extend well known results for the finite dimensional case

convergence of natural adaptive least squares finite element methods the first order div least squares finite element methods provide inherent a posteriori error estimator by the elementwise evaluation of the functional in this paper we prove q linear convergence of the associated adaptive mesh refining strategy for a sufficiently fine initial mesh with some sufficiently large bulk parameter for piecewise constant right hand sides in a poisson model problem the proof relies on some modification of known supercloseness results to non convex polygonal domains plus the flux representation formula the analysis is carried out for the lowest order case in two dimensions for the simplicity of the presentation

efficient stress velocity least squares finite element formulations for the incompressible navier stokes equations abstract in this contribution three different mixed least squares finite element methods lsfems are investigated with respect to accuracy and efficiency with simultaneous consideration of regular and adaptive meshing strategies the deliberations are made for the incompressible navier stokes equations the generally known first order div grad system in terms of the total stress velocity and pressure svp formulation is the basis for two further div grad least squares formulations in terms of total stress and velocity sv whereby both formulations are derived from different fountainheads the extended sv formulation is an enrichment of the known stress velocity formulation proposed by cai et al 2004 the second sv formulation is based on the substitution of the pressure in the svp formulation thus it is based on a discontinuous pressure interpolation advantage of the sv formulations is a smaller system matrix size due to the reduction of the relevant degrees of freedom nevertheless the approximation quality has to last the demand of the established least squares formulations the drawback of a poor mass conservation is well investigated for the lsfem and known to overcome by using high order interpolations for all solution variables therefore the main attention of this contribution is focused on accuracy while aiming for high efficiency which is intrinsic for the fundamental idea in here in the light of efficient lsfem solutions the advantage of the inherent a posteriori error estimator of the method is used for deliberations on different marking strategies in an h type adaptive mesh refinement

combinatorial enumeration and group representations it is shown that in addition to the number of distinct equivalence classes the representation characters of a labelling can also be derived using standard combinatorial methods these characters provide additional information on the structure of the equivalence classes which is relevant in physical applications

dissipativity and the lur e problem for parabolic boundary control systems dissipativity is studied for a class of boundary control systems whose free evolution is described by a holomorphic semigroup it is proved that the system is dissipative if and only if there exists a bounded selfadjoint solution of the corresponding linear operator inequality in addition the lur e problem is solved under particular assumptions

branching processes in the annni model the mean field equations of the simple cubic annni model are studied on finite lattices the results are consistent with the sequence of distinct commensurate phases 2k 13 k 1 2 3 springing from the multiphase point found using low temperature series expansions moreover evidence for new structure combination branching processes is presented which generate phases of type 2l3 m 2l 13 n or 23l m 23l 1 n where l m and n are integers

nonlinear discontinuous petrov galerkin methods the discontinuous petrov galerkin method is a minimal residual method with broken test spaces and is introduced for a nonlinear model problem in this paper its lowest order version applies to a nonlinear uniformly convex model example and is equivalently characterized as a mixed formulation a reduced formulation and a weighted nonlinear least squares method quasi optimal a priori and reliable and efficient a posteriori estimates are obtained for the abstract nonlinear dpg framework for the approximation of a regular solution the variational model example allows for a built in guaranteed error control despite inexact solve the subtle uniqueness of discrete minimizers is monitored in numerical examples

quasi optimal convergence rate for an adaptive finite element method we analyze the simplest and most standard adaptive finite element method afem with any polynomial degree for general second order linear symmetric elliptic operators as is customary in practice the afem marks exclusively according to the error estimator and performs a minimal element refinement without the interior node property we prove that the afem is a contraction for the sum of the energy error and the scaled error estimator between two consecutive adaptive loops this geometric decay is instrumental to derive the optimal cardinality of the afem we show that the afem yields a decay rate of the energy error plus oscillation in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity

the kalman yakubovich popov lemma for pritchard salamon systems in this paper we generalize the kalman yakubovich popov lemma to the pritchard salamon class of infinite dimensional systems i e systems determined by semigroups of operators on a hilbert space with unbounded input and output operators at the same time we prove a comparison theorem for riccati equations of the same class

an adaptive least squares fem for the stokes equations with optimal convergence rates this paper introduces the first adaptive least squares finite element method ls fem for the stokes equations with optimal convergence rates based on the newest vertex bisection with lowest order raviart thomas and conforming p 1 p1 discrete spaces for the divergence least squares formulation in 2d although the least squares functional is a reliable and efficient error estimator the novel refinement indicator stems from an alternative explicit residual based a posteriori error control with exact solve particular interest is on the treatment of the data approximation error which requires a separate marking strategy the paper proves linear convergence in terms of the levels and optimal convergence rates in terms of the number of unknowns relative to the notion of a non linear approximation class it extends and generalizes the approach of carstensen and park siam j numer anal 53 43 62 2015 from the poisson model problem to the stokes equations

unique labelling of many electron states in multicentre systems general methods of generating complete sets of states for l or 2n electrons in s states on n centres are described it is shown that a unique labelling of these states can always be obtained using a fictitious sn symmetry of the n centres explicit descriptions of eight seven and four electrons on eight centres are derived and the problem of determining a unique labelling for these states on rigid eight centre structures with cubic and lower symmetry is solved

the unique labelling of states via mackey s theorem a new approach to finding unique labels for many electron states in multicentre systems is described and a particular example eight electrons in s states on eight centres is worked out in detail this approach is shown to provide a labelling with a more direct physical interpretation than has previously been found

fast computation in adaptive tree approximation adaptive methods of approximation arise in many settings including numerical methods for pdes and image processing they can usually be described by a tree which records the adaptive decisions this paper is concerned with the fast computation of near optimal trees based on n adaptive decisions the best tree based on n adaptive decisions could be found by examining all such possibilities however this is exponential in n and could be numerically prohibitive the main result of this paper is to show that it is possible to find near optimal trees using computations linear in n

monotonicity of maximal solutions of algebraic riccati equations abstract a monotonicity result for the maximal solution of the equation xbb x a x xa q 0 q q a b stabilizable is proved

design of consensus and adaptive consensus filters for distributed parameter systems this work establishes an abstract framework that considers the distributed filtering of spatially varying processes using a sensor network it is assumed that the sensor network consists of groups of sensors each of which provides a number of state measurements from sensing devices that are not necessarily identical and which only transmit their information to their own sensor group a modification to the local spatially distributed filters provides the non adaptive case of spatially distributed consensus filters which penalize the disagreement amongst themselves in a dynamic manner a subsequent modification to this scheme incorporates the adaptation of the consensus gains in the disagreement terms of all local filters both the well posedness of these two consensus spatially distributed filters and the convergence of the associated observation errors to zero in appropriate norms are presented their performance is demonstrated on three different examples of a diffusion partial differential equation with point measurements

convergence and optimality of adaptive least squares finite element methods the first order div least squares finite element methods lsfems allow for an immediate a posteriori error control by the computable residual of the least squares functional this paper establishes an adaptive refinement strategy based on some equivalent refinement indicators since the first order div lsfem measures the flux errors in h div the data resolution error measures the l 2 norm of the right hand side f minus the piecewise polynomial approximation pi f without a mesh size factor hence the data resolution term is neither an oscillation nor of higher order and consequently requires a particular treatment e g by the thresholding second algorithm due to binev and devore the resulting novel adaptive lsfem with separate marking converges with optimal rates relative to the notion of a nonlinear approximation class

classification of configurations of systems of identical particles on finite sets all possible configurations of a system of n identical particles are classified for the case when the configuration space q of a single particle reduces to a finite set the classification is achieved by means of a combinatorial analogue of the duality of weyl two groups the pauli group permuting identical particles and the symmetric group on q act on the nth cartesian power of q these actions mutually commute and their orbit structure is classified by appropriate epikernels moreover each stratum of each group is invariant under the dual action which yields a coarsening of permutation representations acting on appropriate sets of orbits it is shown that such a coarsening recovers the geometric symmetry for the case when particles are indistinguishable

classification and determination of irreducible bases for induced representations for chains of subgroups of finite groups the problem of classification and determination of standard irreducible bases for induced representations of an arbitrary finite group is discussed a general prescription for classification and determination of these bases for an arbitrary induced representation is proposed and an extra independent prescription for the special case of primitive representations is given relations between appropriate transformation coefficients are investigated and in particular a matrix similar to the racah recoupling matrix from theory of multiple coupling of angular momentum is introduced

structure and representations of the symmetry group of the four dimensional cube in this paper we give explicitly the description of the structure the characters and the complete system of irreducible representations of the hyperoctahedral group in four dimensions which we call w4 from the german wurfel in a second step we do the same for the subgroup sw4 which is formed by all pure rotations contained in w4

symmetry coordinates for oh atomic shells the group factorisation and correlation theorem methods used previously to provide a unique labelling of atomic shell symmetry coordinates are shown to give also a simple means for deriving explicit expressions for these coordinates results are given for the 12 atom shell and the two types of 24 atom shells in oh symmetry

unique labelling of symmetry coordinates and cyclic regions in crystals earlier work on unique labelling schemes in oh symmetry is extended to include 48 atom shells and the symmetry coordinates of shells subject to periodicity conditions a new approach to labelling is introduced based on the concept of cyclic regions in a crystal these are described by finite factor groups of the crystallographic space group which contain the isogonal group as a subgroup some examples are worked out in detail

algebraic riccati equation and symplectic algebra questions of existence uniqueness and the parametric dependence of solutions of the algebraic riccati equation are considered the different criteria for the solubility of this equation are obtained with the help of symplectic algebra

minimum phase infinite dimensional second order systems in general better performance can be achieved with a controlled minimum phase system than a controlled nonminimum phase system we show that a wide class of second order infinite dimensional systems with either velocity or position measurements are minimum phase the results are illustrated by two examples

an inexpensive method for the evaluation of the solution of the lowest order raviart thomas mixed method for the model problem operatorname div a nabla u f we show how the approximate solution produced by the raviart thomas lecture notes in mathematics 606 springer verlag berlin 1977 mixed method of lowest degree can be obtained from the solution of the p 1 nonconforming galerkin method modified in a virtually cost free manner

a short note on plain convergence of adaptive least squares finite element methods abstract we show that adaptive least squares finite element methods driven by the canonical least squares functional converge under weak conditions on pde operator mesh refinement and marking strategy contrary to prior works our plain convergence does neither rely on sufficiently fine initial meshes nor on severe restrictions on marking parameters finally we prove that convergence is still valid if a contractive iterative solver is used to obtain the approximate solutions e g the preconditioned conjugate gradient method with optimal preconditioner the results apply within a fairly abstract framework which covers a variety of model problems