parametric output feedback controller design abstract a parameterization of fixed gain output feedback controllers which will assign a complete set of distinct eigenvalues to a given state space system is presented the main result is a compact parametric expression for the output feedback controller gain matrix explicitly characterized by two sets of free parameters the set of n closed loop system eigenvalues and a set of mr n design parameters for n state r input m output systems the free design parameters are determined by a sequential design procedure which automatically satisfies the basic orthogonality constraints on the left and right closed loop eigenvectors under the condition m r n the principal benefits of the explicit characterization of a parametric class of output feedback controllers lie in the ability to directly accommodate various different design criteria an illustration of the parametric output feedback design procedure is given by way of approximating the optimal full state regulator response of a fifth order pilot plant evaporator model

brief paper output regulation for linear distributed parameter systems using finite dimensional dual observers in this article the solution of the output regulation problem is considered for linear infinite dimensional systems where the outputs to be controlled cannot be measured it is shown that this problem can be solved by a finite dimensional dual observer that is directly implementable so that the separation principle can be applied for the stabilization as in finite dimensions a parametric design of these dual observers is proposed for riesz spectral systems that allows to achieve a low controller order and a desired control performance for the closed loop system the presented results are illustrated by determining a finite dimensional regulator for an euler bernoulli beam with kelvin voigt damping that achieves tracking for steplike reference inputs and that asymptotically rejects sinusoidal disturbances

smoothed projections in finite element exterior calculus the development of smoothed projections constructed by combining the canonical interpolation operators defined from the degrees of freedom with a smoothing operator have proved to be an eective tool in finite element exterior calculus the advan tage of these operators is that they are l 2 bounded projections and still they commute with the exterior derivative in the present paper we generalize the construction of these smoothed projections such that also non quasi uniform meshes and essential boundary conditions are covered the new tool introduced here is a space dependent smoothing operator which commutes with the exterior derivative

generalized quantum onsager coefficients from a symmetrized campbell hausdorff expansion the analogue of the classical onsager theory of entropy production is systematically derived for weakly irreversible processes in open quantum systems with finite dimensional hilbert space the dynamics is assumed to be given by a quantum dynamical semigroup with infinitesimal generator of gorini kossakowski sudarshan type the basic spohn formula for entropy production is used to obtain an expansion in terms of powers of the deviation of the initial state relative to the final stationary state of irreversible dynamics to this end an appropriate lie series is constructed from a particular symmetrization procedure applied to the ordinary campbell hausdorff expansion in this way only hermitian contributions by higher order commutators are generated which allow an identification with so called generalized onsager coefficients the explicit derivations concentrate on second third and fourth order coefficients whereas complete detailed expressions are worked out for second and third order in a suitable coherence vector representation of density matrices the results can be given in terms of the dynamical parameters fixing the infinitesimal semigroup generator and in terms of symmetric and antisymmetric structure constants of the lie algebra of su n as an illustration an application to generalized bloch equations for two level systems is studied where the onsager like expansion can be compared with exact results for entropy production we find that convergence is good even for rather large deviations between initial and final state if the calculation includes second and third order coefficients only the formalism presented in this paper generalizes restrictions on admitted final states adopted in much simpler earlier treatments to the most general case of arbitrary unique final states of irreversible processes

an efficient method to factorize fuzzy attribute oriented concept lattices abstract factorization by similarity is a mathematical technique used in formal concept analysis with fuzzy attributes for reducing the complexity of different types of fuzzy concept lattices in this paper we find the structure of the factor lattice of a fuzzy attribute oriented concept lattice namely the intervals representing the blocks of this lattice we provide a procedure for generating the infimum and the supremum concepts of these intervals as fixpoints of a fuzzy closure operator this theoretical result allows to develop a more efficient algorithm for building the factor lattice of the fuzzy attribute oriented concept lattice a comparison between our approach and the existing algorithms is presented

stability of hodge decompositions in finite element spaces of differential forms in arbitrary dimension we elaborate on the interpretation of some mixed finite element spaces in terms of differential forms in particular we define regularization operators which combined with the standard interpolators enable us to prove discrete poincare friedrichs inequalities and discrete rellich compactness for finite element spaces of differential forms of arbitrary degree on compact manifolds of arbitrary dimension

discrete differential forms for 1 1 dimensional cosmological space times in this article we describe applications of the numerical method of discrete differential forms in computational general relativity gr in particular we consider the initial value problem for vacuum space times that admit plane gravitational waves as described in an earlier paper the discrete differential form approach provides accurate results in spherically symmetric static space times r richter j frauendiener and m vogel classical quantum gravity 24 2007 p 433 and it is manifestly coordinate independent here we extend the method to time dependent systems we use the polarized gowdy solution as a testbed for two numerical schemes one scheme reproduces that solution very well in particular it is stable for a long time and converges quadratically

higher order operator splitting methods via zassenhaus product formula theory and applications abstract in this paper we contribute higher order operator splitting methods improved by zassenhaus product we apply the contribution to classical and iterative splitting methods the underlying analysis to obtain higher order operator splitting methods is presented while applying the methods to partial differential equations the benefits of balancing time and spatial scales are discussed to accelerate the methods the verification of the improved splitting methods are done with numerical examples an individual handling of each operator with adapted standard higher order time integrators is discussed finally we conclude the higher order operator splitting methods

on the linearization of regge calculus we study the linearization of three dimensional regge calculus around euclidean metric we provide an explicit formula for the corresponding quadratic form and relate it to the curl t curl operator which appears in the quadratic part of the einstein hilbert action and also in the linear elasticity complex we insert regge metrics in a discrete version of this complex equipped with densely defined and commuting interpolators we show that the eigenpairs of the curl t curl operator approximated using the quadratic part of the regge action on regge metrics converge to their continuous counterparts interpreting the computation as a non conforming finite element method

a parametric approach to finite dimensional control of linear distributed parameter systems this article considers the design of finite dimensional compensators for distributed parameter systems using eigenvalue assignment the proposed compensator consists of an observer estimating additional outputs and a static feedback of the measurable and the estimated outputs since the additional outputs can be asymptotically reconstructed the compensator can be designed using the separation principle i e the closed loop eigenvalues are given by the observer eigenvalues and the eigenvalues resulting from the static output feedback control in order to solve the corresponding eigenvalue assignment problem the parametric approach for the design of static output feedback controllers in finite dimensions is extended to distributed parameter systems by using a parameter optimisation it is possible to assign all closed loop eigenvalues within specified regions of the complex plane in order to stabilise the system and to assure a desired control performance a heat conductor is used to demonstrate the prop

exponential operators and the algebraic description of quantum confined systems we study the relations and transformations produced by exponential operators the argument of which are dependent on the basic algebraic elements of supersymmetric and shape invariant potential systems and obtain explicit expressions we apply our results to self similar potential systems and to a set of translational shape invariant systems including the morse poschl teller scarf and rosen morse potentials and obtain closed form expressions we show that our results reproduce those obtained for the harmonic oscillator in the appropriate limits

maximal reductions in the baker hausdorff formula a preliminary expression for the baker hausdorff formula is found up to ninth order i e a series expansion of z in terms of multiple commutators where ez exey with x and y noncommuting up to ninth degree in x y by means of complete sets of linear relations between multiple commutators maximal reduction of the number of different multiple commutators in the series are obtained

finite dimensional observer based control of linear distributed parameter systems using cascaded output observers the synthesis of compensators for linear distributed parameter systems on the basis of a finite dimensional approximation is a classical technique this so called early lumping approach suffers from the occurrence of spillover which means that the closed loop dynamics is deteriorated by the neglected dynamics in this contribution an early lumping compensator design is presented that overcomes the spillover problem by using certain fictitious outputs which can be reconstructed without spillover and that suppress the contributions of the unmodelled dynamics when the new outputs are used for the compensator instead of the available measurements the perturbation of the closed loop spectrum caused by spillover can be reduced to an arbitrary extent it is shown that the worst case eigenvalue perturbation decreases exponentially with respect to the compensator order so that the spillover can be reduced systematically in addition an a priori estimate for the compensator order that guarantees a prescribed

relating attribute reduction in formal object oriented and property oriented concept lattices attribute reduction is an important step in reducing computational complexity in order to extract information from relational systems three of these systems are the formal object oriented and property oriented concept lattices attribute reduction in the last two concept lattices has recently been studied the relation with the first concept lattice is very important since two important independent tools to extract information from databases the formal concept analysis and rough set theory will be related this paper studies attribute reduction in these three frameworks the main results are that the classification of each attribute into absolutely necessary relatively necessary and absolutely unnecessary attributes is independent of the framework considered and that an attribute reduct in one of these relational systems is also an attribute reduct in the others

fault detection for linear distributed parameter systems using finite dimensional functional observers abstractin this article finite dimensional residual generators are directly designed for riesz spectral systems with bounded input and output operators to detect faults this is achieved by using finite dimensional observers that can estimate linear functionals of the state without spillover these observers allow for a decoupling of the unknown disturbances from the estimation error dynamics under mild assumptions then a finite dimensional residual generator is obtained by approximately decoupling the state from the residual that is generated by the observer states and the outputs it is shown that the resulting approximation error can be made small by increasing the observer order then fault detection with the finite dimensional residual generator can be assured by introducing a time varying threshold a faulty euler bernoulli beam with structural damping illustrates the proposed finite dimensional fault detection approach

on the convergence and optimization of the baker campbell hausdorff formula abstract in this paper the problem of the convergence of the baker campbell hausdorff series for z log e x e y is revisited we collect some previous results about the convergence domain and present a new estimate which improves all of them we also provide a new expression of the truncated lie presentation of the series up to sixth degree in x and y requiring the minimum number of commutators numerical experiments suggest that a similar accuracy is reached with this approximation at a considerably reduced computational cost

attribute oriented fuzzy concept lattices and standard fuzzy concept lattices induce the same similarity on objects abstract g ciobanu c v ideanu 2015 1 define the similarity of objects in attribute oriented fuzzy concept lattice and claim that it is different from the similarity of objects induced by the corresponding fuzzy context we show that this claim is wrong and that the two similarities are equal as a consequence we get that attribute oriented concept lattices and standard concept lattices induce the same similarity on objects

on the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form in the approximation of linear elliptic operators in mixed form it is well known that the so called inf sup and ellipticity in the kernel properties are sufficient and in a sense to be made precise necessary in order to have good approximation properties and optimal error bounds one might think in the spirit of mercier osborn rappaz raviart and in consideration of the good behavior of commonly used mixed elements like raviart thomas or brezzi douglas marini elements that these conditions are also sufficient to ensure good convergence properties for eigenvalues in this paper we show that this is not the case in particular we present examples of mixed finite element approximations that satisfy the above properties but exhibit spurious eigenvalues such bad behavior is proved analytically and demonstrated in numerical experiments we also present additional assumptions fulfilled by the commonly used mixed methods already mentioned which guarantee optimal error bounds for eigenvalue approximations as well

a div curl lemma for edge elements a variant of murat and tartar s div curl lemma is stated and proved for nedelec s edge elements given two sequences of vector fields of this type converging weakly in l2 as the mesh width tends to 0 we prove that their scalar product converges in the sense of distributions when one of the sequences consists of so called discrete divergence free fields whereas the other has relatively compact curl in h 1 the proof uses a uniform norm equivalence related to discrete compactness properties of vector finite element spaces and a super approximation property of scalar finite element spaces

diamond elements a finite element discrete mechanics approximation scheme with guaranteed optimal convergence in incompressible elasticity we present a finite element discretization scheme for the compressible and incompressible elasticity problems that possess the following properties i the discretization scheme is defined on a triangulation of the domain ii the discretization scheme is defined and is identical in all spatial dimensions iii the displacement field converges optimally with mesh refinement and iv the inf sup condition is automatically satisfied the discretization scheme is motivated both by considerations of topology and analysis and it consists of the combination of a certain mesh pattern and a choice of interpolation that guarantees optimal convergence of displacements and pressures rigorous proofs of the satisfaction of the inf sup condition are presented for the problem of linearized incompressible elasticity we additionally show that the discretization schemes can be given a compelling interpretation in terms of discrete differential operators in particular we develop a discrete analogue of the classical tensor differential complex in terms of which the discrete and continuous boundary value problems are formally identical we also present numerical tests that demonstrate the dimension independent scope of the scheme and its good performance in problems of finite elasticity

operator splitting methods via the zassenhaus product formula abstract in this paper we contribute an operator splitting method improved by the zassenhaus product zassenhaus products are of fundamental importance for the theory of lie groups and lie algebras while their applications in physics and physical chemistry are important novel applications in cfd computational fluid dynamics arose based on the fact that their sparse matrices can be seen as generators of an underlying lie algebra we apply this to classical splitting and the novel zassenhaus product formula the underlying analysis for obtaining higher order operator splitting methods based on the zassenhaus product is presented the benefits of dealing with sparse matrices given by spatial discretization of the underlying partial differential equations are due to the fact that the higher order commutators are very quickly computable their matrix structures thin out and become nilpotent when applying these methods to convection diffusion reaction equations the benefits of balancing time and spatial scales can be used to accelerate these methods and take into account these sparse matrix structures the verification of the improved splitting methods is done with numerical examples finally we conclude with higher order operator splitting methods

the baker campbell hausdorff formula and nested commutator identities the coefficients of the baker campbell hausdorff expansion are calculated by using various methods comparison of the results yields several remarkable identities satisfied by multiple commutators which in turn allow us to greatly simplify the form of the expansion up to order eight

finite dimensional dual state feedback control of linear boundary control systems in this article the finite dimensional control of distributed parameter systems with boundary control and unbounded observation is considered this problem is solved by extending the concept of dual observer based compensators to infinite dimensions these compensators have the advantage that on the one hand the separation principle can be used to directly determine a finite dimensional compensator and on the other hand they can be directly applied to boundary control systems with a time derivative of the input in order to achieve a low controller order as well as a desired control performance a parametric approach is proposed to parameterise the degrees of freedom contained in the controller consequently desired design specifications can be achieved by using a parameter optimisation the proposed design of finite dimensional dual observer based compensators is demonstrated by means of a heat conducting system with neumann boundary control and boundary measurements

from time ordered products to magnus expansion recent work by lam on the decomposition of time ordered products appearing in the time evolution operator in terms of sums of products of nested commutators is endowed with a recursive algorithm a sufficient condition for absolute convergence is subsequently obtained connection with magnus expansion is established

an efficient algorithm for computing the baker campbell hausdorff series and some of its applications we provide a new algorithm for generating the baker campbell hausdorff bch series z log exey in an arbitrary generalized hall basis of the free lie algebra l x y generated by x and y it is based on the close relationship of l x y with a lie algebraic structure of labeled rooted trees with this algorithm the computation of the bch series up to degree of 20 111 013 independent elements in l x y takes less than 15min on a personal computer and requires 1 5gbytes of memory we also address the issue of the convergence of the series providing an optimal convergence domain when x and y are real or complex matrices

goldberg s theorem and the baker campbell hausdorff formula abstract goldberg s theorem duke math j 23 1956 13 is applied to expressing the logarithm of exponential product in terms of free lie elements using hyperoperators namely inner derivations a symmetric property of the expansion coefficients in the formal power series is also discussed

rough set driven approach for attribute reduction in fuzzy formal concept analysis abstract the reduction of the set of attributes is an important preliminary challenge in order to obtain information from knowledge systems two remarkable formal tools for extracting such information are rough set theory rst and formal concept analysis fca as well as their fuzzy generalizations this work introduces a new method to reduce attributes in fuzzy fca considering the reduction philosophy given in rst and studies its main properties this method allows us to carry out a deeper study of the relation between these two theories moreover the proposed methodology has been compared with other existing reduction mechanisms

dynkin s method of computing the terms of the baker campbell hausdorff series the infinite series for log exp x exp y for noncommuting x and y is expressible in terms of iterated commutators of x and y except for the linear term x y dynkin derived an explicit expression for the terms as a sum of iterated commutators over a certain set of sequences this paper presents a practical algorithm for applying dynkin s formula and gives several illustrative examples

a simple expression for the terms in the baker campbell hausdorff series a simple expression is derived for the terms in the baker campbell hausdorff series one formulation of the result involves a finite number of operations with matrices of rational numbers generalizations are discussed

the baker campbell hausdorff formula and the convergence of the magnus expansion the authors show that for a wide class of dynamical systems described by hamiltonians of the form usually considered in time dependent perturbation theory the divergence of the magnus expansion in the schrodinger picture for large time intervals is due to pole singularities inherent to the baker campbell hausdorff formula

similarity relations in fuzzy attribute oriented concept lattices the study of concept lattices and oriented concept lattices with fuzzy attributes provides complementary conceptual structures which can be used to search analyze and extract information from large datasets in this paper we put forward new types of similarity relations between objects or attributes in fuzzy attribute oriented concept lattices we analyze how these similarities are related using the definitions and considering the reducibility between antitone and isotone fuzzy concept lattices as well a comparison is made between two relations which measure the similarity of fuzzy oriented concepts these relations are employed in factorizing the fuzzy attribute oriented concept lattice in order to reduce its complexity

consensus based formation control laws for systems on lie groups this paper designs the consensus based formation control laws for multi agent systems defined on lie groups the specific results are illustrated on se 2 and se 3 the special euclidean groups of rigid body motions and each of which shares the geometric structure of a lie group the paper develops the control laws for the problem concerned under the assumption that the communication topologies of the multi agent systems are spanning tree shaped ring shaped and complete respectively at last several examples are included in the numerical simulations to verify the theoretical results

heat kernel regularization of the effective action for stochastic reaction diffusion equations the presence of fluctuations and nonlinear interactions can lead to scale dependence in the parameters appearing in stochastic differential equations stochastic dynamics can be formulated in terms of functional integrals in this paper we apply the heat kernel method to study the short distance renormalizability of a stochastic polynomial reaction diffusion equation with real additive noise we calculate the one loop effective action and its ultraviolet scale dependent divergences we show that for white noise a polynomial reaction diffusion equation is one loop finite in d 0 and d 1 and is one loop renormalizable in d 2 and d 3 space dimensions we obtain the one loop renormalization group equations and find they run with scale only in d 2

concept lattices of isotone vs antitone galois connections in graded setting mutual reducibility revisited it is well known that concept lattices of isotone and antitone galois connections induced by an ordinary binary relation and its complement are isomorphic via a natural isomorphism mapping extents to themselves and intents to their complements it is also known that in a fuzzy setting this and similar kinds of reduction fail to hold in this note we show that when the usual notion of a complement based on a residuum w r t 0 is replaced by a new one based on residua w r t arbitrary truth degrees the above mentioned reduction remains valid for ordinary relations the new and the usual complement coincide the result we present reveals a new deeper root of the reduction it is not the availability of the law of double negation but rather the fact that negations are implicitly present in the construction of concept lattices of isotone galois connections

a note on the zassenhaus product formula we provide a simple method for the calculation of the terms cn in the zassenhaus product ea b ea eb n 2 ecn for noncommuting a and b this method has been implemented in a computer program furthermore we formulate a conjecture on how to translate these results into nested commutators this conjecture was checked up to order n 17 using a computer

variations on a theme of jost and pais abstract we explore the extent to which a variant of a celebrated formula due to jost and pais which reduces the fredholm perturbation determinant associated with the schrodinger operator on a half line to a simple wronski determinant of appropriate distributional solutions of the underlying schrodinger equation generalizes to higher dimensions in this multi dimensional extension the half line is replaced by an open set r n n n n 2 where has a compact nonempty boundary satisfying certain regularity conditions our variant involves ratios of perturbation determinants corresponding to dirichlet and neumann boundary conditions on and invokes the corresponding dirichlet to neumann map as a result we succeed in reducing a certain ratio of modified fredholm perturbation determinants associated with operators in l 2 d n x n n to modified fredholm determinants associated with operators in l 2 d n 1 n 2 applications involving the birman schwinger principle and eigenvalue counting functions are discussed