locally definite normal operators in krein spaces abstract we introduce the spectral points of two sided positive type of bounded normal operators in krein spaces it is shown that a normal operator has a local spectral function on sets which are of two sided positive type in addition we prove that the riesz dunford spectral subspace corresponding to a spectral set which is only of positive type is uniformly positive the restriction of the operator to this subspace is then normal in a hilbert space

on the negative squares of indefinite sturm liouville operators abstract the number of negative squares of all self adjoint extensions of a simple symmetric operator of defect one with finitely many negative squares in a krein space is characterized in terms of the behaviour of an abstract titchmarsh weyl function near 0 and these results are applied to a large class of symmetric and self adjoint indefinite sturm liouville operators with indefinite weight functions

super cocommutative hopf algebras of finite representation type abstract given an algebraically closed field k of characteristic p 5 we classify the finite algebraic k supergroups whose algebras of measures are of finite representation type let g be such a supergroup and g the largest ordinary algebraic k group determined by g we show that both g and u lie g the restricted enveloping algebra of lie superalgebra of g are of finite representation type moreover only some special representation finite algebraic k groups of dimension zero are shown to appear if g g the structure of g is almost determined by g and u lie g the auslander reiten quivers are determined by showing that they are nakayama algebras

the varieties and the cohomology ring of a module let g be a finite group and let k be a field of characteristic p 0 recently several studies have focused on the homological invariants of finitely generated kg modules in l alperin proposed the study of the complexity of a module the complexity is related to the degree of the polynomial rate of growth of the terms in a projective resolution of the module see definition 2 4 in 2 alperin and evens proved that the complexity of a kg module m is equal to the maximum of the complexities of the restrictions of m to elementary abelian p subgroups of g one of the roots of the alperin evens theorem is quillen s dimension theorem see 21 or 22 which with some difficulty can be interpreted as saying the same thing for the special case of the trivial kg module alperin and evens 3 and avrunin 4 have further studied the annihilator in ext k k of the cohomology of m and have produced theorems of a similar nature these are related to the work in 9 on the structure of the cohomology ring ext m m the purpose of this paper is to investigate the relationship between these homological invariants and the structure of the module we concentrate on the case in which g is an elementary abelian p group however using such results as those mentioned above we can apply the present work to more general groups even as the earlier results indicate that much of the structure and cohomology of a module is revealed in the restrictions to elementary abelian p subgroups we show here that in the case of an elementary abelian group much of this information can be found by looking at the restrictions to cyclic p subgroups of the group of units of kg some of the results in this paper were announced in s let g xl xn be an elementary abelian group of order p suppose that k is algebraically closed to any finitely generated kg module m we

a wider class of stable gyroscopic systems abstract it is well known that in a physically interesting sense n n gyroscopic systems of the form l 2 i b c where c 0 and b is indefinite and invertible are stable whenever b ki k 1 c for some real k 0 it is shown that stability is retained under a considerably weaker condition formulated in terms of the spectral radius of b 1 i 1 c

the tame infinitesimal groups of odd characteristic abstract given an infinitesimal group g over an algebraically closed field k of characteristic p 3 we provide criteria for the principal block b 0 g of its algebra of distributions to be of tame representation type these are employed in conjunction with galois coverings to determine the structure of g modulo its multiplicative center as well as the quiver and the relations of the algebra b 0 g

eigenvalues of schrodinger operators with definite and indefinite weights in this paper computational algorithms are presented to compute the eigenvalues of schrodinger operators with definite and indefinite weights the algorithms are based on titchmarsh weyl s theory and can be applied to solve a wide class of problems in quantum mechanics explicit examples are presented some of which recover the known exact solutions finally some analytical bounds have been reviewed for definite and indefinite singular problems weyl s spectral bisection algorithm has been proposed to solve definite problems as algebraic problems open problems have been presented based on the results of the presented algorithms

on the spectral theory of singular indefinite sturm liouville operators we consider a singular sturm liouville differential expression with an indefinite weight function and we show that the corresponding self adjoint differential operator in a krein space locally has the same spectral properties as a definitizable operator

elliptic eigenvalue problems with eigenparameter dependent boundary conditions abstract let a be a uniformly elliptic second order linear operator on a smooth bounded domain r n we study the eigenvalue problem au u subject to boundary conditions b0u b1u on where bj are linear boundary operators the problem is recast in the form a u u in a hilbert or krein space and results are given on the location and type of the spectrum full and half range completeness and regularity of critical points

sturm liouville operators with indefinite weight functions and eigenvalue depending boundary conditions we consider a class of boundary value problems for sturm liouville operators with indenite weight functions the spectral parameter appears nonlinearly in the boundary condition in the form of a function which has the property that 7 is a generalized nevanlinna function we construct linearizations of these boundary value problems and study their spectral properties

on finite rank perturbations of definitizable operators it was shown by p jonas and h langer that a selfadjoint definitizable operator a in a krein space remains definitizable after a finite rank perturbation in resolvent sense if the perturbed operator b is selfadjoint and the resolvent set b is nonempty it is the aim of this note to prove a more general variant of this perturbation result where the assumption on b is dropped as an application a class of singular ordinary differential operators with indefinite weight functions is studied

spectral points of type π and π of self adjoint operators in krein spaces via approximative eigensequences we introduce the notion of spectral points of type and for self adjoint operators in krein spaces they are stable under compact perturbations for real spectral points of type and which are not in the interior of the spectrum we prove that the growth of the resolvent in some neighbourhood of them is of finite order there exists a local spectral function with singularities it turns out that all spectral subspaces corresponding to sufficiently small neighbourhoods of points of type or type are pontryagin spaces

on cocommutative hopf algebras of finite representation type abstract let g be a finite algebraic group defined over an algebraically closed field k of characteristic p 0 such a group decomposes into a semidirect product g g 0 g red with a constant group g red and a normal infinitesimal subgroup g 0 if the principal block b 0 g of the group algebra h g has finite representation type then both constituents have the same property with at least one of them being semisimple we determine the structure of the infinitesimal constituent g 0 up to the classification of v uniserial groups

positive galois coverings of self injective algebras abstract in the article we study the structure of galois coverings of self injective artin algebras with infinite cyclic galois groups in particular we characterize all basic connected self injective artin algebras having galois coverings by the repetitive algebras of basic connected artin algebras and with the galois groups generated by positive automorphisms of the repetitive algebras

tame biserial algebras a similar result is suspected if we drop the property representationfinite in classes 1 and 2 and we will start an investigation of biserial algebras of infinite representation type the key to the result for representation finite algebras was to observe that each representation finite biserial algebra is special 13 see also the basic notations below which means that it can be presented by a quiver with nice relations special algebras play an important role in the modular representation theory of finite groups namely each representation finite block of a group algebra and some of the tame blocks they occur only in characteristic 2 are special 7 10 4 moreover in the complex representation theory of the lorentz group the so called harish chandra modules are defined over tame special algebras 6 in this paper we show that l a 62 for any special algebra a and that special algebras of infinite representation type are tame our paper consists of four parts first we prove that each special algebra is a factor of a special symmetric algebra theorem 1 5 for the latter the methods of gelfand and ponomarev used in the classification of the indecomposable harish chandra modules of the lorentz group apply and furnish a complete list of indecomposable finitely generated modules over special algebras proposition 2 3 presented in section 2 from this list we 480 0021 8693 85 3 00

accumulation of complex eigenvalues of indefinite sturm liouville operators spectral properties of singular sturm liouville operators of the form with the indefinite weight x sgn x on are studied for a class of potentials with lim x v x 0 the accumulation of complex and real eigenvalues of a to zero is investigated and explicit eigenvalue problems are solved numerically

on the structure of tame graded basic hopf algebras ii abstract in this paper firstly we give a complete list of tame local graded frobenius algebras then by the method of bosonization and this list we determine the structure of tame radically graded basic hopf algebras completely

galois actions and blocks of tame infinitesimal group schemes given an infinitesimal group that is defined over an algebraically closed field of characteristic we determine the block structure of the algebra of measures in case its principal block is tame and the height of the factor group of by its multiplicative center is at least two our results yield a complete description of the stable auslander reiten quiver of along with a criterion for the domesticity of

improved cost functions for modelling of noisy chaotic time series abstract standard least squares cost functions yield unbiased results only if the independent variables are noise free which is not the case in time series analysis new minimization problems for the determination of the dynamics underlying noisy chaotic data are formulated which can overcome the problem of noise in the independent variables for a given model these improved cost functions give the chance to estimate its parameters with a strongly reduced bias in the case of large amplitude measurement noise the method is illustrated by the help of numerical and experimental examples results for dynamical noise are discussed

self adjoint analytic operator functions and their local spectral function for a self adjoint analytic operator function a which satisfies on some interval of the real axis the virozub matsaev condition a local spectral function q on the values of which are non negative operators is introduced and studied in the particular case that a i a with a self adjoint operator a it coincides with the orthogonal spectral function of a an essential tool is a linearization of a by means of a self adjoint operator in some krein space and the local spectral function of this linearization the main results of the paper concern properties of the range of q and the description of a natural complement of this range

polyhedral groups mckay quivers and the finite algebraic groups with tame principal blocks given an algebraically closed field k of characteristic p 3 we classify the finite algebraic k groups whose algebras of measures afford a principal block of tame representation type the structure of such a group mathcal g is largely determined by a linearly reductive subgroup scheme hat mathcal g of sl 2 with the mckay quiver of hat mathcal g relative to its standard module being the gabriel quiver of the principal block mathcal b 0 mathcal g the graphs underlying these quivers are extended dynkin diagrams of type tilde a tilde d or tilde e and the tame blocks are morita equivalent to generalizations of the trivial extensions of the radical square zero tame hereditary algebras of the corresponding type

selfinjective algebras with hereditary stable slice abstract we determine the structure of all finite dimensional selfinjective algebras over a field whose auslander reiten quiver admits a hereditary stable slice

a krein space approach to symmetric ordinary differential operators with an indefinite weight function on considere les proprietes spectrales du probleme differentiel l f 1 n p 0 f n n 1 n 1 p 1 f n 1 n 1 p n f rf sur un intervalle fini ou infini a b avec des coefficients reels localement sommables 1 p 0 p 1 p n r sous l hypothese que p 0 0 et que la fonction poids r change de signe sur a b

bounds on the non real spectrum of differential operators with indefinite weights ordinary and partial differential operators with an indefinite weight function can be viewed as bounded perturbations of non negative operators in krein spaces under the assumption that 0 and infty are not singular critical points of the unperturbed operator it is shown that a bounded additive perturbation leads to an operator whose non real spectrum is contained in a compact set and with definite type real spectrum outside this set the main results are quantitative estimates for this set which are applied to sturm liouville and second order elliptic partial differential operators with indefinite weights on unbounded domains

sharp eigenvalue estimates for rank one perturbations of nonnegative operators in krein spaces abstract let a and b be selfadjoint operators in a krein space and assume that the resolvent difference of a and b is of rank one in the case that a is nonnegative and i is an open interval such that a i consists of isolated eigenvalues we prove sharp estimates on the number and multiplicities of eigenvalues of b in i the general result is illustrated with eigenvalue estimates for singular indefinite sturm liouville problems

on auslander reiten components of blocks and self injective biserial algebras we investigate the existence of auslander reiten components of euclidean type for special biserial self injective algebras and for blocks of group algebras in particular we obtain a complete description of stable auslander reiten quivers for the tame self injective algebras considered here

detecting phase synchronization in noisy data from coupled chaotic oscillators two schemes are proposed to detect phase synchronization from chaotic data contaminated by noise the first is a neighborhood based method which links time delay embedding with instantaneous phase estimation the second adopts the local projection method as a preprocessing filter to noisy data both schemes utilize the state recurrence an important feature of chaotic data the proposed schemes are applied to data measured from two typical chaotic systems i e the coupled rossler systems and the coupled lorenz systems respectively the results show that phase synchronization which may be buried by noise is detected even when the noise level is high moreover the overestimation of the degree of phase synchronization which may be introduced by the hilbert transform combined with a traditional linear bandpass filter can be avoided when the data are contaminated by only measurement noise

definitizable operators and quasihyperbolic operator polynomials abstract the main concern of this paper is bounded operators a on a hilbert space with inner product which are selfadjoint in an indefinite scalar product say x y gx y and have entirely real spectrum in addition all points of spectrum are required to have determinate type a notion refining earlier ideas of krein langer et al which implies a strong stability property under perturbations the central result states that the spectrum is of this type if and only if the operator in question is uniformly definitizable i e gp a for some polynomial p as a first application characterizations of compact uniformly definitizable operators on pontrjagin spaces are obtained then the basic ideas are extended to selfadjoint monic operator polynomials via their linearizations in particular a new class of quasihyperbolic polynomials qhp with real and determinate spectrum is introduced it is shown that qhp have nontrivial monic factors the special cases of strongly hyperbolic and quadratic polynomials are also discussed in particular a factorization theorem is proved for a class of gyroscopically stabilized quadratic polynomials which originate in recent investigations of problems in mechanics

brauer type reciprocity for a class of graded associative algebras classical brauer reciprocity can be stated roughly as follows let g be a finite group and let k be an algebraically closed field of characteristic p 0 if s is a simple kg module and p s is its projective cover then the multiplicity of a simple module l as a composition factor of a characteristic zero lift of p s is the same as the multiplicity of s as a composition factor of a modulo p reduction of l all modules in this paper are assumed to be finite dimensional one thinks of the module l as playing an intermediate role between p s and s reciprocities similar to brauer reciprocity here called brauer type reciprocities have subsequently been found to occur in many other settings for instance if g is a classical modular lie algebra definition given in section 2 then there exists a set 9 of g modules with the following property given a simple g module s the projective cover p s of s has a filtration with each successive quotient isomorphic to a module in 5 and for each such filtration the number of times ze 9 occurs is the same as the multiplicity of s as a composition factor of z this was proved for g of type a i by pollack lo and for arbitrary g by humphreys 3 the reciprocity in this setting is often called humphreys reciprocity inspired by humphreys result bernstein gelfand and gelfand sought and found a brauer type reciprocity bgg reciprocity in a certain truncated category their category 8 of modules for a complex semisimple lie algebra see 11 later this was generalized by mirollo and

on a class of j self adjoint operators with empty resolvent set in the present paper we investigate the set j of all j self adjoint extensions of an operator s which is symmetric in a hilbert space h with deficiency indices 2 2 and which commutes with a non trivial fundamental symmetry j of a krein space h sj js our aim is to describe different types of j self adjoint extensions of s which in general are non self adjoint operators in the hilbert space h one of our main results is the equivalence between the presence of j self adjoint extensions of s with empty resolvent set and the commutation of s with a clifford algebra cl2 j r where r is an additional fundamental symmetry with jr rj this enables one to parameterize in terms of cl2 j r the set of all j self adjoint extensions of s with stable c symmetry here an extension has stable c symmetry if it commutes with a fundamental symmetry and in turn this fundamental symmetry commutes with s such a situation occurs naturally in many applications here we discuss the case of indefinite sturm liouville operators and the case of a one dimensional dirac operator with point interaction

selfadjoint operators in s spaces abstract we study s spaces and operators therein an s space is a hilbert space s with an additional inner product given by u where u is a unitary operator in s we investigate spectral properties of selfadjoint operators in s spaces we show that their spectrum is symmetric with respect to the real axis as a main result we prove that for each selfadjoint operator a in an s space we find an inner product which turns s into a krein space and a into a selfadjoint operator therein as a consequence we get a new simple condition for the existence of invariant subspaces of selfadjoint operators in krein spaces which provides a different insight into this well know and in general unsolved problem

an analytic characterization of the eigenvalues of self adjoint extensions let a be a self adjoint extension in k of a fixed symmetric operator a in k k an analytic characterization of the eigenvalues of a is given in terms of the q function and the parameter function in the krein naimark formula here k and k are krein spaces and it is assumed that a locally has the same spectral properties as a self adjoint operator in a pontryagin space the general results are applied to a class of boundary value problems with dependent boundary conditions

spectral enclosures for a class of block operator matrices abstract we prove new spectral enclosures for the non real spectrum of a class of 2 2 block operator matrices with self adjoint operators a and d on the diagonal and operators b and b as off diagonal entries one of our main results resembles gershgorin s circle theorem the enclosures are applied to j frame operators

spectral analysis of singular ordinary differential operators with indefinite weights abstract in this paper we develop a perturbation approach to investigate spectral problems for singular ordinary differential operators with indefinite weight functions we prove a general perturbation result on the local spectral properties of selfadjoint operators in krein spaces which differ only by finitely many dimensions from the orthogonal sum of a fundamentally reducible operator and an operator with finitely many negative squares this result is applied to singular indefinite sturm liouville operators and higher order singular ordinary differential operators with indefinite weight functions

modular representation theory of lie algebras on etudie les representations non necessairement restreintes de l algebre universelle enveloppante u 7b g pour une algebre de lie restreinte 7b g

on domains of cal p cal t symmetric operators related to y x 1 nx2ny x in recent years a generalization of hermiticity has been investigated using a complex deformation h p2 x2 ix of the harmonic oscillator hamiltonian where is a real parameter these complex hamiltonians possessing symmetry the product of parity and time reversal can have a real spectrum we will consider the most simple case even in this paper we describe all self adjoint hermitian and at the same time symmetric operators associated with h p2 x2 ix surprisingly it turns out that there is a large class of self adjoint operators associated with h p2 x2 ix which are not symmetric

dimension of interaction dynamics a method allowing one to distinguish interacting from noninteracting systems based on available time series is proposed and investigated some facts concerning generalized renyi dimensions that form the basis of our method are proved we show that one can find the dimension of the part of the attractor of the system connected with interaction between its parts we use our method to distinguish interacting from noninteracting systems on the examples of logistic and henon maps a classification of all possible interaction schemes is given

spectral theory of elliptic differential operators with indefinite weights the spectral properties of a class of non self adjoint second order elliptic operators with indefinite weight functions on unbounded domains are investigated it is shown under an abstract regularity assumption that the non real spectrum of the associated elliptic operators in l 2 is bounded in the special case where r n decomposes into subdomains and with smooth compact boundaries and the weight function is positive on and negative on it turns out that the non real spectrum consists only of normal eigenvalues that can be characterized with a dirichlet to neumann map

spectral properties of singular sturm liouville operators with indefinite weight sgn x we consider a singular sturm liouville expression with the indefinite weight sgnx to this expression there is naturally a self adjoint operator in some krein space associated we characterize the local definitizability of this operator in a neighbourhood of 1 moreover in this situation the point 1 is a regular critical point we construct an operator a sgnx id 2 dx 2 q with non real spectrum accumulating to a real point the obtained results are applied to several classes of sturm liouville operators

nonlinear noise reduction a case study on experimental data we apply a recently proposed nonlinear noise reduction method to time sequences from two different experiments we demonstrate that it is not difficult to choose the parameters of this algorithm even though we use no other information about the underlying dynamics than the data themselves the noise reduction is very robust with respect to changes in the choice of parameters the reliability of the result is tested by an analysis of the corrections we discuss the effect of noise reduction on estimates of dimensions entropies and liapunov exponents for comparison we process one of the sets densely sampled taylor couette flow data with a global filter based on singular value decomposition