determinative vertices of interval family with ωθ stability abstract in this article we pay attention to the stability of interval polynomials with regard to the region an analysis on how to generate the finite set whose stability implies robust stability of the entire family is given those results are intuitive and convenient to operate

comments with reply on robust stability of an interval plant with respect to a convex region in the complex plane by j shaw and j jayasuriya the commenter points out that the results of j shaw and j jayasuriya ibid vol 38 no 2 pp 284 287 feb 1993 were published by f j kraus and w truol int j contr vol 52 no 4 pp 967 983 1991 for robust stability of polynomial families with affline linear parameter dependence the authors agree with the commenter and point out that their own results derive from a paper they submitted for publication in 1990 but which has not yet been printed which contained arguments similar to those of the commenters

cohomology of hopf algebras and the clifford s extension problem in this paper we continue our previous study of the extension problem for modules in the case of hopf galois extensions let h be a hopf algebra over a field k let arb be a faithfully flat h galois extension and let m m be a right b module we suppose that all of them are k vector spaces of finite dimension we say that there exists an a extension of m if there exists a k linear map m m m a a m such that m m is a right k a module and m m m b s m m m b m g m b g b w x in 8 for an algebraically closed field k a cocommutative hopf algebra h and a simple stable b module m we have shown that the extension problem is equivalent to the vanishing of a certain 2 cohomology class in 2 h h k and moreover the set of isomorphism classes of a extensions is 1 in one to one correspondence with h h k here k is an h module algebra with trivial action and the cohomology groups are defined as in w x 16 the purpose of this paper is to prove these results for any field not only for algebraically closed ones in the first section we shall prove some results which will help us to reduce our problem to the case of a simple ring b first of all we shall prove that the right ideal generated by the annihilator i of m is a two sided ideal of a and a right coideal so aria is a comodule algebra next we shall prove that its coinvariant subalgebra is isomorphic with bri and the extension arai bri is faithfully flat h galois moreover there exists an a extension for m if and only if there exists an arai ex

properties of the entire set of hurwitz polynomials and stability analysis of polynomial families it is proved in this paper that all hurwitz polynomials of order not less than n form two simply connected borel cones in the polynomial parameter space based on this result edge theorems for hurwitz stability of general polyhedrons of polynomials and boundary theorems for hurwitz stability of compact sets of polynomials are obtained both cases of families of polynomials with dependent and independent coefficients are considered different from the previous ones our edge theorems and boundary theorems are applicable to both monic and nonmonic polynomial families and do not require the convexity or the connectivity of the set of polynomials moreover our boundary theorem for families of polynomials with dependent coefficients does not require the coefficient dependency relation to be affine

robust stability of an interval plant with respect to a convex region in the complex plane the determination of robust stability of interval polynomials with respect to an arbitrary closed convex contour in the left half complex plane is considered it is shown that the problem can be solved by checking only the root locations of a subset of the exposed edges of the family the number of exposed edges to be checked depends on the damping margin defined by the contour and the degree of interval polynomials but does not depend on the exact shape of the contour a simple easy to apply graphical approach determines the subset of the exposed edges an example illustrates the merit of the approach

a new self stabilizing maximal matching algorithm the maximal matching problem has received considerable attention in the self stabilizing community previous work has given several self stabilizing algorithms that solve the problem for both the adversarial and the fair distributed daemon the sequential adversarial daemon as well as the synchronous daemon in the following we present a single self stabilizing algorithm for this problem that unites all of these algorithms in that it has the same time complexity as the previous best algorithms for the sequential adversarial the distributed fair and the synchronous daemon in addition the algorithm improves the previous best time complexities for the distributed adversarial daemon from o n 2 and o dm to o m where n is the number of processes m is the number of edges and d is the maximum degree in the graph

invariant blocks under coprime action let a and g be finite groups with a g 1 assume that a acts on g via automorphisms let b be an a invariant block of g if each irreducible character in b is a invariant and b has cyclic defect groups then a centralizes a defect group of b

extension groups between simple mackey functors in order to better understand the structure of indecomposable projective mackey functors we study extension groups of degree 1 between simple mackey functors we explicitly determine these groups between simple functors indexed by distinct normal subgroups we next study the conditions under which it is possible to restrict ourselves to that case and we give methods for calculating extension groups between simple mackey functors which are not indexed by normal subgroups we then focus on the case where the simple mackey functors are indexed by the same subgroup in this case the corresponding extension group can be embedded in an extension group between modules over a group algebra and we describe the image of this embedding in particular we determine all extension groups between simple mackey functors for a p group and for a group that has a normal p sylow subgroup finally we compute higher extension groups between simple mackey functors for a group that has a p sylow subgroup of order p

the schur clifford subgroup of the brauer clifford group abstract we define a schur clifford subgroup of turull s brauer clifford group similar to the schur subgroup of the brauer group the schur clifford subgroup contains exactly the equivalence classes coming from the intended application to clifford theory of finite groups we show that the schur clifford subgroup is indeed a subgroup of the brauer clifford group as are certain naturally defined subsets we also show that this schur clifford subgroup behaves well with respect to restriction and corestriction maps between brauer clifford groups

clifford theory and galois theory i abstract we combine galois theory with crossed products by letting grading groups act on ground fields we show that clifford theory for suitable normal subgroups preserves such actions

representation groups for the schur index fix an algebraic number field k and an irreducible complex character x of some finite group g assume k k x contains the values of x so that the simple component a 4 x k of the group algebra kg determined by 1 is central over k it is a basic consequence of the brauer witt theorem that a is a so called cyclotomic algebra see ls we exhibit here a slightly different point of view namely that of constructing a representation group for a similar to the concept used in clifford theory from class field theory it is known that the order of the class a of a in the brauer group br k equals its index m x it suffices to study the primary components of a so fix a prime p and let a denote the pcomponent of a in br k let k be the field obtained from k by adjoining the nth roots of unity if n exp g then by brauer k is a splitting field for a and g we shall see that there is a field l s k with p power degree over k such that

the mckay conjecture and galois automorphisms the main problem of representation theory of finite groups is to find proofs of several conjectures stating that certain global invariants of a finite group g can be computed locally the simplest of these conjectures is the mckay conjecture which asserts that the number of irreducible complex characters of g of degree not divisible by p is the same if computed in a p sylow normalizer of g in this paper we propose a much stronger version of this conjecture which deals with galois automorphisms in fact the same idea can be applied to the celebrated alperin and dade conjectures

generalized pi controllability integral controllable systems can be stabilized by all sufficiently low gain integral controllers the concept of integral controllability is extended to two generalized stability regions and to proportional controllability and pi controllability with respect to those regions furthermore closed eigenvalue formulae are derived for the maximal low integral gain and the maximal low proportional gain for generalized stability related eigenvalue formulae are then used in combination to find all low gain pi controllers that place the closed loop eigenvalues in the desired region

broue s conjecture holds for principal 3 blocks with elementary abelian defect group of order 9 abstract in representation theory of finite groups there is a well known and important conjecture due to m broue he conjectures that for any prime p if a finite group g has an abelian sylow p subgroup p then the derived categories of the principal p blocks of g and of the normalizer ng p of p in g are equivalent we prove in this paper that broue inage kus conjecture holds for the principal 3 block of an arbitrary finite group g with an elementary abelian sylow 3 subgroup p of order 9 by using initiated works for the case where g is simple which are due to puig okuyama waki miyachi and the authors the result depends on the classification of finite simple groups

extending irreducible representations of normal subgroups let h be a normal subgroup of some finite group g and let v be an irreducible right fh module where f is an arbitrary field it is one of the primary objectives of clifford theory to find necessary and sufhcient conditions so that v can be extended to an fg module an obvious necessary condition is that v has to be g invariant stable sufficiency relies upon various properties of the group extension h g w s where s g h and the module itself the theory is well developed in case v is absolutely irreducible recently isaacs 12 and dade 4 proved generalizing a theorem by gallagher that v can be always extended to an fg module provided h is a hall subgroup of g and v is g invariant the present paper resulted from an effort to understand this remarkable and somewhat surprising result assume in the sequel that v is g invariant let d end v be the opposite endomorphism ring of k basic clifford theory yields a group extension d 52 p p s such that there exists an fg module extending v if and only if 52 52 v splits over the multiplicative group d of the division algebra d see theorem 1 below in general d is noncommutative and so the usual cohomological methods seem to fail dade even remarked in 4 sceptically that none of the nice theorems about extensions of v such as those of gallagher which depend upon calculations in the cohomology groups of s with coefficients in a commutative group can be proved this way in the following situation however one can argue as usual

using g algebras for schur index computation abstract conditions for the existence of a bijective character correspondence that preserves schur indices are given based on the theory of central simple g algebras over fields of characteristic 0 it is then shown how this applies to schur index computation in the case of finite solvable groups

damping margins of interval polynomials the zero locations of interval polynomials are examined in particular it is shown that a family of interval polynomials will have zeros only in the left sector if the real and imaginary parts of four specially constructed complex polynomials have an interlacing real zero property this is significant for the analysis of uncertain systems as the computation cost associated with checking the zero locations of interval polynomials will be greatly reduced the results presented can be readily extended to more general stability regions where the real and imaginary parts of the polar plot are polynomial functions

a class of stability regions for which a kharitonov like theorem holds families of complex polynomials whose coefficients lie within given intervals are discussed in particular the problem of determining if all polynomials in a family have the property that all of their roots lie within a given region is discussed towards this end a notion of a kharitonov region is defined roughly speaking a kharitonov region is a region in the complex plane with the following property given any suitable family of polynomials in order to determine if all polynomials in the family have all of their roots in the region it suffices to check only the vertex polynomials of the family the main result is a sufficient condition for a given region to be a kharitonov region

comments on comments on stability analysis of a family of matrices by j h kim and z bien a flaw is pointed out in the proof of a necessary and sufficient condition for robust stability of a polytope of matrices given by j h kim and z bien ibid vol 387 no 9 pp 1470 1472 sep 1992 in their comments on a paper by the present authors ibid vol 35 no 11 pp 1257 1259 nov 1190 hence the necessary condition as proposed may not be correct

hopf galois extensions and an exact sequence for h picard groups let h be a hopf algebra and a an h galois extension we investigate h morita autoequivalences of a introduce the concept of h picard group and we establish an exact sequence linking the h picard group of a and the picard group of acoh

clifford theory for g functors abstract when does a g functor admit a clifford theory in this paper we give a simple axiomatic property which characterizes the existence of such a theory satisfaction of this condition in the different contexts leads automatically to the satisfaction of the corresponding theorems of clifford we apply it to several examples some of them already known including the character and burnside rings

on the automorphism group of extraspecial 2 groups fix a nonsingular quadratic space u q of dimension 2n over 2 n 1 then either q has witt index n q q or index n 1 2 n q q let e e q and o q be the associated extraspecial 2 group 2 n 1 2 n and full orthogonal group respectively usually one writes e 2 if 1 2 n q q and e 2 otherwise the orthogonal group is written o q 2 n o 2 it is well known that the group of outer automorphisms out e 2 n o q acting on e z e u in the natural way by griess 3 the group extension

actions of picard groups on graded rings throughout this paper g denotes a group with identity e if r is a g graded ring we write r g grg and we refer to re as the coefficient ring we denote the category of graded left r modules i e those left r modules m with a g grading m g gmg such that rhmg mhg by r gr the aim of this paper is to present a method to reduce the study of one strongly graded ring r to the study of another strongly graded ring r that is more tractable this reduction process has two aspects reducing the coefficient ring or reducing the grading since we apply this method in our study of module theoretic properties such as semisimplicity for graded rings we require that this reduction process preserve the category of modules r mod and the category of graded modules r gr this leads us to the notion of graded equivalence in this paper a morita context is the usual tuple a b p q where we assume and are isomorphisms a g graded context is a morita context r r p q where r and r are g graded rings rpr and r qr are graded bimodules i e they have a grading which makes them into both graded r modules and graded r modules on the corresponding side and and are graded bimodule homomorphisms see 14 for details about graded modules and homomorphisms given a functor f r gr r gr we say that f is a graded equivalence of categories if any of the following equivalent conditions hold see for example 8

on equivalences between blocks of group algebras reduction to the simple components abstract a conjecture of michel broue states that if d is an abelian sylow p subgroup of a finite group g and h n g d then the principal blocks of g and h are rickard equivalent the structure of groups with abelian sylow p subgroups as determined by p fong and m e harris raises the following question assuming that broue s conjecture holds for the simple components of g under what conditions does it hold for g itself due to the structure of g this problem requires mainly the lifting of rickard complexes to p extensions of the simple components and the construction of complexes over wreath products we give here these reduction steps which may be regarded as a clifford theory of tilting complexes

corestriction for algebras with group action abstract we define a corestriction map for equivariant brauer groups in the sense of frohlich and wall which contain as a special case the brauer clifford groups introduced by turull we show that this corestriction map has similar properties as the corestriction map in group cohomology especially galois cohomology in particular composing corestriction and restriction associated to a subgroup h g amounts to powering with the index g h

strengthening the mckay conjecture to include local fields and local schur indices abstract a central problem in the representation theory of finite groups is given a prime p the study of relationships between the representations of g and those of p local subgroups of g the simplest of these is perhaps the mckay conjecture which concerns the number of irreducible characters of degree prime to p even though this conjecture remains open in an effort to understand it a number of strengthenings of it have been proposed recently isaacs and navarro and later navarro have proposed interesting stronger conjectures indicating that congruences modulo p of the degree of the characters and the invariance under certain type of fixed galois automorphism could be taken into account as well we propose an even further strengthening of mckay s conjecture which takes into account in addition the p local field of definition of the irreducible characters as well as their p local schur indices

a generalization of the routh hurwitz stability criteria and an application to a problem in robust controller design the classical results of routh and hurwitz on the stability of polynomials are generalized from the open left half plane to certain other subregions g of the complex plane by our method arbitrary conic sections can be handled the results are interpreted as criteria for g stability of linear systems a numerical algorithm to compute the criteria is given an application to a problem in robust controller design is outlined

kernels inflations evaluations and imprimitivity of mackey functors abstract let m be a mackey functor for a finite group g by the kernel of m we mean the largest normal subgroup n of g such that m can be inflated from a mackey functor for g n we first study kernels of mackey functors and relative projectivity of inflated mackey functors for a normal subgroup n of g denoting by p h v g the projective cover of a simple mackey functor for g of the form s h v g we next try to answer the question how are the mackey functors p h n v g n and p h v g related we then study imprimitive mackey functors by which we mean mackey functors for g induced from mackey functors for proper subgroups of g we obtain some results about imprimitive mackey functors of the form p h v g including a mackey functor version of fong s theorem on induced modules of modular group algebras of p solvable groups aiming to characterize subgroups h of g for which the module p h v g h is the projective cover of the simple k n g h module v where the coefficient ring k is a field we finally study evaluations of mackey functors

improved finite word length characteristics in digital control using delta operators this paper examines some of the consequences of finite word lengths in digital control it is shown that in many cases of practical importance the usual shift operator formulation is inferior to an alternative formulation which we designate the delta operator approach this latter approach is shown to give better coefficient representation and less roundoff noise in many cases we thus argue that the shift operator and its associated z transform can be replaced by delta operators and their associated transform which we designate a transform an added advantage of this approach is that discrete designs and transforms converge to their continuous time counterparts as the sampling rate is increased

the brauer clifford group abstract clifford theory provides well behaved character correspondences between different groups which have isomorphic quotients given one such quotient group we define the brauer clifford group we show that each character of the original groups gives rise to a specific element of the brauer clifford group when two characters of different groups yield the same element of the brauer clifford group we obtain a very well behaved character correspondence between the characters of the different groups which preserves not only induction restriction multiplicities but also fields of values for the corresponding characters and schur indices we also show that the brauer clifford group has a natural homomorphism into a brauer group the brauer clifford group can be thought of as a refinement of the previously introduced clifford classes

strong kharitonov theorems for low order polynomials the root clustering property of low order interval polynomials is examined and it is shown that the number of polynomials required to check root clustering within the left sector may be reduced if the order of the interval polynomial is low specifically the reduction is possible if n phi

representation groups of characters the subject we are concerned with has its roots in the work of schur who introduced the notion of a representation group darstellungs gruppe these groups have the property to lift linearize all projective representations of some finite group over an algebraically closed field 16 fong 8 j noticed that schur s method applies in context with clif ford s work exhibiting some single invariant stable character or module the familiar proof for the famous fong swan theorem gives the model for this kind of representation groups see serre 17 or feit 7 throughout we fix a finite group g with normal subgroup h and quotient group s g h we also fix an absolutely irreducible frobenius character x of h and a field f containing the values of x we assume that x is g invariant as passage from the inertia group to the whole group usually is easy to handle this is not too restrictive it is a necessary con dition for x to be extendible to g

on damping ratio of polynomials with perturbed coefficients c b soh and c s berger 1988 derived a sufficient condition for a family of interval polynomials to have a damping ratio of phi using kharitonov s theorem for complex polynomials this paper points out that the transformations used by soh and berger to obtain the sufficient conditions also guarantee a simplification of kharitonov s theorem for complex polynomials that is the number of required polynomials to be hurwitz is half the number specified by soh and berger

clifford theory for mackey algebras we develop a clifford theory for mackey algebras for simple mackey functors using their classification we prove mackey algebra versions of clifford s theorem and the clifford correspondence let r g be the mackey algebra of a finite group g over a commutative unital ring r a nd let 1 n be the unity of r n where n is a normal subgroup of g observing that 1n r g 1n is a crossed product of g n over r n a number of results concerning group graded algebras are extended to the context of mackey algebras including fong s theorem green s indecomposibility theorem and some reduction and extension techniques for indecomposable mackey functors

on clifford theory with galois action abstract let g be a finite group n a normal subgroup of g and irr n let f be a subfield of the complex numbers and assume that the galois orbit of over f is invariant in g we show that there is another triple g 1 n 1 1 of the same form such that the character theories of g over and of g 1 over 1 are essentially the same over the field f and such that the following holds g 1 has a cyclic normal subgroup c contained in n 1 such that 1 n 1 for some linear character of c and such that n 1 c is isomorphic to the abelian galois group of the field extension f f 1 more precisely having the same character theory means that both triples yield the same element of the brauer clifford group brcliff g f defined by a turull

on kharitonov type theorems for real polynomials when is degree drop admissible a root locus approach is adopted to show that extension of kharitonov type theorem for real interval polynomial with degree drop is not only applicable to the open left half plane but also other regions such as the left shifted open left half plane and the left sector

endoisomorphisms yield module and character correspondences abstract in an earlier paper turull 2012 12 the author introduced the concept of endoisomorphisms and showed its natural connection with clifford theory of finite groups associated with clifford theory are module and character correspondences in the present paper the author shows that each endoisomorphism produces a unique module and character correspondence with excellent compatibility results

on the bonnafe dat rouquier morita equivalence abstract we prove that the cohomology group of a deligne lusztig variety defines a morita equivalence in a case which is not covered by the argument in 2 specifically we consider the situation for semisimple elements in type d whose centralizer has non cyclic component group some arguments use considerations already present in an unpublished note by bonnafe dat and rouquier

closed loop hyperstability of interval plants this note examines the conditions for robust bounded input bounded output closed loop stability of interval plants when there is a nonlinear element in the feedback path specifically the authors present the robust generalized circle criterion and popov s criterion

a new analysis of a self stabilizing maximum weight matching algorithm with approximation ratio 2 the maximum weight matching problem is a fundamental problem in graph theory with a variety of important applications recently manne and mjelde presented the first self stabilizing algorithm computing a 2 approximation of the optimal solution they established that their algorithm stabilizes after o 2n resp o 3n moves under a central resp distributed scheduler this paper contributes a new analysis improving these bounds considerably in particular it is shown that the algorithm stabilizes after o nm moves under the central scheduler and that a modified version of the algorithm also stabilizes after o nm moves under the distributed scheduler the paper presents a new proof technique based on graph reduction for analyzing the complexity of self stabilizing algorithms

schur stability of interval polynomials a method for checking the schur stability of interval polynomials is presented the numbers of critical vertex and edge polynomials that are sufficient for inferring robust schur stability are obtained it is shown that the critical edges can be greatly reduced to the order of 2n sup 2

the strengthened alperin weight conjecture for p solvable groups abstract the alperin weight conjecture is a well known conjecture and it is central to the modern representation theory of finite groups it is known to be true for many types of finite groups but it remains open in particular the conjecture is known to be true for finite p solvable groups in this paper we prove that a very strong form of the conjecture holds for all finite p solvable groups it follows that the strengthening of the alperin weight conjecture that includes galois automorphisms over the p adic numbers holds for all finite p solvable groups this strengthening of the alperin weight conjecture was suggested in an earlier work of navarro on the mckay conjecture the present paper provides the first direct evidence beyond the calculation of individual cases and the case of groups of odd order for this strengthening

expansion of det a b and robustness analysis of uncertain state space systems in this note an expansion for the determinant of the sum of two matrices det a b is given a robustness analysis approach for state space systems with uncertain parameters is investigated by using kharitonov s theorem some useful results are obtained several illustrative examples are given

generalization of strong kharitonov theorems to the left sector the zero locations of interval polynomials are examined in particular it is shown that a family of interval polynomials will have only zeros in a certain class of left sector if and only if a finite number of specially chosen vertex polynomials have only zeros in the left sector this finite number of vertex polynomials is dependent on the damping margins of the left sector but is independent of the orders of the polynomials the exact number of vertex polynomials to be checked will depend on the damping margins but it will be independent of the orders of the polynomials these specially chosen vertex polynomials are independent of the order of the polynomials the importance of the result lies with the great reduction in computation cost associated with checking the zero locations of interval polynomials

rational matrix groups of a special type abstract let g be a finite group having a faithful irreducible character for which 1 is prime to g 1 let n q q 1 and assume that the factors are not both even then g can be embedded in gl n q in such a way that its normalizer therein splits over its centralizer

robust hurwitz l sup p stability of polynomials with complex coefficients necessary and sufficient conditions for robust stability of a family of polynomials with complex coefficients are provided this work generalizes a frequency domain criteria for robust stability of continuous linear systems reported on by y z tsypkin and b t polyak 1991

above the glauberman correspondence let s be a finite solvable group and suppose s acts on the finite group n and they have coprime orders then the celebrated glauberman correspondence provides a natural bijection from the set irrs n of irreducible characters of n which are invariant under the action of s to the set irr cn s of all irreducible characters of the centralizer of s in n suppose further that the semidirect product sn is a normal subgroup of a finite group g let irrs n and let irr cn s be its glauberman correspondent we prove that there is a bijection with good compatibility properties from the set irr g of the irreducible characters of g above to irr ng s such that in the case when s is a p group for some prime p it preserves fields of values and schur indices over qp the field of p adic numbers using this result we also prove a strengthening of the mckay conjecture for all p solvable groups

a survey of extreme point results for robustness of control systems abstract this paper surveys a subset of the body of research which was sparked by kharitonov s theorem the focal point is extreme point results for robust stability and robust performance that is we give conditions under which satisfaction of a performance specification is ascertained for a family of systems by only checking a finite subset of the extreme members of this family the results which are surveyed apply mainly to systems with structured real parametric uncertainty in addition a number of counterexamples are given to illustrate cases for which an extreme point result does not hold for such cases a solution via the so called edge theorem is often possible

defects of irreducible characters ofp soluble groups abstract we prove a refinement of the p soluble case of robinson s conjectural local characterization of the defect of an irreducible character

a self stabilizing 23 approximation algorithm for the maximum matching problem the matching problem asks for a large set of disjoint edges in a graph it is a problem that has received considerable attention in both the sequential and the self stabilizing literature previous work has resulted in self stabilizing algorithms for computing a maximal 12 approximation matching in a general graph as well as computing a 23 approximation on more specific graph types in this paper we present the first self stabilizing algorithm for finding a 23 approximation to the maximum matching problem in a general graph we show that our new algorithm when run under a distributed adversarial daemon stabilizes after at most o n2 rounds however it might still use an exponential number of time steps

brauer clifford equivalence of full matrix algebras abstract the brauer clifford group was introduced to describe the clifford theory for finite groups it was proved that it has a natural homomorphism into a brauer group and the kernel of this homomorphism is the set of all equivalence classes of g algebras which are full matrix algebras in this paper we prove that this kernel is isomorphic to a second cohomology group in the clifford theory for finite groups situation we characterize families of characters which yield elements in the full matrix subgroup of the brauer clifford group as those where an appropriate character has schur index one we also show in this case how to compute the element of the second cohomology group associated with this family of characters

schur indices and schur groups w x schur groups were introduced in 8 as a certain kind of smallest groups admitting characters with nontrivial schur index indeed every finite group g having an irreducible character x with m x 1 involves such schur groups they occur as terminal steps by repeated application of the brauer witt theorem over local fields of characteristic zero there is a connection with the schur subgroup s k of the brauer group br k of such a field k this s k has been studied in detail by w x w x yamada 12 and quite recently by adem and milgram 1 from the character theoretic point of view it appears to be of interest to give explicit generators for s k in terms of schur groups see section 6 in the course w x of this investigation the authors noticed that there must be a gap in 8 concerning the exceptional dyadic schur groups yielding nontrivial 2 lo cal index it turned out that the definition of the schur groups of type qd q was too restrictive the correct definition will be given in section 5 below we take the opportunity to discuss these terminal q reduc2 tions once again and the class of 2 groups involved theorem 4 and 5 the schur groups admit faithful irreducible characters which are induced from linear characters of cyclic normal subgroups for the sake of clarity we begin our present work by making some general comments on schur indices in group extensions this should be compared with turull w x 11 our approach is perhaps more elementary and we just demonstrate w x how to use the results on stable clifford theory 7 more details may be w x found in the recent thesis 5 of the first named author

a note on two methods related to stability robustness of polynomials in a sector relative stability two recent results on the stability robustness of uncertain systems polynomials are addressed one gives a vertex test for left sector relative stability where the number of the required vertices is four for a real case and eight for a complex case the other gives an elegant frequency domain graphical hodograph approach for left plane stability where the plotting of only one two for a complex case hodograph plus simple boundary conditions is both necessary and sufficient to obtain the maximum coefficient perturbation bounds it is shown that a similar graphical test exists for the left sector stability case

socles and radicals of mackey functors we study the socle and the radical of a mackey functor m for a finite group g over a field k usually of characteristic p 0 for a subgroup h of g we construct bijections between some classes of the simple subfunctors of m and some classes of the simple kn g h submodules of m h we relate the multiplicity of a simple mackey functor sh vg in the socle of m to the multiplicity of v in the socle of a certain kn g h submodule of m h we also obtain similar results for the maximal subfunctors of m we then apply these general results to some special mackey functors for g including the functors obtained by inducing or restricting a simple mackey functor mackey functors for a p group the fixed point functor and the burnside functor bkg for instance we find the first four top factors of the radical series of bkg for a p group g and assuming further that g is an abelian p group we find the radical series of bkg

generalized edge theorem abstract in this paper we shall show that a family of analytic functions constructed from the convex hull of a finite number of vertex functions will have no zeros within a simply connected region in the complex plane if and only if all its edge functions have no zeros within the simply connected region the result is in fact a generalization of the edge theorem which has been derived for polynomial functions 1 2 7

morita equivalences induced by bimodules over hopf galois extensions let h be a hopf algebra and a b be h galois extensions we investigate the category mbha of relative hopf bimodules and the morita equivalences between a and b induced by them

some discrete time counterparts to kharitonov s stability criterion for uncertain systems in 1 kharitonov gave an elegant and simple stability criterion for continuous time systems this note reports on similar results for discrete time systems

a fault containing self stabilizing 3 2δ 1 approximation algorithm for vertex cover in anonymous networks abstract the non computability of many distributed tasks in anonymous networks is well known this paper presents a deterministic self stabilizing algorithm to compute a 3 2 1 approximation of a minimum vertex cover in anonymous networks the algorithm operates under the distributed unfair scheduler stabilizes after o n m moves respectively o rounds and requires o log n storage per node recovery from a single fault is reached within a constant time and the contamination number is o for trees the algorithm computes a 2 approximation of a minimum vertex cover

the brauer clifford group of g rings abstract in previous work the author introduced the brauer clifford group of certain g algebras this group is useful because to every irreducible character of a normal subgroup of a finite group one can associate a unique element of a specific brauer clifford group and this element controls the clifford theory of this character in its ambient group in the present paper we define the brauer clifford group of g rings this new definition only requires us to discuss tensor products over the underlying g ring and it is simpler than the earlier one we prove that the new definition yields a group which is canonically isomorphic to the brauer clifford group of a corresponding suitable g algebra

extending irreducible modules an important part of clifford s theory does not work properly for arbitrary fields r to see why we must recall the main steps of this theory we are interested in extensions to rh modules of a fixed h invariant irreducible rn module a where n is a normal subgroup of a finite group h we first pass to the factor algebra u rh 3 modulo the ideal 3 3 rh rh3 generated by the annihilator 3 of 8 in rn if sth is given its natural grading by the factor group g h n see 6 2 below then 3 is a graded ideal so that u is a strongly g graded ring in the sense of 3 or section 1 below whose l component u rn d is a finite dimensional simple ralgebra with i as its irreducible module it follows that the extensions of 8 to rh modules are just the pull backs of the extensions of 2i to 2l modules via the natural epimorphism of rh onto u when 52 is algebraically closed or more generally when i is split over r then 2i is the kronecker product

clifford theory and endoisomorphisms this paper describes the relation between clifford theory of finite groups over any field and the brauer clifford group let g and g be finite groups let g g be a surjective group homomorphism with kernel h and let f be any field let k be any extension field of f and let s be an irreducible kh module we show that to s is associated in a natural way a specific element s f of a brauer clifford group defined over and f as a tool to prove the existence of this association and to study its properties we use endoisomorphisms these are simply certain isomorphisms of related endomorphism algebras as g algebras over f we show that two modules for two different finite groups have the same in an appropriate sense element of the brauer clifford group if and only if there exists an appropriate endoisomorphism

a swan length theorem and a fong dimension theorem for mackey algebras we first present some results about mackey algebras of p groups over fields of characteristic p including their primitive idempotents and decompositions of their simple and principal indecomposable modules under restriction we then apply these results together with a green s indecomposability theorem for mackey algebras to obtain mackey algebra versions of some classical results of group algebras which are mostly related to restriction induction and dimensions of modules our results about dimensions include mackey algebra analogues of dickson s theorem swan s theorem and fong s dimension formula

clifford theory of simple modules clifford theory primarily is concerned with the representations of normal subgroups and quotient groups of finite groups the basic contribution being due to a h clifford 3 modern developments founded on the concepts of group graded rings and crossed products have shown close relations with commutative and non commutative galois theory this is most conspicuous in stable clifford theory of simple modules suppose n w h g is an extension of finite groups and v is a simple f n module where f is any field since passage from the inertia group to the whole group usually is easy to handle we may and do assume that v is g invariant or h invariant stable then the annihilator i ann v of v generates a homogeneous ideal z of f h and a f h z is a fully g graded algebra with simple l component r f n z general clifford theory is concerned with such g graded algebras a and their modules cf dade 7 lo assume the l component r a is simple artinian as above then the centre k z r has in a natural way the structure of a g field the main object of study is the endomorphism ring e end v of the a module induced from the simple r module v this is a fully g graded algebra as well generalizing wedderburn s theorem we show that a z m e as g graded algebras where d is the dimension of v over the division algebra d end v theorem 2 1 in particular k z d has the same g structure with respect to e and a and e are morita equivalent in some strong sense now assume that dim r d2m2 is finite and that the schur index m m v is relatively prime to the order of the automorphism group g c k of k induced by g then the graded structure of a e can be described by some distinguished cohomology class w v e h g k more precisely w oa v determines a crossed product sz of k with g