support varieties ar components and good filtrations let g be a reductive group defined over the galois field mathbb f p with p being good for g using support varieties and covering techniques based on grt modules we determine the position of simple modules and baby verma modules within the stable auslander reiten quiver s gr of the rth frobenius kernel of g in particular we show that the almost split sequences terminating in these modules usually have an indecomposable middle term concerning support varieties we introduce a reduction technique leading to isomorphisms mathcal v g r z r lambda cong mathcal v g r d z r d mu for baby verma modules of certain highest weights lambda mu in x t which are related by the notion of depth

on modules with cyclic vertices in the auslander reiten quiver in studying indecomposable modules of a finite dimensional k algebra where k is a field the auslander reiten quiver r has proved to be a powerful tool if the algebra is a group algebra kg or a block b then properties of the group are related to the graph structure of the auslander reiten quiver for example the graph f kg or t b is finite if and only if a sylow p subgroup of g or a defect group of b is cyclic where p is the characteristic of k it is now well known that an indecomposable kg module has a vertex which is a minimal subgroup q of g such that a4 is q projective and q is unique up to g conjugation 7 we ask whether there are constraints on the position of modules with a given vertex in the auslander reiten quiver here we are concerned with this problem for modules with cyclic vertices which are not projective there is not much to say in the case when the defect group of the block is cyclic in this case all its modules have cyclic vertices we assume therefore that the block containing these modules has a non cyclic defect group assume that m is indecomposable with a cyclic vertex and that a4 is not projective it is well known that m is n periodic hence r periodic for group algebras the auslander reiten translate r is the same as 02 where 52 is the usual heller operator by a more general theorem 2 p 1631 the component o say of the stable auslander reiten quiver containing m is a tube 151 given such a tube 0 then the vertices in each row form a single r orbit the end of the tube is the unique row where each vertex has only one predecessor that is if m is a module corresponding to a vertex it41 in 0 and o rm e m o i 289 002 l 8693 86 3 00

the contraction principle for set valued mappings on a metric space with a graph let x d be a metric space and f x x be a set valued mapping we obtain sufficient conditions for the existence of a fixed point of the mapping f in the metric space x endowed with a graph g such that the set v g of vertices of g coincides with x and the set of edges of g is e g x y x y xxx

on simple modules in the auslander reiten components of finite groups let g be a finite group in this paper we consider a block algebra b of g over an algebraically closed field k of characteristic p where p is a prime we denote the stable auslander reiten quiver ar quiver for short ofb by s b each connected component of s b ar component has a tree class which is determined up to quiver isomorphisms and is obtained from it and the ar translates a p block b is wild if its defect group is neither cyclic dihedral generalized quaternion nor semidihedral it is known that if b is wild then any ar component of s b has tree class a 9 that is any ar component is isomorphic to either za or za where is the ar translate the latter is called an infinite tube of rankm since group algebras are symmetric the auslander reiten translate is equal to the composite 2 of two heller translates thus a module lies in a tube if and only if it is periodic with respect to in the case where an ar component has tree class a we say that a module x in lies at the end if there is only one arrow in which goes into x or out of x moreover a module x in is said to lie in the i th row from the end if there is a subquiver x xi xi 1 x1 of such that x1 lies at the end and that xj 2 xj for all j with 1 j i 2 for wild blocks we consider the following problems on simple modules

mizoguchi takahashi s fixed point theorem is a real generalization of nadler s abstract we give an example which says that mizoguchi takahashi s fixed point theorem for set valued mappings is a real generalization of nadler s we also give a very simple proof of mizoguchi takahashi s theorem

multivalued f weakly picard mappings recently m berinde and v berinde m berinde v berinde on a general class of multi valued weakly picard mappings j math anal appl in press extended the notion of weak contraction from single valued mappings to multivalued mappings and obtained some convergence theorems for the picard iteration associated with multivalued weak contractions we further extended the notion of weak contraction and introduced the notions of multivalued f weak contraction and generalized multivalued f weak contraction we obtain some coincidence and fixed point theorems our results extend m berinde v berinde on a general class of multi valued weakly picard mappings j math anal appl in press and those contained therein

fixed points of generalized contractive multi valued mappings in a recent paper n mizoguchi and w takahashi gave a positive answer to the conjecture of s reich concerning the existence of fixed points of multi valued mappings that satisfy a certain contractive condition in this paper we provide an alternative and somewhat more straightforward proof for the theorem of mizoguchi and takahashi also the problems associated with fixed points of weakly contractive multi valued mappings are studied finally we make a few comments that improve other results from their paper j math anal appl 141 1989 177 188

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

new approach to robust model reference adaptive control for a class of plants motivated by recent advances in designing robust adaptive controllers and in dealing with uncertain dynamical systems a new model reference adaptive control which is robust to a class of unmodelled dynamics and bounded output disturbances in the case of relative degree one is presented the implementation of the controllers includes a switching mechanism which plays a crucially important role in functions of stabilizing as well as tracking it is shown that global stability of the overall system is achieved under no assumption of persistency of excitation and tracking errors will converge to a residual set whose size can be directly related to the size of unmodelled dynamics and of output disturbances explicitly in the ideal case the residual set degenerates to a single null point and convergence can be achieved in finite time without any requirement of persistency of excitation

on the vertices of indecomposable modules over dihedral 2 groups abstract let k be an algebraically closed field of characteristic 2 we calculate the vertices of all indecomposable k d 8 modules for the dihedral group d 8 of order 8 we also give a conjectural formula of the induced module of a string module from k t 0 to kg where g is a dihedral group of order 8 and where t 0 is a dihedral subgroup of index 2 of g some cases where we verified this formula are given

nadler s type principle with high order of convergence abstract recently proinov p d proinov a generalization of the banach contraction principle with high order of convergence of successive approximations nonlinear analysis 2006 doi 10 1016 j na 2006 09 008 generalized banach contraction principle with high order of convergence we extend some results of proinov to the case of multi valued maps from a complete metric space x into the space of all nonempty proximinal closed subsets of x our results not only generalize nadler s fixed point theorem in the case when t is a mapping from a complete metric space x into the space of all nonempty proximinal closed subsets of x but also gives high order of convergence as an application we obtain an existence theorem for first order initial value problem

suzuki type fixed point theorems for generalized multi valued mappings on a set endowed with two b metrics abstract in this paper we obtained suzuki type fixed point results for a generalized multi valued mapping on a set equipped with two b metrics as a consequence existence and uniqueness of solution of functional equation arising in dynamical programming is also derived

g algebras jacobson radical and almost split sequences keywords finite group indecomposable kg module sas sequence almost split exact sequences restriction relative jacobson radicals local criterion multiplicity module vertex source indecomposable projective module twisted group algebra induction reference ctg article 1988 006doi 10 1007 bf01393690 record created on 2008 12 16 modified on 2017 05 12

on auslander reiten components and trivial source lattices for integral group rings let g be a finite group o a complete discrete valuation ring of characteristic zero with residue class field o o of prime characteristic and b a block of the group ring og suppose that b is of infinite representation type and o is large enough let b be the auslander reiten quiver of b and a connected component of b in this paper we show that if contains a trivial source og lattice t then the tree class of the stable part of is a and t lies at the end of

endpoints of multi valued generalized weak contraction mappings abstract let x d be a complete metric space and let t x p cl bd x be a multi valued generalized weak contraction mapping then t has a unique endpoint if and only if t has the approximate endpoint property our results extend previous results given by ciric 1971 15 nadler 1969 11 daffer and kaneko 1995 9 and amini harandi 2010 8

fixed point theorems for multi valued contractions in complete metric spaces abstract in this paper the concept of a contraction for multi valued mappings in a metric space is introduced and the existence theorems for fixed points of such contractions in a complete metric space are proved presented results generalize and improve the recent results of y feng s liu y feng s liu fixed point theorems for multi valued contractive mappings and multi valued caristi type mappings j math anal appl 317 2006 103 112 d klim d wardowski d klim d wardowski fixed point theorems for set valued contractions in complete metric spaces j math anal appl 334 2007 132 139 and several others the method used in the proofs of our results is new and is simpler than methods used in the corresponding papers two examples are given to show that our results are genuine generalization of the results of feng and liu and klim and wardowski