graded identities of matrix algebras and the universal graded algebra in the last decade group gradings and graded identities of finite dimensional central simple algebras have been an active area of research we refer the reader to bahturin et al 6 and 7 there are two basic kinds of group grading elementary and fine it was proved by bahturin and zaicev 7 that any group grading of mn c is given by a certain composition of an elementary grading and a fine grading in this paper we are concerned with fine gradings on mn c and their corresponding graded identities let r be a simple algebra finite dimensional over its center k and g a finite group we say that r is fine graded by g if r g grg is a grading and dimk rg 1 thus any component is either 0 or isomorphic to k as a k vector space it is easy to show that supp r the subset of elements of g for which rg is not 0 is a subgroup of g moreover

blocks with abelian defect groups of rank 2 and one simple module abstract in this paper we investigate the block that has an abelian defect group of rank 2 and its brauer correspondent has only one simple module we will get an isotypy between the block and its brauer correspondent it will generalize a result of kessar and linckelmann

boundary fitted coordinate systems for numerical solution of partial differential equations a review a comprehensive review of methods of numerically generating curvilinear coordinate systems with coordinate lines coincident with all boundary segments is given some general mathematical framework and error analysis common to such coordinate systems is also included the general categories of generating systems are those based on conformal mapping orthogonal systems nearly orthogonal systems systems produced as the solution of elliptic and hyperbolic partial differential equations and systems generated algebraically by interpolation among the boundaries also covered are the control of coordinate line spacing by functions embedded in the partial differential operators of the generating system and by subsequent stretching transformation dynamically adaptive coordinate systems coupled with the physical solution and time dependent systems that follow moving boundaries are treated references reporting experience using such coordinate systems are reviewed as well as those covering the system development

on stable equivalences and blocks with one simple module using a stable equivalence due to rouquier we show that alperin s weight conjecture holds for any p block of a finite group with defect 2 whose brauer correspondent has a unique isomorphism class of simple modules

induced characters which are multiples of irreducibles our statement of theorem a assumes slightly less than that g is solvable but we pay a price for that extra generality we must appeal to the classification of simple groups actually the extra work required in this paper for the sake of not quite assuming that g is solvable is negligible we merely quote the theorem 2 that groups of central type are solvable all of the actual use of the list of simple groups and their properties occurs in the proof of that result it would be nice if no solvability assumption at all were needed in

induction and restriction of characters and hall subgroups abstract let h be a hall subgroup of a separable group g given a character irr g we determine when it happens that g for every irreducible constituent of h and dually given a character irr h we determine when it happens that h for every irreducible constituent of g we also study some related questions

numerical solution of the quasilinear poisson equation in a nonuniform triangle mesh a finite difference method using a nonuniform triangle mesh is described for the numerical solution of the nonlinear two dimensional poisson equation s 0 where is a function of or its derivatives sis a function of position and or its normal derivative is specified on the boundary the finite difference equations are solved by successive overrelaxation the triangle mesh which is constructed numerically by solving laplace s equation is easily adapted to nonrectangular boundaries and interfaces examples of numerical results are given for the magnetostatic problem with iron and other possible applications are mentioned

the weyl algebra as a fixed ring abstract we prove that the weyl algebra over c cannot be a fixed ring of any domain under a nontrivial action of a finite group by algebra automorphisms thus settling a 30 year old problem in fact we prove the following much more general result let x be a smooth affine variety over c let d x denote the ring of algebraic differential operators on x and let be a finite group if d x is isomorphic to the ring of invariants of a c domain r on which acts faithfully by c algebra automorphisms then r is isomorphic to the ring of differential operators on a galois covering of x

fully ramified subgroups a configuration that occurs often in character theory of solvable groups and therefore has been the object of much research is that of fully ramified sections see 7 6 4 23 a normal subgroup h of g is said to be fully ramified in g if there exists an irreducible character 0 of h such that 0 ed for x some irreducible character of g and xh e0 the purpose of this paper is to study a generalization of this situation which seems to occur quite often that is when h is not known to be normal a technique is developed to study such situations based on the following result

on fully ramified brauer characters let z be a normal subgroup of a finite group let p 5 be a prime and let ibr z be an irreducible g invariant p brauer character of z suppose that g e for some ibr g then g z is solvable in other words a twisted group algebra over an algebraically closed field of characteristic not 5 with a unique class of simple modules comes from a solvable group

transfinite mean value interpolation in general dimension mean value interpolation is a simple fast linearly precise method of smoothly interpolating a function given on the boundary of a domain for planar domains several properties of the interpolant were established in a recent paper by dyken and the second author including sufficient conditions on the boundary to guarantee interpolation for continuous data a formula for the normal derivative at the boundary and the construction of a hermite interpolant when normal derivative data is also available in this paper we generalize these results to domains in arbitrary dimension

blocks inequivalent to their frobenius twists abstract let k be an algebraically closed field of characteristic p we give a general method for producing examples of blocks b of finite group algebras that are not morita equivalent as k algebras to the frobenius twist b p our method produces non nilpotent blocks having one simple module and elementary abelian defect group these also provide the first known examples of blocks where there is a perfect isotypy at the level of ordinary characters with all the signs positive but no derived equivalence between the blocks we do not know of any examples of blocks b that are not morita equivalent to the second frobenius twist b p 2

on groups of central type a finite group is said to be of central type if it possesses an irreducible complex character which takes the value zero on all noncentral elements equivalently the degree of this character is the square root of the index of the center in 1964 iwahori and matsumoto 15 conjectured that a group of central type must be solvable the paper of liebler and yellen 16 aims to prove this but there is a gap in their proof as explained below nevertheless they do correctly obtain substantial information about a minimal counterexample to the conjecture the aim of this paper is to fill the gap in the liebler yellen proof using as they do the classification of finite simple groups and to provide further information about which solvable groups can have central type we show for instance that every normal subgroup of the central factor group of a solvable group of central type has the property that its system normalizers have square index it is interesting to contrast this restrictive condition with the result of gagola that every solvable group is embeddable in the central factor group of a group of central type see 6 theorem 1 2 the arguments in 16 show that a minimal counterexample g to the iwahori matsumoto conjecture must have the following structure if k z is any minimal normal subgroup of g z where z z g the center then k is abelian k z is a 2 group s cg k is the maximal solvable normal subgroup of g and any minimal normal subgroup his of g s is the direct product of a number l of copies of gl 3 2 indeed the liebler yellen methods can be used to show that the sylow 2 subgroup of k considered as an h s module is a product o f copies of a gl 3 2 module of dimension 4 this module having a unique minimal submodule which is trivial and an irreducible factor module of dimension 3 the minimal submodules are amalgamated in the product

on the monomiality of nice error bases unitary error bases generalize the pauli matrices to higher dimensional systems two basic constructions of unitary error bases are known an algebraic construction by knill that yields nice error bases and a combinatorial construction by werner that yields shift and multiply bases an open problem posed by schlingemann and werner relates these two constructions and asks whether each nice error basis is equivalent to a shift and multiply basis we solve this problem and show that the answer is negative however we find that nice error bases have more structure than one can anticipate from their definition in particular we show that nice error bases can be written in a form in which at least half of the matrix entries are 0

character correspondences in solvable groups our concern in this paper is with the ordinary complex number character theory of solvable groups if the lengths of some of the earlier papers on this subject l 3 8 are a reliable indication this is a surprisingly complicated and nontrivial theory nevertheless some of the important results are considerably less deep than one would be led to believe by the number of pages involved this paper contains no new theorems and few new ideas it is addressed to the reader who while familiar with character theory has not mastered the literature on characters of solvable groups its purpose is to give such a reader a feeling for the subject and to provide him with accessible proofs for a few of the main results although we certainly do not attempt to prove the strongest or most general results possible or known nevertheless the paper does contain a few items which have application outside of solvable groups in particular we have included in section 4 a brief exposition of tensor induction although our main result could have been proved using an ad hoc argument in place of this general technique the author is now convinced after considerable initial resistance that tensor induction is valuable and deserves further exposure naturally a key idea in the study of characters of solvable groups is to consider the relationships between characters and chief sections in particular suppose that k l is an abelian chief section of g and let 9 e irr k be invariant in g then according to the going down theorem theorem 6 18 of 9 one of the following must occur

phragmen lindelof principles on algebraic varieties from several results in recent years starting with h6rmander s characterization of the constant coefficient partial differential equations p d u f that have a real analytic solution u for every real analytic function f it has become clear that certain properties of the partial differential operator p d are equivalent to estimates of phragmen lindelof type for plurisubharmonic functions on the algebraic variety

the algebraic surfaces on which the classical phragmen lindelof theorem holds let v be an algebraic variety in open image in new window we say that v satisfies the strong phragmen lindelof property spl or that the classical phragmen lindelof theorem holds on v if the following is true there exists a positive constant a such that each plurisubharmonic function u on v which is bounded above by z o z on v and by 0 on the real points in v already is bounded by a im z for algebraic varieties v of pure dimension k we derive necessary conditions on v to satisfy spl and we characterize the curves and surfaces in open image in new window which satisfy spl several examples illustrate how these results can be applied

the geometry of analytic varieties satisfying the local phragmen lindelof condition and a geometric characterization of the partial differential operators that are surjective on ℝ the local phragmen lindelof condition for analytic subvarieties of c n at real points plays a crucial role in complex analysis and in the theory of constant coefficient partial differential operators as hormander has shown here necessary geometric conditions for this phragmen lindelof condition are derived they are shown to be sufficient in the case of curves in arbitrary dimension and of surfaces in c 3 the latter result leads to a geometric characterization of those constant coefficient partial differential operators which are surjective on the space of all real analytic functions on r 4

moving mesh partial differential equations mmpdes based on the equidistribution principle this paper considers several moving mesh partial differential equations that are related to the equidistribution principle several of these are new and some correspond to discrete moving mesh equations that have been used by others their stability is analyzed and it is seen that a key term for most of these moving mesh pdes is a source like term that measures the level of equidistribution it is shown that under weak assumptions mesh crossing cannot occur for most of them finally numerical experiments for these various moving mesh pdes are performed to study their relative properties

a fast mesh deformation method using explicit interpolation a novel mesh deformation algorithm for unstructured polyhedral meshes is developed utilizing a tree code optimization of a simple direct interpolation method the algorithm is shown to provide mesh quality that is competitive with radial basis function based methods with markedly better performance in preserving boundary layer orthogonality in viscous meshes the parallelization of the algorithm is described and the algorithm cost is demonstrated to be o nlogn the parallel implementation was used to deform meshes of 100 million nodes on nearly 200 processors demonstrating that the method scales to large mesh sizes results are provided for a simulation of a high reynolds number fluid structure interaction case using this technique

infinite crossed products and group graded rings in this paper we precisely determine when a crossed product r g is semiprime or prime indeed we show that these conditions ultimately depend upon the analogous conditions for the crossed products r n of the finite subgroups n of g and upon the interrelationship between the normalizers of these subgroups and the ideal structure of r the proof offered here is combinatorial in nature using the a methods and is entirely self contained furthermore since the argument applies equally well to strongly g graded rings we have opted to work in this more general context let g be a multiplicative group and let r be a ring with 1 then a crossed product r g of g over r is an associative ring determined by g r and certain other parameters to be more precise for each x e g there exists an element je r g and every element a e r g is uniquely writable as a finite sum

algebraic varieties on which the classical phragmen lindelof estimates hold for plurisubharmonic functions algebraic varieties v are investigated on which the natural analogue of the classical phragmen lindelof principle for plurisubharmonic functions holds for a homogeneous polynomial p in three variables it is shown that its graph has this property if and only if p has real coefficients no elliptic factors is locally hyperbolic in all real characteristics and the localizations in these characteristics are square free the last condition is shown to be necessary in any dimension

galois algebras and monoidal functors between categories of representations of finite groups abstract we use relations between galois algebras and monoidal functors to describe monoidal functors between categories of representations of finite groups we pay special attention to two kinds of these monoidal functors monoidal functors to vector spaces and monoidal equivalences between categories of representations the functors of the second kind induce isomorphisms of character tables we show that pairs of groups with the same character table obtained in this way are a generalization of the construction proposed by b fischer 1988 rend circ mat palermo 2 suppl 19 71 77

mesh adaptation based on transfinite mean value interpolation abstract in this paper the efficiency in mesh updating r adaptivity of the transfinite mean value interpolation tmi and its generalization k tmi are compared on three standardized problems to the well known inverse distance weighted interpolation idw and radial basis function interpolation rbf for unstructured data points and the new k transfinite barycentric interpolation k tbi for structured data points such as for instance curves or surfaces in 3d this is achieved by introducing a dynamical version of these interpolations via an ordinary differential equation that can be solved by standard ode methods that are more economical than for instance solving vector partial differential equations as in the pseudo solid method a review of the very recent mathematical foundations of the k tmi and k tbi constructed from the function alone standard or from the function and its derivatives enhanced is provided in the first part of the paper

beyond stabilizer codes i nice error bases nice error bases have been introduced by knill 1996 as a generalization of the pauli basis these bases are shown to be projective representations of finite groups we classify all nice error bases of small degree and all nice error bases with abelian index groups we show that in general an index group of a nice error basis is necessarily solvable

counterexample to a conjecture about braces abstract we find an example of a finite solvable group in fact a finite p group in which is not possible to define a left brace structure or equivalently which is not an iyb group this answers a question posed by cedo jespers and del rio related to the yang baxter equation our argument is an improvement of an argument of rump using results about hopf galois extensions and lsa structures we explain explicitly the relation between these two areas of mathematics and brace theory hoping that it will be useful in the future

groups with two extreme character degrees and their normal subgroups we study the finite groups g for which the set cd g of irreducible complex character degrees consists of the two most extreme possible values that is 1 and g z g 1 2 we are easily reduced to finite p groups for which we derive the following group theoretical characterization they are the p groups such that ig z g l is a square and whose only normal subgroups are those containing g or contained in z g by analogy we also deal with pgroups such that g z g i p2n 1 is not a square and we prove that cd g 1 pn if and only if a similar property holds for any n g either g n or nz g z g p the proof of these results requires a detailed analysis of the structure of the p groups with any of the conditions above on normal subgroups which is interesting for its own sake it is especially remarkable that these groups have small nilpotency class and that if the nilpotency class is greater than 2 then the index of the centre is small and in some cases we may even bound the order of g

a computational framework for fluid structure interaction finite element formulation and applications abstract this work is concerned with the modelling of the interaction of fluid flow with flexibly supported rigid bodies the fluid flow considered is governed by the incompressible navier stokes equations and modelled by employing stabilised low order velocity pressure finite elements the motion of the fluid domain is accounted for by an arbitrary lagrangian eulerian ale strategy the rigid body motion excited by the fluid flow is restricted by the elastic and damping properties of the supports for the temporal discretisation the discrete implicit generalised method is employed the resulting strongly coupled set of nonlinear equations is solved by means of a novel partitioned solution procedure which is based on the newton raphson methodology and incorporates full linearisation of the overall incremental problem the strong coupling is resolved and optimal convergence of the residuals is achieved several numerical examples are presented to demonstrate the robustness and efficiency of the methodology the examples clearly capture the phenomena of vortex induced oscillations and galloping