endomorphism rings of permutation modules let k be an algebraically closed field of characteristic p and g a finite group then the permutation module kpg of g on the cosets of a sylow p subgroup p is via fitting correspondence strongly related to its endomorphism ring endkg kpg on the other hand each green correspondent in g of a weight module of g occurs as a direct summand of kpg this fact suggests to analyze both structures the permutation module and the associated endomorphism ring towards hints at a proof for alperin s weight conjecture we present a selection of such investigations for different groups and characteristics in particular we focus on the socle and head constituents of the indecomposable direct summands of kpg and of the pims of ee

worst case evaluation complexity for unconstrained nonlinear optimization using high order regularized models the worst case evaluation complexity for smooth possibly nonconvex unconstrained optimization is considered it is shown that if one is willing to use derivatives of the objective function up to order p for p ge 1 p 1 and to assume lipschitz continuity of the p th derivative then an epsilon approximate first order critical point can be computed in at most o epsilon p 1 p o p 1 p evaluations of the problem s objective function and its derivatives this generalizes and subsumes results known for p 1 p 1 and p 2 p 2

dade s invariant conjecture for the chevalley groups of type g2 in the defining characteristic abstract this paper is part of a program to study the conjecture of e c dade on counting characters in blocks for several finite groups in this paper we verify dade s invariant conjecture for the chevalley groups g 2 p a in the defining characteristic when p 2 or 3 this implies dade s final conjecture when p 2 or 3

blocks with trivial intersection defect groups we show that each block whose defect groups intersect pairwise trivially either has cyclic or generalised quaternion defect groups or is morita equivalent to one of a given list of blocks of central extensions of automorphism groups of non abelian simple groups in particular we classify all blocks of automorphism groups of non abelian simple groups whose defect groups are non cyclic and intersect pairwise trivially a consequence is that donovans conjecture holds for blocks whose defect groups intersect pairwise trivially

optimality condition and complexity analysis for linearly constrained optimization without differentiability on the boundary in this paper we consider the minimization of a continuous function that is potentially not differentiable or not twice differentiable on the boundary of the feasible region by exploiting an interior point technique we present first and second order optimality conditions for this problem that reduces to classical ones when the derivative on the boundary is available for this type of problems existing necessary conditions often rely on the notion of subdifferential or become non trivially weaker than the kkt condition in the twice differentiable counterpart problems in contrast this paper presents a new set of first and second order necessary conditions that are derived without the use of subdifferential and reduce to exactly the kkt condition when twice differentiability holds as a result these conditions are stronger than some existing ones considered for the discussed minimization problem when only non negativity constraints are present to solve for these optimality conditions in the special but important case of linearly constrained problems we present two novel interior point trust region algorithms and show that their worst case computational efficiency in achieving the potentially stronger optimality conditions match the best known complexity bounds since this work considers a more general problem than those in the literature our results also indicate that best known existing complexity bounds are actually held for a wider class of nonlinear programming problems this new development is significant since optimality conditions play a fundamental role in computational optimization and more and more nonlinear and nonconvex problems need to be solved in practice

bilinear factor matrix norm minimization for robust pca algorithms and applications the heavy tailed distributions of corrupted outliers and singular values of all channels in low level vision have proven effective priors for many applications such as background modeling photometric stereo and image alignment and they can be well modeled by a hyper laplacian however the use of such distributions generally leads to challenging non convex non smooth and non lipschitz problems and makes existing algorithms very slow for large scale applications together with the analytic solutions to ell p norm minimization with two specific values of p i e p 1 2 and p 2 3 we propose two novel bilinear factor matrix norm minimization models for robust principal component analysis we first define the double nuclear norm and frobenius nuclear hybrid norm penalties and then prove that they are in essence the schatten 1 2 and 2 3 quasi norms respectively which lead to much more tractable and scalable lipschitz optimization problems our experimental analysis shows that both our methods yield more accurate solutions than original schatten quasi norm minimization even when the number of observations is very limited finally we apply our penalties to various low level vision problems e g text removal moving object detection image alignment and inpainting and show that our methods usually outperform the state of the art methods

characters of relative p degree over normal subgroups let z be a normal subgroup of a finite group g let irr z be an irreducible complex character of z and let p be a prime number if p does not divide the integers 1 1 for all irr g lying over then we prove that the sylow p subgroups of g z are abelian this theorem which generalizes the gluck wolf theorem to arbitrary finite groups is one of the principal obstacles to proving the celebrated brauer height zero conjecture

on block induction abstract block induction is a correspondence between certain p blocks of subgroups and blocks of the whole group it is a tool with the help of which one often can reduce the solution of problems in the whole group to smaller groups the literature lists several ways to define induced blocks in the sense of brauer alperin and burry green and wheeler it is well known that any two of these concepts of block induction coincide in their common domain of definition for example each of them is defined if the subgroup h of the group g satisfies dc g d h n g d for a p subgroup d of g we study some other features of blocks which are compatible with the above induction concepts in this sense and formulate them as induction concepts so that one could deal with these and the above concepts in a unified way we compare their fields of definition to the others in general and in the special cases of defect zero blocks and of p groups we give some sufficient conditions so that induction would not be defined in any sense we show that unlike in the case of brauer sense induction block induction in the sense of alperin burry is not always defined from the normalizer of a chain of p subgroups our example also shows that admissible induction and being of multiplicity one are not transitive in the last part of the paper we study the connection of the defect groups of the induced and the original block in the case where the latter group is abelian

adaptive cubic regularisation methods for unconstrained optimization part ii worst case function and derivative evaluation complexity an adaptive regularisation framework using cubics arc was proposed for unconstrained optimization and analysed in cartis gould and toint part i math program doi 10 1007 s10107 009 0286 5 2009 generalizing at the same time an unpublished method due to griewank technical report na 12 1981 damtp university of cambridge an algorithm by nesterov and polyak math program 108 1 177 205 2006 and a proposal by weiser deuflhard and erdmann optim methods softw 22 3 413 431 2007 in this companion paper we further the analysis by providing worst case global iteration complexity bounds for arc and a second order variant to achieve approximate first order and for the latter second order criticality of the iterates in particular the second order arc algorithm requires at most mathcal o epsilon 3 2 iterations or equivalently function and gradient evaluations to drive the norm of the gradient of the objective below the desired accuracy epsilon and mathcal o epsilon 3 iterations to reach approximate nonnegative curvature in a subspace the orders of these bounds match those proved for algorithm 3 3 of nesterov and polyak which minimizes the cubic model globally on each iteration our approach is more general in that it allows the cubic model to be solved only approximately and may employ approximate hessians

neural network for nonsmooth nonconvex constrained minimization via smooth approximation a neural network based on smoothing approximation is presented for a class of nonsmooth nonconvex constrained optimization problems where the objective function is nonsmooth and nonconvex the equality constraint functions are linear and the inequality constraint functions are nonsmooth convex this approach can find a clarke stationary point of the optimization problem by following a continuous path defined by a solution of an ordinary differential equation the global convergence is guaranteed if either the feasible set is bounded or the objective function is level bounded specially the proposed network does not require 1 the initial point to be feasible 2 a prior penalty parameter to be chosen exactly 3 a differential inclusion to be solved numerical experiments and comparisons with some existing algorithms are presented to illustrate the theoretical results and show the efficiency of the proposed network

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

complexity of partially separable convexly constrained optimization with non lipschitzian singularities an adaptive regularization algorithm using high order models is proposed for solving partially separable convexly constrained nonlinear optimization problems whose objective function contains non l

blocks with cyclic defect groups and clifford extensions abstract we prove that the residual clifford extension of any non exceptional character belonging to a p block with cyclic defect groups of a twisted group algebra is naturally isomorphic to the clifford extension of any simple module belonging to that block we also show that dade s inductive conjecture holds for blocks with cyclic defect groups

a reduction theorem for the mckay conjecture the mckay conjecture asserts that for every finite group g and every prime p the number of irreducible characters of g having p degree is equal to the number of such characters of the normalizer of a sylow p subgroup of g although this has been confirmed for large numbers of groups including for example all solvable groups and all symmetric groups no general proof has yet been found in this paper we reduce the mckay conjecture to a question about simple groups we give a list of conditions that we hope all simple groups will satisfy and we show that the mckay conjecture will hold for a finite group g if every simple group involved in g satisfies these conditions also we establish that our conditions are satisfied for the simple groups psl2 q for all prime powers q 4 and for the suzuki groups sz q and ree groups r q where q 2 e or q 3 e respectively and e 1 is odd since our conditions are also satisfied by the sporadic simple group j 1 it follows that the mckay conjecture holds for all primes p for every finite group having an abelian sylow 2 subgroup

neural network based approximate reasoning principles and implementation instead of seeking a structure mapping from a fuzzy reasoning system to a neural network this paper is intended to find a functional mapping from a fuzzy logic based algorithm to the network based approach by viewing the given rule base as defining a global linguistic association constrained by fuzzy sets approximate reasoning is implemented here by a backpropagation neural network bnn with the aid of fuzzy set theory by paying particular attention to the generalization capability of the bnn the underlying principles have been examined in detail using two examples a small demonstration at the linguistic level and a more realistic problem of multivariable fuzzy control of blood pressure the simulation results not only indicate the feasibility of the bnn based approach but also reveal some deeper similarities which exist in the two methods which may have some important implications for future studies into fuzzy control in addition this work may be considered as another application example of t

the mckay conjecture is true for the sporadic simple groups abstract the mckay conjecture states that the number of irreducible complex characters of a group g that have degree prime to p is equal to the same number for the sylow p normalizer in g we verify this conjecture for the 26 sporadic simple groups

optimality conditions and a smoothing trust region newton method for nonlipschitz optimization regularized minimization problems with nonconvex nonsmooth perhaps non lipschitz penalty functions have attracted considerable attention in recent years owing to their wide applications in image restoration signal reconstruction and variable selection in this paper we derive affine scaled second order necessary and sufficient conditions for local minimizers of such minimization problems moreover we propose a global convergent smoothing trust region newton method which can find a point satisfying the affine scaled second order necessary optimality condition from any starting point numerical examples are given to demonstrate the effectiveness of the smoothing trust region newton method

on the complexity of approximating a kkt point of quadratic programming we present a potential reduction algorithm to approximate a karush kuhn tucker kkt point of general quadratic programming qp we show that the algorithm is a fully polynomial time approximation scheme and its running time dependency on accuracy 0 1 is o l log l log log l compared to the previously best known result o l 2 furthermore the limit of the kkt point satisfies the second order necessary optimality condition of being a local minimizer 1998 the mathematical programming society inc published by elsevier science b v

cubic regularization of newton method and its global performance in this paper we provide theoretical analysis for a cubic regularization of newton method as applied to unconstrained minimization problem for this scheme we prove general local convergence results however the main contribution of the paper is related to global worst case complexity bounds for different problem classes including some nonconvex cases it is shown that the search direction can be computed by standard linear algebra technique

dade s conjecture for the simple o nan group let g be the simple o nan group it has order 29 34 5 73 11 19 31 it has schur multiplier of order 3 and aut g g is cyclic of order 2 in this paper we prove that dade s conjecture for g is true here we mean by saying the conjecture the inductive form of the conjecture see 3 according to 3 the inductive form is equivalent to the invariant projective form if the outer automorphism group has cyclic sylow q subgroups for each prime q thus for the simple o nan group it suffices to check the invariant projective form moreover if a defect group of a block is cyclic then the invariant projective form is proved to be true for this block in 4 thus it suffices to treat the primes 2 3 and 7 in this paper a b and a b denote a split and a nonsplit extension of a by b respectively we use n and dn to denote a cyclic and a dihedral group of order n respectively moreover for a prime p an elementary abelian group of order p is denoted by p for an odd prime p we denote by p1 2n an extraspecial group of order p1 2n and exponent p and by 21 2n a minus type extraspecial group of order 21 2n finally sn and an denote the symmetric group and the alternating group of degree n respectively

2 blocks with minimal nonabelian defect groups abstract we study numerical invariants of 2 blocks with minimal nonabelian defect groups these groups were classified by redei see redei 1947 41 if the defect group is also metacyclic then the block invariants are known see sambale 43 in the remaining cases there are only two infinite families of interesting defect groups in all other cases the blocks are nilpotent we prove brauer s k b conjecture and olsson s conjecture for all 2 blocks with minimal nonabelian defect groups for one of the two families we also show that alperin s weight conjecture and dade s conjecture are satisfied this paper is a part of the author s phd thesis

accelerating the cubic regularization of newton s method on convex problems in this paper we propose an accelerated version of the cubic regularization of newton s method nesterov and polyak in math program 108 1 177 205 2006 the original version used for minimizing a convex function with lipschitz continuous hessian guarantees a global rate of convergence of order o big 1 over k 2 big where k is the iteration counter our modified version converges for the same problem class with order o big 1 over k 3 big keeping the complexity of each iteration unchanged we study the complexity of both schemes on different classes of convex problems in particular we argue that for the second order schemes the class of non degenerate problems is different from the standard class

towards donovan s conjecture for abelian defect groups abstract we define a new invariant for a p block of a finite group the strong frobenius number which we use to address the problem of reducing donovan s conjecture to normal subgroups of index p as an application we use the strong frobenius number to complete the proof of donovan s conjecture for 2 blocks with abelian defect groups of rank at most 4 and for 2 blocks with abelian defect groups of order at most 64

non lipschitz ell_ p regularization and box constrained model for image restoration nonsmooth nonconvex regularization has remarkable advantages for the restoration of piecewise constant images constrained optimization can improve the image restoration using a priori information in this paper we study regularized nonsmooth nonconvex minimization with box constraints for image restoration we present a computable positive constant for using nonconvex nonsmooth regularization and show that the difference between each pixel and its four adjacent neighbors is either 0 or larger than in the recovered image moreover we give an explicit form of for the box constrained image restoration model with the non lipschitz nonconvex lp norm 0 p 1 regularization our theoretical results show that any local minimizer of this imaging restoration problem is composed of constant regions surrounded by closed contours and edges numerical examples are presented to validate the theoretical results and show that the proposed model can recover image restoration results very well

verification of dade s conjecture for janko groupj3 abstract in 3 dade made a conjecture expressing the number k b d of characters of a given defect d in a given p block b of a finite group g in terms of the corresponding numbers k b d for blocks b of certain p local subgroups of g several different forms of this conjecture are given in 5 dade claims that the most complicated form of this conjecture called the inductive conjecture 5 8 in 5 will hold for all finite groups if it holds for all covering groups of finite simple groups in this paper we verify the inductive conjecture for all covering groups of the third janko group j 3 in the notation of the atlas 1 this is one step in the inductive proof of the conjecture for all finite groups certain properties of j 3 simplify our task the schur multiplier of j 3 is cyclic of order 3 see 1 p 82 hence there are just two covering groups of j 3 namely j 3 itself and a central extension 3 j 3 of j 3 by a cyclic group z of order 3 we treat these two covering groups separately the outer automorphism group out j 3 of j 3 is cyclic of order 2 see 1 p 82 in this case dade affirms in 5 section 6 that the inductive conjecture for j 3 is equivalent to the much weaker invariant conjecture 2 5 in 5 furthermore dade has proved in 6 that this invariant conjecture holds for all blocks with cyclic defect groups the sylow p subgroups of j 3 are cyclic of order p for all primes dividing j 3 except 2 and 3 so we only need to verify the invariant conjecture for the two primes p 2 and p 3 we do that in theorem 2 10 1 and theorem 3 6 1 below the group out 3 j 3 z of outer automorphisms of 3 j 3 centralizing z is trivial see 1 p 82 in this case dade affirms that the inductive conjecture is equivalent to the projective conjecture 4 4 in 5 again dade has shown in 6 that this projective conjecture holds for all blocks with cyclic defect groups so we only need to verify the projective conjecture for p 2 and p 3 we do that in theorem 4 4 2 and theorem 4 3 1 for the above reasons the four theorems 2 10 1 3 6 1 4 4 2 and 4 3 1 below are sufficient to prove the inductive form of dade s conjecture for all covering groups of j 3 this paper is my ph d thesis it would not have been possible without the help and infinite patience of my advisor everett dade

iterative reweighted minimization methods for l_p lp regularized unconstrained nonlinear programming in this paper we study general l p l p regularized unconstrained minimization problems in particular we derive lower bounds for nonzero entries of the first and second order stationary points and hence also of local minimizers of the l p l p minimization problems we extend some existing iterative reweighted l 1 l 1 mathrm irl 1 irl 1 and l 2 l 2 mathrm irl 2 irl 2 minimization methods to solve these problems and propose new variants for them in which each subproblem has a closed form solution also we provide a unified convergence analysis for these methods in addition we propose a novel lipschitz continuous epsilon approximation to vert x vert p p x p p using this result we develop new mathrm irl 1 irl 1 methods for the l p l p minimization problems and show that any accumulation point of the sequence generated by these methods is a first order stationary point provided that the approximation parameter epsilon is below a computable threshold value this is a remarkable result since all existing iterative reweighted minimization methods require that epsilon be dynamically updated and approach zero our computational results demonstrate that the new mathrm irl 1 irl 1 method and the new variants generally outperform the existing mathrm irl 1 irl 1 methods chen and zhou in 2012 foucart and lai in appl comput harmon anal 26 395 407 2009

a local strategy to decide the alperin and dade conjectures abstract we present a new strategy which exploits both the maximal andp local subgroup structure of a given finite simple group in order to decide the alperin and dade conjectures for this group we demonstrate the computational effectiveness of this approach by using it to verify these conjectures for the conway simple group co2

uno s invariant conjecture for the general linear and unitary groups in nondefining characteristics abstract uno s invariant conjecture which is a combination of dade s invariant conjecture and the isaacs navarro conjecture is proved for general linear and unitary groups in nondefining characteristics

a smoothing sqp framework for a class of composite l_q lq minimization over polyhedron the composite l q 0 q 1 minimization problem over a general polyhedron has received various applications in machine learning wireless communications image restoration signal reconstruction etc this paper aims to provide a theoretical study on this problem first we derive the karush kuhn tucker kkt optimality conditions for local minimizers of the problem second we propose a smoothing sequential quadratic programming framework for solving this problem the framework requires a approximate solution of a convex quadratic program at each iteration finally we analyze the worst case iteration complexity of the framework for returning an epsilon kkt point i e a feasible point that satisfies a perturbed version of the derived kkt optimality conditions to the best of our knowledge the proposed framework is the first one with a worst case iteration complexity guarantee for solving composite l q minimization over a general polyhedron

weights for symmetric and general linear groups abstract an important feature of the theory of finite groups is the number of connections and analogies with the theory of lie groups the concept of a weight has long been useful in the modular representation theory of finite lie groups in the defining characteristic of the group the idea of a weight in the modular representation theory of an arbitrary finite group was recently introduced in alperin proc sympos pure math 41 1987 369 379 where it was conjectured that the number of weights should equal the number of modular irreducible representations moreover this equality should hold block by block the conjecture has created great interest since its truth would have important consequences a synthesis of known results and solutions of outstanding problems in this paper we prove the conjecture first for the modular representations of symmetric groups and second for modular representations in odd characteristic r for the finite general linear groups in the latter case r may be assumed to be different from the defining characteristic p of the group since the result is known when r is p the well known analogy between the representation theory of the symmetric and general linear groups holds here too

the blocks and weights of finite special linear and unitary groups abstract this paper has two main parts first we give a classification of the l blocks of finite special linear and unitary groups sl n q in the non defining characteristic l 3 second we describe how the l weights of sl n q can be obtained from the l weights of gl n q when l gcd n q and verify the alperin weight conjecture for sl n q under the condition l gcd n q as a step to establish the alperin weight conjecture for all finite groups we prove the inductive blockwise alperin weight condition for any unipotent l block of sl n q if l gcd n q

control of ph processes using fuzzy modeling of titration curve abstract model based ph control is a difficult problem due to its inherent nonlinearity and time varying characteristics this is specially the case when the ph regulation of streams consisting hundreds of constituents with varying concentrations is desired the ph process can be modeled by using conservation laws and neutrality condition an alternative to this mechanistic modeling is using nonlinear empirical modeling such as fuzzy modeling this paper proposes a new approach of fuzzy modeling of titration curves which can be used for control purposes the takagi sugeno model was used for the modeling of titration curve two control algorithms which are designed based on the titration curve namely strong acid equivalent sae and multiple model switching and tuning mmst have been used the efficacy of the proposed approach is demonstrated via simulation and experimental study of a ph control system

a reduction theorem for the alperin weight conjecture the alperin weight conjecture was stated by j alperin in 1986 and constitutes one of the main problems in the representation theory of finite groups in this paper we provide a reduction to simple groups and as an application we prove the conjecture in several cases

weights for the steinberg triality groups 3 d 4 q in the paper 1 j l alperin conjectured that for a block b of a finite group g the number of b weights should equal the number of irreducible brauer characters in b due to e c dade s reduction theorem it is important to check the conjecture for simple groups the truth of this conjecture has been proved for several classes of groups in this paper we prove the conjecture for the steinberg triality groups g 3d4 q for all primes r where q is a power of a prime p the block structure of g and the number of irreducible brauer characters for a block were given by deriziots and michler 11 we may always suppose that r is different from p since the result is known when r is p see 1

perfect isometries for principal blocks with abelian defect groups and elementary abelian 2 inertial quotients abstract letbbe the principalp block of a finite groupgwith an abelian defect grouppandea root ofbincg p if the inertial quotiente ng p e p cg p is an elementary abelian 2 group respectively a dihedral group of order 8 andp 3 thenband its brauer correspondent considered as blocks ofgandng p are isotypic and in particular there is a perfect isometry between them

dade s invariant conjecture for steinberg s triality groups d43 2n in defining characteristic abstract in this paper we verify dade s invariant conjecture for steinberg s triality groups d 4 3 2 n in the defining characteristic i e in characteristic 2 together with the results in j an dade s conjecture for steinberg triality groups d 4 3 q in non defining characteristics math z 241 2002 445 469 and j an f himstedt s huang uno s invariant conjecture for steinberg s triality groups in defining characteristic in preparation this completes the proof of dade s conjecture for steinberg s triality groups furthermore we show that the isaacs malle navarro version of the mckay conjecture holds for d 4 3 2 n in the defining characteristic i e d 4 3 2 n is good for the prime 2 in the sense of isaacs malle and navarro

dade s inductive conjecture for the ree groups of type g2 in the defining characteristic abstract we verify the inductive form of dade s conjecture for the finite simple groups 2 g 2 3 2 m 1 where m is a positive integer for the prime p 3 together with work by j an 1994 indian j math 36 7 27 this completes the verification of the conjecture for this series of groups

high order evaluation complexity for convexly constrained optimization with non lipschitzian group sparsity terms this paper studies high order evaluation complexity for partially separable convexly constrained optimization involving non lipschitzian group sparsity terms in a nonconvex objective function we propose a partially separable adaptive regularization algorithm using a pth order taylor model and show that the algorithm needs at most o epsilon p 1 p q 1 evaluations of the objective function and its first p derivatives whenever they exist to produce an epsilon delta approximate qth order stationary point our algorithm uses the underlying rotational symmetry of the euclidean norm function to build a lipschitzian approximation for the non lipschitzian group sparsity terms which are defined by the group ell 2 ell a norm with a in 0 1 the new result shows that the partially separable structure and non lipschitzian group sparsity terms in the objective function do not affect the worst case evaluation complexity order

a unified real time approximate reasoning approach for use in intelligent control part 1 theoretical development taking account of the fuzzy nature of human decision making processes and real time properties this paper establishes a unified approximate reasoning model based on possibility theory rather than on relational matrix computation both fuzzy and random uncertainties can be coped with in the model in the case of sensor based situations a simpler reasoning scheme is derived by introducing the concepts of matching measures the proposed models may provide another possibility for on line reasoning in real time expert system applications in part 1 the theoretical foundations are provided while in part 2 their application to multivariate blood pressure control is described

character correspondences in finite general linear unitary and symmetric groups let r 0 be a prime integer and let g be a finite general linear group gl n q a unitary group u n q with q qg or a symmetric group s n of degree n the partition of the irreducible ordinary characters of s n into r blocks of s n was given by brauer s and robinson s solution of nakayama s conjecture see 7 p 245 in their fundamental paper 5 fong and srinivasan have recently classified the r blocks of gl n q and u n q for all primes r 2 with r q 1 using these classifications of the r blocks of g we show in this article that there is a natural one to one correspondence b between the irreducible characters of height zero of an r block b of g with defect group r and the irreducible characters of height zero of the brauer correspondent b of b in n ng r theorem 4 10 in particular ko b ko b where ko b denotes the number of all irreducible characters of b with height h t z 0 therefore alperin s conjecture on the numbers of irreducible characters of height zero is verified for all general linear unitary and symmetric groups if g s n then theorem 4 10 also holds for r 2 in order to establish the character correspondence three other correspondences are studied the product of which is b in sect 1 we construct for every r block b of g with defect group r a subgroup g of g with a sylow r subgroup r such that there is a natural height preserving one to one correspondence t between the set irr b of all irreducible characters of b and the set irr 0 of all irreducible characters of the principal r block o of d reduction theorems 1 9 and 1 10 the map respects the geometric conjugacy classes of characters if 0 denotes the principal r block of n nd and if b is the brauer correspondent of b in n n6 r then by theorems 3 8 and 3 10 the block ideals o and b are morita equivalent have the same decomposition numbers and there is a natural height preserving one to one correspondence a between irr bo and irr b

uno s invariant conjecture for steinberg s triality groups in defining characteristic abstract we verify uno s invariant conjecture for steinberg s triality groups d 4 3 q q a power of an odd prime p in the defining characteristic p uno s invariant conjecture is a refinement of dade s invariant conjecture

on finite groups of p local rank one and conjectures of dade and robinson abstract we use the classification of finite simple groups to demonstrate that conjecture 4 1 of g r robinson 1996 proc london math soc 3 72 312 330 and dade s projective conjecture hold for finite groups of p local rank one

worst case complexity of smoothing quadratic regularization methods for non lipschitzian optimization in this paper we propose a smoothing quadratic regularization sqr algorithm for solving a class of nonsmooth nonconvex perhaps even non lipschitzian minimization problems which has wide applications in statistics and sparse reconstruction the proposed sqr algorithm is a first order method at each iteration the sqr algorithm solves a strongly convex quadratic minimization problem with a diagonal hessian matrix which has a simple closed form solution that is inexpensive to calculate we show that the worst case complexity of reaching an epsilon scaled stationary point is o epsilon 2 moreover if the objective function is locally lipschitz continuous the sqr algorithm with a slightly modified updating scheme for the smoothing parameter and iterate can obtain an epsilon clarke stationary point in at most o epsilon 3 iterations

the alperin and dade conjectures for ree groups2f4 q2 in non defining characteristics abstract this paper is part of a program to study alperin s weight conjecture and dade s ordinary conjecture on counting characters in blocks for several finite groups the local structures of radical subgroups and certain radical 3 chains of a ree group of typef4are given and the conjectures and a conjecture of olsson are confirmed for the group when the characteristic of the modular representation is distinct from the defining characteristic of the group

modeling ph neutralization processes using fuzzy neural approaches abstract this paper is concerned with the modeling and identification of ph processes via fuzzy neural approaches a simplified fuzzy model acting as an approximate reasoner is used to deduce the model output on the basis of the identified rule base which is derived by using one of the following three network based self organizing algorithms unsupervised self organizing counter propagation network usocpn supervised self organizing counter propagation network ssocpn and self growing adaptive vector quantization sgavq three typical ph processes were treated including a strong acid strong base system a weak acid strong base system and a two output system with buffering taking part in reaction extensive simulations including on line modeling have shown that these nonlinear ph processes can be modeled reasonably well by the present schemes which are simple but efficient

radical 2 subgroups of the monster and the baby monster the radical 2 subgroups of the monster and the baby monster are determined up to conjugacy which completes the classification problem of radical 2 subgroups for all sporadic simple groups

alperin mckay implies brauer s problem 21 there are several outstanding open problems in modular representation w x theory including several on brauer s famous list b one such conjecture w x w x is the alperin mckay conjecture e g a1 f whenever b is a p block of a finite group g and b has defect group d then the number k b of 0 w x 5 irreducible characters x in b with g d x 1 is the same as the p corresponding number k b for the unique p block b of n d with 0 g g w x w x b s b another is r brauer s problem 21 see b f let b be a p block with defect group d and containing k b s k irreducible charac ters is d bounded in terms of k we prove here that a positive answer to brauer s question follows if the alperin mckay conjecture is correct by virtue of zelmanov s solution of w x w x the restricted burnside problem see z1 z2 and the positive solution w x of brauer s problem in the case of p solvable g by the first author k our main result is

large character heights qd p and the ordinary weight conjecture abstract we show that if the ordinary weight conjecture is correct then large character heights for a p block b p an odd prime necessitate the involvement of q d p in the normalizers of alperin goldschmidt b subpairs

complexity analysis of interior point algorithms for non lipschitz and nonconvex minimization we propose a first order interior point algorithm for a class of non lipschitz and nonconvex minimization problems with box constraints which arise from applications in variable selection and regularized optimization the objective functions of these problems are continuously differentiable typically at interior points of the feasible set our first order algorithm is easy to implement and the objective function value is reduced monotonically along the iteration points we show that the worst case iteration complexity for finding an epsilon scaled first order stationary point is o epsilon 2 o 2 furthermore we develop a second order interior point algorithm using the hessian matrix and solve a quadratic program with a ball constraint at each iteration although the second order interior point algorithm costs more computational time than that of the first order algorithm in each iteration its worst case iteration complexity for finding an epsilon scaled second order stationary point is reduced to o epsilon 3 2 o 3 2 note that an epsilon scaled second order stationary point must also be an epsilon scaled first order stationary point

isometries and extra special sylow groups of order p3 abstract a new type of isometry between the sets of irreducible characters of blocks is introduced and a conjecture is stated using the classifications of finite simple groups and saturated fusion systems over an extra special group p of order p 3 and exponent p we check that the conjecture is affirmatively answered for the principal blocks of finite groups with sylow p subgroup p

smoothing methods for nonsmooth nonconvex minimization we consider a class of smoothing methods for minimization problems where the feasible set is convex but the objective function is not convex not differentiable and perhaps not even locally lipschitz at the solutions such optimization problems arise from wide applications including image restoration signal reconstruction variable selection optimal control stochastic equilibrium and spherical approximations in this paper we focus on smoothing methods for solving such optimization problems which use the structure of the minimization problems and composition of smoothing functions for the plus function x many existing optimization algorithms and codes can be used in the inner iteration of the smoothing methods we present properties of the smoothing functions and the gradient consistency of subdifferential associated with a smoothing function moreover we describe how to update the smoothing parameter in the outer iteration of the smoothing methods to guarantee convergence of the smoothing methods to a stationary point of the original minimization problem

radical 3 subgroups and essential 3 rank of f4 q abstract we classify maximal 3 local subgroups and radical 3 subgroups of g f 4 q when q odd and then determine the essential 3 rank of the frobenius category f d g where d is a sylow 3 subgroup of g

recursive trust region methods for multiscale nonlinear optimization a class of trust region methods is presented for solving unconstrained nonlinear and possibly nonconvex discretized optimization problems like those arising in systems governed by partial differential equations the algorithms in this class make use of the discretization level as a means of speeding up the computation of the step this use is recursive leading to true multilevel multiscale optimization methods reminiscent of multigrid methods in linear algebra and the solution of partial differential equations a simple algorithm of the class is then described and its numerical performance is shown to be numerically promising this observation then motivates a proof of global convergence to first order stationary points on the fine grid that is valid for all algorithms in the class

dade s invariant conjecture for general linear and unitary groups in non defining characteristics this paper is part of a program to study the conjecture of e c dade on counting characters in blocks for several finite groups the invariant conjecture of dade is proved for general linear and unitary groups when the characteristic of the modular representation is distinct from the defining characteristic of the groups

asymptotic properties of bridge estimators in sparse high dimensional regression models we study the asymptotic properties of bridge estimators in sparse high dimensional linear regression models when the number of covariates may increase to infinity with the sample size we are particularly interested in the use of bridge estimators to distinguish between covariates whose coefficients are zero and covariates whose coefficients are nonzero we show that under appropriate conditions bridge estimators correctly select covariates with nonzero coefficients with probability converging to one and that the estimators of nonzero coefficients have the same asymptotic distribution that they would have if the zero coefficients were known in advance thus bridge estimators have an oracle property in the sense of fan and li j amer statist assoc 96 2001 1348 1360 and fan and peng ann statist 32 2004 928 961 in general the oracle property holds only if the number of covariates is smaller than the sample size however under a partial orthogonality condition in which the covariates of the zero coefficients are uncorrelated or weakly correlated with the covariates of nonzero coefficients we show that marginal bridge estimators can correctly distinguish between covariates with nonzero and zero coefficients with probability converging to one even when the number of covariates is greater than the sample size

on the evaluation complexity of composite function minimization with applications to nonconvex nonlinear programming we estimate the worst case complexity of minimizing an unconstrained nonconvex composite objective with a structured nonsmooth term by means of some first order methods we find that it is unaffected by the nonsmoothness of the objective in that a first order trust region or quadratic regularization method applied to it takes at most mathcal o epsilon 2 function evaluations to reduce the size of a first order criticality measure below epsilon specializing this result to the case when the composite objective is an exact penalty function allows us to consider the objective and constraint evaluation worst case complexity of nonconvex equality constrained optimization when the solution is computed using a first order exact penalty method we obtain that in the reasonable case when the penalty parameters are bounded the complexity of reaching within epsilon of a kkt point is at most mathcal o epsilon 2 problem evaluations which is the same in order as the function evaluation complexity of steepest descent methods applied to unconstrained nonconvex smooth optimization