on centers of blocks with one simple module abstract let g be a finite group and let b be a non nilpotent block of g with respect to an algebraically closed field of characteristic 2 suppose that b has an elementary abelian defect group of order 16 and only one simple module the main result of this paper describes the algebra structure of the center of b this is motivated by a similar analysis of a certain 3 block of defect 2 in kessar 2012 15

character tables and defect groups abstract let b be a block of a finite group g with defect group d we prove that the exponent of the center of d is determined by the character table of g in particular we show that d is cyclic if and only if b contains a large family of irreducible p conjugate characters more generally for abelian d we obtain an explicit formula for the exponent of d in terms of character values in small cases even the isomorphism type of d is determined in this situation moreover it can read off from the character table whether d d 4 where d denotes the commutator subgroup of d we also propose a new characterization of nilpotent blocks in terms of the character table

nilpotent blocks of quasisimple groups for odd primes we investigate the nilpotent blocks of positive defect of the quasisimple groups for odd primes in particular it is shown that every nilpotent block of a quasisimple group has abelian defect groups a conjecture of puig concerning the recognition of nilpotent blocks is also shown for these groups

on the inductive alperin mckay and alperin weight conjecture for groups with abelian sylow subgroups abstract we study the inductive alperin mckay conjecture the inductive isaacs navarro refinement and the inductive blockwise alperin weight conjecture for groups of lie type in the generic case of abelian sylow l subgroups we also show that the alternating groups the suzuki groups and the ree groups satisfy the inductive condition necessary for spath s reduction of the blockwise alperin weight conjecture to the case of simple groups

controlled 2 blocks of quasisimple groups abstract controlled p blocks introduced by alperin and broue in 1979 were classified for quasisimple groups g for odd primes in 10 we classify the controlled 2 blocks of quasisimple groups g the results imply that every nilpotent 2 block of g has abelian defect groups this agrees with one of the main results proved in 15 we also give an explicit characterization of non controlled 2 blocks of all quasisimple groups this implies that the p 2 version of the block theoretic analogue of glauberman s zj theorem proved by kessar linckelmann and robinson in 27 holds for quasisimple groups g

sharp variational conditions for convex composite nonsmooth functions in this paper we present first and second order variational conditions for a convex composite function g circ f where g is a nonsmooth convex function and f is a vector valued map the first order results which apply to not necessarily locally lipschitz continuous maps f not only recapture the results of the special cases where f is locally lipschitz or g teaux differentiable but also yield sharp necessary variational conditions in these cases the results are achieved by applying a new strengthened notion of approximate jacobian called a g teaux g approximate jacobian without the use of the upper semicontinuity of the approximate jacobian these variational results are generally derived by using a chain rule formula or by constructing upper convex approximations to the composite function these approaches often need the upper semicontinuity requirement of a generalized jacobian map such a requirement not only limits the derivation of sharp optimality conditions as the small approximate jacobians or generalized subdifferentials lack an upper semi continuity property but also restricts the treatment of g teaux differentiable maps f this situation is overcome by the use of g approximate jacobians the second order variational conditions are shown to hold in particular in the case where f is continuously g teaux differentiable

on normal subgroups and simple modules with a given vertex in a p solvable group abstract let n be a normal subgroup of a p solvable group g and let m be a simple fn module where f is an algebraically closed field of characteristic p let q be any p subgroup of g in this paper we count the number of simple fg modules lying over m and having q as a vertex

the inductive blockwise alperin weight condition for psln q with n q 1 1 abstract this paper is a continuation of our previous paper in li and zhang 2018 13 we prove that if n q 1 1 psl n q satisfies the inductive blockwise alperin weight condition for any prime l different from the defining characteristic

the minimal number of characters over a normal p subgroup abstract if n is a normal p subgroup of a finite group g and irr n is a g invariant irreducible character of n then the number irr g of irreducible characters of g over is always greater than or equal to the number k p g n of conjugacy classes of g n consisting of p elements in this paper we investigate when there is equality

counting characters above invariant characters in solvable groups abstract this paper discusses two related questions first given a g invariant character of a normal subgroup n of a solvable group what can we say if the number of characters of g above is in some sense as small as possible isaacs and navarro 5 have shown that under certain assumptions about primes dividing the order of the group one can show that g n must have a very particular structure here we show that these assumptions can be weakened to obtain results about all solvable groups we also discuss a related question about blocks for a prime p and a p block b of g we let k b denote the number of ordinary characters in b it is relatively easy to show that k b is bounded below by k g d which is the number of conjugacy classes of g that intersect the defect group d of b in this paper we ask what can be said if equality is achieved we show that for p solvable groups if k b k g d then b is nilpotent and thus k b irr d in addition we show that this result holds for many blocks of arbitrary finite groups including all blocks of the symmetric groups we also extend a result on fully ramified coprime actions in 5

composite nonsmooth programming with gâteaux differentiability this paper examines constrained nonsmooth optimization problems where the objective function and the constraints are compositions of locally lipschitz functions and g teaux differentiable functions but are not necessarily frechet differentiable or strict differentiable lagrangian necessary and sufficient optimality conditions are presented for various classes of composite programs these are obtained by constructing appropriate convex approximations for composite functions the lagrange multipliers are also characterized in terms of subgradients of the value function under appropriate conditions

a sharp lagrange multiplier rule for nonsmooth mathematical programming problems involving equality constraints it is shown that a lagrange multiplier rule that uses approximate jacobians holds for mathematical programming problems involving lipschitzian functions finitely many equality constraints and convex set constraints it is sharper than the corresponding lagrange multiplier rules for the convex valued subdifferentials such as those of clarke optimization and nonsmooth analysis 2nd ed siam 1990 and michel and penot differential integral equations 5 1992 pp 433 454 the lagrange multiplier result is obtained by means of a controllability criterion and the theory of fans developed by a d ioffe math oper res 9 1984 pp 159 189 math programming 58 1993 pp 137 145 as an application necessary optimality conditions are derived for a class of constrained minimax problems an example is discussed to illustrate the nature of the multiplier rule

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 asymptotics of the arakelov geen s function and faltings delta invariant we study the behavior of the arakelov green s function and faltings delta invariant on degenerating riemann surfaces

height zero characters and normal subgroups in p solvable groups abstract fix a prime number p and let n be a normal subgroup of a finite p solvable group g let b be a p block of n and suppose b is a p block of g covering b let d be a defect group for the fong reynolds correspondent of b with respect to b and let b be the unique p block of n n g d having defect group d and inducing b suppose further that irr b and let irr 0 b be the set of irreducible characters in b of height zero that lie over we show that the number of characters in irr 0 b is equal to the number of characters in t irr 0 b t where t runs through the inertial group t of b in g this result generalizes a theorem of t okuyama and m wajima which confirms the alperin mckay conjecture for p solvable groups

special values of canonical green s functions we give a precise formula for the value of the canonical green s function at a pair of weierstrass points on a hyperelliptic riemann surface further we express the energy of the weierstrass points in terms of a spectral invariant recently introduced by n kawazumi and s zhang it follows that the energy is strictly larger than log 2 our results generalize known formulas for elliptic curves

second order global optimality conditions for convex composite optimization in recent years second order sufficient conditions of an isolated local minimizer for convex composite optimization problems have been established in this paper second order optimality conditions are obtained of aglobal minimizer for convex composite problems with a non finite valued convex function and a twice strictly differentiable function by introducing a generalized representation condition this result is applied to a minimization problem with a closed convex set constraint which is shown to satisfy the basic constraint qualification in particular second order necessary and sufficient conditions of a solution for a variational inequality problem with convex composite inequality constraints are obtained 1998 the mathematical programming society inc published by elsevier science b v

calculus on arithmetic surfaces in a2 and a3 arakelov introduces an intersection calculus for arithmetic surfaces that is for stable models of curves over a number field in this paper we intend to show that his intersection product has a lot of useful properties more precisely we show that the following properties from the theory of algebraic surfaces have an analogue in our situation

variational analysis of non lipschitz spectral functions we consider spectral functions f where f is any permutation invariant mapping from cn to r and is the eigenvalue map from the set of n n complex matrices to cn ordering the eigenvalues lexicographically for example if f is the function maximum real part then f is the spectral abscissa while if f is maximum modulus then f is the spectral radius both these spectral functions are continuous but they are neither convex nor lipschitz for our analysis we use the notion of subgradient extensively analyzed in variational analysis r t rockafellar and r j b wets springer 1998 we show that a necessary condition for y to be a subgradient of an eigenvalue function f at x is that y commutes with x we also give a number of other necessary conditions for y based on the schur form and the jordan form of x in the case of the spectral abscissa we refine these conditions and we precisely identify the case where subdifferential regularity holds we conclude by introducing the notion of a semistable program maximize a linear function on the set of square matrices subject to linear equality constraints together with the constraint that the real parts of the eigenvalues of the solution matrix are non positive semistable programming is a nonconvex generalization of semidefinite programming using our analysis we derive a necessary condition for a local maximizer of a semistable program and we give a generalization of the complementarity condition familiar from semidefinite programming

the weighted fusion category algebra and the q schur algebra for gl 2 q abstract we show that the weighted fusion category algebra of the principal 2 block b 0 of gl 2 q is the quotient of the q schur algebra s 2 q by its socle for q an odd prime power as a consequence we get a canonical bijection between the set of isomorphism classes of simple k gl 2 q b 0 modules and the set of conjugacy classes of b 0 weights in this case

theta functions modular invariance and strings we use quillen s theorem and algebraic geometry to investigate the modular transformation properties of some quantities of interest in string theory in particular we show that the spin structure dependence of the chiral dirac determinant on a riemann surface is given by riemann s theta function we use this result to investigate the modular invariance of multiloop heterotic string amplitudes

the mumford form and the polyakov measure in string theory an explicit formula is derived for the mumford form on the moduli space of algebraic curves according to the belavin knizhnik theorem this gives a formula for the polyakov bosonic string measure

controlled blocks of the finite quasisimple groups for odd primes abstract we classify controlled blocks introduced by alperin and broue in 1979 for all quasisimple groups g for odd primes the results imply that every nilpotent block of g has abelian defect groups which in turn is one of the main results proved in an and eaton 2011 6 we also give an explicit characterization of non controlled blocks of all quasisimple groups g for odd primes this implies the block theoretic analogue of glauberman s zj theorem for g proved by kessar linckelmann and robinson 2002 18

bosonization on higher genus riemann surfaces we prove the equivalence between certain fermionic and bosonic theories in two spacetime dimensions the theories have fields of arbitrary spin on compact surfaces with any number of handles global considerations require that we add new topological terms to the bosonic action the proof that our prescription is correct relies on methods of complex algebraic geometry

approximate jacobian matrices for nonsmooth continuous maps and c 1 optimization the notion of approximate jacobian matrices is introduced for a continuous vector valued map it is shown for instance that the clarke generalized jacobian is an approximate jacobian for a locally lipschitz map the approach is based on the idea of convexificators of real valued functions mean value conditions for continuous vector valued maps and taylor s expansions for continuously g teaux differentiable functions i e c1 functions are presented in terms of approximate jacobians and approximate hessians respectively second order necessary and sufficient conditions for optimality and convexity of c1 functions are also given

convex composite multi objective nonsmooth programming this paper examines nonsmooth constrained multi objective optimization problems where the objective function and the constraints are compositions of convex functions and locally lipschitz and g teaux differentiable functions lagrangian necessary conditions and new sufficient optimality conditions for efficient and properly efficient solutions are presented multi objective duality results are given for convex composite problems which are not necessarily convex programming problems applications of the results to new and some special classes of nonlinear programming problems are discussed a scalarization result and a characterization of the set of all properly efficient solutions for convex composite problems are also discussed under appropriate conditions

constraint qualifications for constrained lipschitz optimization problems and applications to a mpcc abstract three constraint qualifications the weak generalized robinson constraint qualification the bounded constraint qualification and the generalized abadie constraint qualification which are weaker than the generalized robinson constraint qualification grcq given by yen 1997 1 are introduced for constrained lipschitz optimization problems relationships between those constraint qualifications and the calmness of the solution mapping are investigated it is demonstrated that the weak generalized robinson constraint qualification and the bounded constraint qualification are easily verifiable sufficient conditions for the calmness of the solution mapping whereas the proposed generalized abadie constraint qualification described in terms of graphical derivatives in variational analysis is weaker than the calmness of the solution mapping finally those constraint qualifications are written for a mathematical program with complementarity constraints mpcc and new constraint qualifications ensuring the c stationary point condition of a mpcc are obtained

optimality conditions in mathematical programming and composite optimization new second order optimality conditions for mathematical programming problems and for the minimization of composite functions are presented they are derived from a general second order fermat s rule for the minimization of a function over an arbitrary subset of a banach space the necessary conditions are more accurate than the recent results of kawasaki 1988 and cominetti 1989 but more importantly in the finite dimensional case they are twinned with sufficient conditions which differ by the replacement of an inequality by a strict inequality we point out the equivalence of the mathematical programming problem with the problem of minimizing a composite function our conditions are especially important when one deals with functional constraints when the cone defining the constraints is polyhedral we recover the classical conditions of ben tal zowe 1982 and cominetti 1990

second order necessary and sufficient conditions for convex composite ndo in convex composite ndo one studies the problem of minimizing functions of the formf h f whereh m is a finite valued convex function andf n m is continuously differentiable this problem model has a wide range of application in mathematical programming since many important problem classes can be cast within its framework e g convex inclusions minimax problems and penalty methods for constrained optimization in the present work we extend the second order theory developed by a d ioffe in 11 12 13 for the case in whichh is sublinear to arbitrary finite valued convex functionsh moreover a discussion of the second order regularity conditions is given that illuminates their essentially geometric nature

on blocks with one simple module in any brauer correspondent abstract we prove that a block of a finite group such that the defect groups are abelian and all the brauer correspondents the block itself included have a unique isomorphism class of simple modules is nilpotent